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Stresses in a perforated cylindrical shell
ht.
J. Solids
S?rucmres.
1966, Vol.
2, pp. 141 to
156.Pergamon
Press
Ltd. Printed
STRESSES IN A PERFORATED
in Great
Britain
CYLINDRICAL
SHELL*
F. BAUER and E. L. REM Courant Institute of Mathematical Sciences, New York University
AI&met--It is assumed that the deformations of a thin cylindrical elastic sheli, subjected to a uniform lateral pressure, are adequately described by the Donnell shell equations. Tbe surface of the shell is perforated by a rectangular cutout. The tangential membrane d~pla~rnen~ and the normal membrane stresses vanish on the simply supported edges of the hole and the ends of the cylinder. Numerical solutions are obtained by replacing the boundary value problem by a finite. difference approximation and solving the resulting system of linear algebraic equations by an iteration procedure.
1. INTRODUCTION
THE presence of a hole in the wall of a cylindrical elastic shell that is subjected to surface and edge forces, significantly alters the distribution and magnitudes of the stresses. It also complicates the mathematical analysis of the shell problem. Approximate methods of solution, which are restricted in their applicability, have been previously proposed? P-61. In this paper, a numerical method is presented to determine the stresses and displacements in the shell for a specific cutout problem. However, with suitable modifications, it can be extended to include a variety of cutout shapes and other boundary and loading conditions. The procedure we employ is the well-known finite difference method. In this way, the boundary value problem of shell theory is approximated by a system of linear algebraic equations. In principle, there is usually no difficulty in the application of the method. However, for cutout problems, in which the solutions vary rapidly, many mesh points may be required. Roundoff errors and excessive computing time make it impractical to use the “standard” methods (e.g. Gauss elimination) to solve the resulting large system of algebraic equations. We present and successfully apply an iterative method involving two “acceleration” parameters to solve the algebraic equations. The procedure is described in Section 3 and the computational methods required to apply it are outlined in Section 4. For the specific problem investigated the perforation is rectangular, i.e. the edges of -the cutout coincide with generators and parallel circles of the cylinder’s midsurface. The shell is loaded by a uniform pressure applied normal to the surface. On the simply supported boundaries the tangential membrane displacements and the normal membrane stresses vanish. It is assumed that the deformations are adequately described by the Donnell thin shell theory [7,8]. * This report represents resulta obtained at the Courant Institute of Mathematical Sciences, New York University, with the OfIice of Naval Research, Contract No. Nonr-285(42) and tbe United States Atomic Energy Commission, Contract AT(30-l)-1480. It is a shghtly abridged version of the report IMM-~ 320, February 1964. t Reference f4] contains a general discussion of shell cutout problems and references to previous work. 141
142
F. BAUERand E. L. R~lss
In Section 5 the results of the computation are discussed and graphs of stresses and displacements are shown. They indicate that the stresses are large near the corner of the rectangular cutout. These high stresses may have slowed the convergence of the iterative procedure. We therefore conjecture that for problems with smoother cutouts the method may be more efficient. 2. FORMULATION The cylindrical shell is of mean radius R, thickness t and length coordinate system r, 0, z is employed where r and 0 are plane polar is the axial distance measured from one end of the cylinder. Thus the the ranges R-t/2 I I I R + t/2, --7c I 8 I z, -L 5 z I L. The ordinates x and y are defined by the relations x = z/L,
1x12 1,
y = (R/L)&
lyl I (nR/L) = 1.
2L. A cylindrical coordinates and z coordinates are in dimensionless co-
(1)
The wall of the shell is perforated by a rectangular hole whose edges are on the lines x = +a and y = &/?l where 0 I tl, /I < 1. The shell is deformed by a uniform pressure p applied normal to the surface. The ends of the shell and the edges of the hole are simply supported and the normal membrane stresses and the tangential membrane displacements vanish there. The solution is therefore assumed to be symmetric with respect to the planes x = 0 and y = 0. Thus we consider the region D, see Fig. 1, which is one quarter of the cylindrical surface. Symmetry conditions must be specified along the following three boundary lines of D: u I x I 1, y = 0; 0 5 x I 1, y = 1; x = 0, /I1 5 y I 1. The Donnell theory for cylindrical shells [7,8] can be formulated as a boundary value problem for two coupled fourth order partial differential equations in the two dependent variables W(z, 0) and F(z, 0). Here, W is the normal displacement of the midsurface I = R of the cylinder and EF is the membrane stress function where E is Young’s modulus. The positive directions of p and W are outward from the center line of the cylinder. In dimensionless variables the differential equations of the DOMel theory are
L\2W+3= 1 ax2
’
A'f =
T$
FIG. 1. The domain of integration, D, which is one quarter of the cylindrical surface.
Stresses in a perforated cylindrical shell
143
where A2 is the biharmonic operator, A2 z $+*A+<.
ax ay
WI
ay
On the boundary of D we specify that
azw w=p=f=pl,
forx = 1,
w2!&fx&o,
for x = a,
w+f=$zO,
for 0 5 x 5 a,
aw a% af a3f o ay=dy3=ay=ay3=,
(34
for y = 0,
CrIXll,
and y = 1,
OlX
aw a3w af a"f o jy=-j-g=z=ax"=,
for x = 0,
/?I I y I 1.
The dimensionless variables in equations (2) and (3) are defined by
F(z,6’)
w(x, y) = CW(z, e),
CE
f (x, y) = [ 12(1- v2)l*c 7
c 3
E
12(1- v2)pL 0-z
’
z s (12(14)l’$.
where v is Poisson’s ratio. The membrane stresses,c$‘, cr$’and.& and the corresponding dimensionless stresses X,0, C,Oand CoxY are given in terms of the dimensionless stress function f(x, y) by
1,” (x, y)
=
Ao,O(z, S) =
2$+
yy (x, y)
E
c,” (x, y)
Aa,O(z,e) =
=
-9
Ao,O(z, e) =
w, (5)
where A-’ z [12(1-vZ)]+p(p(L/t)2. The quantities, cr:, ai and, o&, , which are proportional to the bending moments, and the dimensionless bending stresses m,, my and mxYare given in terms of w(x, y) by,
F. BAUER and E. L. REM
144
m,(x,Y) = Wk
0) =
a24x,Y) + va2w(x? Y) ay2
ax2
a2w(x, Y)
my(x, y) = Ba;(z, e) = --7+
v
ay
’
a2W(X, Y) ax2
(6)
2
a2w(x, Y)
mxy(x,y) = (1 - v)Bo:&, 0) =
axay ’
where 2
0
BE-L-4
12pL
.
If the displacements of the midsurface of the shell in the z and 6’ directions are U(z, f3) and V(z, 19)respectively, then the dimensionless displacements u(x, y) and u(x, y) are defined by
4% Y) =
0
R W, 0)
z
7
The membrane stress displacement relations of the Donnell theory thus reduce to [7,8],
c; = c:=+Jg+v(g+w)].
*~+w+vg], (7)
( 1 !?!+U.
g=LL 2(i+v)
ay
ax
The relations (5), (6) and (7) and the conditions (3) imply that the required boundary and symmetry conditions* are satisfied. The complete formulation of the boundary value problem consists of the differential equations (2) and the boundary conditions (3). The solutions depend only on the parameters ~1,fi, 1 and r. We obtain approximate solutions for fixed ~1,B, and I and a sequence of r values.
3. NUMERICAL
METHODS
To apply the approximate method the region D is covered with a rectilinear mesh. The mesh is of uniform spacing 6 in the x and y directions. The mesh is chosen so that the boundary lines of D coincide with mesh lines. There are M + 1 mesh lines in the x direction and N + 1 in the y direction so that, 6 = l/M = l/N.
(8)
The mesh points (Xi,yj) are defined by, xi = is,
yj = jS,
i = 0, 1,. . , M;
* The relations actually imply that the membrane However, these constants are set equal to zero.
displacements
j = 0, 1, . . N, tangential
to each boundary
(9) are constants.
Stresses in a perforated cylindrical shell
145
and the integers i and J are defined by, XI = CL,
YJ = Pi*
The mesh region D, is defined as the set of points (Xi, Yj) interior to and on the boundary of D. The mesh region DQ is obtained from D, by excluding the mesh points : (xy , Yj), J‘-0-
,.‘.,
N;(x,,yj),j=O,...,J;(xi,y,),i=0,...,1.
At each point of Db the solutions w(xi, Yj),f(xi, Yj) of equations (2) and (3) are assumed to be approximated by Wij,fii which satisfy the difference equations, A~Wij+ ~L,~j = 1,
Ad”f;i- zL,Wij =L:0,
(xi~Yj) E ob
Here the difference operator Ai, obtained by replacing derivatives operator (2b) by centered difference quotients, is given by
in the differential
Ad’= L; + 2L,L, +‘L;. L, and L,, are the “one-dimensional” centered second difference operators any mesh function* (gijj, by +
(W
U 14
defined, for
1 .j-2gi,jfgi-1,j)(6)-29
Lxgij
=
ki
Lygij
=
(gi,j+ 1 -2gi,j+gi,j-1)(d)-2e
(1lb) The approximating quantities (Wij> and (f;i> must satisfy difference equivalents of the boundary conditions (3) where all derivatives are replaced by centered difference approximations. Fictitious mesh points exterior to D, are introduced to apply the difference equations. Thus a system of 2JM(N+ l)-(I + l)(J+ l)] linear algebraic equations in the 2[M(N+ I)-(f+l)(J+l)] unknowns (Wij> and (&jjl is obtained. We assume that as 6 -+ 0 the solution of the algebraic system converges to the solution of (2) and (3). Hence for a sufficiently small S the solution of the algebraic problem should yield a “close” approximation to the solution of the boundary value problem. For large M and N, it is impractical to solve the algebraic system by a direct method, e.g. Gaussian elimination. Therefore, an iteration procedure is used to obtain approximate solutions of this sytem. With w$) as an initial guess, for 6xed values of a, j3, I and z, a sequence of iterates [wj;),ffjn)] are defined by the recursions
A;f'i,?' = & xw!?' 1J'
A2$~+" _ -rL,f$) 6 11
,!?+r) y -- ~~~~+l)+(l-~)w~) ” *.I
f 1,
I
n=O,l,..., txi i Yj) E Ete*
(12)
Each iterate must also satisfy the difference equivalents of equation (3). The acceleration parameter L should be determined so that the iterations (12) converge as rapidly as possible. Therefore, for each n the solutions of two algebraic systems of the form A;g$+l)
= 4:;’
txi, Yj) E Di
(13)
are required where the mesh function {#) is either {~L,w$‘f or ( - zL,fg'+l> depending upon whether (g$‘> is @J’> or ( w(?+l)) . A line relaxation procedure, which is a simplification of the alternating due&ion method [9, lo] is used to obtain a crude approximation to the solution of (13). A single horizontal and a single vertical sweep of the mesh is * A function fgij) defined only at points of the mesh is called a mesh function.
F. BAIJEX and E. L. REM
146
performed instead of alternating back and forth between horizontal and vertical sweeps.* Thus the approximate solution (gg’ ‘)} of equation (13) is defined as the solution of the one dimensional difference equations [lo] : (1 +pQ$+”
= g{;++’ +pL$g{$‘,
2 (n+9 (l + PL.xkij
= g!?’ 8~- 2&&$’
- @g$’
(Xi,Yj)EDk9
+ pq$’ I
(14)
where (g$+*)} and (g $+i)} must also satisfy the appropriate difference boundary conditions. The matrices corresponding to the one dimensional difference operators in equation (14) are easily inverted by factoring. The number p is a second acceleration parameter and should be determined so that the iterations (12) converge as rapidly as possible. 4. COMPUTATIONAL Application of the numerical procedure An acceptable value of 6 is determined by calculations with successively Iiner meshes. mesh yields sufficiently accurate solutions numerical criterion for convergence is that z(n) z
(
PROCEDURES
requires that the mesh spacing be “small”. comparing the solutions of a sequence of test From these tests we conclude that the 6 = & for the range of parameters considered. The the iterations satisfy the condition,
*ma~DbflWp-w~-l)/,Ify-_fpI}
< 10-k
X,
(15)
The value of k that is employed depends on the magnitudes of the solutions. For all our calculations we used k = 8 or 9. The condition, lim z(“) = 0 is only necessary for convergence. It is found, starting from a given initial iterate, that the number of iterations required to satisfy equation (15) depends on the values of p and t. For fixed a, /?, I and r, the optimal values Lo and p0 are defined as those values which minimize the number of iterations necessary to satisfy equation (15). To obtain estimates for them a series of test calculations are employed. Thus for a fixed configuration, i.e. fixed a, /I and 1, and usually employing a coarse mesh estimates of & and p. are determined for an increasing sequence of r values. These values are used for the finer meshes and 1, and p. for other values of z are estimated by interpolating or extrapolating. In Table 1 representative values are given of the acceleration parameters that were employed. TABLE 1. ACCELERATIONPARAMETERS EMPLOYEDIN THE COMPUTATION
Shell I 1
Shell II 1
P
P
340 680 1020 1360
1.0 0.615 0.30 0.20
24.6 9.7 9.7 9.7
1.0 0.55 0.25
10.0 10.0 9.6 -
1700
0.12
9.1
0.10
9.6
* A limited number of numerical experiments indicate that the convergence does not change if several sweeps are made through the mesh.
rate of equation
(12) essentially
141
Stresses in a perforated cylindrical shell
Numerical solutions are obtained for two configurations, which we call Shells I and II, and a sequence of values of r = 340,680, 1020, 1360, 1700. Shells I and II are defined respectively by, a = 10/24, /.I = 10122, 1 = 22124 and a = 7124, j? = 7122, 1 = 22124. The initial iterate for a given z is taken as the “converged” solution for the nearest z for the same shell. When the iterations converge the dimensionless stresses are computed employing obvious finite difference equivalents of equations (5) and (6). It is observed that the number of iterations necessary for convergence depends on the values of the parameters a, j?, I and r. For example, with a fixed configuration the number increases rapidly as z increases, and with fixed I and r the number increases as a and /_Idecrease. In all cases a relatively large number of iterations is needed to satisfy equation (15). For example using Shell I and r = 340, 492 iterations are needed, and for r = 1020, 2481 iterations are required. However, the computing time per iteration is relatively small, e.g. the above computations required approximately 13 and 65 minutes* respectively. 5. PRESENTATION
OF RESULTS
Representative graphs of the dimensionless stresses and normal displacement obtained from the numerical solution for Shell I with r = 1700 are given in Figs. 2-8. In these 4
qw,v,xd 0
I =
-4
woo
-at FIG. 2 The variation with the axial distance x of the dimensionless axial membrane stress Zz for Shell I for a sequence of y values. The hole occurs at y = 2. * All computations employed v = 0.3 and were performed on the IBM-7090 computer at the A.E.C. Cornputing and Applied Mathematics Center of the Courant Institute of Mathematical Sciences, New York University.
148
-I FIG. 3. The variation
I
with x of the dimensionless hoop membrane stress $’ for Shell I for a sequence y values. The dotted curve is the stress if no hole is present. 20
IO
Z’,y(xrlxlo~
0
----Q-I PI700
-10 FIG. 4. The variation
with x of Cz,, for Shell I for a sequence
of y values.
of
149
Stresses in a perforated cylindrical shell
figures the variations with the axial distance are shown for a sequence of y values. These results indicate that near the corner of the cutout, x = a, y = fil, the numerical stresses are quite large. The hole apparently affects the stress distribution everywhere in the shell. The hoop membrane stress Cy and the axial bending stress YX are influenced more by the hole than the other stresses. If u = /I = 0, the stresses and displacements obtained from the well-known solution of the boundary value problem are independent of y. These quantities are indicated with a super-bar and are given by, S(x) = $ -
cos jx cash fix + tan /I tanh j3 sin /3x sinh /Ix rz cos /I cash p(1 + tan’/3 tanh2P) ’
(16)
8
qxa)xlo4 4
0
I I -4 I--
-I lFIG. 5. The variation with x of the dimensionless axial beading stress Zi l&SheUfer a sequence of y values. For the shell with no cutout, ZX coincides, within the resolution of the graph, with the x-axis for the range 0 I x c 0.833. For 0.833 5 x I 1, Z: is represented by the dotted curve.
150
F. BAUJZRand E. L. REISS
where r2 =
484.
These results are shown by the dotted curves in Figs. 3, 5 and 6. In Fig. 8, the lateral displacement i+(x) coincides, within the resolution of the graph, with the numerical results for the line y = 0. These figures indicate that the cutout and its comer may increase the magnitudes of the stresses and displacements by large factors. Although their magnitudes differ, the form of the stresses and displacements for other values of r and for Shell II are similar to those shown in Figs. 2-8. A simple asymptotic solution is obtained for large r by dividing the differential equations by r and letting r-+co while w(x, y; co) and f(x, y; co) remain finite. Integrating the resulting equations and applying the boundary conditions we obtain w(x, y; co) = j-(x, y; co) = 0. To illustrate this result, graphs are shown in Figs. 9-11 of typical stresses and displacements for Shell II on the line y = 1 for an increasing sequence of z values. All stresses and displacements seem to approach zero as z increases. The decrease in the magnitudes of the solution as r increases is another possible cause for the slow rate of convergence for large r.
16
FIG. 6. The variation pY coincides, within
with x of ZY for Shell I for a sequence of y values. For the shell with no cutout, the resolution of the graph, with the x-axis for the range 0 s x < 0.875. For 0.875 5 x I I, e; is represented by the dotted curve.
Strews
in a perforated cylindrical shell
1.51
I52
I--
0,--
2
1
Stresses in
a
perforated cylindrical shell
SHELL
FIG. 9. The
variation
P
with x of the dimensionless normal displacement w for Shell 11 on the line y = I for a sequence of r values.
153
154
F. BAUERand E. L. Rmss
i-
SHELL II
f-
2--
7.9340
::__
1-r
680
to20
#
i
.3
I.0 x
FIG. 10. The variation with x of Zz for Shell II on the line y = 2 for a sequence of T values.
Stresses in a perforated cylindrical shell
-9
\
SHELL I[
? -6 r-
-2 I-.
--l---l
\\
,
FIG. 11. The variation with x of Zy for Shell II on the line y = I for a sequence of T values.
155
F. BAUER and E. L. REISS
156
REFERENCES [I] A. I. [2] [3]
[4] [5] [6]
[7] [8]
[9] [lo]
LOURYE, Concentration of Stresses in the Vicinity of an Aperture in the Surface of a Circular Cylinder, [MM-NYU 280; Translated from Prikl. Mat. Mekh. 10, 397 (1946). D. WHITHAM, &.-Arch. 28, 435 (1958). D. V. VAINBERC and A. L. SIN~AVSKII, Problems of Continuum Mechanics, SIAM, Philadelphia, Pa. p, 570 (1961). G. N. SAVIN, Problems in Continuum Mechanics, SIAM, Philadelphia, Pa. p. 382 (1961). 0. M. Guz, Prik. Mekh. ANUSSR, Inst. Mekh. 8, 605 (1962). T. MYINT, J. R. -M. RADOK and M. WOLFSON, Numerical Stress Analysis of Circular Cylindrical Shells, PIBAL Report No. 592, Polytechnic Inst. of Bklyn. (1961). E. L. REISS,Q. J. Mech. appl. Math. 15, 325 (1962). J. KEMPNER, J. uppl. Mech. 22, 117, 118 (1955). J. DOUGLAS, JR. and H. H. RACHFORD, JR.,Trans. Am. math. Sot. 82, 421 (1956). S. D. CONTE and R. T. DAMES, Muthl. Tabl. n&n. Res. Coun., Wash. 12, 198 (1958).
(Received 1 February
1965)
R&sum&L’on a assume que les d&formations d’une enveloppe cylindrique klastique et mince, soumise B une pression 1atCrale uniforme, sont adhuatement d&rites par les Cquations de Donnell. La surface de l’enveloppe est perforCe par un rectangle. Les dtplacements tangentiels de la membrane ainsi que les efforts normaw de membrane disparaissent sur les rebords simplement support&s du trou et sur les extrCmitt% du cylindre. Des solutions numtriques sont obtenues en remplapant le probldme B valeurs limites par une approximation de diffkrences finies et en r&solvant le systhme rtsultant d’huations algtbriques lintaires par une mbthode d’iteration.
Zusammenfnasung-Es wird angenommen dass die Verformungen einer diinnen zylindrischen Schale, welche einem gleichmlssigen Seitendruck unterworfen ist, durch die Donnell Schalengleichung hinreichend beschrieben sind. Die Oberfllche der Schale ist mit einem rechteckigen Ausschnitt perforiert. Die tangentiellen Membranverschiebungen und die iiblichen Membranbeanspruchungen verschwinden an den einfach unterstiitzten RLndem des Loches und den Enden des Zylinders. Zahlenmlssige Auflijsungen werden erhalten durch Ersetzung des Randwertproblemes durch eine endliche Differenzenannlherupg und Lljsung des resultierenden Systems von linearen algebraischen Gleichungen durch ein Iterationsverfahren.
A6cTpaKT-~paHHMaCTCa,YTOAC~OpMalIclH PaBHOMepHOM)' IlOnepe’IHOMy AaBJIeHBh3 (Donnell). persble)
l-lOBelJXHpCTb nepeMeI.qewin
IJWIHHApa.
l-lOJIyYeHI,I
KOHe’IHbIM
pa3HOCWM
ypaBHeHk&
IIpOIleCCOM
06OJIOyKki
MeM6paHbI
TOHKO~4H~WHA~WY~CKO~~~~~~O~O60~OYK~,~OABe~~luC~CK COOTBeTCTBeHHO
IlpOCBepJIeHa
wIe3aIoT
Ha o6bIYHO
YHCJTOBbIe lEUIeHHX
3aMeIUeHHeM
Ii
pa3&P%IIaff
~OB~O~HWI.
BbIpaxeHbI
ITpKMOyrOJIbHbIM
lIOJIy%HH)'tOB
yPaBHeHH5lMI.I
IroMepwcHBaeMbIX npo6AeMbI pe3yJIbTaTe
O~OJIOYKH
BblPeJKOM. TaHreIiIJWlbHbIe Kpaax
KpaeBOtt CkiCTeMy
OTB~~CTHII
3aaaWi
fiOHIieJIJIa (none-
a Ha KoIiuax
IIpH6JIHXreHHeM
JIIiHe&HbIX
anre6panrecKax
I’IO
J. Solids
S?rucmres.
1966, Vol.
2, pp. 141 to
156.Pergamon
Press
Ltd. Printed
STRESSES IN A PERFORATED
in Great
Britain
CYLINDRICAL
SHELL*
F. BAUER and E. L. REM Courant Institute of Mathematical Sciences, New York University
AI&met--It is assumed that the deformations of a thin cylindrical elastic sheli, subjected to a uniform lateral pressure, are adequately described by the Donnell shell equations. Tbe surface of the shell is perforated by a rectangular cutout. The tangential membrane d~pla~rnen~ and the normal membrane stresses vanish on the simply supported edges of the hole and the ends of the cylinder. Numerical solutions are obtained by replacing the boundary value problem by a finite. difference approximation and solving the resulting system of linear algebraic equations by an iteration procedure.
1. INTRODUCTION
THE presence of a hole in the wall of a cylindrical elastic shell that is subjected to surface and edge forces, significantly alters the distribution and magnitudes of the stresses. It also complicates the mathematical analysis of the shell problem. Approximate methods of solution, which are restricted in their applicability, have been previously proposed? P-61. In this paper, a numerical method is presented to determine the stresses and displacements in the shell for a specific cutout problem. However, with suitable modifications, it can be extended to include a variety of cutout shapes and other boundary and loading conditions. The procedure we employ is the well-known finite difference method. In this way, the boundary value problem of shell theory is approximated by a system of linear algebraic equations. In principle, there is usually no difficulty in the application of the method. However, for cutout problems, in which the solutions vary rapidly, many mesh points may be required. Roundoff errors and excessive computing time make it impractical to use the “standard” methods (e.g. Gauss elimination) to solve the resulting large system of algebraic equations. We present and successfully apply an iterative method involving two “acceleration” parameters to solve the algebraic equations. The procedure is described in Section 3 and the computational methods required to apply it are outlined in Section 4. For the specific problem investigated the perforation is rectangular, i.e. the edges of -the cutout coincide with generators and parallel circles of the cylinder’s midsurface. The shell is loaded by a uniform pressure applied normal to the surface. On the simply supported boundaries the tangential membrane displacements and the normal membrane stresses vanish. It is assumed that the deformations are adequately described by the Donnell thin shell theory [7,8]. * This report represents resulta obtained at the Courant Institute of Mathematical Sciences, New York University, with the OfIice of Naval Research, Contract No. Nonr-285(42) and tbe United States Atomic Energy Commission, Contract AT(30-l)-1480. It is a shghtly abridged version of the report IMM-~ 320, February 1964. t Reference f4] contains a general discussion of shell cutout problems and references to previous work. 141
142
F. BAUERand E. L. R~lss
In Section 5 the results of the computation are discussed and graphs of stresses and displacements are shown. They indicate that the stresses are large near the corner of the rectangular cutout. These high stresses may have slowed the convergence of the iterative procedure. We therefore conjecture that for problems with smoother cutouts the method may be more efficient. 2. FORMULATION The cylindrical shell is of mean radius R, thickness t and length coordinate system r, 0, z is employed where r and 0 are plane polar is the axial distance measured from one end of the cylinder. Thus the the ranges R-t/2 I I I R + t/2, --7c I 8 I z, -L 5 z I L. The ordinates x and y are defined by the relations x = z/L,
1x12 1,
y = (R/L)&
lyl I (nR/L) = 1.
2L. A cylindrical coordinates and z coordinates are in dimensionless co-
(1)
The wall of the shell is perforated by a rectangular hole whose edges are on the lines x = +a and y = &/?l where 0 I tl, /I < 1. The shell is deformed by a uniform pressure p applied normal to the surface. The ends of the shell and the edges of the hole are simply supported and the normal membrane stresses and the tangential membrane displacements vanish there. The solution is therefore assumed to be symmetric with respect to the planes x = 0 and y = 0. Thus we consider the region D, see Fig. 1, which is one quarter of the cylindrical surface. Symmetry conditions must be specified along the following three boundary lines of D: u I x I 1, y = 0; 0 5 x I 1, y = 1; x = 0, /I1 5 y I 1. The Donnell theory for cylindrical shells [7,8] can be formulated as a boundary value problem for two coupled fourth order partial differential equations in the two dependent variables W(z, 0) and F(z, 0). Here, W is the normal displacement of the midsurface I = R of the cylinder and EF is the membrane stress function where E is Young’s modulus. The positive directions of p and W are outward from the center line of the cylinder. In dimensionless variables the differential equations of the DOMel theory are
L\2W+3= 1 ax2
’
A'f =
T$
FIG. 1. The domain of integration, D, which is one quarter of the cylindrical surface.
Stresses in a perforated cylindrical shell
143
where A2 is the biharmonic operator, A2 z $+*A+<.
ax ay
WI
ay
On the boundary of D we specify that
azw w=p=f=pl,
forx = 1,
w2!&fx&o,
for x = a,
w+f=$zO,
for 0 5 x 5 a,
aw a% af a3f o ay=dy3=ay=ay3=,
(34
for y = 0,
CrIXll,
and y = 1,
OlX
aw a3w af a"f o jy=-j-g=z=ax"=,
for x = 0,
/?I I y I 1.
The dimensionless variables in equations (2) and (3) are defined by
F(z,6’)
w(x, y) = CW(z, e),
CE
f (x, y) = [ 12(1- v2)l*c 7
c 3
E
12(1- v2)pL 0-z
’
z s (12(14)l’$.
where v is Poisson’s ratio. The membrane stresses,c$‘, cr$’and.& and the corresponding dimensionless stresses X,0, C,Oand CoxY are given in terms of the dimensionless stress function f(x, y) by
1,” (x, y)
=
Ao,O(z, S) =
2$+
yy (x, y)
E
c,” (x, y)
Aa,O(z,e) =
=
-9
Ao,O(z, e) =
w, (5)
where A-’ z [12(1-vZ)]+p(p(L/t)2. The quantities, cr:, ai and, o&, , which are proportional to the bending moments, and the dimensionless bending stresses m,, my and mxYare given in terms of w(x, y) by,
F. BAUER and E. L. REM
144
m,(x,Y) = Wk
0) =
a24x,Y) + va2w(x? Y) ay2
ax2
a2w(x, Y)
my(x, y) = Ba;(z, e) = --7+
v
ay
’
a2W(X, Y) ax2
(6)
2
a2w(x, Y)
mxy(x,y) = (1 - v)Bo:&, 0) =
axay ’
where 2
0
BE-L-4
12pL
.
If the displacements of the midsurface of the shell in the z and 6’ directions are U(z, f3) and V(z, 19)respectively, then the dimensionless displacements u(x, y) and u(x, y) are defined by
4% Y) =
0
R W, 0)
z
7
The membrane stress displacement relations of the Donnell theory thus reduce to [7,8],
c; = c:=+Jg+v(g+w)].
*~+w+vg], (7)
( 1 !?!+U.
g=LL 2(i+v)
ay
ax
The relations (5), (6) and (7) and the conditions (3) imply that the required boundary and symmetry conditions* are satisfied. The complete formulation of the boundary value problem consists of the differential equations (2) and the boundary conditions (3). The solutions depend only on the parameters ~1,fi, 1 and r. We obtain approximate solutions for fixed ~1,B, and I and a sequence of r values.
3. NUMERICAL
METHODS
To apply the approximate method the region D is covered with a rectilinear mesh. The mesh is of uniform spacing 6 in the x and y directions. The mesh is chosen so that the boundary lines of D coincide with mesh lines. There are M + 1 mesh lines in the x direction and N + 1 in the y direction so that, 6 = l/M = l/N.
(8)
The mesh points (Xi,yj) are defined by, xi = is,
yj = jS,
i = 0, 1,. . , M;
* The relations actually imply that the membrane However, these constants are set equal to zero.
displacements
j = 0, 1, . . N, tangential
to each boundary
(9) are constants.
Stresses in a perforated cylindrical shell
145
and the integers i and J are defined by, XI = CL,
YJ = Pi*
The mesh region D, is defined as the set of points (Xi, Yj) interior to and on the boundary of D. The mesh region DQ is obtained from D, by excluding the mesh points : (xy , Yj), J‘-0-
,.‘.,
N;(x,,yj),j=O,...,J;(xi,y,),i=0,...,1.
At each point of Db the solutions w(xi, Yj),f(xi, Yj) of equations (2) and (3) are assumed to be approximated by Wij,fii which satisfy the difference equations, A~Wij+ ~L,~j = 1,
Ad”f;i- zL,Wij =L:0,
(xi~Yj) E ob
Here the difference operator Ai, obtained by replacing derivatives operator (2b) by centered difference quotients, is given by
in the differential
Ad’= L; + 2L,L, +‘L;. L, and L,, are the “one-dimensional” centered second difference operators any mesh function* (gijj, by +
(W
U 14
defined, for
1 .j-2gi,jfgi-1,j)(6)-29
Lxgij
=
ki
Lygij
=
(gi,j+ 1 -2gi,j+gi,j-1)(d)-2e
(1lb) The approximating quantities (Wij> and (f;i> must satisfy difference equivalents of the boundary conditions (3) where all derivatives are replaced by centered difference approximations. Fictitious mesh points exterior to D, are introduced to apply the difference equations. Thus a system of 2JM(N+ l)-(I + l)(J+ l)] linear algebraic equations in the 2[M(N+ I)-(f+l)(J+l)] unknowns (Wij> and (&jjl is obtained. We assume that as 6 -+ 0 the solution of the algebraic system converges to the solution of (2) and (3). Hence for a sufficiently small S the solution of the algebraic problem should yield a “close” approximation to the solution of the boundary value problem. For large M and N, it is impractical to solve the algebraic system by a direct method, e.g. Gaussian elimination. Therefore, an iteration procedure is used to obtain approximate solutions of this sytem. With w$) as an initial guess, for 6xed values of a, j3, I and z, a sequence of iterates [wj;),ffjn)] are defined by the recursions
A;f'i,?' = & xw!?' 1J'
A2$~+" _ -rL,f$) 6 11
,!?+r) y -- ~~~~+l)+(l-~)w~) ” *.I
f 1,
I
n=O,l,..., txi i Yj) E Ete*
(12)
Each iterate must also satisfy the difference equivalents of equation (3). The acceleration parameter L should be determined so that the iterations (12) converge as rapidly as possible. Therefore, for each n the solutions of two algebraic systems of the form A;g$+l)
= 4:;’
txi, Yj) E Di
(13)
are required where the mesh function {#) is either {~L,w$‘f or ( - zL,fg'+l> depending upon whether (g$‘> is @J’> or ( w(?+l)) . A line relaxation procedure, which is a simplification of the alternating due&ion method [9, lo] is used to obtain a crude approximation to the solution of (13). A single horizontal and a single vertical sweep of the mesh is * A function fgij) defined only at points of the mesh is called a mesh function.
F. BAIJEX and E. L. REM
146
performed instead of alternating back and forth between horizontal and vertical sweeps.* Thus the approximate solution (gg’ ‘)} of equation (13) is defined as the solution of the one dimensional difference equations [lo] : (1 +pQ$+”
= g{;++’ +pL$g{$‘,
2 (n+9 (l + PL.xkij
= g!?’ 8~- 2&&$’
- @g$’
(Xi,Yj)EDk9
+ pq$’ I
(14)
where (g$+*)} and (g $+i)} must also satisfy the appropriate difference boundary conditions. The matrices corresponding to the one dimensional difference operators in equation (14) are easily inverted by factoring. The number p is a second acceleration parameter and should be determined so that the iterations (12) converge as rapidly as possible. 4. COMPUTATIONAL Application of the numerical procedure An acceptable value of 6 is determined by calculations with successively Iiner meshes. mesh yields sufficiently accurate solutions numerical criterion for convergence is that z(n) z
(
PROCEDURES
requires that the mesh spacing be “small”. comparing the solutions of a sequence of test From these tests we conclude that the 6 = & for the range of parameters considered. The the iterations satisfy the condition,
*ma~DbflWp-w~-l)/,Ify-_fpI}
< 10-k
X,
(15)
The value of k that is employed depends on the magnitudes of the solutions. For all our calculations we used k = 8 or 9. The condition, lim z(“) = 0 is only necessary for convergence. It is found, starting from a given initial iterate, that the number of iterations required to satisfy equation (15) depends on the values of p and t. For fixed a, /?, I and r, the optimal values Lo and p0 are defined as those values which minimize the number of iterations necessary to satisfy equation (15). To obtain estimates for them a series of test calculations are employed. Thus for a fixed configuration, i.e. fixed a, /I and 1, and usually employing a coarse mesh estimates of & and p. are determined for an increasing sequence of r values. These values are used for the finer meshes and 1, and p. for other values of z are estimated by interpolating or extrapolating. In Table 1 representative values are given of the acceleration parameters that were employed. TABLE 1. ACCELERATIONPARAMETERS EMPLOYEDIN THE COMPUTATION
Shell I 1
Shell II 1
P
P
340 680 1020 1360
1.0 0.615 0.30 0.20
24.6 9.7 9.7 9.7
1.0 0.55 0.25
10.0 10.0 9.6 -
1700
0.12
9.1
0.10
9.6
* A limited number of numerical experiments indicate that the convergence does not change if several sweeps are made through the mesh.
rate of equation
(12) essentially
141
Stresses in a perforated cylindrical shell
Numerical solutions are obtained for two configurations, which we call Shells I and II, and a sequence of values of r = 340,680, 1020, 1360, 1700. Shells I and II are defined respectively by, a = 10/24, /.I = 10122, 1 = 22124 and a = 7124, j? = 7122, 1 = 22124. The initial iterate for a given z is taken as the “converged” solution for the nearest z for the same shell. When the iterations converge the dimensionless stresses are computed employing obvious finite difference equivalents of equations (5) and (6). It is observed that the number of iterations necessary for convergence depends on the values of the parameters a, j?, I and r. For example, with a fixed configuration the number increases rapidly as z increases, and with fixed I and r the number increases as a and /_Idecrease. In all cases a relatively large number of iterations is needed to satisfy equation (15). For example using Shell I and r = 340, 492 iterations are needed, and for r = 1020, 2481 iterations are required. However, the computing time per iteration is relatively small, e.g. the above computations required approximately 13 and 65 minutes* respectively. 5. PRESENTATION
OF RESULTS
Representative graphs of the dimensionless stresses and normal displacement obtained from the numerical solution for Shell I with r = 1700 are given in Figs. 2-8. In these 4
qw,v,xd 0
I =
-4
woo
-at FIG. 2 The variation with the axial distance x of the dimensionless axial membrane stress Zz for Shell I for a sequence of y values. The hole occurs at y = 2. * All computations employed v = 0.3 and were performed on the IBM-7090 computer at the A.E.C. Cornputing and Applied Mathematics Center of the Courant Institute of Mathematical Sciences, New York University.
148
-I FIG. 3. The variation
I
with x of the dimensionless hoop membrane stress $’ for Shell I for a sequence y values. The dotted curve is the stress if no hole is present. 20
IO
Z’,y(xrlxlo~
0
----Q-I PI700
-10 FIG. 4. The variation
with x of Cz,, for Shell I for a sequence
of y values.
of
149
Stresses in a perforated cylindrical shell
figures the variations with the axial distance are shown for a sequence of y values. These results indicate that near the corner of the cutout, x = a, y = fil, the numerical stresses are quite large. The hole apparently affects the stress distribution everywhere in the shell. The hoop membrane stress Cy and the axial bending stress YX are influenced more by the hole than the other stresses. If u = /I = 0, the stresses and displacements obtained from the well-known solution of the boundary value problem are independent of y. These quantities are indicated with a super-bar and are given by, S(x) = $ -
cos jx cash fix + tan /I tanh j3 sin /3x sinh /Ix rz cos /I cash p(1 + tan’/3 tanh2P) ’
(16)
8
qxa)xlo4 4
0
I I -4 I--
-I lFIG. 5. The variation with x of the dimensionless axial beading stress Zi l&SheUfer a sequence of y values. For the shell with no cutout, ZX coincides, within the resolution of the graph, with the x-axis for the range 0 I x c 0.833. For 0.833 5 x I 1, Z: is represented by the dotted curve.
150
F. BAUJZRand E. L. REISS
where r2 =
484.
These results are shown by the dotted curves in Figs. 3, 5 and 6. In Fig. 8, the lateral displacement i+(x) coincides, within the resolution of the graph, with the numerical results for the line y = 0. These figures indicate that the cutout and its comer may increase the magnitudes of the stresses and displacements by large factors. Although their magnitudes differ, the form of the stresses and displacements for other values of r and for Shell II are similar to those shown in Figs. 2-8. A simple asymptotic solution is obtained for large r by dividing the differential equations by r and letting r-+co while w(x, y; co) and f(x, y; co) remain finite. Integrating the resulting equations and applying the boundary conditions we obtain w(x, y; co) = j-(x, y; co) = 0. To illustrate this result, graphs are shown in Figs. 9-11 of typical stresses and displacements for Shell II on the line y = 1 for an increasing sequence of z values. All stresses and displacements seem to approach zero as z increases. The decrease in the magnitudes of the solution as r increases is another possible cause for the slow rate of convergence for large r.
16
FIG. 6. The variation pY coincides, within
with x of ZY for Shell I for a sequence of y values. For the shell with no cutout, the resolution of the graph, with the x-axis for the range 0 s x < 0.875. For 0.875 5 x I I, e; is represented by the dotted curve.
Strews
in a perforated cylindrical shell
1.51
I52
I--
0,--
2
1
Stresses in
a
perforated cylindrical shell
SHELL
FIG. 9. The
variation
P
with x of the dimensionless normal displacement w for Shell 11 on the line y = I for a sequence of r values.
153
154
F. BAUERand E. L. Rmss
i-
SHELL II
f-
2--
7.9340
::__
1-r
680
to20
#
i
.3
I.0 x
FIG. 10. The variation with x of Zz for Shell II on the line y = 2 for a sequence of T values.
Stresses in a perforated cylindrical shell
-9
\
SHELL I[
? -6 r-
-2 I-.
--l---l
\\
,
FIG. 11. The variation with x of Zy for Shell II on the line y = I for a sequence of T values.
155
F. BAUER and E. L. REISS
156
REFERENCES [I] A. I. [2] [3]
[4] [5] [6]
[7] [8]
[9] [lo]
LOURYE, Concentration of Stresses in the Vicinity of an Aperture in the Surface of a Circular Cylinder, [MM-NYU 280; Translated from Prikl. Mat. Mekh. 10, 397 (1946). D. WHITHAM, &.-Arch. 28, 435 (1958). D. V. VAINBERC and A. L. SIN~AVSKII, Problems of Continuum Mechanics, SIAM, Philadelphia, Pa. p, 570 (1961). G. N. SAVIN, Problems in Continuum Mechanics, SIAM, Philadelphia, Pa. p. 382 (1961). 0. M. Guz, Prik. Mekh. ANUSSR, Inst. Mekh. 8, 605 (1962). T. MYINT, J. R. -M. RADOK and M. WOLFSON, Numerical Stress Analysis of Circular Cylindrical Shells, PIBAL Report No. 592, Polytechnic Inst. of Bklyn. (1961). E. L. REISS,Q. J. Mech. appl. Math. 15, 325 (1962). J. KEMPNER, J. uppl. Mech. 22, 117, 118 (1955). J. DOUGLAS, JR. and H. H. RACHFORD, JR.,Trans. Am. math. Sot. 82, 421 (1956). S. D. CONTE and R. T. DAMES, Muthl. Tabl. n&n. Res. Coun., Wash. 12, 198 (1958).
(Received 1 February
1965)
R&sum&L’on a assume que les d&formations d’une enveloppe cylindrique klastique et mince, soumise B une pression 1atCrale uniforme, sont adhuatement d&rites par les Cquations de Donnell. La surface de l’enveloppe est perforCe par un rectangle. Les dtplacements tangentiels de la membrane ainsi que les efforts normaw de membrane disparaissent sur les rebords simplement support&s du trou et sur les extrCmitt% du cylindre. Des solutions numtriques sont obtenues en remplapant le probldme B valeurs limites par une approximation de diffkrences finies et en r&solvant le systhme rtsultant d’huations algtbriques lintaires par une mbthode d’iteration.
Zusammenfnasung-Es wird angenommen dass die Verformungen einer diinnen zylindrischen Schale, welche einem gleichmlssigen Seitendruck unterworfen ist, durch die Donnell Schalengleichung hinreichend beschrieben sind. Die Oberfllche der Schale ist mit einem rechteckigen Ausschnitt perforiert. Die tangentiellen Membranverschiebungen und die iiblichen Membranbeanspruchungen verschwinden an den einfach unterstiitzten RLndem des Loches und den Enden des Zylinders. Zahlenmlssige Auflijsungen werden erhalten durch Ersetzung des Randwertproblemes durch eine endliche Differenzenannlherupg und Lljsung des resultierenden Systems von linearen algebraischen Gleichungen durch ein Iterationsverfahren.
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