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On the problem of flexure of anisotropic cylindrical shells
hr.
J. Solids Svucrures,
1973. Vol.
9. pp. I I55 to 1175. Pergamon
Press. Printed
in Great
Britain
ON THE PROBLEM OF FLEXURE OF ANISOTROPIC CYLINDRICAL SHELLS?
Institute of Applied Mathematics, Tsing Hua University, Hsinchu, Taiwan, China
Abstract-The problem of determining stresses and deformations in anisotropic, elastic, axially homogeneous thin-walled shells subjected to equal and opposite transverse end forces, and additional moments to assure overall equilibrium, is studied. The solution procedure is on the basis of a decomposition of stresses and strains in terms of a portion dependent linearly on the axial coordinate and a portion independent of the axial coordinate. The most significant aspect of the work has to do with the analysis of the effect of anisotropy of the material. The general formulae of the theory are illustrated for a class of shells consisting of an ‘.‘ordinary” material. Here explicit formulae are obtained for certain types of open- as well as closed-cross-section shells.
INTRODUCTION paper considers elastic axially homogeneous anisotropic cylindrical shells subject to transverse end forces, together with the corresponding bending moments to assure overall equilibrium, as a problem on extension of the classical St. Venant theory of flexure. The differential equations of equilibrium and compatibility are special cases of general linear shell equations given by Gunther [ 11, and the constitutive equations are special cases of equations given by Reissner [2]. Important characteristics of these equations are the inclusion of moments turning about the normals to the middle surface.of the shell, and incorporation of the effect of transverse shear deformation. The effect of anisotropy on the distribution of stresses, strains and displacements is the particular concern of this paper. In attempting a solution of the flexure problem, we start with a formulation of the overall equilibrium conditions for the shell acted upon by transverse end forces and the associated equilibrium end bending moments. We conclude from these that in its simplest form the solution of the problem should consist of two portions. The first portion, due to the flexural bending moments, depends linearly on the axial coordinate. The second portion due to the flexural forces is independent of the axial coordinate. Investigation of the general system of differential equations and its associated boundary conditions reveals that in fact the two indicated portions are all that is necessary for the solution of the problem of axially homogeneous cylindrical shell flexure, upon appeal to an appropriate version of St. Venant’s principle. In the following we first indicate the procedure of decomposing the problem into the two mentioned portions. We then proceed to indicate the solution procedure for both portions. The result for the axially dependent portion is directly given upon making use THIS
t This report represents a chapter from the author’s Ph.D. dissertation (UCSD, May 1972, Advisor : Prof. E. Reissner). The author’s work has been supported by the Office of Naval Research, Washington. D.C. $ Associate Professor. 1155
W. T. TSAI
Il.56
of an appropriate .interpretation for the result of the problem of pure bending, stretching and twisting, as recently presented by Reissner and Tsai [3]. The result for the axially independent portion is then obtained as function of the quantities of the axially dependent portion.
FUNDAMENTAE
EQUATIONS
We take as curvilinear coordinates the circumferential arc length s and the axial distance z. The coefficients of the linear arc element are a, = ct, = 1, while the curvatures are l/R,, = 1/R = drplds = k and l/R,, = 1/R,, = 0. Designating stress resultants by N,, N,,, N,, N,,, Q,, Q, and stress couples by M,S, M,,, M,,, M,,, P,, P,, with cmresponding strain resultants E,y and strain couples x, I, we have as six equilibrium equations and six compatibility equations in accordance with Gunther [ 11, N,, + N,,, +
x*z*.l -x,,,,kl, = 0, A:.s -~j,z+kx,, = 0,
kQ,= 0,
Q,,,+Q,,,-Ws = 0%
’
&II,S -&,,,,-~;iZ = 0, xZJ,S --x ss.* = 0,
M,,s+M,,,, -Qs = 0, N,,., +Nzz.z = 0,
G&W Oa,bJ Pa,b)
Ezs.3 -E,,,, + ky, - A, = 0,
(5a, b)
yz,s- ys,z- kc,, + x:s - x,, = 0.
@a, W
M,,,+Mzz,:-K-Q,= 0, 2,,, + Pz,zf kM,, + N,, - N,, = 0,
(la, W
The twelve equations fla)-f6b) which involve twelve stress quantities and twelve strain quantities are here complemented by twelve constitutive equations, written in the form
(NS,N,~,,N,,Q,Q,M,,M~~M~*M~~~P=~ = 4~ 55 ELI E51 6zs Y5Y0x5~~z,wzJs~,~~ (71 where A is a 12 x 12 matrix, the elements of which are given functions of the arc length coordinates. In addition we have twelve strain~ispla~ment relations involving t~nslationai displacements uS, IA,, w and rotational displacements (~3,cp,, o. These relations are
and Es, = u z,s -0, x sz
=
9, ,f - ko,
Ezs =
x,,
=
us,,+ a,
yr = W,Z+-9zr
9s.: t
1, =
@b) 0,s + k9,.
Having stated the appropriate system of differential equations, we now consider the associated boundary conditions for ends z = constant of the shell and for boundaries s = constant of the shell. There are altogether twelve boundary conditions for s = constant. For clarity’s sake we consider these conditions separately for open- and closed-cross-section tubes, as follows. For the open-cross-section tube with longitudinal boundaries s = s1 and s = sir
On the problem of flexure of anisotropic cylindrical shells
115-l
we assume that the longitudinal boundaries are traction-free. This means that we prescribe the vanishing of the three stress resultants N,,, N,:, Q,, and of the three stress couples M,,, M,,, P, for s = s1 and for s = s2. For the closed-cross-section tube, we assume that the stress measures N,, N,,, Q,, M,,, M,,, P, and the displacement measures us, u,, w, cps, v,, w altogether are twelve univalued quantities. Since our interest is essentially in the determination of stress and strain quantities, we replace the six displacement univaluedness conditions by six conditions expressed in terms of strain quantities. We find with the help of the strain displacement relations @a, b) that the univaluedness of displacement conditions are equivalent to two conditions of univalued E,,, yz and to four strain integral relations 0=
x,, ds,
0
('%,Y f
=
O-
0 =
Ptx,,xf
w’ + Y,Y’)ds, E,,Y’ +
Y,x’)ds,
[E,, - (x,,y’ + i&,X’)X+(x,*x’ - i,y’)yJ ds.
(9c)
(94
We now turn to the consideration of boundary conditions for sections z = constant. We appeal to St. Venant’s principle and replace the local conditions of prescribed edge stresses by overall equilibrium conditions as follows, 0
(104
Qg=Jt N,,y - Mzrx'+ Pzy’) ds,
(lob)
Qxz=
(1Oc)
0 =
I
(N,,x
CM,, + (N,& - Qzx’)x - (Nz,x’ + Q,Y’)Y] ds,
Q, = JW,g~-
Q, =
+ M _y’ + P:x’) ds,
JVW+
QA ds, Q,Y')
ds.
(104
(lf)e)
t1w
In this, Q, and Q, are the prescribed overall transverse forces, Q,+Tand Q,,z are the prescribed overall bending moments, and equations (lOa, d) are the statement of vanishing of axial force and of axial torque. We note that the six integral relations (1Oa) to (10f) are equivalent to four only, consisting of the relations (lOa, b, c, d). The two relations (lOe, f) may be shown, by means of an introduction of (4a) to (da) into the z-derivatives of (lob, c) and by suitable integrations
W. T. TSAI
1158
by part, upon making use of the previously indicated boundary conditions M,, M,,, P, for s = constant, to be equivalent to equations (lob, c).
for N,,. Q.,,
N,,,
DECOMPOSITION
OF STRESSES
AND STRAINS
Given the form of the boundary conditions (10) and the fact that the differential equations (1) to (6) together with (7) and the remaining boundary conditions are homogeneous, we realize the possibility of the following decomposition of the state of stress and strain (N, Q, M, P, E, y, x, i)(s, z) = z. (N*, Q*, M*, P*, ---+(N,
E*, y*, x*,
i*)(s)
Q, M, P, I, 7. 2, I;)(s).
(11)
Introduction of (11) into (1) to (6) and (lOa, b. c, d) leaves as system of equations for the z-dependent quantities NZ.,+kQ:
= 0,
x& - ki;
= 0,
(la*, b*)
Q:,-kN:,
= 0,
E.$ + kxtz = 0,
(2a*, b*)
M&--Q,*
= 9
(3a*, b*)
E*_ I-.s -i:=o,
N:,., = 0,
x*ZS.S= 0,
(4a*, b*)
M,:,-kP:--Q:
= 0,
E;_ + k-$ - ij’ = 0,
(5a*, b*)
P&+kM:!+N:z-N;s
= 0,
_i:,. - kc;, + x;, - x,: = 0,
(6a*, b*)
and 0 =
N;, ds,
(lOa*)
Q, = j(~:,y-~‘x.+Pfy.lds.
(lob*)
Qx = s(N&x
(lot*)
0 =
s
+ M,:y’ + P:x’) ds,
[M;,+(N&y’-
Q:x’,x -(N,*,x’+
Q:y’)yl ds.
(lOd*)
At the same time we obtain as system of equations for the z-independent -kr:,
quantities
= it;=,
iv,,., + hQr = - NZ,
L
Q,.,-kg,,
= -Q:,
A,,,+ kn,, = A_;,
m,,/-Q,
= -ML
m,=,, = - NZ, J&Z., - kPs--QQ, = -M;,, PS,.,+ k&f,, + IV,, - KJ,, = P:,
&
(l&1;)
(2% 6)
- AZ= E,:,
(39,6,
j&S = x:S,
(4% 6)
E,,,, + k?, - i, = E;, 7z.s - kl,, + x,, - ii,, = y:,
(5% 6) (69,li)
On the problem of flexure of anisotropic cylindrical shells
1159
and 0 = J
0
=
0 =
0 =
r
N,, ds,
s s r J
(102)
(&,y - !i&,x’ + F:y’, ds,
(106)
(m,,x+ M,,y’
(1W
+
Pzx’)ds,
[R,, + (N,,y’ - Qzx’)x- (m,p’ + Q,y')y] ds.
(loa)
It is obvious that equations (la*) to (6b*) and (lOa*, b*, c*, d*) are of exactly the same form as equations (la) to (6b) and (16a, b, c, d) in [3] except that the places of the stretching force N, the bending moments M, and M, and of the twisting moment T in equations (16a, b, c, d) in [3] are taken over by 0, Qy, Q, and 0, respectively. In addition to this, the constitutive equations and the boundary conditions for the starred quantities are also equivalent to the corresponding formulas in [3]. From this follows that the results for the problem of determining the starred measures may be stipulated to be directly given in the same form as the corresponding results for the problem of pure bending, stretching and twisting in [3], upon replacing N, M,, M,, T by 0, Q,,, Q,, 0, respectively. We next turn to the determination of the barred quantities on the basis of equations (la) to (66) and (10% 6, E, dj, together with the appropriate boundary conditions and constitutive equations.
FIRST
INTEGRALS
OF EQUATIONS
Let s0 be a specified arc length coordinate. in the form
FOR BARRED
QUANTITIES
We begin by integrating
equations (4% 6)
s iV,, = s-
N:, ds,
zzs = cl+
I so
x: ds.
(12a b)
where 3 and E are constants of integration. Next, we integrate equations (1% fi) and (2g, b), with the help of the relation ( ),, = 4 ),cp,with equations (5a*) to (6b*), and with appropriate integrations by part. In this way, we find the important result that we can express iJ,, W,,, Q, and 2, explicitly as
iv,,= X,x’ Q, =
-
Nxy' +
ivyy’ -
s*r,- PZ,
m,x’ - S*r, - Mzz,
zsr = x,y' - Zxx' + a*r, + y:,
(13a b) (14a, b)
In this rn 3 xy’ -x’y, r, c xx’ + yy’ ; S* E N,*, and a* = x,: are given constants ; and m,, m,,, ii, and zi, are four additional constants of integration.
W. T. TSAI
1160
We now introduce stants of integration,
(14a, b) into (3% b), and obtain with M and E as two further con-
R,, = ,V,y+R++M-
2,, = z,y+ii,,x+++
’ (M:, + M:, + S*r,) ds,
s SrJ
’ (.s,:+s;,--*r,)ds. 150
(15a)
(15b)
Having the system of first integrals (12a) to (15b), we are left with four first-order differential equations (52) to (66), which involve the eight quantities R,,, P*, Q,, mzs, i,, ,7,, 1, and gsz, and which can not in general be integrated explicitly.
BOUNDARY
CONDITIONS
FOR BARRED
QUANTITIES
Equations (la) to (66) are a twelfth-order system of ordinary differential equations. We look for twelve boundary conditions. We have, in accordance with the previously given fundamental equations, four stress integral relations (log, 6, E, d) and twelve boundary conditions for s = constant. Among these, four of the boundary conditions for s = constant are extraneous, leaving us twelve required boundary conditions. We show this by considering separately the cases of open- and closed-cross-section tubes. For the case of closed-cross-section tubes, we have that S*, Mz and P: are univalued, and f N:, ds and f (M,*,+ Mz’+ S*r,) ds vanish. Introducing this into (12a) to (15a) we find that N,,, m,,, 0, and R, are the measures which automatically satisfy the univaluedness conditions for arbitrary s. Upon excluding the conditions for these four quantities we are left with twelve boundary conditions which consist of the four conditions of univaluedness of R,,, pz,, b,, and 7,, the four stress integral relations (10% 6, c, a), and the four barred strain integral relations in (9a, b, c, d), written here in the form
Pii,,
ds = 0,
(ii,a + &.xX’+ ;I,y’) ds = 0,
(96)
- I,,y' + *yp’) ds = 0,
(9E)
( iirsx
~,[E,-(~~,y’+X,r’)x+(ii,,x’-X,y’)y]ds We note for subsequent reference the following important (1Od) and (9d)
s
(R,, + &if,,+ Sr,) ds =
(9%)
= 0. transformation
NZ, ds + r,M:, - r,,PZ ds,
(9d) of equations
Cloa
1161
On the problem of flexure of anisotropic cylindrical shells
and (&,+E,,-Er,)ds
= +l
rn ’ x,: ds - I& - r,y: J so
Further simplifications are possible upon noting the It remains to formulate the boundary conditions s0 = si and use the facts that S* = 0 throughout, vanishing of m,,, m,,, Q, and R,, for s = si in (12a)
ds.
(9d’)
geometrical formula 2A, = $ rn ds. for open-cross-section tubes. We let and M: = P: = 0 for s = si. The to (15a) is then equivalent to
s = m, = NY = ii;i = 0,
(16)
and we are left with the explicit results
(17) Q, = -M,*,,
R,, = -
’
(M,:+M:,)ds.
I s1
It is apparent that the boundary conditions of vanishing m,,, m,, Q,, R,, for s = s1 are satisfied automatically because of the boundary conditions for the starred quantities for open tubes involving 52 N:, ds =
s SI
” (M: + M:,)ds = 0 I 51
and
M,*,= P$ = 0 for
s=sl.
Recognizing this, we are again left with twelve boundary conditions involving the four and results S = W, = W, = R = 0 in (16), four conditions K&: = ij;, = 0 for s=s, s = s2, and the four conditions (lOS6, E) and (1Od’). We note in conjunction with the result (17) that the quantities m,,, n,,, Q, and &?,, together with the result that Nzz, N$, Q: and Mz vanish throughout, and with the decomposition (11) provide explicit results for the actual quantities N,, N,,, QS and M,, for open tubes, namely N,, = -
’ N,*, ds,
sSI
Qs = -Wz,
N,, = -P$,
(18) M,, = - '(M:,+ M:,)ds. I s1
Equations (18) show that N,,, N,,, Q, and M, are always independent of z. Furthermore, * = 0 in (18) means an automatic satisfaction of the Kirchhoff boundary the relation Q, + M,, = 0 for the case that the transverse shearing strain yz is stipulated condition Q, + M,,,, to vanish.
CALCULATION
OF DISPLACEMENTS
In what follows we consider displacements of the points of the cylindrical she11 surface for the problem of fiexure. As for the problems of pure stretching, bending and twisting we are led to distinguish two types of displacements, those which translate and rotate the
1162
w. T.
fSA1
cross section of the shell as a rigid body and those which represent a distortion of the cross section, axially and laterally. In both types, two different classes of elements are involved. The class of “ordinary” elements which also occur for orthotropic shells and the class of “unusual” elements which are due to the effect of non-orthotropy. We start to calculate displacements by considering the strain displacement relations (8a, b) which with the help of the decomposition (11) assume the form .s$r + E, = u,,, + kw, x2
EDZ + i,, x2
+ %* = rp,s
+ ii,,
= u,*, ,
y:z + 9, = cp,+ w,, - ku,,
= to,,,,
/i;‘?+i,
(194
= O,:’
and
In these ii,, ii,,, X,, E,, are explicitly given by (12b) to (1Sb) and x&, xzz, Jr, E$, with r*, x:, x;, E* as four constants, are x;S = c1*,
x,*, = x;y’-x:x’,
2: = x: y ’ + Xy*X’,
&* ** = x:y + x;x f &*.
(20)
We now introduce (20) together with the first strain integrals (12b) to (15b) and with the strain differential equations (5b*, 6) and (6b*, 6) into the strain displacement relations (19a, b). An observation of the z-independence of all starred and barred strain components leads to the following expressions for the six components of rotational and translational displacement
cpr = +(x;y'-x:x')22+(j2yy'- zxx’ -t a*r,)z + yz + cq(x -x0)x’ +r,jX;dS-x’
j (x,gx - &;y’ + y:x’)
ds - y’
j
+ (y - y,)y’]
(~3 + s&x’+ yty’) ds,
w = +(x:y’+ x;x’)zZ +(ii,y’+ izyx - a*r,)z+ E,, - ji[(x - xo)v’ -(y - yo)x’] -r.jx;ds+y’ and with k,
j (x$x - &:a’ + y:x’) ds - x’
s,, YJ = CG s& ~:)s+&,
j
($y + E:J’ + y:y’) ds,
b, &), P
u, =
+(x:y+ x;x+E*)z2“k(X*y+ a+ + L)z+@(x,y-
ypx)
+
J(E,,+
+ fr& + r,yr) ds + y
E,, -
JW~)ds
(~3 + E$..’ + y:y’) ds, (x&x - s$y’ -I-y:x’) ds -x j j I US= - &x:y’ + $X’)Z3 - &z,y + Z#’ - r*r#Jz2 + E[(x - x,)y’ -(y - yO)x’]z +r,
jx,,ds-y' j(x&-s&+y&)ds+x’
j
(x~fs,~‘+yy’)ds,
(22a)
(22b)
1163
On the problem of flexure of anisotropic cylindrical shells W = -4t$Y’
- X,*x’)z3 -
-r+,ds+x’S
)(X,y’-
S&x + a*r,)z2 - E[(x -x,)x’
(xd - ~,a’ + ysx’) ds + y’
+ (y - yo)y’]~
(x,,y + 8,~’ + y,y’) ds.
(22c)
We note that the translational components u, and w can be resolved into two displacement components u, and oYin x and y directions upon introducing u, = u&Xx’ 4 WY’,
tiY= l&y’- wx’.
(23)
The result is given by u*=
-@z” -&3X,+ a*y)z2 - E(y- y&z - y
f
x, ds +
I
(xd + E&’ + ya’) ds,
(24a)
In equations (24a, b) the terms without integral sign represent displacements with the tube cross section translating and rotating as a rigid body and the terms with integral signs represent the effect of distortion. Furthermore, since@,, E,, v,) I (~2, E& y:)z + (3i,, i35, T,), the axially linear dependence of the effect of distortion involves the well-known effect of transverse shear on the deflection of the beam, as stated by Timoshenko and Goodier [4] for the problem of isotropic sheet flexure. The remaining measures, involving Z,, E, and jj,, are the unusual terms which vanish for the case of orthotropic materials. We also find from the terms without integral sign in (24a, bf that rigid body translation and rotation involve terms proportional to z3, z2 and z. In these, the z3-terms represent the deflection due to flexural bending, and the z-terms represent the deflection due to axial twisting produced by the flexural forces7 [6]. We designate the r2-terms as “unusual” since they represent deflections which vanish for the case of orthotropic materials. We note in conjunction with the above observation for u, and u, that the axial displacement ~1,also involves both ordinary and unusual terms. The z2 and z-independent terms are of course the ordinary terms representing the displacement due to bending and warping. The term linear in z is the unusual term which is again due to anisotropy because the coefficient of this term is, with equation (lSb), the axially homogeneous strain measure & which comes from the constitutive coupling coefficients for anisotropic tubes.
METHOD
OF SOLUTION
FOR THE GENERAL
CASE
Equations (12a) to (15b) express the four stress quantities %,, iJ,, &, If;i, and the four strain quantities ii,,, ji,,, X,, E,, in terms of eight constants of integration. Equations (53) to (66) are four first-order differential equations allowing us to introduce four additional constants of integration. The total of tweIve constants of integration is to be determined in terms of integrals involving starred quantities by means of the four stress integral relations (103, 6, E, a’) together with the four univaluedness conditions for &?,,, g,, 1,,, 7, and the four strain integral relations (9s. 6, E, d’) for closed-cross-section tubes, or tot For the case of orthotropic tubes, axial twisting may be eiiminated by letting the flexural forces act through the center o~twist, Reissner and Tsai [5J.
W. T. ,-SAI
1164
gether with the eight traction conditions along the longitudinal edges, NSZ= N,, = Q, = @, = 0 for s = s1 and R,, = P, = 0 for s = si , and for s = s2, for open-cross-section tubes. In what follows we wish to indicate the solution procedure for obtaining the barred quantities for the case of the closed-cross-section shell, in such a way that the open-crosssection shell solution is included as a special case. Considering the solution procedure for the general case of anisotropy, we first rewrite the constitutive equations (7) for the barred contributions in the form
I
3
+B
(25a b)
In this, the eight quantities B,,, R,,, i,, ii,, and R,,, m,,, Q,, n,, are given by the first integrals (12) to (15). We further write the differential equations (5% 6) and (6&b) in the form
Wa b)
and use this to transform (25b) into
(27)
On the problem of flexure of anisotropic cylindrical shells
1165
We now have (27) as a fourth-order system of simultaneous differential equations for P,, RsZ, 7, and 1,, . The nonhomogeneous part of this involves the known starred quantities Pf, M,*Z, y:, t$ and the giuen barred quantities EZZ,ii,=, %,, &) R,,, m,,, Q,, p,,. Equations (27) are to be solved subject to the four boundary conditions of univaluedness of P_ Ri,,, -7, and i,, for the case of the closed-cross-section shell. Having solved (27) we must introduce the result into the constitutive equations (25a, b) to express, with the help of the first integrals (12a) to (15b), all quantities on the left-hand sides as functions of the eight constants of integration 3, E,, m,,, R, d x,, gi,, i, as well as of certain functions of the starred quantities appearing explicitly and implicitly in (27). These expressions in turn are to be introduced into the set (10% 6, c, a’) together with (98,6, E, a’), thereby producing a system of eight simultaneous linear equations for the determination of the eight constants 3, m,, m,,, R, Cr,x,, z,., i in terms of integrals of starred quantities. The fourth,-order system (27) is reduced to a zeroth-order system if we stipulate, in conformity with observations in [3] that the shearing strain component 7, and the moment stress resultant ps may be set equal to zero. With this, equation (27) reduces to two simultaneous equations for R,, and iZs of the form
(28a)
At the same time, the quantities QZ and A, are given by Q; = R;,+
M:,,
i, = &-&,:.
(28b)
In what follows we extend this general consideration to describe the solution procedure for the case for which the constitutive equations are taken in the form of the socalled conventional case.
METHOD
OF SOLUTION
In using the words “conventional stitutive equations are of the form
FOR CONVENTIONAL
CASE
cases” we have in mind cases for which the con-
iN,,N~NssM-_zM~Mss~ = Ac{~Zr~~&,s~,13(~)(sS),
(29)
together with ,,IZ -- .> Is = 0, Eis = E,, = ET,
P, = P, = 0, M,, = M,, = MT,
(3% b) (3Ia, b)
W. T. TSAI
1166
and with the definitions 2Nr = N,,+ N;,,
(32a, b)
2x, = x,, + XSZ,
Having equations (30a, b) we have from the equilibrium equation (6a), m,, = m,, + k&T,, and from the compatibility equation (6b), R,, = &,-k+, and therewith m p = m,,++klS;i,,
5iT = zLS--+kBT.
(3% b)
We now use the simplified subscript notations (N,, M,, E., x,) = (N,,, M,,, E,,, x,,). With this we introduce (33) into the barred part of (29) and this with the help of a suitable transformation gives
[ii]
=B~E;,,!+B~ [-:;I,[;;I=&[;zj+& [{IT
(34a,b)
where i,, Zz, ii,,, R,, m, and m,, are as in equations (12%) to (156). In order to simplify the result for the barred quantities, we rewrite equations (129) to (156) in the matrix form
-f,.
(35a, b)
where X
1 Y’
0
J 3
(i = 1, 2),
(36)
and where
(3% b)
On the problem of flexure of anisotropic cylindrical shells
Having equations (34) and (35), it remains to determine -grations 6 ii,, ii,,, &, M, N,,R,,and 3.
1167
the eight constants
of inte-
Open-cross-section shells
For an open-cross-section shell, the boundary conditions of vanishing traction for s=si and s= s2 give R = iir, = R,, = s = 0. The corresponding result for F,,, 15, and M, is given in (17) and can, with M,*,= M:, E M$ and P: = 0,and with S* = 0, be written as {MS--&&,}
= --j-i,=
- ~‘{ZM;ON:}ds. S1
It now remains only to determine the four constants of integration Z, and x,,. Evidently, this will be accomplished upon introducing into equations (10% 6, E, a’), written here in the matrix form
(38) E, ii,, ji, and 6 in L,, (34a), (35a) and (38)
(39)
where Ql is transpose of Qz and g, is a vector given by s g1=
ii
‘n
N:ds+r,M:
0 0 0 .
(404
so
Equations (39) directly give a system of equations of starred quantities of the form
for 6 8, ii, and ii, in terms of integrals
(41)
Closed-cross-section
shell
We now have the overall equilibrium R,, = IiTS, = M,, written in the form
equations
‘2A$ 0 0
(lOa, ‘6, E, a’), with P, = 0 and
1 =
Q g,
ds,
(42)
W. T. TSAI
1168
and this, with (34a) and (35a, b), and with four-by-four
matrices Ci, becomes
(43)
At the same time, we define s 92
=
P”
is
x:ds-r,s:
0 0 0 ,
50
1
@‘W
to write the strain integral relations (9% 6, E, al), with 7, = 0 and i,, = i,, = 8,, into the form
(W
Introduction
of ii,, 1, and ET from (34b) and (35a, b) into (44) results in the relations
=f k2 - Q&h
- &_I-,)1 ds.
(45)
Equations (43) and (45) determine the eight constants of integration in terms of integrals of starred quantities which are implicitly given by vectors f;: and gi. We note that the system of equations (43) and (45) for closed tubes may be reduced to a system of equations for open tubes upon appropriately interpreting the constitutive coefficients. Stipulating that an open-cross-section tube may be considered as if it were a closed-cross-section tube with an open part, we set in the constitutive equations (34a, b) the elements of Br equal to zero, and the elements of B, equal to infinite over the open part of the closed-cross-section. When this is done, the elements of C, and of the righthand side of (45) are finitet and the elements of C, become infinite, requiring that the constants 3, !‘@,m,, NY must vanish. Setting s = R = m, = NY = 0 into (43), we directly obtain a system of equations for the remaining four constants & 8, ii, and x,, of exactly the same form as equations (41) which follows from a direct consideration. t In view of equation (45). the statement that the elements of the right side of (45) are finite involves the constitutive matrix B,, the elements of which are stipulated to be infinite over the open part However, we have with the relations S* = I:: Nf ds = I:: ZM’, ds = 0, that the elements off, in (37a) for open-cross-section tubes vanish over the open part. With this, the elements of BJ, may be stipulated to be finite over the open part of the closed-cross-section tube. The integral of Q:BJt on the right-hand side of (45) therefore has a finite value confirming our earlier observations.
On the problem of fiexure of anisotropic cylindrical shells
EXPLICIT
1169
SOLUTION FOR A CLASS OF ANISOTROPIC SHELLS
In order to illustrate the nature of our general results, we consider shells with constitutive equations of the form E, =
M, = LX,
+ I&x, + D,r2~r,
(46) 2~~ =
$+9+$, Tt
in conjunction
Ts
MT
=
DTA + DTS% + DTT~XT,
TT
with P, = P, = 0,
I’* = 1’$= 0,
MS: = M,, z MT,
(47)
E,, = eSz EZET,
as previously done for the problems of stretching, bending and twisting [3]. With this we consider separately solutions for the case of open-cross-section shells and for the case of closed-cross-section shells. For both cases we can assume that the starred (z-dependent) portion of the state of stress and strain is given by corresponding results in [3), upon replacing (N, M,, M,, T) by (0, Qy, Q,, 0). In what follows we determine the barred (z-independent) portion of the stress and strain state, separately for open and for closed-cross-section shells. For the former case we list, in particular, the results for the special case of a flat plate. Open-cross-section shells Since S* = R = m, = is, = S = 0, we have from equation w,, are explicitly given by 5
5 jq, = -
2M F ds, s s1
(35b) that @_ RX and
15, = 0,
fl,, = -
s s1
N: ds.
(48)
We next use -equations (46) together with the relations mr = liT,+$~&?~ and %?r= xZ3-jkE, to express fl,, R, and Rr in terms of L, Z,, R,, and of the quantities given by (48). After some transformations, the barred portion of equations (46) leads to relations which, with a simplification basic on the stipulations D = 0(Eh3), C = O(Eh), and within the range of applicability of shell theory k2h2 C-CI, Ikcz,l<
and Nf ds.
(49d
W. T. TSAI
1170
In this ii,,, ii; and e, are given by (35a), rewritten here in the form Ez = i + ii,y + 2,.x +
’ (2&T- a(*~,)ds, j 51
xz = x,y’ - 2,x + %*I,)
%s = 5+
(50)
S
j s1
x: ds.
Equations (49a, b, c) together with (50) in turn are introduced into (10% 6. c, d’) to form a system of four simultaneous equations for E, $ ii, and ii,. We list here, in particular, the system of equations for a Jut plate, for which s = x, x’ = 1, y = y’ = 0, and s1 = -b, s2 = b, X
b
b
1
C,, x dx = -
C,, dx + ii,
N: dx + C,,
$
si -b
-b
-b
b
sT
j
(51a)
-b
b
i
J
Czz.x dx + x,
4
N: dx + CL: s -b
-b
Dzz-D=I$,d:
-
nZdx-2D,
4sJ’
_b M+,dx
1
dx, (51~)
= j;b[.XM:-4(DTT-~)j~b~:dx-2(D,;-~)1*1 +&
x J 1 M*,dx
D,,
(514
dx.
-b
We note that the constants B and R,, depend only on the stretching stress resultant N: and the shearing strain resultant .sF, while ii, and @depend on the twisting and bending stress couples Mf, MT and the twisting and bending strain couples r*, xt. Having obtained I, ii,, ii, and 6, it becomes a simple matter to determine &, gz and ii,, from (50) and 1Gi,, R, and liErzfrom (49). The remaining quantities ET, 8, and X, which incorporate the same simplification as (49a, b, c) are then obtained from . z, =
-
D,,X,+ D,,2Z,, --
DS
1
*
I
Dam -b
2M*, dx,
(52a)
(52b)
(52~)
On the problem of flexure of anisotropic cylindrical shells
Closed-cross-section
1171
shells
In order to simplify the result for stresses and strains for closed-cross-section shells, we start with the same stipulations as in [3] that we may assume M,, M, and N, negligible in the constitutive equations (46). With this, we have now as effective constitutive equations MT=O,
M, = 0,
1 X, = -MS,
(53a)
D
and
(5W
We note that by letting M, = 0, the possibility of treating the open-cross-section shell as a limiting case of the closed-cross-section shell is precluded. In the following we determine barred stresses and strains in terms of starred stresses and strains. We first have the barred portion of M,, E: and N,,, as in equations (35a. b) with MT = S* = 0.t written here as s
R,
Iv,, = s-
= R+rn,y+iQ,
N: ds.
s SO
(54)
& = I+z,y+x,x+
‘(2+~*r,)ds. i SO We now introduce equations (53% b) together with (54) into the overall equilibrium equations (10% 6, E, a’) and the strain integral relations (98, 6, S, a’) to form, as -- a special case of the system (43) and (45), a system of eight equations for the constants M, N,, n,., S, E, T&, 2, and 6. This system of equations which with coordinate axes chosen so as to make $ C’Jx, ~1:x-y) ds = (0,0,O)appears in the form ~~CI,ds-~~(C,,/C,,)ds
= -$
[~~(28:_-Z’I.)ds+(~~N:ds)/C,i]C,,ds.
.AX C,:L’~ ds - S (C&,,)y P P =U x,
Czzx2 ds - S
P
(554
ds S(2c:-z*r.)ds+(~N:ds)/C,r]C,,Vds, SO
(C,,/C,,)x
(5%)
ds N:ds)jC,,]C,,xds,
(55c)
(55d) t S*( s Nz) = 0 is a direct consequence of equation (48d’) in [3], with T set equal to zro..
w. T.
1172
TSAl
and (%a)
(56~) - 2/Q -t-.i * (C,:,‘C,,) ds i- 2, $ (Cz&,,)y
$
ds + R,
(c=z;cr_)X ds+S$
(1 -$p]g
It is apparent that equation (55d) gives S explicitly in term, of IV!, Introducing this result into (55a, b, c) we obtain B, zX and s$.. We then use equation (56d) to express ji in terms of XT, .$, E$, CL*and NT. With the determined results for S, 1, ;7, and x,, equations -(56a, b, c) then become three equations for M, N, and w,. Having determined the eight constants, it becomes a simple matter to obtain the barred measures of stress and strain upon making use of the first integrals, (54) together with ii,, ii,, and Rj- in (35a, b), and of the constitutive equations (53a, b). Among these results, we are particularly interested in the distribution of the shearing stress resultant %,,( = N,, = N,, because N,*Z= MT = 0) since an explicit result for this has, so far as the author knows, not yet been given. We first introduce S from (55d) into (54) to obtain (57) Here the initial arc length .sc in (57) is arbitrary because N,, is independent of it. We now rewrite equation (57) in more explicit form, upon making use of N: = C,,E: = C&* +x:yf X:X) and of E* = 0, K: f C,,y’ ds = Q, and X: $ Czz,~’ ds = QX.t By doing this, the result, with a suitable integration by part and with Q, set equal to zero for t The values of E*, x$ and x: are obtained from equations (48a’. b’, c’) in [3] with N, M, and M, replaced by 0. Q, and Q,.
On the problem of flexure of anisotropic cylindrical shells
1173
simplicity’s sake, appears in the form
(58) It is of interest to note that N,, in equation (58) depends only on the constitutive coefficient Czz even though a.system of anisotropic constitutive equations (53a, b) has been used. ~n~o~rn closed-cross-secrioi
~ir~~~~rc~~i~~ric~l shell
We set x = asincp, J= -acoscp and ds = a dq, and replace (N, M,, M,, T) in [3] by (0, Q,., 0,O). In this way we obtain directly from equations (48a’, b’, c’, d’) and (51a, b, c, d) in [31, &* =
x*Y = a* -_ 0
Q x$ = ---L-
7cCzza3’
9
N% x -
M* =: Nfr = S* = 0,
Equations quantities
(59a, b) together N,: = 0,
with equations
Qy - D xa2 CszaZ.
(59b)
(47~‘) and (49a’, b’, c’) in [3] give for stress
(NT, N:, M:) =
Q, 7ca3
and for strain quantities
(61)
We now make use of equations (53) to (56) to write the z-independent measures of stress and strain as follows. The constants i, x,, 5, 6!, ii?, R,, R,. and 3 are, with so set equal to zero, given by -
& =
_ x, =cc=(),
(62) Jg_& ‘-xay--p
R=R,=S=O,
D \T
Stress resultants and couples are -
hi; = 0,
~,s,=$!2 za2 ’
jq,=: __!L -Q,xD a
7ra2 CSTa2 ’
(63)
Strain resultants and couples are
(64)
1174
w.
T.
TSAl
With (60), (61), (63) and (64), we have as expressions for measures of stress and strain
(63
and
jczs= -- Qy - x RU3
c,;
The results for NZ and N,, are of exactly the same form as those following from elementary considerations. The quantities N, and M, are of negligible magnitude, the same as for the problem considered in [3]. The quantities eZ, E_ X, and x, consist of two portions, the portion linear on z and the portion independent of z. The latter portion is due to the effect of anisotropy. When z/a >> 1, the effect of anisotropy on eZ, Ed, xZ and x, becomes unimportant. The strain resultant sr also consists of two portions. But in sr the anisotropic effect which occurs in the portion depending on z dominates. The strain couple icZscomes out independent of anisotropic constitutive coefficients. The results which are probably of greatest interest for the present case are expressions for the displacement components u,, u, and uy. With x,, and y, set equal to zero and with uZ,uXandvyassumed tovanishaty = z = O,weobtainfromequations(22a)and(24a, b),
W’a) (6%)
z3
vy =-a
Qy
na3Czz
z
Q,x2; Qy-v RU3Cs,
RU2CsT’
WC)
Equation (67a) indicates that the effect of anisotropy on U, is not so important as the first term when z/a is large. Equation (67b) shows that the leading term of U, depends on z* and is due to anisotropy. Comparing this term with the second term we find that the lateral contraction due to the constitutive coefficient C,, becomes of secondary importance when z/a >> 1. Equation (67~) indicates that the effect of anisotropy on u, represents a distortion which is unimportant insofar as the displacements for the entire shell are concerned. In order to see the effects of Crr and C,, on uyr we first observe equation (67a). Since the last term in (67a) represents a rigid body displacement, we may rotate the entire beam about the -x-axis so as to make u, = 0 at the plane z = 0. When this has been done, lh,,‘i?y
On the problem of flexure of anisotropic cylindrical shells
at z = 0 becomes zero and consequently We then obtain u,, with the distortion
1175
u,,is increased by a quantity Q,,z(l/CT7+ l/C,,)/lra. term in (67~) discarded and with a2 -x2 = y2, as
z3 Q, " = --if 7ra3C,,
w-9
In the parenthesis of equation (68), the second term represents the well-known effect of shearing force. The last term represents the effect of lateral contraction. The displacement component ux, with the distortion term discarded, is listed in the form
Remarkably, the effect of anisotropy on v, is large in comparison with the effect of shearing force on u,,, yet is small in comparison with the first term in (68) when z/a is large. Acknowledgemennl-The
author is deeply indebted to Professor E. Reissner
for his guidance throughout thiswork.
REFERENCES [l] W. GUNTHER.Analoge Systcme von Schalengleichungen. Ing Arch. 30, 160 (1961). [2] E. REISSNER. On the Foundations of the Theory of Elastic Shells. Proc. I Ith fnl. Cong. uppl. Mech. (Munich), pp 20-30. (1966). [3] E. REI~~NER and W. T. TSAI. Pure bending. stretching and twisting of anisoptropic Mech. 39. 148 (1972). [4] S. TIMOSHENKO and J. N. GCKJDIER,Theory of Elasticiry. McGraw-Hill (1970).
cylindrical shells. J. appl.
[5] E. REI~~NERand W. T. TSAI.On the Determination of the Centers of Twist and of Shear of Cylindrical-Shell Beams, J. appl. Mech. Preprint Paper No. 72-APM-XX. [6] A. E. H. LOVE,A Treatise on the Mathematical Theory of Elasriciry. Dover (1944).
(Received 28 Seplember
1972)
A~T~~KF-MCCJICJI~CTC~ 3anaya 0npeneneHHa HanpmKewiA H ne@opMauHFi B aHm30TpHnHblx, ynpymx. ocedo Ol.lHOpOZlHblX,TOHKOCTeHHbIX obonorrax, nOnBepXCeHHblX JlefiCTBHbIpaBHbIX H npOTHBOnOJlO)I(HO HallpaBJleHHblX nOnepe'lHbiXKpaeBblXyCWlHiili JlO6aBO~HbIXMOMCHTOB,~JIa O6CCne'ieHHa~OJIHOrOpaBHOBWHR. MeTon peuiekm oCHOB~H Ha pa3nomeHHu HanpnxeHHB H ne+3puauHiI a YneHbl wmw~o 3amtcmble JlHHefiHO OT OCeBOti KOOpDHHaTbI H 'IaCTHYHO Hc3aBWHMbleOTOCeBOfi KOOpnHHaTM. HaRBonee 3HaYHTenbHbIM acneKToM pa6oTbl im.naeTcfl o6yxneHue s@@eKTa anu30Tponnm MaTepmana. fi.aloTcn unn~TpanmH 06m~x @opMyn TeopHH MR xnacca 06onouer, H3roToaneHHblx w3 "06blKHoBeHHoro" MaTepHana. Llnn 3Toro cnyrar, uonyqaroTcn +opMynbl B SIBH~B $opMe,KoTopwe Kacawcn tieKosopblx THnon o6onoseK, KaKOTKpblTOrO TaK H 3aMKHy~OrO ilOIlepe'lHOrOCe'leHHK.
J. Solids Svucrures,
1973. Vol.
9. pp. I I55 to 1175. Pergamon
Press. Printed
in Great
Britain
ON THE PROBLEM OF FLEXURE OF ANISOTROPIC CYLINDRICAL SHELLS?
Institute of Applied Mathematics, Tsing Hua University, Hsinchu, Taiwan, China
Abstract-The problem of determining stresses and deformations in anisotropic, elastic, axially homogeneous thin-walled shells subjected to equal and opposite transverse end forces, and additional moments to assure overall equilibrium, is studied. The solution procedure is on the basis of a decomposition of stresses and strains in terms of a portion dependent linearly on the axial coordinate and a portion independent of the axial coordinate. The most significant aspect of the work has to do with the analysis of the effect of anisotropy of the material. The general formulae of the theory are illustrated for a class of shells consisting of an ‘.‘ordinary” material. Here explicit formulae are obtained for certain types of open- as well as closed-cross-section shells.
INTRODUCTION paper considers elastic axially homogeneous anisotropic cylindrical shells subject to transverse end forces, together with the corresponding bending moments to assure overall equilibrium, as a problem on extension of the classical St. Venant theory of flexure. The differential equations of equilibrium and compatibility are special cases of general linear shell equations given by Gunther [ 11, and the constitutive equations are special cases of equations given by Reissner [2]. Important characteristics of these equations are the inclusion of moments turning about the normals to the middle surface.of the shell, and incorporation of the effect of transverse shear deformation. The effect of anisotropy on the distribution of stresses, strains and displacements is the particular concern of this paper. In attempting a solution of the flexure problem, we start with a formulation of the overall equilibrium conditions for the shell acted upon by transverse end forces and the associated equilibrium end bending moments. We conclude from these that in its simplest form the solution of the problem should consist of two portions. The first portion, due to the flexural bending moments, depends linearly on the axial coordinate. The second portion due to the flexural forces is independent of the axial coordinate. Investigation of the general system of differential equations and its associated boundary conditions reveals that in fact the two indicated portions are all that is necessary for the solution of the problem of axially homogeneous cylindrical shell flexure, upon appeal to an appropriate version of St. Venant’s principle. In the following we first indicate the procedure of decomposing the problem into the two mentioned portions. We then proceed to indicate the solution procedure for both portions. The result for the axially dependent portion is directly given upon making use THIS
t This report represents a chapter from the author’s Ph.D. dissertation (UCSD, May 1972, Advisor : Prof. E. Reissner). The author’s work has been supported by the Office of Naval Research, Washington. D.C. $ Associate Professor. 1155
W. T. TSAI
Il.56
of an appropriate .interpretation for the result of the problem of pure bending, stretching and twisting, as recently presented by Reissner and Tsai [3]. The result for the axially independent portion is then obtained as function of the quantities of the axially dependent portion.
FUNDAMENTAE
EQUATIONS
We take as curvilinear coordinates the circumferential arc length s and the axial distance z. The coefficients of the linear arc element are a, = ct, = 1, while the curvatures are l/R,, = 1/R = drplds = k and l/R,, = 1/R,, = 0. Designating stress resultants by N,, N,,, N,, N,,, Q,, Q, and stress couples by M,S, M,,, M,,, M,,, P,, P,, with cmresponding strain resultants E,y and strain couples x, I, we have as six equilibrium equations and six compatibility equations in accordance with Gunther [ 11, N,, + N,,, +
x*z*.l -x,,,,kl, = 0, A:.s -~j,z+kx,, = 0,
kQ,= 0,
Q,,,+Q,,,-Ws = 0%
’
&II,S -&,,,,-~;iZ = 0, xZJ,S --x ss.* = 0,
M,,s+M,,,, -Qs = 0, N,,., +Nzz.z = 0,
G&W Oa,bJ Pa,b)
Ezs.3 -E,,,, + ky, - A, = 0,
(5a, b)
yz,s- ys,z- kc,, + x:s - x,, = 0.
@a, W
M,,,+Mzz,:-K-Q,= 0, 2,,, + Pz,zf kM,, + N,, - N,, = 0,
(la, W
The twelve equations fla)-f6b) which involve twelve stress quantities and twelve strain quantities are here complemented by twelve constitutive equations, written in the form
(NS,N,~,,N,,Q,Q,M,,M~~M~*M~~~P=~ = 4~ 55 ELI E51 6zs Y5Y0x5~~z,wzJs~,~~ (71 where A is a 12 x 12 matrix, the elements of which are given functions of the arc length coordinates. In addition we have twelve strain~ispla~ment relations involving t~nslationai displacements uS, IA,, w and rotational displacements (~3,cp,, o. These relations are
and Es, = u z,s -0, x sz
=
9, ,f - ko,
Ezs =
x,,
=
us,,+ a,
yr = W,Z+-9zr
9s.: t
1, =
@b) 0,s + k9,.
Having stated the appropriate system of differential equations, we now consider the associated boundary conditions for ends z = constant of the shell and for boundaries s = constant of the shell. There are altogether twelve boundary conditions for s = constant. For clarity’s sake we consider these conditions separately for open- and closed-cross-section tubes, as follows. For the open-cross-section tube with longitudinal boundaries s = s1 and s = sir
On the problem of flexure of anisotropic cylindrical shells
115-l
we assume that the longitudinal boundaries are traction-free. This means that we prescribe the vanishing of the three stress resultants N,,, N,:, Q,, and of the three stress couples M,,, M,,, P, for s = s1 and for s = s2. For the closed-cross-section tube, we assume that the stress measures N,, N,,, Q,, M,,, M,,, P, and the displacement measures us, u,, w, cps, v,, w altogether are twelve univalued quantities. Since our interest is essentially in the determination of stress and strain quantities, we replace the six displacement univaluedness conditions by six conditions expressed in terms of strain quantities. We find with the help of the strain displacement relations @a, b) that the univaluedness of displacement conditions are equivalent to two conditions of univalued E,,, yz and to four strain integral relations 0=
x,, ds,
0
('%,Y f
=
O-
0 =
Ptx,,xf
w’ + Y,Y’)ds, E,,Y’ +
Y,x’)ds,
[E,, - (x,,y’ + i&,X’)X+(x,*x’ - i,y’)yJ ds.
(9c)
(94
We now turn to the consideration of boundary conditions for sections z = constant. We appeal to St. Venant’s principle and replace the local conditions of prescribed edge stresses by overall equilibrium conditions as follows, 0
(104
Qg=Jt N,,y - Mzrx'+ Pzy’) ds,
(lob)
Qxz=
(1Oc)
0 =
I
(N,,x
CM,, + (N,& - Qzx’)x - (Nz,x’ + Q,Y’)Y] ds,
Q, = JW,g~-
Q, =
+ M _y’ + P:x’) ds,
JVW+
QA ds, Q,Y')
ds.
(104
(lf)e)
t1w
In this, Q, and Q, are the prescribed overall transverse forces, Q,+Tand Q,,z are the prescribed overall bending moments, and equations (lOa, d) are the statement of vanishing of axial force and of axial torque. We note that the six integral relations (1Oa) to (10f) are equivalent to four only, consisting of the relations (lOa, b, c, d). The two relations (lOe, f) may be shown, by means of an introduction of (4a) to (da) into the z-derivatives of (lob, c) and by suitable integrations
W. T. TSAI
1158
by part, upon making use of the previously indicated boundary conditions M,, M,,, P, for s = constant, to be equivalent to equations (lob, c).
for N,,. Q.,,
N,,,
DECOMPOSITION
OF STRESSES
AND STRAINS
Given the form of the boundary conditions (10) and the fact that the differential equations (1) to (6) together with (7) and the remaining boundary conditions are homogeneous, we realize the possibility of the following decomposition of the state of stress and strain (N, Q, M, P, E, y, x, i)(s, z) = z. (N*, Q*, M*, P*, ---+(N,
E*, y*, x*,
i*)(s)
Q, M, P, I, 7. 2, I;)(s).
(11)
Introduction of (11) into (1) to (6) and (lOa, b. c, d) leaves as system of equations for the z-dependent quantities NZ.,+kQ:
= 0,
x& - ki;
= 0,
(la*, b*)
Q:,-kN:,
= 0,
E.$ + kxtz = 0,
(2a*, b*)
M&--Q,*
= 9
(3a*, b*)
E*_ I-.s -i:=o,
N:,., = 0,
x*ZS.S= 0,
(4a*, b*)
M,:,-kP:--Q:
= 0,
E;_ + k-$ - ij’ = 0,
(5a*, b*)
P&+kM:!+N:z-N;s
= 0,
_i:,. - kc;, + x;, - x,: = 0,
(6a*, b*)
and 0 =
N;, ds,
(lOa*)
Q, = j(~:,y-~‘x.+Pfy.lds.
(lob*)
Qx = s(N&x
(lot*)
0 =
s
+ M,:y’ + P:x’) ds,
[M;,+(N&y’-
Q:x’,x -(N,*,x’+
Q:y’)yl ds.
(lOd*)
At the same time we obtain as system of equations for the z-independent -kr:,
quantities
= it;=,
iv,,., + hQr = - NZ,
L
Q,.,-kg,,
= -Q:,
A,,,+ kn,, = A_;,
m,,/-Q,
= -ML
m,=,, = - NZ, J&Z., - kPs--QQ, = -M;,, PS,.,+ k&f,, + IV,, - KJ,, = P:,
&
(l&1;)
(2% 6)
- AZ= E,:,
(39,6,
j&S = x:S,
(4% 6)
E,,,, + k?, - i, = E;, 7z.s - kl,, + x,, - ii,, = y:,
(5% 6) (69,li)
On the problem of flexure of anisotropic cylindrical shells
1159
and 0 = J
0
=
0 =
0 =
r
N,, ds,
s s r J
(102)
(&,y - !i&,x’ + F:y’, ds,
(106)
(m,,x+ M,,y’
(1W
+
Pzx’)ds,
[R,, + (N,,y’ - Qzx’)x- (m,p’ + Q,y')y] ds.
(loa)
It is obvious that equations (la*) to (6b*) and (lOa*, b*, c*, d*) are of exactly the same form as equations (la) to (6b) and (16a, b, c, d) in [3] except that the places of the stretching force N, the bending moments M, and M, and of the twisting moment T in equations (16a, b, c, d) in [3] are taken over by 0, Qy, Q, and 0, respectively. In addition to this, the constitutive equations and the boundary conditions for the starred quantities are also equivalent to the corresponding formulas in [3]. From this follows that the results for the problem of determining the starred measures may be stipulated to be directly given in the same form as the corresponding results for the problem of pure bending, stretching and twisting in [3], upon replacing N, M,, M,, T by 0, Q,,, Q,, 0, respectively. We next turn to the determination of the barred quantities on the basis of equations (la) to (66) and (10% 6, E, dj, together with the appropriate boundary conditions and constitutive equations.
FIRST
INTEGRALS
OF EQUATIONS
Let s0 be a specified arc length coordinate. in the form
FOR BARRED
QUANTITIES
We begin by integrating
equations (4% 6)
s iV,, = s-
N:, ds,
zzs = cl+
I so
x: ds.
(12a b)
where 3 and E are constants of integration. Next, we integrate equations (1% fi) and (2g, b), with the help of the relation ( ),, = 4 ),cp,with equations (5a*) to (6b*), and with appropriate integrations by part. In this way, we find the important result that we can express iJ,, W,,, Q, and 2, explicitly as
iv,,= X,x’ Q, =
-
Nxy' +
ivyy’ -
s*r,- PZ,
m,x’ - S*r, - Mzz,
zsr = x,y' - Zxx' + a*r, + y:,
(13a b) (14a, b)
In this rn 3 xy’ -x’y, r, c xx’ + yy’ ; S* E N,*, and a* = x,: are given constants ; and m,, m,,, ii, and zi, are four additional constants of integration.
W. T. TSAI
1160
We now introduce stants of integration,
(14a, b) into (3% b), and obtain with M and E as two further con-
R,, = ,V,y+R++M-
2,, = z,y+ii,,x+++
’ (M:, + M:, + S*r,) ds,
s SrJ
’ (.s,:+s;,--*r,)ds. 150
(15a)
(15b)
Having the system of first integrals (12a) to (15b), we are left with four first-order differential equations (52) to (66), which involve the eight quantities R,,, P*, Q,, mzs, i,, ,7,, 1, and gsz, and which can not in general be integrated explicitly.
BOUNDARY
CONDITIONS
FOR BARRED
QUANTITIES
Equations (la) to (66) are a twelfth-order system of ordinary differential equations. We look for twelve boundary conditions. We have, in accordance with the previously given fundamental equations, four stress integral relations (log, 6, E, d) and twelve boundary conditions for s = constant. Among these, four of the boundary conditions for s = constant are extraneous, leaving us twelve required boundary conditions. We show this by considering separately the cases of open- and closed-cross-section tubes. For the case of closed-cross-section tubes, we have that S*, Mz and P: are univalued, and f N:, ds and f (M,*,+ Mz’+ S*r,) ds vanish. Introducing this into (12a) to (15a) we find that N,,, m,,, 0, and R, are the measures which automatically satisfy the univaluedness conditions for arbitrary s. Upon excluding the conditions for these four quantities we are left with twelve boundary conditions which consist of the four conditions of univaluedness of R,,, pz,, b,, and 7,, the four stress integral relations (10% 6, c, a), and the four barred strain integral relations in (9a, b, c, d), written here in the form
Pii,,
ds = 0,
(ii,a + &.xX’+ ;I,y’) ds = 0,
(96)
- I,,y' + *yp’) ds = 0,
(9E)
( iirsx
~,[E,-(~~,y’+X,r’)x+(ii,,x’-X,y’)y]ds We note for subsequent reference the following important (1Od) and (9d)
s
(R,, + &if,,+ Sr,) ds =
(9%)
= 0. transformation
NZ, ds + r,M:, - r,,PZ ds,
(9d) of equations
Cloa
1161
On the problem of flexure of anisotropic cylindrical shells
and (&,+E,,-Er,)ds
= +l
rn ’ x,: ds - I& - r,y: J so
Further simplifications are possible upon noting the It remains to formulate the boundary conditions s0 = si and use the facts that S* = 0 throughout, vanishing of m,,, m,,, Q, and R,, for s = si in (12a)
ds.
(9d’)
geometrical formula 2A, = $ rn ds. for open-cross-section tubes. We let and M: = P: = 0 for s = si. The to (15a) is then equivalent to
s = m, = NY = ii;i = 0,
(16)
and we are left with the explicit results
(17) Q, = -M,*,,
R,, = -
’
(M,:+M:,)ds.
I s1
It is apparent that the boundary conditions of vanishing m,,, m,, Q,, R,, for s = s1 are satisfied automatically because of the boundary conditions for the starred quantities for open tubes involving 52 N:, ds =
s SI
” (M: + M:,)ds = 0 I 51
and
M,*,= P$ = 0 for
s=sl.
Recognizing this, we are again left with twelve boundary conditions involving the four and results S = W, = W, = R = 0 in (16), four conditions K&: = ij;, = 0 for s=s, s = s2, and the four conditions (lOS6, E) and (1Od’). We note in conjunction with the result (17) that the quantities m,,, n,,, Q, and &?,, together with the result that Nzz, N$, Q: and Mz vanish throughout, and with the decomposition (11) provide explicit results for the actual quantities N,, N,,, QS and M,, for open tubes, namely N,, = -
’ N,*, ds,
sSI
Qs = -Wz,
N,, = -P$,
(18) M,, = - '(M:,+ M:,)ds. I s1
Equations (18) show that N,,, N,,, Q, and M, are always independent of z. Furthermore, * = 0 in (18) means an automatic satisfaction of the Kirchhoff boundary the relation Q, + M,, = 0 for the case that the transverse shearing strain yz is stipulated condition Q, + M,,,, to vanish.
CALCULATION
OF DISPLACEMENTS
In what follows we consider displacements of the points of the cylindrical she11 surface for the problem of fiexure. As for the problems of pure stretching, bending and twisting we are led to distinguish two types of displacements, those which translate and rotate the
1162
w. T.
fSA1
cross section of the shell as a rigid body and those which represent a distortion of the cross section, axially and laterally. In both types, two different classes of elements are involved. The class of “ordinary” elements which also occur for orthotropic shells and the class of “unusual” elements which are due to the effect of non-orthotropy. We start to calculate displacements by considering the strain displacement relations (8a, b) which with the help of the decomposition (11) assume the form .s$r + E, = u,,, + kw, x2
EDZ + i,, x2
+ %* = rp,s
+ ii,,
= u,*, ,
y:z + 9, = cp,+ w,, - ku,,
= to,,,,
/i;‘?+i,
(194
= O,:’
and
In these ii,, ii,,, X,, E,, are explicitly given by (12b) to (1Sb) and x&, xzz, Jr, E$, with r*, x:, x;, E* as four constants, are x;S = c1*,
x,*, = x;y’-x:x’,
2: = x: y ’ + Xy*X’,
&* ** = x:y + x;x f &*.
(20)
We now introduce (20) together with the first strain integrals (12b) to (15b) and with the strain differential equations (5b*, 6) and (6b*, 6) into the strain displacement relations (19a, b). An observation of the z-independence of all starred and barred strain components leads to the following expressions for the six components of rotational and translational displacement
cpr = +(x;y'-x:x')22+(j2yy'- zxx’ -t a*r,)z + yz + cq(x -x0)x’ +r,jX;dS-x’
j (x,gx - &;y’ + y:x’)
ds - y’
j
+ (y - y,)y’]
(~3 + s&x’+ yty’) ds,
w = +(x:y’+ x;x’)zZ +(ii,y’+ izyx - a*r,)z+ E,, - ji[(x - xo)v’ -(y - yo)x’] -r.jx;ds+y’ and with k,
j (x$x - &:a’ + y:x’) ds - x’
s,, YJ = CG s& ~:)s+&,
j
($y + E:J’ + y:y’) ds,
b, &), P
u, =
+(x:y+ x;x+E*)z2“k(X*y+ a+ + L)z+@(x,y-
ypx)
+
J(E,,+
+ fr& + r,yr) ds + y
E,, -
JW~)ds
(~3 + E$..’ + y:y’) ds, (x&x - s$y’ -I-y:x’) ds -x j j I US= - &x:y’ + $X’)Z3 - &z,y + Z#’ - r*r#Jz2 + E[(x - x,)y’ -(y - yO)x’]z +r,
jx,,ds-y' j(x&-s&+y&)ds+x’
j
(x~fs,~‘+yy’)ds,
(22a)
(22b)
1163
On the problem of flexure of anisotropic cylindrical shells W = -4t$Y’
- X,*x’)z3 -
-r+,ds+x’S
)(X,y’-
S&x + a*r,)z2 - E[(x -x,)x’
(xd - ~,a’ + ysx’) ds + y’
+ (y - yo)y’]~
(x,,y + 8,~’ + y,y’) ds.
(22c)
We note that the translational components u, and w can be resolved into two displacement components u, and oYin x and y directions upon introducing u, = u&Xx’ 4 WY’,
tiY= l&y’- wx’.
(23)
The result is given by u*=
-@z” -&3X,+ a*y)z2 - E(y- y&z - y
f
x, ds +
I
(xd + E&’ + ya’) ds,
(24a)
In equations (24a, b) the terms without integral sign represent displacements with the tube cross section translating and rotating as a rigid body and the terms with integral signs represent the effect of distortion. Furthermore, since@,, E,, v,) I (~2, E& y:)z + (3i,, i35, T,), the axially linear dependence of the effect of distortion involves the well-known effect of transverse shear on the deflection of the beam, as stated by Timoshenko and Goodier [4] for the problem of isotropic sheet flexure. The remaining measures, involving Z,, E, and jj,, are the unusual terms which vanish for the case of orthotropic materials. We also find from the terms without integral sign in (24a, bf that rigid body translation and rotation involve terms proportional to z3, z2 and z. In these, the z3-terms represent the deflection due to flexural bending, and the z-terms represent the deflection due to axial twisting produced by the flexural forces7 [6]. We designate the r2-terms as “unusual” since they represent deflections which vanish for the case of orthotropic materials. We note in conjunction with the above observation for u, and u, that the axial displacement ~1,also involves both ordinary and unusual terms. The z2 and z-independent terms are of course the ordinary terms representing the displacement due to bending and warping. The term linear in z is the unusual term which is again due to anisotropy because the coefficient of this term is, with equation (lSb), the axially homogeneous strain measure & which comes from the constitutive coupling coefficients for anisotropic tubes.
METHOD
OF SOLUTION
FOR THE GENERAL
CASE
Equations (12a) to (15b) express the four stress quantities %,, iJ,, &, If;i, and the four strain quantities ii,,, ji,,, X,, E,, in terms of eight constants of integration. Equations (53) to (66) are four first-order differential equations allowing us to introduce four additional constants of integration. The total of tweIve constants of integration is to be determined in terms of integrals involving starred quantities by means of the four stress integral relations (103, 6, E, a’) together with the four univaluedness conditions for &?,,, g,, 1,,, 7, and the four strain integral relations (9s. 6, E, d’) for closed-cross-section tubes, or tot For the case of orthotropic tubes, axial twisting may be eiiminated by letting the flexural forces act through the center o~twist, Reissner and Tsai [5J.
W. T. ,-SAI
1164
gether with the eight traction conditions along the longitudinal edges, NSZ= N,, = Q, = @, = 0 for s = s1 and R,, = P, = 0 for s = si , and for s = s2, for open-cross-section tubes. In what follows we wish to indicate the solution procedure for obtaining the barred quantities for the case of the closed-cross-section shell, in such a way that the open-crosssection shell solution is included as a special case. Considering the solution procedure for the general case of anisotropy, we first rewrite the constitutive equations (7) for the barred contributions in the form
I
3
+B
(25a b)
In this, the eight quantities B,,, R,,, i,, ii,, and R,,, m,,, Q,, n,, are given by the first integrals (12) to (15). We further write the differential equations (5% 6) and (6&b) in the form
Wa b)
and use this to transform (25b) into
(27)
On the problem of flexure of anisotropic cylindrical shells
1165
We now have (27) as a fourth-order system of simultaneous differential equations for P,, RsZ, 7, and 1,, . The nonhomogeneous part of this involves the known starred quantities Pf, M,*Z, y:, t$ and the giuen barred quantities EZZ,ii,=, %,, &) R,,, m,,, Q,, p,,. Equations (27) are to be solved subject to the four boundary conditions of univaluedness of P_ Ri,,, -7, and i,, for the case of the closed-cross-section shell. Having solved (27) we must introduce the result into the constitutive equations (25a, b) to express, with the help of the first integrals (12a) to (15b), all quantities on the left-hand sides as functions of the eight constants of integration 3, E,, m,,, R, d x,, gi,, i, as well as of certain functions of the starred quantities appearing explicitly and implicitly in (27). These expressions in turn are to be introduced into the set (10% 6, c, a’) together with (98,6, E, a’), thereby producing a system of eight simultaneous linear equations for the determination of the eight constants 3, m,, m,,, R, Cr,x,, z,., i in terms of integrals of starred quantities. The fourth,-order system (27) is reduced to a zeroth-order system if we stipulate, in conformity with observations in [3] that the shearing strain component 7, and the moment stress resultant ps may be set equal to zero. With this, equation (27) reduces to two simultaneous equations for R,, and iZs of the form
(28a)
At the same time, the quantities QZ and A, are given by Q; = R;,+
M:,,
i, = &-&,:.
(28b)
In what follows we extend this general consideration to describe the solution procedure for the case for which the constitutive equations are taken in the form of the socalled conventional case.
METHOD
OF SOLUTION
In using the words “conventional stitutive equations are of the form
FOR CONVENTIONAL
CASE
cases” we have in mind cases for which the con-
iN,,N~NssM-_zM~Mss~ = Ac{~Zr~~&,s~,13(~)(sS),
(29)
together with ,,IZ -- .> Is = 0, Eis = E,, = ET,
P, = P, = 0, M,, = M,, = MT,
(3% b) (3Ia, b)
W. T. TSAI
1166
and with the definitions 2Nr = N,,+ N;,,
(32a, b)
2x, = x,, + XSZ,
Having equations (30a, b) we have from the equilibrium equation (6a), m,, = m,, + k&T,, and from the compatibility equation (6b), R,, = &,-k+, and therewith m p = m,,++klS;i,,
5iT = zLS--+kBT.
(3% b)
We now use the simplified subscript notations (N,, M,, E., x,) = (N,,, M,,, E,,, x,,). With this we introduce (33) into the barred part of (29) and this with the help of a suitable transformation gives
[ii]
=B~E;,,!+B~ [-:;I,[;;I=&[;zj+& [{IT
(34a,b)
where i,, Zz, ii,,, R,, m, and m,, are as in equations (12%) to (156). In order to simplify the result for the barred quantities, we rewrite equations (129) to (156) in the matrix form
-f,.
(35a, b)
where X
1 Y’
0
J 3
(i = 1, 2),
(36)
and where
(3% b)
On the problem of flexure of anisotropic cylindrical shells
Having equations (34) and (35), it remains to determine -grations 6 ii,, ii,,, &, M, N,,R,,and 3.
1167
the eight constants
of inte-
Open-cross-section shells
For an open-cross-section shell, the boundary conditions of vanishing traction for s=si and s= s2 give R = iir, = R,, = s = 0. The corresponding result for F,,, 15, and M, is given in (17) and can, with M,*,= M:, E M$ and P: = 0,and with S* = 0, be written as {MS--&&,}
= --j-i,=
- ~‘{ZM;ON:}ds. S1
It now remains only to determine the four constants of integration Z, and x,,. Evidently, this will be accomplished upon introducing into equations (10% 6, E, a’), written here in the matrix form
(38) E, ii,, ji, and 6 in L,, (34a), (35a) and (38)
(39)
where Ql is transpose of Qz and g, is a vector given by s g1=
ii
‘n
N:ds+r,M:
0 0 0 .
(404
so
Equations (39) directly give a system of equations of starred quantities of the form
for 6 8, ii, and ii, in terms of integrals
(41)
Closed-cross-section
shell
We now have the overall equilibrium R,, = IiTS, = M,, written in the form
equations
‘2A$ 0 0
(lOa, ‘6, E, a’), with P, = 0 and
1 =
Q g,
ds,
(42)
W. T. TSAI
1168
and this, with (34a) and (35a, b), and with four-by-four
matrices Ci, becomes
(43)
At the same time, we define s 92
=
P”
is
x:ds-r,s:
0 0 0 ,
50
1
@‘W
to write the strain integral relations (9% 6, E, al), with 7, = 0 and i,, = i,, = 8,, into the form
(W
Introduction
of ii,, 1, and ET from (34b) and (35a, b) into (44) results in the relations
=f k2 - Q&h
- &_I-,)1 ds.
(45)
Equations (43) and (45) determine the eight constants of integration in terms of integrals of starred quantities which are implicitly given by vectors f;: and gi. We note that the system of equations (43) and (45) for closed tubes may be reduced to a system of equations for open tubes upon appropriately interpreting the constitutive coefficients. Stipulating that an open-cross-section tube may be considered as if it were a closed-cross-section tube with an open part, we set in the constitutive equations (34a, b) the elements of Br equal to zero, and the elements of B, equal to infinite over the open part of the closed-cross-section. When this is done, the elements of C, and of the righthand side of (45) are finitet and the elements of C, become infinite, requiring that the constants 3, !‘@,m,, NY must vanish. Setting s = R = m, = NY = 0 into (43), we directly obtain a system of equations for the remaining four constants & 8, ii, and x,, of exactly the same form as equations (41) which follows from a direct consideration. t In view of equation (45). the statement that the elements of the right side of (45) are finite involves the constitutive matrix B,, the elements of which are stipulated to be infinite over the open part However, we have with the relations S* = I:: Nf ds = I:: ZM’, ds = 0, that the elements off, in (37a) for open-cross-section tubes vanish over the open part. With this, the elements of BJ, may be stipulated to be finite over the open part of the closed-cross-section tube. The integral of Q:BJt on the right-hand side of (45) therefore has a finite value confirming our earlier observations.
On the problem of fiexure of anisotropic cylindrical shells
EXPLICIT
1169
SOLUTION FOR A CLASS OF ANISOTROPIC SHELLS
In order to illustrate the nature of our general results, we consider shells with constitutive equations of the form E, =
M, = LX,
+ I&x, + D,r2~r,
(46) 2~~ =
$+9+$, Tt
in conjunction
Ts
MT
=
DTA + DTS% + DTT~XT,
TT
with P, = P, = 0,
I’* = 1’$= 0,
MS: = M,, z MT,
(47)
E,, = eSz EZET,
as previously done for the problems of stretching, bending and twisting [3]. With this we consider separately solutions for the case of open-cross-section shells and for the case of closed-cross-section shells. For both cases we can assume that the starred (z-dependent) portion of the state of stress and strain is given by corresponding results in [3), upon replacing (N, M,, M,, T) by (0, Qy, Q,, 0). In what follows we determine the barred (z-independent) portion of the stress and strain state, separately for open and for closed-cross-section shells. For the former case we list, in particular, the results for the special case of a flat plate. Open-cross-section shells Since S* = R = m, = is, = S = 0, we have from equation w,, are explicitly given by 5
5 jq, = -
2M F ds, s s1
(35b) that @_ RX and
15, = 0,
fl,, = -
s s1
N: ds.
(48)
We next use -equations (46) together with the relations mr = liT,+$~&?~ and %?r= xZ3-jkE, to express fl,, R, and Rr in terms of L, Z,, R,, and of the quantities given by (48). After some transformations, the barred portion of equations (46) leads to relations which, with a simplification basic on the stipulations D = 0(Eh3), C = O(Eh), and within the range of applicability of shell theory k2h2 C-CI, Ikcz,l<
and Nf ds.
(49d
W. T. TSAI
1170
In this ii,,, ii; and e, are given by (35a), rewritten here in the form Ez = i + ii,y + 2,.x +
’ (2&T- a(*~,)ds, j 51
xz = x,y’ - 2,x + %*I,)
%s = 5+
(50)
S
j s1
x: ds.
Equations (49a, b, c) together with (50) in turn are introduced into (10% 6. c, d’) to form a system of four simultaneous equations for E, $ ii, and ii,. We list here, in particular, the system of equations for a Jut plate, for which s = x, x’ = 1, y = y’ = 0, and s1 = -b, s2 = b, X
b
b
1
C,, x dx = -
C,, dx + ii,
N: dx + C,,
$
si -b
-b
-b
b
sT
j
(51a)
-b
b
i
J
Czz.x dx + x,
4
N: dx + CL: s -b
-b
Dzz-D=I$,d:
-
nZdx-2D,
4sJ’
_b M+,dx
1
dx, (51~)
= j;b[.XM:-4(DTT-~)j~b~:dx-2(D,;-~)1*1 +&
x J 1 M*,dx
D,,
(514
dx.
-b
We note that the constants B and R,, depend only on the stretching stress resultant N: and the shearing strain resultant .sF, while ii, and @depend on the twisting and bending stress couples Mf, MT and the twisting and bending strain couples r*, xt. Having obtained I, ii,, ii, and 6, it becomes a simple matter to determine &, gz and ii,, from (50) and 1Gi,, R, and liErzfrom (49). The remaining quantities ET, 8, and X, which incorporate the same simplification as (49a, b, c) are then obtained from . z, =
-
D,,X,+ D,,2Z,, --
DS
1
*
I
Dam -b
2M*, dx,
(52a)
(52b)
(52~)
On the problem of flexure of anisotropic cylindrical shells
Closed-cross-section
1171
shells
In order to simplify the result for stresses and strains for closed-cross-section shells, we start with the same stipulations as in [3] that we may assume M,, M, and N, negligible in the constitutive equations (46). With this, we have now as effective constitutive equations MT=O,
M, = 0,
1 X, = -MS,
(53a)
D
and
(5W
We note that by letting M, = 0, the possibility of treating the open-cross-section shell as a limiting case of the closed-cross-section shell is precluded. In the following we determine barred stresses and strains in terms of starred stresses and strains. We first have the barred portion of M,, E: and N,,, as in equations (35a. b) with MT = S* = 0.t written here as s
R,
Iv,, = s-
= R+rn,y+iQ,
N: ds.
s SO
(54)
& = I+z,y+x,x+
‘(2+~*r,)ds. i SO We now introduce equations (53% b) together with (54) into the overall equilibrium equations (10% 6, E, a’) and the strain integral relations (98, 6, S, a’) to form, as -- a special case of the system (43) and (45), a system of eight equations for the constants M, N,, n,., S, E, T&, 2, and 6. This system of equations which with coordinate axes chosen so as to make $ C’Jx, ~1:x-y) ds = (0,0,O)appears in the form ~~CI,ds-~~(C,,/C,,)ds
= -$
[~~(28:_-Z’I.)ds+(~~N:ds)/C,i]C,,ds.
.AX C,:L’~ ds - S (C&,,)y P P =U x,
Czzx2 ds - S
P
(554
ds S(2c:-z*r.)ds+(~N:ds)/C,r]C,,Vds, SO
(C,,/C,,)x
(5%)
ds N:ds)jC,,]C,,xds,
(55c)
(55d) t S*( s Nz) = 0 is a direct consequence of equation (48d’) in [3], with T set equal to zro..
w. T.
1172
TSAl
and (%a)
(56~) - 2/Q -t-.i * (C,:,‘C,,) ds i- 2, $ (Cz&,,)y
$
ds + R,
(c=z;cr_)X ds+S$
(1 -$p]g
It is apparent that equation (55d) gives S explicitly in term, of IV!, Introducing this result into (55a, b, c) we obtain B, zX and s$.. We then use equation (56d) to express ji in terms of XT, .$, E$, CL*and NT. With the determined results for S, 1, ;7, and x,, equations -(56a, b, c) then become three equations for M, N, and w,. Having determined the eight constants, it becomes a simple matter to obtain the barred measures of stress and strain upon making use of the first integrals, (54) together with ii,, ii,, and Rj- in (35a, b), and of the constitutive equations (53a, b). Among these results, we are particularly interested in the distribution of the shearing stress resultant %,,( = N,, = N,, because N,*Z= MT = 0) since an explicit result for this has, so far as the author knows, not yet been given. We first introduce S from (55d) into (54) to obtain (57) Here the initial arc length .sc in (57) is arbitrary because N,, is independent of it. We now rewrite equation (57) in more explicit form, upon making use of N: = C,,E: = C&* +x:yf X:X) and of E* = 0, K: f C,,y’ ds = Q, and X: $ Czz,~’ ds = QX.t By doing this, the result, with a suitable integration by part and with Q, set equal to zero for t The values of E*, x$ and x: are obtained from equations (48a’. b’, c’) in [3] with N, M, and M, replaced by 0. Q, and Q,.
On the problem of flexure of anisotropic cylindrical shells
1173
simplicity’s sake, appears in the form
(58) It is of interest to note that N,, in equation (58) depends only on the constitutive coefficient Czz even though a.system of anisotropic constitutive equations (53a, b) has been used. ~n~o~rn closed-cross-secrioi
~ir~~~~rc~~i~~ric~l shell
We set x = asincp, J= -acoscp and ds = a dq, and replace (N, M,, M,, T) in [3] by (0, Q,., 0,O). In this way we obtain directly from equations (48a’, b’, c’, d’) and (51a, b, c, d) in [31, &* =
x*Y = a* -_ 0
Q x$ = ---L-
7cCzza3’
9
N% x -
M* =: Nfr = S* = 0,
Equations quantities
(59a, b) together N,: = 0,
with equations
Qy - D xa2 CszaZ.
(59b)
(47~‘) and (49a’, b’, c’) in [3] give for stress
(NT, N:, M:) =
Q, 7ca3
and for strain quantities
(61)
We now make use of equations (53) to (56) to write the z-independent measures of stress and strain as follows. The constants i, x,, 5, 6!, ii?, R,, R,. and 3 are, with so set equal to zero, given by -
& =
_ x, =cc=(),
(62) Jg_& ‘-xay--p
R=R,=S=O,
D \T
Stress resultants and couples are -
hi; = 0,
~,s,=$!2 za2 ’
jq,=: __!L -Q,xD a
7ra2 CSTa2 ’
(63)
Strain resultants and couples are
(64)
1174
w.
T.
TSAl
With (60), (61), (63) and (64), we have as expressions for measures of stress and strain
(63
and
jczs= -- Qy - x RU3
c,;
The results for NZ and N,, are of exactly the same form as those following from elementary considerations. The quantities N, and M, are of negligible magnitude, the same as for the problem considered in [3]. The quantities eZ, E_ X, and x, consist of two portions, the portion linear on z and the portion independent of z. The latter portion is due to the effect of anisotropy. When z/a >> 1, the effect of anisotropy on eZ, Ed, xZ and x, becomes unimportant. The strain resultant sr also consists of two portions. But in sr the anisotropic effect which occurs in the portion depending on z dominates. The strain couple icZscomes out independent of anisotropic constitutive coefficients. The results which are probably of greatest interest for the present case are expressions for the displacement components u,, u, and uy. With x,, and y, set equal to zero and with uZ,uXandvyassumed tovanishaty = z = O,weobtainfromequations(22a)and(24a, b),
W’a) (6%)
z3
vy =-a
Qy
na3Czz
z
Q,x2; Qy-v RU3Cs,
RU2CsT’
WC)
Equation (67a) indicates that the effect of anisotropy on U, is not so important as the first term when z/a is large. Equation (67b) shows that the leading term of U, depends on z* and is due to anisotropy. Comparing this term with the second term we find that the lateral contraction due to the constitutive coefficient C,, becomes of secondary importance when z/a >> 1. Equation (67~) indicates that the effect of anisotropy on u, represents a distortion which is unimportant insofar as the displacements for the entire shell are concerned. In order to see the effects of Crr and C,, on uyr we first observe equation (67a). Since the last term in (67a) represents a rigid body displacement, we may rotate the entire beam about the -x-axis so as to make u, = 0 at the plane z = 0. When this has been done, lh,,‘i?y
On the problem of flexure of anisotropic cylindrical shells
at z = 0 becomes zero and consequently We then obtain u,, with the distortion
1175
u,,is increased by a quantity Q,,z(l/CT7+ l/C,,)/lra. term in (67~) discarded and with a2 -x2 = y2, as
z3 Q, " = --if 7ra3C,,
w-9
In the parenthesis of equation (68), the second term represents the well-known effect of shearing force. The last term represents the effect of lateral contraction. The displacement component ux, with the distortion term discarded, is listed in the form
Remarkably, the effect of anisotropy on v, is large in comparison with the effect of shearing force on u,,, yet is small in comparison with the first term in (68) when z/a is large. Acknowledgemennl-The
author is deeply indebted to Professor E. Reissner
for his guidance throughout thiswork.
REFERENCES [l] W. GUNTHER.Analoge Systcme von Schalengleichungen. Ing Arch. 30, 160 (1961). [2] E. REISSNER. On the Foundations of the Theory of Elastic Shells. Proc. I Ith fnl. Cong. uppl. Mech. (Munich), pp 20-30. (1966). [3] E. REI~~NER and W. T. TSAI. Pure bending. stretching and twisting of anisoptropic Mech. 39. 148 (1972). [4] S. TIMOSHENKO and J. N. GCKJDIER,Theory of Elasticiry. McGraw-Hill (1970).
cylindrical shells. J. appl.
[5] E. REI~~NERand W. T. TSAI.On the Determination of the Centers of Twist and of Shear of Cylindrical-Shell Beams, J. appl. Mech. Preprint Paper No. 72-APM-XX. [6] A. E. H. LOVE,A Treatise on the Mathematical Theory of Elasriciry. Dover (1944).
(Received 28 Seplember
1972)
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