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Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
ht. J. Engng Sci., 1976. Vol. 14, pp. 99-112.
Pergamoo Press.
Printed in Great Britain
AXIAL EIGENFUNCTIONS FOR THE AXISYMMETRIC PROBLEM OF AN ELASTIC CIRCULAR CYLINDER B. S. RAMACHANDRA RAO and A. K. KANDYA Department of Mathematics, Indian Institute of Technology, Powai Bombay-400076,India and S. GOPALACHARYULU Loughborough University of Technology, Leicestershire, U.K. Abstract-Constructing a biorthogonality relation it is shown how solutions can be obtained for the stress, displacement and mixed problems associated with the axisymmetric deformations in a cylinder whose plane faces are either stress-free or displacement-free. Numerical results are presented for the reproduction of the boundary stresses when a self-equilibrating load is prescribed on the circular boundary of the cylinder whose plane faces are stress-free.
INTRODUCTION THE PROBLEM of
determining the stress and displacement distributions in a solid cylinder undergoing axisymmetric deformations with the circular boundary stress-free and with stresses or displacements prescribed on the plane ends has been the subject of several interesting investigations. The Fourier analysis for obtaining solutions was first given by Pickett [l] and later Valov[2] has shown how Fourier analysis can be used to obtain solutions for the mixed and second fundamental boundary value problems. Eigenfunction expansions have been used by many authors and various approximate methods of satisfying the boundary conditions have been suggested by Lurje [3], Horvay and Mirabal[4], Mendelson and Roberts [5] and a few others. The method of satisfying the boundary conditions through suitable biorthogonality relations have been given by Little and Childs [6], Flugge and Kelkar[7], Nuller [8] and Fama[9]. Relatively few papers however report on the method of obtaining solutions of the axisymmetric problem by developing the eigenfunctions in the axial coordinate. Green[lO] deals with the problem pertaining to a cylinder whose plane faces are stress-free by developing the eigenfunctions in the axial coordinate. He has also shown how general solutions of the equations can be expressed either in terms of Fourier series or power series. The aim of the present work is to provide a biorthogonality relation for the eigenfunctions developed in the axial coordinate and to indicate the use of this biorthogonality for obtaining solutions of the problems associated with a cyclinder. The method of biorthogonal functions developed by Johnson and Little [l l] for the semi-infinite elastic strip problem has been used in the present case. It may be noted that this method has already been used to provide effective solutions of problems in the theory of elasticity[l2-151 and in the theory of flexure of thin plates[l6,17]. Numerical results for the convergence of the eigenfunction expansions have been illustrated for a self equilibrating load prescribed on the circular boundary of a cylinder whose plane faces are stress-free. 2. STATEMENT
AND FORMULATION
OF THE PROBLEM
We will consider the axisymmetric deformations in a cylinder occupying the region r s a, (zJd h with its plane faces z = rh either stress-free or displacement free, i.e. uz=r=O
on
z=kh,
r
(2. la)
u=w=O
on
z=ah,
r
(2.lb)
or
99
B. S. RAMACHANDRA RAO et al.
100
Throughout the paper a,, (re, a, represent the normal stresses, r the non-zero shear stress, u the radial displacement and w the displacement in the axial direction. The constants p and E represent the Poisson’s ratio and the Young’s modulus respectively. On the circular boundary r = a, we will prescribe any one of the following pairs of boundary functions
0;
=
ub
7 =
;
(2.2b)
Tb
(2.2c)
(2.2d) where the subscript b indicates a specified function on the boundary r = I(. The problems (2.2a) and (2.2~) will be referred to as the mixed problems, (2.2b) as the stress problem and (2.2d) as the displacement problem. The governing differential equation for the problems can be stated in terms of the Love strain function x as follows: rda,
AAx=&
lzlch
(2.3a)
where &+la+a’ dr’
r ar
(2.3b)
dz”
We also record the relations ~=+(pk$)x
(2.4a)
G=;
(2.4b)
(&$,)x
(2.4~)
I+,= za [(2--~)A-$]x
r=$ Eu -=-l+P -=Ew 1+FL We
(l-&A--$ E
I
x
8’~
(2.4e)
Wz
2(1-&A--$
(2.4d)
I
x.
(2.4f)
introduce the functions f(‘)(r, z) (i = 1, 2, 3) related to the derivatives of x as follows: (2Sa)
(2Sb)
101
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
(2Sc)
f(3) = 2.
It can be easily shown that (2.6a)
(2.6b)
Using (2Sa), (2.5b) the biharmonic eqn (2.3) can be expressed as
The above relation is identically satisfied by introducing a function f(*)(r, z) satisfying the relations -aft') =
a2
ar
a cf”’
_t f”‘)
(2.6~) $
cf”’
+
f (2))
=
_
+
$
(rf’“‘)*
Eliminating af"'/az between the above relation and (2.6a) we get aft*) _
TF-
_ia rar
of”+
$f”‘)
(2.6d)
(2.6a-2.66) expresses (2.3a) in terms of four partial differential equations of the first order. 3. BOUNDARY CONDITIONS
ON f”’ (i = 1,2,3,4)
Using the relations (2.4e), (2.4f) and (2.5) we get EU lfCL = -fc3)
= 2(1 - p)f(l)+ (1- zp)f(*),
j$
(3.la)
(3.lb)
Using (2Sa), (2.5b), (2.6a) and (2.6d) the relations (2.4~) and (2.46) may be written as a, =
f$
7 = ;
[rCf’“-
(1 - p)f(‘)}]
[(l - &f”’ - pf’2’],
(3.lc)
(3Sd)
Using (2.2d), (3.la), (3.lb) we get & = -fb”
w,,= 2(1 - ,.‘))f’b’+ (1 - 2,.&)f(bz). Using (2.4a) and (2.5) we get
(3.2a) (3.2b)
102
B. S. RAMACHANDRA RAO et al. (z,b =ffP’-$
[(l- /.L)fb”- /L~ufs’].
(3.2~)
Using (2.4d), (2.9, (2.6b) and (2.6~) we get [(l -/.&q?].
Tb = g
(3.2d)
Using the relations (3.2a-3.2d) the functions fb” may be expressed in terms of Indeed we get
Urb,
76,
ub
and
wb.
(3.3a) z
I
fF’= wb +2
(C&b+Ub/a)dZ
(3.3b)
fS’= -ub
(3.3c) (3.3d)
4. EIGEN VECTORS
FOR THE STRESS
FREE PLANE
FACES
We seek the eigenfunction expansions of the functions fci’ (i = 1, 2, 3, 4) in the form
(4.1)
Upon substituting for
fci’(i = 1, 2, 3, 4) in the relations (2.6a-2&i) we get Z(l)’ = * 2’3’ ” “”
(4.2a)
Z(Z)! = _* nn[Z(3) n Z(3)’= * n
+
pq
(4.2b)
n
(4.2~)
z’2’ nn
z(4), [Z’” + Z(2)] n = *nn “.
(4.2d)
It may be noted that throughout the paper, primes denote differentiation with respect to “2”. The relations (4.2) may be written in the matrix form
00 1
Z:,=A,U.% where the matrix U is given by
(4.3a)
10
01 0 0 11 0 0’ u= ! 0 0 -1 -1
(4.3b)
Using relations (3.lc), (3.ld) and (4.1) the boundary conditions on 2. may be written as z’,“-(l-~)zlf’=o
(4.3c)
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
103
and (1 -
p)Z(.I)- pZ'.z' = 0
on
2 = kh.
(4.3d)
Solutions of (4.3a) subject to the conditions (4.3~) and (4.3d) separate into even and odd eigen vectors. The components of .$!”may be listed in terms of a function F, and its derivative as follows: Z’.” = X;F, (4.4a) z’,” = h,F::
(4.4b)
z’,“= A2,F:,
(4.4c)
2’4’
=
_F”
n
_ ”
(4.4d)
h 2Fl nn
where F,, satisfies the differential equation F:+2A:F;+A:F.
=0,
IzIG
(4Sa)
and the boundary conditions pF::-(l-p)hf,Fn
(4Sb)
=0
and (l-p)F’::+(2-p)AfJ7h=O
on
z =+h.
(4.k)
The even solutions of F, are given by eFn = (2~ cos A,h - A,h sin A,h) cos A,z + A.2 cos A.h sin A.2
(4.6a)
where the A,‘s are the roots of sin 2hh - 2Ah = 0.
(4.6b)
oFn= (2~ sin h,h + A,h cos A,h) sin A,z - A,,.zsin A,h cos AJ
(4.6~)
The odd solutions of F, are given by
where the A,% are the roots of sin 2Ah + 2Ah = 0.
(4W
The roots of (4.6b) and (4.6d) to any desired degree of accuracy can easily be determined using their asymptotic forms as the first approximation to the roots and then applying Newton’s iterative technique. These roots are symmetrically located over the four quadrants of the complex plane so that if Ah is a root so are -Ah, A*h and -A *h. Further the eigen-functions corresponding to the eigenvalues Ah and -Ah are not distinct. The summation appearing in (4.1) is restricted over the pairs of eigenvalues lying in the right half of the complex plane. The roots of (4.6b) were 6rst investigated by Hilman and Salzer [18] and those of (4&l) by Robbins and Smith[l9]. It may be noted that A = 0 is a root of (4.6b) and (4.6d). The significance of the root is given in Appendix 1.
5. BIORTHOGONALITY
RELATION
The differential equation adjoint to (4.3a) is given by @:,=-AtUt.
w,
(5.1)
B. S. RAMACHANDRA RAO et al.
104
where * denotes the complex conjugate and t, the conjugate transpose. We have
I
+-h
(WA,‘?.% - @,,,tzb) dz = W, t .z
-h
+h. I -h
Using (4.3) and (5.1) the above relation may be written in the form (A, -A,) f+h W,,,. u. 2, & = Z’,” [W:‘+?
W:‘] + ,?$‘[(I - p)W:‘+
WE’].
(5.2)
-h
It therefore follows that, if W,,,‘s satisfy the following boundary conditions (5.3a) and WE’+(l-p)WE’=O
on z=+h,
(5.3b)
we have the biorthogonality relation
I
+h
-h
(m#
Cii,t.U..%dz=O
n).
(5.4)
Solutions of (5.1) subject to the conditions (5.3a) and (5.3b) separate into even and odd eigen vectors. The components of W? may be listed in terms of a function G,, and its derivatives as follows: (5.5a) wL2’ =
z2G X’
(5.5b)
W’,“= _A*G*“_A*‘G* nn nn
(5.5c)
A
At3Gz
W;‘=
(5.5d)
where G, satisfies the differential equation G:t2A:G:tA:G,
=O,
(z(Gh
(5.6a)
and the boundary conditions (l-/J)G::-pAZ.Gn
=0
(5.6b)
and @;+
(1 t /.~)h:Gh = 0.
(5.6~)
The even and odd solutions of G, are given by eGn = [2(1- p) cos A,h - A,h sin A,h] cos A,z t A,z cos A.h sin A,z
(5.7a)
oGn = [2(1- /.L)sin A.h + A,h cos A,h] sin A,z - A,z sin A.h cos A,z.
(5.7b)
The eigen values appearing in (5.7a) are the roots of (4.6b) and those appearing in (5.7b) are the roots of (4.6d).
105
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
The biorthogonality relation (5.4) may be written in the form
I
+h [z’,”
w$4’
+
Z’.“(
Wf”’
+
j.j.7~“))
+
Z(,3’(
,p
-
wp)
-
z~)W:(~)]
-h
dz = 0. (m # n) (5.8a)
Using (4.4), (4.9, (5.5) and (5.6) it can easily be verified that ,.+h
I [
z’,)wgw
+ w:“‘)]
z’,)(wf)
+
,.+h
dz = 1
[z’,‘y wf)
_ wf2)) - z’.” wf”‘]
dz
J-h
J-h
,
r+h
(A:FnG, - FZG:) dz.
= + j_
n
h
It therefore follows from (5.8a) that
I
+h [z(nl)wg(4)
+ z’,2’(wf)
+
w:“‘)]
dz = K.&n,
(5.8b)
-h
and
I
+h [z’,“(
w$”
_ ,zc2’)
- 2;’
W*,,,
*(2)] dz
= Kn&
(5.8~)
-h
where K. =+, _+h(h4,F.G. - FEGE)dz nI h
(5.8d)
and 8,. denotes the Kronecker delta. The relations (5.8b) and (5.8~) are referred to as the inner orthogonality relations. 6. EIGENFUNCTIONS
It follows from (4.1) that on the boundary we have (6.la) and (6.lb) Using (5.8b) the constants C. in the expansion (6.la) are given by C,
I”ha )
K,
I&La)
=
I_+’
vb” Wf4’ + ffZ)(Wf3) + W$*?] dz.
(6.2a)
Using (5.8~) the constants C,, in the expansions (6.lb) are given by C,Kn =
+hvf’( Wf” - Wf”) _ fb” Wf”] dz.
(6.2b)
Indeed (6.2b) gives a closed form solution to the mixed problem (2.2a). For, using (3.3~)and (3.3d) we get
B.S.RAMACHANDRA RAO etaf.
106
and the constants C. in the expansions (4.1) are given by C”K” = -(1‘I[(/‘Tb _
dz) Wt’“‘+{(l-~)W:“‘-
p)
W’.(“}Ub]dz.
(6.2~)
The solution of the boundary value problems (2.2b-2.2d) can be reduced to solving a system of linear algebraic equations in the infinitely many unknowns C,. For, upon adding (6.2a) and (6.2b) we get
c”K”(l+g$=j-:
up
wy)
+
fry
+ wy’)
,y’
+ ff’(wf”
- w$“)
- f(b4)WfZ’] dz
(6.3) and using (6.3) the solution of the problems (2.2b-2.2d) can be obtained as follows: For the stress problem (2.2b) we get (using (3.2~) and (3.2d))
and using (6.3) we get the following linear algebraic system of equations for the unknowns C,, viz. (6.4a)
C, = G, + x CmRm. where C,=-L,11_:*[W:“(~‘~~sdz)+W*‘2’(~iT~dz)]dz
R,, = L,’
~{W:‘“+(l-~)wf”)}Z~‘+
(6.4b)
(;j=Z:‘dz)
(2- ,.L) W:“‘}Z:‘] dz
+{(I-y)WX(‘)L” =(l-/L)K”[l+$j]. For the mixed problem (2.2~) we take (using (3.2b) and (3.2~)) j.;)=&,,
-+=$j=-
dz++=$$
C-l’Z$‘dz;
fF’ = c c,z?; fY)=w,+2
I
z n,bdz-;zC,
I
Wf’)
1 Z:’ dz ;
f6”’= x c,zF. For the displacement problem (2.2d) we take (using (3.2a) and (3.2b))
(6.4~)
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
107
Usin’: (6.3) it can easily be shown that the problems (2.2~) and (2.2d) can be reduced to linear algebraic system of equations of the same type as (6.4a). The necessary conditions to be imposed on the boundary stresses and displacements in order to ensure the convergence of the stress, displacement and mixed problems are presented in Appendix 2. 7. EIGEN VECTORS FOR THE BUILT-IN
PLANE FACES
When the plane faces are displacement free (i.e. built-in) the boundary conditions u=w=O
on z=+h
are to be satisfied. Using (3.la) and (3.lb) it follows that the above boundary conditions are equivalent to z’.3’=0
(7.la)
and 2(l-P)Z(nl’+(l-2~)Z(n2)=0
on z=*h.
(7.lb)
Solutions of (4.3a) subject to the conditions (7.la) and (7.lb) can again be listed in terms of a function F. and its derivatives. The components of Z? (i = 1,2,3,4) are essentially the same as listed in (4.4a-4.4d). The functions F, are now given by eFn = (sin A,h + h.h cos h,h) cos A,z + A,z sin A,h sin A,z
(7.2a)
where the An’s are the roots of sin 2Ah + - ’
3-4p
2Ah=O
(7.2b)
and oFn = (cos A,h - A,h sin A,h) sin A.z - A,z cos A.h cos A,z
(7.2~)
where the A, ‘s are the roots of sin 2Ah - - ’ 2Ah=O. 3-4/k
(7.2d)
Both (7.2b) and (7.2d) have an infinite number of complex roots. (7.2d) has a real root also. These roots to any degree of accuracy can be determined using their asymptotic form as the tirst approximation to the roots and then followed by Newton’s iterative technique. It is extremely important to note that in the present case the stresses in the neighbourhood of r = a, z = kh become unbounded on account of the singularities which exist around the circumference of the plane ends. The attention to the existence of the stress singularities arising from the various boundary conditions in the angular comers of plates in extension was drawn by
108
B.S.RAMACHANDRA RAO eta!.
Williams[20]. The character of these singularities are of the type a=Real(cp-“)
as p-+0,
O
where cr is a stress along the plane end of the cylinder, p the distance from the circumference, a the exponent of the singularity and c the strength of the singularity. If these singularities are not taken into account then whatever be the method used for the analysis of the problem one has to solve an infinite system of linear algebraic equation and this will invariably pose bad convergence and will not lead to satisfactory results for the stresses in the neighbourhood of the circumference of the plane ends. Benthem[21] and Benthem and Minderhoud[22] have shown the method of introducing these singularities to the analysis of the problems. The introduction of these singularities make the eigenfunctions over complete and the resulting linear algebraic system of equations become ill-conditioned if one uses large number of eigenvalues to obtain the solutions of the problems. However any practical formulation of the problem must take into account of these singularities so that the stresses in the neighbourhood of these singularities are determined fairly accurately using the first few eigenvalues. It again follows from (5.2b) that in order to get the bi-orthogonality relation (5.4), the product ( wrn t . 2”) should vanish at z = +h. It can easily be verified that in order to satisfy this condition the vectors W,,, must satisfy the following boundary conditions: w’,“=o
(7.3a)
and (l-2p)W!!‘-2(1-p)W(,z)=O
on
z=?h.
The biorthogonal vectors are determined on solving (5.1) subject to the above boundary conditions. The components of W,, are again listed out in terms of a function G. and its derivatives. They are essentially the same as listed in (S.Sa-5.5d). The functions G, are given by eGn = sin A.h cos h.z - h,z cos A,h sin h,z
(7.4a)
oGn = cos A.h sin A,z - A,z sin A,h cos A,z.
(7.4b)
The eigenvalues appearing in (7.4a) are the roots of (7.2b) and those appearing in (7.4b) are the roots of (7.2d). The analysis for determining the solutions of the problems (2.2a-2.2d) follow on the same lines as indicated in Section 6.
8.NUMERICAL RESULTS
For numerical results we shall consider the case of a cylinder whose plane faces z = +h are stress free and upon whose circular edge arbitrary, otherwise self equilibrating load is prescribed. Consider the load ub = (z/h)*- l/3 Q, = sin (nzlh). The ratio of the diameter to the height of the cylinder is set equal to 5, i.e. a/h = 5. The Poisson’s ratio p is taken as 0.3. The infinite system (6.4a) is solved using five eigenvalues. It may be remembered that even or odd expressions are taken for the vector G,, and the matrix R,. depending upon whether the deformation is symmetric or antisymmetric with respect to the plane z = 0. The remarkable accuracy in reproducing the boundary stresses through the eigen function series is apparent from the results presented in Table 1. The results are also illustrated graphically in Figs. 1 and 2.
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder Table 1. Convergence of eigenfunction expanisons
r/h 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0
Specified function UIJ
Reproduced gO,using five eigenvalues
-0.333 -0.323 -0.293 -0,243 -0.173 -0.083 0.027 0.157 0.307 0.477 0.667
-0.312 -0.302 -0.284 -0.241 -0.146 -0.053 0.025 0.165 0.347 0.483 0.777
Specified function 7.5
Reproduced rb using five eigenvalues
0.000 0.309 0,588 0.809 0,951 1,000 0.951 0.809 0.588 0.309 O@OO
0.000 0.316 0.581 0,805 0.960 0.995 0.941 0.820 0.578 0,298 O@Xl
I
0
cl1
0.2
0.3
0.4
Fig. 1. Reproduction of the normal stress a, -,
0.5
0.c Z/h
0.7
08
0.9
1.0
-
specified function a,; ---, reproduced (r, for 5 pairs of eigenvalues.
1.0 -
0.5 Z/h
Fig. 2. Reproduction of the shear stress 7. -,
0.6 -
0.7
ae
specified function 7; ---, eigenvalues.
0.9
1.0
reproduced r for 5 pairs of
109
110
B. S. RAMACHANDRA RAO et al. CONCLUSIONS
The paper presents a method for obtaining solutions for the stress, displacement and mixed problems associated with the axisymmetric deformations in a cylinder whose plane faces are stress free. While a closed solution is available for the mixed problem (2.2a), it is shown that all other problems can be reduced to the solution of an infinite system of linear algebraic equations. The authors are not aware of any literature wherein the problems connected with a cylinder have been solved by using a biorthogonality for the eigenfunctions developed in the axial coordinate. To this end it is believed that the analysis presented here can easily be used to obtain solutions of the problems connected with the axisymmetric deformations in a cylinder. REFERENCES [l] G. PICKETT, J. Appl. Mech. 11, 176 (1944). [2] G. M. VALOV, P.M.M. 26, 975 (1%2). [3] A. J. LURJE, Raumliche probleme der Elnstizitatstheorie. Akademic-Verlag (1963). [4] G. HORVAY and J. A. MIRABEL, J. App. Mech. 25, 561 (1958). [5] A. MENDELSON and E. ROBERTS, 8th Midwestern Mechanics Conj. p. 40 (1%3). [6] R. W. LITTLE and S. B. CHILDS, Q. Appl. Math. 25, 261 (1967). [7] W. FLUGGE and V. S. KELKAR, Int. .I. Solids Structures 4, 397 (1%8). [8] B. M. NULLER, P.M.M. 33, 364 (1969). [9] M. E. D. FAMA, Q. J. Maths. Appl. Mech. 25, 479 (1972). [lo] A. E. GREEN, Proc. Roy. Sot. Land. 195, 533 (1949). [II] M. W. JOHNSON and R. W. LITTLE, Q. Appl. Maths. 12, 335 (1964). [12] R. W. LITTLE, J. Appl. Mech. 36, 320 (1969). [13] J. L. KLEMM and R. W. LITTLE, SIAM J. Appl. Maths. 23, 209 (1970). [14] B. S. RAMACHANDRA RAO, C. S. KALE and R. P. SHIMPI, Int. Engng Sci. 11, 531 (1973). [IS] P. V. B. A. S. SARMA,B. S. RAMACHANDRARAO and S. GOPALACHARYULU, fnt. J. Engng Sci. 13,149(1975). [16] P. V. B. A. S. SARMA, B. S. RAMACHANDRARAO and S. GOPALACHARYULU, SIAM. .I.Appl. Maths. 26(3),568 (1974). [17] K. V. RAJAN and B. S. RAMACHANDRA RAO, Int. J. Engng Sci. 12, 1079(1974). [18] A. P. HILMAN and H. E. SALZER, Phil. Msg. Land. 34, 575 (1943). [19] C. S. ROBBINS and R. C. T. SMITH, Phil. Msg. Lond. 39, 1004(1948). [20] M. L. WILLIAMS, .I. Appl. Mech. 19, 526 (1952). [20] J. P. BENTHEM, Q. J. Mech. Appl. Maths. 16, 413 (1963). [22] J. P. BENTHEM and P. MINDERHOUD, Int. J. Solids Structure 8, 1027(1972). [23] M. I. GUSEIN-ZADE. P.M.M. 29(2), 393 (1965). (Received 3 June 1974)
APPENDIX 1 It can easily be verified that A = 0 is a six fold root of the eqn (4.6d)and an eightfold root of the eqn (4.6b).Accordingly we seek the eigenfunctions of the function f”’ in the form
(Al) Upon substituting in the relations (2.6a-2.6d) we get Z”” = 2Z’“’ Z’2”= _2(Z(3l+ Z(4)) Z”” = 2p z’4” = 2(a + 0).
(A2)
Solution for f”’ satisfying the boundary conditions (4.3~)and (4.3d) splits into two parts. Solutions where the function x is
symmetric in z are given by f”’ = 2(1- k)z't pr* f::: = -2(2 - p)z’t (1 - F)r* ;,,, 1 ;(I - Y)Z 2
(A3)
The solutions for which x is antisymmetric in z are given by f”’ = 2(1- p)z f”’ = -2(2 - p)z f”’ = (1 - p)
f”‘= 1.
Using (3.2~)and (3.2d) we get for the solutions (A3)
(A4)
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder fJ,b= -2(1 t /L)z;
111
7b=o
and for the solutions (A4) rl, =o.
orb =-(1+11);
It therefore follows that the even solution of x corresponding to the eigen value A = 0 will give rise to inplane moment and the corresponding odd solution to inplane tension. If the applied load on the boundary is self equilibrating, i.e. ~_~~~~dz=I~~z~,~dz=I_*:7cdz=O
(AS)
then the inplane moment and the inplane tension vanish and hence the eigenvalue A = 0 is excluded from the summation. APPENDIX 2 The conditions necessary to ensure the convergence of the stress, displacement and mixed problems can be obtained analogous to the method used by Gusein_Zade[23].These conditions corresponding to the cases for which the function x is symmetric or antisymmetric in the coordinate z can be obtained as follows. Symmetric case
Using (4.1), (4.4) and (A3) the expansion of the boundary function fb” may be written in the form I&a 1
fb”=C,[2(l-~)z’+~a*]+~
C.I(~)A:F.
I I
~~‘=C0[2(2-p)z’+(l-p)a2]+~C~~A.F: jr’ = C0[2(l- p)az] + 2 C”A2.F:
(B3)
fr’= Co[2az]+~
(B4)
C,(-F’L-Af,i%).
In order to ensure the convergence of the various boundary value problems discussed in Section 6, it is necessary to set the constant C, equal to zero. The constant C, may be evaluated in terms of the boundary functions fb” using the conditions (4Sb) and (4.5~). Multiplying (B4) by pz, (B3) by z, upon adding and integrating the result obtained between the limits -h and h we get (using 3.3~ and 3.3d) (2% +2zu,) dz. Multiplying (Bl) by 1 - p, (B2) by -p, upon adding and integrating the result obtained between the limits -h and th we get (using 3.3a and 3.3b) Co=$j-;z(u,a
+;)dz.
For the stress problem (2.2b), using (AS) we get (B7) as the necessary conditions for convergence. For the mixed problem (2.2a), using (B5) we get
I
h(pz2+2zUb)dz =0 0
as the necessary condition. For the mixed problem (2.2c),using (BS)and (B6) it follows that there is no condition upon the function w,,.The function c,~ satisfies the first of the self-equilibrating conditions in (B7). For the displacement problem (2.2d),it again follows that while there is no condition upon the function w,,,the function us has to satisfy the condition (B8). It may be observed that the eigenfunction expansion X a.2:’ converges to R’- C, - C,z where C, and C, are arbitrary constants putting u.(z)-Zr) and using (3.3~)it follows that
z
aa. (z) =b(z) + C, + Czz.
This may be confirmed by examining the closed form solution (6.2~).The expression on the right hand side of (6.2~)remains unchanged if us is replaced to ub + C, + C2z. For the symmetric case we take
The constant C, is determined using the condition (BS). We get
112
B. S. RAMACHANDRA RAO et al. pz’ 2 a,r.(t) + 2zub dr I
7”
=(l-p)z:“-zy”.
Anrisymmetric case Using (4.1) and (A4) the expansions of the boundary functions fb” may be written in the form fb” = C0[2(1- c()I] t c C”y
L n
Z(.l’
(Cl)
fP’ = q-2(2 - /L)z]+ c C”# z’.” I n fP’ = &[a(1 -IL)]+ 2 C.Z!?
(C3)
ff’ = &[a] t x C.Z!,?
(C4)
Multiplying(C3) by I, (C4) by CL,upon adding and integrating the result obtained between -h and th we get (using 3.3~and 3.3d)
v3 Differentiating (C,), (C,) with respect to z, multiplying the result obtained by (1 -IL) and -p respectively and upon adding and integrating the result obtained between -h and th we get (using 3.3a and 3.3b)
For the stress problem (2.2b), using (AS) we get h
g,b
dz = 0
(C7)
as the necessary condition for convergence. For the mixed problem (2.2a), using (CS) we get
‘(u,+pm*) dz = 0
0)
as the necessary condition. For the mixed problem (2.2c),using (CS)and (C6) it follows that there is no condition upon the function w,. The function cr* satisfies (C7). For the displacement problem (2.2d), it can be argued that the eigenfunction expansion converges to ug t C,. The constant C, is determined using the condition (C8), we get
T”(2) = (1 - p)ZY”
-Z’.“‘.
Pergamoo Press.
Printed in Great Britain
AXIAL EIGENFUNCTIONS FOR THE AXISYMMETRIC PROBLEM OF AN ELASTIC CIRCULAR CYLINDER B. S. RAMACHANDRA RAO and A. K. KANDYA Department of Mathematics, Indian Institute of Technology, Powai Bombay-400076,India and S. GOPALACHARYULU Loughborough University of Technology, Leicestershire, U.K. Abstract-Constructing a biorthogonality relation it is shown how solutions can be obtained for the stress, displacement and mixed problems associated with the axisymmetric deformations in a cylinder whose plane faces are either stress-free or displacement-free. Numerical results are presented for the reproduction of the boundary stresses when a self-equilibrating load is prescribed on the circular boundary of the cylinder whose plane faces are stress-free.
INTRODUCTION THE PROBLEM of
determining the stress and displacement distributions in a solid cylinder undergoing axisymmetric deformations with the circular boundary stress-free and with stresses or displacements prescribed on the plane ends has been the subject of several interesting investigations. The Fourier analysis for obtaining solutions was first given by Pickett [l] and later Valov[2] has shown how Fourier analysis can be used to obtain solutions for the mixed and second fundamental boundary value problems. Eigenfunction expansions have been used by many authors and various approximate methods of satisfying the boundary conditions have been suggested by Lurje [3], Horvay and Mirabal[4], Mendelson and Roberts [5] and a few others. The method of satisfying the boundary conditions through suitable biorthogonality relations have been given by Little and Childs [6], Flugge and Kelkar[7], Nuller [8] and Fama[9]. Relatively few papers however report on the method of obtaining solutions of the axisymmetric problem by developing the eigenfunctions in the axial coordinate. Green[lO] deals with the problem pertaining to a cylinder whose plane faces are stress-free by developing the eigenfunctions in the axial coordinate. He has also shown how general solutions of the equations can be expressed either in terms of Fourier series or power series. The aim of the present work is to provide a biorthogonality relation for the eigenfunctions developed in the axial coordinate and to indicate the use of this biorthogonality for obtaining solutions of the problems associated with a cyclinder. The method of biorthogonal functions developed by Johnson and Little [l l] for the semi-infinite elastic strip problem has been used in the present case. It may be noted that this method has already been used to provide effective solutions of problems in the theory of elasticity[l2-151 and in the theory of flexure of thin plates[l6,17]. Numerical results for the convergence of the eigenfunction expansions have been illustrated for a self equilibrating load prescribed on the circular boundary of a cylinder whose plane faces are stress-free. 2. STATEMENT
AND FORMULATION
OF THE PROBLEM
We will consider the axisymmetric deformations in a cylinder occupying the region r s a, (zJd h with its plane faces z = rh either stress-free or displacement free, i.e. uz=r=O
on
z=kh,
r
(2. la)
u=w=O
on
z=ah,
r
(2.lb)
or
99
B. S. RAMACHANDRA RAO et al.
100
Throughout the paper a,, (re, a, represent the normal stresses, r the non-zero shear stress, u the radial displacement and w the displacement in the axial direction. The constants p and E represent the Poisson’s ratio and the Young’s modulus respectively. On the circular boundary r = a, we will prescribe any one of the following pairs of boundary functions
0;
=
ub
7 =
;
(2.2b)
Tb
(2.2c)
(2.2d) where the subscript b indicates a specified function on the boundary r = I(. The problems (2.2a) and (2.2~) will be referred to as the mixed problems, (2.2b) as the stress problem and (2.2d) as the displacement problem. The governing differential equation for the problems can be stated in terms of the Love strain function x as follows: rda,
AAx=&
lzlch
(2.3a)
where &+la+a’ dr’
r ar
(2.3b)
dz”
We also record the relations ~=+(pk$)x
(2.4a)
G=;
(2.4b)
(&$,)x
(2.4~)
I+,= za [(2--~)A-$]x
r=$ Eu -=-l+P -=Ew 1+FL We
(l-&A--$ E
I
x
8’~
(2.4e)
Wz
2(1-&A--$
(2.4d)
I
x.
(2.4f)
introduce the functions f(‘)(r, z) (i = 1, 2, 3) related to the derivatives of x as follows: (2Sa)
(2Sb)
101
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
(2Sc)
f(3) = 2.
It can be easily shown that (2.6a)
(2.6b)
Using (2Sa), (2.5b) the biharmonic eqn (2.3) can be expressed as
The above relation is identically satisfied by introducing a function f(*)(r, z) satisfying the relations -aft') =
a2
ar
a cf”’
_t f”‘)
(2.6~) $
cf”’
+
f (2))
=
_
+
$
(rf’“‘)*
Eliminating af"'/az between the above relation and (2.6a) we get aft*) _
TF-
_ia rar
of”+
$f”‘)
(2.6d)
(2.6a-2.66) expresses (2.3a) in terms of four partial differential equations of the first order. 3. BOUNDARY CONDITIONS
ON f”’ (i = 1,2,3,4)
Using the relations (2.4e), (2.4f) and (2.5) we get EU lfCL = -fc3)
= 2(1 - p)f(l)+ (1- zp)f(*),
j$
(3.la)
(3.lb)
Using (2Sa), (2.5b), (2.6a) and (2.6d) the relations (2.4~) and (2.46) may be written as a, =
f$
7 = ;
[rCf’“-
(1 - p)f(‘)}]
[(l - &f”’ - pf’2’],
(3.lc)
(3Sd)
Using (2.2d), (3.la), (3.lb) we get & = -fb”
w,,= 2(1 - ,.‘))f’b’+ (1 - 2,.&)f(bz). Using (2.4a) and (2.5) we get
(3.2a) (3.2b)
102
B. S. RAMACHANDRA RAO et al. (z,b =ffP’-$
[(l- /.L)fb”- /L~ufs’].
(3.2~)
Using (2.4d), (2.9, (2.6b) and (2.6~) we get [(l -/.&q?].
Tb = g
(3.2d)
Using the relations (3.2a-3.2d) the functions fb” may be expressed in terms of Indeed we get
Urb,
76,
ub
and
wb.
(3.3a) z
I
fF’= wb +2
(C&b+Ub/a)dZ
(3.3b)
fS’= -ub
(3.3c) (3.3d)
4. EIGEN VECTORS
FOR THE STRESS
FREE PLANE
FACES
We seek the eigenfunction expansions of the functions fci’ (i = 1, 2, 3, 4) in the form
(4.1)
Upon substituting for
fci’(i = 1, 2, 3, 4) in the relations (2.6a-2&i) we get Z(l)’ = * 2’3’ ” “”
(4.2a)
Z(Z)! = _* nn[Z(3) n Z(3)’= * n
+
pq
(4.2b)
n
(4.2~)
z’2’ nn
z(4), [Z’” + Z(2)] n = *nn “.
(4.2d)
It may be noted that throughout the paper, primes denote differentiation with respect to “2”. The relations (4.2) may be written in the matrix form
00 1
Z:,=A,U.% where the matrix U is given by
(4.3a)
10
01 0 0 11 0 0’ u= ! 0 0 -1 -1
(4.3b)
Using relations (3.lc), (3.ld) and (4.1) the boundary conditions on 2. may be written as z’,“-(l-~)zlf’=o
(4.3c)
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
103
and (1 -
p)Z(.I)- pZ'.z' = 0
on
2 = kh.
(4.3d)
Solutions of (4.3a) subject to the conditions (4.3~) and (4.3d) separate into even and odd eigen vectors. The components of .$!”may be listed in terms of a function F, and its derivative as follows: Z’.” = X;F, (4.4a) z’,” = h,F::
(4.4b)
z’,“= A2,F:,
(4.4c)
2’4’
=
_F”
n
_ ”
(4.4d)
h 2Fl nn
where F,, satisfies the differential equation F:+2A:F;+A:F.
=0,
IzIG
(4Sa)
and the boundary conditions pF::-(l-p)hf,Fn
(4Sb)
=0
and (l-p)F’::+(2-p)AfJ7h=O
on
z =+h.
(4.k)
The even solutions of F, are given by eFn = (2~ cos A,h - A,h sin A,h) cos A,z + A.2 cos A.h sin A.2
(4.6a)
where the A,‘s are the roots of sin 2hh - 2Ah = 0.
(4.6b)
oFn= (2~ sin h,h + A,h cos A,h) sin A,z - A,,.zsin A,h cos AJ
(4.6~)
The odd solutions of F, are given by
where the A,% are the roots of sin 2Ah + 2Ah = 0.
(4W
The roots of (4.6b) and (4.6d) to any desired degree of accuracy can easily be determined using their asymptotic forms as the first approximation to the roots and then applying Newton’s iterative technique. These roots are symmetrically located over the four quadrants of the complex plane so that if Ah is a root so are -Ah, A*h and -A *h. Further the eigen-functions corresponding to the eigenvalues Ah and -Ah are not distinct. The summation appearing in (4.1) is restricted over the pairs of eigenvalues lying in the right half of the complex plane. The roots of (4.6b) were 6rst investigated by Hilman and Salzer [18] and those of (4&l) by Robbins and Smith[l9]. It may be noted that A = 0 is a root of (4.6b) and (4.6d). The significance of the root is given in Appendix 1.
5. BIORTHOGONALITY
RELATION
The differential equation adjoint to (4.3a) is given by @:,=-AtUt.
w,
(5.1)
B. S. RAMACHANDRA RAO et al.
104
where * denotes the complex conjugate and t, the conjugate transpose. We have
I
+-h
(WA,‘?.% - @,,,tzb) dz = W, t .z
-h
+h. I -h
Using (4.3) and (5.1) the above relation may be written in the form (A, -A,) f+h W,,,. u. 2, & = Z’,” [W:‘+?
W:‘] + ,?$‘[(I - p)W:‘+
WE’].
(5.2)
-h
It therefore follows that, if W,,,‘s satisfy the following boundary conditions (5.3a) and WE’+(l-p)WE’=O
on z=+h,
(5.3b)
we have the biorthogonality relation
I
+h
-h
(m#
Cii,t.U..%dz=O
n).
(5.4)
Solutions of (5.1) subject to the conditions (5.3a) and (5.3b) separate into even and odd eigen vectors. The components of W? may be listed in terms of a function G,, and its derivatives as follows: (5.5a) wL2’ =
z2G X’
(5.5b)
W’,“= _A*G*“_A*‘G* nn nn
(5.5c)
A
At3Gz
W;‘=
(5.5d)
where G, satisfies the differential equation G:t2A:G:tA:G,
=O,
(z(Gh
(5.6a)
and the boundary conditions (l-/J)G::-pAZ.Gn
=0
(5.6b)
and @;+
(1 t /.~)h:Gh = 0.
(5.6~)
The even and odd solutions of G, are given by eGn = [2(1- p) cos A,h - A,h sin A,h] cos A,z t A,z cos A.h sin A,z
(5.7a)
oGn = [2(1- /.L)sin A.h + A,h cos A,h] sin A,z - A,z sin A.h cos A,z.
(5.7b)
The eigen values appearing in (5.7a) are the roots of (4.6b) and those appearing in (5.7b) are the roots of (4.6d).
105
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
The biorthogonality relation (5.4) may be written in the form
I
+h [z’,”
w$4’
+
Z’.“(
Wf”’
+
j.j.7~“))
+
Z(,3’(
,p
-
wp)
-
z~)W:(~)]
-h
dz = 0. (m # n) (5.8a)
Using (4.4), (4.9, (5.5) and (5.6) it can easily be verified that ,.+h
I [
z’,)wgw
+ w:“‘)]
z’,)(wf)
+
,.+h
dz = 1
[z’,‘y wf)
_ wf2)) - z’.” wf”‘]
dz
J-h
J-h
,
r+h
(A:FnG, - FZG:) dz.
= + j_
n
h
It therefore follows from (5.8a) that
I
+h [z(nl)wg(4)
+ z’,2’(wf)
+
w:“‘)]
dz = K.&n,
(5.8b)
-h
and
I
+h [z’,“(
w$”
_ ,zc2’)
- 2;’
W*,,,
*(2)] dz
= Kn&
(5.8~)
-h
where K. =+, _+h(h4,F.G. - FEGE)dz nI h
(5.8d)
and 8,. denotes the Kronecker delta. The relations (5.8b) and (5.8~) are referred to as the inner orthogonality relations. 6. EIGENFUNCTIONS
It follows from (4.1) that on the boundary we have (6.la) and (6.lb) Using (5.8b) the constants C. in the expansion (6.la) are given by C,
I”ha )
K,
I&La)
=
I_+’
vb” Wf4’ + ffZ)(Wf3) + W$*?] dz.
(6.2a)
Using (5.8~) the constants C,, in the expansions (6.lb) are given by C,Kn =
+hvf’( Wf” - Wf”) _ fb” Wf”] dz.
(6.2b)
Indeed (6.2b) gives a closed form solution to the mixed problem (2.2a). For, using (3.3~)and (3.3d) we get
B.S.RAMACHANDRA RAO etaf.
106
and the constants C. in the expansions (4.1) are given by C”K” = -(1‘I[(/‘Tb _
dz) Wt’“‘+{(l-~)W:“‘-
p)
W’.(“}Ub]dz.
(6.2~)
The solution of the boundary value problems (2.2b-2.2d) can be reduced to solving a system of linear algebraic equations in the infinitely many unknowns C,. For, upon adding (6.2a) and (6.2b) we get
c”K”(l+g$=j-:
up
wy)
+
fry
+ wy’)
,y’
+ ff’(wf”
- w$“)
- f(b4)WfZ’] dz
(6.3) and using (6.3) the solution of the problems (2.2b-2.2d) can be obtained as follows: For the stress problem (2.2b) we get (using (3.2~) and (3.2d))
and using (6.3) we get the following linear algebraic system of equations for the unknowns C,, viz. (6.4a)
C, = G, + x CmRm. where C,=-L,11_:*[W:“(~‘~~sdz)+W*‘2’(~iT~dz)]dz
R,, = L,’
~{W:‘“+(l-~)wf”)}Z~‘+
(6.4b)
(;j=Z:‘dz)
(2- ,.L) W:“‘}Z:‘] dz
+{(I-y)WX(‘)L” =(l-/L)K”[l+$j]. For the mixed problem (2.2~) we take (using (3.2b) and (3.2~)) j.;)=&,,
-+=$j=-
dz++=$$
C-l’Z$‘dz;
fF’ = c c,z?; fY)=w,+2
I
z n,bdz-;zC,
I
Wf’)
1 Z:’ dz ;
f6”’= x c,zF. For the displacement problem (2.2d) we take (using (3.2a) and (3.2b))
(6.4~)
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder
107
Usin’: (6.3) it can easily be shown that the problems (2.2~) and (2.2d) can be reduced to linear algebraic system of equations of the same type as (6.4a). The necessary conditions to be imposed on the boundary stresses and displacements in order to ensure the convergence of the stress, displacement and mixed problems are presented in Appendix 2. 7. EIGEN VECTORS FOR THE BUILT-IN
PLANE FACES
When the plane faces are displacement free (i.e. built-in) the boundary conditions u=w=O
on z=+h
are to be satisfied. Using (3.la) and (3.lb) it follows that the above boundary conditions are equivalent to z’.3’=0
(7.la)
and 2(l-P)Z(nl’+(l-2~)Z(n2)=0
on z=*h.
(7.lb)
Solutions of (4.3a) subject to the conditions (7.la) and (7.lb) can again be listed in terms of a function F. and its derivatives. The components of Z? (i = 1,2,3,4) are essentially the same as listed in (4.4a-4.4d). The functions F, are now given by eFn = (sin A,h + h.h cos h,h) cos A,z + A,z sin A,h sin A,z
(7.2a)
where the An’s are the roots of sin 2Ah + - ’
3-4p
2Ah=O
(7.2b)
and oFn = (cos A,h - A,h sin A,h) sin A.z - A,z cos A.h cos A,z
(7.2~)
where the A, ‘s are the roots of sin 2Ah - - ’ 2Ah=O. 3-4/k
(7.2d)
Both (7.2b) and (7.2d) have an infinite number of complex roots. (7.2d) has a real root also. These roots to any degree of accuracy can be determined using their asymptotic form as the tirst approximation to the roots and then followed by Newton’s iterative technique. It is extremely important to note that in the present case the stresses in the neighbourhood of r = a, z = kh become unbounded on account of the singularities which exist around the circumference of the plane ends. The attention to the existence of the stress singularities arising from the various boundary conditions in the angular comers of plates in extension was drawn by
108
B.S.RAMACHANDRA RAO eta!.
Williams[20]. The character of these singularities are of the type a=Real(cp-“)
as p-+0,
O
where cr is a stress along the plane end of the cylinder, p the distance from the circumference, a the exponent of the singularity and c the strength of the singularity. If these singularities are not taken into account then whatever be the method used for the analysis of the problem one has to solve an infinite system of linear algebraic equation and this will invariably pose bad convergence and will not lead to satisfactory results for the stresses in the neighbourhood of the circumference of the plane ends. Benthem[21] and Benthem and Minderhoud[22] have shown the method of introducing these singularities to the analysis of the problems. The introduction of these singularities make the eigenfunctions over complete and the resulting linear algebraic system of equations become ill-conditioned if one uses large number of eigenvalues to obtain the solutions of the problems. However any practical formulation of the problem must take into account of these singularities so that the stresses in the neighbourhood of these singularities are determined fairly accurately using the first few eigenvalues. It again follows from (5.2b) that in order to get the bi-orthogonality relation (5.4), the product ( wrn t . 2”) should vanish at z = +h. It can easily be verified that in order to satisfy this condition the vectors W,,, must satisfy the following boundary conditions: w’,“=o
(7.3a)
and (l-2p)W!!‘-2(1-p)W(,z)=O
on
z=?h.
The biorthogonal vectors are determined on solving (5.1) subject to the above boundary conditions. The components of W,, are again listed out in terms of a function G. and its derivatives. They are essentially the same as listed in (S.Sa-5.5d). The functions G, are given by eGn = sin A.h cos h.z - h,z cos A,h sin h,z
(7.4a)
oGn = cos A.h sin A,z - A,z sin A,h cos A,z.
(7.4b)
The eigenvalues appearing in (7.4a) are the roots of (7.2b) and those appearing in (7.4b) are the roots of (7.2d). The analysis for determining the solutions of the problems (2.2a-2.2d) follow on the same lines as indicated in Section 6.
8.NUMERICAL RESULTS
For numerical results we shall consider the case of a cylinder whose plane faces z = +h are stress free and upon whose circular edge arbitrary, otherwise self equilibrating load is prescribed. Consider the load ub = (z/h)*- l/3 Q, = sin (nzlh). The ratio of the diameter to the height of the cylinder is set equal to 5, i.e. a/h = 5. The Poisson’s ratio p is taken as 0.3. The infinite system (6.4a) is solved using five eigenvalues. It may be remembered that even or odd expressions are taken for the vector G,, and the matrix R,. depending upon whether the deformation is symmetric or antisymmetric with respect to the plane z = 0. The remarkable accuracy in reproducing the boundary stresses through the eigen function series is apparent from the results presented in Table 1. The results are also illustrated graphically in Figs. 1 and 2.
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder Table 1. Convergence of eigenfunction expanisons
r/h 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0
Specified function UIJ
Reproduced gO,using five eigenvalues
-0.333 -0.323 -0.293 -0,243 -0.173 -0.083 0.027 0.157 0.307 0.477 0.667
-0.312 -0.302 -0.284 -0.241 -0.146 -0.053 0.025 0.165 0.347 0.483 0.777
Specified function 7.5
Reproduced rb using five eigenvalues
0.000 0.309 0,588 0.809 0,951 1,000 0.951 0.809 0.588 0.309 O@OO
0.000 0.316 0.581 0,805 0.960 0.995 0.941 0.820 0.578 0,298 O@Xl
I
0
cl1
0.2
0.3
0.4
Fig. 1. Reproduction of the normal stress a, -,
0.5
0.c Z/h
0.7
08
0.9
1.0
-
specified function a,; ---, reproduced (r, for 5 pairs of eigenvalues.
1.0 -
0.5 Z/h
Fig. 2. Reproduction of the shear stress 7. -,
0.6 -
0.7
ae
specified function 7; ---, eigenvalues.
0.9
1.0
reproduced r for 5 pairs of
109
110
B. S. RAMACHANDRA RAO et al. CONCLUSIONS
The paper presents a method for obtaining solutions for the stress, displacement and mixed problems associated with the axisymmetric deformations in a cylinder whose plane faces are stress free. While a closed solution is available for the mixed problem (2.2a), it is shown that all other problems can be reduced to the solution of an infinite system of linear algebraic equations. The authors are not aware of any literature wherein the problems connected with a cylinder have been solved by using a biorthogonality for the eigenfunctions developed in the axial coordinate. To this end it is believed that the analysis presented here can easily be used to obtain solutions of the problems connected with the axisymmetric deformations in a cylinder. REFERENCES [l] G. PICKETT, J. Appl. Mech. 11, 176 (1944). [2] G. M. VALOV, P.M.M. 26, 975 (1%2). [3] A. J. LURJE, Raumliche probleme der Elnstizitatstheorie. Akademic-Verlag (1963). [4] G. HORVAY and J. A. MIRABEL, J. App. Mech. 25, 561 (1958). [5] A. MENDELSON and E. ROBERTS, 8th Midwestern Mechanics Conj. p. 40 (1%3). [6] R. W. LITTLE and S. B. CHILDS, Q. Appl. Math. 25, 261 (1967). [7] W. FLUGGE and V. S. KELKAR, Int. .I. Solids Structures 4, 397 (1%8). [8] B. M. NULLER, P.M.M. 33, 364 (1969). [9] M. E. D. FAMA, Q. J. Maths. Appl. Mech. 25, 479 (1972). [lo] A. E. GREEN, Proc. Roy. Sot. Land. 195, 533 (1949). [II] M. W. JOHNSON and R. W. LITTLE, Q. Appl. Maths. 12, 335 (1964). [12] R. W. LITTLE, J. Appl. Mech. 36, 320 (1969). [13] J. L. KLEMM and R. W. LITTLE, SIAM J. Appl. Maths. 23, 209 (1970). [14] B. S. RAMACHANDRA RAO, C. S. KALE and R. P. SHIMPI, Int. Engng Sci. 11, 531 (1973). [IS] P. V. B. A. S. SARMA,B. S. RAMACHANDRARAO and S. GOPALACHARYULU, fnt. J. Engng Sci. 13,149(1975). [16] P. V. B. A. S. SARMA, B. S. RAMACHANDRARAO and S. GOPALACHARYULU, SIAM. .I.Appl. Maths. 26(3),568 (1974). [17] K. V. RAJAN and B. S. RAMACHANDRA RAO, Int. J. Engng Sci. 12, 1079(1974). [18] A. P. HILMAN and H. E. SALZER, Phil. Msg. Land. 34, 575 (1943). [19] C. S. ROBBINS and R. C. T. SMITH, Phil. Msg. Lond. 39, 1004(1948). [20] M. L. WILLIAMS, .I. Appl. Mech. 19, 526 (1952). [20] J. P. BENTHEM, Q. J. Mech. Appl. Maths. 16, 413 (1963). [22] J. P. BENTHEM and P. MINDERHOUD, Int. J. Solids Structure 8, 1027(1972). [23] M. I. GUSEIN-ZADE. P.M.M. 29(2), 393 (1965). (Received 3 June 1974)
APPENDIX 1 It can easily be verified that A = 0 is a six fold root of the eqn (4.6d)and an eightfold root of the eqn (4.6b).Accordingly we seek the eigenfunctions of the function f”’ in the form
(Al) Upon substituting in the relations (2.6a-2.6d) we get Z”” = 2Z’“’ Z’2”= _2(Z(3l+ Z(4)) Z”” = 2p z’4” = 2(a + 0).
(A2)
Solution for f”’ satisfying the boundary conditions (4.3~)and (4.3d) splits into two parts. Solutions where the function x is
symmetric in z are given by f”’ = 2(1- k)z't pr* f::: = -2(2 - p)z’t (1 - F)r* ;,,, 1 ;(I - Y)Z 2
(A3)
The solutions for which x is antisymmetric in z are given by f”’ = 2(1- p)z f”’ = -2(2 - p)z f”’ = (1 - p)
f”‘= 1.
Using (3.2~)and (3.2d) we get for the solutions (A3)
(A4)
Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder fJ,b= -2(1 t /L)z;
111
7b=o
and for the solutions (A4) rl, =o.
orb =-(1+11);
It therefore follows that the even solution of x corresponding to the eigen value A = 0 will give rise to inplane moment and the corresponding odd solution to inplane tension. If the applied load on the boundary is self equilibrating, i.e. ~_~~~~dz=I~~z~,~dz=I_*:7cdz=O
(AS)
then the inplane moment and the inplane tension vanish and hence the eigenvalue A = 0 is excluded from the summation. APPENDIX 2 The conditions necessary to ensure the convergence of the stress, displacement and mixed problems can be obtained analogous to the method used by Gusein_Zade[23].These conditions corresponding to the cases for which the function x is symmetric or antisymmetric in the coordinate z can be obtained as follows. Symmetric case
Using (4.1), (4.4) and (A3) the expansion of the boundary function fb” may be written in the form I&a 1
fb”=C,[2(l-~)z’+~a*]+~
C.I(~)A:F.
I I
~~‘=C0[2(2-p)z’+(l-p)a2]+~C~~A.F: jr’ = C0[2(l- p)az] + 2 C”A2.F:
(B3)
fr’= Co[2az]+~
(B4)
C,(-F’L-Af,i%).
In order to ensure the convergence of the various boundary value problems discussed in Section 6, it is necessary to set the constant C, equal to zero. The constant C, may be evaluated in terms of the boundary functions fb” using the conditions (4Sb) and (4.5~). Multiplying (B4) by pz, (B3) by z, upon adding and integrating the result obtained between the limits -h and h we get (using 3.3~ and 3.3d) (2% +2zu,) dz. Multiplying (Bl) by 1 - p, (B2) by -p, upon adding and integrating the result obtained between the limits -h and th we get (using 3.3a and 3.3b) Co=$j-;z(u,a
+;)dz.
For the stress problem (2.2b), using (AS) we get (B7) as the necessary conditions for convergence. For the mixed problem (2.2a), using (B5) we get
I
h(pz2+2zUb)dz =0 0
as the necessary condition. For the mixed problem (2.2c),using (BS)and (B6) it follows that there is no condition upon the function w,,.The function c,~ satisfies the first of the self-equilibrating conditions in (B7). For the displacement problem (2.2d),it again follows that while there is no condition upon the function w,,,the function us has to satisfy the condition (B8). It may be observed that the eigenfunction expansion X a.2:’ converges to R’- C, - C,z where C, and C, are arbitrary constants putting u.(z)-Zr) and using (3.3~)it follows that
z
aa. (z) =b(z) + C, + Czz.
This may be confirmed by examining the closed form solution (6.2~).The expression on the right hand side of (6.2~)remains unchanged if us is replaced to ub + C, + C2z. For the symmetric case we take
The constant C, is determined using the condition (BS). We get
112
B. S. RAMACHANDRA RAO et al. pz’ 2 a,r.(t) + 2zub dr I
7”
=(l-p)z:“-zy”.
Anrisymmetric case Using (4.1) and (A4) the expansions of the boundary functions fb” may be written in the form fb” = C0[2(1- c()I] t c C”y
L n
Z(.l’
(Cl)
fP’ = q-2(2 - /L)z]+ c C”# z’.” I n fP’ = &[a(1 -IL)]+ 2 C.Z!?
(C3)
ff’ = &[a] t x C.Z!,?
(C4)
Multiplying(C3) by I, (C4) by CL,upon adding and integrating the result obtained between -h and th we get (using 3.3~and 3.3d)
v3 Differentiating (C,), (C,) with respect to z, multiplying the result obtained by (1 -IL) and -p respectively and upon adding and integrating the result obtained between -h and th we get (using 3.3a and 3.3b)
For the stress problem (2.2b), using (AS) we get h
g,b
dz = 0
(C7)
as the necessary condition for convergence. For the mixed problem (2.2a), using (CS) we get
‘(u,+pm*) dz = 0
0)
as the necessary condition. For the mixed problem (2.2c),using (CS)and (C6) it follows that there is no condition upon the function w,. The function cr* satisfies (C7). For the displacement problem (2.2d), it can be argued that the eigenfunction expansion converges to ug t C,. The constant C, is determined using the condition (C8), we get
T”(2) = (1 - p)ZY”
-Z’.“‘.