Fundamental collocation method applied to plane thermoelasticity problems

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Pergamon

& Strucrures Vol. 51. No. 5. pp. 795-797. 1995 Copyright 0 1995 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 19.50 + 0.00

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FUNDAMENTAL COLLOCATION METHOD APPLIED PLANE THERMOELASTICITY PROBLEMS

TO

E. Mahajerint and G. Burgessf i-Mechanical Engineering Department, Saginaw Valley State University, 2250 Pierce Road, University Center, MI 48710, U.S.A. SSchool of Packaging, Michigan State University, East Lansing, MI 48824, U.S.A. (Received 2 June 1994)

Abstract-The fundamental collocation method is adapted to plane stress thermoelasticity problems by considering the analogous body force problem. The region is subdivided into cells to handle the body forces and body forces are treated as concentrated loads acting at the centers of the cells. Examples are used to illustrate the method. The results show that the accuracy is good provided that the number of cells is sufficiently large.

aaxvaavv

INTRODUCTION

ax+-+FY=O Thermal stresses that arise from temperature changes in an elastic structure can be substantial. Interest in the computation of these thermal stresses is due to the failure of many industrial components because of them. Generally speaking, we can regard the quantity EcrAT as the order of the magnitude of these stresses where E is the modulus of elasticity, c1is the thermal expansion coefficient, and AT is the temperature change. The yield point of a material may be reached if AT approaches a,/Ea where a, is the flow (yield) stress of the material. The exact thermal stress distribution in a material can be analyzed by solving a body force problem in elasticity. Due to the complexity of the formulation, however, limited computer modeling techniques are available in the literature [1,2]. This paper uses the relatively new approach called the fundamental collocation method (FCM) [3] to analyze planar regions of arbitrary shapes with stress-free boundaries subjected to nonsteady state temperature fields. The thermoelastic problem is decomposed into two problems: one has a known closed form solution and the other involves body forces which are handled numerically using the fundamental solution for a point load in an infinite plate.

ay

V2(a,+a,,+EuT)+(1+v)

a",>

ax+dy

=0(3)

where v is Poisson’s ratio. For the problem with no body forces, stress-free boundary conditions and steady-state temperature field, the solution is 0X.X= b,y = cvv = 0 [5]. For other cases with V2T # 0, we need to reformulate the problem in order to generate solutions that satisfy both the equilibrium and the compatibility conditions.

THE PROBLEM

The problem of interest (Fig. 1) consists of no applied forces with stress-free boundary conditions subjected to a non-steady-state temperature field. The solution to this problem is the superposition of the solutions described below [5]: Problem l-the stress field required to suppress the thermal strain, o XI =ayv= Problem

2-a

EuT -I_v,

and a normal

uxy=o.

body force problem

Eu aT F,=--.-.-i -vax’

FORMULATION

The three necessary and sufficient conditions on the stresses Q, , cxy and avv for any plane stress, thermoelastic problem with pure stress boundary conditions and known body forces F,, Fy and temperature field T(x, y) are [4,5]:

(9%

F,=--

Ea

(4)

with aT

i-vay

(5)

tensile stress on the boundary

equal to

o,=j--$

EciT

(6)

To solve the problem, the region of interest is imagined to be embedded inside an infinite plane of the same material. A series of N point load pairs in the (x, y) directions with magnitudes 795

196

E. Mahajerin and G. Burgess axx=oxy=oyy=0

I-2 ---~-~2a

I

i, Y (_-+ 1-m ~~ + 2b \

‘,>;-

f

I-+x

I /

0

* T=T(x,y)

P=P(y)“L_

$T+O_ axx=axy=ayy=0

j

Fig. 1. A general two-dimensional lem.

thermoelasticity

prob-

(P,, Q,), = 1,2, . , N are applied outside the region at the source points XS,, YS, located at a distance DS from the boundary (Fig. 2). To treat the body forces, subdivision of the interior of the region into cells becomes necessary. Point loads consistent with the magnitudes of the body forces are then placed at the centers of the cells. Using the notation Hap; y as the stress aZp due to a unit point load in the ydirection, we can write the stresses at any internal field point (x, y) as the combined effect of the external and internal point loads. For example, the expression for cr,, at any field point (x, y) will be

Fig. 3. A plate subjected to pressure and internal heat generation. are given in the Appendix. The arguments of H indicate the point of influence (x, y) followed by the point of application of the unit point load. Similar expressions can be written for cr,, and o)r. To determine (P,, Q,), we select N boundary points (xe,, yB,), i = 1,2, , N and force satisfaction of zero traction boundary conditions at these boundary which generates a system of 2N linear points, equations for the determination of (P,, Q,). Once found, the solution for the stresses at any internal field point is obtained via eqn (7) and related expressions for err and o”..

Example

x

H,,,(x,Y;%,

YG,)

+ Q, H,,.,. 6, Y ; XS, > YS,)I.

(7)

Here XC, YG denote coordinates of the centers of m internal cells of areas AA,. The expressions for H,,,; (xGyG),Cell

Consider a rectangular plate 2a x 2b (Fig. 3) with material conductivity k subjected to parabolic pressure P = -EaT,,(l - y*/b’) on the sides and an internal heat generation q. = 2kTo/a2 where To = constant. It can be shown that the corresponding temperature field is T = T,(l - x2/a’). The exact solution for the stresses is determined from (1) (2) - y*/b*) and el, = orv = 0. and (3) as o,, = -E,aT,(l The stresses were computed numerically via eqn (7) for values of E = 30 x 106psi, a = 7.5 x 10m6 in(in OF))‘, To = 100 “F and a = b = 5 in using 40 boundary points and DS = 1 in. The interior of the region was subdivided into 100 internal cells of equal areas. The results for ulr along the diagonal of the region agree well with the exact results (Fig. 4). Values for o,, and u,,~ are not shown because they are zero.

0

‘Z

Center

1

9.

I

.Z -500 1 5 -1000 g

Num.

point

“,,DS <-’

:

:

3.5

4

Exact ;-1500

. source point boundary

point

Fig. 2. Setup for FCM.

z -2500

0

0.5

1

1.5

2

2.5

3

4.5

Position Along Diagonal (x=y), inches

Fig. 4. Results for the rectangular

plate (example

I).

197

Fundamental collocation method I

B I NUM 30 i ‘Z

20 1

;

10 I

d/ B_

p! 0: K,,

'

outside the boundary are sufficient to produce an accuracy on the order of 1% when the number of cells is large. The point load pairs may be placed at virtually any distance DS without affecting accuracy. Reducing the number of cells however can have a profound effect, depending on the problem. This is why a total of 90 x 30 = 2700 cells were required to produce the level of accuracy shown in Fig. 5.

-20

-30 : -40 ’

I

Distance

along the x-axis,

inches

Fig. 5. Results for the circular disk (example 2).

Example

REFERENCES

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

2

A thin circular (E = 30 x 106psi,

disk of radius

“a”,

made of steel

a = 7.5 x 10m6in(in OF))’ with the

temperature field T(r) = 2T,(l - 2r2/a2). The solution for the radial stress cr, and the circumferential stress croO can be obtained analytically [5,6]. The radius was taken to be 5 in, N = 20, and DS = 10 in. The body force sums were evaluated by dividing the interior of the disk into 90 identical sectors (polar triangles). Each sector was subdivided into 30 cells and body forces were placed at the midpoint of the resulting polar rectangles. The resulting stresses are compared to the analytical results in Fig. 5. CONCLUSIONS

The fundamental collocation method is easily adapted to the solution of non-steady state plane stress thermoelasticity problems. It loses some of its simplicity, however, because now the domain must be subdivided in order to handle the body force problem associated with thermal effects. Numerical errors in general are attributed to treating the distribution of body forces as point loads centered on the cells, not to the collocation process itself. Varying the number of cells (M) and concentrated sources (2N) shows that accuracy is affected mainly by M not N. In fact, for many problems, as few as 10 point load pairs

1. Y. Ochiai and R. Ishida, Unsteady thermal stress analysis in axisymmetric by means of boundary element method. J. therm. Stresses 15, 507-518 (1992). 2. S. Sharp and S. L. Crouch, Boundary integral methods for thermoelasticity problems. J. uppl. Mech. 53, 298-302 (1986). 3. G. Burgess and E. Mahajerin, A comparison of the boundary element and superposition methods. Comput. Strucf. 19, 697-705 (1984). 4. B. A Boley and J. H. Weiner, Theory of Thermal Stresses. Krieger, Florida (1985). 5. S. P. Timoshenko and J. N. Goodier, Theory of Elasficity, 3rd Edn. McGraw-Hill, New York (1970). 6. J. N. Goodier, Thermal stresses and deformations. J. appl. Mech. 24, 461474 (1957).

APPENDIX: FUNDAMENTAL SOLUTION CONCENTRATEDLOAD

FOR

(a) Stresses (H,,, H,,,, H,,) due to a unit point load acting at the origin of coordinates in the x-direction in an infinite plane with Poisson’s ratio v [4].

[ii]=

-:~[;~p:+z;;~.

(b) Stresses (H,,, H,, H,) due to a unit point load acting at the origin of coordinates in the y-direction in an infinite plane with Poisson’s ratio Y [4].

where P = (I - v)/2(1 + v), Q = 0.5 + 2P and r* =x2 +y’.