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A parametric study of the axisymmetric full-range response of thickness-tapered circular steel plates
W.(-79.l9W NJ0 + .OO P 19% Pergamon Press Ltd.
A PARAMETRIC STUDY OF THE AXISYMMETRIC FULLRANGE RESPONSE OF THICKNESS-TAPERED CIRCULAR STEEL PLATES G. J. Department
of Engineering.
TURVEY: University of Lancaster,
United Kingdom
and
Department
of Civil
Engineering.
G. T. LIXI$ King’s College. University
of London.
United Kingdom
~/?t~eiwf 3 Ocrober 1984) .Abstract-The incremental equations governing the full-range axisymmetric flexural response of thickness-tapered circular plates and their numerical solution via a finite-difference implementation of the Dynamic Relaxation (DR) technique are briefly outlined. A computer implementation of the DR analISIS is used to carry out a parametric study of the response of thickness-tapered plates subjected to uniform normal pressure loading. Dimensionless results (deflections etc.) spanning the practical range of linear thickness tapers are presented for simply supported and clamped. slender and stocky plates.
c Value of the variable at the plate centre e Value of the variable at the plate edge y Value of the variable at first yield
NOTATION
z,(
2) Extensional
plate stiffnesses 2) Coupling plate stiffnesses 1) Flexural plate stiffnesses e’,’ Radial and circumferential midplane strains 61 Young’s modulus Plate thickness kP. $ Radial and circumferential midplane curvatures M,. Mn Radial and circumferential stress couples = M,r~E,; ‘/I,“). ;i?, Dimensionless radial and circumferential stress couples Nr. NR Radial and circumferential stress resultants Dimensionless radial and circumferential stress resultants Normal pressure Dimensionless normal pressure r, 0. ; Polar coordinates r0 Plate radius F( = r,, /I,- ‘1 Plate slenderness 11 Radial displacement Deflection Dimensionless deflection Thickness-taper ratio Effective slenderness Dimensionless yield stress parameter Y Poisson’s ratio utr Yield stress quantity A( ) Incremental Total derivative with respect ( )’ to the radial coordinate Aij (i, j = I. Eii (i, j = I. Dij (i, j = I. I) e,.
Slrhsc.riplslslrperscriprs b
Value of variable at the end of the previous load increment
t Lecturer in Civil Engineering. $ Research Associate. Presently civil structural engineer, International Development & Consultancy Corporation (PTE) Ltd. Kent Ridge off South Buona Vista Road, Singapore 05 I I.
1. INTRODUCTION
The axisymmetric
of circular plates to has been the subject of continuous research activity throughout this century. This research effort has been stimulated partly by the widespread use of circular plates as components of metal structures, e.g. tank bottoms. etc.. and partly by the need for structural designers to cope with ever more exacting performance and efficiency requirements. Early research on circular plates was primarily concerned with elastic small deflection (linear) response. Many important results of this research are summarised in Ref. [I]. These early studies were followed by elastic large deflection (geometric nonlinear) response studies (see, for example. Refs. [241) and rigid plastic and elastoplastic (material nonlinear) studies (see, for example, Refs. [5-71). The widespread availability of signiticant computing power in the late 1960s and early 1970s not only allowed more extensive studies of this type to be undertaken (see, for example, Refs. [8-IO]), but also saw the first full-range (geometric and material nonlinear) response studies being reported (see, for example, Refs. [I l-161). Thus today’s structural designer has at his disposal much of the data required for metallic circular plate components to be designed to satisfy modern structural performance and efficiency requirements. Nevertheless, it is to be expected that perform-
transverse
459
normal
response
loading
460
G. J. TCRVEYand G. T. LIW
ante and efficicency requirements of structural components will continue to “sharpen” and designers will, doubtless. feel that the existing level of knowledge of plate response, even for circular plates, is inadequate. And in some respects this view will be justified, for all of the research cited above refers exclusively to hjorm thickness circular plates. The freedom to taper the plate thickness offers the possibility of more efficient material distribution and may represent an appropriate means of meeting more stringent performance requirements, especially for metal plate structures. From the practical standpoint, however, it is likely that linear thickness variations will prove adequate. Although design data are available on the linear axisymmetric flexural response of circular plates with linear thickness tapers (see, for example, Refs. [17-?Ol), there is scant information on the nonlinear response. Elastic large deflection studies have been reported by Federhofer and Egger[Zl], Murthy and Sherbourne(221 and Turvey[23], but, as far as the authors are aware, the only elastoplastic thicknesstapered circular plate results reported are due to Turvey[24, 251. The latter studies are based on a deformation-type plasticity model and, therefore, ignore the loading history. Consequently, throughthickness plastic zone regression processes which may arise as the lateral pressure on the plate increases may not be accurately modelled. The analysis used in the present study does not suffer from this defect because it is based on an incremental formulation and is, therefore, able to take account of loading history. Moreover, the material constitutive relationships are based on the Prandtl-Reuss associated flow theory of plasticity and the Von Mises layer yield criterion, both of which are generally accepted as being particularly suitable for modelling the full-range response of metal, especially steel, plate structures. The Dynamic Relaxation (DR) iterative procedure is used to solve the governing equations of the analysis because it is a simple and effective technique. Indeed, the results of uniform pressure tests on uniform thickness circular plates have been accurately modelled by the analysis using the DR method (see Refs. [26,27]). Here, however, the DR method is used to solve the variable thickness circular plate full-range incremental equations and thereby enable a parametric study to be undertaken. The study is concerned with clamped and simply supported plates which are fully restrained in-plane around their circumference and are subjected to uniform nopfnal pressure loading. Linear thickness tapers on(y are considered and the practical range of variations is covered. Although, from the design standpoint load-deflection response is of primary interest, additional results, particularly concerning the development of stress resultants and couples, are included because they shed light on the changing pattern of load supporting mechanisms within the full-range regime. These mecha-
nisms are not only influenced by the edge support conditions and the degree of thickness-taper, but also by the plate slenderness. Accordingly, computations have been undertaken for both slender and stocky plates to show the range of influence of slenderness on the thickness-tapered plate response. A graphical dimensionless results presentation format has been adopted in order to facilitate the application of the parametric results in preliminary design. 2. CIRCULAR
PL;\TE GEOMETRY-THICKNESS-TAPER AND STOCKINESS
The parametric study encompasses two plate geometry characteristics, viz. thickness-taper and slenderness, each of which will now be addressed in detail. Linear thickness-tapers only are considered and the thickness variation over the plate radius is defined as h = h,.(l + arr;
‘)
(OG]r]sro).
(1)
Figure I shows examples of the thickness variations for selected values of the taper ratio, a. Thus, negative values of CLdenote plates in which the thickness decreases from the centre to the edge and vice versa. For the present study computations have been undertaken for four taper ratios, i.e. a
Iz
DC
=
-ve
=
+ve
(al
0c
Fig. I. Geometry of linear thickness-tapered circular plates. (a) Negative thickness-taper ratio. (b) Zero thickness-taper ratio. (c) Positive thickness-taper ratio.
461
Thickness-tapered circular steel plates = 20.1 and kO.5, which span the practical range of values. As pointed out in the previous section, the plate response depends not only on the thickness variation and the edge support condition, but also on the plate slenderness. In order that the latter factor may be taken into account in the study computations have been undertaken for two values of “effective” slenderness, which reflect slender and stocky plate behaviour, respectively. The “effective” slenderness parameter, p, is defined as (2)
P = yr,
in which y = (o&<‘)“‘. The parameter, p, arises as a scaled yield stress in the Von Mises yield criterion as a consequence of the nondimensionalisation of the governing system of equations prior to coding and numerical solution. However, for steel the yield stress, uo, and Young’s modulus, Eo, do not vary significantly and so y may be regarded as reasonably constant. Hence, p in eqn (2) varies directly with F and may, therefore, be regarded as an “effective” slenderness parameter. Two values of F, which correspond to the slender and stocky limits of practical plate slendernesses, have been combined with a y-value for mild steel to provide p-values of 1.732 and 0.633 for the parametric computations. Thickness-tapered plates associated with the larger p-value will, henceforth, be referred to as slender and those associated with the smaller value will be referred to as stocky. 3. AXISYMMETRIC
CIRCULAR
PLATE INCREMENTAL
current load increment and which are assumed to remain constant during the current load increment. 3.2 Incremental compatibility eqrlarions
The incremental equations describing the continuity of displacement/deflection and strain/curvature for axisymmetric deformations are Ae; = AU’ + w~-AH*’ + ~(Aw’)‘. AeP = ~-‘Au,
(4)
A/$ = - A,,,“, Ak; = - r-‘Aw’. Geometric nonlinearity is accounted for by the presence of the third term on the right-hand side of eqns (4) and the superscript, b, has precisely the same significance as in eqns (3). 3.3 Incremental consritutive equarions
The elastic-perfectly plastic incremental constitutive equations, which have been formulated using the Prandtl-Reuss associated flow rule and the Von Mises layer yield criterion may be expressed as AN, = A”AeF
+ A12Aet + BllAk?
AN,, = A’rAeF + AzzAez + B,?Akf AM, = B”Aez
+ BIJki. + BzzAkt,
+ B12Aei + DllAky + D,Jkt,
AMe = B’zAef + Bz2Aei + D,,ik:
+ DZzAkjl, (5)
in which Aij, Bij and Dij (i, j = I, 2) are the elastoplastic plate stiffnesses. Equations (5) decouple for purely elastic material response, since Bij = 0.
EQUATIONS
An incremental formulation has been adopted in order to permit the load-path to be modelled reasonably accurately. The set of equations governing the axisymmetric response of circular plates under transverse normal pressure loading are summarised below for completeness. A more detailed discussion, particularly with regard to the formulation of the constitutive equations, may be consulted in Ref.
3.4 Boundary conditions
As symmetry is exploited in the computer solution of the governing incremental equations. constraints at the plate centre must be imposed in addition to those at the edge. The conditions at the centre, i.e. at r = 0, are Au = Aw’ = 0.
(6)
WI. 3. I Incremental equilibrium equations The pair of ordinary differential equations describing the incremental change in the states of inplane and normal equilibrium are AN,’ + r-‘(AN, AMM,” + r-‘(ZAM,’ + AN,wb”
- AN,,) = 0.
- AMe’)
+ AN,Aw”
+ N,bAw”
+ r-‘(NiAw’
In addition to the constraints implied by eqns (6). A 14must be antisymmetric and the other incremental variables symmetric with respect to the plate centre. At the plate edge two types of flexural condition are considered, viz. simply supported and clamped, in combination with full in-plane fixity. These conditions are (a) Simply supported edge (r = ro)
(3)
AU = AW = AM, = 0.
(7a)
+ ANgnlb’ + AN8Aw’) + Aq = 0. (b) Clamped edge (r = rO) In eqns (3) the superscript, b, denotes quantities associated with the state of equilibrium prior to the
Air = Aw = Aw’ = 0.
(7b)
462
G. J. TURVEY and G. T. LIM
The constraints at the plate centre and edge are adequately defined in terms of incremental rather than total terms, because the total values are updated at the end of each load increment.
.l. COMXIESTSONTHE NUMERICALSOLLTIONOF GOVERNISC EQUATIOSS
Because of its meaningful physical interpretation. its relative simplicity and of their experience of its successful application, the authors have elected to use the Dynamic Relaxation (DR) method to solve eqns (3~45) subject to the constraints given in eqns (6) and (7). The details of the DR algorithm have been adequately described by the originator[28] and are, therefore, not reiterated here. The DR method is. of course, used to solve iteratively the discrete equivalents of eqns (3)-(7). and a finitedifference discretisation has been used. A IOf-interval interlacing mesh over the plate radius [see Fig. 2(a)] has been adopted for the parameter study, since it provides an adequate compromise between the dual requirements of solution accuracy and computational efftciency[26]. Because the more accurate layer yield criterion is used in preference to the full-section criterion the plate has to be subdivided into a number of “notional” through-thickness layers. Based on previous experience[M], six have been used for the computation of the elastoplastic plate stiffnesses.
0
Ibl
Fig. 2. Discretisation for axisymmetric circular plate analysis. (a) Radial interlacing finite-difference meshes. (b) “Notional” through-thickness layers.
0.3 0,6 0.9 12
1.5 1.e
Fig. 3. Uniform normal pressure-centre deflection response for axisymmetric thickness-tapered circular plates (V = 0.3). (a) Slender clamped plate CR = I .731). (b) Slender simply supported plate CR = 1.731). (cl Stocky clamped plate (p = 0.633). (d) Stocky simply supported plate Cp = 0.633).
Thickness-tapered
Table I. First yield pressures clamped
circular
circular
-0.5 -0.1 +0.1 ‘0.5 -0.5 -0.1 +0.1 10.5
6.70 6.45 5.90 5.45 1.80 4.05 5.75 9.45
0.230 0.306 0.340 O.UO 0.195 0.498 0.708 I.233
The degree of approximation from this choice may be judged with reference to Fig. 2(b). To enhance both the stability characteristics of the DR iterative procedure and the rate of solution convergence a unit pseudotime increment and rationally derived fictitious densities have been employed (see Ref. 1231 for a full explanation).
OF PARAMETRIC
RESULTS
The principal results of the parametric study are presented as dimensionless normal pressure-response curves in Figs. 3-10. The four parts to each figure, i.e. (a)-(d), are juxtaposed to permit both the effects of edge support condition and plate slenderness on the plate response to be readily assimilated. Moreover, in each of parts (a)-(d) are shown curves corresponding to the four selected thickness-taper ratios. Thus, full-line, chain-dotted. short-dashed and long-dashed curves correspond to the load-response curves of plates with linear thickness-taper ratios of +O.S, + 0. I, -0.1 and -0.5, respectively. Of course, other results presentation formats could equally well have been used, but the present one serves to emphasise the effects of thickness-taper. Each of the results sets will now be discussed in detail. The transverse normal pressure-central deflection response is shown in Fig. 3. As expected. the deflection tends to reduce as the taper ratio increases, i.e. as the plates thicken towards the support. The effect of the taper ratio on the pressuredeflection response is greater in clamped than in simply suppported plates, especially when the plate is stocky and the pressure is low (see Fig. 3(c)). The pressure-deflection response curves in Fig. 3(c) show a distinct loss of stiffness in the initial elastoplastic regime which is also characteristic of stocky, uniform thickness clamped plates. This contrasts strongly with the corresponding response curves for slender simply supported plates shown in Fig. 3(b). The latter curves suggest that stiffening occurs gime,
in the i.e.
initial
membrane
stages action
463
for uniformly loaded, simply supported plates with linear thickness tapers
First yield pressure (&I Slender plates Stocky platqs cp = 1.731) (p = 0.633)
Thickness-taper ratio (a)
5. DISCUSSION
steel plates
of the
elastoplastic
re-
fully
compensates
for
and
Plate edge support condition Simply
supported
Clamped
the loss of stiffness due to progressive yielding of the plate section. First yield pressures for simply supported and clamped, slender and stocky plates were established during the main parametric computations and are presented in Table 1. With the exception of slender simply supported plates, the first yield pressure increases with increasing thickness-taper ratio. The anomalous situation for slender simply supported plates is believed to be due to the lack of support stiffness and, especially for negative taper ratios, self-stiffness which enables the plate to undergo relatively large deflections at low pressures. This promotes the development of membrane action, so that first yield is no longer predominantly a flexural phenomenon. The central stress resultant (radial and circumferential components equal) versus normal pressure response is shown in Fig. 4. It is evident that the rate of development of the stress resultant increases as the taper ratio decreases. This effect becomes more pronounced with decreasing slenderness and increasing support stiffness. The stocky clamped plate stress resultant curves of Fig. 4(c) show this particularly clearly. Another feature of Fig. 4(c), which is characteristic of stocky plates, is the rapid increase in the stress resultant over a small pressure range in the early part’of the elastoplastic regime. Central stress couple (radial and circumferential components equal)-normal pressure response curves are shown in Fig, 5. They exhibit several features which merit further comment. Firstly, it appears that the peak value of the stress couple increases as the taper ratio increases. Secondly, the gradient of the rising part of each curve increases as the taper ratio decreases-this is most marked in stocky clamped plates. Thirdly. in slender plates the normal pressure corresponding to the peak value of the stress couple is similar for all taper ratios. whereas for stocky plates the pressure varies significantly with taper ratio. And finally. the central stress couple reduces to zero at lateral pressures corresponding approximately to those at which the central stress resultant reaches its maximum value [cf. Figs. 4(a)-(d) with Figs. 5(a)-(d)].
G. J. TURVEY
464
and G. T. LIM
36 3.0
0.6
.12 ’
’
’
’
’
’
’
’
’
’
’
’
I
,lO-
q
0
0.4
08 ~ 1.2
1.6
200
0
0.4
0.8
1.2
1,6
2.0
! Id)
Fig. 4. Centre stress resultant (radial and circumferential)uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate (p = 1.732). (b) Slender simply supported plate (P = I .732). (c) Stocky clamped plate (p = 0.633). (d) Stocky simply supported plate (p = 0.633).
Fig. 5. Centre stress couple (radial and circumferential)uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate (~3 = 1.732). (b) Slender simply supported plate (p = I .732). (c) Stocky clamped plate (p = 0.633). (d) Stocky simply supported plate ()3 = 0.633).
Thickness-tapered
circular
steel plates
465
0
10
20
Ti
3;
10
5:
4
0.8
0.2 0
0
IO
20
4
30
40
50
: 0
10
20
30
a
40
gr,
lb)
.I0 Q8 D6 ,04 92 0
P
a
0.4
0.8
12 il 1,6
2.0
2;
28
IC)
IdI
Fig. 6. Radial edge stress rest&ant-uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate Cp = 1.732). (b) Slender simply supported plate Cp = I .732). (cl Stocky clamped plate Cp = 0.633). (d) Stocky simply supported plate @ = 0.633). CA8
ZllJ-0
(dt
Fig. 7. Circumferential edge stress resultant-uniform normal presw-e response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate (p = I .732). (b) Slender simply supported plate Cp = I .7X!). (c) Stocky clamped plate (~3 = 0.633). (d) Stocky simply sup_ ported plate (J3 = 0.633).
G. J. TURVEYand G. T. LIM
466
0
10
20
T
30
40
50
(bl
Fig. 8. Radial edge stress couple-uniform normal pressure response for axisymmetric thickness-tapered clamped circular plates (V = 0.3). (a) Slender plate (S = 1.732). (b) Stocky plate (~3 = 0.633).
-.lO-
-,oaI
lx -,O6-
The radial and circumferential edge stress resultant-normal pressure responses are shown respectively in Figs. 6 and 7. Most of the remarks made about the central stress resultant response shown in Fig. 4 apply equally to the edge stress resultants. However, the latter do not achieve maximum values at very high loads-they continue rising with increasing load. This is because the presence of full in-plane edge restraint inhibits complete through-thickness plastification. For clamped plates the radial edge stress couple-normal pressure response is shown in Fig. 8. It is clear, both for slender and stocky plates, that the edge stress couple increases significantly as the taper ratio increases, i.e. as the edge thickness of the plate increases. Unlike the central stress couple response shown in Fig. 5, the radial edge stress couples do not “die away” rapidly after reaching their peak values, and only for the case of OL= -0.5 does the stress couple approach a zero value. The circumferential edge stress couples are shown as functions of normal pressure in Fig. 9. For both slender and stocky clamped plates the response is similar to the radial stress couple response shown in Fig. 8, though the peak values are only about one half of the radial couple peak values. However, for slender and stocky simply supported
04
0
0.8
1.2 -1.6 9
2.0
24
20
ICI .24
0
”
”
0.4
”
0.8
”
1.2 4
’
1.6
”
2.0 24
Id)
Fig. 9. Circumferential edge stress couple-uniform normal pressure response for axisymmetric thickness-tapered circular plates (V = 0.3). (a) Slender clamped plate CP = I .732). (b) Slender simply supported plate (p = I .732).(C) Stocky clamped plate (g = 0.633). (d) Stocky simply supported plate (~3 = 0.633).
467
Thickness-tapered circular steel plates
plates the circumferential edge stress couples are of opposite sign to their clamped counterparts, but they also increase as the taper ratio increases. Moreover, as the normal pressure increases they continue to increase or reach a constant value. Only for the case of a = - 0.5 does the circumferential stress couple reach a peak value and then decay to zero. i.e. show a response similar to that of the radial stress couples. The final set of results of the parameter study are presented in Fig. 10, where the percentage of yielded nodes is plotted against normal pressure. It is evident from these curves that once siggnificant yielding of the plate has occurred, then plastification develops more rapidly as the taper ratio and the rotational edge stiffness decrease. 6. CONCLUSIOSS
60 s 40
0
0.L
0.8
1.2
9
1.6
2.0
24
2B
(Cl
0
0.4
0.8
I I I1 I I 1.2 1.6 2.0 q
Fig. 10. Percentage of yielded nodes-uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate Cp = 1.732). ib) Slender simply supported plate i/3 = I .732). (c) Stocky clamped plate ((3 = 0.633). (d) Stocky simply supported
plate Cp = 0.633).
A parametric study of the axisymmetric fullrange response of slender and stocky, simply supported and clamped steel plates with various linear thickness-tapers and subjected to uniform normal pressure loading has been undertaken using the DR method. The more important results of the study may be summarised as follows: 1. The centre deflection reduces as the taper ratio increases. This is especially so in stocky clamped plates at low pressures. 2. Except for slender simply supported plates, for which the inverse situation applies, first yield pressures increase as the taper ratio increases. 3. The central stress resultants and couples develop more rapidly with increasing pressure as the taper ratio decreases. This is more pronounced in stocky plates and under clamped edge conditions. 4. The normal pressure at which the central stress resultant reaches its maximum value corresponds to that at which the central stress couple decays to zero. This appears to be so for all taper ratios. 5. The peak value of the central stress couple increases with increasing taper ratio. This effect is more pronounced in slender plates and the normal pressure associated with the peak value appears to be almost constant and. therefore, independent of the taper ratio. For stocky plates the pressure associated with the peak stress couple increases as the taper ratio increases. 6. The “rate” of plastification of the plate section increases as the taper ratio decreases.
Acknowledgments-The second author wishes to record his gratitude to the trustees of the Vickers Endowment and the Peel and Senate Studentships of the University of Lancaster for providing financial support. Both authors wish to record their appreciation of the support of the Department of Engineering during the course of this research.
G. J. TURVEYand G. T. List
468
15. hf. Tanaka. Large deflection
REFERENCES
2.
3.
4.
5.
6.
7.
8.
S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. McGraw-Hill, New York (1959). r\. Nadai, Nasrische Platten, p. 288. Springer-Verlag, Berlin (1925). S. Way, Bending of circular plates with large deflection. Trans. Am. Sot. Mech. Engrs 56, No. 8, 627636 (1934). H. Nylander, Initially deflected thin plate with initial deflection afftne to additional deflection. Int. Assoc. Bridge Struclrtra/ Engng 11, 347-374 (I95 I). H. G. Hopkins and W. Prager, The load carrying caoacities of circular mates. J. Mech. Phvs. Solids 2, No. I. l-13 (1953). ’ H. G.Hopkins and A. J. Wang, Load-carrying capacities for circular plates of perfectly-plastic material with arbitrarv vield condition. J. Mech. Phvs. So/ids 3, No. 2, I I;-i29 (1954). V. V. Sokolovsky, Elasto-plastic bending of circular and annular plates. Brown University, Division of Applied Mathematics, Technical Report No. 3 (1955). G. D. Galletly and F. Y. Tuma, Large deflection analysis of plates using dynamic relaxation. Proceedings of 4th Canadian
Congress
of Applied
Mechanics,
Montreal (1973). 9. G. W. May and K. H. Gerstle, Elastic-plastic behaviour of axisymmetric plates. Proceedings ofthe ASCE Structural Engineering Conference, Seattle, Preprint 462 (1967). IO. E. P. Popov, M. Khojesteh-Bakht and S. Yaghmai, Analysis -of elastic-p&tic circular plates. 1. Engng Jfech. Div. Proc. Am. Sot. Civil Enars 93. No. EM6, I
-19-65 (1967). II. Y. Ohashi and S. Murakami, The elasto-plastic bending of a clamped thin circular plate. Proceedings of llrh International Conaress of Applied Mechanics,
Munich (1964). pp. 2121223. - . . 12. Y. Ohashi and S. Murakami, Large deflection in elastoplastic bending of a simply supported circular plate under a uniform load. J. Appl. Mech. 33, No. 4, 866870 (1966). 13. Y. Ohashi and S. Murakami, Axisymmetric elastoplastic deformation of circular plates under combined action of lateral load and membrane force. Mem. Fat. Engng Nagoya Univ. 21, No. I, 79-121 (1969). 14. J. Myszkowski, Endliche Durchbiegungen beliebig eingespannter dinner Kreis- und Kreisringplatten im plastischen Materialbereich. Ingen. Arch. 40, No. I, l-13 (1971).
analvsis of elastic-plastic circular plates with combined*isotropic and kinematic hardening. Ingen. Arch. 41, So. 5. 342-356 (197’). 16. G. J. Turvey. Axisymmetric elasto-plastic flexure of circular plates in the large deflection regime. Proceedings of rhe Institution of Civil Engineers (Parr 2: Research and Theory). Vol. 69. pp. 81-92 (1979).
17. R. G. Olsson. Biegung Kreisformiger Platten von radial veranderlicher Dicke. Ittgen. Arch. 8, No. 2. 8198 (1937). IS. H. D. Conway. The bending of symmetrically loaded circular plates of variable thickness. J. Appl. Mech. 15. No. I, l-6 (1948). 19. H. D. Conway, Axially symmetrical plates with linearly variable thickness. J. Appl. Jfech. 18, No. 2. 140-142 (1951). 20. A. N. Sherbourne and D. N. S. Mm-thy. Bending of circular plates with variable protile. Compur. Srrrrct. 11, No. 4, 355-361 (1980). 21. K. Federhofer and H. Egger. Berechnung der diinnen Kreisplatte mit grosser Ausbiegung. Sitxngrber. Akad. Wiss. Vienna Aasrria lSS, No. 2a. 15-43 (1946). 22. S. D. N. Murthy and A. N. Sherbourne, Nonlinear bending of elastic plates and variable profile. J. Engng Mech. Div. Proc. Am. Sot. Civil Engrs 100, No. EM’, 251-265 (1974). 23. G. J. Turvey, Large deflection of tapered annular plates by dynamic relaxation. J. Engng Mech. Div. Proc. Am. Sot. Civil Engrs 104, No. EM2. 351-366 (1978). 74 -7. G. J. Turvey. Thickness-tapered circular plates-An elastoplastic large deflection analysis. J. Srncct. Mech. 7, No. 3, 247-271 (1979). 25 G. J. Turvey, Elasto-plastic analysis of thickness-ta’ pered circular plates. Proceedings of 7th Canadian Congress of Applied Mechanics, Vol. I, pp. 77-78 (1979). 26 G. J. Turvey and G. T. Lim, Axisymmetric full-range analysis of transverse pressure loaded circular plates. Inr. J. Mech. Sci. 26(9/10), 489-502 (1984). 27, G. T. Lim and G. J. Turvey, Full-range response of uniformly loaded clamped circular steel plates-Cornparisons between theory and experiment. Proceedings of the Institution of Civil Engineers (Purr 2: Research and Theory), Vol. 77, pp. 139-165 (1984).
28, A. S. Day, An introduction to dynamic relaxation. Engineer 219, No. 5688, 218-221 (19651.
A PARAMETRIC STUDY OF THE AXISYMMETRIC FULLRANGE RESPONSE OF THICKNESS-TAPERED CIRCULAR STEEL PLATES G. J. Department
of Engineering.
TURVEY: University of Lancaster,
United Kingdom
and
Department
of Civil
Engineering.
G. T. LIXI$ King’s College. University
of London.
United Kingdom
~/?t~eiwf 3 Ocrober 1984) .Abstract-The incremental equations governing the full-range axisymmetric flexural response of thickness-tapered circular plates and their numerical solution via a finite-difference implementation of the Dynamic Relaxation (DR) technique are briefly outlined. A computer implementation of the DR analISIS is used to carry out a parametric study of the response of thickness-tapered plates subjected to uniform normal pressure loading. Dimensionless results (deflections etc.) spanning the practical range of linear thickness tapers are presented for simply supported and clamped. slender and stocky plates.
c Value of the variable at the plate centre e Value of the variable at the plate edge y Value of the variable at first yield
NOTATION
z,(
2) Extensional
plate stiffnesses 2) Coupling plate stiffnesses 1) Flexural plate stiffnesses e’,’ Radial and circumferential midplane strains 61 Young’s modulus Plate thickness kP. $ Radial and circumferential midplane curvatures M,. Mn Radial and circumferential stress couples = M,r~E,; ‘/I,“). ;i?, Dimensionless radial and circumferential stress couples Nr. NR Radial and circumferential stress resultants Dimensionless radial and circumferential stress resultants Normal pressure Dimensionless normal pressure r, 0. ; Polar coordinates r0 Plate radius F( = r,, /I,- ‘1 Plate slenderness 11 Radial displacement Deflection Dimensionless deflection Thickness-taper ratio Effective slenderness Dimensionless yield stress parameter Y Poisson’s ratio utr Yield stress quantity A( ) Incremental Total derivative with respect ( )’ to the radial coordinate Aij (i, j = I. Eii (i, j = I. Dij (i, j = I. I) e,.
Slrhsc.riplslslrperscriprs b
Value of variable at the end of the previous load increment
t Lecturer in Civil Engineering. $ Research Associate. Presently civil structural engineer, International Development & Consultancy Corporation (PTE) Ltd. Kent Ridge off South Buona Vista Road, Singapore 05 I I.
1. INTRODUCTION
The axisymmetric
of circular plates to has been the subject of continuous research activity throughout this century. This research effort has been stimulated partly by the widespread use of circular plates as components of metal structures, e.g. tank bottoms. etc.. and partly by the need for structural designers to cope with ever more exacting performance and efficiency requirements. Early research on circular plates was primarily concerned with elastic small deflection (linear) response. Many important results of this research are summarised in Ref. [I]. These early studies were followed by elastic large deflection (geometric nonlinear) response studies (see, for example. Refs. [241) and rigid plastic and elastoplastic (material nonlinear) studies (see, for example, Refs. [5-71). The widespread availability of signiticant computing power in the late 1960s and early 1970s not only allowed more extensive studies of this type to be undertaken (see, for example, Refs. [8-IO]), but also saw the first full-range (geometric and material nonlinear) response studies being reported (see, for example, Refs. [I l-161). Thus today’s structural designer has at his disposal much of the data required for metallic circular plate components to be designed to satisfy modern structural performance and efficiency requirements. Nevertheless, it is to be expected that perform-
transverse
459
normal
response
loading
460
G. J. TCRVEYand G. T. LIW
ante and efficicency requirements of structural components will continue to “sharpen” and designers will, doubtless. feel that the existing level of knowledge of plate response, even for circular plates, is inadequate. And in some respects this view will be justified, for all of the research cited above refers exclusively to hjorm thickness circular plates. The freedom to taper the plate thickness offers the possibility of more efficient material distribution and may represent an appropriate means of meeting more stringent performance requirements, especially for metal plate structures. From the practical standpoint, however, it is likely that linear thickness variations will prove adequate. Although design data are available on the linear axisymmetric flexural response of circular plates with linear thickness tapers (see, for example, Refs. [17-?Ol), there is scant information on the nonlinear response. Elastic large deflection studies have been reported by Federhofer and Egger[Zl], Murthy and Sherbourne(221 and Turvey[23], but, as far as the authors are aware, the only elastoplastic thicknesstapered circular plate results reported are due to Turvey[24, 251. The latter studies are based on a deformation-type plasticity model and, therefore, ignore the loading history. Consequently, throughthickness plastic zone regression processes which may arise as the lateral pressure on the plate increases may not be accurately modelled. The analysis used in the present study does not suffer from this defect because it is based on an incremental formulation and is, therefore, able to take account of loading history. Moreover, the material constitutive relationships are based on the Prandtl-Reuss associated flow theory of plasticity and the Von Mises layer yield criterion, both of which are generally accepted as being particularly suitable for modelling the full-range response of metal, especially steel, plate structures. The Dynamic Relaxation (DR) iterative procedure is used to solve the governing equations of the analysis because it is a simple and effective technique. Indeed, the results of uniform pressure tests on uniform thickness circular plates have been accurately modelled by the analysis using the DR method (see Refs. [26,27]). Here, however, the DR method is used to solve the variable thickness circular plate full-range incremental equations and thereby enable a parametric study to be undertaken. The study is concerned with clamped and simply supported plates which are fully restrained in-plane around their circumference and are subjected to uniform nopfnal pressure loading. Linear thickness tapers on(y are considered and the practical range of variations is covered. Although, from the design standpoint load-deflection response is of primary interest, additional results, particularly concerning the development of stress resultants and couples, are included because they shed light on the changing pattern of load supporting mechanisms within the full-range regime. These mecha-
nisms are not only influenced by the edge support conditions and the degree of thickness-taper, but also by the plate slenderness. Accordingly, computations have been undertaken for both slender and stocky plates to show the range of influence of slenderness on the thickness-tapered plate response. A graphical dimensionless results presentation format has been adopted in order to facilitate the application of the parametric results in preliminary design. 2. CIRCULAR
PL;\TE GEOMETRY-THICKNESS-TAPER AND STOCKINESS
The parametric study encompasses two plate geometry characteristics, viz. thickness-taper and slenderness, each of which will now be addressed in detail. Linear thickness-tapers only are considered and the thickness variation over the plate radius is defined as h = h,.(l + arr;
‘)
(OG]r]sro).
(1)
Figure I shows examples of the thickness variations for selected values of the taper ratio, a. Thus, negative values of CLdenote plates in which the thickness decreases from the centre to the edge and vice versa. For the present study computations have been undertaken for four taper ratios, i.e. a
Iz
DC
=
-ve
=
+ve
(al
0c
Fig. I. Geometry of linear thickness-tapered circular plates. (a) Negative thickness-taper ratio. (b) Zero thickness-taper ratio. (c) Positive thickness-taper ratio.
461
Thickness-tapered circular steel plates = 20.1 and kO.5, which span the practical range of values. As pointed out in the previous section, the plate response depends not only on the thickness variation and the edge support condition, but also on the plate slenderness. In order that the latter factor may be taken into account in the study computations have been undertaken for two values of “effective” slenderness, which reflect slender and stocky plate behaviour, respectively. The “effective” slenderness parameter, p, is defined as (2)
P = yr,
in which y = (o&<‘)“‘. The parameter, p, arises as a scaled yield stress in the Von Mises yield criterion as a consequence of the nondimensionalisation of the governing system of equations prior to coding and numerical solution. However, for steel the yield stress, uo, and Young’s modulus, Eo, do not vary significantly and so y may be regarded as reasonably constant. Hence, p in eqn (2) varies directly with F and may, therefore, be regarded as an “effective” slenderness parameter. Two values of F, which correspond to the slender and stocky limits of practical plate slendernesses, have been combined with a y-value for mild steel to provide p-values of 1.732 and 0.633 for the parametric computations. Thickness-tapered plates associated with the larger p-value will, henceforth, be referred to as slender and those associated with the smaller value will be referred to as stocky. 3. AXISYMMETRIC
CIRCULAR
PLATE INCREMENTAL
current load increment and which are assumed to remain constant during the current load increment. 3.2 Incremental compatibility eqrlarions
The incremental equations describing the continuity of displacement/deflection and strain/curvature for axisymmetric deformations are Ae; = AU’ + w~-AH*’ + ~(Aw’)‘. AeP = ~-‘Au,
(4)
A/$ = - A,,,“, Ak; = - r-‘Aw’. Geometric nonlinearity is accounted for by the presence of the third term on the right-hand side of eqns (4) and the superscript, b, has precisely the same significance as in eqns (3). 3.3 Incremental consritutive equarions
The elastic-perfectly plastic incremental constitutive equations, which have been formulated using the Prandtl-Reuss associated flow rule and the Von Mises layer yield criterion may be expressed as AN, = A”AeF
+ A12Aet + BllAk?
AN,, = A’rAeF + AzzAez + B,?Akf AM, = B”Aez
+ BIJki. + BzzAkt,
+ B12Aei + DllAky + D,Jkt,
AMe = B’zAef + Bz2Aei + D,,ik:
+ DZzAkjl, (5)
in which Aij, Bij and Dij (i, j = I, 2) are the elastoplastic plate stiffnesses. Equations (5) decouple for purely elastic material response, since Bij = 0.
EQUATIONS
An incremental formulation has been adopted in order to permit the load-path to be modelled reasonably accurately. The set of equations governing the axisymmetric response of circular plates under transverse normal pressure loading are summarised below for completeness. A more detailed discussion, particularly with regard to the formulation of the constitutive equations, may be consulted in Ref.
3.4 Boundary conditions
As symmetry is exploited in the computer solution of the governing incremental equations. constraints at the plate centre must be imposed in addition to those at the edge. The conditions at the centre, i.e. at r = 0, are Au = Aw’ = 0.
(6)
WI. 3. I Incremental equilibrium equations The pair of ordinary differential equations describing the incremental change in the states of inplane and normal equilibrium are AN,’ + r-‘(AN, AMM,” + r-‘(ZAM,’ + AN,wb”
- AN,,) = 0.
- AMe’)
+ AN,Aw”
+ N,bAw”
+ r-‘(NiAw’
In addition to the constraints implied by eqns (6). A 14must be antisymmetric and the other incremental variables symmetric with respect to the plate centre. At the plate edge two types of flexural condition are considered, viz. simply supported and clamped, in combination with full in-plane fixity. These conditions are (a) Simply supported edge (r = ro)
(3)
AU = AW = AM, = 0.
(7a)
+ ANgnlb’ + AN8Aw’) + Aq = 0. (b) Clamped edge (r = rO) In eqns (3) the superscript, b, denotes quantities associated with the state of equilibrium prior to the
Air = Aw = Aw’ = 0.
(7b)
462
G. J. TURVEY and G. T. LIM
The constraints at the plate centre and edge are adequately defined in terms of incremental rather than total terms, because the total values are updated at the end of each load increment.
.l. COMXIESTSONTHE NUMERICALSOLLTIONOF GOVERNISC EQUATIOSS
Because of its meaningful physical interpretation. its relative simplicity and of their experience of its successful application, the authors have elected to use the Dynamic Relaxation (DR) method to solve eqns (3~45) subject to the constraints given in eqns (6) and (7). The details of the DR algorithm have been adequately described by the originator[28] and are, therefore, not reiterated here. The DR method is. of course, used to solve iteratively the discrete equivalents of eqns (3)-(7). and a finitedifference discretisation has been used. A IOf-interval interlacing mesh over the plate radius [see Fig. 2(a)] has been adopted for the parameter study, since it provides an adequate compromise between the dual requirements of solution accuracy and computational efftciency[26]. Because the more accurate layer yield criterion is used in preference to the full-section criterion the plate has to be subdivided into a number of “notional” through-thickness layers. Based on previous experience[M], six have been used for the computation of the elastoplastic plate stiffnesses.
0
Ibl
Fig. 2. Discretisation for axisymmetric circular plate analysis. (a) Radial interlacing finite-difference meshes. (b) “Notional” through-thickness layers.
0.3 0,6 0.9 12
1.5 1.e
Fig. 3. Uniform normal pressure-centre deflection response for axisymmetric thickness-tapered circular plates (V = 0.3). (a) Slender clamped plate CR = I .731). (b) Slender simply supported plate CR = 1.731). (cl Stocky clamped plate (p = 0.633). (d) Stocky simply supported plate Cp = 0.633).
Thickness-tapered
Table I. First yield pressures clamped
circular
circular
-0.5 -0.1 +0.1 ‘0.5 -0.5 -0.1 +0.1 10.5
6.70 6.45 5.90 5.45 1.80 4.05 5.75 9.45
0.230 0.306 0.340 O.UO 0.195 0.498 0.708 I.233
The degree of approximation from this choice may be judged with reference to Fig. 2(b). To enhance both the stability characteristics of the DR iterative procedure and the rate of solution convergence a unit pseudotime increment and rationally derived fictitious densities have been employed (see Ref. 1231 for a full explanation).
OF PARAMETRIC
RESULTS
The principal results of the parametric study are presented as dimensionless normal pressure-response curves in Figs. 3-10. The four parts to each figure, i.e. (a)-(d), are juxtaposed to permit both the effects of edge support condition and plate slenderness on the plate response to be readily assimilated. Moreover, in each of parts (a)-(d) are shown curves corresponding to the four selected thickness-taper ratios. Thus, full-line, chain-dotted. short-dashed and long-dashed curves correspond to the load-response curves of plates with linear thickness-taper ratios of +O.S, + 0. I, -0.1 and -0.5, respectively. Of course, other results presentation formats could equally well have been used, but the present one serves to emphasise the effects of thickness-taper. Each of the results sets will now be discussed in detail. The transverse normal pressure-central deflection response is shown in Fig. 3. As expected. the deflection tends to reduce as the taper ratio increases, i.e. as the plates thicken towards the support. The effect of the taper ratio on the pressuredeflection response is greater in clamped than in simply suppported plates, especially when the plate is stocky and the pressure is low (see Fig. 3(c)). The pressure-deflection response curves in Fig. 3(c) show a distinct loss of stiffness in the initial elastoplastic regime which is also characteristic of stocky, uniform thickness clamped plates. This contrasts strongly with the corresponding response curves for slender simply supported plates shown in Fig. 3(b). The latter curves suggest that stiffening occurs gime,
in the i.e.
initial
membrane
stages action
463
for uniformly loaded, simply supported plates with linear thickness tapers
First yield pressure (&I Slender plates Stocky platqs cp = 1.731) (p = 0.633)
Thickness-taper ratio (a)
5. DISCUSSION
steel plates
of the
elastoplastic
re-
fully
compensates
for
and
Plate edge support condition Simply
supported
Clamped
the loss of stiffness due to progressive yielding of the plate section. First yield pressures for simply supported and clamped, slender and stocky plates were established during the main parametric computations and are presented in Table 1. With the exception of slender simply supported plates, the first yield pressure increases with increasing thickness-taper ratio. The anomalous situation for slender simply supported plates is believed to be due to the lack of support stiffness and, especially for negative taper ratios, self-stiffness which enables the plate to undergo relatively large deflections at low pressures. This promotes the development of membrane action, so that first yield is no longer predominantly a flexural phenomenon. The central stress resultant (radial and circumferential components equal) versus normal pressure response is shown in Fig. 4. It is evident that the rate of development of the stress resultant increases as the taper ratio decreases. This effect becomes more pronounced with decreasing slenderness and increasing support stiffness. The stocky clamped plate stress resultant curves of Fig. 4(c) show this particularly clearly. Another feature of Fig. 4(c), which is characteristic of stocky plates, is the rapid increase in the stress resultant over a small pressure range in the early part’of the elastoplastic regime. Central stress couple (radial and circumferential components equal)-normal pressure response curves are shown in Fig, 5. They exhibit several features which merit further comment. Firstly, it appears that the peak value of the stress couple increases as the taper ratio increases. Secondly, the gradient of the rising part of each curve increases as the taper ratio decreases-this is most marked in stocky clamped plates. Thirdly. in slender plates the normal pressure corresponding to the peak value of the stress couple is similar for all taper ratios. whereas for stocky plates the pressure varies significantly with taper ratio. And finally. the central stress couple reduces to zero at lateral pressures corresponding approximately to those at which the central stress resultant reaches its maximum value [cf. Figs. 4(a)-(d) with Figs. 5(a)-(d)].
G. J. TURVEY
464
and G. T. LIM
36 3.0
0.6
.12 ’
’
’
’
’
’
’
’
’
’
’
’
I
,lO-
q
0
0.4
08 ~ 1.2
1.6
200
0
0.4
0.8
1.2
1,6
2.0
! Id)
Fig. 4. Centre stress resultant (radial and circumferential)uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate (p = 1.732). (b) Slender simply supported plate (P = I .732). (c) Stocky clamped plate (p = 0.633). (d) Stocky simply supported plate (p = 0.633).
Fig. 5. Centre stress couple (radial and circumferential)uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate (~3 = 1.732). (b) Slender simply supported plate (p = I .732). (c) Stocky clamped plate (p = 0.633). (d) Stocky simply supported plate ()3 = 0.633).
Thickness-tapered
circular
steel plates
465
0
10
20
Ti
3;
10
5:
4
0.8
0.2 0
0
IO
20
4
30
40
50
: 0
10
20
30
a
40
gr,
lb)
.I0 Q8 D6 ,04 92 0
P
a
0.4
0.8
12 il 1,6
2.0
2;
28
IC)
IdI
Fig. 6. Radial edge stress rest&ant-uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate Cp = 1.732). (b) Slender simply supported plate Cp = I .732). (cl Stocky clamped plate Cp = 0.633). (d) Stocky simply supported plate @ = 0.633). CA8
ZllJ-0
(dt
Fig. 7. Circumferential edge stress resultant-uniform normal presw-e response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate (p = I .732). (b) Slender simply supported plate Cp = I .7X!). (c) Stocky clamped plate (~3 = 0.633). (d) Stocky simply sup_ ported plate (J3 = 0.633).
G. J. TURVEYand G. T. LIM
466
0
10
20
T
30
40
50
(bl
Fig. 8. Radial edge stress couple-uniform normal pressure response for axisymmetric thickness-tapered clamped circular plates (V = 0.3). (a) Slender plate (S = 1.732). (b) Stocky plate (~3 = 0.633).
-.lO-
-,oaI
lx -,O6-
The radial and circumferential edge stress resultant-normal pressure responses are shown respectively in Figs. 6 and 7. Most of the remarks made about the central stress resultant response shown in Fig. 4 apply equally to the edge stress resultants. However, the latter do not achieve maximum values at very high loads-they continue rising with increasing load. This is because the presence of full in-plane edge restraint inhibits complete through-thickness plastification. For clamped plates the radial edge stress couple-normal pressure response is shown in Fig. 8. It is clear, both for slender and stocky plates, that the edge stress couple increases significantly as the taper ratio increases, i.e. as the edge thickness of the plate increases. Unlike the central stress couple response shown in Fig. 5, the radial edge stress couples do not “die away” rapidly after reaching their peak values, and only for the case of OL= -0.5 does the stress couple approach a zero value. The circumferential edge stress couples are shown as functions of normal pressure in Fig. 9. For both slender and stocky clamped plates the response is similar to the radial stress couple response shown in Fig. 8, though the peak values are only about one half of the radial couple peak values. However, for slender and stocky simply supported
04
0
0.8
1.2 -1.6 9
2.0
24
20
ICI .24
0
”
”
0.4
”
0.8
”
1.2 4
’
1.6
”
2.0 24
Id)
Fig. 9. Circumferential edge stress couple-uniform normal pressure response for axisymmetric thickness-tapered circular plates (V = 0.3). (a) Slender clamped plate CP = I .732). (b) Slender simply supported plate (p = I .732).(C) Stocky clamped plate (g = 0.633). (d) Stocky simply supported plate (~3 = 0.633).
467
Thickness-tapered circular steel plates
plates the circumferential edge stress couples are of opposite sign to their clamped counterparts, but they also increase as the taper ratio increases. Moreover, as the normal pressure increases they continue to increase or reach a constant value. Only for the case of a = - 0.5 does the circumferential stress couple reach a peak value and then decay to zero. i.e. show a response similar to that of the radial stress couples. The final set of results of the parameter study are presented in Fig. 10, where the percentage of yielded nodes is plotted against normal pressure. It is evident from these curves that once siggnificant yielding of the plate has occurred, then plastification develops more rapidly as the taper ratio and the rotational edge stiffness decrease. 6. CONCLUSIOSS
60 s 40
0
0.L
0.8
1.2
9
1.6
2.0
24
2B
(Cl
0
0.4
0.8
I I I1 I I 1.2 1.6 2.0 q
Fig. 10. Percentage of yielded nodes-uniform normal pressure response for axisymmetric thickness-tapered circular plates (v = 0.3). (a) Slender clamped plate Cp = 1.732). ib) Slender simply supported plate i/3 = I .732). (c) Stocky clamped plate ((3 = 0.633). (d) Stocky simply supported
plate Cp = 0.633).
A parametric study of the axisymmetric fullrange response of slender and stocky, simply supported and clamped steel plates with various linear thickness-tapers and subjected to uniform normal pressure loading has been undertaken using the DR method. The more important results of the study may be summarised as follows: 1. The centre deflection reduces as the taper ratio increases. This is especially so in stocky clamped plates at low pressures. 2. Except for slender simply supported plates, for which the inverse situation applies, first yield pressures increase as the taper ratio increases. 3. The central stress resultants and couples develop more rapidly with increasing pressure as the taper ratio decreases. This is more pronounced in stocky plates and under clamped edge conditions. 4. The normal pressure at which the central stress resultant reaches its maximum value corresponds to that at which the central stress couple decays to zero. This appears to be so for all taper ratios. 5. The peak value of the central stress couple increases with increasing taper ratio. This effect is more pronounced in slender plates and the normal pressure associated with the peak value appears to be almost constant and. therefore, independent of the taper ratio. For stocky plates the pressure associated with the peak stress couple increases as the taper ratio increases. 6. The “rate” of plastification of the plate section increases as the taper ratio decreases.
Acknowledgments-The second author wishes to record his gratitude to the trustees of the Vickers Endowment and the Peel and Senate Studentships of the University of Lancaster for providing financial support. Both authors wish to record their appreciation of the support of the Department of Engineering during the course of this research.
G. J. TURVEYand G. T. List
468
15. hf. Tanaka. Large deflection
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2.
3.
4.
5.
6.
7.
8.
S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. McGraw-Hill, New York (1959). r\. Nadai, Nasrische Platten, p. 288. Springer-Verlag, Berlin (1925). S. Way, Bending of circular plates with large deflection. Trans. Am. Sot. Mech. Engrs 56, No. 8, 627636 (1934). H. Nylander, Initially deflected thin plate with initial deflection afftne to additional deflection. Int. Assoc. Bridge Struclrtra/ Engng 11, 347-374 (I95 I). H. G. Hopkins and W. Prager, The load carrying caoacities of circular mates. J. Mech. Phvs. Solids 2, No. I. l-13 (1953). ’ H. G.Hopkins and A. J. Wang, Load-carrying capacities for circular plates of perfectly-plastic material with arbitrarv vield condition. J. Mech. Phvs. So/ids 3, No. 2, I I;-i29 (1954). V. V. Sokolovsky, Elasto-plastic bending of circular and annular plates. Brown University, Division of Applied Mathematics, Technical Report No. 3 (1955). G. D. Galletly and F. Y. Tuma, Large deflection analysis of plates using dynamic relaxation. Proceedings of 4th Canadian
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17. R. G. Olsson. Biegung Kreisformiger Platten von radial veranderlicher Dicke. Ittgen. Arch. 8, No. 2. 8198 (1937). IS. H. D. Conway. The bending of symmetrically loaded circular plates of variable thickness. J. Appl. Mech. 15. No. I, l-6 (1948). 19. H. D. Conway, Axially symmetrical plates with linearly variable thickness. J. Appl. Jfech. 18, No. 2. 140-142 (1951). 20. A. N. Sherbourne and D. N. S. Mm-thy. Bending of circular plates with variable protile. Compur. Srrrrct. 11, No. 4, 355-361 (1980). 21. K. Federhofer and H. Egger. Berechnung der diinnen Kreisplatte mit grosser Ausbiegung. Sitxngrber. Akad. Wiss. Vienna Aasrria lSS, No. 2a. 15-43 (1946). 22. S. D. N. Murthy and A. N. Sherbourne, Nonlinear bending of elastic plates and variable profile. J. Engng Mech. Div. Proc. Am. Sot. Civil Engrs 100, No. EM’, 251-265 (1974). 23. G. J. Turvey, Large deflection of tapered annular plates by dynamic relaxation. J. Engng Mech. Div. Proc. Am. Sot. Civil Engrs 104, No. EM2. 351-366 (1978). 74 -7. G. J. Turvey. Thickness-tapered circular plates-An elastoplastic large deflection analysis. J. Srncct. Mech. 7, No. 3, 247-271 (1979). 25 G. J. Turvey, Elasto-plastic analysis of thickness-ta’ pered circular plates. Proceedings of 7th Canadian Congress of Applied Mechanics, Vol. I, pp. 77-78 (1979). 26 G. J. Turvey and G. T. Lim, Axisymmetric full-range analysis of transverse pressure loaded circular plates. Inr. J. Mech. Sci. 26(9/10), 489-502 (1984). 27, G. T. Lim and G. J. Turvey, Full-range response of uniformly loaded clamped circular steel plates-Cornparisons between theory and experiment. Proceedings of the Institution of Civil Engineers (Purr 2: Research and Theory), Vol. 77, pp. 139-165 (1984).
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