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Spline compound strip analysis of folded plate structures with intermediate supports
00457949/91 $3.00 + 0.00 0 1991 Pergamoa Pm8 plc
Computers& S~mcruresVol. 39, No. 3/4, pp. 369-379, 1991 Printed in Great Britain.
SPLINE COMPOUND STRIP ANALYSIS OF FOLDED PLATE STRUCTURES WITH INTERMEDIATE SUPPORTS C.-J. CHEN,~ R. M. GUTKOWSKI~and J. A. F?JCKETT$ TDepartment of Civil Engineering, Chung-Cheng Institute of Technology, Tahsi, Taoyuan 33509, Taiwan IDepartment
of Civil Engineering, Colorado State University, Ft. Collins, CO 80523, U.S.A.
§Department of Civil Engineering, The University of Wyoming, Laramie, WY 82071, U.S.A. (Received 4 April 1990)
Abstract-A B-spline column element in 3D space is derived and combined with the B-spline compound strip method for the analysis of plate-type structures (e.g. folded plates, box-girders, etc.) with intermediate supports. A direct stiffness method is used. Numerical examples demonstrate the advantages of this method: accuracy, efficiency and simplicity.
INTRODUCTION
The finite strip method (FSM) was first presented by Cheung [l] for the analysis of linear elastic flat plates with two opposite simply supported ends. This method has been extended to include a variety of structures and boundary conditions. For analysis of a structure which has constant depth in the longitudinal direction, the FSM has proven to be the most efficient method. Combined with the flexibility method, the FSM has been used to analyze continuous plate structures that have intermediate supports [2-4]. However, the efficiency and convenience of the FSM was compromised by the repetitive solution passes required for each redundant in a flexibility method. Puckett and Gutkowski [5] developed the compound strip method (CSM) to include interior supporting members, either flexible or rigid, as an extension of the FSM. By doing so, they overcame the need for the computationally cumbersome and inefficient flexibility method. An important advantage of the CSM over the FSM in the flexibility method is its capability to incorporate the effect of support elements in a direct stiffness methodology. This is done by creating a subassembly composed of a strip and its supporting beams, and column elements. Cheung and Li [6,7] introduced the eigenfunctions of a continuous beam as shape functions for the FSM in the longitudinal direction. This technique also avoids the use of the flexibility method but has disadvantages. The free vibration equation of a continuous beam must be solved numerically, and choosing the number of integration points needed in the longitudinal direction requires experience. All the previous studies mentioned above used the Fourier series as shape functions in the longitudinal direction. The fact that the trigonometric series is continuously differentiable becomes a limitation
when dealing with structures that involve abruptly changing properties, concentrated loading, and intermediate supports, etc. The second or the third derivatives of displacement, or both, are discontinuous in those cases. Thus, not only is the rate of the convergence for problems involving these situations slow, but some stress resultants also exhibit ‘Gibbs’ phenomenon’. Another drawback is that the displacement shape function in the longitudinal direction is different for various boundary conditions. In recent years, some contributions to the solution procedure have been made by using spline functions for plate-type structures. Mizusawa et al. [8] studied the buckling behavior of skew plates by applying B-spline functions in both longitudinal and transverse directions. Yang and Chong [9] combined the FSM and X-spline functions for the analysis of plate bending problems with irregular boundaries. Cheung et al. [lo, 1l] combined the FSM and equally spaced B-spline for the analysis of plate- and shell-type structures. Cheung’s method is more flexible than the conventional FSM in the treatment of boundary conditions. Chen et al. [ 12,131 introduced the unequally spaced B-spline compound strip method (BCSM) for the analysis of thin plate-type structures. The basic mathematics of the formulation for flat plates [14] and stiffened plates [12] has been described. The BCSM not only inherits the advantages and the applicability of the conventional compound strip method [S], but also possesses some other important characteristics. The convergence of the BCSM is improved significantly in comparison to the conventional FSM, CSM and the finite element method. The ‘Gibbs’ phenomenon’ is also avoided. The unequally spaced cubic B-splines [12] are made up of simple piecewise cubic polynomials with C2 continuity everywhere (in the case of no multiple knots). In the finite element formulation, quintic polynomials are 369
C.-J. CHENet al.
370
required for the same order of continuity. The use of low-order polynomials by spline interpolation simplifies the computation, reduces the risk of numerical instabilities and reduces the poor approximation which sometimes occurs in higher-order polynomial interpolation. The introduction of unequally spaced knots allows one to describe accurately the response in the region with high stress gradients, or in locations of abrupt change in the order of continuity condition (Co vs C’ vs C2). This occurs because a deflected shape with the required discontinuity conditions and smoothness, where necessary, can be found by arranging the spacing and multiplicity of the knots of the B-spline functions (Fig. 1). In this paper, a space B-spline column element is developed and implemented in the BCSM to analyze foldedplate-type structures that have intermediate supports. FORMULATION
The quantities &, 4ni34,, , 40,~4#, A,, 4wj and & are row matrices of cubic B-spline series which can be directly modified to adapt to various prescribed boundary conditions at the longitudinal ends of thestrip.Forexample,&=[r$_,,&,6, 9.9.t 4”_,, $,, 4, + ,lUi, in which n is the number of sections. Also, 4, vi, wi, O,,i, uj, vj, wj and 0,, are the column matrices of displacement coefficients for nodal lines i and j. For example, UT= [u_, , uo, u, , . . . , u,_ , , u,,
ht+lli. The generalized strain-displacement for the strip element is
relationship
$ = B&?,
(4)
where
OF STRIP ELEMENT
The displacement function of a strip is expressed as the product of the conventional polynomial transverse shape function Ni and the longitudinal cubic E-spline series Cpias shown in Fig. 2. The displacement field of the strip is
sg= 14, Vir Wir eyi, uj, vj9 wj9 eyj]
(5)
and
(1)
4 = ]N416,,
-l
Ni9ui 0
BP=
0 N141i
0
0
OPu,
0
0
0
O
0
0
N29ij
0
0
0 0 Ni4vj N24hj
-Nit& 0
-N;:t$,j 0
-N5t#(Lj 2N;&,,
-2N;4Lj N,qi;;,
-N;& 0
0 0 -N;& NI4Ii Ni#ui 0
1
0
0
00
00 -N4qS;;i 2N;+L
-N3#$ 2N;4;,
(6)
’
L
ui 7 vi wi 0
0
0
N24uj
0
0
0
N24uj
0
0
NjQwi NdQ,i
0
0
eYl
0
0
uj
N,Qw, N69oj
>
Vj wI eYJJ
in which
in which
N, = 1 - r,
N2 = r,
N, = 1 - 3r2 + 2r’,
N, = x(1 - 2r + r*),
N, = (3r* - 2r’),
N6 = x(r* - r),
and r =x/b.
N’ =
N,x N" = N,,,
(3)
4J’= 4.Y 4” = 4,YY.
3
(2)
Spline compound strip analysis of folded plate structures
Cubic B-spline
371
of (1,2,3,4,5)
0.81 0.6.
C2 everywhere
0.4.
0.2.
Cubic B-spline
of (1,4,7,8,9)
Cubic B-spline
0.8-
of (1,2,2,35)
0.6. C’ @ knot 2. C? elsewhere 0.4.
Cubic B-spline
1.2”
of (1,5,5,5,5)
1.0. 0.8. 0.6. 0.4.
Displacement
discontinuity
02. 0
1
2
3
4
5
, 6
Fig. 1. Cubic B-splines with various knot spaces and m~tip~icity for knot sequence (.Y~_~. y,_ , , Y,, Yr+,,Y,+*).
372
C.-J.
(hEN
et al.
a-nseetionsofthe
/
/
UnWally spaced cuMc Bapline series
~~pi7i7c_n knot
nodal line
j J X
Strip element Fig. 2. A typical strip element. The generalized stress-strain strip element now follows:
relationship
Up=D+
for the
(7)
pressive strain in thej-direction produced by a tensile stress in the i-direction. The stiffness and load matrices, KP and FP, of a strip element can be derived by minimization of the potential energy in the usual manner:
where a:=
W, 4
Nxy M, M,
M,,l
K, =
(8)
B,TD,B, dA
(JOa)
[WI*q dx dy,
(Job)
5
and the non-zero elements Dpij in the 6 x 6 symmetric constitutive matrix D, are given as follows:
FP = I
where q is the applied load in a strip. Dp,, = E,tl(l
- VxyVJr FORMULATION
Dpz = Ey t I( 1 - vxyvvxh Dp,z = D.,m =
v&42
=
~,&a
OF COLUMN
ELEMENT
To analyze structures that have intermediate supports, an appropriate model for an arbitrarily oriented column element is derived. The arbitrarily oriented column is treated as a space frame element, with a fixed support at one end and the other end connected rigidly to the strip element. The strain energy U, of the column element is then given by
9
Dp,, = Gt, DH = EXt3/12(1 - vXyvyX), Dpss = Eyt3/12(1 - v,vyJ,
U, = l/W,K,,d,,,
D,AS= D,M = vxy0,~s = vyxDp449 De
= Gt’/12,
(9)
in which E,, E, are Young’s moduli in the X- and y-directions, G is the shearing modulus, and v,, represents Poisson’s ratio, which characterizes the com-
(11)
where K,, is the stiffness matrix of the column element in the column local axis, and d,, is the matrix of column local displacements. d,: = I+ r,/ w,/ @xc,‘&c,‘%,I
(12)
and
E,A,IL, 0
K,, =
1
0 )2E,L,lL;
0
0
0
0
0
0
0
6E,LILf
0
0
0
0
0
0
0
6E,LZlL3
12E,I,./L;
0
0
G,J,ILc
0
0
0
4E, 1, IL,
0
-6E4,lL.f 0
0
-6E,I,./L:
I
0 (13)
373
gpline compound strip analysis of folded plate structures Y
Next, the local column displacements can be expressed in terms of the global system displacements:
X
d,=R,d,.
In Fig. 3 the individual coordinates of a strip are labeled x,y, z and the global coordinates x, Y, z. The transformation of displacements between the two sets of coordinate systems is then given by
,X
where in which ECA,, ECI, and E,Z,, and G, J, are axial, tlexural and torsional rigidities, respectively, and L, is the length of the column. Before the stiffness matrix of column element can be assembled into a strip element stiffness matrix, the column element displacements that have been chosen to ensure compatibility with the strip element displacements must be defined in terms of the strip nodal line displacements 6,. The procedure consists of three steps. The first step is to transform the column local displacements to the global coordinate system. The displacements are then transformed to local strip element coordinates. Finally, these displacements are interpolated in terms of the strip element nodal line displacements.
N,4oi
0
N24oj
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
N244
1
R” ,
c
cos p cos 8 R, =
-cosacosjsin@-sino!sinB sin a cos /3 sin
e - cos a sin fi
;
;sjnj.
(19)
Finally, the displacements in a strip can be expressed in terms of nodal line displacements by the interpolation matrix [N41E:
0
;
(18)
R,’
[ 0
&=[;I;
0
[
Rs0 1
with
0
R,=
with
R2 =
N,Ai
Figure 3 illustrates the column element and the relationship of its local system and global system. The 3D rotation transformation matrix is given as
(17)
d, = R&
Fig. 3. The local (x-y-z) and global (X-Y-Z) coordinates of a column element.
=
(16)
0
sP’
(20)
in which [N#], is calculated at specified x and y values (the coordinates of the point at which the column element intercepts the strip).
(14) sin 0
sin /3 cos 0
cos a cos e
sin a cos j? - cos a sin j!Isin 8
-sinacose
sinasin/?sinO+cosacosj?
.1 (15
C.-J. CHENet al.
314
The stiffness matrix for the column element that contributes to the strip element is then given by K, =
[N~ITRTRTK,,RIR*[N~I,.
(22)
By assembling eqns (10) and (22) for all strips and supporting elements, a global system of linear simultaneous equations may be given as follows:
attached column
K6=F.
(23)
Solution of eqn (23) follows the routine procedure.
Fig. 4. Strip local and global coordinates.
NUMERICAL EXAMPLW
The strain energy for a column element in terms of strip nodal line displacements is
Example 1: barrel-type folded plate continuous over rigid diaphragm
By symmetry (Fig. 5), the BCSM analyses were
(21) performed on one-quarter of the folded plate, with a
Uc= 1/26J[N~ITRTRTK,,R,R,[N~1,6,.
rigid alaphragm
,_.___....__.._...._.___....-.___..,..--......-.--.-....-----...----..
riga diaphragm
___...___.__.___.__.__...___....._
.
rigid diaphragm
_._......____......___.....---.....
L
(a) E-144OOOpsf v- 0.25
plate thickness width vertical load 0.075 psf 1 112’ 3’ l/3’ 10’ 0.080wf 2
I
(b)
Fig. 5. (a) Longitudinal span. (b) Cross-section and material properties. (c) Strip and section divisions.
Spline compound strip analysis of folded plate structures
375
q lBcsM n Theo.
20
B
0
5
10 15 Location (ft)
20
FSM
23
Fig. 6. Longitudinal forces, N,,, at the quarter-span.
BCSM Theo. FSM
0
5
10
15
Location
20
23
(ft)
Fig. 7. Transverse moments, M,, at the quarter-span.
lzl0.1
E
q
Location
Theo. FSM
(ft)
Fig. 8. Deflections, w, at the quarter-span.
20 0 -20 -40
BCSM
-60 FSM
-80 -100 -120 -140 !
1 0
5
10 Location
15
20
23
(ft)
Fig. 9. Longitudinal forces, N,,, at the midspan.
376
C.-J. cHl3N et al. -? rigii diaphragm
rigId
. . . . ..-_._..............--......---.............-.............-._.....
diaphragm point support 7.9 ”
7.9 ”
I
I
I
I
(4
plate thickneee width vertical load 1 0.2” 2” 0.009 psi 0.2” 4” 0.009 psi 0.2” 4” 0.109 psi
Q
-
-_;;.
\.I 1 x
I‘Q
I 2
1Y
(cl
Fig. 10. (a) Longitudinal span. (b) Cross-section and material properties. (C)Strip and discretization of 10 strips and 10 unequally spaced sections (484 complete equations). A closer spacing is used where needed, i.e. at the intermediate supports.
0
2
The rigid diaphragm was modeled as a transverse heam element with high flexural rigidity and column elements with high axial rigidities but no flexural
6
4
Location
(in)
Fig. 11. Transverse moments, M,, at the midspan cross-section.
371
Spline compound strip analysis of folded plate structureS 43-
-6csM 2-
+
Theo. A
Exp.
l-
Location
(in)
Fig. 12. Longitudinal forces, N,, at the midspan cross-section.
The cited theoretical analysis was performed by Pultar et al. [15], based on thin shell theory, with Levy-type solutions for 95 harmonic terms. The FSM was performed by Rao and Rao [4]. To account for the intermediate rigid diaphragm, Rao used the flexi-
rigidities. Incorporation of beam elements has been previously described [12, 131.The results for the longitudinal forces, NY,transverse moments, Af,, and deflections, w,atthequarter-spanarecomparedwiththose from the theoretical analysis and the FSM (Figs 6-8). unit : mm rigid diaphrwm
I I
3500
I
(4
lb)
Fig. 13. (a) Longitudinal span. (b) Cross-section and material properties. (c) Strip and section divisions.
C.-J.
378
CHEN et al.
0.0 i c I!
-0.5 -
*
BCSM
8
-1.0 -
0
FEM(S)
A FM(T) -1.5 -
+
-2.0
I 675
0
Exp.
I 1750
I 2625
I 3500
I 4375
1 5250
I 6125
I 7000
Location (mm)
Fig. 14. Distribution of vertical deflection along the beam.
bility method with the interior supports removed as redundants. The results for the FSM were obtained with discretization of 14 strips and 40 harmonic terms (2400 complete equations). Results for the longitudinal force at the midspan for the BCSM are compared with those from the theoretical analysis and the FSM (Fig. 9). The results for the BCSM, with considerably fewer ‘degrees of freedom than the FSM, are in excellent agreement with the results for the theoretical analysis.
This method eliminates the repetitive solution passes required for each redundant in a flexibility method. The accuracy and efficiency of this method have been tested on a variety of examples with considerably fewer degrees of freedom than the conventional FSM, and the results indicated very good agreement with available experimental results.
Example 2: continuous folded plate with intermediate ball-bearing supports (Fig. 10)
Y. K. Cheung, The finite strip method in the analysis of
By symmetry, analyses were performed on a quarter of the folded plate with discretization of 12 strips and eight equally spaced sections (468 complete equations). In the BCSM, the ball-bearing support was modeled as a column element, with high axial rigidity but no flexural rigidities. Results of the transverse moments, iU,, and longitudinal forces, NY, at the midspan are compared with those from the photoelastic analysis and theoretical analysis of Mark and Riera [ 161, as shown in Figs 11 and 12. As a whole, the results for the BCSM are in good agreement with the theoretical results and experimental values. Example 3: a continuous two-span double-cell box beam under concentrated loaa!s (Fig. 13)
Analyses were performed by using the BCSM with discretization of 22 strips and 10 unequally spaced sections. The vertical deflections at the bottom of the side web along the continuous box beam are plotted (Fig. 14). The FEM(S) and FEM(T) represent the FEM based on the shell analysis and thin-walled box beam analysis in [ 171, respectively. The experimental values are also from [17]. Results are in good agreement. CONCLUSION
A B-spline space column element was derived, and combined the concept of the BCSM for the analysis of plate-type structures with intermediate supports.
REFERENCES
elastic plates with two opposite simply supported ends. Proc. Instn Ciu. Engrs 40, l-7 (1968). 2. F. A. Branco and R. Green, Composite box girder bridge behavior during construction. J. Srrucr. Engng, ASCE 111(3), 577-593 (1985). M. S. Cheung, Y. K. Cheung and A. Ghali, Analysis of slab and girder bridges by the finite strp method. Building Sci. S(2), 95-105 (1970). P. S. Rao and P. S. Rao, Continuous folded plates. Bull. Int. Ass. Shell Spat. Struct., 1%2(67), 37-44 (1978). J. A. Puckett and R. M. Gutkowski. Comnound strin _ method for analysis of plate systems. J. Struct. En&, ASCE 112(l), 121-138 (1986): M. S. Cheung and W. Li, Finite strip analysis of continuous structures. Can. J. Civ. Ennnn -- 15, 424-429 (1988). 7. M. S. Cheung and W. Li, Finite strip analysis of continuous structures. Can. J. Civ. Enema _- 15, 424-429 8. T. Mizusawa, T. Kajita and M. Naruoka, Vibration of skew plates by B-spline functions. J. Sound I’ibr. 301-308 (1979). 9. H. Y. Yang and K. P. Chong, Finite strip method with X-spline functions. Compur. Strucr. HI(l), 127-132 (1984). 10. Y. K. Cheung, S. C. Fan and C. Q. Wu, Spline finite strip in structure analysis. In Proc. Int. Conf. on Finite Element Methods, Shanghai, pp. 704-709. Science Press, Beijing, and Gordon & Breach, New York (1982). 11. Y. K. Cheung and S. C. Fan, Static analysis of right box girder bridges by spline finite strip method. Proc. Insf. Ciu. Engrs, Part 2, 75, 311-323 (1983). 12. C. J. Chen, R. M. Gutkowski and J. A. Puckett, B-Spline compound strip analysis of stiffened plates under transverse loading. Compur. Srruct. 34(2), 337-347 (1990). 13. C. J. Chen, R. M. Gutkowski and J. A. Puckett, The unequally spaced cubic B-spline compound strip method. Struct. Res. Rep. 67, Civil Engineering Department, Colorado State University, Fort Collins (1989).
Spline compound strip analysis of folded plate structures 14. R. M. Gutkowski, C. J. Chen and J. A. Puckett, Plate bending analysis by unequally spaced splines. ThinWalled Smct. (accepted) (1990). 15. M. Pultar, D. P. Billington and J. D. Riera, Folded plates continuous over flexible support. J. Sfruc~. Diu., ASCE 93(STS), 253-277 (1967).
319
16. R. Mark and J. D. Riera, Photoelastic analysis of folded plate structures. J. Engng Me& Div., ASCE 93(EM4), 79-93 (1967). 17. L. F. Boswell and S. H. Zhang, An experimental investigation of the behavior of thin-walled box beams. Thin-Walled Struck 3, 35-65 (1985).
Computers& S~mcruresVol. 39, No. 3/4, pp. 369-379, 1991 Printed in Great Britain.
SPLINE COMPOUND STRIP ANALYSIS OF FOLDED PLATE STRUCTURES WITH INTERMEDIATE SUPPORTS C.-J. CHEN,~ R. M. GUTKOWSKI~and J. A. F?JCKETT$ TDepartment of Civil Engineering, Chung-Cheng Institute of Technology, Tahsi, Taoyuan 33509, Taiwan IDepartment
of Civil Engineering, Colorado State University, Ft. Collins, CO 80523, U.S.A.
§Department of Civil Engineering, The University of Wyoming, Laramie, WY 82071, U.S.A. (Received 4 April 1990)
Abstract-A B-spline column element in 3D space is derived and combined with the B-spline compound strip method for the analysis of plate-type structures (e.g. folded plates, box-girders, etc.) with intermediate supports. A direct stiffness method is used. Numerical examples demonstrate the advantages of this method: accuracy, efficiency and simplicity.
INTRODUCTION
The finite strip method (FSM) was first presented by Cheung [l] for the analysis of linear elastic flat plates with two opposite simply supported ends. This method has been extended to include a variety of structures and boundary conditions. For analysis of a structure which has constant depth in the longitudinal direction, the FSM has proven to be the most efficient method. Combined with the flexibility method, the FSM has been used to analyze continuous plate structures that have intermediate supports [2-4]. However, the efficiency and convenience of the FSM was compromised by the repetitive solution passes required for each redundant in a flexibility method. Puckett and Gutkowski [5] developed the compound strip method (CSM) to include interior supporting members, either flexible or rigid, as an extension of the FSM. By doing so, they overcame the need for the computationally cumbersome and inefficient flexibility method. An important advantage of the CSM over the FSM in the flexibility method is its capability to incorporate the effect of support elements in a direct stiffness methodology. This is done by creating a subassembly composed of a strip and its supporting beams, and column elements. Cheung and Li [6,7] introduced the eigenfunctions of a continuous beam as shape functions for the FSM in the longitudinal direction. This technique also avoids the use of the flexibility method but has disadvantages. The free vibration equation of a continuous beam must be solved numerically, and choosing the number of integration points needed in the longitudinal direction requires experience. All the previous studies mentioned above used the Fourier series as shape functions in the longitudinal direction. The fact that the trigonometric series is continuously differentiable becomes a limitation
when dealing with structures that involve abruptly changing properties, concentrated loading, and intermediate supports, etc. The second or the third derivatives of displacement, or both, are discontinuous in those cases. Thus, not only is the rate of the convergence for problems involving these situations slow, but some stress resultants also exhibit ‘Gibbs’ phenomenon’. Another drawback is that the displacement shape function in the longitudinal direction is different for various boundary conditions. In recent years, some contributions to the solution procedure have been made by using spline functions for plate-type structures. Mizusawa et al. [8] studied the buckling behavior of skew plates by applying B-spline functions in both longitudinal and transverse directions. Yang and Chong [9] combined the FSM and X-spline functions for the analysis of plate bending problems with irregular boundaries. Cheung et al. [lo, 1l] combined the FSM and equally spaced B-spline for the analysis of plate- and shell-type structures. Cheung’s method is more flexible than the conventional FSM in the treatment of boundary conditions. Chen et al. [ 12,131 introduced the unequally spaced B-spline compound strip method (BCSM) for the analysis of thin plate-type structures. The basic mathematics of the formulation for flat plates [14] and stiffened plates [12] has been described. The BCSM not only inherits the advantages and the applicability of the conventional compound strip method [S], but also possesses some other important characteristics. The convergence of the BCSM is improved significantly in comparison to the conventional FSM, CSM and the finite element method. The ‘Gibbs’ phenomenon’ is also avoided. The unequally spaced cubic B-splines [12] are made up of simple piecewise cubic polynomials with C2 continuity everywhere (in the case of no multiple knots). In the finite element formulation, quintic polynomials are 369
C.-J. CHENet al.
370
required for the same order of continuity. The use of low-order polynomials by spline interpolation simplifies the computation, reduces the risk of numerical instabilities and reduces the poor approximation which sometimes occurs in higher-order polynomial interpolation. The introduction of unequally spaced knots allows one to describe accurately the response in the region with high stress gradients, or in locations of abrupt change in the order of continuity condition (Co vs C’ vs C2). This occurs because a deflected shape with the required discontinuity conditions and smoothness, where necessary, can be found by arranging the spacing and multiplicity of the knots of the B-spline functions (Fig. 1). In this paper, a space B-spline column element is developed and implemented in the BCSM to analyze foldedplate-type structures that have intermediate supports. FORMULATION
The quantities &, 4ni34,, , 40,~4#, A,, 4wj and & are row matrices of cubic B-spline series which can be directly modified to adapt to various prescribed boundary conditions at the longitudinal ends of thestrip.Forexample,&=[r$_,,&,6, 9.9.t 4”_,, $,, 4, + ,lUi, in which n is the number of sections. Also, 4, vi, wi, O,,i, uj, vj, wj and 0,, are the column matrices of displacement coefficients for nodal lines i and j. For example, UT= [u_, , uo, u, , . . . , u,_ , , u,,
ht+lli. The generalized strain-displacement for the strip element is
relationship
$ = B&?,
(4)
where
OF STRIP ELEMENT
The displacement function of a strip is expressed as the product of the conventional polynomial transverse shape function Ni and the longitudinal cubic E-spline series Cpias shown in Fig. 2. The displacement field of the strip is
sg= 14, Vir Wir eyi, uj, vj9 wj9 eyj]
(5)
and
(1)
4 = ]N416,,
-l
Ni9ui 0
BP=
0 N141i
0
0
OPu,
0
0
0
O
0
0
N29ij
0
0
0 0 Ni4vj N24hj
-Nit& 0
-N;:t$,j 0
-N5t#(Lj 2N;&,,
-2N;4Lj N,qi;;,
-N;& 0
0 0 -N;& NI4Ii Ni#ui 0
1
0
0
00
00 -N4qS;;i 2N;+L
-N3#$ 2N;4;,
(6)
’
L
ui 7 vi wi 0
0
0
N24uj
0
0
0
N24uj
0
0
NjQwi NdQ,i
0
0
eYl
0
0
uj
N,Qw, N69oj
>
Vj wI eYJJ
in which
in which
N, = 1 - r,
N2 = r,
N, = 1 - 3r2 + 2r’,
N, = x(1 - 2r + r*),
N, = (3r* - 2r’),
N6 = x(r* - r),
and r =x/b.
N’ =
N,x N" = N,,,
(3)
4J’= 4.Y 4” = 4,YY.
3
(2)
Spline compound strip analysis of folded plate structures
Cubic B-spline
371
of (1,2,3,4,5)
0.81 0.6.
C2 everywhere
0.4.
0.2.
Cubic B-spline
of (1,4,7,8,9)
Cubic B-spline
0.8-
of (1,2,2,35)
0.6. C’ @ knot 2. C? elsewhere 0.4.
Cubic B-spline
1.2”
of (1,5,5,5,5)
1.0. 0.8. 0.6. 0.4.
Displacement
discontinuity
02. 0
1
2
3
4
5
, 6
Fig. 1. Cubic B-splines with various knot spaces and m~tip~icity for knot sequence (.Y~_~. y,_ , , Y,, Yr+,,Y,+*).
372
C.-J.
(hEN
et al.
a-nseetionsofthe
/
/
UnWally spaced cuMc Bapline series
~~pi7i7c_n knot
nodal line
j J X
Strip element Fig. 2. A typical strip element. The generalized stress-strain strip element now follows:
relationship
Up=D+
for the
(7)
pressive strain in thej-direction produced by a tensile stress in the i-direction. The stiffness and load matrices, KP and FP, of a strip element can be derived by minimization of the potential energy in the usual manner:
where a:=
W, 4
Nxy M, M,
M,,l
K, =
(8)
B,TD,B, dA
(JOa)
[WI*q dx dy,
(Job)
5
and the non-zero elements Dpij in the 6 x 6 symmetric constitutive matrix D, are given as follows:
FP = I
where q is the applied load in a strip. Dp,, = E,tl(l
- VxyVJr FORMULATION
Dpz = Ey t I( 1 - vxyvvxh Dp,z = D.,m =
v&42
=
~,&a
OF COLUMN
ELEMENT
To analyze structures that have intermediate supports, an appropriate model for an arbitrarily oriented column element is derived. The arbitrarily oriented column is treated as a space frame element, with a fixed support at one end and the other end connected rigidly to the strip element. The strain energy U, of the column element is then given by
9
Dp,, = Gt, DH = EXt3/12(1 - vXyvyX), Dpss = Eyt3/12(1 - v,vyJ,
U, = l/W,K,,d,,,
D,AS= D,M = vxy0,~s = vyxDp449 De
= Gt’/12,
(9)
in which E,, E, are Young’s moduli in the X- and y-directions, G is the shearing modulus, and v,, represents Poisson’s ratio, which characterizes the com-
(11)
where K,, is the stiffness matrix of the column element in the column local axis, and d,, is the matrix of column local displacements. d,: = I+ r,/ w,/ @xc,‘&c,‘%,I
(12)
and
E,A,IL, 0
K,, =
1
0 )2E,L,lL;
0
0
0
0
0
0
0
6E,LILf
0
0
0
0
0
0
0
6E,LZlL3
12E,I,./L;
0
0
G,J,ILc
0
0
0
4E, 1, IL,
0
-6E4,lL.f 0
0
-6E,I,./L:
I
0 (13)
373
gpline compound strip analysis of folded plate structures Y
Next, the local column displacements can be expressed in terms of the global system displacements:
X
d,=R,d,.
In Fig. 3 the individual coordinates of a strip are labeled x,y, z and the global coordinates x, Y, z. The transformation of displacements between the two sets of coordinate systems is then given by
,X
where in which ECA,, ECI, and E,Z,, and G, J, are axial, tlexural and torsional rigidities, respectively, and L, is the length of the column. Before the stiffness matrix of column element can be assembled into a strip element stiffness matrix, the column element displacements that have been chosen to ensure compatibility with the strip element displacements must be defined in terms of the strip nodal line displacements 6,. The procedure consists of three steps. The first step is to transform the column local displacements to the global coordinate system. The displacements are then transformed to local strip element coordinates. Finally, these displacements are interpolated in terms of the strip element nodal line displacements.
N,4oi
0
N24oj
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
N244
1
R” ,
c
cos p cos 8 R, =
-cosacosjsin@-sino!sinB sin a cos /3 sin
e - cos a sin fi
;
;sjnj.
(19)
Finally, the displacements in a strip can be expressed in terms of nodal line displacements by the interpolation matrix [N41E:
0
;
(18)
R,’
[ 0
&=[;I;
0
[
Rs0 1
with
0
R,=
with
R2 =
N,Ai
Figure 3 illustrates the column element and the relationship of its local system and global system. The 3D rotation transformation matrix is given as
(17)
d, = R&
Fig. 3. The local (x-y-z) and global (X-Y-Z) coordinates of a column element.
=
(16)
0
sP’
(20)
in which [N#], is calculated at specified x and y values (the coordinates of the point at which the column element intercepts the strip).
(14) sin 0
sin /3 cos 0
cos a cos e
sin a cos j? - cos a sin j!Isin 8
-sinacose
sinasin/?sinO+cosacosj?
.1 (15
C.-J. CHENet al.
314
The stiffness matrix for the column element that contributes to the strip element is then given by K, =
[N~ITRTRTK,,RIR*[N~I,.
(22)
By assembling eqns (10) and (22) for all strips and supporting elements, a global system of linear simultaneous equations may be given as follows:
attached column
K6=F.
(23)
Solution of eqn (23) follows the routine procedure.
Fig. 4. Strip local and global coordinates.
NUMERICAL EXAMPLW
The strain energy for a column element in terms of strip nodal line displacements is
Example 1: barrel-type folded plate continuous over rigid diaphragm
By symmetry (Fig. 5), the BCSM analyses were
(21) performed on one-quarter of the folded plate, with a
Uc= 1/26J[N~ITRTRTK,,R,R,[N~1,6,.
rigid alaphragm
,_.___....__.._...._.___....-.___..,..--......-.--.-....-----...----..
riga diaphragm
___...___.__.___.__.__...___....._
.
rigid diaphragm
_._......____......___.....---.....
L
(a) E-144OOOpsf v- 0.25
plate thickness width vertical load 0.075 psf 1 112’ 3’ l/3’ 10’ 0.080wf 2
I
(b)
Fig. 5. (a) Longitudinal span. (b) Cross-section and material properties. (c) Strip and section divisions.
Spline compound strip analysis of folded plate structures
375
q lBcsM n Theo.
20
B
0
5
10 15 Location (ft)
20
FSM
23
Fig. 6. Longitudinal forces, N,,, at the quarter-span.
BCSM Theo. FSM
0
5
10
15
Location
20
23
(ft)
Fig. 7. Transverse moments, M,, at the quarter-span.
lzl0.1
E
q
Location
Theo. FSM
(ft)
Fig. 8. Deflections, w, at the quarter-span.
20 0 -20 -40
BCSM
-60 FSM
-80 -100 -120 -140 !
1 0
5
10 Location
15
20
23
(ft)
Fig. 9. Longitudinal forces, N,,, at the midspan.
376
C.-J. cHl3N et al. -? rigii diaphragm
rigId
. . . . ..-_._..............--......---.............-.............-._.....
diaphragm point support 7.9 ”
7.9 ”
I
I
I
I
(4
plate thickneee width vertical load 1 0.2” 2” 0.009 psi 0.2” 4” 0.009 psi 0.2” 4” 0.109 psi
Q
-
-_;;.
\.I 1 x
I‘Q
I 2
1Y
(cl
Fig. 10. (a) Longitudinal span. (b) Cross-section and material properties. (C)Strip and discretization of 10 strips and 10 unequally spaced sections (484 complete equations). A closer spacing is used where needed, i.e. at the intermediate supports.
0
2
The rigid diaphragm was modeled as a transverse heam element with high flexural rigidity and column elements with high axial rigidities but no flexural
6
4
Location
(in)
Fig. 11. Transverse moments, M,, at the midspan cross-section.
371
Spline compound strip analysis of folded plate structureS 43-
-6csM 2-
+
Theo. A
Exp.
l-
Location
(in)
Fig. 12. Longitudinal forces, N,, at the midspan cross-section.
The cited theoretical analysis was performed by Pultar et al. [15], based on thin shell theory, with Levy-type solutions for 95 harmonic terms. The FSM was performed by Rao and Rao [4]. To account for the intermediate rigid diaphragm, Rao used the flexi-
rigidities. Incorporation of beam elements has been previously described [12, 131.The results for the longitudinal forces, NY,transverse moments, Af,, and deflections, w,atthequarter-spanarecomparedwiththose from the theoretical analysis and the FSM (Figs 6-8). unit : mm rigid diaphrwm
I I
3500
I
(4
lb)
Fig. 13. (a) Longitudinal span. (b) Cross-section and material properties. (c) Strip and section divisions.
C.-J.
378
CHEN et al.
0.0 i c I!
-0.5 -
*
BCSM
8
-1.0 -
0
FEM(S)
A FM(T) -1.5 -
+
-2.0
I 675
0
Exp.
I 1750
I 2625
I 3500
I 4375
1 5250
I 6125
I 7000
Location (mm)
Fig. 14. Distribution of vertical deflection along the beam.
bility method with the interior supports removed as redundants. The results for the FSM were obtained with discretization of 14 strips and 40 harmonic terms (2400 complete equations). Results for the longitudinal force at the midspan for the BCSM are compared with those from the theoretical analysis and the FSM (Fig. 9). The results for the BCSM, with considerably fewer ‘degrees of freedom than the FSM, are in excellent agreement with the results for the theoretical analysis.
This method eliminates the repetitive solution passes required for each redundant in a flexibility method. The accuracy and efficiency of this method have been tested on a variety of examples with considerably fewer degrees of freedom than the conventional FSM, and the results indicated very good agreement with available experimental results.
Example 2: continuous folded plate with intermediate ball-bearing supports (Fig. 10)
Y. K. Cheung, The finite strip method in the analysis of
By symmetry, analyses were performed on a quarter of the folded plate with discretization of 12 strips and eight equally spaced sections (468 complete equations). In the BCSM, the ball-bearing support was modeled as a column element, with high axial rigidity but no flexural rigidities. Results of the transverse moments, iU,, and longitudinal forces, NY, at the midspan are compared with those from the photoelastic analysis and theoretical analysis of Mark and Riera [ 161, as shown in Figs 11 and 12. As a whole, the results for the BCSM are in good agreement with the theoretical results and experimental values. Example 3: a continuous two-span double-cell box beam under concentrated loaa!s (Fig. 13)
Analyses were performed by using the BCSM with discretization of 22 strips and 10 unequally spaced sections. The vertical deflections at the bottom of the side web along the continuous box beam are plotted (Fig. 14). The FEM(S) and FEM(T) represent the FEM based on the shell analysis and thin-walled box beam analysis in [ 171, respectively. The experimental values are also from [17]. Results are in good agreement. CONCLUSION
A B-spline space column element was derived, and combined the concept of the BCSM for the analysis of plate-type structures with intermediate supports.
REFERENCES
elastic plates with two opposite simply supported ends. Proc. Instn Ciu. Engrs 40, l-7 (1968). 2. F. A. Branco and R. Green, Composite box girder bridge behavior during construction. J. Srrucr. Engng, ASCE 111(3), 577-593 (1985). M. S. Cheung, Y. K. Cheung and A. Ghali, Analysis of slab and girder bridges by the finite strp method. Building Sci. S(2), 95-105 (1970). P. S. Rao and P. S. Rao, Continuous folded plates. Bull. Int. Ass. Shell Spat. Struct., 1%2(67), 37-44 (1978). J. A. Puckett and R. M. Gutkowski. Comnound strin _ method for analysis of plate systems. J. Struct. En&, ASCE 112(l), 121-138 (1986): M. S. Cheung and W. Li, Finite strip analysis of continuous structures. Can. J. Civ. Ennnn -- 15, 424-429 (1988). 7. M. S. Cheung and W. Li, Finite strip analysis of continuous structures. Can. J. Civ. Enema _- 15, 424-429 8. T. Mizusawa, T. Kajita and M. Naruoka, Vibration of skew plates by B-spline functions. J. Sound I’ibr. 301-308 (1979). 9. H. Y. Yang and K. P. Chong, Finite strip method with X-spline functions. Compur. Strucr. HI(l), 127-132 (1984). 10. Y. K. Cheung, S. C. Fan and C. Q. Wu, Spline finite strip in structure analysis. In Proc. Int. Conf. on Finite Element Methods, Shanghai, pp. 704-709. Science Press, Beijing, and Gordon & Breach, New York (1982). 11. Y. K. Cheung and S. C. Fan, Static analysis of right box girder bridges by spline finite strip method. Proc. Insf. Ciu. Engrs, Part 2, 75, 311-323 (1983). 12. C. J. Chen, R. M. Gutkowski and J. A. Puckett, B-Spline compound strip analysis of stiffened plates under transverse loading. Compur. Srruct. 34(2), 337-347 (1990). 13. C. J. Chen, R. M. Gutkowski and J. A. Puckett, The unequally spaced cubic B-spline compound strip method. Struct. Res. Rep. 67, Civil Engineering Department, Colorado State University, Fort Collins (1989).
Spline compound strip analysis of folded plate structures 14. R. M. Gutkowski, C. J. Chen and J. A. Puckett, Plate bending analysis by unequally spaced splines. ThinWalled Smct. (accepted) (1990). 15. M. Pultar, D. P. Billington and J. D. Riera, Folded plates continuous over flexible support. J. Sfruc~. Diu., ASCE 93(STS), 253-277 (1967).
319
16. R. Mark and J. D. Riera, Photoelastic analysis of folded plate structures. J. Engng Me& Div., ASCE 93(EM4), 79-93 (1967). 17. L. F. Boswell and S. H. Zhang, An experimental investigation of the behavior of thin-walled box beams. Thin-Walled Struck 3, 35-65 (1985).