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Thermal stresses in a cylindrical shell containing a circular hole or a rigid inclusion
Nuclear Engineering and Design 40 (1977) 337-346 © North-Holland Publishing Company
THERMAL STRESSES IN A CYLINDRICAL SHELL CONTAINING A CIRCULAR HOLE OR A RIGID INCLUSION D.H. BONDE and K.P. RAO Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 560012, India
Received 11 March 1976
The thermal stress problem of a circular discontinuity in a cylindrical shell has been solved by continuum approach. Two types of discontinuities are considered: (i) a circular hole and (i.i) a circular regid inclusion. The effect of a uniform temperature or a linearly varying temperature across the thickness has been studied. The problem is converted into an equivalent boundary value problem and boundary conditions are specified around the discontinuity. The results are presented in a graphical form for ready use.
thickness of shell with the middle surface experiencing zero temperature. It is well known that the disturbance due to a discontinuity is confined to a small region in the neighbourhood of the discontinuity. The temperature involved is assumed not high enough to cause any changes in the material properties. The problem of a uniform temperature distribution and a linearly varying temperature across the thickness is converted into an equivalent boundary value problem, with forces specified around the discontinuity and the solution is obtained by a continuum approach. Superposition of this solution on the solution of an uncut shell gives the total solution.
1. Introduction A clear understanding of the structural behaviour under thermal loads is of primary importance in many nuclear engineering and aerospace structures. Thermal stresses around a circular discontinuity in a circular cylindrical shell is one of the interesting problems. Cylindrical shells are used in nuclear power plants, aerospace structures, marine, chemical and several other industries. Cutouts are introduced in these structures to fulfil functional requirements. These cutouts give rise to stress concentrations around the boundary. The understanding of the structural behaviour around these openings is necessary for efficient design. Hoffman and Ariman [1,2] have solved the problem of thermal bending of finite plates having circular holes. They also determined thermal and mechanical stresses in nuclear reactor vessels. Prakash and Rao [3] have considered thermal stresses around a circular hole in a spherical shell. The circular hole and rigid inclusion problems in a pressurized cylindrical shell have been solved by Van Dyke [4,5]. The present work deals with the thermal stresses around a circular hole or a circular rigid inclusion in a circular cylindrical shell. The problem is solved using thin shallow shell theory. Any linearly varying temperature distribution across the thickness of the shell can be considered as the combination of a uniform temperature and a linearly varying temperature across the
2. Governing differential equation and its solution Fig. 1 shows the configuration of a cylindrical shell with a circular hole. The equations that govern the behaviour of the region of the shell, influenced by the presence of a discontinuity, are assumed to be the shallow, thin shell equations. These were written in general, nonlinear form by Marguerre [6] and are simplified to the linear equations for the case of a cylindrical shell [7]. The stress conditions in the shell, as described by the stress resultants and the displacements tangential and normal to the middle surface, are shown in fig. 2. 337
338
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
- - -
Shellmiddle
~fi,'
The primes indicate that the quantities are dimensional. The non-dimensionalization of these parameters is now carried out and non-dimensional coordinates are defined as ~ = x/a, rl = y/a. The membrane stresses are non-dimensionalized with respect to a reference membrane stress Nre f. Defining
/ '~
-,.-/&:./ =
~
7
/
~
"= r COSt r sin
e
F-
F' a2Nref
Fig. 1. Cylindrical shell configuration. and The linear shallow shell equations for a cylindrical shell geometry involving only membrane stress function F' and normal displacements W' are [4] v 4 If' + F ; x x / R D : p / D
(1)
W'Et 2
W=
a2Nref[12{1 _ u2)] 1/2 and choosing the complex potential function q~, such that
and
(s)
~=W-iF, v 4 F ' - tEW',xx/R = 0 ,
(2)
where
',/4{b + 8i/32q5,~ = 8j32 .
Et 3 D - 1 2 ( 1 _ u 2) '
V4W '
eqs. (1) and (2) can be combined to give [4]
= Wt,xxxx
3W' W " x - 3x ' t
+
2W,xxyy
+
W,yyyy
(3) .
(4)
(6)
The right-hand side of this equation is present if the shell is loaded by uniform pressure and is zero otherwise. And [32 = (a2 /8R T) [12(1 _ u2)] 1/2 ,
f-
STRESSCOUPLES & ROTATIONS
N~
I I
/ f
where a is the radius of the circular discontinuity; R is the radius of the cylinder at middle surface; and t is the thickness of the cylinder. The stress and displacement functions can be expressed as F=F +F*
m/,~ Q:~/
I Q{~ZN~.~ STRESS RESULTANTS wl/~-~"U{ /'-'-'~ & DISPLACEMENTS I
I! ~"
Fig. 2. Stress couples and resultants, rotations and displacements.
(7)
and
W = W + W*
where F and W are the solutions corresponding to a shell without a discontinuity and F* and W* are the perturbation components caused by the presence of the discontinuity and which vanish at distances far from the discontinuity. The residual problem then reduces to solving the homogeneous differential equation v4O * + 8i/32q~*~ = O,
(8)
where q~* = W* - iF*, which vanishes far from the discontinuity. The solution which is symmetric in ~ and r/then be-
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell comes [41 ¢ = (El -- iE2)~ symm + (E3 - iE4)@anti-symm '
(9)
where E1 _ i g 2 = 1 {exp [(1 ,-i)/3~] + exp [ - ( 1 - i ) ~ ] ) , E 3 - iE 4 = ~(1 + i){exp[(1 - i ) ~ ] = r cos 0
and
exp[-(1 -
r~ = r sin 0 ,
(10)
v 2 ~ + 2i/32ff = O.
(11)
The solutions relevant to the present problems are oo
0,2,4 ....
radially without developing any stresses, irrespective of the shape of opening in the shell. Hence we consider only the case when the temperature varies linearly across the thickness of the shell. An uncut shell subjected to linearly varying temperature across its thickness experiences a uniform bending moment, which is given by
i)t3~] ) ,
ffsymm and ffantisymm are the symmetric and antisymmetric parts of the solution of the equation
~symm
339
Mrr =Moo = aD(1 + v)r = M 0 ,
where D is flexural rigidity and r is the temperature gradient and no deflections or direct stresses occur, i.e. W = F = 0. The boundary conditions at the free hole edge are given by
N'rr=N~r+N'r~:O, r
(A n + iBn)Hl[~r(2i) 1/2 ] cos nO ,
(14)
--r
N'ro=Nro
+
,
~r0 = 0 , (15)
M'rr=.M;r + M'rr* = 0 , ffantisymm =
~ (An + iBn)Hl[ ~r(2i)1/2] cos nO , 1,3,5 .... (12)
where unknown coefficients are to be determined from the boundary conditions. Stress and moment resultants are given b y
Nrr = F,r/r + F,oo/r 2 ,
Noo = F,rr ,
t
Qr = Q'r + Q'r* = 0 Using Nre f = 6Mo/t and non-dimensionalizing the above boundary conditions we get Nr*r : 0 ,
Nr*0 = 0 ,
/~rr = --~ [12(1 -- V2)] 1/2,
Q* = O.
(16)
Nro = -(F,o/r),r , 3.2. Cylindrical shell with a rigid circular &clusion r
Mrr = -
r
W'°
[
W,rr + V r W'r +
W'°°
,r , When the shell is heated to a uniform temperature a circle drawn on its side will try to expand. This expansion will be prevented by the rigid inclusion. Thus, the following displacement boundary conditions t are to be satisfied at the rigid inclusion-shell junction
'
141,'r
1
Moo = - [ V W , r r + ( 1 W , r + - ~ W , o o ) ] , Mro = - ( 1 - v) [(1/r)W,o ] ,r"
=0
e 0' = 0 , (13)
t
W, o = 0 ,
x; = 0 ,
(17)
where e 0 is the strain in the 0 direction, and ×o is the in-plane change of curvature. The strain and change of curvature can be expressed in terms of stresses as (see
3. B o u n d a r y c o n d i t i o n s
3.1. Cylindrical shell with a circular hole When an unrestrained cylindrical shell is heated uniformly to a constant temperature the shell expands
t A similar set of boundary conditions was originally proposed by Murthy [8] when solving the problem of a rigid elliptic inclusion in a pressurized cylindrical shell.
340
D.IL Bonde, K.P. Rao / Thermal stresses in a cylhMrieal shell
the appendix) e'o = (1/Et)(N'oo
the quadrant 0 = 0 to rr/2. At each point there are four boundary conditions to be satisfied, which gives four equations for the unknown coefficients. The series is terminated at ( 2 m 1) terms, where m is the nmnber of points chosen. For a given/~, successively higher number of points are chosen until the values of the stresses remained essentially the same. For ~ = 4 the maximum number of points needed are 14, where a good convergence is obtained. The number of terms needed in the series for convergence to occur was essentially the same as used by Van Dyke [4].
VHrr)
and t
t
t
Xo = ( 1 / E t ) [(N'rr - uNoo)
(18)
- ( X ' o o , r , -UN'rr, r , ) + 2(1 +u)N'ro,o ] .
Eqs. (17) on non-dimensionalization with respect to Nre f = E t a T 0 , give W, r = 0 ,
W, 0 = 0 ,
eo = - 1 ,
X0 = - 1
4. l. Cylindrical shell w i t h a circular c u t o u t
.
(19) Fig. 3 shows the tangential membrane stress variation with respect to 0 for various values of/3, at the hole edge. Maximum Noo occurs at 0 = 0 for ~ values less than 2, beyond which the angle at which maximum stress exists starts shifting towards 90 ° . Fig. 4 shows the variation of maximum principal stress with respect to ~ at four different values of 0. It is observed from the graph that the maximum principal stress starts decreasing with increasing/3. The flat plate values are achieved in the limiting case of/3 -+ O. The decay in principal stress is faster for 0 = 90 ° than for
A linearly varying temperature across the thickness with the middle surface experiencing zero temperature causes a uniform bending moment in the shell given by eq. (14). This moment will be reacted at the rigid inclusion, causing no extra stresses.
4. Resultsand discussion The boundary conditions are satisfied by a collocation procedure at discrete points on the boundary in
0.7
0.6
I
N ~ ° O, 4 -
T~
Nref 0.3
0.5
O.
0.1 0
I 10
I 20
I 30
I 40
I 50
I 60
I 70
I 80
I 90
(9 ( decjrees )
Fig. 3. Tangential membrane stress variation with respect to 0 (circular cutout).
D.H. Bonde, K.P, Rao / Thermal stresses in a cylindrical shell
341
2-2 TOP SURFACE = TIe
BOTTOM
[
ec,e
)
~0 1 . 8
"~1.4 O
90 °
. i 1.2
~
E
0-8 0
I
I
I
1'0
2.0
3.0
I
4.0
/3 Fig. 4. Maximum principal stress around a circular cutout in a cylindrical shell.
0.7 TOP SURFACE = T~ BOTTOM SURFACE = T~ 0'5
6 Mo Nref
=
t
0.3 Nee N ref 8 = 90 °
0.1
-o-I I" 1.0
~
0° J
I
I
I
1.2
1.4
1'6
-
I
I
1'8
2.0
Fig. 5. Tangential membrane stress NO0 variation with respect to r (circular cutout).
342
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell 1-00 0=90 °
~
J
Nee
0°
0-75 \
Nref : Eo~t TO
-Nee Nref
\
\
\
\
\
0.50
\
Rigid inclusion
\
Nrr Nref
\ ~0
.... - -
o
Nrr Nee
9 0 °`````` 0.25 - ~--~
Nrr
I
I
I
I
1.0
2-0
3.0
4.0
FIG. 6"
MEMBRANE
STRESSES IN THE SHELL AT THE INTERFACE
Fig. 6. Membrane stresses in the shell at the interface.
2 2 F 2" 01--
~--~'~..,. ~
f -'=4
Bottom surface ( Tensile )
~
Top surface ( Compressive )
2
~
;-
- - - - ~
0
10
20
30
40
50
60
70
80
O (degrees) Fig, 7. Maximum principal stressesin the shell at the interface,
90
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
343
0 -0,1
-0-3 Nee N r ef
Nre f =Ea(tTo
-0.5
-0.7 -0.9 1.0
I
I
I
1.5
2.0
2.5
3.0
r
Fig. 8. Tangential membrane stress variation with r[(2)l/2j3 = 2.8].
any other angle. It is also seen that for higher/3 values the location of maximum principal stress shifts from 0 = 0 ° to some higher 0 value. Maximum principal stress is observed to occur at the top surface at all angles around the hole and is tensile. Fig. 5 shows the variation of Noo along the distance from the hole edge, for (2)1/2/3 = 2.8. The maximum stress decreases with increasing angle and so also does the decay rate. The stress value reduces to about 1% for 0 = 0 ° at r = 2.0, while for 0 = 900 the same value is reached at about r = 2.7. The reference stress value chosen here is 6Mo/t.
4.2. Circular cylindrical shell with a rigid circular inclusion (uniform temperature) Fig. 6 shows the variation of Nrr and Noo with respect to/3. Increasing the value of/3 leads to a higher compressive Noo and decreasing Nrr. In the limiting case of/3 ~ 0 the stress values are seen to tend to the flat plate solution, i.e. Nrr = 0.75 and Noo = - 0 . 7 5 . Fig. 7 shows the variation of maximum principal stresses with respect to 0 for different/3 values. For all angular positions the maximum principal stresses are seen to increase up to/3 = 2. But then the maximum
Nref. =
Eo(tT o
Fig. 9. Bending stress variation with r [(2)1/2j3 = 2.8].
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
344
6.0 Rigid inclusion Nre f =Eot I T o
5.0 0°
c~
4.O
o 3.o u L. Y
2.0
1. O
~o o
0 1.
I
3.0
-10 Fig. 10. Radial shear variation with r [(2)1/2j3 = 2.8].
principal stress at 0 = 0 starts decreasing and slowly the maximum principal stress point shifts away from 0 = 0 °. The maximum principal stress (in magnitude) occurs at the bottom surface and is tensile in the neighbourhood of 0 = 0 °, while it occurs at the top surface and is compressive in the neighbourhood of 0 = 90 °. Figs. 8--10 show the variation of membrane stress Noo, bending stress Nrb and Kirchoff's shear Qr with respect to r at (2)1/2~ = 2.8. The decay to uncut shell values is seen to occur when r ~> 2.5 in most of the cases except for Nrb (bending stress) at 90 ° which is slightly slower. The reference stress value chosen in this case is Nre f = E~tTO. All the plots are for nondimensional quantities.
5. Conclusions The problems of a circular hole and a rigid circular inclusion in the cylindrical shell subjected to uniform
temperature and a linearly varying temperature across the thickness is solved. The solution for a pressurized shell with thermal loading can be obtained by superposing the two solutions corresponding to pressure loading and thermal loading. The results presented here are in non-dimensional form and are valid for temperatures not high enough to cause changes in material properties.
Acknowledgements The authors wish to thank the Aeronautics Research and Development Board, Ministry of Defence, Government of India, for sponsoring a research project on 'Elastic shell structures with cutouts and reinforcements subjected to static and thermal loadings', of which the work reported herein is a part. The authors also wish to thank Mr. M.V.V. Murthy, Structures Division, National Aeronautical Laboratory, Bangalore, India, for many helpful discussions.
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
345
r # 1 sin 2 0 3w' + ~~ 0 - + R ~a w ' sin 0 cos 0 .
A p p e n d i x : c h a n g e in c u r v a t u r e o f t h e c i r c u l a r b o u n dary
A t the b o u n d a r y , r = 1, using t h e fact t h a t Fig. A1 shows the initial and d e f o r m e d p l a n f o r m o f a circular b o u n d a r y w i t h the d i s p l a c e m e n t c o m p o n e n t s . T h e change in c u r v a t u r e o f t h e circle is given b y [9]
F
t
W,o = W, r = 0 we get
,
1(
XO = - tt' + r'2 \ 302 i
(
'
×'o
w h e r e u ' is the t o t a l d i s p l a c e m e n t in r d i r e c t i o n including the c o m p o n e n t due to n o r m a l d i s p l a c e m e n t w', +u w
+~-
3~/'rO]
Expressing in t e r m s o f stresses we have ' , , r ,) Xo' = ( l / E t )[N'rr - vNoo - (Noo,r, - uNrr,
Wrf t
u =u
~e;
= e'r- -3r - r + ~ 0 ]"
sin20 + 2(1 + v)N'~o,o__ ]
I
and f
wtr t
u' = v' + o w
+ ~--
sin 0 cos 0
Notations
Also
,
3U'
W'
er = ~ / r ' + R- sin20 '
,
1 ~v'+w'
e ° - ~r '- - -
1 _ 1 3u-_
7ro
10w'
+u'
R c°s20
r +R
~
sinOcosO,
30-- - 7 U p + W 1 sin(20)
r' 30 + 3 /
~W
a D E F ff F*
= radius of circular discontinuity = flexural rigidity of shell = Et3/12(1 - v2) = modulus of elasticity of the shell material = non-dimensional Airy-stress function = asymptotic value of F far from the discontinuity = F - F, perturbation stress function due to the presence of discontinuity H n1 = Hankel function of first kind and order n
i
= (--1) I/2,r =r'/a
r'
= distance from origin measured along the surface of the wall = non-dimensional principal stress = thickness of the shell = non-dimensional normal displacement = asymptotic value of W far from the discontinuity = W w, perturbation componant of deflection due to the presence of discontinuity = curvature parameter, defined as/32 = (a 2/8Rt)[12(1 v2)] 1/2
Np t W
AI
w* ~3
---
Undeformed Deformed
/
R
v r 0 0" 7~
= Poisson's ratio = temperature gradient (T 1 - T2)/t = complex potential function = W - iF = perturbation function = W* - iF* = temperature of the outer surface temperature of the inner surface.
References ~r,u
KC = v~v
--~-e,U
Fig. A1. Deformation of circular boundary due to shell expansion.
[ 1 ] R.E. Hoffman and T. Ariman, Thermal bending of plates with circular holes, Nucl. Eng. Des. 14 (1970) 231-238. [2] R.E. Hoffman and T. Ariman, Thermal and mechanical stresses in nuclear vessels, Paper G1/2*, 1st Int. Conf. on
346
[3]
[4] [5 ] [6]
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
Struct. Mech. in Reactor Technol., Berlin, Sept. 1971; Nucl. Eng. Des. 20(1972) 31 55. P. Prakash and K.P. Rao, Thermal stresses around circular holes in spherical shells, Nucl. Eng. Des. 30 (1974) 8 3 87. P. Van Dyke, Stresses about a circular hole in a cylindrical shell, AIAA J. 3 (9), Sept. (1965) 1733-1742. P. Van Dyke, Stresses in a cylindrical shell with rigid inclusion, AIAA J. 5 (1), Jan. (1967) 125-137. K. Marguerre, Zur Theorie der gekrt~mmten Platte grosser FormSnderung, Proc. of Fifth International Congress of
Applied Mechanics, John Wiley and S(~ns Inc., New York (1938) 93 99. [7] A.L. [.urie, Statics of thin-walled elastic shells, transl, Atomic Energy Commission AEC-TR-3798 (1959). [8] M.V.V. Nurthy, On the stress problem of elliptical holes and inclusions and straight line cracks in cylindrical shells, Ph.D. Thesis, Indian Institute of Science, Bangak~re (1975) 108--115. [91 W. Fltigge, Stresses in Shells, Springer-Verlag, Berlin (1967)
THERMAL STRESSES IN A CYLINDRICAL SHELL CONTAINING A CIRCULAR HOLE OR A RIGID INCLUSION D.H. BONDE and K.P. RAO Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 560012, India
Received 11 March 1976
The thermal stress problem of a circular discontinuity in a cylindrical shell has been solved by continuum approach. Two types of discontinuities are considered: (i) a circular hole and (i.i) a circular regid inclusion. The effect of a uniform temperature or a linearly varying temperature across the thickness has been studied. The problem is converted into an equivalent boundary value problem and boundary conditions are specified around the discontinuity. The results are presented in a graphical form for ready use.
thickness of shell with the middle surface experiencing zero temperature. It is well known that the disturbance due to a discontinuity is confined to a small region in the neighbourhood of the discontinuity. The temperature involved is assumed not high enough to cause any changes in the material properties. The problem of a uniform temperature distribution and a linearly varying temperature across the thickness is converted into an equivalent boundary value problem, with forces specified around the discontinuity and the solution is obtained by a continuum approach. Superposition of this solution on the solution of an uncut shell gives the total solution.
1. Introduction A clear understanding of the structural behaviour under thermal loads is of primary importance in many nuclear engineering and aerospace structures. Thermal stresses around a circular discontinuity in a circular cylindrical shell is one of the interesting problems. Cylindrical shells are used in nuclear power plants, aerospace structures, marine, chemical and several other industries. Cutouts are introduced in these structures to fulfil functional requirements. These cutouts give rise to stress concentrations around the boundary. The understanding of the structural behaviour around these openings is necessary for efficient design. Hoffman and Ariman [1,2] have solved the problem of thermal bending of finite plates having circular holes. They also determined thermal and mechanical stresses in nuclear reactor vessels. Prakash and Rao [3] have considered thermal stresses around a circular hole in a spherical shell. The circular hole and rigid inclusion problems in a pressurized cylindrical shell have been solved by Van Dyke [4,5]. The present work deals with the thermal stresses around a circular hole or a circular rigid inclusion in a circular cylindrical shell. The problem is solved using thin shallow shell theory. Any linearly varying temperature distribution across the thickness of the shell can be considered as the combination of a uniform temperature and a linearly varying temperature across the
2. Governing differential equation and its solution Fig. 1 shows the configuration of a cylindrical shell with a circular hole. The equations that govern the behaviour of the region of the shell, influenced by the presence of a discontinuity, are assumed to be the shallow, thin shell equations. These were written in general, nonlinear form by Marguerre [6] and are simplified to the linear equations for the case of a cylindrical shell [7]. The stress conditions in the shell, as described by the stress resultants and the displacements tangential and normal to the middle surface, are shown in fig. 2. 337
338
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
- - -
Shellmiddle
~fi,'
The primes indicate that the quantities are dimensional. The non-dimensionalization of these parameters is now carried out and non-dimensional coordinates are defined as ~ = x/a, rl = y/a. The membrane stresses are non-dimensionalized with respect to a reference membrane stress Nre f. Defining
/ '~
-,.-/&:./ =
~
7
/
~
"= r COSt r sin
e
F-
F' a2Nref
Fig. 1. Cylindrical shell configuration. and The linear shallow shell equations for a cylindrical shell geometry involving only membrane stress function F' and normal displacements W' are [4] v 4 If' + F ; x x / R D : p / D
(1)
W'Et 2
W=
a2Nref[12{1 _ u2)] 1/2 and choosing the complex potential function q~, such that
and
(s)
~=W-iF, v 4 F ' - tEW',xx/R = 0 ,
(2)
where
',/4{b + 8i/32q5,~ = 8j32 .
Et 3 D - 1 2 ( 1 _ u 2) '
V4W '
eqs. (1) and (2) can be combined to give [4]
= Wt,xxxx
3W' W " x - 3x ' t
+
2W,xxyy
+
W,yyyy
(3) .
(4)
(6)
The right-hand side of this equation is present if the shell is loaded by uniform pressure and is zero otherwise. And [32 = (a2 /8R T) [12(1 _ u2)] 1/2 ,
f-
STRESSCOUPLES & ROTATIONS
N~
I I
/ f
where a is the radius of the circular discontinuity; R is the radius of the cylinder at middle surface; and t is the thickness of the cylinder. The stress and displacement functions can be expressed as F=F +F*
m/,~ Q:~/
I Q{~ZN~.~ STRESS RESULTANTS wl/~-~"U{ /'-'-'~ & DISPLACEMENTS I
I! ~"
Fig. 2. Stress couples and resultants, rotations and displacements.
(7)
and
W = W + W*
where F and W are the solutions corresponding to a shell without a discontinuity and F* and W* are the perturbation components caused by the presence of the discontinuity and which vanish at distances far from the discontinuity. The residual problem then reduces to solving the homogeneous differential equation v4O * + 8i/32q~*~ = O,
(8)
where q~* = W* - iF*, which vanishes far from the discontinuity. The solution which is symmetric in ~ and r/then be-
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell comes [41 ¢ = (El -- iE2)~ symm + (E3 - iE4)@anti-symm '
(9)
where E1 _ i g 2 = 1 {exp [(1 ,-i)/3~] + exp [ - ( 1 - i ) ~ ] ) , E 3 - iE 4 = ~(1 + i){exp[(1 - i ) ~ ] = r cos 0
and
exp[-(1 -
r~ = r sin 0 ,
(10)
v 2 ~ + 2i/32ff = O.
(11)
The solutions relevant to the present problems are oo
0,2,4 ....
radially without developing any stresses, irrespective of the shape of opening in the shell. Hence we consider only the case when the temperature varies linearly across the thickness of the shell. An uncut shell subjected to linearly varying temperature across its thickness experiences a uniform bending moment, which is given by
i)t3~] ) ,
ffsymm and ffantisymm are the symmetric and antisymmetric parts of the solution of the equation
~symm
339
Mrr =Moo = aD(1 + v)r = M 0 ,
where D is flexural rigidity and r is the temperature gradient and no deflections or direct stresses occur, i.e. W = F = 0. The boundary conditions at the free hole edge are given by
N'rr=N~r+N'r~:O, r
(A n + iBn)Hl[~r(2i) 1/2 ] cos nO ,
(14)
--r
N'ro=Nro
+
,
~r0 = 0 , (15)
M'rr=.M;r + M'rr* = 0 , ffantisymm =
~ (An + iBn)Hl[ ~r(2i)1/2] cos nO , 1,3,5 .... (12)
where unknown coefficients are to be determined from the boundary conditions. Stress and moment resultants are given b y
Nrr = F,r/r + F,oo/r 2 ,
Noo = F,rr ,
t
Qr = Q'r + Q'r* = 0 Using Nre f = 6Mo/t and non-dimensionalizing the above boundary conditions we get Nr*r : 0 ,
Nr*0 = 0 ,
/~rr = --~ [12(1 -- V2)] 1/2,
Q* = O.
(16)
Nro = -(F,o/r),r , 3.2. Cylindrical shell with a rigid circular &clusion r
Mrr = -
r
W'°
[
W,rr + V r W'r +
W'°°
,r , When the shell is heated to a uniform temperature a circle drawn on its side will try to expand. This expansion will be prevented by the rigid inclusion. Thus, the following displacement boundary conditions t are to be satisfied at the rigid inclusion-shell junction
'
141,'r
1
Moo = - [ V W , r r + ( 1 W , r + - ~ W , o o ) ] , Mro = - ( 1 - v) [(1/r)W,o ] ,r"
=0
e 0' = 0 , (13)
t
W, o = 0 ,
x; = 0 ,
(17)
where e 0 is the strain in the 0 direction, and ×o is the in-plane change of curvature. The strain and change of curvature can be expressed in terms of stresses as (see
3. B o u n d a r y c o n d i t i o n s
3.1. Cylindrical shell with a circular hole When an unrestrained cylindrical shell is heated uniformly to a constant temperature the shell expands
t A similar set of boundary conditions was originally proposed by Murthy [8] when solving the problem of a rigid elliptic inclusion in a pressurized cylindrical shell.
340
D.IL Bonde, K.P. Rao / Thermal stresses in a cylhMrieal shell
the appendix) e'o = (1/Et)(N'oo
the quadrant 0 = 0 to rr/2. At each point there are four boundary conditions to be satisfied, which gives four equations for the unknown coefficients. The series is terminated at ( 2 m 1) terms, where m is the nmnber of points chosen. For a given/~, successively higher number of points are chosen until the values of the stresses remained essentially the same. For ~ = 4 the maximum number of points needed are 14, where a good convergence is obtained. The number of terms needed in the series for convergence to occur was essentially the same as used by Van Dyke [4].
VHrr)
and t
t
t
Xo = ( 1 / E t ) [(N'rr - uNoo)
(18)
- ( X ' o o , r , -UN'rr, r , ) + 2(1 +u)N'ro,o ] .
Eqs. (17) on non-dimensionalization with respect to Nre f = E t a T 0 , give W, r = 0 ,
W, 0 = 0 ,
eo = - 1 ,
X0 = - 1
4. l. Cylindrical shell w i t h a circular c u t o u t
.
(19) Fig. 3 shows the tangential membrane stress variation with respect to 0 for various values of/3, at the hole edge. Maximum Noo occurs at 0 = 0 for ~ values less than 2, beyond which the angle at which maximum stress exists starts shifting towards 90 ° . Fig. 4 shows the variation of maximum principal stress with respect to ~ at four different values of 0. It is observed from the graph that the maximum principal stress starts decreasing with increasing/3. The flat plate values are achieved in the limiting case of/3 -+ O. The decay in principal stress is faster for 0 = 90 ° than for
A linearly varying temperature across the thickness with the middle surface experiencing zero temperature causes a uniform bending moment in the shell given by eq. (14). This moment will be reacted at the rigid inclusion, causing no extra stresses.
4. Resultsand discussion The boundary conditions are satisfied by a collocation procedure at discrete points on the boundary in
0.7
0.6
I
N ~ ° O, 4 -
T~
Nref 0.3
0.5
O.
0.1 0
I 10
I 20
I 30
I 40
I 50
I 60
I 70
I 80
I 90
(9 ( decjrees )
Fig. 3. Tangential membrane stress variation with respect to 0 (circular cutout).
D.H. Bonde, K.P, Rao / Thermal stresses in a cylindrical shell
341
2-2 TOP SURFACE = TIe
BOTTOM
[
ec,e
)
~0 1 . 8
"~1.4 O
90 °
. i 1.2
~
E
0-8 0
I
I
I
1'0
2.0
3.0
I
4.0
/3 Fig. 4. Maximum principal stress around a circular cutout in a cylindrical shell.
0.7 TOP SURFACE = T~ BOTTOM SURFACE = T~ 0'5
6 Mo Nref
=
t
0.3 Nee N ref 8 = 90 °
0.1
-o-I I" 1.0
~
0° J
I
I
I
1.2
1.4
1'6
-
I
I
1'8
2.0
Fig. 5. Tangential membrane stress NO0 variation with respect to r (circular cutout).
342
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell 1-00 0=90 °
~
J
Nee
0°
0-75 \
Nref : Eo~t TO
-Nee Nref
\
\
\
\
\
0.50
\
Rigid inclusion
\
Nrr Nref
\ ~0
.... - -
o
Nrr Nee
9 0 °`````` 0.25 - ~--~
Nrr
I
I
I
I
1.0
2-0
3.0
4.0
FIG. 6"
MEMBRANE
STRESSES IN THE SHELL AT THE INTERFACE
Fig. 6. Membrane stresses in the shell at the interface.
2 2 F 2" 01--
~--~'~..,. ~
f -'=4
Bottom surface ( Tensile )
~
Top surface ( Compressive )
2
~
;-
- - - - ~
0
10
20
30
40
50
60
70
80
O (degrees) Fig, 7. Maximum principal stressesin the shell at the interface,
90
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
343
0 -0,1
-0-3 Nee N r ef
Nre f =Ea(tTo
-0.5
-0.7 -0.9 1.0
I
I
I
1.5
2.0
2.5
3.0
r
Fig. 8. Tangential membrane stress variation with r[(2)l/2j3 = 2.8].
any other angle. It is also seen that for higher/3 values the location of maximum principal stress shifts from 0 = 0 ° to some higher 0 value. Maximum principal stress is observed to occur at the top surface at all angles around the hole and is tensile. Fig. 5 shows the variation of Noo along the distance from the hole edge, for (2)1/2/3 = 2.8. The maximum stress decreases with increasing angle and so also does the decay rate. The stress value reduces to about 1% for 0 = 0 ° at r = 2.0, while for 0 = 900 the same value is reached at about r = 2.7. The reference stress value chosen here is 6Mo/t.
4.2. Circular cylindrical shell with a rigid circular inclusion (uniform temperature) Fig. 6 shows the variation of Nrr and Noo with respect to/3. Increasing the value of/3 leads to a higher compressive Noo and decreasing Nrr. In the limiting case of/3 ~ 0 the stress values are seen to tend to the flat plate solution, i.e. Nrr = 0.75 and Noo = - 0 . 7 5 . Fig. 7 shows the variation of maximum principal stresses with respect to 0 for different/3 values. For all angular positions the maximum principal stresses are seen to increase up to/3 = 2. But then the maximum
Nref. =
Eo(tT o
Fig. 9. Bending stress variation with r [(2)1/2j3 = 2.8].
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
344
6.0 Rigid inclusion Nre f =Eot I T o
5.0 0°
c~
4.O
o 3.o u L. Y
2.0
1. O
~o o
0 1.
I
3.0
-10 Fig. 10. Radial shear variation with r [(2)1/2j3 = 2.8].
principal stress at 0 = 0 starts decreasing and slowly the maximum principal stress point shifts away from 0 = 0 °. The maximum principal stress (in magnitude) occurs at the bottom surface and is tensile in the neighbourhood of 0 = 0 °, while it occurs at the top surface and is compressive in the neighbourhood of 0 = 90 °. Figs. 8--10 show the variation of membrane stress Noo, bending stress Nrb and Kirchoff's shear Qr with respect to r at (2)1/2~ = 2.8. The decay to uncut shell values is seen to occur when r ~> 2.5 in most of the cases except for Nrb (bending stress) at 90 ° which is slightly slower. The reference stress value chosen in this case is Nre f = E~tTO. All the plots are for nondimensional quantities.
5. Conclusions The problems of a circular hole and a rigid circular inclusion in the cylindrical shell subjected to uniform
temperature and a linearly varying temperature across the thickness is solved. The solution for a pressurized shell with thermal loading can be obtained by superposing the two solutions corresponding to pressure loading and thermal loading. The results presented here are in non-dimensional form and are valid for temperatures not high enough to cause changes in material properties.
Acknowledgements The authors wish to thank the Aeronautics Research and Development Board, Ministry of Defence, Government of India, for sponsoring a research project on 'Elastic shell structures with cutouts and reinforcements subjected to static and thermal loadings', of which the work reported herein is a part. The authors also wish to thank Mr. M.V.V. Murthy, Structures Division, National Aeronautical Laboratory, Bangalore, India, for many helpful discussions.
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
345
r # 1 sin 2 0 3w' + ~~ 0 - + R ~a w ' sin 0 cos 0 .
A p p e n d i x : c h a n g e in c u r v a t u r e o f t h e c i r c u l a r b o u n dary
A t the b o u n d a r y , r = 1, using t h e fact t h a t Fig. A1 shows the initial and d e f o r m e d p l a n f o r m o f a circular b o u n d a r y w i t h the d i s p l a c e m e n t c o m p o n e n t s . T h e change in c u r v a t u r e o f t h e circle is given b y [9]
F
t
W,o = W, r = 0 we get
,
1(
XO = - tt' + r'2 \ 302 i
(
'
×'o
w h e r e u ' is the t o t a l d i s p l a c e m e n t in r d i r e c t i o n including the c o m p o n e n t due to n o r m a l d i s p l a c e m e n t w', +u w
+~-
3~/'rO]
Expressing in t e r m s o f stresses we have ' , , r ,) Xo' = ( l / E t )[N'rr - vNoo - (Noo,r, - uNrr,
Wrf t
u =u
~e;
= e'r- -3r - r + ~ 0 ]"
sin20 + 2(1 + v)N'~o,o__ ]
I
and f
wtr t
u' = v' + o w
+ ~--
sin 0 cos 0
Notations
Also
,
3U'
W'
er = ~ / r ' + R- sin20 '
,
1 ~v'+w'
e ° - ~r '- - -
1 _ 1 3u-_
7ro
10w'
+u'
R c°s20
r +R
~
sinOcosO,
30-- - 7 U p + W 1 sin(20)
r' 30 + 3 /
~W
a D E F ff F*
= radius of circular discontinuity = flexural rigidity of shell = Et3/12(1 - v2) = modulus of elasticity of the shell material = non-dimensional Airy-stress function = asymptotic value of F far from the discontinuity = F - F, perturbation stress function due to the presence of discontinuity H n1 = Hankel function of first kind and order n
i
= (--1) I/2,r =r'/a
r'
= distance from origin measured along the surface of the wall = non-dimensional principal stress = thickness of the shell = non-dimensional normal displacement = asymptotic value of W far from the discontinuity = W w, perturbation componant of deflection due to the presence of discontinuity = curvature parameter, defined as/32 = (a 2/8Rt)[12(1 v2)] 1/2
Np t W
AI
w* ~3
---
Undeformed Deformed
/
R
v r 0 0" 7~
= Poisson's ratio = temperature gradient (T 1 - T2)/t = complex potential function = W - iF = perturbation function = W* - iF* = temperature of the outer surface temperature of the inner surface.
References ~r,u
KC = v~v
--~-e,U
Fig. A1. Deformation of circular boundary due to shell expansion.
[ 1 ] R.E. Hoffman and T. Ariman, Thermal bending of plates with circular holes, Nucl. Eng. Des. 14 (1970) 231-238. [2] R.E. Hoffman and T. Ariman, Thermal and mechanical stresses in nuclear vessels, Paper G1/2*, 1st Int. Conf. on
346
[3]
[4] [5 ] [6]
D.H. Bonde, K.P. Rao / Thermal stresses in a cylindrical shell
Struct. Mech. in Reactor Technol., Berlin, Sept. 1971; Nucl. Eng. Des. 20(1972) 31 55. P. Prakash and K.P. Rao, Thermal stresses around circular holes in spherical shells, Nucl. Eng. Des. 30 (1974) 8 3 87. P. Van Dyke, Stresses about a circular hole in a cylindrical shell, AIAA J. 3 (9), Sept. (1965) 1733-1742. P. Van Dyke, Stresses in a cylindrical shell with rigid inclusion, AIAA J. 5 (1), Jan. (1967) 125-137. K. Marguerre, Zur Theorie der gekrt~mmten Platte grosser FormSnderung, Proc. of Fifth International Congress of
Applied Mechanics, John Wiley and S(~ns Inc., New York (1938) 93 99. [7] A.L. [.urie, Statics of thin-walled elastic shells, transl, Atomic Energy Commission AEC-TR-3798 (1959). [8] M.V.V. Nurthy, On the stress problem of elliptical holes and inclusions and straight line cracks in cylindrical shells, Ph.D. Thesis, Indian Institute of Science, Bangak~re (1975) 108--115. [91 W. Fltigge, Stresses in Shells, Springer-Verlag, Berlin (1967)