Computer generated elasto-plastic design data for pressure loaded circular plates

ecus-7949/91$3.00 + 0.00 Q 1991 Civil-cOmp Ltd and Ftrgmon Press pk

Computera ci Structures Vol. 41, No. 6.99. 1329-1340, 1991 Printed in Great Britain.

COMPUTER GENERATED ELASTO-PLASTIC DESIGN DATA FOR PRESSURE LOADED CIRCULAR PLATES G. J. TURVEYand M. SALEHI Department of Engineering, University of Lancaster, Bail+,

Lancaster JAI 4YR, U.K.

Ahatrac-Two finite-difference computer programs for the non-axisymmetric elasto-plastic large deflection analysis of circular plates are described. One of the programs uses a constitutive model based on the Ilyushin full-section yield criterion, whereas the other uses the von Mises layered yield criterion. The programs have been used to model the full-range response including multi-stage unloading, of uniformly loaded circular plates with various boundary conditions for a range of slendemesses. From the computed results design charts of 6rst yield over-pressure versus maximum and residual centre deflection have been compiled. A numerical example of the use of these charts in the design of an end closure to a cylindrical pressure vessel is presented. Finally, sample charts for ‘pre-dished’ plates are presented as an illustration of the further potential of these programs for design data generation.

M-L

NOTATION

WI i?e

000

e,wh E ;T;I &o k0 .-* 8, rl) &MB,& N,,N,,N, 4 f ;=zqr;E-‘h,4)

, , r0 i

(=rr;‘)

u, v W

KJ(=wh;‘)

{

I(#lE-1/2)

(=rohi

I 00 @,r,fJB,Ql# (I?(

1

extensional stiffness matrix flexural stiffness matrix strain vector plate mid-plane strain components Young’s modulus yield function plate thickness curvature vector plate mid-plane curvature components stress couples stress resultants transverse pressure dimensionless transverse pressure polar co-ordinates plate radius dimensionless plate radius radial and circumferential plate mid-plane displacement components plate deflection dimensionless plate deflection dimensionless plate slenderness increment of variable plastic strain multiplier yield stress stress components derivatives with respect to r and 0, respectively

Subscriptslsuperscripts c e

CL-IP-FI F-S

variable associated with the plate centre elastic part of variable initial value of variable maximum value of variable plastic part of variable residual value of variable first yield value of variable

data for clampled-in-plane 6xed edge plate data derived from Ilyushin fullsection yield constitutive model

SS-IP-FI

data derived from von Mises layer yield constitutive model data for simply supported-inplane fixed edge plate

INTRODUCITON

The end closure of a cylindrical tank or storage vessel may, for the purpose of its structural design, be

idealised as a flat or ‘dished’ uniform thickness circular plate subjected to uniform pressure loading. In consequence, the analytical and/or numerical solution of this idealised design problem is of practical significance. Exact solutions for uniformly loaded circular plates with simply supported and clamped edges are given in Timoshenko and Woinowsky-Krieger’s classic text [l]. Similar circular and annular plate solutions are presented in the form of practical design formulae in Roark’s text [2]. These solutions/ formulae, which have enjoyed widespread application to design, are really only valid when the plate deflection is small (less than about one quarter of the plate thickness) and the plate material retains its original elastic integrity at maximum pressure. These two restrictions imply that an end closure design based on the formulae in (1,2], i.e. on the basis of small deflection response, is likely to be unnecessarily thick. If, in the interests of achieving improved structural efficiency and greater economy in the use of material, a smaller plate thickness is desirable, then design on the basis of small deflection response has to be abandoned. When relatively thin plates are used deflections in service and at maximum pressure may be of the order of or greater than the plate thickness. In these circumstances, elastic large deflection analysis provides an alternative basis for design. Indeed,

1329

1330

G. J. TURVEY and M.

design on this basis results in the use of thinner plate sections due to the transverse stiffening effect of ‘membrane action’ which increases as the plate deflection increases. Exact and approximate elastic large deflection solutions for uniformly loaded flat circular plates have been developed by several researchers during the course of this century and many of them are presented in a format suitable for design application in Aalami and Williams’ text on the design of elastic plated structures [3]. This design data might usefully be extended to the design of ‘dished’ circular plates by making use of the affine transformation technique described by Nylander [4]. Although design based on elastic large deflection analysis leads to the use of thinner plate sections, when compared with design based on small deflection analysis, the full thickness reduction benefits may not always be realisable in practice simply because there is no guarantee that the elastic integrity of the plate is preserved when the deflection is large. The loss of elastic integrity, caused by yielding of the plate material, produces a reduction in transverse stiffness, so that a somewhat larger plate thickness, than predicted by elastic large deflection theory, is required to support the pressure loading and keep the maximum plate deflection within the design limits. There is, therefore, a need to develop elasto-plastic large deflection design data to supplement the purely elastic data of [3,4]. The authors have recently developed two computer programs for the non-axisymmetric elasto-plastic large deflection analysis of circular plates. The programs utilise the Dynamic Relaxation (DR) method in conjunction with graded finite-differences to obtain numerical solutions of the plate equations. The programs differ from one another primarily in the form of the elasto-plastic constitutive relations employed. One uses constitutive relations based on the Ilyushin full-section yield criterion and the other constitutive relations based on the von Mises layer yield criterion. Although the programs were primarily developed for studying non-axisymmetric elasto-plastic large deflection circular plate response, they may readily be used to model the simpler axisymmetric response. Thus, the purpose of the present paper is to outline the underlying theoretical and numerical background to the development of the programs and to describe their use in producing the supplementary design data alluded to above, ie maximum and residual deflection versus uniform pressure data, for a range of plate slendernesses and edge support conditions. The practical application of the design data is also illustrated via a simple numerical example. The paper is concluded with the presentation of ‘sample’ design data for ‘pre-dished’ circular plates subjected to uniform pressure loading in order to illustrate the further practical potential of these computer programs.

SALEHI

PLATE

GEOMETRY

AND LOADING

SITUATIONS

The geometric and co-ordinate details of a uniformly loaded axisymmetric circular plate are shown in Fig. l(a). In the figure simple support edge conditions are shown for illustrative purposes only. The loading magnitude and the corresponding plate deformation state at the onset of yielding, which in this case arises in the lower surface at the plate centre, is shown in Fig. l(b). The plate centre deflection, w:, at first yield may be either small or large depending on the physical slenderness, r,,/h,, of the plate. Irrespective of whether the deflection is small or large, in this situation the plate is elastic and design data for the computation of w{ is readily available in, for example, [2,3]. However, if the load on the plate is increased beyond the yield value, qy, to some maximum value, qmax , then the most highly stressed points yield (see shaded region of the plate cross-section shown in Fig. l(c)) and the plate responds elasto-plastically. At the maximum pressure, qmax,the centre deflection increases to WY. This latter deflection may only be determined accurately from an elasto-plastic large deflection analysis. Whilst a few such analyses of uniformly loaded circular plates have been undertaken, see for example [S], no serious attempt has been made to present the output from such analyses in a design useable format. If, subsequently, the maximum pressure loading is removed from the plate, then the maximum plate deflection, WY, reduces to a residual value, WY, which is sometimes referred to as the ‘permanent set’ deflection. Here again, a few analyses of the ‘springback’ or residual deflections that may arise in circular

‘PLASTIC

Fig. 1. Simply supported circular plate: (a) details of geometry and loading, (b) deformation state at first yield, (c) deformation state at maximum pressure, (d) deformation state after unloading.

Elasto-plastic design data for circular plates

steel plates have been undertaken (see [6-g]), but these analyses have been concerned with metal forming and stamping applications for which both the loading and plate edge conditions differ from the present application, viz. storage vessel end closures. As yet, no design data are available for the latter application.

1331

and the mid-plane curvature and twist components are: k; = -_w” ki = -r-‘(r-lw”

kf+ = -2r-‘(w*’

NON-AXISYMMETRIC ELASTO-PLASTIC LARGE DEFLECTION PLATE EQUATIONS

The two programs, as mentioned earlier, were originally developed in order to study non-axisymmetric large deformation plate response, but are used here only to generate design data for axisymmetric response. It is, therefore, appropriate to present the full set of non-axisymmetric plate equations first and then to describe later how they are used to produce degenerate axisymmetric results. It is convenient to present the equilibrium, strain/curvature and constitutive equations separately, especially as their separate identities are preserved in the DR solution algorithm. Equilibrium equations The large deflection circular plate equilibrium equations have been taken from [9] and may be expressed as follows: N;+r-‘(N,-N,)+r-‘N&=0 N,+r-‘Nh+2r-IN,=0

+

w)

- r-‘w’).

(2b)

Elasto-plastic incremental constitutive equations

The stress resultants and stress couples of Eqs. (1) and the strain and curvature components of Eqs. (2) are related via the plate constitutive equations. Moreover, because the material state changes from elastic to perfectly plastic as the pressure on the plate increases, it is necessary to use an incremental form of the constitutive equations. Two sets of constitutive equations have been developed-one for the Ilyushin full-section yield criterion and the other for the von Mises layered yield criterion. The latter set is the more accurate, since it is capable of modelling the spread of plasticity through the thickness of the plate (which is ‘notionally’ divided into a small number of layers) as the loading is increased. In contrast, the constitutive equations based on the Ilyushin criterion, though rather simpler, require that, at any radial position, the plate section is either entirely elastic or entirely plastic. The intermediate elasto-plastic state is not accounted for when the Ilyushin full-section constitutive model is used. Ilyushin full-section yield criterion and incremental constitutive equations

x (2M; + 2r -‘MA - MB) + N, w” +N,(r-

Iw. +r-

2

The Ilyushin full-section criterion may be expressed in terms of the stress resultants and stress couples as follows:

wII )+2N,,

x(r-‘w.‘-r-2w’)+q=0.

~=a+e,+cvJr,f2,,=

(1)

The first two of Eqs. (1) describe the in-plane equilibrium in the radial and circumferential directions respectively and the third describes the equilibrium in the transverse or z-direction (see Fig. l(a)).

in which 0” = N;‘(N;

em=

The direct and shear strain components of the plate mid-plane are given as:

- v) + 0’ + r-‘w,w’

M;‘N,‘(M,N,+

MBN,,-O.S(M,N,

and

ei = r-‘(u + v’) + (l/2)r-2w’2 es = r-‘(u’

+ N; - N, NO+ 3N;)

&, = Mo2(A4; + M: - M,MO + 3M;)

Strain and curvature equations

ef = u’ + (1/2)wa2

N,, = a,, h, (24

1

MO= (1/4)a,h;

(3)

G. J. TURVNand M, SALBHI

1332

The well known elastic isotropic thin plate constitutive equations may be written in incremental polar format as,

in which 1, the plastic strain multiplier, is positive for plastic straining to occur. It is readily shown that 1 may be evaluated as:

(AN) = [A](Ae’) (AM) = [D]{Ak’}

(4) Von Mises layered yield criterion and incremental constitutiveequations

in which ~~~=~~~};

The von Mises yield criterion is expressed in terms of the direct and shear stress com~nents as:

~A~~={~~};

F = a,2(a: + a; - (T,bgf 36;) = 1. i-l={::}:

(9)

Following a procedure similar to that outlined under the preceding section, the plastic strain increments are evaluated as,

iAkj={$};

and (AeP> = 1 (cW/C%} [A] = Ek,(l vz)-Ir

x

(li,w

and the incremental is,

-1

elastic stress-strain

(10) relationship

[D] = (~k~~lZ)[~].

{Arr) = [A](Ae’)

If, however, under any particular load increment the plate yields, then the incremental strains and curvatures, (de) and {Ak 1, at the yield locations must consist of both elastic and plastic components. Hence, the elastic components arising under the incremental loading may bc expressed as,

and, moreover, recognising that the total strain increments can be separated into elastic and plastic components (see first of Eqs. (5)), then using Eqs. (10) the alternative form of Eq. (11) becomes,

(11)

(Au j = [A](Ae - 1(W/&)).

(12)

{Ace} = {de} - {AeJ’} {Ak”) = (Ak) - {AkJ’).

(5)

Now the plastic strain and curvature increments may be evaluated using the Prandtl-Reuss associated flow rule and the derivatives of the Ilysuhin yield function as: (AeP) = A(F,,)

Furthermore, from the requirement that during plastic straining the plastic stress increments must remain on the yield surface, i.e. 6F = 0, it may be shown that the expression for the plastic strain multiplier becomes,

Hence, substituting Eq. (13) into Eqs. (12) the layered form of the incremental constitutive equations becomes, {Au) = [AP]{Ae)

in which

(14)

and

Hence, on substituting Eqs. (6) into Eqs. (5) and the result into Eqs. (4) the final form of the incremental constitutive equations becomes:

[#]=[A]-

[~l~(~F~a~)~ ~~~F/~~)~‘~~ 1 f(aF/aa)~‘[Al((aF/aa))

*

Boundaryconditions

(AN} = [A]{Ae - IF,,} (AM) = [D](Ak - nr;,>

in which

(7)

Four extreme cases of edge boundary conditions have been selected for the generation of design data.

Elasto-plastic design data for circular plates They are: simply supported-in-plane lixed and free and clamped-in-plane 6xed and free. These four sets of edge conditions imply that the following constraints are imposed on the stress resultants etc. at the edge, r =ro (a) simply supported-in u=t)=

-plane fixed

(154

w =M,=O

(b) simply supported-in-plane

free

N,=N,=w=M,=O (c) clam~d-in

-plane jxed u=l_J=

(d) clamped-in

Wb)

w=w’=O

(154

-plane free Nr=Nd=w

=w*=O.

Because use is made of a non-axisymmetric analysis to model an axisymmetric deformation state, it is also necessary to impose symmetry conditions along the two radial ‘edges’ in the quarter plate analysis (see Fig. 2(a)). In addition, appropriate symmetry and antisymmetry conditions are enforced at the plate centre. NUMERICAL SOLUTION OF THE PLATE EQUATIONS

In order to facilitate the numerical solution of the governing system of plate equations, i.e. Eqs. (l)-(9) and (13)-(15), they are first of all discretised using graded interlacing finite-differences and then rearranged to suit the application of the Dynamic Relaxation (DR) algorithm. No details of these procedures are presented here, but the interested reader may consult [lo, 111, which respectively provide detailed accounts of the application of the DR procedure to the solution of elastic axisymmetric and non-axisymmetric circular plate bending problems.

However, a few remarks concerning the particular tinitedifference mesh used for the numerical computations are in order. The mesh arrangement, ie four interlacing meshes extending over a quadrant of the plate, is shown in Fig. 2(b). The distribution of the problem variables (stress resultants etc.) between the nodes of the four meshes is also shown in Fig. 2(b). In the numerical analysis no use was made of the mesh grading capability. All computations were made using equi-spaced mesh points. In the radial direction 8f sub-divisions were used and only two sub-divisions were used in the circumferential direction. For axisymmetric analysis, the size of the mesh sub-division in the circumferential direction is unimportant and for convenience the relatively large value of A0 = 45” was used. In order to model the through-thickness spread of plasticity in the plate using the von Mises layered yield constitutive model, it is necessary to introduce a ‘notional’ layering or sub-division of the plate thickness. Five interior layers bounded by upper and lower half-thickness face layers was found to be a satisfactory arrangement. Indeed, trial computations with significantly more ‘notional’ layers produced only a slight change in the elasto-plastic load-deflection response. In order to improve the stability and convergence of the DR algorithm, use was made of a unit time increment in conjunction with fictitious densities, as originally advocated by Cassell and Hobbs [12]. Details of the fictitious density expressions are given in [ll]. SCOPE AND ORGANISATION OF THR DESIGN DATA

The two DR computer programs have heen used to model the complete elasto-plastic loading and unloading response of uniformly loaded circular plates. They incorporate the facility to unload from any value of the applied pressure as well as a restart facility using the elasto-plastic solution corresponding to the unloading pressure. The latter feature allows computations for the second and subsequent

EOGEBOUNOARYCONOITIONS lE~sl1511 APPLY

CONDITIONS

1333

APPLY 0

:

w, “r, N,,Hr, “,

. (‘3 (4 Fig. 2. plate quadrant for axisymmetric analysis: (a) geometric and boundary condition details, (b) finite difference mesh details. ”

G. J.

1334

TURVEY and

pressures to be made more efficiently. Thus, the programs have been used to model the pressur&eflection response including multiple unloading stages. From these response curves maximum and residual deflections are readily abstracted for each unloading pressure. This type of data, as already pointed out, is not presently available to designers, but is essential for the rational design of circular end closures. In order to minim% the computation required to generate the circular plate design data, a non-dimensional approach was adopted. Advantage was taken of the fact that most of the plate material and geometric properties may be combined into a single parameter, /I, which takes the form

(16) For steels of various types, the variation in the ratio of the yield stress to the Young’s modulus is relatively small when compared with the variation in the radius to thickness ratio. Hence, fl is usually thought of as a non-dimensional slenderness parameter. It may be shown for steel plates that the practical range of /?* varies from’0.4 + 3 or 4. Clearly, this is quite a small range and may be exploited in as much as it is only necessary to compute the complete pressure-deflection response (including multiple unloading states) for a small number of 8’ values in order to characterise the complete range of variation of the design data. Hence, the circular plate pressure-deflection response was modelled for three values of fl*, viz. 0.4, 1.5 and 3, corresponding to stocky, intermediate and slender plates respectively, and for each set of edge conditions. From this mass of computer data the relevant maximum and residual deflection design data was abstracted. The next step was to decide how to present this data. A non-dimensional format is obviously to be preferred and recognising that the plate response is

M. SALEHI

elastic up to first yield and, moreover, that unloading from any pressure less than or equal to the 6rst yield pressure produces no residual deflection, it was decided to nondimensional%. the applied pressure with respect to the first yield pressure computed on the basis of small deflection theory. The following simple expressions apply to simply supported and clamped plates irrespective of the type of in-plane restraint, gy = (8/3)fl*(3 + v)-’ (simply supported)

(17a)

& = (4/3)j2(1 -v

(17b)

+ v*)-“* (clamped)

The deflections, maximum and residual, are nondimensionalised with respect to the plate thickness. Thus, the design data are conveniently presented in graphical format for each of the /I2 values as first yield over-pressure, q/g,,, versus centre deflection, 6’c (maximum or residual) in Figs. 3-8. In each of the figures the two curves defined by solid and open rectangles represent the data based on the Ilyushin full-section yield model and the pair of curves defined by the solid and open circles represent the corresponding data based on the von Mises layered yield model. The use of open and solid symbols on these curves denotes maximum and residual deflection data respectively. For convenience the design data is arranged such that Figs. 3-5 apply to plates with in-plane fixed and Figs. 6-8 to plates with in-plane free edges. Moreover, part (a) of each figure applies to plates with simply supported and part (b) to plates with clamed edges. DESIGN DATA-GENERAL

OBSERVATIONS

There are a number of general features of the first yield over pressure-maximum and residual deflection design data presented in Figs. 3-8 which merit detailed comment. Firstly, if one examines the maximum deflection data (open circles and rectangles) for

3.0

2.5

4 9

2.0 r

1.0 0 (4

I.”

)

0.50

0.i5 w c

h 0

~-

0.00

c

_r

0.25

~

(

~~

,

0.75

0.50

l.dO

1.;5

W

(b)

c

h

0

Fig. 3. PressurAeflection design data for uniformly loaded stocky circular plates with full in-plane edge restraint (8’ = 0.4; Y = 0.3), (a) simply supported, (b) clamped.

1335

Elasto-plastic design data for circular plates

l.Oi 0.0

0 W _c h 0

(a)

:

.Il-L.

!

OLI_L

1.0

1.5

0.5

0.0

W c h

(W

1.0

1.5

Fig. 4. Pressuredeflection design data for uniformly loaded intermediate circular plates with full in-plane edge restraint (/I* = 1.5; v = 0.3), (a) simply supported, (b) clamped.

0.0

0.k

1.b

1.;

:

W __C h 0

(4

cW

@I

h

0

Fig. 5. Pressurtieflection design data for uniformly loaded slender circular pIates with full in-plane edge restraint (J2 = 3; v = 0.3), (a) simpiy supported, (b) clamped.

3.0

2.5

s 6.

2.0

1

1.5 wF_S. fir-5.

.u 1.0 (

0.5

(4

Fig. 6. Pressure-deflection

1.0 "c h 0

1.5

1.0 0

2 @)

0.5

1.0

1.5

2.0

:

Wc h

0

design data for uniformly loaded stocky circular plates with in-plane free edges CjY2 = 0.4; v = 0.3), (a) simply supported, (b) clamped.

G. J.

1336

0.0

0.5

1.0

1.5

2.0

2.5

TIJRVEY

and M. SALEHI

3.0

0.0

w,

0.5

1.0

1.5

2.0

:

W

_c

h

h

0

Fig. 7. Pressure-deflection design data for uniformly loaded intermediate circulir plates with in-plane free edges (8’ = 1.5; v = 0.3), (a) simply supported, (b) clamped.

clamped plates, ie the (b) parts of Fig. 3-8, it appears that irrespective of the type of in-plane edge restraint there is but little difference between the Ilyushin full-section and the von Mises layered yield values. The differences are greatest for stocky plates (B* = 0.4). A similar observation may be made in respect of intermediate and slender simply supported-in-plane fixed and slender simply supported-in-plane free plates, but not for the remaining categories of simply supported plate (see Figs. 3(a), 5(a) and 6(a)). Indeed, for simply supported-inplane free plates of intermediate slenderness corresponding Ilyushin and von Mises design values differ significantly. At very high first yield over-pressures there is some evidence (see Figs. 7(a), 7(b) and 8(a)) that the Ilyushin constitutive model leads to larger maximum deflection predictions than the von Mises model. Turning now to the residual deflection or ‘springback’ design data (see solid circles and rectangles on Figs. 3-8), it is evident that the differences between the Ilyushin and von Mises residual deflection data

4.0

5. 4

"

/ 1

are generally greater than the corresponding differences between maximum deflection data. When the plate edges are simply supported the springback predictions obtained from the two types of plasticity model are very marked indeed. For example, in the case of a slender (fl* = 3) plate with simply supported-in-plane fixed edges (see Fig. 5(a)) which has been subjected to a pressure equal to five times its first yield pressure and then unloaded, the von Mises layered yield model indicates a residual centre deflection roughly equal to one quarter of the plate thickness, whereas the Ilyushin full-section model implies no residual deflection whatsoever, i.e. the plate returns to its original flat configuration because it has not yielded inspite of being loaded to five times the small deflection first yield pressure. The third general observation also concerns the residual deflection design data. It is apparent that only in the case of stocky plates (8* = 0.4) does the residual deflection data start at the origin of the graph, i.e. Ce = O,q/& = 1. This is because, as the plate becomes more and more slender, the first

.

/

/ 3.0 -. c'.

2.0-O !

2.b

3.b

4

WC

(4

h

0

Fig. 8. Pressure=4eflection design for uniformly loaded slender circular plates with in-plane free edges (j* = 3; v = 0.3), (a) simply supported, (b) clamped.

1337

Elasto-plastic design data for circular plates yield pressure computed on the basis of small deflection plate theory becomes increasingly conservative. Thus, for a slender plate with simply supported-inplane fixed edges membrane action causes a two and a half or a five fold increase in the yield pressure over that predicted by small deflection theory (see Fig. 5(a)) depending on whether the von Mises or Ilyushin constitutive model has been used. That the Ilyushin model predicts such a high first yield pressure is a consequence of the inherent assumption of full-section rather than surface plastification at yield. The latter, of course, corresponds to reality and is accurately simulated with the von Mises layer model. The same reasoning may be used to explain why there is a significant difference between the residual deflection data derived from the two constitutive models. Finally, the relatively close agreement of the maximum deflection values predicted by the two constitutive models, especially for slender plates, is because the load supporting mechanisms are ‘membrane action’ dominated and both constitutive models give similar results in these circumstances. Indeed, it may be shown that both models are identical for pure membrane stress states. USE OF DESIGN

DATA

Although this paper is primarily concerned with presenting and demonstrating the use of elasto-plastic design data for uniformly loaded circular plates, it was deemed sensible, for the sake of completeness, to include elastic large deflection design data as well. This is presented as a series of dimensionless centre deflection versus pressure plots for simply supported and clamped edge conditions in Figs. 9(a) and 9(b) respectively. On each of the design curves in these figures open and solid circles are marked which correspond to first yield pressures and deflections for specific values of the plate slenderness (/Iz). As far as the thrust of this paper is concerned, the principal use

envisaged for Figs. 9(a) and 9(b) is as a means of rapidly checking whether an elastic design is feasible or not. The design example which follows demonstrates this type of use. Numerical example

The example chosen to illustrate the application of the elasto-plastic design data of Figs. 3-g is the design of a suitable end closure for a cylindrical steel pressure vessel. It is assumed that the end closure may be modelled as a uniformly loaded circular plate and the following data are already determined. Material and geometric data

Plate radius, r, = 400 mm Yield stress, a, = 200 N/mm2 Young’s modulus, E = 210 kN/mm’ Poisson’s ratio, v = 0.3. Solution of design problem In this instance the solution of the design problem is

provided by the answers to the following questions:

(1) Assuming that the maximum working pressure is 0.25 N/mm’, what is a suitable thickness, &, for the plate? (2) What is the plate centre deflection at the maximum working pressure and what is the residual centre deflection when this pressure is removed? (3) If, due to unforseen circumstances, the pressure is doubled, what changes in the maximum and residual deflections arise? The answers to these questions are obtained following the design procedure set out below. Preliminary remarks

Before embarking on the design calculation procedure, it is necessary to exercise engineering judgement in order to decide which of the four sets of

3.c

1.2

0.9

2.0 W, h

WC

i-0

0

0.6

1.0

4.0

(a)

p”

q E;’

EOh,*

Lh:

6.0

E

Fig. 9. JXlection-pressure design data for uniformly loaded circular plates (elastic large deflection response; v = 0.3): (a) simply supported-in-plane

CAS 4,160

by

fkd or free edges, (b) clamped-in-plane edges.

tixed or free

1338

G.

J.

TURV@Y and

M. SALWII

ideal&d boundary conditions (see Eqs. (15)) approximate most closely the real situation. In this case, clamped-in-plane fixed edge conditions (Eq. 15(c)) are deemed most relevant, but a check is also made for simply supported-in-plane fixed edges (Eq. 15(a)). Design procedure

1, Guess a plate thickness, say ho = 10 mm. 2. Evaluate

12. 13.

/I2 = (ao/E)(roho)2 = (200/210000)(400/10)2 = 1.52 (say 1.5).

14.

3. Evaluate 4 = (4/E)(r0/h0)4

15.

= (0.25/210000)(400/10)4 = 3.05. 4. Check using Fig. 9(b) whether an elastic design is feasible. The full-line curve on this figure applies to clamped-in-plane fixed edge conditions. By inspection, it is clear that the first yield pressure, &, corresponding to /I2 = 1.5 is about 2.4 (< 3.05). Hence, an elastic design using a 10 mm thick plate is not feasible. It is, therefore, necessary to use the elasto-plastic design data in Fig. 4(b) in order to establish whether this plate thickness may, nevertheless, constitute a satisfactory design. 5. Using Eq. 17(b) evaluate, & = (4/3)/P(l - v + v2)-“2 = (4/3) x 1.5 x (1 - 0.3 + 0.32))“2 = 2.25. 6. Hence, evaluate the first yield over-pressure, (g/&,) = (3.05/2.25) = 1.36. I. Now using Fig. 4(b) locate the point corresponding to (q/CT,)= 1.36 on the ordinate axis and draw from it a line parallel to the abscissa to intersect the von Mises data (open circle curve). The abscissa value of the intersection point gives @y = 0.48 or w$ax= 3yaxh0 = 0.48 x 10 = 4.8 mm. Similarly, the abscissa of the intersection point with the residual deflection data (solid line curve) gives KJ~ * 0, so that the plate is, for all practical purposes, flat on unloading. 8. Now, using Fig. 4(a), check the design against the von Mises data (open circle curve) for simply supported-in-plane fixed edges. 9. Using Eq. 17(a) evaluate, $ = (8/3)f12(3 + v)-’ = (8/3) x 1.5 x (3 + 0.3))’ = 1.21. 10. Hence, evaluate the first yield over-pressure, (q&y) = (3.05/1.21) = 2.52. 11. Now, using Fig. 4(a), locate the point corresponding to (4/&) = 2.52 on the ordinate axis and draw from it a line parallel to the abscissa to intersect the von Mises data (open circle curve).

16.

The abscissa value of the intersection point gives *,P = 0.88 or WY = @:“& = 0.88 x 10 = 8.8 mm. Similarly, the abscissa of the intersection point with the residual deflection data (solid circle curve) gives +y = 0.06 or w: = fiyh,, = 0.06 x 10 = 0.6 mm. Doubling the pressure for the clamped-in-plane fixed case gives (q/q,,) = 2 x 1.36 = 2.72. Hence, following the procedure described under Step 7 on Fig. 4(b) gives @I:- = 1.21 or WY = K$%, = 1.21 x 10 = 12.1 mm. Similarly, @y = 0.69 or WY = Oye6h0 = 0.69 x 10 = 6.9 mm. Now check for the simply supported-in-plane fixed edge case; doubling the pressure gives (q/g,,) = 2 x 2.52 = 5.04 (say 5). Hence, following the procedure described under Step 11 on Fig. 4(a) gives rt,” = 1.31 or w:” = *Faxho = 1.31 x 10 = 13.1 mm. Similarly, @y = 0.63 or WY = @y/r,, = 0.63 x 10 = 6.3 mm. Hence, a design using a 1Omm thick plate is suitable for normal maximum operating conditions having a maximum deflection lying in the range: 4.8 mm+8.8 mm (corresponding to deflection-span ratios of l/166.7+1/90.9) and a residual deflection in the range ==0+0.6mm (the upper limit corresponding to a deflection-span ratio of l/1333.3). The design is probably also satisfactory for the overload situation as well.

Further development of the design data

In the preceding sections of this paper the development and application of the DR circular plate design data has been described. All of this data is based on the assumption that the plate is initially Jut. In practice, of course, such plates may be ‘dished’ rather than flat. Therefore, an obvious extension of the design data is for it to account for initial deflections. In order to illustrate the type of data that might be produced some preliminary elasto-plastic analyses of uniformly loaded circular plates with axisymmetric initial deflections have been undertaken with the DR program based on the von Mises layered yield criterion. The axisymmetric initial deflection profiles used in the program are Wi= w,(l - O.lP - 0.9f2) (simply supported)

(18a)

“3,= w, cos(O.kf)

(18b)

(clamped).

The first set of data produced with the program are shown in Fig. 10 as plots of first yield pressure versus initial deflection amplitude, w,, for clamped and simply supported-in-plane fixed stocky and slender plates. Again, the primary purpose of this large deflection data is to allow the designer to determine rapidly whether or not he is dealing with an elastic or an elasto-plastic design situation. However, the data also illustrate that for stocky plates (see Fig. 10(a)) provided the initial deflection amplitude is less than about 0.75, then the first yield pressure of a clamped

1339

Elasto-plastic design data for circular plates

wt

(4

WI

(b)

Fig. 10. First yield versus imperfection amplitude for uniformly loaded circular plates with clamped or simply supported-in-plane fixed edges (v = 0.3): (a) stocky plates (B2 = 0.4), (b) slender plates (/I2 = 3).

q _.

,-

3.c

3.0

2.5

2.5

3.

2.0

cr I

2.0

B Y

2 / //

!/ -.

1.5

1.0 (

.I.

iI.9

1.0 0.3

(a)

0.6

1.b

c

0.9

0.3

o

W

_c

(b)

3

0.6

h 0

h

Fig. 11. Pressure-deflection design data for uniformly loaded imperfect stocky circular plates (8’ = 0.4; w, = 0.0, 0.1, 0.5 and 1.0; v = 0.3): (a) simply supported-in-plane tixed edges, (b) clamped-in-plane fixed edges.

7.0

6.0

2.0 0.0

(a)

0.5

1.0

1.5

;

0.0

W -G h 0

0.5

1.0

1.5

:

W c @I

h 0

Fig. 12. Pressure-deflection design data for uniformly loaded imperfect slender circular plates (8* = 3; wi = 0.0, 0.1, 0.5 and 1.O; v = 0.3): (a) simply supported-in-plane tied edges, (b) clamped-in-plane fixed edges.

G. 3. Tuavn~ and M. SALEH~

1340

plate is always greater than that of a simply supported plate. In contrast, Fig. IO(b) shows that the first yield pressure of a simply supported slender plate is always greater than that of a similar clamped plate irrespective of the initial imperfection amplitude. The sample elasto-plastic design data produced with the DR program is shown in Figs. 11 and 12 for stocky and slender plates respectively. The data apply only to clamped and simply supported plates with in-plane fixed edges. Moreover, only four initial im~rf~tion ~plitudes have been considered, viz. wi = 0.0, 0.1, 0.5 and 1.0. Whilst the data does not purport to be comprehensive and, therefore, any general observations must be treated with some caution, it does, nevertheless, appear that ‘springback’ (the difference between the maximum and the residual deflection) is reduced as the im~rf~tion amplitude increases. It is, of course, envisaged that this data would be applied to design in a manner similar to that described for flat plates.

generated design data to ‘predished’ plates is briefly illustrated. Acknowledgements--The authors wish to record their ap preciation to the Department of Enginezzing for supporting this research. SpeciaI thanks are due to Mrs Kathryn Rucastle, who typed the manuscript, and to Mrs Audrey Parker, who prepared the tracings for Figs. I and 2. REFERENCES

1. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Hates and Shells, pp. 51-78. Meow-Hail, New York (1959). 2. R. J. Roark, Formulasfor Stress and Strain, 4th P&ion. McGraw-Hill, New York (1965). 3. B. Aalami and D. G. Williams, Thin Plate Designfor Transverse Loading, pp. 104-108. Crosby Lockwood

Staples, London (1975).

4. H. Nylander, Initi~ly deflected thin plate with initial deflection alline to additions deflection. Int. Assoc. Bridge Struct. Engng 11,347-374 (1951). 5. G. J. Turvey and G. T. Lim, Axisymmetric full-range analysis of transverse pressure loaded circular plates. Int. J. Mech. Sci. 26, 489-502 (1984).

6. W. Johnson and A. N. Singh, Springback in circular CONCLUSIONS

The background to the two DR computer programs for the elasto-plastic large deflection analysis of circular plates has been described. The programs have been used to develop maximum and residual deflection data for the design of uniformly loaded plates undergoing axis~et~c defo~ations and taking full account of the reduction in transverse stiffness due to continuing yielding of the plate material. A numerical design example has been presented which provides an illustration of the design of a steel plate which forms the end closure of a cylindrical pressure vessel. Finally, the further development of this type of computer

blanks. Metallurgia 47, 275-280 (1980).

7. T. X. Yu and W. Johnson, The large elastic-plastic

8. 9. 10. 11.

12.

deflection with springback of a circular plate subjected to circumferential moments. J. Appl. Mech. 49,507~515 (1982). T. X. Yu, W. Johnson and W. J. Stronge, Stamping and springback of circular plates deformed in hemispherical dies. Znt. J. Neck. Sci. 26, 131-148 (1984). C.-Y. Chia, Nonlinear Analysis of Plates, pp. 291-295. Meow-Hill, New York (1980). G. J. Turvey, Large deflection of tapered annular plates by dynamic relaxation. J. Engng Mech. Div. Proc. Am. Sot. Civil Engrs. 104,351-366 (1978). G. J. Turvey and M. Salehi, DR large deflection analysis of sector nlates. Cornnut. Strut. 3d 101-112 11990). A. C. &sell and R.-E. Hobbs, Numerical stability’of dynamic relaxation analysis of non-linear structures. Int. J. Numer. Meth. Engng 10, 1~7-~410

(1976).