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Static and free vibration analysis of variable-depth bridges of arbitrary alignments using the isoparametric spline finite strip method
ELSEVIER
0263-8231
Thin-Walled Structures 24 (1996) 19-51 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/96/$15.00 (95)00038-0
Static and Free Vibration Analysis of Variable-depth Bridges of Arbitrary Alignments Using the Isoparametric Spline Finite Strip Method
F. T. K. Au & Y. K. Cheung Department of Civil and Structural Engineering,Universityof Hong Kong, Hong Kong (Received 19 October 1994; accepted 21 February 1995)
ABSTRACT The isoparametric spline finite strips for shells are employed to solve static and free vibration problems of variable-depth bridges of arbitrary alignments. Using this approach, a continuous bridge is first split up into substructures. Each substructure is modelled as an assemblage of isoparametric spline finite strips. Compatibility between substructures is ensured by suitable transformation at the interface. The presence of support diaphragms and bearings can also be accounted for. This method retains the computational efficiency of the spline finite strip method while it is much more flexible in geometric modelling. Solutions of this method are compared with other available solutions, and good agreement is observed.
1 INTRODUCTION The finite strip method can be considered as a special form of the finite element procedure 1'2 using the displacement approach. The semianalytical finite strip method, pioneered by Cheung, 3 is an effective method of structural analysis for engineering structures with constant geometrical properties along a particular direction. The method is applicable to single-span and multiple-span bridges with straight or circular alignment in plan. It is especially efficient in the analysis of single span simply supported right box girders for which the Fourier sine series is 19
F. T. K. Au, Y. K. Cheung
20
a suitable set of displacement functions. However, in the solution of problems with concentrated forces, multiple spans, discrete supports at strip ends etc., difficulties were encountered. The spline finite strip method 4 was later developed to overcome the above difficulties. The method was first successfully applied to static and free vibration of thin rectangular plates 4' 5 and subsequently extended to parallelogram skew plates. 6' 7 Plates of arbitrary shapes 8"9 were also solved utilizing the mapping concept. In this method, the plate of arbitrary shape is mapped into a regular square domain which is discretized into spline finite strips for analysis. Subsequent developments of the method also cater for shells of arbitrary shapes, l°'lt box girder bridges of curved alignments, ~2'13 etc. In the above developments, the thin plate or shell approach was employed. In addition, both semi-analytical and spline finite strip methods had been successfully applied to the analysis of rectangular and circular Mindlin plates.~4" t5 In order to satisfy modern traffic requirements, highway bridges are often curved to tie in with the approach roads. Moreover, as the highway authorities are increasingly conscious of the maintenance costs of bridges, many of them are specifying that all future bridges shall be designed with a minimum of bearings and movement joints. In other words, continuous bridges will become more and more popular in future and many such bridges are in the form of variable-depth 'sucker' bridges. In an attempt to respond to such complex requirements, the isoparametric spline finite strip method was introduced and the flexibility to model various geometric shapes was further enhanced. The method was first developed for plane structures ~6 and later extended to shells. ~7 It was also reported that the semi-analytical finite strip method was extended to solve right continuous haunched plate-girder bridges, t8 In this paper, the isoparametric spline finite strip method is applied to the static and free vibration analysis of variable-depth bridges of arbitrary alignments.
2 ISOPARAMETRIC SPLINE FINITE STRIP A typical isoparametric spline finite strip for shell is shown in Fig. 1. The geometry of the strip is described with respect to the global coordinate set (X, Y and Z) by the product of two functions: spline functions along the length and Lagrange polynomials transversely, i.e. m+l
{P((,~/)} = ~
p
y~[NIj((,T/)] {Q/j}
i=-Ij=l
(1)
Vibration analysis of variable-depth bridges
21
/ y /
/
/
/ /
/
x Fig. l. Typical segment of finite strip for a shell with coordinate systems.
where {P((, rt) } is the position vector of a point on the mid-surface of the strip domain, rn is the number of segments in the strip, p is the number of nodal lines defining the strip, {Q;j} is the ith curve defining vector of the j t h nodal line of the strip and [Nij ((, z/)] is the shape function matrix associated with {Qij}- The shape function matrix [N,-j((, ~/)] is given by [N~j((,~)] =
N~j((,g)[I]= LP(()4)i(g)[I]
(2)
in which LP(O is the j t h Lagrange polynomial, ~)i(7]) is the Ba-spline function of unit section length with the section-knot 7/= 7/,. as the centre and [I] is an identity matrix of appropriate order. Figure 1 also shows the local coordinate set (x~,y~ and z~) of an arbitrary point on the strip and the nodal coordinate set (x.,y. and z.) at a nodal line. They are so defined that the z~ and z. axes are normal to the mid-surface of the strip and the y~ and y. axes are tangential to the ¢ = constant curves. The displacement vector {6} defined as
22
F. T. K. Au, Y. K. Cheung
{6} = [u v w Ox~, Oy~,]T
(3)
consists of the three translational degrees of freedom (DOFs) u, v and w along the global X, Y and Z axes, respectively, and two rotational DOFs 0x~ and 0y~ about the local x~ and y~ axes, respectively, in the right-hand sense. The displacement {6((, r/)} of point {P((, r/)} on the strip is similarly expressed in terms of the displacement parameters {6ij} associated with curve defining vector {Q/j} m+l
{6(¢,r/)} = Z
p
Z [Nij(~'r/)] {6(j}.
(4)
i=-lj=l
To facilitate subsequent evaluation of strip matrices and load vectors, the translational displacements u,, v, and w~ along the local x,, y~ and z~ axes are expressed in terms of the respective global translational displacements using the transformation matrix [R,] v~ w~
=
m2~ n2~/ [_/3~ m3~ n3~J
/ 12~
v
w
= [R~]
v w
(5)
where (ll~,ml~ and nl,), (12~,m2~ and n2,) and (/3~,m3~ and n3a), are the directional cosines of the unit vectors [,,j~ and k~ along the x~, y, and z, axes of the local coordinate set with respect to the unit vectors [, j and/~ along the global X, Y and Z axes, respectively. Once the displacement functions have been chosen, the strip characteristics can be obtained in line with the standard finite element method and these shell strips will be the basic tools for subsequent analysis. 3 GEOMETRIC DESCRIPTION OF BRIDGE DECK Apart from the relatively rare major bridge crossings, the vertical profile and horizontal alignment of a highway bridge are invariably governed by traffic requirements. Normally straight lines and parabolic curves are used in vertical proffl,es, whereas straight lines, circular curves and spiral curves are employed in horizontal alignments. The vertical alignment of a bridge seldom affects its structural behaviour. However, it is important that the horizontal alignment be accurately modelled in structural analysis. An isoparametric spline finite strip is essentially a C2 shell surface. A longitudinal kink, i.e. in the r/direction, between adjacent strips is allowed and this is achieved by proper description of strip geometry. If the profile of a bridge consists of breaks in slope or curvature along its length, it has to be divided into substructures. Figure 2 shows a number of typical
23
Vibration analysis of variable-depth bridges
SS
I
•L
SS
2
~I~
SS
_J
3
(a)
SS
1
_.
SS
2
LI
ss3
(b)
l SS 1
J SS 2
~ - SS 3
SS 4
SS
7~
(c)
JrJr
Scheme A SS 1 _L s s 2
__ ss 3
I
SS 4
I
SchemessB 1 SS 2
(d)
Fig. 2. Typical continuous bridges and schemes of sub-division into substructures (SS). (a) Constant depth; (b) variable depth with curved soffit; (c) partial length of constant depth with straight haunches; (d) partial length of constant depth with parabolic haunches.
continuous bridges. Both the bridge of constant cross section (shown in Fig. 2(a)) and the variable-depth bridge (shown in Fig. 2(b)) can be conveniently divided into three substructures for analysis so that each substructure is smooth in the r/direction. The bridge with straight haunches (shown in Fig. 2(c)) should be divided into seven substructures for analysis. Figure 2(d) shows a continuous bridge with parabolic haunches. Theoretically speaking, there is a break in curvature at the junction between a straight section and the adjacent parabolic haunch, and hence the structure should be divided into seven substructures for exact geometric modelling, as shown in Scheme A. A more convenient arrangement is shown in Scheme B in which the structure is divided into three
24
F. T. K. Au, Y. K. Cheung
substructures for analysis, although in this case some approximation in geometric modelling is involved.
4 EFFECTS OF D I A P H R A G M S A N D BEARINGS A box girder structure consists of top and bottom flanges connected by vertical or inclined webs to form a cellular section. It is one of the most popular forms of highway bridges, primarily because of its high flexural and torsional rigidities. However, the thin-walled section is rather weak in resisting distortion and therefore stiffening is often required in sections subjected to large concentrated loads, especially at supports. The stiffening system may be in the form of a truss, a frame or most frequently a solid diaphragm with or without opening. The stiffening system is normally designed to have its primary strength within the plane of the cross section. Hence a diaphragm is normally assumed to be perfectly rigid in its plane and perfectly flexible out of plane. Bridges are usually supported on bearings. In normal practice, bearings are set horizontal to restrain any vertical deflection. Horizontal movements at a bearing can be controlled. A bearing can be free sliding in which horizontal movement is unrestrained. It can be guided sliding so that horizontal movement is only allowed in a certain direction. It can also be fixed so that no horizontal movement is allowed. Rotations at the bearings are usually taken to be free. A bridge deck is considered torsionally restrained at a support with a diaphragm if two or more bearings are provided because any rotation of the diaphragm within its own plane is restrained, as shown in Fig. 3(a). If a single bearing is provided as shown in Fig. 3(b), rotation of the diaphragm within its own plane is allowed but only about the bearing.
5 T R A N S F O R M A T I O N FOR STRIPS IN EACH SUBSTRUCTURE In the formulation of stiffness matrices, mass matrices and load vectors, the global translational DOFs u, v and w and the local rotational DOFs 0x~ and Oy~ have been used. Apart from the simplest case of slab bridges in which all strips are coplanar, a 6-DOF analysis is preferred incorporating the additional drilling DOF Oz~,. The global translational DOFs u, v and w are used throughout each substructure to obviate the need for transformation. However transformation is frequently required for the rotational DOFs. The transformation consists of expressing all the rotational DOFs at a nodal line, in terms of the rotational DOFs according to the nodal set,
Vibration analysis of variable-depth bridges
25
(a)
!
J
(b)
Fig.
3. Typical supports for box girder bridges. (a) Supported on two bearings; (b) supported on a single bearing.
which is the local coordinate set of a chosen adjoining strip, and the procedure has been described in Ref. 15.
6 T R A N S F O R M A T I O N FOR COMPATIBILITY BETWEEN SUBSTRUCTURES At the interface between adjacent substructures, compatibility must be ensured for all DOFs. The displacement of each point on a strip within a substructure is expressed in terms of the displacement parameters associated with the curve defining vectors of the nodal lines defining the strip. First of all, it is necessary to express the physical displacement at each end knot of a substructure in terms of the displacement parameters of the corresponding nodal line. Concentrating on the deflection of the j t h nodal line and dropping the index j for simplicity, the deflection w~ at the ith
F. T. K. Au, Y. K. Cheung
26
knot point depends only on the displacement parameters w;_ 1, wi and w~+ i associated with three vertices of the curve defining polygon, i.e.
I Wi+l } [Wi]
Wi- 1
I Wi+l } wi = wi- 1
Wi+ Wi Wi_
2 3
[i °
(6)
0
The index i may take the value of zero or m depending on which end of the substructure is being considered. In this manner, all displacement parameters associated with the knot points 0 and m of the nodal lines are replaced by the corresponding physical displacements. As the global translational DOFs u, v and w are used for all substructures, transformation is again not necessary. However the nodal sets, according to which the rotational DOFs are defined, are often different for the adjacent substructures as shown in Fig. 4, and hence transformation is necessary. The transformation for rotational DOFs for nodal line j is as shown below
(7)
{6~}j2 = [Rs]j{ar}j,
where {6r}jl and {6r}j2 are the rotational DOFs at nodal line j at the interface with respect to the nodal sets of the left a n d right adjacent substructures, respectively.
{al} = E0x~0y~0=]T,
(8) i.2 •
[Rs] = | j . 2
lnl
Jn2 "Jnl
jn2 1~,,,
,
(9)
L/~.2 " '.,'~ /~.2 "L, /~.2 •/~., ^
^
i.l, J.l and/~.l are the unit vectors along the Xnl, Ynl and ZnJ axes of the nodal coordinate set of the left adjacent substructure and i.2, £2 and/~.2 are the unit vectors along the Xn2, Yn2 and zn2 axes of the nodal coordinate set of the right adjacent substructure.
7 T R A N S F O R M A T I O N F O R D I A P H R A G M S A N D BEARINGS The presence of a diaphragm at an interface between substructures requires special treatment. Figure 5 shows a typical support provided with a diaphragm to constrain some of the nodal lines. The diaphragm is assumed to be planar, perfectly rigid in its plane and perfectly flexible out
Vibration analysis of variable-depth bridges
27
Y
x
Fig. 4. Relationship between nodal coordinate sets for adjacent substructures at the interface.
of plane. It is further assumed that only the translational DOFs of the bridge are constrained by the diaphragm. One of the nodes constrained by the diaphragm is selected as the master node while all the others are taken as slave nodes. The in-plane displacement of any point lying on the diaphragm is therefore dependent only on two translational DOFs and one rotational DOF of the master node in the plane of the diaphragm. For convenience, the master node is often located at one of the bearing positions. If the diaphragm is supported on more than one bearing, this rotational D O F is set to zero as no rotation within the plane of the diaphragm is allowed.
28
F. T. K. Au, Y. K. Cheung
J"
Master
J
J
J
node
Fig. 5. A support provided with a diaphragm showing the master node and one of the slave nodes. Figure 6 illustrates the relationship between a slave node S at nodal line j and the master node M in a diaphragm. The orientation of the diaphragm is defined by a set of axes XM, YM and ZM in which the YM axis is normal to the plane of the diaphragm. The displacement of the diaphragm is characterized by the translational DOFs UM and WM along the XM and ZM axes, respectively, and the rotational D O F OyM about the YM axis in the right-hand sense. As the translational displacement at the slave node S is given by the global translational DOFs Us, Vs and Ws, it is necessary to transform such DOFs to USM, VSM and WSM which are given with reference to the XM, YM and ZM axes, respectively, as shown below {6~) = [Ro] {6~M}
(10)
where {6~} = [Us Vs Ws]T,
(11)
{6~M} = [USMVSMWSM]T,
(12)
[i. [M iM j'J i.j ]'h,, i. M 1 [Rol----/)"
(13)
[k.iM I~'jM k'kM i, j and/~ are the unit vectors along the global X, Y and Z axes and iM, jM and kM are the unit vectors along the XM, YM and ZM axes, respectively.
Vibrationanalysisof varhTble-depthbridges
29
z(w $ )
z
\d./.Yfvs) S ~N~-,.~''- xM / /"-..
y
/
xCu s)
\ ~-----~'~
XM(UM)
x
Fig. 6. Relationship between master node M and slave node S at a diaphragm. D O F vSM is unaffected by the presence of the diaphragm and DOFs USM and wSM are related to the DOFs at the master node M as follows =
WSM
i,0 01
WM
--(Ps -- PM)" iM
(14)
0yM
in which PM and Ps are the position vectors of master node M and slave node S, respectively. The orientation of the supporting bearing system is defined by a set of axes XB, YB and zB in which ZB axis is along the direction of major reaction as shown in Fig. 7. As the orientation of the bearing may be different from that of the diaphragm, further transformation is necessary before the boundary condition for the support is introduced. It is assumed that a bearing is attached to the diaphragm at the master node M. The displacement { ~ } with respect to the set of axes xa, YB and zB for the bearing system is related to the displacement {6h } at the master node by {6h} = [RB] {6~}
(15)
where
{•h} = [UM VMWM]T'
(16)
30
F. T. K. Au, Y. K. Cheung
Z
z M
Fig. 7, Relationship between coordinate set (XM,)'M and ZM) defining orientation of a diaphragm and coordinate set (xa, YB and zB) defining orientation of a supporting bearing system at master node M.
(17)
[RB]=
lM'lB "'B
1M'SB iM' B JM'JB jM "kH ,
kkM " iB /~M "jB
(18)
/~M'/~B
and iB, jB and /CB are the unit vectors along the XB, )'B and ZB axes, respectively. 8 TRANSFORMATION FOR SYMMETRY If a structure possesses symmetry in its geometry, support condition and loading, its displacement pattern should also be symmetrical. If the global coordinate set is so orientated that the X and Z axes are lying in the plane of symmetry, DOFs u, w and 0y should be symmetrical with respect to r/, as in Fig. 8(a), and DOFs v, 0x and 0z should be antisymmetrical with respect to rl, as in Fig. 8(b). Advantage of symmetry can of course be taken but there should be additional measures to ensure a fast rate of convergence. In the vicinity of the plane of symmetry, the function is given by
Vibration analysis of variable-depth bridges
31
f (a}
tO))
y
(b)
Fig. 8. (a) Symmetrical case. (b) Antisymmetrical case. 1
f(rl) = Z
ci ~i (7/).
(19)
i=-I
To ensure t h a t s y m m e t r y a n d a n t i s y m m e t r y are satisfied properly, it is necessary to specify c-1 = cl
(20)
for the symmetrical case a n d co = 0
(21)
F. T. K. Au, Y. K. Cheung
32
and c1=-cl
(22)
for the antisymmetrical case. These additional conditions should therefore be introduced to the displacement parameters accordingly.
9 N U M E R I C A L EXAMPLES Several numerical examples are included to demonstrate the accuracy and versatility of the present method. Solutions of this method are compared with other available solutions. Unless otherwise stated, quadratic strips were used and selective integration (on membrane and shear effects) was employed. In the following numerical examples, E is Young's modulus, v is Poisson's ratio and p is the density. Tensile stresses are taken to be positive. Imperial units are also replaced by S.I. units where appropriate.
9.1 Example 1: Three-span haunched girder reinforced concrete bridge The continuous three-span haunched girder reinforced concrete bridge (shown in Fig. 9) has been analysed by Cheung and Lau 18 using the semianalytical finite strip method with variable-width folded plate strips. The span lengths are 13.411 m, 16.764 m and 13-411 m, respectively. The cross section comprises a 0.159 m thick top flange and two girders 2.438 m apart. Each girder has a thickness of 0.425 m and a height of 0-770 m at the end supports and 1.329 m at the interior supports. The haunches are parabolic with vertical axes and they start at a distance of 3.962 m from the interior supports. Transverse diaphragms are provided at all supports as implied by the choice of displacement functions by Cheung and Lau.~S It is also assumed that the roller supports are constrained to move in the longitudinal direction. In the reference solution,~8 imperial units were used and Young's modulus E and Poisson's ratio v were given hypothetical values of 1-0 and 0.0, respectively. With the conversion to S.I. units, a more realistic value for Young's modulus E of 30 k N / m m 2 was assumed and the reference solution was amended accordingly. An OHBD design truck according to Ontario Highway Bridge Design Code 19 was applied symmetrically about the longitudinal centre line of the bridge deck to represent one of the loading conditions and Fig. 9(a) shows the truck loading on one longitudinal half of the bridge deck. The axle spacings assumed by Cheung and Lau ~s have been employed so that subsequent comparison can be performed on the same basis and hence, on conversion
Vibration analysis of variable-depth bridges m A
i_
12192
_h
~
33
OHBD truck load
_1
_
6004
7193
_1
H
13962] 39621
1 39621 3962] 16764
13411
i 12134 i: 13411
.I
(a)
1219
1219
.J
i
425
1219
= 899
899
~
.
1219
l
Data: E = 30 k N / m m 2
u = 0.0
.~_
.[,1,425
Section at the end
Section at i n t e r i o r support
(b)
Fig. 9. Three-span haunched girder reinforced concrete bridge. (a) Elevation and loading; (b) cross section (dimensions in ram).
to S.I. units, there are some slight discrepancies between the assumed values and the recommended metric values in the Code. m Symmetry of the problems allows analysis to be carried out on one half of the structure. Two discretization schemes have been used in the present analysis. Figure 10(a) shows Scheme A in which the bridge is divided into seven substructures to achieve an exact geometric model. In Scheme B, as shown in Fig. 10(b), each span is treated as one single substructure and the soffit line is approximated by a spline curve. In both schemes, five finite strips and 20 segments were used. In terms of the required number of DOFs, Scheme B (1782 DOFs) is more efficient than Scheme A (2310 DOFs) because there are fewer substructures and hence fewer interfaces. Each substructure is divided into segments of equal length for convenience.
34
F. T. K. A u, Y. K. Cheung
I
I I ~
[ I
1I
SS 4
(a)
I I I_L LI
I I
SS 1
SS 2 I
-
(b)
I.
609
. I_
I
609
= 320 320 _1_290,.Z9oj
Equa 1
9
(c)
;I
~
Equal I ual
11
10. Discretization schemes for a three-span haunched girder reinforced concrete bridge. (a) Scheme A: 7 substructures (SS); (b) scheme B: 3 substructures; (c) cross section
Fig.
(dimensions in ram).
As only average displacements were quoted by Cheung and Lau, Is a separate finite element analysis of a total of 3234 DOFs was carried out using COSMOS/M 2° and 9-node isoparametric shell elements with assumed strain stabilization. Results of the displacements of the top flange at various sections are presented in Tables 1-5 and compared with the semianalytical finite strip results using 30 terms and the A D I N A results quoted by Cheung and Lau. is Results of membrane forces from Scheme A at various cross sections are presented in Figs 11-15 and compared with the finite element solution from COSMOS/M. 2° Good agreement is observed.
35
Vibration analysis of variable-depth bridges
TABLE 1 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section B (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
-0.894 -0.895 -0.893 -0.873 -0.859 -0.859 -0.859
Present method
Average values 18
Scheme A
Scheme B
30 terms
ADINA
-0.887 -0.893 -0-896 -0.886 -0.878 -0-876 -0-877
-0-872 -0-881 -0.885 -0-876 -0-869 -0.869 -0.870
-0.888
-0.871
TABLE 2 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section D (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
1.865 1.970 2.084 2-189 2-279 2-316 2-329
Present method
Average values TM
Scheme A
Scheme B
30 terms
ADINA
1-836 1.954 2-083 2-190 2-285 2-328 2.341
1.820 1.934 2.058 2.165 2.259 2.301 2.314
2.080
2. i 14
TABLE 3 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section E (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
2.912 3.057 3.216 3.332 3.431 3.476 3.490
Present method
Average values 18
Scheme A
Scheme B
30 terms
AD I NA
2.776 2.937 3.113 3.243 3.353 3.402 3.418
2-742 2.900 3.068 3.185 3.282 3.326 3.341
3.104
3.141
36
F. T. K. Au, Y. K. Cheung
TABLE 4 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section F (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
4.532 4.710 4.901 5.038 5.148 5.193 5-207
Present method
Average values 18
Scheme A
Scheme B
30 terms
AD1NA
4-508 4-696 4.892 5.014 5.113 5.160 5.175
4.48 I 4.668 4.863 4.981 5.076 5.120 5.134
4.897
4.935
TABLE 5 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section H (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
-2.051 -2.053 -2.055 -2.055 -2-056 -2.056 -2-056
Present method
Average values 18
Scheme A
Scheme B
30 terms
A DINA
-2.049 -2.052 -2.054 -2.055 -2.056 -2.057 -2.057
-2.036 -2-038 -2-041 -2.042 -2.043 -2-044 -2-044
-2.084
-2.045
9.2 Example 2: Single-span concrete bridge with variable-depth integral parapets Figure 16 shows a single-span concrete bridge with variable-depth integral parapets supported on four bearings. It is a half-through bridge and hence there is no diaphragm at each support. The bearing layout, together with the allowable movement directions at the bearings, is a decisive factor of its dynamic behaviour especially for the higher modes. A similar structure has been studied by Graves-Smith a n d Walker. 2~ However, direct comparison with it is not possible as full information on its assumptions is not available. Free vibration analysis was carried out using four quadratic isoparametric spline finite strips (one for each web and two for the slab) and ten segments of equal length with a total of 810 DOFs. The bridge was also
Vibration analysis of variable-depth bridges
37
Finite element20_ Isoparametric spline finite strip
t
I
I
N/mm
Fig. !1. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section B.
I
Finite
+
element k
Isoparametric
finite strip
'
(
.
~
2
+
:
O
!
~
I
I
,
spl ine /1 ( ~ _ ~
o
°oo
? N/mmUnl ~,
Fig. 12. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section D.
o t
i
i
N/mm
element~° Isoparametric spline finite strip Finite
Fig. 13. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section E.
[
I
+
-I-
4-
+
+l
o
o
O
tn
t
i
CD t
N/mm Finite element20 +
Isoparametric strip
spl ine
finite
Fig. 14. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section F.
38
F. T. K. ,4u, Y. K. Cheung ".
Finite
m . . . .
i
e l e m e n t 20
Isoparametric finite strip
o
spline
o i
u~° , N/mm
o r
Fig. 15. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section H.
F t A
21000
Data:
,. (a)
E = 30 k N / m m 2 v
=
0.15
p
=
2400
oo o
o tn
B
21000
L
kg/m 3
I 42000
J _
(b)
80~_~_
•t.800
E 0 0 0 N
E
O
I
9000 (c)
Fig. 16. Single-span concrete bridge with variable-depth integral parapets. (a) Elevation; (b) plan showing bearing layout; (c) cross section (dimensions in ram).
Vibration analysis of variable-depth bridges
39
TABLE 6 Single-span Concrete Bridge with Variabledepth Integral Parapets
Mode
Natural frequency (Hz)
number
COSMOS/M2O
Present method
1 2 3 4 5 6 7 8 9 10
2.905 3.550 6-299 7.697 8.541 10.520 11.465 11-685 14-654 15.741
2.912 3.638 6.398 7.556 8.643 10.533 11.412 11-619 14.685 15-787
analysed by COSMOS/M 2° using 9-node isoparametric shell elements with assumed strain stabilization and a total of 2574 DOFs. The lowest ten natural frequencies are given in Table 6 and the results from the present method compare favourably with the finite element solution. The corresponding mode shapes are also presented in Fig. 17.
9.3 Example 3: Three-span variable-depth box girder bridge with curved alignment A three-span variable-depth single-cell box girder bridge is shown in Fig. 18. Figure 18(b) shows the developed sectional elevation of the bridge along the centre line. The soffit lines are parabolic with vertical axes. The horizontal alignment consists of a circular curve in the central span and two straight tangents in the end spans. Three different values of horizontal curvature in the central span have been considered, as shown in Table 7. The bearing layout for the bridge is shown in Fig. 18(c). Two bearings are provided at each support and it is assumed that they are directly under the webs. Support B is provided with a fixed bearing to serve as the expansion centre and a guided sliding bearing with allowable movement directed towards the expansion centre. Each of the other supports is provided with a guided sliding bearing with allowable movement directed towards the expansion centre, as well as a free sliding bearing. The cross sections at an interior support and middle of the central span are shown in Fig. 19.
40
F. T. K. Au, Y. K. Cheung
B Undeformed mesh
Mode 2
Mode 1
Mode 3
Mode 5
Mode 4
Fig. 17. Caption opposite.
As one of the loading conditions, 45 units of abnormal vehicle HB loading to BS 5400:Part 2:197822 was applied at maximum eccentricity on the bridge deck. Figure 20 shows the plan and axle arrangement of the loading on the bridge. A total load of 1800 kN is assumed to be equally distributed among the 16 wheels on four axles. Static analysis was carried out using six quadratic isoparametric spline finite strips (three for the top flange, one for each web and one for the bottom flange) and a total of 18 segments, including ten for the central span and four for each end span involving a total of 1800 DOFs. The reference solution was obtained by
Vibration analysis of variable-depth bridges
Mode 6
Mode 7
Mode 8
Mode 9
41
Mode 10
Fig. 17. Mode shapes for a single-span concrete bridge with variable-depth integral parapets.
COSMOS/M 2° using 9-node isoparametric shell elements with assumed strain stabilization and a total of 5160 DOFs. The results indicate that the twist of the bridge generally increases with the horizontal curvature but it is mainly confined to the central span. This can be attributed to the torsional restraint provided at Supports B and C. The web under the abnormal vehicle is more heavily loaded around midspan but at Supports B and C both webs are roughly sharing the same load. The effect of changes in horizontal curvature on longitudinal membrane stresses is most noticeable in the web on the outer curve of the central span. With the increase in horizontal curvature, higher long-
42
F. T. K. Au, Y. K. Cheung Span 2
.rj
Data: E = 30 kN/mm z v = 0.15 p = 2400 kg/m3
~ /~
lncreaslnE chainage
(a) ~" o
45 unitsllof IIHBloading A
B
~ C
D
i, 1+
4oooo
1
+oooo
°oooo
_
(b)
A
B _ ~ . ~ ~ - _
C D
- -
- ~ -
"
It)
~
2~00i
( n o t to s c a l e )
Fig. 18. Three-span variable-depth box girder bridge with curved alignment. (a) Plan;
(b) developed sectional elevation along centre line; (c) bearing layout (dimensionsin mm).
itudinal membrane stresses are attracted to Supports B and C while it is relieved around the mid-span. The deflections at the tips of side cantilevers and top flange-web junctions at mid-span positions of the three cases investigated are given in Tables 8-10. The vertical deflections at the tips of side cantilevers for Case 3 are presented in Figs 21 and 22. The corresponding membrane forces at various sections of the central span are shown in Figs 23-25. G o o d agreement is observed. Free vibration analysis was also carried out using six quadratic isoparametric spline finite strips (three for the top flange, one for each web and one for the bottom flange) and a total of 14 segments, including six for the central span and four for each end span involving a total of 1512
Vibration analysis of variable-depth bridges 500 Q o
43
500
--t-t-
-'7 o
o
4000
4000 I
Ca)
500
SO0
°
I
O O 00
?
I I
24oo
,ooo
_i
jl
2~oo
J
°ooo'
(b)
L
Fig. 19. Three-span variable-depth box girder bridge with curved alignment. (a) Cross section at middle of central span and end supports; (b) cross section at interior supports (dimensions in ram). DOFs. The problem w a s also solved by COSMOS/M 2° using 5160 DOFs for comparison. The lowest four natural frequencies for the three cases investigated are given in Table I 1 and the corresponding mode shapes for Case 3 are also presented in Fig. 26. G o o d agreement is again observed. 10 C O N C L U S I O N S The static and free vibration analysis of variable-depth bridges of arbitrary alignments was successfully carried out using the isoparametric spline finite strip method. The presence of haunches, support diaphragms and bearings can all be accounted for. It retains the computational
44
F. T. K. Au, Y. K. Cheung TABLE 7
Three-span Variable-depth Box Girder Bridge With Curved Alignment. Horizontal Curvature of Central Span (centre line)
I
1800
I~
Case
Radius of curvature (m)
Subtended angle (radian)
1 2 3
300 100 60
0-2 0.6 1-0
~
3000
I
J
3000
I
~"
1800
×
o
o
9? o
i
Oo~tn
i
o!
.i
i
/~i transverse 9f c e n t r a l
centre span
line
i
~
longitudinal
oI centr
line
c~ o
Fig. 20. Three-span variable-depth box girder bridge with curved alignment, plan and axle
arrangement of HB loading (dimensions in mm). efficiency o f the spline finite strip method, while it is much more flexible in geometric modelling. The accuracy and efficiency o f the method were tested on a variety o f examples.
REFERENCES 1. Zienkiewicz, O. C. & Taylor, R. L., The Finite Element Method, VoL 1 Basic Formulation and Linear Problems, 4th edn. McGraw-Hill, New York, 1989.
Vibration analysis of variable-depth bridges
45
TABLE $ Three-span Variable-depth Box Girder Bridge With Curved Alignment. Deflections of Top Flange at Mid-span Positions for Case 1 (mm)
Position
Mid-span 1 Mid-span 2 Mid-span 3
Method
Tip of left s i d e cantilever
Present COSMOS/M 2° Present COSMOS/M 2° Present COSMOS/M 2°
-5.598 -5.640 19.587 19.650 -6.000 -5.893
Left flange-web junction -5.567 -5.610 18.698 18.924 -5.967 -5.858
Right flange-web junction -5.565 -5.609 16.667 16.798 -5.966 -5-835
Tip of right side cantilever -5-591 -5-635 16.216 16.435 -5.992 -5.860
TABLE 9 Three-span Variable-depth Box Girder Bridge With Curved Alignment. Deflections of Top Flange at Mid-span Positions for Case 2 (mm)
Position
Mid-span 1 Mid-span 2 Mid-span 3
Method
Tip of left s i d e cantilever
Present COSMOS/M 2° Present COSMOS/M 2° Present COSMOS/M 2°
-5-830 -5.870 20.955 21.014 -6.241 -6-133
Left flange-web junction -5-798 -5-838 19.664 19.891 -6.208 -6.098
Right .flange web junction -5.797 -5.838 16.937 17.055 -6.208 -6.076
Tip of right side cantilever -5.824 -5.865 16.133 16-336 -6.237 6-103
TABLE 10 Three-span Variable-depth Box Girder Bridge With Curved Alignment. Deflections of Top Flange at Mid-span Positions for Case 3 (mm)
Position
Mid-span 1 Mid-span 2 Mid-span 3
Method
Tip of left s i d e cantilever
Present COSMOS/M 2° Present COSMOS/M 2° Present COSMOS/M 2°
-6.149 -6.188 22.566 22.633 -6.564 -6.459
LeJt Right flange web flange-web .junction junction -6.116 -6.155 20.887 21.124 -6.532 -6-424
-6.115 -6.155 17.467 17.579 -6.533 -6-402
Tip of right side cantilever -6.144 -6.184 16.324 16-517 -6-566 -6-431
Vertical deflection (mm)
strip
ill
(m)
Fig. 21. Three-span variable-depth box girder bridge with curved alignment (Case 3), vertical deflection at outer curve.
20
10
-I0
-20
:x
Vertical
deflection
(mm)
strip
Cm)
Fig. 22. Three-span variable-depth box girder bridge with curved alignment (Case 3), vertical deflection at inner curve.
20
I0
-10
-20
48
F. T. K. Au, Y. K. Cheung
®
0
1
i
I
2 1
3
I
N/mm 2
Finite element 2° Isoparametric finite strip
spline
Fig. 23. Three-span variable-depth box girder bridge with curved alignment (Case 3), longitudinal membrane stresses in middle of central span.
I
1
2
3
i
i
r
N/mm 2
® Finite element 20 Isoparametric finite strip
spline
Fig. 24. Three-span variable-depth box girder bridge with curved alignment (Case 3), longitudinal membrane stresses at support B of central span. 2. Zienkiewicz, O. C. & Taylor, R. L., The Finite Element Method, Vol. 2 Solid and Fluid Mechanics, Dynamics and Non-linearity, 4th edn. McGraw-Hill, New York, 1991. 3. Cheung, Y. K., Finite Strip Method in Structural Analysis. Pergamon Press, Oxford, England, 1976. 4. Cheung, Y. K., Fan, S. C. & Wu, C. Q., Spline finite strip in structural analysis. Proc. of Int. Conf. on Finite Element Method, Shanghai, China, 1982, pp. 704-709. 5. Fan, S. C. & Cheung, Y. K., Flexural free vibration of rectangular plates with complex support conditions. J. Sound Vib., 93(1) (1984) 81-94. 6. Chen, M. J., Tham, L. G. & Cheung, Y. K., Spline finite strip for parallelo-
Vibration analysis of variable-depth bridges
49
TABLE 11 Three-span Variable-depth Box Girder Bridge With Curved Alignment. Natural Frequencies (Hz)
Case
Case I Case 2 Case 3
Method
Mode number
Present COSMOS/M2° Present COSMOS/M20 Present COSMOS/M2°
1
2
3
4
2.235 2.221 2.220 2.206 2.194 2.181
4.032 3.988 3.921 3.871 3.582 3.525
4-902 4-817 4.445 4.383 4.158 4.114
5.562 5.382 5.553 5.370 5.524 5-340
® +_---
0 i
1 i
2 i
3 I
N/mm 2
Finite +
e l e m e n t 20
Isoparametric finite strip
spline
Fig. 25. Three-span variable-depth box girder bridge with curved alignment (Case 3), longitudinal membrane stresses at support C of central span.
7. 8. 9. 10. 11.
gram plate. Proc. of Int. Conf. on Accuracy Estimates and Adaptive Refinements in Finite Element Computations (ARFEC), Lisbon, Portugal, 1984, Vol. 1, pp. 95-104. Tham, L. G., Li, W. Y., Cheung, Y. K. & Chen, M. J., Bending of skew plates by spline-finite-strip method. J. Comput. Struct., 22(1) (1986) 31-8. Cheung, Y. K., Tham, L. G. & Li, W. Y., Application of spline-finite-strip method in the analysis of curved slab bridge. Proc. Inst. Cir. Engrs, Part 2, 81 (1986) 111-124. Cheung, Y. K., Tham, L. G. & Li, W. Y., Free vibration and static analysis of general plate by spline finite strip. Computat. Mech., 3 (1988) 187-97. Cheung, Y. K., Li, W. Y. & Tham, L. G., Free vibration analysis of singly curved shell by spline finite strip method. J. Sound Fib., 128(3) (1989) 411 422. Li, W. Y., Tham, L. G., Cheung, Y. K. & Fan, S. C., Free vibration analysis of doubly curved shells by spline finite strip method. J. Sound Vib., 140(1) (1990) 39-53.
50
F. T. K. A u, Y. K. Cheung
B A
Undeformed
Mode 2
mesh
Mode
1
Mode 3
Mode 4
Fig. 26. Mode shapes for three-span variable-depth box girder bridge with curved alignment (Case 3).
Vibration analysis of variable-depth bridges
51
12. Li, W. Y., Tham, G. & Cheung, Y. K., Curved box-girder bridges. J. Struct. Engng, ASCE, 114(6) (1988) 1324-1338. 13. Cheung, Y. K. & Li, W. Y., Free vibration analysis of longitudinal arbitrarily curved box-girder structures by spline finite strip method. Proc. of Asian Pacific Conf. on Computational Mechanics, Hong Kong, 1991, Vol. 2, pp. 1139-44. 14. Onate, E., Suarez, B. & Hinton, E., Mindlin finite strip and axisymmetric finite element shell analysis. In Finite Element Software for Plates and Shells, eds E. Hinton & D. R. J. Owen. Pineridge Press, 1984, pp. 49-156. 15. Dawe, D. J., The analysis of shear-deformable plates using finite strip elements. In Finite Element Methods for Plate and Shell Structures, Volume 2." Formulations and Algorithms, eds T. J. R. Hughes & E. Hinton, Pineridge Press, 1986, pp. 102-126. 16. Au, F. T. K. & Cheung, Y. K., Isoparametric spline finite strip for plane structures. J. Comput. Struct., 48(1) (1993) 23-32. 17. Cheung, Y. K. & Au, F. T. K., Isoparametric spline finite strip for degenerate shells. Thin-Walled Struct., 21(1) (1995) 65 92. 18. Cheung, M. S. & Lau, W., Analysis of continuous, haunched, plate-girder bridges by the finite strip method. Proc. of the Second East Asia-Pacific Conference on Structural Engineering & Construction, Chiang Mai, Thailand, 1989, pp. 1091-1098. 19. Ontario Highway Bridge Design Code (OHBDC), 2nd edn. Ministry of Transportation and Communications, Downsview, Ontario, Canada, 1983. 20. Lashkari, M., C O S M O S / M User Guide, 7th edn. Structural Research and Analysis Corporation, 1992. 21. Graves-Smith, T. R. & Walker, B. D., Dynamic analysis of partially prismatic thin-walled structures. Thin-Walled Struct., 5 (1987) 39-54. 22. BS 5400 Steel, Concrete and Composite Bridges, Part 2 Specification for Loads. British Standards Institution, Linford Wood, Milton Keynes, UK, 1978.
0263-8231
Thin-Walled Structures 24 (1996) 19-51 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/96/$15.00 (95)00038-0
Static and Free Vibration Analysis of Variable-depth Bridges of Arbitrary Alignments Using the Isoparametric Spline Finite Strip Method
F. T. K. Au & Y. K. Cheung Department of Civil and Structural Engineering,Universityof Hong Kong, Hong Kong (Received 19 October 1994; accepted 21 February 1995)
ABSTRACT The isoparametric spline finite strips for shells are employed to solve static and free vibration problems of variable-depth bridges of arbitrary alignments. Using this approach, a continuous bridge is first split up into substructures. Each substructure is modelled as an assemblage of isoparametric spline finite strips. Compatibility between substructures is ensured by suitable transformation at the interface. The presence of support diaphragms and bearings can also be accounted for. This method retains the computational efficiency of the spline finite strip method while it is much more flexible in geometric modelling. Solutions of this method are compared with other available solutions, and good agreement is observed.
1 INTRODUCTION The finite strip method can be considered as a special form of the finite element procedure 1'2 using the displacement approach. The semianalytical finite strip method, pioneered by Cheung, 3 is an effective method of structural analysis for engineering structures with constant geometrical properties along a particular direction. The method is applicable to single-span and multiple-span bridges with straight or circular alignment in plan. It is especially efficient in the analysis of single span simply supported right box girders for which the Fourier sine series is 19
F. T. K. Au, Y. K. Cheung
20
a suitable set of displacement functions. However, in the solution of problems with concentrated forces, multiple spans, discrete supports at strip ends etc., difficulties were encountered. The spline finite strip method 4 was later developed to overcome the above difficulties. The method was first successfully applied to static and free vibration of thin rectangular plates 4' 5 and subsequently extended to parallelogram skew plates. 6' 7 Plates of arbitrary shapes 8"9 were also solved utilizing the mapping concept. In this method, the plate of arbitrary shape is mapped into a regular square domain which is discretized into spline finite strips for analysis. Subsequent developments of the method also cater for shells of arbitrary shapes, l°'lt box girder bridges of curved alignments, ~2'13 etc. In the above developments, the thin plate or shell approach was employed. In addition, both semi-analytical and spline finite strip methods had been successfully applied to the analysis of rectangular and circular Mindlin plates.~4" t5 In order to satisfy modern traffic requirements, highway bridges are often curved to tie in with the approach roads. Moreover, as the highway authorities are increasingly conscious of the maintenance costs of bridges, many of them are specifying that all future bridges shall be designed with a minimum of bearings and movement joints. In other words, continuous bridges will become more and more popular in future and many such bridges are in the form of variable-depth 'sucker' bridges. In an attempt to respond to such complex requirements, the isoparametric spline finite strip method was introduced and the flexibility to model various geometric shapes was further enhanced. The method was first developed for plane structures ~6 and later extended to shells. ~7 It was also reported that the semi-analytical finite strip method was extended to solve right continuous haunched plate-girder bridges, t8 In this paper, the isoparametric spline finite strip method is applied to the static and free vibration analysis of variable-depth bridges of arbitrary alignments.
2 ISOPARAMETRIC SPLINE FINITE STRIP A typical isoparametric spline finite strip for shell is shown in Fig. 1. The geometry of the strip is described with respect to the global coordinate set (X, Y and Z) by the product of two functions: spline functions along the length and Lagrange polynomials transversely, i.e. m+l
{P((,~/)} = ~
p
y~[NIj((,T/)] {Q/j}
i=-Ij=l
(1)
Vibration analysis of variable-depth bridges
21
/ y /
/
/
/ /
/
x Fig. l. Typical segment of finite strip for a shell with coordinate systems.
where {P((, rt) } is the position vector of a point on the mid-surface of the strip domain, rn is the number of segments in the strip, p is the number of nodal lines defining the strip, {Q;j} is the ith curve defining vector of the j t h nodal line of the strip and [Nij ((, z/)] is the shape function matrix associated with {Qij}- The shape function matrix [N,-j((, ~/)] is given by [N~j((,~)] =
N~j((,g)[I]= LP(()4)i(g)[I]
(2)
in which LP(O is the j t h Lagrange polynomial, ~)i(7]) is the Ba-spline function of unit section length with the section-knot 7/= 7/,. as the centre and [I] is an identity matrix of appropriate order. Figure 1 also shows the local coordinate set (x~,y~ and z~) of an arbitrary point on the strip and the nodal coordinate set (x.,y. and z.) at a nodal line. They are so defined that the z~ and z. axes are normal to the mid-surface of the strip and the y~ and y. axes are tangential to the ¢ = constant curves. The displacement vector {6} defined as
22
F. T. K. Au, Y. K. Cheung
{6} = [u v w Ox~, Oy~,]T
(3)
consists of the three translational degrees of freedom (DOFs) u, v and w along the global X, Y and Z axes, respectively, and two rotational DOFs 0x~ and 0y~ about the local x~ and y~ axes, respectively, in the right-hand sense. The displacement {6((, r/)} of point {P((, r/)} on the strip is similarly expressed in terms of the displacement parameters {6ij} associated with curve defining vector {Q/j} m+l
{6(¢,r/)} = Z
p
Z [Nij(~'r/)] {6(j}.
(4)
i=-lj=l
To facilitate subsequent evaluation of strip matrices and load vectors, the translational displacements u,, v, and w~ along the local x,, y~ and z~ axes are expressed in terms of the respective global translational displacements using the transformation matrix [R,] v~ w~
=
m2~ n2~/ [_/3~ m3~ n3~J
/ 12~
v
w
= [R~]
v w
(5)
where (ll~,ml~ and nl,), (12~,m2~ and n2,) and (/3~,m3~ and n3a), are the directional cosines of the unit vectors [,,j~ and k~ along the x~, y, and z, axes of the local coordinate set with respect to the unit vectors [, j and/~ along the global X, Y and Z axes, respectively. Once the displacement functions have been chosen, the strip characteristics can be obtained in line with the standard finite element method and these shell strips will be the basic tools for subsequent analysis. 3 GEOMETRIC DESCRIPTION OF BRIDGE DECK Apart from the relatively rare major bridge crossings, the vertical profile and horizontal alignment of a highway bridge are invariably governed by traffic requirements. Normally straight lines and parabolic curves are used in vertical proffl,es, whereas straight lines, circular curves and spiral curves are employed in horizontal alignments. The vertical alignment of a bridge seldom affects its structural behaviour. However, it is important that the horizontal alignment be accurately modelled in structural analysis. An isoparametric spline finite strip is essentially a C2 shell surface. A longitudinal kink, i.e. in the r/direction, between adjacent strips is allowed and this is achieved by proper description of strip geometry. If the profile of a bridge consists of breaks in slope or curvature along its length, it has to be divided into substructures. Figure 2 shows a number of typical
23
Vibration analysis of variable-depth bridges
SS
I
•L
SS
2
~I~
SS
_J
3
(a)
SS
1
_.
SS
2
LI
ss3
(b)
l SS 1
J SS 2
~ - SS 3
SS 4
SS
7~
(c)
JrJr
Scheme A SS 1 _L s s 2
__ ss 3
I
SS 4
I
SchemessB 1 SS 2
(d)
Fig. 2. Typical continuous bridges and schemes of sub-division into substructures (SS). (a) Constant depth; (b) variable depth with curved soffit; (c) partial length of constant depth with straight haunches; (d) partial length of constant depth with parabolic haunches.
continuous bridges. Both the bridge of constant cross section (shown in Fig. 2(a)) and the variable-depth bridge (shown in Fig. 2(b)) can be conveniently divided into three substructures for analysis so that each substructure is smooth in the r/direction. The bridge with straight haunches (shown in Fig. 2(c)) should be divided into seven substructures for analysis. Figure 2(d) shows a continuous bridge with parabolic haunches. Theoretically speaking, there is a break in curvature at the junction between a straight section and the adjacent parabolic haunch, and hence the structure should be divided into seven substructures for exact geometric modelling, as shown in Scheme A. A more convenient arrangement is shown in Scheme B in which the structure is divided into three
24
F. T. K. Au, Y. K. Cheung
substructures for analysis, although in this case some approximation in geometric modelling is involved.
4 EFFECTS OF D I A P H R A G M S A N D BEARINGS A box girder structure consists of top and bottom flanges connected by vertical or inclined webs to form a cellular section. It is one of the most popular forms of highway bridges, primarily because of its high flexural and torsional rigidities. However, the thin-walled section is rather weak in resisting distortion and therefore stiffening is often required in sections subjected to large concentrated loads, especially at supports. The stiffening system may be in the form of a truss, a frame or most frequently a solid diaphragm with or without opening. The stiffening system is normally designed to have its primary strength within the plane of the cross section. Hence a diaphragm is normally assumed to be perfectly rigid in its plane and perfectly flexible out of plane. Bridges are usually supported on bearings. In normal practice, bearings are set horizontal to restrain any vertical deflection. Horizontal movements at a bearing can be controlled. A bearing can be free sliding in which horizontal movement is unrestrained. It can be guided sliding so that horizontal movement is only allowed in a certain direction. It can also be fixed so that no horizontal movement is allowed. Rotations at the bearings are usually taken to be free. A bridge deck is considered torsionally restrained at a support with a diaphragm if two or more bearings are provided because any rotation of the diaphragm within its own plane is restrained, as shown in Fig. 3(a). If a single bearing is provided as shown in Fig. 3(b), rotation of the diaphragm within its own plane is allowed but only about the bearing.
5 T R A N S F O R M A T I O N FOR STRIPS IN EACH SUBSTRUCTURE In the formulation of stiffness matrices, mass matrices and load vectors, the global translational DOFs u, v and w and the local rotational DOFs 0x~ and Oy~ have been used. Apart from the simplest case of slab bridges in which all strips are coplanar, a 6-DOF analysis is preferred incorporating the additional drilling DOF Oz~,. The global translational DOFs u, v and w are used throughout each substructure to obviate the need for transformation. However transformation is frequently required for the rotational DOFs. The transformation consists of expressing all the rotational DOFs at a nodal line, in terms of the rotational DOFs according to the nodal set,
Vibration analysis of variable-depth bridges
25
(a)
!
J
(b)
Fig.
3. Typical supports for box girder bridges. (a) Supported on two bearings; (b) supported on a single bearing.
which is the local coordinate set of a chosen adjoining strip, and the procedure has been described in Ref. 15.
6 T R A N S F O R M A T I O N FOR COMPATIBILITY BETWEEN SUBSTRUCTURES At the interface between adjacent substructures, compatibility must be ensured for all DOFs. The displacement of each point on a strip within a substructure is expressed in terms of the displacement parameters associated with the curve defining vectors of the nodal lines defining the strip. First of all, it is necessary to express the physical displacement at each end knot of a substructure in terms of the displacement parameters of the corresponding nodal line. Concentrating on the deflection of the j t h nodal line and dropping the index j for simplicity, the deflection w~ at the ith
F. T. K. Au, Y. K. Cheung
26
knot point depends only on the displacement parameters w;_ 1, wi and w~+ i associated with three vertices of the curve defining polygon, i.e.
I Wi+l } [Wi]
Wi- 1
I Wi+l } wi = wi- 1
Wi+ Wi Wi_
2 3
[i °
(6)
0
The index i may take the value of zero or m depending on which end of the substructure is being considered. In this manner, all displacement parameters associated with the knot points 0 and m of the nodal lines are replaced by the corresponding physical displacements. As the global translational DOFs u, v and w are used for all substructures, transformation is again not necessary. However the nodal sets, according to which the rotational DOFs are defined, are often different for the adjacent substructures as shown in Fig. 4, and hence transformation is necessary. The transformation for rotational DOFs for nodal line j is as shown below
(7)
{6~}j2 = [Rs]j{ar}j,
where {6r}jl and {6r}j2 are the rotational DOFs at nodal line j at the interface with respect to the nodal sets of the left a n d right adjacent substructures, respectively.
{al} = E0x~0y~0=]T,
(8) i.2 •
[Rs] = | j . 2
lnl
Jn2 "Jnl
jn2 1~,,,
,
(9)
L/~.2 " '.,'~ /~.2 "L, /~.2 •/~., ^
^
i.l, J.l and/~.l are the unit vectors along the Xnl, Ynl and ZnJ axes of the nodal coordinate set of the left adjacent substructure and i.2, £2 and/~.2 are the unit vectors along the Xn2, Yn2 and zn2 axes of the nodal coordinate set of the right adjacent substructure.
7 T R A N S F O R M A T I O N F O R D I A P H R A G M S A N D BEARINGS The presence of a diaphragm at an interface between substructures requires special treatment. Figure 5 shows a typical support provided with a diaphragm to constrain some of the nodal lines. The diaphragm is assumed to be planar, perfectly rigid in its plane and perfectly flexible out
Vibration analysis of variable-depth bridges
27
Y
x
Fig. 4. Relationship between nodal coordinate sets for adjacent substructures at the interface.
of plane. It is further assumed that only the translational DOFs of the bridge are constrained by the diaphragm. One of the nodes constrained by the diaphragm is selected as the master node while all the others are taken as slave nodes. The in-plane displacement of any point lying on the diaphragm is therefore dependent only on two translational DOFs and one rotational DOF of the master node in the plane of the diaphragm. For convenience, the master node is often located at one of the bearing positions. If the diaphragm is supported on more than one bearing, this rotational D O F is set to zero as no rotation within the plane of the diaphragm is allowed.
28
F. T. K. Au, Y. K. Cheung
J"
Master
J
J
J
node
Fig. 5. A support provided with a diaphragm showing the master node and one of the slave nodes. Figure 6 illustrates the relationship between a slave node S at nodal line j and the master node M in a diaphragm. The orientation of the diaphragm is defined by a set of axes XM, YM and ZM in which the YM axis is normal to the plane of the diaphragm. The displacement of the diaphragm is characterized by the translational DOFs UM and WM along the XM and ZM axes, respectively, and the rotational D O F OyM about the YM axis in the right-hand sense. As the translational displacement at the slave node S is given by the global translational DOFs Us, Vs and Ws, it is necessary to transform such DOFs to USM, VSM and WSM which are given with reference to the XM, YM and ZM axes, respectively, as shown below {6~) = [Ro] {6~M}
(10)
where {6~} = [Us Vs Ws]T,
(11)
{6~M} = [USMVSMWSM]T,
(12)
[i. [M iM j'J i.j ]'h,, i. M 1 [Rol----/)"
(13)
[k.iM I~'jM k'kM i, j and/~ are the unit vectors along the global X, Y and Z axes and iM, jM and kM are the unit vectors along the XM, YM and ZM axes, respectively.
Vibrationanalysisof varhTble-depthbridges
29
z(w $ )
z
\d./.Yfvs) S ~N~-,.~''- xM / /"-..
y
/
xCu s)
\ ~-----~'~
XM(UM)
x
Fig. 6. Relationship between master node M and slave node S at a diaphragm. D O F vSM is unaffected by the presence of the diaphragm and DOFs USM and wSM are related to the DOFs at the master node M as follows =
WSM
i,0 01
WM
--(Ps -- PM)" iM
(14)
0yM
in which PM and Ps are the position vectors of master node M and slave node S, respectively. The orientation of the supporting bearing system is defined by a set of axes XB, YB and zB in which ZB axis is along the direction of major reaction as shown in Fig. 7. As the orientation of the bearing may be different from that of the diaphragm, further transformation is necessary before the boundary condition for the support is introduced. It is assumed that a bearing is attached to the diaphragm at the master node M. The displacement { ~ } with respect to the set of axes xa, YB and zB for the bearing system is related to the displacement {6h } at the master node by {6h} = [RB] {6~}
(15)
where
{•h} = [UM VMWM]T'
(16)
30
F. T. K. Au, Y. K. Cheung
Z
z M
Fig. 7, Relationship between coordinate set (XM,)'M and ZM) defining orientation of a diaphragm and coordinate set (xa, YB and zB) defining orientation of a supporting bearing system at master node M.
(17)
[RB]=
lM'lB "'B
1M'SB iM' B JM'JB jM "kH ,
kkM " iB /~M "jB
(18)
/~M'/~B
and iB, jB and /CB are the unit vectors along the XB, )'B and ZB axes, respectively. 8 TRANSFORMATION FOR SYMMETRY If a structure possesses symmetry in its geometry, support condition and loading, its displacement pattern should also be symmetrical. If the global coordinate set is so orientated that the X and Z axes are lying in the plane of symmetry, DOFs u, w and 0y should be symmetrical with respect to r/, as in Fig. 8(a), and DOFs v, 0x and 0z should be antisymmetrical with respect to rl, as in Fig. 8(b). Advantage of symmetry can of course be taken but there should be additional measures to ensure a fast rate of convergence. In the vicinity of the plane of symmetry, the function is given by
Vibration analysis of variable-depth bridges
31
f (a}
tO))
y
(b)
Fig. 8. (a) Symmetrical case. (b) Antisymmetrical case. 1
f(rl) = Z
ci ~i (7/).
(19)
i=-I
To ensure t h a t s y m m e t r y a n d a n t i s y m m e t r y are satisfied properly, it is necessary to specify c-1 = cl
(20)
for the symmetrical case a n d co = 0
(21)
F. T. K. Au, Y. K. Cheung
32
and c1=-cl
(22)
for the antisymmetrical case. These additional conditions should therefore be introduced to the displacement parameters accordingly.
9 N U M E R I C A L EXAMPLES Several numerical examples are included to demonstrate the accuracy and versatility of the present method. Solutions of this method are compared with other available solutions. Unless otherwise stated, quadratic strips were used and selective integration (on membrane and shear effects) was employed. In the following numerical examples, E is Young's modulus, v is Poisson's ratio and p is the density. Tensile stresses are taken to be positive. Imperial units are also replaced by S.I. units where appropriate.
9.1 Example 1: Three-span haunched girder reinforced concrete bridge The continuous three-span haunched girder reinforced concrete bridge (shown in Fig. 9) has been analysed by Cheung and Lau 18 using the semianalytical finite strip method with variable-width folded plate strips. The span lengths are 13.411 m, 16.764 m and 13-411 m, respectively. The cross section comprises a 0.159 m thick top flange and two girders 2.438 m apart. Each girder has a thickness of 0.425 m and a height of 0-770 m at the end supports and 1.329 m at the interior supports. The haunches are parabolic with vertical axes and they start at a distance of 3.962 m from the interior supports. Transverse diaphragms are provided at all supports as implied by the choice of displacement functions by Cheung and Lau.~S It is also assumed that the roller supports are constrained to move in the longitudinal direction. In the reference solution,~8 imperial units were used and Young's modulus E and Poisson's ratio v were given hypothetical values of 1-0 and 0.0, respectively. With the conversion to S.I. units, a more realistic value for Young's modulus E of 30 k N / m m 2 was assumed and the reference solution was amended accordingly. An OHBD design truck according to Ontario Highway Bridge Design Code 19 was applied symmetrically about the longitudinal centre line of the bridge deck to represent one of the loading conditions and Fig. 9(a) shows the truck loading on one longitudinal half of the bridge deck. The axle spacings assumed by Cheung and Lau ~s have been employed so that subsequent comparison can be performed on the same basis and hence, on conversion
Vibration analysis of variable-depth bridges m A
i_
12192
_h
~
33
OHBD truck load
_1
_
6004
7193
_1
H
13962] 39621
1 39621 3962] 16764
13411
i 12134 i: 13411
.I
(a)
1219
1219
.J
i
425
1219
= 899
899
~
.
1219
l
Data: E = 30 k N / m m 2
u = 0.0
.~_
.[,1,425
Section at the end
Section at i n t e r i o r support
(b)
Fig. 9. Three-span haunched girder reinforced concrete bridge. (a) Elevation and loading; (b) cross section (dimensions in ram).
to S.I. units, there are some slight discrepancies between the assumed values and the recommended metric values in the Code. m Symmetry of the problems allows analysis to be carried out on one half of the structure. Two discretization schemes have been used in the present analysis. Figure 10(a) shows Scheme A in which the bridge is divided into seven substructures to achieve an exact geometric model. In Scheme B, as shown in Fig. 10(b), each span is treated as one single substructure and the soffit line is approximated by a spline curve. In both schemes, five finite strips and 20 segments were used. In terms of the required number of DOFs, Scheme B (1782 DOFs) is more efficient than Scheme A (2310 DOFs) because there are fewer substructures and hence fewer interfaces. Each substructure is divided into segments of equal length for convenience.
34
F. T. K. A u, Y. K. Cheung
I
I I ~
[ I
1I
SS 4
(a)
I I I_L LI
I I
SS 1
SS 2 I
-
(b)
I.
609
. I_
I
609
= 320 320 _1_290,.Z9oj
Equa 1
9
(c)
;I
~
Equal I ual
11
10. Discretization schemes for a three-span haunched girder reinforced concrete bridge. (a) Scheme A: 7 substructures (SS); (b) scheme B: 3 substructures; (c) cross section
Fig.
(dimensions in ram).
As only average displacements were quoted by Cheung and Lau, Is a separate finite element analysis of a total of 3234 DOFs was carried out using COSMOS/M 2° and 9-node isoparametric shell elements with assumed strain stabilization. Results of the displacements of the top flange at various sections are presented in Tables 1-5 and compared with the semianalytical finite strip results using 30 terms and the A D I N A results quoted by Cheung and Lau. is Results of membrane forces from Scheme A at various cross sections are presented in Figs 11-15 and compared with the finite element solution from COSMOS/M. 2° Good agreement is observed.
35
Vibration analysis of variable-depth bridges
TABLE 1 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section B (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
-0.894 -0.895 -0.893 -0.873 -0.859 -0.859 -0.859
Present method
Average values 18
Scheme A
Scheme B
30 terms
ADINA
-0.887 -0.893 -0-896 -0.886 -0.878 -0-876 -0-877
-0-872 -0-881 -0.885 -0-876 -0-869 -0.869 -0.870
-0.888
-0.871
TABLE 2 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section D (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
1.865 1.970 2.084 2-189 2-279 2-316 2-329
Present method
Average values TM
Scheme A
Scheme B
30 terms
ADINA
1-836 1.954 2-083 2-190 2-285 2-328 2.341
1.820 1.934 2.058 2.165 2.259 2.301 2.314
2.080
2. i 14
TABLE 3 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section E (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
2.912 3.057 3.216 3.332 3.431 3.476 3.490
Present method
Average values 18
Scheme A
Scheme B
30 terms
AD I NA
2.776 2.937 3.113 3.243 3.353 3.402 3.418
2-742 2.900 3.068 3.185 3.282 3.326 3.341
3.104
3.141
36
F. T. K. Au, Y. K. Cheung
TABLE 4 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section F (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
4.532 4.710 4.901 5.038 5.148 5.193 5-207
Present method
Average values 18
Scheme A
Scheme B
30 terms
AD1NA
4-508 4-696 4.892 5.014 5.113 5.160 5.175
4.48 I 4.668 4.863 4.981 5.076 5.120 5.134
4.897
4.935
TABLE 5 Three-span Haunched Girder Reinforced Concrete Bridge. Deflection of Top Flange at Section H (mm) Nodal line
C O S M O S / M 2°
1 2 3 4 5 6 7
-2.051 -2.053 -2.055 -2.055 -2-056 -2.056 -2-056
Present method
Average values 18
Scheme A
Scheme B
30 terms
A DINA
-2.049 -2.052 -2.054 -2.055 -2.056 -2.057 -2.057
-2.036 -2-038 -2-041 -2.042 -2.043 -2-044 -2-044
-2.084
-2.045
9.2 Example 2: Single-span concrete bridge with variable-depth integral parapets Figure 16 shows a single-span concrete bridge with variable-depth integral parapets supported on four bearings. It is a half-through bridge and hence there is no diaphragm at each support. The bearing layout, together with the allowable movement directions at the bearings, is a decisive factor of its dynamic behaviour especially for the higher modes. A similar structure has been studied by Graves-Smith a n d Walker. 2~ However, direct comparison with it is not possible as full information on its assumptions is not available. Free vibration analysis was carried out using four quadratic isoparametric spline finite strips (one for each web and two for the slab) and ten segments of equal length with a total of 810 DOFs. The bridge was also
Vibration analysis of variable-depth bridges
37
Finite element20_ Isoparametric spline finite strip
t
I
I
N/mm
Fig. !1. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section B.
I
Finite
+
element k
Isoparametric
finite strip
'
(
.
~
2
+
:
O
!
~
I
I
,
spl ine /1 ( ~ _ ~
o
°oo
? N/mmUnl ~,
Fig. 12. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section D.
o t
i
i
N/mm
element~° Isoparametric spline finite strip Finite
Fig. 13. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section E.
[
I
+
-I-
4-
+
+l
o
o
O
tn
t
i
CD t
N/mm Finite element20 +
Isoparametric strip
spl ine
finite
Fig. 14. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section F.
38
F. T. K. ,4u, Y. K. Cheung ".
Finite
m . . . .
i
e l e m e n t 20
Isoparametric finite strip
o
spline
o i
u~° , N/mm
o r
Fig. 15. Three-span haunched girder reinforced concrete bridge, longitudinal membrane forces at section H.
F t A
21000
Data:
,. (a)
E = 30 k N / m m 2 v
=
0.15
p
=
2400
oo o
o tn
B
21000
L
kg/m 3
I 42000
J _
(b)
80~_~_
•t.800
E 0 0 0 N
E
O
I
9000 (c)
Fig. 16. Single-span concrete bridge with variable-depth integral parapets. (a) Elevation; (b) plan showing bearing layout; (c) cross section (dimensions in ram).
Vibration analysis of variable-depth bridges
39
TABLE 6 Single-span Concrete Bridge with Variabledepth Integral Parapets
Mode
Natural frequency (Hz)
number
COSMOS/M2O
Present method
1 2 3 4 5 6 7 8 9 10
2.905 3.550 6-299 7.697 8.541 10.520 11.465 11-685 14-654 15.741
2.912 3.638 6.398 7.556 8.643 10.533 11.412 11-619 14.685 15-787
analysed by COSMOS/M 2° using 9-node isoparametric shell elements with assumed strain stabilization and a total of 2574 DOFs. The lowest ten natural frequencies are given in Table 6 and the results from the present method compare favourably with the finite element solution. The corresponding mode shapes are also presented in Fig. 17.
9.3 Example 3: Three-span variable-depth box girder bridge with curved alignment A three-span variable-depth single-cell box girder bridge is shown in Fig. 18. Figure 18(b) shows the developed sectional elevation of the bridge along the centre line. The soffit lines are parabolic with vertical axes. The horizontal alignment consists of a circular curve in the central span and two straight tangents in the end spans. Three different values of horizontal curvature in the central span have been considered, as shown in Table 7. The bearing layout for the bridge is shown in Fig. 18(c). Two bearings are provided at each support and it is assumed that they are directly under the webs. Support B is provided with a fixed bearing to serve as the expansion centre and a guided sliding bearing with allowable movement directed towards the expansion centre. Each of the other supports is provided with a guided sliding bearing with allowable movement directed towards the expansion centre, as well as a free sliding bearing. The cross sections at an interior support and middle of the central span are shown in Fig. 19.
40
F. T. K. Au, Y. K. Cheung
B Undeformed mesh
Mode 2
Mode 1
Mode 3
Mode 5
Mode 4
Fig. 17. Caption opposite.
As one of the loading conditions, 45 units of abnormal vehicle HB loading to BS 5400:Part 2:197822 was applied at maximum eccentricity on the bridge deck. Figure 20 shows the plan and axle arrangement of the loading on the bridge. A total load of 1800 kN is assumed to be equally distributed among the 16 wheels on four axles. Static analysis was carried out using six quadratic isoparametric spline finite strips (three for the top flange, one for each web and one for the bottom flange) and a total of 18 segments, including ten for the central span and four for each end span involving a total of 1800 DOFs. The reference solution was obtained by
Vibration analysis of variable-depth bridges
Mode 6
Mode 7
Mode 8
Mode 9
41
Mode 10
Fig. 17. Mode shapes for a single-span concrete bridge with variable-depth integral parapets.
COSMOS/M 2° using 9-node isoparametric shell elements with assumed strain stabilization and a total of 5160 DOFs. The results indicate that the twist of the bridge generally increases with the horizontal curvature but it is mainly confined to the central span. This can be attributed to the torsional restraint provided at Supports B and C. The web under the abnormal vehicle is more heavily loaded around midspan but at Supports B and C both webs are roughly sharing the same load. The effect of changes in horizontal curvature on longitudinal membrane stresses is most noticeable in the web on the outer curve of the central span. With the increase in horizontal curvature, higher long-
42
F. T. K. Au, Y. K. Cheung Span 2
.rj
Data: E = 30 kN/mm z v = 0.15 p = 2400 kg/m3
~ /~
lncreaslnE chainage
(a) ~" o
45 unitsllof IIHBloading A
B
~ C
D
i, 1+
4oooo
1
+oooo
°oooo
_
(b)
A
B _ ~ . ~ ~ - _
C D
- -
- ~ -
"
It)
~
2~00i
( n o t to s c a l e )
Fig. 18. Three-span variable-depth box girder bridge with curved alignment. (a) Plan;
(b) developed sectional elevation along centre line; (c) bearing layout (dimensionsin mm).
itudinal membrane stresses are attracted to Supports B and C while it is relieved around the mid-span. The deflections at the tips of side cantilevers and top flange-web junctions at mid-span positions of the three cases investigated are given in Tables 8-10. The vertical deflections at the tips of side cantilevers for Case 3 are presented in Figs 21 and 22. The corresponding membrane forces at various sections of the central span are shown in Figs 23-25. G o o d agreement is observed. Free vibration analysis was also carried out using six quadratic isoparametric spline finite strips (three for the top flange, one for each web and one for the bottom flange) and a total of 14 segments, including six for the central span and four for each end span involving a total of 1512
Vibration analysis of variable-depth bridges 500 Q o
43
500
--t-t-
-'7 o
o
4000
4000 I
Ca)
500
SO0
°
I
O O 00
?
I I
24oo
,ooo
_i
jl
2~oo
J
°ooo'
(b)
L
Fig. 19. Three-span variable-depth box girder bridge with curved alignment. (a) Cross section at middle of central span and end supports; (b) cross section at interior supports (dimensions in ram). DOFs. The problem w a s also solved by COSMOS/M 2° using 5160 DOFs for comparison. The lowest four natural frequencies for the three cases investigated are given in Table I 1 and the corresponding mode shapes for Case 3 are also presented in Fig. 26. G o o d agreement is again observed. 10 C O N C L U S I O N S The static and free vibration analysis of variable-depth bridges of arbitrary alignments was successfully carried out using the isoparametric spline finite strip method. The presence of haunches, support diaphragms and bearings can all be accounted for. It retains the computational
44
F. T. K. Au, Y. K. Cheung TABLE 7
Three-span Variable-depth Box Girder Bridge With Curved Alignment. Horizontal Curvature of Central Span (centre line)
I
1800
I~
Case
Radius of curvature (m)
Subtended angle (radian)
1 2 3
300 100 60
0-2 0.6 1-0
~
3000
I
J
3000
I
~"
1800
×
o
o
9? o
i
Oo~tn
i
o!
.i
i
/~i transverse 9f c e n t r a l
centre span
line
i
~
longitudinal
oI centr
line
c~ o
Fig. 20. Three-span variable-depth box girder bridge with curved alignment, plan and axle
arrangement of HB loading (dimensions in mm). efficiency o f the spline finite strip method, while it is much more flexible in geometric modelling. The accuracy and efficiency o f the method were tested on a variety o f examples.
REFERENCES 1. Zienkiewicz, O. C. & Taylor, R. L., The Finite Element Method, VoL 1 Basic Formulation and Linear Problems, 4th edn. McGraw-Hill, New York, 1989.
Vibration analysis of variable-depth bridges
45
TABLE $ Three-span Variable-depth Box Girder Bridge With Curved Alignment. Deflections of Top Flange at Mid-span Positions for Case 1 (mm)
Position
Mid-span 1 Mid-span 2 Mid-span 3
Method
Tip of left s i d e cantilever
Present COSMOS/M 2° Present COSMOS/M 2° Present COSMOS/M 2°
-5.598 -5.640 19.587 19.650 -6.000 -5.893
Left flange-web junction -5.567 -5.610 18.698 18.924 -5.967 -5.858
Right flange-web junction -5.565 -5.609 16.667 16.798 -5.966 -5-835
Tip of right side cantilever -5-591 -5-635 16.216 16.435 -5.992 -5.860
TABLE 9 Three-span Variable-depth Box Girder Bridge With Curved Alignment. Deflections of Top Flange at Mid-span Positions for Case 2 (mm)
Position
Mid-span 1 Mid-span 2 Mid-span 3
Method
Tip of left s i d e cantilever
Present COSMOS/M 2° Present COSMOS/M 2° Present COSMOS/M 2°
-5-830 -5.870 20.955 21.014 -6.241 -6-133
Left flange-web junction -5-798 -5-838 19.664 19.891 -6.208 -6.098
Right .flange web junction -5.797 -5.838 16.937 17.055 -6.208 -6.076
Tip of right side cantilever -5.824 -5.865 16.133 16-336 -6.237 6-103
TABLE 10 Three-span Variable-depth Box Girder Bridge With Curved Alignment. Deflections of Top Flange at Mid-span Positions for Case 3 (mm)
Position
Mid-span 1 Mid-span 2 Mid-span 3
Method
Tip of left s i d e cantilever
Present COSMOS/M 2° Present COSMOS/M 2° Present COSMOS/M 2°
-6.149 -6.188 22.566 22.633 -6.564 -6.459
LeJt Right flange web flange-web .junction junction -6.116 -6.155 20.887 21.124 -6.532 -6-424
-6.115 -6.155 17.467 17.579 -6.533 -6-402
Tip of right side cantilever -6.144 -6.184 16.324 16-517 -6-566 -6-431
Vertical deflection (mm)
strip
ill
(m)
Fig. 21. Three-span variable-depth box girder bridge with curved alignment (Case 3), vertical deflection at outer curve.
20
10
-I0
-20
:x
Vertical
deflection
(mm)
strip
Cm)
Fig. 22. Three-span variable-depth box girder bridge with curved alignment (Case 3), vertical deflection at inner curve.
20
I0
-10
-20
48
F. T. K. Au, Y. K. Cheung
®
0
1
i
I
2 1
3
I
N/mm 2
Finite element 2° Isoparametric finite strip
spline
Fig. 23. Three-span variable-depth box girder bridge with curved alignment (Case 3), longitudinal membrane stresses in middle of central span.
I
1
2
3
i
i
r
N/mm 2
® Finite element 20 Isoparametric finite strip
spline
Fig. 24. Three-span variable-depth box girder bridge with curved alignment (Case 3), longitudinal membrane stresses at support B of central span. 2. Zienkiewicz, O. C. & Taylor, R. L., The Finite Element Method, Vol. 2 Solid and Fluid Mechanics, Dynamics and Non-linearity, 4th edn. McGraw-Hill, New York, 1991. 3. Cheung, Y. K., Finite Strip Method in Structural Analysis. Pergamon Press, Oxford, England, 1976. 4. Cheung, Y. K., Fan, S. C. & Wu, C. Q., Spline finite strip in structural analysis. Proc. of Int. Conf. on Finite Element Method, Shanghai, China, 1982, pp. 704-709. 5. Fan, S. C. & Cheung, Y. K., Flexural free vibration of rectangular plates with complex support conditions. J. Sound Vib., 93(1) (1984) 81-94. 6. Chen, M. J., Tham, L. G. & Cheung, Y. K., Spline finite strip for parallelo-
Vibration analysis of variable-depth bridges
49
TABLE 11 Three-span Variable-depth Box Girder Bridge With Curved Alignment. Natural Frequencies (Hz)
Case
Case I Case 2 Case 3
Method
Mode number
Present COSMOS/M2° Present COSMOS/M20 Present COSMOS/M2°
1
2
3
4
2.235 2.221 2.220 2.206 2.194 2.181
4.032 3.988 3.921 3.871 3.582 3.525
4-902 4-817 4.445 4.383 4.158 4.114
5.562 5.382 5.553 5.370 5.524 5-340
® +_---
0 i
1 i
2 i
3 I
N/mm 2
Finite +
e l e m e n t 20
Isoparametric finite strip
spline
Fig. 25. Three-span variable-depth box girder bridge with curved alignment (Case 3), longitudinal membrane stresses at support C of central span.
7. 8. 9. 10. 11.
gram plate. Proc. of Int. Conf. on Accuracy Estimates and Adaptive Refinements in Finite Element Computations (ARFEC), Lisbon, Portugal, 1984, Vol. 1, pp. 95-104. Tham, L. G., Li, W. Y., Cheung, Y. K. & Chen, M. J., Bending of skew plates by spline-finite-strip method. J. Comput. Struct., 22(1) (1986) 31-8. Cheung, Y. K., Tham, L. G. & Li, W. Y., Application of spline-finite-strip method in the analysis of curved slab bridge. Proc. Inst. Cir. Engrs, Part 2, 81 (1986) 111-124. Cheung, Y. K., Tham, L. G. & Li, W. Y., Free vibration and static analysis of general plate by spline finite strip. Computat. Mech., 3 (1988) 187-97. Cheung, Y. K., Li, W. Y. & Tham, L. G., Free vibration analysis of singly curved shell by spline finite strip method. J. Sound Fib., 128(3) (1989) 411 422. Li, W. Y., Tham, L. G., Cheung, Y. K. & Fan, S. C., Free vibration analysis of doubly curved shells by spline finite strip method. J. Sound Vib., 140(1) (1990) 39-53.
50
F. T. K. A u, Y. K. Cheung
B A
Undeformed
Mode 2
mesh
Mode
1
Mode 3
Mode 4
Fig. 26. Mode shapes for three-span variable-depth box girder bridge with curved alignment (Case 3).
Vibration analysis of variable-depth bridges
51
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