8/3/2019 5tk

1/9

Compufns & Sfr ucrur cs. Vol. 14. No. 3-4. PP. 195-203. 1981 CM-7949/81 03019549$02.00/ 0Printed in G eat Brit ain. Pergamon Press Ltd.

NU MERICAL ANALYSIS OF ELASTIC PLATES WITHTWO OPP OSITE SIMP LY SUPPORTED EN DS BY

SEGMENTATION METHOD

TARUN KANTtDepartment of Civil Engin eering, Indian Insti tute o f Technol ogy, Bombay 400 076, India

(Received 24 June 1980; received for pub lication I December 1980)Abstract-A method for the numerical analysis o f elastic plates with two opposite simply sup ported ends ispresented. A variety of boundary condition s includi ng the mixed and the nonhomog eneous types can be prescribedalong either of the remaining two oppo site edges. Numerical results are presented for the three examples. Based onthe comparisons made with the results available elsewhere, it is concluded that the present method is efficient,economi cal, reliable and very accurate.

INTRODUCTIONThe present paper is concerned with the numericalanalysis of rectangular plates with different boundaryconditions. A less known formulation which is originallydue to Goldberg et a/.[11 is used in the present study.The system of equations are then numerically integratedby using a so-called segmentation method which iscomprehensively documented in a recent publication[2].This method is found to be efficient, reliable, accurateand computationally economical for a certain class ofplate problems. The plate is assumed to be simply sup-ported along two opposite edges. Different boundaryconditions are then prescribed along the other twoopposite edges.

PROBLEM FORMULATIONThe present study is based on the thin plate theory[3]due to Kirchhoff with the following assumptions:(1) Material is homogeneous, isotropic and linearelastic.(2) Deflections are small compared to the thickness ofthe plate.(3) Normals to the reference surface before defor-mation remain straight and normal to the deformedreference surface and their length remain unchanged(rxz = ?YZ G = 0).(4) Transverse normal stress components acting onplanes parallel to the reference surface is neglectedcompared to other stress components (uz = 0).The well-known governing equations of such a theorywhich defines a boundary value problem are summarisedin Appendix A. Numerical integration of such a boundaryvalue problem by the segmentation method[2] which isoriginally due to Goldberg et a/.[11 involves first al-gebraic manipulation of the basic equations so as toobtain a set of first order differential equations-calledthe intrinsic equations involving only some particulardependent variables-called the intrinsic variables, thenumber of which equals the order of the partial differen-

tial equation system of such a theory (fourth order inthe present case). Then out of the two independentcoordinates which describe the problem, one is chosen tobe the preferred one. In the present analysis, x-coor-dinate is selected as the preferred one. Intrinsic equa-tions are then derived consisting of a system of firsttAssistant Professor: Presently a Visitor at the Deoartment ofCivil Engineering , University of kales, Swansea SA28PP (U.K.)under the Jawaharlal Nehru Memorial Trust (U.K.) Scholar-ship-1979 Award.

order partial differential equations each of which con-tains necessarily the first derivative with respect of x ofone of the so-called intrinsic dependent variables whichappear naturally in the boundary conditions on the edgex = a constant. In the present analysis, y defining thevector of intrinsic variables, consist of the dependentvariables IV,&, V, and M,. After the required manipula-tions, the system of equations are obtained in the fol-lowing form:

(lb)~=D(,-v2)$,~ -(Pz+PZ-+Ph) (14

+$=-20(1-v)+ V,.The other dependent variables are expressed as func-tions of intrinsic variables by simple algebraic reiations,called the auxiliary relations in the following form:

MY=-D(I-V)+MxM,, = D(l - V)2

Qx=D(l-@t$$Qy=-D(bv)$+V,=-D(1-u)$+(2-+ (2e)

The generalised displacement conponents and the cor-responding stress resultants which form the vector j ofthe intrinsic variables are functions of x and y, and for aplate with two opposite edges, y = 0 and y = b as simplysupported, these may be represented in the form of aFourier series which automatically satisfies both thedisplacement and the force boundary conditions alongthese edges, to any desired degree of accuracy as fol-lows:w(x, y) = z w,(x) sin (2m - 1)7m (34

I95

8/3/2019 5tk

2/9

196 T. KANT&(x,y)=z &,(x)sin(2m-I)? (3b)mV,(x,y)=z V,,(x)sin(2m-I)? (3c)m

M,(x,y)=~M,,(X)sin(2m-1)~. (3d)mSubstitution of the expansions (3ad) in the system ofdifferential equations (lad) and analytic integration ofthese equations with respect to the indpendent variable(coordinate) y coupled with the use of the orthogonalityconditions of the basic beam functions used in the y-direction in the aforesaid expansions reduces the set ofpartial differential equations (lad) into the following setof simultaneous first order ordinary differential equations(say for the mth harmonic) involving only the fourintrinsic variables. It may be mentioned that it is notnecessary to express the external loads in the form of aFourier series in the y-direction unlike the analyticalO]and the semi-analytical[4] methods. Further it is notedthat the series uncouple with respect to the harmonic mleading to a term-by-term analysis which enables storageof only the final discrete values of the intrinsic and theauxiliary dependent variables corresponding to a particular harmonic analysis to be added to the values of thesubsequent harmonic analyses.

dw t-$- = - (44%=-,(2m-I)

D(1 - V2)(2m Ir(f)w, t v(2m - I)x 9 *MxMxm0b - q&j (Pl + pr- + ph) (4c)

$$ = 2D(l- v)(2m - I)($&, + V,, (44and the auxiliary relations (2a-e) take the form:

MY= 2 [D(l- v2)(2m - I)* 5 W,m 0t vM,,] sin(2m- I)? (5a)

MXY z D(1 - v)(2m - 1) (a) L cos (2m - I]? (Sb)QX= 2 [&I- v)(2m -l)*(~)fLm

+ VXln1in (2m- 1) y (54Q, = C [D(l- u)(2m - l)*(~)w,,, + AL,,] (2m 1)

mx f cos(2m-I)?0 (5d)V, = 2 [ D( I - u)*(2m - I) (5) w, + (2 - v)M,.]m

X(2m-I) f cos(2m-I)?0 (Se)tFor the sake of brevity W,(X), &,(x), etc. are hereafterwritten Fimplyas w,, M,,, etc.

The equations (4ad) are numerically integrated by thesegmentation method[2] for th mth harmonic at a timeand the discrete point values of all the dependent vari-ables are obtained by summing the corresponding valuesgot for the given number of harmonics as given by therelations (3a-d) and (Sa-e).NUMERICAL EXAMPLES

Numerical results are presented for a square plate ofside a and thickness h, simply supported along thetwo opposite edges, y = 0 and y = a with the boundaryconditions w, = MY,,, 0 and loaded with a uniformlydistributed load pz+. Discrete numerical values of thedependent variables are presented in the non-dimensionalform as follows:wP$ @a)

M, = /3pz+a2 (6b)MY= &pz+a2 (64MXY 5 pz+a2 (64

Ox = m + a 6 3Q, = y I p l + a ( 6 0V, = 6ppl+a (63V, = S,p,+a. (W

The same geometric and material properties, viz, a//r =50 and v =0.3 are used throughout. Five equal seg-ments and five subdivisions within each segment for theRunge-Kutta-Gill algorithms[5] have been found suit-able for the half plate analysis in the x-direction takingadvantage of the symmetric conditions along the centreline in all the examples considered in this section. On anedge with outward unit normal vector n and unit tangentvector f, the following nomenclature has been used forthe designation of the prescribed boundary conditons.

S for w=M,,=O (74C for w = 0. =0 (7b)

F for V. = M. = 0. (7c)Results obtained in the present study are compared withthose available elsewhere [3]. Most of the discrete valuesof the dependent variables are tabulated for the first 20harmonics. Convergence is seen to be excellent. It isobserved that while the value of w converges at thesecond or the third harmonic, the values of M,, MY,MXY,QX and V, take six to seven harmonics to converge.Convergence of the values of Q, and V, is seen to beslow.

Example 1. Plate with boundary conditions s alongall the four edgesThe maximum values of w, M,, MY, MY, Ox, ,, ,and V, are presented in Table 1 while the variation of M,and MY along the centre line y/a = 0.5 is tabulated inTable 2. Variations of QX and V, along the centre liney/a = 0.5 and that of Mxy, , nd V, along the supportededge y/a = 0.0 are plotted in Figs. l-3.Example 2. Hate with boundary conditions C alongx = 0, a and S along y = 0, a

8/3/2019 5tk

3/9

Numerical analysis of elastic plates 197Table 1. Maximum values of transverse deflection and stress resultants in a square plate simply supported on all edges(W= M, = 0 along I= 0, w = MY= 0 along y = 0, a) under U.D.L. based on Kirchhoff plate theory formulation ofsegmentation method-a convergence study (v = 0.3, a/h = 50)

1 . 0 0 4 1 0 . 0 4 9 1 5 . 0 5 1 6 42 . 0 0 4 0 5 . 0 4 7 6 03 . 0 0 4 0 6 . 0 4 7 9 14 . 0 0 4 0 6 . 0 4 7 6 05 . 0 0 4 0 6 . 0 4 7 6 5

. 0 4 7 0 9

. 0 4 6 1 2

. 0 4 7 7 4

. 0 4 7 9 26 . 0 0 4 0 6 . 0 4 7 8 37 . 0 0 4 0 6 . 0 4 7 6 4a . 0 0 4 0 6 . 0 4 7 6 39 . 0 0 4 0 6 . 0 4 7 6 4

1 0 . 0 0 4 0 6 . 0 4 7 6 31 5 . 0 0 4 0 6 . 0 4 7 6 42 0 . 0 0 4 0 6 . 0 4 7 8 4

. 0 4 7 6 2

. 0 4 7 6 6

. 0 4 7 6 4

. 0 4 7 8 7

. 0 4 7 6 5

. 0 4 7 8 6

. 0 4 7 6 6

- . 0 3 0 1 2- . 0 3 1 7 9- . 0 3 2 1 5- . 0 3 2 2 6- . 0 3 2 3 4- , 0 3 2 3 6- . 3 3 2 4 0- . 0 3 2 4 1- . 0 3 2 4 2- . 0 3 2 4 3- . 0 3 2 4 4- . 0 3 2 4 5

. 3 7 1 4 5 . 2 4 3 6 6 . 4 6 6 1 2 . 3 2 5 0 3

. 3 2 6 4 6 . 2 6 7 6 5 . 4 0 5 4 1 . 3 7 0 5 5

. 3 4 2 6 6 - 3 0 4 0 4 . 4 2 7 2 0 . 3 6 6 7 7

. 3 3 4 4 0 . 3 1 2 3 0 . 4 1 6 1 2 . 3 9 5 0 4. 3 3 9 3 9 . 3 1 7 3 0 . 4 2 2 0 7 . 4 0 0 0 4. 3 3 6 0 5 . 3 2 0 6 5. 3 3 . 3 4 4 . 3 2 3 0 4

. 4 1 6 3 5 . 4 0 3 3 9

. 4 2 1 5 9 . 4 0 5 7 6. 3 3 6 6 4 . 3 2 4 6 4 . 4 1 9 1 6 . 4 0 7 5 6. 3 3 6 0 5 . 3 2 6 2 4 . 4 2 1 0 5 . 4 0 6 9 6. 3 3 6 9 2 . 3 2 7 3 7 . 4 1 9 5 4 . 4 1 0 1 0. 3 3 7 6 5 . 3 3 0 7 3 . 4 2 0 5 2 . 4 1 3 4 7. 3 3 7 3 0 . 3 3 2 4 2 . 4 2 0 0 4 . 4 1 5 1 6

. 0 0 4 0 6 a . 0 4 7 9 . 0 4 7 9 . 0 3 2 &i . 3 3 6 . 3 3 6 . 4 2 0 . 4 2 0

a Va l u e s q u o t e d i n R e f . 3 b a s e d o n Ki r c h h o f f P l a t e Th e o r y

Table 2. Bending moments in a square plate simply supported on all edges (w = M, = 0 along x = 0, a; w = MY= 0along y = 0, a) under U.D.L. based on Kirchhoff plate theory formulation of segmentation method-a convergencestudy (V = 0.3, a/h = 50)

m a = .1 ; = 0.2 F. = 0 . 3a ; = 0 . 4 : = 0 . 5 f = 0 . 1 f = 0 . 2 f = 0 . 3 ; = 0 . 4 G = 0 . 5r j x = Bp z a ' . f = 0 . 5 f l y = 6 + m2 . f = 0 . 5

-1 . 0 2 2 1 1 . 0 3 5 7 5 . 0 4 3 7 2 . 0 4 7 6 72 . 0 2 0 6 3 . 0 3 4 0 3 . 0 4 2 0 8 . 0 4 6 3 03 . 0 2 0 9 9 . 0 3 4 3 6 . 0 4 2 4 0 . 0 4 6 6 14 . 0 2 0 6 6 . 0 3 4 2 6 . 0 4 2 2 9 . 0 4 6 5 05 . 0 2 0 9 2 . 0 3 4 3 1 . 0 4 2 3 4 . 0 4 6 5 56 . 0 2 0 6 6 . 0 3 4 2 8 . 0 4 2 3 1 . 0 4 6 5 27 . 0 2 0 9 0 . 0 3 4 3 0 . 0 4 2 3 3 . 0 4 6 5 46 . 0 2 0 6 9 . 0 3 4 2 9 . 0 4 2 3 2 . 0 4 6 5 39 . 0 2 0 9 0 . 0 3 4 3 0 . 0 4 2 3 3 . 0 4 6 5 4

1 0 . 0 2 0 8 9 . 0 3 4 2 9 . 0 4 2 3 2 . 0 4 6 5 31 5 . 0 2 0 9 0 . 0 3 4 2 9 . 0 4 2 3 2 . 0 4 6 5 32 0 . 0 2 0 9 0 . 0 3 4 2 9 . 0 4 2 3 2 . 0 4 6 5 3

. 0 4 9 1 5 . 0 1 8 6 3 . 0 3 3 1 4 . 0 4 3 4 5 .04960 . 0 5 1 6 4

. 0 4 7 6 0 . 0 1 6 3 3 . 0 2 9 5 6 . 0 3 9 2 6 . 0 4 5 1 3 . 0 4 7 0 9

. 0 4 7 9 1 . 0 1 7 0 3 . 0 3 0 5 2 . 0 4 0 2 7 . 0 4 6 1 5 . 0 4 8 1 2

. 0 4 7 6 0 . 0 1 6 7 3 . 0 3 0 1 5 . 0 3 9 8 9 . 0 4 5 7 8. 0 4 7 6 5 . 0 1 6 6 9 . 0 3 0 3 3 . 0 4 0 0 7 . 0 4 5 9 5 . 0 4 7 9 2. 0 4 7 6 3 . 0 1 6 6 0 . 0 3 0 2 3 . 0 3 9 9 7 . 0 4 5 6 6 . 0 4 7 6 2. 0 4 7 6 4. 0 4 7 6 3. 0 4 7 6 4. 0 4 7 6 3. 0 4 7 6 4. 0 4 7 8 4

. 0 1 6 8 5 . 0 3 0 2 9 . 0 4 0 0 3 . 0 4 5 9 2 . 0 4 7 6 8. 0 1 6 6 2 . 0 3 0 2 5 . 0 3 9 9 9 . 0 4 5 6 6 . 0 4 7 6 4. 0 1 6 8 4 . 0 3 0 2 6 . 0 4 0 0 2. 0 1 6 6 2 . 0 3 0 2 6. 0 1 6 6 3 . 0 3 0 2 7. 0 1 6 6 3 . 0 3 0 2 6

. 0 4 0 0 0. 0 4 0 0 1. 0 4 0 0 1

. 0 4 5 9 0

. 0 4 5 6 8

. 0 4 5 9 0

. 0 4 5 6 9

. 0 4 7 7 4

. 0 4 7 6 7

. 0 4 7 8 5

. 0 4 7 6 6

. 0 4 7 6 6

. 0 2 0 s a . 0 3 4 3 . 0 4 2 4 . 0 4 6 6 . 0 4 7 9 . 0 1 6 6 . 0 3 0 3 . 0 4 0 0 . 0 4 5 9 . 0 4 7 9

%a l u e s q u o t e d i n R e f . 3 b a s e d 0 " Ki r c h h o f f P l a t e T h e o r y

8/3/2019 5tk

4/9

198 T. KANTI42-a

042-038-1 040-038-036-034-032-

o+o+o K~hhott theory

X/~----,Fig. 1. Variationof Q, and V, along y/a =O.S.

4038

-0036 1-0034-OO32-PO-o. o\QOSJ- %

-0028--0026--0024- \ 0

JN _- 0:; ;Y>~ott theary

0251\\ 0

\\\\ c.+?CO Klrchhoftheory-0008-

-0006- %-ocn4- \ -0 002 -\

-0 01 02 03 or4 05$--w

Fig. 2. Variation of M,, along the suppo rted edge (y/a = 0).

Discrete numerical values of all the dependent vari-ables are tabulated in Tables 3 and 4, while the variationsof QX,V,, M,,, Q, and V, are plotted in Figs. 4-6. The

Fig. 4. Variation Q, and V, along y/ a = 0.5.occurrence of a maxima of Q, at the corner of the plateis unrealistic. But this is not due to the method used in y/a = 0.5, and Mxy, (IY and V, along the supported edgey/a =O.Oare shown m Figs. 7-9. These plots are quitethe present study. This is due to the inherent limitation revealing. It gives a feel for the variation of the quan-of the Kirchhoff plate theory itself. This point has been tities. The value of QX is seen to be inconsistent at theclarified in a recent paper [61. free edge. This is due to the well-known problem ofsatisfaction of the boundary conditions at a free edge inExample 3. Plate with boundary conditions F along a Kirchhoff plate, wherein a new quantity V, is intro-x = 0, a and S along y = 0, a duced in the theory. In a recent publication[6] it isRelevant discrete numerical values are presented in shown that barring its value at the free edge the variationTables 5 and 6. Plots of QXand V, along the centre line of QX is realistic in a physical situation. Thus it is

002-0 01 02 03 04 05XIo--+

Fig. 3. Variation of Q, and V, along the supported edge (y/a =0).

0 5 5 1050

,045 jc,

040Y

0354 i \/ / \

8/3/2019 5tk

5/9

Numerical analysis of elastic plates 199Table 3. Maximum values of transverse deflection and stress resultants in a square plate with two opposite edgessimply supported and the other two edges clamped ( w = 0 , = 0 along x = 0, a; w = My = 0 along y = 0, a ) under U.D.L.based on Kirchhoff plate theory formulation of segmentation method-a convergence study (V= 0.3, a / h = SO)

w [ r l x lma x ma x = B p z a [ MYI ma x = B , p z a 2 mx y 1 + ma x [ Q x l r n a x [ O y l t i ma x u / x l ma x [ V y l + + " a xm = p z a '" D P O S N E G P O S N E G = 5 p z a 2 = YDZ a = Y, , P Z a = 6 p z a * 6 , p z ' a

89

I O1 52 0

. 0 0 1 3 6. 0 0 1 9 1. 0 0 1 9 2. 0 0 1 9 2. 0 0 1 3 2. 0 0 1 9 2.00192.00192. 0 0 1 9 2. 0 0 1 9 2. 0 0 1 9 2. 0 0 1 9 2

.03459 - . 0 7 3 7 6 . 0 2 6 0 0 - . 0 2 2 1 3 - . 0 1 3 3 2 . 5 8 4 1 3.0x299 - . 0 6 9 0 2 . 0 2 3 6 2 - . 0 2 0 7 1 - . 0 1 4 2 6 . 4 9 4 2 9. 0 3 3 3 0 - . 0 7 0 0 5 . 0 2 4 6 4 - . 0 2 1 0 2 - . 0 1 4 3 6 . 5 2 6 6 9. 0 3 3 1 9 - . 0 6 9 8 7 . 0 2 4 2 7 - . 0 2 0 9 0 - . 0 1 4 3 7 . 5 1 0 1 6. 0 3 3 2 4 - . 0 6 9 8 5 . 0 2 4 4 4 - . 0 2 0 9 6 - . 0 1 4 3 6 . 5 2 0 1 6. 0 3 3 2 1 - . 0 6 9 7 5 . 0 2 4 3 5 - . 0 2 0 9 3 - . 0 1 4 3 6 . 5 1 3 4 7. 0 3 3 2 3 - . 0 6 9 8 1 n o 2 4 4 1 - . 0 2 0 9 4 - . 0 1 4 3 8 . 5 1 8 2 6. 0 3 3 2 2 - . 0 6 9 7 8 . 0 2 4 3 7 - . 0 2 0 9 3 - . 0 1 4 3 8 . 5 1 4 6 6. 0 3 3 2 2 - . 0 6 9 8 0 . 0 2 4 3 3 - . 0 2 0 9 4 - . 0 1 4 3 8 . 5 1 7 4 6. 0 3 3 2 2 - . 0 6 9 7 6 . 0 2 4 3 8 - . 0 2 0 9 3 - . 0 1 4 3 8 . 5 1 5 2 2. 0 3 3 2 2 - . 0 6 3 7 9 . 0 2 4 3 9 - . 0 2 0 3 4 - . 0 1 4 3 8 . 5 1 6 6 7. 0 3 3 2 2 - . 0 6 9 7 9 . 0 2 4 3 6 - . 0 2 0 9 4 - . 0 1 4 3 6 . 5 1 5 9 7

- . 2 3 1 8 7 . 5 6 4 1 3 - . 3 9 4 1 8- . 2 7 6 7 3 . 4 9 4 2 9 - . 4 7 0 4 4- . 2 9 2 9 3 . 5 2 6 6 9 - . 4 9 7 9 8- . 3 0 1 1 9 . 5 1 0 1 6 - . 5 1 2 0 3- . 3 0 6 1 9 . 5 2 0 1 6 - . 5 2 0 5 3- . 3 0 9 5 4 . 5 1 3 4 7 - . 5 2 6 2 2- . 3 1 1 9 4 . 5 1 8 2 6 - . 5 3 0 2 9- . 3 1 3 7 4 . 5 1 4 6 6 - . 5 3 3 3 5- . 3 1 5 1 4 . 5 1 7 4 6 - . 5 3 5 7 3- . 3 1 6 2 6 . 5 1 5 2 2 - . 5 3 7 6 4- . 3 1 9 6 3 . 5 1 6 6 7 - . 5 4 3 3 7- . 3 2 1 3 1 . 5 1 5 3 7 - . 5 4 6 2 3

- . 0 0 1 9 2 a - . 0 6 9 7 . 0 2 4 4

%a l u e s q u o t e d i n R e f . 3 b a s e d o n Kl r c h h o f f P l a t o T h e o r yt Oc c u r s a l o n g t h e s u p p o r t e d e d g e a t ; = 0 . 2 a n d f = 0 ; . + tOc c u r s a t c o r n e r s o f t h e s u p p o r t e d e d g a s .

Table 4. Bending moments in a square plate with two opposite edges simply supported and the other two edgesclamped (w = 0, = 0 along x = 0, a; w = My = 0 along y = 0, a ) under U.D.L. based on Kirchhoff plate theoryformulation of segmentation method-a convergence study (V= 0.3, a/h = 50)rlx = @a*. y / a = 0 . 5 My = 6,pza2, Y/ a - 0 . 5

Ill i - 0.0 t - 0 . 1 3 = 0 . 2 f - 0 . 3 ; = 0 . 4 i = 0 . 5 t = 0 . 0 c = 0 . 1 f = 0 . 2 ; = 0 . 3 9 = 0 . 4 i - 0 . 51 - . 0 7 3 7 8 - . 0 2 6 4 4 . 0 0 3 6 4 n o 2 1 8 5 . 0 3 1 5 5 .034592 - . 0 6 9 0 2 - a 0 2 6 6 9 . 0 0 2 1 9 . 0 2 0 2 0 . 0 2 9 9 3 .032993 - . 0 7 0 0 5 - . 0 2 6 4 2 . 0 0 2 5 3 . 0 2 0 5 3 . 0 3 0 2 4 . 0 3 3 3 04 - . 0 6 9 6 7 - . 0 2 6 5 4 . 0 0 2 4 1 . 0 2 0 4 2 . 0 3 0 1 3 . 0 3 3 1 95 - . 0 6 9 8 5 - . 0 2 6 4 8 . 0 0 2 4 7 . 0 2 0 4 7 . 0 3 0 1 6 . 0 3 3 2 46 - . 0 6 9 7 5 - . 0 2 6 5 2 . 0 0 2 4 4 a 0 2 0 4 4 . 0 3 0 1 5 . 0 3 3 2 17 - . 0 6 9 8 1 - . 0 2 6 5 0 . 0 0 2 4 6 . 0 2 0 4 6 . 0 3 0 1 7 . 0 3 3 2 36 - . 0 6 9 7 8 - . 0 2 6 5 1 . 0 0 2 4 5 . 0 2 0 4 5 . 0 3 0 1 6 . 0 3 3 2 29 - . 0 6 9 8 0 - . 0 2 6 5 0 . 0 0 2 4 5 . 0 2 0 4 6 . 0 3 0 1 7 . 0 3 3 2 2

1 0 - . 0 6 9 7 8 - . 0 2 6 5 1 . 0 0 2 4 5 . 0 2 0 4 5 . 0 3 0 1 6 . 0 3 3 2 21 5 - . 0 6 9 7 9 - . 0 2 6 5 0 . 0 0 2 4 5 . 0 2 0 4 5 . 0 3 0 1 7 . 0 3 3 2 22 0 - . 0 6 9 7 9 - . 0 2 6 5 0 . 0 0 2 4 5 . 0 2 0 4 5 . 0 3 0 1 6 . 0 3 3 2 2

- . 0 2 2 1 3 - . 0 0 5 4 0 . 0 0 8 7 5 . 0 1 9 3 2 . 0 2 5 8 1- . 0 2 0 7 1 - . 0 0 6 5 2 a 0 0 5 8 9 . 0 1 5 5 1 . 0 2 1 5 6- . 0 2 1 0 2 - . 0 0 6 0 1 a 0 0 6 7 6 . 0 1 6 5 0 . 0 2 2 5 8- . 0 2 0 9 0 - . 0 0 6 2 6 . 0 0 6 4 1 . 0 1 6 1 2 . 0 2 2 2 0- . 0 2 0 9 6 - . 0 0 6 1 2 . 0 0 6 5 8 . 0 1 6 3 0 . 0 2 2 3 8- . 0 2 0 9 3 - . 0 0 6 2 1 . 0 0 6 4 8 . 0 1 6 2 0 . 0 2 2 2 8- . 0 2 0 9 4 - . 0 0 6 1 5 . 0 0 6 5 4 . 0 1 6 2 6 . 0 2 2 3 4- . 0 2 0 9 3 - . 0 0 6 1 9 . 0 0 6 5 0 . 0 1 6 2 2 . 0 2 2 3 0- a 0 2 0 9 4 - . 0 0 6 1 6 . 0 0 6 5 3 . 0 1 6 2 5 . 0 2 2 3 3- . 0 2 0 9 3 - . 0 0 6 1 6 . 0 0 6 5 1 . 0 1 6 2 3 . 0 2 2 3 1- . 0 2 0 9 4 - . 0 0 6 1 7 . 0 0 6 5 2 . 0 1 6 2 4 . 0 2 2 3 2- . 0 2 0 9 4 - . 0 0 6 1 8 . 0 0 6 5 2 . 0 1 6 2 4 . 0 2 2 3 2

. 0 2 8 0 0

. 0 2 3 6 2

. 0 2 4 6 4

. 0 2 4 2 7- 0 2 4 4 4S O 2 4 3 5. 0 2 4 4 1. 0 2 4 3 7- 0 2 4 3 9

S O 2 4 3 9S O 2 4 3 6

- .0697= - - . 0 3 3 2 . 0 2 4 4=Val".s u o t e d i n R e f . 3 b a s e d o n Ki r c h h o f f P l a t e T h e o r y

8/3/2019 5tk

6/9

200 T. KANT

Table 5. Maximumvaluesof transverse deflection and stress resultants ina square plate with two opposite e d g e s simplysupported and the other two edges free (M, = V, = 0 along x = 0, a; w = MY= 0 along y = 0, a) under (V = 0.3,a/h = 50)w+mm (Mx)max my1+max [MXYImax (Qxlnl,x (Qylnlax [VJ+max maxVy l. Q . g I B P & = B , p z a Z = 4 pza2 = Y p z a = Yl p z a = & p m = 6 , p z a

ml

1 . 0 1 5 0 4 . 0 2 8 2 5 . I 3 5 1 9 . 0 2 2 8 0 . 0 7 1 6 72 . 0 1 4 9 8 . 0 2 6 8 2 . I 3 0 1 1 . 0 2 3 6 7 . 0 6 3 4 93 . 0 1 4 9 8 . 0 2 7 1 3 . 1 3 1 2 1 . 0 2 3 8 6 . 0 6 6 4 34 . 0 1 4 9 0 . 0 2 7 0 2 . I 3 0 8 1 . 0 2 3 9 3 9 0 6 4 9 35 . 0 1 4 9 8 . 0 2 7 0 7 . I 3 1 0 0 . 0 2 3 9 6 . 0 6 5 8 46 . 0 1 4 9 8 . 0 2 7 0 4 . I 3 0 8 9 . 0 2 3 9 8 . 0 6 5 2 37 . 0 1 4 9 8 . 0 2 7 0 6 . I 3 0 9 6 . 0 2 3 9 9 . 0 6 5 6 68 . 0 1 4 9 8 . 0 2 7 0 5 . I 3 0 9 1 . 0 2 3 9 9 . 0 6 5 3 49 . 0 1 4 9 8 . 0 2 7 0 6 . I 3 0 9 4 . 0 2 4 0 0 . 0 6 5 5 9

1 0 . 0 1 4 9 8 . 0 2 7 0 5 . I 3 0 9 2 . 0 2 4 0 0 . 0 6 5 3 91 5 . 0 1 4 9 8 . 0 2 7 0 5 . I 3 0 9 3 . 0 2 4 0 1 . 0 6 5 5 22 0 . 0 1 4 9 8 . 0 2 7 0 5 . I 3 0 9 3 . 0 2 4 0 1 . 0 6 5 4 6

. 3 7 3 8 4

. 4 1 8 6 9

. 4 3 4 a a

. 4 4 3 1 5

. 4 4 8 1 5

. 4 5 1 4 9. 4 5 3 8 9. 4 5 5 6 9. 4 5 7 0 9. 4 5 8 2 1. 4 6 1 5 6

. 0 0 5 1 4 . 3 5 0 4 9

. 0 0 4 3 3 . 3 9 5 3 8. 0 0 4 4 7 . 4 1 1 5 8. 0 0 4 4 4 . 4 1 9 8 4. 0 0 4 4 5 . 4 2 4 8 4. 0 0 4 4 4 . 4 2 8 1 9. 0 0 4 4 4 . 4 3 0 5 8. 0 0 4 4 4 . 4 3 2 3 8. 0 0 4 4 4 . 4 3 3 7 0. 0 0 4 4 4 . 4 3 4 9 1. 0 0 4 4 4 . 4 3 8 2 6. 0 0 4 4 5 . 4 3 9 9 6

1 Oc c u r s a t a = 0 a n d i = 0 . 5t t Oc c u r s a t $ = 0 . 2 a n d s = 0 . 5

Table 6. Bending moments in a square plate with two opposite edges simply supported and the other two edges freeV, = M, = 0 along x = 0, a; w = MY= 0 along y = 0, a) under U.D.L. based on Kirchhoff plate theory formulation ofsegmentation method-a convergence study (V = 0.3, a/h = 50)Mx = B P & , y / a = 0 . 5 My = B I P z a * . y / a = 0 . 5

m5 = 0 . 1a t = 0 . 2 ; = 0 . 3 a = 0 . 4 $ = 0 . 5 5 = 0 . 0a i = 0 . 1 f = 0 . 2 i = 0 . 3 t = 0 . 4 ; = 0 . 5

1 . 0 1 1 8 6 . 0 1 9 7 2 . 0 2 4 6 7 . 0 2 7 3 3 . 0 2 8 2 5 . I 3 5 1 3 . I 3 1 1 9 . I 2 8 7 5 . I 2 7 3 4 . I 2 6 6 0 . I 2 6 3 82 . 0 1 0 8 7 . 0 1 8 4 2 . 0 2 3 2 7 . 0 2 5 9 6 . 0 2 6 8 2 . I 3 0 1 1 . I 2 6 4 1 . I 2 4 0 2 . I 2 2 6 0 . I 2 1 8 6 . I 2 1 6 33 . 0 1 1 1 4 . 0 1 8 7 2 . 0 2 3 5 8 . 0 2 6 2 7 . 0 2 7 1 3 . I 3 1 2 1 . 1 2 7 4 4 . I 2 5 0 5 . I 2 3 6 3 . I 2 2 8 9 . I 2 2 6 64 . 0 1 1 0 3 . 0 1 8 6 1 . 0 2 3 4 7 . 0 2 6 1 6 . 0 2 7 0 2 . I 3 0 8 1 . I 2 7 0 6 . I 2 4 6 7 . I 2 3 2 5 . I 2 2 5 1 . I 2 2 2 65 . 0 1 1 0 8 . 0 1 8 6 6 . 0 2 3 5 2 . 0 2 6 2 1 . 0 2 7 0 7 . I 3 1 0 0 . I 2 7 2 4 . I 2 4 8 5 . I 2 3 4 3 . I 2 2 6 9 . I 2 2 4 66 . 0 1 1 0 5 . 0 1 0 6 3 . 0 2 3 4 9 . 0 2 6 1 8 . 0 2 7 0 4 . I 3 0 8 9 . I 2 7 1 4 . I 2 4 7 5 . I 2 3 3 3 . I 2 2 5 9 . I 2 2 3 67 . 0 1 1 0 7 . 0 1 8 6 5 . 0 2 3 5 1 . 0 2 6 2 0 . 0 2 7 0 6 . I 3 0 9 6 . I 2 7 2 0 . I 2 4 8 1 . I 2 3 3 3 . I 2 2 6 5 . I 2 2 4 20 . 0 1 1 0 6 . 0 1 ' 3 6 4 . 0 2 3 5 0 . 0 2 6 1 9 . 0 2 7 0 5 . I 3 0 9 1 . I 2 7 1 6 . I 2 4 7 7 . I 2 3 3 5 . I 2 2 6 1 . I 2 2 3 89 . 0 1 1 0 7 . 0 1 8 6 5 . 0 2 3 5 1 . 0 2 6 2 0 . 0 2 7 0 6 . I 3 0 9 4 . I 2 7 1 9 . I 2 4 8 0 . I 2 3 3 8 . I 2 2 6 4 . I 2 2 4 1

I O . 0 1 1 0 6 . O l B 6 4 . 0 2 3 5 0 . 0 2 6 1 9 . 0 2 7 0 5 . I 3 0 3 2 . I 2 7 1 7 . I 2 4 7 8 . I 2 3 3 6 . I 2 2 6 2 . I 2 2 3 91 5 . 0 1 1 0 6 . 0 1 6 6 5 . 0 2 3 5 0 . 0 2 6 1 9 . 0 2 7 0 5 . I 3 0 9 3 . I 2 7 1 8 . I 2 4 7 9 . I 2 3 3 7 . I 2 2 6 3 . I 2 2 4 02 0 . 0 1 1 0 6 . 0 1 8 6 4 . 0 2 3 5 0 . 0 2 6 1 3 . 0 2 7 0 5 . I 3 0 3 3 . I 2 7 1 8 . I 2 4 7 9 . I 2 3 3 7 . I 2 2 6 3 . I 2 2 4 0

8/3/2019 5tk

7/9

Numerical analysis of elastic plates 201

8/3/2019 5tk

8/9

Fig. 8. Variation of MXy long the supported edge (y/a = 0).

o-0-0 Ktrchho ff theo ry

T. KANTAcknowledgements-The author is thankful to Dr. E. Hinton forhelpful discussions. Sincere thanks are expressed to Mrs. JeanDavies for her very patient and excellent typing work. Initialsupport of this work by grants BRNS/ENGG/l7/75 andHCS/DST/198/76 s gratefully acknowledged.

1.

2.

J. E. Boldberg, A. V. Setlur and D. W. Alspaugh, Computeranalysis of non-circular cylindrical shells. Proc., IASS Symp.Shell Structures in EngineeringPractice, Budapest, Hungary(1%5).T. Kant and C. K. Ramesh, Numerical integration of linearboundary value problems in solid mechanics by segmentationmethod. Int. J. Numerical Methods Engng(to appear).S. Timoshenko and S. Woinowsky-Krieger, Theory of Platesand Shells, 2nd Edn. McGraw-Hill, New York (1959).Y. K. Cheung, Finite Strip Method in Structural Analysis.Pergamon Press, Oxford (1976).A. Ralston and H. S. Wilf (Editors), Mathematical Methodsfor Digital Computers,pp. 95-120. Wiley, New York (l%O).T. Kant and E. Hinton, Analysis of rectangular Mindlinplates bythe segmentation method, Report C/R/365/80,Civil Engg.Uni-versity of Wales, Swansea (1980).7. 0. C. Zienkiewicz, The Finite EIement Method.McGraw-Hill,New York (1977).

REFERENCES

V,=6,P,a

i0 1

Fig. 9. Variation of Q, and V, along the supported edge (y/a =0).

concluded that it is QX and not V, which should be usedin designs in such situations.CONCLUSIONS

A less known formulation for economical numericalanalysis of elastic plates is presented. The method isfound to be very efficient and accurate. The numericalresults obtained show excellent comparison with those ofTimoshenko[3] which are presumably based on analy-tical solution. The main use of the method seems to lie inits adoption in design offices for preparation of reliabledesign charts at reasonable costs. Although the methodhas limitations in its applications to general problemswhen compared with the versatile finite elementmethod[7], it appears to be superior to the finite stripmethod [4] because of its relatively accurate mathematicalmodel.

APPENDIX ABasic equations of Kirchhoff plate theoryDisplacement model (Ref. Fig. Al)U(x, Y, 2) = z&k Y)ux, Y. 2) = 20,(x, Y)

w, 4 2) = wk Y)Strain-displacement relations

d2Wex=-z,id2Wey=-zdyz

yxy = - 2z a!!a x a y

Equilibrium equations (Ref. Fig. A2-4)$$+$$+p,++p,-+ph=O (Ma)

Fig. Al. Positive set of displacement components.

(Ala)(AIN(Ale)

Wa)(A%)642~)(A24NW

8/3/2019 5tk

9/9

Numerical analysis of elastic plates 203

etcFig. A2. Positive set of stress components. Fig. A4. Positive set of stress resultants-couples.

Force-displacement relations

Reference surface

Q:=Q,*dy, etc

Kirchhoff shear and boundary conditionsV,=p,+$)

(A4a)Wb)(A4c)

(ASa)W-b)

Fig. A3. Positive set of stress resultants-forces. Wa)

ah& aM,,_ax+ ay ox=0on edge x = const. and Wb)(A3b) o=G or v,= 5

0, = I?~ or My= MY(A34 (A7a)on edge y = const. (A7b)