Behaviour of concrete box girder bridges of deformable cross-section
Proc. Instn Ciu. Engrs Sfructs & Bldgs, 1993,99,
B. Kermani, BSc, PhD, MIStructE,and P,Waldron, BSc, PhD, DIC, MICE, MIStructE
Structural Board Paper 9940
H Distortion of the cross-section, w h i c h commonly occurs in thin-walled box girder bridges that contain f e w interm e d i a t e diaphragms, may modify the distribution o f stress throughout the s t r u c t u r e to a significant degree. A is presented which method of analysis allowsfor the effectsof cross-sectional distortion in addition to those o f torsional warping. The a p p r o a c h is a rational extension of thin-walled beam t h e o r y and m a y be applied to continuous or simplys u p p o r t e d box g i r d e r s o f either straight or curved configurationin plan. A s f e w test d a t a o f continuous girders w e r e available against which the accuracy Of the method could be verified, two models-one straight* the Other curved in fabricated for this p u ~ o s e The tion and t e s t i n g of these models under a variety Of 'Oncentrated and distributed torsional loads are briefly described. from these tests show general agreement w i t h those f r o m the proposed analyticalmethod.
Notation cross-sectionalarea section breadth torsional bimoment distortional bimoment modulus of elasticity torsional warpingfunction shear flow ~~~~.~ shear modulus of elasticity section height flexural second moment of area torsional warpingconstant distortional warping constant St Venant torsional constant distortional frame stiffness elastic foundation stiffness transverse distortional moment per unit length uniformly distributed distortional moment per unit length distortional moment bending moment radius of curvature torsional sectorial shear function distortional sectorial shear function uniformly distributed torsional moment per unit length torsional moment warping torsion vertical displacement shear force distortional warping displacement ~~~
W,
ya
6 9
B p ed
e, T~
4 & & f~
May, 109-122
torsionalwarpingdisplacement distortional angle wall thickness sectional constant bending rotation torsionalwarping shear parameter distortional warpingdirect stress torsional warping direct stress distortional warping shear stress torsional warping shear torsionalrotation torsionalwarpingsectorialcoordinate distortionalwarpingsectorialcoordinate twice the enclosedcell area
~
~
~
Written discussion closes 15July I993
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Box girders have evolved into a highly efficient and aesthetically pleasing solution for medium span and long span bridges. As span length increases into the range where dead load domi. nates, saving in self-weight becomes important. It is here that the efficient use of the box section, which possesses considerable flexural and torsional stiffness, permits a reduction in overall section size witha consequential saving in weight. 2. One type of configuration which merits particular attention is the curved box girder bridge which feature essential is an of most modern highway interchanges and urban motorways. A characteristic featureof curved girders isrelatively the high level of torsion to whichsubjected they arecontinthe owing to uous interaction between flexural and torsional B. Kermani, moments along the entire man. Box girders withtheirhightorsionalstiffnessare ideally Bridge Engineer* Acer Consultants suited for these applications where signficant levels of torsion mav beinduced even bv the self-weight of the structurealone. 3. For the purposes of analysis, box girders may be divided into two groups according to their behaviour under torsionalloads. The first group contains girders assumed to have a rigid crosscross-section which do not change sectional shapewhen rotated about their longitudinal axis (Fig. l(a)). This is the case for concrete box girders with relatively thick walls, in which transverse frame action may be sufficient to maintain the original cross-section. In P. Waldron, Professor of other cases, closely spaced diaphragmsor cross-bracing may be required to satisfy this Structural Engineering, condition. The second group comprises Department ofCiuil deformable girders which possess sufficient and Structural transverse flexibility to undergo distortion Engineering. of the cross-section under torsional loading Uniuersify of (Fig. l(b)). Sheffield
-
d
KERMANI ANDWALDRON
t
t Fig. 1. Cross-sectional deformation of typical boxgirder subjected to torsion: (a)-rigid cross-section; ( b )deformable cross-section
Fig. 2. Torsional warping displacements occurring in a typical box section: (a) unrestrained at both ends; (b) fully restrained at one end only
l
I
/
.
Structural response of thin-walled box girders wall elements are accompanied by a degree of shear deformation to ensure continuityof axial 4. The behaviour ofthin-walled beams under flexural loading is essentially the same displacement around the closed perimeter, as for solid and thick-walled sections. It is resulting in a system of circulatory shear usually satisfactory, by assuming that plane stresses in accordance with St Venant theory. sections remain plane after loading, to use The shearflow generated is of constant magnisimple beam theory to predict the level of flextude and, in box girders with side cantilevers, ural stress around the cross-section and the dis- occurs only around the perimeterof the closed tribution of bending moment at anysection cell, as shown in Fig. 3(a). along the girder. Shear lag effectsmay become 7. If the torsional warping displacements important in certain thin-walled girders but are are now restrained, a s by some form of physical not considered further here. restraint as shown in Fig. 2(b), the individual wall elements are no longer free to rotate as 5. The response of thin-walled box beams to torsional loading, however, may differ consider- rigid bodies but are subjected to bending about their own major axes. This results in a system ably from that observed for solid or thickwalled sections, owing to the formation of more of axial direct stresses referred to as longitudisignificant out-of-plane warping deformations. nal torsional warping stresseswhich have the These axial deformations invalidate the usual distribution around the section shownin Fig. assumption of plane sections adoptedin simple 3(b). These are self-equilibrating and have no resultant component of either direct forceor theory and require special consideration.In general, two components of warping displacebending moment. ment arise from torsional loading: the first, 8. A complementary system of shear flows referred to as torsional warping, occursin is also created which extends over the entire response to the applicationof pure St Venant's cross-section and constitutes a system in equitorsion to the undeformed section; the second librium (Fig. 3(c)). In thin-walled beams of arises from distortion of the cross-section itself. certain proportions, the stresses due to restraining torsional warping may become 6. Consider first a box beam of nondeformable cross-section under the application significant and should be fully considered in of an equal but opposite torque at both ends the analysis. (Fig. 2(a)). Under this loading, the distribution 9. In practice, the number of diaphragms of pure torsional warping displacement around along the span are kept to the minimum since the cross-section is identical at all positions they induce additional dead load and cause disalong the beam. Rigid-body rotations of the ruption and delays in the casting cycle. If the intermediate diaphragms arefew in number, the cross-section of thin-walled girders may distort, resulting in the transverse bendingof the walls shown in Fig. l(b). Apartfrom the transverse flexural deformations, longitudinal out-of-plane displacements are also induced, referred to as distortional warpingtud. These axial displacements occur in addition to the torsional warping displacementsW , shown in Fig. 2. The nature of the loading and beam geometry is such that the distributionof distortional warping varies continuously along the span, causing additional direct and shear stresses to be developed. The distortional direct stresses, with the typical cross-sectional distribution shown in Fig. 4(a), are self-equilibrating and do +lWt not interact with the other direct stress resultants, However, they may modify significantly
CONCRETE BOXGIRDER BRIDGES OF DEFORMABLE CROSS-SECTION
the state of stress due to primary bending, pure torsion and torsional warping. The complementary distortional shear flows shownin Fig. 4(b) are also self-equilibrating and provideno resistance to the applied torsionalmoment. 10. As a direct result of cross-sectional distortion, transverse bending moments are also produced by frame action around the box. These have the distribution shownin Fig. 4(c), and the resulting stresses canbe of the same order as the longitudinal stresses associated with bending and torsional and distortional warping. In such cases, the longitudinal and transverse direct stresses cancombine such that Poisson's ratio effectsmay be significant and should not be neglected in design.
already used in simple beam theory. Vlasov's original work has since been reformulated and generalized by Dabrowski3 for curved girders subjected to non-uniform torsion, and by Kollbrunner and Basler5 andHeilig6 for multicell boxes with arbitrary cross-sections. 13. In the case of non-uniform torsion in closed box sections, which occurs either as a result of variations in the level of torsion along the span or as a resultof some form of physical restraint, the longitudinal distributionof torsional warping displacementW , may be expressed in terms of a dimensionless warping function f such that W,=
-f'&
Fig. 3. Typical cross-sectional distributions of: ( a )pure St Venant's shear flow Fsv; (b)axial stress a, due to the restraint of warping; (c)corresponding shear flow F,
(1)
The term (5 is called the torsional warping sectorial coordinate (units:L') and represents the 11. A generalized theory for the analysis of level of out-of-plane warping due to a unit rate thin-walled beams havebeen developed by of twist. Subsequently, the longitudinal disVlasov' and extended to curved girders by Dab- tribution of the stress resultantsB and T, r ~ w s k iThis . ~ approach enables the distribution along the beam are obtained from of stress at anypoint in the girder tobe deterB = -EI,f" (2) mined from the familiar expression usedin simple beam theory, but extended to account and for the effects of torsional warping and distorT , = B' = -EI+f"' (3) tion of the cross-section. For thin-walled behaviour to be assumed with the contextof Vlasov's where I , is a torsional warping constant (units: theory, the section mustbe sufficiently thin for L6) defined as variations in stress across the thicknessof the walls to be neglected. This condition, which is I, = (5'dA met by the large majority of concrete box girder bridges found in practice4 and virtually all 14. The direct stress and Shear flow steel and composite sections, permits the levels resulting from restraint to torsional warping at of direct and shear stresses on the median line any point on the medial lineof the section may of the cross-section tobe used throughout the then be expressed a s Fig. 4 . Typical analysis. cross-sectional distributions of: Torsional warping ( a )distortional axial 12. For thin-walled, thick-walled and solid warping stress a,; members to be accommodated by the same ( b )corresponding general beam theory, Vlasov' introduced new distortional shear cross-sectional functions called torsional secflow Fd; (c) transverse in which S, (units: L4) is a further sectorial torial properties. New types of stress resultant distortional moment function used to describe the distributionof were also created, denoted torsionalbimoment B and warping torsion T,, to supplement those shear flow due to torsional warping. The exten- per unit length mdb
Thin-walled beam theory
I
KERMANIANDWALDRON rigidity against differential bendingof the top and bottom slabs of the box provides a continuous elastic support for each half of the section. Subsequently, the theory hasbeen developed and generalized by others-notably Dabrowski,3 Wrightet al.,' Steinle' and Zhang"-for box girder bridges of different geometrical configurations. 18. In developing the theory of distortion, it is assumed that in-plane displacementof the cross-section is accompanied by sufficient which is equally valid for straight and curved warping to annul the shear strain. The approximate method based on the beam on elastic thin-walled beams. However, in the case of foundation analogy enables use tobe made of straight beams, the flexural and torsional the expressions derived previously and applied effects are uncoupled since the torsional component of the load due to longitudinal bending, commonly in the design of such beams." 19. Consider the typical single cell box given by the term(MJr),is zero. In equation beam of trapezoidal cross-section shown in Fig. (7), J is the familiar secondmoment of area for 5, loaded with external twisting momentst, and pure torsion, and p, called the torsional t , per unit length. These loadsmay be divided warping shear parameter, is a measure of the into two components: pure torsional loads; variation in shear stiffness around the closed section deforming loads. At any section, the section. The derivation of the various sectional and sectorial properties introduced here may be uniformly distributed torque t and the uniformly distributed distortionalmoment md may found in detail elsewhere7 for the full range of then be defined'* in terms of the horizontal and thin-walled sections used in bridge construcvertical components of twisting moment as tion.
sion of simple beam theory to include torsional warping effects is then made possible by simply adding the terms defined in equations (5) and (6) to the usual equations for direct stress and shearflow. 15. The fundamental partial differential equation governing torsional warpingin thinwalled closed beams may be expressed in terms of the warping functionf as
Fig. 5. Force components in wall elements due to (a) vertical, and ( b )horizontal antisymmetrical loading
Distortion 16. The theory of torsion described above is based on the assumption that the beam crosssection preserves its original shape under torsional loading. This isnot the usual behaviour of box girder bridges found in practice where the cross-section may deform, resulting in additional distortional effects such as distortional warping and transverse bending. 17. Distortion of thin-walled beams has been considered previously by a numberof investigators. Vlasov2 was the first to draw an analogy between the response of a box beam subjected to distortional loading and thatof a beam on an elastic foundation. The out-of-plane
I
112
btt, l fl
t
= t,
+ t,
(84
and
m
-
1{h t, - th} b,
d-2
20. In box girder bridges subjected to gravity loads, the vertical componentof twisting moment t, usually predominates. In such cases, the effect of sloping webs on the distortional behaviour is clear from equation (8),i.e. the component of distortional moment reduces with increased slope a s a result of a corresponding increase in frame stiffness around the box.
CONCRETE BOX GIRDER BRIDGES OF
21. The level of cross-sectional deformation which occurs as a result of the distortional forces acting on the box may be expressed in terms of a distortional angle y d . This isdefined a s the change in the angle between the top flange and the inclined side web, on the assumption that they are straight. The other form of distortional displacement is the distortional warping w d which occurs in the longitudinal direction and may be expressed in terms of the distortional angle yd a s
expression
in which the alternative sign relates to the transverse bending moments at the top and bottom of the webs, and the term is a simple geometrical constant for the section. The distortional frame stiffnessK , may be evaluated by consideration of a unit lengthof box beam loaded by diagonal forces witha unit horizontal w d = -7;h (9) component. For the purposesof this calculation, the frame may be considered to be in which ii, is called the distortional coordinate restrained at thelower corner points in both the (units: L'). vertical and horizontal directions. By using the 22. Since distortional warping displaceflexibility method, it is then possible to determents are rarely constant along the beam, addi-mine Kdfor single cell cross-sections. tional systems of distortional direct and shear 24. The differential equation governing the stresses are created. Corresponding stress distortional behaviour of a single cell box beam D resultants known as distortional bimoment may be expressed as and distortional moment M d are formed, which EZ, 7: + K , Yd = ( m d M J 2 Y ) (16) may be defined in terms of the distortional angle thus This is the general expression for girders
+
D
= -EI,y:
in which
ZiG = j+lil is called the distortional warping constant (units: L6).The distortional warping direct stress b d and shear flow Fdmay then be expressed in the following form
(10)
curved circularly in plan, but it is equally valid for straight beamsfor which the final term disappears asY tends to infinity. It can be seen'' that this is directly analogous to the differential equation governing displacementy in a beam with bending stiffness EI, supported on an elastic foundationof stiffness K , and under applied loadingp , given by
EIxy"
+ K,y
=p
(17)
25. For simple structural arrangements and loading conditions, closed form solutions exist for equations (7) and (16) that describe the response of box girder bridges to torsional warping and distortion.I3 These are suitable for hand calculation. However, for the global analysis of thin-walled structures incorporating features such as continuity, curvature, variations in cross-section and complex systems of loading and restraint, computer-based methods are usually necessary.One such approach, the equivalent beam method, is described later.
in which the term S, (units: L4) is called the distortional sectorial shear function. There is a clear similarity between the section properties and functions which havebeen introduced here for distortion and the corresponding expressions derived previously for the theoryof Experimental investigation warping torsion. 26. Only a very limited number of experi23. Full or partial restraint of the axial mental investigations of box beams of deformable cross-section have been reported.I4-l6 warping displacements arising from crossTherefore, in order to widen the study of box sectional distortion are an effective means of girder bridges to include alternative geometrireducing the level of distortion that would cal loading and restraint conditions, further otherwise have occurred. Frame action detailed experimental investigation was necesresulting in the creation of transverse bending moments around the box provides another form sary, resulting in the construction, instrumentation and testing of two model box beams. The of resistance to distortion of the cross-section. The resistance is caused by the transverse flex- models, one straight and the other curved in plan, were both continuousover two spans and ural stiffness of the walls forming the closed of similar proportions. The principal dimencell of the box when the section is subjected to sions of the models were selected from a feature distortional loading. The resulting transverse survey by Swann4 tobe representative of actual distortional bending moment per unit lengthof concrete box girder bridges. The nominal the box can be obtained from the following
113
KERMANIANDWALDRON
Fig. 6. ( a ) Typical cross-section of both models, together with plan view of: (b) straight model; (c)circularly curved model
dimensions and general arrangementsof the models are shown in Fig.6. 27. A number of materials may be used for modelling concrete girders within the elastic range. For this investigation, it was decided to use a sand and epoxy resin mixture,as some experience had already been gained by others in its use.15.16 The epoxy resin employed provided a useful pot life of 20 hours and a curing time of seven days after mixing room at temperature. As the models were tobe instrumented on the inside as well as on the outside surfaces, they were designed to be cast as two matching channel sections, each comprising a flange and portionsof the webs and diaphragms. The joint between the two channel sections was chosen tobe 20 mm above middepth of the web so as to avoid the point at which the shear stresses were likely be to at a maximum (Fig. 6(a)). 28. One tensile test specimen was cast from each of the eight mixesused in the construction of the models in order to obtain the elastic properties of the material. The average values
r
114
215
of Young’s modulus determined at the time of testing were 16.2 kN/mm2 and17.6 kN/mm* for the straight and curvedmodels respectively. Poisson’s ratio was found tobe 0.35 for both models.
Instrumentation and testing 29. Both models were instrumented for the measurement of strain, displacement and support reactions, using a totalof 376 electrical resistance strain gauge elementsof 6 mm gauge length, four displacements transducers and six load cells. Strain gauges were positioned at all five of the sections (A-E) along one spanof the straight model, as identified in Fig. 7(a). At each of these sections, four single element strain gaugeswere placed on the outside faces of the webs on the centre-line of the top and bottom flanges (Fig.7(b)). These were used to measure the extreme fibre stressesat the corners of the box section in the longitudinal direction. Two critical sections were then selected for full instrumentation, one near the midspan (section C), the other adjacent to the central support (section E). Rosette gauges were bonded to both the inside and the outside surfaces of the box walls in order to separate in-plane and flexural componentsof longitudinal, transverse and shear strain at of allthe gauge locations around the cross-section shown in Fig. 7(c). The curved model was instrumented in an identical fashion, with the same separation between the five instrumented sections measured along the curved centre-line. 30. Displacements were measured at a single section at anyone time with four linear variable displacement transducers to provide measurements of vertical displacement, torsional angle and distortional angle. The transducers were supported on an independent frame which could be positioned at anysection at which measurements were required. By repeating the load test several times with the transducer frame located at different sections, it was possible to generate a longitudinal profileof the various componentsof deformation along the girder. 31. Six load cells were manufactured for the measurement of support reactions during the tests. Each load cell consisted of a 1 mm thick cylindrical mild steel section instrumented with of four linear strain gauge elements. The bases the load cells were boltedat the appropriate positions on top of a support rigwhich itself consisted of three independent steel frames bolted to the laboratory floor. Loading system and testing procedure 32. Each model was subjected separately to a concentrated midspan loadof 518 N and a uniformly distributed lineload of 0.468 N/mm. In both cases, the load was applied eccentrically over one web within one span only.
CONCRETE BOXGIRDER BRIDGES OF DEFORMABLECROSS-SECT10 Although only two alternativeload systems were available, several differentload cases and structural configurations could be tested. The model could be either torsionally restrained at its centre, by using twoload cells, one under each web, or restrained against vertical displacement only, by placingone load cell centrally. By applying the load to a single span only at any one time, use could be made of the symmetry of the structure to estimate the effects of full loading on both spansby superimposing the results. 33. A data logger was used in conjunction with a microcomputer to read andrecord the results from the instrumentation. A totalof four readings was recorded for each channel from which the mean and standard deviation was found and displayed automatically. A facility was provided for the user to either accept or reject each setof readings and in this way it was possible to control the accuracy of the data recorded.
Presentation of experimental results 34. Results from the load cells, displacement transducers and the strain gauges were all tabulated and presented diagrammatically for each individual load case and restraintcondition.” From the corner strain gauges, extreme fibre stresseswere obtained at all five instrumented sections. From the results obtained from the rosette gauges on the inside and outside surfaces of the box walls at sections C and E, it was possible to estimate the shear stress and the in-plane and out-of-plane components of the axial and transverse direct stresses.
Least-squares analysis of results 35. While the four corner gauges at each section provided a unique estimateof the four stress resultants causing axial direct stress (axial force, bi-axial bending and thecombined torsional and distortional bimoment), the rosette gauges at thetwo fully instrumented sections provided redundant information.A least-squares analysis wasperformed on these gauge results tominimize experimental errors and to obtain animproved estimate of the various stress distributions and stress resultants at thesetwo sections. 36. From thin-walled beam theory, the distribution of the longitudinal stressa and the shear flow F at any pointon the median line of the cross-section maybe expressed as
B
A
C
D
5
l
“t
a
b
A
I
i
torsional and distortional warping terms defined in equations (5), (6),(13)and (14). In addition, the transverse bendingmoment per unit length due to distortionof the crosssection may be obtained from the measured transverse bending stressadbfrom the expression
Fig. 7. ( a )Location of instrumented sections in both models, showingpositions of: ( b ) single strain gauge elements at all five sections; (c) rosette strain gauges at sections C and E only
38. The theoretical distributions for the axial direct stress and transverse bending moment vary linearly around the cross-section between adjacent corner values. For shear flow it varies according to the sectional and sectorial functions given in equation (19). 39. Consider a best-fit line to the experimental points a , ( = i 1, 12) from the twelve strain gauges located at both sectionsC and E. If the values of this best-fit function are given by f, at the twelve strain gauge positions, the discrepancy between the experimental and best fit values at each experimental point is(ai - fi). The least-squares criteriamay then be expressed as 12
S
=
1 ( a ,-
fi)2
i= 1
in which S, the overall squared error, is tobe minimized. 40. In the case of the axial direct stress and the transverse bendingmoment per unit length, which both vary linearlybetween box corners, the value of the best fitline at thetwelve strain gauge pointsf i may be written in terms of the N M,y M,n B& D& g=-+-+-+-+(18) four corner values F j ( j = 1, 4). Minimizing S A I. I, I; I; with respect toF j yields four equations in terms of the four unknowncorner values. The best estimates of the corner values thus determined were then used to evaluate the valuesfi at each of the strain gauge positionsby linear 37. These expressions are the familiar equainterpolation, and to predict the value of the tions for simple beam theory modified by the
115
KERMANIANDWALDRON axial force N, the bi-axial bending moments M,, M,, and the totalbimoment (B + D) due to the torsional and distortional warping restraint effects combined. 41. Since a least squares treatment was not possible on the four gaugeslocated at the corners of each instrumented section, the results were used to establish four simultaneous equations from which unique values for the various stress resultantswere obtained at each of the five instrumented sections. 42. For the shear flows, a similar leastsquare approach was possible although the values of f i had now to be expressed in terms of the various sectional and sectorial coordinates given in equation (19) rather than the corner values. The overall squared error S was then minimized with respect to the various stress resultants directly. By solving the resulting simultaneous equations, least-squares estimates of the various stress resultantswere found which were then used to obtain the distribution of shear flow around the cross-section.
Equivalent beam method of analysis 43. The most appropriate methods of analysis for box girder bridges of complex configuration are thosein which the structure may be envisaged as an assemblage of structural members connected at discrete points. The most powerful and versatile tool for structural analysis is the finite-element (FE) method which can accommodate all the special features encountered in the box girders. However, while the FE method is applicable, in principle, to the analysis of any bridge typeof any configuration, its practical application isoften restricted by the large amountof input/output data and computertime required for solution. For reasons of economy, a need therefore exists for simpler methodsof analysis which consider all the important structural actions associated with thin-walled box girder bridges with sufficient accuracy, particularly during the preliminary analysis and conceptual design stages. 44. A method of elastic analysis has been developed which is suitable for rapid solution by computer." Based on the stiffness approach, it is equally applicable to both straight and curved girders of deformable cross-section. The approach is a continuationof the work by Waldron" on torsional warping but extended to account for the distortional effectsin single cell box girder bridges. In addition to the four degrees of freedom system developed previously, which considers rotation due to uniaxial bending (about the horizontal axis), vertical displacement, twist and torsional warping, two further degrees of freedom have been incorporated in the formulation to account for the distortionaleffects. The additional degrees of freedom are the distortional angleyd
116
and the rateof change of distortional angle y: along the lengthof the girder. 45. The stiffness method adopted makes use of discrete beam elements and istherefore applicable to complex structural systems. The principal requirement of the method is to find a relationship between the end forces and end displacements for each individual beam member. This may be expressed in the usual form as
{PI = [ K I W
(22)
where the load vector { p }contains the sixcomponents of load at each end, thus {p}T =
{MxTVBDM,}
(23)
and the corresponding componentsof end displacement are given by
P I T= {e4?mYd)
(24)
46. The member stiffness matrix [ K ] is a symmetrical 12 X 12 matrix and is determined initially for each element in local member coordinates. Assembly of the stiffness matrix and load vector for the entire structure follows in the usual way after transformation into afixed global coordinate system. Theoverall structural stiffness matrixwill be singular in this form as no account has yetbeen taken of the various restraint conditions, and an infinitenumber of rigid body displacements are possible without violation of the general force/displacement expressions. Accordingly, the appropriate rows and columns of the assembled system matrix and vectors mustbe modified or removed to take account of the various restraint conditions before the reduced stiffness matrix canbe inverted to yield a solution for the end displacements associated with each load case. 47. The stiffness method has been incorporated into a Fortrancomputer program" to analyse singlecell box girder bridges of deformable cross-section. The program permits two types of load to be applied to the structure: concentrated loads at the nodes;uniformly distributed loads between nodes. The fixed-end forces attributable to theuniformly distributed shear, torsional and distortional loads are evaluated, and after transformation into system coordinates areapplied as nodal forces. Unlike the FE method, the equivalent beam method treats the longitudinal distributionof the various stress resultant as continuous functions and not as a series of discontinuous values at the nodal points.Hence the accuracy of the solution does not depend on the degreeof refinement adopted in the initial idealization of the structure. By adopting combinations of straight or curved beam elements, all normal bridge configurations can be analysed accurately with a minimum number of elements. As relatively few beam elements are required, a facility has been included in the program which
CONCRETE BOX GIRDER BRIDGES OF DEFORMABLECROSS-SECT10 calculates the various stress resultant at any number of intermediate points between nodes.
Analysis of model beams 48. Taking full advantage of the efficiency of the proposed method, the two model box girders described previously were discretized with a total of only four elements each, as shown in Fig. 8. For the simple singlecell rectangular box section without side cantilevers selected for the models, the various sectorial functions for torsional warping and distortional warping are identical. Thesemay be calculated by hand or bycomputer’ and are shownin Fig. 9 for themodel section. 49. When the sectional and sectorial properties of the box section and its geometry are known, the next stepin the analysis is to compile the stiffness matrix [ K ] for each member. On account of the complexity of some of the distortional termsin [ K ] , this is best achieved numerically, which is the approach adopted by the computer program described previously.” The various element stiffness matrices may thenbe assembled into a global stiffness matrix [K,] for the structure which, for both the straight and curved four-element models, has a symmetrical 30 X 30 form. However, for the type of line structure considered here, [ K,] is very sparselyfilled with all the non-zero elements banded about the leading diagonal. The rank and fileof [K,] is reduced further by five or six elements (depending on the nature of the central support) to take accountof the support restraints. 50. The inversion of [K,] follows which, for a problem of this size, is a trivial task for any modern desktop or personal computer. This yields a solution for the various nodal displacements which may then be inserted into equation (22) to provide valuesfor the unknown member forces { p }at each node. Once these nodal forces have been determined, a subroutine within the computer program then calculates intermediate values of the various stress resultants alongeach of the individual beam elements a s required.
Discussion of results 51. A comparison is presented here between the results obtained both experimentally and theoretically for each of the two models. The behaviour of the individual models has been studied with reference to both the global and cross-sectional distributions of the various stresses and stress resultants. 52. The experimental programme incorporated eight different tests on each model, comprising two different loading arrangements in each span independently for each of the two alternative central support configurations. Full details of the tests on both the straight and curved models are given el~ewhere.’~*” Some
representative results are presented here to Fig. 8. Four-element verify the accuracy of the proposed analytical idealization adopted method and to provide an insight into the struc-for: (a) straight tural behaviourof single cell box girder bridges model; ( 6 ) circularly of deformable cross-section. curved model
Straight model 53. In order to assess the quality of the data obtained from the load cells, the total applied load was compared with the sumof the reactions. The maximum error in this equilibrium check for the four test configurations was found to be 2.97%, with an average errorof 2.13%. The difference between the total reactions at eachof the three supports, determined experimentally and analytically, has alsobeen assessed. The maximum error, found to be 1.70% of the total appliedload with an average of value of 0.74%,indicated good agreement with the total theoretical reactions at each support. 54. A comparison of the longitudinal direct stresses obtained theoretically and experimentally at section E are shownin Fig. 10 forthe two alternative central support arrangements when the model was subjected to concentrated loading at midspan. Three different quantities are compared: the experimental results; the best estimate of the experimental resultsfrom the least-squares analysis; the theoretical values obtained from the equivalent beam method. 55. The comparison shows that the values obtained from the equivalent beam analysis are in close agreement with the experimental results, particularly for the support configuration allowing torsional rotationat the intermediate support. In this case (Fig. 10(a)), the values of stress at the four corners obtained from the least-squares analysisof the experimental results were all within 2.5% of those
Fig. 9. Cross-sectional distributions of (a) sectorial coordinates h, 15,and ( 6 ) sectorial shear functionsS+, S;, adopted f o r analysis of both models
KERMANIANDWALDRON
O
0.268 (A) 0.261 (B) \
J
0.296 (A) 0.302 (B)
0
f
m
I
t 4
H
0 U
-0.268 (A) -0.262 (B)
0-247 (A) 0.249 (B)
m
-0.247 (A) -0.224 (B)
Fig. 10. Comparison of longitudinal direct stresses (N/mm2) at section E of straight model with central support: (a) torsionally free; ( b ) torsionally restrained
118
-0.316 (A) -0.318 (B)
E<
-
(A) Analystical results (B) -------- Best-fit results 0
Experimental results
m
P
=
518 N
E d Test configuration
obtained by analysis. For the torsionally restrained model (Fig. 10(b)),these corner values agreed within 10%. However, perhaps a better measure of this agreement isa comparison of the various stress resultants obtained from analysis andfrom the best estimate of the experimental results. The average valuesof the bending moment obtained from experimental results were, for example,96.2% of those derived from the theory. In particular, for the two cases shown in Fig. 10, these ratios were 99.6% and 97.9% respectively. 56. For the concentrated load case, the effect of restraining torsion at the central support resulted in an increase of 7.2% in the ratio of warping stress to bending stress at the section corners (Fig.10(b)). Asimilar increase of 8.8% was observed for the case of uniformly distributed loading, not shownhere. However, for both load cases, distortion was found to be the main source of warping stress, and this can
form a significant addition to the crosssectional distribution of longitudinal bending stress. Maximum distortional warping stresses for the straightmodel were found tobe 35.6% and 24.7% of the bending stresses for the concentrated load and uniformly distributed load cases respectively. 57. The values of the transverse bending moments per unit length around sectionC of the model are plotted in Fig. 11, for the case when the model is subjected to a uniformly distributed line load placed eccentrically over one web. Positive bending moments are defined here as those which produce tension on the inner surface of the cross-section. The results show good agreement between the theoretical values and the least-squares estimates using the experimental values. Moreover, these results have verified that the distortional and torsional effects donot interact in straight box beams and that distortion canbe analysed in
CONCRETE BOXGIRDER BRIDGES OF DEFORMABLE CROSS-SECTTOF
+
*-
-6.413 (A)
6.413 (A)
-6,413 (A)
6,554 (B)
-8.033 (B)
t""' -6.413 (A) -7.142 (B)
(A) (B)
6.413 (A)
-6.413 (A)
7.157 (B)
-7.993 (B)
-Analysticalresults -------0
Best-fitresults Experimental results
UDL configuration
isolation. The calculated transverse bending moments per unit length caused by distortion of the cross-section have alsobeen shown tobe independent of the conditions of torsional restraint at the intermediate support. This is verified by the experimental results obtained for the two different support conditions represented in Fig. 11. 58. It is notoriously difficult to achieve a good agreement between experimental measurements and analytical predictionsof the magnitude and distributionof shear around box C and E sections. The shear flows at sections
=
0.468 Nimm
Test
Fig. 1 l . Comparison of transoerse bending moments per unit length (Nmmlmm) at section C of straight model with central support: ( a ) torsionally free; ( b ) torsionally restrained
are shownin Fig. 12 for one test configuration only as an exampleof the quality of the results obtained in this study. Loading consistsof the eccentric concentrated load applied within the instrumented span of the model which is torsionally restrained at the central support. Although the shear strains measured experimentally were very small in relation to the sensitivity of the instrumentation and data acquisition system, the stress resultants obtained from experimental and theoretical results are generallyin good agreement. The values of the vertical shearforce obtained from
119
KERMANIANDWALDRON
-0.484 -0.501
-0.484 -0.438
c' -0.579 (A)
-0.266 (B)
-0.579 (A) -0.335 (B) (b)
Fig. 12. Comparison of shear flows (Nirnm) in the straight model torsionally restrained at the intermediate support at: (a) section C; ( b )section E
(A)
- Analytical results
(B) --------
o
Best-fit results Experimental results
1 Test conflguration
the best-fit analysis are88.3% and 80.1% of the theoretical values derivedfrom the equivalent beam method for sectionsC and E respectively (Fig. 12). The corresponding valuesof total torque at these two sections are 95.5% and 96.6%.
Curved model 59. A comparison between the theoretical and experimental results for the curvedmodel shows that the equivalentbeam method has resulted in a general underestimationof axial
120
stress around the cross-section. The average value of the bending moment obtained from the analysis is 86.5% of that obtained from a leastsquares analysis of the experimental results. As an example, the resultsof the rosette strain gauges atsection C for the in-plane component of the axial direct stress is presented in Fig. 13. The beam is subjected to a uniformly distributed line loading applied within the instrumented span. 60. It has been assumed in the analysis of curved girders that the cross-sectional distribu-
CONCRETE BOX GIRDER BRIDGES OF DEFORMABLECROSS-SECTIO)
-0.426 (A)
-0.455 (A)
ty0L473 (A) -0.501 (B)
-0.486 (B)
Analytcal results
(B) -------- Best-fltresults 0
Experimental results
c UDL
=
0.468 Nlmm Test configuratlon
tion of direct stress for any particular loading is identical to that foran equivalent straight beam. This assumption is generally acceptable in box girder bridges of typical proportions, where the radius of curvature is very much greater than the section breadth.However, in the case of the curved box girdermodel considered here, the curvature is relatively large and it may be necessary to consider the effects of curvature on the transverse distributionof stress in the analysis. W a l d r ~ n states '~ that the
maximum error arising in the transverse distribution of the bending stressis approximately fb/2r, where b is the section breadth andr is the radius of curvature of the section. In the case of the curved model considered here, this error is equal to about10% across the breadth of the section. Additional errors may also arise as a result of a change in the positionof the shear centre axis in highly curved sections away from the geometrical centroid. 61. The results obtained for the curved
Fig. 13. Comparison of longitudinal direct stresses ( N / m m 2 a) t section C of curved model with central support: ( a )torsionally free; ( b ) torsionally restrained
121
KERMANI AND WALDRON 2. VLAWVV. 2. Thin-walled elastic beams. National Science Foundation, Washington,DC, 1961. 3. DABROWSKI R. Curved thin-walled girders.Cement and Concrete Association, Wexham Springs,1972, Pub. No. 144 (translation from German byC. V. Amerongen). 4. SWANN R. A. A feature survey of concrete box spine beam bridges. Cement and Concrete Association, Wexham Springs,1972, June, Pub. No. 42.469. 5. KOLLBRUNNER C. F. and BASLER K. Torsion in structures-an engineering approach. SpringerVerlag, New York, 1969. 6. HEILIGR. A contribution to the theory of box girders of arbitrary cross-sectional shape.Cement and Concrete Association, Wexham Springs, 1971, Trans. No. 145 (translation from German by C. V. Amerongen). 7. WALDRON P. Sectorial properties of straight thinwalled beams. Computers & Structures, 1986, 24, NO. 1, 147-156. 8. WRICHTR. N. et al. BEF analogy for analysisof Conclusions box girders.J. Struct. Diu. Am. Soc. Civ. Engrs, 62. A method has been developed for the 1968, 94, July, NO.ST7, 1719-1743. elastic analysis of single cell box girder 9. STEINLE A. Torsion and cross-sectional distortion bridges, with at least one axisof symmetry. It of the single cellbox beam. Beton und Stalbetonis equally applicable to both straight and bau, 1970, 65, Sept., No. 9, 215-222. curved bridges. The method, basedon the 10. ZHANGS. H. The finite element analysis of thinstiffness approach with six degreesof freedom walled box spine-beam bridges.The City Uniat each node, is capableof analysing deformversity, London, 1982, PhD thesis. 11. HETENYI M. Beams on elastic foundation.The Uniable box girder bridges which havecomplex versity of Michigan Press, 1946. geometry, loading and restraint conditions. The 12. KERMANI B. Single cell boxgirder bridges of performance and accuracyof the equivalent deformable cross-section.University of Bristol, beam method has been shown to compare 1988, PhD thesis. favourably with the resultsof tests on two con13. MAISEL B. I. and ROLLF. Methods of analysis and tinuous model box beams. design of concrete box beams with side cantile63. In straight box beams, distortional vers. Cement and Concrete Association, Wexham behaviour can be analysed independently from Springs, 1974, Nov., Pub. No. 42.494. bending and torsion. Conversely, in curved box 14. BOSWELL L. F. and ZHANC S. H. An experimental beams, additional distortional forcesoccur investigation of the behaviour of thin-walled box beams. Thin- Walled Structs,1985, 3, No. 1, owing to the radial componentof the longitudi35-65. nal bending stress. Since bending and torsion C. J. and DOWLING P. J. Bifurcated eleinteract with each other along the entire length 15. BILLINGTON vated highways-construction, instrumentation of curved beams, this results in full interaction and testing offour linearly elastic models, between bending, torsion and distortion. Imperial College, London, Engineering Structures 64. Out-of-plane warping resulting from Laboratories, 1972, Jan., ReportBB1. torsion and distortion is an important featureof 16. EVANS H. R. and AL-RIFAIE W. N. An experimental box girder bridges. If these warping displaceand theoretical investigationof the behaviour of ments are restrained, systemsof shear and box girders curved in-plan.Proc. Instn Civ. axial stresses are set up which may modify sigEngrs, Part 2,1975,59, June, 323-352. nificantly the state of stress resulting from 17. KERMANI B. and WALDRON P. Distortion of box girders, l :Construction, instrumentation and primary bending throughout the structure. It testing of two sandlaraldite models;2: Results of was found that warping effects due to distorstraight model test;3 : Results of curved model tion of the cross-section predominate, resulting test. University of Bristol, Department of Civil in significantly greater warping stresses in Engineering, 1987, June, ReportNo. thin-walled box girder bridges of deformable UBCE/C/87/3. cross-section. 18. WALDRON P. Equivalent beam analysisof thinwalled beam structures. Computers & Structures, 1987,26, NO. 4,609-620. References 19. WALDRON P. Elastic analysis of curved thin1. WALDRON P. The significance of warping tension walled girders including the effects of warping in the design of straight box girder bridges. J. restraint. Engng Structs, 1985, 7, Apr., 93-104. Can. Soc. Civ. Engrs,1988, 15, Oct., 879-889.
model indicate once again that distortion was the principal source of warping stress.For the case in which torsion was fully restrained at the central support, the distortional warping stresses were found to be up to three times larger than torsional warping stresses at the same sections. Unlike the case of straight beams, flexural, torsional and distortional actions interact along the entire lengthof the curved thin-walled girder. Thus, by restraining one action, all the other actions are influenced, modifying the whole state of stress throughout the girder. In curved beams, additional distortional forces occur owing to the radial component of the longitudinal bending stress, which must be included in the distortional calculations even when the load is applied through the shear centre.
