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Circular elastic inclusions in cylindrical shells
CIRCULAR ELASTIC INCLUSIONS IN CYLINDRICAL SHELLS D. H. Department
of Aeronautical
BoNDEt and K. P. RAO$
Engineering, Indian Institute of Science. Bangalore 560012, India
(ReceivedI September1977; receivedfor pub~~caIio~ 5 huory
1978)
Abstract-The problem of a circular elastic inclusion in a cylindrical shell subjected to internal pressure or thermal loading is studied. The two shallow-shell equations governing the behaviour of a cylindrical shell are transformed into a single differential equation involving a curvature parameter and a complex potential function in a non-dimensional form. In the shell region, the solution is represented by Hankel functions of first kind, whereas in the inclusion region it is represented by Bessel functions of first kind. Boundary conditions at the shell-inclusion junction are expressed in a simple form involving in-plane strains and change in curvature. The effect of such inclusion parameters as extensional rigidity, bending rigidity, and thermal expansion coefficients on the stress concentrations has been determined. The results are presented in non-dimensional form for ready use.
INTRODUCTION Elastic
inclusions
are
introduced
in a cylindrical shell structure for accessibility and visibility, Such structures are widely used in such fields as aerospace, nuclear, chemical and marine engineering. The discontinuities thus introduced give rise to stress concentrations. Understanding of these stresses is necessary for an efficient and reliable design. Internal pressure and thermal loadings are two important types of loadings. The problem of a circular hole in a cylindrical shell was discussed by Lurie [ 11.His solution is valid for small curvature parameter /3[/3*= values of the (ff2~S~f)[I& I - Y2)p2, where a is the hole radius, R and t are shell radius and thickness, and Y is Poisson’s ratio]. Later Withum[2], Lekkerkerker[31 and Eringen ef al. 141 employed numerical procedures to extend this analysis for fi values up to 2 for torsion, tension, and internal pressure loadings. Van Dyke[5] extended the analysis up to J? = 4. Rao[7] used similar approach to solve the problem of edge reinforced cutouts in cylindrical shells. Van Dyke[6] inclusion in
also solved
the problem
of a rigid circular
problem describing the effect of discontinuity. Using the total solution, the stresses are obtained and plotted in non-dimensional form for ready use. The tem~ratures involved are assumed not to be high enough to cause any changes in material properties. 2.GOVERNING Mm L&l.EQUATION AND ITS SOLUTION Figure 1 shows a cylindrical shell with a circular inclusion. The equations governing the behaviour of the region of the shell influenced by the presence of the discontinuity are assumed to be the shallow, thin shell equations. The stress resultants and displacements in the shell tangential and normal to the middle surface are shown in Fig. 2. The linear shallow shell equations for cylindrical shell geometry involving only the membrane stress function F and normal displacements W’ are151 V4W’+F,../RD=pID
(1)
and
a cylindrical shell for static Ioading. He splits the stresses and displacements into periodic components, and specifies the boundary conditions on inplane displacements u and u by integrating the strain displacement relations. This method is tedious and involved. Murthy[8] showed that the conditions used by Van Dyke are equivalent to specifying the boundary conditions on normal displacements, circumferential strain, and change in in-plane curvature. He used these conditions successfully to obtain the stresses around an elliptical inclusion in a cylindrical shell. The present work is concerned with the problem of a circular elastic inclusion in a cylindrical shell. Two types of loadings are considered; (a) internal pressure and (b) thermal loading represented by a linearly varying temperature across the thickness. The problem is solved using thin shallow shell theory together with simplified boundary conditions. The problem is split into a uniform state of stress problem for uncut shell and a residual
vF’-
tEW,x.JR = 0
(2)
where
v4w = w’,,,, + 2w’,XX,,+ w’,,,,,. -. -
Shell middle
7 = r scn 8
tProject Assistant. *Assistant Professor.
Fig. I. Cylindrical 349
shell configuration.
D. H. BONDEand
STRESS COUPLES
K. P. RAO
where P and @ are the solutions corresponding to the shell without discontinuity and F* and W* are perturbation components which are caused by the presence of the discontinuity and which vanish at great distances from it. The residual problem then reduces to solving the homogeneous differential equation V44* + 8i/32cl),*ee =0
(8)
where d* = W* - iF* vanishes far away from the discontinuity. With the polar coordinates, 4 = r cos B and n = r sin 8, the solution of eqn (8) for the shell region is m
dt = (E, - iEz).O,& (A, + iB,)H:[/3r(2i)“21 cos no STRESS RESULTANTS AND DISPLACEMENTS
2
+ (5 - i&)
1.3.5.
(A, + i~“)~~[~~(2i)“2] cos n# (9)
where (El - i.&) and (Es - iEd) are Krylov functions[S] and for the inclusion region, is m Fig. 2. Stress Couples-resultants and displacements, The primes indicate dimensional quantities. Non-dimensional parameters are now introduced. the non-dimensional coordinates being defined as 6 = x/a, q = y/a. The membrane stresses are non-dimensionalised with respect to a reference membrane stress N,e,, where
Nre,= PR
for internal pressure loading: for uniform temperature distribution:
= E,t,a,T,,
=6a,D~[ltv,l
Tz- Tt \ t2
+
( E3 - iE,)
for linearly varying temperature across thickness.
With
where A,. B,, C. and D,, are constants which are to be determined to satisfy the boundary conditions. Once the complex stress functions are known, stresses and displacements can easily be computed everywhere in the shell or inclusion. The elastic inclusion is bonded to the shell in such a manner that its rotations and displacements match those of shells at the junction. Thus, for an eccentric inclusion, the following boundary conditions will be applicable[Fig. 31: NLr, = N:,
F
Et2 and W= ~~2N~,~,2~,-~2)~1/~
(10)
3. BOUNDARY CONDITIONS
(3)
F=-
2 fC, + iD, V, [Br(2i)“21cos ntl
IA’.
(4)
; NX,=N:,
Q:, = QL,.+ e,N&,e
; M:,, = ML,, +e,N:,<;
; WL = W:, : Wb = WX,;
e;, = 4:‘ : s&i,= JI&
(11) (12) (13)
and the complex potential function 4= W-iF
(5)
eqns (t and 2) can be combined to yield[5]
where e, is the eccentricity (see Fig. 3). For a symmetric inclusion, these boundary conditions are readily modified by putting e, = 0. Au quantities in eqn (13) can be directly expressed in terms of normal displacements W: and W: and Airy’s stress functions FL and F:. 4. EVALUATION OF CONSTANTS
where 2
@2=&[12(1-
v2 )I l/2
(7)
The right hand side of this equation is or is not zero according to whether the shell is or is not loaded by uniform pressure. The stresses and displacement functions can be expressed as F=t’+F*
and
W=I?i+W*
The boundary conditions are satisfied by a collocation procedure at discrete points on the boundary in the quadrant B = 0 to r/2. The points chosen are equidistant. At each point, there are eight boundary conditions to be satisfied, which give eight equations for the unknowns. The series is terminated at (2m - I) terms where m is the number of points, thus giving 8m equations for 8m unknowns. For a given /3 value, successively higher numbers of points are chosen and series terminated at higher numbers of terms accordingly, until values of stresses remain essentially the same. For a symmetric
351
Circular elastic inclusions in cylindrical shells e, = ECCENTRICITY
Fig. 3. Forces at shell-inclusion junction. inclusion for small & values. As pV approaches zero, the shell stresses obtained by the present analysis are found to tend to those in a plate. Figure 4 shows the variation of maximum principal stress at B = 0, with extensional rigidity ratio CL<,and bending rigidity ratio CL,,for q/(2)r3S = 2.8. With increasing rigidity of the shell, the stresses in the shell are seen to be increasing. The bending rigidity, however, does not affect the stresses much, particularly when the exten-
at & = 3, the change in the value of maximum principal stress for m = 9 and m = 10 was less than I%. All numerical calculations have been performed with V, = vC = l/3 on IBM 370115.5 computer. (see Fig. 10).
inclusion
5. RESULTSAND DWUWON Internal
pressure
loading
Table I shows the comparison of maximum principal stresses for a flat plate and a shell with a circular elastic
Table I. Stresses around a circular elastic inclusion in a cylindrical shell
E.t,IE,t, = 2.0,DJD,= 1.5. EC,= 0 9 In a plate+ 8< =0 In a cylindrical shell (Present analysis) & = 0.35 tobtained
0
IO
20
30
40
50
60
70
80
90
1.375
1.352
1.285
I.182
I.061
0.933
0.827
0.767
0.749
0.747
1.402
1.373
1.293
I.177
1.049
0.932
0.841
0.780
0.746
0.735
using[9].
Fig. 4. Circular elastic inclusion in a pressurised cylindrical shell, variation of maximum principal stress on outer surface of shell.
352
D. H. BONDEand K. P. Rqo
sional rigidity ratio is around 1. With decreasing rigidity ratio, the stresses tend to the rigid inclusion value of 0.1788. Figure 5 shows the effect of hLL. and CL,,on the maximum principal stresses in the inclusion for t/‘(2) & = 2.8 and B = 0. Here the stresses decrease with increasing pC,i.e. with decreasing rigidity of inclusion the
stresses decrease. The bending rigidity again has negligible effect. particularly when pI is around 1. The effect of eccentricity is shown in Fig. 6. The maximum principal stress at the outer surface decreases slightly for small positive eccentricities. At the inner surface, however, it is lowest for slightly negative ec-
SCALE /-----!%=I-4 i--“--fib
=1--i
Nref = PR
Fig. 5. Circular elastic inclusion in a Dressurised cylindrical shell. variation of maximum arincipal stress at outer surf&e of inclusion.
Nref =pR
EiX =%
r
Inner sufoce
\ \
e -1.0
-0.5
/
-
0
_
0.5
Ecc Fig. 6. Eccentriccircular etastic inclusion in a pressurised ~ylindri~~ with eccentricity.
A
’
Q~+er
surface
I.0 shell, kriation
of maximum principal stress
Circular elastic inclusions
p
IR
cylindrical
shells
353
05-
bi
+-Pa I--Kq.05-f
Fig. 7. Uniformly
heated cylindrical
= 1.0-l
shell with a circular elastic inclusion, variation of principal stress on outer surface of she&
SCALE
Nref
= 6QsO,(l+~s)~~
h-T
a5
Fig. 8. Cylindrical
shell with a circular elastic inclusion subjected to a linearly varying temperature across thickness (/.&I,= 1.5).
K,.20
f’e’05 SCALE
A
Fig. 9. Cylindrical
shell with a circular elastic inclusion subjected to a linearly varying temperature across thickness (& = 0.5).
D. H. BONDE and K. P. RAO
354
N = 2M-1
J. GENERATE
BESSELS
FUNCTIONS
-
GENERATE HANKELS FUNCTIONS H,1(&r~/2i)
J,(B,rfi) I
-i
CHANGE TO NEXT POINT/ INCREASE 0
*
CALCULATE B FOR THE COLLOCATION POINT AND CALCULATE KRYLOV FUNCTIONS (El - iE,) and (E3- iE.,)
4 n=l n VARIES FROM
FOR BOUNDARY
‘I
I
EVALUATE DERIVATIVES OF c: and rp; FORMULATE MULTIPLIERS OF THE UNKNOWNS A,, B,, C, and D,
SOLVE FOR UNKNOWNS
I
I
f~ FIND MAXIMUM PRINCIPAL STRESS AT r = 1
FIND PERCENTAGE CHANGE IN STRESSES
_
t
1 + YES *
PERCENTAGE CHANGES1
STORE A,. B,, C,, D, J CALCULATE STRESSES DIFFERENT POINTS
AT
4 PRINT REQUIRED
STRESSES
+ STOP
Fig.
10. Flow
chart
NO *
STORE STRESSES
Circular elastic inclusions in cylindrical shells
centricity. The crossover of maximum principal stress from outer to inner surface at about zero eccentricity suggests that symmetric inclusion is the best choice.
355
The effect of eccentricity has also been studied and it has been found that symmetric inclusion is the best choice. in the case of pressure loading, extensional rigidity ratio Jo*plays a predominant role and the influence of bending rigidity ratio &c is only marginal.
Thermaf Loading
The thermal loading is assumed to be given by uniform temperatures on the inner and outer surfaces of the shell and the inclusion, but it may vary linearly across the thickness. Such a loading can be treated as a superposition of (i) a uniform temperature and (ii) a linearly varying temperature with the middle surface experiencing no change in temperature. (i) Uniform temperature distribution. The variation of the maximum principal stress with p@and K,, the ratio of expansion coefficient of the inclusion (I= and the expansion coefficient of the shell arr is seen in Fig. 7. There are no stresses in the shell or inclusion when K__= 1.0, i.e. when the expansion coefficients are the same. For any other value of Km the stresses are higher. The variation with K, is linear. The principal stresses increase with increasing extensional rigidity ratio for K, < 1.0 and decrease with increasing pp for K, > 1.0. (ii) Linearly varying temperature. Figure 8 shows the variation of the maximum principal stress at the outer surface with fiC and Km. The minimum principal stress occurs in this case when Icz = pc ph. It is found that when such a relation is satisfied, the discontinuity moments at the junction vanish. Thus, minimum stress concentration, i.e. 1.0 is obtained. The bending rigidity however has a predominant effect in this case as can be seen by comparing Figs. 8 and 9. CONCLUSION
The effect of a circular elastic inclusion in a cylindrical shell on stress concentrations around the interface has been studied. Internal pressure and thermal loadings are considered. The results are found to tend to the plate solution as the curvature p~ameter becomes very small.
CAS Vol
a. NOS3 B 4-n
Acknowled~emenfs-The authors wish to thank Dr. M. V. V. Murthy. Structures Division, National Aeronautical Laboratory. Bangalore. India and Prof. A. V. Krishna Murty. Department of Aeronautical Engineering, Indian institute of Science. Bangalore. India. for their many helpful discussions. Thanks are also due to the Aeronautical 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.
St&s oi thin wallede/astir shells. (Transl. Atomic Energy Commission. AEC-TR-3798 (1959). D. Withum. The cylindrical shell with a circular hole under torsion. Ingr-Arch.26. 435-446(1958). 1. G. Lekkerkerker. Stress concentrations around circular holes in cylindrical shells. Proc. I lrh inf. Gong. of Appi. hfech. ( 1964). 4. A. C. Eringen, A. K. Naghdi. S. S. Mahmood, C. C. Thiel and T. Ariman, Stress concentrations in two normally intersecting cylindrical shells subject to internal pressure. Welding ResearchCouncilBulletinNo. 139. t-34 tl%9). P. Van Dyke. Stresses about a circular hole in a cylindrical shell. AIAA J. 8 (91. 1733-1745(1965). P. Van Dyke. Stresses in a cylindrical shell with rigid inclusion. AlAA J. 5. (It. 12S-t37(I%?). K. P. Rao and G. A. 0. Davies, Reinforced circular holes in cylindrical shells. Apron.f. RO.VU/ Aeron. Sot. 74. (153-155) (Feb. 1970). 8. M. V. V Murthy. On the stress problem of elliptical holes and inclusions and straight line cracks in cylindrical shells. Ph.D. Thesis. Indian institute of Science, Bangalore. (1975). 9. S. Timoshenko and J. N. Goodier. Theory of Elasticity. McG~w-Hill. New York (1951). A. 1. Lurie,
of Aeronautical
BoNDEt and K. P. RAO$
Engineering, Indian Institute of Science. Bangalore 560012, India
(ReceivedI September1977; receivedfor pub~~caIio~ 5 huory
1978)
Abstract-The problem of a circular elastic inclusion in a cylindrical shell subjected to internal pressure or thermal loading is studied. The two shallow-shell equations governing the behaviour of a cylindrical shell are transformed into a single differential equation involving a curvature parameter and a complex potential function in a non-dimensional form. In the shell region, the solution is represented by Hankel functions of first kind, whereas in the inclusion region it is represented by Bessel functions of first kind. Boundary conditions at the shell-inclusion junction are expressed in a simple form involving in-plane strains and change in curvature. The effect of such inclusion parameters as extensional rigidity, bending rigidity, and thermal expansion coefficients on the stress concentrations has been determined. The results are presented in non-dimensional form for ready use.
INTRODUCTION Elastic
inclusions
are
introduced
in a cylindrical shell structure for accessibility and visibility, Such structures are widely used in such fields as aerospace, nuclear, chemical and marine engineering. The discontinuities thus introduced give rise to stress concentrations. Understanding of these stresses is necessary for an efficient and reliable design. Internal pressure and thermal loadings are two important types of loadings. The problem of a circular hole in a cylindrical shell was discussed by Lurie [ 11.His solution is valid for small curvature parameter /3[/3*= values of the (ff2~S~f)[I& I - Y2)p2, where a is the hole radius, R and t are shell radius and thickness, and Y is Poisson’s ratio]. Later Withum[2], Lekkerkerker[31 and Eringen ef al. 141 employed numerical procedures to extend this analysis for fi values up to 2 for torsion, tension, and internal pressure loadings. Van Dyke[5] extended the analysis up to J? = 4. Rao[7] used similar approach to solve the problem of edge reinforced cutouts in cylindrical shells. Van Dyke[6] inclusion in
also solved
the problem
of a rigid circular
problem describing the effect of discontinuity. Using the total solution, the stresses are obtained and plotted in non-dimensional form for ready use. The tem~ratures involved are assumed not to be high enough to cause any changes in material properties. 2.GOVERNING Mm L&l.EQUATION AND ITS SOLUTION Figure 1 shows a cylindrical shell with a circular inclusion. The equations governing the behaviour of the region of the shell influenced by the presence of the discontinuity are assumed to be the shallow, thin shell equations. The stress resultants and displacements in the shell tangential and normal to the middle surface are shown in Fig. 2. The linear shallow shell equations for cylindrical shell geometry involving only the membrane stress function F and normal displacements W’ are151 V4W’+F,../RD=pID
(1)
and
a cylindrical shell for static Ioading. He splits the stresses and displacements into periodic components, and specifies the boundary conditions on inplane displacements u and u by integrating the strain displacement relations. This method is tedious and involved. Murthy[8] showed that the conditions used by Van Dyke are equivalent to specifying the boundary conditions on normal displacements, circumferential strain, and change in in-plane curvature. He used these conditions successfully to obtain the stresses around an elliptical inclusion in a cylindrical shell. The present work is concerned with the problem of a circular elastic inclusion in a cylindrical shell. Two types of loadings are considered; (a) internal pressure and (b) thermal loading represented by a linearly varying temperature across the thickness. The problem is solved using thin shallow shell theory together with simplified boundary conditions. The problem is split into a uniform state of stress problem for uncut shell and a residual
vF’-
tEW,x.JR = 0
(2)
where
v4w = w’,,,, + 2w’,XX,,+ w’,,,,,. -. -
Shell middle
7 = r scn 8
tProject Assistant. *Assistant Professor.
Fig. I. Cylindrical 349
shell configuration.
D. H. BONDEand
STRESS COUPLES
K. P. RAO
where P and @ are the solutions corresponding to the shell without discontinuity and F* and W* are perturbation components which are caused by the presence of the discontinuity and which vanish at great distances from it. The residual problem then reduces to solving the homogeneous differential equation V44* + 8i/32cl),*ee =0
(8)
where d* = W* - iF* vanishes far away from the discontinuity. With the polar coordinates, 4 = r cos B and n = r sin 8, the solution of eqn (8) for the shell region is m
dt = (E, - iEz).O,& (A, + iB,)H:[/3r(2i)“21 cos no STRESS RESULTANTS AND DISPLACEMENTS
2
+ (5 - i&)
1.3.5.
(A, + i~“)~~[~~(2i)“2] cos n# (9)
where (El - i.&) and (Es - iEd) are Krylov functions[S] and for the inclusion region, is m Fig. 2. Stress Couples-resultants and displacements, The primes indicate dimensional quantities. Non-dimensional parameters are now introduced. the non-dimensional coordinates being defined as 6 = x/a, q = y/a. The membrane stresses are non-dimensionalised with respect to a reference membrane stress N,e,, where
Nre,= PR
for internal pressure loading: for uniform temperature distribution:
= E,t,a,T,,
=6a,D~[ltv,l
Tz- Tt \ t2
+
( E3 - iE,)
for linearly varying temperature across thickness.
With
where A,. B,, C. and D,, are constants which are to be determined to satisfy the boundary conditions. Once the complex stress functions are known, stresses and displacements can easily be computed everywhere in the shell or inclusion. The elastic inclusion is bonded to the shell in such a manner that its rotations and displacements match those of shells at the junction. Thus, for an eccentric inclusion, the following boundary conditions will be applicable[Fig. 31: NLr, = N:,
F
Et2 and W= ~~2N~,~,2~,-~2)~1/~
(10)
3. BOUNDARY CONDITIONS
(3)
F=-
2 fC, + iD, V, [Br(2i)“21cos ntl
IA’.
(4)
; NX,=N:,
Q:, = QL,.+ e,N&,e
; M:,, = ML,, +e,N:,<;
; WL = W:, : Wb = WX,;
e;, = 4:‘ : s&i,= JI&
(11) (12) (13)
and the complex potential function 4= W-iF
(5)
eqns (t and 2) can be combined to yield[5]
where e, is the eccentricity (see Fig. 3). For a symmetric inclusion, these boundary conditions are readily modified by putting e, = 0. Au quantities in eqn (13) can be directly expressed in terms of normal displacements W: and W: and Airy’s stress functions FL and F:. 4. EVALUATION OF CONSTANTS
where 2
@2=&[12(1-
v2 )I l/2
(7)
The right hand side of this equation is or is not zero according to whether the shell is or is not loaded by uniform pressure. The stresses and displacement functions can be expressed as F=t’+F*
and
W=I?i+W*
The boundary conditions are satisfied by a collocation procedure at discrete points on the boundary in the quadrant B = 0 to r/2. The points chosen are equidistant. At each point, there are eight boundary conditions to be satisfied, which give eight equations for the unknowns. The series is terminated at (2m - I) terms where m is the number of points, thus giving 8m equations for 8m unknowns. For a given /3 value, successively higher numbers of points are chosen and series terminated at higher numbers of terms accordingly, until values of stresses remain essentially the same. For a symmetric
351
Circular elastic inclusions in cylindrical shells e, = ECCENTRICITY
Fig. 3. Forces at shell-inclusion junction. inclusion for small & values. As pV approaches zero, the shell stresses obtained by the present analysis are found to tend to those in a plate. Figure 4 shows the variation of maximum principal stress at B = 0, with extensional rigidity ratio CL<,and bending rigidity ratio CL,,for q/(2)r3S = 2.8. With increasing rigidity of the shell, the stresses in the shell are seen to be increasing. The bending rigidity, however, does not affect the stresses much, particularly when the exten-
at & = 3, the change in the value of maximum principal stress for m = 9 and m = 10 was less than I%. All numerical calculations have been performed with V, = vC = l/3 on IBM 370115.5 computer. (see Fig. 10).
inclusion
5. RESULTSAND DWUWON Internal
pressure
loading
Table I shows the comparison of maximum principal stresses for a flat plate and a shell with a circular elastic
Table I. Stresses around a circular elastic inclusion in a cylindrical shell
E.t,IE,t, = 2.0,DJD,= 1.5. EC,= 0 9 In a plate+ 8< =0 In a cylindrical shell (Present analysis) & = 0.35 tobtained
0
IO
20
30
40
50
60
70
80
90
1.375
1.352
1.285
I.182
I.061
0.933
0.827
0.767
0.749
0.747
1.402
1.373
1.293
I.177
1.049
0.932
0.841
0.780
0.746
0.735
using[9].
Fig. 4. Circular elastic inclusion in a pressurised cylindrical shell, variation of maximum principal stress on outer surface of shell.
352
D. H. BONDEand K. P. Rqo
sional rigidity ratio is around 1. With decreasing rigidity ratio, the stresses tend to the rigid inclusion value of 0.1788. Figure 5 shows the effect of hLL. and CL,,on the maximum principal stresses in the inclusion for t/‘(2) & = 2.8 and B = 0. Here the stresses decrease with increasing pC,i.e. with decreasing rigidity of inclusion the
stresses decrease. The bending rigidity again has negligible effect. particularly when pI is around 1. The effect of eccentricity is shown in Fig. 6. The maximum principal stress at the outer surface decreases slightly for small positive eccentricities. At the inner surface, however, it is lowest for slightly negative ec-
SCALE /-----!%=I-4 i--“--fib
=1--i
Nref = PR
Fig. 5. Circular elastic inclusion in a Dressurised cylindrical shell. variation of maximum arincipal stress at outer surf&e of inclusion.
Nref =pR
EiX =%
r
Inner sufoce
\ \
e -1.0
-0.5
/
-
0
_
0.5
Ecc Fig. 6. Eccentriccircular etastic inclusion in a pressurised ~ylindri~~ with eccentricity.
A
’
Q~+er
surface
I.0 shell, kriation
of maximum principal stress
Circular elastic inclusions
p
IR
cylindrical
shells
353
05-
bi
+-Pa I--Kq.05-f
Fig. 7. Uniformly
heated cylindrical
= 1.0-l
shell with a circular elastic inclusion, variation of principal stress on outer surface of she&
SCALE
Nref
= 6QsO,(l+~s)~~
h-T
a5
Fig. 8. Cylindrical
shell with a circular elastic inclusion subjected to a linearly varying temperature across thickness (/.&I,= 1.5).
K,.20
f’e’05 SCALE
A
Fig. 9. Cylindrical
shell with a circular elastic inclusion subjected to a linearly varying temperature across thickness (& = 0.5).
D. H. BONDE and K. P. RAO
354
N = 2M-1
J. GENERATE
BESSELS
FUNCTIONS
-
GENERATE HANKELS FUNCTIONS H,1(&r~/2i)
J,(B,rfi) I
-i
CHANGE TO NEXT POINT/ INCREASE 0
*
CALCULATE B FOR THE COLLOCATION POINT AND CALCULATE KRYLOV FUNCTIONS (El - iE,) and (E3- iE.,)
4 n=l n VARIES FROM
FOR BOUNDARY
‘I
I
EVALUATE DERIVATIVES OF c: and rp; FORMULATE MULTIPLIERS OF THE UNKNOWNS A,, B,, C, and D,
SOLVE FOR UNKNOWNS
I
I
f~ FIND MAXIMUM PRINCIPAL STRESS AT r = 1
FIND PERCENTAGE CHANGE IN STRESSES
_
t
1 + YES *
PERCENTAGE CHANGES1
STORE A,. B,, C,, D, J CALCULATE STRESSES DIFFERENT POINTS
AT
4 PRINT REQUIRED
STRESSES
+ STOP
Fig.
10. Flow
chart
NO *
STORE STRESSES
Circular elastic inclusions in cylindrical shells
centricity. The crossover of maximum principal stress from outer to inner surface at about zero eccentricity suggests that symmetric inclusion is the best choice.
355
The effect of eccentricity has also been studied and it has been found that symmetric inclusion is the best choice. in the case of pressure loading, extensional rigidity ratio Jo*plays a predominant role and the influence of bending rigidity ratio &c is only marginal.
Thermaf Loading
The thermal loading is assumed to be given by uniform temperatures on the inner and outer surfaces of the shell and the inclusion, but it may vary linearly across the thickness. Such a loading can be treated as a superposition of (i) a uniform temperature and (ii) a linearly varying temperature with the middle surface experiencing no change in temperature. (i) Uniform temperature distribution. The variation of the maximum principal stress with p@and K,, the ratio of expansion coefficient of the inclusion (I= and the expansion coefficient of the shell arr is seen in Fig. 7. There are no stresses in the shell or inclusion when K__= 1.0, i.e. when the expansion coefficients are the same. For any other value of Km the stresses are higher. The variation with K, is linear. The principal stresses increase with increasing extensional rigidity ratio for K, < 1.0 and decrease with increasing pp for K, > 1.0. (ii) Linearly varying temperature. Figure 8 shows the variation of the maximum principal stress at the outer surface with fiC and Km. The minimum principal stress occurs in this case when Icz = pc ph. It is found that when such a relation is satisfied, the discontinuity moments at the junction vanish. Thus, minimum stress concentration, i.e. 1.0 is obtained. The bending rigidity however has a predominant effect in this case as can be seen by comparing Figs. 8 and 9. CONCLUSION
The effect of a circular elastic inclusion in a cylindrical shell on stress concentrations around the interface has been studied. Internal pressure and thermal loadings are considered. The results are found to tend to the plate solution as the curvature p~ameter becomes very small.
CAS Vol
a. NOS3 B 4-n
Acknowled~emenfs-The authors wish to thank Dr. M. V. V. Murthy. Structures Division, National Aeronautical Laboratory. Bangalore. India and Prof. A. V. Krishna Murty. Department of Aeronautical Engineering, Indian institute of Science. Bangalore. India. for their many helpful discussions. Thanks are also due to the Aeronautical 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.
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