Guidebook-2 Design of Bridges

Guidebook 2

DESIGN of BRIDGES Pietro Croce et al.

Guidebook 2

DESIGN of BRIDGES Pietro Croce et al.

Editor: Pietro Croce, University of Pisa, Department of Civil Engineering, Structural Division Authors: Pietro Croce, University of Pisa, Department of Civil Engineering, Structural Division Milan Holický, Czech Technical University in Prague, Klokner Institute Jana Marková, Czech Technical University in Prague, Klokner Institute Angel Arteaga, E. Torroja Institute of Construction Sciences, CSIC, Madrid Ana de Diego, E. Torroja Institute of Construction Sciences, CSIC, Madrid Peter Tanner, E. Torroja Institute of Construction Sciences, CSIC, Madrid Carlos Lara, E. Torroja Institute of Construction Sciences, CSIC, Madrid Dimitris Diamantidis, University of Applied Sciences in Regensburg, Faculty of Civil Engineering Ton Vrouwenvelder, Netherlands Organisation for Applied Scientific Research, Delft

Guidebook 2 Design of B ridges ISBN: 978-80-01-04617-3 E dite d by : Piet ro Cro c e, Universit y of Pis a Publishe d by : Cz e ch Te chnic a l Universit y in Prague, K lok ner Inst itute Š olínova 7, 166 08 Prague 6, Cz e ch R epublic Pages: 230 1 st e dit ion

Foreword

FOREWORD The Leonardo da Vinci Project CZ/08/LLP-LdV/TOI/134020 “Transfer of Innovations Provided in Eurocodes, addresses the urgent need to implement the new system of European documents related to design and construction work and products. These documents, called Eurocodes, are systematically based on the recently developed Council Directive 89/106/EEC “The Construction Product Directive” and its Interpretative Documents ID 1 and ID 2. Implementation of Eurocodes in each Member State is a demanding task as each country has its own long-term tradition in design and construction. The project should enable an effective implementation and application of the new methods for designing and verification of buildings and civil engineering works in all the partner countries (CZ, DE, ES, IT, NL) and in other Member States. The need to explain and effectively use the latest principles specified in Eurocodes standards is apparent from enterprises, undertakings and public national authorities involved in construction industry and also from university and colleges. Training materials, manuals and software programmes for education are urgently required. The submitted Guidebook 2 completes the set of two guidebooks intended to provide required manuals and software products for training, education and effective implementation of Eurocodes: Guidebook 1: Load Effects on Buildings Guidebook 2: Design of bridges. It is expected that the Guidebooks will address the following intents in further harmonization o f European construction industry: -

reliability improvement and unification of the process of design, development of a single market for product and for construction services, improvement of the competitiveness of the European industries in the global world market; new opportunities for trained primary target groups in the labour market.

The Guidebook 2 is focused on determining load effects on road, railway and pedestrian bridges and special civil structures. The following main topics are discussed in particular: -

basic requirements on bridges, basis of structural design, traffic loads for static and fatigue assessment and climatic actions, accidental actions, combination rules for bridges, examples and case studies.

Annex A to Guidebook 2 concerns new traffic trends in European countries and their consequences on load models and on assessment of existing bridges; Annex B provides basic information about action and combination rules for special structures, like cranes, masts, towers and pipelines.

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Foreword

The Guidebook 2 is written in a user-friendly way employing only basic mathematical tools, supplemented by examples and case studies developed in detail. A wide range of potential users of the Guidebooks and other training materials includes practising engineers, designers, technicians, experts of public authorities, young people – high school and university students. The target groups come from all territorial regions of the partner countries. However, the dissemination of the project results is foreseen to be spread into all Member States of CEN and other interested countries.

Pisa 2010

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Contents

GUIDEBOOK 2 – DESIGN OF BRIDGES

CONTENTS

Foreword

3

Contents

5

Chapter 1: Basic requirements

7

Chapter 2: Basis of design – methodological aspects Appendix A to Chapter 2 – Principles of probabilistic optimization

15 27

Chapter 3: Static loads due to traffic Appendix A to Chapter 3 – Development of static load traffic models for road bridges of EN 1991-2

33 63

Chapter 4: Fatigue loads due to traffic Appendix A to Chapter 4 – Vehicle interactions and fatigue assessments

79 99

Chapter 5: Non traffic actions

105

Chapter 6: Accidental actions

119

Chapter 7: Combination rules for bridges in Eurocodes

134

Chapter 8: Case study - Design of a concrete bridge

147

Chapter 9: Case study – Design of a steel bridge

165

Chapter 10: Case study - Design of a composite bridge

191

Annex A: Effects of LHVs on road bridges and EN1991-2 load models

209

Annex B: Actions and combination rules for cranes, masts, towers and pipelines

221

5

Contents

6


CHAPTER 1: BASIC REQUIREMENTS Angel Arteaga1, and Ana de Diego1 1

E. Torroja Institute of Construction Sciences, CSIC. Madrid. Spain

Summary The Eurocode system establishes a series of basic requirements that must be met by all structures to ensure their suitability for their intended use and durability. Those requirements, based on European Commission Directives and other construction standards in place, are reviewed and explained in Chapter 1 of Guidebook 1. This first chapter of Guidebook 2 describes the specific requirements applicable to bridges.

1

INTRODUCTION

1.1

Background documents All the essential requirements to be met by any construction work, bridges included, are laid down in Eurocode EN 1990 [1]. The serviceability limits specifically applicable to bridges, in which user safety and comfort are the prime concerns, are set out in EN 1990/A1 (EN 1990 Annex 2) [2]. That standard also specifies the combinations of actions to be considered when verifying ultimate and serviceability limit states in bridges.

1.2

General principles Chapter 1 of Guidebook 1 on buildings [3] provides a detailed account of the requirements that, pursuant to the Construction Products Directive (CPD) [4], must be met by construction products for their free circulation on the European construction products market. The provisions of the CPD are applicable not only to buildings, but to construction works in general. The definition of construction works contained in Interpretative document No 1: Mechanical resistance and stability [5], expressly includes bridges. Hence, the entire content of that Chapter 1 is relevant to bridges. Readers who wish to consult the general requirements for bridges are referred to the aforementioned chapter of Guidebook 1: the present chapter of Guidebook 2 is limited to questions exclusively pertinent to bridges. In its Annex I [4], the CPD lists six essential requirements that must be met by all construction products and works, as follows: 1. 2. 3. 4. 5. 6.

Mechanical resistance and stability Safety in case of fire Hygiene, health and the environment Safety in use Protection against noise Energy economy and heat retention

Of these, only the first two are generally related to structural behaviour and consequently only they are covered by structural Eurocodes.

7


The provisions of essential requirement 2, safety in case of fire, include a number of structural behaviour-related issues. Indeed, while Part 1.2 of all the Eurocodes, from EN 1991 on actions in general to EN 1992 to EN 1999 on structural materials, deal with structural behaviour in the event of fire, this question was not addressed in Guidebook 1 nor is it included in this Guidebook 2. Consequently, the present publication and specifically this chapter cover only the first of the essential requirements, mechanical resistance and stability. Inasmuch as the third through the sixth requirements do not involve structural behaviour, compliance therewith cannot, generally speaking, be ensured under the provisions of the structural Eurocodes. For that reason, they are not considered in either Guidebook 1 [3] or the present text. That notwithstanding, Annex 2 of EN 1990 [2] lists user safety and comfort requirements that bridges must meet in connection with the fourth essential requirement, safety in use. Since [2] relates these requirements to structural response, they are dealt with in this Guidebook as part of the discussion on serviceability criteria.

2

BASIC REQUIREMENTS

2.1

General The scope of EN 1990 Annex A2 [2] and EN 1991 Part 2 [6 ] covers road, rail and foot bridges. By contrast, certain special kinds of bridges, such as moveable bridges, aqueducts and combination road and railway bridges are excluded. EN 1990 Annex A2 [2] also lists criteria for the combination of actions to be applied to verify ultimate limit states (ULS), serviceability limit states (SLS), partial factors (γ values) and combination coefficients (ψ values). These issues are addressed in detail in Chapter 6 hereunder. Annex A2 also lays down procedures and methods for verifying SLS when such limit states are not related to the structural materials. 2.2

Permanent design situation requirements All the provisions of [3] on ULS and SLS requirements are applicable to bridges and nothing more is needed to say. Only to highlight that, in fact, the indication on the Guidebook 1 on the greater influence, today, of SLS than ULS requirements is particularly pertinent to bridges, whose longer spans and stronger and lighter materials intensify their susceptibility to deformation and vibrations. As a result, serviceability limit states may be more quickly reached in such structures. In addition to that technological challenge, designers are faced with an aesthetic issue: in bridges, beauty is essentially a result of the structure itself, which means that possible flaws cannot be cloaked for want of a superstructure. 2.3

Transient situation requirements In general, the load conditions affecting bridges during construction (see figure 1) can vary in an important way from the conditions prevailing during normal use. The structural strength scheme may also differ. Piers and decks that are designed to form portal frames in the finished structure, for instance, may work like cantilevered beams with wholly different stress distributions during construction. Therefore construction stage conditions must be specially taken into account and the construction phases planned with particular care. One condition, whose verification is normally very important in bridges, but much less so in buildings, is the static equilibrium during construction. In certain stages of the construction, actions or limit states may be of greater influence than in the finished structure. The EQU limit states in these transient situations are often relevant and meticulous planning

8


is required to prevent EQU failure. In such cases, the possible position and values of selfweight and loads at different construction times must be taken into consideration, in the full understanding that the values of characteristic actions, partial factors and combination coefficients may differ from the permanent situation values. In this phase, the weight of concrete cast on decks, for instance, is regarded to be not a permanent action as in the finished structure, but a variable action. As a result, the γ-factor applicable will be γQ equal to 1.5 rather than γG equal to 1.35. The values of these parameters applicable to transient situations are given in Eurocode EN 1991-1-6: Actions during execution [7].

Figure 1. Bridge under construction 2.4

Accidental situation requirements Of the accidental actions listed in EN 1991-1-7 [8], only impacts and gross errors are generally applicable to bridges. Impact may be the result of collision either under or on the bridge. These situations are dealt with separately in the Eurocodes: the former in [8] and the latter in [6] as traffic loads. In bridges, clearly, other strategies besides design can be implemented to protect the structure from major impact damage. Such alternative strategies include: - the use of low sensitivity, highly robust structural typologies, such as redundancies - the use of structural systems that warn of collapse i.e., ductile members - the prevention or reduction of possible hazards by providing suitable clearance, ample distance between lane centrelines and bridge members liable to be impacted (bollards...) and so on. The effect of impact on lightweight structures, such as some footbridges, is not covered by [7]. In these cases, the aforementioned alternative strategies would be ever more relevant. Accidental actions affecting bridges are discussed in depth in Chapter 5 hereunder.

3

SERVICEABILITY REQUIREMENTS

3.1

General As specified in [3], most serviceability criteria defined in terms of structural materials are equally applicable to bridges and buildings.

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In addition to general serviceability criteria, however, certain specific criteria are in place for bridges in connection with user safety and comfort, more specifically to avoid excessive deck vibration or deformation. 3.2

Serviceability criteria for road bridges Table 1 gives the design values for the combinations of actions to be considered in SLS verification. While the Eurocode recommends partial factors, γ, equal to 1,0 when calculating the design values for actions from their characteristic values, this criterion may be modified in national annexes.

Table 1. Design values for use in combinations of actions Combination Permanent actions Gd Prestress Unfavourable Favourable Characteristic Gkj,sup Gkj,inf P

Variable actions Qd Leading Others Qk,1 ψ0,i Qk,i

Frequent

Gkj,sup

Gkj,inf

P

ψ1,1 Qk,1

ψ2,i Qk,i

Quasi-permanent

Gkj,sup

Gkj,inf

P

ψ2,1 Qk,1

ψ2,i Qk,i

Generally speaking, EN 1990 [1] recommends the use of characteristic and frequent combinations for irreversible and reversible SLS, respectively, and the quasi-permanent combination for long-term effects and structural aesthetics. Certain specific SLS for road bridges address durability or user comfort and safety. Damage to structural load bearings before the end of their design working life must be prevented by limiting the amplitude of deck vibration over the supports. Another solution is to adopt for these elements, if replaceable, a shorter service life than for other members: 15-25 years, instead of the 100 years normally established for bridges. There are SLS directly related to user comfort and safety, such as uplift of the deck in the supports (and, which would be, as noted, also cause damage to the bearings). Since these limit states are related to human safety may be demanding higher safety levels than other SLS. Wind- or traffic-induced deck vibrations may also have to be limited to ensure user comfort. 3.3

Serviceability criteria for footbridges The pursuit of ever lighter weight and more engaging bridge designs has led to some well-known cases of footbridges subject to excessive vibrations, causing extreme user discomfort and forcing to take important measures to improve its conditions. Depending of the different situations, Annex 2 [2] recommends considering different load values: -

-

10

for persistent design situations, the presence of a group of about 8 to 15 people (depending on the deck area) walking at a normal pace for other traffic categories, depending on the design situations: permanent, transient or accidental, or the deck area or part of it in consideration, specific load cases should be considered, when relevant: presence of streams of pedestrians (significantly more than 15 persons), or occasional festive or choreographic events EN 1990 Annex 2 [3] recommends the following maximum deck acceleration values: 0.7 m/s2 for vertical vibrations


0.2 m/s2 for horizontal vibrations due to normal use 0.4 m/s2 for exceptional crowd conditions. Such verifications are generally necessary only in footbridges with low natural frequencies, i.e., under 5 Hz for vertical and under 2.5 Hz for horizontal (lateral) and torsional vibrations. These low natural frequencies often occur in light footbridges. 3.4

Serviceability criteria for railway bridges As in road bridges, the SLS for railway bridges are related to durability and user comfort or safety, all in the context of excessive deck deformation or vibration and ultimately of bridge stiffness. Traffic may be compromised by excessive bridge deformation, which generates unacceptably large vertical and horizontal variations in track geometry and vibrations in bridge members. Such excessive vibrations may also cause ballast instability, inadmissible reductions in wheel-rail contact forces or rail fatigue, not to mention passenger discomfort. Given that user safety is highly sensitive to track conditions, the following checks should be performed: -

vertical acceleration in the deck, to prevent ballast instability; vertical deflection in the deck throughout each span; unrestrained uplift at the bearings, to avoid premature bearing failure; twist of the deck measured along the centre line of each track on the approaches to and passage over bridges to minimise the risk of train derailment; - horizontal transverse deflection; - limits to the first natural frequency of lateral span vibration, to prevent resonance between the bridge and the lateral motion of vehicles on their suspension.

Some of the above phenomena also affect passenger comfort due to excessive vertical or horizontal acceleration. To determine the effect of the actions on the bridge could be necessary to perform a dynamic analysis, the EN 1991-2 [6] gives the conditions when this analysis is needed. In general is needed in bridges serving lines with Maximum Line Speed in site bigger than 200 km/h, with no simple structure, spanning more than 40 m and with first natural torsional frequency more than 1,2 times the first natural bending frequency. That document indicates also the way to perform a dynamic analysis, but this is out of the scope of this Guidebook. If this dynamic analysis is not needed, static load effects are enhanced by a dynamic factor Φ. This factor assume the value Φ2 or Φ3 depending on the track conditions, it results 1,44 Φ2 = + 0.82 1.00 ≤ Φ2 ≤ 1.67 (1) LΦ − 0.2 for carefully maintained track and 2,16 Φ3 = + 0.73 1.00 ≤ Φ3 ≤ 2.0, (2) LΦ − 0.2

being LΦ the determinant length (length associated withΦ), which is given in EN 1991-2 [6] paragraph 6.4.5.3 depending on kind and dimensions of the elements. EN 1991/A1 [2] gives the criteria regarding the traffic safety limiting vertical acceleration in deck, deck twist and vertical deformation of the deck: - Vertical acceleration in deck: acceleration is limited to prevent track instability and thereby ensure traffic safety. The maximum design values specified for frequencies of

11


up to 3.5 Hz or 1.5 times the frequency of the fundamental mode of vibration of the member considered are 3.5 m/s2 in ballasted track or 5 m/s2 for decks in which the elements supporting the track are secured directly. - Deck twist: maximum track twist must be limited. For tracks with gauge s [m] of 1.435 m, t [mm/3m] measured over a length of 3 m (see figure 2) should not exceed the values given in Table 2.

Figure 2 – Definition of deck twist Table 2. Recommended maximum values for deck twist Speed range V (km/h) V ≤ 120 120 < V ≤ 200 V > 200

Maximum twist t (mm/3m) t ≤ 4,5 t ≤ 3,0 t ≤ 1,5

- Vertical deformation of the deck: Passenger comfort depends on vertical acceleration inside the coach during travel on, approach to, passage over and departure from the bridge. Vertical acceleration must be limited, therefore, to ensure acceptable comfort levels, according to Table 3. Table 3. Recommended comfort levels Comfort level Vertical acceleration bv (m/s2) Very high 1.0 High 1.3 Acceptable 2.0

4

DESIGN WORKING LIFE AND RELIABILITY MANAGEMENT

One of the first things a designer must know when designing a structure is its projected working life. The indicative design working life categories listed in Eurocode EN 1990 [2] and given in Table 4 below may be modified in the national annexes. In bridges, as noted earlier, not all members need be designed to the same working life: some of them, which are more or less readily replaceable, like bearings, may be classified under category 2 and designed for shorter working lives than the main members. In Table 4 bridges are included under category 5, with a design working life of 100 years. These values are indicative only, and in each specific case subject to an agreement between the owner (usually the authorities) and the designer, in which bridge characteristics play a significant role: traffic density, accessibility, existence of alternative routes and so on. For instance, for a bridge in a main road the 100 years design working life appears adequate, but for a minor bridge serving an area with little traffic with alternative paths a 25 or 50 years would be more adequate design working life.

12


Table 4. Design working life. Indicative values Design working Indicative design Examples life category working life (years) 1 10 Temporary structures (1) 2 10-25 Replaceable structural parts, e.g. gantry girders, bearings 3 15-30 Agricultural and similar structures 4 50 Building structures and other common structures 5 100 Monumental building structures, bridges, and other civil engineering structures (1) Structures or parts of structures that can be dismantled with a view to being re-used should not be considered as temporary. Closely reliability level and its attainment are associated with design working life. In the Eurocodes, partial factors and characteristic values are based on a design working life of 50 years and a reliability index of β = 3,8, assuming normal consequences. In bridges, the β value and consequently the partial factors and characteristic values listed in the Eurocodes derived from there would seemingly have to be revised to adapt them to a 100-year design working life and potentially sizeable economic consequences. This option shall be necessary in works of major importance. In most common bridges, however, this approach to raising the reliability index is not always justified. Rather, greater reliability should be attained via suitable quality control during construction and satisfactory inspection and maintenance policies throughout the working life. The aim of this approach is to reduce the dispersion of material strength values, lower the probability of gross errors and raise the likelihood of detecting minor flaws. These requirements are more commonly met and more readily assumed in bridges than buildings.

5

DURABILITY

Durability is a major issue in bridges. They normally have a fairly long design working life (100 years) and are directly exposed to environmental conditions, for they have not protective superstructure. Furthermore, in cold climates de-icing salts, which cause aggressive corrosion in steel, are frequently strewn over bridges. In any event, the presence of water always intensifies durability problems in any material. The requirements for long durability can be summarised as follows: - appropriate choice of materials, in keeping with the environmental conditions - careful design from the standpoint of durability: speedy evacuation of rainwater, for instance - quality control measures - inspection and maintenance programme tailored to the prevailing conditions.

6

REFERENCES

[1] EN 1990 Eurocode - Basis of structural design. CEN, Brussels, 2002. [2] EN 1990/A1 – Application for bridges. CEN, Brussels, 2002. 13


[3] Milan Holický et al., Guidebook1: Load Effects on Buildings. Leonardo da Vinci Project, CTU, Klokner Institute, Prague, 2009 [4] Construction Products Directive (Council Directive 89/106/EEC). European Commission, Enterprise Directorate-General, 2003 http://ec.europa.eu/enterprise/construction/internal/cpd/cpd.htm [5] Interpretative document No. 1: Mechanical resistance and stability. European Commission, Enterprise Directorate-General, 2004 http://ec.europa.eu/enterprise/construction/internal/intdoc/idoc1.htm [6] EN 1991-2 – Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges. CEN, Brussels, 2003. [7] EN 1991-1-6 – Eurocode 1: Actions on structures – Part 6: Actions during execution. CEN, Brussels, 2003. [8] EN 1991-1-7 – Eurocode 1: Actions on structures – Part 7: Accidental actions. CEN, Brussels, 2006.

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Chapter 2: Basis of design – methodological aspects

CHAPTER 2: BASIS OF DESIGN – METHODOLOGICAL ASPECTS Milan Holický1 and Dimitris Diamantidis2 1

Klokner Institute, Czech Technical University in Prague, Czech Republic 2 University of Applied Sciences, Regensburg, Germany

Summary Uncertainties affecting structural performance can never be entirely eliminated and must be taken into account when designing any construction work. Various design methods and operational techniques for verification of structural reliability have been developed and worldwide accepted in the past. The most advanced operational method of partial factors is based on probabilistic concepts of structural reliability and risk assessment. General principles of structural reliability and risk assessment can be used to specify and further calibrate partial factors and other reliability elements. Moreover, developed calculation procedures and convenient software products can be used directly for verification of structural reliability using probabilistic concepts and available experimental data.

1

INTRODUCTION

1.1

Background materials Basic concepts of structural reliability are codified in a number of national standards, in the new European document EN 1990 [1] and the International Standard ISO 2394 [2]. Additional information may be found in the background document developed by JCSS [3] and in recently published handbook to EN 1990 [4]. Guidance for application of probabilistic methods of structural reliability may be found in working materials provided by JCSS [5] and in relevant literature listed in [4] and [5]. 1.2

General principles General principles of structural reliability are described in both the international documents EN 1990 [1] and ISO 2394 [2]. Basic requirements on structures are specified in Section 2 of EN 1990 [2]: a structure shall be designed and executed in such a way that it will, during its intended life, with appropriate degrees of reliability and in an economic way - sustain all actions and influences likely to occur during execution and use; - remain fit for the use for which it is required. It should be noted that two aspects are explicitly mentioned: reliability and economy (see also Guidebook 1 [6]). However, this Guidebook shall be primarily concerned with reliability of bridge structures, which include - structural resistance; - serviceability; - durability.

15


Additional requirements may concern fire safety of structures (see Handbook 5) or other accidental design situations. In particular it is required by EN 1990 [1] that in the case of fire, the structural resistance shall be adequate for the required period of time. To verify all the aspects of structural reliability implied by the above-mentioned basic requirements, an appropriate design lifetime, design situations and limit states should be considered. Note that the basic lifetime for a common building is 50 years and that, in general, four design situations are identified: permanent, transient, accidental and seismic. Two types of limit states are normally verified: ultimate limit states and serviceability limit states.

2

UNCERTAINTIES

2.1

Classification of uncertainties It is well recognised that construction works including bridges are complicated technical systems suffering from a number of significant uncertainties in all stages of execution and use. Depending on the nature of a structure, environmental conditions and applied actions, various types of uncertainties become more significant than the others. The following types of uncertainties can be identified in general: - natural randomness of actions, material properties and geometric data; - statistical uncertainties due to a limited size of available data; - uncertainties of the resistance and load effect models due to simplifications of actual conditions; - vagueness due to inaccurate definitions of performance requirements; - gross errors in design, during execution and use; - lack of knowledge concerning behaviour of new materials and actions in actual conditions. The order of the listed uncertainties corresponds approximately to the decreasing level of current knowledge and available theoretical tools for their description and consideration in design (see following sections). It should be emphasized that most of the above listed uncertainties (randomness, statistical and model uncertainties) can never be eliminated absolutely and must be taken into account when designing any construction work. 2.2

Available tools to describe uncertainties Natural randomness and statistical uncertainties may be relatively well described by available methods provided by the theory of probability and mathematical statistics. In fact the EN 1990 [1] gives some guidance on available techniques. However, lack of credible experimental data (e.g. for new materials, some actions including environmental influences and also for some geometrical properties) causes significant problems. In some cases the available data are inhomogeneous, obtained under different conditions (e.g. for material resistance, imposed loads, environmental influences, for inner dimensions of reinforced concrete cross-sections). Then it may be difficult, if not impossible, to analyse and use them in design. The uncertainties of computational models may be to a certain extent assessed on the basis of theoretical and experimental research. EN 1990 [1] and materials of JCSS [5] provide some guidance. The vagueness caused by inaccurate definitions (in particular of serviceability and other performance requirements) may be partially described by the means of the theory of fuzzy sets. However, these methods have a little practical significance, as suitable experimental data are rarely available. The knowledge of the behaviour of new materials and

16


structures may be gradually increased through theoretical analyses verified by experimental research. The lack of available theoretical tools is obvious in the case of gross errors and lack of knowledge, which are nevertheless often the decisive causes of structural failures. To limit gross errors due to human activity, a quality management system including the methods of statistical inspection and control may be effectively applied. Various design methods and operational techniques, which take these uncertainties into account, have been developed and worldwide used. The theory of structural reliability provides background concept techniques and theoretical bases for description and analysis of the above-mentioned uncertainties concerning structural reliability.

3

RELIABILITY

3.1

General The term "reliability" is often used very vaguely and deserves some clarification. Often the concept of reliability is conceived in an absolute (black and white) way – the structure either is or isn’t reliable. In accordance with this approach the positive statement is understood in the sense that “a failure of the structure will never occur“. This interpretation is unfortunately an oversimplification. Although it may be unpleasant and for many people perhaps unacceptable, the hypothetical area of “absolute reliability” for most structures (apart from exceptional cases) simply does not exist. Generally speaking, any structure may fail (although with a small or negligible probability) even when it is declared as reliable. The interpretation of the complementary (negative) statement is usually understood more correctly: failures are accepted as a part of the real world and the probability or frequency of their occurrence is then discussed. In fact in the design it is necessary to admit a certain small probability that a failure may occur within the intended life of the structure. Otherwise designing of civil structures would not be possible at all. What is then the correct interpretation of the keyword “reliability” and what sense does the generally used statement “the structure is reliable or safe” have? Several bridge failures have been occurred in the past. Frequent causes of bridge failures are floods, collisions and fatigue problems. Figure 1 shows the failure of the bridge over the Mississippi River in central Minneapolis, which collapsed in 2007. The bridge had an age of 40 years, not many compared to the 100 years desired lifetime of bridge. Therefore the reliability of a bridge should be focussed through all phases i.e. design, construction, operation, maintenance and upgrading. Under this basic consideration it appears even more important to implement reliability concepts from the very initial design stage and consequently to use modern reliability based design elements. Therefore a scientific definition of reliability and an associated derivation of the reliability based design elements are necessary. Such steps have been implemented in the Eurocodes and are explained and illustrated next. 3.2

Definition of reliability A number of definitions of the term “reliability” are used in literature and in national and international documents. ISO 2394 [2] provides a definition of reliability, which is similar to the approach of national standards used in some European countries: reliability is the ability of a structure to comply with given requirements under specified conditions during the intended life, for which it was designed. In quantitative sense reliability may be defined as the complement of the probability of failure.

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Figure 1. Failure of the Mississippi River bridge in Minneapolis, August 2007. Note that the above definition of reliability includes four important elements: -

given (performance) requirements – definition of the structural failure, time period – assessment of the required service-life T, reliability level – assessment of the probability of failure Pf, conditions of use – limiting input uncertainties.

An accurate determination of performance requirements and thus an accurate specification of the term failure are of uttermost importance. In many cases, when considering the requirements for stability and collapse of a structure, the specification of the failure is not very complicated. In many other cases, in particular when dealing with various requirements of occupants’ comfort, appearance and characteristics of the environment, the appropriate definitions of failure are dependent on several vaguenesses and inaccuracies. The transformation of these occupants' requirements into appropriate technical quantities and precise criteria is very hard and often leads to considerably different conditions. In the following the term failure is being used in a very general sense denoting simply any undesirable state of a structure (e.g. collapse or excessive deformation), which is unambiguously given by structural conditions. The same definition as in ISO 2394 is provided. In Eurocode EN 1990 [1] including note that the reliability covers the load-bearing capacity, serviceability as well as the durability of a structure. Fundamental requirements include the statement (as already mentioned) that ”a structure shall be designed and executed in such a way that it will, during its intended life with appropriate degrees of reliability and in an economic way sustain all actions and influences likely to occur during execution and use, and remain fit for the use for which it is required”. Generally a different level of reliability for load-bearing capacity and for serviceability may be accepted for a structure or its parts. In the documents [1] and [2] the

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probability of failure Pf (and reliability index β) are indicated with regard to failure consequences (see Guidebook 1 [6]). 3.3

Probability of failure The most important term used above (and in the theory of structural reliability) is evidently the probability of failure Pf. In order to defined Pf properly it is assumed that structural behaviour may be described by a set of basic variables X = [X1, X2, ... , Xn] characterizing actions, mechanical properties, geometrical data and model uncertainties. Furthermore it is assumed that the limit state (ultimate, serviceability, durability or fatigue) of a structure is defined by the limit state function (or the performance function), usually written in an implicit form as Z(X) = 0

(1)

The limit state function Z(X) should be defined in such a way that for a favourable (safe) state of a structure the function is positive, Z(X) ≥ 0, and for a unfavourable state (failure) of the structure the limit state function is negative, Z(X) < 0 (a more detailed explanation is given in the following Chapters of this Guidebook 2). For most limit states (ultimate, serviceability, durability and fatigue) the probability of failure can be expressed as Pf = P{Z(X) < 0}

(2)

The failure probability Pf can be assessed if basic variables X = [X1, X2, ... , Xn] are described by appropriate probabilistic (numerical or analytical) models. Assuming that the basic variables X = [X1, X2, ... , Xn] are described by time independent joint probability density function ϕX(x) then the probability Pf can be determined using the integral Pf =

∫ ϕ X ( x )dx

(3)

Z( X ) < 0

More complicated procedures need to be used when some of the basic variables are time-dependent. Some details concerning theoretical models for time-dependent quantities (mainly actions) and their use for the structural reliability analysis are given in other Chapters of this Guidebook 2. However, in many cases the problem may be transformed to a timeindependent one, for example by considering in equation (2) or (3) a minimum of the function Z(X) over the reference period T. Note that a number of different methods [2] and software products [8, 9, 11] are available to calculate failure probability Pf defined by equation (2) or (3). 3.4

Reliability index An equivalent term to the failure probability is the reliability index β, formally defined as a negative value of a standardized normal variable corresponding to the probability of failure Pf. Thus, the following relationship may be considered as a definition

β = −ΦU−1 ( Pf )

(4)

Here Φu−1 ( p f ) denotes the inverse standardised normal distribution function. At present the reliability index β defined by equation (4) is a commonly used measure of structural reliability in several international documents [1], [2], [5]. It should be emphasized that the failure probability Pf and the reliability index β represent fully equivalent reliability measures with one to one mutual correspondence given by equation (4) and numerically illustrated in Table 1.

19


In EN 1990 [1] and ISO 2394 [2] the basic recommendation concerning required reliability level is often formulated in terms of the reliability index β related to a certain design working life. Table 1. Relationship between the failure probability Pf and the reliability index β. Pf 10−1 10−2 10−3 10−4 10−5 10−6 10−7 1.3 2.3 3.1 3.7 4.2 4.7 5.2 β 3.5

Time variance of failure probability When the vector of basic variables X = X1, X2, ... , Xm is time variant, then the failure probability p is also time variant and should be always related to a certain reference period T, which may be generally different from the design working life Td. Considering a structure of a given reliability level, the design failure probability pd = pn related to a general reference period Tn = n T1 can be approximately assessed as pd ~ n p1 corresponding to the period T1 where T1 denotes the basic period, for example 1 year.

4

RELIABILITY TARGETS

4.1

General on risk acceptance Risk acceptance criteria are introduced in the EN 1990 [1] in terms of target and acceptable (i.e. design) failure probabilities and associated reliability indices. They are used in order to obtain safety factors for design purposes. The values have been derived through long studies by combining the various approaches reviewed in the previous paragraph. The values reflect the possible failure consequences, the reference time period and are valid for component failures. Special attention must be given to global failure conditions and to target reliability criteria for existing structures. 4.2

Reliability classes Design failure probabilities pd are usually indicated in relation to the expected social and economical consequences in order to reflect the aforementioned risk acceptance criteria. Table 2 shows classification of target reliability levels provided in EN 1990 [1]. Reliability indexes β are given for two reference periods T (1 year and 50 years) only, without any explicit link to the design working life Td. The values are based on calibration and optimization and reflect results from several studies. It is noted that similar β-values as in Table 1 are given in other national and international guidelines (see for example [2], [3]). It should be underlined that a couple of β values (βa and βd) specified in Table 2 for each reliability class (for 1 year and 50 years) corresponds to the same reliability level. Practical application of these values depends on the time period Ta considered in the verification, which may be connected with available statistical information concerning time variant actions (wind, earthquake, etc.) and the related vector of basic variables X = X1, X2, ..., Xn. By considering for example a 50 years design working period then the reliability index βd = 4.3 should be used in the verification of structural reliability. The same reliability level corresponding to class 3 is achieved when the time period Ta = 1 year and βa = 5.2 is used, again in case of limit states dominated by time varying actions. If the working life is 100 years as it usual for bridges then in case of time dependent actions the lifetime target reliability index (i.e. for T= 100years) can be obtained from the formulae discussed in §3.3 of Guidebook 1 [6] and results approximately as βd = 4,1 (for T = 100 years)

20


Table 2. Reliability classification in accordance with EN [1] Reliability Consequences for Reliability index β classes loss of human life, βa for Ta= βd for Td= economical, social 1 year 50 years and environmental consequences RC3 – high High 5.2 4.3 RC2 – normal

Medium

4.7

3.8

RC1 – low

Low

4.2

3.3

Examples of buildings and civil engineering works

Important bridges, public buildings Residential and office buildings Agricultural buildings, greenhouses

Important bridges are in general important structures with considerable consequences of failure and should be in general RC3 (reliability class 3) structures. However bridges can be also classified in the three categories of Table 2 according to the consequences of failure. A small bridge in a rural area with limited traffic can be for example associated to RC1. The target reliability can be obtained also on the basis of an optimisation procedure as illustrated in Appendix A of this chapter. 4.3

Global failure – robustness Structures are composed of various elements such as columns, beams, plates etc. The aforementioned target reliability values are valid for components, since structural design is based on design of components. The global reliability i.e. the reliability against collapse of the entire bridge system or a major part of it is a function of the reliability of all the elements against local failure but also of the system response to local failure. The assumption that a consistent level of reliability of a structural system is reached by an adequate reliability of its members is not generally valid. Especially for bridges structures subjected to extreme environmental loads such as earthquake or wind or accidental loads such as explosion or impact is not sufficient. Therefore structural codes have additional requirements regarding global failure or progressive failure of the structure. The associated requirements in the Eurocodes are discussed in Chapter 5 of Guidebook 1 [6].

5

DESIGN METHODS IN PRACTICE

5.1

General During their historical development the design methods have been closely linked to the available empirical, experimental as well as theoretical knowledge of mechanics and the theory of probability. The development of various empirical methods for structural design gradually crystallized in the twentieth century in three generally used methods, which are, in various modifications, still applied in standards for structural design until today: the permissible stresses method, the global factor and partial factor methods. All these methods are often discussed and sometimes reviewed or updated. The following short review of historical development illustrates general formats of above mentioned design methods and indicates relevant measures that are applied to take into account various uncertainties of basic variables and to control resulting structural reliability. In addition a short description of probabilistic methods of structural reliability and their role

21


in further development of design procedures is provided. Detailed description of probabilistic methods of structural reliability is given in Chapter 2, Chapter 3 and in Annex B of Guidebook 1 [6]. 5.2

Permissible stresses The first of the worldwide-accepted design methods for structural design is the method of permissible stresses that is based on linear elasticity theory. The basic design condition of this method can be written in the form

σmax < σper, where σper = σcrit / k

(6)

The coefficient k (greater than 1) is the only explicit measure supposed to take into account all types of uncertainties (some implicit measures may be hidden). Moreover, only a local effect (a stress) σmax is compared with the permissible stress σper and, therefore, a local (elastic) behaviour of a structure is used to guarantee its reliability. No proper way is provided for treating geometric non-linearity, stress distribution and ductility of structural materials and members. For that reasons the permissible stress method leads usually to conservative and uneconomical design. However, the main insufficiency of the permissible stress method is lack of possibility to consider uncertainties of individual basic variables and computational models used to assess load effects and structural resistances. Consequently, reliability level of structures exposed to different actions and made of different material may be not only conservative (uneconomical) but also considerably different. 5.3

Global safety factor The second widespread method of structural design is the method of global safety factor. Essentially it is based on a condition relating the standard or nominal values of the structural resistance R and load effect E. It may be written as s = R / E > s0.

(7)

Thus the calculated safety factor must be greater than its specified value s0 (for example s0=1,9 is commonly required for bending resistance of reinforced concrete members). The global safety factor method attempts to take into account realistic assumptions concerning structural behaviour of members and their cross-sections, geometric non-linearity, stress distribution and ductility; in particular through the resulting quantities of structural resistance R and action effect E. However, as in the case of the permissible stresses method the main insufficiency of this method remains a lack of possibility to consider the uncertainties of particular basic quantities and theoretical models. The probability of failure can, again, be controlled by one explicit quantity only, by the global safety factor s. Obviously, harmonisation of reliability degree of different structural members made of different materials is limited. 5.4

Partial factor method At present, the most advanced operational method of structural design [1, 2] accepts the partial factor format (sometimes incorrectly called the limit states method) usually applied in conjunction with the concept of limit states (ultimate, serviceability or fatigue). This method can be generally characterised by the inequality Ed (Fd, fd, ad, θd) < Rd (Fd, fd, ad, θd)

(8)

where the design values of action effect Ed and structural resistance Rd are assessed considering the design values of basic variables describing the actions Fd = ψ γF Fk, material

22


properties fd = fk /γm, dimensions ad + ∆a and model uncertainties θd. The design values of these quantities are determined (taking into account various uncertainties) using their characteristic values (Fk, fk, ak, θk), partial factors γ, reduction factors ψ and other measures of reliability [1, 2, 3, 4], Thus the whole system of partial factors and other reliability elements may be used to control the level of structural reliability. Detailed description of the partial factor methods used in Eurocodes method is provided in Guidebook 1. Compared with previous design methods the partial factor format obviously offers the greatest possibility to harmonise reliability of various types of structures made of different materials. Note, however, that in any of the above listed design methods the failure probability is not applied directly. Consequently, the failure probability of different structures made of different materials may still considerably vary even though sophisticated calibration procedures were applied. Further desired calibrations of reliability elements on probabilistic bases are needed; it can be done using the guidance provided in the International standard ISO 2394 [2] and European document EN 1990 [1]. 5.5

Probabilistic methods The probabilistic design methods introduced in the International Standard [2] are based on a requirement that during the service life of a structure T the probability of failure Pf does not exceed the design value pd or the reliability index β is greater than its design value βd Pf ≤ Pd or β > βd

(9)

In EN 1990 [1] the basic recommended reliability index for ultimate limit states βd = 3.8 corresponds to the design failure probability Pd = 7.2 × 10-5, for serviceability limit states βd = 1.5 corresponds to Pd = 6.7 × 10-2. These values are related to the design working life of 50 years that is considered for building structures and common structures. In general greater β - values should be used when a short reference period (one or five years) will be used for verification of structural reliability. It should be mentioned that probabilistic methods are not yet commonly used in design praxis. However, the developed calculation procedures and software products (for example [8, 9] and [11]) already enable the direct verification of structural reliability using probabilistic concepts and available experimental data. Recently developed software product CalCode [11] is primarily intended for calibration of codes based on the partial factor method. 5.6

Risk assessment The risk assessment of a system consists of the use of all available information to estimate the risk to individuals or populations, property or the environment, from identified hazards. The risk assessment further includes risk evaluation (acceptance or treatment). The whole procedure of risk assessment is typically an iterative process as indicated in Figure 2. The first step in the risk analysis involves the context (scope) definition related to the system and the subsequent identification of hazards. The system is understood as a bounded group of interrelated, interdependent or interacting elements forming an entity that achieves in its environment a defined objective through the interaction of its parts. In the case of technological hazards related to civil engineering works, a system is normally formed from a physical subsystem, a human subsystem, their management, and the environment. Note that the risk analysis of civil engineering systems (similar to the analysis of most systems) usually involves several interdependent components (for example human life, injuries, and economic loss). Any technical system may be exposed to a multitude of possible hazard situations. In the case of civil engineering facilities, hazard situations may include both environmental effects (wind, temperature, snow, avalanches, rock falls, ground effects, water and ground

23


water, chemical or physical attacks, etc.) and human activities (usage, chemical or physical attacks, fire, explosion, etc.). As a rule, hazard situations due to human errors are more significant than hazards due to environmental effects. S tart

Probability analysis

Conse que nce a na lysis

Ris k es tima tion

Risk assessment

Ha za rd ide ntifica tion

Risk analysis

Definition of the system

R isk e va lua tion

No R is k tre a tm e nt

Acce pta ble risk? Yes S top

Figure 2. Flowchart of iterative procedure for risk assessment.

6

CONCLUDING REMARKS

The basic concepts of the probabilistic theory of structural reliability are characterized by two equivalent terms, the probability of failure Pf and the reliability index β. Although they provide limited information on the actual frequency of failures, they remain the most important and commonly used measures of structural reliability. Using these measures the theory of structural reliability may be effectively applied for further harmonisation of reliability elements and for extensions of the general methodology for new, innovative structures and materials. Historical review of the design methods worldwide accepted for verification of structural members indicates different approaches to considering uncertainties of basic variables and computational models. The permissible stresses method proves to be rather conservative (and uneconomical). The global safety factor and partial factor methods lead to similar results. Obviously, the partial factor method, accepted in the recent EN documents, represents the most advanced design format leading to a suitable reliability level that is

24


relatively close to the level recommended in EN 1990 (β = 3.8). The most important advantage of the partial factor method is the possibility to take into account uncertainty of individual basic variables by adjusting (calibrating) the relevant partial factors and other reliability elements. Various reliability measures (characteristic values, partial and reduction factors) in the new structural design codes using the partial factor format are partly based on probabilistic methods of structural reliability, partly (to a great extent) on past empirical experiences. Obviously the past experience depends on local conditions concerning climatic actions and traditionally used construction materials. These aspects may be considerably different in different countries. That is why a number of reliability elements and parameters in the present suite of European standards are open for national choice. It appears that further harmonisation of current design methods will be based on calibration procedures, optimisation methods and other rational approaches including the use of methods of the theory of probability, mathematical statistics and the theory of reliability and risk assessment. The probabilistic methods of structural reliability provide the most important tool for gradual improvement and harmonisation of the partial factor method for various structures from different materials. Moreover, developed software products enable direct application of reliability methods for verification of structures using probabilistic concepts and available data. Design of a structure assisted by risk assessment may be effectively used when there is a need to consider failure consequences of a system containing the structure and costs of safety measures. Probabilistic optimisation of the system utility may provide valuable information concerning the optimum target reliability level. Finally it should be mentioned that many famous structures have been designed according to the Eurocodes [12], and a typical case is the famous bridge in Millau, France, which is illustrated in Figure 3.

Figure 3. Millau bridge in France

25


7.

REFERENCES

EN 1990 Eurocode - Basis of structural design. CEN, Brussels, 2002. ISO 2394 General principles on reliability for structures, ISO, 1998. JCSS: Background documentation, Part 1 of EC 1 Basis of design, 1996. Gulvanessian, H. – Calgaro, J.-A. – Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, ISBN: 07277 3011 8, 192 pp. [5] JCSS: Probabilistic model code. JCSS working materials, http://www.jcss.ethz.ch/, 2001. [6] Milan Holický et al., Guidebook1: Load Effects on Buildings. Leonardo da Vinci Project, CTU, Klokner Institute, Prague, 2009 [7] EN 1991-1-1 Eurocode 1 Actions on structures. Part 1-1 General actions. Densities, selfweight, imposed loads for buildings, CEN, Brussels, 2002 [8] VaP, Variable Processor, version 2.3, Petschacher Software and Project, Feldkirchen, 2009. [9] COMREL, version 7.10, Reliability Consulting Programs, RCP MUNICH, 1999. [10] ISO 13822. Basis for design of structures - Assessment of existing structures, ISO 2001. [11] CodeCal, Excel sheet developed by JCSS, http://www.jcss.ethz.ch/. [12] de Ville de Goyet, V. Important Structures designed Using the Eurocodes, Workshop EU-Russia cooperation on standardisation for construction, Moscow, October 2008. [13] Diamantidis, D. - P. Bazzurro, Target Safety Criteria for Existing Structures, Workshop on Risk Acceptance and Risk Communication, Stanford University, CA, USA, March 2007. [14] Joint Committee on Structural Safety (JCSS), Assessment of Existing Structures, RILEM Publications S.A.R.L., 2000. [15] Allen, D.E., 1993, Safety Criteria for the Evaluation of Existing Structures, Proceedings IABSE Colloquium on Remaining Structural Capacity, Copenhagen, Denmark. [16] CSA S6-1990, 1990, Design of Highway Bridges: Supplement No. 1-Existing Bridge Evaluation, Canadian Standards Association, Ottawa, Ontario. [1] [2] [3] [4]

26


Appendix A to Chapter 2 – Principles of probabilistic optimization

A.1

General principles

Principles of probabilistic optimization are illustrated considering the objective function in a basic form of the total cost Ctot(x,q,n) as n

Ctot(x,q,n) = Cf ∑ Pf ( x,i ) Q(q,i ) + C0 + x C1

(A.1)

i =1

Here x is a decision parameter of the optimization (a parameter of structure resistance), q is annual discount rate (e.g. 0.03, an average long run value of the real discount rate in European countries), n the number of years of a considered design working life (e.g. 50, 100), Pf(x,i) failure probability at the year i, Cf malfunctioning costs (due to loss of structural utility), Q(q,i) discount factor dependent on the annual discount rate q and number of years i, C0 initial cost independent of decision parameter x, and C1 cost per unit of the decision parameter x. Note that the design working life is considered here as a given determistic quantity. In reality the working life for a given design is a random quantity depending on social and physical factors. The design itself may aim at some optimum. This, option, however, is neglected in this appendix. Assuming independence, the annual probability of failure Pf(x,i) at the year i is given by the geometric sequence Pf(x,i) = p(x) (1 − p(x))i−1

(A.2)

where p(x) denotes the initial probability of failure that is dependent on the decisive parameter of structural resistance x. Then the failure probability Pfn(x) during n years can be estimated by the sum of the sequence Pf(x,i) given as Pfn(x) = 1 – (1 – p(x))n ≈ n p(x)

(A.3)

Where the approximation indicated in equation (A.3) is acceptable for small probability p(x) < 10−3. The discount factor of the expected future costs at the year i is considered in a usual form as Q(q,i) = 1/ (1+q)i

(A.4)

Here q denotes discount rate. Thus, the cost of malfunctioning Cf is discounted by the factor Q(q,i) depending on the discount rate q and the point in time (number of year i) when the loss of structural utility occurs. The necessary conditions for the minimum of the total cost follow from equation (A.1) as n ∂C tot ( x, q, n) ∂P ( x, i ) (A.5) = C f ∑ Q ( q, i ) f + C1 = 0 ∂x ∂x i =1 thus n ∂P ( x, i ) C (A.6) Q (q, i ) f =− 1 ∑ ∂x Cf i =1 Equation (A.6) represents a general form of the necessary condition for the minimum of total cost Ctot(x,q,n) and the optimum value xopt of the parameter x. It generates also the

27


optimum (target) probability of failure and corresponding target value of the reliability index β. Considering equation (2) and (4) the total costs Ctot(x,q,n) described by equation (A.1) may be written as n

 (1 − p( x))  1−  (1 + q)  Ctot(x,q,n) = Cf p( x)  + C 0 + x C1 (1 − p( x)) 1− (1 + q)

(A.7)

Note that the total sum of expected malfunction costs during the period of n years is dependent on the product of the one-time malfunction Cf , initial probability p(x) and a sum of the geometric sequence having the quotient (1− p(x)/(1+ q). Thus the total malfunction cost Ctot(x,q,n) depends on the annual probability of failure p(x), discount rate q and on number of years n. For small probabilities of failure p(x), the cost given by equation (A.7) may be well approximated as Ctot(x,q,n) ≈ Cf p(x) PQ(x,q,n) + C0 + x C1

(A.8)

where the time factor PQ(x,q,n) depends primarily on the number of years n and discount rate q. It is almost independent of the structural parameter x and, for small probability p(x) may be approximate as n

n

 (1 − p( x))   1  1−  1−   (1 + q)  (1 + q)    = PQ(q,n) PQ (x,q,n) = ≈ (1 − p( x)) 1 1− 1− (1 + q) (1 + q)

(A.9)

For given q and n the simplified time factor PQ(q,n) is independent of x; for q = 0.03 and n = 50, the simplified time factor PQ(q,n) ~ 26.5, for q = 0.03 and n = 100 PQ(q,n) ~ 32,5. If the discount rate is small, q ~ 0, then the simplified time factor converts to number of years n, PQ(q,n) ~ n. The necessary condition for the minimum of the total costs then follows from equations (A.8) and (A.9) as dp( x) C1 =− (A.10) dx C f PQ(q, n) Equation (A.10) can be used for assessing the target (optimum) value pt(x) of the initial annual probability p(x).

A.2

A special case

A special case concerns structures when a failure is dominated by a load like wind with an exponential distribution. Then the failure probability p(x) may be expressed as p(x) = exp{-x/a } (A.11) where a is an appropriate statistical parameter. Then the target (optimum) probability pt(q,n) follows from equation (A.10) as a C1 (A.12) pt ( q , x ) = Cf PQ ( q, n)

28


The corresponding optimum structural parameter xopt(q,n) may be then written in an explicit form as a C1 (A.13) xopt ( q, n) = −a ln Cf PQ ( q, n) Note that the simplified time factor PQ(q,n) is independent of x. Obviously the optimum structural parameter xopt(q,n) and the target probability pt(q,n) depends on the cost ratio C1/ C, number of years n and discount rate q. Given the optimum value for x we may find the optimum value for the annual failure probability as: aC1 p( x) opt = (A.14) C f PQ(q, n) The optimum probability for the total design working life Td = n years is then: Pfn ( x) opt ≈ np ( x) opt =

aC1 n PQ(q, n) C f

(A.15)

The value of n/PQ(q,n) runs for q = 0.03 from 1.1 for n = 1 to 3.0 for n=100. Note that for small discount rates q ~ 0, the value n/PQ(q,n) = 1. This means that the optimum failure probability is almost independent of n and, thus, of the design working life. Consider for instance the case that for q = 0.03 and a certain value of aC1/Cf, the optimum failure probability for a design life time of 50 years is 7.2 10-5, which corresponds to β = 3.8. If the design life time is 100 years, the optimum failure probability decreases to 10-4, or β = 3.7; if the design life time is 10 years, the optimum β increases to 3.9. The difference between 3.7 and 3.9 is small enough to be neglected in practical design.

A.3

An example

The following example illustrates the general principles and special case of probabilistic optimization described above. To simplify the analysis the total costs Ctot(x,q,n) given by equation (A.1) is transformed to the standardized form κtot(x,q,n) as

κtot(x,q,n) =

Ctot ( x, q, n) − C0 = p( x) PQ( x, q, n) +x C1/ Cf Cf

(A.16)

Obviously, both costs Ctot(x,q,n) and κtot(x,q,n) achieve the minimum for the same parameter xopt. The exponential expression for the probability p(x) in equation (A.11) is simplified assuming a = 1. Further it is considered that the discount rate q = 0.03 and the total period of time is n = 50 years. Under this assumptions Figure A.1 shows variation of the total standardized costs κtot(x,q,n) (given by equation (A.14)), and the reliability index β corresponding to the probability Pfn(x) (given by equation (A.3)), with structural parameter x for selected costs ratio C1/Cf. The optimum values xopt(q,n) of the structural parameter x are indicated by the dotted vertical lines. Figure A.2 shows variation of the optimum structural parameter xopt(q,n) with the costs ratio C1/Cf , again for q =0.03, n =50. The optimum parameter xopt(q,n) may be obtained from general condition (A.6) or from simplified expression (A.13) for the simplified time factor Q(q,n) ~ 26.5.

29


κtot(x,q,n)

β 6

0.04

5

β

0.03

C1/ Cf =0.002

4

0.02

C1/ Cf =0.001

3

C1/ Cf =0.0001 C1/ Cf =0.00001

0.01

0

6

8

10

12

14

16

x

18

2

1 20

Figure A.1. Variation of the total standardized costs κtot(x,q,n) and the reliability index β with structural parameter x for q =0.03, n = 50 and selected costs ratios C1/Cf. xopt(q,n) for q =0.03, n =50 16 14 12 10 8

C1 / Cf 6 5 1 .10

1 .10

4

1 .10

3

0.01

Figure A.2. Variation of the optimum structural parameter xopt obtained from equation (4) or (13) with the costs ratio C1/Cf for q =0.03, n =50. The optimum reliability index βopt is generally a function of a number of basic variables. In the fundamental case considered above, the optimum reliability index βopt = βopt(q,n,C1/Cf) depends particularly on the discount rate q, design working life n and the cost ratio C1/Cf. However, the index βopt is primarily dependent on the cost ratio C1/Cf and its dependence on the discount rate q and the design working life n seems to be insignificant. This is well illustrated by Figure A.3 that shows variation of the optimum reliability index βopt with the cost ratio C1/Cf for selected design working life n = 10, 50, 100, and the discount rate q = 0.03. It follows from Figure A.3 that with increasing working life n (and increasing discount rate q) the optimum reliability index βopt slightly decreases. For very small discount rate q ∼ 0, the value n/PQ(q,n) = 1 and the index βopt is independent of n.

30


5

βopt n = 10 50 100

4

3

C1/Cf 2 6 . 1 10

1 .10

5

1 .10

4

1 .10

3

0.01

Figure A.3. Variation of the optimum reliability index βopt with the cost ratio C1/Cf for selected design working life n = 10, 50, 100, and the discount rate q = 0.03.

A.4

Concluding remarks

Probabilistic optimization may provide valuable background information concerning reliability differentiation by assessing the target (optimum) probability of failure or the reliability index. It appears that the target reliability index and corresponding resistance parameter depends on - the ratio of cost per unit of structural parameter and cost of structural failure (malfunctioning costs), - the statistical parameter of failure probability, - discount rate and design working life. Results obtained from analyzed example indicate more specific conclusions, validity of which should be conditioned by the accepted assumptions concerning the objective function and annual failure probability. It appears that with increasing malfunctioning cost, the target reliability index and the optimum structural resistance increase (Figure A.1 and A.2). The design working life seems to have a very limited influence on the optimum life time reliability, particularly for small discount rates (Figure 3). For practical purposes the optimum target reliability index and the corresponding structural parameter can be well assessed considering reasonable lower bounds for the design working life (say 50 years) and the discount rate (say 0.02). Available experience indicates that applications of the optimization approach in practice should be primarily based on properly formulated objective functions, and on credible estimates for the cost per unit of structural parameter and cost of structural failure (malfunctioning costs).

31


32

Chapter 3: Static loads due to traffic

CHAPTER 3: STATIC LOADS DUE TO TRAFFIC Pietro Croce1 1

Department of Civil Engineering, Structural Division - University of Pisa

Summary In contemporary codes for bridges, load traffic models for static verification aim to reproduce the real values of the effects induced in the bridges by the real traffic, i.e. the effects having specified return periods; therefore they are artificial models, generally not representing real vehicles. Static traffic load models of EN1991-2 are illustrated and their origin is discussed.

1

INTRODUCTION

Whilst in traditional bridge codes static loads were represented by real vehicles, in modern codes, static verifications are performed through artificial models, resulting in the same values of the effects induced in the bridges by the real traffic. Static traffic load models for road, pedestrian and railway bridges of the new Eurocode EN 1991-2 [1] are illustrated, stressing the background philosophy and the applied methodological criteria. Calibration of traffic models for road bridges was based on real traffic data recorded in two experimental campaign performed in Europe between 1980 and 1994 and mainly on the traffic recorded in may 1986 in Auxerre (F) on the motorway Paris- Lyon. The Auxerre traffic was identified, on the basis of the available data, as the most representative European continental traffic in terms of composition and severity, also taking into account the expected traffic trends. This conclusion was confirmed by more recent studies [2]. The calibration is discussed in much more detail in Appendix A to the present chapter.

2

THE EN 1991-2 LOAD TRAFFIC MODELS FOR ROAD BRIDGES

The static load model for road bridges of the EN 1991-2 is illustrated in the following. As the load model is calibrated for road bridges having carriageway width smaller than 42 m and span length up to 200 m, it cannot be used, in principle, outside the above mentioned field. Anyhow, it results generally safe-sided for bigger spans. 2.1

Division of the carriageway and numbering of notional lanes The carriageway is defined as the part of the roadway surface sustained by a single structure (deck, pier etc.). It includes all the physical lanes (marked on the roadway surface), the hard shoulders, the hard strips and the marker strips. Its width w should be measured between the kerbs, if their height is greater than 100 mm, or between the inner limits of the safety barriers, in all other cases. The carriageway width does not include, in general, the distance between fixed safety barriers or kerbs of a central reservation nor the widths of these barriers.

33


The carriageway is divided in notional lanes, generally 3 m wide, and in the remaining area, according to Table 1, as reported, for example, in figure 1. If the carriageway is physically divided in two parts by a central reservation, then: -

each part, including all hard shoulder or strips, should be separately divided in notional lanes, if the parts are separated by a fixed safety barrier; the whole carriageway, central reservation included, should be divided in notional lanes, if the parts are separated by demountable safety barriers or another road restraint system.

Table 1. Subdivision of the carriageway in notional lanes Carriageway width w

Number of notional lanes nl

Width of a notional lane

Width of the remaining area

w<5.4 m

1

3m

w-3 m

5.4 m ≤w<6 m

2

0.5 w

0

6 m ≤w

Int(w/3)

3m

w-3×nl

Remaining area 3.0

Notional lane n. 1 Remaining area

w

3.0


3.0


Figure 1. Example of lane numbering The location of the notional lanes is not linked with their numbering, so that number and location of the notional lanes should be chosen each time in order to maximize the considered effect. In particular cases, for example for some serviceability limit states or for fatigue verifications, it is possible to derogate from this rule and to consider less severe locations of the notional lanes. In general, the notional lane that gives the most severe effect is numbered lane n. 1 and so on, in decreasing order of severity. The numbering of the carriageway depends on the element under consideration. When the carriageway is made by two separate supported by a unique deck, the lane numbering should regard the entire carriageway, considering, obviously, that lane n. 1 can be alternatively on the two parts (figure 2). When, instead, carriageway consists of two separate parts on two independent decks supported by the same abutments or the same piers, it needs to distinguish two cases: for deck design purposes, each part is considered and numbered independently, while, on the contrary, for abutment or pier design the two parts are considered and numbered together (figure 3). 34


Deck design: one notional lane numbering

Pier design: one notional lane numbering

Figure 2. Lane numbering in case the entire carriageway is supported by a single deck

Deck design: two separate notional lane numbering

Pier design: one notional lane numbering

Figure 3. Lane numbering in case the carriageway consists of two separate parts supported by two separate decks 2.2

Load models for vertical loads Load models representing vertical loads are intended for the evaluation of road traffic effects associated with ULS verifications and with particular serviceability verifications. Four different load models are considered: -

load model n. 1 (LM1) generally reproduces traffic effects to be taken into account for global and local verifications; it is composed by concentrated and uniformly distributed loads; load model n. 2 (LM2) reproduces traffic effects on short structural members; it is composed by a single axle load on specific rectangular tire contact areas; load model n. 3 (LM3), special vehicles, should be considered only when requested, in a transient design situation; it represents abnormal vehicles not complying with national regulations on weight and dimension of vehicles. The geometry and the axle loads of the special vehicles to be considered in the bridge design should be assigned by the bridge owner; load model n. 4 (LM4), a crowd loading.

-

2.3

Load model n. 1 Load model n. 1 consists of two subsystems: -

a system of two concentrated axle loads, representing a tandem system weighing 2⋅αQ⋅Qk (see Table 2), whose geometry is shown diagrammatically in figure 4; a system of distributed loads having a weight density per square meter of αq⋅qk (see Table 2).

The adjustment factors αQ and αq depend on the class of the route and on the expected traffic type: in absence of specific indications, they are assumed equal to 1. The characteristic loads values on the notional i-th lane are indicated αQi⋅Qki and αqi⋅qki while on the remaining area the weight density of the uniformly distributed load is expressed as αqr⋅qkr. 35

0.4

0.4


0.4

0.4

1.6

Longitudinal axis of the bridge

2

1.2

0.4

0.8

0.4

Figure 4. Tandem system of LM1 Table 2. Load model n. 1 – characteristic values Position

Tandem system – Axle load Qik [kN]

Uniformly distributed load qik [kN/m2]

Notional lane n. 1

300

9.0

Notional lane n. 2

200

2.5

Notional lane n. 3

100

2.5

Other notional lanes

0

2.5

Remaining area

0

2.5

For bridges without road signs restricting vehicle weights, it should be assumed αQ1≥0.8 for the tandem system on the first notional lane, while for i≥2, αqi≥1.0 except for the remaining area. The load model n. 1 should apply according to the following rules (see figure 5): -

-

36

in each notional lane only one tandem system should be considered, situated in the most unfavourable position; the tandem system should be considered travelling in the direction of the longitudinal axis of the bridge, centred on the axis of the notional lane; when present, the tandem system should be considered in full, i.e. with all its four wheels; the uniformly distributed loads apply, longitudinally and transversally, only on the unfavourable parts of the influence surface; the two load systems can insist on the same area; the impact factor is included in the load values αQi⋅Qki and αqi⋅qki; when static verification is governed by combination of local and global effects, the same load arrangement should be considered for calculation of local and global effects; when relevant, and only for local verifications, the transverse distance between adjacent tandem systems should be reduced, up to a minimum of 0.4 m.


Q ik

Q ik qik

0.5

Q1k =300 kN q1k =9.0 kN/m

Lane n. 1

2.0 0.5 0.5

w

Q 2k=200 kN 2 q 2k=2.5 kN/m

Lane n. 2

2.0 0.5 0.5 2.0

Q 3k=100 kN 2 q 3k=2.5 kN/m

Lane n. 3

0.5

Remaining area

qrk =2.5 kN/m2

Figure 5. Example of application of load model n.1 Load model n. 2 The local load model n. 2, LM2 (figure 6), consists of a single axle load βQ⋅Qak with Qak=400 kN, dynamic amplification included. Unless otherwise specified βQ=αQ1.

2

0.6


1.4

0.6

2.4

0.35

Figure 6. Load model n. 2 (single axle) The load model, which is intended only for local verifications, should be considered alone on the bridge, travelling in the direction of the longitudinal axis of the bridge. The model should be applied in any location on the carriageway and, if necessary, only one wheel load of βQ⋅200 kN should be considered. If not otherwise specified, the contact surface of each wheel is rectangle, whose dimensions are 0.35 m×0.6 m. 2.5

Load model n. 3 - Special vehicles Besides the above mentioned load models, the Eurocode also foresees the possibility to consider special vehicles, whose transit on the road network and in particular on the bridges

37


is subject to special authorisation, because they exceed the legal limits in length, in width and/or in mass. These special vehicles are represented by a set of standardised arrangements of axle loads, where the bridge owner can pick-up, according to his specific necessities, one or more vehicles to be taken into account in the bridge design. The load model should be considered only if expressly required and its application should regard only the selected special vehicles. A useful reference is represented by the set of standardized special lorries given in the informative Appendix A of EN 1991-2, which is reported in Tables 3.a and 3.b. The nominal values of the axle loads of the special lorries are associated exclusively to transient design situations. Each axle load is considered uniformly distributed over two or three narrow rectangular surfaces 1.20 m long and 0.15 m wide. Axles weighing 150 or 200 kN are considered distributed on two surfaces, axles weighing 240 kN are considered distributed on three surfaces, as illustrated in figure 7. Special vehicles characterised by axle loads in the interval 150 to 200 kN occupy the notional lane n. 1, while special vehicles characterized by 240 kN axle loads occupy two adjacent notional lane, lanes n. 1 and n. 2 (figure 8). The lanes are situated in the most unfavourable position, at most excluding hard shoulders, hard strips and marker strips. More favourable positions can be considered, if transit is allowed only under special limitations. Table 3.a. Special vehicles with axle weighing 150 and 200 kN 150 kN axle loads

200 kN axle laods

Vehicle weight

Geometry

Axle loads

Vehicle type

600 kN

3×1.5 m

4×150 kN

600/150

900 kN

5×1.5 m

4×150 kN

900/150

1200 kN

7×1.5 m

4×150 kN

1500 kN

9×1.5 m

1800 kN

11×1.5 m

Geometry

Axle loads

Vehicle type

1200/150

5×1.5 m

6×200 kN

1200/200

4×150 kN

1500/150

7×1.5 m

1×100+7× 200 kN

1500/200

4×150 kN

1800/150

8×1.5 m

9×200 kN

1800/200

2400 kN

11×1.5 m

12×200 kN

2400/200

2400 kN

5×1.5+12+5×1.5 m

12×200 kN

2400/200/200

3000 kN

14×1.5 m

15×200 kN

3000/200

3000 kN

7×1.5+12+6×1.5 m

15×200 kN

3000/200/200

3600 kN

17×1.5 m

18×200 kN

3600/200

Table 3.b. Special vehicles with axles weighing 240 kN 240 kN axle loads Vehicle weight

Geometry

Axle loads

Vehicle type

2400 kN

8×1.5 m

10×240 kN

2400/240

3000 kN

12×1.5 m

1×120+12×200 kN

3000/240

3600 kN

14×1.5 m

15×240 kN

3600/240

3600 kN

7×1.5+12+6×1.5 m

15×200 kN

3600/240/240

38


150 kN or 200 kN axle weight 1.2

1.2

of the bridge

Longitudinal axis

0.3

240 kN axle weight 1.2

1.2 0.3

1.2 0.3

Figure 7. Axle lines and wheel contact areas for special vehicles

4.20

Figure 8. Arrangement of special vehicle on the carriageway Since special vehicles are assumed to move at low speed (5 km/h), dynamic effects are not significant; therefore dynamic magnification is considered included in the nominal values of the axle loads. As a rule, concomitance of the special vehicles with the load model n. 1 is taken into account considering that the lane (lane n. 1) or the two adjacent lanes (lanes n. 1 and 2), occupied by the standardized special vehicle, are not subjected to additional traffic loads in a range of 25 m each side from the front axle and the rear axle of the special vehicle itself, measured in the longitudinal direction as shown in figure 9. According to the aforementioned general rules, the remaining parts of the notional lanes and of the carriageway are loaded with the frequent values of load model n. 1. 2.6

Load model n. 4 – Crowd loading The uniformly distributed load model n. 4, the crowd loading, is particularly significant for bridges situated in urban areas and it should be considered only when expressly required. 39


Figure 9. Simultaneity of special vehicles and load model n. 1 The nominal value of the load, including dynamic amplification, is equal to 5.0 kN/m2, while the combination value is reduced to 3.0 kN/m2, even if it seems that calculations are considerably simplified adopting a value of 2.5 kN/m2, like in lanes 2 and 3 and in the remaining area. The crowd loading should be applied on all the relevant parts of the length and width of the bridge deck, including the central reservation, if necessary. 2.7

Characteristic values of horizontal actions

2.7.1 Braking and acceleration forces The braking or acceleration force, denoted by Qlk, should be taken as a longitudinal force acting at finished carriageway level. The characteristic values of Qlk depends on the total maximum vertical load induced by LM1 on notional lane n. 1, as follows 180 ⋅ α Q1 kN ≤ Q lk = 0.6 ⋅ α Q1 ⋅ (2 ⋅ Q1k ) + 0.10 ⋅ α q1 ⋅ q1k ⋅ wl ⋅ L ≤ 900 kN ,

(1)

being w1 is the lane width and L the length of the loaded area. This force, that includes dynamic magnification, should be considered located along the axis of any lane. When the eccentricity is not significant, the force may be considered applied along the carriageway axis and uniformly distributed over the loaded length.

40


2.7.2 Centrifugal force The centrifugal force Qtk is a transverse force acting at the finished carriageway level and perpendicularly to the axis of the carriageway. Unless otherwise specified, Qtk should be considered as a point load at any deck cross section. The characteristic value of Qtk, including dynamic magnification, depends on the horizontal radius r [m] of the carriageway centreline and on the total maximum weight of the vertical concentrated loads of the tandem systems of the main loading system Qv Qv = ∑i α Qi ⋅ (2 ⋅ Qik ) ,

(2)

and it is given by Qtk = 0.2 ⋅ Qv [kN], r<200 m; Qtk = 40 ⋅

Qv [kN], 200 m≤r≤1500 m; r

(3)

Qtk = 0 , r>1500 m. 2.8

Groups of traffic loads on road bridges When simultaneity of traffic actions with non-traffic actions is significant, the characteristic values of the traffic actions can be determined considering the five different, and mutually exclusive, group of loads reported in table 4, where the dominant component action is underlined. Each groups of loads indicated in the table should be considered as defining a characteristic action for combination with non-traffic loads, but they can be used also to evaluate infrequent and frequent values. To obtain infrequent combination values it is sufficient to replace in table 4 characteristic values with the infrequent ones, leaving unchanged the others, while frequent combination values are obtained replacing characteristic values with the frequent ones and setting to zero all the others. The ψ-factors for traffic loads on road bridges are reported in table 5. Table 4. Assessment of characteristic values of multi-component actions for traffic loads on road bridges Footways and cycle tracks

Carriageway Vertical loads Group of loads

Main load model

1

Characteristic values

2

Frequent values

Special vehicles

Horizontal loads Crowd loading

Centrifugal force

Uniformly distributed loads Combination value

Characteristic Characteristic values values

3


4 5

Braking force

Vertical loads only

Characteristic values see §2.5 and figure 9



41


Table 5. Recommended values of ψ- factors for traffic loads on road bridges ψ0

ψ1infq

ψ1

ψ2

Tandem System

0.75

0.80

0.75

0

UDL

0.40

0.80

0.40

0

gr1b (single axle)

0

0.80

0.75

0

Traffic loads

gr2 (Horizontal Forces)

0

0

0

0

(see table 9)

gr3 (Pedestrian loads)

0

0.80

0

0

gr4 (LM4 – Crowd loading))

0

0.80

0.75

0

gr5 (LM3 – Special vehicles))

0

1.0

0

0

Action

Symbol gr1a (LM1)

The values of ψ0, ψ1, ψ2 for gr1a, referring to load model n.1 are assigned for routes with traffic corresponding to adjusting factors αQi, αqi, αqr and βQ equal to 1, while those relating to UDL correspond to the most common traffic scenarios, in which an accumulation of lorries can occur, but not frequently. Other values may be envisaged for other classes of routes or of other classes of expected traffic, according to the relevant α factors. For example, for traffic situations characterised by severe presence of continuous traffic, like for bridges in urban areas, a value of ψ2 other than zero may be envisaged for the UDL system of LM1 only. The factors for the UDL, given in table 5, apply not only to the distributed part of LM1, but also to the combination value of the pedestrian load mentioned in table 5.

3

ACTIONS ON FOOTBRIDGES

The section of EN1991-2 concerning actions on footbridges covers explicitly actions on footways, cycle tracks and footbridges and it is specifically devoted only to footbridges. The uniformly distributed load qfk and the concentrated load Qfwk given in the following, where relevant, can be also used for parts of road and railway bridges accessible to pedestrian. Load models and their representative values include dynamic amplification effects and should be used for all serviceability and ultimate limit state static calculations, excluding fatigue limit states. When vibration assessments based on specific dynamic analysis are necessary, ad hoc studies should be performed. Some guidance about vibration check of footbridges is given in EN1990-A2 [3] as summarized in the following §3.6 3.1

Vertical load models Three different vertical load models can be envisaged for footbridges: 1. an uniformly distributed load representing the static effects of a dense crowd; 2. one concentrated load, representing the effect of a maintenance load; 3. one or more, mutually exclusive, standard vehicles, to be taken into account when maintenance or emergency vehicles are expected to cross the footbridge itself.

3.1.1 Uniformly distributed load The crowd effect on the bridge is represented by a uniformly distributed load. When risk of dense crowd exists or if required, Load Model 4 for road bridges should be considered also for footbridges.

42


On the contrary, if the application of the aforesaid Load Model 4 is not required, a uniformly distributed load qfk, to be applied to the unfavourable parts of the influence surface, should be considered. Value of qfk depends on the loaded length L [m] and it is given by (4)

120 ≤ 5.0 kN/m 2 . L + 30

2,5 kN/m 2 ≤ q fk = 2.0 +

In road bridges supporting footways or cycle tracks, the characteristic value 5 kN/m2 or the combination value (2.5 kN/m2) should be considered, according to figure 10.

2

q fk =5.0 kN/m

Figure 10. Characteristic load on a footway (or cycle track) of a road bridge 3.1.2 Concentrated load For local assessments, a 10 kN concentrated load Qfwk, representing a maintenance load should be considered on the bridge, acting on a square surface of sides 0.1 m. The concentrated load will not be combined with other variable non-traffic loads. Obviously, when the service vehicle described in §3.1.3 is taken into account, Qfwk should be disregarded. 3.1.3 Service vehicle Service vehicles for maintenance, emergencies (e.g. ambulance, fire) or other services can be assigned when necessary, depending on the particular situation. When no special information is available and no permanent obstacle prevents the transit of vehicles on the bridge deck, the special service vehicle defined in figure 11 should be considered in transient design situations. When consideration of the service vehicle is not required, the transit of the vehicle shown in figure 11 should be considered as accidental.

0.2

Q sv2=40 kN

0.2

Q sv1=80 kN

0.2

1.3

0.2 Longitudinal axis of the bridge 0.2

0.2

3

Figure 11. Service or accidental vehicle

43


3.2

Characteristic values of horizontal forces For footbridges, it should be considered a horizontal force Qflk, acting simultaneously with the corresponding vertical load, whose characteristic value is equal to the greater of: -

10% of the total load corresponding to the uniformly distributed load or 60% of the total weight of the service vehicle, when relevant.

The horizontal force, which does not coexists with the concentrated load Qfwk, acts along the bridge deck axis at the pavement level on a square surface of sides 0.1 m and it is normally sufficient to ensure the horizontal longitudinal stability of the footbridge.

3.3

Groups of traffic loads on footbridges Vertical loads and horizontal forces due to traffic should be combined, when relevant, considering the groups of loads defined in table 6. Each one of these mutually exclusive groups defines a characteristic action for combination with non – traffic loads. When combinations of traffic loads with actions of different nature need to be considered, any group of loads in table 6 should be considered as one action.

Table 6. Definition of groups of loads (characteristic values) Load type Load system

Vertical forces Uniformly Service distributed load vehicle

Horizontal forces

Groups

gr1

Fk

0

Fk

of loads

gr2

0

Fk

Fk

Wind and snow are not considered to act simultaneously with traffic loads on footbridges, except on roofed bridges, which are considered according to the appropriate rules given in EN 1991-1-3. Wind and thermal actions should not be considered as simultaneous.

3.4

Application of the load models The traffic models described in the previous paragraphs, with the exception of the service vehicle model, may also be used for pedestrian and cycle traffic areas of the deck of road bridges limited by parapets and not included in the carriageway, as well as on footpaths of railway bridges. These actions are free, so that the vertical loads should be applied anywhere within the relevant areas, in order to obtain the most adverse effects.

3.5

Verifications of traffic induced deformations and vibrations for footbridges As known, traffic induced deformations and vibrations of footbridges strongly influence the serviceability level. The main structure is generally affected by vertical and horizontal vibrations, as well as torsional vibrations, either alone or coupled with vertical and/or horizontal vibrations The design situations to be studied depend on the type of pedestrian traffic admitted on individual footbridge during its design working life and on the level of regulation, authorisation and control of the traffic itself. In general, the following design situations could be taken into account:

44


1. A persistent design situation considering the simultaneous presence of a group of about 8 to 15 persons walking normally; 2. the simultaneous presence of streams of pedestrians (significantly more than 15 persons), which could be persistent, transient or accidental depending on boundary conditions, like location of the footbridge in urban or rural areas, the possibility of crowding due to the vicinity of railway and bus stations, schools, important building with public admittance, relevance of the footbridge itself and so on; 3. occasional sports, festive or choreographic events, requiring ad hoc investigations. 3.5.1. Bridge-traffic interaction A pedestrian normally walking exert on the bridge vertical periodic forces, with a frequency ranging between 1 and 3 Hz, and horizontal periodic forces, perfectly synchronised with the vertical ones, with a frequency ranging between 0.5 and 1.5 Hz, but forces exerted by several persons are generally not synchronised and characterised by different frequencies. When the frequency of the periodic forces normally exerted by pedestrians is close to a natural frequency of the deck, it commonly happens that the subjective perception of the bridge oscillation induces the pedestrian to synchronise their steps with the vibrations of the bridge, so that resonance occurs, increasing considerably the response of the bridge. The number of pedestrians participating to the resonance is highly random and depends on the number of persons on the bridge: when the number of persons on the bridge is bigger than 10, the number of the participating persons is a decreasing function of their number, but correlation between forces exerted by pedestrians themselves may increase with movements. 3.5.2. Dynamic models of pedestrian loads The studies about dynamic modelling of pedestrian loads for footbridge design are still in progress, also in consideration of the serviceability failures produced by vibrations in recently built footbridges (Millennium bridge in London, Solferino bridge in Paris), so that what given in this chapter should be considered as merely informative. Dynamic pedestrian loads could be defined by means of two separate models consisting of: 1. a concentrated force (Fn), representing the excitation by a limited group of pedestrians, which should be systematically used for the verification of the comfort criteria; 2. a uniformly distributed load (Fs), representing the excitation caused by a continuous stream of pedestrians, which should be used separately from Fn where specified. Load model Fn, which should be placed in the most adverse position on the bridge deck, is represented by one pulsating force with a vertical component Fn,v Fn,v = 280 k v ( f v ) sin(2π f v t ) [N]

(5)

and an horizontal component Fn,h Fn,h = 70 k h ( f h ) sin(2π f h t ) [N],

(6)

to be considered separately. In equations (5) and (6) fv is the natural vertical frequency of the bridge closest to 2 Hz, fh is the natural horizontal frequency of the bridge closest to 1 Hz, t is the time in s and kv(fv) and kh(fh) are suitable coefficients, depending on the frequency according to figure 12. For the evaluation of fv, fh and of the inertia effects, Fn should be associated, if unfavourable, with a static mass equal to 800 kg, applied at the same location. 45


The uniformly distributed load model Fs, to be applied on the whole deck of the bridge, consists in a uniformly distributed pulsating load with vertical component Fs ,v = 15 k v ( f v ) sin(2π f v t ) [N/m2],

(7)

and horizontal component Fs ,h = 4 k h ( f h ) sin(2π f h t ) [N/m2],

(8)

to be considered separately. For the evaluation of fv, fh and of the inertia effects, Fs should be associated, if unfavourable, with a static mass equal to 400 kg/m2, applied at the same area. In special cases, likes relevant footbridges, it may be possible to increase the reliability degree of the assessments, by specifying to apply Fs on limited unfavourable areas (e.g. span by span) or with an opposition of phases on successive spans.

3

kv(f v)

kh(f h)

3

2

2

1

1

Vertical vibrations 0

1

2

3

4

f v [Hz]

Horizontal vibrations 5

0

1

2

3

4

f h [Hz]

5

Figure 12. Relationships between coefficients kv(fv), kh(fh) and frequencies fv, fh 3.5.3. Comfort criteria In order to ensure pedestrian comfort, the maximum acceleration of any part of the deck should not exceed -

0,7 m/s2 for vertical vibrations; or 0,15 m/s2 for horizontal vibrations.

The assessment of comfort criteria should be performed for natural vertical frequency of the footbridge up to 5 Hz or horizontal and torsional natural frequencies up to 2.5 Hz. In the evaluation of natural frequencies fv or fh the mass of any permanent load should be taken into account and the stiffness parameters of the deck should be calculated using the short term dynamic elastic properties of the structural material and, if significant, of the parapets. It must be noted that generally the mass of pedestrians is relevant only for very light decks. If comfort criteria cannot be satisfied with a significant margin, the possible installation of dampers in the structure after its completion should be envisaged in the design.

46


Evaluation of accelerations shall take into account the damping of the footbridge, through the damping factor ζ referring to the critical damping, or the logarithmic decrementδ, which is equal to 2πζ. For rather short spans, when calculations are performed using the groups of pedestrians given before, the effect of the damping on the acceleration can be considered through the reduction factors: -

k n,v = 1 − exp(−2π nζ ) for vertical vibrations or

-

k n,h = 1 − exp(−π nζ ) for horizontal vibrations,

where n is the number of steps necessary to cross the span under consideration. For a simply supported bridge, the design value of the vertical acceleration a1d due to the group of pedestrians may then be assumed as: a1d = 165 k v ( f v )

1 − exp(−2π nζ ) [m/s2], Mζ

(9)

where M is the total mass of the bridge, f is the relevant, i.e. the determining, fundamental frequency, and kv(fv) is given in figure 12.

4

ACTIONS ON RAILWAY BRIDGES

Like road bridge load models, also railway bridges load models of EN 1991-2 do not describe actual loads, although weight and geometry of trains are often exactly known. Load models for railway bridges have been set-up in such a way that their effects, amplified by the dynamic coefficients, which in this case are given separately, represent the characteristic effects of the most severe train traffic expected on the European railways network. The rail traffic within the scope in EN1991-2 concerns standard track gauge and wide track gauge of the European mainline network. In general, the load models given here are not applicable to narrow-gauge railways, tramways and other light railways, preservation railways, rack and pinion railways, funicular railways and so on, that require specific loading models, to be specifically defined. Of course, when other traffic conditions need to be considered, which are outside the scope of the load models specified in EN 1991-2, specific alternative load models and combination rules should be defined for the particular case under consideration.

4.1

Representation of actions and nature of rail traffic loads Actions due to normal railway operations are usually represented by: -

vertical loads, vertical loading for earthworks, dynamic effects, centrifugal forces, nosing forces, traction and braking forces, combined response of a structure and track to variable actions, aerodynamic effects from passing trains,

47


-

4.2

actions due to overhead line equipment and other railway infrastructure and equipment.

Vertical loads In EN 1991-2 five load models are given for railway loading: 1. 2. 3. 4. 5.

Load Model 71, representing normal rail traffic on mainline railways; Load Model SW/0, which could be relevant for continuous bridges; Load Model SW/2, representing heavy trains; Load Model HSLM, representing high speed (>200 km/h) passenger trains; Load Model “unloaded train”, representing the effect of an unloaded train.

The loadings can vary depending on the nature, the volume and maximum weight of rail traffic on different railways, as well as on different qualities of track. 4.2.1 Load Model 71 Load Model 71, representing the vertical static effect of normal rail traffic, is composed by a 4-axles vehicle weighing 1000 kN and by a uniformly distributed load equal to 80 kN/m, not limited in extension, as illustrated in figure 13. Q vk=250 kN 250 kN 250 kN 250 kN

q vk =80 kN/m

0.8

1.6

1.6

1.6

0.8

q vk =80 kN/m

no limitation in extension

no limitation in extension

Figure 13. Load Model 71 Heavier or lighter rail traffics can be taken into account multiplying the characteristic values of loads given in figure 13 by a factor α, which should assume following values: 0.75; 0.83; 0.91; 1.10; 1.21; 1.33; 1.46, being α=1.0 the α factor for normal traffic. In any case, the same factor α should be considered to evaluate equivalent vertical loading for earthworks and earth pressure effects, centrifugal, traction and braking forces, combined response of structure and track to variable actions, accidental actions and Load Model SW/0 for continuous span bridges.

4.2.2 Load Models SW/0 and SW/2 Load Model SW/0 represents the static effect of normal traffic on continuous beams, while Load Model SW/2, which represents the static effect of heavy rail traffic, should be taken into account only where heavy rail traffic is foreseen. LM SW/0 and SW/2 are represented by two uniformly distributed loads of length a, spaced by c, as reported in table 7 and illustrated in figure 14.

Table 7. Characteristic values of vertical loads for Load Models SW/0 and SW/2

48

Load

qvk

a

c

Model

[kN/m]

[m]

[m]

SW/0

133

15.0

5.3

SW/2

150

25.0

7.0


q vk

q vk

a

c

a

Figure 14. Load Models SW/0 and SW/2 4.2.3 Load Model “unloaded train” The so called unloaded train is a particular load model consisting of a vertical uniformly distributed load with a characteristic value of 10.0 kN/m, which could be used for some particular verifications. 4.2.4 Eccentricity of vertical load models 71 and SW/0 The eccentricity of vertical load due to lateral displacement to be considered for static verifications assessments can be taken into account considering the ratio of wheel loads on all axles as up to 1.25:1.00 on each track, so that it results the eccentricity e shown in figure 15, which should be not greater than r/18, being r the transverse distance between the wheel loads. The loads to be taken into account are the appropriate uniformly distributed and concentrated loads pertaining to LM71 and SW0 when required. The load eccentricity e may be neglected in fatigue verifications

qv1 +q v2 Qv1 +Qv2 qv2 e

qv1 Qv1

qv2

qv1

Qv2

Qv2 Qv1

≤1.25 ≤1.25

r e≤ 18 r Figure 15. Eccentricity of vertical loads 4.2.5

Distribution of axle loads Distribution of axle loads by the rails, sleepers and ballast, for all kind of trains and verifications, including fatigue, can be taken into account -

in the longitudinal direction considering that a point force or an axle load is distributed by the rail over three adjacent sleepers, being the loaded one subjected to the 50% of the load and each of the two adjacent one subjected to the 25% of the load as indicated in figure 16; for local verifications a load dispersal with 4:1 slope

49


-

through the ballast can be considered according to figure 17, where a represents the sleeper’s spacing; in the transverse direction depending on the track configuration: the actions should be distributed transversely according to figure 18 for bridges with ballasted track without cant, according to figure 19 for bridges with ballasted tracks with cant and for full length sleepers, where the ballast is only consolidated under the rails, or for duo-block sleepers; according to figure 20 for bridges with ballasted tracks with cant and full length sleepers; finally, in case of bridges with ballasted track and cant and for full length sleepers, where the ballast is only consolidated under the rails, or for duo-block sleepers, figure 20 should be suitably modified to take into account the transverse load distribution under each rail shown in figure 19.

Qvi

Qvi 4

a

Qvi 2

a

a

Qvi 4

a

Figure 16. Longitudinal distribution of concentrated loads

Qvi

sleeper

4:1

reference plane

Figure 17. Longitudinal dispersal of sleeper loads through the ballast 4.2.6 Equivalent vertical loading for earthworks and earth pressure effects To determine earth pressure effects or to design earthworks under or adjacent to the track, an equivalent vertical loading due to rail traffic actions can be considered for the evaluation of global effects, represented by the appropriate load of model LM71 or SW/2 50


uniformly distributed over a width of 3.0 m at a level 0.70 m below the running surface of the track. Dynamic effects can be disregarded. For local elements close to a track (e.g. ballast retention walls and so on), the maximum local vertical, longitudinal and transverse loadings on the element due to rail traffic actions should be evaluated. 4.2.7 Footpaths and general maintenance loading For pedestrian and cycle paths and for general maintenance loads a uniformly distributed load with a characteristic value qfk=5 kN/m² should be taken into account, while for design of local elements a concentrated load Qk=2.0 kN acting alone should be applied on a square surface with a 200 mm side.

Qh

Qv

Qr

h

4 :1

A

R M

reference plane

B

σB

σA σM

Figure 18. Transverse distribution of action for ballasted tracks without cant Qh

0.6

Qr

Qv

h

σA

4:1

4:1

A

running plane

0.6

R M

reference plane

B

σB

Figure 19. Transverse distribution of action for duo-block sleepers

51


Qh

Qr

Qv

h

running plane u

4:1

A

R

M

reference plane

B

σB

σA σM

Figure 20. Transverse distribution of action for ballasted tracks with cant Dynamic magnification factors Φ (Φ2, Φ3) Provided that risks of resonance effects and excessive vibrations of the bridge are negligible, dynamic magnification of stresses and vibration effects can be taken into account through the dynamic factor Φ. On the contrary, when risks of resonance or excessive vibrations exist, a suitable dynamic analysis is necessary. In these cases static load effects multiplied by the dynamic factor Φ are unable to predict resonance effects from high speed trains, therefore dynamic analysis techniques, taking into account the time dependant nature of the loading from the High Speed Load Model (HSLM) and Real Trains (e.g. by solving equations of motion) are required for predicting dynamic effects at resonance. The dynamic factors can be applied also to structures with more than one track.

4.3

4.3.1. Definition of the dynamic factor Φ The dynamic factor Φ which increases the static load effects induced by Load Models 71, SW/0 and SW/2 depends on the level of maintenance of tracks. For carefully maintained track, it is 1,44 + 0.82 ≤1.67, 1.00≤ Φ2 = (10) LΦ − 0.2 while for standard maintained track, it is 2,16 1.00≤ Φ3 = + 0.73 ≤ 2.00, (11) LΦ − 0.2 being LΦ the “determinant” length associated with Φ in [m]. For the most common and practically relevant cases, LΦ is defined in tables 8.a, 8.b and 8.c. For cases not covered by the tables, a satisfactory estimate of LΦ can be obtained evaluating LΦ itself as the base length of the influence line for the deflection of the member under consideration. The dynamic factor Φ shall not be used with the loading due to Real Trains, Fatigue Trains, Load Model HSLM and load model “unloaded train”. When the resultant stress in a structural member depends on several effects, each of which relates to a separate structural behaviour, each effect should be calculated using the appropriate determinant length. 52


Table 8.a. Determinant length LΦ Case Structural element Determinant length LΦ Steel deck plate: closed deck with ballast bed (orthotropic deck plate) (global and local transverse stresses) Deck with cross girders and continuous longitudinal ribs 1.1

Deck plate (for both directions)

3 times cross girder spacing

1.2

Continuous longitudinal ribs (including small cantilevers up to 0.5 m – cantilevers greater than 0.5 m require ad hoc studies)

3 times cross girder spacing

Twice the length of the cross girder

1.3 Cross girders

3.6 m (it is recommended adoption of Φ3)

1.4 End cross girders Deck plate with cross girder only 2.1

Deck plate for both directions

Twice the cross girder spacing + 3 m

2.2

Cross girders

Twice the cross girder spacing + 3 m

2.3

End cross girders


Steel grillage: closed deck with ballast bed (orthotropic deck plate) (global and local transverse stresses)

3.1

Rail bearers: - as an element of continuous grillage - simply supported

3 times cross girder spacing Cross girder spacing + 3 m

3.2

Cantilever of the rail bearers


3.3

Cross girders (as part of cross girder/continuous rail bearers grillage)

Twice the length of the cross girder

3.4

End cross girders


For arch bridges and concrete bridges of all types with a cover of more than 1.00 m, Φ2 and Φ3 may be reduced according to the formula red Φ2,3 = Φ2,3 -

h - 1.00 ≥ 1.0 , 10

(12)

being h [m] the height of cover from the top of the deck to the top of the sleeper, including the ballast, or, in case of arch bridges, from the crown to the extrados. Rail traffic actions on columns with a slenderness <30, abutments, foundations, retaining walls and ground pressures may be calculated disregarding dynamic effects. Bridges sensitive to dynamic effects and in any case bridge on high speed lines (V≥200 km/h) require specific dynamic analysis considering Real trains or High Speed Load 53


Model, according to the specific application rules. The question is outside the scope of the present Guidebook and it will not be discussed here.

Table 8.b. Determinant length LΦ Case Structural element Determinant length LΦ Concrete deck slab with ballast (global and local transverse stresses) 4.1

Deck slab as part of box girder or upper flange of the main beam: - spanning transversally to the main girders - spanning in longitudinal direction

3 times span of deck plate 3 times span of deck plate Twice the length of the cross girder

-

cross girders

-

transverse cantilever supporting railway loadings

e

-

If e≤0.5 m, three times the distance between the webs If e>0.5 m ad hoc studies are necessary

4.2 Deck slab continuous (in main girder direction) over the cross girders

Twice the cross girder spacing

4.3

4.4

Deck slab for half through and through bridges : - spanning perpendicular to the main girders - spanning in longitudinal direction

4.5

Deck slab spanning transversely between longitudinal steel beams in filler bridge decks

Twice span of deck slab + 3 m Twice span of deck slab Twice the determinant length in the longitudinal direction

Longitudinal cantilevers of deck slab 4.6 End cross girders or trimmer beams

If e≤0.5 m, 3.6 m (it is recommended adoption of Φ3) If e>0.5 m ad hoc studies are necessary


Note: For cases 1.1 to 4.6 inclusive LΦ cannot exceed the determinant length of the main girders

54


Table 8.c. Determinant length LΦ Case Main girders

Structural element

Determinant length LΦ

5.1

Simply supported girders and slabs (including steel beams embedded in concrete)

Span in the main girder direction

5.2

Girders and slabs continuous over n spans with Lm=(L1+ L2+....+ Ln)/n

LΦ=min[(1+0.1⋅n)⋅Lm; Li,max]

5.3

Portal frames and closed frames or boxes: - single span

-

spanning in longitudinal direction

5.4

Consider as three span continuous beam (use expression in 5.2 with horizontal and vertical lengths of members of frame or box) Consider as multi span continuous beam (use expression in 5.2 with lengths of end vertical members and members) Half span of the bridge

5.5

Single arch, arch rib, stiffened girder of bowstring

5.6

Series of arches with solid spandrels retaining fills

Twice the clear opening

4 times the longitudinal spacing of suspension bars

Suspension bars (in conjunction with stiffening girders) Structural supports 6

Columns, trestles, bearings, uplift bearings, tension anchors and for calculation of contact pressure under bearings

Determinant length of the supported member

Application of traffic loads on railway bridges In designing a railway bridge, the number and the position of the loaded tracks should determined first, according to the relevant influence surfaces. In each verification the greatest number of tracks geometrically and structurally possible should be considered in the less favourable position, irrespective of the effective positions of the intended tracks, according to the given minimum spacing between centre-lines of adjacent tracks. Actions of different nature should be combined considering traffic loads and forces placed in the most unfavourable positions. For the determination of the most adverse load effects, the following rules apply: when LM 71 is considered,

4.4.

-

any number of uniformly distributed loads qvk should be applied on the track and up to four concentrated loads Qvk should be applied once per track,

55


-

for elements carrying two tracks, Load Model 71 shall be applied to either track or both tracks, for bridges carrying three or more tracks, LM 71 should be applied if loaded tracks are less than three, while 0.75 times LM71 should be applied for three or more loaded tracks;

when LM SW/0 is considered, -

-

the loading SW/0 should be applied once to a track; for elements carrying two tracks, LM SW/0 should be applied to either track or both tracks; for bridges carrying three or more tracks, LM SW/0 should be applied if loaded tracks are less than three, while 0.75 times LM SW/0 should be applied for three or more loaded tracks; continuous beam bridges designed for LM71 must be checked for LM SW/0 too;

finally, when LMSW/2 is considered, -

the loading SW/2 shall be applied once to a track, for elements carrying more than one track, LM SW/2 shall be applied to any one track only, with LM71 or LMSW/0 applied to the other tracks, in accordance with the specific application rules.

When relevant, the following rules apply for the Load Model “unloaded train”: -

the Load Model “unloaded train” shall only be considered in the design of structures carrying one track. any number of lengths of the uniformly distributed load qvk shall be applied to a track.

For serviceability assessments concerning deformations, LM 71 and, if relevant, Load Models SW/0 and SW/2 increased by the dynamic factor Φ should be applied. For vibrations or resonance assessments, the above mentioned Load Models for High Speed trains (HSLM) or Real Trains (RT) should be also taken into account. The number of loaded tracks to be loaded when checking the limits of deflections and vibration is given in table 9.

4.5

Horizontal forces - characteristic values

4.5.1 Centrifugal forces When the track is curved over the whole or part of the length of the bridge, the centrifugal force and the track cant should be considered. The centrifugal forces act outwards in a horizontal direction and are applied 1.80 m above the running surface, considering the Maximum Line Speed V allowed on the bridge, except for Load Model SW/2, for which a maximum speed of 80 km/h may be assumed. Said Qvk and qvk the characteristic values of the vertical load components of the railway load models, LM 71, SW/0, SW/2 and “unloaded train”, any enhancement for dynamic effects is disregarded, the characteristic values Qtk, qtk of the corresponding centrifugal forces are given by

56


Qtk =

v2 V2 ( f Qvk ) = ( f Qvk ) [kN] gr 127 ⋅ r (13)

qtk =

2

2

v V ( f qvk ) = ( f qvk ) [kN/m], gr 127 ⋅ r

being f a reduction factor, given in the following.

Table 9. Number of tracks to be loaded for checking limits of deflection and vibration Limit States Checks and associated acceptance criteria Traffic Safety Checks:

1

2

≥3 1 or 2 or 3 or more *) 1 or 2 or 3 or more *) 1 or 2 or 3 or more *)

–

Deck twist (EN 1990 Annex 2 A2.4.4.2.2)

1

1 or 2 *)

–

Deformation of the deck (EN 1990 Annex 2 A2.4.4.2.3)

1

1 or 2 *)

–

Horizontal deflection of the deck (EN 1990 Annex 2 A2.4.4.2.4) Combined response of structure and track to variable actions including limits to vertical and longitudinal displacement of the end of a deck ( 6.5.4) Vertical acceleration of the deck (EN1991-2 6.4.6 and EN 1990 Annex 2 A2.4.4.2.1)

1

1 or 2 *)

1

1 or 2 *)

1 or 2 *)

1

1

1

1

1

1

1

1 or 2 *)

1 or 2 or 3 or more *)

–

–

–

– *)

Number of tracks

SLS Checks: Passenger comfort criteria (EN 1990 Annex 2 A2.4.4.3) ULS Checks Avoidance of unrestrained uplift at bearings

whichever is critical (see multi-component actions in §4.6)

In equations (13), v in m/s and V in km/h are the maximum line speed, g is the gravity acceleration and r is the radius of curvature in m. In case of varying radius, r could suitably set to its mean value. Centrifugal forces should be combined with the pertinent vertical traffic load. The centrifugal force shall not be multiplied by the dynamic factor Φ2 or Φ3. The factor f takes into account the reduced mass of higher speed trains, therefore, as for short loaded lengths the magnitude of centrifugal forces is dictated by faster light vehicles, for Load Model 71 (and where significant Load Model SW/0) the cases considered in table 10 shall be considered, depending on the line speed V and on the adjustment factor α. For Load Model 71 (and LM SW/0, if relevant) the reduction factor f is given by:

 V − 120  814 2.88    ≤1.0, f = 1 − + 1 . 75 1 −   1000  V L f    

(14)

where V is the maximum line speed in km/h and Lf is the influence length in m of the loaded part of curved track on the bridge, which is most unfavourable for the design of the structural element

57


under consideration. For Load Model 71 (and where significant Load Model SW/0) and V>120 km/h, two cases should be taken into account: -

-

case a: in this case Load Model 71 (and where relevant Load Model SW/0) is taken into account with its dynamic factor and the centrifugal force is evaluated according to equations (13) setting V=120 km/h, so that the latter is not reduced (f = 1); case b: in this case Load Model 71 (and where relevant Load Model SW/0) is taken into account with its dynamic factor and the centrifugal force is evaluated according to equations (13) considering the maximum speed V=120 km/h and the related reduction factor f, evaluated according (14).

Table 10. Load cases for centrifugal force evaluation Value of α

α<1

Max Line Speed [km/h] >120

Centrifugal force based on V [km/h]

α

f

V

1

f

1⋅f⋅(LM71”+”SW/0)

Associated vertical traffic based on

Φ⋅α⋅1⋅(LM71”+”SW/0) or Φ⋅α⋅0.5⋅(LM71”+”SW/0) when vertical traffic actions are favourable

≤120

α =1

>120

≤120

α >1

>120*)

≤120

α

1

α⋅1 (LM71”+”SW/0)

0

---

---

---

V

α

1

α⋅1 (LM71”+”SW/0)

0

---

---

---

V

1

f

1⋅f⋅(LM71”+”SW/0)

Φ⋅1⋅1⋅(LM71”+”SW/0) orΦ⋅1⋅0.5⋅(LM71”+”SW/0) when vertical traffic actions are favourable

120

1

1

1⋅1 (LM71”+”SW/0)

0

---

---

---

V

1

1

1⋅1 (LM71”+”SW/0)


0

---

---

---

V

1

f

1⋅f⋅(LM71”+”SW/0)


120

α

1

α⋅1 (LM71”+”SW/0)

0

---

---

---

Φ⋅α⋅1⋅(LM71”+”SW/0) or Φ⋅α⋅0.5⋅(LM71”+”SW/0)

V

α

1

α⋅1 (LM71”+”SW/0)

0 ------* valid only if maximum speed of heavy freight traffic limited to 120 km/h )

58

Φ⋅α⋅1⋅(LM71”+”SW/0) or Φ⋅α⋅0.5⋅(LM71”+”SW/0)

120

when vertical traffic actions are favourable

when vertical traffic actions are favourable


When the line speed V is bigger than 300 km/h and the influence length Lf is bigger than 2.88 m a lower bound exists for f, f (V=300 km/h). For Load Models SW2 and unloaded train f=1.0. 4.5.2 Nosing force The nosing force Qsk, to be always combined with the vertical traffic load, is represented by a concentrated force acting horizontally, applied at the top of the rails, normally to the centre-line of track, on both straight and curved track. For rail traffic with maximum axle load of 250 kN, the characteristic value should be taken as Qsk=100 kN and it should be multiplied by the dynamic magnification factor Qsk. Nosing force should by multiplied by the adjustment factor α only when α>1. 4.5.3 Actions due to traction and braking Traction and braking forces are commonly considered as uniformly distributed over the corresponding influence length La,b for traction and braking effects related to the structural element considered. Traction force is indicated with Qlak and braking force is indicated with Qlbk. Traction and braking forces, applied the top of the rails in the longitudinal direction of the track, should be determine according to the following expressions, which are applicable to all types of track construction, e.g. continuous welded rails or jointed rails, with or without expansion devices, being La,b in m: Traction force:

Qlak = 33 La,b [kN]≤1000 [kN] for Load Models 71, SW/0, SW/2, “unloaded train” and HSLM;

Braking force:

Qlbk = 20 La,b [kN]≤6000 [kN] for Load Models 71, SW/0 and Load Model HSLM Qlbk = 35 La,b [kN] for Load Model SW/2 Qlbk = 2.5 La,b [kN] for Load Model “unloaded train”.

(15)

(16)

In some special case, like for lines carrying special traffic (restricted to high speed passenger traffic for example) the traction and braking forces may be taken as equal to 25% of the sum of the axle-loads of the Real Train acting on the influence length of the action effect of the structural element considered, with an upper limits of 1000 kN for Qlak and 6000 kN for Qlbk.

4.6

Multicomponent traffic actions

4.6.1 Characteristic values of multicomponent actions The simultaneity of the above mentioned traffic loadings may be taken into account by considering the mutually exclusive groups of loads given in EN 1991-2 and reported in table 11. Each groups of loads reported in the table defines a single variable characteristic action, that should be taken into account for combination with non-traffic loads and should be considered as a single variable traffic action. The number of group loads is high; therefore it implies a considerable increase of the number of the load combinations to be considered in the analyses. Anyhow, the number of the load groups can be considerably reduced, making some little safe-sided simplification, so that only four relevant group of loads need to be taken in account, according to table 12.

59


Table 11. Groups of traffic loads (characteristic values of multicomponent actions) Groups of loads Reference EN 1991-2

n= 1

2

≥ 3

Number of tracks

Horizontal forces 6.5.1 6.5.2 Centrifugal Nosing force(1) force

1

11

T1

1

1 (5)

0.5 (5)

0.5 (5)

1

12

T1

1

0.5 (5)

1 (5)

1 (5)

1

13

T1

1 (4)

1

0.5 (5)

0.5 (5)

1 1

14 15

T1 T1

1 (4)

0.5 (5) 0.5 (5)

1 1 (5)

1 1 (5

1 1 2

16 17 21

T1 T1 T1 T2

1 1

1 (5) 0.5 (5) 1 (5) 1 (5)

0.5 (5) 1 (5) 0.5 (5) 0.5 (5)

0.5 (5) 1 (5) 0.5 (5) 0.5 (5)

2

22

T1 T2

1 1

0.5 (5) 0.5 (5)

1 (5) 1 (5)

1 (5) 1 (5)

2

23

2

24

1 (4) 1 (4) 1 (4) 1 (4)

2

26

T1 T2 T1 T2 T1 T2 T1 T2 Ti

1 1 0.5 (5) 0.5 (5) 1 (5) 1 (5) 0.5 (5) 0.5 (5) 0.75 (5)

0.5 (5) 0.5 (5) 1 1 0.5 (5) 0.5 (5) 1 (5) 1 (5) 0.75 (5)

0.5 (5) 0.5 (5) 1 1 0.5 (5) 0.,5 (5) 1 (5) 1 (5) 0.75 (5)

2 ≥3

(1) (2) (3) (4) (5) (6) (7)

Vertical forces

6.3.2/6.3.3 6.3.3 6.3.4 6.5.3 (1) Load Loaded LM 71 SW/2 Unload Traction, group track SW/0 (1), (2) (1),(3) ed train Braking(1) HSLM(6)(7)

27 31

1

1 1

1 1 1 1 0.75

Comment

Max. vertical 1 with max. longitudinal Max. vertical 2 with max. transverse Max. longitudinal Max. lateral Lateral stability with “unloaded train” SW/2 SW/2 Max. vertical 1 with max longitudinal Max. vertical 2 with max. transverse Max. longitudinal Max. lateral SW/2 SW/2 Additional load case

All relevant factors (α, Φ, f, ...) shall be taken into account. SW/0 shall only be taken into account for continuous span bridges. SW/2 needs to be taken into account only if it is stipulated for the line. Factor may be reduced to 0.5 if favourable effect, it cannot be zero. In favourable cases these non dominant values shall be taken equal to zero. HSLM and Real Trains where required in accordance with 6.4.4 and 6.4.6.1.1. of EN 1991-2 If a dynamic analysis is required in accordance with 6.4.4 of EN 1991-2 also see 6.4.6.5(3) of EN 1991-2. Leading component action as appropriate to be considered in designing a structure supporting one track (Load Groups 11-17) to be considered in designing a structure supporting two tracks (Load Groups 11-27 except 15). Each of the two tracks shall be considered as either T1 (Track one) or T2 (Track 2) to be considered in designing a structure supporting three or more tracks; (Load Groups 11 to 31 except 15. Any one track shall be taken as T1, any other track as T2 with all other tracks unloaded. In addition the Load Group 31 has to be considered as an additional load case where all unfavourable lengths of track Ti are loaded.

60


Table 12. Assessment of simplified groups of traffic loads Load type

Vertical load

Horizontal load

Notes

Group of loads

Vertical load (1)

Unloaded train (1)

Traction Braking

Centrifugal force

Nosing force

Group 1 (2)

1.00

----

0.5 (0)

1.00 (0)

1.00 (0)

Maximum vertical and lateral action

Group 2 (2)

----

1.00

0

1.00 (0)

1.00 (0)

Lateral stability

Group 3 (2)

1.00 (0.5)

----

1.00

0.5 (0)

0.5 (0)

Maximum longitudinal action

Group 4 (3)

0.8 (0.6, 0.4)

0.8 (0.6, 0.4)

0.8 (0.6, 0.4)

0.8 (0.6, 0.4)

Concrete cracking

Leading action

(1) (2) (3)

Including all his coefficients φ, α and so on Simultaneity of several actions with their characteristic values, even if improbable, is considered for groups 1, 2 and 3, considering that the consequences of this assumptions are not relevant Group 4 should be considered only for concrete cracking (values in parentheses are to be considered when more than 1 track is loaded: 0.6 for two loaded tracks and 0.4 for three or more loaded tracks.

4.6.2 Other representative values of the multicomponent actions The frequent values of multicomponent actions can be determined using the same rules given above: for each group of loads, instead of characteristic values, the factors given in table 11 affect the frequent values of the relevant actions considered. When the reduced number of groups of loads is considered, according to table 12, only group 4 should be taken into account. Quasi-permanent traffic actions shall be taken as zero.

6

CONCLUDING REMARKS

In recent years, bridge codes have considerably evolved. This evolution is the result of the positive interaction of several factors, namely -

recent progresses of the probabilistic theory of structural reliability; improved knowledge of the statistical parameters governing the extreme value distributions of climatic actions; and, finally, availability of large databases regarding in situ traffic measurements all around the world.

The most evident outcome of the modern code development is the “artificial” nature of traffic load models. Since they aim to reproduce to real traffic effects characterised by specified return period or by given probability to be exceed in the design working life of the bridge, traffic load models can differ considerably by real vehicles, in terms of silhouette and axle’s arrangement as well as in terms of axle and vehicle weights. 61


Amongst the contemporary codes for bridge design, Eurocode emerges for its primary importance, profiting of out to date background and ad hoc studies. It must be highlighted that Eurocode has been widely checked and it is successfully applied in current design practice in Europe. In the present chapter the static traffic load models for road, pedestrian and railway bridges of EN 1991-2 have been discussed, highlighting the application rules and the group of traffic loads to be considered in combination with non-traffic actions. When relevant, particular attention has been devoted to background information, also aiming to suggest safe-sided simplified assumptions. In the appendix A to this chapter, calibration study of the traffic load models for road bridges are illustrated in details, while in the Annex A to the present Guidebook, the future traffic trends of the lorry traffic in European road network is discussed and their consequences on EN 1991-2 load models are analysed. Non traffic actions are illustrated in chapter 5 and load combinations in chapter 7. The practical application of the load models for road bridges is better detailed in chapters 8, 9 and 10, where three case studies are developed.

7

REFERENCES

[1] EN1991-2, Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges. CEN, Brussels, 2003. [2] O’Connor, A.J. et al., Effects of traffic loads on road bridges – Preliminary studies for the re-assessment of the Eurocode 1, Part 3. Proceedings of the 2nd European Conference on Weigh-in-motion of road vehicles. Lisbon, 1998 [3] EN1990-A2, Eurocode: Basis of structural design – Annex A2: Applications for bridges. CEN, Brussels, 2005 [4] Croce, P. & Sanpaolesi, L., Design of bridges. Pisa: TEP, 2004. [5] O’Brien, E.J. et al., Bridge applications of weigh-in-motion. Paris: Laboratoire Central des Ponts et Chaussées, 1998. [6] Bruls, A et al., ENV1991 Part 3: The main model of traffic loads on road bridges. Background studies. Proceedings of IABSE Colloquium on Basis of Design and Actions on Structures. Background and Application of Eurocode 1. Delft, 1996. [7] Croce, P. & Salvatore, W., Stochastic model for multilane traffic effects on bridges. Journal of Bridge Engineering, ASCE, 6(2): 136-143, 2001 [8] Tschemmernegg, F. & al, Verbreiterung und Sanierung von Stahlbrücken. Stahlbau n. 9, 1989. [9] Sedlacek, G. & al. 1991. Eurocode 1 - Part 12. Traffic loads on road bridges. Definition of dynamic impact factors. Report of subgroup 5.

62


Appendix A to Chapter 3 – Development of static load traffic models for road bridges of EN 1991-2

A.1

General principles

Static load models for road bridges of EN 1991-2 have been developed considering that an up to date structural code should -

-

be easy to use; be applicable independently on the static scheme and on the span length of the bridge; reproduce as accurately as possible the real load effects induced on the bridge by all possible flowing and jammed traffic scenarios, that can occur on the bridge during its design working life; the real load effects are characterised by a specified return period or by a given probability to be exceeded during the design working life; include the dynamic magnification due to the road-vehicle and to the bridgevehicle interactions in load values; allow combinations of local and global effects of actions; be unambiguous, covering all the cases that could occur in the design practice.

Obviously, as load models are to be defined and calibrated referring to traffic effects having specified return periods, pre-normative studies had to deal with complicated theoretical and methodological problems. Among these, especially significant were those concerned with the extrapolation to very long time periods of effects due to flowing traffics, recorded on the slow lane for few days or few weeks, taking into account the most severe flowing and/or congested traffic scenarios that could happen on one or on several lanes.

A.2

Static load model philosophy

As a rule, the evaluation of real traffic effects and the subsequent drafting and calibration of the load model can be carried out by analytical and numerical methodologies consisting of: -

identification of the most significant real traffic records; choice of the static schemes and spans of the relevant bridges; specification of the influence surfaces the most significant effects; elaboration of the traffic data and their manipulation to obtain jammed, slowed down and flowing traffic types; determination of the histograms of the extreme values of the effects induced by the transit of the different traffic typologies on the influence surfaces; simulation of extreme scenarios for multilane traffic; elaboration and extrapolation of the histograms of the extreme values of the effects to evaluate the reference values, characterized by specified return period; correction of the reference values to take into account the dynamic effects due to road-vehicle and to vehicle-structure interactions; drafting and calibration of the load model; applicative trials;

63


-

A.3

model refinement.

Statistical analysis of European traffic data

As said, the first phase of the study was devoted to statistic analysis of European traffic data, in order to select the most representative traffics, in terms of the expected flow and composition. Available European traffic’s data were mainly the result of two large measurement campaigns performed, respectively, within 1977 and 1982 on bridges situated in France, Germany, Great Britain, Italy and Holland and within 1984 and 1988 on several roads all around the Europe. Recorded daily flows on the slow lane were varying between 1000 and 8000 lorries on motorways, and between 600 and 1500 lorries on main roads, while fast lane daily flows on motorway and slow lane daily flows on secondary roads resulted drastically reduce to 100-200 lorries. [4, 5]. Statistical analyses, that allowed to know the distributions of the most significant traffic parameters, like traffic composition, inter-vehicle distances, axle’s spacing, weight, length and speed of each lorry, essentially was limited to data recorded in Italy, France and Germany; in fact, UK data appeared poorly representative of the continental situation, while Spanish and Dutch data seemed excessively influenced by the respective road systems peculiarities. Significant data, derived from long distance motorway traffics (Auxerre (F), Garonor (F), Brohltal (D), Fiano Romano (I), Sasso Marconi (I) and Piacenza (I)), are summarized in tables A.1÷A.5. Table A.1 shows the daily flows of cars and lorries per lane and the percentage of inter-vehicular distances smaller than 100 meters; table A.2 illustrates the traffic compositions in terms of standardized lorries, while table A.3 illustrates the composition of the entire fleets of circulating commercial vehicles in the three above mentioned Countries. Finally, daily flows of axles heavier than 10 kN and lorries, together with the respective values of statistical parameters are shown in table A.4 and A.5, respectively. Generally, the analysis of the European traffic data shows that -

mean values of axle-loads and total weight of heavy vehicles strongly depend on the traffic typology, i.e. on the road classification; they are usually very scattered: the statistical distribution of the axle-load is generally unimodal, with the mode around 60 kN, while the statistical distribution of the total weight is bimodal with the first mode around 150 kN and the second mode around 400 kN;

Table A.1. Daily traffic flows per lane Cars

Lorries

% intervehicle distance<100 m

11126

4793

26.7

Garonor (F – 1982)

--

2570

32.6

Garonor (F – 1984)

--

3686

32.3

Auxerre (slow lane) (F)

8158

2630

18


1664

153

8.5

Fiano R. (I)

8500

4000

26.1

Piacenza (I)

8500

5000

30.9

Sasso M. (I)

7500

3500

24.3

Brohltal (D)

64


Table A.2. Composition of the commercial traffic Lorries (%) (2 Axles) 16.6

Lorries (%) (>2 Axles) 1.6

Articulated lorries (%) 40.2

Lorries with trailer (%) 41.6

Garonor (F - 1982)

38.6

2.6

47.6

11.2

Garonor (F - 1984)

47.5

2.2

44.3

6.0


22.7

1.3

65.2

10.8

Auxerre (fast lane) (F)

27.6

3.5

58.4

10.5

Fiano R. (I)

41.4

7.0

29.0

22.6

Piacenza (I)

35.3

7.5

35.8

21.4

Sasso M. (I)

40.1

10.0

30.2

19.7

Brohltal (D)

Table A.3. Composition of the circulating lorry fleets Germany

France

Italy

2 axles

17.0

32.0

38.67

3 axles

5.0

5.8

9.0

4 axles

25.0

25.0

10.0

5 axles

52.0

33.2

33.0

6 axles

1.0

4.0

8.0

> 6 axles

--

--

1.33

Table A.4. Daily flows and statistical parameters of axles heavier than 10 kN and lorries ALL AXLES Flow Pmean σ [kN] [kN]

TANDEM AXLES Pmax [kN]

TRIDEM AXLES

σ Pmax Flow Pmean Flow Pmean [kN] [kN] [kN] [kN]

σ [kN]

Pmax [kN]

Brohltal (D)

19970 59.0 28.4 165.0 1977 116.5 54.6 260.0 1035

60.0

230.0 355.0

Garonor (F) 1982

8470

126.3 49.3 340.0 303

90.0

200.0 295.0

Garonor (F) 1984

11593 59.3 30.0 195.0 1016 132.1 58.1 290.0 489

90.0

200.0 320.0

57.6 27.6 180.0

712

Auxerre (F) 10442 82.5 35.2 195.0 slow lane

844

165.6 54.0 305.0 961

130.0 250.0 390.0

Auxerre (F) fast lane

47

141.2 63.9 275.0

120.0 250.0 390.0

581

73.1 41.2 200.0

51

Fiano R. (I) 15000 56.8 32.9 142.0 2000 115.2 45.5 245.0 900 Piacenza (I)

80.0

260.0 360.0

20000 61.8 31.0 135.0 2500 127.0 44.1 260.0 1500 100.0 220.0 365.0

Sasso M. (I) 13000 61.9 30.8 135.0 1600 136.4 49.5 260.0 800

110.0 250.0 375.0

65


Table A.5. Daily flow and total weights of commercial vehicles Pmean σ Pmax Flow [kN]

[kN]

[kN]

Brohltal (D)

4793

245.8

127.3

650.0

Garonor (F) 1982

2570

189.8

107.5

550.0

Garonor (F) 1984

3686

186.5

118.0

560.0

Auxerre (F) slow lane

2630

326.7

144.9

630.0

Auxerre (F) fast lane

153

277.2

163.6

670.0

Fiano R. (I)

4000

204.5

130.3

590.0

Piacenza (I)

5000

235.2

140.0

630.0

Sasso M. (I)

3500

224.9

149.0

620.0

-

-

-

-

-

on the contrary, daily maxima are much less sensitive to traffic composition and they vary between 130 and 210 kN for single axles, between 240 and 340 kN for two axles in tandem, between 220 and 390 kN for three axles in tridem, and between 400 and 650 kN for the total lorry weight; daily maxima of axle-loads and of total weight of the vehicle largely exceed the legal values; in consequence of industrial choices of lorry manufacturers, vehicle geometries have remained practically unchanged since the 1980’s: the inter-axle distance distribution strongly results trimodal: the first mode, a little scattered, is located around 1.30 m, corresponding to the usual axle’s spacing for tandem and tridem arrangements of axles, the second mode, also characterized by low scattering, is located around 3.20 m, a typical value for tractors of articulated lorries, while the third one, located around 5.40 m, is much more dispersed; long distance continental Europe traffic data are sufficiently homogeneous; the heavy traffic composition evolved in a very straightforward way during the 1980’s: the percentage of articulated lorries stepped up despite a strong reduction in the less commercially profitable trailer trucks, in conjunction with a contraction of the number of single lorries, whose use is increasingly limited to local routes; in consequence of a better and more rational management of the lorry fleets, the number of empty lorry passages has been strongly reduced and often limited to the sole tractor unit in case of articulated lorries, , so raising the mean vehicle loads; long distance traffics are much more aggressive than local traffics; generally lorry flows tend to increase, even if the absolute maximum flow was recorded in 1980 in Germany on the Limburger Bahn (8600 lorries per day on the slow lane).

On the basis of the above mentioned considerations, the studies for calibration of EN 1991-2 load models for road bridges were based on the traffic recorded in Auxerre (France), on the motorway A6 Paris-Lyon. The Auxerre traffic is very severe and summarizes effectively the main characteristics of the long distance European traffic, especially in terms of composition. Other traffic data have been used only for checking the reliability of the results obtained with Auxerre data. The most relevant parameters of the slow lane Auxerre traffic are summarized in figures A.1÷A.6. More precisely, in figures A.1, A.2 and A.3 are shown the histograms of

66


vehicle speeds, inter-vehicle distances and axle loads, respectively, referring to the total vehicle’s flow (lorries plus cars), while in figures A.4, A.5 and A.6 are reported the analogous histograms referring only to the lorry flow. The statistical analyses allow to conclude that speed and length of vehicles are poorly correlated and, from the probabilistic point of view, practically independent on the axle-loads and on the total weight of the vehicles. It must be stressed, finally, that European traffics exist which are more aggressive than the Auxerre traffic, like the one recorded in Paris on the Boulevard Périférique. Such traffics, nevertheless, are not very significant, since they depend on local situations and are hard to generalize. 0.08 0.07

Slow lane 0.06

Auxerre (F)

0.05 0.04 0.03 0.02 0.01 0.00 6

11

16

21

26

31

36

41

46

51

speed - [m/s]

Figure A.1. Histogram of the vehicle speed frequency – Auxerre – total flow

0.40 Slow lane

0.30

Auxerre (F)

0.20

0.10

0.00 1.36 10

2

5.07 10

3

1.00 10

4

4

4

1.50 10 2.00 10 Inter-vehicle distance - [m]

Figure A.2. Histogram of the inter-vehicle distance frequency – Auxerre – total flow

67


0.45 0.40 Slow lane

0.35

Auxerre (F)

0.30 0.25 0.20 0.15 0.10 0.05 0.00

12

25

38

51

64

77

90

103 116 129 142 155 168 181 Axle load - [kN]

Figure A.3. Histogram of the axle load frequency – Auxerre – total flow

0.07 0.06

Slow lane

0.05

Auxerre (F)

0.04 0.03 0.02 0.01 0.00

6

11

16

21

26

31

36

41

46 51 speed - [m/s]

Figure A.4. Histogram of the vehicle speed frequency – Auxerre – lorries

A.4

Traffic scenarios

The evaluation of the reference values of the real traffic effects induced on the bridge by the recorded traffic is not trivial. Traffic records generally refer to normal flowing situations; they are often inadequate to represent the most severe situations, which can happen in disturbed traffic scenarios. For this reason, in order to consider extreme traffic situations as well, traffic data have been opportunely manipulated, considering deterministic traffic scenarios being representative of some relevant real situation [6], [7]. Concerning the single lane, four different types of traffic models have been developed as follow: flowing, slowed down, and congested with/or without cars. 68


0.30 0.25

Slow lane 0.20

Auxerre (F)

0.15 0.10 0.05 0.00

8.9 101

5.83 103

1.16 104 Intervehicle -distance - [m]

Figure A.5. Histogram of the inter-vehicle distance frequency – Auxerre – lorries

0.07 0.06

Slow lane

0.05

Auxerre (F)

0.04 0.03 0.02 0.01 0 10

37

64

91

118

145 172 Axle load - [kN]

Figure A.6. Histogram of the axle load frequency – Auxerre – lorries The flowing traffic is represented by the traffic as recorded. Flowing traffic, to which a suitable dynamic coefficient must be associated, is particularly important for bridges spanning up to 30 to 40 m, when characteristic values are sought. If frequent values are wanted, flowing traffic is relevant in a much wider span range. Slowed down traffic is significant when infrequent loads are sought. It can be easily obtained considering the vehicles in the recorded order and reducing the distance among adjacent axles of two consecutive vehicles to a suitable minimum value to simulate vehicle convoys in braking phase. The minimum distance can be generally set to 20 m. The congested traffic, which is relevant when the bridge span is greater than 50 m, can be finally extracted from the recorded traffic reducing to 5 m the distance between the adjacent axles of two consecutive vehicles, so reproducing a traffic column in slow (stop and go) motion.

69


The traffic scenarios are particularly influenced by the driver behaviours, therefore, among the congested traffic configurations, it is particularly meaningful that one characterized by the presence on the slow lane of lorries only. In fact, when the traffic slows down, the drivers of lighter and faster vehicles tend to change lane to overtake heaviest vehicles. This effect is very well represented in classical photos, like that reported by Tschemmenerg & al. [8], relative to traffic jams on the Europa bridge (figure A.7). Obviously, the congested traffic without cars can be simply obtained disregarding light vehicles, below 35 kN.

Figure A.7. Traffic jam on the Europa Bridge (from [8]) In defining the target values of traffic effects, extreme situations characterised by flowing or jammed traffic on one or several lanes have been modelled considering several deterministic traffic scenarios, as synthesized below, considering Auxerre traffic. Flowing multilane traffics were obtained considering the following effects: -

70

for the most loaded lane, the first lane, the extrapolated effect induced by the slow lane traffic as recorded; for the second lane, the daily maximum effect (not extrapolated), induced by the slow lane traffic as recorded; for the third lane, the mean daily effect induced by the slow lane traffic as recorded;


-

for the fourth lane, the mean daily effect induced by the fast lane traffic as recorded.

Jammed traffic scenarios took into account: -

for the most loaded lane, the first lane, the extrapolated effect induced by the congested traffic without cars, deduced from the slow lane traffic; for the second lane, the daily maximum effect induced by the congested traffic with cars, deduced from the slow lane traffic; for the third lane, the daily maximum effect induced by the slow lane traffic as recorded; for the fourth lane, the mean daily effect induced by the slow lane traffic as recorded.

Target values have then been evaluated referring to a considerable number of bridge spans and influence surfaces. In particular, nine cylindrical influence surfaces have been considered for simply supported as well as continuous bridges, spanning between 5 and 200 m.

A.5

Extrapolation methods

As mentioned above, the choice of the main load model and its calibration requires the preliminary knowledge of the reference values, which are the relevant values of the effects characteristic, infrequent, frequent, and quasi-permanent - to be reproduced through the load model itself. Obviously, even considering deterministic traffic scenarios, the methodology to evaluate the reference values cannot be taken for granted, so that suitable numerical procedures, based on appropriate extrapolation methods of the histograms of the traffic induced effects must be set-up. The relationship between the return period and the distribution fractile can be easily determined, assuming a uniform flow of lorries on the bridge. Under this hypothesis the distance amongst two vehicles can be considered as equivalent to unit time interval, so that the vehicles are described by a stationary time series X1, X2.…, Xi,…,Xn, being Xi the weight of the i-th vehicle entering the bridge at time i. If the weights Xi are independent and distributed according to the same cumulative distribution function F(x), the return period Rx of the x value of Xi, which is defined as R x = E [N x ] , where N x = inf {n | X 1 < x, X 2 < x,...., X n−1 < x, X n ≥ x},

(A.1)

can be derived from R x = [1 − F ( x)] . −1

(A.2)

If the time series is replaced by a stationary stochastic process {Xt, t>0}, then R x = E [Tx ], where Tx = inf {t | X t ≥ x ∧ X u < x, ∀u < t}.

(A.3)

If YN is the maximum value of Xi and N is the total number of vehicles crossing the bridge during Rx, then it is YN = max{X i ,0 < i ≤ N }.

(A.4)

71


As the Xi are independent and identically distributed, the cumulative distribution function F[YN] of YN itself is given by F [YN ] = [F (x )] .

(A.5)

n

In conclusion, said yp the upper p-fractile of YN, p is

( )

p = 1− F yp ,

(A.6)

and, for N→∞ and T→∞, R results

( )

R = R yp = −

T T ≅ , where 0 < p << 1 . ln(1 − p ) p

(A.7)

The expression (A.7), which does not depends on yp and on the distribution of X, associates the return period and the fractile. For example, when the design life is 50 year, the 5% fractile (p=0.05) matches the value having a return period R=974.78≈1000 years. In general, to evaluate the extreme values of the effects induced by the traffic, three different methods of extrapolation have been employed in the framework of EN 1991-2 works. The methods are based on the half-normal distribution, on the Gumbel distribution and the Monte Carlo method, respectively. It is important to stress, however, that the characteristic values of real traffic effects resulted practically independent on the extrapolation method. A.5.1 Half-normal distribution extrapolation method In the half-normal distribution extrapolation method the upper tail of the extreme values distribution of the stochastic variable X is approximated by a normal distribution, through an opportune choice of two parameters of the normal distribution. The value xR, having a return period R, is given then by x R = x0 + σ ⋅ z R ,

(A.8)

where x0 is the last mode of the distribution and zR is the upper p-fractile of the standardized normal variable Z,

p = (2 ⋅ N )−1 ,

(A.9)

being N the total number of lorries during the reference period R. A.5.2 Gumbel distribution extrapolation method Under hypotheses similar to those illustrated in the previous paragraph, the extreme values distribution can be represented through a two parameters Gumbel (type I) distribution. The parameters u and α’, which represent, respectively, the mode and the scattering of the distribution, can be derived from the extreme values histogram as u = m − 0.45 ⋅ σ , α ' = (0.7797 ⋅ σ ) , −1

(A.10)

where m and σ are the mean value and the standard deviation of the histogram itself. The value xR is then

xR = u + y ⋅α ' , being

[

(A.11)

]

y = − ln − ln(1 − R )−1 ,

72

(A.12)


the reduced variable of the Gumbel distribution. An example of application of this extrapolation method on a Gumbel chart for a generic effect T(y) is synthetically illustrated in figure A.8.

Gumbel chart 200 190

Effect T(y)

180

170 160 150 -2

0

2

4

8

6

Reduced variable y Figure A.8. Example of data extrapolation on a Gumbel chart A.5.3 Monte Carlo extrapolation method Numerical extrapolation procedures based on the Monte Carlo method make use of automatic generations of a suitable set of extreme traffic situations, derived from the recorded traffic data, in such a way that a suitable population of extreme values is obtained. This population is elaborated in turn with an appropriate extrapolation method. The population can be produced in several ways. The simplest and intuitive procedure consists in the repeated application of the Monte Carlo method. The vehicles crossing the bridge are randomly selected amongst a complete set of standard vehicles, representing the most common real lorries. Lorry silhouettes, axle loads, axle spacings and the inter-vehicle distances are generated according to the relevant statistic parameters of the recorded traffic. An alternative procedure, more complicated but also much more effective, has been proposed and adopted for calibrating the reference values of real traffic effects [7]. In this methodology the crude Monte Carlo method is employed to obtain representative statistics of the effects, which are the input data for the calculation of the statistical parameters of the Gumbel distribution. This latter method allowed also to underline that reference values are poorly dependent upon the traffic jam frequency, at least for the most loaded lane [7].

A.6

Definition of dynamic magnification factors

In addition to the extrapolated static effects, the target values evaluation requires also specific knowledge about the dynamic effects, due to vehicle-bridge interactions, to be considered in calibration studies [9].

73


A.6.1 The inherent impact factors Since the recorded traffic data refer to flowing traffic, they contain some dynamic effects and they must be corrected through the so-called inherent impact factor, which is intrinsic in the measurements and can be evaluated simulating numerically the weighing in motion measurements. For the purposes EN 1991-2, the Auxerre weighing in motion device has been simulated considering the lorries, represented by a sequence of axles with shock-absorbers having suitable dynamic characteristics, running on good roughness pavement resting on rigid foundation. In this way it was stated that the characteristic values, which are relevant for the ultimate limit states, are affected by an inherent impact factor ϕin=1.10. When serviceability limit states and fatigue are considered, instead, and the range between the 10% and the 90% fractiles is taken into account, static and dynamic effects practically coincide and ϕin=1.00. A.6.2 The impact factor The impact factor depends on several parameters, like type, static scheme, span, natural frequency and damping coefficient of the bridge, dynamic characteristics and speed of the lorries, roughness of the road pavement and so on. Generally, it results greater when the natural frequency of the bridge is close to the natural frequencies of axles (10÷12 Hz) and lorries (1÷2 Hz). In the framework of EN 1991-2 pre-normative studies, in order to determine global local impact factors, a number of numerical simulations have been performed considering several bridge schemes with varying traffic scenarios. Concerning global effects, medium or good road pavement roughness has been considered in turn. Local dynamic effects have been studied taking into account the presence of a stepped irregularity, 30 mm height and 500 mm wide, simulating a road surface discontinuity, like that caused by a damaged expansion joint, a pothole or an ice sheet. The result of each numerical simulation was a time history of the considered effect, like the one reported in figure A.9, from which the so-called physical impact factor ϕ can be derived. Physical impact factor is the ratio between the maximum dynamic response and the maximum static response of the bridge

ϕ=

max dyn max st

(A.13)

.

The physical impact factor refers to a well precise load configuration and it depends on such a quantity of parameters that cannot to be directly employed for load model calibration. Besides, heaviest vehicles, which mainly influence the extreme values of the dynamic distribution of traffic effects, are generally slow and are characterised by small value of ϕ factors. For calibration purposes, two alternative approaches can be adopted: - the first approach takes into account dynamic effects referring directly to the dynamic effects distribution; - an alternative method takes into account the static effect distribution multiplied by a suitable calibration value of the impact factor, ϕcal. ϕcal can be defined as the ratio between the dynamic value Edyn(p-fractile) and the static value Est(p-fractile) corresponding to the same assigned p-fractile

ϕ cal =

74

E dyn( p − fractile) E st ( p − fractile)

.

(A.14)


Effect

Dynamic oscillogram

Static oscillogram

N0

t

Dynamic increment

t

Figure A.9. Definition of the physical impact factor ϕ Obviously, due to its conventional nature, ϕcal doesn't have a precise physical meaning; in fact the static and dynamic x-fractiles don’t correspond to the same load configuration. The characteristic values of the calibration impact factors ϕcal, derived from Auxerre traffic and employed in EN1991-2, are synthesized in figure A.10, depending on the span length L. The target dynamic values Edyn(x-fractile) can be finally evaluated through the expression

E dyn( x − fractile) =

ϕ cal ⋅ ϕ local ⋅ E st ( x − fractile) . ϕ in

(A.15)

where ϕlocal represents the local impact factor, when relevant. Local impact factor ϕlocal takes into account concentrated irregularities of the roadway surface.

75


1.8 1.7

Bending 1 lane

1.6

Bending 2 lanes

φ cal

1.5 1.4 1.3

Bending 4 lanes

1.2

Shear

1.1 1 0

100

200

L [m] Figure A.10. Calibration value of the impact factors ϕcal (EN 1991-2). A.7

Concluding remarks

Traffic load models for road bridges of EN 1991-2 have been defined and calibrated step by step balancing demand for accuracy and demands for ease of use. Preliminary calibrations highlighted that load models best fitting the target values should consist of concentrated loads and distributed loads: -

at least two concentrated loads should be considered in each relevant lane; Introduction of more than two concentrated loads doesn’t affect the precision of the results. the intensity of the uniformly distributed loads result slowly decreasing functions of the loaded length L.

The preliminary outcome has been successively modified to simplify the structure and the application rules of the load model, mainly to eliminate any reason for ambiguity, finally arriving to a load model characterised by: -

load values independent from the loaded length; dynamic effects include in load values; coexistence of concentrated and distributed loads on the same loaded area; aptitude to evaluate local and global, even simultaneous, effects; width of the notional lane equal to 3.0 m.

For the sake of model coherence, it has been established that, when relevant, the entire carriageway width can be loaded, i.e. not only the part occupied by the physical lanes, but also that one remaining. In order to reproduce the real traffic effects in secondary elements, characterized by influence surfaces with very small base length, it has been also introduced a local load model, constituted by a single axle, which should be considered alone on the bridge. Once opportunely calibrated, the so defined load model constitutes the load model of EN 1991 - 2, which is illustrated more precisely in §2 of chapter 3.

76


Besides characteristic loads, having a probability of about 2% to be exceed in 50 years design working life, i.e. about 1000 years return period, other relevant values of real traffic effects exist, like infrequent, frequent and quasi-permanent values, which are particularly relevant for SLS assessments. Infrequent and frequent can be identified by one year or one week return period, respectively. Quasi-permanent values result generally negligible and can be set zero, except for particular cases, like, for example, bridges in the urban zone. Frequent and infrequent values of traffic effects can be determined resorting to methods substantially analogous to those used for the evaluation of characteristic values. Their detailed illustration is omitted here, stressing only some relevant conclusion, on which the relevant parts of EN 1991-2 are based: -

characteristic values of traffic effects increase slowly with the return period, in fact taking into account a medium roadway roughness, infrequent values of traffic effects are about 90% of the corresponding characteristic values; taking into account a good roadway roughness, infrequent values reduce a little and become about 80% of the corresponding characteristic values; frequent values of traffic effects are 70%÷80% of the corresponding characteristic values; since the frequent values of traffic effects depend substantially on flowing traffic, as the span increase frequent values tend to precise lower limits, which are approximately 40%÷50% of the corresponding characteristic values.

77


78

Chapter 4: Fatigue loads due to traffic

CHAPTER 4: FATIGUE LOADS DUE TO TRAFFIC Pietro Croce1 1


Summary Fatigue performance of bridges is becoming more and more important due to the growth of traffic flows, the increase of mean weight of the Heavy Good Vehicles, the advances in conception and plan of bridges, the refinement of stress analysis techniques and the deeper knowledge of the mechanical properties of the materials. In many cases fatigue assessments strongly influence the design of new bridges and the assessment of existing bridges. Fatigue traffic load models of EN1991-2 [1] for road and railway bridges are illustrated and their background is discussed.

1

INTRODUCTION

In the last years the advances in conception and designing of bridges, the refinement of stress analysis techniques and the deeper knowledge of the mechanical properties of the materials have determined a relevant improvement of the bridge design as well as of the bridge performances. At the same time, as the traffic flows and the mean weight of the Heavy Good Vehicles have increased considerably, the fatigue resistance demand of modern bridges have so significantly risen, that in many cases fatigue assessments strongly influence the design. On the base of the aforesaid considerations, in modern bridge codes sophisticated fatigue load models should be given aiming to reproduce as well as possible the fatigue induced by real traffic. In the following fatigue traffic load models of EN1991-2 for road and railway bridges are illustrated and their background is discussed. The knowledge of the actual road traffic is affected by high uncertainties, on the contrary, railway traffic is not only better known, but can be managed much more easily. For this reason the calibration of fatigue traffic models for road bridges, mainly based on the real traffic recorded in 1986 in Auxerre (F) on the motorway Paris- Lyon, is discussed in much more detail.

2

DEVELOPMENT OF THE FATIGUE ROAD LOAD MODELS OF EN 1991-2

In the following the pre-normative background studies which have been carried out in the framework of EN 1991-2 to define fatigue loads models for road traffic is discussed, together with the main features of the models themselves. 2.1

Modelling of fatigue loads As known, the ISO definition states that fatigue is the progressive, localised and permanent structural change occurring in a material subjected to conditions that produce

79


fluctuating stresses and strains at some point or points and that may culminate in cracks or complete fracture after a sufficient number of fluctuations. In engineering structures, fatigue is induced by actions and loads varying with time and/or space and/or by random vibrations. Thus fatigue can be originated by natural events, like waves, wind and so on, or by loads deriving from the normal service of the structure itself. Among the civil structures exposed to fatigue, bridges occupy a prominent position, as they are subjected to the fluctuating action of lorries or trains crossing the bridges themselves. The assignment of appropriate fatigue load models is therefore a key topic in contemporary bridge design codes of practice. In principle, modelling of fatigue loads asks for the complete knowledge of the socalled load spectrum, expressing the load variations or the number of recurrences of each load level during the design working life of the structure. Load spectrum is generally given in terms of an appropriate function, graph, histogram or table. The load spectrum is often deduced from recorded data, referring to relatively short time intervals. In this case, additional problems must be faced regarding the statistical processing, the reliability over longer periods of the available data and the future trends of traffic. Whenever the real load spectrum results so complicated that cannot be directly used for fatigue checks, as it happens for bridge, it is replaced by some conventional load spectrum, aimed to reproduce the fatigue induced by the real one. The evaluation of conventional load spectra is particularly problematic, because it requires to consider the actions also from the resistance point of view. In fact, fatigue depends on the nature of the varying actions and loads, and additionally on structural material details, through the shape and the properties of the relevant S-N curves. Problems become even tougher when details exhibit endurance (fatigue) limit. As fatigue limits under constant amplitude represents a threshold value for the damaging stress range, it needs to distinguish between equivalent load spectra, aiming to reproduce the actual fatigue damage, and frequent load spectra, aiming to reproducing the maximum load range to be taken into account for fatigue assessments. Since fatigue verifications are performed in different ways, depending on the necessity to assess fatigue damage or boundless fatigue life, the distinction between equivalent and frequent spectra appears quite obvious. Moreover, the powerful methods of the stochastic process theory, often used in defining fatigue load spectra in other engineering structures, cannot be applied to bridges, as road traffic loads induce broad band stress histories. All that implies that the link between the action and the effect cannot be expressed by simple formulae, while further difficulties arise when vehicle interactions, whether due to simultaneity or not, become significant. Nevertheless, provided that vehicle interaction problems can be solved in some way, as shown in the Appendix A to the present chapter, it is intuitive enough to think that fatigue load spectra for bridges are composed by suitable sets of standardized lorries, where each lorry is identified by its own relevant properties, i.e. relative frequency, number of axles, axle loads, axle’s spacing, deduced processing appropriate traffic records. At this stage, it appears quite evident that definition of load spectra for bridges requires careful consideration of fatigue assessment methodology, to assure that assessments based on conventional spectra or on real spectra lead to the same results. 2.1.1 Fatigue verification methods The preliminary explanation of fatigue assessment methodology based on conventional load spectra is a crucial question in studying fatigue load models.

80


It can be easily recognised that fatigue verification methods goes along with a welldefined procedure, characterised by the following steps 1 2 3 4

assignment of fatigue load spectra, discriminating, if necessary, equivalent ones from frequent ones; detection and classification of structural details most vulnerable to fatigue cracking and selection of the appropriate S-N curves; choice of the pertinent partial factors γM; evaluation, for each detail, of the appropriate influence surface.

At this stage, the assessment methodology splits up in two different branches, accordingly as fatigue verification is devoted to compute fatigue damage or to assess boundless fatigue life. Damage computation procedure 5.a calculation of the design stress history σ=σ(t) produced in the detail by the equivalent load spectrum travelling over the influence surface; 6.a analysis of the stress history by means of a suitable cycle counting method, like the reservoir method or the rainflow method, to obtain the stress spectrum, where the number of occurrences of each stress range in the design working life is associated with the stress range itself; 7.a computation of the cumulative damage D using the Palmgren-Miner rule: if D≤1 the fatigue check is satisfied, otherwise, it is necessary to raise the fatigue strength of the detail. Fatigue resistance can be enhanced both reducing the stress range, i.e. enlarging the dimensions, or increasing fatigue category, i.e. adopting more refined workmanship or details. Boundless fatigue life assessment 5.b calculation of the design stress history σ=σ(t) produced in the detail by the frequent load spectrum transiting over the influence surface; 6.b computation of the maximum stress range ∆σmax=σmax-σmin, being σmax and σmin, respectively, the absolute maximum and the absolute minimum of the stress history; 7.b boundless fatigue life assessment. If the verification is not satisfied, it is possible to improve fatigue resistance using the provisions described in 7.a, or to attempt to go through fatigue damage computation. Obviously, in bridges exposed to high-density traffic, concrete slabs and orthotropic steel deck details are subject to such a huge number of stress cycles, that boundless fatigue life assessment using frequent load spectra becomes quite obligatory. 2.1.2 Reference traffic measurements Also the fatigue load models of Eurocode 1 have been defined and calibrated on the basis of the two large traffic measurement campaigns carried out in several European countries in years 1977 to 1982 and 1984 to 1988, which have been discussed in Annex A to chapter 3. Unlike static loads, which depend only on the upper tail of the traffic load distribution, fatigue loads are influenced by the entire distribution.

81


For this reason, fatigue models have been refined, supplementing the main calibration, based on Auxerre traffic data, with supplementary studies, based on different traffic data, in order to enlarge their field of application,. These supplementary calibrations regarded not only motorway traffics - Brothal (D), Piacenza, Fiano Romano, Sasso Marconi (I) – but also local traffic on secondary roads (Epone (F)). In effect, long distance traffics, typical of motorways and main roads, are characterised by high percentage of heavy vehicles, while local traffics, typical of secondary roads, are lighter and composed mostly by two axle lorries. Besides, it should be considered that, as confirmed by recent traffic data, European traffics show a trend characterised by - marked increase of the number of articulated lorries vis-à-vis the simultaneous reduction of the number lorries with trailer; - reduction of the number of three axle lorries for the benefit of two axle lorries; - increase of the average load per lorry. 2.2

The fatigue load models of EN 1991-2 The calibration method, the underlying philosophy, the methodological approach and the main features of the fatigue models of Eurocode 1 are summarised in the following. 2.2.1 Calibration method Fatigue load models have been calibrated referring to reference influence surfaces relative to simply supported and continuous bridges, spanning in the range 3 m to 200 m. In agreement with the general fatigue assessment procedures, calibration has been setup according to the following scheme, -

choice of the most significant European traffic data; selection of appropriate S-N curves; evaluation of the stress histories in reference bridges; cycle counting and stress spectra computation; first identification of fatigue models; definition of standardised lorry silhouettes; calibration of frequent load models, best fitting the maximum stress range ∆σmax induced by real traffic; - calibration of equivalent load models, best fitting the fatigue damage D induced by the traffic.

2.2.2 Reference S-N curves Reference S-N curves pertain to steel details, characterised by endurance limit. As known, in the logarithmic S-N chart these curves are represented by a bilinear curve, characterised by a sloping branch of constant slope, m=3, (figure 1), or by a trilinear curve, characterised by two sloping branches, m=3 and m=5, (figure 2), according as boundless fatigue life or fatigue damage is to be assessed. Since steel details exhibit fatigue limit ∆σD, two cases can be envisaged, according to the maximum stress range ∆σmax of the real stress history is higher than ∆σD or not. To be significant for fatigue, ∆σmax must be exceeded several times during the design working life of the bridge and its definition is not trivial. Two different approaches have been proposed, leading to similar results: in the former, ∆σmax is defined as the stress range such that the 99% of the total fatigue damage results from all stress ranges below ∆σmax; in the latter ∆σmax is the stress range exceeded approximately 5⋅104 times during the bridge life.

82


S

S m

m =3

=3

∆σ

∆σ

O

m= 5

D

D

∆σL ?σ

5⋅106

O

N

?σ

5⋅106

108

N

Figure 2. Trilinear S-N curve

Figure 1. Bilinear S-N curve

This last definition implies that the return period of ∆σmax is about half a day, giving so direct explanation of frequent load spectrum denomination. In EN 1991-2 studies, to derive equivalent load spectra independently from the fatigue classification of details, cumulative damage has been computed referring generally to simplified S-N curves with unique slope, in turn m=3 (figure 3) or m=5 (figure 4). S-N curves with double slope (figure 5) and without endurance limit have been used for some additional calculations. Some comparisons show that load spectra obtained using the simplified curve m=5 are free from significant errors and reproduce generally well the actual fatigue damage.

S

S m =3

m= 5

N O Figure 3. Single slope S-N curve (m=3)

N O Figure 4. Single slope S-N curve (m=5)

S m =3

∆σ

m= 5

D

O

?σ

5⋅106

N

Figure 5. Double slope m=3- m=5 S-N curve without endurance limit

83


2.2.3 Fatigue load models From the above-mentioned considerations, it derives that at least two conventional fatigue load models must be considered: the one for boundless fatigue life assessments, the other for fatigue damage calculations. Besides, since an adequate fitting of the effects induced by the real traffic requires very sophisticated load models, whose application is often difficult, the introduction of simplified and safe-sided models, to be used when sophisticated checks are unnecessary, seems very opportune. For this reason in EN 1991-2 two fatigue load models are foreseen for each kind of fatigue verification: the former is essential, safe-sided and easy to use, the latter is more refined and accurate, but also more complicated. In conclusion, four conventional models are given: - models 1 and 2 for boundless fatigue checks; - models 3 and 4 for damage calculations. Fatigue load model 1 is extremely simple and generally very safe-sided. It directly derives from the main load model used for assessing static resistance, where the load values are simply reduced to the frequent ones (figure 6.a), multiplying the tandem axle loads Qik by 0.7 and the weight density of the uniformly distributed loads qik by 0.3. Obviously, for local verifications, the fatigue load model n. 1 is constituted by the isolated concentrated axle weighing Q=280 kN (frequent value - figure. 6.b). Q ik

Qik qik

0.5

Lane n. 1

2.0

Q1k =210 kN q1k =2.7 kN/m2

0.5 0.5

w

Lane n. 2

2.0

Q 2k=140 kN q 2k=0.75 kN/m2

0.5 0.5 2.0

Lane n. 3

Q 3k=70 kN q 3k=0.75 kN/m2

0.5

Remaining area

qrk =0.75 kN/m2

Figure 6.a. Fatigue load model n. 1

Figure 6.b. Fatigue load model n. 1 for local verifications

The verification consists of checking that the maximum stress range ∆σmax induced by the model is smaller of the fatigue limit ∆σD. The application rules for the load model n. 1 agree exactly with those given for the main load model, so that the absolute minimum and maximum stresses correspond as rule to different load configurations. The model allows making “coarse” verifications also in multi-lane configurations, generally resulting extremely safe-sided. The simplified fatigue model n. 3, conceived for damage computation, is constituted by a symmetrical conventional four axle vehicle, also said fatigue vehicle (figure 7). The equivalent load of each axle is 120 kN. This model is accurate enough for spans bigger than 10 m, while for smaller spans it results safe-sided.

84


Figure7. Fatigue load model n. 3 Fatigue load models n. 2 and n. 4 are the most refined one and they are load spectra constituted by five standardised vehicles, representative of the most common European lorries. Fatigue load model n. 2, which is a set of lorries with frequent values of axle loads, and fatigue model n. 4, which is a set of lorries with equivalent values of the axle loads, are illustrated in tables 1 and 2, respectively. They allow to perform very precise and sophisticated verifications, provided that the interactions amongst vehicles simultaneously crossing the bridge are negligible or opportunely considered. Table 1. Fatigue load model n. 2 – frequent set of lorries LORRY SILHOUETTE

Axle spacing [m]

Frequent axle loads [kN]

Wheel type (see table 3)

4.5

90 190

A B

4.20 1.30

80 140 140

A B B

3.20 5,20 1.30 1.30

90 180 120 120 120 90 190 140 140 90 180 120 110 110

A B C C C A B B B A B C C C

3.40 6.00 1.80 4.80 3.60 4.40 1.30

85


Table 2. Fatigue load model n. 4 – equivalent set of lorries LORRY SILHOUETTE Long distance LORRY

TRAFFIC TYPE Medium distance

Local traffic

Lorry percentage

Lorry percentage

Axle

Equivalent

Lorry

spacing

Axle loads

percentage

[m]

[kN]

4.5

70 130

20.0

40.0

80.0

4.20 1.30

70 120 120

5.0

10.0

5.0

3.20 5.20 1.30 1.30

70 150 90 90 90

50.0

30.0

5.0

3.40 6.00 1.80

70 140 90 90

15.0

15.0

5.0

4.80 3.60 4.40 1.30

70 130 90 80 80

10.0

5.0

5.0

The types of wheels pertaining to each standardised lorries of fatigue load models n. 2 and n. 4 are indicated in table 1, referring to table 3. The number of lorries to be taken into account for damage assessments depends on the traffic category: indicative values of Nobs, representing the number of lorries of year per slow lane, are given in table 4. The additional traffic on the fast lane can be assumed to be 10% of the slow lane traffic. In fact, in EN 1991-2 a further general purpose fatigue model is anticipated too, denominated fatigue model n. 5. This model is constituted by a sequence of consecutive axle loads, directly derived from recorded traffic, duly supplemented to take into account vehicle interactions, where relevant. Fatigue model n. 5 is aimed to allow accurate fatigue verifications in particular situations, like suspended or cable-stayed bridges, important existing bridges or bridges carrying unusual traffics, whose relevance justifies ad hoc investigations [2]. 2.2.4 Accuracy of fatigue load models In the following, some significant results obtained using the fatigue load models are compared with those pertaining to the reference traffic, allowing to highlight the accuracy and the field of application of the each conventional model. 86


Table 3. Definition of wheels and axles of standard lorries Wheel axle type


Geometrical definition

A

0.22

0.32

0.32

1.78 0.22

0.22 0.22 2

0.22

0.32

C

1.73 0.27

0.27

0.32


0.22

0.54

0.32

0.54

B


0.32

2

2

Table 4. Indicative number of lorries expected per year and for a slow lane Traffic categories 1 2 3 4

Roads and motorways with 2 or more lanes per direction with high flow rates of lorries Roads and motorways with medium flow rates of lorries Main roads with low flow rates of lorries Local roads with low flow rates of lorries

Nobs per year and per slow lane 2.0⋅106 0.5⋅106 0.125⋅106 0.05⋅106

Essentially, the comparison concerns the four influence surfaces shown in figure 8, for bridges span L varying between 3 m and 100 m. The influence surfaces pertain to bending moment M0 at midspan of simply supported beams, bending moments M1 and M2 at midspan and on the support, respectively, of two span continuous beams and bending moment M3 at midspan of three span continuous beams.

87


Figure 8. Reference influence lines In figures 9 to 14 the outcomes of different fatigue load models for the aforesaid influence lines are compared with the real traffic (Auxerre traffic) effects, in function of the span length L, considering only one notional lane. For unlimited fatigue life assessments, the ratios between the maximum bending moment ranges due to fatigue load model 1, ∆Mmax,LM1, and fatigue load model 2, ∆Mmax,LM2, and the maximum bending moment ranges due to real traffic, ∆Mmax,real, are diagrammatically reported in figures 9 and 10, respectively. For fatigue damage assessments reference can be made to the equivalent bending moment range corresponding to 2⋅106 cycles. The ratios between the equivalent bending moment ranges due to fatigue load model 3, ∆Meq,LM3, and the equivalent bending moment ranges due to real traffic, ∆Meq,real, are shown in figures 11 and 12, considering one slope S-N curves characterised by m=3 and m=5, respectively. Analogous diagrams for equivalent bending moment ranges due to fatigue load model 4, ∆Meq,LM4, are illustrated in figure 13, for m=3 S-N curve.

Figure 9. Accuracy of fatigue load model n. 1

88


Figure 10. Accuracy of fatigue load model n. 2

Figure 11. Accuracy of fatigue load model n. 3 – m=3

Figure 12. Accuracy of fatigue load model n. 3 – m=5

89


Figure 13. Accuracy of fatigue load model n. 4 Analysis of the diagrams shows that: - as just said, fatigue load model n. 1 appears very safe-sided, especially for short spans; - load model n. 2 results much more reliable; values little below the actual ones are obtained for M2 in the span range 20 to 50 m, but this depends on the particular shape of the influence line; - as expected, model n. 4 fits very good actual results for short influence lines; - fatigue model n. 3 looks unsafe for M2 influence lines when spans are above 30 m, in particular for higher m values. To solve the problem it has been proposed to modify the model n. 3 taking into account an additional fatigue vehicle each time that the influence surface exhibits two contiguous areas of the same sign. This additional fatigue vehicle, having equivalent axle loads set to 40 kN, should run on the same lane of the basic fatigue vehicle, 40 m away from it. The adoption of such an additional vehicle should mitigate the error in computation of ∆M2,eq, as it appears evident in figure 14, where damage calculations for M2 influence line and for m=3 and m=9 S-N curves, considering additional fatigue vehicle.

Figure 14. Accuracy of improved fatigue load model n. 3 – 2 vehicles

90


The λ-coefficient method Besides the usual damage computations based on Palmgren-Miner rule, EN 1991-2 also foresees a conventional simplified fatigue assessment method, said λ-coefficient method, based on λ damage equivalent factors, which are dependent on the material. The method, derived originally for railway bridges, is based for road bridges on fatigue model n. 3 (fatigue vehicle) and it is aimed to bring back fatigue verifications to conventional resistance checks, comparing a conventional equivalent stress range, ∆σeq, depending on appropriate λ-coefficients, with the detail category [3], [4]. The equivalent stress range ∆σeq is given by 2.3

∆σ eq = λ1 ⋅ λ 2 ⋅ λ3 ⋅ λ 4 ⋅ ϕ fat ⋅ ∆σ p = λ ⋅ ϕ fat ⋅ ∆σ p ,

(1)

where -

∆σ p = σ p , max − σ p , min is the maximum stress range induced by fatigue model n. 3;

- λ1 is a coefficient depending on the shape and on the base length of the influence surface, i.e. on the number of secondary cycles in the stress history; - λ2 is a coefficient allowing to pass from reference traffic, used in fatigue model calibration, to expected traffic; - λ3 depends on the design life of the bridge; - λ4 takes into account vehicle interactions amongst lorries simultaneously crossing the bridge; - ϕfat is the equivalent dynamic magnification factor for fatigue verifications.

λ1, represented in graphical or tabular form, is calculated in the calibration phase, comparing the damage due to the fatigue vehicle with the damage produced by a single stress cycle having the maximum stress range ∆σp. If m is the slope of S-N curve, it is  ∑ ni ⋅ ∆σ im λ1 =  i  ∆σ mp 

1

m  .  

(2)

λ2 depends on the annual lorry flow and on traffic composition. In general, said N1 and Qm1 the flow and the equivalent weight of the actual traffic,

∑n ⋅Q ∑n i

Qm1 = m

m i

i

(3)

,

i

i

and N0 and Q0 the flow and the equivalent vehicle weight of the reference traffic, it results 1

Q λ2 = k ⋅ m1 Q0

 N m ⋅  1  .  N0 

(4)

In equation (4) k represents a conversion parameter, given by k=

Def Dv

⋅

Q0 , Qm1

(5)

91


where Dv is the damage produced by N0 fatigue vehicles and Def is the damage produced by N0 real lorries. For Auxerre traffic it ensues Q0 = 480 kN and N 0 = 2 ⋅106 lorries per year. λ3 is given by

λ3 = m

T , TR

(6)

where TR is the reference design working life (TR=100 years) and T is the actual design working life. λ4, which, as said, takes into account vehicle interactions, can be expressed as

λ4 (l , N1 ) = m

N* N1* + ∑ i N1 N i   1

η  ⋅  i   η1 

m

N  + ∑  comb  i  N1

η  ⋅  comb   η1 

m

, 

(7)

where N1 is the lorry flow (number of the lorries) on the main lane, Ni the lorry flow on the ith lane, ηi the max ordinate of the influence surface corresponding to i-th lane, N i* the lonely, i.e. not interacting, lorry flow on the i-th lane, Ncomb the number of interacting lorries and ηcomb the overall ordinate of the influence surface for the ”interacting” lanes, being the second summation extended to all relevant combinations of lorries on several lanes. An appropriate closed form expression for λ4 can be theoretically derived for two simultaneously loaded lanes, as shown in the Appendix A to the present chapter. The equivalent impact factor ϕfat, finally, is the ratio between the damage due to the dynamic stress history and the damage due to the corresponding static stress history

ϕ fat = m

∑ ni,dym ⋅ (∆σ i,dym )m ∑ ni,stat ⋅ (∆σ i,stat )m

.

(8)

In conclusion, said ∆σc the detail category, the fatigue assessment reduces to check that expression ∆σ eq = λ ⋅ ϕ fat ⋅ ∆σ p ≤ ∆σ c

(9)

is satisfied.

Partial factors γM The partial factors γf, regarding the action aspect, and γm, regarding the fatigue resistance aspect, cover uncertainties in the evaluation of loads and stresses as well as fatigue strength scattering. According to the experience from steel structures, these partial factors affecting stress ranges are generally combined in a unique factor γ M = γ f ⋅ γ m . Beside the material, the 2.4

numerical values of γM depend on the possibility to detect and repair fatigue cracks and on the consequences of fatigue failure and are given ion the relevant Eurocodes.

2.5

Effects of future traffic trends In the next future, the traffic composition could be sensibly modified. In fact, in order to improve the organization of the European transportation network, the European Commission enacted the 96/53/EC Directive limiting the total mass of Heavy Good Vehicles (HGV) to 44 t, but admitting, on a parity basis, the possibility to allow the

92


circulation of Long and Heavy Vehicles (LHV), characterised by mass up to 60 t and length till 25 m. This possibility has been admitted in Germany and in northern Countries, in particular Sweden, Finland, Denmark and The Netherlands. For this reason in these Countries a significant increase of the number of LHVs in long distance traffic has been experienced. Despite of their effectiveness in terms of decrease of pollutant emissions and cost reduction, LHVs could result too much demanding for existing infrastructures, in particular for bridges, so that their impact requires careful examination. In order to evaluate the aptness of EN 1991-2 fatigue load models to cover also the effects of LHVs, some additional studies has been performed on relevant bridge schemes and spans comparing the Auxerre traffic effects with those induced by a more recent one, containing a relevant number of LHVs, recorded with a WIM device at the Moerdijk site in the Netherlands in April 2007. These studies are illustrated in the Annex A to the present Guidebook.

3

FATIGUE LOAD MODELS FOR RAILWAY BRIDGES OF EN 1991-2

For fatigue damage assessments of railway bridges subjected to normal railway traffic based on characteristic values of Load Model 71, including the dynamic factor Φ, EN 1991-2 assigns three different fatigue load spectra. These load spectra refer to three different traffic mixes, usual traffic mix, traffic with 250 kN-axles mix or light traffic mix, depending on whether the structure carries mixed (usual) traffic, predominantly heavy freight traffic or lightweight passenger traffic. The above mentioned load spectra are based on an annual traffic tonnage of 25 Mt per each track. If not otherwise specified, fatigue damage should be assessed taking into account that: - in structures carrying multiple tracks, fatigue loadings shall be applied to a maximum of two tracks, which should be the most unfavourable; - the structural design working life of the bridge is 100 years; - when dynamic effects can be considered through static dynamic factors, static dynamic factors should be determined according to the method illustrated in § 3.2; - bridges requiring dynamic analysis could necessitate of additional investigations and/or recommendations.

3.1

Train types for fatigue As just said, three different load spectra are given on the basis of three traffic mixes, standard, heavy and light, characterized by illustrated respectively in tables 5.a, 5.b and 5.5 and in figures 15.a, 15.b and 15.b, where also train type are indicated. The load spectra refer to adjustment factors α=1.0. When relevant, the axle loads should be multiplied by the appropriate adjustment factor α.

3.2

Dynamic magnification factors for fatigue assessments The static dynamic factors Φ2 and Φ3 are applied to load models LM 71, SW/0 and

SW/2. Since the static dynamic factors are intended for static assessments of bridge members, they are evaluated considering extreme loading cases that can occur in the design working life. For this reason, they would be excessively onerous if applied to Real Trains included in fatigue load spectra.

93


Table 5.a. Standard traffic mix with axles ≤225 kN Train type

Number of trains/day

Mass of train [t]

Traffic volume [Mt/year]

1

12

663

2.90

2

12

530

2.32

3

5

940

1.72

4

5

510

0.93

5

7

2160

5.52

6

12

1431

6.27

7

8

1035

3.02

8

8

1035

2.27

Total

67

24.95

Table 5.b. Heavy traffic mix with 250 kN axles Train type


Mass of train [t]


5

6

2160

4.73

6

13

1431

6.79

11

16

1135

6.63

12

16

1135

6.63

Total

51

24.78

Table 5.c. Light traffic mix with axles ≤225 kN Train type


Mass of train [t]


1

10

663

2.4

2

5

530

1.0

5

2

2160

1.4

9

190

296

20.5

Total

207

25.3

In reality, in fatigue verifications only the equivalent dynamic effect over the assumed 100 years design working life needs to be considered, therefore the dynamic enhancement for each Real Train can be reduced, for Maximum Permitted Vehicle Speeds up to 200 km/h, to 1 1  1 +  ϕ '+ ϕ "  2 2 

(10)

where L2

− K 100 " 0 , 56 ϕ'= and ϕ = e 1− K + K 4

94

(11)


being v the Maximum Permitted Vehicle Speed in m/s, L the determinant length LΦ in m of v v for L≤20 m and K = for L>20 m. the structural member and K = 160 47,16 L0, 408 2.6

11.5

2.6

3.6

2.6

11.5

2.6

3.2

2.2 2.2

6.9

2.2

2.2

6x225 kN

12x(4x110 kN)

4x225 kN

2.5

16.5

2.5

5.0

2.5

16.5

2.5

2.9

3.3

3.3

6.7

Train type 1 - Passenger train - V=200 km/h - P=6630 kN - n=12/d

10x(4x110 kN)

4x20 t

3.0

8.46

3.0

4.45

2.5

4.95

2.5

16.5

2.5

4.45

3.0

3.0

8.46


4x20 t

13x(4x150 kN)

4x170 kN

3x170 kN

3x170 kN

8x(2x170 kN)

3.0

11.0

3.0 3.3 3.0

15.7

3.0

3.0

15.7

3.0

15.7

3.0 3.3 3.0

3.0

11.0

Train type 3 - High speed train - V=250 km/h - P=9400 kN - n=5/d

4x170 kN

Train type 4 - High speed train - V=250 km/h - P=5100 kN - n=5/d 1.8 1.8

5.7

1.8 1.8

5.0

1.8 1.8

5.7

1.8 1.8

4.0

15x(6x225 kN) 2.1 2.1

2.1 2.1

4.4

6x225 kN

A

B

1.8 3.2

8.0

3.5 1.8

12.8

A

4x225 kN 6.5

4x225 kN 1.8 3.7

2x70 kN

3.7 1.8

6.5

3.8

2x70 kN 3.9

2.1 2.1

4.4

2.1 2.1

6x225 kN

6.5

Train type 5 - Freight train - V=80 km/h - P=21600 kN - n=7/d

C

A

CABBBAAC+ CABCAACCB


1.8

11.0

1.8

3.2

1.8

11.0

1.8

3.0

2.2 2.2

6.9

2.2

2.2

6x225 kN

10x(4x225 kN)


5.5

4.2

5.5

4.2

5.5

3.5

2.2 2.2

6.9

2.2

2.2

6x225 kN

20x(2x225 kN)

Train type 8 - Freigth train - V=100 km/h - P=10350 kN - n=6/d

Figure 15.a. Fatigue load spectrum for normal traffic mix

95


1.8 1.8

5.7

1.8 1.8

5.0

1.8 1.8

5.7

1.8 1.8

4.0

15x(6x225 kN) 2.1 2.1

2.1 2.1

4.4

6x225 kN

A

B

A

1.8 3.2

8.0

3.5 1.8

6.5

12.8

A

4x225 kN

1.8 3.7

4x225 kN

3.7 1.8

2x70 kN 3.8

6.5

3.9

4.4

2.1 2.1

2x70 kN

2.1 2.1

6x225 kN

6.5


C

CABBBAAC+ CABCAACCB


1.8

11.0

1.8

3.2

1.8

11.0

1.8

3.0

2.2 2.2

6.9

2.2

2.2

6x225 kN

10x(4x250 kN)


5.5

4.2

5.5

4.2

5.5

3.5

2.2 2.2

6.9

2.2

2.2

6x225 kN

20x(2x250 kN)


Figure 15.b. Fatigue load spectrum for heavy traffic mix

2.6

11.5

2.6

2.6

3.6

11.5

2.6

3.2

2.2 2.2

6.9

2.2

2.2

6x225 kN

12x(4x110 kN)

4x225 kN

2.5

16.5

2.5

5.0

2.5

16.5

2.5

2.9

3.3

3.3

6.7


10x(4x110 kN)

Train type 2 - Passenger train - V=160 km/h - P=5300 kN - n=5/d 1.8 1.8

5.7

1.8 1.8

5.0

1.8 1.8

5.7

1.8 1.8

4.0

15x(6x225 kN) 2.1 2.1

2.1 2.1

4.4

6x225 kN

4x130 kN

4x110 kN

2x(4x130 kN)

4x110 kN

4x130 kN

Train type 9 - Suburban train - V=120 km/h - P=2960 kN - n=190/d

Figure 15.c. Fatigue load spectrum for light traffic mix

96

2.5

14.0

4.3 2.5

2.5

11.5

2.5 4.3 2.5

14.0

4.3 2.5

2.5

11.5

2.5 4.3 2.5

14.0

2.5



The λ-factor design method Also for railway bridges, the fatigue assessment can be simplified and reduced to an equivalent stress range verification, determined according to the relevant Eurocodes (EN 1992-2, EN 1993-2, EN 1994-2), on the basis of the so-called λ-method. For example, for steel bridges, the safety verification shall be carried out by ensuring that the following condition is satisfied:

3.3

γ Ff λΦ 2 ∆σ 71 ≤

∆σ c

γ Mf

(12)

where γFf is the partial factor for fatigue loading, usually set to 1.00, λ is the damage equivalent factor for fatigue, which takes account the expected railway traffic on the bridge and the span of the member, Φ2 is the dynamic factor, ∆σ71 is the maximum stress range due to the Load Model 71 (and where required SW/0), ∆σC is the reference value of the fatigue strength, i.e. the class of the detail, and γMf is the partial factor for fatigue strength. Load model should be placed in the most unfavourable position for the element under consideration, disregarding any adjustment factor.

4

CONCLUDING REMARKS

Fatigue verifications are decisive for designing new road and railway bridges as well for assessing existing ones. In up to date structural codes fatigue loads are given through suitable load spectra, deduced from real traffic data, recorded using weighing in motion devices. In theory, load spectrum can be directly deduced from real traffic data, provided that they are representative of the traffic concerning the bridge, during its design working life. In practice, the management of real load spectrum is very complicated and it requires a huge amount of calculations; therefore, its application is justified only for particularly important bridges. Usually, in structural codes fatigue loads are assigned through conventional load spectra, which reproduce the fatigue effects induced by the real traffic. Since fatigue effects depend not only on the actions but also on the material properties, through the appropriate S-N curve, the definition and the use of conventional load spectra is not trivial. Duly taking into account the theoretical differences that exist between equivalent load spectra, intended to reproduce fatigue damage, and frequent load spectra, intended to reproduce the maximum load range for fatigue assessments, in EN 1991-2 five load spectra are assigned for road bridge assessments and three load spectra are assigned for railway bridges assessments. In addition to the usual damage computations, based on Palmgren-Miner rule, EN 1991-2 allows to adopt also a conventional simplified fatigue assessment method, based on λ damage equivalent factors, which are dependent on the material. This method brings back fatigue verification to conventional resistance check, where an appropriate equivalent stress range, ∆σeq, is compared with the detail category. In the present chapter, background information and main features of EN 1991-2 load spectra have been illustrated, discussing their possibilities and their fields of application and highlighting the results of pre-normative calibration studies. When vehicle’s interactions are significant, EN 1991-2 fatigue load models cannot be used, unless additional information is available. Vehicle’s interactions problems are tackled

97


from the theoretical point of view in Appendix A to the present chapter, where simplified formulae are also given for the evaluation of the pertinent damage equivalent factor λ4. Finally, in Annex A to the present Guidebook, the aptness of EN 1991-2 fatigue load models to face the actual trends of road traffic and in particular the effects of Long and Heavy Vehicles, allowed by the 96/53/EC Directive is discussed and additional studies are illustrated.

5

REFERENCES

[1] EN1991-2, Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges. Brussels: CEN, 2003 [2] Caramelli, S. & Croce, P., Messina bridge: testing assisted deck fatigue design, International Institute of Welding Conference on Welded Constructions: Achievements and perspectives for the new millennium. Florence, 2000. [3] Croce, P., Background to Fatigue Load Models for Eurocode 1: Part 2 Traffic Loads. Progress in Structural Engineering and Materials 1(3:4): 250-263, 2001 [4] Bruls A et al., Part 3: Traffic loads on bridges. Calibration of road load models for road bridges. Proceedings of IABSE Colloquium on Basis of Design and Actions on Structures. Background and Application of Eurocode 1. Delft, 1996. [5] Ventsel E.S., Probability theory. Moscow: MIR, 1983.

98


Appendix A to Chapter 4 – Vehicle interactions and fatigue assessments

A.1

General principles

When vehicle interaction is relevant, stress histories cannot be determined using conventional fatigue models or recorded traffic data, unless appropriate additional information are available. The achievement of general theoretical results in modelling vehicle interactions could sensibly enlarge the field of application of the fatigue load models and it represents a main objective in the improvement of EN 1991-2.

A.2

Modelling of vehicle interactions

The probability that several vehicles are running simultaneously on the bridge, on the same lane or on several lanes, can be found theoretically in the framework of the queuing theory. The bridge can be assimilated to a service system, with or without waiting queue, and the stochastic processes can be modelled as Markov processes. This allows to arrive to a suitably modified load spectrum, composed by single vehicles or vehicle convoys travelling alone on the bridge so that the complete stress history results a random assembly of their individual stress histories. A.2.1 Basic assumptions Let the load spectrum consisting in a set of q types of lorries and be Nij the number of i-th vehicle per year (annual flow) on the j-th lane. The total flow on the j-th lane is then q

N j = ∑ N ij .

(A.1)

i =1

Obviously, the probability that several lorries are simultaneously travelling on the bridges, which is negligible for L<40 m, becomes more and more relevant as the characteristic length L of the influence line increases. Basic hypotheses of the theory are that the vehicle arrivals are distributed according a Poisson law and that the transit time Θ on L is exponentially distributed. A.2.2 Interaction between lorries simultaneously travelling on one lane The probability Pn that n lorries are simultaneously travelling on L can be calculated considering the bridge as a single channel system with a waiting queue, in which the waiting time, depending on the number of requests in the queue, and the number of the request in the queue itself are limited. In fact, as there is a minimum value for the time interval Ts between two consecutive lorries, the waiting time for the i-th vehicle in the queue is given by Ti = Θ − i ⋅ Ts and the

(

)

number of requests in the queue is limited to w = int Θ ⋅ Ts−1 − 1 . Under the assumption that each Ti is distributed with an exponential law whose parameter is ϕ i = Ti −1 , the problem can be solved in a closed form [5]. The probability Pn to have n vehicles on the lane, i.e. n-1 requests in queue, is then given by

99


 w  δ   δ Pn =   ⋅ 1 + + ∑ δ i  α   α i=2    n

−1  

i −1  s   ⋅ α ⋅ ∏ α + ∑ϕ j     s =1  j =1   

−1

   

(A.2)

n=0, 1,

and by −1 −1  n  n−1  s w   i −1  s    δ     δ i Pn =  ∏  α + ∑ ϕ j   ⋅ 1 + + ∑ δ  α ∏  α + ∑ ϕ j            j =1 j =1  α  s =1     α i = 2   s =1     

−1

2≤n≤w,

(A.3)

where δ represents the lorry flow density and α = Θ −1 . The annual number of interactions between n vehicles i1, .., in on the j-th lane can be then obtained substituting these formulae in the general equation, n

N (i1 , i 2 , ...., i n ), j

Nj P = n ⋅ ⋅ 1 − P0 n

∏ Ni

∑ q

j

∑  ∏ N i qn

where

k

k =1  n

ts

 s =1

(A.4) j

  

indicates the sum over all the possible choices with repetitions of n elements

2

among q. In the practice, the problem is reduced to consider the simultaneous presence of two lorries r and t only, so that it results

 δ P0 = 1 +  α

−1

 δ  δ  δ2  , P2 = ⋅ 1 + ⋅ 1 + α ⋅ (α + ϕ1 )  α  α + ϕ1 

 δ   ⋅ 1 +  α + ϕ1 

−1

(A.5)

and the annual number of interactions becomes N ( r , t), j =

N rj ⋅ N tj ⋅ δ  2  (δ + α + ϕ1 ) ⋅ ∑  ∏ N its j   q 2  s =1

⋅

Nj 2

.

(A.6)

When a single vehicle model is given, equation (A.6) simplifies further into N (1, 1), j =

N j ⋅δ

2 ⋅ (δ + α + ϕ1 )

.

(A.7)

A.2.3 Interaction between lorries simultaneously travelling on several lanes Under the aforementioned hypotheses, interactions between lorries simultaneously travelling on several lanes can be tackled in analogous way. The bridge is considered as a multiple channel system without waiting queue, where new requests are refused if all channels are occupied. In this case the probability Pk to have simultaneously vehicles on k lanes, i.e. k occupied channels, can be deduced solving an Erlang type system. Said µ the density of the total flow N* and recalling that α = Θ −1 , it results

100


µk

 m µi   ⋅∑ Pk = k α ⋅ k!  i = 0 α i ⋅ i! 

−1

0≤k≤m.

(A.8)

Substituting (A.8) in the general expression k

N i1 h1 , i 2 h2 , ...., i k h k

P = k 1 − P0

 k N i jhj ⋅ ∏  j =1 N h j 

∏ N hj  N* j =1 ⋅ ⋅  k  k  ∑  ∏ N hts  m   s =1  

(A.9)    

k 

where

∑

represents the sum over all the possible choices of k elements among m, it is

 m k   

possible to derive the annual number of interactions of k lorries, i1 on the h1-th lane,....., ik on the hk-th lane, k

µk



N i jhj k = mα ⋅ kj! ⋅  ∏  j =1 N h µ j ∑ α j ⋅ j!  j =1

N i1 h1 , i 2 h2 , ...., i k h k

k

∏ N hj  N* j =1 ⋅ ⋅  k  k  ∑  ∏ N hts  m   s =1  

   

.

(A.10)

k 

As said before, usually only the case in which two lorries r and t are simultaneously present on the h-th and the j-th lane is relevant, so that it results

µ2 P2 = 2 ⋅α 2 N r h, t j

N rh ⋅ N tj

 2 µi  ⋅  ∑ i   i = 0 α ⋅ i! 

µ2 = ⋅ N h ⋅ N j 2 ⋅α 2

−1

and −1

Nh + N j  2 µi  ⋅  ∑ i  ⋅ , 2  i =1 α ⋅ i! 

(A.11)

or, simply, when a single type of vehicle is considered, N h,

j

µ2 = 2 ⋅α 2

 2 µi  ⋅  ∑ i   i =1 α ⋅ i! 

−1

⋅

Nh + N j 2

.

(A.12)

A.2.4 The time independent load spectrum The procedures described above allow to obtain the so-called lonely vehicles spectrum, which is time-independent being composed by individual vehicles and by vehicle convoys travelling alone on the bridge. Generally, the evaluation of the lonely vehicles spectrum requires to resort to both the above mentioned procedure: - the simultaneous transit on the same lane is considered first, obtaining a new load spectrum, composed by individual vehicles and by vehicle convoys travelling alone on the lane; - new load spectra are then applied on different lanes to solve the multilane case.

101


A.2.5 Time independent interactions Once the lonely vehicle spectrum is known, the complete stress history can be derived as a random assembly of the individual stress histories. Unfortunately, the stress spectrum cannot be determined, in general, as a pure and simple sum of the individual stress spectra. In fact when maximum and minimum stresses are given by different members of the spectrum, the individual stress histories can combine, depending on the cycle counting method adopted, originating some kind of time independent interaction. If cycles are identified using the reservoir method or the rainflow method, the problem can be solved in the general case. The demonstration is out of the scope of this Guidebook and it will be shown only the main results. Two individual stress histories σ Ai and σ A j interact if and only if max σ Ai ≤ max σ A j ∧ min σ Ai ≤ min σ A j

(A.13)

max σ A j ≤ max σ Ai ∧ min σ A j ≤ min σ Ai

(A.14)

or

If the couples of interacting histories are sorted in such a way that the corresponding ∆σ max are in descending order, the number of the combined stress histories as well as the residual numbers of each individual stress history can be computed in a very simple recursive way, as follows. In general, an individual stress history can interact with several others; therefore the number of combined stress histories Ncij, obtained as h-th combination of the stress history σ Ai and as k-th combination of the stress history σ A j is given by ( h −1)

N cij = where

( h −1)

N i and

( k −1)

( h −1)

N i ⋅( k −1) N j

N i + ( k −1)N j

(A.15)

,

N j are the number of the individual stress histories σ Ai and σ A j

which are not yet combined and being

(0)

N i = N Ai and

(0)

N j = N A j the number of

repetitions of σ Ai and σ A j in the lonely vehicle spectrum. The actual number of individual stress histories σ Ai , which do not combine with other stress histories, is given by ( p)

N i = ( 0) N i − ∑ ( N ik + N ki ) , k ≠i

(A.16)

being the sum extended to all the stress histories σ Ak , which combine with σ Ai itself. In conclusion, a new modified load spectrum is obtained, whose members, represented by the lonely individual vehicles and convoys and by their time independent combinations, are interaction free, so that it can be defined as interaction-free vehicle spectrum.

A.6

Concluding remarks The above mentioned methods allow the derivation of some important general results.

102


The methods can be used to tackle relevant questions concerning the calculation of the maximum length of the influence line for which lorry interaction on the same lane can be disregarded or with the calibration of damage equivalent λ4-factor accounting for multilane effect in λ-coefficient method. The analysis, shortly illustrated below, can be performed taking into account: - S-N curve in characterised by on slope m=5; - four different annual lorry flow rates, N1=2.5×105; N2=5.0×105; N3=1.0×106; N4=2.0×106, distributed over 280 working days; - constant lorry speed v=13.89 m/sec. Assuming an inter-vehicle interval Ts=1.5 s, application of (A.7) allows to calculate, for example, how many vehicles per years are travelling simultaneously on the same lane, in function of the annual vehicle flow and on the considered length L, as summarised in table A.1.

Table A.1. Number of yearly interacting vehicles on one lane L (m)

N1

N2

N3

N4

40

1190

4729

18566

71605

50

1690

6670

25987

98813

60

2165

8515

32940

123618

75

2858

11177

42796

157689

100

3978

15423

58110

208240

These theoretical results, which are in good agreement with numerical simulations, confirm that simultaneous presence of several lorries on the same lane is generally not relevant for spans below 75 m. On the contrary, when bending moment on support of two span continuous beams is considered under high traffic flows, simultaneity results significant starting from 30 m span. Closed form expression for calculation of λ4 coefficients can be obtained resorting to equation (A.12) referring to two lanes carrying equal lorry flows per year, which is the most relevant case for practical applications. The results are summarized in table A.2 for different traffic flows and span length.

Table A.2. Number of yearly interacting vehicles on two lanes carrying equal lorry flows L (m)

N1

N2

N3

N4

10

1846

7331

28901

112358

20

3666

14450

56179

212764

30

5458

21367

81966

303028

50

8967

34626

129532

458712

75

13213

50200

182480

617280

100

17312

64766

229356

746264

150

25100

91240

308640

943390

200

32383

114678

373132

1086953

103


Taking into account lorry interactions in all possible relative positions of the two lorries, equivalent stress ranges ∆σeq, can be easily evaluated from table A.2, provided that influence coefficient pertaining to each lane is known. Obviously, said ∆σ1 the equivalent stress range corresponding to one lane flow only, the required λ4 coefficient is simply given

λ4 =

∆σ eq

∆σ 1

(A.17)

.

If the two lanes have the same influence coefficient, i.e. the influence surface is cylindrical, λ4 values result those indicated in table A.3, being 1.149 ≈ 5 2 (m=5) the λ4 basic value, corresponding to zero interactions.

Table A.3. λ4-factors for two lanes carrying lanes carrying equal lorry flows L (m)

N1

N2

N3

N4

10

1.156

1.162

1.174

1.197

20

1.162

1.174

1.197

1.234

30

1.168

1.186

1.217

1.264

50

1.180

1.207

1.250

1.310

75

1.194

1.230

1.283

1.351

100

1.207

1.250

1.310

1.381

150

1.230

1.283

1.351

1.423

200

1.250

1.310

1.381

1.450

These results demonstrate that λ4, which takes into account globally vehicle interactions, is a quasi-linear function of Θ ⋅ N , which can be expressed in closed form as

λ4 = 5

η1 + η 2 η1

L⋅N  ⋅ 1.03 + 0.01 ⋅ v ⋅ 10 6 

 . 

(A.18)

where L is in m and v in m/s, being η1 and η2, η1≥η2, the influence coefficients related to the two interacting lanes.

104


CHAPTER 5: NON TRAFFIC ACTIONS Pietro Croce1 1


Summary In bridge design, climatic, geotechnical and environmental actions should be considered, like wind, snow, temperature, earth pressure, water actions, uneven settlements and so on. Variable climatic actions for bridges are discussed, stressing the peculiarities of the relevant load models.

1

INTRODUCTION

Besides dead loads and imposed loads, other climatic, geotechnical and environmental actions should be considered in bridge design, like wind, snow, temperature, earth pressure, water actions, uneven settlements and so on. In this chapter, the climatic actions are discussed, devoting special attention to peculiarities of the relevant load models for bridges.

2

WIND ACTIONS

Wind actions on bridges are specified in EN1991-1-4 [1]; here only peculiarities concerning bridges themselves will be taken into account, as general information are just given in the chapter 4 of Guidebook 1 [2]. Strictly speaking, EN1991-1-4 specifications are applicable only to girder bridges spanning up to 200 m with a constant cross section and one or more spans . Cross section can be boxed, mono or multi-cell, or open with two or more longitudinal beams, which can be made, in turn, by open or box sections or by truss, with a single deck (upper or lower). In any case, it must be stressed that EN1991-1-4 rules can be easily extended to variable cross sections, to double deck bridges as well as to other bridge types, provided that wind-structure interactions are not relevant. Bridges characterised by multiple or significantly curved decks, roofed bridges and movable bridges could require some additional studies. Lower or intermediate deck arch bridges or suspended and cable stayed bridges call for specific studies, since for them wind-structure interactions cannot be disregarded. In general, wind is considered blowing in two horizontal directions, x and y, being y the longitudinal axis of the bridge and x the transversal axis (figure 1), originating forces in x, y and z direction. Forces induced by wind blowing in direction x can be considered not simultaneous with forces induced by wind blowing in direction y and vice versa; on the contrary, wind forces acting in z direction should be considered acting simultaneously with the corresponding x or y force. In some of particular orography, it could be necessary to consider some inclination of the wind directions, out of the horizontal plane.

105


B z y

L

x

D

Figure 1. Reference system for wind actions on bridges. 2.1

Wind forces on bridges For the evaluation of wind actions on bridges, two different load scenarios are taken into account, depending on the compatibility of strong wind with traffic on bridge. In fact, if the traffic is not protected against the wind, for example by wind shields, road, pedestrian or railway traffic is not possible when the fundamental value of basic wind velocity vb,0 attains a particular threshold value, equal to vb*, 0 = 23 m/s ,

(1)

vb**, 0 = 25 m/s ,

(2)

for road bridges, and to

for railway bridges. When bridge in unloaded, the wind forces Fwk, evaluated as indicated in the following, should be considered. If the bridge is not protected against the wind, the combination value of the wind actions is given by ' Fwk = ψ 0 Fwk ,

(3)

for road bridges, where ψ0=0.6 for persistent design situations and ψ0=0.8 for actions during execution, Fwk'' = ψ 0 Fwk ,

(4)

for railway bridges, where ψ0=0.75, and Fwk''' = ψ 0 Fwk ,

(5)

for footbridges, where ψ0=0.30. If the bridge is protected against the wind, i.e. wind shields are foreseen on the bridge or the bridge is covered, the combination value of the wind should not exceed

for road bridges, and

Fwk* = Fwk vb*, 0 ,

( )

(6)

( )

(7)

Fwk** = Fwk vb**, 0 ,

106


for railway bridges, while for footbridges no additional limitation is necessary as ψ0 is low enough. If dynamic analysis is not required, as it happens in normal bridges, for example bridge spanning up to 40 m, whatever the construction material used, the structural factor cs⋅cd, can be assumed cs⋅cd=1.0, being cs the size factor and cd the dynamic factor. 2.1.1 Wind forces on the deck in the x-direction Wind forces in the x-direction can be evaluated using the expression Fwk =

1 ρ vb2 C Aref , x , 2

(8)

where ρ=1.25 kg/m3 is the air density, vb is the basic wind speed for the site under consideration, Aref,x is the reference area and C is the wind load factor for bridges. In absence of traffic, reference area Aref,x should be evaluated taking into account: - in case of plain (web) beams, the total height d of the projection on a vertical plane of all the main beams, including the part of one cornice or footway or ballasted track projecting over the front main girder, (see figure 2), plus the sum d1 of the heights of solid parapets, noise barriers, wind shields and open safety barriers installed on the bridge; - in case of truss beams, the total height d of the projection on a vertical plane of all the trusses, including the part of one cornice or footway or ballasted track projecting over the front main girder, or the projection of the contour of the solid section, whichever is less, plus the sum d1 of the heights of solid parapets, noise barriers, wind shields, and open safety barriers installed on the bridge.

d1

Solid safety barrier or parapet or noise barrier

0.3

Open safety barrier

Open parapet

The height of open safety barrier is set to 0.3 m, so that the reference heights to be considered in same relevant case can be derived from table 1.

d

Fig. 2. Depth to be used in evaluation of Aref,w. Table 1. Depth to be considered in evaluation of Aref,w Road restraint systems and shields

On one side

On both sides

Open parapet or open safety barrier

d+0.3 m

d+0.6 m

Solid parapet or solid safety barrier

d+d1

d+2 d1

Open parapet and open safety barrier

d+0.6 m

d+1.2 m

107


During execution, finishing can be disregarded in the evaluation of Aref,x and, prior of the placement of the carriageway, the surface of two main beams should be considered. In presence of traffic, reference area Aref,x should be assumed as the larger between the area evaluated considering absence of traffic and the area obtained taking into account the presence of traffic. Lateral surface of vehicles exposed to wind is represented - in road bridges, with a rectangular area, 2 m in height, starting from the carriageway level, on the most unfavourable position, independently of the location of the vertical traffic loads; - in railway bridges, with a rectangular area, 4 m in height, starting from the top of the rail, on the whole length of the bridge. Calculation of wind load factor C The wind load factor C is given by C = ce c f , x ,

(9)

where ce is the exposure coefficient for kinetic pressure and cf,x is the force coefficient, which is equal to cf,x0, being cf,x0 the force coefficient or drag coefficient without free end flow. The exposure coefficient could be evaluated considering a reference height ze given by the distance from the lowest point of the ground and the centre of the bridge beck, disregarding additional parts, parapets, barriers and so on, included in the reference area. The force coefficient cf,x can be assumed equal to 1.30 for normal bridges, or can be determined using the expression

  b   c f , x = min 2.4; max 2.5 − 0.3 ; 1 , d tot    

(10)

for bridge with solid parapets and/or solid barriers and/or traffic, and using the expression    b c f , x = min 2.4; max 2.5 − 0.3 ; 1.3   , d tot   

(11)

for the construction phase and/or bridges with open parapets. In expressions (10) and (11) b represents the total width of the bridge and dtot the height considered in the evaluation of Aref,x, Aref,x=dtot⋅L, except for truss girder where dtot does not include the truss height, so that truss girder should be considered separately. Two generally similar decks, being at the same level and separated transversally by a gap not significantly greater than 1 m, can be considered as a unique structure, when windward structure forces are to be calculated. In other cases special studies are necessary. If particular orography determines an incoming wind inclined more than 10° in the vertical plane, the drag coefficient may be derived from special investigation. When the windward face of the section is inclined to the vertical of an angle α1, the drag coefficient cf,x0 may be decreased by a factor η1,

η1 = max(1 − 0.005 α 1 ; 0.7 ) ,

(12)

but this reduction does not affect Fw. If the bridge is sloped transversely by an angle α2, the drag coefficient cf,x0 should be increased by a factor η2,

η 2 = max(1 + 0.03 α 1 ; 1.25) .

108

(13)


Provided that dynamic analysis is not necessary, the wind load factor C can be also evaluated in a more simple way using table 2. In the table the force factors C refer to terrain category II, Area with low vegetation such as grass and isolated obstacles (trees, buildings) with separations of at least 20 obstacle height. For intermediate values of ze and/or b/dtot, linear interpolation is permitted.

Table 2. Force factors C for bridges b/dtot

ze≤20 m

ze=50 m

≤0.5

6.7

8.3

≥4.0

3.6

4.5

2.1.2 Wind forces on deck in the y-direction When relevant, longitudinal wind forces in y-direction can be assumed equal to 25% of wind forces in x-direction for plated bridges and to 50% of wind forces in x-direction for truss bridges. 2.1.3 Wind forces on deck in the z-direction Upward and downward vertical wind forces in z-direction can be determined using the lift force coefficients cf,z, which should not be used to calculate vertical vibrations of bridge deck. If suitable wind tunnel tests are not available, it can be assumed cf,z=±0.9, so considering globally the influence of a possible transverse slope of the deck, of the slope of terrain and of fluctuations of the angle of the wind direction with the deck due to turbulence. Considering the symbols indicated in figure 3, alternative values of cf,z, can be derived from the diagrams reported in figure 4. These diagrams are valid in the range -10°≤θ≤10°, where θ is the sum of α, which is the inclination of the wind direction in the vertical plane, and β, which is the superelevation of the deck. When using the diagrams, the total depth dtot may be limited to the depth of the deck structure, disregarding the presence of traffic and any bridge equipment. The reference height can be assumed be ze, as in previous cases.

Fz

β θ

α

e

Aref,z=bL dtot

b

θ=α+β on i t c re β=superelevation d di win α=angle of the wind with the horizontal Figure 3. Transversal slope and wind inclination for z-direction wind forces For hilly terrain, when the bridge deck is at least 30 m above ground, and in every case for flat and horizontal terrain, the angle α of the wind with the horizontal, due to turbulence, may be taken as ±5°.

109


Figure 4. Force coefficients cf,z for bridges with transversal slope and wind nclination The reference area Aref,z should be set equal to the planar area of the bridge, Aref,z=b⋅L, being b the total width and L the length of the bridge. The vertical force Fz, which is relevant only if it is of the same order of magnitude of the dead load, can be considered applied with an eccentricity e=b/4, if not otherwise specified.

2.2

Wind forces on piers Wind actions for the entire bridge should be evaluated considering the most unfavourable wind direction for bridge and supporting piers. Wind forces on piers should be determined using the general formulae and the relevant force and pressure coefficients. During construction phases, intermediate situations can occur where horizontal transmission or redistribution of wind actions by the deck are not granted: these transient design situations could result the most severe for the piers and for some particular type of deck. For this reason, besides persistent design situations, also transient design situations for wind actions during execution should be assessed.

110


3

SNOW LOADS

Snow loads on bridges should be determined according the general procedure of Eurocode EN 1991-1-3 [3], just described in Guidebook 1, therefore only some additional information about simultaneity of snow load with other actions is given here. In general, except for roofed bridges and for bridges situated in particular geographic area, snow loads should not be combined with traffic actions

4

THERMAL ACTIONS

As known, in EN 1991-1-5 [4] a generic temperature profile in the cross section of a structural element can be obtained by superposition of four essential components, as illustrated in figure 5.

Figure 5. Essential constituents of a temperature profile These components correspond to -

a uniform temperature component ∆Tu; a linearly variable component about the z-axis ∆TMy; a linearly variable component about the y-axis ∆TMz; a non linear component ∆TE, which results in a self-equilibrated system of stesses.

4.1

Temperature changes on bridges The temperature changes on bridges are given in terms of uniform temperature component, vertical difference component, which includes non linear component also, and, when relevant, a horizontal difference component, which can be assumed linearly varying. The temperatures in the bridge depend not only by the shade air temperature and solar radiation, but also on the scheme, on the cross section, on the mass and on the material. Therefore, bridges can be classified in terms of categories and subcategories as follows: 1.

Steel bridge:

2. 3.

Composite bridge Concrete bridge:

steel box girder steel truss or plate girder; concrete slab concrete beam concrete box girder.

111


4.1.1 Uniform temperature component The uniform temperature component depends on the maximum and minimum temperature, Te,max and Te,min, that the bridge can attain during its working life. Once determined the maximum and minimum shade air temperatures of the site characterized by 50 years return period, Tmax and Tmin, the uniform temperature components Te,max and Te,min can be determined according to the diagrams of figure 6, where Te,max and Te,min, in °C, are expressed in terms of Tmax and Tmin, in °C, for each bridge category recalled before. Values of Te,max values for truss or plated steel bridges (category 1) can be reduced by 3 °C. 70 60 1 50 2

40

Te,max

3 30 20 10

Te,min

0 -10 3 -20 2 -30

1

-40 -50 -50

-40

-30

-20

-10

Tmin

0

10

20

30

40

50

Tmax

Figure 6. Correlation between shade air temperature (Tmin, Tmax) and uniform components of the bridge temperature (Te,min, Te,max) If T0 is the initial bridge temperature, i.e. the temperature of the bridge at the time when it is restrained, the variation of the uniform bridge temperature ∆Tu is given by ∆Tu = Te,max - Te,min = ∆TN ,esp + ∆TN ,con ,

(14)

where ∆TN,exp=Te,max-T0

and ∆TN,con=T0-Te,min

(15)

are the temperature variations to be considered when the bridge expands or contracts, respectively. Assessing bearing displacements it can be assumed ∆TN,exp=Te,max-T0+20°C and ∆TN,con=T0-Te,min+20°C.

112


4.1.2 Vertical temperature varying component In consequence of the different heating and cooling of the top and bottom surfaces of the bridge, vertical temperature variations can occur. These variations correspond to maximum heating, when the top surface is warmer than bottom surface, and maximum cooling, when the bottom surface is warmer than the top surface. The vertical temperature profiles can be defined under two different hypothesis, according as non-linear temperature profile ∆TE is disregarded or not: in the former case a simplified equivalent linear profile can be considered, while in the latter one a non linear profile, including ∆TE, is taken into account. Equivalent linear vertical temperature profile When equivalent linear vertical temperature profile is adopted, the maximum temperature differences corresponding to maximum heating or maximum cooling, denoted as ∆TM,heat and ∆TM,cool, for the different bridge categories can be deduced by table 3. The values given in table 3 represent upper bound values of the temperature differences for road and railways bridges carrying a 50 mm surfacing. For different thickness of the surfacing, values of table 3 should be multiplied by the adjustment factors ksur given in table 4.

Table 3. Equivalent linear vertical temperature variations for bridges Type of deck

Top warmer than bottom ∆TM,heat [°C]

Bottom warmer than top ∆TM,cool [°C]

Type 1: Steel deck

18

13

Type 2: Composite deck

15

18

Type 3: Concrete deck - concrete box girder - concrete beam - concrete slab

10 15 15

5 8 8

Table 4. Adjustment factors ksur for road, foot and railway bridges Type 1 Surface thickness [mm]

Type 2

Type 3

Top warmer than bottom

Bottom warmer than top





insurfaced

0.7

0.9

0.9

1.0

0.8

1.1

water-proofed

1.6

0.6

1.1

0.9

1.5

1.0

50

1.0

1.0

1.0

1.0

1.0

1.0

100

0.7

1.2

1.0

1.0

0.7

1.0

150

0.7

1.2

1.0

1.0

0.5

1.0

ballast (750 mm)

0.6

1.4

0.8

1.2

0.6

1.0

113


Non linear vertical temperature profiles When more refined analyses are necessary, the vertical temperature profile can be assumed as non linear, considering in the heating and cooling conditions the temperature profiles given in tables 5, 6 and 7 for steel bridges, concrete bridges and composite bridges, respectively. For composite bridges two alternative profiles, normal and simplified, are given in tables 7.a and 7.b. The simplified profile is generally safe-sided. The temperature differences ∆T given in tables 5 to 7 include both the vertical temperature component ∆TM and the non linear temperature component ∆TE, together with a little part of uniform component ∆TN, just considered in the uniform temperature component ∆TN, given in §4.1.1. The temperatures for other surfacing depths of bridge decks of type 1 to 3 are given in Tables B.1 to B.3 of EN 1991-1-5, Annex B. 4.1.3 Horizontal temperature varying component Horizontal temperature differences in bridges can be generally disregarded, except in special cases, for example when one side of the bridge is much more exposed to the sunlight of the other one. When horizontal component must be taken into account, a linear variation of 5 °C can be assumed.

Table 5. Non linear vertical temperature differences for steel bridges Temperature difference (∆T)

Type of construction

Cooling

24 °C 14 °C 8 °C 4 °C

-6 °C

0.5 m

h

Steel deck with 40 mm surfacing

0.1 m

-5 °C

h

h

0.5 m

h

21 °C

h

0.3 m 0.2 m

h

0.1 m

Heating

Steel deck on truss or plate girder with 40 mm surfacing

114


Table 6. Non linear vertical temperature differences for concrete bridges Temperature difference (∆T) Type of construction

Heating

Cooling

h

100 mm surfacing

Concrete slab

h

100 mm surfacing

Concrete beam

h

100 mm surfacing

Concrete box girder

Table 7.a. Non linear vertical temperature differences for composite bridges, normal profile Temperature difference (∆T) Type of construction

Heating

Cooling

h

100 mm surfacing

Concrete deck on box, truss or plate girder

115


Table 7.b. Non linear vertical temperature differences for composite bridges, simplified profile Temperature difference (∆T) Type of construction

Heating

Cooling

h

100 mm surfacing

Concrete deck on box, truss or plate girder

On the contrary, special attention should be paid for concrete multicell box girder where temperature of the inner webs can differ significantly (around 15°C) from the temperature of the outer ones.

5.

CONCLUSIONS

Effects of variable climatic actions, wind, snow and temperature given in Eurocodes EN 1991-1-x have been illustrated, with special emphasis on their application on bridges, discussing peculiarities, application rules and possible simplification of the relevant load models. Wind specifications are applicable only to girder bridges spanning up to 200 m with a constant cross section and one or more spans, but they can be extended variable cross sections, to double deck bridges as well as to other bridge types, provided that wind-structure interactions are not relevant. Bridge types which are sensitive to wind-structure interactions, like lower or intermediate deck arch bridges or suspended and cable stayed bridges, call for specific studies, duly supported by wind tunnel tests. Simultaneity of snow loads with traffic actions is generally not significant, except in very particular cases, as roofed bridge, and can be disregarded. Air shade temperature variations and solar radiation result in temperature fields in the bridge, depending on the bridge location and on the structural material. These temperature fields are typically non linear and have been described in detail, but often it is possible to refer to simplified and safe-sided linear distributions. Seismic actions have been not considered here, as their illustration is beyond the scope of the present Guidebook.

6.

REFERENCES

[1] EN1991-1-4, Eurocode 1: Actions on structures - Part 1-4: General actions – Wind actions. Brussels: CEN, 2005

116


[2] Holicky, M et al., GB1: Basis of design and actions on structures, Leonardo Project number: CZ/08/LLP-LdV/TOI/134020, Prague, CTU, 2010 [3] EN1991-1-3, Eurocode 1: Actions on structures - Part 1-3: General actions – Snow loads. Brussels: CEN, 2004 [4] EN1991-1-5, Eurocode 1: Actions on structures - Part 1-5: General actions – Thermal actions. Brussels: CEN, 2004

117


118


CHAPTER 6: ACCIDENTAL ACTIONS Ton Vrouwenvelder1, Dimitris Diamantidis2 1

2

TNO, Delft, Netherlands University of Applied Sciences, Regensburg, Germany

Summary The accidental actions covered by Part 1.7 of EN 1991 are discussed and guidance for their application in design calculations is given. A short summary is presented of the main clauses in the code for collisions due to trucks. After the presentation of the clauses an example is given in order to get some idea of the design procedure and the design consequences.

1

INTRODUCTION

1.1

General General principles for classification of actions on structures, including accidental actions and their modelling in verification of structural reliability, are introduced in EN 1990 Basis of Design. In particular EN 1990 defines the various design values and combination rules to be used in the design calculations. A detailed description of individual actions is then given in various parts of Eurocode 1, EN 1991 [2]. Part 1.7 of EN 1991 covers accidental actions and gives rules and values for the following topics: - -impact loads due to road traffic - -impact loads due to train traffic - -impact loads due to ships It should be kept in mind that the loads in the main text are rather conventional. More advanced models are presented in annex C of EN 1991-1-7. Apart from design values and other detailed information for the loads mentioned above, the document EN 1991, Part 1-7 also gives guidelines how to handle accidental loads in general. In many cases structural measures alone cannot be considered as very efficient. 1.2

Background Documents Part 1.7 of EN 1991 is partly based on the requirements put forward in the Eurocode on traffic loads (ENV 1991-3) and some ISO-documents. For the more theoretical parts use has been made of prenormative work performed in IABSE [5] and CIB [6]. Specific backgrounds information can be found in [7] and [8].

119


2

BASIC FRAMEWORK

In order to reduce the risk involved in accidental type of load one might, as basic strategies, consider probability reducing as well as consequence reducing measures, including contingency plans in the event of an accident. Risk reducing measures should be given high priority in design for accidental actions, and also be taken into account in design. Design with respect to accidental actions may therefore pursue one or more as appropriate of the following strategies, which may be mixed in the same design: 1.

2. 3. 4. 5.

preventing the action occurring or reducing the probability and/or magnitude of the action to a reasonable level. (The limited effect of this strategy must be recognised; it depends on factors which, over the life span of the structure, are normally outside the control of the structural design process) protecting the structure against the action (e.g. by traffic bollards) designing in such a way that neither the whole structure nor an important part thereof will collapse if a local failure (single element failure) should occur designing key elements, on which the structure would be particularly reliant, with special care, and in relevant cases for appropriate accidental actions applying prescriptive design/detailing rules which provide in normal circumstances an acceptably robust structure (e. g. tri-orthogonal tying for resistance to explosions, or minimum level of ductility of structural elements subject to impact). For prescriptive rules Part 1.7 refers to the relevant ENV 1992 to ENV 1999.

The design philosophy necessitates that accidental actions are treated in a special manner with respect to load factors and load combinations. Partial load factors to be applied in analysis according to strategy no. 3 are defined in Eurocode, Basis of Design, to be 1.0 for all loads (permanent, variable and accidental) with the following qualification in: "Combinations for accidental design situations either involve an explicit accidental action A (e.g. fire or impact) or refer to a situation after an accidental event (A = 0)". After an accidental event the structure will normally not have the required strength in persistent and transient design situations and will have to be strengthened for a possible continued application. In temporary phases there may be reasons for a relaxation of the requirements e.g. by allowing wind or wave loads for shorter return periods to be applied in the analysis after an accidental event. As an example Norwegian rules for offshore structures are referred to. The typical difference between permanent, variable and accidental loads is shown in Figure 1 depending on time. Permanent loads are always present (e.g permanent weight of the construction). Variable loads are nearly always present even its value may be small for a considerable part of time. However values which are nonzero will occur many times during the design life of the structure (traffic, snow, wind). Accidental loads, on the contrary, usually never occur during the lifetime of a structure. But if they are present, it takes only a short time. The duration depends on the manner of load. For example: Explosions take a shorter time (seconds) than floods (some days).

120


Force (a) time

Force

(b) time

Force (c)

time

Figure 1. Typical time characteristics of (a) accidental, (b) variable and (c) permanent loads

3

IMPACT DUE TO ROAD TRAFFIC (VEHICLE COLLISIONS)

3.1

Impact on substructure Impacts on the substructure of bridges by road vehicles are a relatively frequent occurrence and may have considerable consequences. Specific provisions are consequently specified in EN 1991 Part 1-7. In the case of soft impact, design values for the horizontal actions due to impact on vertical structural elements (e.g. columns, walls) in the vicinity of various types of internal or external roads may be obtained from Table 1. Soft impact means that the impacting body consumes most of the available kinetic energy. The forces Fdx and Fdy denote respectively the forces in the driving direction and perpendicular to it. There is no need to consider them simultaneously. The collision forces are supposed to act at 1.25 m above the level of the driving surface (0.5 m for cars). The force application area may be taken as 0.25 m (height) by 1.50 m (width) or the member width, whichever is the smallest. In addition to the values in this Table 1 the code specifies more advanced models for nonlinear and dynamic analysis in an informative annex. The values of design impact forces given in Table 1 are left open for the national choice as Nationally Determined Parameters.

121


Table 1. Horizontal static equivalent design forces due to impact on supporting substructures of structures over roadways Type of road

Type of vehicle

Force Fd,x [kN]

Force Fd,y [kN]

Motorway Country road Urban area Courtyards/garages Courtyards/garages

Truck Truck Truck Passengers cars only Trucks

1000 750 500 50 150

500 375 250 25 75

The probabilistic methods of the reliability theory have been used in [9] to determine the impact forces due to vehicle impact. Two alternative procedures given in EN 1991-1-7 [1], Annexes B and C have been analysed. The following assumptions have been taken: 1. 2. 3.

The probability of a structural member being impacted by a lorry leaving its traffic lane is 0.01 per year. The target failure probability for a structural member, given a lorry hits the substructure of the bridge is 10-4/10-2 = 0.01 [3, 4]. The probabilistic models given by the working documents of JCSS [10] have been implemented

The values of accidental impact forces have been computed in [9] and are shown here in Table 5 for the three assumed distances d of the substructure from the road. The resulting impact forces determined on the basis of above introduced alternative probabilistic procedures (see Table 2) are considerably higher than the minimum (indicative) requirement for impact forces given Section 4 of EN 1991-1-7 [1] (see Table 1). This is mainly due to the rather high probability of a structural member being impacted by a lorry leaving its traffic lane, i.e. 0.01 per annum, which represents a conservative assumption. For roadways, the impact forces are in a range from 2.9 to 2.8 MN, for roads in urban areas, the impact forces are in a broader range from 1.9 to 1.4 MN (depending on the applied probabilistic approach) for three study cases of distances d from 3 to 9 m. The study [9] indicates that for the design of structural members located nearby the traffic routes the upper bound of the accidental impact forces should be recommended in the National annex of EN 1991-1-7 [1] provided that no other safety measures are implemented. However it is stated here that the values of Table 2 are relatively high and that they may be recomputed based on recorded statistics in a certain region or for certain road types. Table 2: Design values of impact forces based on the probabilistic approach. Type of road Roadways Urban areas

Design impact force Fd,x [kN] d=3m d=6m d=9m 2910 2850 2810 1580 1500 1430

Design Example Consider the reinforced concrete bridge pier of Figure 2. The cross sectional dimensions are b = 0.50 m and h = 1.00 m. The column height h = 5 m and it is assumed to be hinged to both the bridge deck as to the foundation structure. The reinforcement ratio ρ is 0.01 for all four

122


groups of bars as indicated in figure 4.1, right hand side. Let the steel yield stress be equal to 300 MPa and the concrete strength 50 MPa. The column will be checked for impact by a truck under motorway conditions.

x

H

h y

Fdy a b

Figure 2. Bridge pier under impact loading According to the code, the forces Fdx and Fdy should be taken as 1000 kN and 500 kN respectively and act at a height of a = 1.25 m. The design value of the bending moments and shear forces resulting from the static force in longitudinal direction can be calculated as follows: Mdx =

a( H − a ) 1.25(5.00 − 1.25) Fdx = 1000 = 940 kNm H 5.00

Qdx =

H −a Fdx H

=

5.00 − 1.25 1000 5.00

= 750 kN

Similar for the direction perpendicular to the diving direction: Mdy=

a( H − a ) 1.25(5.00 − 1.25) Fdy = 500 H 5.00

= 470 kNm

Qxy =

H −a Fdy H

= 375 kN

=

5.00 − 1.25 500 5.00

Other loads are not relevant in this case. The self-weight of the bridge deck and traffic loads on the bridge only lead to a normal force in the column. Normally this will increase the load bearing capacity of the column. So we may confine ourselves to the accidental load only. Using a simplified model, the bending moment capacity can conservatively be estimated from: MRdx = 0.8 ρ h2 b fy = 0.8 0.01 1.002 0.50 300 000 = 1200 kNm

123


MRdy = 0.8 ρ h b2 fy = 0.8 0.01 1.00 0.502 300 000 = 600 kNm As no partial factor on the resistance need to be used in the case of accidental loading, the bending moment capacities can be considered as sufficient. The shear capacity of the column, based on the concrete tensile part (say fctk = 1200 kN/m2) only is approximately equal to: QRd = .0.3 bh fctk = 0.3 1.00 0.50 1200 = 360 kN. This is almost sufficient for the loading in y-direction, but not for the x-direction. Additional shear force reinforcement is necessary.

3.2

Impact on superstructure Design values for actions due to impact from lorries and/or loads carried by the lorries on members of the superstructure should be defined unless adequate clearances or suitable protection measures to avoid impact are provided. The recommended value for adequate clearance, excluding future re-surfacing of the roadway under the bridge, to avoid impact is in the range 5.0 m to 6.0 m. The following scenarios are considered: a)

impact on restraint system on the superstructure (see Figure 3) Indicative equivalent static design forces are given in the Eurocode 1 part 1.7 for that scenario and are reported here in Table 3. The forces apply perpendicular to the direction of normal travel.

Figure 3. Vehicle impact on restraint system

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Table 3 - Indicative equivalent static design forces due to impact on superstructures. Category of traffic

Equivalent static design force Fdx a [kN]

Motorways and country national and main roads Country roads in rural area Roads in urban area Courtyards and parking garages

500 375 250 75

a

x = direction of normal travel.

b)

impact forces on the underside surfaces of bridge decks The same impact forces as given in Table 3 with an upward inclination are considered as also shown in Figure 4.

F(h')

F(h)

F(h)

10°

10°

h' h

drivig direction

h

Figure 4. Impact forces on underside surfaces of superstructure

4

LOADS DUE TO RAILWAY COLLISION

EN 1991 1-7 classifies structures that maybe subject to impact from derailed railway traffic according to Table 4. Bridges belong consequently to class B. For that class each requirement should be specified. To some extent it is questionable whether an analysis for horizontal impact should be made at all since the probability of such an event is very small. The probability depends on: - Likelihood of train derailment - Likelihood of train colliding with the bridge given train derailment The likelihood of train derailment depends on the derailment rate, the number of trains per day and the critical distance. The likelihood of collision depends on the lateral distance from the structure and on the train velocity [11]. A risk analysis approach can be found in [12].

125


Table 4. Classes of structures subject to impact from derailed railway traffic. Class A

Class B

Structures that span across or near to the operational railway that are either permanently occupied or serve as a temporary gathering place for people or consist of more than one storey. Massive structures that span across or near the operational railway such as bridges carrying vehicular traffic or single storey buildings that are not permanently occupied or do not serve as a temporary gathering place for people.

Recent studies in Switzerland have investigated the impact force on bridges after train derailment. The impact force is a function of the speed and direction of impact, which depends mainly on the train velocity vE and the distance from the point of derailment to the point of impact, as well as on the dynamic friction coefficients. Before the engine or the rest of the train impacts on a structure after derailment, the train, or a part of it, travels a certain distance across the ballast and the platform. Thus, part of the kinetic energy is dissipated before impact. Thereby a number of different cases have been treated. Some of the more important ones are shown in Figure 5 and are: 1. 2. 3.

Derailment of the engine at the front end of the train. Derailment of a carriage at the front end of the train (engine at rear) Derailment of two carriages at an arbitrary position of the train. y vE L

w

θ0

1)

x

derailed vehicle vehicle on the railway track

y vE w

L 2)

derailed vehicle

vE 3)

θ0 θ0

θ0

x L = engine w = carriage v = velocity at moment of derailment

x

derailed vehicles

Cases of derailment engine Re 6/6:

length=19.3 m; width=3.0 m; height=4.5 m; mass=120 t

carriage EW IV:

length=26.4 m; width=2.8 m; height=4.1 m; mass=41/50 t

Figure 5. Cases of train derailment

126


The results of the studies have shown that the impact forces depend on the scenario and can reach values up to 30 MN. Impact functions of train engines on stiff structures are reported in Figure 6 as function of the engine velocity at moment of derailment. It is recommended here to perform a risk analysis to define the impact forces in case that this accidental load is considered, i.e. has an annual probability of occurrence greater than 10-6. Protection measures are also recommended to reduce the risk. F [MN]

F [MN]

v ≤8 m/s

F [MN]

8 m/s
E

v >12 m/s

E

E

30

30

30

20

20

20

10

10

10

O

2

4

6

8

10

t [s/100]

O

2

4

6

8

t [s/100]

O

2

4

6

8

t [s/100]

Figure 6. Impact functions of train engines on stiff structures

5

LOADS DUE TO SHIP IMPACT

Ship impact accidents (see Figure 7) have occurred several times in the past with considerable consequences. Table 5 shows the most important casualties of accidents involving ships and bridges, [13].

Table 5. Fatalities in ship-bridge collisions (1960-2002)

127


Figure 7. Ship impact on bridges The probability of a ship colliding with a particular object in the water (here bridge deck or bridge piers) depends on the intended course of the ship relative to the object and the possibilities of navigation or mechanical errors. In order to find the total probability of an object being hit, the total number of ships should be taken into account. Finally, the probability of having some degree of structural damage also depends on the mass, the velocity at impact, the place and direction of the impact and the geometrical and mechanical properties of ship and structure. When discussing ship collisions, it is essential to make a distinction between rivers and canals on the one side and open water areas like lakes and seas on the other. On rivers and canals the ship traffic patterns can be compared to road traffic. On open water, shipping routes have no strict definitions, although there is a tendency for ships to follow more or less similar routes when having the same destination. A typical possible model for the ship distribution within a traffic lane is presented in Figure 8. In general it will be possible to model the position of a ship in a lane as a part with some probability density function. Details will of course depend on the local circumstances. It should be noted that sometimes the object under consideration might be the destination of the ship, as for instance a supply vessel for an offshore structure. Navigation errors are especially important for collisions at sea. Initial navigation errors may result from inadequate charts, instrumentation errors and human errors. The probabilistic description of these errors depends on the type of ship and the equipment on board, the number of the crew and the navigation systems in the sea area under consideration. Given a ship on collision course, the actual occurrence of a collision depends on the visibility (day or night, weather conditions, failing of object illumination, and so on) and on possible radar and warning systems on the structure itself. Mechanical failures may result from the machinery, rudder systems or fire, very often in connection with bad weather conditions. The course of the ship after the mechanical failure is governed by its initial position and velocity, the state of the (blocked) rudder angle, the current and wind forces, and the possibility of controlling the ship by anchors or tugs. These parameters 128


together with the mass and dimensions of the ship should be considered as random. Given these data, it is possible to set up a calculation model from which the course of the ship can be estimated and the probability of a collision can be found. Such models have been applied several times in ship collision specific studies for important bridges.

Figure 8. Ingredients for a ship collision model The ship bridge accidents can be in general divided into three cases (Figure 9): A. bow collision with bridge pillar, B. side collision with bridge pillar, C. deckhouse (superstructure) collision with bridge span.

Figure 9. Frequent types of ship bridge accidents: A) bow collision with bridge pillar; B) side collision with bridge pillar; C) deckhouse (superstructure) collision with bridge span The most important and frequent in scope of energy distributed during collision are bow collisions. The occurrence of a mechanical or navigation error, leading to a possible collision with a structural object, can be modelled as an (inhomogeneous) Poison process. Given this Poison

129


failure process with intensity λ(x), the probability that the structure is hit at least once in a period T can be expressed as:

Pc (T) = nT( 1- P a ) ∫ ∫ λλ(x) Pc (x,y) f s (y) dx dy whereas: T = n = λ(x) = Pc(x,y) = fs(y) = Pa =

(1)

period of time under consideration number of ships per time unit (traffic intensity) probability of a failure per unit travelling distance conditional probability of collision, given initial position (x,y) distribution of initial ship position in y direction the probability that a collision is avoided by human intervention.

For the evaluation in practical cases, it may be necessary to evaluate Pc for various individual object types and traffic lanes, and add the results in a proper way at the end of the analysis. To give some indication for λ, in the Nieuwe Waterweg near Rotterdam in the Netherlands, 28 ships were observed to hit the river bank in a period of 8 years and over a distance of 10 km. Per year 80 000 ships pass this point, leading to λ=28/(10×8×80000) = 10-6 per ship per km. For practical applications mechanical models rules have been developed to calculate the part of the total energy that is transferred into the structure. Some of these rules are based on empirical models, others on a static approximation, starting from so-called load indentation curves (F-u diagrams) for both the object and the structure. According to this model the interaction force during collapse is assumed to raise form zero up to the value where the sum of the energy absorption of both ship and structure equal the available kinetic energy at the beginning of the impact. Design values can be then defined then from the collision model. The occurrence of a mechanical or a navigation error, leading to a possible collision with a structural object, can be modelled as a Poisson process. If data about types of ships, traffic intensities, error probability rates and sailing velocities are known, a design force could be found from:

P(F>Fd ) = nT(1-p a ) ∫ ∫ λλ(x) P[vx,y) ( km) > F d ] f s (y) dx dy

(2)

Given target reliability and estimates for the various parameters in (2) design values for impact forces may be derived. The values in Tables 4.5 and 4.6 of EN 1991 Part 1.7, however, have not been derived on the basis of explicit target reliability. For inland ships the values in Table 4.5 in [2] have been chosen in accordance with ISO DIS 10252. For a particular design it should be estimated which size of ships on the average might be expected, and on the basis of those estimates, design values for the impact forces can be found. Table 6 shows a comparison between: - the values in Table 4.5 of EN 1991-1-7; - the values based on Annex C of EN 1991-1-7, equation (C1); - the values based on Annex C of EN 1991-1-7, equation (C9). The masses for the inland waterways ships should been taken in the middle of the class. The velocity used is 3 m/s and the equivalent stiffness k = 5 MN/m.

130


Table 6. Design forces Fd for inland ships m [ton]

v k [m/s] [MN/m]

300 1250 4500 20000

3 3 3 3

5 5 5 5

Fd Fd [MN] Fd [MN] [MN] Table 4.5 of eq (C.1) of EN eq (C.9) of EN EN 1991-1-7 1991-1-7 1991-1-7 2 4 5 5 8 7 10 14 9 20 30 18

For sea going vessels values in Table 7 are based on equation (2), with v=3 m/s and k0=15 MN/m for the smallest ship category and 60 MN/m for the heaviest category. Table 7. Design forces Fd for seagoing vessels m k [ton] v [m/s] [MN/m]

3000 10000 40000 100000

6

5 5 5 5

15 30 45 60

Fd [MN] Table 4.6 of EN 1991-1-7 50 80 240 460

Fd Fd [MN] [MN] eq(C.1) of EN eq (C.11) of 1991-1-7 EN 1991-1-7 34 33 87 84 212 238 387 460

DISCUSSION ON ANNEX C

The informative Annex C of EN 1991 Part 1-7 gives the designer information on background information for dynamic calculations in the case of impact loading. A correct impact assessment s requires a nonlinear dynamic analysis of a model that comprises both the structure as the impacting body. The annex demonstrates the principles of such an analysis using simple empirical models. It should be noted that more advanced models might be appropriate in special cases or background studies. In the assumption that the structure is rigid and immovable and the colliding object deforms linearly during the impact phase and remains rigid during unloading, the maximum resulting dynamic interaction force is given by: F = vr k m

(3)

where vr is the object velocity at impact; k is the equivalent elastic stiffness of the object (i.e. the ratio between force F and total deformation); m is the mass of the colliding object. The stiffness, of course, is some kind of an averaged equivalent value, incorporating all kind of geometrical and physical nonlinearities in the mechanics of the collision process. Some reasonable estimates for these quantities are shown in Table 8:

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Table 8. Statistical parameters for input values

m v k

mean value 20 ton 80 km/hr 300 kN/m

mass velocity equivalent stiffness

standard deviation 12 ton 10 km/hr

As we are considering a loading situation conditional upon the accidental event of collision, there is no need to use extreme fractiles of these distributions. In many cases one chooses to use the mean plus one standard deviation. In this case this leads to m=32 ton and v= 90 km/hr=25 m/s and from there we find: F = 25 (300 ×32)0.5 = 2400 kN Compared to the F = 1000 kN in table 3.1 this is a large number. However, we should keep in mind that the load in table 3.1 is intended as a static value, where the force acts only over a short period of time. The shape of the force due to impact can usually be assumed as a rectangular pulse and the duration of the pulse is then given by: ∆t = m / k

(4)

In the given example the duration would be 0.3 s. Another point is that the vehicle usually looses speed between the point where it leaves the track and the point where it hits the structure (see Figure 10). d

d structure structure road

v0

d

s

road

ϕ road

structure d

road structure

Figure 10. Situation sketch for impact by vehicles (top view and cross sections for upward slope, flat terrain and downward slope)

For a given deceleration a, the velocity vr after a distance s from the critical point is: vr = (v02– 2 a s )0.5

132

(5)


Using a = 4 m/s2 we arrive at a distance s = 80 m. This means that the force will be zero if the distance between the centre line of the track and the structural element is about 20 m. Here it has been assumed that the angle ϕ = 15o. For intermediate distances one may use the expression: F = Fo 1 − d / d b (for d < db).

(6)

Note that the value of db may be adjusted because of the terrain characteristics. The force of eq. (3) is the force at the impact surface between the structure and the impacting vehicle. Inside the structure this load will lead to dynamic effects. As long as the structure behaves elastically there may be some the dynamic amplification (one may think of 40 percent). However, due to elastic-plastic effects stresses may be reduced.

7

RISK ANALYSIS

For important bridges risk analyses are performed in order to compute the risk associated to impact forces and especially due to ship collision. Site specific data of the type of traffic are combined with probabilistic analyses as mentioned above in order to define: - Design impact force associated to a target probability level - Protection measures - Robustness measures to avoid global failure (see for example [14]) The risk analysis scheme given in the Eurocodes 1991 Part 1.7 can be useful in order to analyse the aforementioned aspects. It is reported here in Figure 11. Decision measures are taken based on such a procedure.

Figure 10. Risk analysis procedure in the Eurocodes

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8

CONCLUSIONS

Accidental actions on bridges as specified in the Eurocode 1991 Part 1.7 have been reviewed in this chapter. The following topics have been covered: - -impact loads due to road traffic - -impact loads due to train traffic - -impact loads due to ships The models presented in Annex C of EN 1991-1-7 have been discussed. Apart from design values and other detailed information for the loads mentioned above, the document EN 1991, Part 1-7 also gives guidelines how to handle accidental loads in general. In many cases structural measures alone cannot be considered as very efficient.

9

REFERENCES

[1] EN 1990 Eurocode - Basis of structural design. European Comittee for Standardisation, 04/2002. [2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Comittee for Standardisation, 04/2002. [3] ISO 2394, General principles on reliability for structures. 1998. [4] I SO 3898, Bases for design of structures – Notations - General Symbols, 1997. [5] Larsen, O.D.: Structural Engineering Documents “Ship Collision with bridges”, The interaction between Vessel traffic and Bridge structures [6] CIB: Actions on structures impact, CIB Report, Publication 167, CIB, Rotterdam 1992 [7] Vrouwenvelder, T.: Stochastic modelling of extreme action events in structural engineering, Probabilistic Engineering Mechanics 15 (2000) 109-117 [8] Vrouwenvelder, T.:”Design for ship impact according to Eurocode 1, Part 2.7”, Ship collision analysis, Gluver and Olson, 1998 Balkema, ISBN 9054109629 [9] Markova, J. and K. Jung: “Alternative procedures for impact forces in Eurocodes” Journal of KONBIN, In: Proceeding of the 4th International Conference on Safety and Reliability, Wydawnictwo Instytutu Technicznego Wojsk Lotniczych, 30 May-2 June 2006, ISSN 18958281, pp 175-182

[10] Joint Committee on Structural Safety (JCSS), Probabilistic Model Code, www.jcss.ethz.ch [11] UIC Code 777-2: Structures Built over Railway lines, Paris, 2003. [12] Jung, K. and J. Markova: “Risk assessment of structures exposed to impact by trains" In: Walraven, Blaauwendraad, Scarpas & Snijder (eds.), Proceedings of 5th International PhD Symposium in Civil Engineering; 16-19 June 2004, Delft, The Netherlands, A.A. Balkema Publishers, ISBN 90 5809 676 9; pp. 1057-1063 [13] Proske,D.:Ein Beitrag zur Risikobeurteilung von alten Brücken unter Schiffsanprall, Dissertation, TU Dresden, 2003. [14] Starossek, U., Progressive Collapse of Structures: Nomenclature and Procedure, Structural Engineering International Vol. 2, 2006.

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CHAPTER 7: COMBINATION RULES FOR BRIDGES IN EUROCODES Milan Holický1, Jana Marková1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary The combination rules for bridges introduced in this Chapter are based on Eurocode EN 1990/A1. The combinations of traffic loads with non-traffic actions and alternative procedures for load combinations are presented here. An example of verification of the bridge cantilever for the limit state of static equilibrium is included. Furthermore, selected results of application of alternative combination rules for the design of prestressed concrete highway bridge in the Bohemia are presented. Comparison of obtained action effects indicates that alternative combination rules may lead to considerably diverse load effects. Further harmonisation based on calibrations of partial factors and other safety elements is needed.

1

INTRODUCTION

1.1

Background documents EN 1990/A1 [2] provides basis for determination of combinations of actions for ultimate and serviceability limit state verifications of bridges. The aim of this Chapter is to describe principles of load combinations. Permanent actions, traffic loads and climatic actions due to wind, snow and temperature are considered in accordance with relevant Parts of EN 1991. Supplementary information on the traffic load models provided in EN 1991-2 [3] is given in the Background document [4] which is expected to be available on the JRC web site. 1.2

General principles EN 1990/A1 [2] gives rules focused on the application of basis provided in EN 1990 [1] for the bridge design. The critical load cases should be determined for the selected design situations and identified limit states. Similarly as for buildings (see Annex A1 in [1]) the alternative combination rules for the ultimate limit states provided in EN 1990/A1 [2] may lead to significantly diverse load effects as illustrated in Section 5.

2

COMBINATION OF ACTIONS

2.1

General The effects of actions that cannot occur simultaneously due to physical or functional reasons should not be considered together in combinations of actions. In case when specific measures are provided preventing some actions to act simultaneously, these combinations need not be considered in analysis (e.g. when it is assured that some construction loads are not simultaneously acting during a specific construction phase). The expressions 6.9a to 6.12b in EN 1990 [1] are applied for the verification of ultimate limit states and the expressions 6.14a to 6.16b are applied for the verification of the

135


serviceability limit states in bridge design. 2.2

Fundamental combination of actions

Three alternatives procedures for fundamental combination of actions is provided for bridges, similarly as for buildings. The choice of combination is the National Determined Parameter (NDP) which may be nationally selected. For example the fundamental combination of actions in bridge design in the Czech Republic is based on the twin expressions (6.10a), (6.10b) given as

∑ γ G , j Gk , j "+" γ P P "+"γ Q,1ψ 0,1 Qk ,1"+" ∑ γ Q,iψ 0,i Qk ,i

(1)

∑ ξ γ G , j Gk , j "+" γ P P "+"γ Q,1 Qk ,1 "+" ∑ γ Q,i ψ 0,i Qk ,i

(2)

j ≥1

i >1

j ≥1

j ≥1

where the less favourable expression needs to be considered. Where relevant, the favourable or unfavourable design values of permanent actions Gd,sup or Gd,inf should be considered. The application of the combination of actions according to the twin expressions (6.10a), (6.10b) gives in common cases more uniform reliability level of bridges for various ratios of the characteristic values of variable loads and permanent loads. Furthermore, it was also decided in the Czech Republic to allow application of the unique expression (6.10) ∑ γ G , j Gk , j " +" γ P P " +" γ Q ,1 Qk ,1" +" ∑ γ Q ,i ψ 0 ,i Qk ,i j ≥1

j ≥1

(3)

Selected results of application of alternative combination rules for a prestressed concrete road bridge is shown in the example 5.2 at the end of this Chapter. The rules for simultaneous combinations of individual traffic load models and their arrangements are provided in EN 1991-2 [3]. Five different and mutually exclusive groups of traffic loads as given in [3] are shown in Table 1. Any group of traffic loads should be taken into account as one variable action which is acting in combination with other variable actions. Table 1. Assessment of groups of traffic loads (characteristic values of the multicomponent action) Carriageway Vertical loads Groups of loads gr1 gr2

Main load model LM1

Special vehicles LM3

Horizontal loads Crowd loading LM4

Characteristic values Frequent values

gr3 Characteristic values

gr4 gr5

See EN 19912

Braking, acceleration forces

Centrifugal. transverse force



Footways and cycle tracks Vertical loads only Uniformly distributed loads

Characteristic values Characteristic values


The rules for combinations of construction loads during execution for bridges were transferred from EN 1991-1-6 [6] to EN 1990/A1 [2]. However, some rules for the

136


verification of ultimate and serviceability limit states during execution and for combination of construction loads with other variable loads remain till now in Annex A2 of EN 1991-1-6 [5]. It should be taken into account in appropriate cases that construction loads Qc act simultaneously with other types of actions. Different construction loads (Qca to Qce) should be considered according to the project conditions as one single action, or also as several individual construction loads that are combined with other variable actions. In some cases it need not be considered in one combination some variable actions. For example, it is rather unlikely the simultaneous occurrence of construction load Qca due to working personnel with small site equipment together with maximum wind or snow actions. For individual project however, it may need be considered in combination snow and wind simultaneously with other types of construction loads, e.g. with cranes. The characteristic values of climatic actions may be reduced for short-time construction periods on the basis of EN 1991-1-6 [6]. Where relevant, the thermal actions and water loads should be considered simultaneously with construction loads. The various parameters governing water actions and components of thermal actions should be taken into account when identifying appropriate combinations with construction loads. The selection of actions to load combinations need to be considered according to the conditions of individual project. 2.3

Combinations of actions for road bridges Suplementary rules are introduced for load combinations on road bridges. The snow load and wind actions need not to be combined with - the braking and acceleration forces or the centrifugal forces or the associated group of loads gr2 - the loads on footways and cycle tracks or with the associated group of loads gr3, - the crowd loading (LM 4) or the associated group of loads gr4. In common cases it is not necessary to consider the snow loads together with models LM1 and LM2. However, in some mountain areas it may be also necessary to consider the combination of snow with traffic. LM1 or group of loads gr1a need not to be considered with wind actions greater than FW* or ψ 0 FWk . For certain serviceability limit states of concrete bridges the infrequent combination of actions is also recommended in EN 1990/A1 [2] given as

{

E d = E G k , j ; P; ψ 1,infq Qk ,1 ; ψ 1,i Qk ,i

}

j ≥ 1; i > 1

(4)

where the combination of actions in braquet may be expressed as

∑ Gk , j "+" P"+"ψ 1,infq Qk,1 "+" ∑ψ 1,i Qk,i j ≥1

(5)

i >1

The infrequent value of the traffic load corresponds to the mean return period of one year which is based on the product of the characteristic value of variable load and factor ψ1,infq. The recommended value of the factor for traffic loads ψ1,infq = 0.8. 2.4

Combination rules for footbridges Infrequent combination of variable actions is not considered in footbridge design. The concentrated load (wheel load) Qfwk need not to be combined with any other variable actions than those due to traffic. In general, wind loads and thermal actions need not be taken into account simultaneously in common cases.

137


Snow loads need not be combined with groups of loads gr1 and gr2 for footbridges unless otherwise specified for particular geographical areas and certain types of footbridges. In case that fotbridges provide protection of the pedestrians and cyclists against all kinds of unfavourable weather, the specific load combinations may be determined. The combination similar to actions on buildings may be applied in which instead of relevant category of imposed load the specific group of traffic loads is applied. 2.5

Combinations of actions for accidental design situations One accidental action should be considered only in accidental load combination which need not be combined with snow load or wind actions. Models of accidental actions on bridges are given in EN 1991-2 [3], models of impact forces due to cars, heavy vehicles and vessels on bridge substructures in EN 1991-1-7 [7]. Additional combinations of actions for other accidental design situations (e.g. combination of road or rail traffic actions with avalanche, flood or scour effects) may be agreed for the individual project. Values of ψ factors ψ factors for traffic loads, wind, thermal actions and construction loads are given for road bridges in the following Table 2. The recommended values of ψ factors are given for the road traffic corresponding to adjusting factors αQi, αqi, αqr and βQ equal to 1. The recommended ψ0 value for thermal actions may in most cases be reduced to 0 for ultimate limit states EQU, STR and GEO. The characteristic values of wind actions and snow loads during execution are defined in EN 1991-1-6 [6]. Representative values of water loads are not given in EN 1990/A1 [2]. They may be defined as NDP in the National Annex or for the individual project.

2.6

Table 2. Recommended values of ψ factors for road bridges. Type of action ψ0 gr1a (LM1 + pedestrian or cycle-track 1) loads)

ψ1,infq

ψ1

ψ2

TS (tandem system)

0.75

0.8

0.75

0

UDL (uniform)

0.40

0.8

0.40

0

pedestrian or cycletrack loads 2)

0.40

0.8

0.40

0

gr1b (single axle)

0

0.8

0.75

0

gr2 (horizontal forces)

0

0

0

0

gr3 (pedestrian loads)

0

0.8

0.40

0

gr4 (LM4 – crowd loading)

0

0.8

0.75

0

gr5 (LM3 – special vehicles)

0

0

0

0

0.6 0.8

0.6 -

0.2 -

0 0

Fw*

1.0

1

-

-

Thermal actions

Tk

0.6

0.8

0.6

0.5

Snow loads

QSn,k – transient design situation

0.8

-

-

-

Construction loads

Qc

1.0

-

-

1.0

Traffic loads

Wind actions

138

Fw

– persistent design situation – transient design situation


The recommended values of ψ0, ψ1 and ψ2 for gr1a and gr1b are given for road traffic corresponding to adjusting factors αQi, αqi, αqr and βQ equal to 1. Those relating to UDL correspond to common traffic scenarios, in which a rare accumulation of lorries can occur. Other values may be expected for other classes of routes, or expected traffic, related to the choice of the corresponding α factors. Recommended ψ factors for footbridges are given in Table 3. The combination value of the pedestrian and cycle-track load, mentioned in Table 4.4a of EN 1991-2 [3] is a reduced value to which the factors ψ0 and ψ1 may be used. The recommended ψ0 value for thermal actions may in most cases be reduced to 0 for ultimate limit states EQU, STR and GEO. Table 3. Recommended values of ψ factors for footbridges. ψ0

ψ1

ψ2

gr1

0.40

0.40

0

Qfw

0

0

0

gr2

0

0

0

Wind forces

Fw

0.3

0.2

0

Thermal actions

T

0.6

0.6

0.5

Snow loads

QSn (transient design situation)

0.8

-

0

Construction loads

Qc

1.0

-

1.0

Action

Traffic loads

3

ULTIMATE LIMIT STATES

3.1

Design values of actions in persistent and transient design situations The design values of actions and recommended partial factors given in EN 1990/A1 [2] for the ultimate limit states (EQU) in the persistent and transient design situations are given in Table 4. The design values of actions and recommended partial factors given in EN 1990/A1 [2] for the ultimate limit states (STR) in the persistent and transient design situations are given in Table 5. The design values of actions and recommended partial factors given in EN 1990/A1 [2] for the ultimate limit states (STR/GEO) in the persistent and transient design situations are given in Table 6. For the design of geotechnical structures (e.g. footings, piles, piers, abutments) where geotechnical actions and the resistance of the ground are involved the following three alternative approaches recommended in EN 1990/A1 [2] - Approach 1: Applying in separate calculations design values from Table 6(C) and Table 5(B) to the geotechnical actions as well as the actions from the structure; - Approach 2: Applying design values of actions from Table 6(B) to the geotechnical actions as well as the actions from the structure; - Approach 3: Applying design values of actions from Table 6(C) to the geotechnical actions and, simultaneously, applying design values of actions from 5(B) to the actions from the structure.

139


Table 4. Design values of actions (EQU) (Set A) Persistent and transient design situation Eq. (6.10)

Permanent actions

Prestress

Leading variable action

Accompanying variable actions main others (if any)

γP P

γQ,1 Qk,1

γQ,iψ0,iQk,i

unfavourable favourable

γGj,sup Gkj,sup

γGj,inf Gkj,inf

The recommended set of partial factors: γGj,sup=1.05 for unfavourable actions and γGj,inf =0.95 for favourable permanent actions, γQi=1.35 for unfavourable actions due to road and pedestrian traffic, γQ=1.45 for unfavourable actions due to railway traffic, reduced to γQ=1.20 for load models SW/2 and unloaded train, γQi=1.5 for other variable actions, 0 for favourable variable actions. The values of partial factors for prestressing γP are given in the relevant design Eurocodes. For unfavourable construction loads in transient design situations γQ=1.35. When a counterweight is used, partial factor for its weight can be assumed γG,inf =0.8. For the verification of uplift of bearings of continuous bridges or in cases where the verification of static equilibrium also involves the resistance of structural members (e.g. where the loss of static equilibrium is prevented by stabilising systems or devices, e.g. anchors, stays or auxiliary columns), as an alternative to two separate verifications based on Tables 4 (Set A) and 5 (Set B), a combined verification, based on Table 4 (Set A), may be applied. The following values of γ factors are recommended in the combined verification: γG,sup=1.35, γG,inf=1.25 for permanent actions, γQ =1.35 for unfavourable actions due to road traffic and pedestrians, γQ= 1.45 for unfavourable actions due to railway traffic, reduced to γQ=1.20 for load models SW/2 and unloaded train; γQ =1.50 for other variable actions and γQ=1.35 for construction loads. For favourable variable actions, γQ=0, provided that applying γG,inf =1,00 both to the favourable and unfavourable part of permanent actions does not give more unfavourable effect.

Table 5. Design values of actions (STR) (Set B) Persistent and transient design situation

Permanent actions

Prestress



γQ,1 Qk,1

γQ,iψ0,iQk,i


Exp. (6.10)

γGj,supGkj,sup

γGj,infGkj,inf

γPP

Exp. (6.10a)

γGj,supGkj,sup

γGj,infGkj,inf

γPP

Exp. (6.10b)

ξγGj,supGkj,sup

γGj,infGkj,inf

γPP

γQ,1ψ0,1Qk,1 γQ,1 Qk,1

γQ,iψ0,iQk,i γQ,iψ0,iQk,i

The choice between exp. 6.10, or 6.10a and 6.10b may be decided in the National Annex. Recommended values: γGj,sup = 1.35 for unfavourable and γGj,inf = 1.0 for favourable permanent actions; γQ = 1.35 for unfavourable actions due to road or pedestrian traffic; γQ = 1.45 for unfavourable actions due to railway traffic, reduced to γQ=1.20 for load models SW/2 and unloaded train; For favourable variable actions, γQ = 0. The characteristic values of all permanent actions from one source may be multiplied by γG,sup if the total resulting action effect is unfavourable and γG,inf if the total resulting action effect is favourable. For particular verifications, the values for γG and γQ may be subdivided into γg and γq and the model uncertainty factor γSd (a value of γSd in recommended in the range 1.0 to 1.15).

140


Table 6. Design values of actions (STR/GEO) (Set C) Persistent and transient design situation Exp. (6.10))

Permanent actions

Prestress


γP P

γQ,1 Qk,1


γGj,supGkj,sup

γGj,infGkj,inf


γQ,iψ0,iQk,i

Recommended set of partial factors for actions:

γGj,sup=γGj,inf =γGset=1.0 for permanent actions and actions due to settlements, γQ =1.15 for unfavourable actions due to road and pedestrian traffic, γQ =1.15 for unfavourable actions due to railway traffic, γQ = 1,30 for the variable part of horizontal earth pressure from soil, ground water, free water and ballast, γQ = 1.30 for all other unfavourable variable actions (0 for favourable).

The selection of the geotechnical approach is a NDP which may be given the National Annex. For example it was decided in the Czech Republic to recommend the Approach 2 for footings, piles, anchors, underground walls etc., and the Approach 3 is recommended to be applied for the stability of the slopes. The design values of actions for the ultimate limit states in the accidental and seismic design situations are given in Table 7. Table 7. Design values of actions for use in accidental and seismic combinations Persistent and transient design situation

Permanent actions unfavourable favourable

Prestress

Accidental or seismic action

Accidental exp. (6.11a/b)

Gkj,sup

Gkj,inf

P

Ad

Seismic Exp. (6.12a/b)

Gkj,sup

Gkj,inf

P

γIAEk or


ψ11 Qk1 or ψ21Qk1

ψ2,i Qk,i ψ2,i Qk,i

AEd

In the case of accidental design situations, the main variable action may be taken with its frequent or quasi-permanent values. The choice is given in the National Annex depending on the accidental action under consideration.

For execution phases during which there is a risk of loss of static equilibrium, the combination of actions is given as

∑G j ≥1

kj, sup

"+" ∑ G kj,inf "+"P"+" Ad "+"ψ 2 Qc,k

(6)

j ≥1

where Qc , k is the characteristic value of construction loads as defined in EN 1991-1-6 [6], i.e. the characteristic value of the relevant combination of groups Qca, Qcb, Qcc, Qcd, Qce and Qcf.

4

SERVICEABILITY LIMIT STATES

4.1

Design values of actions for serviceability verifications Three combinations of actions are recommended for verification of serviceability criteria. The design values of actions for the serviceability limit states are given in Table 8.

141


Several serviceability criteria are recommended in EN 1992 to EN 1999 including cracking in concrete structures. Table 8 Design values of actions for use in serviceability limit states Combination Characteristic

Permanent actions Gd Unfavourable Favourable Gk,j,sup Gk,j,inf

Prestress P

Variable actions Qd Leading Others Qk,1 ψ0,iQk,i

Frequent

Gk,j,sup

Gk,j,inf

P

ψ1,1Qk,1

ψ2,iQk,i

Quasi-permanent

Gk,j,sup

Gk,j,inf

P

ψ2,1Qk,1

ψ2,iQk,i

5

EXAMPLES

Example 5.1 Verification of the ultimate limit states (EQU) during execution The stability of a bridge cantilever during execution should be verified. The self-weight G, construction loads Qc and wind actions W are acting on a bridge cantilever. The scheme is illustrated in Figure 1.

gd,sup

wd

qcb,d

qca,d

Qcc,d

gd,inf

wd b

a

Figure 1. Actions on a bridge cantilever during an execution phase It is assumed that the coefficient of variation of self-weight g is very small and therefore, the mean value of the self-weight may be applied. Three construction loads are present on the right-side of the bridge cantilever: the uniformly distributed load due to working personnel with small site equipment qca, the movable storage of materials qcb and moveable heavy devices Qcc. Moreover, it is assumed a non-favourable effect of the wind forces w acting on the bridge cantilever (on the right side in the downwards direction, on the left side in the upward direction. The following condition given in (6.7) of EN 1990 [1] is applied for the verification of static equilibrium Ed, dst ≤ Ed, stb

The destabilizing effects of actions are determined on the basis Table 4 Ed,dst = 1.05 gk b2/2 + 1.35 (qca,k b2/2 + qcb,k b2/2 + Qcc,k b2) + 1,5 ψ0 wk b2/2

and stabilizing effects of actions are given as Ed,dst = 0.95 gk a2/2 –1.5 wk b2/2

142

(7)


where the lengths of bridge cantilevers are a = 24 m and b = 27 m. The self-weight of the prestressed concrete cantilever is determined on the basis of nominal dimensions of the box girder cross-section considering the mean value of density. The bridge cross-sectional area is A = 7.6 m2 and the density of prestressed concrete γc = 25 kN/m3. The self-weight is determined as gk = Ac γc = 7.6 × 25 = 190 kN/m

The heavy construction device Qk,cc (50 kN), construction loads due to working personnel qca (1 kN/m2) and movable storage of material qcb (0.2 kN/m2) are applied considering the recommended values given in ČSN EN 1991-1-6 [6]. The wind action per 1 metre of bridge length is determined as wk = ±6.9 kN/m (the procedure for specification of wind actions is not included here). The destabilising effects in case that the leading construction load (dominant) is present may be determined Ed,dst=1.05×190×272/2+1.35×(50×272+1×272/2+0.2×272/2)+1.5×0.8×6.9×272/2=117.4 MNm

and in case that the wind is a leading variable action Ed,dst = 1.05×190×272/2+1.5×6.9×272/2+1.35×(50×272+1×272/2+0,2×272/2) = 118.2 MNm

and the stabilizing effects of actions are given as Ed,stb = 0.95 × 190 × 242/2 – 1.5 × 6.9 × 242/2 = 101 kNm.

The condition given by expression (7) is not satisfied and therefore, for assurance of the cantilever stability it is necessary to accept additional measures. In case that the contra-weight is applied, the uncertainties in the position of the contra-weight need to be considered or the recommended partial factor γg,inf = 0.8 applied according to Table 4. Example 5.2 Selected results of application of alternative combination rules Selected results of the application of alternative load combinations given by expressions (6.10) or twin expressions (6.10a), (6.10b) for the analysis of internal forces of the highway prestressed concrete bridge in Čekanice are presented here. The bridge is a 13 span continuous beam, length of spans from 22 m to 40.5 m. The first part of bridge is built of beams (3 spans above the railway), characterised by the cross sections reported in Figure 2.a, the second part is built of box girder, see Figure 2.b. The self-weight, permanent loads, the group of traffic loads gr1 and thermal actions are taken into account. The two alternative approaches for the vertical difference component of thermal actions provided in EN 1991-1-5[5] are also considered here (linear approach 1 and non-linear approach 2). The approach 2 is selected in the National Annex of the Czech Republic.

Figure 2.a. Cross-sections of the Čekanice bridge on highway D3 (Prague - Tábor).

143


Figure 2.b. Box girder of the Čekanice bridge on highway D3 (Prague - Tábor). Selected results of comparative studies of the highway bridge in Čekanice are given in Tables 9 to 12. Table 9. Moments based on alternative procedures given in EN 1990/A1[2] and EN 1991-1-5 [5], hogging cross-sections. Type of crosssection, EN 19911-5 approach No. Beam - 2 Box girder - 1 Box girder - 2

Moments in MNm (6.10) Q1 -12.47 -36.26 -31.39

(6.10) T1 -9.15 -32.67 -24.55

(6.10a) -7.68 -27.88 -23.01

(6.10b) Q1 -8.67 -28.92 -24.05

(6.10b) T1 -5.36 -25.32 -17.20

Table 10. Moments based on alternative procedures given in EN 1990/A1 [2] and EN 1991-1-5 [5], sagging cross-sections. Type of crossMoments in MNm section, EN 1991-1-5 approach No.

Beam - 2 Box girder - 1 Box girder - 2

(6.10) Q1 14.48 34.97 31.99

(6.10) T1 13.42 34.65 29.69

(6.10a) 10.83 27.61 24.64

(6.10b) Q1 12.59 30.60 27.63

(6.10b) T1 11.54 30.28 25.32

Table 11. Upper stresses based on alternative procedures given in EN 1990/A1 [2] and EN 1991-1-5 [5], hogging cross-sections Type of crosssection, EN 1991-1-5 approach No.

Beam - 2 Box girder - 1 Box girder - 2

144

Stresses in MPa

(6.10) Q1 1.20 1.23 2.85

(6.10) T1 1.08 0.85 3.55

(6.10a) -0.51 0.34 1.96

(6.10b) Q1 -0.16 0.45 2.07

(6.10b) T1 -0.28 0.07 2.77


Hogging and sagging moments based on alternative combination rules given by expression (6.10) and twin expressions (6.10a,6.10b) of EN 1990/A1 [2] considered for two types of bridge cross-sections are shown in Tables 9 and 10, for stresses in Tables 11 and 12. The leading variable action is noted as Q1 for traffic loads or T1 for thermal actions. Table 12 Lower stresses based on alternative procedures given in EN 1990/A1 [2] and EN 1991-1-5 [5], hogging cross-sections Type of crossStresses in MPa section, EN 1991-1-5 approach No. (6.10) Q1 (6.10) T1 (6.10a) (6.10b) Q1 (6.10b) T1 Beam - 2 6.34 3.26 2.82 4.52 1.44 Box girder - 1 3.54 3.48 2.27 2.78 2.73 Box girder - 2 1.81 0.57 0.53 1.05 -0.16

6

REFERENCES

[1] EN 1990 Eurocode - Basis of structural design. CEN, Brussels, 2002. [2] EN 1990/A1 – Application for bridges. CEN, Brussels, 2002. [3] EN 1992-1 – Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges. CEN, Brussels, 2003. [4] Sedlacek G. et al., Background document to EN 1991-2 Traffic loads for road bridges – and consequences for the design, JRC Report, 2009 [5] EN 1991-1-5 – Eurocode 1: Actions on structures – Part 5: Thermal actions. CEN, Brussels, 2003 [6] EN 1991-1-6 – Eurocode 1: Actions on structures – Part 1: Actions during execution. CEN, Brussels, 2005 [7] EN 1991-1-7 – Eurocode 1: Actions on structures – Part 1: Accidental actions. CEN, Brussels, 2006.

145


146


CHAPTER 8: CASE STUDY - DESIGN OF A CONCRETE BRIDGE Pietro Croce1 1

Department of Civil Engineering, Structural Division, University of Pisa, Italy

Summary In this chapter an example of a simply supported prestressed concrete road bridge with open cross section. The load analysis is performed according to the provisions of EN 1990 and EN 1991, with special emphasis on traffic loads given in EN 1991-2. Aim of the case study is to clarify load application and load combinations, taking into account their influence on the local and global behaviour of the bridge members. Static and fatigue assessments are out of the scope of the present paper and are not considered here.

1

INTRODUCTION

In the present case study, the design of a prestressed concrete road bridge is discussed, with special emphasis on loads and load combinations. Loads are determined according to EN 1991-1-1 [1], EN 1991-1-4 [2], EN 1991-1-5 [3], EN 1991-2 [4], and load combinations are derived from EN 1990 [5]. The simply supported bridge, which covers an effective span of 45.0 m (figure 1), is located in a urban area and it is characterised by an open cross section composed by four precast pre-stressed concrete longitudinal beams set at constant spacing of 2.95 m (figure 2), connected by four stiff transverse beams. The transverse beam spacing is 15.0 m. The upper flanges of the precast longitudinal beams are duly connected to a 0.30 mthick concrete slab, cast in situ in a second phase. The concrete slab is not prestressed. Only end transverse beams (diaphragms) are connected to the concrete slab. End transverse beam

15

Transverse beams

End transverse beam

15

15

45

Figure 1. Static scheme of the bridge

2

THE SIMPLY SUPPORTED BRIDGE

2.1

General The total width of the bridge is 11.8 m. The carriageway, 7.50 m wide, is separated from the two walkways, each one 1.50 m wide, by means of two fixed safety barriers. The height of the longitudinal beams, whose geometry is represented in figure 3, is 2.76 m; therefore the total height of the cross section is 3.06 m. The distance between the bridge’s intrados and an underlying roadway is 6.0 m. The surfacing is made by a 60 mm thick asphalt layer.

147


11.8 1.5

0.5

7.5

0.5

1.5

Figure 2. Cross section of the bridge 11.8

3.06

0.35 0.2 0.6

0.8 0.98

0.3

1.81

0.82 0.9

0.6

2.76

2.95

0.35

2.95

0.3

2.95

Figure 3. Geometry of the cross section 2.2

Mass properties of the longitudinal beams The area A of each precast longitudinal beam is 1.27 m2. The centroid of the precast beam is located 1.24 m away from its intrados and the principal moments of inertia of the beam about its centroidal axes are Jx=1.099 m4 about the horizontal axis x and Jy=0.045 m4 about the vertical axis y. Once cast the slab, the mass properties of each beam modify as follows: area A=2.153 m2; centroid located 1.93 m away from the intrados; principal moments of inertia of the beam about its centroidal axes Jx=2.564 m4 about the horizontal axis x and Jy=0.687 m4 about the vertical axis y. The moment of inertia Jy of the whole cross section about its centroidal vertical axis y is finally 96.453 m4. 2.3

148

Structural materials Materials are chosen according to EN 1992-1-1 [6] and EN 1992-2 [7] The strength class of structural normal weight concrete is C50/60. Reinforcing steel is B450C. Prestress is obtained using post-tensioned prestressing tendons.


3

LOAD ANALYSIS

3.1

Structural self-weight Considering a density γ =25.0 kN/m3 for reinforced concrete, the nominal self-weight of each precast beam is: g k ,1b = γ A = 25.0 kN/m 3 × 1.24 m 2 = 31.0 kN/m .

(1)

The self-weight of the cast-in-situ r.c. slab pertaining to each longitudinal beam is g k ,1s = γ A = 25.0 kN/m 3 × 0.3 m × 2.95 m = 22.13 kN/m .

(2)

The total self-weight of each transverse beam is Gk ,1t = γ Vt = 25.0 kN/m 3 × 4.69 m 3 = 11.72 kN ,

(3)

being Vt its volume. 3.2

Self weight of non structural elements Self weights of non structural elements to be considered are those due to the waterproofing, to the 60 mm asphalt surfacing, to the safety barriers and to the parapets. For global verifications it is a reasonable approximation to consider these loads distributed per unit surface, by “spreading out” the weight of safety barriers and parapets. All things considered, the equivalent uniformly distributed loads corresponding to the self-weight of non structural elements gk,1p is about 2.2 kN/m2, so that the load per unit length of the bridge gk,2 is g k , 2 = 2.2 kN/m 3 × 11.80 m = 25.96 kN/m .

(4)

3.3

Traffic loads According to EN 1991-2 [4], traffic loads should be applied on the carriageway, longitudinally and transversally, in the most adverse position, according to the shape of the influence surface, in order to maximize or minimize the considered load effects. The first operation to be performed consists of determining the width w of the carriageway and the number of notional lanes. The carriageway width w depends, first of all, on whether the walkways are accessible to vehicular traffic or not. In the present case, as walkways are protected by fixed safety barriers, only crowd loading needs to be considered on them. The carriageway width w is given by the clear distance between the safety barriers, therefore w=7.50 m. As w>6.0 m, each notional lane is 3.0 m wide and the maximum number of notional lanes nl which can be considered is given by

 w   7.50  nl = Int  = Int  =2,  3 m   3 

(5)

and the maximum width of remaining area wr is 1.50 m (figure 4), wr = w − nl ⋅ 3.0 m = 7.50 m − 2 ⋅ 3.0 m = 1.50 m .

(6)

As known, EN 1991-2 calls for four separate static load models, being the single axle load model n. 2 (LM2) devoted only to local verifications.

149

3.0

notional lane n. 1

1.5

notional lane n. 2

7.5

3.0


remaining area

Figure 4. Notional lane arrangement on the carriageway For global verifications of the urban bridge in question, only load model n. 1 (LM1) and the expressly required crowd loading, load model n. 4 (LM4), are relevant. Load model n. 3 corresponding to special vehicles (LM3) is not considered, as the bridge is not concerned with special vehicle transit. On the i th notional lane, the main load model LM1 provides for a tandem system of axles weighing αQi Qik, accompanied by a uniformly distributed load αqi qik, being αQi and αqi the adjustment factors. In the present work it has been assumed αQi=αqi =1.0 for each lane, while the values Qik and qik are summarized in table 1. Table 1. Characteristic values for load model n. 1 (LM1) Notional lane Qk [kN] qk [kN/m2] Lane 1 300 9.0 Lane 2 200 2.5 Remaining area 0 2.5 Regarding the crowd loading, EN 1991-2 prescribes a nominal value of 5.00 kN/m2, and a combination value of 3.0 kN/m2 (2.5 kN/m2 in the Italian National Annex). LM1 and LM4 loads must be distributed in the most unfavourable way (both transversally and longitudinally) for the effect under consideration, bearing in mind, however, that a single lane cannot hold more than one tandem system, and that the tandem system, if present, must be considered in full, that is to say, with all the four wheels. 3.4

Wind actions Wind actions can be represented by vertical and horizontal equivalent static forces. Vertical force is orthogonal to the roadway plane, while horizontal forces are represented by two components, parallel and orthogonal to the bridge’s longitudinal axis, respectively. The equivalent pressure exerted by the wind can be calculated through the expression q p (z e ) = c e ( z e )

ρ 2

vb2 ,

(7)

in which ρ is the air density, which is assumed to be constant and equal to 1.25 kg/m3, vb is the basic wind velocity and ce(ze) is the so-called exposure coefficient, given by ce ( ze ) = cr2 (ze ) ⋅ c02 ( ze ) ⋅ [1 + 7 ⋅ I v (ze )] ,

(8)

where ze stands for the reference altitude over the ground. Expression (8) depends on the roughness coefficient cr, on the orography factor c0 and on the turbulence intensity Iv. The orography factor, taking into account any significant local

150


variations in the site’s orography, can usually be assumed equal to 1.0. Iv and cr are defined by the following expressions

ki    z   I v ( z e ) =  c0 ( z e ) ⋅ ln   z0   I v ( z min )

  z  k r (z e ) ⋅ ln  cr ( z e ) =   z0   cr ( z min )

if z min < z ≤ 200 m (9)

,

if z min ≥ z

if z min < z ≤ 200 m

,

(10)

if z min ≥ z

6

ze = 7.53

1.53

3.06

where ki is the turbulence factor, usually set to 1.0. In formulae (9) and (10), the terrain factor kr, the roughness length z0 and the minimum height zmin depend on the terrain category. As said, the bridge in question is located in an urban area which can be classified in terrain category IV, that is an area in which at least 15 % of the surface is covered with buildings whose average height exceeds 15 m. For terrain category IV it results z0=1.0 m, zmin=10.0 m, kr=0.234. The reference height ze represents the distance between the lowest ground level to the centre of the bridge deck structure, disregarding other parts (e.g. parapets) of the reference areas Recalling that the intrados of the structure is 6.0 m above ground level, it is ze=(6.0+0.5⋅3.06) m=7.53 m (figure 5).

Figure 5. Evaluation of the reference height ze for wind actions

151


As ze
I v ( ze ) = I v ( z min ) =

1 = 0.434 ,  10 m  1.0 ⋅ ln   1.0 m 

(11)

 10 m  cr (ze ) = cr (z min ) = 0.234 ⋅ ln  = 0.539 ,  1.0 m 

(12)

ce (z e ) = ce (z min ) = 0.539 2 ⋅ 1.0 2 ⋅ [1 + 7 ⋅ 0.434] = 1.176 .

(13)

so that

Basic wind velocity vb is function of geographic site. Here we assume vb=27 m/s, obtaining an equivalent static pressure

q p (z e ) = 1.176 ⋅

1.25 ⋅ 27.0 2 = 535.9 N/m 2 ≅ 0.54 kN/m 2 . 2

(14)

Said x the horizontal direction orthogonal to the bridge’s axis, the force Fwk,x is Fwk , x = q p ( z e ) ⋅ c f , x ⋅ Aref , x ,

(15)

where the coefficient cf,x depends on the ratio between the deck’s width b and the total deck’s height dtot exposed to wind. When the bridge is unloaded the exposed height is 4.26 m, as the presence of two open safety barriers and two open parapets is equivalent to an increase of 1.2 m in the exposed height. For unloaded bridge, the coefficient cf,x is    b 11.8 c f , x = min 2.4; max 2.5 − 0.3 ; 1.3   = 2.5 − 0.3 = 1.669 . d 4 . 26 tot   

(16)

If must be noted that the simplified approach proposed in EN1991-1-4 [2] allowing to set cf,x=1.3 is generally unsafe-sided. Using (16), the force Fwk,x for unloaded bridge is then (figure 6)

1.2

Fwk , x = q p ( z e )c f , x Aref , x = 0.54 ⋅ 1.669 ⋅ 4.26 = 3.84 kN/m .

Fwk,x

Figure 6. Equivalent static force Fwk,x (unloaded bridge) 152

(17)


When the bridge is loaded, the exposed height increases by 2.0 m (3.0 m in the Italian National Annex), so it becomes 5.06 m (6.06 m). In that case the coefficient cf,x is then    b 11.8 c f , x = min 2.4; max 2.5 − 0.3 ; 1.0   = 2.5 − 0.3 = 1.80 , d tot 5.06   

(18)

or, according to the Italian National Annex,    11.8 b ; 1.0   = 2.5 − 0.3 = 1.916 , c f , x = min 2.4; max 2.5 − 0.3 6 . 06 d tot    

(19)

and also for loaded bridge the simplification cf,x=1.3 results unsafe-sided. Since ψ0w=0.6, the combination values ψ0wFwk,x for the equivalent wind force for loaded bridge result (figure 7)

ψ 0 w Fwk , x = ψ 0 w q p (z e )c f , x Aref , x = 0.6 ⋅ 0.54 ⋅ 1.8 ⋅ 5.06 = 2.95 kN/m ,

(20)

ψ 0 w Fwk , x = ψ 0 w q p (z e )c f , x Aref , x = 0.6 ⋅ 0.54 ⋅ 1.916 ⋅ 6.06 = 3.76 kN/m ,

(21)

respectively.

2 m EN 1991-1-4 3 m Italian National Annex

wind action ψ Fwk,x 0

Figure 7. Equivalent static force ψ0wFwk,x (loaded bridge) Regarding the vertical action, lacking more precise data from wind tunnel tests, the coefficient cf,z can be set to c f , z = ± 0. 9 ,

(22)

where the sign is determined by the most unfavourable situation. As in that case the reference area is the horizontal projection of the bridge deck, Fwk,z is

Fwk , z = q p ( z e ) ⋅ c f , z ⋅ Aref , z = 0.54 ⋅ (± 0.9 ) ⋅ 11.80 m = ±5.74 kN/m ,

(23)

applied with an eccentricity, e, with respect to the longitudinal axis of the bridge

e=

d 11.80 m = = 2.95 m . 4 4

(24)

153


The vertical force Fwk,z is much smaller than the permanent loads, therefore, according to EN 1991-1-4, it could be disregarded. Finally, in the longitudinal direction y, the action to be considered is 25% of that for the direction orthogonal to the axis, but it is not relevant in this case.

3.5

Thermal actions As we are dealing with a statically determined structure, uniform and linear thermal variations should be taken into account only to calculate deformations of the structure and, if relevant, effects of friction forces in bearings. Non linear thermal variations cause stresses, but they are not particularly relevant in the present example. Assuming that in the site under consideration the maximum and minimum air shade temperatures with an annual probability of being exceeded of 0.02 are Tmax=40 °C and Tmin=10 °C, respectively, the uniform bridge temperature components for a concrete bridge result Te,max=41.7 °C and Te,min=-1.8 °C (figure 8). 70 60 1 50 2

40

Te,max

3 30 20 10

Te,min

0 -10 3 -20 2 -30

1

-40 -50 -50

-40

-30

-20

Tmin

-10

0

10

20

30

40

50

Tmax

Figure 8. Evaluation of Te,max and Te,min Setting the initial bridge temperature T0 to 20 °C, the characteristic values of the maximum expansion and contraction ranges, ∆TN,exp and ∆TN,con, result so

∆TN,exp = Te,max − T0 = 41.7 − 20 = 21.7 °C ,

(25)

∆TN,con = T0 − Te,min = 20 − (−1.8) = 21.8 °C .

(26)

Linearly varying vertical temperature difference components can be set to ∆TM,heat=+15 °C for top warmer than bottom and to ∆TM,cool=+8 °C for bottom warmer than top (figure 9). Deeper investigations are out of the scope of the present example.

154


15 °C

8 °C

Figure 9. Simplified temperature distributions along the height of the cross section

4

STRESS CALCULATION

4.1

Global behaviour of the structure Since transverse beams are much stiffer than longitudinal beams, transverse load distribution can be studied through the classical Courbon-Engesser theory, which assumes that transverse beams can be modelled as rigid beams resting on linear elastic supports, corresponding to the longitudinal beams. When the cross section is made of n longitudinal beams, the reaction Ri induced in the i-th longitudinal beam by a concentrated load P, applied with an eccentricity e1 with respect to the vertical centroidal axis of the deck, is given by

  d i e1 1  , + n Ri = P J i 2  n J ∑ j =1 J j d j   ∑ j =1 j

(27)

where di is the distance of the i-th longitudinal beam from the vertical centroidal axis and Ji is the moment of inertia of the i-th longitudinal beam. If the n longitudinal beams are identical, expression (27) simplifies in

  d i e1  1  Ri = P + n , 2 n d ∑ j   j =1

(28)

that clearly indicates that the most heavily stressed beams are the external ones. When, as in the present example, the longitudinal beams are also equally spaced, expression (28) can be further simplified and for the external beams it becomes

1 6 e1  R1 = P  + ,  n n (n + 1) ∆ 

(29)

said ∆ the spacing of the longitudinal beams. Setting P=1, the above mentioned formulae allow to determine the influence lines of the load pertaining to each beam. In figures 10 and 11 are illustrated the influence lines concerning loads on the external beam n. 1 and on the internal beam n. 2, respectively. From these influence lines it can be easily derived the influence lines for shear forces and bending moments acting on relevant cross sections of the transverse beam. For example, in figure 12 it is illustrated the influence line for bending moment in the cross section A-A of the transverse beam.

155


2

3

4

-0.35

0.1

-0.2

0.4

0.7

0.85

1

Figure 10. Influence line for load on the external longitudinal beam (n. 1)

1

4

3

0.05

0.1

0.2

0.3

0.45

0.4

2

Figure 11. Influence line for load on the internal longitudinal beam (n. 2) 4.2

Local effects on the slab Since the concrete slab behaves like a continuous plate supported by the longitudinal beams, main stresses in it are in the transverse direction, except that in the neighbourhood of end diaphragms, where the slab is connected to end diaphragms themselves. For the evaluation of the local effects, the wheel contact pressures can be determined resorting to the well known hypothesis of 45° diffusion of the loads through the surfacing and the slab. In this way, recalling that the surfacing thickness is 60 mm and the slab thickness is 300 mm, the equivalent contact area dimensions result 820×820 mm2 for the wheels of the tandem axle systems of LM1 (figure 13) and 1020×770 mm2 for the wheels of the isolated axle of LM2 (figures 14), respectively.

156


A

∆

∆

-0.20 ∆

A

4

0.4 ∆

0.1 ∆

∆

3

-0.35 ∆

2

-0.65 ∆

-0.30 ∆

1

Figure 12. Influence line for bending moment in section A-A of the transverse beam 600

300

300

60

60

400

° 45

° 45

820

1020

Figure 13. Load dispersal for the wheel of the tandem system of LM 1

Figure 14. Load dispersal for the wheel of the single axle of LM 2

Under these hypotheses, the relative contact pressures result pQ1k =

150 = 233.1 kN/m 2 , 0.82 ⋅ 0.82

(30)

for a single wheel of the heaviest tandem system of LM1 and pQak =

200 = 254.6 kN/m 2 , 1.02 ⋅ 0.77

(31)

for a single wheel of the isolated axle load of LM 2.

157


Calculation of the main structure under traffic actions

4.3 4.3.1

Effects of the self-weight and dead loads Recalling expressions (1) and (2), the characteristic value of the uniformly distributed self-weight is Gk = g k ,1b + g k , s = 53.13 kN/m ,

(32)

while the effect of the weight of each transverse beam on each longitudinal beam is a concentrated load of 2.93 kN, placed 15.0 m away from the support. From expression (4) the dead load of non structural parts pertaining to each longitudinal beam is 6.49 kN/m. Under permanent loads, the maximum bending moment occurs at midspan (cross section C) and it is given

Gk1,t L  1 1  M g (C )k = Gk L2 + ⋅ =  ⋅ 59.62 ⋅ 45.0 2 + 2.93 ⋅ 15.0  kNm = 15135.3 kNm (33) 8 4 3 8  , while the maximum shear force occurs at support (cross section A) and it results

V g ( A )k =

Gk1,t  1 1  Gk L + =  ⋅ 59.62 ⋅ 45.0 + 2.93  kN = 1344.4 kN . 2 4 2 

(34)

Effects of traffic loads on main beams The influence lines to be considered for transversal distribution of traffic loads have those previously discussed, shown in figures 10 and 11. For an external beam, that is the most heavily loaded, the most unfavourable load arrangements are the one represented in figure 15 for traffic loads and the one represented in figure 16 for crowd loading.

4.3.2

Q1k=300 kN

Q1k=300 kN

Q2k=200 kN

2

Q2k=200 kN

2 2

qrk =2.5 kN/m 2

q1k =9 kN/m 2

2

q fk =3.0 kN/m

q 2k=2.5 kN/m

4

3

-0.35

-0.2

0.01

0.1

0.17

0.4

2

0.48

0.7

0.76

0.85

1

Figure 15. Most unfavourable traffic load arrangement for beam n. 1 158


2

2

q fk =5.0 kN/m

q fk =5.0 kN/m

3

4

-0.35

-0.2

0.31

0.1

0.4

2

0.7

0.85

0.76

1

Figure 16. Most unfavourable LM4 (crowd loading) arrangement for beam n. 1 When traffic loads are taken into account (figure 15), to beam n. 1 pertain a concentrated load Qk = 2 ⋅ 300 ⋅ 0.48 + 2 ⋅ 200 ⋅ 0.17 = 356 kN ,

(35)

and a uniformly distributed load q k = 1.5 ⋅ 3.0 ⋅ 0.76 + 3.0 ⋅ 9.0 ⋅ 0.48 + 3.0 ⋅ 2.5 ⋅ 0.17 + 0.21 ⋅ 2.5 ⋅ 0.01 = 17.66 kN/m .

(36)

Considering the tandem system as a unique concentrated load (knife load), it is

Q L 1 1 45.0  M q (C )k max = q k L2 + k =  ⋅ 17.66 ⋅ 45.0 2 + 356 ⋅  = 8475.2 kNm , 8 4 4  8

(37)

and Vq ( A)k max =

1 1  q k L + Qk =  ⋅ 17.66 ⋅ 45.0 + 356  = 753.3 kN . 2 2 

(38)

When crowd loading (LM4) is considered instead (figure 16), to beam n. 1 pertains only a uniformly distributed load q k = 1.5 ⋅ 5.0 ⋅ 0.76 + 6.2 ⋅ 5.0 ⋅ 0.31 = 15.31 kN/m ,

(39)

whose effects are, in the present example, less severe than those caused by lorry traffic. 4.3.3

Effects of traffic loads on transverse beams Stresses in transverse beams are caused only by traffic loads. Considering the just mentioned influence line (figure 12), the load arrangements that determine maximum and minimum bending moment in cross section A-A of the transverse beam are those illustrated in figures 17 and 18. With the load arrangement of figure 17, the bending moment in section A-A results

300(0.33 + 0.23) + 9 ⋅ (0.8 ⋅ 0.305 + 2.2 ⋅ 0.29 ) ⋅  M q ( AA)k max = 2.95  = 922.2 kNm , (40) ( ) ⋅ + ⋅ + ⋅ ⋅ ⋅ + ⋅ 15 200 0 . 07 2 . 5 15 3 0 . 07 0 . 89 0 . 11  

159


while with the crowd loading arrangement of figure 18 it is

 1.5 ⋅ (0.43 + 0.26) + 0.57 ⋅    ⋅ 15 = −254.7 kNm . M q ( AA)k min = −2.955 ⋅      ⋅ 0.07 + 1.27 ⋅ 0.06 Q1k=300 kN

Q1k=300 kN

Q2k=200 kN

2 2

qrk =2.5 kN/m

2 2

A

q 2k=2.5 kN/m

3

0.02 ∆

0.1 ∆

∆

0.18 ∆

∆

0.23 ∆

0.33 ∆

0.21 ∆

0.11 ∆

0.4 ∆

A

∆

4

-0.35 ∆

2

-0.20 ∆

1

-0.30 ∆

Q2k=200 kN

2

q1k =9 kN/m

-0.65 ∆

(41)

Figure 17. Load arrangement maximizing M(AA) in the transverse beam 2

2

q fk =5.0 kN/m

q fk =5.0 kN/m

∆

-0.20 ∆

∆

0.1 ∆

0.4 ∆

-0.07 ∆

A

4

3

-0.35 ∆

2

∆

-0.30 ∆ -0.43 ∆

q fk =5.0 kN/m

-0.06 ∆

1

-0.65 ∆

2

A

-0.26 ∆

2

q fk =5.0 kN/m

Figure 18. Load arrangement minimizing M(AA) in the transverse beam 160


4.3.4

ULS combinations for permanent and traffic loads Adopting expression (6.10) of EN 1990 and recalling that in ULS structural verifications (STR) partial factor for permanent loads, γG, is set to 1.35 for unfavourable effects and to 1.0 for favourable effects, and partial factor for traffic loads, γQ, is set to 1.35 for unfavourable effects and to 0 for favourable effects, design values of stresses can be obtained. Maximum bending moment at midspan M(C)dmax and maximum shear force at end support V(A)dmax are M (C )d max = 1.35 M g (C ) k + 1.35 M q (C ) k = 31874.2 kNm .

(42)

V ( A)d max = 1.35 V g ( A) k + 1.35 Vq (Q) k = 2831.9 kN .

(43)

and

Design value of braking or acceleration forces depends on the vertical loads applied on notional lane n. 1 and it results Qld max = 1.35 (0.6 ⋅ 2 Q1k + 0.1 q1k w1 L) = 1.35 ( 360 + 2.7 ⋅ 45) = 650.0 kN .

(44)

This value must be combined with an appropriate combination value of vertical traffic load, corresponding to its frequent value. Recalling that ψ1=0.75 for the tandem systems of LM1 and ψ1=0.4 for the uniformly distributed load component of LM1, loads pertaining to beam n. 1 (see expressions (35) and (36)) become Qk = 0.75 (2 ⋅ 300 ⋅ 0.48 + 2 ⋅ 200 ⋅ 0.17 ) = 267 kN ,

(45)

q k = 0.4 (1.5 ⋅ 3 ⋅ 0.76 + 3.0 ⋅ 9.0 ⋅ 0.48 + 3 ⋅ 2.5 ⋅ 0.17 + 0.21 ⋅ 2.5 ⋅ 0.01) = 7.06 kN/m ,

(46)

and

so that maximum bending moment in C and maximum shear force in A result

 7.06 ⋅ 45 2 276 ⋅ 45  Q L 1  = 6469.0 kNm M q (C )d max = 1.35  q k L2 + k  = 1.35  + 4  8 4  8 

(47)

and

1   7.06 ⋅ 45.0  Vq ( A)d max = 1.35  q k L + Qk  = 1.35  + 267  = 575.0 kN . 2 2   

(48)

When the leading traffic loads are vertical ones, the accompanying value of the braking and acceleration forces are to be defined in National Annex and can be set to zero. 4.3.5

SLS combinations for permanent and traffic loads Concerning SLS verifications, combinations of permanent and traffic load are generally relevant only for characteristic and frequent load combinations, as quasi-permanent values of traffic loads are zero, except in very particular cases. Significant load arrangements to be considered look very similar to the ones just illustrated regarding ULS verifications, so they will not be further discussed. As just recalled, it is necessary to stress that frequent values of traffic loads are obtained via the ψ1 factors, which depend on the nature of the load: in fact ψ1=0.75 for the

161


tandem systems of LM1, for the isolated single axle (LM2) and for crowd loading (LM4), while ψ1=0.40 for the uniformly distributed load component of LM1.

4.4

Wind effects Partial factor γQ for wind actions in ULS combinations is γQ=1.50 if unfavourable and γQ=0 if favourable. Design effects of vertical wind actions The vertical component of the pressure exerted by the wind, as discussed in §3.4, is a uniformly distributed load acting on the entire length of the bridge with an eccentricity e=2.95 m. For unloaded bridge it results

4.4.1

Fwd , z = γ Q Fwk , z = ±1.5 ⋅ 5.74 kN/m = ±8.61 kN/m ,

(49)

so that the design wind actions on the external beams become

 8.61 6 ⋅ 8.61 ⋅ 2.95  + q wd = ±  = ±4.74 kN/m . 20 ⋅ 2.95   4

(50)

For loaded bridge, the combination value ψ0wqwk =±2.84 kN/m should be considered, in place of qwd. 4.4.2

Design effects of horizontal wind actions Design values of horizontal wind action should derived from expressions (17) for unloaded bridge and from expressions (20) or (21) for loaded bridge. For unloaded bridge, the design load is Fwd , x = γ Q Fwk , x = 1.5 ⋅ 3.84 kN/m = 5.76 kN/m ,

(51)

applied with an eccentricity of ez=2.13 m with respect to the bearing’s plane. For loaded bridge, instead, the design load is

ψ 0 w Fwd , x = γ Qψ 0 w Fwk , x = 1.5 ⋅ 2.95 kN/m = 4.43 kN/m ,

(52)

applied with an eccentricity of ez=2.53 m, according to EN 1991-2, or

ψ 0 w Fwd , x = γ Qψ 0 w Fwk , x = 1.5 ⋅ 3.76 kN/m = 5.64 kN/m ,

(53)

applied with an eccentricity of ez=3.03 m, according to Italian National Annex. Horizontal actions determine, as well as bending moments in the horizontal plane Mz, total horizontal support reaction at each end given by Rd , x = 0.5 ⋅ Fwk , x ⋅ L or Rd , x = 0.5 ⋅ψ 0 w Fwk , x L ,

(54)

and vertical reactions on the supports of the external beams given by

 6 ⋅ 0.5 ⋅ Fwk , x L e z Rd , z = ± 20 ⋅ 2.95  4.5

  6 ⋅ 0.5 ⋅ψ 0 w Fwk , x L e z  or Rd , z = ± 20 ⋅ 2.95  

  . 

(55)

Effects of thermal variations As said, uniform and linearly varying temperature variations induce only displacements and secondary stress states due to friction in bearings, which can be determined

162


considering that the coefficient of thermal expansion for prestressed concrete is αT=10⋅10-6 °C.

5

FINAL REMARKS

In the present chapter, effects of loads and load combinations on a simply supported concrete bridge with open cross section are discussed. The road bridge is located in an urban area, so that also crowd loading needs to be explicitly taken into account. Application of permanent, climatic and traffic actions, derived from the relevant parts of Eurocode 1, is illustrated in detail, paying special attention to traffic loads. As the transverse beams are much stiffer than the longitudinal beams, transverse load distribution has been studied resorting to the Courbon-Engesser theory. Load combinations for ultimate and serviceability limit state assessments are determined according to EN 1990 rules, highlighting specific features of local or global behaviour and their consequences as well as possible simplifications. The example confirms that Eurocodes are very appropriate for bridge design.

6

REFERENCES

[1] EN 1991-1-1 Eurocode 1 Actions on structures. Part 1-1 General actions. Densities, selfweight, imposed loads for buildings, CEN, Brussels, 2002. [2] EN 1991-1-4 Eurocode 1 Actions on structures. Part 1-4 General actions. Wind actions, CEN, Brussels, 2005. [3] EN 1991-1-5 Eurocode 1 Actions on structures. Part 1-5 General actions. Thermal actions, CEN, Brussels 2004. [4] EN 1991-2 Eurocode 1 Actions on structures. Part 2 Traffic loads on bridges, CEN, Brussels, 2003. [5] EN 1990 Eurocode - Basis of structural design. CEN, Brussels, 2002. [6] EN 1992-1-1 Eurocode 2 Design of concrete structures. Part 1-1 General rules and rules for buildings, CEN, Brussels, 2004. [7] EN 1992-2 Eurocode 2 Design of concrete structures. Part 2 Concrete bridges. Design and detailing rule, CEN, Brussels, 2005.

163


164


CHAPTER 9: CASE STUDY – DESIGN OF A STEEL BRIDGE Pietro Croce1 1

Department of Civil Engineering, Structural Division, University of Pisa, Italy

Summary In the present chapter it is discussed the design of an orthotropic steel deck bridge according to Eurocodes EN 1990 and EN 1991, in particular referring to traffic loads given in EN 1991-2. The case study refers to a three span continuous bridge with box cross section. Aim of the case study is to clarify the influence of load application and load combinations on the local and global behaviour of bridge members.

1

INTRODUCTION

In the present case study, the design of an orthotropic steel deck bridge is discussed, with special emphasis on loads and load combinations. The steel bridge considered here is a three span continuous bridge on four supports. The clear length of each span is 120 m, so that the length of the bridge is 360 m (figure 1). Loads are determined according to EN 1991-1-1 [1], EN 1991-1-4 [2], EN 1991-1-5 [3], EN 1991-2 [4], and load combinations are derived from EN 1990 [5]. Fatigue aspects are out of the scope of the present paper; therefore they are not discussed here.

2

THE THREE SPAN CONTINUOUS BRIDGE

2.1

General The bridge’s structure is made up of an orthotropic steel deck, with closed trapezoidal longitudinal stiffeners, sustained by a box girder, 3800 mm height (figure 2), whose current geometry is described below. The deck is made up by an 18 mm upper flange stiffened by 8 mm thick trapezoidal stiffeners. The trapezoidal stiffeners, which have a lower flange 200 mm wide and are 270 mm in height, are characterised by a spacing of 600 mm (300+300 mm) and are continuous through I-shaped transverse beams, 750 mm height. The span of the longitudinal stiffeners is 3000 mm, i.e. the transverse beam’s spacing. The webs of the box girder are 12 mm thick, while the lower flange is 34 mm thick. The webs and the lower flange are stiffened by L-shaped 200x100x14 longitudinal stiffeners. The carriageway is composed by two physical lanes, each one 3.75 m wide, and by two walkways, each one 1.50 m wide, so that the total width of the carriageway between the safety barriers is 10.50 m and the overall width of the bridge is 11.60 m. The walkways are at the same level of the physical lanes, from which they are separated only by the road signs. The bridge is located in an extra-urban area and the distance between the bridge intrados and the underlying ground is 20.0 m.

165


120

120

120

Figure 1. Static scheme of the bridge

Figure 2. Cross section of the bridge 2.2

Mass properties of the cross section The total area A of the cross section is 0.584 m2 and its centroid is 1.54 m from the deck extrados. The principal moments of inertia about the centroidal axis are Jx=1.665 m4 about the horizontal axis x and Jy=3.711 m4 about the vertical axis y. The strength moduli are then W’x=-1.081 m3 and W”x=0.737 m3, about the x-axis, and W’y=-0.639 m3 and W”y=0.639 m3, about the y-axis. 2.3

Material Materials are chosen according to EN 1993-1-1 [6] and EN 1993-2 [7] The structural steel is S355J2 grade.

3

LOAD ANALYSIS

3.1

Structural self-weight Since the steel density is γ =78.5 kN/m3, the nominal self-weight of the bridge is: g k ,1b = γ A = 78.5 kN/m 3 × 0.584 m 2 = 45.84 kN/m .

(1)

To take into account the weight of other structural parts (transverse beams, bracings and so on), the value of gk,1b, is increased of about 6%, so that the self-weight gk,1 results g k ,1 = 1.06 g k ,1b ≅ 48.6 kN/m .

166

(2)


Self weight of non structural elements Self weights of non structural elements to be considered are those due to the waterproofing, to the 60 mm thick asphalt surfacing and to the safety barriers. For global verifications it is a reasonable approximation to consider these loads distributed per unit surface, by “spreading out” the weight of safety barriers. All told, the equivalent uniformly distributed loads corresponding to the self-weight of non structural elements gk,1p is about 2.2 kN/m2, so that the load per unit length of the bridge gk,2 is

3.2

g k , 2 = 2.2 kN/m 3 × 11.60 m = 25.52 kN/m .

(3)

3.3

Traffic loads According to EN 1991-2 [1], traffic loads should be applied on the carriageway, longitudinally and transversally, in the most adverse position according to the shape of the influence surface, in order to maximize or minimize the considered load effect. For example, to determine the maximum sag moments in the spans and the hog moment at the supports, the relevant influence surfaces, illustrated in figures 3, 4 and 5, must be considered. According to EN 1991-2, it is necessary first to determine the total width of the carriageway w and the number of conventional lanes. The width w depends on whether the walkways are isolated from vehicular traffic by fixed safety barriers or by kerbs of sufficient height (>100 mm) or not. In the present case, walkways are potentially interested by vehicle traffic, as they are separated from the physical lanes only by road signs. For this reason, the width w is represented by the inner distance between the safety barriers, and therefore w=10.50 m. 5 0 -5 0

51.36 60

120

180

240

300

360

-10 -15 -20 -25

-24.5898

Figure 3. Influence line for max sag moment in span 1 [m] 5

180

0 -5 0

60

120

180

240

300

360

-10 -15 -20

-21

-25

Figure 4. Influence line for max sag moment in span 2 [m]

167


Influence Line for Hog Moment at Support 2

15

12.3168 10 5 69.24

0 -5

0

60

120

180

240

300

360

Figure 5. Influence line for hog moment at support 2 [m] As the influence surfaces for bending in box girders are cylindrical, i.e. they have rectangular cross section, to maximize bending moments the entire carriageway should be loaded. The number of notional lanes nl is then given by

 w  10.50  nl = Int  = Int  = 3,  3 m   3 

(4)

and the remaining area is 1.50 m wide, wr = w − nl ⋅ 3.0 m = 10.50 m − 3 ⋅ 3.0 m = 1.50 m .

(5)

notional lane n. 3

3.0

remaining area

1.5

10.5

notional lane n. 2

3.0

notional lane n. 1

3.0

Obviously, to maximize the torque coexisting with the maximum bending moment, it is necessary to maximize the load eccentricity, so obtaining the notional lane arrangement illustrated in figure 6. Clearly, when load conditions maximizing the torque are explored, load eccentricities should be maximized and different lane arrangement should be considered, like the one illustrated in figure 7, where only two notional lanes need to be loaded.

3.0

notional lane n. 2

Figure 7. Notional lane arrangement on the carriageway (torque calculation) 168

10.5

notional lane n. 1

3.0

Figure 6. Notional lane arrangement on the carriageway (bending moment calculation)


As known, EN 1991-2 calls for four separate static load models, being the single axle load model n. 2 (LM2) devoted only to local verifications. For global verifications of the bridge in question, only load model n. 1 (LM1) is relevant. In fact, load model n. 3 corresponding to special vehicles (LM3) and load model n. 4, crowd loading (LM4), are not accounted for, as the bridge is not interested by special vehicle transit and it is located in an extra-urban area. In this regard, it must be recalled that load models LM4 and LM3 need to be considered only when expressly required. On the i th notional lane, the main load model LM1 provides for a tandem system of axles weighing αQi Qik, accompanied by a uniformly distributed load αqi qik, being αQi and αqi the adjustment factors. In the present work it has been assumed αQi=αqi =1.0 for each lane, while the values Qik and qik are summarized in table 1. Only one tandem system should be considered per lane, placed in the most unfavourable position. Table 1. Characteristic values for load model n. 1 (LM1) Notional lane Qk [kN] qk [kN/m2] Lane 1 300 9.0 Lane 2 200 2.5 Lane 3 100 2.5 Remaining area 0 2.5 As said, when seeking a determined effect on the bridge, the LM1 must obviously be arranged in the most unfavourable position and the tandem systems, when present, need to be considered in full, that is, with all their four wheels. By way of example, possible arrangements of the static traffic loads are represented in figures 8 and 9, corresponding to notional lane numberings discussed below and illustrated in figures 6 and 7, respectively. Q1k=300 kN

Q1k=300 kN 2000

Q2k=200 kN

Q2k=200 kN Q3k=100 kN 2000

Q3k=100 kN 2000

2

q1k =9 kN/m

2

q2k=2.5 kN/m

2

q3k =2.5 kN/m

2

qrk=2.5 kN/m

Figure 8. Load condition corresponding to notional lane numbering in figure 6

169


Q1k=300 kN

Q1k=300 kN 2000

Q2k=200 kN

Q2k=200 kN 2000

2

q1k =9 kN/m

2

q2k=2.5 kN/m

Figure 9. Load condition corresponding to notional lane numbering in figure 7 3.4

Wind actions Wind actions can be represented by vertical and horizontal equivalent static forces. Vertical force is orthogonal to the roadway plane, while the horizontal forces can be represented by two components, parallel and orthogonal to the bridge’s longitudinal axis, respectively. The equivalent pressure exerted by the wind can be calculated through the expression q p (z e ) = c e ( z e )

ρ 2

(6)

vb2 ,

in which ρ is the air density, which is assumed to be constant and equal to 1.25 kg/m3, vb is the basic wind velocity and ce(ze) is the so-called exposure coefficient, depending on the reference altitude over the ground, ze, and given by ce ( ze ) = cr2 (ze ) ⋅ c02 ( ze ) ⋅ [1 + 7 ⋅ I v (ze )] .

(7)

Expression (7) depends on the roughness coefficient cr, on the orography factor c0 and on the turbulence intensity Iv. The orography factor, taking into account any significant local variations in the site’s orography, can usually be assumed equal to 1.0. Iv and cr, instead, are defined by the following expressions

ki    z   I v ( z e ) =  c0 ( z e ) ⋅ ln   z0   I v ( z min )

  z  k r (z e ) ⋅ ln  cr ( z e ) =   z0   cr ( z min )

170

if z min < z ≤ 200 m (8)

,

if z min ≥ z

if z min < z ≤ 200 m if z min ≥ z

,

(9)


where ki is the turbulence factor, usually set to 1.0. The terrain factor kr, the roughness length z0 and the minimum height zmin depend on the terrain category. As said, the bridge in question is located in an extra-urban area which can be classified in terrain category II, i.e. an area with low vegetation, such as grass, and isolated obstacles (trees, buildings) with separations of at least 20 obstacle heights. For terrain category II it results z0=0.050 m, zmin=2.0 m, kr=0.19. The reference height ze represents the distance between the lowest ground level to the centre of the bridge deck structure, disregarding other parts (e.g. parapets) of the reference areas Recalling that the intrados of the structure is 20.0 m above ground level, it is ze=20.0+(3.80/2) m= 21.90 (figure 10).

Figure 10. Evaluation of the reference height ze for wind actions As ze>zmin, it results I v ( ze ) =

1 = 0.164 ,  21.90 m  1.0 ⋅ ln   0.05 m 

(10)

 21.90 m  cr ( z e ) = 0.19 ⋅ ln  = 1.156 ,  0.05 m 

(11)

ce ( z e ) = 1.156 2 ⋅1.0 2 ⋅ [1 + 7 ⋅ 0.164] = 2.869 .

(12)

whence

Considering for the site a basic wind velocity vb=27 m/s, qp(ze) results q p ( z e ) = 2.869 ⋅

1.25 ⋅ 27.0 2 = 1307 N/m 2 . 2

(13)

171


According to EN 1991-2, the y-axis is assumed parallel to the bridge axis, x-axis is assumed horizontal and perpendicular to the y-axis, while the z-axis lies in the vertical plane containing the y-axis. The equivalent static force Fwk,x in the x direction is given by Fwk , x = q p ( z e ) ⋅ c f , x ⋅ Aref , x ,

(14)

where the coefficient cf,x is a function of the ratio between the total deck’s width b and the total deck’s height dtot exposed to wind. When the bridge is unloaded the exposed height is 4.4 m, as the presence of two open safety barriers determines an increase of 0.6 m in the exposed height. When the bridge is loaded the exposed height increases by 2.0 m, so becoming 5.8 m. For unloaded bridge, the coefficient cf,x is

   b 11.6 c f , x = min 2.4; max 2.5 − 0.3 ; 1.3   = 2.5 − 0.3 = 1.709 . d tot 4.4   

(15)

If must be noted that the simplified approach proposed in EN1991-1-4, suggesting to assume cf,x=1.3, is generally unsafe-sided. Since the webs of the box girder are inclined by the angle α=10° with respect to the vertical (figure 11), the coefficient cf,x calculated in (15) can be reduced by the factor η1

η1 = max(1 − 0.005 α1 ; 0.7 ) = 0.95 ,

(16)

Fwk , x = q p ( z e )c f , xη1 Aref , x = 1.307 ⋅ 1.709 ⋅ 0.95 ⋅ 4.4 = 9.34 kN/m .

(17)

obtaining (figure 12)

Figure 11. Web inclination allowing a reduction of the coefficient cf,x For loaded bridge, the coefficient cf,x is    b 11.6 c f , x = min 2.4; max 2.5 − 0.3 ; 1.0   = 2.5 − 0.3 = 1.90 , d tot 5.8   

(18)

and also for loaded bridge the simplification cf,x=1.3 proposed in EN1991-1-4 is unsafe-sided. Considering the reduction factor η1 calculated before, the combination value ψ0wFwk,x for the equivalent wind force for loaded bridge results (figure 18)

ψ 0 w Fwk , x = ψ 0 w q p (z e )c f , xη1 Aref , x = 0.6 ⋅1.307 ⋅1.9 ⋅ 0.95 ⋅ 5.8 = 8.21 kN/m ,

172

(19)


being ψ0w=0.6. It is interesting to note that in the Italian National Annex the exposed height of lorries it has been set equal to 3.0 m, instead of 2.0 m. In this case, as dtot=6.8 m, cf,x=1.988 and

ψ 0 w Fwk , x = ψ 0 w q p ( ze )c f , xη1 Aref , x = 0.6 ⋅ 1.307 ⋅ 1.988 ⋅ 0.95 ⋅ 6.8 = 10.07 kN/m .

(20)

Figure 12. Equivalent static force Fwk,x (unloaded bridge)

Figure 13. Equivalent static force ψ0wFwk,x (loaded bridge) Concerning the vertical action, lacking more precise data from wind tunnel tests, a value of ±0.9 can be assumed for the force coefficient cf,z. Since the reference area Aref,z is the horizontal projection of the bridge deck, Aref,z=b=11.6 m2/m, the equivalent static force for unit length Fwk,z is then Fwk , z = q p ( z e ) ⋅ c f , z ⋅ Aref , z = 1.307 kN/m 2 ⋅ (± 0.9) ⋅11.6 m = ±13.64 kN/m .

(21)

to be applied with eccentricity e=0.25 b=0.25⋅11.6 m=2.9 m with respect to the longitudinal axis of the bridge. According to EN1991-2, as Fwk,z is much lower than the permanent load (65.52 kN/m), it could be disregarded. Finally, when relevant, equivalent static forces in the longitudinal y-direction (the bridge’s longitudinal axis) should be considered, which can be set equal to 25% of the forces in the x-direction. 173


3.5

Thermal actions As the structure under consideration is a continuous beam resting on four supports, thermal actions induce displacements and stresses. In order to account for thermal variations, two different contributions must be distinguished. A uniform thermal variation along the cross section and a non-uniform temperature variation along the section’s height, corresponding to situations where the top and the bottom of the bridge are at different temperatures, due to differential heating or cooling effects. The former contribution does not provoke any stresses as long as the bridge can slide horizontally in correspondence to its supports and it will cause only a shortening or elongation of the structure’s line of axis (i.e. it is relevant only for design of bearings and expansion joints). A typical uniform temperature variation ∆TU can be derived from EN 1991-1-5. Assuming that in the site under consideration the maximum and minimum air shade temperatures with an annual probability of being exceeded of 0.02 are Tmax=40 °C and Tmin=10 °C, respectively, the uniform bridge temperature components for a steel bridge result Te,max=56.6 °C and Te,min=-13.3 °C (figure 14). Setting the initial bridge temperature T0 to 20 °C, the characteristic values of the maximum expansion and contraction ranges, ∆TN,exp and ∆TN,con, result so

∆TN,exp = Te,max − T0 = 56.6 − 20 = 36.6 °C ,

(22)

∆TN,con = T0 − Te,min = 20 − (−13.3) = 33.3 °C .

(23)

70 60 1

50

2

40

Te,max

3 30 20 10

Te,min

0 -10 3

-20

2 -30

1

-40 -50 -50

-40

-30

-20

Tmin

-10

0

10

20

30

Tmax

Figure 14. Evaluation of Te,max and Te,min 174

40

50


The second non uniform contribution is clearly very significant for the bridge under consideration. EN 1991-1-5 offers two possible procedures to deal with it, provided that the surfacing thickness is not less than 40 mm. The first, more accurate one, calls for applying rather complex thermal variation laws along the cross section’s height (figure 15), while the second instead makes use of simpler linear variations. Consequently, while the first variation laws require employing dedicated software for the structural analysis, simplified linear variations enable even manual calculations, at least up to a certain degree. In case of steel deck structures, simplified linear variations correspond to a raise in temperature of 18 °C for top warmer than bottom, ∆TM,heat=+18 °C, and to an increase of 13 °C for bottom warmer than top, ∆TM,cool=+13 °C, (figure 16).

Figure 15. Accurate temperature distributions along the height of the cross section

Figure 16. Simplified temperature distributions along the height of the cross section

4

STRESS CALCULATION

4.1

Behaviour of the structure The structural behaviour of the orthotropic steel deck bridges is usually described taken into account three separate static functions for the stiffened plate. In effect, the stiffened plate participates to three different resisting systems, each one characterised by precise stress and deformation patterns. In the local resisting system the deck plate directly sustains the traffic loads and transfers them to the orthotropic plate system, composed by the deck plate, the longitudinal stiffeners and the transverse beams. In the intermediate resisting system the orthotropic system transmits the loads to the main system.

175


Finally, in the global resisting system, the stiffened plate represents the upper flange of the main girders. The resultant stress pattern in the deck plate can be so obtained by the superposition of the individual stress patterns induced by each one of the three above mentioned static systems.

4.2

Local effects on the deck plate The stress pattern of the local resisting system is often not considered in the static checks, as the combined effect of local plastic deformations and membrane-type behaviour limits the stress peaks. In fact, when it collapses, the deck plate behaves mainly like a tensile cylindrical membrane restrained by two longitudinal hinges located at the connections with the stiffener’s webs: in this mechanism normal stresses in longitudinal direction are not relevant. However, if an evaluation of such a stresses is required, the wheel contact pressure can be determined resorting to the well known hypothesis of 45° diffusion of the loads through the surfacing and the deck plate. In this way, recalling that the surfacing thickness is 60 mm and the deck plate thickness is 16 mm, the contact area dimensions result 536×536 mm2 for the wheels of the tandem axle systems of LM1 (figure 17) and 736×486 mm2 for the wheels of the isolated axle of LM2 (figures 18), respectively.

Figure 17. Load dispersal for the wheel of the tandem system of LM 1

Figure 18. Load dispersal for the wheel of the single axle of LM 2

Thus, the relative contact pressure for a single wheel of the heaviest tandem system of LM1 is pQ1k =

150 = 522.1 kN/m 2 0.536 ⋅ 0.536

(24)

and for a single wheel of the isolated axle load of LM 2 is pQak =

200 = 523.6 kN/m 2 . 0.736 ⋅ 0.486

(25)

Local stresses are very important when fatigue assessments are concerned. As known, fatigue assessments can be determinant in designing deck plates, ribs and transverse beams details, but they are outside the scope of the present example.

176


Orthotropic deck plate analysis The analysis of the orthotropic deck plate, whose function is to transfer the traffic loads to the main girder, can be performed through analytical or numerical methods. The analytical methods are based on the Hüber equation for the orthotropic plate. Although all the calculation procedures available in the literature are based on defining an equivalent orthotropic deck, they can be divided into two classical families, depending upon whether both longitudinal ribs and transverse beams are spread out in calculating the equivalent deck’s stiffness, or only longitudinal ribs are considered. The former methods include, for instance, the Cornelius method, while the Pelikan-Esslinger method represents a refined example of the second approach, largely applied, especially in studying influence surfaces. A detailed illustration of the analytical methods is beyond the scope of the present example, but it is important to stress that in both cases local plate behaviour cannot be determined, while the effective state of stress and deformation in the longitudinal stiffeners can be derived only adopting the latter ones. A very refined approach, able also to reproduce the local behaviour, consists in the implementation of a finite-element model, made up essentially of isotropic shell elements, which reproduces a significant portion of the deck. In the FE model the upper plate, the stiffeners and the transverse beams are appropriately meshed as shown in figures 19 and 20. 4.3

Figure 19. FE mesh of an orthotropic deck

Figure 20. FE mesh detail

In order to reduce the degrees of freedom of the problem, alternative simplified FE models can be implemented, using equivalent orthotropic shell elements, based on one of the two analytical approaches illustrated before. Since the wheel loads influence only few longitudinal stiffeners (three to five), it is necessary to model only a limited deck width, not bigger than the width of a notional lane, regardless of the calculation method adopted (FEM or numerical analysis), so limiting the problem’s complexity. The uniformly distributed loads to be considered can be determined, considering their dispersal through the surfacing and the deck plate, as illustrated in §4.2.

4.4

Calculation of the main girder under vertical loads The main girder is calculated simply as a continuous beam resting on four supports, characterised by constant flexural stiffness EJx (Jx=1.151 m4). 4.4.1

Effects of the self-weight and dead loads Applying the single source principle of EN 1990 to the structural self-weight and to dead loads, their effects can be calculated considering a uniformly distributed load 177


Gk = g k ,1b + g k , 2 = 74.12 kN/m .

(26)

Said A the cross section of the first span where bending moment attains its local maximum, and denoted with B and C the intermediate supports, Gk yields the symmetrical diagram of bending moments shown in figure 21, where M g ( A)k = M g (B )k = −

2 2 Gk L2 = 74.12 ⋅120.0 2 kNm = 85381.8 kNm , 25 25

(27)

1 1 Gk L2 = − 74.12 ⋅120.0 2 kNm = −106727.4 kNm , 10 10

(28)

M g (C )k =

Sec. B

1 1 Gk L2 = 74.12 ⋅120.0 2 kNm = 26681.8 kNm . 40 40

(29)

Sec. C

Sec. A

Figure 21. Bending moment diagrams for permanent loads. 4.4.2 Effects of traffic loads As said, since the bending moment influence surfaces of the box girder are cylindrical, the most severe transversal load arrangement for bending moment is the one previously illustrated in figure 8, where all three notional lanes as well as the residual area of the carriageway are loaded. For this reason, the total uniformly distributed load of LM1 to be applied on the bridge is given by

qk = (9 + 2.5 + 2.5) kN/m 2 × 3.0 m + 2.5 kN/m 2 ×1.5 m = 45.75 kN/m ,

(30)

while the global effects of the tandem systems of LM 1 can be determined considering in the worst longitudinal position a single concentrated load (knife load) Qtk = 2Q1k + 2Q2 k + 2Q3k = (600 + 400 + 200) kN = 1200 kN ;

(31)

this is the sum of the axle loads of the three tandem systems applied on the three notional lanes. Referring to the bending moment in the side spans, in the central span and at the intermediate supports, the most unfavourable load arrangements should be determined according to the influence lines illustrated in figures 3, 4 and 5, respectively. For example, to maximize the bending moment in the first span due to traffic loads MQ(A’)kmax, the uniformly distributed load should be applied on the first and on the third span as indicated in figure 22, while the concentrated load should be applied on the section A’, located around 51.4 m from the first support, so that

M Q ( A')k max = M q ( A')k max + M Qt ( A')k max = 66548.9 + 29507.7 = 96056.6 kNm .

178

(32)


Figure 22. Arrangement of UDL to maximize bending moment in lateral spans To maximize the bending moment at midspan (section C) due to traffic loads, MQ(C)kmax, the central span should be loaded with uniformly distributed traffic load (figure 23), being the concentrated load located at midspan

M Q (C )k max = M q (C )k max + M Qt (C )k max = 49410.0 + 25200.0 = 74610.0 kNm .

(33)

Figure 23. Arrangement of UDL to maximize bending moment in the central span To minimize the bending moment MQ(B)kmin due to traffic loads on cross section B, corresponding to the second support, the first and the second span should be loaded with uniformly distributed load (figure 24), while the concentrated load should be placed 69.24 m away from the first support

M Q (B )k min = M q (B )k min + M Qt (B )k min = −78860.0 − 14780.2 = −91640.2 kNm .

(34)

Figure 24. Arrangement of UDL to minimize bending moment on the second support When minimum traffic load effects are investigated, different load conditions are to be considered, as indicated in the following. The minimum bending moment in the section A’ of the first span, MQ(A’)kmin, is obtained applying the uniformly distributed load on the central span, as just indicated in figure 22, and the concentrated load around 49.9 m away from the second support, so that

M Q ( A')k min = M q ( A')k min + M Qt ( A')k min = −14109.3 + −4915.3 = −19024.6 kNm . (35) The minimum bending moment at midspan (section C), MQ(C)kmin, is obtained applying the uniformly distributed load on the lateral spans, as indicated in figure 22, and the concentrated load around 70.7 m away from the first or the last support, so that

179


M Q (C )k min = M q (C )k min + M Qt (C )k min = −32940.0 + −5539.1 = −38479.1 kNm .

(36)

The maximum bending moment at intermediate support (section B), MQ(B)kmax, is obtained applying the uniformly distributed load on the third span, and the concentrated load around 50.5 m away from the third support, so that

M Q (B )k max = M q (B )k max + M Qt (B )k max = 10980.0 + 3695.0 = 14675.0 kNm .

(37)

4.4.3 ULS combinations for permanent and traffic loads Applying the single source principle for permanent loads and recalling that in ULS structural verifications (STR) partial factor for permanent loads, γG, is set to 1.35 for unfavourable effects and to 1.0 for favourable effects, and partial factor for traffic loads, γQ, is set to 1.35 for unfavourable effects and to 0 for favourable effects, design values of bending moments are obtained as follows, considering expression (6.10) of EN 1990. The maximum bending moment in the first span M(A’’)dmax is obtained applying the permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the lateral spans and the design concentrated traffic load Qd=1.35 Qk =1620 kN on the section A’’, around 51.0 m away from the first support, so that M ( A' ')d max = 244421.0 kNm .

(38)

The corresponding bending moment and shear force diagrams are illustrated in figures 25 and 26, respectively.

Figure 25. Bending moment diagram corresponding to M(A’’)dmax in the first span Even if, in general, cross section A’’ differs from A’, in the present example the difference is so little that it could be disregarded. In fact, considering A’, it would result M ( A')d max = 244364.0 kNm .

Figure 26. Shear force diagram corresponding to Mdmax in the first span

180

(39)


The maximum bending moment at midspan M(C)dmax is obtained applying the permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN on section C, so that M (C )d max = 136742.0 kNm .

(40)

The corresponding bending moment and shear forces diagrams are illustrated in figures 27 and 28, respectively.

Figure 27. Bending moment diagram corresponding to Mdmax at midspan

Figure 28. Shear force diagram corresponding to Mdmax ad midspan The minimum bending moment on the second support M(B)dmax is obtained applying the permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the first and the second span and the design concentrated traffic load Qd=1.35 Qk =1620 kN approximately around 69.24 m away from the first support, obtaining M (B )d max = −267796.0 kNm .

(41)


Figure 29. Bending moment diagram corresponding to Mdmin on the second support 181


Figure 30. Shear force diagram corresponding to Mdmin on the second support The minimum bending moment in the first span M(A’)dmin is obtained applying the permanent design load Gd=1.0 Gk =74.12 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN on the second span, approximately around 49.9 m away from the second support, so that M ( A')d min = −59275.4 kNm .

(42)

The corresponding bending moment and shear force diagrams are illustrated in figures 31 and 32, respectively. The minimum bending moment at midspan M (C)dmin is obtained applying the permanent design load Gd=1.0 Gk =74.12 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the lateral spans and the design concentrated traffic load Qd=1.35 Qk =1620 kN approximately 70.7 m away from the first or the last support, so M (C )d min = −25261.7 kNm .

Figure 31. Bending moment diagram corresponding to Mdmin in section A’

Figure 32. Shear force diagram corresponding to Mdmin in section A’

182

(43)



Figure 33. Bending moment diagram corresponding to Mdmin in section C

Figure 34. Shear force diagram corresponding to Mdmin in section C The maximum bending moment on the second support M(B)dmax is obtained applying the permanent design load Gd=1.0 Gk =74.12 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the third span and the design concentrated traffic load Qd=1.35 Qk =1620 kN approximately around 50.5 m away from the third support, obtaining M (B )d max = −86922.2 kNm .

(44)


Figure 35. Bending moment diagram corresponding to Mdmax in section B Particularly relevant cases concern minimum design shear force at the right hand end of first span (section B-), maximum design shear force at the left hand end of the central span (section B+) and maximum and minimum shear forces at midspan (sections C- and C+). Minimum design shear force in section B- V(B-)dmin is obtained applying the permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed design traffic

183


load qd=1.35 qk =61.76 kN/m on the first and the third span and the design concentrated traffic load Qd=1.35 Qk =1620 kN immediately left hand of section B-, obtaining the shear force diagram illustrated in figure 37, where

( )

V B−

d min

= −12890.0 kN .

(45)

Figure 36. Shear force diagram corresponding to Mdmax in section B

Figure 37. Shear force diagram corresponding to Vdmin in section BMaximum design shear force in section B+ V(B+)dmax is obtained applying the permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN immediately right hand of section B+, obtaining the shear force diagram illustrated in figure 38, where

( )

V B+

d max

= 11329.2 kN .

(46)

Figure 38. Shear force diagram corresponding to Vdmax in section B+ Minimum design shear force in section C-, V(C-)dmin is obtained applying the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the left hand half of the central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN immediately left hand of section C-, obtaining the shear force diagram illustrated in figure 39, where

184


( )

V C−

d min

= −2276.0 kN .

(47)

5 4

V [MN]

3 2 1 0 -1 -2 -3 -4 0

120

x [m]

240

360

Figure 39. Shear force diagram corresponding to Vdmin in section CFinally, maximum design shear force in section C-, V(C-)dmax is obtained applying the uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the right hand half of the central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN immediately right hand of section C-, obtaining the shear force diagram illustrated in figure 40, where

( )

V C−

d max

= 2276.0 kN .

(48)

Figure 40. Shear force diagram corresponding to Vdmax in section C4.4.4 SLS combinations for permanent and traffic loads Concerning SLS verifications, combinations of permanent and traffic load are generally relevant only for characteristic and frequent load combinations, as quasi-permanent values of traffic loads are zero, except in very particular cases. Significant load arrangements to be considered look very similar to those illustrated before regarding ULS verifications, so they will not be discussed in detail. It is just necessary to recall that frequent values of traffic loads are obtained via the ψ1 factors, which depend on the nature of the load: in fact, ψ1=0.75 for the tandem systems of LM1, for the isolated single axle of LM2 and for crowd loading (LM4), while ψ1=0.40 for the uniformly distributed loads of LM1. 4.4.5

Wind effects When relevant, the vertical component of the pressure exerted by the wind should be considered as uniformly distributed load acting on the entire length of the bridge. The bending moment diagram is then analogous to that due to dead loads and therefore, recalling expression (21), for unloaded bridge it results

185


M wz ( A)k =

2 2 Fwk , z L2 = ± 13.64 ⋅120.0 2 kNm = ±15713.3 kNm , 25 25

(49)

M wz (B )k = −

1 1 Fwk , z L2 = m 13.64 ⋅120.0 2 kNm = m19641.6 kNm , 10 10

(50)

M wz (C )k =

1 1 Fwk , z L2 = ± 13.64 ⋅120.0 2 kNm = ±4910.4 kNm , 40 40

(51)

while for loaded bridge, the combination value ψ0wFwk,z=±8.18 kN/m, ψ0w=0.6, should be considered, in place of Fwk,z. Partial factor γQ for wind actions in ULS combinations is γQ=1.50 if unfavourable and γQ=0 if favourable.

4.5

Design values of horizontal wind actions The design values of horizontal wind action are derived from expressions given in §3.4, recalling that γQ=1.50 if the effect is unfavourable and γQ=0 if the effect is favourable. 4.6

Effects of thermal variations As said, in a continuous bridge the uniform temperature variation ∆TU induces only longitudinal displacements. Considering that the coefficient of thermal expansion for steel is αT=12⋅10-6 °C-1, the longitudinal displacements uy associated with the uniform temperature variations ∆TN,exp=36.6 °C and ∆TN,exp=33.3 °C result u y ,exp = 36.6 ⋅ 12 ⋅ 10 −6 ⋅ 1000 = 0.439 mm/m ,

(52)

u y ,con = 33.3 ⋅ 12 ⋅ 10 −6 ⋅ 1000 = 0.40 mm/m .

(53)

The vertical temperature differences, assumed to be linear through the cross section’s height, produce a bending moment diagram that is linear in the two side spans and constant in the central one. In figure 41 is reported the bending moment diagram for heating (top warmer then bottom).

120

120

120

Figure 41. Bending moment diagram for vertical temperature differences (heating) Recalling that, for the continuous beams considered in the present example, the extreme values of bending moments are given by

α ∆T 6 M ∆T = − EJ T , 5 h it results

186

(54)


6 12 ⋅10 −6 ⋅18 = −23849.8 kNm , M ∆T = − ⋅ 2.1⋅108 ⋅1.665 ⋅ 5 3.8

(55)

in case of heating (top warmer then bottom), and M ∆T

6 12 ⋅ 10 −6 ⋅ (− 13) 8 = − ⋅ 2.1 ⋅ 10 ⋅ 1.665 ⋅ = 17224.9 kNm . 5 3.8

(56)

in case of cooling (bottom warmer then top). When, depending on the local site conditions, the adoption of a more precise law of variation for vertical temperature differences is required, as described in §3.5, more refined analyses should be performed.

4.7

Assessment of the members of the cross section To avoid use of complicated models, the assessment of members of the cross section of the box girder can be performed in a very simplified way considering the classical superposition of two structural systems. In the first structural system, where member loads are considered, the cross section is suitably modified so that it results kinematically determinate or restrained. In this restrained structure the member-end forces due to member applied loads are calculated and the equivalent fixed-end structure forces are determined. In the second structural system, the static equilibrium in the kinematically released structure is restored by applying the negative of the equivalent fixed-end structure forces on the actual structure. The application of this method to the relevant cases discussed in §3.3 and illustrated in figures 8 and 9 is summarized in figures 42 and 43, respectively.

M1

R1

M2

R2

M2

M1 R1

R2

Figure 42. Structural systems for assessment of cross section members (ref. figure 8)

187


M1

R1

M2

R2

M2

M1 R1

R2

Figure 43. Structural systems for assessment of cross section members (ref. figure 9)

5

FINAL REMARKS

In the present chapter, a relevant example is developed concerning steel bridges. The design of a three spans continuous bridge with orthotropic steel deck and box cross section is illustrated, paying special attention to the evaluation of extreme effects induced by permanent, climatic and traffic actions according to EN 1991. The road bridge, whose total length is 360 m, is located in an extra-urban area, so that also crowd loading can be disregarded. Starting from the usual distinction between local and global static behaviour, different methodological approaches for orthotropic deck bridges analysis are illustrated, stressing their theoretical bases. Discussion of load combinations according to EN 1990 rules demonstrates also in this case the soundness of Eurocode system for bridge design.

6

REFERENCES

[1] EN 1991-1-1 Eurocode 1 Actions on structures. Part 1-1 General actions. Densities, selfweight, imposed loads for buildings, CEN, Brussels, 2002. [2] EN 1991-1-4 Eurocode 1 Actions on structures. Part 1-4 General actions. Wind actions, CEN, Brussels, 2005. [3] EN 1991-1-5 Eurocode 1 Actions on structures. Part 1-5 General actions. Thermal actions, CEN, Brussels 2004. [4] EN 1991-2 Eurocode 1 Actions on structures. Part 2 Traffic loads on bridges, CEN, Brussels, 2003.

188


[5] EN 1990 Eurocode - Basis of structural design. CEN, Brussels, 2002. [6] EN 1993-1-1 Eurocode 3 Design of steel structures. Part 1-1 General rules and rules for buildings, CEN, Brussels, 2005. [7] EN 1993-2 Eurocode 3 Design of steel structures. Part 2 Steel bridges, CEN, Brussels, 2006.

189


190


CHAPTER 10: CASE STUDY - DESIGN OF A COMPOSITE BRIDGE Peter Tanner1, Carlos Lara1 and Angel Arteaga1 1

Institute of Construction Science, IETcc – CSIC, Madrid, Spain

Summary The identification of all the actions and action effects likely to arise during construction and future use is a crucial step in bridge design. Actions and effects that go unrecognised in this stage and are consequently ignored in further analyses may result in structural designs with an unacceptably low level of reliability. With proper detection, on the contrary, suitable safety measures can be readily adopted to ensure the required levels of reliability are reached. This chapter deals with the application of the structural Eurocodes, particularly EN-1991, “Actions on Structures”, to the analysis of composite decks on road bridges. The effects of actions and combinations of actions relevant to the verification of the ultimate and serviceability limit states in bridge decks are also studied.

1

INTRODUCTION

The primary purpose of the present chapter is to illustrate how Eurocode EN 1991, “Actions on Structures” and other structural Eurocodes can be applied to the analysis of composite steel and concrete decks on road bridges. The specific aims sought are: 1.

2.

to define the site-specific actions and environmental effects included in the EN 1991 models, as well as the combinations of actions that must be taken into account when verifying conformity to structural safety and serviceability requirements. to identify the action effects to be used to verify bridge deck conformity to structural safety and serviceability requirements for each relevant combination of actions.

EN 1990/A1 [2] provides basis for determination of combinations of actions for ultimate and serviceability limit state verifications of bridges. The aim of this Chapter is to describe principles of load combinations. Permanent actions, traffic loads and climatic actions due to wind, snow and temperature are considered in accordance with relevant Parts of EN 1991. Supplementary information on the traffic load models provided in EN 1991-2 [3] is given in the Background document [4] which is expected to be available on the JRC web site. 1.1

Scope Although bridge deck design normally covers serviceability, structural safety, fatigue resistance and durability of all structural members, i.e., the bridge deck, piers, abutments and foundations, the present chapter deals with the structural safety and serviceability of the bridge deck only. The verification of ultimate, serviceability and fatigue limit states is not explicitly covered. The example used illustrates only the questions relating to overall structural analysis for determining the relevant action effects.

191


In the example introduced in this chapter, adapted from [1], the bridge deck consists of two steel girders and a concrete slab. This example was chosen because as a result of the ample coverage of this type of deck in Eurocode 4, Part 2 on the design of composite steel and concrete bridges [2], Eurocode provisions can be applied rather straightforwardly. For the intents and purposes of this example, certain simplifying assumptions have been made, particularly respecting construction stages (§2.4) and the design situations considered in the verification of ultimate limit states (§3.1). Nonetheless, the deck dimensions, materials used and assumptions hereunder are realistic and the deck in the example meets all the applicable structural safety, serviceability, fatigue and durability requirements laid down in the relevant structural Eurocodes. 1.2

Chapter organisation The introduction, the first of this chapter, describes the aims and scope of the example. It is followed by a review of the bridge deck, including geometric characteristics, material properties and the construction sequence. Section 3 identifies all the actions and their effects likely to arise during bridge deck construction and future use, along with the specific load models for the actions that should be considered in analysis and design calculations. A number of the features of the structural model used for the analysis of the composite bridge deck are described in Section 4. The values to be used for internal forces and moments and the displacements found with bridge deck analysis are summarised in section 5. Section 6 reviews the partial factors used to calculate the design values of the effects of actions and resistance thereto, while Section 7 discusses the combinations of actions applied to verify structural conformity to safety and serviceability standards. The chapter concludes with an eighth section containing general remarks on composite bridge deck analysis as stipulated in structural Eurocodes. 2

BRIDGE DECK DESCRIPTION

2.1

Geometry The solution adopted is a continuous composite bridge deck carrying three lanes of road traffic, with a constant cross-sectional height of 2,15 m and a total width of 10 m. The deck cross-section comprises an in situ concrete slab 0,25 m deep and two 1,9-m deep welded steel girders, set at a distance of 5,0 m (Figure 1). The bridge has a total length of 103,5 m and three spans: 30,0 +43,5+30,0 m (Figure 2). 10,0 m 0,5 m

1,90 m

0,25 m

0,5 m

2,5 m

5,0 m

2,5 m

Figure 1. Composite deck cross-section for a road bridge

192


30,0 m

43,5 m

30,0 m

103,5 m

Figure 2. Static system 2.2

Material properties Structural steel: As per EN 1993-1-1 [3], §3.2:

Steel grade

Nominal thickness t [mm]

Yield strength fy [N/mm2]

S355

t ≤ 40 mm

355

S460

40 < t ≤ 80 mm

430

Modulus of elasticity Ea [kN/mm2] 210

Concrete: As per EN 1992-1-1 [4], §3.1: Strength class Characteristic cylinder strength Secant modulus of elasticity

C30 fck = 30 N/mm2 Ecm = 33 kN/mm2

Reinforcing steel 1: As per EN 1992-1-1 [4], §3.2 and Annex C: Steel grade B 500 Specified yield strength fsk = 500 N/mm2 Modulus of elasticity Es = 200 kN/mm2 In composite structures, the design value of the modulus of elasticity may be taken to be equal to the value for structural steel: Es = 210 kN/mm2 ([2], §3.2.2). Stud connectors 2: Nominal ultimate strength Diameter Height

fu = 450 N/mm2 φ = 19 mm h = 125 mm.

1

While in EN 1992-1-1 [4] the yield strength of reinforcing steel is symbolized as fyk,, in EN 1994-1-1 [5] it is shown fsk to distinguish it from structural steel.

2

In the context of the material recommended for stud connectors, reference is made in [2], §3.4.2.1, to EN13918.

193


2.3

Steel plate thickness, stiffeners and diaphragms According to the distribution of stiffeners and diaphragms and the steel plate thickness set out in Figure 3, referred to the bridge deck surface, a total of 100 kg/m2 of structural steel is needed. SL ST

ST

DI

ST

ST

DI

ST

DI

SL

DB ST

DI

ST

DI

ST

3,125

6,25

3,125

6,25

3,125 6,25

DI

21,75 m

30,00 m 2,08 2,08 2,08 3,125

ST

DI

3,00

3,00

2,50

6,25

2,50

5,00

2,50

2,50

5,00

3,35

3,35 6,70

3,35

3,35

6,70

3,35 3,35

450x25 TOP FLANGE tw = 12 mm 600x25

tw = 15 mm

600x40

WEB

600x60 (S460M) BOTTOM FLANGE

tw = 12 mm 600x40

ST: TRANSVERSE STIFFENER; DI: INTERMEDIATE DIAPHRAGM; DB: BEARING DIAPHRAGM; SL: LONGITUDINAL STIFFENER

Figure 3. Distribution of stiffeners and diaphragms 2.4

Construction For the purpose of structural analysis, bridge construction is assumed to be divided into the following stages: - Erection of the steel structure. - Casting of the in situ concrete in a single lift across the entire length of the bridge without temporary supports. - Simultaneous application of all dead loads, in particular the vehicle restraint system and the asphalt layer, two weeks after the in situ concrete is poured.

3

ACTIONS

3.1

Introduction Structural reliability is closely related to the recognition of the actions and effects to which the structure may be exposed during construction and use. The goal is to identify all actions and effects likely to arise. Only then can a solution be found that meets the basic requirements laid down in EN 1990 [6], §2.1. In light of the importance of this step, the actions and action effects that might be relevant to the bridge deck in the example are described below. Only persistent and transient design situations are considered for ultimate limit state verification. In other words, the analysis does not address the effects of either accidental or seismic actions. Loads affecting only one half of the bridge deck are considered, however. 3.2

Permanent actions

Self-weight of steel structure The self-weight of the steel structure is sustained by the steel structure alone without temporary supports: g s = 1, 0 kN/m 2 ⋅ 5, 0 m = 5, 0 kN/m

194


Self-weight of in situ concrete slab Like the self-weight of the steel structure, the weight of the in situ concrete is borne by the steel structure alone [7]: g c = 0, 25 m ⋅ 25, 0 kN/m 3 ⋅ 5, 0 m = 31, 25 kN/m

Dead loads The dead loads consist primarily of the vehicle restraint system and the asphalt layer and are borne by the composite structure. g dl = 1, 6 kN/m 2 ⋅ 5, 0 m = 8, 0 kN/m

3.2

Creep and shrinkage

Shrinkage Pursuant to EN 1994-2 [2] §3.1(3) and EN 1992-1-1 [4] §3.1.4 and Annex B, total shrinkage strain has two components, drying and autogenous shrinkage strain. Taking ambient relative humidity to be 70 % and assuming that the concrete is manufactured with Class N cement:   

 70    100 

3

β70% = 1,55 ⋅ 1 − 

  = 1, 018 . 

The basic drying shrinkage strain, εcd,0, is calculated as: 

 

38  

ε cd ,0 = 0,85 ( 220 + 110 ⋅ 4 ) ⋅ exp  −0,12 ⋅   ⋅10−6 ⋅1, 018 = 362 ⋅10−6 10 



2 Ac 2 ⋅ 2,5 = = 0, 244 m u 20,5 where Ac is the cross-sectional area and u the perimeter of the member in contact with the atmosphere. The notional height is:

h0 =

β ds (t , ts ) =

10000 10000 + 0, 04 ⋅ 2443

= 0,985

Drying shrinkage strain develops over time as:

ε cd (t ) = 0, 985 ⋅ 0,85 ⋅ 362 ⋅10−6 = 303 ⋅10 −6

β as (t ) = 1 − exp ( −0, 2 ⋅100000.5 ) = 1, 0 ε ca (∞) = 2,5 ⋅ (30 − 10) ⋅10−6 = 50 ⋅10−6 The autogenus shrinkage strain is:

ε ca (∞) = 1, 0 ⋅ 50 ⋅10−6 = 50 ⋅10−6 and the total shrinkage strain at t = ∞ is:

195


ε cs∞ = (303 + 50) ⋅10−6 = 353 ⋅10−6 Creep Given that all dead loads are applied 15 days after the in situ concrete is poured and that the ambient relative humidity is 70 %, it follows that ([2], §2.3.3 and §3.1; [4], §3.1.4 and Annex B):

ϕ70% = 1 + β ( f cm ) = β (t0 ) =

1 − 70 /100 = 1, 48 0,1⋅ 3 244

16,8 16,8 = = 2, 73 f cm 30 + 8

1 1 = = 0,55 0,2 0,1 + t0 0,1 + 150,2

where t0 is the age of concrete at first loading in days. The notional creep coefficient ϕ0 is:

ϕ0 = 1, 48 ⋅ 2, 73 ⋅ 0, 55 = 2, 22 18 β H = 1,5 ⋅ 244 ⋅ 1 + ( 0, 012 ⋅ 70 )  + 250 = 632 ≤ 1500





 10000  β c ( ∞, t 0 ) =   632 + 10000 

0.3

= 0,982

The final creep coefficient is:

ϕ ( ∞, t0 ) = ϕ0 ⋅ β ( ∞, t0 ) = 2, 22 ⋅ 0,982 = 2,18 3.3

Variable actions

Traffic loads. Further to EN 1991-2 [8], §4.3.2, only Load Model 1 is applied. In this model, used for general verification calculations, both concentrated two-axle tandem loading and uniformly distributed loads are considered. The carriageway width w is assumed to be equal to the distance between the inner limits of the vehicle restraint system, therefore w=9,0 m. As w>6,0 m, the number of conventional lanes, each of which is wl=3,0 m wide, is afforded by the relation (Figure 4):  w  9, 0  nl = int   = int   = 3. 3  3 

Since the span length is greater than 10 m, each of the three tandem load systems may be replaced by a one-axle load equal to the total load exerted by the two axles constituting the system [8], §4.3.2 (6). Uniformly distributed loads:

196

Lane 1 q1k = 9,0 kN/m2 other notional lanes qi,k = 2,5 kN/m2


remaining area Lane 1 Lane 2 Lane 3 other notional lanes

Tandem system

wl = 3,0 m Lane 1

Q1k = 300 kN

wl = 3,0 m Lane 2

qrk = 2,5 kN/m2 Q1k = 300 kN (1 axle of 600 kN) Q2k = 200 kN (1 axle of 400 kN) Q3k = 1 00 kN (1 axle of 200 kN) Qik = 0. wl = 3,0 m Lane 3

Q1k = 300 kN Q2k = 200 kN

Q2k = 200 kN Q3k = 100 kN Q3k = 100 kN 2,0 m 2,0 m

2,0 m q1k = 9,0 kN/m2

q2,3k = 2,5 kN/m2

w = 9,0 m

Figure 4. Traffic loads on the bridge deck Due to the absence of specific indications related to the expected traffic, the adjustment factors αQ and αq are assumed to be equal to 1,0. Temperature Linear temperature variation from the upper to the lower face of the bridge deck (see §6.1.4.1 of [9] and §4 of chapter 5): Top surface warmer (heating) Bottom surface warmer (cooling)

∆T = +15 ºC (7 ºC/m) ∆T = −18 ºC (-8,4 ºC/m)

The same linear thermal expansion coefficient is assumed for the steel and the concrete, namely α = 10·10-6 ºC-1 ([2], §3.1(1) and §5.4.2.5(3); [4], §3.1.3(5)). The value specified in EN 1991-2 [8], section 5, for uniform thermal variation is disregarded because it is irrelevant to the present study. Construction loads during the casting of concrete Pursuant to EN 1991-1-6 [10], §4.11.2, the following construction loads are taken into account simultaneously in the calculations to verify steel structure conformity. These loads are intended to be positioned to cause the maximum effects, which may be symmetrical or not. - Actual area: Self-weight of the formwork:

qcc,k = 0,5 kN/m2

197


Weight of the fresh concrete (density 26 kN/m3): qcf,k=0,25 26=6,5 kN/m2 - Outside the working area: Working personnel with small site equipment: qca,k = 0,75 kN/m2 - Inside the working area (3 m×3m): 10 % of the self-weight of fresh in situ concrete, but 0,75 ≤ qca,3·3 ≤ 1,5 kN/m2: qca,3·3=0,1 6,5= 0,65 kN/m2<0,75 kN/m2 therefore, in the present case qca,3·3 = 0,75 kN/m2.

4

STRUCTURAL ANALYSIS

4.1

Effective width

4.1.1 General remarks The effective widths calculated in the items below, which are valid for verifying the ultimate and serviceability limit states, refer to the concrete slab only. The reduction set out in EN 1993-1-5 [11] should be applied to the steel flanges, in the present case ψel = 1. 4.1.2 Global bridge deck analysis Each span is assumed to have a constant effective width across its entire length, which is taken to be the mid-span value calculated as described in item 4.1.2 ([2], §5.4.1.2(4)). 4.1.3 Effective width for verifying cross-sections The effective width of the concrete slab for verifying cross-sectional conformity to requirements is determined as specified in EN 1994-2 [2], §5.4.1.2. At mid-span or over an internal support, the total effective width beff is determined from. beff = b0 + Σbei

b eff b e1

where: bei = Le / 8 ≤ bi

bo

be2

bi is the actual width; Le is the equivalent span (approximate distance between points of zero bending moment); b0 is the distance between the centres of the outstand shear connectors. In the present case it is assumed b0 = 0,20 m. The effective width at the end supports is corrected by applying the βi factor: beff ,0 = b0 + Σβ i ⋅ bei

where:

βi = (0, 55 + 0, 025 ⋅ Le / bei ) ≤ 1, 0 , Consequently:

βi = (0, 55 + 0, 025 ⋅ 25, 5 / (2,5 − 0,1) = 0,82

198


beff ,0 = 0, 2 + 2 ⋅ 0,82 ⋅ 2, 5 = 4,3 m .

The resulting distribution of equivalent spans and effective widths is shown in Figure 5.

Le = 18,4 m

Le = 18,4 m

Le = 25,5 m

Le = 30,5 m

30,0 m

b eff

7,5 m

15,0 m

4,3 m

5,0 m

Le = 25,5 m

43,5 m 7,5 m 10,9 m

21,7 m

4,8 m

30,0 m 10,9 m 7,5 m

5,0 m

4,8 m

15,0 m

7,5 m

5,0m

4,3 m

Figure 5. Distribution of equivalent spans and effective widths 4.1.4 Construction Given the construction sequence described in §3.4, the procedure to verify steel structure conformity with the legislation must cover the actions generated by its self-weight, the weight of the in situ concrete and the construction loads ([2], §5.4.2.4). The construction sequence is also used as a factor in composite deck verification. 4.1.5 Creep According to [2], §5.4.2.2, modular ratios nL for the concrete may be used to calculate the effects of creep. Depending on the type of loading, the modular ratios are given by: nL = n0 (1 + ψ Lϕt ) where:

n0 ψL φt Ecm

is the modular ratio Ea /Ecm for short-term loading; is the creep multiplier depending on the type of loading; is the creep coefficient; is the secant modulus of elasticity of the concrete for short-term loading.

The modular ratio n0 for analysing the structure when exposed to traffic loads and temperature (and dead loads, where the analysis is performed for t = 0) is: n0 = Ea / Ecm = 210 / 33 = 6, 4 The ratio for permanent loads3 is:

3

In this case, the dead load only, because the construction entails no shoring.

199


n = 6.4 ⋅ (1 + 1,1 ⋅ 2,18) = 21, 7

The ratio for primary and secondary shrinkage effects is: n = 6, 4 ⋅ (1 + 0, 55 ⋅ 2,18) = 14 .

4.2

Concrete cracking Given that the ratio between lengths of adjacent continuous spans is 30, 0 / 43, 5 = 0, 7 > 0, 6 , the effect of cracking is calculated by applying the flexural stiffness of the cracked section to 15% of the span on both sides of each internal support ([2], §5.4.2.3). This simplification is acceptable for all calculations with the exception of connection design, where the result might err on the unsafe side. Further to the previous assumption, the length affected by cracking is 0,15×30,0 = 4,5 m in the lateral spans and 0,15×43,5=6.5 m in the central span, both adjacent to each support. Since the thickness of the web and the bottom flange is designed to change at a distance of 5,0 m from the support on each side, for the sake of simplification, this is the distance used as the region subject to cracking in the three spans.

4.3

Mechanical properties of cross-sections As discussed in §4.1.2 and §4.1.3, a constant effective width of beff = 5 m along the entire deck is assumed for structural analysis, obtaining the section properties reported below. Type 1 section: over piers (5 m on both sides) 0,25 m

5,0 m

1,90 m

= 450 x 25

Ø 20 a 20 (s) Ø 16 a 20 (i)

= 1815 x 15 = 600 x 60 (S 460)

Figure 6. Type 1 section Table 1. Type 1 section properties Structural steel section

Transformed section n = 6.4

n = 14

n = 21,7

Cracked Section

Area [m2]

0,074

0,270

0,164

0,132

0,087

Inertia [m4]

0,041

0,143

0,118

0,102

0,063

v [m]

1,247

0,504

0,749

0,898

1,291

0,653

1,646

1,401

1,252

0,859

v’ [m] NOTE 1:

NOTE 2:

200

v is the distance between the uppermost fibre in the section and the centroid. v’ is the distance between the centroid and the lowermost fibre in the section. In the cracked section, the upper face of the concrete slab is regarded to be the uppermost fibre. The contribution of the reinforcing steel is disregarded in the calculations to determine the characteristics of the transformed section.


Type 2 section: over abutments (5 m from bearing)

0,25 m

5,0 m

1,90 m

= 450 x 25 = 1850 x 12 = 600 x 25



n = 14

n = 21,7

Area [m2]

0,048

0,244

0,138

0,106

4

Inertia [m ]

0,029

0,082

0,071

0,065

v [m]

1,023

0,353

0,529

0,649

v’ [m]

0,877

1,797

1,621

1,501

Type 3 section: span 0,25 m

5,0 m

1,90 m

= 450 x 25 = 1835 x 12 = 600 x 40



n = 14

n = 21,7

Area [m2]

0,057

0,253

0,147

0,115

4

Inertia [m ]

0,034

0,108

0,092

0,082

v [m]

1,153

0,415

0,624

0,762

v’ [m]

0,747

1,735

1,526

1,388

201


4.4

Structural model The structural model is a continuous beam whose characteristics are shown in Figure

9. 5,0 m Type of section

20,0 m

10,0 m

33,5 m

10,0 m

20,0 m

3

1

3

1

3

2

30,0 m

43,5 m

5,0 m 2

30,0 m

103,5 m

Figure 9. Structural model adopted The characteristics considered for each section type are given in Table 4, depending on the applied action.

Table 4. Characteristics by section type and applied action TYPE 1 SECTION

TYPE 2 SECTION

TYPE 3 SECTION

Self-weight of steel structure

Structural steel

Structural steel

Structural steel

Self-weight of in situ concrete

Structural steel

Structural steel

Structural steel

t=0

Cracked section

Transformed n=6,4

Transformed n=6,4

t=∞

Cracked section

Transformed n=21,7

Transformed n=21,7

Traffic load

Cracked section

Transformed n=6,4

Transformed n=6,4

Temperature

Cracked section

Transformed n=6,4

Transformed n=6,4

Shrinkage

Cracked section

Transformed n=14,0

Transformed n=14,0

Dead load

5

ACTION EFFECTS

5.1

Internal forces and moments Elastic analysis is used to calculate the action effects in the sections over the pier and at mid-span in the intermediate bay. The findings are given in Table 5. The values for shear refer to the section over the pier in the intermediate span. The effect of thermal action is calculated by entering the gradient set out in §3.4 above in the structural model. Primary and secondary shrinkage-induced effects are estimated as follows ([2], §5.4.2.2): - in light of the properties of the sections where n = 14, the forces and moments applied at the two ends of the deck are as follows (figure 10) 6 Ecm   −6  33 ⋅ 10  N = Ac ⋅ ε cs∞ ⋅  = ⋅ ⋅ ⋅ ⋅ 5 0.25 353 10   = 6649 kN   1 + 0.55 ⋅ 2.18   2.19 

0.25   M = N ⋅v −  = 6649 ⋅ ( 0.529 − 0.125 ) = 2686 kN ⋅ m * 2   202


where the value of v is that pertaining to the section over the abutment; - in addition to the effects discussed above, tensile stresses equal to N/Ac (using the aforementioned values of N and Ac) must be considered to estimate the shrinkageinduced stress in the slab.

Table 5. Internal forces and moments Over pier

At mid-span

M (kN·m)

V (kN)

M (kN·m)

V (kN)


-744

+109

+439

0

Self-weight of in situ concrete

-4648

+680

+2744

0

-1011 -1087

+174 +174

+881 +805

0 0

-4214

+707

+4380

0

Mmax = -2760 Mconc = 0

Vconc = +519 Vmax = +800

+5938

+400

-1709 +1424

0 0

-1709 +1424

0 0

-394

0

-394

0

t=0 t=∞

Dead load

Uniformly distributed load Tandem system Cooling Heating

Temperature Shrinkage

N

N M

30,0 m

43,5 m

30,0 m

M

103,5 m

Figure 9. Forces and moments to be applied to estimate shrinkage effects 5.2

Vertical displacements The following mid-span displacements in the intermediate bay are obtained from the model using the properties of the transformed sections where the effective width of the concrete is reduced, according to the modular ratio related with the type of loading. The mechanical property values used to calculate the deflection induced by distributed traffic loads and tandem system are the values for the short-term loads at t = 0 and t = ∞.

Table 6. Displacements at mid-span in the central bay f [mm] (t = 0)

f [mm] (t = ∞)


8

8

Self-weight of concrete

51

51

Dead load

6

7

Shrinkage

0

5

Total traffic load

22

22

Distributed load on the central span

34

34

Tandem system

33

33

Load state

203


6

DESIGN VALUES

6.1

Effects of actions In serviceability limit state verifications, the partial factors for the actions are assumed to be equal to 1,0 ([6], §A1.4.1 (1)). In ultimate limit state verifications, the partial factors used are given in [12] (§A.2.3) and listed in Table 7 below:

Table 7. Partial factors for actions, ultimate limit state calculations Type of action

Partial factor

Permanent loads

γG = 1,35

Traffic loads

γQ = 1,35

Thermal action

γQ = 1,5

Shrinkage

γGsc = 1,0 ([2] §2.3.3)

6.2

Material properties In serviceability limit state verifications, the partial factors for material properties are assumed to be 1,0 ([6], section 6.5.4(1)). In ultimate limit state verifications, the partial factors used are given in [2] (§2.4) and listed in Table 8 below:

Table 8. Partial factors for material properties, ultimate limit state calculations Material Structural steel

Partial factor γM0 = 1,0 γM1 = 1,1

Concrete

γc = 1,5

Reinforcing steel

γs = 1,15

The values used for partial factors γM0 and γM1 are as recommended in EN 1993-2 [13].

7

COMBINATIONS OF ACTIONS

7.1

ψ factor values The ψ factors for the action combinations considered in this study, given in the following table 9, are the factors recommended in EN 1990/A1 (Annex A2, on application to bridges) [12].

7.2

Combinations for the verification of ultimate limit states4 (ULS) The combinations of actions used for ULS verification, applying the above action combination and partial factors, are:

4

Only combinations for t = ∞ are used in the present example.

204


Table 9. Combination factors used for ultimate limit state calculations Type of action

Ψ0

Ψ1,infq

Ψ1

Ψ2

Uniformly distributed load

0,4

0,8

0,4

0,0

Tandem system

0,75

0,8

0,75

0,0

Thermal action

0,0 (ULS) 0,6 (SLS)

0,8

0,6

0,5

a) b) where:

1.35 ⋅ G p + Gsc + 1.35 ⋅ Q

1.35 ⋅ G p + Gsc + 1.5 ⋅ T + 1.35 ⋅ (ψ 0 ⋅ QTS + ψ 0 ⋅ QUDL )

ψ0 = 0,75 ψ0 = 0,40 Gp Gsc Q QTS QUDL T

for tandem system (TS) for uniformly distributed loads (UDL) permanent loads shrinkage traffic loads traffic loads due to tandem system uniformly distributed traffic loads temperature

7.2.1 Mid-span section Where traffic loads are assumed to be the predominant variable action: 1.35 ⋅ G p + Gsc + 1.35 ⋅ Q

MEd , max = 1.35 · ( 439 + 2744 + 805 ) − 394 + 1.35 ⋅ (4380 + 5938) = 18919 kN ⋅ m VEd = 1.35 · 400 = 540 kN

Where thermal action is regarded to be the predominant variable action: 1.35 ⋅ G p + Gsc + 1.5 ⋅ T + 1.35 ⋅ (ψ 0 ⋅ QTS + ψ 0 ⋅ QUDL )

M Ed,max = 1.35 ⋅ (439 + 2744 + 805) − 394 + 1.5 ⋅1424 + 1.35 ⋅ (0.4 ⋅ 4380 + 0.75 ⋅ 5938) = 15503 kN ⋅ m VEd = 1.35 · 0.75 ·400 = 405 kN

7.2.2 Support section The two combinations studied are: minimum bending moment and concomitant shear, and maximum shear and concomitant bending moment.

- 1st combination: MEd,min - VEd,conc Where traffic loads are assumed to be the predominant variable action: 1.35 ⋅ G p + Gsc + 1.35 ⋅ Q

205


M Ed,min = 1.35 ⋅ (-744 − 4648 − 1087) − 394 + 1.35 ⋅ (-4214 − 2760) = −18556 kN ⋅ m VEd ,conc = 1.35 ⋅ (109 + 680 + 174) + 1.35 ⋅ (707 + 519) = 2955 kN

Where thermal action is regarded to be the predominant variable action: 1.35 ⋅ G p + Gsc + 1.5 ⋅ T + 1.35 ⋅ (ψ 0 ⋅ QTS + ψ 0 ⋅ QUDL )

M Ed,min = 1.35 ⋅ (-744 − 4648 − 1087) − 394 + 1.5 ⋅ (-1709) + 1.35 ⋅ (0.4 ⋅ (-4214) + 0.75 ⋅ (-2760)) = -16774 kN ⋅ m VEd ,conc = 1.35 ⋅ (109 + 680 + 174) + 1.35 ⋅ (0.4 ⋅ 707 + 0.75 ⋅ 519) = 2207 kN .

- 2nd combination: MEd,conc - VEd,max Where traffic loads are assumed to be the predominant variable action: 1.35 ⋅ G p + Gsc + 1.35 ⋅ Q M Ed ,conc = 1.35 ⋅ (-744 − 4648 − 1087) − 394 + 1.35 ⋅ (-4214) = -14829 kN ⋅ m VEd ,max = 1.35 ⋅ (109 + 680 + 174) + 1.35 ⋅ (707 + 800) = 3334 kN

Where thermal action is regarded to be the predominant variable action: This combination does not condition the results. Table 10 summarises the action effects found for the mid-span and support sections.

Table 10. Moments and shear forces acting on the mid-span and support sections Type of section Mid-span

Action effects

M [kN·m]

V [kN]

MEd,max - VEd,con

18919

540

MEd,min - VEd,con

-18556

2955

MEd,conc – VEd,max

-14829

3334

Support

7.3

Combinations for verifying serviceability limit states (SLS) The following combinations of actions are established for verifying SLS ([6], [8] and

[13]): -

Characteristic combination:

Frequent combination:

(a)

G p + Gsc + Q + 0.6 ⋅ T

(b)

G p + Gsc + T + ψ 0 ⋅ QTS + ψ 0 ⋅ QUDL

(a)

G p + Gsc + ψ 1 ⋅ QTS + ψ 1 ⋅ QUDL + 0.5 ⋅ T

(b)

G p + Gsc + 0.6 ⋅ T

Quasi-permanent combination: where:

206

ψ0 = ψ1 = 0,75 ψ0 = ψ1 = 0,40

G p + Gsc + 0.5 ⋅ T

for tandem system (TS) for uniformly distributed loads (UDL).


8

FINAL REMARKS

In this chapter, the provisions of a number of items in EN-1991, “Actions on Structures”, along with specific stipulations for composite steel and concrete bridges and other structural Eurocodes are applied to analyse a composite steel and concrete bridge deck for vehicle traffic. The following concluding remarks are in order: - The study discussed hereunder focuses on the application, as reliably as possible, of the rules laid down in the structural Eurocodes referenced. - In keeping with the purpose and scope of this chapter, all the relevant actions and their effects likely to arise during bridge deck construction and future use are identified. The specific load models for the actions that should be considered in analysis and design calculations are also determined. - Further to the requirements for safety and serviceability of composite steel and concrete structures, the relevant action effects are calculated and the combinations of actions for verifying ultimate and serviceability limit states are determined for the example chosen.

9 REFERENCES [1] Monografía M-10. Comprobación de un Tablero Mixto. Comisión 5, Grupo de Trabajo 5/3 “Puentes Mixtos”, Asociación Científico-técnica del Hormigón Estructural, Madrid, 2006, ISBN 84-89670-47-1. [2] EN 1994-2 Eurocode 4: Design of composite steel and concrete structures. Part 2: Rules for bridges. CEN, Brussels, 2005. [3] EN 1993-1-1 Eurocode 3: Design of steel structures. Part 1-1: General rules and rules for buildings. CEN, Brussels, 2005. [4] EN 1992-1-1 Eurocode 2: Design of concrete structures. Part 1-1: General rules and rules for buildings. CEN, Brussels, 2004. [5] EN 1994-1-1 Eurocode 4: Design of composite steel and concrete structures. Part 1-1: General rules and rules for buildings. CEN, Brussels, 2004. [6] EN 1990 Eurocode - Basis of structural design, CEN, Brussels, 2002. [7] EN 1991-1-1 Eurocode 1 - Actions on structures. Part 1-1: General actions – Densities, self-weight, imposed loads for buildings, CEN, Brussels, 2002. [8] EN 1991-2 Eurocode 1: Actions on structures. Part 2: Traffic loads on bridges. CEN, Brussels, 2003. [9] EN 1991-1-5 Eurocode 1: Actions on structures. Part 1-5: General actions – Thermal actions. CEN, Brussels, 2003. [10] EN 1991-1-6 Eurocode 1: Actions on structures. Part 1-6: General actions – Actions during execution. CEN, Brussels, 2005. [11] EN 1993-1-5 Eurocode 3: Design of steel structures. Part 1-5: Plated structural elements. CEN, Brussels, 2006. [12] EN 1990:2002/A1:2005 Eurocode - Basis of structural design. Annex A2: Application for bridges. CEN, Brussels, 2005. [13] EN 1993-2 Eurocode 3: Design of steel structures. Part 2: Steel bridges. CEN, Brussels, 2006.

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208


ANNEX A: EFFECTS OF LHVs ON ROAD BRIDGES AND EN1991-2 LOAD MODELS Pietro Croce1 1


Summary To improve the European transportation network, the European Commission issued the 96/53/EC directive, that besides limiting the total mass of Heavy Good Vehicles (HGV) to 44 t, allowed, on a parity basis, the possibility to permit the circulation of Long and Heavy Vehicles (LHV), with total mass up to 60 t and length till 25 m. Northern European countries took advantage of this possibility, experiencing a significant increase of LHVs on long distance traffic. Since the static and fatigue models for road bridges of EN1991-2 have been calibrated on the traffic recorded in Auxerre (F) in 1986, where LHVs were not included, the effect of the introduction of LHVs could be excessively demanding, with disproportionate increase of costs. Recent studies concerning this relevant question are discussed.

1

INTRODUCTION

In order to improve the organization of the European transportation network, the European Commission issued the 96/53/EC Directive [1] limiting the total weight of Heavy Good Vehicles (HGV) to 44 t, but admitting, on a parity and not discriminating basis, the possibility to allow the circulation of Long and Heavy Vehicles (LHV), with a total mass up to 60 t and with length till 25 m. Some northern European countries, in particular Sweden, Finland, The Netherlands and Germany, permitted the LHVs, so experiencing a significant increase of the number of LHVs in long distance traffic. The introduction of LHVs opens new prospects for traffic management thanks to of their effectiveness in terms of decrease of pollutant emissions and cost reduction. However, it could lead on the other hand to excessive surcharge of bridges, with disproportionate increase of costs. Since the load models for static and fatigue assessments of road bridges given in EN19912 [2] have been calibrated using the traffic recorded in Auxerre (F) in 1986, obviously not including LHVs,, the impact of LHVs on existing infrastructures as well as on the load models themselves could be very relevant. In order to clarify this problem and to draw some preliminary conclusion, additional studies have been performed on relevant bridge schemes and spans comparing the Auxerre traffic effects with those induced by the traffic recorded in April 2007 in Moerdijk, the Netherlands, characterised by an high percentage of LHVs.

209


2

LONG AND HEAVY VEHICLE TRAFFIC MEASUREMENT

As just said, at present the results of a wide campaign of in-situ measurements concerning typical LHV traffic in the Netherlands are available, which have been used in the present study. The above mentioned measurements have been performed in the first week of April 2007 in Moerdijk (NL), using a state of art weighing in motion (WIM) device, in the framework of the studies concerning the assessment of equivalent fatigue loads for bridge decks, made by van Bentum and Dijkstra [3]. In the records vehicles travelling with speed greater than 33 m/s were disregarded, considering that in this case the reliability is not granted. 2.1

Characteristics of Moerdijk traffic Analysis of Moerdijk traffic allows classifying commercial lorries in 53 relevant subclasses, characterized by axle number varying between 2 and 9, described in figure 1. It must be noted that lorries with up to 13 axles appear in the records and that the maximum recorded lorry load was about 1140 kN, pertaining to a ten axle lorry type O13411, 19.5 m long. Preliminary analysis of the records revealed that in few cases measurements were inaccurate, so that it was necessary to eliminate wrong data from traffic database. In fact, the four records summarized in table 1 and relative to two axle lorries appear largely unrealistic, also due to excessive speed or length. Table 1. Inaccurate data for 2 axle lorries in Moerdijk records Speed [m/s]

Length [m]

Weight [kN]

30.6 3.3 3.3 27.2

21.02 13.05 62.02 11.32

707 613 684 689

1st axle Load [kN] 398 208 318 353

2nd axle load [kN] 309 405 366 336

Disregarding wrong data, the maximum recorded axle load results about 292 kN, pertaining to the 3rd axle of a T12O3 lorry, whose total weight is 636 kN, while the maximum uniformly distributed load is about 63 kN/m, pertaining to a T12O21 silhouette weighing 813 kN in total. Moerdijk traffic measurements, amended according to the aforesaid considerations, were taken into account for the evaluation of static and fatigue effects on reference bridge schemes and spans, to be compared with those induced by the Auxerre traffic, as well as with those induced by EN1991-2 load models, as described in the following.

3 3.1

HGV TRAFFIC AND LHV TRAFFIC COMPARISON

Axle and lorry loads Static effects can be compared in a very simple way in terms of load spectra. This is highlighted in particular in figures 2, 3, 4 and 5, which refer to comparison of single, tandem, tridem axle’s weight spectra and total lorry weight spectra of Auxerre and Moerdijk, respectively. 210


Figure 1. Lorry subclasses and symbols for Moerdijk (NL) traffic

211


Figure 2. Comparison of single axle load spectra for Auxerre and Moerdijk traffics.

Figure 3. Comparison of tandem axle load spectra for Auxerre and Moerdijk traffics

Figure 4. Comparison of tridem axle load spectra for Auxerre and Moerdijk traffics

212


Figure 5. Comparison of total lorry weight spectra for Auxerre and Moerdijk traffics. 3.2

Preliminary conclusions about static loads The examination of the traffic records and the load spectra comparison shown above highlight that: - in consequence of the new traffic trend, the average axle number of the commercial vehicle tends to increase significantly; - total weight of LHVs could raise very high level, but usually this level is associated with axle loads close to the legal limits; - despite that occasionally axle loads can reach about 300 kN, Moerdijk traffic appears, in general, less severe than the Auxerre traffic; - it seems, therefore, that EN 1991-2 load models for static verifications cover also Moerdijk traffic effects, so confirming, at this stage, its effectiveness; - clearly, to draw more definitive conclusions it is necessary to enlarge the field of investigation, also considering different traffic records. It must be stressed, moreover, that Auxerre traffic data, obtained with less refined WIM devices, probably are affected by a systematic overestimate.

4

FATIGUE DAMAGE

4.1

Reference traffics and equivalent load spectra Besides in terms of static assessment, it is necessary to ascertain the aggressiveness of traffic in terms of fatigue damage. For this purpose four different traffic samples, composed by 10 000 vehicle each, have been taken into account: two real ones, directly derived from Auxerre traffic measurements used for calibration of EN 1991-2 models and from the new Moerdijk traffic measurements, respectively, and two conventional ones, suitably derived from the fatigue load model n. 4 for long distance traffic. The aforesaid conventional traffic models were obtained taking into account an annual flow of 2×106 standard LM 4 lorry silhouettes on the slow lane. These models, obtained

213


through a Monte Carlo simulation, differ on the inter-vehicle distances, that in the former case are as simulated, in order to consider also interaction between vehicles simultaneously present on the bridge, and in the latter case they are suitably increased in such a way that only isolated lorries can cross the bridge, so avoiding interaction. It must be also stressed that EN 1991-2 states that, as rule, fatigue load models cannot be used directly when vehicle interactions become significant, unless adequate additional ad hoc studies are performed. The choice of these four reference traffics is particularly appropriate, because it allows to compare the fatigue damage induced by the Auxerre traffic not only with the damage induced by Moerdijk one, but also with those induced by the equivalent load spectra of EN 1991-2. 4.2

Reference influence lines and spans for bridges Preliminarily, five influence surfaces for simply supported and continuous beams have been selected to perform the fatigue damage assessment. The influence surfaces, illustrated in figures 6 to 10, refer to the bending moment at midspan of a simply supported beam (Mo), to the bending moment at intermediate support (M1) and at section located 0.432·L from the first support, where the bending moment attains the maximum value(M2), in a two span continuous beam and to the to the bending moment at the third support (M3) and at midspan (M4) of a five span continuous beam.

0

0.5

1

Figure 6. Influence line for bending moment M0 at midspan of simply supported bridge

0

1

2

Figure 7. Influence line for hog moment M1 at intermediate support of two span continuous bridge

0

0.432

1

2

Figure 8. Influence line for max sag moment M2 in the section located 0.432·L from the support in two span continuous bridge

214


0

1

2

3

4

5

Figure 9. Influence line for hog moment M3 at the third support of five span continuous bridge

0

1

2

3

4

5

Figure 10. Influence line for max sag moment M4 at midspan of five span continuous beam bridge 4.3

Comparison of fatigue damage For each influence line, nine different span lengths have been considered, varying from 3 m to 100 m (3 m, 5 m, 10 m, 20 m, 30 m, 50 m, 70 m, 100 m), and the bending moment histories induced in each of them, in turn, by the four relevant traffics have been determined. Stress spectra have been derived from the above mentioned time histories using the rainflow cycle counting method. Finally, fatigue damage has been evaluated using the Palmgren Miner rule. In the damage assessment, consistently with the assumptions made in EN1991-2 background studies, simplified single slope S-N curves, without fatigue limit, have been taken into account, assuming, in turn, a slope m=3 and m=5. These statements are fully justified, as they simplify significantly fatigue assessments, introducing negligible errors. The fatigue damage induced by each relevant traffic is then compared with the one induced by the Auxerre traffic, both for m=3 or m=5, and the aggressiveness of each traffic is finally derived in terms of the equivalence factor Keq,t for the actual traffic, given by

K eq ,t

 Dt =  D Aux

1

∆σ eq t m  = ∆σ eqAux 

(1)

where Dt is the fatigue damage induced by the actual traffic, DAux is the fatigue damage induced by the Auxerre traffic, m is the slope of the S-N curve adopted for the evaluation of Dt and DAux, ∆σeq,t is the equivalent value of the stress range for the actual traffic and ∆σeq,Aux is the equivalent value of the stress range for the Auxerre traffic. Obviously, higher values of Keq,t correspond to more aggressive traffics. Keq,t is a characteristic traffic parameter and it represents a concise way to compare different traffics: in fact, it can be interpreted as an adjustment factor for which the axle load

215


values of Auxerre traffic must be multiplied to reproduce the fatigue damage induced by the actual traffic. The Keq,t curves, pertaining to Moerdijk traffic as well as to conventional traffics derived from fatigue load model LM4 of EN 1991-2, with and without vehicle interaction, are plotted for each relevant influence line, in terms of span length, in figures 11 to 15 for m=3 and in figures 16 to 20 for m=5. More precisely figures 11 and 16 refer to M0, figures 12 and 17 to M1, figures 13 and 18 to M2, figures 14 and 19 to M3 and figures 15 and 20 to M4.

Figure 11. Keq curves for bending moment M0 (m=3)



216





217





218


Figure 20. Keq curves for bending moment M4 (m=5) Critical examination of these Keq,t curves allows to conclude that: - load model LM 4 represents very satisfactorily the actual fatigue damage induced by the Auxerre traffic and is generally safe-sided; - despite of EN 1991-2 statement that establishes that isolated standard lorries cannot be used when vehicle interactions are relevant (i.e. when L > 30 m), the use of conventional often allows to approximate real traffic effects better than resorting to improved load model LM4, taking into account interactions; - conventional load model LM4, disregarding vehicle interactions, results unsafe-sided for bending moment at intermediate supports of continuous beams: in this case, on the contrary, the use of improved load model LM4 leads to significant overestimates of fatigue damage, with Keq values raising up 1.25; - the aforesaid phenomena can be explained considering that influence line of intermediate support is characterized by two adjacent zones where the ordinates of the influence line are comparable, so that from one side the stress range induced by isolated vehicles, which affect only one zone, is too low, while, from the other side, interacting LM4 vehicles, which affect both adjacent zones, determine too high stress range, as their equivalent axle loads values, calibrated considering short and medium span bridges, are similar; - the Moerdijk traffic is characterized by Keq factors generally ranging between 0.8 and 0.85 except the cases M1 and M3 for span length L > 50 m and m=5. 5

CONCLUSIONS

The impact of LHVs on design of bridges in terms of static and fatigue assessments has been discussed, comparing the effects induced by the Moerdijk (NL) traffic, characterized by high percentage of LHVs, with those induced by the Auxerre traffic, used as reference traffic in background of EN1991-2. The fatigue assessments have been supplemented also considering two conventional traffics, deduced by the fatigue load model 4 of EN1991-2, constituted by equivalent lorries.

219


Results of the study demonstrate that EN 1991-2 load models adequately cover the effect induced by the LHVs, as included in Moerdijk measurements. This can be explained considering, on the one hand, that overloads of single axles of LHVs are usually not so relevant as for HGVs, on the other hand, that Auxerre data, obtained in 1986 with a less refined WIM device, could be affected by some systematic overestimate. Clearly, these results need to be supplemented and improved as they concern specific traffic measurement; therefore further studies are necessary enlarging the field of investigation and considering various traffic measurements.

6.

REFERENCES

[1] Directive 96/53/EEC, OJ L.235, 17/09/1996. [2] EN1991-2, Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges. Brussels: CEN 2003 [3] van Bentum, C.A. & Dijkstra, O.D. Process description of equivalent fatigue load on bridge decks. TNO report 366 B UK. Delft: TNO, 2008.

220


ANNEX B: ACTIONS AND COMBINATION RULES FOR CRANES, MASTS, TOWERS AND PIPELINES Milan Holický1, Jana Marková1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary Relevant loads due to cranes and machinery need to be considered in the design of towers, masts and pipelines. For the selected design situations and identified limit states, the critical load cases should be assessed. The combination rules for actions, partial factors and other reliability elements provided in EN 1990 are supplemented for the purposes of cranes and machinery in EN 1991-3, for towers and masts in EN 1993-3-1 and for pipelines in EN 1993-4-3. It is foreseen that during the planned revision of EN 1990 the basis of design for these structures will be transferred to a new Annexes of EN 1990.

1

INTRODUCTION

1.1

Background documents The aim of this Chapter is to introduce an overview of actions and supplementary rules for their combinations needed for the design of masts, towers, pipelines and crane supporting structures. The basic procedures for the determination of characteristic and design values of actions are given in EN 1990 [1] and in various Parts of EN 1991. Some Parts of EN 1993 provide rules for specification of actions and their combinations for masts, towers and pipelines. 1.2

Basic principles EN 1990 [1] gives basic principles for the design of construction works. EN 1991-3 [6] provides supplementary rules for the design of cranes and machinery, EN 1993-3-1 [7] for masts and towers and EN 1993-4-3 [8] for pipelines.

2

ACTIONS AND COMBINATION RULES FOR CRANES AND MACHINERY

2.1

General EN 1991-3 [6] give guidance for specification of actions induced by cranes on runway beams and by rotating machines on supporting structures. 2.2

Actions due to cranes Actions due to cranes include hoist loads, inertial forces caused by acceleration/deceleration and by other dynamic effects. Following vertical and horizontal components are distinguished: – variable vertical crane actions caused by the self-weight of the crane and the hoist load

221


– variable horizontal crane actions caused by acceleration/deceleration or by skewing or by other dynamic effects. The various representative values of variable crane actions are characteristic values composed of a static and a dynamic component. Dynamic components (induced by vibration due to inertial and damping forces) are in general taken into account by dynamic factors ϕ to be applied to the static values of actions Fϕ,k = ϕ i Fk

(1)

Where Fϕ,k is the characteristic value of a crane action, ϕ i is the dynamic factor and Fk is the characteristic static component of a crane action. The various dynamic factors and their application are listed in Table 2.1 of EN 1991-3 [6]. The simultaneity of crane load components may be taken into account by considering groups of loads. Each of these groups should be considered as defining one characteristic crane action for the combination with non-crane loads. Cranes can also evoke accidental actions due to collision with buffers or collision of lifting attachments with obstacles. These actions should be considered for the structural design where appropriate protection is not provided. They are represented by various load models defining design values in the form of equivalent static loads. 2.3

Actions due to machinery Structures supporting rotating machines are commonly loaded by permanent actions due to the self-weight of all fixed or movable parts and static actions due to operations. Variable actions due to machinery during normal service are dynamic actions caused by accelerated masses including – periodic frequency-dependent bearing forces due to eccentricities of rotating masses in all directions, mainly perpendicular to the axis of the rotors – periodic actions due to service depending on the type of machine that are transmitted by the hull or bearings to the foundations – free mass forces or mass moments – forces or moments due to switching on/off or other transient procedures such as synchronisations. Moreover, accidental actions may occur due to e.g. accidental magnification of the eccentricity of masses (for instance by fracture of brakes), by short circuit or out of synchronisation of the generators and machines. 2.4

Combinations of actions Effects of actions that cannot exist simultaneously due to physical or functional reasons should not be considered together in combination of actions. When combining a group of crane loads together with other actions, the group of crane loads should be considered as one action. For runways outside buildings the wind actions, snow and temperature should be considered. The maximum wind force FW* compatible with crane operations need to be considered (the recommended value is FW* = 20 m/s). The combination of actions for ultimate limit states is based on expressions (6.9a) to (6.12b) and for serviceability limit states on expressions (6.14a) to (6.16b) according to EN 1990 [1].

222


Where an accidental action is to be considered, no other accidental action nor wind nor snow actions need be considered to occur simultaneously. Recommended values of ψ-factors are given in Table 1. Table 1. Recommended values of ψ-factors Action ψ0 Single crane or groups of loads induced by cranes

1,0

ψ1

ψ2

0,9

(*)

(*) The recommended value is the ratio between the permanent crane action and the total crane action.

2.5

Ultimate limit states For each critical load case, the design values of the effects of actions should be determined by combining the values of actions which occur simultaneously in accordance with EN 1990 [1]. Static equilibrium (EQU) and uplift of structural bearings for structures supporting cranes and machinery should be verified using the design values of actions given in Table 2. Table 2. Design values of actions (EQU) (Set A) P/T situations

Eq. (6.10)

Permanent actions

Unfavourable γGj,supGkj,sup

Favourable γGj,infGkj,inf

Leading variable action (*)

γQ,1 Qk,1

Accompanying variable actions

Main (if any)

Others γQ,iψ0,iQk,i

NOTE 1 The recommended set of values for γ are γGj,sup = 1,05 γGj,inf = 0,95 γQ = 1,35 for unfavourable variable crane actions

γQ = 1,00 for favourable variable crane actions, where crane is present γQ = 0 for favourable variable crane actions, where crane is not present γQ = 1,5 for other unfavourable variable actions (0 where favourable) NOTE 2 For verification of uplift of structural bearings or in cases where the verification of static equilibrium also involves the resistance of structural members (e.g. where loss of static equilibrium is prevented by stabilizing systems or devices, e.g. anchors, stays or auxiliary columns) as an alternative to two separate verifications based on Tables 2.2 and 2.3 a combined verification based on Table 2.2 may be adopted with the following recommended values γGj,sup = 1,35 γGj,inf = 1,25

γQ,1 = 1,35 for unfavourable variable crane actions γQ,1 = 1,00 for favourable variable crane actions where crane is present γQ,1 = 0 for favourable variable crane actions where crane is not present γQ = 1,5 for other unfavourable variable actions (0 where favourable) provided that applying γGj,inf = 1,00 both to the favourable part and to the unfavourable part of permanent actions does not give a more unfavourable effect.

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The design values of actions and recommended partial factors based on EN 1990 [1] and EN 1991-3 [6] for the ultimate limit states (STR) in the persistent and transient design situations are given in Table 2.3. Table 3. Design values of actions (STR) (Set B) P/T situations

Permanent actions

Prestress


Accompanying variable actions Main (if any) others

γQ,1 Qk,1

γQ,iψ0,iQk,i

unfavourable favourable Exp. (6.10)

γGj,supGkj,sup

γGj,infGkj,inf

γP Pk

Exp. (6.10a)

γGj,supGkj,sup

γGj,infGkj,inf

γP Pk

Exp. (6.10b)

ξγGj,supGkj,sup

γGj,infGkj,inf

γP Pk

γQ,1ψ0,1Qk,1 γQ,1 Qk,1

γQ,iψ0,iQk,i γQ,iψ0,iQk,i

Recommended values: γGj,sup = 1,35 for unfavourable and γGj,inf = 1,0 for favourable permanent actions γQ = 1,35 for unfavourable crane actions, γQ = 1,00 for favourable crane actions, where crane is present, γQ = 0 for favourable crane actions, where crane is not present. γQ = 1,5 for other unfavourable variable actions, γQ = 0 for favourable variable actions where crane is not present ξ = 0,85 (so that ξγGj,sup = 0,85×1,35 = 1,15) γP = recommended values are specified in the relevant design Eurocode. The characteristic values of all permanent actions from one source may be multiplied by γG,sup if the total resulting action effect is unfavourable and γG,inf if the total action effect is favourable. For particular verifications, the values for γG and γQ may be subdivided into γg and γq and the model uncertainty factor γSd (a value of γSd in recommended in the range 1,0 to 1,15).

The design values of actions for the ultimate limit states in the accidental and seismic design situations are given in Table 4. Table 4 Design values of actions for use in accidental and seismic combinations P/T situations

Permanent actions unfavourable favourable

Prestress

Accidental or seismic action

Accidental Exp. (6.11a/b)

Gkj,sup

Gkj,inf

Pk

Ad

Seismic Exp. (6.12a/b)

Gkj,sup

Gkj,inf

Pk

γIAEk or

Accompanying variable actions main (if any) others

ψ11 Qk1 or ψ21Qk1

ψ2,i Qk,i

ψ2,i Qk,i

AEd

In case of accidental design situations, the main variable action may be taken with its frequent or quasi-permanent values. The choice is given in the National Annex depending on the accidental action under consideration.

2.6

Serviceability limit states Three combinations of actions are recommended for verification of serviceability criteria. The design values of actions for the serviceability limit states are given in Table 2.5. The serviceability criteria should be defined in relation to the serviceability requirements in accordance with EN 1992 to EN 1999. Deformations should be calculated in accordance with relevant EN 1991 to EN 1999 by using the appropriate combinations of

224


actions according to expressions (6.14a) to (6.16b) taking into account the serviceability requirements and the distinction between reversible and irreversible limit states. Table 5. Design values of actions for use in serviceability limit states Combination Characteristic Frequent Quasi-permanent

3

Permanent actions Gd Unfavourable Favourable Gk,j,sup Gk,j,inf Gk,j,sup Gk,j,inf Gk,j,sup Gk,j,inf

Prestress Pk Pk Pk

Variable actions Qd Leading Others Qk,1 ψ0,iQk,i ψ1,1Qk,1 ψ2,iQk,i ψ2,1Qk,1 ψ2,iQk,i

ACTIONS AND COMBINATION RULES FOR MASTS AND TOWERS

3.1

General The lattice towers and guyed masts which are within the scope of EN 1993-3-1 [8] should be designed in accordance with general rules of EN 1990 [1] for the basis of design and EN 1993-1-1 [7] for design of steel structures. Three reliability levels are distinguished for the ultimate limit state verifications of structures, depending on the possible economic and social consequences of their collapse. 3.2

Actions and environmental influences Models of actions are given in relevant Parts of EN 1991. Self-weight should be determined in accordance with EN 1991-1-1 [2]. Imposed load is recommended to be considered on members inclined within 30° to horizontal to carry the weight of a workman making possible the maintenance. Wind loads need to be specified according to EN 1991-1-4 [4] however, using the wind force coefficients and supplementary rules for wind actions given in Annex B of EN 1993-1-1 [7]. Basic guidance for modelling of climatic actions [3,4] are provided in Annex B and for ice loads in Annex C [7], more details for icing are given in ISO 12494 [10]. Hazard situations, proposed strategies for risk reduction, accidental actions and guidance for their application are provided in EN 1991-1-7 [5]. 3.3

Ultimate limit states Permanent, transient and accidental design situations are distinguished. The basis for specification of characteristic and design values of actions are given in EN 1990 [1]. The values of partial factors and ψ-factors for actions in the ultimate limit states are recommended in Annex A to EN 1993-1-1 [7]. 3.4

Serviceability limit states The serviceability limit states which need to be considered within the scope of structures given in EN 1993-3-1 [8] include – deflections or rotations that may adversely affect the use of the structure – vibration, oscillation or sway that cause loss of transmitted signals – deformations, deflections, vibration, oscillation or sway leading to damage to nonstructural elements. 3.5

Reliability differentiation Three reliability classes 1 to 3 are distinguished in EN 1993-3-1 [8] related to the consequences of structural failure as illustrated in Table 6. The values of partial factors γG and γQ recommended in EN 1993-3-1 [8] for the ultimate limit state verifications considering three reliability classes are introduced in Table 7.

225


Table 6. Reliability differentiation for towers and masts. Reliability Class 3

2 1

Examples of structures towers and masts erected in urban locations, or where their failure is likely to cause injury or loss of life; towers and masts used for vital telecommunication facilities; other major structures where the consequences of failure would be likely to be very high all towers and masts that cannot be defined as class 1 or 3 towers and masts built on unmanned sites in open countryside; towers and masts, the failure of which would not be likely to cause injury to people

Table 7. Partial factors for actions Type of action effect unfavourable favourable

Reliability Class

γG for permanent

γQ for variable

actions 1,2 1,1 1,0 1,0

actions 1,6 1,4 1,2 0,0

3 2 1 all classes

However, the values of partial factors of actions appear to be considerably lower than those recommended for structures in the basic Eurocode EN 1990 [1], Annex A1. Presently, the target values of reliability index βt are not recommended in EN 1993-3-1 [8]. 3.6

Ice load and combinations with wind The basis for specification of ice load is given in ISO 12494 [10] to which EN 1993-31 [8] make reference. The ice loads are based on Ice Classes IC for rime (IR) and glaze (IG). The specification of relevant Ice Class for the site location is left for national decision. Combination of ice with wind can often govern the design of masts and towers. Two combinations of wind and snow need to be considered for both symmetrical icing and asymmetrical icing: – leading ice load and accompanying wind

γG Gk + γice Qk,ice + γW k ψW Qk,w

(2)

– leading wind and accompanying ice

γG Gk + γW k Qk,w + γice ψice Qk,ice

(3)

where k is given in ISO 12494 [10] dependent on Ice Class and the values of reduction coefficients ψI = ψW = 0,5 are recommended in EN 1993-3-1 [8]. 4 4.1

ACTIONS AND COMBINATION RULES FOR PIPELINES

General EN 1993-4-3 [9] provides principles and application rules for the structural design of buried cylindrical steel pipelines for the transport of liquids or gases or mixtures of liquids and gases at ambient temperatures. The design of pipelines should be in accordance with provisions of EN 1990 [1] for the basis of design and EN 1991 for actions.

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Different levels of reliability may be adopted for different types of pipelines, depending on possible economic and social consequences of their collapse. Each CEN Member State may define the minimum level of reliability. The recommended values given in EN 1993-4-3 [9] are intended for medium safety requirements. 4.2

Actions and environmental influences Actions which should be considered include – – – – – – –

internal and external pressure self weight of the pipeline and its content (transported product) soil loads, imposed deformation (due to settlements, landslides) traffic loads temperature variations construction loads seismic loads.

4.3

Ultimate limit states The basis for specification of characteristic and design values of actions are provided in EN 1990 [1]. The values of partial factors shall be based on the required reliability level. The following combinations of actions for ultimate limit states should be considered a) Internal pressure: The difference between the maximum internal pressure and the smallest external pressure. This limit state is generally used first for the determination of the wall thickness. b) Internal pressure plus other relevant loads: The internal and external pressure conditions defined in (a), with the other relevant design loads added. This limit state is generally used to check critical strains c) External pressure plus other relevant loads: The difference between the maximum external pressure and the smallest internal pressure, with the other relevant design loads added. This limit state is generally used to check ovalisation, critical strains, local buckling etc. d) Temporal variations in pressure plus other relevant design loads. This case is concerned with cyclic actions on the pipe. This limit state is generally used last to check for fatigue. 4.4

Serviceability limit states For serviceability limit states the following combinations should be considered a) Internal pressure plus other relevant loads: The difference between the maximum internal pressure and the smallest external pressure with the other relevant design loads. b) External pressure plus other relevant loads: The difference between the maximum external pressure and the smallest internal pressure, with the other relevant design loads added.

4.5

Reliability differentiation Pipelines usually comprise several associated facilities such as pumping stations, operation centres, maintenance stations, etc., each of them housing different sorts of mechanical and electrical equipment. Since these facilities have a considerable influence on 227


the continued operation of the system, it is necessary to give them adequate consideration in the design process aimed at satisfying the overall reliability requirements. Different levels of reliability may be adopted for different types of pipelines, depending on possible economic and social consequences of their collapse. The choice of the target reliability should be agreed between the designer, the client and the relevant authority.

5

AN EXAMPLE OF SPECIFICATION OF ACTIONS

A simply supported steel beam span of 3 m of a transmitting tower is located at high of 14 m above the terrain of category 2 where the basic wind velocity is vb0 = 27,5 m/s, see Fig. 1. The steel beam (cross-sectional shape I, in mutual distances of 1 m) is loaded by the self-weight of a steel deck thickness of 0,01 m and by average permanent load 3 kN/m2 due to the transmitting facilities. The tower is located in region with Ice Class IR5. The self-weight of a beam is 0,1 kN/m, the permanent load g = 3 kN/m2, wind pressure w at high 14 m w = 0,6 kN/m2 (or suction -1,8 kN/m2) and icing i = 0,5 kN/m2. i

w g

L=3m Figure 1. Scheme of a structural member and its loading. The fundamental combination and reliability elements given in EN 1990 [1] is considered for a common structural reliability class RC2. The maximum moment for the simple beam is determined for an icing considered as a leading action M = 1/8 L2 (1,35 g + 1,5 i + 1,5 ψW k w) M = 1/8×32 (1,35×3,1 + 1,5×0,5 + 1,5×0,6×0,6×0,6) = 5,9 kNm where ψW = 0,6, k = 0,6 [10], and for the wind considered as a leading action M = 1/8 L2 (1,35g + 1,5 w + 1,5 ψI i) M = 1/8×32 (1,35×3,1 + 1,5×0,6 + 1,5×0,5×0,5) = 6,1 kNm where ψI = 0,5. The values of reliability elements (partial factors and reduction factors) are based on EN 1990 [1]. However, when the values of reliability elements given in EN 1993-3-1 [8] are considered then the following result will be obtained if icing is a leading action M = 1/8×32 (1,1×3,1 + 1,4×0,5 + 1,4×0,6×0,6×0,6) = 5 kNm and for the wind considered as a leading action M = 1/8×32 (1,1×3,1 + 1,4×0,6 + 1,4×0,5×0,5) = 5,2 kNm

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Thus, the application of the reliability elements given in EN 1993-3-1 [8] leads to about 15 % reduction of internal moment. In case that the reliability elements recommended in the CENELEC standard EN 50341-1 [11] should be applied then the resulting moments decrease about 30 % in comparison with results based on EN 1990 [1]. The reliability of structural members designed considering reliability elements according to EN 50341-1 [11] appears to be significantly low. 8

CONCLUDING REMARKS

The design of structures supporting cranes and machinery, towers and masts and pipelines are based on the load combinations provided in EN 1990 [1]. Supplementary rules for the specification of loads and load effects are given in relevant Parts of EN 1991 and EN 1993. It is expected that within the revision of EN 1990 similar Annexes A for cranes and machinery, masts and towers and pipelines will be further developed. However, proposed values of partial factors and other reliability elements should be further calibrated and harmonised.

9

REFERENCES

[1] EN 1990 Eurocode – Basis of design. CEN, Brussels, 2006. [2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. CEN, Brussels, 2002. [3] EN 1991-1-3 Eurocode 1: Actions on structures – Part 1-3: Snow loads. CEN, Brussels, 2003. [4] EN 1991-1-4 Eurocode 1: Actions on structures – Part 1-4: Wind loads. CEN, Brussels, 2005. [5] EN 1991-1-7 Eurocode 1: Actions on structures – Part 1-7: Accidental actions. CEN, Brussels, 2006. [6] EN 1991-3 Eurocode 1 – Actions on structures Part 3: Actions induced by cranes and machinery. CEN, Brussels, 2006. [7] EN 1993-1-1 Eurocode 3: Design of steel structures – Part 1.1: General rules and rules for buildings. CEN, Brussels, 2005. [8] EN 1993-3-1 Eurocode 3: Design of steel structures – Part 3.1: Towers, masts and chimneys – Towers and masts. CEN, Brussels, 2006. [9] EN 1993-4-3 Eurocode 3: Design of steel structures – Part 4.3: Pipelines. CEN, Brussels, 2007. [10] ISO 12494 Atmospheric icing on structures. ISO/TC98, 2001. [11] EN 50341. Overhead electrical lines exceeding AC 45 kV – Part 1: General requirements – Common specifications, CENELEC, Brussels, 2001.

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Guidebook-2 Design of Bridges

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