Large deflection analysis of elastic sector Mindlin plates

Compufers & Slructures Vol. 52. No. 5. pp. 987-998, 1994 Elsevier Science Ltd

Pergamon 00457949(!24)EOO81-C Printed in Great Britain. 0045-7949/w 57.00 + 0.00

LARGE DEFLECTION ANALYSIS OF ELASTIC SECTOR MINDLIN PLATES

M. Salehi and A. Shahidi

Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

(Received 8 January 1993)

Abstract-For the first time governing equations for the elastic large deflection of uniformly pressure loaded sector Mindlin plates are presented. The solution of these non-linear equations using the Dynamic Relaxation (DR) Technique together with the central finite differences is given. For this purpose, a FORTRAN computer program is developed for non-interlacing finite differences which produces numerical results. In order to validate the accuracy of the computer program, a series of comparison tables for elastic large and small deflection thin plate results is presented. This also proves the accuracy of the DR technique.

A D e;.ei, e5

ho k2 k;, 4, k;

k,,k,,k,,k,>k,

%w4,&

47 4.

NOTATION

plate extensional stiffness plate flexural stiffness radial, circumferential and shear strain components of plate mid- plane transverse shear components of plate mid-plane Young’s modulus plate thickness Shear Correction Factor (SCF) radial, circumferential curvature and twisting components of plate mid-plane in-plane, out-of-plane and rotational damping factors radial, circumferential and twisting stress couples non-dimensional&d stress couples

radial, circumferential and shear stress resultants non-dimensionalized stress resultants transverse pressure transverse shear resultants plate thickness non-dimensionalized radial coordinate polar coordinates radial and circumferential in-plane displacements deflection non-dimensionalized deflection sector angle radial finite difference mesh length time increment angle subtended at the plate vertex by the circumferential finite difference mesh length rotation of the original normal to the plate mid-plane about l3 and r respectively Poisson’s ratio

in-plane, out-of-plane and rotational fictitious densities partial derivatives with respect to the radial and circumferential coordinate directions, respectively derivative with respect to time

1. INTRODUCTION

The analysis of thin plates started as long ago as the end of the last century. Since then many techniques and methods have been developed for this purpose. Although the small deflection theory (SDT) gives satisfactory results, in practice, more often it happens that the deflections are much larger than predicted by the SDT. Consequently this necessitates the develop- ment of a large deflection theory (LDT) for the analysis of plates. The classical governing differential equations for the large deflection of rectangular plates were first developed by Von Karman [I]. How- ever, in the case of sector plates, Nadai [2] appears to be the first to have carried out sector plate analysis. This analysis is for uniform pressure loading when the sector is completely simply supported or when the curved edge is clamped and the radial edges are simply supported. For the sake of brevity, a complete literature survey on sector plates may be found in [3].

For more than 40 years, it has been recognized that response predictions based on thin plate theory may begin to be inaccurate when the ratio of plate thickness to plate radius (h,/r,) is greater than about 0.1. This ratio is known as the ‘thickness ratio’ [4]. When the thickness ratio is greater than 0.1 then the shear deformation of the plate cross-section becomes sig- nificant and the thin plate theory ignores this effect. Reissner [S] appears to be the first researcher to have

987

988 M. Salehi and A. Shahidi

taken into account the effect of the transverse shear deformability of the cross-section of the plate. Later, Mindlin [6] also carried out a similar analysis and both analyses were for rectangular plates. Recently, the non-linear analysis of rectangular Mindlin plates has become of interest [7, 81. With the advent of digital computers, numerical methods have gained momentum for the non-linear analysis of plates. Recently, Reddy and Chao [7] presented results for the non-linear bending of thick rectangular plates. Azizian and Dawe [8] used the finite strip method to solve the same problem, but the plate material is isotropic. Very recently, Turvey and Osman [4] reported numerical results for the large deflection analysis of isotropic rectangular Mindlin plates. It appears that no work, numerical or classical, has been done on the large deflection analysis of moderately thick sector plates.

With the introduction of digital computers into the world of engineering, engineers recognized that there are several mathematical methods which were almost impossible or impractical to apply to engineering problem solving but can now be easily utilized with the help of these computers. Consequently, numerical methods become very popular in engineering applications and several different techniques have been used, namely, finite difference, finite element, finite strip, relaxation, etc., to solve very complicated engineering design problems.

In the present analysis. the dynamic relaxation (DR) method is used together with central finite differences. This technique has been previously used to solve various plate and shell problems. The advan- tages of this technique have been shown in [3,4,9]. So far, the application of the DR method has been extended to include elastic large deflection analysis of thick rectangular plates [4]. The main objectives of this research work are first to extend the application of the DR method even further to include the analysis of elastic large deflections of sector Mindlin plates, and secondly, to produce new results for deflections and stresses for sector plates.

The sector plate analysis is based on the isotropic, non-axisymmetric elastic large deflection equations derived for the first time. There are five sets of equations, three of which are equilibrium equations in the radial, circumferential and transverse directions and the last two are the transverse shear resultants. These equations are first converted to quasi-dynamic by introducing darning and inertia terms to the right-hand-side of the equations.

With the use of a simple approximation for the velocity and acceleration terms, the equations are expressed in initial value format. A finite difference technique is then applied to obtain approximate equations and their numerical solution is achieved by means of the DR method. Finally, in order to verify the numerical results obtained, small and large deflection analyses comparisons are carried out to present both the accuracy and convergence of the DR results.

2. PLATE GEOMETRY AND EQUATIONS

2.1. Geometry

The plate geometry used in the analysis is shown in Fig. 1. The loading on the plate is a uniform pressure. For solving the computational effects, only one half of the plate is considered in the analysis and this is possible because the plate geometry and loading are symmetrical. Therefore, only the AOB section of the plate is analysed. The positive coordinate system and the corresponding displacements in the given directions are presented in Fig. 1.

2.2. Equations

The governing system of non-axisymmetric large deflection thick sector plate equilibrium equations are derived and the final form of the equations is given. The other plate equations comprise compatibility (strain and curvature-displacement), constitutive (stress resultant-strain and stress couple-curvature) and finally boundary conditions relationships.

2.2.1. Equilibrium equations. There are five equations describing the plate static equilibrium state with both in- and out-of-plane deformations. The equations are deduced from a consideratoin of trans- lational equilibrium in the r, fl and z directions and rotational equilibrium about the r and 0 axes. Figure 2 shows the ‘force’ system used to derive the equilibrium equations [eqn (I)]

(N,)“+r~‘(N,)‘+r~‘(N,-N,)=O (la)

(N,o)"+r-'(No)'+2r-'Nd=0 (lb)

(M,0)“+r-‘(M,)‘+2r~‘Md-Q,=0 (Ic)

(M,)“+r-‘(M,)‘+r~‘(M,-M,)-Q,=O (14

r-IQ, + (&?,I” + r -'(Qo)' + N,(w)“”

+ N0(r-‘(w)“+r~2(w)“)

+ 2N,,(r-‘(w)“’ - F2(w)‘) + q = 0. (le)

2.2.2. Compatibility equations. The mid-plane strain and curvature-displacement expressions which

Fig. 1. Plate geometry and positive coordinate system.

Analysis of elastic sector Mindlin plates 989

(b) (4 t

Ne N er

N,

aNre ar

aN + *dr

dr

aM + -$ dr

aMre Mre + ar dr dr

Fig. 2. Plate ‘force’ system: (a) stress resultants; (b) stress couple and transverse resultants.

account for direct bending, stretching and rotation of and k* is the Shear Correction Factor (SCF) which the plate mid-plane are as follows: according to [5] can be taken as 76.

2.2.4. Boundary conditions. With regard to the e: = (u)” + O.S(w)o’ (2a) applications of such plates, two cases of edge con-

ditions, considered to be appropriate, are applied to es = r -‘(u + (u)‘) + 0.5r -‘(w)‘* (2b) the edges of the plate and they are: simply supported

and clamped with in-plane movements fixed. These es = r-‘((u)’ - 0) + (v)o + r-‘(WY(w)’ (2c) constraints are applied along OA and AB, and along

OB symmetrical edge conditions are applied (Fig. 1).

eL = d, + (w) (2d) The displacement and force constraints imposed in each case are given below:

e& = ds + r-‘(w)’ (2e) (a) On the radial edge OA (0 = 0”)

and the curvature expressions which apply for small rotations are:

for a simply supported edge:

u =o =w =f#+=MO=O (4a) k; = (4,) (2f)

for a clamped edge: u = v = w = 4, = & = 0.

G = r-l@, + (4,)‘) (2g) (4b)

k~=r-I(~,)‘+(~,)“-r-‘~, (2h) (b) On the circumferential edge AB (r = r,)

2.2.3. Constitutive equations. For an isotropic plate, for a simply supported edge:

which is considered here, these equations are given in simple forms as follows: u=v=w=M,=&J=o (4c)

N,=A(eF+vei) (3a) for a clamped edge: u=v=w=$,=&=O.

(M) N, = A (es + ve;) (3b) (c) On the symmetrical edge OB (0 = a/2):

Nro = 0.5(1 - v)Ae$ (3c) v=r$,=N,=M,=O. (44

M,=D(k;+vk;) (34 The edges can be either in-plane fixed or in-plane free. If the circumferential edge is in-plane free, then

Mu = D(k; + vk;) (3e) N,, = 0, otherwise N, # 0. If the radial edges are in-plane free, then N, = 0 otherwise N, # 0.

Mti = 0.5( 1 - v)Dk% (30 3. TRANSFORMATION FOR DR NUMERICAL SOLUTION

Q, = 0.5(1 - v)k2AeFz (3g) 3.1. Equilibrium equations

Q. = 0.5(1 - v)k2Aeiz, (3h) In order to facilitate the integration of the equilibrium equations, eqns (1), they must be transformed

in which to a set of quasi-dynamic velocity equations by

A = E/z/(1 - v*), adding inertia and damping terms to the right-hand-

D = Ah*/12 = Eh3/(12(l -v*)) sides of the equilibrium equations. As an example,


eqn (la) is transformed to a quasi-dynamic equation (1) Set all variables except the transverse pressure, as follows: q, to zero.

(N,)“fr-‘(N,)‘+r-‘(N,-N,)=p;u,,,$k;u,,

in which pu and k, are mass density and damping factor respectively.

The following simple approximations for the velocity and acceleration terms are utilized to compute the velocities:

(2) Calculate the velocities utilizing eqns (6). (3) Integrate the velocities to give the displacements

and rotations using eqn (7). (4) Apply displacement boundary conditions at the

radial, circumferential and symmetry edges of the plate using the relevant parts of eqns (4).

(5) Compute the mid-plane strains and curvatures using eqns (2).

U,t = o.s[u; + Upi] Pa)

U,f, = ru,: - $l/Jt. Gb)

Therefore, eqn (la) becomes:

U; = (1 + k,*)-‘((1 - k:,u,“, + Stp;’

x [(N,)” + r-‘(N,,)’ + r -‘(N, - No)]}. @a)

The superscipts ‘a’ and ‘b’ refer to the values of the velocities after and before the time step 6t respectively, and

k,* = O.Sk;Gt -p;’ .

(6) Compute the stress resultants, stress couples and transverse shear resultants using eqns (3).

(7) Apply stress resultants, stress couples and transverse shear resultant boundary conditions utilizing the relevant parts of eqns (4).

(8) Check if the velocities are negligibly small, i.e., less than 10m6 everywhere, and the static solution has been obtained.

For the remainder of the equilibrium equations the subscript u is replaced by the appropriate subscripts as follows:

(9) If step (8) is satisfied print out the results, otherwise return to step (2) and repeat the sequence.

The above procedure is coupled with a suitable selection of ps and ks in eqns (6). These variables were determined by ‘trial and error’ until Welch [lo] suggested the use of fictitious densities. This method was used to solve a non-linear structure [l 11. Also, it was used in [3] and it proved to be very successful. To estimate the values of the damping factors, ks, a simple expression originated from [4] which in terms of the minimum fictitious densities and a factor F is used as follows:

v; = (1 + k:)-‘{(l - k,t)u,‘, + 6tp;’

x [(NJ’+ r-‘(No)’ + 2r-‘N,]} (6b)

4% = (1 + k&d-‘W - kZ>#lt + %c&’

1 f 1 x IGhY + r- (61 + 2r- 40 - Qd tW

566, = (1 + k;, )-‘((I - k$, M;, + Qi,’ KM,)

+ r -‘(M,,j’ + r-W, - MO) - Q,]] (6d)

w;=(l +k;)-‘((i -k~)w,~+~?~~’

x [r -‘Q, + (Ql)” + r-‘(Q,,)’ + N,(w)00

k,= (~uhnin/F.

The factor F is in the range specified below:

0.001 < F < 2.

For the remaining ks the appropriate subscripts in place of u are used.

4. FINITE DIFFERENCE DISCRETIZATION OF PLATE RQUATIONS

+ N,(r-‘(w)” + rF2(w)“)

+ 2N~(r-‘(~~ - r-‘(w)‘) + q]). ( W

With the use of the following simple integration rule

ua = ub +&u;, (7)

the displacement is computed where ‘a’ and ‘b’ are defined as previously.

In order to obtain numerical solutions for the plate equations, the DR equations are first non- dimensionalized and then eqns (7), (6), (2) and (3) are discretized. Various techniques may be used, i.e., finite element, finite difference, boundary element, etc. The method used here is the finite difference which suits the DR method and has also been previously used in [3,4] and others, some of which are given in [3]. In most applications an interlacing finite difference is used, here however, it was decided to use non-interlacing meshes, since the difference in the results is small [4].

The remaining displacements and rotations are computed with the use of eqn (7) by replacing u with the relevant variable. To apply the DR technique the following sets of operations are required:

5. NUMERICAL RESULTS AND DISCUSSION

Because of the large quantity of comparison and new results produced in the course of the programme

Analysis of elastic sector Mindlin plates 991

Table 1. A comparison of DR and Kobayashi [12] and Turvey and Saiehi’s [3] small deflection results for 60” sector plates simply supported at the radial and circumferential

edges (v = 0.3, 4 = 1)

0.~

0.0588

0.1765

0.2941

0.4118

0.5294

0.6471

0.7647

0.8824

1.0000

0.5294

0.5294

0.5294

0.5294

0.5294

0.5294

30

30

30

30

30

30

30

30

30

30

0

6

12

18

24

30

O.OOOOO 0.00000 0.00000

0.~ 0.00003 0.00006

0.00088 0.00079 0.00080

0.003 14 0.00296 0.00319

0.00635 0.00618 0.00654

0.00937 0.09930 0.00976

0.01087 0.01094 0.01146

0.00981 0.00997 0.01043

0.00589 0.00605 0.00644

0.~ 0.00000 0.00000

0.00000 0.00000 0.~00

0.00297 0.00296 0.00311

0.00561 0.00577 0.00585

0.00765 0.00759 0.00797

0.00893 0.~886 0.00930

0.00937 0.00930 0.00976

0.0000 0.0000 0.0000

- 0.0039 - 0.0034 -0.0037

-0.0051 -0.0052 -0.0045

- 0.0008 -0.0008 -0.0001

0.0083 0.0075 0.0081

0.0170 0.0165 0.0167

0.0232 0.0231 0.0229

0.0240 0.0241 0.0237

0.0168 0.0170 0.0169

0.0000

YI:E

0.0000 0.0000 0.0009

0.0057 0.0056 0.0057

0.0105 0.0102 0.0103

0.0140 0.0137 0.0138

0.0162 0.0158 0.0160

0.0170 0.0165

0.0000 0.0000 0.0000

0.0049 0.0023 O.0058

0.0130 0.0113 0.0109

0.0199 0.0183 0.0194

0.0241 0.0232 0.0240

0.0257 0.0253 0.0259

0.0246 0.0246 0.0250

0.0207 0.0208 0.0210

0.0140 0.0140 0.0141

0.0044 0.0042 0.0044

0.0000 O.OOOfl 0.0000

0.0101 0.0099 0.0101

0.0173 0.0171 0.0174

0.022 1 0.0218 0.0222

0.0248 0.0245 0.0250

0.0257 0.0253 0.0259

0.0000

O.?&OO

0.~ ?

o.oooo 0.0000

? 0.0000

0.0000 ?

0.0000

0.0000 ?

0.0000

0.0000 ?

0.0000

o.oooo ?

o.oooo o.oooo ?

o.oooo 0.~~ ?I

O.OOOO$

0.0000 ?

0.0000

-0.0017 ?

-0.0018

-0.0014

-O.?bolS

-0.0009 ?

-0.0009

-0.0005 ?

-0.0005

-0.0002 ?

- 0.0002

0.0000 ?

0.0000

t Results from 1121.

0.0167

. _ $ Results from [3]. 5 DR results for ho/r, = 0.05 (8.5 x 16 mesh arrangement). ? The value is not specified in the original paper.


Table 2. A comparison of DR and Kobayashi [12] and Turvey and Salehi’s [3] small deflection results for 60” sector plates simply supported on the radial edges and clamped

at the circumferential edge (v = 0.3, 4 = 1)

0.0588

0.1765

0.2941

0.4118

0.5294

0.6471

0.7647

0.8824

1.0000

0.5294

0.5294

0.5294

0.5294

0.5294

0.5294

30

30

30

30

30

30

30

30

30

30

0

6

12

18

24

30

0.00000 0.00000 0.00000

0.00004 0.00003 o.OOoO5

0.00074 0.00068 0.00068

0.00252 0.00245 0.00263

0.00477 0.00484 0.0051 I

0.00647 0.00678 0.00711

0.00659 0.00717 0.00749

0.00478 0.00550 0.00578

0.00181 0.00238 0.00253

0.00000 0.00000 0.00000

O.OOOOO 0.00000 0.00000

0.00207 0.00218 0.00299

0.00390 0.00408 0.00429

0.00530 0.00551 0.00583

0.00617 0.00647 0.00679

0.00647 0.00678 0.00711

0.0000 o.oooo 0.0000

-0.0032 -0.0033 -0.0031

-0.0035 - 0.0038 -0.0032

0.0018 0.0010 0.0015

0.0093 0.0086 0.0090

0.0156 0.0153 0.0151

0.0172 0.0176 0.0170

0.0111 0.0122 0.0112

-0.0056 -0.0036 - 0.0048

- 0.0340 -0.0315 -0.0331

0.0000 0.0000 0.0000

0.0053 0.0052 0.0052

0.0097 0.0095 0.0094

0.0129 0.0127 0.0125

0.0149 0.0146 0.0145

0.0156 0.0153 0.0151

0.0000 0.0000 0.0000

0.0043 0.002 I 0.0050

0.0115 0.0101 0.0095

0.0167 0.0159 0.0166

0.0194 0.0193 0.0199

0.0194 0.0199 0.0202

0.0166 0.0174 0.0174

0.0108 0.0117 0.0115

0.0194 0.0028 0.0024

0.0000t

O.O&

0.0000 7

o.odoo 0.0000

? 0.0000

0.0000 ?

0.0000 0.0000

? 0.0000

0.0000 ?

0.0000

0.0000 ?

0.0000

0.0000 ‘I

o.odoo

-0.0102 -0.0095 - 0.0099

0.0000 0.0000 0.0000

0.0081 0.0082 0.0083

0.0000t ?I

o.oooog 0.0000

? 0.0000

0.0007 ?

0.0003

0.0136 0.0139 0.0140

0.0009 ?

0.0005

0.0011 1

0.0008

0.0170 0.0010 0.0174 ?

0.0176 o.odo7

0.0189 0.0006 0.0193 ? 0.0196 0.0004

0.0194 0.0199 0.0202

0.0000 ?

0.0000

t Results from [12]. $ Results from [3]. 0 DR results for ho/r, = 0.05 (8.5 x 16 mesh arrangement). ?The value is not specified in the original paper.

Analysis of elastic sector Min~lin plates 993

Table 3. A comparison of DR and Turvey and Salehi [3] and Morely’s [13] small deflection results for 90” sector plates

clamped at all edges (v = 0.3,4 = 1)

0.~

0.0588

0.1765

0.2941

0.4118

0.5294

0.647 1

0.7647

0.8824

l.OOoO

45

45

45

45

4.5

45

45

45

4.5

45

0.~~ 0.~ o.ooooo 0.00000

? 0.00004

0.00061 0.~55 0.~58

0.00254 0.00240 0.00268

0.00532 0.0049 1 0.0@552

0.00755 0.00688 0.00761

0.0079 1 0.00732 0.00788

0.00594 0.00546 0.00590

0.00250 0.00186 0.00250

0.~ 0.~~ 0.00000

0.0000 0.~ o.oooo

-0.0029 - 0.0028 - 0.0048

-0.0069 -0.0077 -0.0063

-0.0022 -0.0019 -0.0014

0.0075 0.0074 0.0080

0.0159 0.0160 0.0149

0.0184 0.0180 0.0164

0.0119 0.0110 0.0099

-0.0051 -0.0077 -0.0060

-0.0331 - 0.0350

o.oooot 0.~~ O.ooooC

-0.0020 -0.0028 -0.0010

-0.0045 -0.0044

0.0022

0.0098 0.0100 0.0091

0.0139 0.0140 0.0136

0.0157 0.0150 0.0148

0.0144 0.0140 0.0133

0.0099 0.0096 0.0088

0.0019 0.0013 0.0014

-0.0099 -0.0099

- 0.0326 - 0.0098 0.0588 60 t Results from [3]. $ Results from [12]. $ DR results for A&, =0.05 (8.5 x 16 mesh arrange-

ment). ?The value is not specified in the original paper.

of the work, here, it is decided to only present the comparison results and a limited number of new large deflection results for thickness ratios of 0.01, 0.05,O.l and 0.2. The comparison results are grouped under the type of analysis, i.e., thick or thin plate, and small or large deflection results.

5.1. Small deflection thin plate results

Table 1 gives a set of results for simply supported sector plates with a sector angle of 60” and g = 1. The results correspond to dimensionless deflections (Gt), radial stress couples (JZr), circumferential stress couples (li;i) and twisting stress couples (li;i,). The angle ~9 represents the radial line of symmetry and the results are along this line at different mesh points and the mesh size is 8.5 divisions in the radial direction and 16 angular divisions in the circumferential directions. Also in Table 1, the results at F = 0.5294 and different sector angles for the above mentioned quan- tities are given. The mesh point at P = 0.5294 ap- proximately gives the maximum deflection, consequently, the numerical values of the other quan-

tities are given at this point. The question mark means that the value is not specified in the original paper. As seen from a comparison of the three sets of results, the differences between the results are only minor. The results in the second row are taken from f3] in which the same technique was used as in the present analysis but the analysis was for thin plates. Here, however, the ratio of h,/r, (plate thickness to plate radius) specifies how thick or thin the plate is. This ratio for this set of results in the third row is given as 0.05. Therefore, it is assumed that the plate is thin. The results given in the first row are derived from an exact solution [12] as claimed by the author. One reason for the minor differences could be the effect of the value of h,/r,. In most cases the values obtained in the present analysis are slightly higher than in the other two. In general, however, there is a good agreement between the three sets of results for all the variables specified.

Table 2 gives exactly the same set of results but for a circumferentially clamped and radially simply sup ported plate. Here again, the values for the variables

Table 4. A comparison of DR and Turvey and Salehi [3] and Merely’s [13] small deflection results for 120” Sector plates

clamped at all edges (v = 0.3,4 = 1)

P 0” P J& Ic3, 0.0000 60 O.OOOOO

0.1765 60

0.~ 0.~

0.00013 0.00016 0.00019

0.00217 0.00207 0.00204

-0.0432

o.oooo 0.~ 0.~

-0.0132 -0.0140 -0.0154

-0.0123 -0.0110 -0.0103

0.0031 0.0036 0.0042

0.0177 0.0170 0.0177

0.0252 0.0240 0.0233

0.0232 0.0230 0.0212

0.0109 0.0100 0.0090

-0.0116 -0.0100 -0.0121

-0.0438 -0.0440

0.0000t 0.~~ 0.~

-0.0028 ?

-0.0040

-0.0059 -0.0061

0.0027

0.2491 60

0.4118 60

0.00655 0.00622 0.00709

0.01104 0.01016 0.01130

0.01342 0.01201 0.01360

0.01252 0.01092 0.01254

0.00857 0.00753 0.00857

0.00331 0.00273 0.00335

O.OOOOO 0.~ 0.~

0.0125 0.0120 0.0130

0.0168 0.0160 0.0166

0.9254 60 0.0181 0.0180 0.0171

0.6471 60

0.7647 60

0.8824 60

l.OoOO 60

-0.0130

0.0160 0.0150 0.0150

0.0103 0.0100 0.0093

0.0006 ?

0.0003

-0.0131 -0.0140

t Results from [3]. $ Results from [13] estimated from graphs. 4 Results for ho/r, = 0.05 (8.5 x 16 mesh arrangement). ?The value is not specified in the original paper.


Table 5. A comparison of DR and Turvey and Salehi [3] and Merely’s [13] small deflection results for 180” sector plates

clamned at ednes (v = 0.03. d = 1)

0.0588

0.1765

0.2941

0.4118

0.5294

0.6471

0.6747

0.8824

1.0000

0.00000 o.ooooo o.ooooo 0.00074 0.00148 0.00149

0.00710 0.00797 0.00794

0.01615 0.01529 0.01691

0.02197 0.02075 0.02232

0.02307 0.02184 0.02298

0.01933 0.01747 0.01935

0.01221 0.01092 0.01205

0.00433 0.00415 0.00435

0.00000 0.00000 0.00000

-0.0749 -0.0730 - 0.0809

-0.0484 -0.0440 - 0.0494

-0.0065 -0.0050 -0.0041

0.0191 0.0190 0.0177

0.0340 0.0330 0.0328

0.035 1 0.0340 0.0316

0.0256 0.0250 0.0240

0.0064 0.0061 0.0052

-0.0215 -0.0260 -0.0218

-0.0573 - 0.0580 - 0.0546

-0.0263t - 0.0220$ - 0.03035

-0.0148 -0.0130 -0.0172

-0.0024 -0.0025

0.0014

-0.0121 -0.0110

0.0100

0.0182 0.0170 0.0171

0.0200 0.0190 0.0180

0.0174 0.0170 0.0162

0.0103 0.0100 0.0095

-0.0013 -0.0012 -0.0016

-0.0172 -0.0180 -0.0016

t Results from [3]. $ Results from [13] estimated from graphs. §DR results for ho/r, = 0.05 (8.5 x 16 mesh arrange-

ment). ?The value is not specified in the original paper.

estimated by the present analysis are slightly higher than the values given in [3] and [12].

Further small deflection result comparisons are carried out for clamped plates with sector angles of 90”, 120” and 180”. The results are presented in Tables 3-5 for I?‘, li;i, and M0 along the radial line of symmetry. A comparison of the results is with two other sources. The first row of results are taken from [3] and the second row of the results are presented in graphical form in [13]. As before, the third row shows the results from the present analysis for the ratio of h,/r, = 0.05 and the same mesh size as before. The present deflection results are again slightly higher than the other two sets, but generally in good agreement. The stress couple results vary, at some mesh points the present results are higher than the other two or vice versa.

5.2. Large dejection thin plate results

Large deflection comparison result sets are tabu- lated in different forms. Tables 6 and 7 give the

approximate maximum deflections at i = 0.6471 for sector plates with sector angles of 30”, 60”, 90” and 120” and at P = 0.5294 for a sector angle of 180”. The tables give the results of the present analysis for h,/r, = 0.01,0.05,0.1 and 0.2 and compare them with the thin plate results reported in [3]. As can be seen the present results for h,/r, = 0.01 which represent a thin plate, are in good agreement with the results given in [3].

These are the only large deflection sector plate results which the authors were able to find in the literature. As is observed, the non-dimensional deflections for a thick plate are larger than those for a thin plate. This is because of the nature of the non-dimensionalization of the variables. The actual dimensional load, for example, for the ratio of h,/r, = 0.2 is larger than that of h,/r, = 0.1, although the dimensionless loads are equal but the dimensional loads are not. Therefore, the higher the ratio of h,/r, gets the larger the actual loads become which results in higher dimensionless deflections (see Fig. 3).

The second set of comparison tables are Tables 8 and 9. Here, three sets of results are compared. The first row in the tables for different qs corresponds to the results reported in [3]. The results in the second row are obtained by the ANSYS finite element computer code [14] as reported in [3]. The remaining rows for each q correspond to the present analysis for h,/r, = 0.01, 0.05, 0.1 and 0.2. The numerical results given are for Wn at r = 0.6471 specified as point D, i%if and @ at ? = 1 specified as point B, as well as p and &’ at point D and also li;i, and I%!?, at points B and D. The plate specificatons are a 60” sector angle with simply supported (Table 8) and clamped (Table 9) edges. Here again, the results are in good agreement with the other two sets of results and as the ratio of h,/r, increases so do the values of the deflections and stress resultants, but the stress couples decrease.

9 8

6

4

2

Fig. 3. q (=cjE,hi/rt) versus ho/r, assuming 4 = I and 2, = 104.

Analysis of elastic sector Mindlin plates

Table 6. Near centre deflection versus pressure results for simply supported plates of various sector angles (v = 0.3. i = 0.5294 for 180” sector plates and i = 0.6471 for all

others

995

4 ho/r, 30 60” 90 120 180

10 0.01 0.05 0.1 0.2

20 0.01 0.05 0.1 0.2

30 0.01 0.05 0.1 0.2

40 0.01 0.05 0.1 0.2

50 0.01 0.05 0.1 0.2

0.00943 0.09047 0.2332 0.3756 0.5765t 0.00989 0.07169 0.2426 0.3578 0.5213$ 0.01553 0.10893 0.2474 0.3649 0.5271 0.01946 0.12011 0.2541 0.3666 0.5252 0.03200 0.15078 0.2865 0.3948 0.5441

0.01884 0.1736 0.4009 0.5860 0.8069 0.01837 0.1400 0.4158 0.5583 0.7596 0.03224 0.2118 0.4203 0.5633 0.7630 0.03924 0.2273 0.4283 0.5655 0.7616 0.06345 0.2708 0.4599 0.5864 0.7735

0.02820 0.2460 0.5222 0.7271 0.9585 0.02630 0.2140 0.5399 0.6924 0.9164 0.04885 0.2996 0.5438 0.696 1 0.9188 0.05881 0.3164 0.5510 0.6980 0.9171 0.09320 0.3624 0.5781 0.7140 0.9252

0.03749 0.3085 0.6169 0.8349 1.075 0.05968 0.3709 0.6355 0.7931 1.035 0.06530 0.3739 0.6395 0.7977 1.038 0.07801 0.3906 0.6458 0.7992 1.036 0.12084 0.4353 0.6688 0.8118 1.042

0.04670 0.3628 0.6949 0.9233 1.170 0.07770 0.4344 0.7148 0.8774 1.134 0.08152 0.4375 0.7182 0.8813 1.137 0.09675 0.4536 0.7236 0.8823 1.135 0.14631 0.4958 0.7434 0.8924 1.139

t Turvey and Salehi’s results [3] for thin sector plates. $ DR results for ho/r, = 0.01, 0.05, 0.1 and 0.2 (8.5 radial sub-divisions and about 4”

for 60).

Table 7. Near centre deflection versus pressure results for clamped plates of various sector angles (v = 0.3). i = 0.5294 for 180” sector plates and P = 0.6471 for all others

(7 0, 30” 60” 90 120” 180”

10

20

30

40

50

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.00208 0.02627 0.0747 1 0.00325 0.02470 0.05784 0.00472 0.03718 0.08387 0.00848 0.05049 0.10442 0.02213 0.09569 0.17606

0.00416 0.05245 0.1481 0.00650 0.05007 0.1169 0.00983 0.07601 0.1690 0.01716 0.10165 0.2080 0.04567 0.18849 0.3206

0.1312 0.22547 0.1213 0.2177$ 0.1316 0.2300 0.1519 0.2478 0.2391 0.3451

0.2562 0.4220 0.2378 0.4090 0.2548 0.4264

0.00624 0.07849 0.2191 0.00976 0.07635 0.1773 0.01498 0.11433 0.2498 0.02588 0.15229 0.2942 0.06941 0.27059 0.4425

0.00832 0.1043 0.2870 0.01303 0.1034 0.3035 0.02012 0.1520 0.3286 0.03468 0.2017 0.3928 0.09264 0.3423 0.5481

0.2944 0.4682 0.4229 0.5970

0.3709 0.5845 0.3433 0.5645 0.3655 0.5858 0.4185 0.6325 0.5638 0.7776

0.4745 0.7192 0.4352 0.6921 0.4633 0.7178

0.01039 0.1299 0.3512 0.01630 0.1345 0.3690 0.02526 0.1888 0.3987 0.04355 0.2494 0.4720

0.5257 0.7716 0.6770 0.9174

0.5675 0.8336 0.5185 0.8043 0.5500 0.8277 0.6190 0.8885 0.7715 1.0322 0.11468 0.4055 0.6332

t Turvey and Salehi’s results [3] for thin sector plates. $ DR results for ho/r, = 0.01, 0.05, 0.1 and 0.2 (8.5 radial sub-divisions and about 4”

for SO).


Table 8. A comparison of DR and Turvey and Salehi [3] and ANSYS’s[l4] large deflection results for 60” sector plates with simply supported edges (v = 0.3)

4 h, ir, mn P , E P I &?

100

200

300

400

500

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.6614 2.557 0.7671 2.464 2.712t 0.6698 2.879 0.8304 2.680 2.806$ 0.6609 2.281 0.6842 2.466 2.74Q 0.6647 2.321 0.6963 2.500 2.777 0.6751 2.437 0.7310 2.592 2.878 0.7057 2.804 0.8413 2.883 3.199

0.9311 5.169 1.551 4.921 5.407 0.9441 5.816 1.685 5.367 5.613 0.9271 4.614 1.384 4.894 5.423 0.9290 4.666 1.400 4.926 5.457 0.9343 4.814 1.444 5.014 5.552 0.9525 5.251 1.575 5.305 5.870

1.109 7.428 2.228 1.014 7.697 1.125 8.372 2.427 7.671 8.019 1.103 6.654 1.996 6.956 7.709 1.104 6.712 2.014 6.956 7.709 1.107 6.873 2.062 7.072 7.823 1.119 7.342 2.203 7.351 8.126

1.245 9.46 2.838 8.882 9.739 1.260 10.65 3.078 9.696 10.120 1.239 8.50 2.551 8.817 9.752 1.239 8.56 2.596 8.841 9.776 1.240 8.73 2.620 8.916 9.856 1.248 9.22 2.766 9.179 10.142

1.357 11.33 3.400 10.60 11.61 1.373 12.70 3.694 11.55 12.06 1.351 10.22 3.065 10.52 11.63 1.350 10.28 3.084 10.54 11.65 1.350 10.45 3.135 10.61 11.72 1.356 10.95 3.285 10.86 12.00

4 100

ho lr,

0.01 0.05 0.1

200

0.01 0.05 0.1 0.2

300

0.01 0.05 0.1 0.2

400

0.01 0.05 0.1 0.2

500

0.01 0.05 0.1

0.0000 0.0000 0.0000 o.oooo 0.0000

o.oooo 0.0000 o.oooo 0.0000 o.oooo o.oooo 0.0000 0.0000 o.oooo 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 o.oooo o.oooo 0.0000

0.2588 1.314 ? ?

0.2697 1.267 0.2644 I .245 0.2448 1.181

1.41 It

2 1.397 1.373 1.303

0.3687 ?

0.3847 0.3752 0.3486 0.2836

1.769 ‘I

1.706 1.676 1.588 1.308

1.915 ?

1.888 1.855 1.575 1.444

0.4427 2.046 2.26 ? ? ?

0.4615 1.980 2.19 0.4491 1.948 2.16 0.4152 1.851 2.05 0.3222 1.528 1.69

0.5002 2.251 ? ?

0.5208 2.187 0.5063 2.154 0.4668 2.053 0.3601 1.699

2.456 ?

2.423 2.387 2.273 1.876

0.5481 2.470 2.642 ? ? ?

0.5699 2.356 2.610 0.5536 2.323 2.574 0.5095 2.220 2.457

0.2 0.0000 0.3917 1.842 2.033

t Results from [3] for thin sector plates. $ Results from [14] as reported in [3] for thin sector plates. $ DR results for ho/r, = 0.01, 0.05, 0.1 and 0.2 (8.5 x 8 mesh arrangement). ?The value is not specified in the original paper.

Analysis of elastic sector Mindlin plates

Table 9. A comparison of DR and Turvey and Salehi [3] and ANSYS’s [14] large deflection results for 60” sector plates with clamped edges (v = 0.3)

991

100

200

300

400

500

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.01 0.05 0.1 0.2

0.3439 0.4190 0.3095 0.4729 0.3296 0.3941 0.3601 0.3720 0.4576 0.3483 0.6404 0.6909

0.1 57 0.1 68

& 0.11182 0.116 0.1045 0.2073

0.5946 0.6509t 0.5777 0.6183$ 0.6297 0.6168$ 0.7174 0.7000 1.0317 0.9424 2.0599 1.6546

0.6197 1.382 0.4145 1.913 2.091 0.561 I 1.584 0.4554 1.873 2.001 0.5641 1.251 0.3753 1.912 1.949 0.6292 1.116 0.3348 2.197 2.126 0.7544 1.060 0.3180 2.926 2.562 0.9260 1.812 0.5436 4.471 3.599

0.8331 2.537 0.7611 3.423 3.737 0.7738 3.010 0.8758 3.553 3.792 0.7489 2.240 0.6719 3.338 3.444 0.8310 1.925 0.5774 3.830 3.668 0.9629 1.887 0.5662 4.849 4.172 1.1167 2.867 0.8601 6.577 5.232

1 II050 3.743 1.123 4.935 5.379 0.9408 4.505 1.313 5.217 5.560 0.8964 3.247 0.974 4.742 4.930 0.9919 2.728 0.818 5.447 5.171 1.1246 2.734 0.820 6.666 5.688 1.2640 3.842 1.153 8.476 6.887

1.148 4.950 1.081 6.029 1.019 4.240 1.126 3.512 1.258 3.567 1.386 4.753

1.485 6.399 1.758 6.865 I.272 6.089 1.054 7.011 1.070 8.375 1.426 10.229

6.964 7.304 6.367 6.606 7.111 8.334

Q ho ho lw I E w , lizi

100 - 2.018 - 0.6055 1.005 1.08lt ? ? ? ?$

0.01 - 1.989 - 0.5968 0.967 1.0305 0.05 -2.159 - 0.6478 0.949 1.058 0.1 - 2.399 - 0.7198 0.881 1.066 0.2 - 1.953 - 0.5859 0.698 0.840

200 - 3.720 - 1.116 I .745 1.883 ? ? ? ?

0.01 - 3.438 - 1.031 1.672 1.777 0.05 - 3.875 - 1.163 1.586 1.772 0.1 -4.152 - 1.246 1.383 1.652 0.2 - 2.823 - 0.847 1.017 1.182

300 - 5.119 - 1.536 2.255 2.441 ? ? ? ?

0.01 - 4.493 - 1.348 2.181 2.314 0.05 - 5.231 - 1.569 2.012 2.253 0.1 - 5.397 - 1.619 1.711 2.027 0.2 - 3.393 - 1.018 1.229 1.407

400 - 6.302 - 1.891 2.622 2.848 ? ? ? ?

0.01 - 5.314 - 1.594 2.568 2.719 0.05 - 6.351 - 1.907 I.959 2.305 0.1 - 6.358 - 1.907 I.959 2.305 0.2 - 3.831 - 1.149 1.329 1.581

500 - 7.331 - 2.199 2.903 3.160 ? ? ? ?

0.01 - 5.988 - 1.796 2.878 3.041 0.05 - 7.311 - 2.193 2.564 2.877 0.1 - 7.145 - 2.143 2.160 2.527 0.2 -4.193 - I.258 1.527 1.724

t Results from [3] for thin sector plates. # Results from [14] as reported in [3] for thin sector plates. Q DR results for ho/r, = 0.01, 0.05, 0.1 and 0.2 (8.5 x 8 mesh arrangement). ? The value is not specified in the original paper.


6. CONCLUSIONS

A geometrically non-linear formulation is developed for the large deflection analysis of moderately thick sector plates which takes into account the effect of shear defo~ation. The Dynamic Relaxation method together with the finite difference discretization technique is used to solve the partial differential equations of the plate. A FORTRAN computer program is then developed to carry out the numerical calculations for the deflections, stress resultants and stress couples. In order to verify the validity of the numerical results comprehensive comparison tables are produced and the present DR results are in good agreement with the other two sets of results from two independent sources.

~ck~o~~e~ge~enr_Both authors wish to acknowledge the support and encouragement for this research work provided by the Mechanical Engineering Department.

I.

2.

REFERENCES

T. von Karman, Festjgkatsprobieme in M~hinenbaa. E~~yk~opuedic der ~athe~at~chen ~isse~sc~~ten 4, 348 (1910). A. Nadai, 2. Verein deutscher ingenieure 59,169 (1915). [Cited in S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, p. 295. McGraw-Hill, New

3. G. J. Turvey and M. Salehi, DR large deflection analysis of seetor plates. Comput. Sfruct. 34(l), 101-l 12 (1990).

4. G. J. Turvey and M. Y. Osman, Elastic large deflection analysis of isotropic rectangular Mindlin plates. Int. J. Mech. Sci. 32(4), 315-328 (1990).

5. E. Reissner, The effect of transverse shear defo~ation on the bending of elastic pIates. J. appl. Mech. 12, A69-A77 (1945).

6. R. D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. appl. Mech. 18, 31-38 (1951).

7. J. N. Reddy and W. C. Chao, Non-linear bending of thick rectangular, laminated composite plates. Int. J. eon-linear Mech. 16, 291 (1981).

8. Z. G. Azizian and D. J. Dawe, Geometricahy non- linear analysis of rectangular Mindlin plates using the finite strip method. Compur. Strucr. 21, 423-436 (1985).

9. K. R. Rushton, Large deflection of variable-thickness plates. Znf. J. Mech. Sci. 10, 723 (1968).

10. A. K. Welch, Discussion on dynamic relaxation. Proc. fnst. Civ. Engrs 37, 723-750 (1967).

I I. A. C. Cassell and R. E. Hobbs, Numerical stability of dynamic relaxation analysis of non-linear structures. ht. J. Numer. Meth. Engng 10, 1407-1410 (1976).

12. H. Kobayashi, Private communication. Department of Civil Engineering, Osaka City University, Osaka, Japan (1988).

13. L. S. D. Merely, Variational reduction of the ciamped plate to two successive membrane problems with an application to uniformly loaded sectors. Q. J. Mech. appl. Math. 16, 451-471 (1963).

14. ANSYS, Swanson Analysis System Inc., P.O. Box 65, York (1959)]. Houston, PA 15342, U.S.A.

Large deflection analysis of elastic sector Mindlin plates

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