STRESSES AROUND TWO EQUAL CIRCULAR ELASTIC INCLUSIONS IN A PRESSURISED CYLINDRICAL SHELL D. H. BONDE? and K. P. RAOS Department of Aeronautics, Indian Institute of Science, Bangalore 560012,India (Received 5 October 1978;received for publication 31 January 1979) Abstract-The stress problem of two equal circular elastic inclusions in a pressurised cylindrical shell has been solved by using single inclusion solutions together with Graf’s addition theorem. The effect of the inter-inclusion distance on the interface stresses in the shell as well as in the inclusion is studied. The results obtained for small values of curvature parameter fi @*=(a*/8Rt) [12(1-v*)]“*,a, R, t being inclusion radius and shell radius and thickness) when compared with the flat-plate results show good agreement. The results obtained in non-dimensional form are presented graphically.
A., Bn,AZ,B!: CmDm,CX,DX
XI, YI,nr X2,
H”’ h B :
t sutxx s suffix c S 0’ V*
NOMENCLATURE w: - iF:
to determine the stresses around two equal reinforced circular holes in a plate under biaxial stresses. Yu and wf-iF! Sendeckyj  have given the mathematical formulation unknown constants in the series of Q? for multiple circular inclusions in plane elastostatics. unknown constants in the series of Q! They have found some results for two equal circular coordinate system corresponding to inclusion 1 elastic inclusions in a plate. coordinate system corresponding to inclusion 2 The problem of multiple circular discontinuities in Hankel function of first kind and order n shells has not been discussed much in literature. The Bessel function of first kind and order II curvature parameter, fl* = (a2/8Rt) [12(1- v*)]“* problem of two equal circular holes located along the generator has been solved by Seide and HaGz, for radius of the inclusion axial tension case. The solution has been obtained by radius of the shell thickness superposition of individual solutions corresponding to indicates quantities referring to the shell each hole considered independently. Hanzawa et al. [7,8] indicates quantities referring to the inclusion also solved the problem of two circular holes as well as half the distance between centers of two intwo rigid circular inclusions located on a generator or in elusions the circumferential direction in a cylindrical shell. They 02. v2 used Green’s general biharmonic analysis and Graf’s a2 a2 addition theorem to get a new potential function describZQ ing the stress and displacement state in the shell. strain in tangential direction In this paper we present an analytical solution of the in-plane change in curvature normal displacement, normal to shell mid-surface problem of two equal circular elastic inclusions in a cylindrical shell, lying along a generator of the cylincrical Airy stress function . . . . Es& shell. The results presented here are for a pressurised extensional [email protected]
ratlo = E,I, cylindrical shell. However, the method can be applied for bending rigidity ratio = D./D, eccentric inclusions as well as thermal loading problems Young’s modulus by changing the boundary conditions for the particular Et’ cases following the lines discussed by authors in Ref. [ 11. bending rigidity = l2(* _ v2) In the present work Green’s general biharmonic analysis is used as in  and the boundary conditions are satisfied by least square collocation procedure.
1.INTRODUCTION is in continuation of the work done by
authors on circular elastic inclusions in cylindrical shells [ 11,where the problem of one elastic inclusion in a pressurised cylindrical shell was considered. The stress concentrations around an inclusion changes if another discontinuity exists in its vicinity. Several investigations have been carried out on the two discontinuities in plates. Lii using bipolar coordinates solved the problem of two circular holes in a plate. Sampath and Hulbert have analysed multiple holes in orthotropic laminated plates. Ling’s method was used by Dhir[41 tResearch student. SAssistant Professor. CW Vd. II. No. 4-A
2. THECOORDINATE SYSTEM general coordinates are assumed to be lying on the shell mid-surface with the origin at the mid-point “0” of the two inclusions (Fig. 1). For each inclusion a separate coordinate system is defined as shown in Fig. 1. The boundary conditions are satisfied at the shell-inclusion interface at either of the inclusions. While satisfying the boundary conditions symmetry with respect to both the axes through mid-point “0” of the two inclusions is made use of. The
3. G0-G MFFwENTuLEQUATION ANDSOLU’l’ION The governing differential equation of the residual problem, solution of which together with uncut shell
D. A. BONDE
F, = (E, - iE& =
2 ew~6x2+ e-_(l--im2
F2 = (El - i&)2 =
G2 = (ES - iE& = CVLINMIICAL SMELL WITH DlSCONflNUlflES~
V. = H,‘(~(2i)/3,rl) cos nfh
ZS = 2 HL+, (Q(2i)& 2S)J1 (d(2i#%rd cos &. ,=-m
For the inch&on region the solution having SYmme~Y about the generator is given as
Fig. 1. Coordinate system for two discontinuities along logitudinal axis. solution gives the total stresses in the shell or the inclusion is given by [l]
V4’p*t 8i/3’tp :e = 0
where~=rcos~,~=rsine~d 2 p2=&12(1-Y
SlNGLF..V.UUEDNESS OF THFi
= ( - #(A:, + Azl+, - &+I) = 0. c=ci
Solutionfor the shell region having two inclusions can be written as
Qf= QS,+ Qf,
where (~3,and cp&are given by
The boundary conditions for a single eccentric inclusion in a pressurised cylindricai shell are given in Ill. We write here the boundary conditions for the case of a symmetric elastic inclusion.
Q:, = a,,
N:es = N&c, M:, = M:+, w:, = w:,,
r& = c&c, = $& cos n&
t (E3 - iE& 2 (At + iB~~~“‘~~(2~)~r2)cosn&. n-0
Using Graf’s addition theorem  P~=“~~(A.+iB.)(FrVn+~~i + 2 (AZ + iBZ)(Gr V. + GL) n=O
This condition has to be satisfied by the constants Al, A. and B, in addition to the boundary conditions at the shell-inclusion interface.
N& = Xc,,,
c~$ = (Et - iE& x (A, + iB,)H.‘(~(2i)gr3 PI-Cl
Since the shell in this case is a multiply connected region, it is necessary to satisfy the condition that the displacements u’ and u’ are single-valued. These conditions have been ~nvesti~ted in detail by LekkerkerkerflO]. Using a similar procedure it can be shown that stress function cp? satisfies single-valuedness condition for u’ and the condition for single-vahredness of u’ is given by
where 4 is the tangential strain and 4; is the inplanechange in curvature. These conditions can be expressed in terms of qf and rpr. Satisfaction of boundary conditions (eqn 9) and eqn (8) gives the constants A,,, B,, AZ, Bf, C,,, A, CX and DX. Thus & and QZ are known, which together with uncut shell solution gives the stresses in the shell and the inclusion. 6. mBmRtc.u PnoCEnunE The Setief of rpt and Q% (eqns 5 and 7) are terminated at (n - 1)th term giving 8n unknowns in the solutions. The boundary conditions (eqn 9) are satisfied at m(m b n) points in the arc 0 < 0 < 180”along the interface. The
Stresses around two equal circular elastic inclusions in a pressurised cylindrical shell points
are chosen equidistant. Thus 8m equations are obtained. In addition we have to satisfy eqn (8). Thus 8m t 1 equations are solved for 8n unknowns by leastsquare method. m and n are successively increased till the variation in maximum principal stress remained less than 2% between successive approximations. For v(2)& = 1.4, m = 9 and n = 7 were sufficient. Computer programme follows a flow-chart similar to that given in Ul. The series for 2. (eqn 6) represents the quantity H.‘(q(2i)/3r2) cos n&. The convergence of this series was tested by comparing with the values of If.‘(v(2i)/3rz) cos n&, for some points. It was found that good convergence was possible when I varied from - 15 to t 15.
Figure 2 shows the effect of inter-inclusion distance on principal stresses around the interface in the shell, for typical values of shell and inclusion parameters. It is seen that at 6 = 0, particularly for small values of S, the effect is most predominant. As one moves away from this point along the interface the effect reduces. The maximum stress increases with decreasing S, i.e. as the inclusions move closer. It can be seen that the interaction is negligible for S > 2.0. Thus the inclusions can be treated as equivalent to single inclusion if S > 2.0. The effect of distance on the tangential membrane stress is shown in Fig. 3. Here also the effect is mostly confined to the region around 0 = 0” and is negligible for 8 > 40”. The effect on radial membrane stress was found to be very marginal. The tangential bending stresses are presented in Fig. 4. The effect of distance “S” in more pronounced in this case and is felt over a larger zoue. At 8 = 0 the tangential bending stresses become more compressive with decreasing “S”. Figure 5 shows the radial bending stress variation along the boundary. Figures 2-5 show that the effect of decreasing S is more pronounced on bending stresses than on membrane stresses. Figure 6 shows the variation of principal stress with 0 and S in the inclusion. The effect is seen to be pronounced in the region 0~ 0 SW. As in the shell region the effect of S is negligible for S > 2.0. Figures 7 and 8 show the variation of tangential membrane stress and tangential bending stress respectively along the in-
7. RESUL.‘TSAND DlSCUSSlON
Table 1 shows the comparison of principal stresses for the case of a flat-plate with a single inclusion, a cylincrical shell with a single inclusion and a cylindrical shell with two inclusions for a small value of &. It is seen that the present results are in good agreement with those of a flat plate and shell with a single inclusion. Using the plots given in, it is possible to obtain the stresses NM corresponding to the case of two elastic inclusions in an infinite plane-sheet. The results thus obtained (Table 2) when compared with the present solution for d(2)rR = 0.1 show good agreement.
Table I. Comparison with a flat plate containing a single elastic inclusion
Flat Plate (Single elastic inclusion)
s N ref
p< = 2.0 70
Cylindrical shell (Single elastic inclusion) V/(2)&
Cylindrical shell (Two elastic inclusions)
V/(2)/3,= 0.1, s=5.0
4 N rcf
Table 2. Comparison with a flat plate containing two elastic inclusions 0
NJN,r Flat plate with two elastic inclusions [S] s = 2.0, /& = 3.0
NJNrrl Cylindrical shell with two elastic inclusions d\/(2)8‘ = 0. I S = 2.0, fi. = 3.0
D. H. BONDEand K. P. RAO
s1.4 , p, z2.0 , pb .lS,r.lX)
Fig. 2. Variation of maximum principal stress in the shell along interface with centre distance S.
3 z_l 14
J5 (r, El.4 ,
0/Jr = 2.0 ,
/Jb .I.5 s r =’
Fig. 3. Variation of tangential membrane stress in the shell with respect to 0, for different S.
Stresses around two equal circular elastic inclusions in a pressurised cylindrical shell
20 40“LB6Q m8m--‘P xl.4
Fig. 4. Variation of tangential bending stress in the shell with centre distance S’. .l-
.l , I
N,,JPR .2 -
f.4,:l.S . rz1.0
Fig. 5. Variation of radial bending stress in the shell with centre distance S.
D. H. BONNand K. P. RAO
_ scale : b-8.201( I.-5.14
Fig. 6. Variationof principalstress in inclusionwith 0 and S (Top surface).
I!33 ‘ckg I
. /lo s2.0
1.5 , r .l
Variationof tangentialmembranestress in inclusion.
Stresses aroundtwo equal circular elastic inclusions in a pressurised cylindrical shell
Fig. 8. Variation of tangential bending stress
terface in the inclusion.As in case of shell the effect on tangential membrane stress is limited to B<40” and is maximumat e = 00. In the inclusionalso, it is seen that the variation of “5””leads to larger variationsin bending stresses as compared to membranestresses. The stress problem of two equal circular elastic inclusions in a pressurised cylindrical shell has been solved. The effect of the ~ter-inclusion distance on the stresses in the shell as well as inclusionhas been studied. It is observed that the interaction between the two inclusions is more pronounced at B = 0” than any other angle. The rn~~urn shell stress increases as the inclusions come closer. The effect of inter-inclusiondistance on bending stresses is more pronounced than on membranestresses. The two ~ciusions can be treated as equivalent to two single inclusions when the distance between their centres is more than two times the diameterof the inclusion.All the computationshave been carried out on IBM 36OH4System at Indian Institute of Science, Bangalore.The results are nondimensionalized and presented graphically.
REFJmENcEs I. D. H. Bonde and K. P. Kao, Circularelastic inclusions in cylindricalshells. Comput.Structures8,34%355 (1978).
2. 0. B. Lii, Gn the stresses fn a plate containingtwo holes. I. AupfiedPhvsics 19.77-82 (19481 3. S.*b. Sampath ani L. E. ‘Hull&r&Analysis of multiholed o~o~pic taminated plates by the ~~-~~t-l~stsquare method. f. Pressure Vessel Technology 97(2), 1M-112 (May 1975). 4. S. K. Dhir, Stresses around two equal reinforced openisgs in a *-pIate. Int. J. solids Sff7fcfuns 4,109~1106 (1968). 5. I. W. Yu, and G. P. Sendeckyj, Multiple circular inclusion problem in plane elastostatics. 1. Appf. Me&. 41, 215-221 (1974). 6. P. Seide and A. S. Ha&, Stress concentrationsin a stressed cylindricaIshell with equal circular holes. J. Appf. Mech. 105-109(Mar. 1975). 7. H. Hanzawa, M. Kishkla, M. Murai and K. Takashina, Stresses in cylindrical shell having two circular c&x& Buff. SME 151851.781-795 11972). . 8. H. Hanzawa, M. Kishida ani M_‘Murai Stresses in a circular cyfi&i& shell having two circular r&d inclusions. ~~~~ of JSME 19(113),731-738(July 1976). 9. Handbook of Mathanuticaf Functions (Edited by M. Abramowitsand I. A. Stegun)AMS 55. National Bureauof Standards, U.S. Department of Commerce, When (Mar. 1965). 10. J. 0. Lekkerkerker,The determinationof elastic stresses near cylbrder-to-cylinderintersections. Nuclear Engfneerfng II
11. P. Van Dyke, Stress about a circular hole in a cylindrical shell. AJAA J. 3(f), 1733-1742(Sept. 1965).