Analysis of continuous box girder bridges including the effects of distortion

Computers d Sm~rures Vol.47.N~. 3, pp. 427-440, 1993

0045-7949193$6.00+0.00

Printed in GreatBritain.

0 1993 Pergamon PressLtd

ANALYSIS OF CONTINUOUS BOX GIRDER BRIDGES INCLUDING THE EFFECTS OF DISTORTION B. $Department

KERMANI~

and P.

WALDRON~

tAcer Consultants Ltd, Bristol, U.K. of Civil and Structural Engineering, University of Sheffield, PO Box 600, Mappin St, Sheffield, Sl 4DU, U.K. (Received

6 March

1992)

Altstraet-Torsional warping and distortion of the cross-section are important features of thin-waii~ beams and must be considered fully in the design of box girder bridges. A method of elastic analysis is developed based on the stiffness approach which includes the effects of warping torsion and distortion in addition to the more familiar actions of bending moment and torsion. The method, which is applicable to straight single cell box girders with at least one axis of symmetry, is demonstrated here in the analysis of three different box girder models for which experimental or analytical results were already available. The method is shown to be easy and economical to use and provides a physical insight into the structural response of thin-walled box girder bridges under general loading conditions.

NOTATION cross-sectional area section breadth torsional bimoment distortional bimoment modulus of elasticity torsional warping function shear flow 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 tangential radius measured from the shear centre torsional sectorial shear function distortionai sectorial shear function uniformly distributed torsional moment per unit length applied torsional moment warping torsion vertical disp~cement shear force distortional warping displacement torsional warping displacement section shape function distortional angle wall thickness cross-sectional constant torsional warping shear parameter bending rotation distortional warping direct stress torsional warping direct stress distortional warping shear stress torsional warping shear stress torsional rotation transformation angle between member and system axes

6

D Q

torsional warping sectorial coordinate distortional warping sectorial coordinate twice the enclosed cell area

INTRODUCTION Structural design of box girder bridges presents many difficulties because of the complex interaction of the

individual structural effects. For thin-walled girders these include torsional warping and distortion of the cross-section in addition to the usual flexural and torsional actions. Although a wide range of analytical methods is available to estimate the level of these individual effects, many have severe limitations particularly with regard to the structural arrangements that may be considered. A three-dimensional (3-D) finite element method may offer the most comprehensive treatment. but the computational costs involved are high and arc rarely justifiable, particularly in the preliminary analysis and conceptual design stage. Furthermore. such an analysis may be used indiscriminately without a physical understanding of the structural behaviour and significant structural actions may be missed. Therefore a need exists for a simple method of elastic analysis of thin-wailed box girders which has the generality necessary to cope with the complex geometry of modern elevated highways whilst retaining a degree of accuracy sufficient for design. This

paper concentrates on the linear elastic problems associated with straight box girder bridges, including the effects of distortion of the cross-section. In general, any system of eccentric point loads applied at a section can be divided into its component parts. For the case of box girders with deformable cross-sections, these components consist of bending, torsion and distortion (Fig. I). In addition to these 427

B. KERMANIand P. WALDRON

428

_________-----------T-------. _+

(a)

e.

-&--=g (b)

(c)

Fig. 1. Components of deformation due to eccentric loading: (a) bending; (b) torsion and (c) distortion.

actions, there are some further structural characteristics which are peculiar to thin-walled beams and which may require special attention under certain circumstances (e.g. shear lag). Such effects are not considered further here but may be assessed separately and superimposed on the final solution. The behaviour of thin-walled beams under bending and axial loading is essentially the same as for solid and thick-wailed sections. For these actions it is usually satisfactory to assume that plane sections remain plane after loading. However, under eccentric loading, thin-walled beams behave differently from solid or thick-walled sections and large out-of-plane warping displacements may occur. In general, there are two different forms of out-of-plane displacements referred to as torsional warping, w,, and distortional warping, wd. The former component arises as a result of pure twist and is calculated assuming the cross-section remains rigid; the latter is the additional component of axial warping displacement which occurs if the section is permitted to deform. Under pure torsion, the cross-sectional distribution of torsional warping displacements w, is identical at all positions along the beam, their magnitude being proportional to the rate of twist (Fig. 2a). A system of circulatory shear stresses is created to resist the applied torque in accordance with St Venant’s theory, in exactly the same way as for solid and thick-walled sections even though the plane sections no longer

remain plane. This may be expressed as a shear flow F, of constant magnitude around the closed cell of the box section. If, however, the torsional warping displacements are in any way restrained, for instance by a heavy transverse diaphragm or a built-in end, as illustrated in Fig. 2(b), the individual wall elements will be subjected to bending about their own major axes. This results in a system of direct stresses, called longitudinal torsional warping stresses Q,., which is in equilibrium and has no resultant component of direct force or bending moment. A complementary system of torsional warping shear flow F,? is also created which acts in conjunction with the constant St Venant’s shear flow to resist the applied torsional moment. Except where there are rigid diaphragms or crossbracing, the cross-section of a box girder may distort under torsional loading, the magnitude of this distortion depending to a large extent upon the transverse flexibility of the inidividual wall elements. As a direct result of cross-sectional distortion, transverse bending moments are produced aroung the box section by frame action. Apart from the transverse flexural deformation, axial distortional warping displacements wd are also induced. Since these warping displacements are not constant between points of restraint, additional direct stresses ud and complementary shear flows Fd are developed. These are

04

Fig. 2. Torsional warping displacements occurring in a typical box section (a) unrestrained at both ends and (b) restrained fully at one end only.

Analysis of continuous box girder bridges ~If-equilibrating but may significantly modify the final distribution of stress around the box section. In the majority of steel and composite box girders, wall thicknesses are generally such that torsional and distortional effects are important. The present tendency in concrete box beam design is to use thinner webs and flanges to reduce self-weight. This is having the effect of increasing the significance of warping torsion and distortion in this type of section. While the causes of warping torsion and distortion are very different, both effects may become significant and should be considered fully in the analysis of such thin-walled beams. Unlike the distribution of torque and bending moment, which may be simply obtained from a consideration of statics, the effects of torsional warping restraint and distortional warping restraint are indeterminate and can only be evatuated by taking account of the overall state of deformation in the member.

429

point on the median line of the cross-section to be expressed in the following form N

M,y

a=-+ A

vs

vs

F= -z...z+~ 1,’

1,

OF THIN-WALLED

L&i

T

T, Si.

II&S:.

+n-I,.--&7

(1)

(2)

Torsional ~eh~io~r

In eqns (1) and (2), ci, is called the torsional warping sectorial coordinate, given by

BEAMS

Q=

s s 0

Vlasov [I] is generally considered to be the first to have presented a rigorous theory for thin-walled beams. The major attraction of this approach is that thin-walled, thick-walled and solid members are a11 treated as special cases of the same general theory. In order to explain torsional warping, Vlasov divided torsionaf moment into pure and flexural components which correspond to the St Venant and torsional warping shear flows, respectively. For this to be possible, new stress resultants were introduced, referred to as torsional bimoment B and warping torsion T,,., in addition to the usual stress resultants of bending moment M, torque T and flexural shear I’. The analysis of torsion for thin-walled beams was later reformulated and generalized by Benscoter [2], Kollbrunner and Basler [3] and Heihg [4] for multicell boxes with arbitrary cross-sections and by Dabrowski [5] for box girders curved in plan. An analogy between the differential equation describing the response of a box girder to the distortional component of the loading and that of a beam on an etastic foundation was also proposed by Vlasov [I]. The analogy arises from the out-of-plane rigidity against differential bending of the top and bottom slabs of the box girder which provides a continuous elastic support for each half of the section. Subsequently, Wright et al. [6] developed the analogy for the distortional analysis of single-cell box girders with lon~tudinally and transversely stiffened plates. Steinle [7] derived the differential equation governing the distortional behaviour of a rectangular single-cell box girder. The distortional stress resultants are represented by the distortional bimoment D and distortional moment Md and the expressions for the distortional stresses are analogous to those found in warping torsion theory. Thin-walled heam theory permits the distribution of longitudinal direct stress CTand shear how Fat any

&i,

This approach enables the dist~bution of direct and shear stresses (on the median lines of the walls) to be described by the familiar expressions used in simple beam theory, but modified by two additional terms to account for the effects of warping torsion and distortion of the cross-section.

I

THEORY

M,x

I,+I,+I,+r,

‘ds 0%’

(3)

where r, is a radius measured from the shear centre S to the peripheral coordinate s on the median line of the wall, and R represents twice the enclosed area of the cell (Fig. 3a). The torsional warping sectorial coordinate tit represents the level of the out-of-plane warping due to a unit rate of twist and has zero value at points on the cross-section which display zero warping. In the above expression, the first term is the sectorial coordinate for an equivalent open section formed by introducing an imaginary cut in the closed cell and is evaluated around the entire section. The second term applies only to the closed part of the section effectively reducing the level of warping displa~ment by restoring compatibility at the imaginary cut. The cross-sectional distribution of Q

@I

Fig. 3. Torsional warping scctorial coordinate 6: (a) method of determination and (b) distribution around a typical single cell trapezoidal ‘box girder with side cantilevers.

B. KERMANI and P. WALDRON

430

for a typical single cell box section is shown in Fig. 3(b). The torsional bimoment B and warping torsion T,, which are the resultants of the direct and shear stresses caused by restraint to torsional warping, respectively, may be defined in terms of a non-dimensional torsional warping function f as B = -E&j-”

is the familiar second moment of area for pure torsion. The term S;, in eqn (2) is a further sectorial function describing the shear distribution around the section due to torsional warping restraint and is given by

S,=&-$

(4)

and

S,.r,ds ss

(11)

in which S+ is obtained from T,,,= B’ =

-El,f”

(5)

in which I$, is the torsional warping constant by

(6) The warping functionfis Q, by the expression

S<.=

given

related to the angle of twist

iidA. I0

The derivation of the various sectorial and geometrical properties of the section permits the differentiat equation describing torsional warping in thin-wailed beams to be expressed in the following general form EK,f i”- pGJf” = pt.

in which 1, is called the central second moment of area given by I,=

rfdA.

(8)

(12)

(13)

Dabrowski [St presents solutions for the various stress resultants in terms of the applied loads and the boundary conditions at end 1 which, for straight girders under general loading conditions, are shown in Fig. 4. The longitudinal distribution of torsional bimoment B and warping torsion T, may be expressed as

f The term p is called the torsional parameter and is defined as p = 1 -J/I,,

warping

shear

(9)

B= -f

-Eii,

GJfi sinh kz + B, cash kz + i T, sinh kz

Tisinhkz

- $,$,

t,(cosh kzj, -cash

where

kzi2) (14)

J2.-, fs

(10)

and T, = B’.

Fig. 4. Straight member subjected to generalized torsional and distortional loading.

(15)

Analysis of continuous box girder bridges

The term k in the above equation is a measure of the rate of decay of warping restraint effect along the beam and is given by k2 = pGJ/EI, .

(16)

Distortional behaviour The single-cell box girder of deformable cross-section distorts under eccentric load into the shape shown in Fig. 5. The comer points of the box are displaced vertically by an amount + v, and by a,, ub in the horizontal direction at the top and bottom, respectively. A characteristic distortional angle yd may be defined representing the change in angle between the top flange of the cross-section and the inclined side web as described by the box comers. This may be written as Yd=YF+Ywl

(17)

where yp and yW are the rotations of the top flange and side web of the cross-section, respectively, obtained from the following equations YF =

Wb,

Yw=(u,+dlh.

(18)

431

in which ui is called the distortional sectorial coordinate. For a singly symmetrical cross-section, the distribution of the distortional warping coordinate is anti-symmetrical, as shown in Fig. 6(a), and the distortions1 warping displacement is zero on the axis of symmetry. The distortional warping displacements are not generally constant along the beam since they are accompanied by some form of axial restraint. This results in the formation of distortional direct stresses which are self-equilibrating and, as in the case of warping torsion, there are no resultant bending moments and direct forces at any section. This condition may be expressed in terms of ~5 as

From the distribution of ~5 shown in Fig. 6(a), the second and third conditions in eqn (21) are clearly satisfied. The first condition yields a relationship between the distortional sectorial coordinates at the top and bottom of the webs. For the top and bottom comer values of the righ-hand web, for example, (3 may be expressed [8] as

(19) (55= -/Id&,

The other form of displacement which occurs as a result of distortion of the cross-section is distortional warping displacement w,. To develop an approximate theory for dis?ortion of the cross-section, it may be assumed that this component of warping is proportional to the rate of change of the distortional angle yd and that the in-plane displacement is accompanied by sufficient warping to annul the shear strain. This yields an expression for the longitudinal distribution of the distortional warping displacement thus wd= -y;ui b.

(20) b,

(22)

where B = (1 + 2b,/bA3b: t, + b,. tw(26, + bb) b: t, + b, t,.(2b, + b,) ’

(23)

Furthermore, an explicit expression may be obtained for the distortional sectorial coordinate (s by considering the free body diagram of webs and flanges of a box beam and considering the local forces and displacements associated with each element [8]. For the general trapezoidal box beam, the distortional I_

b.

1

Fig. 5. Deformed shape of a typical single cell box girder subjected to torsional load.

B. KERMANIand P. WALDRON

432

Transverse frame action around the box is another form of resistance to distortion of the cross-section which adds to the resistance caused by the distortional warping restraint. This is caused by the transverse flexural stiffness of the walls forming the closed cell of the box when the section is subjected to distortional loading. The distribution of transverse bending moments for a typical single cell box-beam is given in Fig. 6(b) where the transverse bending moments at the top and bottom of the webs are given by

(4

5

6

rn&pP)

(b)

m,(bottom) Fig. 6. Distribution of (a) distortional sectorial coordinate (s and (b) transverse distortional bending moment m6, around a single cell trapezoidal box section.

sectorial coordinate at node 3 (Fig. 6a) is found to take the following form hb;b,

_

03=2(b,+b,,)(/?b,+bb)’

The distortional bimoment D and distortional moment Md, defined in eqns (1) and (2) are given by D = -E&y;

(2%

and M,=D’=

-j&y;

= y

(I - q).

Wb)

In the above expressions, r~is a cross-sectional constant and Kd denotes the distortional frame stiffness of the box. Values of Kd may be evaluated using the method of influence coefficients by considering a unit length box beam loaded by diagonal forces with a unit horizontal component of load [8]. Using energy methods, the differential equation describing the response of a box girder to distortion may be derived in the following form EI,y:

+ Kdyd= md

(31)

in which the applied load consists of pure distortional forces md only. For a box beam loaded with horizontal and vertical components of torsional load t,. and t,,, as shown in Fig. 7, the distortional component md of the load is defined as

(26) M,, = (t,b,/b, - t,,)/2.

(32)

in which I$=

G2dA sA

(27)

is called the distortional warping constant. The term .S, in eqn (2) is a further sectorial function describing the shear distribution due to distortional warping around the section and is obtained from

s

S;=S,--$S,r,&,

\d3 / \

I

(28)

5

where SC=

‘GdA. i (I

(2%

It is apparent that there is a close analogy between the sectorial properties and functions which have been introduced in this section with those developed earlier for the case of warping torsion.

Fig. 7. Vertical and horizontal components loading.

of torsional

Analysis of continuous box girder bridges For a solution of eqn (31), an analogy can be drawn between the differential equation for distortion of a single-cell box girder and the beam-on-elastic foundation (BEF). This enables the expressions given by Hetenyi [9] to be used with the various bending terms replaced by the appropriate distortional properties. For the straight girder shown in Fig. 4, let us consider only the components of concentrated distortional moment Mdi (i = 1, m) and uniformly distributed distortional moment md, (j = 1, n) applied at various positions along the beam. The solution for distortional bimoment D and distortional moment Md in terms of initial boundary conditions at end 1 under the genera1 loading condition are given by

433

joint compatibility. The reverse order is adopted for the stiffness method. However, the major requirement for both methods is the determination of the genera1 relationship between member end-loads and the corresponding end-displacements. For the member fully fixed at end 1 shown in Fig. 8, the unknown displacements at end 2 may be expressed in terms of the applied end loads in the following flexibility equation (37)

@Z1 = P%* 13

where {e2} and {pr} are the vectors containing the various displacement and load terms corresponding to the six degrees of freedom system considered here (38a)

{e,Y= {&W*f;Y:*Yd2} {pzjT=

(38b)

{%T,V,B,D,&J.

The flexibility matrix [JJ contains individual flexibility coefficients f,, (m = l,6; n = 1,6) given by

“L” =

ap

“;p,

IV&= D’

(34)

in which 1, a distortional decay parameter, is given by A4 = K,/4EI,.

and the various following form:

geometrical

(35) functions

take

the

(3%

3

2m

”

where U is the total strain energy stored in the linearly elastic system expressed [8] in terms of the various end loads contained within {p2}. In the stiffness method the displacements are considered to be the basic unknowns and the relationship must be established between the loads {p,}, {pz} and the displacements {d,}, {d2} at both ends of the element shown in Fig. 9. This may be expressed in the following familiar form

F, (AZ) = (cash 1z cos AZ)

FZ(AZ) = (cash AZsin It + sinh Iz cos 1z)/2 F,(lz) = (sinh lz sin 1z)/2 F4(dz) = (cash llz sin Lz - sinh lz cos 12)/4. ANALYSIS

OF COMPLEX

(36)

STRUCI-IJRES

For box girder bridges with complex structural arrangements, the boundary conditions required to yield a solution for the effects of warping torsion and distortion are difficult to formulate. A more generalized approach is therefore required in which the structure is envisaged as an assemblage of elastic structural members connected together at discrete points. The flexibility and the stiffness methods are both suitable for the analysis of structures idealized in this way [lo], the essential theoretical difference between them being the order in which the conditions of joint compatibility and equilibrium are applied. In the flexibility approach conditions of equilibrium are employed first which then give rise to equations of

in which the various elements of the stiffness matrix are defined in a local coordinate system and depend on the geometry of the member and its material properties. The great attraction of the flexibility method is that the number of equations to be solved is equal to the number of redundant forces. In most cases of spine beam construction, these are considerably fewer than the total number of degrees of freedom at all the nodes. In the stiffness approach, the displacements at the nodes are the unknowns which generally results in a much greater computational effort. However, with the capacity and speed of modem computers,

End 1

End 2

Fig. 8. Thin-walled beam element fully fixed at end I

B. KERMANI

434

and

P. WALDRON

may then be inverted and be substituted in eqn (41) to yield a solution for the various member submatrices. Assembly of the system stiffness matrix Fig. 9. Basic

end load/displacement system.

solution of a large system of equations is a relatively trivial part of the complete analysis and the automatic procedure of the stiffness method has become more attractive to use. The equivalent beam method A method of elastic analysis has been developed, based on the stiffness approach, which is suitable for rapid solution by computer. It is equally applicable to both straight and curved thin-walled beams and is an extension of the work by Waldron [I l] which considers torsional warping only. In addition to the four degrees of freedom system developed previously to consider moment, shear, pure torque and warping torsion, two more degrees of freedom have been incorporated in the formulation to account for the distortional effects. The additional degrees of freedom are the distortional angle Y,, and the rate of change of distortional angle ~2. The stiffness method adopted makes use of discrete beam elements and is thus applicable to structures incorporating features such as changes of section, complex systems of loading and support, curvature, skew and other irregularities. The member stiffness submatrices in eqn (40) may be obtained from the member flexibility matrix in eqn (37) from the following relationship

K,l= [-a~l-‘~N+

So far, all the forces and displacements have been established with reference to local member coordinate systems defined by the orientation of the member ends. For the purposes of assembling the global structure stiffness matrix from the individual member stiffness matrices, it is first necessary to define common system axes. In the case of straight spine structures, both local and system axes may be selected to coincide. However, this may not be the case for the more general situation in which the structural configuration may include bifurcations, horizontal curvature in plan or discontinuous changes in direction. Where the local member and global system axes do not coincide, load and displacement vectors may now be expressed in system coordinates as

Wd

where the transformation matrices [T,] and [T,], for an angle $ between member and system axes, are given by

b-f1 [T,I=[T,I= i

cos If5

sin $

0

0

0

0

-sin+

cosJI

0

0

0

0

0”

1 0 0 0 I . (44) 0 1 0 0

0 0

0 0

0 0

[&,I = -VT’[JY’ Ll=

VT’>

(41)

where [A] and [E] are compatibility and equilibrium matrices respectively relating the forces at end 1 to those at end 2, thus IP, 1+ VUPJ

-

[Al{d, I= 0.

1

0”

0 0

1 0 0 1

By substituting for {p,}, {pz}, {d,} and {d2} from eqns (43) into (40), the basic stiffness relationship may be re-expressed in system coordinates as

(42)

In straight thin-walled beams, the basic structural actions do not interact and the coefficients associated with bending, torsion and distortion in the flexibility matrix [Fj become uncoupled. In this case, the member flexibility matrix may be expressed in explicit form [8]. However, the terms associated with the distortion of the cross-section are very tedious to determine explicitly in comparison to the rest of the terms in the flexibility matrix and are best calculated numerically. Once a solution is obtained for [J’j it

As a result of the orthogonal nature of the transformation matrices this can be written more simply as

{PI = mwImw

= mm.

(46)

The overall stiffness matrix of the complete structure may now be formed from the stiffness matrices of the

Analysis

of continuous

individual members in the usual way. For a structure containing n nodes, the six degrees of freedom element described here results in 6n load/displacement relationships, which may be written more simply as

box girder

bridges

435

in which the overall structure stiffness matrix [K,] is a 6n x 6n symmetrical matrix. For spine beam structures this matrix will be sparse consisting mostly of elements concentrated in a narrow band around the leading diagonal.

modified program WARPY [8] is written in Fortran 77 and is suitable for solution on personal computers of modest capacity. Two types of load may be applied to the structure, either concentrated loads at the nodes, or uniformly distributed loads between nodes. In order to be able to use long beam elements in the idealization of the structure, a subroutine has been included in the program which calculates the various stress resultants at any number of intermediate positions between the nodes. The input data required for solution is as follows:

System restraints

(a) basic parameters

It is apparent from the transformation matrices given in eqn (44) that, apart from the two rotational terms about the two orthogonal axes, the remaining terms are unaltered when transformed into system coordinates. It is therefore possible to restrain these components of deformation by the removal of the appropriate rows and columns from eqn (47) in the usual way. For the two rotational components, the procedure is not as straightforward and requires a more complex modification of [K,] before removal of the appropriate rows and columns [12]. Having modified eqn (47) to account for all the boundary conditions, the system stiffness matrix is no longer singular and may be inverted. In this way a solution can be found for the unknown displacement terms in the vector {d,} expressed in system coordinates. Subsequently these may be converted into local member coordinates by use of eqn (43b) which in turn may be fed into eqn (40) to provide the unknown member forces. Fixed-end forces due to uniformly distributed loading

In a matrix analysis of this type, loads must be applied at the nodal points. For uniformly distributed loads one solution is to approximate the loading by using a large number of beam elements and to replace the distributed load with equivalent sets of concentrated loads applied at the nodes. An alternative requiring considerably fewer elements is possible if the uniformly distributed load is applied directly as equivalent fixed-end forces. This latter approach has been considered here [8] since, in the first method, the size of the analytical model increases significantly thereby eliminating much of the computational benefit of the method. COMPUTER

PROGRAM

The stiffness approach is essentially a computerbased analytical method and is unsuitable for hand calculations, even for simple structural configurations. A computer program WARPIT [l l] exists which has been developed to calculate the effects of torsional warping restraint. This has been modified and extended by adding two extra degrees of freedom to account for distortion of the cross-section. The

(number of beam elements, nodes, restraints, etc.), (b) node orientation (with respect to the system coordinates), (4 member properties (geometry and section stiffnesses, etc.), (4 restraints (node reference number and restraint type), and (4 magnitude of loading (either uniformly distributed or concentrated). The output contains the following data for each load case: (a) joint deformations (in system coordinates), (b) joint forces (in local member coordinates for each member), (c) system restraint forces, and (d) distribution of forces along each member (optional). APPLICATION

OF EQUIVALENT

BEAM METHOD

The proposed equivalent beam method is examined here in two ways. Firstly, the inclusion of distortional effects based on the BEF analogy in the element formulation and computer program is verified by comparison with the exact solution for a simply supported beam subjected to concentrated mid-span load. Subsequently the method is applied to a similar simply supported single-cell box beam subjected to uniformly distributed torsional loading and compared with published solutions obtained by other analytical methods. Finally, the method is tested against a two-span continuous box girder for which results have been obtained experimentally by the authors. Simply supported straight box beam with concentrated loading

Consider the example of a straight, simply supported box beam with the uniform cross-section shown in Fig. 10, adopted by Maisel and Roll [13]. The beam has a span of 30 m and is subjected to a concentrated load of 1000 kN placed eccentrically over one web at mid-span. For the purposes of the equivalent beam method, the structure was simply idealized as two beam

B. KERMANIand P. WALDRON

436

1MX)KN

-,-

I-

-I-

,

Fig. 10. Loading and cross-sectional dimensions (mm) of the straight beam model adopted by Maisel and Roll [13].

elements by placing one node under the load at mid-span and one further node at each end. The option available in the program of obtaining intermediate values to describe the distribution of the various stress resultants along the length of the elements was exercised. The input data for the member properties, considering Young’s modulus and Poisson’s ratio of the material to be 34.5 kN/mm2 and 0.15, respectively, are as follows:

tortional moment for the box beam example are given in Fig. 11. These results are identical to the exact solution obtained by solving the appropriate equations given by Maisel and Roll [13]. It may therefore be concluded that the formulation and

‘?-

LOOkN.m

(a) Torque

EZ, = 5.527 x lOr3kN mm2

GJ = 4.888 x 10” kN mm2

600

1

454kN.m*

4w EI-W’ = 2 .736 x 10” kN mm4

p = 0.365

200 i 0

/\

1

(b) Torsional

1

bimoment

EZ-H’= 3 .495 x 10” kN mm4 Kd = 1.942 x 10s kN.

The sectional and sectorial properties of the box have been obtained using the Fortran computer programs SECTOR [14] and DISPRO [8]. Diaphragms are located only at the two ends and are assumed to offer no resistance to warping. However, they are infinitely rigid in their own plane and prevent distortion of the two ends. In other words, the exact boundary conditions were obtained by ensuring that the displacements u, 4 and yd at the two ends were zero. The magnitude of the shear, torsion and distortional moment for .the concentrated load condition considered are as follows:

1500

7

1

(c) Distortional

1285 kN.m2

bimoment

500 kN.m 500 250

n

I’=lOOOkN T=2OOOkNm M,,= 1OOOkNm. The results obtained for the distribution of torque, torsional bimoment, distortional bimoment and dis-

(d) Distortional

moment

Fig. 11. Longitudinal distributions of the various stress resultants for the simply supported box beam under eccentric point load at mid-span.

Analysis of continuous box girder bridges

computational procedure associated with the calculation of the distortional effects by the equivalent beam method are correct.

used in the equivalent follows:

beam

analysis

were as

EI x-- 1429 x 1O’rkN mm* .

Simply supported straight box beam under uniformly distributed loading

A simply supported box beam was analysed by Kristek [15] as an example of his proposed theory based on the folded plate approach. The span of the beam was 80 m and the cross-section was rectangular without any side cantilevers, as shown in Fig. 12(a). The beam was loaded by equal but opposite uniformly distributed line loads applied vertically at each corner resulting in total uniformly distributed torsional and distortional moments of 209.6 and 104.8 kN m/m, respectively. This example has been solved using the equivalent beam method using two beam elements in order to get the deformations and stress resultants at mid-span. Young’s modulus was taken to be 31.05 kN/mm* and Poisson’s ratio to be 0.15, as in Kristek’s example. The various section properties

437

GJ= 1.525 x 10” kN mm2 EZ5--7609x .

lO*‘kNmm’

p = 0.378 EI-W= 5 *023 x 102’kN mm’ K,, = 9.202 x 10’ kN.

Figure 12(b) indicates the shape of the deformed cross-section and the comer displacements at midspan; and Figs 12(c) and (d) show the distributions of warping stresses (torsional and distortional combined) and transverse bending moments respectively at mid-span. The distribution of the shear stresses

p = 9.828 N/mm

P

-----m 10.66Sm

.

4e

------1

I

_____I

:______

0))

1

-TV

&‘I

Fig. 12. Comparison of results for a simply supported box beam under uniformly distributed torsional loading.

B.

438

KERMANI

and P. WALDRON

Table I. Comparison of results obtained from various analytical methods

4

Kristek [I 51 Mikkola and Paavola 114 Boswetl and Zhang [ 171 This study

4

aw

(mm)

(mm)

(N/mm*)

I.245

3.734

0.340

1.257 1.303

3.746 3.886 1.245 3.787

(kNLz/mm)

0.344 0.359 0.350

around

the cross-section over the supports is also in Fig 12(e). The same example has also been considered by Mikkola and Paavola 1161 and Boswell and Zhang [17]. Their results have been tabulated for comparison with those calculated by the equivalent beam method in Table 1. In their finite element idealization, Mikkola and Paavola used ten elements whereas Boswell and Zhang used two elements for each half of the beam. The agreements between the results obtained by the different methods is quite satisfactory, the maximum difference between the results obtained by the equivalent beam method and the others being approximately 4%. illustrated

This final example demonstrates the application of the equivalent beam method to continuous structures. The experimental model described here, the dimensions of which are shown in Fig. 13, was conceived expressly to validate the proposed analytical method. Its construction, instrumentation and testing are described in detail elsewhere [8]. The single cell box beam model was continuous over two spans and constructed from a sand and epoxy resin mixture with a modulus of elasticity of 16.2 kN/mm* and a Poisson’s ratio of 0.35. Diap~a~s were focated at the three supports only.

27.2

27.4 28.6 27.6

(N$m*)

(N$mZ)

(N/zm*)

0.297

0.189

0.148 0.151

0.302

-

0.197

74 W/mm21

0.206 0.210

Full instrumentation was provided at two sections which were located near the mid-span (section A) and adjacent to the central support (section B), as shown in Fig. 14. Rosette gauges were bonded to both the inside and outside surfaces of the box walls at these two sections to enable the in-plane and flexural components of strain to be separated. A number of different load types and support con~gurations were investigated. However, for the purposes of this paper results are presented for a single case in which the model was fixed against torsional rotation and vertical displacement at all three supports and subjected to a uniformly distributed line load of 0.468 N/mm placed eccentrically over one web in one span only. Support diaphragms were assumed to prevent distortion of the crosssection but to offer no restraint to torsional and distortional warping displacements. The various section properties used in the equivalent beam analysis were as follows: EZx = 2.023 x lo* kN mm2 GJ= 1.798 x 10s kN mm2 EZ.W= 2 *359 x 10” kN mm4 j.f= 0.250

Fig. 13. General arrangement of the straight model (dimensions in mm).

Analysis of continuous box girder bridges

439

UN

Fig. 14. Strain gauge positions for the straight model (dimensions in mm). E& = 2.359 x 10” kN mm4

I(*=1.606 x lo* kN. Figure 15 shows the longitudinal distribution of the bending moment, torsional bimoment and distortional bimoment obtained analytically. These values were substituted into eqn (1) to obtain the cross-sectional distribution of direct axial stress at any position along the beam and compared with those obtained experimentally from the two fully instrumented sections. Figure 16(a) corresponds to section A when the load was applied over the instrumented span; Fig. 16(b) to section B when the load was applied over the other span. Values of transverse bending moment per unit length around section A of the model when that span was loaded are also indicated in Fig 17. Here the positive values of bending moment are those corresponding to tensile stresses on the inside face of the box. These comparisons show that the theoretical values obtained from the equivalent beam method are in close agreement with the experimental data. Other test results not reported here for concentrated loading and for alternative central support arrangements compared equally favourably with the proposed theory.

number of elements. External loading may include concentrated loads applied at the nodal points and uniformly distributed load applied between the nodes.

0.658 x lO”N.mm

-~~ (a) Bending moment

(b)Torsionalbimometi

CONCLUSIONS

A method has heen developed for the elastic analysis of single cell box girders with at least one axis of symmetry. The method, based upon the stiffness approach with six degrees of freedom at each node, may be applied to all straight thin-walled box girder bridge configurations with a minimum

(c) DistoftiDnel bimomeni

Fig. 15. Longitudinal distributions of various stress resultants along the straight beam model.

B.

KERMANI and UDL +

(a)

P.

WALDRON

The performance and accuracy of the equivalent beam method has been shown to compare favourably with some reported studies of simply supported and continuous single cell box beams subjected to a variety of different loadings. The method may therefore be used to predict the behaviour of deformable single cell box girders with confidence particularly during the conceptual design stage when a full 3-D finite element analysis is not justifiable.

REFERENCES UDL t

Vlasov, Thin-wailed Elastic Beams. National Science Foundation, Washington, DC (1961). 2. S. U. Benscoter, A theory of torsion bending for multicell beams. J. Appl. Mech., ASME Tram 76,25-34 I. V. Z.

(1954).

3. C. F. Kollbrunner and K. Basler, Torsion in Structures-An Engineering Approach. Springer, New York (1969). 4. R. Heilig, A Contribution to the Theory of Box Girders of Arbitrary Cross -sectional Shape (Translated from the German by C. V. Amerongen), Publication No. 145. Cement and Concrete Association (1971). 5. R. Dabrowski, Curved Thin-wailed Girders (Translated from the German by C. V. Amerongen), Publication No. 144. Cement and Concrete Association (1972). 6. R. N. Wright, S. R. Abdel-Samad and A. R. Robinson, BEF analogy for analysis of box girders. J. Struct. Div.,

(‘4

1.O N/mm2 I-l SCALE __ 0

AnalyticalResults Experimentalresults

ASCE 94, 1719-1743 (1968).

Fig. 16. Comparison of measured and predicted direct stresses at two instrumented sections under eccentric uniformly distributed loading. UDL

c

7. A. Steinle, Torsion and cross-sectional distortion of the single cell box beam. Beton und Stalbetonbau 65, 215-222 (1970).

8. B. Kermani, Single cell box girder bridges of deformable cross-section. Ph.D. thesis, University of Bristol (1988). 9. M. Hetenyi, Beams on Elastic Foundation. University of Michigan Press (1946). 10. R. K. Livesley, Matrix Methods of StructureAnalysis. Pergamon Press, London (1964). II. P. Waldron, Equivalent beam analysis of thinwalled beam structures. Comput. Struct. 26, 609620 (1987). 12. P. Waldron, Stiffness analysis of thin-walled girders. Proc. Am. Sot. Civil Enars. _ Struct. Div. 112, 13661384

5.0 NmnVmm SCALE b-i -

AnalyticalResults L)

Experimentalresults

Fig. 17. Comparison of transverse distortional moments around the box at section A under eccentric uniformly distributed loading.

(1986). 13. B. I. Maisel and F. Roll, Methods of Analysis and Design of Concrete Box Beams with Side Cantilevers. Publication No. 42.494. Cement and Concrete Association (1974). 14. P. Waldron, Sectorial properties of straight thin-walled beams. Compur. Struct. 24, 147-156 (1986). 15 V. Kristek, Box girders of deformable cross-sectionsome theory of elasticity. Proc. Inst. Civ. Engrs 239-253 (1970). 16 M. J. Mikkola and J. Paavola, Finite element analysis of box girders. Am. Sot. Civil Engrs, Struct. Div. 106, 1343-1357 (1980).

17. L. F. Boswell and S. H. Zhang, A box beam finite element for the elastic analysis of thin-walled structures. Thin-Walled Struct. 1, 353-383 (1983).