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Analysis of haunched, continuous bridges by the finite strip method
Compurers & Structures Vol. 28. No. J, pp. 621-626, Printed in Gnat Britain.
1988
53.00 + 0.00 0045-7949/M Pmpmon Pnw plc
ANALYSIS OF HAUNCHED, CONTINUOUS BRIDGES BY THE FINITE STRIP METHOD M. S. CHEUNG and
WENC~NG
LI
Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada KIN 6N5 (Receiued 6 April 1987) Ahatraet-The eigenfunctions of a contmuous beam are derived by numeric& methods. The folded plate type finite strip with intermediate supports is formmated by combining this function in the longitudinal direction with an appropriate finite element shape function in the transverse direction The strip with variable width in a curvilinear coordinate system is also established in order to analyse the web of a haunched continuous box-girder bridge. The numerical examples given in this paper, relating to ~Rti~uo~ beams and box-girder bridges, d~onstmt~ the advantages of this method, which are its simplicity, accuracy and convenience.
INTRODUCTION
For analysis of a structure which has constant depth in the longitudinal direction, the finite strip method has proven to be the most efficient numerical method which uses beam eigenfunctio~ series in the longitudinal direction, thus reducing the number of ~mensions of analysis by at least one [l-3]. Combined with the flexibility method, the finite strip method also can be used to analyse continuous plates or bridges with intermediate columns[4] or continuous line support [5]. However, in this case the convergence of the eigenfuction series is much slower than it is in the simply supported case. In a modified form of finite strip analysis 161, the free vibration equation of a ~~tinuo~ beam is solved by a numerical method, its eigenfunctions are then directly used as the shape functions in the longitudinal direction for finite strip analysis of structures with intermediate line supports. The convergence of this modified strip is improved significantly in comparison to the ~nve~tional ‘flexibility approach’. In the case of a continuous box-girder bridge, the depth of the box girder usually varies in the longitudinal direction. In order to analyse this type of structure, a strip with variable width in the curvilinear coordinate system is developed This makes the finite strip method suitable not only for the structure with constant depth, but also for ones with variable cross-sections. The numerical examples of continuous beams and haunched continuous box-girder bridges demonstrate the convenient, accuracy and efIiciency of this method.
strip shape functions in the longitudinal direction. These functions constitute a complete orthogonal series, which is necessary for convergence to the exact solution when successively increasing the number of terms. Since the functions are o~hogon~, the following integrations over the whole beam vanish: Y,Y,dy=O
if
m icn
Y~Y~dy=O
if
mjtn.
This orthogonal property greatly enhances the efficiency of the method. The beam vibration ~fferential equation [7, S] is
where P2& iu4= EIg EI is flexural rigidity A is the cross-sectional area y is the weight of material of the beam per unit volume g is the gravity acceleration and p is the frequency of the beam. Because all the spans of a continuous beam vibrate at the same frequency p, p is the same for all spans, The general solution of this equation can then be expressed as Y(y) = C, sin fiy + C, cos py
EIGENFUNCTIONS
The eigenfunctions Y,cV), obtained from the general sofution of the gove~ing differential equation of a continuous beam, are applied directly to the finite
+ C, sinh my + C4 cash my, (2) in which C, . ,. C4 are constants which are to be determined in each particular case from the end conditions of each span. 621
622
M. S.
CHEUNG and WENCHANGLI
Table 1. The proper number of segments and Gauss points b constant
Fig. 1. Folded plate strip.
Solutions of the above equation for any number of spans have been discussed in detail in [6], and will not be repeated here. FOLDED PLATE FLANGE STRIP
Each folded plate strip is subject to in-plane stresses and transverse. bending forces [ 1,2,9] (Fig. 1). The nodal displacements and forces of the strip corresponding to the mth series term are
% I,2 3,4 5,6 7,8 9 IO,11 12-14 15-17 18 19-24 25-30
nOavrr
“=a 1
8 10 8 10 10 8 10 10 10 10 10
1 2 2 2 3 3 4 4 5 6
u2 u2 w2 w:
(4)
F,=(U,V,W,M,V,V,W,M,)~. The
displacement a = $
field within a plate strip is
((1 - X)C + X~~)YA_v)
v = 5 ((1 - X)o;l+ Xo~)Y:Cy)/& m=l w= f
(3)
(6) (6)
((1 -3X2+2X3)wy
m=l
- 2x + x2)ec;,
+x(1
+ (3X2 - 2X3)wT +x(X2
- aw~J_Y),
where X = x/b. The strain-displacement
(7)
relations are
au Ex = ax au CY = ay au a0 Yxy= ay + z xx= -p xv=-ayz
alW
Substituting eqn (8)
(8)
eqns (S)-(7) into eqn (8), we can rewrite
E =
f
m=,
B,,J,.
I 1 2 2 3 ; 4 5 5 6
%?,“S 8 10 8 10 10 10 10 10 10 10 10
The stiffness matrix of the strip corresponding m th and n th series terms is B; DB, dx dy. ISL
to the
(10)
b
In the x direction, the integration can be implemented analytically; however, in the y direction Gaussian integration has to be used because of the complexity of the integration in the stiffness matrix. Since JL Y:, Y:dy and SLY; Y,dy are not equal to zero if m is not the same as n, the terms of the series are coupled. In evaluating the stiffness matrix, we should choose the proper number of Gauss integration points to achieve the desired accuracy and efficiency. When necessary, every span can be divided into several segments. In each segment 8-10 integration points can be used. The selection of segment size and Gauss integration points is mainly dependent on the highest order harmonic, nP = &.x Ii/n, to be evaluated in the stiffness matrix. Based on experience, the best selection which will achieve a reasonable balance between accuracy and efficiency of the solution is given in Table 1. Using the procedures commonly used in finite element or finite strip methods [2, lo], it is not difficult to obtain the final solution required, up to the distribution of displacements and stresses within the whole folded plate structure.
VARIABLE DEPTH WEB STRIP
azW
azW &Y = 2-. axay
%
nr = p,lJn. nscg: number of segments in span i. nOausa:numberof Gauss points in a segment.
K,= 6, = (% 01WI0,
b variable
Web depth, b, of a haunched bridge usually varies in the longitudinal direction as shown in Fig. 2. The web of a haunched girder is subjected to both inplane and out-of-plane bending actions. However, it is mainly subjected to in-plane bendingi It is convenient to use curvilinear orthogonal coordinates (< -u) for this kind of web strip (Fig. 3). The web can be divided into a number of equal or unequal width strips. The width of each strip is taken as c x b, whereO
Analysis of bridges by finite strip method b I
The above shape func~ons guarantee that the shear strain, ya, will always be approximately equal to zero at any section of the girder. Substituting eqns (14) and (15) into simplified eqns (1 If-(13), and simplifying further, we obtain the following B matrix of
Y
17 1
I 8
I1
i
I2
623
in-plane
strain:
Fig. 2. Haunched bridge.
For in-plane action the ~lationships placement and strain are ft
&E!!+Y cb@
r,
u
au
r2
atl
LgX --+au
ia0
between dis-
(11)
(12) tl
u (13)
Y’.=~+~dT,-,+r,*
In the case of tl P cb, u % u and r2 9 r~, o/r, may be neglected in eqn (11) and v/r, in eqn (13). The deformation of a girder web is caused mainly by shear forces. Therefore, the axial stress resultant
The B matrix for out-of-plane bending is the same as that for the ordinary folded plate strip in the previous section, except for the substitution of the current variable width for the original constant width. The numerical examples given in the next section show that the above-mentioned simplification not only gains efficiency but also assures the accuracy of the analysis. Arising from the ~mpatibility ~q~~ent, we must use the same displacement function v in the Q direction for the top and bottom flanges which are directly connected to the bridge web strip, so that the B$ matrix of the flange strip must be modified as follows:
0
Y,,,lb
0
(1 - X)b, YIP,,,
0
~b~Y~l~~
-b,
y6/&,,
in any cross-section of the girder-web is approximately equal to zero. This is because in bridge structures usually only one support is restrained in the longitudinal direction. It follows that the term u/r2 in eqn (12) can be safely discarded without sacrificing too much in accuracy. If we assume I~ = b/(~~d~~ then the in-plane displacement shape function can be written as: (141
XC
9
(17)
b, Yblb~u, !
where X = x/b; b is the width of flange strip and b, is the depth of the web. NUMERICAL
EXAMPLES
1. Continous beam Beams A and B are shown in Fig. 4, Each beam is divided into only one strip. The deflection is antisymmetrical with respect to the intermediate support, therefore we use only the antis~et~~l modes as the shape functions in the longitudinal direction. The deflection at midspan and the longitudinal stresses at several points along the bottom nodal line are given in Table 2. The numerical results confirm the correctness of formula (16). 2. Continuous concrete box-girder bridge [9]
X’
‘2
Fig. 3. Curvilineai coordinate system.
Two multi-cell box-girder bridges that are continous over two equal spans are shown in Fig. 5. Bridge 2 is haunched over the interior supports, with c = 0.60 m. The bridge span 1 is chosen to be 40 m, spacing between webs b2 = 0.0751= 3.00 m; thickness of top slab, h, = 0.0051= 0.20 m; thickness of bottom of web, slab, h2 = 0.004 I = 0.16 m; thickness
624
M. S. CHEUNGand WENCHANGLI
Beam A
b z 0
y - y2)/100
Beam B
b I l+O.l y
OlyllO
ForBeamsA and B, 21. I, z 20, B
q
106, Yz 0
Fig. 4. Continuous beams.
above the central pier. However, in the analysis, the deflection of the intermediate support and cross girder in the transverse direction of the bridge are disregarded.
b, = 0.011251= 0.45 m; length of haunched portion (if any) is considered to cover a distance equal to l/5 = 8.00 m. Total depth of the box-girder (excluding the haunch) is h = 1.20 m. There is a cross girder
Table 2. Deflection and longitudinal stresses of continuous beams Beam
Number of antisym. terms
WC
% y=2.5
A
B
1 5 10 15 20
0.03096 0.0328 1 0.03389 0.03489 0.03531
-
Beam theory
0.03425
-
y=5.0
y = 7.5 338.8 278.8 287.7 281.3 306.7
y = 10.0
300.0
300.0
300.0 1946 1682 1492 1500
231.3 309.9 284.8 279.2 326.1
-
1 5 10
0.080 11 0.08405 0.09270
465 977 983
1032 1337 1319
y = 8.0 1665 1471 1471
Beam theory
0.08178
960
1333
1481
I - 40m
,
I:
379.9 327.8 324.7 280.3 283.3
40m
1.2m
I
htc
b2=3.00m y = 24kN/m3, E = 25 GPa, YZ 0.2 Fig. 5. Continuous box-girder bridge.
Analysis of bridges by finite strip method
Centre of
line
midspan, the longitudinal stress at the top and bottom of the web at midspan and over the intermediate support are listed in Table 3. The distribution of the longitudinal stress over the centre line of the intermediate support is depicted in Fig. 7, in which the dashed line expresses the value from beam theory. From Table 3 it can be seen that only three symmetrical terms are required to obtain quite accurate values of deflections and stresses (with an error of less than 3%), this being mainly due to the fact that the chosen shape functions coincide well with the actual deflection pattern of the structure. For the web of the haunched bridge (bridge 2), the strip stiffness was derived from a curvilinear coordinate system. The division of strips is the same as in bridge 1, as shown in Fig. 6. Strictly speaking, the bottom flange of a haunched bridge should be analysed as a thin shell rather than a flat plate. However, in bridge 2, the bottom flange is only slightly curved in the haunched portion, and its is slope insignificantly small (only 0.6/8.00 = 0.075). Therefore, in this situation, it is possible to use flat plate strips to model the entire bottom flange without sacrificing the overall solution accuracy. However, in order to maintain the compatibility requirement between the web and the bot-
line mid-way between two webs
Web
625
Fig. 6. Division of strips
Taking advantage of symmetry, we analysed only half a cell which is divided into 11 strips, as shown in Fig. 6. Each bridge is loaded by the self-weight corresponding to the specific weight of material equal to 24 kN/m3. The material properties are E = 25 Gpa, v = 0.2. Because the geometry of the bridge and applied loads are symmetrical with respect to the intermediate support, the deflections must also be symmetrical. Therefore we take only the symmetrical modes as the shape functions in the longitudinal direction. For bridge 1 (no haunch) the deflection at
Table 3. Deflection and longitudinal stresses of bridge 1 (no haunch) Section t-t over central support
Section s--s at midspan Number of sym. terms 1 2 3 4 5 Solution of beam theorv
(2)
UYr (Mpa)
=Yn (Mpa)
% (Mpa)
% (Mpa)
0.05974 0.06001 0.05956 0.05969 0.05964 0.06131
- 5.592 -5.681 - 5.288 - 5.266 -5.381 -5.374
6.624 6.710 6.288 6.276 6.290 6.284
9.708 10.003 10.584 10.644 10.802 10.748
-11.340 -11.671 - 12.331 - 12.422 - 12.600 - 12.567
10.603 10.802 v
-4__
10.553 /7 10.748 beam theory
computer 10.802 10.748
-12.6
-12.567 ---
beam
theory f -12.600
&554--12.413
-12.567 beam theory -12.365
Fig. 7. Longitudianal stress at section over the central support of bridge 1 0-mhaunch). CA
s. 28,SE
M. S. CHEUNGand WENCHANGLI
626
Table 4. Deflection and longitudinal stresses of bridge 2 (with haunch) Section s--s at midspan
Section I-t over central support
Number of sym. terms
(2)
cJr (Mpa)
c@ (Mpa)
% (Mpa)
cYs (Mpa)
1 3 5 7 10
0.04373 0.04592 0.04645 0.04638 0.04646
-4.162 -4.481 -4.517 -4.428 - 4.493
4.819 5.222 5.263 5.158 5.234
11.328 8.803 7.661 7.678 7.908
- 13.018 - 10.080 -8.731 - 8.758 - 9.024
7.651 7.908
7.589
7.846
c 0 .,2 0
08
.62
.024
0.16
-9.024
18.969
-8.796
-8.738
Fig. 8. Longitudinal stress at section over the central support of bridge 2 (with haunch).
tom flange over the haunched portion, displacement parameters are expressed in terms of the curvilinear coordinate system as used in the web strip. The deflections and longitudinal stresses at the same position as in bridge 1 are shown in Table 4. It can be seen that the changes of deflections and stresses are insignificant when the number of the symmetrical terms used in the analysis is more than 5. The solution converges very quickly for this type of loading. The distribution of the longitudinal stress over the cross-section at the centre line of the intermediate support is depicted in Fig. 8, in which the values of stresses are obtained by using 10 symmetrical terms. Comparing Table 4 with Table 3, the maximum longitudinal stress is reduced by 28%, and the deflection at midspan by 24%, as a result of the haunch.
Acknowledgement-The financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
REFERENCES 1. Y. K. Cheung, The finite strip method in the analysis of
elastic plates with two opposite simply supported ends. Proc. Inst. Civ. Engrs 40, 1-7 (1968). 2. M. S. Cheung, Finite strip analysis of structures. Ph.D.
Thesis, University of Calgary, Alberta (1971). 3. M. S. Cheung and M. Y. T. Chan, Three dimensional
finite strip analysis of elastic solid. Compur. Strucf. 9, 629-638 (1978). 4. M. S. Cheung, Y. K. Cheung and A. Ghali, Analysis of slab and girder bridges by the finite strip method. Building Sci. 5, 95-104 (1970). 5. A. Chali and G. S. Tadros, On finite strip analysis of
continuous plates. Research Report No. CE-72-11, Department of Civil Engineering, University of Calgary (1972). M. S. Cheung and W. Li, Finite strip analysis of continuous structures. Can. J. Ciu. Engng to appear. D. J. Gorman, Free Vibration Analysis of Beams and Shafts. John Wiley, New York (1975). S. Timoshenko and D. H. Young, Vibration Problems in Engineering, 4th Edn. D. Van Nostrand, Toronto, CA (1974). 9. A. Ghali, M. S. Cheung, W. H. Dilger and M. Y. T. Chan, Longitudinal stress over supports of concrete box-girder bridges. Can. J. Civ. Engng 155-164 (1981). 10. Y. K. Cheung, Finite Strip Method in Structural Analysis. Pergamon Press, Oxford (1976).
1988
53.00 + 0.00 0045-7949/M Pmpmon Pnw plc
ANALYSIS OF HAUNCHED, CONTINUOUS BRIDGES BY THE FINITE STRIP METHOD M. S. CHEUNG and
WENC~NG
LI
Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada KIN 6N5 (Receiued 6 April 1987) Ahatraet-The eigenfunctions of a contmuous beam are derived by numeric& methods. The folded plate type finite strip with intermediate supports is formmated by combining this function in the longitudinal direction with an appropriate finite element shape function in the transverse direction The strip with variable width in a curvilinear coordinate system is also established in order to analyse the web of a haunched continuous box-girder bridge. The numerical examples given in this paper, relating to ~Rti~uo~ beams and box-girder bridges, d~onstmt~ the advantages of this method, which are its simplicity, accuracy and convenience.
INTRODUCTION
For analysis of a structure which has constant depth in the longitudinal direction, the finite strip method has proven to be the most efficient numerical method which uses beam eigenfunctio~ series in the longitudinal direction, thus reducing the number of ~mensions of analysis by at least one [l-3]. Combined with the flexibility method, the finite strip method also can be used to analyse continuous plates or bridges with intermediate columns[4] or continuous line support [5]. However, in this case the convergence of the eigenfuction series is much slower than it is in the simply supported case. In a modified form of finite strip analysis 161, the free vibration equation of a ~~tinuo~ beam is solved by a numerical method, its eigenfunctions are then directly used as the shape functions in the longitudinal direction for finite strip analysis of structures with intermediate line supports. The convergence of this modified strip is improved significantly in comparison to the ~nve~tional ‘flexibility approach’. In the case of a continuous box-girder bridge, the depth of the box girder usually varies in the longitudinal direction. In order to analyse this type of structure, a strip with variable width in the curvilinear coordinate system is developed This makes the finite strip method suitable not only for the structure with constant depth, but also for ones with variable cross-sections. The numerical examples of continuous beams and haunched continuous box-girder bridges demonstrate the convenient, accuracy and efIiciency of this method.
strip shape functions in the longitudinal direction. These functions constitute a complete orthogonal series, which is necessary for convergence to the exact solution when successively increasing the number of terms. Since the functions are o~hogon~, the following integrations over the whole beam vanish: Y,Y,dy=O
if
m icn
Y~Y~dy=O
if
mjtn.
This orthogonal property greatly enhances the efficiency of the method. The beam vibration ~fferential equation [7, S] is
where P2& iu4= EIg EI is flexural rigidity A is the cross-sectional area y is the weight of material of the beam per unit volume g is the gravity acceleration and p is the frequency of the beam. Because all the spans of a continuous beam vibrate at the same frequency p, p is the same for all spans, The general solution of this equation can then be expressed as Y(y) = C, sin fiy + C, cos py
EIGENFUNCTIONS
The eigenfunctions Y,cV), obtained from the general sofution of the gove~ing differential equation of a continuous beam, are applied directly to the finite
+ C, sinh my + C4 cash my, (2) in which C, . ,. C4 are constants which are to be determined in each particular case from the end conditions of each span. 621
622
M. S.
CHEUNG and WENCHANGLI
Table 1. The proper number of segments and Gauss points b constant
Fig. 1. Folded plate strip.
Solutions of the above equation for any number of spans have been discussed in detail in [6], and will not be repeated here. FOLDED PLATE FLANGE STRIP
Each folded plate strip is subject to in-plane stresses and transverse. bending forces [ 1,2,9] (Fig. 1). The nodal displacements and forces of the strip corresponding to the mth series term are
% I,2 3,4 5,6 7,8 9 IO,11 12-14 15-17 18 19-24 25-30
nOavrr
“=a 1
8 10 8 10 10 8 10 10 10 10 10
1 2 2 2 3 3 4 4 5 6
u2 u2 w2 w:
(4)
F,=(U,V,W,M,V,V,W,M,)~. The
displacement a = $
field within a plate strip is
((1 - X)C + X~~)YA_v)
v = 5 ((1 - X)o;l+ Xo~)Y:Cy)/& m=l w= f
(3)
(6) (6)
((1 -3X2+2X3)wy
m=l
- 2x + x2)ec;,
+x(1
+ (3X2 - 2X3)wT +x(X2
- aw~J_Y),
where X = x/b. The strain-displacement
(7)
relations are
au Ex = ax au CY = ay au a0 Yxy= ay + z xx= -p xv=-ayz
alW
Substituting eqn (8)
(8)
eqns (S)-(7) into eqn (8), we can rewrite
E =
f
m=,
B,,J,.
I 1 2 2 3 ; 4 5 5 6
%?,“S 8 10 8 10 10 10 10 10 10 10 10
The stiffness matrix of the strip corresponding m th and n th series terms is B; DB, dx dy. ISL
to the
(10)
b
In the x direction, the integration can be implemented analytically; however, in the y direction Gaussian integration has to be used because of the complexity of the integration in the stiffness matrix. Since JL Y:, Y:dy and SLY; Y,dy are not equal to zero if m is not the same as n, the terms of the series are coupled. In evaluating the stiffness matrix, we should choose the proper number of Gauss integration points to achieve the desired accuracy and efficiency. When necessary, every span can be divided into several segments. In each segment 8-10 integration points can be used. The selection of segment size and Gauss integration points is mainly dependent on the highest order harmonic, nP = &.x Ii/n, to be evaluated in the stiffness matrix. Based on experience, the best selection which will achieve a reasonable balance between accuracy and efficiency of the solution is given in Table 1. Using the procedures commonly used in finite element or finite strip methods [2, lo], it is not difficult to obtain the final solution required, up to the distribution of displacements and stresses within the whole folded plate structure.
VARIABLE DEPTH WEB STRIP
azW
azW &Y = 2-. axay
%
nr = p,lJn. nscg: number of segments in span i. nOausa:numberof Gauss points in a segment.
K,= 6, = (% 01WI0,
b variable
Web depth, b, of a haunched bridge usually varies in the longitudinal direction as shown in Fig. 2. The web of a haunched girder is subjected to both inplane and out-of-plane bending actions. However, it is mainly subjected to in-plane bendingi It is convenient to use curvilinear orthogonal coordinates (< -u) for this kind of web strip (Fig. 3). The web can be divided into a number of equal or unequal width strips. The width of each strip is taken as c x b, whereO
Analysis of bridges by finite strip method b I
The above shape func~ons guarantee that the shear strain, ya, will always be approximately equal to zero at any section of the girder. Substituting eqns (14) and (15) into simplified eqns (1 If-(13), and simplifying further, we obtain the following B matrix of
Y
17 1
I 8
I1
i
I2
623
in-plane
strain:
Fig. 2. Haunched bridge.
For in-plane action the ~lationships placement and strain are ft
&E!!+Y cb@
r,
u
au
r2
atl
LgX --+au
ia0
between dis-
(11)
(12) tl
u (13)
Y’.=~+~dT,-,+r,*
In the case of tl P cb, u % u and r2 9 r~, o/r, may be neglected in eqn (11) and v/r, in eqn (13). The deformation of a girder web is caused mainly by shear forces. Therefore, the axial stress resultant
The B matrix for out-of-plane bending is the same as that for the ordinary folded plate strip in the previous section, except for the substitution of the current variable width for the original constant width. The numerical examples given in the next section show that the above-mentioned simplification not only gains efficiency but also assures the accuracy of the analysis. Arising from the ~mpatibility ~q~~ent, we must use the same displacement function v in the Q direction for the top and bottom flanges which are directly connected to the bridge web strip, so that the B$ matrix of the flange strip must be modified as follows:
0
Y,,,lb
0
(1 - X)b, YIP,,,
0
~b~Y~l~~
-b,
y6/&,,
in any cross-section of the girder-web is approximately equal to zero. This is because in bridge structures usually only one support is restrained in the longitudinal direction. It follows that the term u/r2 in eqn (12) can be safely discarded without sacrificing too much in accuracy. If we assume I~ = b/(~~d~~ then the in-plane displacement shape function can be written as: (141
XC
9
(17)
b, Yblb~u, !
where X = x/b; b is the width of flange strip and b, is the depth of the web. NUMERICAL
EXAMPLES
1. Continous beam Beams A and B are shown in Fig. 4, Each beam is divided into only one strip. The deflection is antisymmetrical with respect to the intermediate support, therefore we use only the antis~et~~l modes as the shape functions in the longitudinal direction. The deflection at midspan and the longitudinal stresses at several points along the bottom nodal line are given in Table 2. The numerical results confirm the correctness of formula (16). 2. Continuous concrete box-girder bridge [9]
X’
‘2
Fig. 3. Curvilineai coordinate system.
Two multi-cell box-girder bridges that are continous over two equal spans are shown in Fig. 5. Bridge 2 is haunched over the interior supports, with c = 0.60 m. The bridge span 1 is chosen to be 40 m, spacing between webs b2 = 0.0751= 3.00 m; thickness of top slab, h, = 0.0051= 0.20 m; thickness of bottom of web, slab, h2 = 0.004 I = 0.16 m; thickness
624
M. S. CHEUNGand WENCHANGLI
Beam A
b z 0
y - y2)/100
Beam B
b I l+O.l y
OlyllO
ForBeamsA and B, 21. I, z 20, B
q
106, Yz 0
Fig. 4. Continuous beams.
above the central pier. However, in the analysis, the deflection of the intermediate support and cross girder in the transverse direction of the bridge are disregarded.
b, = 0.011251= 0.45 m; length of haunched portion (if any) is considered to cover a distance equal to l/5 = 8.00 m. Total depth of the box-girder (excluding the haunch) is h = 1.20 m. There is a cross girder
Table 2. Deflection and longitudinal stresses of continuous beams Beam
Number of antisym. terms
WC
% y=2.5
A
B
1 5 10 15 20
0.03096 0.0328 1 0.03389 0.03489 0.03531
-
Beam theory
0.03425
-
y=5.0
y = 7.5 338.8 278.8 287.7 281.3 306.7
y = 10.0
300.0
300.0
300.0 1946 1682 1492 1500
231.3 309.9 284.8 279.2 326.1
-
1 5 10
0.080 11 0.08405 0.09270
465 977 983
1032 1337 1319
y = 8.0 1665 1471 1471
Beam theory
0.08178
960
1333
1481
I - 40m
,
I:
379.9 327.8 324.7 280.3 283.3
40m
1.2m
I
htc
b2=3.00m y = 24kN/m3, E = 25 GPa, YZ 0.2 Fig. 5. Continuous box-girder bridge.
Analysis of bridges by finite strip method
Centre of
line
midspan, the longitudinal stress at the top and bottom of the web at midspan and over the intermediate support are listed in Table 3. The distribution of the longitudinal stress over the centre line of the intermediate support is depicted in Fig. 7, in which the dashed line expresses the value from beam theory. From Table 3 it can be seen that only three symmetrical terms are required to obtain quite accurate values of deflections and stresses (with an error of less than 3%), this being mainly due to the fact that the chosen shape functions coincide well with the actual deflection pattern of the structure. For the web of the haunched bridge (bridge 2), the strip stiffness was derived from a curvilinear coordinate system. The division of strips is the same as in bridge 1, as shown in Fig. 6. Strictly speaking, the bottom flange of a haunched bridge should be analysed as a thin shell rather than a flat plate. However, in bridge 2, the bottom flange is only slightly curved in the haunched portion, and its is slope insignificantly small (only 0.6/8.00 = 0.075). Therefore, in this situation, it is possible to use flat plate strips to model the entire bottom flange without sacrificing the overall solution accuracy. However, in order to maintain the compatibility requirement between the web and the bot-
line mid-way between two webs
Web
625
Fig. 6. Division of strips
Taking advantage of symmetry, we analysed only half a cell which is divided into 11 strips, as shown in Fig. 6. Each bridge is loaded by the self-weight corresponding to the specific weight of material equal to 24 kN/m3. The material properties are E = 25 Gpa, v = 0.2. Because the geometry of the bridge and applied loads are symmetrical with respect to the intermediate support, the deflections must also be symmetrical. Therefore we take only the symmetrical modes as the shape functions in the longitudinal direction. For bridge 1 (no haunch) the deflection at
Table 3. Deflection and longitudinal stresses of bridge 1 (no haunch) Section t-t over central support
Section s--s at midspan Number of sym. terms 1 2 3 4 5 Solution of beam theorv
(2)
UYr (Mpa)
=Yn (Mpa)
% (Mpa)
% (Mpa)
0.05974 0.06001 0.05956 0.05969 0.05964 0.06131
- 5.592 -5.681 - 5.288 - 5.266 -5.381 -5.374
6.624 6.710 6.288 6.276 6.290 6.284
9.708 10.003 10.584 10.644 10.802 10.748
-11.340 -11.671 - 12.331 - 12.422 - 12.600 - 12.567
10.603 10.802 v
-4__
10.553 /7 10.748 beam theory
computer 10.802 10.748
-12.6
-12.567 ---
beam
theory f -12.600
&554--12.413
-12.567 beam theory -12.365
Fig. 7. Longitudianal stress at section over the central support of bridge 1 0-mhaunch). CA
s. 28,SE
M. S. CHEUNGand WENCHANGLI
626
Table 4. Deflection and longitudinal stresses of bridge 2 (with haunch) Section s--s at midspan
Section I-t over central support
Number of sym. terms
(2)
cJr (Mpa)
c@ (Mpa)
% (Mpa)
cYs (Mpa)
1 3 5 7 10
0.04373 0.04592 0.04645 0.04638 0.04646
-4.162 -4.481 -4.517 -4.428 - 4.493
4.819 5.222 5.263 5.158 5.234
11.328 8.803 7.661 7.678 7.908
- 13.018 - 10.080 -8.731 - 8.758 - 9.024
7.651 7.908
7.589
7.846
c 0 .,2 0
08
.62
.024
0.16
-9.024
18.969
-8.796
-8.738
Fig. 8. Longitudinal stress at section over the central support of bridge 2 (with haunch).
tom flange over the haunched portion, displacement parameters are expressed in terms of the curvilinear coordinate system as used in the web strip. The deflections and longitudinal stresses at the same position as in bridge 1 are shown in Table 4. It can be seen that the changes of deflections and stresses are insignificant when the number of the symmetrical terms used in the analysis is more than 5. The solution converges very quickly for this type of loading. The distribution of the longitudinal stress over the cross-section at the centre line of the intermediate support is depicted in Fig. 8, in which the values of stresses are obtained by using 10 symmetrical terms. Comparing Table 4 with Table 3, the maximum longitudinal stress is reduced by 28%, and the deflection at midspan by 24%, as a result of the haunch.
Acknowledgement-The financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
REFERENCES 1. Y. K. Cheung, The finite strip method in the analysis of
elastic plates with two opposite simply supported ends. Proc. Inst. Civ. Engrs 40, 1-7 (1968). 2. M. S. Cheung, Finite strip analysis of structures. Ph.D.
Thesis, University of Calgary, Alberta (1971). 3. M. S. Cheung and M. Y. T. Chan, Three dimensional
finite strip analysis of elastic solid. Compur. Strucf. 9, 629-638 (1978). 4. M. S. Cheung, Y. K. Cheung and A. Ghali, Analysis of slab and girder bridges by the finite strip method. Building Sci. 5, 95-104 (1970). 5. A. Chali and G. S. Tadros, On finite strip analysis of
continuous plates. Research Report No. CE-72-11, Department of Civil Engineering, University of Calgary (1972). M. S. Cheung and W. Li, Finite strip analysis of continuous structures. Can. J. Ciu. Engng to appear. D. J. Gorman, Free Vibration Analysis of Beams and Shafts. John Wiley, New York (1975). S. Timoshenko and D. H. Young, Vibration Problems in Engineering, 4th Edn. D. Van Nostrand, Toronto, CA (1974). 9. A. Ghali, M. S. Cheung, W. H. Dilger and M. Y. T. Chan, Longitudinal stress over supports of concrete box-girder bridges. Can. J. Civ. Engng 155-164 (1981). 10. Y. K. Cheung, Finite Strip Method in Structural Analysis. Pergamon Press, Oxford (1976).