CHAPTER 5: THE LOAD PATH AND LOAD DISTRIBUTION IN BRIDGE SUPERSTRUCTURES An efficient design of a bridge’s superstructure is essential to achieving overall economy in the whole bridge structure, in that the superstructure deadweight may form a significant portion of the total gravity load that the bridge must transmit to the foundation. What is bridge superstructure? It is an integrated body of various members of reinforced concrete, prestressed concrete, or steel in the form of slabs, stringers, floor beams, diaphragms, etc. Load Distribution is often used in a generic sense to denote superstructural analysis. Methods for their analysis, some demanding considerable mathematical theory, have been the focus of numerous studies and research efforts throughout the world in the past several decades and have been detailed in several books such as; Timoshenko and Woinsky-Krieger,1959; Rowe,1962 -AISC,1963; Bares and Massonet,1968; Bakht and Jaeger,1985; Hendry and Jaeger,1959; Balas and Hanuska,1964; ACI,1969,1977; Hambly,1991; O’Connor,1971; Rusch and Hergenroder,1961; Puncher,1964; Cusens and Pama,1975; Troitsky,1967 BRIDGE GEOMETRY Longitudinal is used to denote a direction parallel to traffic Transverse denotes a direction perpendicular to it. Normal (or right) bridge -
are those in which the longitudinal axis of the bridge, which is parallel to the longitudinal axes of the slab, and the supporting beams are normal to the centerlines of supports (abutments and or piers).
Skew bridge -
simple or continuous, is characterized by its longitudinal axis, which forms an acute angle, instead of a right angle, with the centerlines of the supports.
Angle of skew (or skew angle) -
is defined as the angle between the centreline of the supports and the normal to the axis of the bridge.
The skew angle can have a considerable effect on the shear and bending moments in the girders. It has been suggested that, for skew angles not exceeding 20o (30o for slab-on-beam bridges). For the larger skew angles, the torsional moments, which are not calculated directly in simplified analyses, are large and would invalidate the results from the simplified analyses( Kostem,1984; Bakht and Jaeger,1985; Hambly,1991). Curved bridge –
also referred to as horizontally curved bridges, have become almost standard features of highway interchanges and urban expressways in recent times.
–
comprised a series of straight girders used as chords in forming a curved alignment.
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Are characterized by their out- of-straightness alignment, as viewed in the plan.
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Is result from several factors, such as the design requirements for interchanges, the need for smooth dissemination of congested traffic, right-ofway limitations, local topography and foundation conditions, and aesthetics.
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Generally, curved bridges can be analysed by the grillage method, In which the curved members are idealized as curved strings of straight members, or by the space frame model and by computer methods.
Diaphragms -
are short structural members positioned transversely to and between adjacent stringers at various intervals and abutments.
-
usually consist of channels, W shapes, cross frames, or solid vertical slabs (In the case of concrete beams)
-
the purpose of providing diaphragms is to ensure lateral distribution of live loads to various adjacent stringers, which depends on both the stiffness of the diaphragms relative to the connected stringers and the method of connectivity.
Load Path (Load-transfer Mechanism) The Load Path and Load Distribution in Bridge Superstructure A path that seismic forces pass through to the foundation of the structure and, ultimately, to the soil.
Load Path Concepts Materials chosen with sufficient strength Connections made with sufficient size, strength, and number Any weak link could be a failure point Load path requires appropriate soil bearing capacity Soil must be able to maintain bearing capacity A bridge deck is the medium through which all bridge loads are transferred to other components.
Bridge Deck A bridge deck is the medium through which all bridge loads are transferred to other components. The most commonly used bridge deck can be classified as: Slab decks Beam-and-slab decks Beam decks Cellular decks Grid decks
Slab Decks The slab deck is the most commonly used type of deck for short-span bridges. The load carrying mechanism of a slab is analogous to that of plate, which is characterized by its ability to transfer bending and twisting in its own plane owing to continuity in all directions. Application of a load on a portion on a slab causes it to deflect locally in a “dish” causing a two-dimensional system of bending and twisting moments through this mechanism the load is transferred to the adjacent parts of the deck, which are less severely loaded. Usually, slabs are poured in place. For analytical purposes, they are said to be isotropic if they have similar stiffnesses in both the longitudinal and transverse direction, and orthotropic otherwise. Slab decks are not economical for a spans exceeding 50ft or more. As a twodimensional
solution,
the
governing
equation for the lateral deflection of a loaded plate, developed by S. D. Poisson
(1781-1840), with boundary conditions modified by G. R. Kirchhoff (1824-1887) [Timoshenko, 1953], is
Where:
w = lateral deflection of the plate q = intensity of load D = flexural rigidity of the plate
Grillage Method for analytical purposes, the real continuous plated structure (slab) is idealized by a series of discrete, orthogonally intersecting beams. Beam-and-Slab Decks Comprise a number of usually equally spaced (generally 6 - 12 ft. apart) beams spanning longitudinally between supports with a thin, structurally continuous slab spanning transversely across the top. The design is referred to as spaced beam-and-slab decks, or parallel girder systems. The slab can be noncomposite or composite. Concrete spine beams are more closely spaced than steel spine beams. In a two-spine steel beam deck, the spacing of beams can be more than 40-ft, whereas the spacing of solid concrete spines is generally in the 24-ft range. A beam with an infinitely wide flange is subjected to flexure, shear strains in the wide flange cause
nonlinear stress distributions of bending stresses, a phenomenon known as shear lag. Shear Lag When a beam with an infinitely wide flange is subjected to flexure, shear strains in the wide flange cause nonlinear stress distributions of bending stresses. Beam Deck A bridge deck can be assumed to behave as a beam when its length-to-width ratio is such that, under loads, its cross sections displace bodily but without any distortion.
Cellular Decks Also referred to as a box beam bridges Cellular decks have the configuration of a closed cross section comprising a number of thin slabs and thick or thin webs. Their large bending and torsional stiffness, owing to the deep closed-sectional configuration, make them efficient and the preferred type of deck for spans over 100-ft, for reasons of economy Usually multicellular structures whose responses depend on the provision and spacing of stiffening diaphragms. Closely spaced diaphragms prevent cross-sectional distortion, in which case the deck is analyzed as a beam if it is narrow or as a slab if it is wide.
Grid Decks A grid deck consists of a slab supported by a grid of two or more
longitudinal beams and transverse beams or diaphragms. In essence, it resembles a T-beam deck with several closely spaced diaphragms. Bending of a long longitudinal Beam under loads also leads to Bending and
twisting of the
Transverse beams, a phenomenon That causes load distribution in various members of the deck. Methods of analysis of bridge decks can be classified as;
Classical Methods. Such as grillage method, folded plate method, & the equivalent orthotropic plate method.
Computer Methods. Led to the development of a number of computer programs in the field of structural analysis such as SAP (System Analysis and Program Development), STRUDL (Structural Design Language), FINITE, GENDEK, CELL-4, LANELL, CURVBRG, POWELL, SALOD, and MUPDI, MERLIN DASH (for design and analysis of straight highway bridge systems), DESCUS (for curved bridges), PA; MDX’s AASTHO composite steel girder design program, Line Girder System, Bridge Design System, AASTHO BRADD-2 (a bridge automated design and drafting system), CBRIDGE a three-dimensional analysis program for straight or curved bridges, CRVBRG-C, STRESS, and CUGAR2. The Concrete Reinforcing Steel Institute (CRSI) has developed programs such as SLABBRDG for concrete decks
Simplified Methods. The OHBDC (Ontario Highway Bridge Design Code) and the method of distribution coefficients (or the AASHTO method) such as the one specified by AASHTO [1992, 1994a,b]. These methods do not offer very efficient solutions, particularly for longhand calculations, and therefore they are not very appealing to design engineers.
AASHTO METHOD OF LIVE LOAD DISTRIBUTION- SLABS AND BEAMS Support conditions for slab Two cases are considered, based on the direction of span of the slab: 1. The deck consists only of a reinforced concrete slab supported on abutments and/or peirs. The main reinforcement in this case runs parallel to traffic. 2. The deck slab is supported over a number of parallel steel, concrete, or timber beams. The main reinforcements in this case are oriented perpendicular to traffic (and beams). Two key design forces to be determined are: (1) Moment (2) Shear Empirical formulas for calculating slab moments: Based on assumed position of wheel load, One ft from the curb, or one ft from the rail if a sidewalk or curb is not provided.
Determination of moments and shear in slabs CASE 1. Slab supported on an abutment and/or piers AASHTO 3.24.3.2 ("Case B-Main Reinforcement Parallel to Traffic”) Moment Computation The span (S) is defined as the distance between the centers of the supports, but S need not exceed the length of the clear span plus the thickness of the slab.(AASHTO 3.24.1.1) Support conditions for a concrete slab: a. Reinforced concrete slab bridge b. Reinforced concrete T-beam bridge c. Reinforced concrete slab supported on steel stringers d. Reinforced concrete slab supported on prestressed concrete girders
Design moment for simple spans, Dead-load moment = where w= dead load/ft2 of slab and L= span (S, as previously defined) Live-load moment (LLM) Two methods in determining slab moments: (AASHTO 3.24.3.2)
Method 1: Moment is calculated as if it were caused by concentrated load acting on a simple span. Method 2: Alternatively, for HS20 loading, the maximum live-load moment (LLM) per foot width of slab Empirical formulas: •
For spans up to 50 ft, LLM = 900S ft-lb
•
For spans > 50 to 100 ft, LLM = 1000 (1.3S – 20.0) ft-lb
Stiffened section at the edges of the slab Essentially acts as an L-beam whose neutral axis is higher than that of the slab. Bakht and Jaeger (1985) & Hambly (1991) For continuous spans, the simple-span moment can be reduced by 20 percent, both for positive and negative moments. CASE 2: Slab Supported on beams and stringers AASHTO 3.24.3.1, “Case A-Main Reinforcement
Perpendicular to Traffic”)
SPAN: The slab is usually continuous over several parallel stringers or beams, which can be reinforced concrete (T-beams), prestressed concrete, steel, or timber. AASHTO 3.24.1.2 1. When the supporting beams are made from reinforced concrete, the slab is usually cast monolithic with them, resulting in a T-beam section. In such cases, S equals the clear span.
2. When the slab is supported on steel stringers, S equals the clear span plus half the width of the stringer flange. In a case where the widths of the top and the bottom flanges are different (in a composite plate girder), the top flange width should be considered in computing S. Design moment Dead load moment = MD =
or
where w= dead load/ft2 of slab and L= effective span (S,)
Live-load moment (LLM) LLM: LLM =
ft-lb
Where P1= P20 =16,000 lb (load on one rear wheel of an HS20 truck) = P15 =12,000 lb (load on one rear wheel of an HS15 truck)
DISTRIBUTION OF LIVE LOAD IN FLOOR BEAMS The floor beams considered here are those bending members that are positioned perpendicular to the direction of traffic. Live load can be transmitted to floor beams in one of the two ways, depending on the support conditions of the deck slab: 1. It can be supported over stringers which may, in turn, be supported by the floor beams 2. It may be supported directly by the floor beams.
Loads on the floor beams in the two cases are calculated by different methods. Generally speaking, two schemes are used for supporting stringers: 1. Supported directly on abutment and/or piers 2. Supported on traversely positioned floor beams. Latter practice is used for steel stringers (W shapes) used for truss bridges, girder bridges, and other types of medium- span bridges. LOAD ON FLOOR BEAMS TRANSMITTED BY STRINGERS Dead Load is transmitted to floor beams in the form of concentrated loads (reactions from stringers). Dead-Load moment can be determined by treating the dead load due to slab, stringer, and floor beam as uniformly distributed load over the floor beam.
LIVE LOAD TWO-STEP METHOD Maximum stringer reactions are calculated by the appropriate positioning of the moving load in the longitudinal direction (the direction of traffic or along the longitudinal axis of stringer). To obtain the maximum moment in the floor beam, careful consideration should be given to the longitudinal position of the wheel loads of the truck, as well as to the lateral positions of the axles of different trucks.
Positions of truck loads in a two-lane bridge for the maximum live-load moment: a) position of HS20 truck for maximum stringer reactions on the floor beam. b) position of loads for maximum live load moment in the floor beam
POSITIONS OF TRUCK LOADS ON A TWO-LANE HIGHWAY FOR MAXIMUM FLOOR BEAM MOMENTS
LOAD ON FLOOR BEAMS TRANSMITTED DIRECTLY BY SLAB
Open and filled steel grid decks are often used for medium and long-span bridges Two main advantages: 1. They are lighter than the conventional reinforced concrete decks. 2. The open grid minimizes wind loads on the deck. Steel Grid Decks -
they are often used in bridge rehabilitation projects that warrant replacement of old and deteriorated reinforced concrete decks.
-
reduce the dead load on the bridge, resulting in the increased live-load capacity of the superstructure.
-
they are also used for decks of long-span bridges to reduce the effects of wind loads.
Open Steel Grid Decks -
consists of perpendicularly placed and interlocking steel ribs and are available in the form of panels of standard sizes which can be bolted and welded to stringers.
Deck Slab Overhangs It is common for the single-cell or multicell box grinder bridges and spine-beam
bridges
(steel
or
concrete) to have cantilevered decks, with their longitudinal free edges usually stiffed with monolithically cast cuebs or parapets
Distribution of Live Load in Parallel Stringers (Parallel to Traffic) Interior stringers, general case: Bending Moment
In calculating bending moments in longitudinal beams or stringers, no longitudinal distribution of wheel loads shall be assumed. Stringers are generally equally spaced.
Interior stringers, general case: Shear End shears
and
reactions in
transverse
Floor beams and stringers shall be calculated Assuming that the wheel or axle load adjacent to the end being analyzed is not distributed longitudinally.
Wheel loads away from ends of member
Special cases: Box Girder Bridges: Prestressed Concrete Spread Box Girders
Special cases: Box Girder Bridges: Prestressed Concrete Adjacent Box Beams
The value of St. Venant’s torsional constant, J, for various cross section may be computed from the following expressions;
For box beam, the value of J may be computed with references to the following figure:
Steel Box Girders
Exterior stringers: Steel, Concrete, Timber beams and T-beams •
The live-load bending moment in the outside (or the exterior) roadway stringers is determined by applying to them the reactions of the wheel loads obtained by assuming the deck is a simple span between the exterior and the adjacent interior stringers, this method is referred to as the lever rule.
Limitations on the Applicability of AASHTO Load: Distribution Factors The method of distribution factors is not applicable to bridges that •
have deck slabs of nonuniform width
•
have slabs with a few isolated supports
•
have slabs with a skew angle exceeding 20o
•
have curved decks
•
have nonuniform cross sections or nonuniform sections in the middle half of the span
•
are slab-on-beam bridges with three or fewer beams
•
have cellular girders with fewer than three cells
Three separate requirements are stipulated for determining the size of the exterior stringer; the largest size determined is the final choice: 1. If the exterior stringer has to carry both sidewalk and traffic live load(and all dead loads), its size can be determined by allowing 25 percent overstress based on service load (or allowable stress) design. 2. The size can be determined ignoring the sidewalk (and hence no sidewalk dead and live loads). 3. The size of the exterior stringer must be at least as large as the interior stringer. Forces in Stringers Stringers are the longitudinal beams supported on abutments and piers. They receive their loads directly from the slab which they support. The two outside stringers on each side of the bridge deck are called the exterior stringers: the remaining stringers are the interior stringers. Dead-load Forces Dead load is distributed to various stringers in proportion to their tributary widths. In most cases, the stringers are spaced equally, resulting in equal tributary widths for all interior stringers. Consequently, all interior stringers are assumed to carry equal amounts of dead load. Essentially, the dead load consists of the deck slab, the wearing surface, sidewalks, curbs, parapets, and railings. Recent studies (Bishara and Soegiarso,1993) have questioned the validity of this assumption by suggesting that. 1. For the right bridges, 80% of the sidewalk and parapet loads are taken by the exterior beams, and 20% by the interior beams; 2. The asphalt wearing surface load is distributed to each beam in the ratio of its moment of inertia to the total moment of inertia of all the beams. In such cases, the load on the exterior stringers can be calculated as a reaction by considering the slab (with an overhang) as simply supported on both the exterior and the first interior stringer. Live-load Forces The problem of determining shears and moments in stringers due to live lads received from the slab is highly indeterminate owing to the fact that moving loads, generally speaking, can literally occupy any position on the deck; the only exception to this is the portion of the slab covered by the sidewalk, curb, and parapet.
The conclusion is that the live load shared by the exterior stringer would be different from that shared by the interior stringer. This is evident because the exterior stringers support the portion of the slab covered by the sidewalk, curb, and railing, thereby precluding the possibility of the live load occupying this portion of the slab. Owing to the presence of the slab, he loads are not applied directly to the stringers; rather, the stiffness of the slab causes lateral distribution of the moving loads to the adjacent stringers. The method to determine the lateral distribution of moving loads to both exterior and interior stringers. Is the distribution of loads – although simple, it does have some limitations. According to this method of analysis, shear and moments in stringers are obtained first as if they are directly loaded by the axle (or wheel) loads. These values are then multiplied by the appropriate live-load distribution factors, DF and by the impact factors (1+ I) It is incumbent on a designer to apply bridge live loads on the deck in specific manners outlined in the AASHTO specifications. The following points must be noted in this context: 1. In the case of single-span bridges, regardless of the span length of the bridge, only one H or HS truck is assumed to occupy the bridge. The distance between the two rear axles is kept as a variable between 14 and 30ft,as shown Fig. 4.15; this distance should be chosen so as to cause maximum stresses in the supporting members. 2. In the case of lane load, the entire span is assumed to be occupied by uniform lane loading in the designated lane. The line due to concentrated loads is to be positioned, along with the uniform live load, to cause the most critical stress conditions in the member under consideration. 3. In the case of continuous span, they should be loaded so as to cause maximum effects (stresses and deflections) in the member under consideration. Limitations on the Applicability of AASHTO Load: Distribution Factors The method of distribution factors is not applicable to bridges that ( Bakht and Jaeger, 1985, Hambly, 1991) -have the deck slabs of nonuniform width -have slabs with a few isolated supports -have slabs with a skew angle exceeding 20o -have curved decks
-have nonuniform cross sections or nonuniform sections in the middle half of the span. - are slab-on-beam bridges with three or fewer beams -have cellular girders with fewer than three cells The distribution factors are applicable for: 1. Specific truck types- the AASHTO H and HS trucks 2. Single and multilane loading 3. The same type of AASHTO trucks in all lanes; not applicable if one permit truck in one traffic lane combined with an AASHTO truck in another lane 4. Bridges of common types and dimensions-where the width of the deck and the superstructure cross section are constant, and where beams are parallel and have approximately the same stiffness 5. Unless otherwise specified, bridges where the number of beams is not less than four 6. Unless otherwise specified, cases where the roadway part of the overhang does not exceed 3.0ft 7. Bridges where the curvature in the plan is less than the specified limits: a. Segments of horizontally curved superstructure with torsionally stiff closed sections whose central angle subtended by a curved span is less than 12o b. For open sections, the radius is such that the central angle subtended by each span is less than the value given Table 4.2.
