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Analysis of plated structures with rectangular cutouts and internal supports using the spline finite strip method
Pergamon
Computers& StructuresVol. 52, No. 2, pp. 277-286, 1994 Copyright Q 1994Else&r Science Ltd Printed in Great Britain. All rights resewd iws-7949/94 $7.00+ 0.00
~5-7~~~)E~l-3
ANALYSIS OF PLATED STRUCTURES WITH RECTANGULAR CUTOUTS AND INTERNAL SUPPORTS USING THE SPLINE FINITE STRIP METHOD C.
M.
MADA~AMY
and V.
KALYANARAMAN
Department of Civil Enginee~ng, Structural Engineering Laboratory, I.I.T., Madras-~
036, India
(Received 1 February 1993)
Abstract-Analysis of plated structures with rectangular cutouts and internal supports by the finite eiement method is tedious and time consuming. The spline finite strip method has been developed and applied to solve this problem efficiently.
1. INTRODUCTION
used in engineering applications have openings for functional purposes. The elastic analysis of fIoor slabs with rectangular openings or wide column supports by classical plate theory predicts singular moments and shear forces at the corners of such openings and supports. Although the finite element method guarantees a solution, the rate of convergence may be significantly reduced. To overcome this, either the finite element mesh is refined locally or a special-purpose finite element model based on hybrid stress formulation [1] is used. The first option leads to a large number of degrees of freedom, thereby increasing the computation time. The second option is more complicated and time consuming than the displacement based finite element method. On the other hand, the higher order fmite element method and classical finite strip method (CFSM) face difficulty in modelling the problems with steep stress gradients, due to the higher order continuity of their displacement functions. In order to overcome these problems, the spline finite strip method (SFSM), based on classical thin plate theory is used in this present work to model the cutouts efficiently. The spline finite strip method was developed by Cheung and Fan [2] as an alternative to the classical finite strip method. The unknowns in the SFSM are the displacement parameters at the intersection knots of the lon~tu~nal nodal lines and transverse sections. The SFSM adopts B3-splines as displacement functions in the longitudinal direction and cubic hermite shape functions in the transverse direction. The B3-splines are continuous over four sections and the overall displacement representation in the longitudinal direction is given by a linear combination of the local Bfsplines. If the section knots are evenly spaced, the B3-spline function has C2 continuity and this order of continuity can be lowered to C’ or Co near the stress concentration regions by using uneven spacing of the section knots. Structures
The rate of convergence of the deflections and stresses can be increased by using unequal section knots of B3-splines in plate bending analysis using the SFSM and this has been demonstrated by Chen et al. [3] for regular rectangular plates. The SFSM results indicate monotonic convergence of stresses instead of oscillatory convergence (Gibb’s phenomena) encountered while using the CFSM, and the SFSM results are more accurate with a fewer global degrees of freedom compared to the FEM. The advantages of the SFSM over the CFSM and the FEN are discussed by Cheung and Fan [Z]. The application of the SFSM has been extended to the bending analysis of skew plates by Cheung et al. [4] using skew coordinate transformation and to general plates by Cheung et at. [S] using subparametrie mapping. The use of a three-noded higher order spline strip by introducing one auxiliary nodal line in the two-noded lower order strip finds applications near the vicinity of the concentrated loads to reduce the free edge residual moments and in box-type structures [6]. But the higher order strip requires additional computation to condense out the internal degrees of freedom. A Bgspline compound strip method has been developed by Chen et al. [7] for analysing plates with stiffeners, with intermediate supports [8] and with bracing elements [9]. The SFSM based on Mindlin plate theory using a linear Lagrangian polynomial for transverse bending shape functions and subparametric mapping was developed by Tham [lo] for analysing non-prismatic space structures. The plane stress and plate bending problems with openings can be efficiently and accurately analysed using the flexibility of modelling in the SFSM. In this paper the SFSM is extended to analyse such problems, using appropriate unequal section knots and amendments of the splines at the boundaries. The membrane and bending stress analysis using the SFSM has been programmed in a PC-AT in 277
C.
278
M. MADASAMY and V.
where k is a fixed positive integer and y is a bi-finite sequence of real numbers. From the above general expression, the cubic B3-spline as shown in Fig. l(a) can be generated as
Fortran language. The results of the SFSM analysis of structures with cutouts experiencing plane stress and/or plate bending are compared with those available in the literature.
1 1 Ai(Y
-Yi-213t
Ai(Y -Yi-2)3+
My)=
yE[yi-*vYi-11 ci(Y -Yi-A3*
Bi(yi+2-y)3+Di(y.
Y EIYi-ltyil
I+ ,-y)3,
Y EIYi,Yi+ll
mYi+*-Y)3T
2. DISPLACEMENT
Y E[Yi+,7Y,+21
FIELD
where
The term spline function was introduced by Schoenberg [ 111 for solving certain data fitting problems. Classes of spline functions possess many good structural properties as well as excellent approximation powers. They are easy to store, evaluate and manipulate in a computer. The main drawback of polynomials is that the class is relatively inflexible. In order to overcome the severe oscillations which often appear in the relatively inflexible polynomial function in larger intervals, polynomials of low degree have been used to divide up the interval into smaller pieces leading to piecewise polynomial functions. Spline functions maintain the flexibility of piecewise polynomials while at the same time achieve some degree of smoothness. The cubic B-spline has C2 continuity, whereas the cubic Lagrange and cubic Hermite have only Co and C’ continuity, respectively. The B-splines have a small local support and particularly the B3-spline
4i(Y)
=$
1 (Yi+l
+Yi-Z)(Yi-Yi-2)(Yi-I-Yi-2)
1
B, =
(Yi+Z-Yt-l)(Yi+2-Yi)(Yi-2-Yi+I)
ci = -(Y,+Z-Yi-2) (Yi+2-Yr-l)(Yi+l
D;
-Y,-,)(Yi-Yi-,)(Yi-l-Y,-2)
= -(Yi+z-Yi-2) (Yi+,-Yi-2)(Yi+I-Yi-I)(Yi+I-Yi)(Yi+Z-YY,+,)
The general B3-spline expression reduces to the following simple expression (eqn (4)) for B3-splines with equal section knot spacings shown in Fig. l(b). Y EIYi-lvYi-ll
h3+3h2(y-yi_~)+3h(y-yi_~)2-3(y-y,_~)3~
Y E[yi-19Yi+ll
h3+3h2(Yi+I-Y)+3h(yj+~-y)2-3(yi+~-y)3~
Y E[YttY,+,l
(Yi+2-Y13
Y E[Yi+I,Yi+21
1
function is non-zero over only four consecutive sections. The B3-splines with a local hill-shape lead to narrow bandedness in the stiffness matrix, whereas the cardinal splines yield full matrix and natural splines lead to unstable answers [12]. Hence in this work B3-splines have been used. The unequal B-splines of any order can be derived from the normalized B-spline functions as the kth divided difference of the Green’s function G defined as
B,
Ai =
(Y -Yi-2)3.
(y -yJk-‘,
Gk(Y;Yi)=(Y -Yt)+k-l=
=(-l)k[~l,...,yi+klGk(~;~i) 1.k (Yi+k
-Yi)
(3)
’
Y >Yi y
’
(l) (2)
.
(4)
In order to incorporate the appropriate boundary conditions in the spline function, three local splines have to be amended at the boundary and to simplify the amended spline expression the three section knots adjacent to the boundary have to be of equal spacing as shown in Fig. l(c). The equal section B3-spline function and its derivatives at section knots are used to satisfy the boundary conditions. 3. STIFFNESS MATRIX FORMULATION
A two-noded lower order strip as shown in Fig. 2(b) which has four degrees of freedom (u, a, w, 0) per nodal line is used to model prismatic plated members. The assumed displacement function of the strip is a product of a B3-spline function along the longitudinal direction and a Hermitian function
Analysis of plated structures using finite strip method
53-z
Yi
yi-1
Yi-2
Yr-1
Yi
Yi+l
yi*1
Yi+2
yi*2
(b) zw 1,
k-l
k
279
N&l
fine j with d.o.f up vi, wp ej
t SCCtiOZt
k+l
4-h--t--h+
Fig, I {a). Unequd section B3-spline. @) Equal section B3-spline. (cf Equal section knot adjacent to boundary ‘k’.
i
with d.o.f ui, vi, wi, 8,
Knot
in the transverse direction. The transver~ Hermitian function is a linear polynomial for inplane displacements (u and u) and a cubic polynomial for bending displacements (w and 8). The displacement functions are given by
The parameters NWj,N,, N,, ND!,N,, Neil Nwi, and NBi are transverse shape functions and #Ui, rp,i, &, Qjsi,rbUj,rpVj,&j and bej are B3-spline re~r~sentatjons” The expressions for the transverse shape functions are given by Nut= NV,= l-1 NVj=Noi=” NWi= 1 -3Z2+2f3 N& = b(n - 22.2-+-9)
Fig. 2 (a). Di~reti~tion of plate into finite strips. (b) Typical spline finite strip.
Nu$= 3.~2~ - 2i’
NBi= b(B3 - i?),
(6)
where f = x/b.
The nodal lines and section knots for a thin-walled structure discretized using a spline strip is shown in Fig. 2(a). The structure is divided using n nodal lines I@ - 1) strips] transversely and rptsections Iongitudinalfy, To fully defme the overall representation of the B3-splines, one additional section knot (&., and $,,, + l ) is required at each end of the strips. Thus the total number of degrees of freedom is 4n(rn + 3) for
C.
280
M. MADASAMY
and V.
KALYANARAMAN
sphnes for u, v, w and 8 for various boundary conditions are given in Cheung and Fan [2]. The generalized inplane and bending displacement parameters at nodal lines i and j and ujr Q, wi, Bi, u,, nj, w,, 0, and the column vectors of each displa~ment have m f 3 parameters corresponding to m + 3 section knots. curvatureand strain-displacement The displacement relations for the membrane stress state and plate bending based on classical thin-plate theory are given by
(a)
(c)
Fig. 3 (a). B3-spline series. (b) B3-spline series for simply supported boundary condition. (c) Amendment for point support at knot k.
--a*w
~1~ ax2
Kx
lT2W
=
=
“.v ----z combined bending and membrane actions and aY K1.Y 2n(m + 3) for only bending or plane stress problems. 2 ah The spline function matrices [$.,I, [&I, [4wilt [40~1, ax ay [&,j], [$oj], [&I and [4oj] are row vectors, each having m + 3 local B3-splines as shown in Fig. 3(a). In genera1 these relationships Before the boundary amendement the row vectors of 4 for any ith nodal line displacement representation has the following form and is shown in Fig. 3(a).
mJ1&1.
(11)
can be written as
where
After the application of hinged support boundary conditions as shown in Fig. 3(b) to the spline functions, the row vectors of 4 are given by
t&j = {# The constitutive equation for linear elastic orthotropic material can be summarised as follows for membrane stress state
where 4 represent the amended functions. If there is any intermediate support at k as shown in Fig. 3(c) it is only necessary to modify the spline at k and adjacent splines of knot k. This will have the corresponding change in the row matrices of d, as
(12)
where D, , = E, t/( 1 - v, v, ); D,2 = D2, = vxDz2= v,D,, and 4k+l>#k+2r.*.r
4m-Zt &-ir
L
&l+,I‘
(9)
To model the cutouts, the splines passing over the cutouts are tailored to account for the zero thickness. For a wide column support, the local splines near the coiumn corners are to be amended and the splines for column sides also have to be amended. The amended
moment-curvature is
relation
Dz2= ep t/f I - V,vy); D,, = G,t, and the
for plate bending
case
‘Dee04s 0 D54
Dss
0
0
0
Ds I
(13)
Analysis of plated structures using finite strip method DU = E,~‘/l2(1 - v,~v,,.); where (1 - v,~v?); D,, = D, = v,~D~~= 3 D, G,,.t3/12. In general
D,, = E,.t3/12 and D, =
{‘J}={k;); [D]=k’ ffl] and
Using the principle of minimum total potential energy, the stiffness matrix and load vector can be derived as follows:
I-I = ;
sc
-
{s}T[B]T[D][B]{G}
dV
@IT[41TP’lT{q 1dA
- {~IT[41T[NTPI. Taking the first variation 6 II = 0 yields
(14)
of eqn (14), and setting
s
281
longitudinal directions to minimize the half-bandwidth for a flat plate to be equal to 4n for longitudinal sequential numbering and 4(m + 3) for transverse sequential numbering. It can also model strips of different lengths. For modeling cutouts, the splines in the cutouts and some sphnes near the cutouts have to be removed depending on the size of the opening. The tailoring of the splines does not mean that the function is discontinuous over the entire strip as long as the number of sections in the opening is less than four. If the number of sections in the opening is more than or equal to four then the entire strip is treated as two separate strips in the longitudinal direction. In the case of a large number of openings, a reduction in unnecessary computation of the intermediate splines can be achieved by introducing additional strips in the longitudinal direction. For modeling internal supports such as wide column supports, the strips connected to the internal supports have to be amended. For inner simple support the amendment for 4, is the same as that of the clamped support and C#J~ is amended similar to two continuous supports at the column corners. In the case of in plane boundary conditions similar treatment as for the relevant degrees of freedom has been done for incorporating the openings. The amendment functions for various boundary conditions are discussed in greater detail by Fan [ 121.
PITWl PI{6 1d v = [dlTWIT{q 1dA Y sA 5. NUMERICAL
+ MITWIT~~~~(15)
RESULTS
The performance of the SFSM developed for including the effect of cutouts and internal supports has been tested for problems of plate bending and membrane stress states and the results are discussed below.
or simply
where [K] is the linear elastic stiffness matrix and {F} is the load vector given by
5.1. Plates with cutouts and internal supports under UDL
b
[K] = t
I
ss0 0
PITPWl
dx dy dx dr + MITIWT{f’I.
P’I =
The spline function products are numerically integrated using Gauss quadrature in the above equation. In this work four Gauss points are used for one section. The strip stiffness matrix is calculated section knot by section knot and assembled in a skyline fashion for storage of the global stiffness matrix, so as to always keep the strip sectionwise stiffness matrix at a constant size (8 x 8). 4. MODELING
CUTOUTS
AND INTERNAL
SUPPORTS
The program developed for the SFSM analysis has the numbering feature in both transverse and CAS J2,2-”
The dimensions of a square plate with concentric cutout are shown in Fig. 4(a). The material properties are chosen such that qa4/D = 1. Only a quarter of the plate has been modelled for the analysis, due to symmetry. The deflections at the selected points are obtained for different mesh sizes and are plotted against the number of degrees of freedom. The variation of normahsed bending moments M/qa* are also drawn for a section along y = 0.25a for various boundary conditions. The moments plotted by Hrudey and Hrabok [l] are averaged at the nodes for comparison. The unequal spline finite strip model is generated from the 6 x 6 equal mesh by sub-dividing one or more of the strips and sections adjacent to the cutout into two equal parts. Further, one section at the beginning of the cutout is sub-divided into two equal parts in order to obtain four equal sections at the cutouts, thus facilitating the amendment of the splines at the cutouts.
C. M. MADASAMYand V. KALYANARAMAN
282
t
-
SFSM - w,
---
SFSM-w3
A
aI2
0.300 ow
150
A
FEM - SAPXV
9
Hmdey (19S6)
0
USPSM- w,
0
USPSM-aj
300
450
600
Number of D.0.F
0.230 * h a 3
0.223
W
(cl 0
+,“oi-+T-0 /+Jo 0
0.04
A
A
0.03
8 g ‘5 :: c 3 B 2
“e
0.220 A
I! *
SPSM- w* A
FEM - SAPIV
0
Hrudoy (1986)
+
USPM - “2
0.02
3
fl m
0.215
0.01
c
0
I
0.210 0
0.048
I
I 300
150 Number
t
SFSM - 6x6 (DOF 74) Hsmdey(19116)- 6x6 (DOF 87)
-
2
I
I 0.2
I
O.o(
600
0.1
I 0.3
Diruncc X/a from
of D.0.F
I 0.5
left edge
(0
0,140
(d
I 0.4
t P CI 8 0.135
=f
n
3
0.047
8
!! j
-
SFSM - w,
---
SPSM-w, 0
Hntdoy (1986)
:: w i G
A
PEM - SAPIV
e
0.130
-
0.125
SFSM-
ws
A
FEH - SAPIV
0
Hrudsylt. al
2 0.044
I 200 Number
I 400
I
I
600
800
0.120
0
400
200 Number
of D.0.F Fig. 4 (a to f)
of D.0.F
600
800
Analysis
of plated structures using finite strip method
.
283
(h)
-0.03 0 I s
-0.06
-
$J A
0 m -0.09
5
SFSM - 6x6 (DDF 76)
0
SFSM - v, SFSM-w6
--A 0
0.090
FEM- SAPIV Hrudey (1986)
E
Hrudey (1986) - 6x6 @OF 93)
g -0.12
L
I
I
I
I
I
0
0.1
0.2
0.3
0.4
0.5
Distance
0.17r
0.084 U
Number
xh
0.07 -
(i)
IF
800
of D.0.F
(j)
0
0
600
400
200
0
A
P t
P
A
-0.07
00
-
4E
0
m -0.14
-
SFSM - 6x6 @OF 84) 0
Hrudsy (1986) (DDF 99)
-Oe21 00.1 Number
o-o43o r
0.0290
(k)
-
SFSM - w1 SFSM-wI FEM - SAPIV USFSM - w1 USFSM - w3
---
0.0425
-
A
A
0 0
A
0.0420
I 150
I 300
0.5
(1)
0.0285
o.02*oc~A
A
,iEilii!y ,
~~~~
0.0410 * 0
0.4
Distance x/a
of D.0.F
I 450
I 600
Number of D.0.F
0
150
300
Number of D.0.F Fig. 4 (g to 1)
450
600
284
C. M. MADASAMYand V. KALYANARAMAN 2.475
(m) *
$
de=----
P b
A
l
8
A
A
2.450
. (n)
.r
A
A A
A
A
A
-
SFSM - w,
- --
SFSM-w3
A 0
EM-SAPIV USFSM - w1
0
USFSM-w,
-
4 1) t 3
2.400
.
SFSM FEM - SAPIV USFSM
A
E g
I
1
I
I
200
400
600
800
Number
A 0
2.315 L 0
of D.0.F
200
400
Number of
600
800
D.0.F
Fig. 4 (m and n) Fig. 4 (a). Square plate with cutout. (b) Deflection of simply supported square plate (case 1). (c) Deflection of simply supported square plate (case 1). (d) Bending moment hfr/qa2 along Y = 0.25a (case I). (e) Deflection of clamped square plate (case 2). (f3 Deflection of clamped square plate (case 2). (g) Bending moment M,./qa2 along j’ = 0.2% (case 2). (h) Deflection of simply supported square plate (case 3). (i) Deflection of simply supported square plate (case 3). (j) Bending moment M,/qd along y = 0.25~ (case 3). (k) Deflection of clamped square plate (case 4). (1) Deflection of square plate (case 4). (m) Deflection of corner supported piate (case 5). (n) Deflection of corner supported plate (case 5).
The plate has been analysed for the following cases: Case 1. All outer sides simply supported and free. Case 2. All outer edges free and inner clamped. Case 3. All the outer edges free and inner simply supported. Case 4. All outer edges clamped and inner free. Case 5. All outer corners supported and sides free.
inside edges edges edges inner
The results are compared for different levels of diseretization with that of Hrudey and Hrabok(l] based on hybrid stress formulation and that obtained from the SAPIV program. The normalized deflections at the critical points and the normalized bending moment are plotted in Figs 4(a)-4(m). It is found that although a difference in the deflection of similar points exists in the SFSM due to a difference in the deflected shape along and across the strips, the difference reduces with finer discretization. The convergence is better with the unequal spline finite strip method (USFSM). In all cases the USFSM and SFSM indicate faster convergence. The bending moments compare well. 5.2. Piane stress problems Case 1. Plane shear wall with very large opening. The dimensions, material properties and loading conditions of a shear wall with a cutout are given in
Fig. S(a). The variation of horizontal displacement u along the height of the wall is plotted in Fig. S(b). The USFSM results agree very well with the Tham and Cheung 1131 finite element results even with fewer degrees of freedom. The vertical stress G?is plotted at the base of wall in Fig. 5(c) and is compared with beam theory and the FEM of Tham and Cheung [ 131. It seen that the 102 DOF SFSM results show a larger deformation at the location of the openings when compared to the FEM results for essentially the same DOF. Case 2. Plane shear wall supported on two piers. The details of the problem are shown in Fig. 6(a). The variation in horizontal displacement is plotted in Fig. 6(b) along the height of the wall. Unequal section knot spacing is adopted in the SFSM. The USFSM results compare well with those of Tham and Cheung [13] based on the FEM even with fewer degrees of freedom. 6. CONCIXJSIONS The analysis of plates and plane stress problems with cutouts using the spline finite strip method has been demonstrated. An unequally spaced B3-spline is adopted in this work and it is very useful in discretizing sections finer in the regions of cutouts. The cutouts have been modelled by tailoring the splines passing over the opening. In the case of wide internal supports, in addition to tailoring, an amendment of internal boundary splines has been carried out. The program developed has the ability to handle strips with differing lengths so that in the case of very large
Analysis of plated structures using finite strip method
285
(4)
+1+2+2+ -P=l
t
16
0
t
12
+ 16
E = 1000.0 u = 0.0 t= 1.0
2s r M 2 20 15 0 0 A 0
10 5
USFSM 7x3 (DOF 56) USFSM 7x5 (DOF 84) CheuES - Anrt. (1983) CheunS (1983)-FIPd-ISW4 (DOF 184) CheunS (1983)~FEM-ISWE @OF 400)
0 0
0.4
1.6
1.2
0.8
Horizontal
2.0
2.4
disphxmcnt
2.8
-10
-
_,3
-
SFSM 22x3 (DOF
A 0
I 1
-20 .
0
3.2
USFSM 7x3 @OF 56) USFSM 7x5 (DOF 84) Cheung (198$FEM-hWlf @OF 82) Chsung(1983) (Beam theory)
I
I
I
I
I
2
3
4
5
6
Horizontsl
u x 10J
distance
Fig. 5 (a). Shear wail with opening. (b) Horizontal displacement plane shear wall with opening. (c) Stress at the base of the wall.
22
loo
(4 stt3
c
Ib)
20 I8
3-t1t
16 14
t
I
6 = 6.7% x 10’ i) = 0.375 t = 0.04t7
.$ 1
12
g
10 8
0
!F 09000
0.002 Horizontal
-
USFSM 9r4 @OF 86)
~~~.~o~ . displacement
(u)
Fig. 6 (a). Wall supported on two piers. (b) Horizontal displacement of plane shear wall supported on piers.
C. M. MADASAMY and V. KALYAVAKACIA~
286
openings unnecessary evaluation of intermediate splines can be avoided. It is also possible to do number sequencing in either direction so that the bandwidth can be minimized. The analysis results of the plates and two-dimensional shear walls with cutouts show good agreement with available solutions. The analysis of plates with wide column supports has been modelled similar to cutouts and the results show faster convergence. In all these cases comparable results were obtained with fewer degrees of freedom in the USFSM than the comparable FEM. Thus the computational time and storage required are significantly reduced in the SFSM. Acknowledgemenf-The research reported in this paper was supported by a project sponsored by Aeronaufical Research and Development Board, Government of India.
REFERENCES
1. T. M. Hrudey and M. M. Hrabok, Singularity finite elements for plate bending. J. Engng. Mech. 112(7). 666-681 (1986). 2. Y. K. Cheung and S. C. Fan, Static analysis of right box girder bridges by spline finite strip method. Proc. Inst~ Ciu. Engng 75(2), 31 l-323 (1983).
and J. t\. Puckctt, Plate 3. C. J. Chen, R. M. Gutkowski bending analysis of unequally spaced splints. T/IVii,u//ecl Struc,c. II, 409~~430 (1991). 4. Y. K. Cheung. W. Y. Li and M. J. Chen. Bending of skew plates by spline finite strip method. Con~pu~. Srrucr. 22, 3 I 3X (1986). 5. Y. K. Cheung, W. Y. Li and L. G. Tham. Spline finite strip analysis of general plates. J. Enpng hfrch. Dir.. ASCE 112, 43 54 (1986). 6. Y. C. Loo, T. W. Sau and B. Y. Joo, Higher order spline finite strip method. IN. J. S/rut/. S(l),-4549 (1983). and J. A. Puckett. 7. C. J. Chen. R. M. Gutkowski B-spline compound strip analysis of stiffened plates under transverse loading. Comput. Strut./. 34(2). 337 347 (1990). 8. C. J. Chcn. R. M. Gutkowski and J. A. Puckett. B-spline compound strip analysis of folded plate structures with intermediate supports. Comput. Struct. 39(3r’4), 369-379 (1991). 9. C. J. Chen, R. M. Gutowski and J. A. Puckett, B-spline compound formulation for braced thin-walled structures. J. Sfrlrc/. Eng,lg, ASCE 117(5). 1303-1317 (1991). IO. L. G. Tham. Application of spline finite strip method in the analysis of space structures. Thin-walled Swurf. 10, 235 246 (1990). II. I. J. Schoenberg, Contributions to the problem 01 approximation of equidistant data by analytic functions. Quurr. J. uppl. Math. 4, 37~41 (1946). analysis. 12. S. C. Fan, Splme finite strip in structural Ph.D. thesis, University of Hongkong, Hong Kong (1982). 13. L. G. Tham and Y. K. Cheung, Approximate analysis of shear wall assemblies with openings. The S/ruc.t. Eng 618(2). 41 45 (1983).
Computers& StructuresVol. 52, No. 2, pp. 277-286, 1994 Copyright Q 1994Else&r Science Ltd Printed in Great Britain. All rights resewd iws-7949/94 $7.00+ 0.00
~5-7~~~)E~l-3
ANALYSIS OF PLATED STRUCTURES WITH RECTANGULAR CUTOUTS AND INTERNAL SUPPORTS USING THE SPLINE FINITE STRIP METHOD C.
M.
MADA~AMY
and V.
KALYANARAMAN
Department of Civil Enginee~ng, Structural Engineering Laboratory, I.I.T., Madras-~
036, India
(Received 1 February 1993)
Abstract-Analysis of plated structures with rectangular cutouts and internal supports by the finite eiement method is tedious and time consuming. The spline finite strip method has been developed and applied to solve this problem efficiently.
1. INTRODUCTION
used in engineering applications have openings for functional purposes. The elastic analysis of fIoor slabs with rectangular openings or wide column supports by classical plate theory predicts singular moments and shear forces at the corners of such openings and supports. Although the finite element method guarantees a solution, the rate of convergence may be significantly reduced. To overcome this, either the finite element mesh is refined locally or a special-purpose finite element model based on hybrid stress formulation [1] is used. The first option leads to a large number of degrees of freedom, thereby increasing the computation time. The second option is more complicated and time consuming than the displacement based finite element method. On the other hand, the higher order fmite element method and classical finite strip method (CFSM) face difficulty in modelling the problems with steep stress gradients, due to the higher order continuity of their displacement functions. In order to overcome these problems, the spline finite strip method (SFSM), based on classical thin plate theory is used in this present work to model the cutouts efficiently. The spline finite strip method was developed by Cheung and Fan [2] as an alternative to the classical finite strip method. The unknowns in the SFSM are the displacement parameters at the intersection knots of the lon~tu~nal nodal lines and transverse sections. The SFSM adopts B3-splines as displacement functions in the longitudinal direction and cubic hermite shape functions in the transverse direction. The B3-splines are continuous over four sections and the overall displacement representation in the longitudinal direction is given by a linear combination of the local Bfsplines. If the section knots are evenly spaced, the B3-spline function has C2 continuity and this order of continuity can be lowered to C’ or Co near the stress concentration regions by using uneven spacing of the section knots. Structures
The rate of convergence of the deflections and stresses can be increased by using unequal section knots of B3-splines in plate bending analysis using the SFSM and this has been demonstrated by Chen et al. [3] for regular rectangular plates. The SFSM results indicate monotonic convergence of stresses instead of oscillatory convergence (Gibb’s phenomena) encountered while using the CFSM, and the SFSM results are more accurate with a fewer global degrees of freedom compared to the FEM. The advantages of the SFSM over the CFSM and the FEN are discussed by Cheung and Fan [Z]. The application of the SFSM has been extended to the bending analysis of skew plates by Cheung et al. [4] using skew coordinate transformation and to general plates by Cheung et at. [S] using subparametrie mapping. The use of a three-noded higher order spline strip by introducing one auxiliary nodal line in the two-noded lower order strip finds applications near the vicinity of the concentrated loads to reduce the free edge residual moments and in box-type structures [6]. But the higher order strip requires additional computation to condense out the internal degrees of freedom. A Bgspline compound strip method has been developed by Chen et al. [7] for analysing plates with stiffeners, with intermediate supports [8] and with bracing elements [9]. The SFSM based on Mindlin plate theory using a linear Lagrangian polynomial for transverse bending shape functions and subparametric mapping was developed by Tham [lo] for analysing non-prismatic space structures. The plane stress and plate bending problems with openings can be efficiently and accurately analysed using the flexibility of modelling in the SFSM. In this paper the SFSM is extended to analyse such problems, using appropriate unequal section knots and amendments of the splines at the boundaries. The membrane and bending stress analysis using the SFSM has been programmed in a PC-AT in 277
C.
278
M. MADASAMY and V.
where k is a fixed positive integer and y is a bi-finite sequence of real numbers. From the above general expression, the cubic B3-spline as shown in Fig. l(a) can be generated as
Fortran language. The results of the SFSM analysis of structures with cutouts experiencing plane stress and/or plate bending are compared with those available in the literature.
1 1 Ai(Y
-Yi-213t
Ai(Y -Yi-2)3+
My)=
yE[yi-*vYi-11 ci(Y -Yi-A3*
Bi(yi+2-y)3+Di(y.
Y EIYi-ltyil
I+ ,-y)3,
Y EIYi,Yi+ll
mYi+*-Y)3T
2. DISPLACEMENT
Y E[Yi+,7Y,+21
FIELD
where
The term spline function was introduced by Schoenberg [ 111 for solving certain data fitting problems. Classes of spline functions possess many good structural properties as well as excellent approximation powers. They are easy to store, evaluate and manipulate in a computer. The main drawback of polynomials is that the class is relatively inflexible. In order to overcome the severe oscillations which often appear in the relatively inflexible polynomial function in larger intervals, polynomials of low degree have been used to divide up the interval into smaller pieces leading to piecewise polynomial functions. Spline functions maintain the flexibility of piecewise polynomials while at the same time achieve some degree of smoothness. The cubic B-spline has C2 continuity, whereas the cubic Lagrange and cubic Hermite have only Co and C’ continuity, respectively. The B-splines have a small local support and particularly the B3-spline
4i(Y)
=$
1 (Yi+l
+Yi-Z)(Yi-Yi-2)(Yi-I-Yi-2)
1
B, =
(Yi+Z-Yt-l)(Yi+2-Yi)(Yi-2-Yi+I)
ci = -(Y,+Z-Yi-2) (Yi+2-Yr-l)(Yi+l
D;
-Y,-,)(Yi-Yi-,)(Yi-l-Y,-2)
= -(Yi+z-Yi-2) (Yi+,-Yi-2)(Yi+I-Yi-I)(Yi+I-Yi)(Yi+Z-YY,+,)
The general B3-spline expression reduces to the following simple expression (eqn (4)) for B3-splines with equal section knot spacings shown in Fig. l(b). Y EIYi-lvYi-ll
h3+3h2(y-yi_~)+3h(y-yi_~)2-3(y-y,_~)3~
Y E[yi-19Yi+ll
h3+3h2(Yi+I-Y)+3h(yj+~-y)2-3(yi+~-y)3~
Y E[YttY,+,l
(Yi+2-Y13
Y E[Yi+I,Yi+21
1
function is non-zero over only four consecutive sections. The B3-splines with a local hill-shape lead to narrow bandedness in the stiffness matrix, whereas the cardinal splines yield full matrix and natural splines lead to unstable answers [12]. Hence in this work B3-splines have been used. The unequal B-splines of any order can be derived from the normalized B-spline functions as the kth divided difference of the Green’s function G defined as
B,
Ai =
(Y -Yi-2)3.
(y -yJk-‘,
Gk(Y;Yi)=(Y -Yt)+k-l=
=(-l)k[~l,...,yi+klGk(~;~i) 1.k (Yi+k
-Yi)
(3)
’
Y >Yi y
’
(l) (2)
.
(4)
In order to incorporate the appropriate boundary conditions in the spline function, three local splines have to be amended at the boundary and to simplify the amended spline expression the three section knots adjacent to the boundary have to be of equal spacing as shown in Fig. l(c). The equal section B3-spline function and its derivatives at section knots are used to satisfy the boundary conditions. 3. STIFFNESS MATRIX FORMULATION
A two-noded lower order strip as shown in Fig. 2(b) which has four degrees of freedom (u, a, w, 0) per nodal line is used to model prismatic plated members. The assumed displacement function of the strip is a product of a B3-spline function along the longitudinal direction and a Hermitian function
Analysis of plated structures using finite strip method
53-z
Yi
yi-1
Yi-2
Yr-1
Yi
Yi+l
yi*1
Yi+2
yi*2
(b) zw 1,
k-l
k
279
N&l
fine j with d.o.f up vi, wp ej
t SCCtiOZt
k+l
4-h--t--h+
Fig, I {a). Unequd section B3-spline. @) Equal section B3-spline. (cf Equal section knot adjacent to boundary ‘k’.
i
with d.o.f ui, vi, wi, 8,
Knot
in the transverse direction. The transver~ Hermitian function is a linear polynomial for inplane displacements (u and u) and a cubic polynomial for bending displacements (w and 8). The displacement functions are given by
The parameters NWj,N,, N,, ND!,N,, Neil Nwi, and NBi are transverse shape functions and #Ui, rp,i, &, Qjsi,rbUj,rpVj,&j and bej are B3-spline re~r~sentatjons” The expressions for the transverse shape functions are given by Nut= NV,= l-1 NVj=Noi=” NWi= 1 -3Z2+2f3 N& = b(n - 22.2-+-9)
Fig. 2 (a). Di~reti~tion of plate into finite strips. (b) Typical spline finite strip.
Nu$= 3.~2~ - 2i’
NBi= b(B3 - i?),
(6)
where f = x/b.
The nodal lines and section knots for a thin-walled structure discretized using a spline strip is shown in Fig. 2(a). The structure is divided using n nodal lines I@ - 1) strips] transversely and rptsections Iongitudinalfy, To fully defme the overall representation of the B3-splines, one additional section knot (&., and $,,, + l ) is required at each end of the strips. Thus the total number of degrees of freedom is 4n(rn + 3) for
C.
280
M. MADASAMY
and V.
KALYANARAMAN
sphnes for u, v, w and 8 for various boundary conditions are given in Cheung and Fan [2]. The generalized inplane and bending displacement parameters at nodal lines i and j and ujr Q, wi, Bi, u,, nj, w,, 0, and the column vectors of each displa~ment have m f 3 parameters corresponding to m + 3 section knots. curvatureand strain-displacement The displacement relations for the membrane stress state and plate bending based on classical thin-plate theory are given by
(a)
(c)
Fig. 3 (a). B3-spline series. (b) B3-spline series for simply supported boundary condition. (c) Amendment for point support at knot k.
--a*w
~1~ ax2
Kx
lT2W
=
=
“.v ----z combined bending and membrane actions and aY K1.Y 2n(m + 3) for only bending or plane stress problems. 2 ah The spline function matrices [$.,I, [&I, [4wilt [40~1, ax ay [&,j], [$oj], [&I and [4oj] are row vectors, each having m + 3 local B3-splines as shown in Fig. 3(a). In genera1 these relationships Before the boundary amendement the row vectors of 4 for any ith nodal line displacement representation has the following form and is shown in Fig. 3(a).
mJ1&1.
(11)
can be written as
where
After the application of hinged support boundary conditions as shown in Fig. 3(b) to the spline functions, the row vectors of 4 are given by
t&j = {# The constitutive equation for linear elastic orthotropic material can be summarised as follows for membrane stress state
where 4 represent the amended functions. If there is any intermediate support at k as shown in Fig. 3(c) it is only necessary to modify the spline at k and adjacent splines of knot k. This will have the corresponding change in the row matrices of d, as
(12)
where D, , = E, t/( 1 - v, v, ); D,2 = D2, = vxDz2= v,D,, and 4k+l>#k+2r.*.r
4m-Zt &-ir
L
&l+,I‘
(9)
To model the cutouts, the splines passing over the cutouts are tailored to account for the zero thickness. For a wide column support, the local splines near the coiumn corners are to be amended and the splines for column sides also have to be amended. The amended
moment-curvature is
relation
Dz2= ep t/f I - V,vy); D,, = G,t, and the
for plate bending
case
‘Dee04s 0 D54
Dss
0
0
0
Ds I
(13)
Analysis of plated structures using finite strip method DU = E,~‘/l2(1 - v,~v,,.); where (1 - v,~v?); D,, = D, = v,~D~~= 3 D, G,,.t3/12. In general
D,, = E,.t3/12 and D, =
{‘J}={k;); [D]=k’ ffl] and
Using the principle of minimum total potential energy, the stiffness matrix and load vector can be derived as follows:
I-I = ;
sc
-
{s}T[B]T[D][B]{G}
dV
@IT[41TP’lT{q 1dA
- {~IT[41T[NTPI. Taking the first variation 6 II = 0 yields
(14)
of eqn (14), and setting
s
281
longitudinal directions to minimize the half-bandwidth for a flat plate to be equal to 4n for longitudinal sequential numbering and 4(m + 3) for transverse sequential numbering. It can also model strips of different lengths. For modeling cutouts, the splines in the cutouts and some sphnes near the cutouts have to be removed depending on the size of the opening. The tailoring of the splines does not mean that the function is discontinuous over the entire strip as long as the number of sections in the opening is less than four. If the number of sections in the opening is more than or equal to four then the entire strip is treated as two separate strips in the longitudinal direction. In the case of a large number of openings, a reduction in unnecessary computation of the intermediate splines can be achieved by introducing additional strips in the longitudinal direction. For modeling internal supports such as wide column supports, the strips connected to the internal supports have to be amended. For inner simple support the amendment for 4, is the same as that of the clamped support and C#J~ is amended similar to two continuous supports at the column corners. In the case of in plane boundary conditions similar treatment as for the relevant degrees of freedom has been done for incorporating the openings. The amendment functions for various boundary conditions are discussed in greater detail by Fan [ 121.
PITWl PI{6 1d v = [dlTWIT{q 1dA Y sA 5. NUMERICAL
+ MITWIT~~~~(15)
RESULTS
The performance of the SFSM developed for including the effect of cutouts and internal supports has been tested for problems of plate bending and membrane stress states and the results are discussed below.
or simply
where [K] is the linear elastic stiffness matrix and {F} is the load vector given by
5.1. Plates with cutouts and internal supports under UDL
b
[K] = t
I
ss0 0
PITPWl
dx dy dx dr + MITIWT{f’I.
P’I =
The spline function products are numerically integrated using Gauss quadrature in the above equation. In this work four Gauss points are used for one section. The strip stiffness matrix is calculated section knot by section knot and assembled in a skyline fashion for storage of the global stiffness matrix, so as to always keep the strip sectionwise stiffness matrix at a constant size (8 x 8). 4. MODELING
CUTOUTS
AND INTERNAL
SUPPORTS
The program developed for the SFSM analysis has the numbering feature in both transverse and CAS J2,2-”
The dimensions of a square plate with concentric cutout are shown in Fig. 4(a). The material properties are chosen such that qa4/D = 1. Only a quarter of the plate has been modelled for the analysis, due to symmetry. The deflections at the selected points are obtained for different mesh sizes and are plotted against the number of degrees of freedom. The variation of normahsed bending moments M/qa* are also drawn for a section along y = 0.25a for various boundary conditions. The moments plotted by Hrudey and Hrabok [l] are averaged at the nodes for comparison. The unequal spline finite strip model is generated from the 6 x 6 equal mesh by sub-dividing one or more of the strips and sections adjacent to the cutout into two equal parts. Further, one section at the beginning of the cutout is sub-divided into two equal parts in order to obtain four equal sections at the cutouts, thus facilitating the amendment of the splines at the cutouts.
C. M. MADASAMYand V. KALYANARAMAN
282
t
-
SFSM - w,
---
SFSM-w3
A
aI2
0.300 ow
150
A
FEM - SAPXV
9
Hmdey (19S6)
0
USPSM- w,
0
USPSM-aj
300
450
600
Number of D.0.F
0.230 * h a 3
0.223
W
(cl 0
+,“oi-+T-0 /+Jo 0
0.04
A
A
0.03
8 g ‘5 :: c 3 B 2
“e
0.220 A
I! *
SPSM- w* A
FEM - SAPIV
0
Hrudoy (1986)
+
USPM - “2
0.02
3
fl m
0.215
0.01
c
0
I
0.210 0
0.048
I
I 300
150 Number
t
SFSM - 6x6 (DOF 74) Hsmdey(19116)- 6x6 (DOF 87)
-
2
I
I 0.2
I
O.o(
600
0.1
I 0.3
Diruncc X/a from
of D.0.F
I 0.5
left edge
(0
0,140
(d
I 0.4
t P CI 8 0.135
=f
n
3
0.047
8
!! j
-
SFSM - w,
---
SPSM-w, 0
Hntdoy (1986)
:: w i G
A
PEM - SAPIV
e
0.130
-
0.125
SFSM-
ws
A
FEH - SAPIV
0
Hrudsylt. al
2 0.044
I 200 Number
I 400
I
I
600
800
0.120
0
400
200 Number
of D.0.F Fig. 4 (a to f)
of D.0.F
600
800
Analysis
of plated structures using finite strip method
.
283
(h)
-0.03 0 I s
-0.06
-
$J A
0 m -0.09
5
SFSM - 6x6 (DDF 76)
0
SFSM - v, SFSM-w6
--A 0
0.090
FEM- SAPIV Hrudey (1986)
E
Hrudey (1986) - 6x6 @OF 93)
g -0.12
L
I
I
I
I
I
0
0.1
0.2
0.3
0.4
0.5
Distance
0.17r
0.084 U
Number
xh
0.07 -
(i)
IF
800
of D.0.F
(j)
0
0
600
400
200
0
A
P t
P
A
-0.07
00
-
4E
0
m -0.14
-
SFSM - 6x6 @OF 84) 0
Hrudsy (1986) (DDF 99)
-Oe21 00.1 Number
o-o43o r
0.0290
(k)
-
SFSM - w1 SFSM-wI FEM - SAPIV USFSM - w1 USFSM - w3
---
0.0425
-
A
A
0 0
A
0.0420
I 150
I 300
0.5
(1)
0.0285
o.02*oc~A
A
,iEilii!y ,
~~~~
0.0410 * 0
0.4
Distance x/a
of D.0.F
I 450
I 600
Number of D.0.F
0
150
300
Number of D.0.F Fig. 4 (g to 1)
450
600
284
C. M. MADASAMYand V. KALYANARAMAN 2.475
(m) *
$
de=----
P b
A
l
8
A
A
2.450
. (n)
.r
A
A A
A
A
A
-
SFSM - w,
- --
SFSM-w3
A 0
EM-SAPIV USFSM - w1
0
USFSM-w,
-
4 1) t 3
2.400
.
SFSM FEM - SAPIV USFSM
A
E g
I
1
I
I
200
400
600
800
Number
A 0
2.315 L 0
of D.0.F
200
400
Number of
600
800
D.0.F
Fig. 4 (m and n) Fig. 4 (a). Square plate with cutout. (b) Deflection of simply supported square plate (case 1). (c) Deflection of simply supported square plate (case 1). (d) Bending moment hfr/qa2 along Y = 0.25a (case I). (e) Deflection of clamped square plate (case 2). (f3 Deflection of clamped square plate (case 2). (g) Bending moment M,./qa2 along j’ = 0.2% (case 2). (h) Deflection of simply supported square plate (case 3). (i) Deflection of simply supported square plate (case 3). (j) Bending moment M,/qd along y = 0.25~ (case 3). (k) Deflection of clamped square plate (case 4). (1) Deflection of square plate (case 4). (m) Deflection of corner supported piate (case 5). (n) Deflection of corner supported plate (case 5).
The plate has been analysed for the following cases: Case 1. All outer sides simply supported and free. Case 2. All outer edges free and inner clamped. Case 3. All the outer edges free and inner simply supported. Case 4. All outer edges clamped and inner free. Case 5. All outer corners supported and sides free.
inside edges edges edges inner
The results are compared for different levels of diseretization with that of Hrudey and Hrabok(l] based on hybrid stress formulation and that obtained from the SAPIV program. The normalized deflections at the critical points and the normalized bending moment are plotted in Figs 4(a)-4(m). It is found that although a difference in the deflection of similar points exists in the SFSM due to a difference in the deflected shape along and across the strips, the difference reduces with finer discretization. The convergence is better with the unequal spline finite strip method (USFSM). In all cases the USFSM and SFSM indicate faster convergence. The bending moments compare well. 5.2. Piane stress problems Case 1. Plane shear wall with very large opening. The dimensions, material properties and loading conditions of a shear wall with a cutout are given in
Fig. S(a). The variation of horizontal displacement u along the height of the wall is plotted in Fig. S(b). The USFSM results agree very well with the Tham and Cheung 1131 finite element results even with fewer degrees of freedom. The vertical stress G?is plotted at the base of wall in Fig. 5(c) and is compared with beam theory and the FEM of Tham and Cheung [ 131. It seen that the 102 DOF SFSM results show a larger deformation at the location of the openings when compared to the FEM results for essentially the same DOF. Case 2. Plane shear wall supported on two piers. The details of the problem are shown in Fig. 6(a). The variation in horizontal displacement is plotted in Fig. 6(b) along the height of the wall. Unequal section knot spacing is adopted in the SFSM. The USFSM results compare well with those of Tham and Cheung [13] based on the FEM even with fewer degrees of freedom. 6. CONCIXJSIONS The analysis of plates and plane stress problems with cutouts using the spline finite strip method has been demonstrated. An unequally spaced B3-spline is adopted in this work and it is very useful in discretizing sections finer in the regions of cutouts. The cutouts have been modelled by tailoring the splines passing over the opening. In the case of wide internal supports, in addition to tailoring, an amendment of internal boundary splines has been carried out. The program developed has the ability to handle strips with differing lengths so that in the case of very large
Analysis of plated structures using finite strip method
285
(4)
+1+2+2+ -P=l
t
16
0
t
12
+ 16
E = 1000.0 u = 0.0 t= 1.0
2s r M 2 20 15 0 0 A 0
10 5
USFSM 7x3 (DOF 56) USFSM 7x5 (DOF 84) CheuES - Anrt. (1983) CheunS (1983)-FIPd-ISW4 (DOF 184) CheunS (1983)~FEM-ISWE @OF 400)
0 0
0.4
1.6
1.2
0.8
Horizontal
2.0
2.4
disphxmcnt
2.8
-10
-
_,3
-
SFSM 22x3 (DOF
A 0
I 1
-20 .
0
3.2
USFSM 7x3 @OF 56) USFSM 7x5 (DOF 84) Cheung (198$FEM-hWlf @OF 82) Chsung(1983) (Beam theory)
I
I
I
I
I
2
3
4
5
6
Horizontsl
u x 10J
distance
Fig. 5 (a). Shear wail with opening. (b) Horizontal displacement plane shear wall with opening. (c) Stress at the base of the wall.
22
loo
(4 stt3
c
Ib)
20 I8
3-t1t
16 14
t
I
6 = 6.7% x 10’ i) = 0.375 t = 0.04t7
.$ 1
12
g
10 8
0
!F 09000
0.002 Horizontal
-
USFSM 9r4 @OF 86)
~~~.~o~ . displacement
(u)
Fig. 6 (a). Wall supported on two piers. (b) Horizontal displacement of plane shear wall supported on piers.
C. M. MADASAMY and V. KALYAVAKACIA~
286
openings unnecessary evaluation of intermediate splines can be avoided. It is also possible to do number sequencing in either direction so that the bandwidth can be minimized. The analysis results of the plates and two-dimensional shear walls with cutouts show good agreement with available solutions. The analysis of plates with wide column supports has been modelled similar to cutouts and the results show faster convergence. In all these cases comparable results were obtained with fewer degrees of freedom in the USFSM than the comparable FEM. Thus the computational time and storage required are significantly reduced in the SFSM. Acknowledgemenf-The research reported in this paper was supported by a project sponsored by Aeronaufical Research and Development Board, Government of India.
REFERENCES
1. T. M. Hrudey and M. M. Hrabok, Singularity finite elements for plate bending. J. Engng. Mech. 112(7). 666-681 (1986). 2. Y. K. Cheung and S. C. Fan, Static analysis of right box girder bridges by spline finite strip method. Proc. Inst~ Ciu. Engng 75(2), 31 l-323 (1983).
and J. t\. Puckctt, Plate 3. C. J. Chen, R. M. Gutkowski bending analysis of unequally spaced splints. T/IVii,u//ecl Struc,c. II, 409~~430 (1991). 4. Y. K. Cheung. W. Y. Li and M. J. Chen. Bending of skew plates by spline finite strip method. Con~pu~. Srrucr. 22, 3 I 3X (1986). 5. Y. K. Cheung, W. Y. Li and L. G. Tham. Spline finite strip analysis of general plates. J. Enpng hfrch. Dir.. ASCE 112, 43 54 (1986). 6. Y. C. Loo, T. W. Sau and B. Y. Joo, Higher order spline finite strip method. IN. J. S/rut/. S(l),-4549 (1983). and J. A. Puckett. 7. C. J. Chen. R. M. Gutkowski B-spline compound strip analysis of stiffened plates under transverse loading. Comput. Strut./. 34(2). 337 347 (1990). 8. C. J. Chcn. R. M. Gutkowski and J. A. Puckett. B-spline compound strip analysis of folded plate structures with intermediate supports. Comput. Struct. 39(3r’4), 369-379 (1991). 9. C. J. Chen, R. M. Gutowski and J. A. Puckett, B-spline compound formulation for braced thin-walled structures. J. Sfrlrc/. Eng,lg, ASCE 117(5). 1303-1317 (1991). IO. L. G. Tham. Application of spline finite strip method in the analysis of space structures. Thin-walled Swurf. 10, 235 246 (1990). II. I. J. Schoenberg, Contributions to the problem 01 approximation of equidistant data by analytic functions. Quurr. J. uppl. Math. 4, 37~41 (1946). analysis. 12. S. C. Fan, Splme finite strip in structural Ph.D. thesis, University of Hongkong, Hong Kong (1982). 13. L. G. Tham and Y. K. Cheung, Approximate analysis of shear wall assemblies with openings. The S/ruc.t. Eng 618(2). 41 45 (1983).