A combined analytical and numerical approach for the solution of an edge loaded semi-infinite elastic sheet

ABSTRACT

Edge loading of an semi-infinite elastic sheet is of interest to many engineering applications. In this paper the penetra-tion problem involving a rigid indentor and an elastic semi-infinite sheet of uniform thickness is addressed using a com-bined analytical and numerical approach. The tenth-order approximate theory of stretching of an isotropic sheet is applied to formulate the governing differential equations. Solutions arc then obtained using Fourier transforms for vari-ous loading conditions and numerical schemes are employed to calculate the three dimensional state of stress throughout the sheet. The influence of the aspect ratio on the resulting stress state is studied. Limit solutions for thin sheet and thick sheet are presented. Finite element analyses of the same load-ing conditions are also performed. Results are compared with those of the tenth-order theory and ft.e stress distribution assumptions of the tenth-order theory are examined.

INTRODUCTION

Solution for the state of stress in an edge loaded scnhi-inIi-nile plate or sheet is a Lical elasticity prohlem and it is also of interest to many engineering applications such as ollshore engineering and environmental engineering, For example. the solution 01' this prohleni cotid ' elastic

apprxiia-Xin Sun

Edison Welding Institute Columbus, Ohio 43220

U.S.A.

Chunhui Han

Liaoning Institute of Environmental Protection Shenyang, Liaoning 110031

PR. China

Dale G. Karr

Department of Naval Architecture and Marine Engineering

Ann Arbor, Michigan 48 109-2145

U.S.A.

tion for the indentation process between ice sheets and off-shore structures (Kaff and Sun. 1995). When the plate is extremely thin or thick, the problem is often idealized to a two dimensional plane stress or plane strain problem and the solu-tions are readily presented in most elasticity text books such as Elasticity (Barber, 1992). Here, we consider the three dimensional indentation problem in which the homogeneous and isotropic sheet is bounded by two parallel planes z = ±c which are considered traction free, see Fig. I. The tenth-order theory derived by Reissner and Clark (Reissner, 1942;Clark and Reissner, 1984) uses a set of two-dimensional differential equations for approximating the three dimensional Poisson's ratio corrections to the generalized plane stress theory of stretching of sheets. The problem with sheet subjected to nor-mal varying sinusoidal loading has been studied by Clark (1985) using the tenth-order theory. Kaff and Choi (19S9 solved the problem with the applied stress being constant

through the thickness of the sheet and extending from to x = a . In this study. a similar approach is used and more general loading conditions are considered. Solutions arc pre-sented in the general t'orm of Fourier transforms. Numerical solutions for three loading conditions are presented with solu-(ions involving three Bessel Functions of the first kind. More accurate schemes arc applied to evaluate the Fourier trans-fornì and its inverse transtorrii m order to get the converged three dimensional state of stress,

Proceedings of The 1996 ASME Design Engineering Technical Conferences and

Computers ¡n Engineering Conference

,'

'

--r. August 18-22, 1996, Irvine, California

96-DETC/CIE-1 628

A COMBINED ANALYTICAL AND NUMERICAL APPROACH FOR

_THE

A study is presented of numerical results for the stresses that develop for various aspect ratios and various loading con-ditions. Solutions (for the stresses at the center of the

load-ings) are also obtained in the limit as the aspect ratio approaches zero. Clark (1985) obtained solutions for sinusoi-daily varying edge loading; analytic solutions for this loading were obtained for the limit condition as the thickness to wave length ratio approaches infinity. This limit condition is analo-gous to the condition for vanishing aspect ratio in this study. In contrast with what was presented in Kan and Choi (1989) for the limit solution, this study shows that the tenth-order theory is less accurate as the plate thickness increases (or as the width of loading decreases, i.e. knife loading). This is because the assumed stress variation across the thickness is no longer true, and therefore the limit solution does not converge to the plane strain solution.

GOVERNING DIFFERENTIAL EQUATIONS

In order to generalize the two dimensional plane stress solution to _{three dimensional sheet, Clark and Reissner} (1984) assumed the stresses vary across the plate thickness in a parabolic variation. Since the upper and lower surfaces of the sheet are traction free, this assumption is compatible with equilibrium requirement and the boundary conditions. Based on these assumptions, the stress distributions are expressed in terms of stress resultants as:

.

t)

N)

(R , RxY)7,() (1) =

_rZ().

(2) - 2c (r1, tr.) = -(S, S)Z'(z), where, Z(z) =

_(i

z2]2 Z(z) = z[l

-The resulting governing differential equation for the stress field can he reduced to three uncoupled partial differential equations involving three stress functions p. 1 and s (Clark,

1985):

Z(z) = 1 - 3.

C

given i n the tollowi ng (Clark. and Rcissncr. 1984):

A0=,A1

=(6970j.A2=_.

(6)

in which y is the Poisson's ratio. The stress resultants on the right hand side of equations (1) and (2) are found by first introducing the auxiliary functions:

2v 2 K(x,y) =

ç--c w.

l-69v/70

2 2 ' V 2

(x,y) = [(2_v)W _{3(1 +v)} C V \11J

6(1 +v)V P(8)

The equilibrium requirements for the stress resultants are then:

=

N_

a K_2' N Y

2

2a 4c2a2û _2ax 4c2a2c

R_X

= i,-c

2

ay . ax2 axay'

2 '

2ax

2c'(aC1 a&

=

axay+

I,ji -J'

S_x =

-!_,S

=_+,T=_V2w

ax ay ay ax

From these equations we can see that, for a thin sheet, if one formally puts c = O, then from equations (4) and (5), = 0, w = O and all quantities except N1, vanish. K = q is a biharmonic function which represents the Airy stress function of elementary plane stress. As discussed in the following sections, edge loadings are considered which arc applied over an edge distance of 2a (Fig. 1). The aspect ratio is defined as the ratio of the width of the loading to the sheet thickness, y a/c. For a thick plate, i.e. c » a. the aspect ratio y approaches zero. Numerical solutions for a thick sheet

have been obtained in this study as y - O.

APPLIED TRACTIONS AND BOUNDARY CONDI-TI O NS

We consider three kinds 01 loadings as illustrated in Fig. 2. General loading conditions can also he applied to the edge using the formulation shown below. The stress distributions along the plate edge corresponding to the three loading condi-tions are: (7) (9) (IO) 74(p _{= O.} (3)

4V2

-

= O. (4) Aie4 V4 ₊

Ae

\7 _{= (5)}

22

{_o(1_x /a

P1(x)

O (Ixl>a)

2 205

P2(x)

{_oU_x

la)

(k1a)

(13)

O (Ix>a)

0(1x/a)

P3(x)

{ 2 2

O _(Ixt>a)

The boundary conditionsarethe followings:

O, z) = P.(x)

_{j =}

1, 2, 3 (15) 3(1+

v)'iI +

2 _{2(1_69v2/70)-2 2]}

t(x.O.z)

= 0,

_t(x.0,z)

= 0. (16)

+[1+(2_v

63(1+v)

D()

The edge tractions in equations (12)-(14) represent two dimensional parabolic stress distribution, the distribution cor-responding to a frictionless rigid cylindrical indentor(2-D Hertz problem) and the distribution corresponding to a rigid flat indentor respectively (Timoshenko and Goodier, 1970; Barber, 1992).

Substituting of equation (1) into equation (15) yields:

O)

R(x, O)Z"(z)

= P.(x)

j =

_{1,2,3 (17)}

Hence we have:

N(x,

0) = 2c P1(x), (18)

l-2 " - T-2 - -

-(19)

+ iD()

+ + 1iE(E) - i,C(Ej = 0. (27)

R(x.o)

= O.

Equations (1 8)-(22) represent live boundary conditions i n terms of the two dimensional stress resultants.

RESULTING SYSTEM OF EQUATIONS

The method of Fourier transform is applied to a smi il ai prohlcnì with constant edge traction by Karr and Choi (I 989). Here, a more general loading coud1(1011 ¡'(y) can he applied and we use the three edge loadings discussed above as

humer-(jxja)

(1x1 a) (14)

ical examples. The development of the governing equations

(12) _{for general edge load can be performed in a similar manner;}

for completeness of this presentation the Fourier transforms of the stress resultants are presented in the Appendix.

Substituting equations (A.l)-(A.9) in the Appendix to the boundary conditions (18-22), a system of five by five equa-tions can he derived in terms of the normalized variables:

A()+2[D()+E()j

= -2 2(1 69v2/70) +[1 + (2 - v) 63(1+ V) 2ft2]E() = O,

_i3(1)B() +(i

+ +

_{+i [21(2}

21 2 V) 2(1 - 69v2/70) 2 63(1 +v) 2 + 2r_[21(22 V) 2(1 - 69v _"70lî21E(Ej = 0, 63(1 +v)

j

2v-2 I-2 ,

--

B()

+

[

+

D()

2 +

2E()]

= 0,(26) (23)

in which, E is the variable introduced through the Fourier transform, p and 1 are complex conjugates as defined in Clark (1985), R. is the Fourier transform of the edge loading

P,. and the overlined variables are the variables normalized with respect to a or c:

= eE,

i

= x/a. = y/a.

Evaluating the integrals for the Fourier transforms 01 the three edge loadings (Gradshteyn and Ryzhik. 1980). we have:

R1 _{J P1(x)e"d.t} =

_a(1.[_)J(u):

₍₂₈₎

R f !'2(x)e" dt = -(l(1,1t (29)

In addition, the shear stresses and

t

must vanish on the boundary y=0 and therefore:

N(x. 0)

= 0. (20)

R(x,O)

= 0. (21)

A() +

2[D() +E()]

= -alt

J1(y)

-(Ey) (23 b)

A()

+

j2ED()

+ E()] =

_{-alti (Y)}

_(Thc)

Equations (23 a,b.c)-(27) are the five normalized boundary condition equations involving five unknown functions

A(Ej, ß(E), C(), D(), E().

NUMERICAL SCHEMES AND RESULTS

The five by five system of equations (23-27) is solved in terms of E explicitly using the symbolic software MAPLE. Having A(e), B(E), C(), D(), E(Ej, the Fourier transform of

the stress resultants _{Ñ,.. and}

Ï'(see equations A.l-A.9) can be evaluated numerically fora

certain y location in the sheet for all E values in the integra-tion range. Theoretically the range of integraintegra-tion for Fourier

transform

should be - <E

< , but practically a smaller

range can be used based on the shape of the integrand. Karr and Choi (1989) carried out their computation for the range of -60 <60 and their solution is not stable as y approaches zero. In this study, convergence study has been performed and it is found that the minimum converged upper and lower bounds for integrations are = ±1000. The stress resultants are then determined by integrations of the inverse Fourier transforms and finally the stresses throughout the entire sheet can be calculated using equations (I) and (2). Numerical inte-gration s employed to evaluate the inverse Fourier

trans-turms. The Bessel functions on the right hand sides of

equations (23a.h.c) are also evaluated lr each integration point (Watson. 1966). The integrals tor the first two kinds of loadings are relatively easy to evaluate and the Computation tinte used is moderate. The third kind of loading, because of the singularity of the stress distribution at x = ±a . needs

more integration points inside the upper and lower hounds and theretore requires more computation tinte to ohiatu satis-tactory answers. Several Fortran codes have been developed to pertorni the computations.

.1

The numerical results for the stress (î, . 0) are shown in Fig. 3 for cases with the aspect ratio y = 1.5 and Poisson's ratio y = 0.3 . Figure 3(a) shows the results for the parabolic distribution of the applied traction stress. Results for loading conditions P2(x) and P3(x) are shown in Fig. 3(b) and 3(c) respectively. The stress in the middle layer of the sheet , 0) decays more rapidly with respect to for the

para-bolic loading than for the loadings P2(x) and P3(x). lt is also obvious that with the improved integration range for E, the calculated stress distribution presented here are more accurate than that of Karr and Choi (1989) and the stress distributions

for = O converge to those prescribed in equations (12)-(14).

The results for the stress 0) are shown in Fig. 4. For all these loading cases, P1(x), P2(x) and P3(x), the stresses i decay more rapidly with respect to than do the stresses It is also interesting to note the zone of tensile stress at the edge of the sheet ( = _{O )just beyond the edge of}

the loaded area. These tensile stresses are in the order ofone

tenth of the applied traction stress.

Figure 5(a,b,c) shows the transverse normal stress , 0) developed for the three kinds of loading conditions for an aspect ratio y = 1.5 and Poisson's ratio V = 0.3 . It is worthwhile to notice that a significant amount of tensile stress builds up near the location of (i= ±1, = 0, =0) on the edge of the sheet for all the three kinds of loadings. Thus if the sheet is made of brittle material, a spalling mode of failure may initiate at these locations. It is also noted that ,(.t, j, 0) decays more rapidly with respect to than the other two nor-mal stresses and therefore the state of stress away from the indentor is very closed to that of plane stress. This indicates that the tenth-order theory correction to a plane problem rep-resent an edge zone correction which is only important within a certain distance from the indentor.

En order to verify the results obtained by the tenth-order theory, finite element analyses of the same problem have also been performed using the commercial FEA code ABAQUS. Eight node three dimensional brick element C3D8 is used

near the contact region. The far field is modeled with an eight node infinite element CIN3DS. Only one-quarter of the semi-infinite sheet _{is modeled and symmetry conditions are} imposed on the planes of x = _{O and z=O. The model is shown}

in Fig. 6. iThe above discus sed three loading conditions are applied as pressure distributions along the plate edge y=O. Results compared with those of the tenth-order theory are

plotted as discrete points in Fig. 3a.h,c)-Fig. 5(a.b.c). As shown in Fig. 3(a.h.c) and Fig. 4(a.h.c). very good correla-lions are achieved between the tenth-order theory and the huile clenueiit analysis tor normal stresses and on the symmetry laiuc z=O However. results tir between tite t\vo nethods are not consistent uit niagiu itude. as showui un Fii

R3 _{J P3(x)e} dx = _{-aa0tJ0(Ea). (30)}

In equations(28)-(30), J312, J1 and J0 represent the first kind Bessel functions of orders 3/2, 1 and O. Substituting equations (28)-(30) into equation (23) we get:

5(a,h,c) This is due to the fact that only 4 elements are used in the through thickness direction. Should more elements be used in the z direction, more accurate results will be achieved.

By examining the FEA results, it is also found that the assumed parabolic distributions of and o along z direc-tion are very close to the true stress distribudirec-tions, with the magnitude of being very small throughout the sheet and eventually becoming zero at the plate edge y=O. This is con-sistent with equation (19) and it ensures that the resulted stress distribution on edge y=O converges to the prescribed condition of c being constant through the thickness.

CONCLUSIONS

We now discuss the effect of aspect ratio y. 1f the plate thickness 2c is much smaller than the loading region 2a, goes to infinity and the solution degenerates to the solution of a plane stress case as discussed previously. On the other hand, if the plate thickness 2c is very large or if the loading region is very small, i.e. a « c, ygoes to zero. Under such conditions, the assumed stress distribution in the z-direction no longer represents the true stress distribution and therefore the limit-ing solution should not converge to a plane strain case. This conclusion is in contrast to Kan and Choi's y -*0 conclusion. In this study, larger integration limit and more integration points are used to achieve more stable and accurate numerical results as y approaches zero. Moreover, as shown in the fol-lowing discussions, the same limit solution is obtained for dif-ferent loading conditions.

The stress 0, 0) at the center of the edge of the sheet for four loading cases is shown in Fig. 7 as a function of the aspect ratio

ï.

Poisson's ratio is y = 0.3 . The uniform load-ing case addressed by Karr and Choi (1989) is shown together with the three loading cases addressed here. The peak stresses occur near an aspect ratio of -y 0.5 for all loading cases; the maximum compressive stresses are approximately 20% higher than the value obtained for plane stress and plane strain in which the value of 0, 0) is unity (Timoshenko and Goodier, 1970). For all loading conditions the limit value of

(O. 0,0) = -1.09 is approached as y -*0. Since larger E, range and more integration points have been used in this study, the limit stress values as y approaching zero can he car-ned oui very close to -y=O without the numerical instabilities experienced by Karr and Choi(l989).

Results for the stress at the corner ut the plate. (0, O. ±e) arc shown in Fig. X. The minimum compressive stresses occur in the range ut y between 0.3 and 0.8 depending on the load-ing conditions. These minimum compressive stresses range Ironi about fi5(% of the applied stress to about 56% oF the applied stress tor load condition P( y) . The limit condition

for *0 is a(0,0,±c) = -0.815 for all three loading condi-tions.

The transverse normal stress at the center of the loading is shown as a function of in Fig. 9. No significant tensile stresses develop at this point for loading conditions P (i) or P2(x). The limiting value for ,(0, 0, 0) is -0.809. Significant transverse stresses essentially vanish for aspect ratios above 3 or 4.

In addressing the transition from plane stress, three dimensional effects regarding are initially more pro-nounced for loading condition P1 (x), followed by P2(x), the uniform loading, and the condition P3(x). However, as the aspect ratio continues to decrease, below say y = 1.0, the rel-ative influence of the three dimensional effects changes. The loading condition P3(x) shows the most significant three dimensional effects on 0, ±c) and (0, 0. 0). Deviation from plane stress occurs less rapidly (as 'y decreases) for OZ(0. 0, 0) than for 0, ±c) or 0, 0). For example, for loading condition P(x), a change in of 10% of the applied stress occurs for an aspect ratio of 3; a change in Z(0' o, O) of 10% of the applied stress occurs near a value of y = 1.5.

The solutions for 0. 0) and (O, 0, 0) obtained in Fig. 7, 8 and 9 for vanishing aspect ratio are different from the plane strain solutions suggested by Kan and Choi (1989). Again, the underlying assumptions of the tenth-order theory involve a prescribed parabolic distribution of the transverse stress which vanishes at z = ±c. These assumptions preclude obtaining the plane strain condition for which it is assumed that there is no functional dependency with respect to the z coordinate.

REFERENCES

Barber. J.R.. 1992. Elasticity. Kluwer Academic Publish-ers, the Netherlands.

Clark. R.A.and Reissner. E.. 1984, "A tenth-order theory of stretching of transversely isotropic sheets". Z4MP. Vol. 35. pp. 883-889.

Clark. R.A.. 1885. "Three-dimensional corrections tor a plane stress problem". liii. J. Solids Structures.. Vot. 21. PP.

3-10.

Gradshteyn. IS. and Ryihik. 1M., 1980. Table of inte-grals. series and prodwis. ('orrected and enlarged edition. Academic Press Inc.. New York.

Kan, DG. and Choi, S.K., 1989, "Three-dimensional Additionally,

elasticity solutions for edge loaded semi-infinite sheets", i 2 V _11ß()-RIY+

2c

-Z.angew.Math.Mech, Vol. 69, pp. 329-337.

Kan, DG. and Sun, X., 1995. "Damage Evolution During Impact of an Ice Bar with Lateral Confinement", InI. J. of Off-shore and Polar Engineering, Vol. 5, No. 1, pp. 23-31.

Reissner, E., 1942. "On the calculation of three-dimen-sional corrections for the two-dimenthree-dimen-sional theory of plane stress", In: Proc. 15th Semi-Annual Eastern Photoelasticity conference. pp. 23-3 1.

Sneddon, IN., 1951, Fourier transforms. 2nd edition, McGraw Hill Book Co., Inc., New York.

Timoshenko, SP. and Goodier, J.N., 1970, Theory of elas-licity. 3rd edition, McGraw Hill Book Co., mc, New York

Watson, ON., 1966, A treatise on ¡he theory of Bessel functions, Cambridge, Cambridge U.K.

APPIWflIX muitr TRANSFORM MFTI-1Ofl

The Fourier transform and its inverse transform are defined + C22[(2- V) 1 - 69V2/70ft21E()_Yj2

(/)2

to be (Sneddon, 1951):

21 3(1 +v) _J]

y)

=

J

e'q(x, y)dx (p(x, y) 2 J

eIXj

y)d '

4c 2

As shown in Kan and Choi (1989), the Fourier transforms + 2c

of the stress resultants expressed in terms of the five functions

A(E),B(E), C(E), D(Ej and E(e) are: _{1xy =}

_{_2i26(lVV)c2B()et+}

N0 _{2v 2}

Ñ. )

=

2B()l - 2IA()

+ B()y]

}e

- -c N0

J

[2

+ 21D()e + (Íc)

[2 (: + (/c)

} + R

)J

L cj

(Al)

Ñ(E,,v)

=

N0[A() vB()le+ c22No

[D()eI

+

E()e

+ (O/ei] (A.2)

2

22

+11--c

21

4C2J2+2C()4

+ 2c 2 V R_YY -2c = 6(1 + V)

B()e

+

J2 2

1 -69v2/70 2 + C 3(1 +v)

]}D()e

2Oj (A.4) (A.5)

+

_ic2»

₊

E()[2

_y _69V2/702]YJ+(/C)2

(A.7)

222

l-c

f 1C E +

CI

6(1 + V)

692-lV

₇₀ 2

Il-(2-v)

_3(1+v) 2\1 C

+fl(2_v)

69 2

1--v

70 21 E(E)e ()2 3(1 +v)

j

Ñ = -

_{- II[A(}

+ vB()j

Il

and = +I'Cl + E()e ' +

2C22C)

j

_(A.6)

f 21

-yIç +-j

JY = -

iC()e

q 2c

/2

+

(2D

+ (

J2

+

()2E(

-J

()2 (A.8) + * c) O

:'

0.0 -0.2 t) L

- -0.4

e o =

i[2D()Y

+(Ic) g

- ..-

/

-0.o --.----

r

In equations (A.1)-(A.9), the notations N0

= 2c0,

2

= (-

A2+iJ4A - A)/(2A ) have been introduced and

is the complex conjugate of .t2.

Fig. I Semi-infinitesheet withcompceive edge liding.

/

(a) a --X Paz) i 4

Fie. 2 Thro. kini s( ls.Iine cadiuons P,(x).

7

z

-0.8

-,.-o

z

M b 0.0 Ib

a -0.2

I, t) L -0.4 e o -0.6 (I - -0.8 e o z -1.0 o b 0.2 0.2 2.0 -2.0 L a,

- -4.0

e o f., -6.0 V -8.0 e o z -10.0 1=0.0 7=1.0_ 113 M VEA =0.0

U It 13

y ItA _t0 G ItA yi.5 12 1.4 16

FIg. 3(a) (iy.0) for y=1.5 and x'=0.3 for loading condition P(x).

-M FEA

U FFA 1C3

V FU. 1=1.0

a FU. =1.5

Fig. 3(b) ,(f yo) for y=l.5 and v=O.3 for loading condition P,(x).

VO.0

--VEA 10.0 u VtA i=0.5 a fT4 =1O a FT. =15 0 02 0.4 0.6 0.8 12 14 16 X

FIg. 3(c) &,(íy.0) for y=L.5 and v=O.3 for loading condition P,(x). -1.0 o 02 0.4 0.6 0.8 X 16 02 0.4 0.6 0.8 1 12 1.4 X -x

ttd

lei

-.4-'

-

---.

- -0.5 o M) o a

-1.0

o o

z

-1.5

o 2.0 b 0.0 X Ib E -8.0 o o

z

-10 0 o M I I 0.2 04 0.6 02 0.4 0.6 0.8 X

-

__. - - - -.

0.8 X y=o.o y=1,o . . MFFAy.O PEA '=O.5 PEA y=iO o ItA =L5 y,o.o

-M FA2.O.O PEA o PEA o PEA 7=1.5 1 12 1.4 16 y=o.o 7=15 O PEA 7=0.0 u PEA 7=0.5 V PEA 7=1.0 o PEA 7=1.5 1.2 14 16 li V I--J

j 0.05

0.00 V E

i-0.05

-0. 10 E -4.0 -o o o

z

-6.0 LEA 7=0.0 PEA -0.5 o PEA y-i3O O PEA 7-1.5 M

_-_..r-_.._.__:;_

, o O 02 0.4 0.6 0.8 X y15 M VEA 7=00 s A 7=00 ° VEA y-LO 'EA =I.5 I I J I 0.2 0.4 0.6 0.8 X 1 12 1.4 16

Thm

12 1.4 16 b M b 0.3 Ib r e a 0.2 y=Lo M YEA 7=0.0 -I e YEA 7=05 -o o r o u 0.1 _{° YEA 7=1.0} YEA 7=1.5 E o o

z

0.0 -0. 1 J I I I I b 2.0 M b

0.0 -

S S Ib e

r

u I. --2 O Y-0,0 b X b b 0.0 0.5

-

r-M)

-.-M) a _{__.- -s - -S- -} -S-M)

-

_z o o -0.5 M) o o

z

-1.0 -1.5 0.5 b 0.20 b X b _{b 0,15} =o o 'IX _{O O} M 0 Ib M) M) M_. .,.,,,.... 0.10 7-Le o 02 0.4 0.6 0.8 1 12 1.4 1.6 _{o 02 0.4 0.6 0.8 16} X _X

FIg. 4(a) o(iy0) for 7=1.5 and ¡'=0.3 for loading condition P1(x). _{FIg. 5(a) 5(f.70) for y=1ß and v.3 for loading condition P1(x).}

FIg. 4(b) 51(i.0) for 7=1.5 and ¡'=0.3 for loading condition P0(x). Fig. 5(s) (17.0) for 7=1.5 and ¡'=0.3 for loading condition P0(x).

Fig. 6 Finite element mesh used for analysis.

-0 5

b

-0.6

X Ib

-0.7

j

Urnjadflg

Load in(x) Pz(xL Load ing P(x) 0.2 b b" 0.0 -0.2 a -0.4 o -0.6 L E -0.8 -1.0 f

7

f

/ z..

f

/

í''

'z

- __..-.

-- :

- -.- -

:7/7

rrnj Loading P2(x) Uniform loadg Loading P2(x) Loading P3(x) _{Loading P3(x)} 6 2 3 4 5 Aspect ratio, y

Fig.? ff5(0,0.0) versus aspect ratio y for i=0.3.

0 2 3 4 5 6

Aspect ratio, y

Fig 9 a(0,O,0) Versus aspect ratio 7 for v=0.3.

0 2 3 4 5 E

Aspect ratio. y

rig.8 ä1(0,0.±c) versus aspect ratio y for vO.3.

-0.95 b

f -Loo

X -1.05 a

ï

-1.25 C)