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Application of the compound strip method for the analysis of slab-girder bridges
Compurerr & Srrsc,,,rr, Rinled in Great Bntain.
Vol.
2.
Yjo. 6. pp. 939-986.
1906
APPLICATION OF THE COMPOUND STRIP METHOD THE ANALYSIS OF SLAB-GIRDER BRIDGES Depanment
of Civil
Engineering.
J. 4. PUCKETT The University of Wyoming.
Laramie.
FOR
WY 82071. U.S.A.
(Received 19 Febrrmp 1985) Abstract-The compound strip method is illustrated for the analysis of slab-girder bridges modeled as a linear elastic plate continuous over deflecting supports. This approach incorporates the effects of support elements in a direct stiffness methodology by creating a substructure composed of plate. beam, and column elements which is termed a “compound strip.” The theory and application of the compound strip method is presentxl. The finite element and compound strip methods are compared in an illustrative analysis for a slab-+ler bridge. The results of the compound strip analysis compare well with the finite element method. The methodology presented herein can be used to efficiently model any slab-girder bridge configuration. Typically. the compound strip method requires significantly less computational resources than does the finite element method and is well suited for use on today’s microcomputers.
analysis to a series of one-dimensional analyses, which results in fewer equations with a The contemporary approach to the analysis of flat relatively small half-band width. Cheung and others plate structures, such as slab-girder bridges, is the report excellent comparison with classical solutions finite element method[3, 341. However, the necesand significant reduction in computational effort vis-h-vis several FEM models[ 181. sity of employing a relatively tine mesh to obtain Considerable research was conducted from 1968 an accurate model can lead to a large number of simultaneous equations. Design engineers may find to the present to advance Cheung’s original work. the FEM to be prohibitive for continuous plate sys- The FSM has been successfully applied to a variety tems of reasonable size. The costs involved can be of plate systems, including flat plates, folded plates, high, particularly for dynamic analysis. Further, the and box girders[4-8, I l-17, 19-20, 23-301. Despite these successes, an impasse had been reached in computing resources for an accurate FEM analysis of a large plate system can exceed the capacity of attempting to apply the FSM method to one imthe available hardware (particularly microcompuportant category of plate problems: any system ters). This may be true even when the most effkient which is continuous over flexible beams and.ior colof numerical techniques are employed. umns. Some progress was made by researchers Several numerical techniques are available to re- using a flexibility formulation[j, 141, but the effiduce core storage and computational time; e.g. use ciency and convenience of the FSLMrelative to the of higher-order finite element models to improve FEM had to be compromised. Additional advances convergence and/or reduce the mesh size[34], or were made by Delcourt, Wu, and Cheung[20, 321 use of macroelements which, by themselves, can who employed eigenfunctions for analyzing multirepresent full-size structural components[22, 241. span systems continuous over rigid supports. Alternatively, for plate continua with regular geThis paper describes the development and use ometry and certain boundary conditions, the ver- of compound strip elements for the analysis of slabsatility of the conventional FEM is not required. girder bridges. Key features of this method are the Instead, an alternative methodology developed by inclusion of (I) flexural stiffness of beams. (2) torCheung[9, 101and termed the “finite strip method” sional stiffness of beams, (3) axial stiffness of colcan be used. Because the plane geometry of a plate umns, and (4) flexural stiffness of columns in a diis discretized in one principal direction only, the rect stiffness formulation. The versatility and FSM considerably shortens and simplifies the an- capability of the FSM are particularly enhanced by alyst’s input and output effort, reduces computer incorporating the second and fourth features cited memory capacity and time requirements, and offers above. Applications presented herein employ the greater computational efficiency than the EEM for stiffness matrices for transverse beams and colthose types of problems geometrically suited to umns. These elements, in particular, enhance the both methods. In effect, the FSIM reduces a twoFSM for the analysis of slab-girder bridges. 1. Ih’TRODlJCTION
dimensional
979
2. FISITE
STRIP
METHODS-OVERVIEW
equation
The compound strip method (CSXl) can be considered as a special form of the finite element method. which is discretized in one direction. as shown in Fig. I. The CSM employs a displacement function with simple polynomial functions and continuous differentiable smooth series in orthogonal directions. The series functions satisfy the boundary conditions at the end of the elements or strips. For a single strip, the displacement function can be tvritten as it’ =
i X,,,(.r) Y,,, (_v). I,,= I
(I)
where r is the largest term of the truncated series, X,,,(.r) is the polynomial expression with the undetermined constants (termed displacement coefticients) for the mfh term of the series, and Y,,,(y) are predetermined series which satisfy the end conditions and the deflected shapefl8j. The compound strip method is especially adaptable for the analysis of prismatic thin-wall structures. For the analysis of flat plate structures, only transverse deflections are considered. The strip displacement function for the deformation must satisfy the following requirements: I. The Y,,(v) component of the displacement function should satisfy the boundary conditions. 7 The X,,(.r) part of the displacement _. function must be able to represent a constant generalized strain in the transverse (x) direction. This guarantees the generalized strain will converge to the true one as the mesh is refined. function must satisfy the 3. The displacement compatibility of displacements along the boundaries of the strips. The Y,,,(v) series most commonly used are eigenfunctions derived from the solution of the differential equation for beam vibration[ 181. Y(v)”” = $Y(,),
is
where Cr. C:. C?, C, are constants based on the boundary conditions. Y,,(_v)terms are mode shapes which are orthogonal to each other. The orthogonality has signi~cant implications regarding the computational effort required to employ this displacement function in a finite element formulation. The solutions of eqn (3) and for the various boundary conditions are well known[:! I. 3 I], Many of the computational aspects involving the use of eqn (3) are described elsewhere[:!l, 331). The necessary calculations, integral evaluations, etc., can be performed in closed form. obviating costly numerical algorithms such as quadratic integration, etc. The X,&r) function is a polynomial associated with a nodal displacement parameter and describes the co~esponding displacement field in the transverse f.~) direction of the strip. For example. the out-of-plane displacement function for bending a plate strip is It’ =
z
[(I - 3?
+ 2.r3) N’,,n + Is - 27x
m= I
where a is the width of the strip, .\- = .\-in, W’S are the deflections, and B’s are the rotations of the nodal lines along the strip boundary oriented in the y-direction. Equation (4) can be written as]181 It’
=
i
[C,
21
;I
C,] {
WI=I
,n
Y,,,
(2)
where p. is a parameter and h is the length of the strip[fS]. The general solution of this differential
-X
p . EquationfS] can be differ{ tm I to give the curvature-displacement rela-
where {S,), = entiated tionship
where B,. B2 are the appropriate derivative functions. By minimization of the total potential energy, the stiffness matrix can be obtained with the relationship] 181 Fie. I. Compound strip idealization.
Analysis of slab-girder bridges Compound Strip-Additional
where [B] is the generalized
strain matrix, [Dl is and A., is the area of the
the constitutive matrix, strip. Due to a loading q(.r, y). the load vector
F,,,
981 Capobilllies
for
the mrh term is[18] {F,,,1 =
I,, WIT
d-r* Y)
a.
(8)
The strip stiffness and load vectors can be assembled to give the equilibrium equation ]W{W = IF]. 2.2 Compound
strip
(9)
method
The compound strip is a substructuring technique which allows beam and column elements to be attached to a finite strip. This substructure is illustrated in Fig. 1. The displacement function for the attached elements is consistent with eqn (1). For example, consider the strip element shown in Fig. 2. A spring element is attached to the strip at the local coordinates xr,yc.. The strain energy in the spring is U,. = fK/, [N-(X,,y,.)]‘.
Each minimization results ina 4x4 stiffness matrix adds directly to the strip rtiffnarc matrix of the plate
(10)
Fig. 3. Compound strip capabilities.
where K,, is the spring stiffness. Substituting eqn (5) into eqn (10) gives
where Cl-C4 are the polynomials given in eqn (4) evaluated at x,., and Y,,,, Y,,are the boundary funcUC = ;K,, (11) tions evaluated at yc. This 4 x 4 matrix adds directly to the conventional strip matrix prior to assembly. Thus, the mawhere the expressions contained in [N], are evaltrix given in eqn (12) can easily be incorporated into uated at .r,,y,.. Minimizing the strain energy with existing finite strip programs for flat plate analysis. respect to the displacement coefficients 6 yields a Other compound strip stiffness matrices can be stiffness matrix for the spring element which is conformulated for concentrated rotational stiffnesses, sistent with the matrix given in eqn (7). The matrix and beam elements oriented in the transverse is and longitudinal directions. The capabilities are graphically illustrated in Fig. 3. The procedure to CIC2 CIC3 CIC4 arrive at stiffness matrices similar to eqn (12) is discZc4 ctc2 c2c3 I K&n = K,, Y, cussed in detail elsewhere[28]. The stiffness mac3c3 c3c4 ’ Wm) trices of all attached elements are added directly to c4c4 the plate stiffness matrix, eqn (71, prior to assem(12) bly.
1
-x
/ Y
Fig. 2. Compound strip--linear
spring element.
3.SLAB-CIRDERBRIDGEANALYSIS
The bridge shown in Fig. 4 was subjected to typical truck loading and analyzed by using the CSM and the FEM, and the results were compared. The bridge geometry is typical of an interstate overpass. The 7-in. deck is supported by four plate girders uniformly spaced. The deck and the girders are prismatic in the logitudinal direction. Presently, the CSM formulation does not incorporate in-plane deformation; therefore, the girders were assumed concentric in both the CSM and FEM models. Composite behavior was modeled per AASHTO specifications]1 I.
J. A. PLCKETT
982
ILLUSTRATIVE COLUMN
EXAMPLE
CAP
.. :.,;..*. I== 0 ;a
:
.*.:I.;. ‘*
:;
:
..*;. .; : :j. . . . ..-. ,a‘. ;. 0’
‘,
WE3
COVER
L
54;
#. /2.x
COLUMN I
.
.O....’
COLUMNS
COMPOUND
NO.
, .
9/s*’
Fig. 4. Example
STRIP
:
l-M"1
%?”
GIRDERS
.
!
I
..‘, ;: :
;
slab-girder
STRIP
CAPS
bridge.
MODEL 4 GlROERS
CAP
I\ I/
A
2
ot 9’.6”
/
‘II/
3 4
,\ I
5
‘I’
6
I
7
32”
-6’
_32K
.a*
,I,
t\I
A ,r
6
72’.0’,60’-0’
60°.0”L.
192’. 0”
STRIP
No.
TYPE
I
STD. (LO21
2
COMPOUND
3
STD.
4
COMPOUND
5
STD.
6
COMPOUND
7
STD.
8
COMPOUND
Fig. 5. Compound
strip model
38’-0’
Analysis
983
of slab-girder bridges Table
I. Analysis summary FEM
Maximum Deflection, in. Maximum Support Reaction, kip. Maximum Girder Moment. kin.
0.126 0.128 30.1 37_._ ’
- 2.00 IS00 Maximum Transverse Moment, kin./in. 3.47
of Degrees of Freedom No. of Lines of Data input Execution Time NO.
CSM
675 64
--BOO
I’70 3.59
270 31
(Not Available)
The CSM model incorporated axial stiffness eIements for the column, and longitudinal and transverse beam elements for the diaphragms and the girders. The compound strip layout is illustrated in Fig. 5. The finite element model is shown in Fig. 6. The program ANSYS was employed for the FEM. analysis[21. The ANSYS flat shell and beam elements were employed. 3.1 Comparison of the methods The two techniques compare very well throughout the structure. For example, Figs. 7 and 8 show
the detection along the first interior girder and the de8 ection in the transverse direction under the section where the load is applied. respectively. These deflection plots are typical for the entire structure. The maximum deflection in the CSM was 0.128, and 0.126 with the FEM-an approximate 2% diference. The maximum values of interest are given in Table 1. A comparison for the transverse bending moment in the deck is illustrated in Fig. 9. The girder moment for the first interior girder is shown in Fig. 10. Both of these figures show the fine agreement between the two methods. The maximum actions are given in Table 1. The maximum girder moments differ by 22010,and maximum support reaction by 7%. This difference could be decreased by increasing the number of harmonics considered. Note. the FEM required the solution of 675 equilibrium equations, as compared to the CSMs 270, a significant computational saving. In addition, the associated cafculations, such as integrations, are significantly less for the CSM. Because two different computers were used, a comparison of the execution terms was not available. SUMbIARY
The compound strip method was illustrated for the analysis of a slab-girder bridge modefed as a linear elastic plate continuous over deflecting supports. This approach incorporates the effect of support elements in a direct stiffness methodology by creating a substructure composed of plate, beam, and column elements. The compound strip and finite element methods compared favorably for the illustrated example and may be employed to model rectangular slab-girder bridges.
J..4. PWKETI
98-1
I
I
8.60
4.76
I
I
I
14.26
19.00
r
I
23.76
26.60
33.26
3s.00
Distance from Node 91. ft. Fig. 7. Transverse
deflection.
I
0 - finite Element Method 0 - Compound Strip Method
I
I
I
I
I
I
I
I
I
,
I
I
I
,
1
/
12.0
24.0
36.0
46.0
60.0
72.0
64.0
66.0
106.0
120.0
132.0
144.0
166.0
166.0
160.0
Distance from Node 2, ft. Fig. 8. Longitudinal deflection.
985
Analysis of slab-girder bridges
00
4.76
0.60
WI.26
1e.00
23.76
Zd.60
3i.26
3 00
Distance from Node 91, ft. Fig. 9. Transverse
bending moment in deck.
0 = Finite Ekmont Method IJ- CompoundStrip Method
.O
24.0
40.0
uj.0
D%nce
frirt+lodo ?g.
Fig. IO. Longitud~na[ bending moment in the girder.
143.0
142.0
J. A.
986
kCKEll
REFERESCES
I. American Association of State Highway and Transportation Officials. Srtrnd~rd Sprcifictrrions&r High~IUV Eridpcs. 12th Ed. Association General Office. Washington. D.C. (1977). Ancllysis S~srems Drer’s ,tlt~n2. dNSKS Engineering M/. Revision 4.0. Swanson Analysis. Inc.. Houston. PA 119821. 3. K. Bathe. E. L. Wilson and F. E. Peterson. SAP II’. .-t Sam-rrrrcrl omLysis program for sfcttic and dmrrmic response oflinecrr systems. Report No. EERC 73-l I.
University of California, Berkelev. CA (197-t). 4. T. G. Brown and A. Ghali. Finite strip analysis of quadrilateral plates in bending. J. Sfrrrctltrcll Divisiott. ASCE 104, EM2 (1978). 5. M. S. Cheung and M. Y. T. Chan. Analysis of continuous curved box-girder bridges by the finite strip method. ASCE Preprint 3515. Boston (1979). 6. Sl. S. Cheung, Y. K. Cheung and A. Ghali. Analysis of slab and girder bridges by the finite strip method. Build. Sci. 5 (1970). of 7. Xi. S. Cheung. and Y. K. Cheung. “Analysis Curved Box Girder Bridges by the Finite Strip Method.” Proc. Inr’l. Assoc. Bridge cmd Srr. Engr.. 31-I. 1971. 8 M. S. Cheung and Y. K. Cheung. Static and dynamic behaviour of rectangular plates using higher order tinite strips. Build. SFi. 7 (‘1972). 9 Y. K. Cheung. Finite strip method analysis of elastic slabs. J. Engng. Mech. Di\j. ASCE 94, No. EM6 ( 1968). 10. Y. K. Cheung, The finite strip method in the analysis of elastic plates with two opposite supported ends. Proc. Inst. Civ. Engng. 40 ( 1968). II. Y. K. Cheung. Analysis of box girder bridges by the finite strip method. Publ. SP 26. Amer. Cont. Inst.. Detroit. MI (1969). 12. Y. K. Cheung. The analysis of cylindrical orthotropic curved bridge decks. Proc. Inr. Assoc. Bridge nnd Srrucf. Engng. 29-11 (1969). 13. Y. K. Cheung. Folded plate structures by the finite strip method. J. Stmctural Ditvision. ASCE 95, ST12 (1969). II. Y. K. Cheung, Orthotropic right bridges by the finite strip method. Publ. SP 26. Amer. Cont. Inst.. Detroit MI (1969). 15. Y. K. Cheung, Analysis of simply supported thick. layered plates. J. Engng. Mech. Div.. ASCE 97. EM3 (1971). 16 Y. K. Cheung. Flexural vibrations of rectangular and other polygonal plates. J. Engng. Meclr. Div., ASCE 97, (1971). 17 Y. K. Cheung and M. S. Cheung. Free vibration of
18. 19. 20.
y
22.
23.
24.
15.
26.
‘7. 28.
29.
30.
curved and straight beam-slab or box girder bridges. Proc. In!. Assoc. Bridge clnd Srrtrcr. Engng. 32-11 (1972). Y. K. Cheung. Finire Srrip .Llethod in Strtrctuml Anr~lysis. Pergamon Press. Oxford (1976). A. Coull and P. C. Das. Analysis of curved bridge decks. Proc. Insr. Ci\,. Engnp. 37 (1967). C. Delcourt and Y. K. Cheung. Finite strip analysis of continuous folded plates. Proc. 1~. Assoc. Bridge md Strlrcr. Engng. May (1978). R. P. Felgar. Formulas for integrals containing characteristic functions of vibratine beam. Circular I-I. Bureau of Engineering Research. University of Texas. Austin. TX (1950). R. M. Gutkowski. The finite panel method for the elastic analysis of continuous rectangular plate systems. Ph.D. dissertation. University of Wisconsin. Madison. WI (1974). R. M. Gutkowski and C. K. Wang. Continuous plate analysis by finite panel method. J. SrrlrctrtralDi~,ision. ASCE 102, 629-643 t 1976). Y. C. Loo and A. R. Curens. Developments of the finite strip method in the analysis of orthotropic plates. Det,elopments in Bridge Design and Cottstrlrciion. Crosby Lockwood (I971 1. L. C. Maugh and C. W. Pan. Moments in continuous rectangular slabs on rigid supports. ,-ISCE Trrrns. I I I8 (1942). M. S. Mawenya and J. D. Davies. Finite strip analysis of plate bending including transverse shear effects. &i/d. Sci. 9 (19741. M. Petyt. Finite strip analysis of flat skin-stringer structures. J. Sound Vibration 51 (1977). J. A. Puckett, Compound strip method for the analysis of continuous elastic plates. Doctoral Thesis, Colorado State University, Ft. Collins. CO (1983). R. G. Sisodiya. A. Ghali. and \I’. K. Cheung. Diaphragms in single and double-cell box girder bridges with varying angle of skew. AC/J. July (1972). J. W. Smith, Finite strip analysis of the dynamic response of beam and slab highway bridges. Emrhquuke Engng.
and Strucr.
Dynmrics
1
( 1973).
31. S. Timoshenko
and S. Woinowsky-Krieger. Tlteorv sf Plates nnd Shells. 2nd Ed. McGraw-Hill (1971). 32. C. 1. Wu and Y. K. Cheung. Frequency analysis of rectangular plates continuous in one or two directions. Earthquake
Engng.
bind Strrrct.
DJnnmics
3
( 1974).
33. D. Young and R. P. Felgar. Tables of characteristic functions representing normal modes of vibrations of a beam. Publication No. 4913. University of Texas. Austin, TX. July (1949). 3.4. 0. C. Zienkiewicz. Tile Finire Nemenr Merhod in Engineering Science. 3rd Ed. McGrabv-Hill (1977).
Vol.
2.
Yjo. 6. pp. 939-986.
1906
APPLICATION OF THE COMPOUND STRIP METHOD THE ANALYSIS OF SLAB-GIRDER BRIDGES Depanment
of Civil
Engineering.
J. 4. PUCKETT The University of Wyoming.
Laramie.
FOR
WY 82071. U.S.A.
(Received 19 Febrrmp 1985) Abstract-The compound strip method is illustrated for the analysis of slab-girder bridges modeled as a linear elastic plate continuous over deflecting supports. This approach incorporates the effects of support elements in a direct stiffness methodology by creating a substructure composed of plate. beam, and column elements which is termed a “compound strip.” The theory and application of the compound strip method is presentxl. The finite element and compound strip methods are compared in an illustrative analysis for a slab-+ler bridge. The results of the compound strip analysis compare well with the finite element method. The methodology presented herein can be used to efficiently model any slab-girder bridge configuration. Typically. the compound strip method requires significantly less computational resources than does the finite element method and is well suited for use on today’s microcomputers.
analysis to a series of one-dimensional analyses, which results in fewer equations with a The contemporary approach to the analysis of flat relatively small half-band width. Cheung and others plate structures, such as slab-girder bridges, is the report excellent comparison with classical solutions finite element method[3, 341. However, the necesand significant reduction in computational effort vis-h-vis several FEM models[ 181. sity of employing a relatively tine mesh to obtain Considerable research was conducted from 1968 an accurate model can lead to a large number of simultaneous equations. Design engineers may find to the present to advance Cheung’s original work. the FEM to be prohibitive for continuous plate sys- The FSM has been successfully applied to a variety tems of reasonable size. The costs involved can be of plate systems, including flat plates, folded plates, high, particularly for dynamic analysis. Further, the and box girders[4-8, I l-17, 19-20, 23-301. Despite these successes, an impasse had been reached in computing resources for an accurate FEM analysis of a large plate system can exceed the capacity of attempting to apply the FSM method to one imthe available hardware (particularly microcompuportant category of plate problems: any system ters). This may be true even when the most effkient which is continuous over flexible beams and.ior colof numerical techniques are employed. umns. Some progress was made by researchers Several numerical techniques are available to re- using a flexibility formulation[j, 141, but the effiduce core storage and computational time; e.g. use ciency and convenience of the FSLMrelative to the of higher-order finite element models to improve FEM had to be compromised. Additional advances convergence and/or reduce the mesh size[34], or were made by Delcourt, Wu, and Cheung[20, 321 use of macroelements which, by themselves, can who employed eigenfunctions for analyzing multirepresent full-size structural components[22, 241. span systems continuous over rigid supports. Alternatively, for plate continua with regular geThis paper describes the development and use ometry and certain boundary conditions, the ver- of compound strip elements for the analysis of slabsatility of the conventional FEM is not required. girder bridges. Key features of this method are the Instead, an alternative methodology developed by inclusion of (I) flexural stiffness of beams. (2) torCheung[9, 101and termed the “finite strip method” sional stiffness of beams, (3) axial stiffness of colcan be used. Because the plane geometry of a plate umns, and (4) flexural stiffness of columns in a diis discretized in one principal direction only, the rect stiffness formulation. The versatility and FSM considerably shortens and simplifies the an- capability of the FSM are particularly enhanced by alyst’s input and output effort, reduces computer incorporating the second and fourth features cited memory capacity and time requirements, and offers above. Applications presented herein employ the greater computational efficiency than the EEM for stiffness matrices for transverse beams and colthose types of problems geometrically suited to umns. These elements, in particular, enhance the both methods. In effect, the FSIM reduces a twoFSM for the analysis of slab-girder bridges. 1. Ih’TRODlJCTION
dimensional
979
2. FISITE
STRIP
METHODS-OVERVIEW
equation
The compound strip method (CSXl) can be considered as a special form of the finite element method. which is discretized in one direction. as shown in Fig. I. The CSM employs a displacement function with simple polynomial functions and continuous differentiable smooth series in orthogonal directions. The series functions satisfy the boundary conditions at the end of the elements or strips. For a single strip, the displacement function can be tvritten as it’ =
i X,,,(.r) Y,,, (_v). I,,= I
(I)
where r is the largest term of the truncated series, X,,,(.r) is the polynomial expression with the undetermined constants (termed displacement coefticients) for the mfh term of the series, and Y,,,(y) are predetermined series which satisfy the end conditions and the deflected shapefl8j. The compound strip method is especially adaptable for the analysis of prismatic thin-wall structures. For the analysis of flat plate structures, only transverse deflections are considered. The strip displacement function for the deformation must satisfy the following requirements: I. The Y,,(v) component of the displacement function should satisfy the boundary conditions. 7 The X,,(.r) part of the displacement _. function must be able to represent a constant generalized strain in the transverse (x) direction. This guarantees the generalized strain will converge to the true one as the mesh is refined. function must satisfy the 3. The displacement compatibility of displacements along the boundaries of the strips. The Y,,,(v) series most commonly used are eigenfunctions derived from the solution of the differential equation for beam vibration[ 181. Y(v)”” = $Y(,),
is
where Cr. C:. C?, C, are constants based on the boundary conditions. Y,,(_v)terms are mode shapes which are orthogonal to each other. The orthogonality has signi~cant implications regarding the computational effort required to employ this displacement function in a finite element formulation. The solutions of eqn (3) and for the various boundary conditions are well known[:! I. 3 I], Many of the computational aspects involving the use of eqn (3) are described elsewhere[:!l, 331). The necessary calculations, integral evaluations, etc., can be performed in closed form. obviating costly numerical algorithms such as quadratic integration, etc. The X,&r) function is a polynomial associated with a nodal displacement parameter and describes the co~esponding displacement field in the transverse f.~) direction of the strip. For example. the out-of-plane displacement function for bending a plate strip is It’ =
z
[(I - 3?
+ 2.r3) N’,,n + Is - 27x
m= I
where a is the width of the strip, .\- = .\-in, W’S are the deflections, and B’s are the rotations of the nodal lines along the strip boundary oriented in the y-direction. Equation (4) can be written as]181 It’
=
i
[C,
21
;I
C,] {
WI=I
,n
Y,,,
(2)
where p. is a parameter and h is the length of the strip[fS]. The general solution of this differential
-X
p . EquationfS] can be differ{ tm I to give the curvature-displacement rela-
where {S,), = entiated tionship
where B,. B2 are the appropriate derivative functions. By minimization of the total potential energy, the stiffness matrix can be obtained with the relationship] 181 Fie. I. Compound strip idealization.
Analysis of slab-girder bridges Compound Strip-Additional
where [B] is the generalized
strain matrix, [Dl is and A., is the area of the
the constitutive matrix, strip. Due to a loading q(.r, y). the load vector
F,,,
981 Capobilllies
for
the mrh term is[18] {F,,,1 =
I,, WIT
d-r* Y)
a.
(8)
The strip stiffness and load vectors can be assembled to give the equilibrium equation ]W{W = IF]. 2.2 Compound
strip
(9)
method
The compound strip is a substructuring technique which allows beam and column elements to be attached to a finite strip. This substructure is illustrated in Fig. 1. The displacement function for the attached elements is consistent with eqn (1). For example, consider the strip element shown in Fig. 2. A spring element is attached to the strip at the local coordinates xr,yc.. The strain energy in the spring is U,. = fK/, [N-(X,,y,.)]‘.
Each minimization results ina 4x4 stiffness matrix adds directly to the strip rtiffnarc matrix of the plate
(10)
Fig. 3. Compound strip capabilities.
where K,, is the spring stiffness. Substituting eqn (5) into eqn (10) gives
where Cl-C4 are the polynomials given in eqn (4) evaluated at x,., and Y,,,, Y,,are the boundary funcUC = ;K,, (11) tions evaluated at yc. This 4 x 4 matrix adds directly to the conventional strip matrix prior to assembly. Thus, the mawhere the expressions contained in [N], are evaltrix given in eqn (12) can easily be incorporated into uated at .r,,y,.. Minimizing the strain energy with existing finite strip programs for flat plate analysis. respect to the displacement coefficients 6 yields a Other compound strip stiffness matrices can be stiffness matrix for the spring element which is conformulated for concentrated rotational stiffnesses, sistent with the matrix given in eqn (7). The matrix and beam elements oriented in the transverse is and longitudinal directions. The capabilities are graphically illustrated in Fig. 3. The procedure to CIC2 CIC3 CIC4 arrive at stiffness matrices similar to eqn (12) is discZc4 ctc2 c2c3 I K&n = K,, Y, cussed in detail elsewhere[28]. The stiffness mac3c3 c3c4 ’ Wm) trices of all attached elements are added directly to c4c4 the plate stiffness matrix, eqn (71, prior to assem(12) bly.
1
-x
/ Y
Fig. 2. Compound strip--linear
spring element.
3.SLAB-CIRDERBRIDGEANALYSIS
The bridge shown in Fig. 4 was subjected to typical truck loading and analyzed by using the CSM and the FEM, and the results were compared. The bridge geometry is typical of an interstate overpass. The 7-in. deck is supported by four plate girders uniformly spaced. The deck and the girders are prismatic in the logitudinal direction. Presently, the CSM formulation does not incorporate in-plane deformation; therefore, the girders were assumed concentric in both the CSM and FEM models. Composite behavior was modeled per AASHTO specifications]1 I.
J. A. PLCKETT
982
ILLUSTRATIVE COLUMN
EXAMPLE
CAP
.. :.,;..*. I== 0 ;a
:
.*.:I.;. ‘*
:;
:
..*;. .; : :j. . . . ..-. ,a‘. ;. 0’
‘,
WE3
COVER
L
54;
#. /2.x
COLUMN I
.
.O....’
COLUMNS
COMPOUND
NO.
, .
9/s*’
Fig. 4. Example
STRIP
:
l-M"1
%?”
GIRDERS
.
!
I
..‘, ;: :
;
slab-girder
STRIP
CAPS
bridge.
MODEL 4 GlROERS
CAP
I\ I/
A
2
ot 9’.6”
/
‘II/
3 4
,\ I
5
‘I’
6
I
7
32”
-6’
_32K
.a*
,I,
t\I
A ,r
6
72’.0’,60’-0’
60°.0”L.
192’. 0”
STRIP
No.
TYPE
I
STD. (LO21
2
COMPOUND
3
STD.
4
COMPOUND
5
STD.
6
COMPOUND
7
STD.
8
COMPOUND
Fig. 5. Compound
strip model
38’-0’
Analysis
983
of slab-girder bridges Table
I. Analysis summary FEM
Maximum Deflection, in. Maximum Support Reaction, kip. Maximum Girder Moment. kin.
0.126 0.128 30.1 37_._ ’
- 2.00 IS00 Maximum Transverse Moment, kin./in. 3.47
of Degrees of Freedom No. of Lines of Data input Execution Time NO.
CSM
675 64
--BOO
I’70 3.59
270 31
(Not Available)
The CSM model incorporated axial stiffness eIements for the column, and longitudinal and transverse beam elements for the diaphragms and the girders. The compound strip layout is illustrated in Fig. 5. The finite element model is shown in Fig. 6. The program ANSYS was employed for the FEM. analysis[21. The ANSYS flat shell and beam elements were employed. 3.1 Comparison of the methods The two techniques compare very well throughout the structure. For example, Figs. 7 and 8 show
the detection along the first interior girder and the de8 ection in the transverse direction under the section where the load is applied. respectively. These deflection plots are typical for the entire structure. The maximum deflection in the CSM was 0.128, and 0.126 with the FEM-an approximate 2% diference. The maximum values of interest are given in Table 1. A comparison for the transverse bending moment in the deck is illustrated in Fig. 9. The girder moment for the first interior girder is shown in Fig. 10. Both of these figures show the fine agreement between the two methods. The maximum actions are given in Table 1. The maximum girder moments differ by 22010,and maximum support reaction by 7%. This difference could be decreased by increasing the number of harmonics considered. Note. the FEM required the solution of 675 equilibrium equations, as compared to the CSMs 270, a significant computational saving. In addition, the associated cafculations, such as integrations, are significantly less for the CSM. Because two different computers were used, a comparison of the execution terms was not available. SUMbIARY
The compound strip method was illustrated for the analysis of a slab-girder bridge modefed as a linear elastic plate continuous over deflecting supports. This approach incorporates the effect of support elements in a direct stiffness methodology by creating a substructure composed of plate, beam, and column elements. The compound strip and finite element methods compared favorably for the illustrated example and may be employed to model rectangular slab-girder bridges.
J..4. PWKETI
98-1
I
I
8.60
4.76
I
I
I
14.26
19.00
r
I
23.76
26.60
33.26
3s.00
Distance from Node 91. ft. Fig. 7. Transverse
deflection.
I
0 - finite Element Method 0 - Compound Strip Method
I
I
I
I
I
I
I
I
I
,
I
I
I
,
1
/
12.0
24.0
36.0
46.0
60.0
72.0
64.0
66.0
106.0
120.0
132.0
144.0
166.0
166.0
160.0
Distance from Node 2, ft. Fig. 8. Longitudinal deflection.
985
Analysis of slab-girder bridges
00
4.76
0.60
WI.26
1e.00
23.76
Zd.60
3i.26
3 00
Distance from Node 91, ft. Fig. 9. Transverse
bending moment in deck.
0 = Finite Ekmont Method IJ- CompoundStrip Method
.O
24.0
40.0
uj.0
D%nce
frirt+lodo ?g.
Fig. IO. Longitud~na[ bending moment in the girder.
143.0
142.0
J. A.
986
kCKEll
REFERESCES
I. American Association of State Highway and Transportation Officials. Srtrnd~rd Sprcifictrrions&r High~IUV Eridpcs. 12th Ed. Association General Office. Washington. D.C. (1977). Ancllysis S~srems Drer’s ,tlt~n2. dNSKS Engineering M/. Revision 4.0. Swanson Analysis. Inc.. Houston. PA 119821. 3. K. Bathe. E. L. Wilson and F. E. Peterson. SAP II’. .-t Sam-rrrrcrl omLysis program for sfcttic and dmrrmic response oflinecrr systems. Report No. EERC 73-l I.
University of California, Berkelev. CA (197-t). 4. T. G. Brown and A. Ghali. Finite strip analysis of quadrilateral plates in bending. J. Sfrrrctltrcll Divisiott. ASCE 104, EM2 (1978). 5. M. S. Cheung and M. Y. T. Chan. Analysis of continuous curved box-girder bridges by the finite strip method. ASCE Preprint 3515. Boston (1979). 6. Sl. S. Cheung, Y. K. Cheung and A. Ghali. Analysis of slab and girder bridges by the finite strip method. Build. Sci. 5 (1970). of 7. Xi. S. Cheung. and Y. K. Cheung. “Analysis Curved Box Girder Bridges by the Finite Strip Method.” Proc. Inr’l. Assoc. Bridge cmd Srr. Engr.. 31-I. 1971. 8 M. S. Cheung and Y. K. Cheung. Static and dynamic behaviour of rectangular plates using higher order tinite strips. Build. SFi. 7 (‘1972). 9 Y. K. Cheung. Finite strip method analysis of elastic slabs. J. Engng. Mech. Di\j. ASCE 94, No. EM6 ( 1968). 10. Y. K. Cheung, The finite strip method in the analysis of elastic plates with two opposite supported ends. Proc. Inst. Civ. Engng. 40 ( 1968). II. Y. K. Cheung. Analysis of box girder bridges by the finite strip method. Publ. SP 26. Amer. Cont. Inst.. Detroit. MI (1969). 12. Y. K. Cheung. The analysis of cylindrical orthotropic curved bridge decks. Proc. Inr. Assoc. Bridge nnd Srrucf. Engng. 29-11 (1969). 13. Y. K. Cheung. Folded plate structures by the finite strip method. J. Stmctural Ditvision. ASCE 95, ST12 (1969). II. Y. K. Cheung, Orthotropic right bridges by the finite strip method. Publ. SP 26. Amer. Cont. Inst.. Detroit MI (1969). 15. Y. K. Cheung, Analysis of simply supported thick. layered plates. J. Engng. Mech. Div.. ASCE 97. EM3 (1971). 16 Y. K. Cheung. Flexural vibrations of rectangular and other polygonal plates. J. Engng. Meclr. Div., ASCE 97, (1971). 17 Y. K. Cheung and M. S. Cheung. Free vibration of
18. 19. 20.
y
22.
23.
24.
15.
26.
‘7. 28.
29.
30.
curved and straight beam-slab or box girder bridges. Proc. In!. Assoc. Bridge clnd Srrtrcr. Engng. 32-11 (1972). Y. K. Cheung. Finire Srrip .Llethod in Strtrctuml Anr~lysis. Pergamon Press. Oxford (1976). A. Coull and P. C. Das. Analysis of curved bridge decks. Proc. Insr. Ci\,. Engnp. 37 (1967). C. Delcourt and Y. K. Cheung. Finite strip analysis of continuous folded plates. Proc. 1~. Assoc. Bridge md Strlrcr. Engng. May (1978). R. P. Felgar. Formulas for integrals containing characteristic functions of vibratine beam. Circular I-I. Bureau of Engineering Research. University of Texas. Austin. TX (1950). R. M. Gutkowski. The finite panel method for the elastic analysis of continuous rectangular plate systems. Ph.D. dissertation. University of Wisconsin. Madison. WI (1974). R. M. Gutkowski and C. K. Wang. Continuous plate analysis by finite panel method. J. SrrlrctrtralDi~,ision. ASCE 102, 629-643 t 1976). Y. C. Loo and A. R. Curens. Developments of the finite strip method in the analysis of orthotropic plates. Det,elopments in Bridge Design and Cottstrlrciion. Crosby Lockwood (I971 1. L. C. Maugh and C. W. Pan. Moments in continuous rectangular slabs on rigid supports. ,-ISCE Trrrns. I I I8 (1942). M. S. Mawenya and J. D. Davies. Finite strip analysis of plate bending including transverse shear effects. &i/d. Sci. 9 (19741. M. Petyt. Finite strip analysis of flat skin-stringer structures. J. Sound Vibration 51 (1977). J. A. Puckett, Compound strip method for the analysis of continuous elastic plates. Doctoral Thesis, Colorado State University, Ft. Collins. CO (1983). R. G. Sisodiya. A. Ghali. and \I’. K. Cheung. Diaphragms in single and double-cell box girder bridges with varying angle of skew. AC/J. July (1972). J. W. Smith, Finite strip analysis of the dynamic response of beam and slab highway bridges. Emrhquuke Engng.
and Strucr.
Dynmrics
1
( 1973).
31. S. Timoshenko
and S. Woinowsky-Krieger. Tlteorv sf Plates nnd Shells. 2nd Ed. McGraw-Hill (1971). 32. C. 1. Wu and Y. K. Cheung. Frequency analysis of rectangular plates continuous in one or two directions. Earthquake
Engng.
bind Strrrct.
DJnnmics
3
( 1974).
33. D. Young and R. P. Felgar. Tables of characteristic functions representing normal modes of vibrations of a beam. Publication No. 4913. University of Texas. Austin, TX. July (1949). 3.4. 0. C. Zienkiewicz. Tile Finire Nemenr Merhod in Engineering Science. 3rd Ed. McGrabv-Hill (1977).