The influence of complex initial deflection modes on the behaviour and ultimate strength of rectangular plates in compression

J. Construct. Steel Research 5 (1985) 265-302

The Influence of Complex Initial Deflection Modes on the Behaviour and Ultimate Strength of Rectangular Plates in Compression Y. U e d a * a n d T. Y a o t *WeldingResearch Institute, Osaka University, 11-1 Mihogaoka, Ibaraki, Osaka 567, Japan. tFaeulty of Engineering, Hiroshima University, Hiroshima, Japan

SYNOPSIS Detailed theoretical analyses have been performed on the elasto-plastic large-deflection behaviour of rectangular plates with complex modes of initial imperfection loaded in compression. The influence of the initial deflection on the performance of the plate, especially on the ultimate strength has been clarified. In the case of thin plates, one component of the initial deflection which becomes stable above the buckling load plays an important role in the collapse, and in the case o f thick plates, the maximum curvature o f the initial deflection influences the plastification process below the buckling load. Based on these observations, two methods, a deflection method for thin plates and a curvature method for thick plates, have been formulated and they have been shown to predict the ultimate strength o f plates with very good accuracy.

1 INTRODUCTION O n e of the most c o m m o n types of structure is a box girder such as in a ship's hull, or certain types of bridge. Generally, these box-girder type structures are in service under lateral loads, and their deck plates, which are the main structural members, are subjected to in-plane tensile and/or compressive loads. The strength of the deck plates under compression is increased by the use of stiffeners, which are attached by fillet welding. 265 J. Construct. Steel Research 0143-974X/85/$03-30(~) Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

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Y. Ueda, T. Yao

Consequently, the deck plate exhibits initial imperfections comprising residual stresses and distortion due to welding. These imperfections often reduce the strength and rigidity of the plate. When the ultimate strength of the deck plate is considered, knowledge of the contribution of these subdivided plate elements is essential. Many investigations have been performed both theoretically and experimentally into the compressive buckling and ultimate strengths of unstiffened and stiffened plates with initial imperfections. In this paper, attention will be restricted to unstiffened plates. The effects of welding residual stresses on elastic buckling strength were studied by Yoshiki et al. l and Okerblom. 2 Their effect on elastoplastic and plastic buckling strengths have been studied in a series of theoretical and experimental investigations carried out at Lehigh University, USA, and reported by Ueda, 3 Nishino, 4 Ueda and Tall, 5 Nishino et al., 6and others. The same problem has also been investigated at Cambridge University. 7.s In the above investigations related to buckling strength, analytical methods were employed for theoretical analysis. It is almost impossible, however, to calculate ultimate strength analytically, since both geometrical and material nonlinearities must be considered. For such problems, the finite element method is useful. Application of the finite element method to elasto-plastic large-deflection analysis has been performed by Bergan, 90htsubo, 10Arai 11and Ueda et al. 12Applying the finite element method, the compressive ultimate strength of plates has been calculated by Ohtsubo, to Komatsu et al., 13the authors, i,.-~7and others, taking into account the influence of initial imperfections due to welding. Recently, Dow and Smith ~sdiscussed the effect of localized imperfections on the compressive ultimate strength of plates. An alternative method for evaluating the compressive ultimate strength of rectangular plates has used the concept of an effective width. Early work by Schnadel, 19Schumann and Back, ~° Karmfin et al. 2' has been followed more recently by Dwight and Moxham 2: and Becker et al. 2~-~s Formulae for calculating the compressive ultimate strength have been summarized by Faulkner. 26 The authors have also investigated this problem. ~4-~7.27-3~In this paper, the effects of initial imperfections due to welding are considered and the compressive ultimate strength of rectangular plates is discussed, based on results of elasto-plastic large-deflection analysis by the finite element method. First, the ultimate strength is calculated assuming a single mode

267

Behaviour of rectangular plates in compression

of initial deflection. Existing modes of initial deflection have been measured on 33 panels of ships' deck plates, and the general features of these modes are illustrated. Adopting the measured modes of initial deflection, the strength of rectangular plates with multi-mode initial deflection is calculated, and the elasto-plastic large-deflection behaviour up to collapse is investigated. From these results, the characteristics of the behaviour of the plate are studied and the influence of the initial deflections on the ultimate strength clarified. Based on this information, methods are proposed for the evaluation of ultimate strength with multimode initial deflections. The accuracy of the methods is confirmed by comparing the results with the finite element evaluations.

2 LONG RECTANGULAR PLATES CONSIDERED IN THE ANALYSES

2.1 Long rectangular plates and boundary condition A plate element from a stiffened plate assemblage is considered in the analyses. The plate has an initial deflection which is generally of a complex mode accompanied by welding residual stresses, as illustrated in Fig. 1. To represent the assemblage continuity the plate is assumed to be simply supported along all edges (i.e. no out-of-plane deflection and no constraint moments along its four edges). All edges are assumed to remain straight while subjected to in-plane movements. Y

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2.2 Initial d e i k c t i o n o f deck plates

Many measurements have been carded out on various locations in ship structures. 32'33Based on the measured data and actual practice, some quality standards including a limitation of the maximum initial deflections were published. 33-z Few measurements, however, have been carried out as to the mode of the initial deflection. 37.~ In this paper, a series of measurements of the mode of initial deflection is described which has been performed on the deck plates of ships. The measurements were carried out on 21 panels of the upper deck plate of a bulk carrier of 60 000 tons (dead weight), and 12 panels of the upper and main deck plates of a car carrier with 5500 cars on board. The aspect ratio, a/b, of the panels is between 2.65 and 4.41 and the breadth-to-thickness ratio, b/t, between 23-19 and 97.50, as indicated in Table 1. The initial deflection, w0, may be expressed in the following general form, with reference to Fig. 1, w0 = ~

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values of m from 1 to 11. The results are summarized in Table 1 and some examples are plotted in Fig. 2. It should be pointed out that the coefficients of o d d terms are greater than those of even terms in most cases. This results in the hungry-horse mode of initial deflection. These initial deflections may be idealized as illustrated in Fig. 3. For such idealized modes of initial deflection, the coefficients of the deflection c o m p o n e n t s in eqn (3) are calculated taking m as o d d integers up to 21. T h e results are also plotted in Fig. 2 (O) and summarized in Table 2. It may be said that the idealized initial deflection demonstrates the averaged features of the measured ones, except for panels n u m b e r e d 9 and 12 of the car carrier.

2.3 Weidiag residmfl stresses The residual stresses in the plate are produced by fillet welding of the stiffeners along the edges y = 0 and y = b, and their distribution is assumed to be rectangular, which may be valid when the continuity condition between the plate elements of the stiffened panels is considered. If the residual stresses are assumed to be self-equilibrating in the

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2bttr, = ( b - 2b,)o-~

3 METHOD OF THEORETICAL ANALYSIS It is well known that an initially flat plate undergoes a primary buckling under compression. After buckling takes place the lateral deflection increases. If the plate is imperfect, the lateral deflection increases from the beginning of loading, but the rate of increase of deflection is small until the load is near the buckling load. To analyze such behaviour, the geometrical nonlinearity must be taken into account. For a further increase of load, plastification gradually takes place, and the plate finally reaches its ultimate strength. Material nonlinearity must also be taken into account during this phase of behaviour. For the analyses in this paper, therefore, the incremental form of the fundamental equations of the finite element method is used, derived by considering both material and geometrical nonlinearities. The detailed process of derivation of the fundamental equations is given in References 12 and 15.

4 D E F O R M A T I O N AND PLASTIFICATION OF A SHORT R E C T A N G U L A R PLATE IN COMPRESSION WITH SIMPLE MODES OF INITIAL DEFLECTION When performing the elasto-plastic large-deflection analyses, the compressive ultimate strength of a short rectangular plate is calculated for the following initial deflection

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4.1 Initial deflection of single mode, one half-wave, (A02 = 0) By assuming an initial deflection of single mode in a half-wave, and changing the magnitude of the initial deflection, A 0,, ultimate strengths have been calculated for aspect ratios less than 1.2. When a plate is perfectly flat (A0, = 0), it buckles in one half-wave and deflects in this same mode until collapse. The ultimate strength is shown for A 0Jt equal to 0-01, 0.25 and 0.5 in Fig. 5. It is well known that the buckling strength of a simply supported long rectangular plate under uni-axial compression is a minimum for integer aspect ratios. However, the ultimate strength is not necessarily a minimum for these aspect ratios. The aspect ratio giving the minimum ultimate strength is approximately 0.7 of the integer multiple for the size of plate shown in Fig. 5. This aspect ratio increases up to 1.0 times an integer multiple as the slenderness ratio, b/t, decreases. 39This depends also on the magnitude of the initial deflection. 29 4.2 Initial deflection of single mode, two half-waves, (A,, = 0) Ultimate strengths have been calculated assuming an initial deflection of single mode in two half-waves, for aspect ratios larger than 0-8, and the results are also shown in Fig. 5. Although the buckling mode should be

276

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one half-wave, the ultimate strengths of the plates with the initial deflection of two half-waves is lower than that of the plate with the same maximum magnitude of the initial deflection of one half-wave, for aspect ratios larger than approximately 1.0.

4.3 Initial ~ of double mode, t'ombimdion of o ~ and two lullwaves, (A0, ~ A02 ~ 0) The initial deflection of eqn (5) is assumed, and the following two specific cases, (A) and (B), are analyzed. (,4) A o J t = A J t = O . O l , a / b = 1-36and 1-38 Taking a/b as 1-36 and 1-38, the ultimate strength is calculated for the same initial deflection, and is shown in Fig. 5. The ultimate strength decreases suddenly when the aspect ratio changes from 1.36 to 1.38. The result indicates the possibility of a sudden change of ultimate strength with respect to a/b.

Behaviour of rectangularplates in compression

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(B) Aodt = A d t = 0-1, a/b = 1.25 Ao~lt = 0.5, AoYt = 0-1, a/b = 1.25 Keeping the same aspect ratio of 1-25, two kinds of shape and magnitude of initial deflection are used. The maximum deflection of the first is smaller than that of the second which has a larger two half-wave component. Their calculated ultimate strengths are also shown in Fig. 5. In the first case, the plate deflects and collapses in two half-waves in the loading direction, while in the second case, it collapses in one half-wave. The selection of the stable deflection mode depends on the magnitude of the components of the initial deflection and their ratios. This ratio is defined as the critical ratio of the components of initial deflection. 27'28 Although the second plate has a larger initial deflection, its ultimate strength is higher than the first. From these examples, the following deductions can be made: 1. The deflection mode at buckling does not necessarily correspond with the mode producing the minimum ultimate strength. 2. A larger initial deflection does not necessarily reduce the ultimate strength more; the reduction depends on the magnitude and shape of the initial deflection. The initial deflections in actual plates are complex as is their behaviour to collapse. In the following sections, these problems will be investigated in detail.

5 D E F O R M A T I O N AND PLASTIFICATION OF LONG R E C T A N G U L A R PLATES WITH M U L T I - M O D E INITIAL DEFLECTIONS

5.1 Perfectly-elasticlarge-deflectionbehaviour W h e n a long rectangular plate with multi-mode initial deflection is subjected to compressive loading, its behaviour is somewhat different from that of a short one. This section discusses this behaviour. The effect of welding residual stresses is not taken into consideration and it should be noted that the calculated nodal displacements of the finite

Y. Ueda, T. Yao

278

elements are interpolated and the deflection is expressed as the following series w

. zry, = /..,X7Amsin m l r x sm a b

(6)

where values of m are taken from 1 to 11 in this study. First, an elastic large-deflection analysis is presented performed on the panel no. 6 of the car cani'er, which is one with a common mode of initial deflection as shown in Table 1. The relationship between the nondimensionalized compressive mean stress divided by the elastic buckling stress, o-/tr~, and the non-dimensionalized coefficients of the deflection components divided by the thickness, A d t , are represented by dashed lines in Fig. 6a. As this figure indicates, it should be noted that if the mode of initial deflection, the ratio of the maximum magnitude of initial deflection to the plate thickness and the aspect ratio of the plate are specified, the elastic large-deflection behaviour of the plates is described uniquely in these non-dimensionalized coordinates, regardless of the size. Figure 6b shows changes of the modes of deflection and curvature along the centreline, y = b / 2 , of the plate. All component modes of the initial deflection increase until the load approaches the buckling one, but all except the five half-wave mode diminish above the buckling load. In this case, the five half-wave mode becomes stable, though the aspect ratio of this plate is 4-41, and the buckling mode is four half-waves in the loading direction. It is known that the stable deflection mode of an initially-deflected plate above the buckling load is often higher than the buckling mode by one or two degrees, depending on the mode and the magnitude of initial deflection, v.~.40.4~

5.2 Eiasto-plastic large-deflection behaviour of thin plates Next, the elasto-plastic large-deflection analysis is shown of the same plate as above, taking the yield stress, (rv, as 280 N/mm 2. The breadth of the plate is 780 mm and the thickness is chosen as 8 mm, 15 mm and 34.5 ram. The relationship between the non-dimensionalized mean compressive stress and the coefficients of the deflection components are illustrated by solid lines in Fig. 6a, and the spread of the plastic zone together with the modes of deflection and curvature in Fig. 6c.

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280

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Behaviour of rectangularplates in compression

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In the case of the thin plate 8 mm thick, the compressive stresses do not increase at the central portion of the plate when deflections become large above the buckling load. This is due to the tensile membrane stresses caused by the large deflection. Consequently, the compressive load is sustained mainly near the supporting edges, y = 0 and y = b, where the deflection is small. For further load increments, the deflection mode is stable and initial plastification takes place at the supporting edge. The plastic zone spreads as the load increases, and the plate finally collapses when plasticity has spread along the edge. The maximum magnitude of the initial deflection does not influence the ultimate strength, only the magnitude of the mode corresponding to the stable deflection. 5.3 Elasto-plastie large-deflection behaviour of thick plates In contrast, in the case of a thick plate 34.5 mm thick, initial plastification takes place at a certain point along the centreline, y = b/2. At this load, as the deflection is small, the mode is almost the same as that of the initial deflection, and the effect of large deflections is not apparent. Therefore, the in-plane stress is almost the same all over the plate, though the bending stress varies depending on the curvature. Consequently, initial yielding takes place at the point where the maximum bending stress, i.e. the maximum curvature, is produced. This point is usually located along the centreline of the plate just near the point where the curvature of the initial deflection is a maximum. For further increases of load, the plastic zone spreads from the centreline, and the compressive load does not increase in this zone. Consequently, the additional compressive load is carried near the supporting edges, and the plate finally collapses when these edge portions become fully plastic. In this case, the modes of deflection and the location of the maximum curvature at collapse are almost the same as for the initial deflection. Accordingly, the maximum magnitude of initial deflection does not control the ultimate strength, the latter being dependent on the maximum curvature of the initial deflection.

5.4 Elasto-plastic large-deflection behaviour of plates of medium thickness In the case of a plate with a medium thickness of 15 mm, initial plastification takes place at a certain position along the centreline during the

282

Y. Ueda, 7". Yao

course of a mode change from the initial to the stable mode. Although the fundamental behaviour is almost the same as for the thick plate, this interaction between the component modes may accelerate or decelerate the plastification and this greatly influences the resulting ultimate strength.

6 COMPRESSIVE ULTIMATE STRENGTH OF R E C T A N G U L A R PLATES WITH VARIOUS KINDS OF INITIAL DEFLECTIONS 6.1 Classification ofbeimviour to

In the previous section, the deformation and plastification of a long rectangular plate with various types of initial deflection was studied. The ultimate strength may be evaluated more easily by classifying the type of behaviour according to the initial deflection as follows. 1. Uni-modal initial deflection whose mode is the same as that of the buckling mode (this includes the case of no initial deflection). 2. Uni-modal initial deflection ofdifferent mode to that of the buckling mode. 3. Multi-mode initial deflection for a thin plate. 4. Multi-mode initial deflection for a thick plate. 6.2 Ultimate strength for a uni-modal ~ same as that of the budding mode

deflection whose mode is the

Taking the plate width, b, the thickness, t, and the yield stress of the material, o-v, as 1000 mm, 12 mm and 280 N/mm 2, respectively, the compressive ultimate strength is calculated, by the elasto-plastic largedeflection analysis of the finite element method. For a uni-modal initial deflection, where the mode coincides with the buckling mode, the ultimate strength is represented by solid and chain lines in Fig. 7, with the deflection mode stable up to collapse. As shown in Fig. 7, abrupt changes exist in the ultimate strength curves at the aspect ratios where the buckling modes change. For example, the buckling mode changes from one half-wave to two half-waves at an aspect ratio of ~/2.

Behaviour of rectangularplates in compression

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285

6.3 Ultimate strength with a uni-modai initial deflection of different mode to that of the budding mode The uni-modal initial deflection is specified and the deflection mode is assumed to be stable until collapse. The magnitude of the initial deflection should satisfy the requirement relating to the critical ratio of initial deflection components in order that the specified mode is stable. In Fig. 8, the ultimate strength is represented by thin solid lines, and the minimum ultimate strength from those obtained for the same maximum magnitude of uni-modal initial deflection is represented by bold solid lines. The lowest ultimate strength is almost constant for a specified magnitude of initial deflection, irrespective of the aspect ratio.

6.4 Minimum ultimate strength For a plate with complex modes of initial deflection, the minimum ultimate strength must be conservative. It may be estimated by the methods discussed in the following sections. A series of elasto-plastic large-deflection analyses have been performed, taking account of initial deflection and welding residual stresses. The magnitude of the tensile residual stress, trt, in Fig. 4 is taken as the yield stress, try. The calculated minimum ultimate strengths are plotted against the non-dimensional parameter, (b/t)X/(trv/E), in Figs 9a, b and c, in which the magnitude of residual stresses varies, 2b Jb being 0.0, 0.1 and 0.2, respectively. In these figures, the minimum ultimate strengths are represented by the solid lines and expressed by the following simple formulae, which were derived by applying the method of least squares to the calculated values. (a) 2bJb = 0.0 (i) 0.8 <_ ~:_ 2-0 o%/trv = (-2-431,12 + 1.6826-0- 0-2961)(~2- 4-0) + (7.2745-02- 4.7431-0 + 0.6709)(s~- 2.0) + z~ z l = (-0.3597-02 + 0.1748-0 + 0.8598)/(2.2432r~ + 1.3322) + 0-0373-0+ 0.2481

(7) (8)

(ii) 2.0 < s~ <--3.5 o ' J t r v = (-0.3597-02 + 0.1748-0 + 0.8598)/(~ + 2-2432-0 - 0.6678) + 0.0373-0 + 0-2481

(9)

0.6 0.4

= I

_

~ -.....~.

,

0 ; FEManalysis (wo/t = 0.01) ; FEM analysis (wolt • 0.25) 0 ; FEManalysis (wo/t = 0.5) Buckling strength/~ - - ; 6qs.(5) and (7) fly ; Yield stress

0

I

I

I

0.5

1.0

1.5

1.0

~

0-u/0"Y

°6

~

I:L

~

I

I

"2.0

.?.5

~,~--

I

3.15

3.0

bit / ~

O;

~,\

~

fErt analysis (w~/t : 0 . 0 1 ) FEH analysis (wolt=O.l) O ; FEH analysis (wo/t = 0.2)

A;

"~,

~-~.(B,~l._~ ~

b = lO00 mm O'y = 28 kq/r~ 2

o, T

,

t

~i

"-

0.2

•

strenqth

ls

2!o

' 8ucklin(l

"

free .. "~-,... .- . ~/ U. . C KIIn 9 from reslauat stresses Str_en~ ; Ultimatestrenqth free from iresidual stresses (Eqs.(5) and ( 7 ) ) I

o

o s

lo

~.\,

0"ul(TY wdt=

O.

2!5 A

"'-~, D ~,.,0

3'ob / t

,r0"~

3t

FEM analysis(w./t-0.1)

FEM analysis(wolt= 0.2) FEM analysis (wdt-O.3)

O'y = 28 kg/mm)

]~ H r ~ ~ L, ~ ~ L '~- ,l°_ . _ 11~ ~ °~IlNI /JLL-~lr~u~,~J~

I 0

I

~

a

,

|

0.5

'

"lI

l.O

' ; Eqs.(ll)and (13)

\\'. ,ockl,., .ren,t. free ,romresidoa,stres,es

....

BUCsktrenltl ni ~ h

-.

Ultimatestrengthfree from residual stresses(Eqs.(5)and (7))

• I

1.5

!

2.0

I

2.5

"~

Jl

3.0

I

3.5

b/t

Hg, 9. Minimum compressive ultimate strengths of rectangular plates with uni-modal imperfections. (a) Rectangular plates free from residual stresses. (b) Rectangular plates with residual stresses (2bdb = 0.1). (c) Rectangular plates with residual stresses (2bdb = 0.2).

Behaviour of rectangular plates in compression

287

(b) 2 b d b = 0.1

(i) 0 - 8 ~ ~-< 1"6 tr,/trv = (-0"398*/2 + 0.4339~ - 0.1342)(~ 2 - 2.56) + (1.0814./2 - 0-7551,/+ 0.1020)(~ - 1.6) + z2 z2 = (0"4974*/2 + 0.8281./+ 1.0171)/(2.7942./+ 1.2908) 0.1849-0 + 0-1571 -

(10) (11)

(ii) 1-6 < ~: --< 3.5 trJtrv = (0"4974*/2 + 0.8281"0 + 1-0171)/(~: + 2.7942"0 - 0.3092) 0.1849~ + 0.1571 -

(12)

(c) 2 b d b = 0.2

(i) 0.8 <_ ~: - 1.5 trJtrv = (-0"3317./2 + 0.6314./- 0.2656)(~ 2 - 2.25) + (0.5369-02 - 0-7798*/+ 0.2854)(~ - 1.5) + zs z3 = (0"292*/2 + 1.29360 + 0.7471)/(2.897./+ 1-1189) - 0.2715~ + 0-2057

(13) (14)

(ii) 1"5 < ~: < 3-5 o'Jo'v = (0"292*/2 + 1.2936,0 + 0.7471)/(~: + 2.897*/- 0.3811) - 0.2715./+ 0.2057

(15)

where = (b/t)x/(o'v/E) n = wo/t

(16)

(17)

T h e s e equations will be used later in Section 7.5 to predict conservative u l t i m a t e strengths for rectangular plates with multi-modal initial deflection.

7 METHODS FOR PREDICTING THE ULTIMATE STRENGTHS OF LONG RECTANGULAR PLATES WITH UNI- AND MULTI-MODAL INITIAL DEFLECTIONS

7.1 Equivalent rectangular plates In this section, simple methods will be proposed to predict the compressive ultimate strength of a long rectangular plate with multi-modal initial deflections. An equivalent rectangular plate with uni-modal initial

288

Y. Ueda, T. Yoo

deflections is considered, which exhibits the same behaviour as that of the collapsing portion of the rectangular plate with the multi-modal initial deflections. Taking the same width and thickness as the original plate and denoting the length, width, thickness and the maximum magnitude of initial deflection of this equivalent rectangular plate by a0, b, t and A0, respectively, the initial deflection is expressed as w0 = A0sin rrx sin"~'y a0 b

(18)

In the following sections, two different methods will be proposed to determine the length, a0, and imperfection magnitude, A0, of the equivalent plate; the 'deflection' method for thin plates, and the 'curvature' method for thick plates.

7.2 Deaection method (for tbln ldstes) As described in Section 5.1, for a thin plate only a particular deflection mode becomes predominant and stable above the buckling load, depending on the aspect ratio of the plate, and the mode and magnitude of initial deflection. It is postulated that an equivalent plate may show almost the same behaviour as the original if only a component of stable deflection is assumed as an initial deflection and all other components neglected. This method is developed paying attention to the stable deflection mode above the buckling load, and will subsequently be called the 'deflection method'. The calculation has been performed on panel no. 11 of the car carrier. Figure 10a shows the relationship between the mean compressive stress and the coefficients of the deflection components. The solid line represents the results of the actual panel (Case A). In Case A, the collapse of the plate takes place at a section near the right-hand side of the loading edge, where the length of one half-wave is the shortest (Fig. 10b). T w o equivalent plates, Case B and Case C, are considered. They have the same m a x i m u m magnitude of initialdeflection, A0, which is equal to the coefficient of the fifthdeflection component, A o5,of the actual initial one (Case A). In Case B, the length of the equivalent plate, a0, is one half-wave length of the stable deflection mode, which is exactly one-fifth of the original plate length, a/5. In Case C, the length, a0, is equal to the length of one half-wave, I, at collapse, which is measured from the right-hand side edge.

Beimviour of rectangularplates in compression

289

Ao3 Ao. Ao~ Ao6 Ao7 AOB Ao9 Aom Aou 1 '.~ Aol Ao2 CASEA -0.497-0.338 0.339-0.159 0.200-0.252 0.154-0.032-0.031 -0.120 0.0171 CASE B

0.0

CASE A,

0.0'"

O.O

0.0

0.200 0.0

a

0.0

"i

.o:

I~ b: 780nml

t

0.0

0.0

0.0

( CASEA, CASEB) Initial deflection :

CASEB

y~..._

0.0

!lh-I ~

(Aomsin-)sinZ

Deflection underthrust : w: ( ~Amsin-~- ) sin-~

X

CASEC

(a)

Wo=AoSsin-~ sin-~ 0.51 0.4 A

°if

A~L'~0.1

(b)

Deflection :w=Assin~sin_~C underh trust : CASEA . . . . : CASEB ~ - ~ : CASEC O A [:3 : Initial yieldin9

A

O.

I~., -0.2

Z~A

0

Deflection n~desat collapse ~

' 0.2

~

' 0.4

' ///"~ '"

' 0.6

' 0.8

I '.0

'

' 1.2

" ,,z'"""I

t l .4

I'.6 1~8 Am/t

[ ] : Plastification from face [ ] : Plastification fromback I : Full section plastic

1~. 10. Behaviour of actual and equivalent thin plates with initial deflections. (a) Relationship between deflection coefficients and mean compressive stress. (b) Spread of plastic zone at collapse (Case A).

290

Y. Ueda, T. Yao

The elasto-plastic large-deflection analysis is performed on these two equivalent plates (Cases B and C), and the results are represented in Fig. 10a by dashed and chain lines, respectively. It should be noted that the following two relationships are the same in both cases; the relation between the mean compressive stress and the central deflection, and the relation between the mean compressive stress and the coefficient of the fifth deflection component, A~, of the actual plate, except for ultimate strength. The ultimate strength of the actual plate (Case A) is the lowest, that of Case C is close to this, and that of Case B is the highest. For accurate estimation, as for Case C, the length of the equivalent rectangular plate should be taken as the length of one half-wave of the deflection mode where the collapse is observed, but for simplicity with fairly good accuracy (Case B) one half-wave length of the stable deflection mode may be used. In order to estimate the stable deflection mode, an elastic largedeflection analysis at least, on the actual plate with complex initial deflection, is necessary. In order to avoid this analysis, the following method is proposed to obtain an approximate ultimate strength: (a) The length of the equivalent plate is assumed to be either one half-wave length of the buckling deflection mode, one mode higher, or two modes higher than the buckling one, with the maximum initial deflection equal to the coefficient of the assumed component of the initial deflection. (b) The ultimate strength is calculated by the elasto-plastic large-deflection analysis on those three equivalent plates. (c) The lowest ultimate strength of the three plates can be regarded as most appropriate. This method has been applied to twelve panels of the car carrier and three panels, nos. 13, 14 and 15, of the bulk carder. The resulting ultimate strengths are shown in Fig. 11 compared with the exact ones (as calculated). The predictions are very accurate.

7.3 Curvature method (for thick plates) In the case of thick plates, initial plastification takes place just near where the curvature of the initial deflection takes its absolute maximum value. The plastic zone spreads with further load increase, and the plate finally collapses with almost the same deflection mode as the initial deflection. Considering such behaviour, the length, a0, and the maximum magnitude of initial deflection, A0, of an equivalent plate will be determined by

Behaviour of rectangularplates in compression

291

U”C : Calculated ultimate

1.0 l-

strength

fJ"'uc/$

U"P

: Predicted ultimate strength

0.8 -

0.6 -

0.4 -

a

0.2 -

/

b

t

V

: 344Ox78Ox8r1nx

0

: 344Ox78Oxllnun

A

: 28OOx8OOx15rm1

/ 1

0

0.2

I

1

I

0.4

0.6

0.8

J

1.0 (J"V$

Fig.

11.Comparison of calculated and predicted compressive ultimate strengths of thin plates (deflection method).

relating them to the maximum magnitude of the curvature of the initial deflection. Therefore, this method will be called the ‘curvature method’. When the initial deflection is expressed by eqn (3), its curvature in the x-direction along the centreline becomes pax’

a*%, lip,, =

=

-

-$xm*A,,*inF

(19)

If the absolute maximum value of the curvature is obtained at x = x0, it is expressed in the following form

a*wo

( 1 --2-

=

(1IPIJmax = c (UP,,),

= x,,

w

“ax

where

(21)

292

Y. Ueda, T. Yao

Curvature method I For the equivalent plate, the same deflection mode may be used as that of the maximum component of the maximum curvature of the initial deflection, and should indicate the same sign as the maximum curvature. One half-wave length of this deflection component is taken as the length of the equivalent plate. If this component is the kth, the length, a0, should be a/k. Next, the maximum magnitude of initial deflection, A0, is determined so that the maximum curvature of the initial deflection of the equivalent rectangular plate is equal to that of the actual plate considered. From eqn (18), the maximum curvature of the equivalent plate is -Ao(cr/ao) 2, for x = ad2 and y = b/2, and this should be equal to (1/p0)m,. This yields Ao= -

and a o = a / k

(22)

Curvature method H There exists two consecutive points where the curvature is zero in the collapsing portion of the plate. The point of maximum curvature is located between these. In this method, the distance between these two points, l, is taken as the length of an equivalent rectangular plate. These two points are represented by G and F in Fig. 12. The maximum magnitude of initial deflection, A0, is taken as -(ll~r)2(llpo)~, so that the maximum magnitude of initial curvature is the same as that of the original plate. In summary, they are A0 = -(l/¢r)2(1/po)m,

and a o = l

(23)

To confirm the validity of curvature methods I and II, several analyses have been performed on panel no. 1 of the bulk carder, assuming the maximum magnitude of initial deflection equal to half the plate thickness. In Fig. 12, Case A corresponds to the actual panel, and Case B and Case C to the equivalent rectangular plates of curvature method I and curvature method II, respectively. For this panel, the predominant curvature component is the 7th, and the length of the equivalent rectangular plate is taken as a/7 in curvature method I. It should be noted that the relationship between the mean compressive stress and the curvature is almost the same for all cases. Contrary to this, the relationship between the mean compressive stress and the deflection

Behaviour of rectangularplates in compression a

I_

F-

CASE A

•

........................ . .....

~ ......

--.

.---,,

G ,

: Deflection : Curvature

.....

b : 800mm t : 34.5mm (CASE B)

J

I

, F

"T

,

CASE B

P = (llPo)max

(CASE C)

q:p

q:p

ao = a / 7

ao : Z : CASE A

--. . . . o,

~l

293

CASE C

: CASE C : Relative deflection, 6, in CASE A

yielding

A, o : Initial

l.o

r,

0.8

;aod C

r

!

0.6

il

0.4

il

0.2

,

,

4.0

2.0

Curvature

,

0 ( x I0-")

0.05

J~L O.l

,

0.i5'b.4 0.45 0.5 w/t

Fil . 12. B e h a v i o u r o f actual a n d e q u i v a l e n t thick p l a t e s w i t h initial d e f l e c t i o n s .

differs. However, the curvature exerts more effect on plastification than the deflection, and the ultimate strength is almost the same in all cases. In Fig. 12, the chain line with two dots represents the relative deflection, 8, of the actual plate, which is measured with respect to the line between points G and F. It might be worth noting that this deflection is very close to that of the equivalent rectangular plate calculated by curvature method II.

Y. Ueda, 1". Yao

294 1,0

%C/y

(Tuc : Calculated ultimate strength

O'uP : [ [ ; : ~ ; : : d

~I timate

A

A ~ -

r~-

~"

0.8

..............

0.6

0.4 a

0.2

0 (a) 1.0

CuC/Cy

b

t

z~

: 2800x 800 x 15 mm

~

: 2100x 800 x 19ram

0

: 2800x 800x 19 mm

I::1

: 2400x 800x 34.5 mm

,t

i

!

I

0.2

0.4

0.6

0.8

,I

~uPl~l.0

O'uc : Calculated ultimate strength

E ~

O-uP : Predicted ultimate strength .$~.~I'0"~ 0.8

Ell

.........

.......

1E]

0.6

0.4

/ 0.2

0

z~

a b t : 2800xSOOx15ram

~ O D

: 2100 x 800 x 19ram : 2800 x ~ x19 n~ : 2400x 8O0x 34 .5 •n

'

I

I

i

,,~t

0.2

0.4

0.6

O, '~

1,0

CTuP/(TY

(b)

Fig. 13. Comparison between calculated and predicted compressive ultimate strengths o! thick plates. (a) Curvature method I. (b) Curvature method II.

Behaviour of rectangularplates in compression

295

Applying these two methods, the compressive ultimate strengths of the panels of the bulk carrier are calculated. The results are plotted in Figs 13a and b, and compared with the accurate values (as calculated). From these figures, it may be said that both methods predict the ultimate strength fairly well, especially curvature method II.

7.4 Predictions for plates of medium thickness For plates of medium thickness, the mode of the initial deflection may change as the deflection increases. In this process, progress of plastification may be accelerated or decelerated by interaction between the component modes. Therefore, prediction of the ultimate strength by the 'curvature method' may be lower or higher than the actual one. In contrast, in the 'deflection method', local plastification is not allowed for until the stable deflection mode is attained. The prediction, therefore, should be higher than the true behaviour. As will be shown in Figs 15a and b, both prediction methods are not very good for plates of medium thickness, but may be applicable. The average of the two results should be an improvement on their individual predictions.

7.5 Estimation of ultimate strength using the formulae giving the minimum strength of the plates In the previous sections, it was necessary to perform an elasto-plastic large-deflection analysis on an equivalent rectangular plate. However, if such an analysis cannot be performed, eqns (7) to (15) may be used. For this, the plate parameter, ~: = (b/t)X/(trv/E), eqn (16), is determined from the plate dimensions. In the case of thin plates, the magnitude of initial deflection, w0, in eqn (17) is taken as the maximum value of the coefficients of the components of the initial deflection corresponding to the buckling mode, or one or two higher modes. In the case of thick plates, w0 is determined according to curvature method II (or I). Then, 71 =wo/t (eqn (17)) is determined. With these values of sr and 7/, the ultimate strength may be calculated although the results may be conservative because of the nature of the approach. For the various panels indicated in Table 1, the compressive ultimate strength (ignoring residual stresses) has been calculated using eqns (7),

296

K Ueda, T. Yao 1.0

%c/o"v

O'up : Predicted ultimate strenath CT u

p " I~~

: Calculated ultimate

strenoth

0.8

@,

~,~

~

A~/"

[]

A"

E3r, ....

-, ....

I ~11

"-7

r

0.6

Ell, .......... a

,f l

I ~

,,~"

,IE3._i_ i

"1

/

0.4

: 3440x 780xBmm

0 : 3440x 780x II mm 02

A

' 2ROOx800x lSmm

'~

:

0

: 2~00x800x

[]

: 2 4 0 0 x 8 0 0 . x 3 4 . 5 n•r,

I

I

I

0.2

0.4

o..6

2100x800x

,

lgnln

lgn~:~

I

I

0.8

l.o

ffuPl O-v

Fig. 14. Comparison of calculated compressive ultimate strengths with those predicted by the simple formulae.

(8) and (9). The values are plotted in Fig. 14 and compared with the calculated ultimate strengths using the finite element approach. It can be seen that all the predicted ultimate strengths show good correlation with the calculated values.

7.6 Limit of application of the approxfmmte methods Applying both the deflection and curvature methods to the plates ot Table 1 produces ratios between predicted ultimate strengths and calculated strengths which are plotted against the non-dimensionalparameter, (b/t)x/(trv/E) in Figs 15a and b. The deflection method is better than the curvature method for thin plates, while the curvature method is better than the deflection method for thick plates. However, for plates of medium thickness both methods are not very good. Considering the physical background behind the methods, the limit of application of both methods may be taken as (b/t)X/(orv/E)= 1"9. This dimension corresponds to the transition from elastic to plastic buckling of a square

Behaviour of rectangularplates in compression

297

1.2

CuP/Cuc l.l 0 o

o

8

l.O Cuc : Calculated ultimate strength

0.9

Cup : Predicted ultimate strenqth

0.8

Plastic buckling

Elastic bucklinQ

[ L [-

l!

J

a

-I

I

I

I

I

I

I

I

I

0.5

l .0

l .5

2.0

2.5

3.0

3.5

4.0

b/t-V~

(a)

l.l

CuPWuc

o o

§

l.O ~ c : Calculated ultimate strength 0.9

~P : Predicted ultimate strength

o

o

o o o

0.8 Plastic buckling m

0.5

Elastic buckl in o

i

)

I

I

I

I

I

1.0

1.5

2.0

2.5

3.0

3,5

4.0

b/t'/y~

(b) Fig. 15. Applicability of prediction methods. (a) Deflection method. (b) Curvature method.

298

Y. Ueda, T. Yao

plate. In the range (b/t)x/(trv/E) >- 1.9, the deflection method should be applied, while the curvature method should be applied for (b/t)x/(trv /E) <- 1.9.

8 CONCLUSIONS In this paper, the compressive ultimate strength of a rectangular plate with initial imperfections due to welding is investigated using elastoplastic large-deflection finite element analyses. The following conclusions can be drawn. 1. The behaviour of the elasto-plastic large-deflection response of plates may be summarized as follows. All component modes of the initial deflection increase until the load approaches the buckling load, irrespective of the size of the plate. In the case of thick plates, below the buckling load, the position where the largest curvature of the initial deflection is located controls the spread of plasticity, leading to collapse. This maximum curvature has more effect on ultimate strength than the maximum deflection. When the plate is thin, the deflection increases further before plasticity occurs. Above the buckling load, only one mode component of the initial deflection is amplified and becomes stable as the deflection increases. This mode is not necessarily the buckling mode, but is usually one or two modes higher. Only the component of the stable deflection mode influences ultimate strength. This component is usually significantly smaller than the maximum initial deflection. 2. The aspect ratio of the single half-wave of a deflection mode which gives the minimum ultimate strength does not coincide with that of the minimum buckling strength. This aspect ratio is usually less than 1-0. The ultimate strength of a rectangular plate shows a saw-tooth appearance with respect to the aspect ratio of the plate when the deflection corresponding to the buckling mode is maintained up to collapse. Simple formulae are proposed for the calculation of a conservative value of ultimate strength for rectangular plates with initial deflection and welding residual stresses.

Behaviour of rectangularplates in compression

299

3. The in-plane rigidity of the plate decreases as the load increases. Increases of compressive load are not supported by the central part of the plate due to large deflection in the case of thin plates, or due to the effect of plastification by bending in the case of thick plates. 4. The mode of initial deflection has been measured on 33 panels of the deck plates of two ships as indicated in Table 1, and the coefficients of the initial deflection modes, expressed by sine series, have been calculated, taking the components up to the 11th term. 5. Two methods have been proposed to predict the compressive ultimate strength of rectangular plates with multi-modal initial deflections. These methods are called the 'deflection method' for thin plates and the 'curvature method' for thick plates. The usefulness of these methods is confirmed by comparing their predictions with accurately calculated ultimate strengths.

ACKNOWLEDGEMENT The authors wish to express their sincere gratitude to K. Nakacho, Osaka University, Y. Tanaka, graduate student, Osaka University and K. Handa, graduate student, Hiroshima University for their assistance in the measurement of the initial deflections of ship structures and for analysis of the results.

REFERENCES 1. Yoshiki, Y., Fujita, Y. and Kawai, T., Influence of residual stresses on the buckling of plates, J. Soc. Naval Arch. of Japan, 107 (1980) 187-94 (in Japanese). 2. Okerblom, N. O., The influence of residual stresses on stability of welded structures and structural members, Commission X. IIW, Li/~ge, 1960. 3. Ueda, Y., 'Elastic, elasto-plastic and plastic buckling of plates with residuai stresses', Ph.D. Thesis, Lehigh University, USA, 1962. 4. Nishino, F., 'Buckling strength of columns and their component plates', Ph.D. Thesis, Lehigh University, USA, 1964. 5. Ueda, Y. and Tall, L., Inelastic buckling of plates with residual stresses, IABSE, Zurich, 1967. 6. Nishino, F., Ueda, Y. and Tall, L., Experimental investigation of the buckling of plates with residual stresses, Symposium on Test Method for Compression Members, Atlantic City, 1966, and ASTM, Special Technical Publication No. 419, 1967, pp. 12-30.

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