Behaviour of Concrete Box Girder Bridges of Deformable Cross-Section

Behaviour of concrete box girder bridges of deformable cross-section B. Kermani, BSc, PhD, MIStructE, and P, Waldron, BSc, PhD, DIC, MICE, MIStructE

H Distortion of the cross-section, which W , torsional warping displacement commonly occurs in thin-walled box ya distortional angle girder bridges that contain few inter- 6 wall thickness media te diaphragms, may modify the 9 sectional constant distribution of stress throughout the B bending rotation

s t r u c t u r e to a significant degree. A p torsional warping shear parameter

method of analysis is presented which ed distortional warping direct stress e, torsional warping direct stress

distortion in addition to those of torsional torsional warping shear a l lows for the effects of cross-sectional T~ distortional warping shear stress

warping. The approach is a rational exten- 4 torsional rotation

be applied to continuous or simply- sion of thin-walled beam theory and m a y & torsional warping sectorial coordinate

suppor ted box gi rders o f either straight or f~ twice the enclosed cell area curved configurat ion in plan. As few test d a t a o f continuous girders were available ~ ~ ~ ~ ~ d ~ ~ ~ i ~ ~ against which the accuracy Of the method Box girders have evolved into a highly efficient could be verified, two models-one and aesthetically pleasing solution for medium

fabricated for this p u ~ o s e - The increases into the range where dead load domi. straight* the Other curved in span and long span bridges. As span length

variety Of 'Oncentrated and distributed important. It is here that the efficient use of the torsional loads are briefly described.

from these tests show ural and torsional stiffness, permits a reduction box section, which possesses considerable flex-

general agreement wi th those f r o m the proposed analyt ical method.

in overall section size with a consequential saving in weight.

particular attention is the curved box girder Notation bridge which is an essential feature of most

cross-sectional area section breadth modern highway interchanges and urban

torsional bimoment motorways. A characteristic feature of curved distortional bimoment girders is the relatively high level of torsion to modulus of elasticity which they are subjected owing to the contin- torsional warping function shear flow

uous interaction between flexural and torsional moments along the entire man. Box girders

& distortional warping sectorial coordinate

tion and tes t ing of these models under a nates, saving in self-weight becomes

2. One type of configuration which merits

~~~~.~ ~~~

shear modulus of elasticity section height flexural second moment of area torsional warping constant 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

Engrs Sfructs & Proc. Instn Ciu.

Bldgs, 1993,99, May, 109-122

Structural Board Paper 9940

closes 15July I993 Written discussion

B. Kermani, - with their high torsional stiffness are ideally Bridge Engineer* suited for these applications where signficant Acer Consultants

levels of torsion mav be induced even bv the self-weight of the structure alone. 3. For the purposes of analysis, box girders

may be divided into two groups according to

group contains girders assumed to have a rigid their behaviour under torsional loads. The first

cross-section which do not change cross- sectional shape when rotated about their longi-

concrete box girders with relatively thick walls, tudinal axis (Fig. l(a)). This is the case for

in which transverse frame action may be suffi- cient to maintain the original cross-section. In other cases, closely spaced diaphragms or cross-bracing may be required to satisfy this condition. The second group comprises deformable girders which possess sufficient

of the cross-section under torsional loading transverse flexibility to undergo distortion

(Fig. l(b)).

P. Waldron, Professor of Structural Engineering, Department ofCiuil and Structural Engineering.

Sheffield Uniuersify of

KERMANI AND WALDRON

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

Structural response of thin-walled box girders 4. The behaviour ofthin-walled beams

under flexural loading is essentially the same as for solid and thick-walled sections. It is usually satisfactory, by assuming that plane sections remain plane after loading, to use simple beam theory to predict the level of flex- ural stress around the cross-section and the dis- tribution of bending moment at any section along the girder. Shear lag effects may become important in certain thin-walled girders but are not considered further here.

5. The response of thin-walled box beams to torsional loading, however, may differ consider- ably from that observed for solid or thick- walled sections, owing to the formation of more significant out-of-plane warping deformations. These axial deformations invalidate the usual assumption of plane sections adopted in simple theory and require special consideration. In general, two components of warping displace- ment arise from torsional loading: the first, referred to as torsional warping, occurs in response to the application of pure St Venant's torsion to the undeformed section; the second arises from distortion of the cross-section itself.

deformable cross-section under the application of an equal but opposite torque at both ends (Fig. 2(a)). Under this loading, the distribution of pure torsional warping displacement around the cross-section is identical at all positions along the beam. Rigid-body rotations of the

6. Consider first a box beam of non-

l+- Wt

l I /

t

.

wall elements are accompanied by a degree of shear deformation to ensure continuity of axial displacement around the closed perimeter, resulting in a system of circulatory shear stresses in accordance with St Venant theory. The shear flow generated is of constant magni- tude and, in box girders with side cantilevers, occurs only around the perimeter of the closed cell, as shown in Fig. 3(a).

7. If the torsional warping displacements are now restrained, as by some form of physical restraint as shown in Fig. 2(b), the individual wall elements are no longer free to rotate as rigid bodies but are subjected to bending about their own major axes. This results in a system of axial direct stresses referred to as longitudi- nal torsional warping stresses which have the distribution around the section shown in Fig. 3(b). These are self-equilibrating and have no resultant component of either direct force or bending moment.

8. A complementary system of shear flows is also created which extends over the entire cross-section and constitutes a system in equi- librium (Fig. 3(c)). In thin-walled beams of certain proportions, the stresses due to restraining torsional warping may become significant and should be fully considered in the analysis.

9. In practice, the number of diaphragms along the span are kept to the minimum since they induce additional dead load and cause dis- ruption and delays in the casting cycle. If the intermediate diaphragms are few in number, the cross-section of thin-walled girders may distort, resulting in the transverse bending of the walls shown in Fig. l(b). Apart from the transverse flexural deformations, longitudinal out-of-plane displacements are also induced, referred to as distortional warping tud. These axial displacements occur in addition to the tor- sional warping displacements W , shown in Fig. 2. The nature of the loading and beam geometry is such that the distribution of 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 not interact with the other direct stress result- ants, However, they may modify significantly

CONCRETE BOX GIRDER BRIDGES OF DEFORMABLE CROSS-SECTION

the state of stress due to primary bending, pure torsion and torsional warping. The complemen- tary distortional shear flows shown in Fig. 4(b) are also self-equilibrating and provide no resistance to the applied torsional moment.

tortion, transverse bending moments are also produced by frame action around the box. These have the distribution shown in Fig. 4(c), and the resulting stresses can be 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 can combine such that Poisson's ratio effects may be significant and should not be neglected in design.

10. As a direct result of cross-sectional dis-

Thin-walled beam theory 11. A generalized theory for the analysis of

thin-walled beams have been developed by Vlasov' and extended to curved girders by Dab- r ~ w s k i . ~ This approach enables the distribution of stress at any point in the girder to be deter- mined from the familiar expression used in simple beam theory, but extended to account for the effects of torsional warping and distor- tion of the cross-section. For thin-walled behav- iour to be assumed with the context of Vlasov's theory, the section must be sufficiently thin for variations in stress across the thickness of the walls to be neglected. This condition, which is met by the large majority of concrete box girder bridges found in practice4 and virtually all steel and composite sections, permits the levels of direct and shear stresses on the median line of the cross-section to be used throughout the analysis.

Torsional warping 12. For thin-walled, thick-walled and solid

members to be accommodated by the same general beam theory, Vlasov' introduced new cross-sectional functions called torsional sec- torial properties. New types of stress resultant were also created, denoted torsional bimoment B and warping torsion T,, to supplement those

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 Koll- brunner and Basler5 and Heilig6 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 result of some form of physical restraint, the longitudinal distribution of torsional warping displacement W , may be expressed in terms of a dimensionless warping function f such that

W , = -f '& (1)

The term (5 is called the torsional warping sec- torial coordinate (units: L') and represents the level of out-of-plane warping due to a unit rate of twist. Subsequently, the longitudinal dis- tribution of the stress resultants B and T, along the beam are obtained from

B = -EI,f" (2)

and

T , = B' = -EI+f"' (3)

where I , is a torsional warping constant (units : L6) defined as

I , = (5'dA I 14. The direct stress and Shear flow

resulting from restraint to torsional warping at any point on the medial line of the section may then be expressed as

in which S, (units: L4) is a further sectorial function used to describe the distribution of shear flow due to torsional warping. The exten-

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,

Fig. 4 . Typical cross-sectional distributions of: ( a ) distortional axial warping stress a,; ( b ) corresponding distortional shear flow Fd; (c) transverse distortional moment per unit length mdb

KERMANI AND WALDRON

Fig. 5. Force components in wall elements due to (a) vertical, and ( b ) horizontal antisymmetrical loading

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 shear flow.

15. The fundamental partial differential equation governing torsional warping in thin- walled closed beams may be expressed in terms of the warping function f as

which is equally valid for straight and curved thin-walled beams. However, in the case of straight beams, the flexural and torsional effects are uncoupled since the torsional com- ponent of the load due to longitudinal bending, given by the term (MJr), is zero. In equation (7), J is the familiar second moment of area for pure torsion, and p, called the torsional warping shear parameter, is a measure of the variation in shear stiffness around the closed section. The derivation of the various sectional and sectorial properties introduced here may be found in detail elsewhere7 for the full range of thin-walled sections used in bridge construc- tion.

Distortion

based on the assumption that the beam cross- section preserves its original shape under tor- sional loading. This is not the usual behaviour of box girder bridges found in practice where the cross-section may deform, resulting in addi- tional distortional effects such as distortional warping and transverse bending.

17. Distortion of thin-walled beams has been considered previously by a number of investigators. Vlasov2 was the first to draw an analogy between the response of a box beam subjected to distortional loading and that of a beam on an elastic foundation. The out-of-plane

16. The theory of torsion described above is

I btt, l fl

rigidity against differential bending of the top and bottom slabs of the box provides a contin- uous elastic support for each half of the section. Subsequently, the theory has been developed and generalized by others-notably Dabrow- ski,3 Wright et al.,' Steinle' and Zhang"-for box girder bridges of different geometrical con- figurations.

18. In developing the theory of distortion, it is assumed that in-plane displacement of the cross-section is accompanied by sufficient warping to annul the shear strain. The approx- imate method based on the beam on elastic foundation analogy enables use to be made of the expressions derived previously and applied commonly in the design of such beams."

beam of trapezoidal cross-section shown in Fig. 5, loaded with external twisting moments t , and t , per unit length. These loads may be divided into two components: pure torsional loads; section deforming loads. At any section, the uniformly distributed torque t and the uni- formly distributed distortional moment md may then be defined'* in terms of the horizontal and vertical components of twisting moment as

19. Consider the typical single cell box

t = t, + t , (84

and

m - 1 {h t , - th} d - 2 b ,

20. In box girder bridges subjected to gravity loads, the vertical component of twist- ing moment t , usually predominates. In such cases, the effect of sloping webs on the distor- tional behaviour is clear from equation (8), i.e. the component of distortional moment reduces with increased slope a s a result of a corres- ponding increase in frame stiffness around the box.

- 112

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 is defined as 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 distor- tional warping w d which occurs in the longitu- dinal direction and may be expressed in terms of the distortional angle yd as

w d = -7;h (9)

in which ii, is called the distortional coordinate (units: L').

22. Since distortional warping displace- ments are rarely constant along the beam, addi- tional systems of distortional direct and shear stresses are created. Corresponding stress resultants known as distortional bimoment D and distortional moment M d are formed, which may be defined in terms of the distortional angle thus

D = -EI,y: (10)

in which

ZiG = j+lil

is called the distortional warping constant (units: L6). The distortional warping direct stress b d and shear flow Fd may then be expressed in the following form

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 have been introduced here for distortion and the corresponding expres- sions derived previously for the theory of warping torsion.

23. Full or partial restraint of the axial warping displacements arising from cross- sectional distortion are an effective means of reducing the level of distortion that would otherwise have occurred. Frame action resulting in the creation of transverse bending moments around the box provides another form of resistance to distortion of the cross-section. The resistance is caused by the transverse flex- ural stiffness of the walls forming the closed cell of the box when the section is subjected to distortional loading. The resulting transverse distortional bending moment per unit length of the box can be obtained from the following

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 dis- tortional frame stiffness K , may be evaluated by consideration of a unit length of box beam loaded by diagonal forces with a unit horizontal component. For the purposes of this calcu- lation, the frame may be considered to be restrained at the lower corner points in both the vertical and horizontal directions. By using the flexibility method, it is then possible to deter- mine Kd for single cell cross-sections.

24. The differential equation governing the distortional behaviour of a single cell box beam may be expressed as

EZ, 7: + K , Yd = ( m d + M J 2 Y ) (16)

This is the general expression for girders curved circularly in plan, but it is equally valid for straight beams for which the final term dis- appears as Y tends to infinity. It can be seen'' that this is directly analogous to the differen- tial equation governing displacement y in a beam with bending stiffness EI, supported on an elastic foundation of stiffness K, and under applied loading p , given by

EIxy" + K,y = p (1 7)

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, varia- tions 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.

Experimental investigation 26. Only a very limited number of experi-

mental investigations of box beams of deform- able cross-section have been reported.I4-l6 Therefore, in order to widen the study of box girder bridges to include alternative geometri- cal loading and restraint conditions, further detailed experimental investigation was neces- sary, resulting in the construction, instrumen- tation and testing of two model box beams. The models, one straight and the other curved in plan, were both continuous over two spans and of similar proportions. The principal dimen- sions of the models were selected from a feature survey by Swann4 to be representative of actual concrete box girder bridges. The nominal

113

KERMANI AND WALDRON

- 114

Fig. 6. ( a ) Typical cross-section of both models, together with plan view of: (b) straight model; (c ) circularly curved model

dimensions and general arrangements of 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 pro- vided a useful pot life of 20 hours and a curing time of seven days after mixing at room tem- perature. As the models were to be instru- mented 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 portions of the webs and dia- phragms. The joint between the two channel sections was chosen to be 20 mm above mid- depth of the web so as to avoid the point at which the shear stresses were likely to be at a maximum (Fig. 6(a)).

28. One tensile test specimen was cast from each of the eight mixes used in the construction of the models in order to obtain the elastic properties of the material. The average values

r 21 5

of Young’s modulus determined at the time of testing were 16.2 kN/mm2 and 17.6 kN/mm* for the straight and curved models respectively. Poisson’s ratio was found to be 0.35 for both models.

Instrumentation and testing 29. Both models were instrumented for the

measurement of strain, displacement and support reactions, using a total of 376 electrical resistance strain gauge elements of 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 span of the straight model, as identified in Fig. 7(a). At each of these sections, four single element strain gauges were 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 stresses a t the corners of the box section in the longitudinal direction. Two critical sections were then selec- ted 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 components of longitudi- nal, transverse and shear strain at all of the gauge locations around the cross-section shown in Fig. 7(c). The curved model was instru- mented in an identical fashion, with the same separation between the five instrumented sec- tions measured along the curved centre-line.

30. Displacements were measured at a single section at any one time with four linear variable displacement transducers to provide measurements of vertical displacement, tor- sional angle and distortional angle. The trans- ducers were supported on an independent frame which could be positioned at any section at which measurements were required. By repeat- ing the load test several times with the trans- ducer frame located at different sections, it was possible to generate a longitudinal profile of the various components of deformation along the girder.

measurement of support reactions during the tests. Each load cell consisted of a 1 mm thick cylindrical mild steel section instrumented with four linear strain gauge elements. The bases of the load cells were bolted at the appropriate positions on top of a support rig which 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 load of 518 N and a uniformly distributed line load of 0.468 N/mm. In both cases, the load was applied eccentri- cally over one web within one span only.

31. Six load cells were manufactured for the

CONCRETE BOX GIRDER BRIDGES OF DEFORMABLE CROSS-SECT10

Although only two alternative load systems were available, several different load cases and structural configurations could be tested. The model could be either torsionally restrained at its centre, by using two load cells, one under each web, or restrained against vertical dis- placement only, by placing one load cell cen- trally. 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 spans by super- imposing the results.

33. A data logger was used in conjunction with a microcomputer to read and record the results from the instrumentation. A total of 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 set of 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, displace-

ment transducers and the strain gauges were all tabulated and presented diagrammatically for each individual load case and restraint con- dition.” From the corner strain gauges, extreme fibre stresses were 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 sec- tions 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 estimate of the four stress resultants causing axial direct stress (axial force, bi-axial bending and the combined torsional and distortional bimoment), the rosette gauges at the two fully instrumented sections provided redundant information. A least-squares analysis was performed on these gauge results to minimize experimental errors and to obtain an improved estimate of the various stress distributions and stress result- ants at these two sections.

36. From thin-walled beam theory, the dis- tribution of the longitudinal stress a and the shear flow F at any point on the median line of the cross-section may be expressed as

N M,y M,n B& D& g=-+-+-+-+- (18)

A I. I , I; I;

37. These expressions are the familiar equa- tions for simple beam theory modified by the

A B C D 5

“t i

a b A l I

torsional and distortional warping terms defined in equations (5), (6), (13) and (14). In addition, the transverse bending moment per unit length due to distortion of the cross- section may be obtained from the measured transverse bending stress adb from the expres- sion

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 experimen- tal points a , ( i = 1, 12) from the twelve strain gauges located at both sections C and E. If the values of this best-fit function are given by f, at the twelve strain gauge positions, the discrep- ancy between the experimental and best fit values at each experimental point is (ai - fi). The least-squares criteria may then be expressed as

12

S = 1 ( a , - fi)2 i = 1

in which S, the overall squared error, is to be minimized.

40. In the case of the axial direct stress and the transverse bending moment per unit length, which both vary linearly between box corners, the value of the best fit line at the twelve strain gauge points f i may be written in terms of the four corner values F j ( j = 1, 4). Minimizing S with respect to F j yields four equations in terms of the four unknown corner values. The best estimates of the corner values thus deter- mined were then used to evaluate the valuesfi at each of the strain gauge positions by linear interpolation, and to predict the value of the

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

115

KERMANI AND WALDRON

axial force N, the bi-axial bending moments M,, M,, and the total bimoment (B + D) due to the torsional and distortional warping restraint effects combined.

41. Since a least squares treatment was not possible on the four gauges located at the corners of each instrumented section, the results were used to establish four simultan- eous equations from which unique values for the various stress resultants were obtained at each of the five instrumented sections.

42. For the shear flows, a similar least- square 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 resultants were found which were then used to obtain the distribution of shear flow around the cross-section.

Equivalent beam method of analysis

analysis for box girder bridges of complex con- figuration are those in 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 type of any configu- ration, its practical application is often restricted by the large amount of input/output data and computer time required for solution. For reasons of economy, a need therefore exists for simpler methods of analysis which consider all the important structural actions associated with thin-walled box girder bridges with suffi- cient accuracy, particularly during the prelimi- nary 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 continuation of the work by Waldron" on torsional warping but extended to account for the distortional effects in single cell box girder bridges. In addition to the four degrees of freedom system developed pre- viously, which considers rotation due to uni- axial 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 distortional effects. The additional degrees of freedom are the distortional angle yd

43. The most appropriate methods of

and the rate of change of distortional angle y: along the length of the girder.

of discrete beam elements and is therefore 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

45. The stiffness method adopted makes use

{PI = [ K I W (22) where the load vector { p } contains the six com- ponents of load at each end, thus

{p}T = {Mx TVBDM,} (23)

and the corresponding components of end dis- placement are given by

P I T = { e 4 ? m Y d ) (24) 46. The member stiffness matrix [ K ] is a

symmetrical 12 X 12 matrix and is determined initially for each element in local member coor- dinates. Assembly of the stiffness matrix and load vector for the entire structure follows in the usual way after transformation into a fixed global coordinate system. The overall structur- al stiffness matrix will be singular in this form as no account has yet been taken of the various restraint conditions, and an infinite number 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 must be modified or removed to take account of the various restraint conditions before the reduced stiffness matrix can be inverted to yield a solution for the end displace- ments associated with each load case.

47. The stiffness method has been incorpo- rated into a Fortran computer program" to analyse single cell 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 dis- tributed loads between nodes. The fixed-end forces attributable to the uniformly distributed shear, torsional and distortional loads are evaluated, and after transformation into system coordinates are applied as nodal forces. Unlike the FE method, the equivalent beam method treats the longitudinal distribution of 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 degree of 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 accu- rately with a minimum number of elements. As relatively few beam elements are required, a facility has been included in the program which

116

CONCRETE BOX GIRDER BRIDGES OF DEFORMABLE CROSS-SECT10

calculates the various stress resultant at any number of intermediate points between nodes.

Analysis of model beams

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 single cell rec- tangular box section without side cantilevers selected for the models, the various sectorial functions for torsional warping and distortional warping are identical. These may be calculated by hand or by computer’ and are shown in Fig. 9 for the model section.

ties of the box section and its geometry are known, the next step in the analysis is to compile the stiffness matrix [ K ] for each member. On account of the complexity of some of the distortional terms in [K] , this is best achieved numerically, which is the approach adopted by the computer program described previously.” The various element stiffness matrices may then be 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 con- sidered here, [ K,] is very sparsely filled with all the non-zero elements banded about the leading diagonal. The rank and file of [K,] is reduced further by five or six elements (depending on the nature of the central support) to take account of 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 displace- ments which may then be inserted into equation (22) to provide values for 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 result- ants along each of the individual beam ele- ments a s required.

48. Taking full advantage of the efficiency

49. When the sectional and sectorial proper-

Discussion of results

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 incorpo- rated eight different tests on each model, com- prising 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

51. A comparison is presented here between

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 behaviour of single cell box girder bridges model; ( 6 ) circularly of deformable cross-section. curved model

Straight model

obtained from the load cells, the total applied load was compared with the sum of the reac- tions. The maximum error in this equilibrium check for the four test configurations was found to be 2.97%, with an average error of 2.13%. The difference between the total reac- tions at each of the three supports, determined experimentally and analytically, has also been assessed. The maximum error, found to be 1.70% of the total applied load 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 experimen- tally at section E are shown in Fig. 10 for the 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 results from 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 configu- ration allowing torsional rotation at the inter- mediate support. In this case (Fig. 10(a)), the values of stress at the four corners obtained from the least-squares analysis of the experi- mental results were all within 2.5% of those

53. In order to assess the quality of the data

Fig. 9. Cross-sectional distributions of (a) sectorial coordinates h, 15, and ( 6 ) sectorial shear functions S+, S;, adopted f o r analysis of both models

KERMANI AND WALDRON

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

O J 0.268 (A)

f \ m

0.302 (B) 0.261 (B)

0.296 (A) 0

I t 4 H

0 U

-0.268 (A) -0.262 (B)

0-247 (A) 0.249 (B) m

-0.247 (A) -0.316 (A) -0.224 (B) -0.318 (B)

(A) - Analystical results (B) -------- Best-fit results

0 Experimental results

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 is a compari- son of the various stress resultants obtained from analysis and from the best estimate of the experimental results. The average values of 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)). A similar increase of 8.8% was observed for the case of uniformly distributed loading, not shown here. However, for both load cases, distortion was found to be the main source of warping stress, and this can

E <

m

P = 518 N E d

Test configuration

form a significant addition to the cross- sectional distribution of longitudinal bending stress. Maximum distortional warping stresses for the straight model were found to be 35.6% and 24.7% of the bending stresses for the con- centrated load and uniformly distributed load cases respectively.

57. The values of the transverse bending moments per unit length around section C of the model are plotted in Fig. 11, for the case when the model is subjected to a uniformly dis- tributed 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 do not interact in straight box beams and that distortion can be analysed in

CONCRETE BOX GIRDER 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)

6.413 (A)

7.157 (B)

-6.413 (A)

-7.993 (B)

(A) - Analystical results (B) -------- Best-fit results

0 Experimental results

isolation. The calculated transverse bending moments per unit length caused by distortion of the cross-section have also been shown to be 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 rep- resented in Fig. 11.

58. It is notoriously difficult to achieve a good agreement between experimental measure- ments and analytical predictions of the magni- tude and distribution of shear around box sections. The shear flows at sections C and E

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 UDL = 0.468 Nimm

Test configuration restrained

are shown in Fig. 12 for one test configuration only as an example of the quality of the results obtained in this study. Loading consists of the eccentric concentrated load applied within the instrumented span of the model which is tor- sionally restrained at the central support. Although the shear strains measured experi- mentally 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 generally in good agreement. The values of the vertical shear force obtained from

119

KERMANI AND WALDRON

-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

(A) - Analytical results

(B) -------- Best-fit results

at the intermediate o Experimental results support at: (a) section C; ( b ) section E 1 Test conflguration

the best-fit analysis are 88.3% and 80.1% of the theoretical values derived from the equivalent beam method for sections C and E respectively (Fig. 12). The corresponding values of 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 curved model shows that the equivalent beam method has resulted in a general underestimation of axial

stress around the cross-section. The average value of the bending moment obtained from the analysis is 86.5% of that obtained from a least- squares analysis of the experimental results. As an example, the results of the rosette strain gauges at section C for the in-plane component of the axial direct stress is presented in Fig. 13. The beam is subjected to a uniformly distri- buted line loading applied within the instru- mented span.

60. It has been assumed in the analysis of curved girders that the cross-sectional distribu-

120

CONCRETE BOX GIRDER BRIDGES OF DEFORMABLE CROSS-SECTIO)

-0.426 (A)

-0.455 (A) ty0L473 (A) -0.486 (B) -0.501 (B)

(B) -------- Best-flt results

0 Experimental results

Analytcal results c tion of direct stress for any particular loading is identical to that for an 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 girder model con- sidered here, the curvature is relatively large and it may be necessary to consider the effects of curvature on the transverse distribution of stress in the analysis. W a l d r ~ n ' ~ states that the

UDL = 0.468 Nlmm

Test configuratlon

maximum error arising in the transverse dis- tribution of the bending stress is approximately f b/2r, where b is the section breadth and r is the radius of curvature of the section. In the case of the curved model considered here, this error is equal to about 10% across the breadth of the section. Additional errors may also arise as a result of a change in the position of 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

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 length of 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 distor- tional forces occur owing to the radial com- ponent of the longitudinal bending stress, which must be included in the distortional cal- culations even when the load is applied through the shear centre.

Conclusions 62. A method has been developed for the

elastic analysis of single cell box girder bridges, with a t least one axis of symmetry. It is equally applicable to both straight and curved bridges. The method, based on the stiffness approach with six degrees of freedom at each node, is capable of analysing deform- able box girder bridges which have complex geometry, loading and restraint conditions. The performance and accuracy of the equivalent beam method has been shown to compare favourably with the results of tests on two con- tinuous model box beams.

behaviour can be analysed independently from bending and torsion. Conversely, in curved box beams, additional distortional forces occur owing to the radial component of the longitudi- nal bending stress. Since bending and torsion interact with each other along the entire length of curved beams, this results in full interaction between bending, torsion and distortion.

64. Out-of-plane warping resulting from torsion and distortion is an important feature of box girder bridges. If these warping displace- ments are restrained, systems of shear and axial stresses are set up which may modify sig- nificantly the state of stress resulting from primary bending throughout the structure. It was found that warping effects due to distor- tion of the cross-section predominate, resulting in significantly greater warping stresses in thin-walled box girder bridges of deformable cross-section.

63. In straight box beams, distortional

References 1. WALDRON P. The significance of warping tension

in the design of straight box girder bridges. J. Can. Soc. Civ. Engrs, 1988, 15, Oct., 879-889.

2. VLAWV V. 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 by C. V. Amerongen).

4. SWANN R. A. A feature survey of concrete box spine beam bridges. Cement and Concrete Associ- ation, Wexham Springs, 1972, June, Pub. No. 42.469.

5. KOLLBRUNNER C. F. and BASLER K. Torsion in structures-an engineering approach. Springer- Verlag, New York, 1969.

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 thin- walled beams. Computers & Structures, 1986, 24,

6. HEILIG R. A contribution to the theory of box

NO. 1, 147-156. 8. WRICHT R. N. et al. BEF analogy for analysis of

box girders. J. Struct. Diu. Am. Soc. Civ. Engrs,

9. STEINLE A. Torsion and cross-sectional distortion of the single cell box beam. Beton und Stalbeton- bau, 1970, 65, Sept., No. 9, 215-222.

10. ZHANG S. H. The finite element analysis of thin- walled box spine-beam bridges. The City Uni- versity, London, 1982, PhD thesis.

11. HETENYI M. Beams on elastic foundation. The Uni- versity of Michigan Press, 1946.

12. KERMANI B. Single cell boxgirder bridges of deformable cross-section. University of Bristol, 1988, PhD thesis.

13. MAISEL B. I. and ROLL F. Methods of analysis and design of concrete box beams with side cantile- vers. Cement and Concrete Association, Wexham Springs, 1974, Nov., Pub. No. 42.494.

14. BOSWELL L. F. and ZHANC S. H. An experimental investigation of the behaviour of thin-walled box beams. Thin- Walled Structs, 1985, 3, No. 1,

1968, 94, July, NO. ST7, 1719-1743.

35-65. 15. BILLINGTON C. J. and DOWLING P. J. Bifurcated ele-

vated highways-construction, instrumentation and testing offour linearly elastic models, Imperial College, London, Engineering Structures Laboratories, 1972, Jan., Report BB1.

16. EVANS H. R. and AL-RIFAIE W. N. An experimental and theoretical investigation of the behaviour of box girders curved in-plan. Proc. Instn Civ. Engrs, Part 2,1975,59, June, 323-352.

17. KERMANI B. and WALDRON P. Distortion of box girders, l : Construction, instrumentation and testing of two sandlaraldite models; 2: Results of straight model test; 3: Results of curved model test. University of Bristol, Department of Civil Engineering, 1987, June, Report No. UBCE/C/87/3.

walled beam structures. Computers & Structures, 18. WALDRON P. Equivalent beam analysis of thin-

1987,26, NO. 4,609-620. 19. WALDRON P. Elastic analysis of curved thin-

walled girders including the effects of warping restraint. Engng Structs, 1985, 7, Apr., 93-104.