Rayleigh-Ritz and modified Rayleigh-Ritz procedures are used to construct approximate solutions for the response of a thick-walled sphere to uniform pressure loads and an arbitrary radial temperature distribution. The thermoelastic properties of the sphere are assumed to be transversely isotropic and nonhomogeneous; variations in the elastic stiffness and thermal expansion coefficients are taken to be an arbitrary function of the radial coordinate and temperature. Numerical examples are presented which illustrate the effect of the temperature-dependence upon the thermal stress field. A comparison of the approximate solutions with a finite element analysis indicates that Ritz methods offer a simple, efficient, and relatively accurate approach to the problem

Tauchert, T. R.

English

NASA Technical Reports Server

THERMAL STRESSES IN A SPHERICAL PRESSURE VESSEL HAVING 
TEMPERATURE-DEPENDENT, TRANSVERSELY ISOTROPIC, ELASTIC PROPERTIES 
T. R. Tauchert 
University of Kentucky 
SUMMARY 
Rayleigh-Ritz and modified Rayleigh-Ritz procedures are used to construct 
approximate solutions for the response of a thick-walled sphere to uniform pres- 
sure loads and an arbitrary radial temperature distribution. The thermoelastic 
properties of the sphere are assumed to be transversely isotropic and nonhomo- 
geneous; variations in the elastic stiffness and thermal expansion coefficients 
are taken to be an arbitrary function of the radial coordinate and temperature. 
Numerical examples are presented which illustrate the effect of the temperature- 
dependence upon the thermal stress field. A comparison of the approximate solu- 
tions with a finite element analysis indicates that Ritz methods offer a simple, 
efficient, and relatively accurate approach to the problem. 
INTRODUCTION 
Modern engineering structures are often subject to thermal environments in 
which the temperature causes significant variations in the thermal and mechani- 
cal properties of the material. Over certain temperature ranges the material 
may behave elastically, but have variable stiffness and thermal expansion char- 
acteristics. In addition, modern materials of construction (e.g. composites) 
often possess anisotropy and nonhomogeneity. While most classical thermoelas- 
tic solutions are not applicable to situations involving temperaixre-dependent 
anisotropic behavior, some progress has been made in this direction. 
ple, the problem of a hollow sphere with temperature-sensitive isotropic elas- 
tic properties has been studied by Nowinski (ref. 1) and Stanisic and McKinley 
(ref. 2). More recently Hata and Atsumi (ref. 3) investigated the response of 
a transversely isotropic sphere exposed to a sudden temperature rise on its in- 
ternal surface. 
For exam- 
In the present paper a transversely isotropic hollow sphere having temper- 
ature sensitivity and/or initial nonhomogeneity is considered. The variability 
of the thermoelastic properties may result from manufacturing processes, in 
which case the properties depend upon position but not temperature, or the non- 
homogeneity may be a consequence of the materials’ temperature sensitivity. 
639 
https://ntrs.nasa.gov/search.jsp?R=19770003347 2020-03-22T12:11:09+00:00Z
FORMULATION OF THE PROBLEM 
Consider a hollow elastic sphere 
exposed to a temperature distribution 
and p respectively. pressures, pI 11’ 
2’ of inner radius r and outer radius r 
T(r) in addition to internal and external 
Owing to the spherical symmetry of the 
1 
problem, the nonvanishing strain components depend upon the radial displacement 
u according to the relations 
Assuming transverse isotropy, the thermal stresses are related to the strains 
and temperature rise by 
in which A..(T,r) denote the elastic stiffnesses and Bi(T,r) are the stress- 
temperature coefficients. Alternatively, the strains may be expressed in terms 
of the stresses and temperature as 
1J 
where a (T,r) and a.(T,r) are the compliances and the coefficients of thermal ij 1 
expansion, respectively. 
For convenience in later operations the following dimensionless quantities 
are introduced : \ 
640 
in which (3 
T represents some reference temperature. 
denotes an arbitrary reference stress-temperature coefficient and 
0 
0 
In formulating the problem through the use of energy principles, we re- 
quire specification of the total potential energy of the sphere. 
of quasi-static loading, the total  potential energy II consists of the strain 
energy U plus the potential V 
the strain energy in anisotropic, temperature-sensitive, elastic bodies are 
given in reference 4. Based upon these expressions the total potential energy 
for a transversely isotropic sphere with strains given by equation (1) is 
For the case 
of the external forces. General expressions for E 
in which the integral expressions constitute the strain energy, and the terms 
involving qI and qII represent the potential of the pressure loads. 
rather than displacements represent the varied quantities, involves the total  
conrplementary energy. When tractions are specified over the entire boundary 
of the body, the total complementary energy lI* is equal to the complementary 
strain energy U*. 
shown that for the sphere 
A complementary variational approach to the problem, in which stresses 
From the general results given in reference 4 it can be 
+b t t + t  
0 23 @@ 80 PP 
Before developing approximate solutions to the problem, it is noted that 
the governing differential equation and natural boundary conditions can be de- 
rived through direct application of the principZe ofmin<mwn potential energy. 
Requiring that the first variation of the total potential energy be equal to 
zero ( 6 I T = O ) ,  and performing integration by parts, one obtains the displacement 
64 1 
equation of equilibrium 
+-- 2 dv -2(B22+B23-B12) -+--+2 V dBll dv -- dB12 v 
dP dP dp P '11[3 P dP] P2 
and the natural boundary conditions 
( 8 )  
dv(P1) V(P,) @(P1) 
+2B12(P1) -- .f Yl(@,Pl)d@= -qI 
P1 0 B1l(P1) dp 
@ (1) 
Bl1(1) dvO+ 2B12(1) v(1) - yl(O,l)dO= -qII 
0 dP 
Finding an exact solution to these equations does not appear possible for a 
sphere of general nonhomogeneity. 
RAYLEIGH-RITZ METHOD 
In the Rayleigh-Ritz method a kinematically admissible displacement field 
is assumed, and the principle of minimum potential energy is used to determine 
unknown coefficients in the assumed solution. Here we shall represent the rad- 
ial displacement v(p) by the power series 
i -m n n 
i=-m 
v= 1 aip = a-mp -t . . . + ao+ . . . + anp 
in which the number of nonzero coefficients a is arbitrary. Although it is. 
only necessary to satisfy displacement boundary conditions when applying the 
Rayleigh-Ritz method, generally it is desirable to satisfy traction conditions 
as well. Relations (8) will be satisfied identically by the displacement field 
( 9 )  if the coefficients ai satisfy 
i 
@(P1) 
i- 1 i- 1 
fl(ai) = B ~ ~ ( P ~ >  1 i ai p1 +2B12(P1) aiP - .f Y1(O,Pl)dO+ ql= 0 
i 0 
f (a)= Bll(l> 1 i ai+2B12(1) 1 a -1  yl(O,l)dO+qII=O 
i i i o  J 
642 
i These equations can be used in order to eliminate two of the coefficients a 
from the assumed solution. Alternatively, equation (9) can be retained in its 
original form and conditions (10) satisfied by the method of Lagrange multi- 
pliers, as described in reference 4. In this case the restrictions (10) are 
written in terms of Lagrange multipliers A and X as 1 2 
Necessary conditions for a minimum value of the total potential energy 11, sub- 
ject to the subsidiary conditions (ll), then are given by 
where 
0 ( s =  1,Z) all -=0 (j=-m, ..., n), -= 
S 
aa ax 
j 
-.. 
I I = I I + x  f +X2f2 1 1  
Substituting the assumed solution (9) into the potential 
and differentiating if with respect to a as indicated in 
j 
energy expression (S) , 
equation (12), gives 
n 2 
=H. (j=-m y...,n) 1 G * *  ai+ C gjs s 
J i=-m J 1  s=l 
in which 
= ~ l ~ B l l +  2(ij)B12+ 2(BZ2+ Bz3) piSSdp 
Gji P, 3 
The Ritz coefficients ai are then found by solving the algebraic equations.(14) 
together with the constraint equations (10). 
MODIFIED UYLEIGH-RITZ METHOD 
The modified Rayleigh-Ritz method consists of assuming a state of stress 
which satisfies equilibrium and traction boundary conditions, and then deter- 
mining unknown coefficients in the assumed solution by applying the principle 
of m i n i m  complementary energy. 
643 
It is easily verified that equilibrium is satisfied if the dimensionless 
stress components are expressed in terms of a stress function J, as 
In this case the total complementary energy becomes 
We choose to represent the stress function J, by the power series 
n i -m n 
$ =  1 a2p =a* p +...+ a*+ ...+ a:p i=-m -m 0 
in which the number of nonzero coefficients a* is arbitrary. i pression (18) yields stresses which satisfy the traction boundary conditions (8), 
the coefficients ai must satisfy the relations 
In order that ex- 
* 
Proceeding as in the standard Rayleigh-Ritz technique outlined earlier, condi- 
tions (19) are next written in terms of the Lagrange multipliers X.f and 
Application of the principle of minimum complementary energy then leads to the 
set of equations 
. 
where 
G* = 1' Fll+ (2+i+j)b12+ 3 (l+i) (l+j) (b22+ b23 
ji 
P1 
644 
The coef f ic ien ts  a i  and the  Lagrange mul t ip l ie rs  A" then are found by solving 
equations (19) and (20). 
S 
FINITE ELEMENT TECHNIQUE 
The energy formulation developed earlier a l s o  provides a convenient bas i s  
f o r  constructing a f i n i t e  element so lu t ion  t o  the  problem. 
sphere is idealized as a series of N hollow spher ica l  subregions. 
element j has an inner radius  p 
i a l  displacement components are denoted by vi and v 
are taken t o  be (t ) .  and (t ) respectively.  
I n  t h i s  case the  
A t yp ica l  
and an outer radius  p the  corresponding rad- i 
l i d  t he  r ad ia l  stresses 
j '  
PP 1 PP j '  
It is  assumed t h a t  the displacement var ies  l i nea r ly  with p within each 
element, so tha t  
The thermoelastic proper t ies  are 
which case the  following average 
taken t o  be constant over each element, i n  
values w i l l  be used 
(23) 
0 
W P j )  - E k =  %[\ ck(Q,pj)dQ+ 
By analogy wieh equation (5), t h e  t o t a l  po ten t i a l  energy fo r  element j is  
Subst i tut ing equation (22) i n t o  ( 2 4 ) ,  and minimizing II(j) with respect t o  .vs 
and v gives 
2 
j 
- -Pi ( tPPI i  
2 3  
(Pi3 -3PiPi +2Pi 1 
3 (Pj-Pi) 
645 
3 2 3  (Pi -3PiPj +2Pj - (Pi -Pi - 3 3  
=k v +k v~ (25b) r2 12 i 22 j r +  
2 
Pj ( W j +  3(pj-pij 1 3(Pj-Pi) 
where the element stiffness coefficients k are 
ij 
\ 
u p  'P 1 3 2 3  U P j  -3PIPi +ai 1(Pj -Pi - 3 3  ii +a (ii22+ii23) 
kll = 3 (P j -Pi> 2 B1l- 3 (P j-Pi) 2 12 3 
(P -P ) - 3 3  (Pj  -Pi 1 
3 (Pj--Pi> 
% +- (B22+x23- B12) 
2 11 3 k12 =- 
Application of equations (25) to each of the N elements provides a system of 
2N linear equations for the N+l displacement components and the N-1 interface 
stresses. 
equations for the unknown displacement components. 
The interface stresses can be eleminated, resulting in a set of N+1 
NUMERICAL EXAMPLES 
To illustrate the influence of temperature-dependent material properties 
upon the thermoelastic response, and at the same time to demonstrate the appli- 
cability of Ritz methods in thermal stress problems, numerical results are pre- 
sented for a sphere subject to various temperature and pressure conditions. 
The ratio of the sphere's inner and outer radii is taken to be p =0.8. It is 
assumed that the body is initially homogeneous, and that the thermal expansion 
coefficients vary linearly with temperature, while the elastic stiffnesses ex-' 
hibit a quadratic variation. 
1 
In particular we let 
E = E  0 (l+bO), B =B.. 0 (I-& 2 ) 
i i  ij 1 3  
in which b and c are constants. 
coefficients are taken to be 
The initial (zero-temperature) thermoelastic 
= 31.0~ 10 0 4 0 4 0 0 B22 +B23 B12 =1.0 x 10 Bll =3.0 x 10 
0 0 y1 =1.0 y2 =1.5 
646 
These values are representative of certain fiber reinforced composite materials, 
reinforced in the circumferential ((I and 0 )  directions. 
A s  a first example let us consider a sphere subject to a uniform tempera- 
ture rise 0 = 1  and zero internal and external pressure. Values of the thermal 
displacements and stresses found using the Rayleigh-Ritz and the modified Ray- 
leigh-Ritz methods are compared with the exact solution (ref. 5 )  for the limit- 
ing case of temperature-independent properties (b=c=O) in Table I. It is evi- 
dent that the accuracy of the approximate solutions generally improves as addi- 
tional terms are included in the assumed solution. When the Rayleigh-Ritz ap- 
proximation contains 3 independent coefficients (i.e., a total of the 5 coeffi- 
cients a,2,a,l, ao,al,a2 of which 2 may be eliminated using the boundary condi- 
tions), the value of the maximum stress amplitude 1 t+(1(0.8) 1 exceeds the exact 
value by 0.9%. For 5 independent coefficients the error is reduced to 0.3%. On 
the other hand the maximum stresses predicted by the modified Rayleigh-Ritz pro- 
cedure using 3 and 5 independent coefficients are 2.3% and 1.6% smaller than the 
exact value. 
Ritz approximation are taken to be -5, 4 ,  and 1, the computed values of the dis- 
placements and stresses are exact, since the assumed solution then has precisely 
the form of the exact solution. 
When the powers of p in either the standard or modified Rayleigh- 
Results of finite element analyses are compared with the exact solution. to 
this same problem in Table 11. 
solutions improves as the number of independent displacement components is in- 
creased. 
nent components (2 elements), the maximum stress It +(0.8) I exceeds the exact 
value by 2.6%. Thk error is reduced to 0.7% when 1 s displacement components 
(12 elements) are used. However for this problem it was found that the compu- 
tations required to achieve a given level of accuracy were less time consuming 
#hen one of the Ritz methods was used than when the finite element technique 
#as applied. 
To demonstrate the inf hence of temperature-dependent behavior upon the 
Naturally the accuracy of the finite element 
When the finite element solution is based upon 3 independent displace 
Zircumferential stress in the sphere, Ritz solutions based upon 5 independent 
:oefficients are plotted in figures 1-3. 
listributions associated with various values of the temperature-dependent pa- 
-ameters b and c for temperature alone and for combined temperature plus inter- 
tal pressure. 
-ise 0-1. 
cnd 0=-4+5p are given in figure 2 and 3,  respectively. Each of the Ritz solu- 
.ions plotted in the figures was compared with a finite element solution based 
ipon a 12-element model. Agreement between the values of the maximum absolute 
,tress predicted by the two methods varied between 0.1% and 1.6%, with one ex- 
eption. 
iaximum stress was relatively small, and the discrepancy was nearly 5.0%. 
Each of the figures shows 'the stress 
Figure 1 shows the stresses induced by a uniform temperature 
Results for the linearly varying temperature distributions 0=5-5p 
In the case of 0=-4+5p and zero internal pressure 41'0 (fig. 3) the 
As would be expected for the purely temperature loadings (qI=qII=O), the 
aximum stresses diminish with increasing values of c(i.e., with decreasing 
tiffness), whereas they become larger with increasing values of b (increasing 
hermal expansion). The influence of temperature sensitivity is less predict- 
ble in the case of combined temperature and pressure, since both the pressure- 
nduced and temperature-induced stresses are affected by the nonhomogeneity. 
647 
REFERENCES 
1. Nowinski, J.: Thermoelast ic  Problem f o r  a n  I s o t r o p i c  Sphere w i t h  Temperatur 
Dependent P rope r t i e s .  Z e i t s c h r i f  t f c r  Angewandte Mathematik und Physik,  
V O ~ .  10, 1959, pp. 565-575. 
2. S t a n i s i c ,  M.M.;and McKinley, R.M.: The Steady-State  Thermal S t r e s s  F i e l d  i n  
an I s o t r o p i c  Sphere w i t h  Temperature Dependent P r o p e r t i e s .  Ingenieur-Arch 
V O ~ .  31, 1962, pp. 241-249. 
3.  Hata, T.;and Atsumi, A.: T rans i en t  Thermoelast ic  Problem f o r  a Transverse ly  
Aniso t ropic  Hollow Sphere wi th  Temperature-Dependent P r o p e r t i e s .  B u l l e t i n  
of Japan Soc ie ty  of Mechanical Engineers ,  vo l .  1 2 ,  1969, pp. 445-452. 
4 .  Tauchert ,  T.R.: Thermal S t r e s s e s  i n  an Or tho t rop ic  Cyl inder  w i t h  Temperatur 
Dependent Elast ic  P r o p e r t i e s .  Developments i n  T h e o r e t i c a l  and Applied Mec 
a n i c s ,  vo l .  8, V i r g i n i a  Polytech.  I n s t .  and S t a t e  Univ., 1976, pp.201-212 
5. Bi rger ,  B . I . :  Temperature Stresses i n  Aniso t ropic  Bodies. Sovie t  Applied 
Mechanics, vo l .  7, 1971, pp. 292-236. 
648  
Table I. Ritz approximations for the thermal displacements and 
stresses caused by a uniform temperature rise 0 = 1  when b -  c=O. 
Powers of p 
No. of indep. coefs. 
v(0.8) Radial displ. 
107 v(O.9) 
v(1.0) 
t (0.8) 
Radial stress t (0.9) 
t (1.0) 
PP 
PP 
t (0.8) 
Circumf. stress t (0.9) 
@@ 
@$ 
(1.0) 
Ray leigh-Rit z Modified Rayleigh-Ritz Exact 
-2,-1,0,1,2 -3,-2,-1, -2,-1,0,1,2 -3,-2,-1, -5,4,1 
0,19293 0,1,2,3 
3 5 3 5 3 
.058 .059 .066 .064 .060 
I355 * 355 .354 .354 .354 
.636 .634 .633 .632 634 
0 0 0 0 
-. 084 -. 091 -. 092 - 4 2  -. 091 
0 0 0 0 0 
-. 948 -.943 -. 918 -.925 -. 940 
.001 - .001 - ,006 -. 003 -. 003 
.762 .757 .755 .750 .758 
Table 11. Finite element solutions f o r  the thermal displacements and 
stresses caused by a uniform temperature rise 0 = 1  when b =  c =  0. 
No. elements 
No. indep. displ. 
comps. 
v(0.8) 
v(O.9) 
v(1.0) 
Radial displ. 
107 
t (0.8) 
Radial stress t (0.9) 
PP 
PP 
tpp (1.0) 
t (0.8) 
Circumf. stress t (0.9) 
to$ (1.0) 
$4 
$4 
F 1 1 
Finite Element Exact 
2 4 6 12 - 
3 5 7 13 - 
.062 .060 .060 .060 ,060 
.356 ,355 .355 .355 .354 
.636 ,635 .635 .635 .634 
-. 10s -. 064 -. 046 -. 025 0 
-. 073 -.092 -.092 -. 091 -.091 
-. 033 -. 022 -. 016 - .008 0 
-. 965 -.959 -. 954 -. 947 - .940 
.012 . 001 -. 001 - .001 -. 003 
.750 ,752 m 753 .756 .758 
649 
4 
-- 4 -
-- 2 
Q 
9 - 
6 50 
Q 
65 1 


Thermal stresses in a spherical pressure vessel having temperature-dependent, transversely isotropic, elastic properties

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