Marine Structures 5 (1992) 7184
Stability of Initially Stressed Square Plates with Square Openings
R. J w a l a m a l i n i , R. S u n d a r a v a d i v e l u , C. P. V e n d h a n & C. G a n a p a t h y Ocean Engineering Centre, Indian Institute of Technology, Madras36, India (Received 8 January 1991:revised version received 7 June 1991:accepted 20June 1991)
ABSTRACT The stability of a simply supported square plate with openings under inplane loading is analysed using a Finite Element program BUCSAP (BUCkling Structural Analysis Program). The openings are considered as square and central for the main study but rectangular and central for comparison with other work. Different magnitudes of tension and compression are assumed as initial prestress in the transverse direction before the longitudinal stress is applied. Two load cases have been considered. The longitudinal stress is assumed as uniform compression for Case I and is assumed as trapezoidal compression representing hydrostatic loading for Case 2. The results have been used to plot design charts. Key words: stability, plates, openings, initial stress, uniform compression, hydrostatic compression.
1 INTRODUCTION Plated structures such as aeroplane fuselages, plate a n d box girders and ship structures are generally provided with openings for serving various purposes. In ship structures, hatch openings are provided in the deck of ships for facilitating the storage of cargo and m a n h o l e s are provided in the solid floors in the bottom structure of the ship for access and 71 Marine Structures 09518339/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain.
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inspection. The presence of these openings causes a redistribution of membrane stresses accompanied by a significant change in the buckling and ultimate strength characteristics. The development of a direct matrix method for the stability analysis of square plates with central square opening subjected to biaxial loading has been presented by Yettram and Brown ~'2. The effect of the presence of openings on the stability of plates for various loading conditions has been studied by Brown et al. 3. The elastic stability of plates with circular and square openings has been presented by Sabir and Chow 4. Ultimate strength tests conducted on thinwalled steel web plates containing circular and square cutouts have been reported by Narayanan and Chow 5. In ship structures, structural components like the deck plates are under combined stress. Compression in the transverse direction due to hydrostatic pressure from the sides and in the longitudinal direction stress varies from tension, when the ship is subjected to a hogging bending moment, to compression, when the ship is subjected to a sagging bending moment. Similarly the bottom plates are also subjected to compression in the transverse direction due to hydrostatic pressure and tension and compression in the longitudinal direction due to sagging and hogging bending moment conditions, respectively. The transverse bulkheads are subjected to trapezoidal compression due to hydrostatic pressure from the sides and compressive stress in the other direction. In this paper, the analysis of simply supported square plates with a central square opening subjected to the above load cases are presented.
2 METHOD OF ANALYSIS A finite element program BUCSAP 6 developed for linear buckling analysis modifying the Structural Analysis Program, SAPIV, has been used for this study. BUCSAP has been implemented in the SIEMENS main frame E7580 in BS2000 operating system. BUCSAP has the capability to study the buckling of plates subjected to initial prestress, before the stress is applied in the direction of buckling that is, nonproportional loading. The analysis for nonproportional loading consists of two stages. The first stage is the prebuckling analysis in which two load cases are considered. The first load case corresponds to the load in the buckling direction and the second load case corresponds to an initial prestress in the orthogonal direction. The elastic stiffness matrix IKEI, prebuckling stress (trb) and the initial stress (cri) are calculated. In the second stage, the geometric stiffness matrices corresponding to the
Stabili~. of initially stressed square plates
73
buckling load [Kc(crb)] and initial stress [Kc(tri)] are calculated. The linear bifurcation buckling formulation employing the finite element method is used. This leads to a linear algebraic eigenvalue problem represented as [ge + ga(cri)l {u} = Alga(o'~)l lu}
(1)
where {u } is the displacement vector and g. is the eigenvalue. The eigenvalues are then determined by solving the following determinantal equation using the subspace iteration method. lIKE + KG(oi)l  Z [Kc(cr0)] I = 0
(2)
The lowest among the eigenvalues represents the critical eigenvalue (A,). The value ofthe force applied per unit length at the critical load, Nor, is obtained using the following equation N¢, = A~robt
(3)
where t is the thickness of the plate. The critical stress, o , , is given by tr~ 
t
(4)
The plate buckling coefficient, K is then calculated from the following definition Nc, b 2
K = ~
(5)
where b is the width ofthe plate and D is the flexural rigidity of the plate given by Et 3
D = 1 2 ( 1  v 2)
(6)
where E and v are the modulus of elasticity and Poisson's ratio, respectively.
3 CONVERGENCE STUDY A simply supported square plate without opening and with an opening size of 0.5 hole to plate ratio, subjected to uniaxial compression have been analysed for three mesh sizes. The size of the opening is represented as the ratio of the hole dimension in one direction to the plate dimension in the same direction. Since the plate under uniaxial compression is symmetrical, a quarter of the plate has been used for the analysis. The
74
1~ Jwalamalini et al.
three mesh sizes are 8 by 8, 10 by 10 and 12 by 12 with the number of elements being 128, 200 and 288, respectively, for the plate without opening. The mesh discretisation for the plates with openings using 8 by 8, 10 by 10 and 12 by 12 is given in Fig. l(a)(c). The CPU (Central Y S
f.
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ssimply supported
Y
Y
S
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(c) Fig. 1. Convergence study meshes. (a) 8 by 8; (b) 10 by 10; (c) 12 by 12.
Stability of initially stressed square plates
75
TABLE 1
ConvergenceStudy of K Mesh size
Plate without hole
Plate with hole to plate ratio of O.5
8 by 8 10by 10 12by 12
4.010 6 4.006 7 40048
3.086 5 3.041 9 3.011 4
Processing Unit) time for the three cases are 146, 23.2 and 34.6 s. The plate buckling coefficients obtained from the three analyses are given in Table 1. The results indicate satisfactory convergence. A mesh size of 8 by 8 is adopted for the parametric study since its results differ by only 232% in comparison with those of the 12 by 12 mesh, whereas the CPU time for the 8 by 8 mesh is only 42.2% of that of the 12 by 12 mesh. The mesh discretisation along with node numbering and element numbering of a quarter plate of mesh 8 by 8 is shown in Fig. 2. A Poisson's ratio of 03 is used throughout the analyses for plates with and without openings.
4 COMPARISON OF STABILITY OF SQUARE PLATES WITH RECTANGULAR OPENINGS Simply supported square plates without openings and with centrally located openings of various sizes, subjected to uniaxial compression, have been analysed with 8 by 8 mesh size. The plate has been analysed for hole to plate ratios of 0125, 0.25, 0.375, 0.5 and 0.625 in both the X and Y directions. The plate buckling coefficient, K, is computed for each case. The plate buckling coefficient versus the hole to plate ratio in the Y direction for various hole to plate ratios in the X direction have been plotted. The results compare well with those of Brown et al. 3 for opening sizes up to 0.375 (Fig. 3).
5 CASE STUDIES Parametric studies have been carried out on simply supported square plates with central square openings. Two load cases are examined. Case l pertains to the analysis of a plate subjected to uniform initial stress varying from tension to uniaxial critical compression in one direction and uniform compressive stress in the other direction. Case 2 pertains to
R. Jwalamalini et al.
76
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75
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Fig. 2. D i s c r e t i s a t i o n o f a q u a r t e r p l a t e .
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Stability of initially stressed square plates 900

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77
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o
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Computed ~olues us,ng BUCSAP B r o w n et.o' volues
600 
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70
Fig. 3. Comparison of buckling coefficients.
the analysis of a plate subjected to uniform initial stress varying from zero to uniaxial critical compression in one direction and linearly varying compressive stress in the other direction. The plate subjected to load Case 1 is symmetrical and hence, a quarter plate is considered for the analysis. The plate subjected to load Case 2 is unsymmetrical and hence a full plate is considered. Case 1 The quarter plate has been divided into 128 elements. The plate subjected to uniaxial compression, i.e. with zero initial stress is analysed for hole to plate ratios of 0.0, 0125, 0.25, 0.375 and 0.5. The uniaxial critical compressive stress (or,c,) and the K value are obtained for each analysis. The parametric study for each hole to plate ratio has been carried out by varying the initial prestress from tension to uniaxial critical compression and determining in each case the critical buckling stress. In all, 39 analyses have been carried out and the K values have been presented in Table 2.
78
R. Jwalamalini et al. TABLE 2 Parametric Study for Case 1
x=y_ a 0.0
tr2'
O. 125
K
O ucr
0.83 055 028 0.0 0.28 0.55 0.83 1.10
°i
0.25
K
O ucr
0.69 !.8 2.9 4.01 5.12 622 7.33 8.44
0.88 0.59 (~29 0.0 0.29 0.59 0.88 118
b
tr;
0.375
K
¢~ u~r
0.44 1.55 2.66 3.76 486 596 7.05 8.13
0.99 0.66 033 0.0 0.33 066 0.99 I.33
cri
0.5
K
a ucr
0.01 1.14 2.25 3.33 4.35 5.42 641 7.32
0.85 0.71 0.36 00 0.36 0.71  1.07 i.42
o~
K
O ucr
0.49 0.95 2.06 3. I 1 4.09 4.95 568 6.20
0.86 0.36 0.0 036 072  1.08  1.43
0.5 209 3.09 3.90 4.43 4.60 4.52
Case 2 The full plate has been divided into 512 elements. The plate subjected to uniaxial compression varying from uniform (P2/P~ = 1.0) to hydrostatic compression (P2/P~ = 00) in steps of 025 with zero initial stress has been analysed to determine the uniaxial critical compressive stress (tr.,c,). P2 and p~ are the stresses applied at the top and bottom of the plate edges respectively (see Fig. 5). The uniaxial critical stress for uniform compression (P2/Pl = 10) is the m i n i m u m and for hydrostatic compression (P2/Pl = 00) is the m a x i m u m . The value of K obtained is 78 for a plate subjected to hydrostatic compression. This correlates well with the K value of 78 given by T i m o s h e n k o 7. The above analyses were repeated for hole to plate ratios of 025 and 05. The analysis was then c a r d e d out on these plates by increasing the initial stress from zero to the uniaxial critical compressive stress (o.,. ,,c,) and determining in each case the critical buckling stress and the K value. tr,. ~c, is the critical stress which is obtained when only the initial stress is acting on the plate and this is the m a x i m u m value of uniform initial stress. However the uniform initial stress (tr;) is normalised with the linearly varying uniaxial critical stress (cr.,~,) for each P2/Pl ratio. This initial stress ratio (o;/tr,c,) is m a x i m u m w h e n ai is equal to tr,.u~,. The m a x i m u m initial stress ratio is equal to 1.0 for uniform compression (P2/P~ = 1.0). This ratio reduces to 0.51 for hydrostatic compression (P2/P~ = 0.0). The K value for these m a x i m u m ratios is equal to zero and
79
Stability of initially stressed square plates
t h e p late b u c k l e s in u n i a x i a l c o m p r e s s i o n a n d d o e s n o t h a v e a n y reserve c a p a c i t y in the o t h e r di r e c t i on. T h e plates w i t h o u t o p e n i n g s a n d with h o l e to p l a t e ratios o f 0.25 a n d 05 are a n a l y s e d f o r p 2 / p ~ ratios v a r y i n g f r o m 0.0 to 1.0 in steps o f 025. E i g h t y  o n e a n a l y s e s h a v e b e e n c a r r i e d o u t a n d the results are p r e s e n t e d in T a b l e 3.
RESULTS AND DISCUSSIONS Case 1 T h e b u c k l i n g coefficient, K versus the ratio o f initial stress to u n i a x i a l critical c o m p r e s s i v e stress (o;/trucr) is p l o t t e d for h o l e to plate ratios o f 0.0, 0.125, 0.25, 0.375 a n d 0.5 in Fig. 4. T h e results indicate: TABLE 3 Parametric Study for Case 2 P~ pl 1.0 °i
0 75 K
O ucr
ai
t~ 5 K
U uer
tr2i
0.25 K
O ucr
oi
0.0 K
U ucr
tr2i
K
O ucr
x/a  0.0
0.00 028 0.55 083 100
4.01 2.91 1.8 069 0.00
0.0 0.24 0.48 0.73 0.87
4.58 3.33 2.06 0.79 0.00
00 021 0.42 0.62 0.75
5.34 389 241 0.93 0.0
00 017 035 052 0.63
6.37 4.68 2.90 I. 12 0.0
00 014 0.28 042 0.5
7.84 5.84 3.63 1.40 0.0
3.34 253 1.7 0.86 0.0
00 02 04 06 088
3.82 2.97 2.11 123 0.0
0.0 015 0.3 045 0.6 075
4.45 3.60 2.72 183 093 0.0
00 01 0.2 03 05 0.63
5.83 452 369 2.85 113 0.0
00 0.1 0.2 0.3 0.5 0.51
6.61 537 4I 2.79 0.07 0.0
3.09 249 1.8 098 0.0
00 0.2 04 0.6 088
3.50 2.90 2.21 1.4 00
00 0.15 03 0.45 06 0.76
4.05 3.46 2.8 204 115 0.0
00 0.1 02 0.3 05 0.65
4.76 4.25 367 3.03 i49 0.0
0.0 01 02 03 045 0.54
5.76 5.02 4.18 3.21 1.38 0.0
x/a = 0.25
0.0 025 0.5 0.75 1.0 x / a = 0.5
00 0.25 05 0.75 1.0
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0.0 0.125 0.25 0.375 0.5
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 1.25
 1.00
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Fig. 4. K versus
0.00 15i /
0.25
0.50
(3ucr)
(ai/a,,c,)
for Case 1.
(I) An increase in initial tensile stress increases K for both plates with and without openings. (2) The rate of increase of K is nearly constant for plates without openings while it decreases for plates with openings, as initial tensile stress increases. (3) An increase in initial compressive stress decreases K for both plates with and without openings. (4) The rate of decrease of K is nearly constant for plates without openings while it increases for plates with openings, as initial compressive stress increases. (5) The K value reduces to zero as the initial stress is increased to uniaxial critical compression. This is obvious because the plate will buckle in uniaxial compression and has no reserve capacity in biaxial compression. (6) K reduces as the opening size increases. Case 2
The buckling coefficient, K versus the ratio of initial stress to uniaxial critical compressive stress (tr,./G,c,) has been plotted for various P2/P~ ratios ranging from 0.0 to 1.0 in steps of 025 for plates without openings (Fig. 5). Similar graphs are plotted for hole to plate ratios of 0.25 and 05 (Figs 6 & 7). The results indicate that:
Stability of initially stressed square plates 9.00
81
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0.20
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0.40
0.50
0.60
0.70
0.80
0.90
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(R/o.,) Fig. 5. g versus ( o i / o u , ) for Case 2. x / a = 0.0. g.~
o,
8.00
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T ~
x. 6 . 0 0
.R
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t
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i
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i
1 1 1 1 1 1 1 1 1 1
0.10
0.20
i i i
ii
0.30
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0.40
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0.50
i
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ii
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i
i
i
0.70
lf
i
j
i
i
0.80
(oJo.~) Fig. 6. K versus (oi/o,c,) for Case 2. x/a = 0.25.
i
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i
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i
i
i

i
1.00
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. ~ 4.00
e 3.00
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0.10
0.20
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0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
(oJo,,) Fig. 7. K v e r s u s (Oi/¢~ucr) for Case 2, x/a = 05. (1) K decreases as the initial stress (oi) increases from 00 to the m a x i m u m initial stress ratio. (2) K reduces to zero as the initial stress is increased to the m a x i m u m initial stress ratio. (3) T h e rate of decrease of K is constant for plates without o p e n i n g s while it increases for plates with o p e n i n g s as initial stress increases. (4) K reduces as o p e n i n g size increases. T h e variation of K with variation ofp2/pi ratio for (a;lauc,) = 00 is plotted in Fig. 8 from which it can be observed that K reduces as thep2/pi ratio increases from 0.0 to 1.0 for plates without openings. K reduces in a similar fashion for plates with openings.
CONCLUSIONS T h e stability analysis of simply s u p p o r t e d square plates with central square o p e n i n g s subjected to n o n p r o p o r t i o n a l l o a d i n g has be=n carried out using a finite element approach. T h e effect o f different m a g n i t u d e s o f
Stability of initially stressed square plates
83
8.00
X/O
7.00 ~
; "; "~~;~ 0 " 2 5 ..~,~,w 0 . 5
'~ 6.00
._~ ~ 5.00 0 0
~
4.00
0 . ~ 3.013
13_
2.oo 1.00
0.00
illllllllllll
0.00
ill
0.20
i i l l l l l l l l l l l l l l l l l l l l
0.40
0.60
ii
iiii
0.80
iiii111111
1.00
(P2/P,) Fig. 8. K versus
@2/Pl ) for
(a,./a,~,) = 00.
initial prestress in the transverse direction is reported for plates with different hole to plate ratios for loading in the longitudinal direction representing uniform compression and hydrostatic compression. For the uniform compression case, the prestress varies from tension to critical uniaxial compression. For the hydrostatic compression case, the prestress varies from zero to the critical uniaxial value in compression. The results are presented in tabular and graphical form for openings up to 0.5 of the plate width. REFERENCES 1. Yettram, A. L. & Brown, C. J., The elastic stability of square perforated plates. Comput. Struct., 21(6) (1985) 126772. 2. Yettram, A. L. & Brown, C. J., The elastic stability ofsquare perforated plates under biaxial loading. Comput Struct., 22(4) (1986) 58994.
84 3.
R Jwalamalini et al.
Brown, C. J., Yettram, A. L. & Bumett, M., Stability ofplates with rectangular holes. J. Struct. Eng., 118(5) (1987) 11116. 4. Sabir, A . B . & C h o w , F.Y.,Elasticbucklingofflatpanelscontainingcircular and square holes. In Instability and plastic collapse of steel structures, ed L. J. Morris, Granada, London, 1983, pp. 31121. 5. Narayanan, R. & Chow, F. Y., Experiments on perforated plates subjected to shear. J. Strain Anal., 20(1) (1985) 2334. 6. Vendhan, C.P.&Palaninathan, R.,BUCSAPAFiniteElementProgramfor Linear Buckling Analysis  Project Report No. 29, Ocean Engineering Centre, Indian Institute of Technology, Madras, 1988. 7. Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, McGraw Hill, 1961, pp. 348439.