Elasto-plastic shear buckling of square plates with circular holes

ELASTO-PLASTIC

SHEAR BUCKLING OF SQUARE PLATES WITH CIRCULAR HOLES M. UENOYA

Department of Civil Engineering, Hiroshima University, Japan

and

R.G. REDWOOD Department of Civil Engineering and Applied Mechanics, McGill University, Canada (Receiued9 November1976;receivedfor p&biica~jo~ 1 June 1977) Ahstraet-With in-piane stresses calculakd by finite element analysis, critical loads are obtained by the RayleighRitz method for a square plate subjected to uniform edge shear stress and containing centrally located circuiar holes. Elastic and elasto-plastic buckling is examined for clamped and simply supported plates, and results are compared with previous analyses and experiments for various sized holes. The range of hole sizes considered is extended to include larger holes than previously examined, and for small holes, the results suggest that the critical stress is higher than previously thought. For elasto-plastic buckling, critical shear stresses are given for the full range of app~p~te slenderness. Ex~r~en~ results for the cases of simply supine plates support the analytical results, whereas vexation for clamped plates remains inconclusive on account of limited reliable test data. 1. INTRODUCTION

The work described herein was motivated by the need for a bifurcation analysis of perforated elasto-plastic plates under various unctions of toad. A~ention is restricted to square perforated plates subjected to uniform edge shear stresses, and other plate configurations and loading conditionswill be treated subsequently. The finite element method can be used to solve the problems under consideration, but in view of the nonlinear material properties and the fineness of mesh commonly required for accurate ~~ur~tion analysis using this method, it was decided to approach the problem by means of an energy approach in combinationwith the finite element method used only to evaluate in-plane stresses. This was first proposed by Kawai and Ohtsubo[ l] for elastic plates under uniform compression, and extended by Fujita et aL[Z] to elasto-plasticplates under the same loadingcondition.The method is ou~ined in the following,and particular reference is made to the choice of deflection functions, the particular features influencingthe efficiencyof numericalintegrationand the elasto-plasticstress analysis. Results for a wide variety of hole sizes and plate slendernessare given, and where possible comparisonsare made with both anaIyticaland ex~riment~ work described elsewhere.

x and y, D is the flexural rigidity of the plate per unit width, N,, NYand NXY are the plane stress resultants. The strain energy of the plate at bucklingis

dx dy

(2)

where Y is Poisson’sratio, and the work done by external loads is

The total potential energy of the plate is u= u,+u,

(4)

and the required solution is obtained by determiningthe function w in such a way that U is a minimum S(U,+UJ=0 or

(51

2.METHODOFANALYSIS Rayleigh-Ritz method for plate buckling problems In the case of no lateral load, the differential equation of the deflection surface when a plate buckles slightly

under the forces applied in the middle plane is given in Cartesiancoordinates by the foI~owingequation: +ZN:,g%

where w is the deflection function expressed in terms of

where N, = AN:, NY= ANi and N*, = hNz9

291

1dxdy=O

M. UENOYA and R. G. REDWOOD

292

In the Rayleigh-Ritz method, it is assumed that the deflection function is represented by the series w(x,y)

= &fi(x,y)

(7)

where Ai are undetermined coefficients and fJx,y) are functions which satisfy the geometric boundary conditions. After substituting eqn (7) into eqn (6), and taking derivatives with respect to each coefficient, the following equations are obtained: @+hY)A=O

(8)

where Cp= [4ij]*Y = [$ij] and A = {A,}

dx dy (9)

After evaluating eqns (9) and (1l), which include the derivatives and the double integrals of the deflection functions, the buckling problem of the plate is reduced to the eigenvalue problem expressed as eqn (8). Elasto-plastic stress analysis The finite element method using constant stress triangular elements is employed for the in-plane stress analysis of a square plate with a circular hole. The initial stress method[3] is applied for the elasto-plastic stress analysis assuming von Mises’ yield criterion and no strain hardening. Iteration at each incremental load is continued until stresses of all yielded elements converge to the yield surface within 0.5% of the yield stress. The collapse load of the plate is determined as that producing no convergence with 20 iterations. Numerical integration For the evaluation of the double integrals in eqns (9) and (1l), Gaussian quadrature is used in each triangular element. The accuracy of Gaussian quadrature for the Fourier series depends on the number of Grass points and the size of the triangular element. For the area subdivided as shown in Fig. 1, this accuracy has been examined for various size of elements with 3 or 7 integration points and for the following trigonometric functions which usually appear in the energy integration: rnrx , m?rx cos sm cos m?ry sin ydr 1 1 1

112

dy (12)

For stresses obtained by the finite element method using constant stress elements, eqn (10) can be rewritten as follows:

2 cos* 5

1

cos 2 &Y 1

cos2 ~-cOS2

I

mlry

Tdxdy

(13) where m is the number of half-waves in the length 1.The results for m = l-4 are shown in Table 1 in which the error is defined by the following: where the summation is taken over the Ne elements. Using constant stress elements gives a significant advantage in the evaluation of eqn (11) because the double integrals evaluated once can be used through every load increment in an elasto-plastic problem.

Error = I”“; - LX.,, X 100 exact

where I.,,,, is either I, or Z, and I,,,, is the value obtained mathematically. It can be concluded that the

3 Gauss

Points

7 Gauss

Y

4x4 Fig. 1. Idealization

(14)

for numerical integration and Gauss points.

Points

Elasto-pi~tic shear buckli~ of square plates with circular holes

293

Table 1. Numericalerror of Gaussianquadrature

4

-266.2874

-0.0000

0.0000

0.0000

0.0287

\ maximum errOr for the numerical integration is O-74%, and most of the errors are within 0.1% under the conditions involving 7 integration points and the element size being less than half of a half-wave length of the trigonometric function in the coordinate direction.

For a plate clamped on its edges 2rx

w(x,y) = cos -

cos2+0+

amn

cos

x

cos

(2n- ‘h + 1

A2m,2n

sin

1

t/2

1 1*

I

f

mm

II.

The critical value of the shear stress can be represented in the following form: 117) T,, = kc, where

t2

?r2E *~=12(1-~2~

?!!!Gsin Z!!! sin Znlry 1

x n=t,2.

~~os~t~~~sin-sin~ I

I

Under pure shear as shown in Fig. 2(a), the deflection function for a square plate without a hole for symmetric buckling is given by a Fourier series as follows: For a simply supported plate

x *=1.2.

1

0

(18)

7

in which k is the buckling coefficient which is dependent on the boundary conditions and aspect ratio of the plate, and f is the plate thickness. The buckling coefficients k obtained herein are compared with accurate solutions which are k = 9.35141 and 14.71IS] for the simply supported and clamped plate respectively. The convergence of the critical stresses in the symmetric buckling mode is shown in Fig. 3, in which a

c 12 !

2

3

4 Number

Fig.

Fig. 2(b). 2. Perforatedplate subjectedto shear.

5 of

l

Simply

l

Clamped

6

Supported

7

0

Terms

Fig, 3. Convergence of stresses with number of terms in the deflection function.

2pQ

M. UENOYA

and

combination of 8 terms is chosen such that the best convergence can be obtained within four half-ware lengths as follows: For the simply supported plate W(X,Y) = A,, ~0s SOS I

y

+ A42sin *sin 1

t

2n.x . 2?ry AZ2sin 7 sa 7

2my -j-tA,sin~sm-.

4lrx . 4lry ’

(19)

For the clamped plate w(x,y) = co8

FY 7 cos2 -j-

aoo+ b, , sin 4 sin 7

+n,,cos~cos~tb22sin~sin~ + b33 sin&

sin -j-tQ,$OS 3*y

%osY I

(20)

R.G.

~DW~D

the extension to take into account the spread of plasticity is discussed. Elastic buckling HoIe edge stresses obtained from the plane stress finite element analysis are compared with available theory of elasticity results on Fig. 5; due to &heuse of constant stress elements, some averaging and extrapolation was necessary to do this. Circumferential stresses u@in Fig. 5 are plotted along the line AE, shown in Fig. 4, and represent average values mid way between the element centroids (points B,C,D and E). Extrapolation to values at the hole edge is i~erently d~cuIt because of the high stress gradients in this region. However, edge stress results are available from Wang[ci] for the smaller holes (d/l = 0.15, 0.3, 0.5) and it is clear that extrapolation to values close to Wang’s results is feasible. These results were obtained from a limited number of stress functions, which satisfy the boundary conditions approximately around the external boundary, and are limited to small holes of size up to d/l = 0.5. No results for larger holes are available for comparison. Elastic buckling coefficients for the clamped and simply supported cases are shown on Fig. 6, and are compared with a number of other theoretical, and some

I

37rx + u33cos--cos I

37ry I+

4lrx . 4?ry sm I .

ba sin I

The terms are listed in eqns (19) and (20) in order of decreasing con~bution to the convergence of the critical stresses. It can be seen that errors in the critical stress are reduced to under 2% for both cases when the number of terms is 5. Thereafter little improvement is evident for increasing numbers of terms in the series.

Analyses were performed of square plates under uniform edge shear stress containing centrally located circular holes, as shown in Fii. 2(b). Holes with diameter d from 0.151 to 0.91 are considered, where 1 is the length of the plate edge, as well as the limiting case with no hole, Plate thickness is also a variable which is considered. The finite eiement analysis for the in-plane stresses was performed on one quarter of the plate. A typical idealization of this is shown in Fig. 4. This same mesh was used with varying amounts of distortion to accommodate all of the hole sizes. It should be noted that this stress analysis is independent of the flexural boundary conditions on the plate, and consequen~y one analysis serves for the buckling analysis of either the clamped or simply supported case. The deflection functions were chosen such that for the simply supported case M = 1,2, n = 1,2, and for the clamped case, m = n = 1,2,3,4 for a,,,_,,,,_, and b,,.. These were chosen after consideration of the convergence of the case with no hole, and correspond in both cases to an eight term series. It has been assumed throughout that the same form of deflection function is appropriate to the case with or without the hole, and for elasto-plastic behaviour. In the following, results of analyses of elastic buckling of the square shear plate are first discussed and are compared with previous elastic analyses. Subsequently,

Fig. 4.

Typicalfinite element mesh.

FEM Average

Ekrsticity Sotutloi

Z

5

t G

A

B

c

D

E

Fig. 5. Circumferen~al elastic stresses on line A-E (see Fig. 4).

Elasto-plastic

shear buckling of square plates with circular holes

295

k

6

Authors

-+-Rockry 0

et al (F.E.M)

Rockeyet

-.-Yoshiki

al (Exp~) et al (Energy)

0

Yoshiki

.

Kroll

et al (Exp.) (Energy)

A Allmon

(F.E.MM)

Classical Solution Ref. (4) a(5) _

4

-

0

0.1

Simply

Suppotrd

0.2

0.3

Plate

0.4

0.5 Hole

0.6

0 7

Size

0.8

0.9 d 7

1.0

Fig. 6. Variation of elastic buckling coefficients with hole size. experimental,

results.

Theoretical

and experimental

stu-

dies of the elastic buckling of square perforated shear plates have been carried out by Rockey et a[.[71 and by Yoshiki et a1.[8]; theoretical studies have also been presented by Kroll[9] and by Allman[ 101. Kroll[9] and Yoshiki et al. [8] use energy methods with the in-plane stress distribution based on results from the theory of elasticity, which restrict the application to relatively small holes. These studies are further limited to cases of simple support. Agreement between these results and the results obtained herein is good for small holes, when d/l ~0.25. For larger holes, the method of Yoshiki predicts’s higher buckling coefficient than predicted herein. Both Yoshiki’s results and the present work show excellent convergence to the classical solution for the case of no hole as the hole becomes small. Rockey et al. [7] and Allman[ 101have obtained critical loads for the square shear plate with both clamped and simple supports by using the finite element method directly. Of these, the later work of Allman used a more sophisticated element which showed superior results for the case of the unperforated plate, however he gives very limited results for plates with holes, only d/l = 0.5. The results of Rockey et al. comprehensively cover the range of d/l = 0 to 0.5. For d/l = 0.5, the method presented herein gives buckling coefficients which are close to the results of both of these investigators, as shown on Fig. 6; for small holes, on the other hand, considerable divergence occurs between the authors’ and Rockey’s results. In view of the relatively crude element used in the latter study and the excellent agreement with the classical solutions for the unperforated plates given by the element used by Allman and by the method presented herein, these may be preferred over the analysis of Rockey et al. for small holes. Results of carefully performed experiments are reported by Rockey et al. in Ref.[7]. These were based on tests of mild steel square plates with clamped boundaries

and with slenderness ratios I/t about 200. For holes with d/l >0.25, yielding was observed to precede buckling, and since the present discussion relates to elastic buckling, such results are not shown in this Figure. Results where buckling was elastic are shown on Fig. 6, and it can be seen that these results, while showing considerable scatter, predominantly lie close to the finite element results given in the same paper. While this result appears to conflict with the conclusion reached from theoretical considerations, the method by which the experimental buckling loads were determined deserves close attention. The difficulty in this determination is well known and arises because a true bifurcation seldom if ever occurs, and instead the behaviour is dominated by initial imperfections, postbuckling strength and the possibility of non-linear material behaviour. Various experimental methods have been critically examined (see for example Shestedtll] and Mori and Matoba[l2]), but the choice of a particular method remains somewhat arbitrary. The method used by Rockey et al. was to obtain directly from a load vs lateral deflection plot, the intersection of two well defined tangents. One of these corresponded to prebuckling loads, and the other to higher loads representing a transition to the post-buckling characteristic. Of the various possible methods, this corresponds to the one which will predict the lowest value of the critical load; of the others, one which might be preferred is that in which the load is taken as corresponding to the inflection point in the transition part of the curve. Clearly this would predict a higher value than those given in Ref.171. In view of these uncertainties which must be associated with the experimental determination of buckling loads, it is considered that the experimental results of Ref.[7] shown on Fig. 6 do not serve to indicate which of the two curves relating to clamped boundary conditions is the more correct. In fact, the experimental points probably underestimate the critical load, and also because of

M. UENOYAand R. G. REDWOOD

2%

the superior convergence to the unperforated plate solution given by the upper curve, this might be preferred. Additional experimental evidence is provided by Yoshiki et al.[8], who tested short beams under central loads, with holes in both shear spans as shown in Fig. 7. While there was a resulting small bending moment, the predominant loading was shear. The web of each half of the beam consisted of a square panel with a central hole and the boundaries were attached to flanges or web stiffeners. These were sufficiently slender that the plate boundaries were considered to be simply supported.

plate thickness t. Thus each calculated critical load corresponds to a particular plate thickness, and the resulting relationships between critical loads r,, expressed as a proportion of the yield stress in shear TV,are given in Figs. 9 and 10 for the simply supported and clamped cases respectively. t-

Fig. 8(a).

Fig. 7. Test arrangementof Yoshiki et al. (Ref. [S]). The determination of critical load was from a plot of load vs the square of lateral deflection. The intercept with the ordinate of the tangent to this curve at the higher deflections was taken as the critical load. Four results are shown on Fig. 6 for d/l = 0, 0.2, 0.5 and 0.8, and fairly good agreement with the predicted results is evident. The analysis predicts that the buckling coefficient for the clamped plate increases with hole size when the hole exceeds d/l = 0.75. While the reason for this is not fully understood, there is a possibility that for such large holes, a higher buckling mode, which is not described by the deflection series used, predominates over the fundamental mode considered, with lower critical loads resulting. Experimental and theoretical evidence of this in the case of clamped square perforated plates under uniaxial compression has been given by Kumai[l3], and for this loading condition, this occurred for plates in the range d/l > 0.33. Elasto-plastic

buckling

The procedure in this case was to carry out the stepby-step finite element analysis, and for each step in which the yielded zone had been extended to embrace additional elements, the critical load was calculated. The progression of yielding is illustrated in Fig. 8, which shows one-eight of the plate, and it can be seen in the case of the smaller holes the yielded zones differ considerably from the case of larger ones. For the smaller holes, the capacity of the plate is reached when yielding spreads from the hole edge to the nearest point on the plate edge, whereas for the larger holes, the narrow strip of plate is flexing in double curvature, and two sections yield simultaneously approximately at the quarter points of the sides. For the critical load calculated at any step, the stresses given by a finite element analysis can be matched to the corresponding yield pattern by suitable choice of the

Q

0.500

@

0.580

@

0.550

/I

@I 0.660 0 0.650

Fig. 8(b). 7 @

T 0.275

@ @

0.325 0.375

@

0.400

8I7

0.425 0.450 0.463

Fig. 8(c).

Elasto-plastic shear buckling of square plates with circular holes

@

0.215

@

0.218

Fig. 8(e). Fig. 8(d). Fig. 8. Spread of Plasticity (a) d/l = 0.15 (b) d/l = 0.3 (c) d/f = 0.5 (d) d/l = 0.7 (e) d/l = 0.9.

-

Elaato-ptoatie

---

Etadic

-

6uckIIng

Initiation d T v 0 A 0.165 o 0.370 CJ 0.556

A 0.2 l

Ruckling

of

Yielding

1

Mori a Matobao2)

.Yoahiki et a](8)

0.5

Fig. 9. Elasto-plastic buckling of simply supported square panels.

297

298

M. UENOYA and R. G. REDWOOD

0.8

-

Ela~ic-~~a~ic Buckling

-.-

Elastic Buckling

-

lnltlatlon of Yltldtng

A

G

I

I

2

3

,

4

5

I

,

t

6

8

9

I

IO

II

Fig. 10. Elasto-plasticbucklingof clampedsquare panels. As analysed, the spread of yiekiing takes place as finite areas, comprising one or more finite elements, yield under each load increment. For stocky plates, which do not buckle before a complete yielding failure occurs, the accuracy of evaluation of the collapse load is therefore dependent on the size of the load increments. The horizontal lines on Figs. 9 and 10 represent collapse loads which are the average of the last load for which convergence of the finite element stress analysis occurred and the subsequent load for which convergence was not attained. It can be noted from Fiis. 9 and 10 that for d = 0.151 essentially full yielding on the minimum cross section is attained (r/r, = 0.848), and for larger holes the collapse shear stress falls increasingly below the full yielding value as the holes get bigger. For large holes (d/l > 0.70), initiation of yielding was followed so closely by yielding collapse that no meaningful results could be obtained for the intermediate (elasto-plastic) range of slenderness ratio. These may therefore be assumed to behave in an elastic-perfectly plastic manner. The buckhng modes are ~ustrated by means of deflection contours on Figs. 11 and 12. Figure 11 shows results for the simply supported plate with d/l= 0.50 for two load levels, one in the efastic range and one corresponding to a load near failure, that is, well into the plastic range. The mode shape is very similar for elastic and elasto-plastic buckling, the difference being in a slightly more elongated wave form in the latter case. Similar results were obtained for clamped boundary

conditions and are shown in Fig. 12 for a larger hole. In addition to the experimental investigations of elastic buckling previously referred to, Mori and Matoba[l2] have performed experiments to determine the shear buckling loads of square, simply supported, perforated steel plates in the plastic range. These results are shown on Fig. 9, and while considerable scatter is evident, they conform quite well to the predicted curves. Exceptions to this are some results for plates with no hole, for which failure occurred at stresses considerably in excess of the yield stress in shear. This may be due either to strainhardening, or the development of post-buckling strength. These results represent stocky plates which nearly approach their yielding collapse loads, whereas the results of Yoshiki et al.[8]are also shown and which were observed to be elastic, have slenderness ratios such that their buckling appears to take place close to the initiation of plasticity. No experimental results are known which provide information in the range of slenderness intermediate to these tests. For clamped pIates, the only experimental results available appear to be those reported by Rockey et al. [7] in which some yielding was observed prior to buckling. These are plotted on Fig. 10, and like the elastic buckling test results reported in the same reference, the critical loads are considerably lower than the predicted values. Also, it is apparent that, in most cases, elastic buckling is predicted for these plates. The reason for the observed yielding cannot be determined, but the comments made above concerning the elastic buckling results of this test

Elasto-plastic shear buckling of square plates with circular holes

+

r s 0.7

7;;

=

0 1.00

(a) B------b--__z

I I I I -II -“----

.z 8 0.450 ‘Y fb) Fig. 11. Buckling modes-simply supported plate (a) elastic: (b) elasto-plastic. +

4 CONCLUDING REMUtKg

Using in-plane stresses obtained from an elasto-plastic finite element analysis, critical loads have been obtained by an energy method for a square perforated plate under uniform edge shear stress. The form of the displacement function and its convergence were examined with reference to an unperforated elastic plate, for which close agreement with previously published theoretical results were obtained. Critical stresses for elastic buckling of plates with varying size central holes were obtained and compared with results of several previous studies of this problem. Vol. 8, No. 2-J

-c-----v-

0.5

series, apply equally to the inelastic results. In the latter case, yielding will only serve to further reduce the critical load ob~~ned in the manner described in Ref. (71.

CM

in

tb) Fig. 12. Buckling modes-clamped plate (a) elastic; (b) elastoplastic.

The range of hole sizes considered is extended by this study to include larger holes up to about 41 = 0.75 and, in addition, the results suggest that the critical load for small holes is higher than previously thought. There is other evidence to support this in the case of simply supported plates. For elasto-plastic buckling, critical shear stresses are given for different sized holes over the full range of appropriate plate slenderness. Experimenta results for cases of simply supported plates support the analytical results, whereas verification for clamped plates remains inconclusive on account of limited reliable test data. The method described is efficient in terms of computer operations, especially in that the stress analysis is independent of the plate boundary conditions and each load increment in the plastic range provides a result valid for a particular plate slenderness.

M.uENOYh andR.G. &XlWDDD

300

Acknowledgements-The work described in this paper was supported by the National Research Council of Canada, Grant A-3366. Grateful acknowledgement is also made to Hiroshima University for partial support of one of the authors.

-cEs

1. T. Kawai and hf. Ohtsubo, A method of solution for the complicated buckling problems of elastic problems of elastic plates with combined use of Rayleigh-Ritz’s procedure in the finite element method. Froc. 2nd Air Force Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, AF~L-TR~l50,%7-994 (Oct. 196Q. 2. Y. Fujita, K. Yoshida and H. Arai, Ins~b~ty of plates with holes (2nd report). (in Japanese). 1 Sot. Naval Arch. Japan, 126,285-294 (Nov. 1969). 3. 0. C. Zbnkiewicz, S. Valliappan and I. P. King, Elastoplastic solutions of engineering problems; initial stress finite element approach. ht. 1. Num. Meth. Engng 1, 75-100 (1%9). 4. M. Stein and J. Neff, Buc~ng stresses of simply supported rectangular Rat plates in shear. N.A.C.A. T.N. No. 1222 (1947). 5. B. Budiansky and R. W. Connor, Buckling stresses of clamped rectangular flat plates in shear. N.A.C.A. T.N. No. 1559 (1948).

6. C. K. Wang, Theoretical analysis of perforated shear webs. J. Appl. Mech. Trans., A.S.M.E. l3 (2), A-77-A-84 (June 1946). 7. K. C. Rockey, R. G. Anderson and Y. K. Cheung, The behaviour of square shear webs having a circular hole. Symp. on Thin Walled Steet Structures, University College of Swansea (Sept. l%7). 8. M. Yoshiki, Y. Fujita, A. Kawamura and H. Arai, Ins~bility of plates with holes (1st report). (in Japanese). J. Sot. Nauui Arch. lapan, 122, 137-145(1%7). 9. W. D. Kroll, Instability in shear of simply supported square plates with reinforced hole. I. Res. Natl. Eur. Stand. 43, 465-472(Nov. 1947). 10. D. L. Allman, Calculation of the elastic buckling loads of thin flat reinforced plates using ~~~ finite elements. Int. 1. Num. Meth. Engng 9,415-432 (1975). Il. 3. She&d, Experimental buckling criteria for thin pierced plates. Thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Civil Engineering, Rice University, Houston, Texas (May 1971). 12. M. Mori and M. Matoba, Experimental study on the shear buckling of pieced square plates in plastic range. (in Japanese). J. Sot. Naval Arch. West Japan, 33, 259-284 (1966). 13. T. Kumai, Elastic stability of the square plate with a central circular hole under edge thrust. Proc. 1st Japan Nat/. Congr. Appl. Mech. 81-i% (195I).