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M I N I S T R Y OF A V I A T I O N

R. & M. No . 332S:

AERONAUTICAL RESEARCH COUNCIL REPORTS AND MEMORANDA

The Bending under Normal Loading of Plates Tapered in Planform

By G. G. POPE, M.Sc . (Eng . )

t

LONDON: HER MAJESTY'S STATIONERY OFFICE

i963 N I N E S H I L L I N G S N E T

The Bending under Normal Loading of Plates Tapered in Planform By G. G. POPE, M.Sc.(Eng.)

COMMUNICATED BY THE DEPUTY CONTROLLER AIRCRAFT (RESEARCH AND DEVELOPMENT),

MINISTRY OF AVIATION

Reports and Memoranda No. 3325*

April, ±962

Summary.

An analysis is given of the deformation under normal loading of a plate tapered symmetrically in planform with edges either clamped or simply-supported. The thickness may be constant or may vary along the plate as a power of the width. Numerical results are given for plates of constant and linearly varying thickness under uniformly distributed load.

1. Introduction.

~Ii-approximate analysis is given in this report of the small-deflection deformation under normal

loading of a thin elastic plate tapered symmetrically in planform with edges either clamped or

simply-supported. The thickness of the plate may be constant or may vary in the direction of taper

as any power of the width.

The analysis is applicable to loadings under which the deflected shape across the plate normal to

the direction of taper may be represented adequately anywhere along the plate in terms of a few

known functions. The deflection of the plate can then be expressed in terms of the sum of the

products of each of these functions with unknown functions representing the deflected shape in the

direction of taper. Linear homogeneous differential equations are derived for these functions using

the Rayleigh-Ritz method 2 and closed-form solutions are obtained. Similar solutions for rectangular

plates of constant thickness have previously been obtained by Kantorovich 1. The amount of

computation increases rapidly with the number of functions used, and in the applications given here

two functions only are used to represent the deflected shape across the plate.

If the shape of the load distribution across the plate does not vary and if the load intensity varies as a power of the width, it is shown that for a long plate (length/maximum width greater than about 2)

an 'optimum' thickness variation can be derived in which the stress distribution on the surface does not vary along the plate except near the ends.

Williams 8 has examined the local bending behaviour of the corners of plates and has shown that infinite stresses are produced in theory at obtuse corners joining simply-supported edges. The

~" Replaces R.A.E. Report No. Structures 273--A.R.C. 23,975.

present analysis is therefore not valid in the immediate vicinity of such corners. A similar singularity at corners joining simply-supported and clamped edges only occurs when the included angle exceeds 129 ° , which is larger than would be met in this context.

The general analysis is illustrated by application to a plate under uniformly distributed loading with the same boundary conditions along opposite edges. All such combinations are considered in turn. The deflected shape across the plate is here represented by the deflected shape across a parallel infinite strip under the same loading and boundary conditions, together with a second polynomial of higher power. It is shown that in this example an analysis using the first function alone, which could easily be performed on a desk calculating machine, will often give adequate results even for the maximum stress in the plate. Curves showing the variation with the plate geometry of the maximum deflection and the maximum value of the yon Mises stress away from the corners are given both for a plate of constant thickness and for a linear thickness variation, the latter being the 'optimum' thickness variation for this loading.

2. Analysis.

2.1. Basic Theory.

2.1.1. Some general results.--The strain energy of bending U of a thin plate in the x, y plane undergoing a small lateral deflection w(x, y) is given by the expression 2

i I(Ww) - 2(i- F ( U = ~ f f D [_~x ~ ~y~ \ ~ - ~ ] ] l dxdy'.

The increment of the strain energy due to an infinitesimal arbitrary variation ~w of the deflection is thus

Moreover, provided the total work done on the edges of the plate by the variation 8w is zero, and provided the flexural rigidity D is a function of x only, the following expression can be obtained by integrating equation (1) twice by parts.

8U = ~w DV4w + dx (V~w) + ~ \Sx ~ + v ~y~]j dxdy. (2)

The line integrals around the edges of the plate which arise in the process of integrating by parts correspond to work done by forces or moments acting on the edges and here vanish due tO the restrictions imposed on 8w.

The work done on the plate due to the variation ~w of the displacement is given by

(3)

where q(x, y) is the distributed load. Now, by the principle of virtual displacements, the total work done by the infinitesimal variation

of the displacements is equal to the change in the strain energy, so that

8T = 8U.

2

Hence equating expressions (2) and (3), we obtain

dD 0 (Ww)+ + v - q(x, y) dxdy = 0. (4) ~w DV4w + 2 ~ ~x ~ \ Ox ~ Oy ~]

If the variation ~w were completely arbitrary within the plate," equation (4) would yield the exact equation for the bending of the plate. In the present problem however an approximate solution is found by considering only a restricted variation of w.

2.1.2. Application to tapered plates.--The axes and notation used here are shown in Fig. 1. The plate is symmetrically positioned relative to the x axis, the parallel ends being defined by the lines x = 0 and x = a. It is convenient to express the problem in terms of the following

non-dimensional symbols W .7_,0 x y , = - , X = l + p - , y = 2

a a

a Y b2 I , /~ = ~ . 0 = ~ , p - h i

The symbols bl and b2 represent the widths of the plate at x = 0 and x = a respectively (b I > b~). The co-ordinate X represents in non-dimensional form the distance along the plate measured from

the point at which the sides would meet if produced. The analysis given here is immediately applicable when the shape of the load distribution across

the plate does not vary along the plate. More complicated problems may be considered by adding

together a number of such load distributions, which can each be expressed in the form

q = ~,o4X)~(O)

where q0 is a constant and the functions a and ]3 express the shape of the load distribution. • The most general variation of the flexural rigidity to which this analysis is applicable can be

expressed in the form D = D1X ~

where D 1 and r are constants. Re-expressing equation (4) in the notation of this section, we obtain

r (r -1)p 2 [ ~ W O2W] qoa%~(X)fi(O)] d X d g O. + X 2 \P2 + 4vv~ = (5)

7 X ~ a y2 ] D1X ~ .J

An approximate solution is obtained to this equation by assuming that W can be expressed in the f o r m

w = (6) 1

where the functions ~.(0) are chosen to represent approximately the deflected shape normal to the X axis and the functions fj(X) are unspecified. Simultaneous differential equations for these functions are obtained by substituting equation (6) in equation (5) and considering in turn the following m restricted variations of W.

3W~ = %a h . (7)

3

(87i22) A 2

The integration sign with respect to X may then be removed from equation (5) because the variations ~fs are completely arbitrary within the plate. If the resulting equations are then divided through by the relevant variations 3fi and integrated With respect to O, the following set of equations is obtained.

k=~ (i=4 difl~ ) 12(1 _ v~)cq~a~F~ R E E Pski Xi + PskofT~ = (8)

7~=1 ,~=1 ~ X ' -~ where

f+1_1 q o_ l(bll" ~Fj = 30sdO, R = 1:;

Psz:o = pa {lsk ~ + 121Ska + 36/~.k2 + 241Sla = 2r(ljr:a + 6lj7~. + 6/Ski ) +

+ r(r-- 1) (ljk 2 + 2ljkl) } + 4p2/z 2 {2(sj.k2 + 6s~1~1 + 6sjl~o ) --

-- 2r(sjza+Zss~o ) + r(r--1)vs~l~o } + 16/~m~k,

Pskl = P~ { -- 4(ls~,:a + 6I~kz + 6l~a) + 6r( l~ + 2/~va) -- 2r(r -- 1)ls~a} +

+ 4p~/z~ { _ 4(s~ 1 + 2s~o ) + 2rs~o} ' (9)

p'~ {6(l~,~z + 2l~,a) - 6rlsk 1 + r ( r - 1)l~ko} + 8pzt~ssko,

and

Pjk2 =

Pjka =

Pjz~¢ =

ljki =

- 2p~(2ljla - rljl~o),

P jl~o

- 1 0 i ~ -707- dO,

f +l di+2(Dk -10i(~j dOi+2 dO,

-I-1 1 f d407 ~ m~l ~ • s ~ dO .

--1

(lO)

2.1.3. Solution of equations.--As equations (8) are of linear and homogeneous form, they can be solved by standard methods. In the numerical examples given in this report two unknown functions only are considered. For simplicity of presentation the subsequent analysis is therefore restricted to assumed deflected shapes of the form

W = f~(X)01(O ) + f2(X)q)2(O ) . (11)

The complementary function of the solution to equations (8) is then found by substituting

f l = A X ~

f2 = BX'~

and equating the left-hand sides of these equations to zero, giving

and Ah11()t)+Bhl~(A)=O0} (12)

Ahl~(h ) + Bh~z(A) = where

hjz~(Y) = y4Pjk<i + 78( - 6Psic4 +Pj'ka) + 7z( 1 lPjk4 - 3PsT~a +Pjl~2) +

+ y( - 6PjI~4 + 2PsI~a-Pjl,2 +Pjza) + Pjko. (13)

4

Equations (12) are only consistent if the determinant A(A) of the coefficients of A and B is zero,

that is A(~t) = hll(;~)h22(~ ) -h12(/~)h21(Z ) = 0.

The complete solution is then given by

i=8 t f l = f l ( x ) + Z A i X ~ i=1 (14) i=8

.f~ F2(X) + E B¢X ai] i=1 ]

where F 1 and F 2 are the relevant particular integrals and the constants A i and B i are related by the

expression

A~ hl~(Ai) h2~(Ai)

The eight arbitrary coefficients in the above complete solution can be found from the boundary

conditions on each of the functions f~ at X = 1 and X = 1 + p. The constants A i and B i and the

indices )ti are in general complex. If the load intensity at constant 0 along the plate varies as X ~, the particular integrals in expressions

(14) are given by 12(1 - v2)l~aR

F~(X) = -A((4~ r - ~ ) - (tF~h2~(4- r + v) - ~F2h12(4- r + v)} X 4-r+v

(15). 12(1-~)~R { % h ~ # - r + v ) %h~(4-r+v)}X~-~+~J

F~(x) = A(4- r + v)

or, if only one function f is used in the analysis,

12(1 - v2)/z~R WX,_~+~. F(X) - h ( 4 - r + v)

2.1.4. Moments and stresses in p Ia te . - -The bending moments M x and M v and the twisting

moment M~j are given by the expressions,

_ M . = - D \ ~ x ~ +Vay~]

D1X~ [ 2 a2W a=W~

M v = - D \Oy2 + v O x 2] (16)

DxX ~ / 02W ~9'W] - i~ba ~ 41~ ~ + up s OX ~ ] '

m~u = D(1 - v) Ox ~y

D1X~ (1 - v)p ~q4z = 2 ~ OXOY"

5

Substitution of the assumed deflected

~2 W j=2 1 - - = E~ OX2 j = 1

~2 W j'=2 1

32W ~=2 1 - - = E X ~ 3 y2 ~=1

dfj and ' d(I)j. wheref j ' = ~ (I)j - dO"

form (11) into the derivatives of W gives

[~,jx%." - 2o%'xL.' + o(o~/' + 2%')f~], I

[q~/Xfj' - (O(I)j" + ¢P/)fJ] ' 1 (17)

,gj"h j

The value of the von Mises stress at on the surface of the plate, which is here assumed to govern the yielding of the plate, is given by

~/ = V(% 2 + % z - %% + 3T=U z)

6 - tl~X~,,m ~ / (Mx 2 + My2 - M ~ M u + 3M~u ~) (18)

where t 1 is the thickness of the plate at X = 1.

2.1.5. Boundary conditions.

(i) Boundary conditions at ends . - -The boundary condition of zero deflection at the ends of the plate is satisfied completely by making

[L.]o~ = [ G - ] o ~ = o. (19)

The further condition necessary for the virtual displacement 3Wy to do no work on the ends is given by

-1 ~ X P2 dO = O. (20) ~ X ~ + 4v~ ~ ~ y2 ] ona

Substituting equations (17) and (19) in equation (20), we obtain

3f /P ~ Z kofk" -- 2 Z,.I = O. i k = l / - / e n d

The second boundary condition for a clamped end is thus satisfied completely by making

[L.']ona = [~L.']ona = 0

and that for a simply-supported end is represented approximately by

[k=2 2 l j/elf//) ] 0 .

..o o =

(ii) Boundary conditions at clamped s ides . - -The deflection functions Oi can be chosen to satisfy the boundary conditions along clamped sides completely. If a complete series of functions ~j were used and if the ends of the plate were also clamped this method would thus give a solution which would converge to the exact solution as the number of functions was increased.

(iii) Boundary conditions at simply-supported s ides . - -The boundary conditions cannot be satisfied purely by appropriate choice of the functions ~j if the sides are simply-supported,

because the expression for the bending moment about the sides also involves the unknown functions ft" The deflected shape across the plate is here represented approximately by functions which would satisfy these boundary conditions completely if the sides were parallel. It can be shown 4 that the total work done on the sides of the plate in this deflected shape is zero if the plate thickness is constant and the derivation of equations (8) is then strictly valid. Errors introduced by using this derivation when the thickness varies should be small compared with those due directly to the moments along the sides which would be strictly necessary to maintain this deflected form irrespective of the thickness variation. These moments, which are illustrated for a specific example in Fig. 12, are usually small but can lead to inaccurate results for large angles O f taper.

2.2. Some Special Cases.

2.2.1. Plates of constant thickness.--For a plate of constant thickness

r = O

and the coefficients of equation (8) are given by

Pjko = p~(ljk4 + 12l~1~3 + 361jk~ + 24li7~) + 8P~t~2(sjz~,2 + 6sjkl + 6Sjko) + 16/z4mjz~,

p kl = - 4p (l , 3 + 61jk + - 16p (s kl+ 2S ko),

Pjk2 = 6p~(ljl~2 + 2Ij1~1) + 8p2t*Zs~t~o,

Pjk3 = - 4041Jl~,l,

P k4 = p%ko.

2.2.2. Optimum thickness variation when load at constant 0 varies as Xv.- -The particular integrals given in equations (15) can be interpreted physically as representing the deflection of a semi-infinite tapered strip under the same loading with the relevant boundary conditions along the sides. To these are added the complementary functions which here represent the additional deflection necessary to make the strip satisfy the boundary conditions at the ends of the plate. Thus in a relatively long plate the influence of these complementary functions is only significant near the ends and elsewhere the plate behaves as a semi-infinite tapered strip. Now by substituting the particular integrals, equations (15), in equations (17) it is seen that for such a strip the second derivatives of W with respect to X and Y vary as X 2+v-r. Substituting these successively into equations (16) and (18), the yon Mises stress ~1 is seen to vary as X 2+v-zm. Hence the optimum thickness variation of the semi-infinite strip in which the surface stresses at all chordwise sections

are the same is given by 3 ( . + 2 ) . r = ~

Thus for a relatively long plate this represents a near opt imum thickf~ess variation.

3. Application to Plates under Uniformly Distributed Loading.

3.1. Specific Examples Chosen.

The preceding analysis has been computed for plates under uniform loading with opposite pairs of edges either" clamped or simply-supported. The plate thickness is either constant or linearly proportional to the width, the latter being the optimum thickness variation for long plates in the

sense described in Section 2.2.2.

7

The functions chosen to describe the deflected shape across the plate are given in Section 3.2

together with the constants and equations derived using them. In each case the first function chosen

is the deflected shape across an infinitely long strip of constant thickness under uniform loading

with the relevant boundary conditions. In long plates the influence of the second function is only

significant near the ends and the computation of the two-funct ion analysis becomes impracticable.

Equations are given therefore both for one-function and for two-function analyses.

The figures plotted are listed in Section 3.3 and the results are discussed in Section 3.4.

3.2. Transverse Deflection Functions and Related Constants.

3.2.1. Plate with clamped sides.--The transverse deflection functions are chosen as

(I)1 = 0 4 - 202 + 1,

( D 2 = 0~(04--202+ 1).

The related constants are shown in the following tables:

i 315/11 i 3465112 i 3465121i 45045/22 i

0 1 2 3 4

+ 256 - 128

0 + 384 + 384

+ 256 + 128 - 768 + 1152 + 20352

+ 256 - 384 + 256 + 1920 + 1920

+ 768 - 384 - 3584 + 17280 + 151680

105sn i

-- 256 + 384 + 384

315s~2 i

0 - 384 -k, 3456

315S2i i

0 + 384 + 384

3465s2~

-- 768 + 1152 + 20352

5mll 35mi~ 35m21 315rtz2~

128 128 128 3456

15q) 1 105q52

16 16

Substituting in equations (8), the following equations are obtained for f l and f2.

(i) Plate of constant thickness.

(a) Single-function analysis.

4"9 ~3~,,,- X4f,,,,) 3(5p4+72p2tx2+336t~4)f+ 1202(p~+4/~z)(Xf'-X~f") + 2p (~.,t 7 t = 28.665/~3RX 4

where a dash denotes differentiation with respect to X and v = 0- 3.

(b) T w o - f u n c t i o n analysis.

33(5p4 + 72p2/~ 2 + 336/x4)f1 + 132p3(p 3 + 4~ 2) ( X f l ' - X2 f l ") +

+ 22p4(2Xaf~" + X*f~ "") + 3(25p* + 264p3/z 2 + 528/~*)f 3 +

+ 12p2(7p 2 + 44tz2)Xfz ' _ 2p4(12X2f3" + 9Y~¢ ,,, - ~ .l~ -- X*f~"") = 315" 315/zaRX 4,

39(65p4 + 616p2t* 2 + 528t**)A - 156p3(3p 2 + 44t~3)xf~ ' - 2p4(156x2f~" -

- 78x~f~ ' ' - 13x4f~"") + (1725p4 -4-18408pz/x 3 q- 61776~*)f~ +

+ 4p2(51 p2 + 156~a)xf~. ' - 4p3(5 lp2 + 156t,3)x2f2" + 6p*(2xaf3 " + x , f 2 " ")

= 585. 585/zaRX *.

(ii) Plate with thickness proportional to width.

(a) S ingle- funct ion analysis.

(210 ' + 244.8p3/x 2 + 1008~*)f - 12p3(p 3 + 8t~3)Xf ' + 6p2(3p 2 - 8lx2)X2f " +

+ 204(8Xaf " + X 4 f ' ' ) = 28. 665/~8RX.

(b) T w o - f u n c t i o n analysis.

11(21p4 + 244.8p3/x 3 + 1008/~4)f 1 - 132p~(p ~ + 8lx2)Xf~ + 66p~(302 _ 8lz2)X3f~,, +

+ 22p*(8Xaf~"+X4f~ ' ' ) + (177p4+ 1584p3/23 + 1584tz*)f2 + 528p31x2Xf2 ' -

- 2p*( lSXy2" - 4XZf2 ' ' - X4f2 ' ' ) = 3 1 5 . 3 1 5 / ~ R X ,

13(117p* + 105603/~ 2 + 1584/~4)f~ - 312p3(3p 3 + 22/~2)Xf~ ' + 2604(21X~fl" +

+ 12Xaf~" + X' f1 " ) q- (1827p 4 + 18782.4p3/, 3 + 61776fz*)f~ - 1202(3 lp 3 + 104/22)Xf3 ' _

2p3(57p 3 + 312/z2)X3fz" + 6p*(8X~f3 '' + X~f3"') = 585- 585/z~RX.

3.2.2. Plate with s imply-supporteds ides . - -The t ransverse deflection funct ions are chosen as

q ~ l = 0 ' - 6 0 3 + 5 ,

(I) 2 = 0~(50 * - 1403+9) .

T h e related constants are s h o w n in the fo l lowing tables:

i 315~ t [ 3465~2 i 3465~t i 45045~2~

0 1

2 3 4

+ 7936 -- 3968 - - 2688 + 3840 + 3840

+ 18176 + 6272 - - 75648 - - 15360 + 1136640

+ 18176 - - 24448 - 14208 + 30720 + 30720

+ 105728 - 51840 - 706944 + 456960 + 12552960

+ +

35slli

2176 1024 1024

315s12i

- 4224 - 33024 + 158976

315hl i

- 4224 + 5376 + 5376

3465s22i

- 107904 - 59904 + 1476096

5m~ 35m~2 35m21 315mz~

768 1536 1536 209664

5Wz 35~F2

32 64

Subs t i t u t ing in equat ions (8), the fo l lowing equat ions are ob ta ined for f l and f~.

(i) Plate o f constant thickness.

(a) S ing le - func t ion analysis.

3( - 185p 4 - 552p~/~ ~ + 1 0 0 8 ¢ ) f + 12p2(47p ~ + 156/z2)Xf ' -

- 3p2(83p ~ + 2041~2)X~f " + 31p~(2X3f '' + X 4 f ' ' ) = 85 "995/zaRX 4 .

(b) T w o - f u n c t i o n analysis.

3 3 ( - 185p 4 - 552p~/~ + 1008/z~)f~ + 132p~(47p2 + 156/~)Xf~ ' -

- 33p~(83p~+ 204t,~)X2f~ " + 341p*(2Xafl '' +X4f~"") _ 6(1055p4 +

+ 3696p2/z ~ - 1584/~*)f~ + 2402(281p ~ + 11881x~)Xf( - 3p~(493p ~ + 484~)x~f~" - - p * ( 9 8 x a f ( " - 7 1 x ~ f ( ' ' ) = 945.945t~aRX ~ ,

78( - 455p4 + 70402t~ ~ + 1584~) f~ + 312p2(13 lp~ + 8 8 ~ ) X f ~ ' -

- 39p~(493p ~ + 4 8 4 ~ ) X ~ L " + p*(4966x~L" + 9 2 3 x 4 f l ' ' ) +

+ 3( - 11273p * + 63544p2/z ~ + 624624~) f~ + 12p~(5333p ~ + 1 8 6 6 8 ~ ) x f ( -

- 3p2(6333p ~ + 14612/~)x2f2" + p 4 ( 8 1 0 x ~ f ( " + 413x~f~"") = 3513.5 I I~aRX ~ .

(ii) Plate wi th thickness proport ional to width.

(a) S ing le - func t ion analysis.

(42p ~ + 601.202/~ ~ + 3024 /~ ) f + 3p~(02 + 12t ,~)Xf ' +

+ 4p~(5402 - 153tz2)X2f " + 3104(SX3f '' + X a f ' ' ) = 85.995l~3RX.

(b) T w o - f u n c t i o n analysis.

11(42p ~ + 601.2p~p7 ' + 3024~4)/1 + 3302(p ~ + 12 /~ )Xf l ' + 44p2(5402 - 153/~)X~ft" +

+ 341p4(8Xaf~" + X4f~ "") + (230704 + 19285.2p~p, ~ + 9504/x~)f,z +

+ pz(2013pz + 24156,az)Xf2 ' - p~(1494pz + 1452p,~)X~f,~ " + p4(328XSfz"+ 71X4/~ ' ' )

= 945 .945&3RX,

13(507p 4 + 6085-202p, 2 + 9504~4)fl - 39p~(49p 2 + 748/z~)Xfl ' +

+ 78p~( 11 lp~ - Z42/zz)X2f~ " + 13 p~(X3f~ " + X~fx "") +

+ (4317604 + 487203.6p2/z 2 + 1873872p,~)f2 + 3p~(3143p ~ q; 3 0 8 3 6 / ~ ) X f ( -

- 12p~(1073p ~ + 3653/zz)XZf~," + p~(3288X~f ( " + 413Xaf~"") = 3513 .5 I ~ 3 R X .

10

3.3. Results.

The variations of the maximum deflection and the maximum value of the yon Mises stress ~1 with a/b I are plotted in the figures listed below for a series of values of bl/b ~. The corresponding curves for rectangular plates which are also given in these figures have been obtained from results given by Timoshenko and Woinowsky-Krieger 2.

Thickness

Constant

Proportional to width

Boundary conditions

Sides

Clamped Clamped

Simply-supported Simply-supported

Clamped Clamped

Simply-supported Simply-supported

Ends

Clamped Simply-supported

Clamped Simply-supported

Clamped Simply-supported

Clamped Simply-supported

Maximum % occurs on

Sides Sides

Wide end Centre-line e

Sides Sides

Narrow end Centre-line*

Fig. No.

2 3 4 5

* large bending stresses in the immediate vicinity of the obtuse corners are neglected.

The following examples are plotted in more detail. The deflection w and the von Mises stress aj are plotted in each case along the centre-line of the plate. The yon Mises stress along the clamped sides and the error bending stress normal to the simply-supported sides are also plotted.

Fig. 10. Comparison of specimen results of the two analyses. Sides and ends clamped, thickness constant, a/b 1 = 2, bdb 1 = 0 . 4 .

Fig. 11. Comparison of specimen results for long plates of constant and linearly varying thickness. Sides and ends clamped, a/b I = 4, b2/b 1 = 0.4 .

Fig. 12. Specimen results for simply-supported plates, a/b I = 2 , bJb 1 = 0 . 4 .

3.4. Discussion of Results.

3.4.1. Comparison of one-function and two-function analyses . - - I f the results of the two- function analyses plotted in Figs. 2 to 9 are compared with the corresponding results of single- function analyses, the following observations can be made.

(i) Def lec t ions . - -The maximum deflection results are virtually indistinguishable except for the shortest plates where differences of up to 2~o occur.

(ii) S t resses . - -When the maximum yon Mises stress is not at the ends of the plate, the results of the two analyses converge as the length of the plate is increased. When a/b 1 is greater than 2 the difference in the results is less than 3 ~ for a plate of constant thickness with clamped sides and is less than 1 ~/o for all the other examples calculated. When the maximum von Mises stress is at either end of the plate, the results, which do not converge as the length of the plate is increased, differ by

11

up to 3}/o for plates of constant thickness and .~-zt~°//o for plates with thickness proportional to width. The two-function analyses can be used however on plates sufficiently long for the stress distribution at the ends under this loading to depend purely on the angle of taper and not on the length of the

plate.

3.4.2. Opt imum thickness var ia t ion . - - I f a clamped plate such that a/b 1 = 4 and b~/b 1 = O. 4

is designed to a maximum stress specification under uniform load, a weight saving of the order of 149/o is obtained by using the optimum linear thickness variation rather than a constant thickness.

This confirms the usefulness of the optimum thickness variation for a plate under uniform loading.

12

a, b

t

x, y

go

P

/z

X

Y

W

0

E

V

D

q

qo, ~,/~

R

ax, %, %v

%

r

v

L.

N O T A T I O N

Suffices 1 and 2 indicate values at x = 0 and x = a respectively.

Length and width of plate

Plate thickness

Cartesian co-ordinates

Deflection

Maximum value of w

b2 b 1

a

bl

x = l + p -

a

2y bl

Y X

Young's modulus

Poisson's ratio (taken as 0.3 in computations)

Flexural rigidity = Eta~12(1 - v ~)

Normal loading

q = q o a ( X ) ~ ( O ) where a and fi are dimensionless functions

_ q0 (bzl~

Stresses

Yon Mises stress

Maximum value of cr t

Bending and twisting moments per unit length

Index such that D -- D 1 X r

Index such that ~ -- X ~

Unspecified function of X

13

~j

Wj

P j l~i

ljk I SJki J

F,(X)

A

Ai, Bi

T

U

V ~

V ~

NOTATI ON--continued

Known function of 0

f +l d 0 - 1

Defined by equations (9)

Defined by equations (10)

Indices in complementary function in equation (14)

Particular integral

Defined by equation (13)

Determinant of hjk terms

Arbitrary constants in equations (14)

Work done by normal loading

Strain energy of bending

Laplacian operator

Biharmonic operator.

No. Author

1 L.V. Kantorovich and V. I. Krylov

S. P. Timoshenko and S. Woinowsky-Krieger

3 M.L. Williams . . . . . .

4 G.G. Pope . . . . . . . .

REFERENCES

Title, etc.

Approximate methods of higher analysis. Chapter 4. English translation by C. D. Benster of 3rd Russian

edition. Noordhoff, 1958.

Theory of plates and shells. Chapters 4 and 10, 2nd edition, McGraw-Hill, 1959.

Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending.

Proc. 1st U.S. Nat. Congress of Applied Mech., A.S.M.E., pp. 325 to 329. 1952.

The buckling of plates tapered in planform. A.R.C.R. & M. 3324. April, 1962.

14

r m

=m O.

FIO. 1. Axes and notation.

~2

15

O~

O 'O3

O-OZ

~" E O 5~ 4

o-01

0-~

o - 4

O-Z

I 2 (3,

4,

/ I

4

1 . 0

o 8 ~z

0 " 4 •

FIG. 2.

maximum

f f

2 3

~r,

Curves of maximum deflection and or/. Sides and ends clamped, thickness constant.

,¢

I ' 0

0 " 8 ~'2

o-~ -~ 0 .4

0 " 0 3

0 "OZ

E~ 3

%~.4

0 - o l

0 - 6

Y J

J

m

J J

3 -4

I-0

0"8 ~Z

0.6 ~-~

0-4

o.,4

a-z

0 0

f f 0 - 8 ~-z 0.6 0.4

Z 3

FIG. 3. Curves of maximum deflection and maximum G I. Sides clamped, ends simply-

supported, thickness constant.

h.4

O. IZ

0 ' 0 8

~ . E t 3 ~,4

0"0.4

/

J

2 ck

f

/ I I

/ /

I.O

0 , 8

o~6

0.,4,

0 . 6

o.4

0 " 2

0 0

/ f I

f

1 .0

0-8 ~r z 0 . 6 . - - 0 . 4 ~'j

2 3 4

It,

FIC. 4. Curves of maximum deflection and maximum a/. Sides simply-supported,

ends clamped, thickness constant,

O-IZ:

0 . 0 8

I=t= 3

o 'o4"

0

0

/

/

f

/

I /

2 3

~ g z

0 ' 6

0 . 4

0 " 2

0 0

i

/

f

f

/

/

2 3 4

SL

FIG. 5. Curves of maximum deflection and maximum c~/. Sides and ends simply-

supported, thickness constant,

I ' 0

0.8 ~-2

&j 0 . 6

0 . 4

I ' 0

o. 8 .~r 2

0 .0 ~rl

0 . 4

oo

0"02 - -

~. E.~, ~

O - O l

0 0 2 ,.'3

o-. m

0 " 6

0 ' 4

0"2.

4-

I ' 0 O, 8, "~'z 0"~ 0"4 -~1

o:=,/ l ~.o/

i

I i 2 3 O,

FIG. 6. Curves of maximum deflection and maximum G? Sides and ends clamped,

thickness proportional to width.

4-

0 " 0 3 - -

0 " 0 2 - -

E+.I 3

0 -01

, ~ 2 - ~ . L - - -

0 0 Z 3

o.

I .O o .~ , ~'2

0",4, ~l

,:1.

0 . 5

o . 4

0 ' 8

J

.D.2 0"4 ~ - -

"~1 0'6 0"8 / I , O

I I 0 I 2 ~3 4-

o,,

FIG. 7. Curves of maximum deflection and maximum cr I. Sides clamped, ends simply-supported, thickness proportional

to width.

IIL ',.0

O q 5

O'lO

0.05

0 0

0"8

0"6

0'4

0 " 2

O O

/ / .

/

Z 3 o .

&l

f

~'2 o.~ 1 ¢ ~-I 0'8

I .O

4.

I .O

O'B

I 2 ,~ 4

FIG. 8. Curves of maximum deflection and maximum crj. Sides simply-supported, ends clamped, thickness proportional to

width.

0.15

O ' 1 0

0.05

I

2 O,

&l

f

f

f /

3 4

36,*

O ' ~

O-,O,

0 " 2

O O

1"0

~'z 0 '8 "~l 0"6

0 '4

/ / / /

2 3 4 o,,

FIG. 9. Curves of maximum deflection and maximum a 1. Sides and ends simply- supported, thickness proportional to

width.

I , O

0 . 8 ~2 o* 6 ~'I 0-4

bO

E~ 5

o-3: B 2

c l bl2

O ' ol

0 - 0 0 5

o o 0.25 O. 50 O-75

D E F L E C T I O N O F C E N T R E - L I N E . P.,ESULTS oF -r i le TWO ANALYSE5 ARE INDISTINGUISHABLE.

SHAPE o~" PLATE

: o . 4 ,-~ =Z

0 , 3

°2 / o, I /Y~=~'- //

O o. 2.5 0'50 0-75 I

:~- TWO-FUNCTION ANA LYSI$ ONE- FUNCTION ANALYS 15

V O N MISES STRESS O N SURFACE O F P L A T E .

FIG. 10. Comparison of specimen results of the two analyses. Sides and ends

clamped, thiclmess constant.

O'OZ

0-01

0

0 . 4

0 , 2 .

/ \ \

O 0.25 o.50 0.75 X

D E F L E C T I O N O F C E N T R E - L I N E .

SHAPE OF PLATE.

PLATE OF CONSTANT THICKNESS

PLATE WITH THICKNESS ' P R O P O R T I O N A L TO WIDTH

/ /

f -

CENTrE-LINE /

- T / - - - -" OBLIQUE. 51DES '\ / \

\

O O'Z5 0.50 0.75 9(: -&-

VON MISES STRESS AT SURFACE.

FIG. 11. Comparison of specimen results for long plates with thickness constant or proportional to width. Sides and ends

clamped.

~ E 4 : z

S J, 4

0'I0

0'05

/ /

/ /

Y

f

\ -\

\ \ \ \

\

0 0'25 0 ' 5 0 '75 I

DEFLECTION OF CENTRE-L INE . PLATE OF CONSTANT THICKNESS

PLATE. WITH THICKNESS PROPORTIONAL TO WIDTH

SHAPE OF" PLATE. ]

J E = o-4, ~..; = 2

0 , 6

o . 4

0 ' 2

/ /

/ VQN MIS ES ---.--~ / STRESS ALON( \

/ CENTRE- LINE \ \ \ _<,

\ E~R0~ ~ T ~ ' - . . \ NORMAL TO ~ . - oBuouE EO~ES \

0.25 0-5

STRESSES AT SURFACE OF PLATE.

Fla. 12. Specimen results of two-function analysis of simply-supported plate.

(87122) Wt. 64/1857 K.5 6/63 Hw.

21

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