Thermal stress analysis of underground openings

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 13, 41 1425 (1989)

THERMAL STRESS ANALYSIS OF UNDERGROUND OPENINGS

P. KUMAR

Structural Engineering, Central Building Research Institute, Roorkee, India

AND

B. SINGH

Civil Engineering, University of Roorkee, Roorkee, India

SUMMARY

In the use of underground space for storage of solar energy, storage of excess heat from thermal power plants and containment of nuclear waste, the host medium is subjected to a thermal field accompanied by a thermal stress field. These stresses may endanger the stability and integrity of the structure. In this paper, the stress field, in circular and spherical openings, due to a given thermal field is analytically determined and numerically verified. Both lined and unlined openings are considered. The numerical and analytical results are in a satisfactory agreement. The lining stresses, support pressure and radial interface displacement depend only on the thermal field in the lining. A simplified method of stress computation applicable to a thin lining is also presented.

INTRODUCTION

Thermal stresses develop in a solid body whenever expansion or contraction of a differential volume element, that results from a change in temperature, is prevented. Such stresses may upset stability of rock mass by causing movements along the existing discontinuities as well as by creating new discontinuities. In the case of water-bearing rock mass, the creation of new discontinuities may alter the flow pattern. These aspects are important when underground space is to store solar energy in the form of heated rocks, excess heat from thermal power plants and in the containment of high level radioactive nuclear waste. In the last application, hydraulic trans- missivity of the storage medium plays a vital role as ground water transport is the most likely mechanism by which radionuclides might escape containment. The determination of the thermal stress field is the first step in major studies like migration of chemical forms, hydrodynamics of a fractured medium and heat induced geomechanical and hydrodynamic effects.

In this paper, the cylindrical and spherical underground openings lined or unlined, are analysed by theory of elasticity to determine thermal stresses for a given thermal field. The analytical expressions are numerically verified by a finite/infinite/interface element method. The numerical model employs infinite elements to simulate the unbounded extent of the analysis domain and interface elements to simulate the lining-rock interface. A simplified method of stress computation applicable to thin lining is also presented.

THEORY OF THERMAL STRESSES

In general, the problems of thermal field and thermal stress field are coupled. As the thermal field grows with time, the thermal stress field also grows. Furthermore, the material properties of the medium are also temperature dependent. An analytical o r numerical solution of such coupled problems is not straightforward. An alternative could be to solve the two problems independently

0363-9061/89/040411-15$07.50 0 1989 by John Wiley & Sons, Ltd.

Received 8 February 1988 Revised 17 June 1988

412 P. KUMAR AND 8. SINGH

followed by an iterative refinement. The thermal field can be obtained by using the theory of heat conduction in solids which is described by Timoshenko and Goodier,' Carslaw and Jaeger,' Nowacki3 and P a r k ~ s . ~ The thermal stress field may be obtained as described in the following sections.

In most of the available research on the subject of thermal stresses, the problems of heat conduction and thermal stress analysis associated with the underground medium are assumed to be uncoupled, i.e., temperature influences stresses but not vice versa. The available research may be categorized as follows.

In the first category, the thermo-elastic equations are solved for a given thermal field, irrespective of its origin, to obtain displacement and stress fields. The materials are assumed to have time and temperature independent properties. In the second category of problems, the materials involved have time and temperature dependent properties. While maintaining the uncoupled nature of problem, the analysis proceeds by first solving the equations of heat conduction taking into account the properties of the heat source and thermal conduction characteristics of the medium. This thermal field is then substituted into the thermo-elastic equations to obtain displacement and stress fields.

The first category solutions were obtained by Mykle~tad ,~ Mindlin and Cooper,6 Edwards7 and, Florence and Goodier.' Most of these solutions are concerned with an embedded inclusion with properties different from that of the medium. When this system is subjected to a thermal field, stresses develop in the inclusion and in the matrix. Such solutions are of little use in the application to the problem of underground openings.

In the second category of problems, FEM has been used by Wai et aL9 and, Duddeck and Nipp." Booker and Carter" and, Small and Booker" employed a finite layer method while calculating the elastic response of ground containing a heat source. Sugano13 used a finite difference technique. It may be noted that in the solution with FEM, the material properties are essentially taken as constant until these are updated during the next time step. A close-form solution to a heated, spherical, unlined cavity is also presented by Rehbinder14 while considering time and temperature independent material properties.

The problem considered in this paper falls in the first category and is relevent in the case of underground openings. No solution for this problem appears to be available. The assumptions of the solution method are the same as those of the available solutions of the other problems mentioned above.

Theoretically, it is possible to do a highly sophisticated FEM analysis, however, there still remains an important place for the simple approach. This is essentially true for underground structures for which the degree of confidence in the properties of the ground, including provisions for time and temperature dependence, rarely justifies the more prodigious exercise in number- crunching. A special virtue of the simple method is that it quickly indicates the sensitivity of the solution across the range of possible ground parameters. The analysis in this paper derives importance from this point.

In the analysis of linear-elastic materials, the principle of superpositiQn is valid. Consequently, in the subsequent analysis the rock mass is assumed to be stress free at a large distance from the opening. The effect of an initial stress field may be calculated separately and superposed on the thermal load effects.

THERMAL STRESSES IN CIRCULAR CYLINDRICAL OPENINGS

It is assumed that the temperature distribution does not vary along the axis of a cylinder, therefore, stresses and displacements due to heating also do not vary in an axial direction. The radial stress or

THERMAL ANALYSIS OF OPENINGS 413

and oe satisfy the equation of equilibrium (equation (1)). The stress-strain relationships are as given in equations (2) where E, and cg are the radial and circumferential strains; E is the modulus of elasticity; v is the Poisson ratio and a is the coefficient of thermal expansion. The strain- displacement relationships are given in equation (3) where u is radial displacement.

do, or - 00 -+- = o dr r

V E, - ( 1 + v)aT = - - v2 (or - G o o )

Eg - ( 1 + v)aT = - 1-v2(oe-&or) E

E

du U E, =- and cg =- dr r (3)

A general solution of the problem defined by equations ( 1 H 3 ) is given by Timoshenko and Goodier' and is given in equations (4) where C , and C , are the constants of integration and the lower limit in the integrals can be chosen arbitrarily.

l + v a u = ___-

( c' f : ) or = ---I: aE 1 T(r,O)rdr+-

crg = --I: aE 1

l-vr' l + v 1-2v

aE T( r, 0) 1 -v r2 1 - v

(4)

aET(r,O) ~ v E C , OZ = - + 1 - v ( l+v ) ( l -2v )

Unlined cylindrical opening in sound rock

Let a be the radius of the opening. The boundary conditions of the problem are

a , = O a t r = a and a t r = o o

Therefore, from the second equation of equations (4), C , = 0 and C , = 0. The complete solution is then given in equations (5)

aE T( r,O) 1 - v

6, = -

414 P. KUMAR AND B. SINGH

Lined cylindrical opening in sound rock

The geometry of the lined opening is shown in Figure 1. Let p be the pressure at the rock-lining interface. El, v, and a, are the modulus of elasticity, Poisson ratio and coefficient of thermal expansion respectively of the lining material while E,, v2 and a2 are the similar properties of the rockmass. The boundary conditions of this problem are,

a , = O a t r = b and a t r = c o

a, = p at r = c

where b and c are the internal and external radii of the lining. For c < r < co, the constants of integration appearing in equations (4) are given by,

(1 + V 2 k 2 P

E2 C, = O and C2 = -

Therefore, the radial displacement at the rock-lining interface is,

For b < r < c, the constants of integration appearing in equations (4) are,

T(r,B)rdr (1 +v,)(l-2v,) c2 c, = c2 - b2

(8) E*

T(r,B)rdr 1 + v , b2c2 c El c2-b2 1 - v , c2 2 -

With these values of integration constants, the radial displacement at the rock-lining interface is given by,

( l + v l ) c [c2(1-2vl)+ b 2 ] p E , ( c 2 - b 2 )

u(r = c) = f " ' ~ ~ l c 1: T(r,B)rdr - (9)

Figure 1. Geometry of a lined opening


Equate the radial displacements at the rock-lining interface given by equations (7) and (9) and solve for p. This leads to equation (10).

2a1E1 rT( r ,B) rd r J b P = 1 + v , El

1 + v , E2 ~’(1-2v1) + b’ + ___- (c’ - b‘)

The stress and displacements can now be obtained from equations (4) by substituting C , and C , from equation (6) for rock and from equation (8) for lining. Thus, the complete solution of thermal stresses in lined, circular, cylindrical openings is given by equation (10H12). Equation (1 1) is for rockmass while equation (1 2) is for the lining.

When r > c;

C’P or= -__- l q r , O)rdr+, 1-v2r2 r

@,El 1 T( r, O)r dr u = --I: 1 + v , a1 T(r,B)rdr + (1 + Vl)C2 [(l-2v1)r+-][-p+--j’ b2 ] 1-v, r E1(c2 - b’) r 1 -v , c’

bg = - T(r,0) r d r + (1 + 5) 1: T(r,B)r dr ] 1 - v l

6, = - T( r, O)r dr

THERMAL STRESSES IN SPHERICAL OPENINGS

In the situation of spherical symmetry, the temperature is a function of radial distance only. The equation of equilibrium is given in equation (13) while strain-stress relations are given in equations (14).

do, 2 dr r - +-(or - a,) = 0


1 E, - a T = -(or - 2v0,)

E

1 E

(14) E, - ~1 T = -( O, - VO, - VO,)

A general solution of the problem defined by equations (3), (13) and (14) is derived by Timoshenko and Goodier' and is given in equations (15). The solutions for lined and unlined openings are derived from it as follows.

c2 u = __- '+' a j: T(r)r'dr+C,r+, l -vr2 r

EC, 2EC, 1 g =---

r 2aE j: T(r)r2 dr + __ - __- 1-vr3 1-2v 1 + v r 3

aE 1 EC, EC 1 aET(r) T(r)r2 dr + __ + 2- -~ 0, = __-

1-2v i + v r 3 I - ~

Unlined spherical opening in sound rock

Let a be the radius of the opening. The boundary conditions are,

~ , = O a t r = a and a t r = c o

Substituting the boundary conditions in the second equation of equations (15) C, = 0 and C, = 0, means that the complete solution is then as given in equations (16).

u = __- '+' a 1: T(r)r2dr I-vr2

1 -' r3 1: T(r)r2 dr 2aE

(J = -__- r

aE 1 aET(r) l - v

Lined spherical opening in sound rock

the pressure at rock-lining interface. The boundary conditions are, The geometry of the lined opening in axisymmetric formulation is shown in Figure 1. Let p be

c , = O a t r = b and a t r = c o ; a , = p a t r = c

For c < r < 00, the constants of integration appearing in equations (15) are given by,

(1 + v2)c3p 2E2

C, = O and C2 = -

The radial displacement at the interface can be evaluated by equation (18).


For b < r < c, the constants of integration appearing in equations (15) are given by,

c, = c3(1 -2v,) [-,+--I: 2a,E, 1 T(r)r’dr]

c, = (1 +v,)b3c3 [-,+--I: 2a,E, 1 T(r)r’dr]

E,(c3 - b3) 1-v, c3

2E,(c3 - b3) 1 - v , c3

The radial displacement at the interface with these values of integration constants is given by equation (20).

(20) ‘’ +v1)b3] + 3Ci, C T(r)r2 dr

2 u(r = c) = - c ~ [ 2 ( 1 -2v1) +

E,(c3 - b3)

Equate the radial displacements, at the interface, given by equations (18) and (20) and solve for p . The support pressure is, thus, obtained as in equation (21).

The stresses and displacements in the spherical opening can now be derived from equations (15) by substituting for C , and C, from equation (17) for the rock and from equation (19) for the lining. The complete solution is given in equations (22) and (23) which apply to rockmass and lining, respectively. Thus, when r > c;

When r < c;

(1-2vl)r+(l +vl)- 2r b3 1 c3 T(r)r2 dr + E,(C3 - b3) 1 -v l r2

[ --p+ --I: 2a,E, 1 T(r)r’dr] 1-v, c3

0,. = - x - j : 2 a E I-v, r3 1 T(r)r2dr+- (1 -vl)(c3 2a1E1 - b3) (1 - ~ ) j ~ , T ( r ) r z d r - - c3 c3p - b3 (I -5) (23)

(it = ~- j: T(r)r2dr- l -v , r3

+ (1 -vl)(c3 - b3) ( 2 + 5 ) j : T(r)r2dr

%El T(r) --(l+$) C 3 P

1-v, c3-b3

The expressions derived for circular cylindrical and spherical openings are verified in the next


section through finite element analysis to ensure that other opening shapes can also be analysed. The numerical model and results of the analyses are subsequently described.

DESCRIPTION OF NUMERICAL MODEL

A lined circular opening with lining thickness O.lR, where R is the opening radius is analysed for thermal loads. The displacement based finite element method of analysis is used. The FEM model is shown in Figure 2(a). It consists of 8-noded isoparametric quadrilateral finite elements, 5-noded infinite elements and 6-noded interface elements. The finite elements, infinite elements and interface elements simulate near-field, far-field and rock-lining interface, respectively. The stiffness properties of these elements are evaluated by Gauss-Legendre numerical integration of orders 3 x 3, 2 x 2 and 3rd order, respectively.

The infinite elements are described by Kumar.l' The applicability of the infinite elements beyond the homogeneous and isotropic range of material behaviour has also been demon- strated.16 The interface elements were proposed by Buragohain and Shah.17 A complete formulation and implementation is given by Kumar and Singh.'* The value of interface stiffness is placed at 1.0 x lo8 kg/cm2.

This numerical model is analysed in a plane strain formulation for cylindrical opening and in an axisymmetric formulation for the spherical opening. Because this problem is doubly symmetric, only one quadrant is discretized. The numerical values of the material constants are given in Table I. The thermal loads applied to the model are shown in Figure 2(b). With load 1, the lining of the opening is subjected to a uniform temperature rise above the base temperature while the rockmass is maintained at the base temperature. With load 2, the lining as well as some part of the rockmass is subjected to a uniform temperature rise of the same magnitude as with load 1. The rest of the rockmass is maintained at the base temperature. The FEM analysis for thermal loads is described by Hinton and OwenIg which is also illustrated here by an example.

a. FEM model of lined tunnel

T c-!o:!- 1 - - 17

load 2 I I T c -_ - - - _ _ _ - - - - b. Thermal loads

Figure 2. FEM model of lined opening and loads


Table I. Material constants for FEM analysis

E Q Material (Kg/cm2) V per "C

Concrete 20000000 0.15 0000005 Rock 20000.00 0.20 0.000005

Table 11. Support pressure and radial displacement for 100°C temperature rise in a circular cylindrical opening

Numerical value Analytical

value Load 1 Load2

Support pressure (Kg/cm2) 5.44 5.430 5.422 Radial interface displacement (mm) 6.48 6529 6.537

Table 111. Support pressure and radial displacement for 100°C temperature rise in a spherical opening

Numerical value Analytical

value Load 1 Load2

Support pressure (Kg/cm2) 9.8067 9.8410 9.7800 Radial interface displacement (mm) 5.9100 5.9300 5.9630

RESULTS O F ANALYSIS

The results of the analysis are given in Tables 11-VI. A satisfactory agreement between numerical and analytical values establishes correctness of the analytical derivation. Furthermore, it should be possible to analyse any opening shape by the finite/infinite/interface element method.

SIMPLIFIED METHOD O F THERMAL STRESS COMPUTATION

The expressions derived in the preceding section are inconvenient to apply except when an exact stress distribution is needed. For an order-of-magnitude estimation, it is possible to simplify these expressions by assuming that the thickness of the lining is small, therefore, temperature distribution within the lining is approximately uniform. The Poisson ratio of concrete and rock is taken as 0.15 and 0-2, respectively. Also, let t ( = c - b ) be the thickness of the lining and s = t/c. The following simplifications can be done.

Circular cylindrical opening

The support pressure and the radial and hoop stresses can be written as in equations 24-26. To enable quick computation, the factorsf, andf, are plotted in Figures 3-5. For radial stress,f, is


Table IV. Stress in lining at springline for 100°C temperature rise (Kg/cm2)

Numerical value

Analytical value Load 1 Load 2 Opening

shape Node 0, GI Qr 01 Qr Ql

inside 0.OOOO 57.1580 0.1540 57.3290 01660 57.1900 2.9290 54.2290 2.8920 54.3070 2.8340 54.1910 5.4400 51.7280 5.5970 51.8090 5.5940 51.7130

inside 0.OOOO 545200 0.4200 54.7900 04340 54.5060 Sphere middle 5.4420 51.8000 5.2220 51.8990 5.2110 51.6310

outside 9.8500 49.5950 10.2480 49.7530 102090 49.5000

Circular middle cylinder outside

Table V. Stresses in rockmass due to a thermal field in circular cylindrical lined opening (Kg/cm2)

Thermal load 1 Thermal load 2 Distance from Numerical Analytical Numerical Analytical interface (MI Qr Ql Qr Ql 0, Ql Qr Ql

0.00 -5.2841 5.4833 -5.44 5.44 -5.4617 -7.0514 -5.44 - 7.060 2-50 -4.3736 4.2822 -430 4.30 -5.5974 -6.8746 -5.61 - 6.890 5.00 -3'3500 3.5144 -3.48 3.48 -5.7473 -67624 -5.736 -6.768 8.75 -2.6958 2.6167 -2.63 2.63 -5.8391 -6.6300 -5.858 -6.642

12-50 -1.9481 2.0851 -2.06 2.06 -9,1200 -7'3495 -5.943 -6.557

Table VI. Stresses in rockmass due to a thermal field in lined spherical opening (Kg/cm2)

Thermal load 1 Thermal load 2 Distance from Numerical Analytical Numerical Analytical interface (MI Qr 01 Qr Ql a r Ql Qr Ql

0.00 -9.3142 5.0771 -9.806 4.903 -9'7418 -7.5034 -9'850 -7.650 2.50 -7.1664 3.4099 - 6'890 3.445 -9.4075 - 7.7657 -9.398 -6'597 5~00 -4.6679 2.6203 -5'020 2.510 -9-0184 -7'8910 -9'1082 -5.939 8.75 -3.4789 1.6226 - 3.300 1.650 - 8.8336 - 8'0455 - 8'6842 - 5.490

1250 -2'0470 1'2114 -2.280 1'140 -12.1061 -8.9838 -8.539 -5'121

read from Figure 4 while for hoop stress, it is read from Figure 5. T is the uniform temperature rise in the lining.

THERMAL ANALYSIS OF OPENINGS 42 1

I

Figure 3. Factorf, for

0 b . i ' ' 0.03 ' ' ' ' 0.05 - !

computation of support pressure in thin lining of cylindrical opening

- -4

C

Figure 4. Factorf, for computation of radial stress in thin lining of circular cylindrical openings

2 100 80 1 2 j y - - 7 L = 0.010

e LO

_ - 5 - 0.05\ 20

1.0 1.02 1.OL 1.06 c l r -

Figure 5. Factorf, for computation of hoop stress in thin lining of circular cylindrical openings

1 - (1 - s )2 (cz / r2 ) c J * = - p = -Pfz s(2 - s )

1 + (1 - s ) 2 ( c 2 / r Z ) no= - p = -Pf z s(2 - s )

Spherical openings

Following the above approach, the support pressure and lining stresses are written as in equations (27H29). The factors fi and fz are plotted in Figures 6-8. For radial stress, f2 is read

422

N

'60. L 0 - 8

LO.

20.

t t

L

4 = 0.01

= 0.025

I =0.05 O . . F . . ' ' *

P. KUMAR AND B. SlNGH

0.01 0.03 0.05 - L -

Figure 6. FactorJ, for computation of support pressure in thin lining of spherical openings

N

- 2

- 3

I Figure 7. Factorf, for computation of radial stress in thin lining of spherical openings

from Figure 7 while for tangential stress it is read from Figure 8.

4 -a,E, T 3 4alE1 T

fi =- El 1.4 + 1*15(1 - s ) ~ 3OOO

1.2- + E ,

P =

S(S' - 3s + 3)

THERMAL ANALYSIS OF OPENINGS

1 - ( 1 - s ) ~ ( c ~ / ~ ~ ) o r = - p = -Pf ,

s(s2 - 3s + 3)

1 + (1 - ~ ) ~ ( c ~ / r ~ ) a,= - p = - P f 2

s(sZ - 3s + 3)

APPLICATION EXAMPLE

Let E , / E , = 10, t / c = 0.025, E l = 200000 Kg/cm2, T = 200°C, a, = O.OOOOl/.C

Circular cylindrical opening:

fi = 23.0 (Figure 3) Support pressure = 9.2 Kg/cm2 f , = 0.222 (Figure 4) Radial stress = 2.042 Kg/cm2 f 2 = 40.28 (Figure 5) Hoop stress = 370.6 Kg/cmZ

Spherical opening:

fl = 21.8 (Figure 6) Support pressure = 11.627 Kg/cm2 f , = 0.224 (Figure 7) Radial stress = 2-6 Kg/cmZ f , = 20.4 (Figure 8) Tangential stress = 237.2 Kg/cm2

The support pressure due to a uniform temperature rise of 200°C is of the order of 10 Kg/cmZ and the lining stresses due to the same temperature field are of the order of 300 Kg/cmZ. These values are high. It may be possible to control such high stresses by permitting free expansion of the lining by including a compressible layer between lining and rockmass. The material for this compressible layer should have a low ratio of elastic modulus to its compressive strength so that it can easily deform without fracturing. Also, this material should be able to resist the temperature rise without disintegrating. Investigations should be initiated to identify such materials and to study the mechanics of a compressible layer sandwiched between lining and rockmass in an underground opening subjected to a thermal field.

CONCLUSION

This paper presents an exact elastic analysis of thermal stresses in an underground opening. The analysis shows that the support pressure and lining stresses depend upon the temperature distribution within the lining only. For a thin lining, a simplified procedure is also presented which enables a quick solution. The analytical solution is numerically verified by finite/infinite/interface element analysis. The numerical and analytical results are in satisfactory agreement. It should also be possible to apply the numerical method to other opening shapes and a variety of thermal loads may be applied.

NOTATION

U Radius of unlined opening b Internal radius of lining C Outer radius of lining C, , C , Constants of integration E Modulus of elasticity

P. KUMAR AND B. SINGH

Factor for support pressure Factor for lining stresses Support pressure Radial distance Derived quantity ( = t / c ) Lining thickness Intensity of thermal field Radial displacement Coefficient of thermal expansion Poisson ratio Radial strain Circumferential strain Radial stress Tangential stress Yield stress of steel Longitudinal stress Hoop stress

Subscripts

‘1’ For lining properties ‘2’ For rockmass properties

Conversion factor from metric to SI units

10Kg/cm2 = 1 Mpa

REFERENCES

1. S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd Edition, McGraw Hill Book Co., 1951, chap. 14. 2. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Clarendon Press, Oxford, 1959. 3. W. Nowacki, Therrnoelasticity, International Series of Monographs in Aeronautics and Astronotics, Pergamon Press,

4. H. Parkus, Thermoelasticity, 2nd Edition, Springer-Verlag, 1968. 5. N. D. Myklestad, ‘Two problems of thermal stress in the infinite solid’, Journal of Applied Mechanics, ASME, 9,

6. R. D. Mindlin and H. L. Cooper, ‘Thermoelastic stress around a cylindrical inclusion of elliptical cross-section’,

7. R. H. Edwards, ‘Stress concentrations around spheroidal inclusions and cavities’, Journal of Applied Mechanics,

8. A. L. Florence and J. N. Goodier, ‘Thermal stress at spherical cavities and circular holes in uniform heat flow’, Journal of Applied Mechanics, ASME, 26, 293-294 (1959).

9. R. S. C. Wai, K. Y. Lo and R. K. Rowe, ‘Thermal stress analysis in rock with nonlinear properties’, International Journal of Rock Mechanics and Mining Science, 19, 211-220 (1982).

10. H. W. Duddeck and H. Nipp, ‘Time and temperature dependent stress and displacement fields for salt domes’, in Proceedings of 23rd US Symposium on Rock Mechanics, Berkeley, USA, 1982, pp. 596-603.

11. J. R. Booker and J. P. Carter, ‘Steady state response of the elastic ground containing a heat source’, in Proceedings of 9th Australasian Conference on Mechanics of Structures and Materials, Melbourne, Australia, 1984, pp. 86-91.

12. J. C. Small and J. R. Booker, ‘The behaviour of layered soil or rock containing a decaying heat source’, Int. J . numer. and anal. methods geomech., 10, 501-520 (1986).

13. Y. Sugano, ‘On a finite difference method of plane thermo-elastic problem in multiply connected region exhibiting temperature dependency of material properties’, in Computational Techniques and Applications; CTAC-83 (J. Noye and C. Fletcher, Eds.), North Holland Publishers, 1984. pp. 615-624.

14. G. Rehbinder, ‘Stresses and strains around a heated spherical cavity in an elastic ground‘, Rock Mechanics and Rock Engineering, 18, 213-218 (1985).

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A S M E , 18, 19-29 (1951).


15. P. Kumar, ‘Static infinite element formulation’, Journal of Structural Engineering, ASCE, I l l , 2355-2372 (1985). 16. P. Kumar, ‘Nonhomogeneous and cross-anisotropic infinite elements’, Computers and Structures, 28, 327-333 (1988). 17. D. M. Buragohain and V. L. Shah, ‘Curved interface elements for interaction problems’, International Symposium on

Soil-Structure Interaction, Department of Civil Engineering, University of Roorkee, Roorkee, India, 1977, pp.

18. P. Kumar and B.Singh, ‘Pressure on lining due to initial stress field by finite/infinite/interface element method‘, Rock

19. E. Hinton and D. R. J. Owen, Finite Element Programming, Academic Press, London, 1977.

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Mechanics and Rock Engineering, 21, 219-228 (1988).

Thermal stress analysis of underground openings

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