I L L I NOI
S
UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
PRODUCTION NOTE
University of Illinois at
UrbanaChampaign Library
Largescale Digitization Project, 2007.
Theory of NonHomogeneous
Anisotropic Elastic Shells
Subjected to Arbitrary
Temperature Distribution
by
R. E. Miller
ASSISTANT PROFESSOR OF THEORETICAL AND APPLIED MECHANICS
Sponsored by
OFFICE OF NAVAL RESEARCH, DEPARTMENT OF NAVY
DEPARTMENT OF THEORETICAL AND APPLIED MECHANICS
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION BULLETIN NO. 458
© 1960 BY THE BOARD OF TRUSTEES OF THE
UNIVERSITY OF ILLINOIS
UNIVERSITy
(72549) , P 0RESS:
ACKNOWLEDGMENT
This investigation has been carried out in the
Department of Theoretical and Applied Mechanics,
University of Illinois, of which Professor T. J.
Dolan is head. The research was conducted in co
operation with the Office of Naval Research under
Contract NR 1834(14), Project NR 064413. The
author is deeply indebted to Professors H. L.
Langhaar and A. P. Boresi for their extensive as
sistance, both during the investigation and in the
preparation of the paper. The equations were
checked by Mr. Chu Shiang Chen.
CONTENTS
SUMMARY 7
INTRODUCTION 9
NOTATIONS 11
I. GEOMETRICAL PRELIMINARIES
A. Geometry of a Surface 13
B. Geometry of a Shell 14
II. GENERAL ANISOTROPIC SHELLS
A. Equilibrium Relations 13
B. StrainDisplacement Relations 16
C. StressStrainTemperature Relations 17
D. Strain Energy of the Shell 18
III. SPECIALIZATIONS OF THE GENERAL THEORY
A. Cylindrical Shell 20
B. Conical Shell of Circular Cross Section 21
C. Spherical Shell 23
D. Axially Symmetrical Orthotropic Shell of Revolution 25
IV. SPECIAL PROBLEMS OF CYLINDRICAL SHELLS
A. Axially Symmetrical Deformation of an
Orthotropic Circular Cylinder 29
B. Circular Cylinder Subjected to Temperature Which
Varies Through the Thickness 30
C. SemiInfinite Cylinder with End Moments and Shears 30
D. Numerical Example 31
V. DISCUSSION OF RESULTS 35
VI. REFERENCES CITED 36
SUMMARY
This paper presents a general theory for small deformations
of anisotropic elastic shells subjected to arbitrary temperature
distribution.
The shells are assumed to be homogeneous through the thick
ness except for the coefficient of thermal expansion, but are
otherwise unrestricted in homogeneity and isotropy.
The theory is specialized for arbitrary cylinders, circular
cones, spheres, and axially symmetrical orthotropic shells of
revolution. In addition, two special problems are analyzed.
INTRODUCTION
The behavior of structural elements subjected to heating has recently
assumed importance in many fields. Although most materials now in use
are at least approximately isotropic in nature, there are instances where
more general analyses based on considerations of anisotropy and non
homogeneity are necessary. Materials often become anisotropic in manu
facturing processes such as cold rolling or stretching. In addition,
temperature gradients may cause the elastic constants to vary, thus
introducing nonhomogeneity.
The theory presented in this paper is restricted to small deflections
of thin elastic shells. The Kirchhoff assumption, normals to the middle
surface remain straight, normal, and inextensional during the deforma
tion, is used throughout. Although they are assumed to be constant
through the thickness, the elastic coefficients of the material are arbi
trary functions of the surface coordinates x and y of the shell. On the
other hand, the thermal coefficients and the temperature distribution are
unrestricted.
NOTATIONS
X, Y, Z = Rectangular coordinates.
i, j, k= Unit vectors along the X, Y, Z
axes.
= Position vector of the point X,
Y, Z.
x, y, z = Shell coordinates based on the
lines of principal curvature. See
Section I B. The symbols x and
y are also used as arbitrary curvi
linear coordinates in Section I A.
s = Arc length.
E, F, G = Coefficients in the "first funda
mental form" of a surface. See
Eqs. (3) and (4).
S = Surface area.
A = Unit normal vector to a surface
at any point.
e, f, g = Coefficients in the "second funda
mental form" of a surface. See
Eqs. (8) and (9).
ri, r2 = Principal radii of curvature de
fined by Eq. (11).
M = Mean curvature of a surface de
fined by Eq. (10).
K = Gaussian curvature of a surface
defined by Eq. (10).
A, B = Positive functions of x and y de
fined by Eq. (13).
a, gf, 7 = Lam6 coefficients for orthogonal
curvilinear coordinates. See Eqs.
(19) and (20). The symbol a is
also used as the apex angle of a
cone in Section III B, but no con
fusion should result.
h = Thickness of the shell.
Ox, y, oz]x
Txy, Tyz, TzX\
= Stress components at point (x, y,
z). See Fig. 1.
N, . . . , N
Qx, Qy = Tensions, shears, twisting mo
MX, ..., Mlyxj ments, and bending moments de
fined by Eq. (25).
Px, Py, Pz = Components of resultant external
force per unit area on middle sur
face of the shell.
Rx, R, = Components of the external cou
ple per unit area on the middle
surface of the shell.
, y, Ez = Strain components at point (x,
.xyl, Yyz, 'zxJ y, z).
U, V, W = Components of the displacement
vector in the directions of the co
ordinate lines.
u, v, w = Components of the displacement
vector of the middle surface in
the directions of the coordinate
lines.
P1, Q1, Ri
P2, Q2, R2 =
J, K, L, M
Coefficients defined by Eq. (35).
T = Temperature measured above an
arbitrary zero.
bii = Elastic coefficients. See Eq. (36).
ci = Thermal coefficients. See Eq.
(36).
C1, C2 = Constants defined by Eq. (38).
Uo = Strain energy density.
U = Strain energy of the shell.
I = Function defined by Eq. (44).
0 = Shell coordinate used in place of y
for the cone and the shell of revo
lution. See Sections III B and D.
s, c, t = Abbreviations for sin a, cos a, and
tan a respectively in the equa
tions of Section III B. The sym
bols s and c are also used as
abbreviations for sin x and ctn x
in Section III C.
r = Coordinate used for shell of revo
lution. See Fig. 5.
Vi, V2 = Alternative set of displacement
components for shell of revolu
tion. See Fig. 6.
I. GEOMETRICAL PRELIMINARIES
The theory of shells presented here is based
upon the differential geometry of surfaces and
elasticity theory. The geometrical topics discussed
in this chapter are presented in books such as those
by Struik(i)* or Graustein.(2) The following treat
ment is essentially the same as that presented by
Langhaar and Boresi.3)
A. GEOMETRY OF A SURFACE
A point in space may be located with reference
to a rectangular cartesian coordinate system (X,
Y, Z). The radius vector T from the origin to the
point may be written as follows:
S= iX +jY + Z (1)
where i, j, k are unit vectors directed along the
positive X, Y, Z axes respectively.
A surface in space is represented by the equa
tions X=X(x, y), Y= Y(x, y), Z=Z(x, y), where
x, y are arbitrary curvilinear coordinates. Then a
point on the surface is denoted by the relation
y=r(x, y).
The square of the infinitesimal distance ds be
tween two neighboring points of the surface is
ds2 = dT dr = (T.dx + Fy, dy)2
ds2 = E dx2 + 2F dx dy + G dy2.
where subscripts x and y denote partial derivatives.
By comparison of Eqs. (2) and (3), we obtain
E = . = Xz2 ± Yx 2 + Z2
F = r, r, = XXy, + YxYy + ZýZy  (4)
G = yr, = X,2 + YV2+ Zy2.
Equation (3) is called the "first fundamental form"
of the surface.(1) Equation (4) shows that E and
G are positive. Since ds2 is positive, Eqs. (3) and
(4) yield EGF2 > O.
An infinitesimal element of area on the surface
is given by the formula
* Superscripts in parentheses refer to references listed
in the Bibliography.
dS = V EG  F2 dx dy.
Hence, the area of any part of the surface is given
by the relation
S = ffEG  F2 dx dy.
The unit vector A normal to the surface is given
by the formula
, x y_
S=Tr xr X fy
Tx ry 
V EG  F2
This equation fixes the positive sense of A as well
as its direction.
Another important relation in the theory of
surfaces is the following:
dr . di = e dx2 + 2f dx dy + g dy2.
Equation (8) is known as the "second fundamental
form" of a surface. The coefficients e, f, and g are
given by the following formulas:
1
e  EG_  F2
/ EG  F2
XxX YXX ZXX
XX Y, ZX
X, Y, Zy
X Yxy Z
Xi YX ZX
X, Y, Z,
1 Xy Yyy Zy,
9 7 XXx Yx ZI
EGF , Yy Zy
The extreme values of the curvatures of normal
sections of the surface at a given point, called the
"principal curvatures" of the surface, are denoted
by 1/ri and 1/r2. The mean curvature M and the
Gaussian curvature K are defined as follows:
M = (1/ri + 1/r2), K = 1/(r, r2). (10)
In the following, it is assumed that lines of principal
curvature coincide with coordinate lines x, y. With
this restriction, f = F = 0. Then the principal cur
vatures are determined by the following formulas:
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
1/ri = e/E, 1/r2 = g/G.
(11)
Equation (11) determines the signs of ri and r2 as
well as their magnitudes.
A relation of differential geometry which finds
application in shell theory is Rodrigues' theorem.(1)
For f= F = 0, it may be expressed in the form
a9A 1 9r an 1 af
ax ri 9x' Oy r2 ay
(12)
For orthogonal coordinates, it is convenient to
introduce the notations
A2 = E, B2 = G.
(13)
It is shown in differential geometry that the
functions E, G, e, and g satisfy three differential
equations of compatibility, known as the Gauss
Codazzi equations. For orthogonal surface coor
dinates (F = 0), the Gauss equation is
KAB= a (B) + ay ), (14)
where K is the Gaussian curvature. The Gaussian
curvature is a bending invariant. For f=F = 0
(that is, for coordinate lines which coincide with
lines of principal curvature), the Codazzi equa
tions are
A (A> aA (B _ 1 aB (15)
ay r r2 y ' ax r2 ri Ox '
B. GEOMETRY OF A SHELL
It was noted in Section A that f = r(x, y) repre
sents a surface in space. In addition to coordinates
x, y, a third coordinate z is required to represent
a shell. Thus, a point in a shell may be represented
by the relation f= f(x, y, z). If coordinates x, y, z
are orthogonal at every point, they are called
orthogonal curvilinear coordinates. Then, since
the vector derivatives fx, ýy, f, are tangent to their
respective coordinate lines,
f,  , = f  f, = r,  x = 0.
The square of the infinitesimal distance be
tween two neighboring points is
ds2 = dr  dr = (f, dx + f, dy + r, dz)2. (17)
With Eq. (16), Eq. (17) may be written simply
ds2 = x  r, dx2 + r, r dy2 + f  rf dz2. (18)
Let the following notations be introduced:
a2 = f o, ft2 = ry, E 2' = f, . fc. (19)
With this notation, Eq. (18) becomes
ds2 = a2 dx2 + 32 dy2 + 7 dz2.
(20)
The coefficients a, (3, y are functions of x, y, z.
They are called "Lam4 coefficients."
A special type of orthogonal curvilinear coor
dinates is employed in shell theory. Let x and y
denote surface coordinates on the middle surface
of the shell. Let z denote the distance of a point
measured from the middle surface in the direction
of the normal to the middle surface. Positive z is
measured in the positive sense of the unit normal
A of the middle surface (see Eq. 7). The lateral
boundary of the shell is z = + h/2, where h is the
thickness of the shell. In general, h is a function
of x and y. When x, y, and z are defined in this
way, they are called shell coordinates.
If the shell coordinates are orthogonal, the co
ordinate lines on the middle surface coincide with
the lines of principal curvature. This follows from
a theorem of Dupin which states: The surfaces of a
triply orthogonal system intersect on their lines of
principal curvature. For orthogonal shell coordi
nates, the square of the infinitesimal distance ds
between two neighboring points of the middle
surface is
ds2 = A2 dx2 dy2.
(21)
For orthogonal shell coordinates, the Lame co
efficients (Eq. 19) become
a = A(1  z/r), (3 = B(1  z/r2), y = 1, (22)
where ri and r2 are the principal radii of curvature
of the middle surface (see Eq. 11).
(16)
II. GENERAL ANISOTROPIC SHELLS
A. EQUILIBRIUM RELATIONS
The stress notations used are shown in Fig. 1.
The stress ox is normal to a plane which is perpen
dicular to the xaxis, and the stresses rxy, rT, are
tangent to this plane and directed in the y and z
directions, respectively. It is shown in the theory
of elasticity(4) that the shearing stresses obey the
following relations:
VuY = Tyx, TXz = Trz, Tyz = Tzy. (23)
Love(4) has derived the differential equations of
equilibrium for any orthogonal coordinates. For
shell coordinates (7= 1) in the absence of body
forces Love's equations are
Sa (a a3a
+ 9 Ty + 0 (9z  o =0
9y az  ax o
(9 +± (a cy) + ± z (a o3 ,,)
+ x ' + a
(9x "
O9z Tyz  ax = 0
x (Orx) + a (ar,) + az (jo)
Oa Oao
z a a = 0
These equations are exact only if x, y, z are orthog
onal shell coordinates for the deformed shell. For
small displacements, the effects of deformation on
the equilibrium relations are usually neglected.
The tensions N,, Ny, shears Nxy, Nyx, Qx, Qy,
twisting moments Mxy, Myx, and bending moments
M,, My per unit length of the middle surface
may be expressed in terms of the stress com
ponents ax, oa, ry, r7x, Tyz by considering an in
finitesimal element of the shell (Fig. 2). The
positive senses of Nx, Ny, Nxy, Ny, Qx, Qy are
indicated in Fig. 2, where doubleheaded arrows
denote moments with positive sense given by the
righthandrule convention.
In Fig. 2, the total tensile force on the differ
ential element in the x direction is NB dy, where
N, is the tension per unit length on the plane
perpendicular to the xaxis. Also, with the nota
tion of Fig. 1, the total tensile force may be written
in the form
ax 3 dy dz = dy f ax dz.
(24) Hence,
N. = 13 o dz = OX (1  z/r2) dz.
B h/2
Similarly, Ny, Nxy, . . . , Qy may be expressed.
Thus, we obtain the following equations:
h1 1
Ný = /(72x (1  z/r2) dz, (25)
h/2 I
Figure I Figure 2
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
N, = fhJa, (1  z/rl) dz,
h/2
N,, fh_2i*7y (1  z/r2) dz,
= A/2
N, = f x ,, (1  z/rl) dz,
[ = h/ 1 z (1  z/ri) dz,
h/2
My = / z (1  z/r2) dz,
h/2
r (/2
MXY = h/ T z (1  z/r2) dz,
My, = L rh2 z (1  z/ri) dz,
h/2
QX = zh/2r (1  z/r2) dz,
h , (1  z
Qy = / ^ (1  z/ri) dz.
Jh/2
These representations of tensions, shears, and
ments are similar to those used by Fltigge.(5)
The equilibrium equations for a shell are
rived by Langhaar and Boresi.3) The result
repeated here.
S(BN) + (ANx) + Nx A N,
ax ay ay
ABQ + ABP = 0
ri
(BNY) + a (AN,) +Nyx
x ay ax9
ABQ + ABPy = 0
r2
a (BQX) +
ax
 Nx
O
a AB R
S(AQy) + rNx
ay r
AR
+ AB Ny+ABP, = 0
r2
S(BMx)+ a (AMx) + M, A
ax ±y ay
 My a ABQx+ABR,=O
a (BMy) + (AMy) + M a
aA
 M ABQ,ABR=0O
ay
My My = N  Nxy.
In these equations Px, P,, Pz denote the compo
nents of the resultant external force per unit area
on the middle surface, and Rx and Ry denote
components of the external couple on the middle
surface of the shell. If R, and Ry are neglected,
the moment equilibrium equation is
{iI [ (BMx) + (AMy.)
ax axA ay
(25)
" M, QA _ mB aB
ay yax
+ a i [(Bixy) + (AMiy)
SMyX 9B aA M I
ax May f
AB AB
+ A N + Ny + ABP, = 0.
rL Tr
B. STRAINDISPLACEMENT RELATIONS
(32)
mo Let U, V, W denote the components of the dis
placement vector in the directions of the coordinate
de lines. The general expressions for the strains ex,
s are EY, ez, 'Yxy, ^yz, YX, in terms of U, V, W, have been
derived by Novozhilov(6) and Shaw.(7) However,
the objective of the theory of shells is to reduce
B the problem of threedimensional elasticity to two
)X dimensions. To attain this goal, we employ the
.(26) Kirchhoff assumption, which states that under a
deformation, normals to the middle surface remain
A straight, normal, and inextensional. This implies
 that ez= yz, = Yz 0. It follows that the displace
ment vector U, V, W is a linear function of the
(27) thickness coordinate z. Also, for small deflections,
quadratic terms in U, V, W are usually neglected.
With these approximations, the general formu
las for the strain components ex, ey, Txy at any
point in the shell in terms of the displacement
(28) components u, v, w of the middle surface have been
derived by Love(4) and by Langhaar and Boresi.(31
The results3) are
(30)
(31) y = + (1  z/r)) UB 
(33)

I
z(t1z/r'^^z
EX = U + (1  z/rl)l ( "
A AB rB
 z (I  z/rl)l[ U a
[A Ox r i
S1 __ ( _wx ) vAY+ Ay~w,,
+ A ax \A A ABr2 AB2
II. GENERAL ANISOTROPIC SHELLS
+1 a (w ,)+ uB. + .
BJ ay B / ABr, A2B B'
B ( v
x = (1  z/r,)1 (
_ (1 z/r2) A a (u
 z (B  z/ra)1 ,(
Ar2 9x9 B
+ A x \aB / A2B
,Z ( I  A 9 u 4
Brl ay
t a wXA BXW
+ Y ay A / AB2 *
(33)
For the purposes of this paper, it is convenient to
rewrite Eq. (33) in the following form:
X = Pi+ Q1 zR
1  z/r, 1  z/r1
02 zR2
E = P2 ÷ 2 2
1  z/r2 1  z/r2'
S(34)
J K
Yv 1  z/r, 1 z/r2
zL zM
1  z/r 1  z/r2'
_ _ 1 . . .
wnere
P, = Ux P
P A ' 2 B
_ vA, w _ uB.
AB r,' Q AB
w
r2'
u 9 1 ) + 1 a ( W )
A ax rI A ax
vAy Azwy
ABr+ + AB2 '
B a v \
v 0 (1 1 8 / _
B2 B ry + B y B
uBx Bew
 ABr, A2B'
K A 9 ( u
B1 _yA
(35)
SB a ( v
Ar92 a B )
1 a ( w _ Aywx
A ax B A2B '
M A a (u"
Br ay A
+ Bl Oay() AB2
(35)
C. STRESSSTRAINTEMPERATURE RELATIONS
For linearly elastic anisotropic material, the
plane stressstraintemperature relations for a
shell are
oi = bi, Ei  ci T
(36)
where bi6=bjl are elastic constants, and ci are
thermal coefficients.4) A repeated index is to be
summed from one to three. In Eq. (36), the fol
lowing notations have been used:
01 = o, U2 = Ory, 03 = Txy,
EI =E x, 2 = Ey, E3 = "Yxy.
To obtain tensions, shears, and moments in
terms of u, v, w, we substitute Eq. (34) into Eq.
(36) and then substitute the result into Eq. (25).
Thus, we obtain
Nx = h {bn [Pi + Qi (1  C1) + rCIR]
+ b12 [P2 + Q2] + b13 [K + J (1  CQ)
h/2
+ LriCi]} f ciT (1  z/r2) dz,
Ny = h {b22 [P2 + Q2 (1  C2) + r2C2R2]
+ b12 [P1 + Q] 4 b23 [J + K (1  C2)
+ Mr2C2]} _ c2T (1  z/rl) dz,
h/2
Ný, = h {bi3 [Pi + Qi (1  CQ) + rCiRl]
+ b23 [P2 + Q2] + b33 [K + J (1  C1)
+ LrCi]}  h12 T (1  z/r) dz,
Nyx = h {b23 [P2 + Q2 (1  C2) + r2C2R2]
+ b,3 [Pi + Qi] + b33 [J + K (1  C,)
1h/2
+ Mr2Cs]}  a/cT (1  z/r,) dz,
 {V 12
M  = bn 1Pl + fQ1Clrlr2
(37)
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
+ R6r (1  2r 2C,
b12 [P2 + R2r2] + b13 [F JClr r2
+ Mr2 + Lri 
12rir2Ci I
h2 )If
f/cTz (1  z/r2) dz,
h/2
M = 12 {b22 P + 1 Q2C2rlr2
" R2r2 12rir2C2)
12 [
+ b12 [PI + Riri] + b23  KC2rlr2
+ Lri + Mr(1 
12rr2C2]
h2 1_iJ
h/2
 h2 Tz (1  z/r)dz,
= h3 bt[P1 ±12
12r2 Q1 r2
+ iri(1 
12rlr2C1
h2
+ b32 [P2 + R2r2] + b33 j hCirr
+ Mr2+ Lrl ( 12rlr2C1)
h/2
 caT z (1  z/r2) dz,
M h= = {b32 [P2 + 2 Q2C2rir2
M 1=_i2rni h f
+ R2r2 
12rir2C2
+ b13 [P1 + Rir,] + b33 [2 KC2rlr2
+ Lr + Mr2 (1 
12rlr2C )
h2 2C2
h/2
 cLh/T z (1  z/ri) dz,
Jh/2
where
C1 = (1  ri/r2) 1 In
C2 = (1  r2/r1)  In
 n/z r1
(37)
D. STRAIN ENERGY OF THE SHELL
To obtain solutions of shell problems by energy
methods, the strain energy of the shell in terms of
the displacement components of the middle sur
face is required. The sum of the strain energy inte
gral and the potential energy of the external loads
equals the total potential energy of the system. By
minimization of the total potential energy, expres
sions for u, v, w are obtained. When u, v, w are
known, strain components, stress components, and
tractions may be computed by Eqs. (34), (36), and
(37). Since exact solutions for the displacement
components u, v, and w are often difficult to obtain,
approximate methods, such as the RayleighRitz
procedure,(8) are frequently employed.
The expression for the strain energy of the shell
is obtained by integrating the strain energy density
throughout the volume. The stress components
are related to the strain energy density(4) by the
relations
aUo
(7i = .
(39)
Substituting Eq. (36) into Eq. (39) and inte
grating, we obtain
1
Uo = bijie  cjEjT + C(T).
2
(40)
Since the arbitrary function C(T) is immaterial in
the application of energy methods, it is disre
garded. Thus we obtain
1
Uo = 2 bi exia  ceiET.
The total strain energy is
U = ff Uo af dx dy dz.
(41)
(42)
Integration with respect to z may be performed
readily with the assumption that bij are inde
pendent of z. Thus, substituting Eq. (34) into
Eq. (41), substituting the results into Eq. (42),
and integrating through the thickness, we obtain
U = ffh b (PI + Q1)2 + (2 + Q2)2
+ b12 (PI + Q1) (P2 + Q2)
+ b (K +J)2+ b13 (P1 + Q3) (J + K)
+ b23 (P2 + Q2) (J + K) AB dx dy
II. GENERAL ANISOTROPIC SHELLS
+ fh bl (P +± r1R,)2
J12 1 2 L rir2
12C, + b22 (P2 +r2 2)2
 h (Ql,RI) +2
S2 rL r2
12C2 M(QrR)2] + a F (r2M+rL)2
h 2 L r 12
 (J  riL)2  (K  r2 M)2
+ b12 (P1 + riRi) (P2 + r2R2)
b l2 
+ bi3 [ (rlL + r2M) (Pi + rR,)
I rir2
h12C1 11
 2 (J  riL) (QI  r1o)
+ b23 F (r2AI ± r1L) (P2 + r2R2)
r1r2
12C2
S(Kr2M) (Q2r2R)]} ABdxdy
+ff I AB dx dy
where
hA/2
I =  /2[(1  z/r) (1  z/r2) (cIPI + c2P2)
Ih/2
+ (1  z/rI) (c3K + c2Q2)
+ (1  z/r2) (c3J + ciQ1)
 z  z/r) (c2R2 + caM)
 z (1  z/r2) (caL + ciRi)] T dz.
(43)
Equation (43) is the general expression for the
strain energy of an anisotropic, nonhomogeneous,
elastic shell with arbitrary temperature distribu
tion. It is restricted to small deflections. For an
isotropic shell, Eq. (43) reduces to the result
obtained by Langhaar and Boresi.(3)
(44)
III. SPECIALIZATIONS OF THE GENERAL THEORY
In this chapter, the equations developed in
Chapter II are specialized for several common
types of shells.
A. CYLINDRICAL SHELL
In this section, equations are presented for an
arbitrary cylindrical shell oriented with its genera
tors parallel to the Zaxis. Then X=X(y), Y=
Y(y), Z =x, where x is distance measured along a
generator and y is arc length around the girth of
the cylinder.
By Eqs. (4), (9), (11), and (13),
E = 1, F = 0, G = 1, A = 1,1
B = 1, 1/r = 0, r2 = r(y). J
With Eq. (45), Eqs. (35) and (38) become
Pi = ux, P2 = vy, Q1 = 0, Q2 = w/r,
RI = wax, R2 = wy,  vry/r2, J =x,
K= uy, L = vl/r + wxy, M = wxy,
1 r ( 1 + h/2r=
C1 = 0, C2 = 1 ? In 1 h2r = C.
h 1  h/2r
(45)
,(46)
Substitution of Eq. (46) into Eq. (37) yields formu
las for tensions, shears, and moments:
N = = h {bn [u + h2 W] + b,2 [v  w/r]
h/2
+ b,3 1Uy + vx + f (vx/r + wxy)
 ' ciT (1  z/r) dz,
Jh/2
N, = h {b22 [vh,w/r+C(w/r+rwyvr,1/r)]
+ b,2 ux + b23 [v. + Uy + C(rwxy  MU,)]
_h/2
 f_. cTdz,
(47)
Ns, = h {b33 [uy + vx + h2 (v+ + rwxy)/12r2]
+ ba3 [u. + h2wx/1l2r] + b23 [vy  w/r]}
_h/2
 _c8T (1  z/r) dz,
Nyx = h {b33 [Vx + uy + C (rWx,  Uy)] + b13 uý
+ b23 [vyw/r+C(w/r+rwyyvry/r)]}
 / cT dz,
 vry/r] + 2b13 [vx + rwxy]}
&h/2
 h/2 cTz (1  z/r) dz,
Jh/2
M =  b22 (12C/h2) (vry  r2w',  w)
+ bl2Wxx, b23 [2 (ruy  r2Wxy)
+ V +wy  d c2Tzdz,
r J ,h/2
hMX =  {2a33 [vx+rwxj + ba3 [ux+rwxx]
+ b23 [vy + rwy  vry/r]}
fh/2
 cTz (1  z/r) dz,
Mx =  {b33 [v./r + wy + (12Cr/h2) (u,
 rwxy)] + b13 w , + b23 (12C/h2) (vy
S w r)} cTzdz/2
 r2Wyy  w + Vro I C3Tz dz.
'(47)
III. SPECIALIZATIONS OF THE GENERAL THEORY
By Eqs. (26) to (31) and Eq. (45), the equilibrium
equations are
0N + ay + P. = 0
Ox y
ONly + Ny Q, + P, = 0
Ox Oy T
x +x Q, +NY
aOx y 'r + Pz = 0
aM aM,,  (48)
ax + y Qx + R, = 0
Ox y  Rx= 0
Nx  Nx = Mx
T
Also, Eqs. (32)
O2M,
Qx2
and (45) yield
+2 Myx 2 Mxy
OxOy OxOy
+ +2N
+ 02M + 7 + Pý = 0.
ay r 0
(49)
+ wx,) (v,/r + wyy  vry/r2)
+ (12C/h2) (u,  rwx) (rw,, + w/r
 vry/r)]} dx dy + ffI dx dy,
where
(51)
hl/2
= I/ [(1  z/r) (cIu,  c2v + c3vx)
+ c3u,  c2w/r  z (1  z/r) (c3wxy
+ c3vx/r + cw,,x)  z (cw,,, + cswy
 c2ry,,r2)] T dz. (52)
B. CONICAL SHELL OF CIRCULAR CROSS SECTION
The middle surface of the shell is given by the
equations X=x cos a, Y=x sin a cos 0, Z= x sin
a sin 0, where x, 0 are shell coordinates indicated
in Fig. 3.
4
1
The strain components, by Eqs. (34) and (46), are
Ex = Ux  ZWxxl
zr
Ey = v, w (rz) (wy, Vryr2),
Y = vx + ruy/(r  z)  z (v,/r + wx,) 50)
By Eqs. (43) and (46), the strain energy for the
cylindrical shell is
U = h {bll ux2 + b22 (vy  w/r)2 + bs3 (vx
+ uy)2 + 2bl2u (vy  w/r) + 2b13u, (v,
+ uy) + 2b23 (Vy  w/r) (uy + v.)} dx dy
+ f f bi (w2 + 2uWx/r)
 b22 (12C/h2) (w/r + rwyY  vr, /r)2
+ b3s [3 (v,/r + wY)2  (12C/h2) (u,
 rwY)2 + 2b]2 [wxv/r + wxx (wyy
 Vr,/r2)] + 2b,3 [(u,/r) (v,/r + wy)
+ 2wx, (v,/r + wy)] + 2b23 [(vi/r
Figure 3
By Eqs. (4), (9), (11), and (13),
E = 1, F = 0, G = (xs)2, A = 1,
B = xs, 1/r1 = 0, r x = xt = r,
where s=sin a, c=cos a, t=tan a. With Eq. (53),
Eqs. (35) and (38) become
P, = ux, P,= O 1
xs
Q1 = 0, Q2 u w
R, = wx, R2 = Woo+ Wx
x ýs x
J=vx  , K= _
x' xs '
L = vx v WX Wo
xt x2t X8 X28'
(54)
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
M W.x Wo
xs x2s'
Ci = 0,
xt ( 1 + h/2xt
C2 In 1  h/2xt =
(54)
C.
Substituting Eq. (54) into Eq. (37), we obtain
tensions, shears, and moments as follows:
Nx = h {bllI [  wxx] + b2 [
+ s xt ]+ b' v   x s
X8 xt x xS
h2 / v Wxj
+ 12x2t2 vx x c
rh/2
 cf T (1  z/xt) dz,
Jh/2
No = h b12 ux + b22  (1
I L xs
 C) x + ct wx + sO]
+ v  x + o (1  C)
+ C ( wso\)]} ri2
+  w e!  L[c2T dz,
No = h b3 u + hw2 +.] +2[
I 12xt zs
+  ]+b33 [ +vx
x xt xs
h2 v Wxo
+ 12x2t2 (vx c +
weC
~ c)\
 ch/23T (1  z/xt) dz,
No = h b13 x + b23 + (
C)x xt + tC W xs
+ b33 vx  + (1  C) uo
+ b x xs
+  {Wx _  cs3T dz,
Mx = h bi [ux + xtwx]
b12+woo
+ bl  t lwx  7s
L xs \ .s2 J
.(55)
h/2
 T h/2 z (1  z/xt) dz,
_ h [ 12tC r w
12 [2 h u t
we xtwx] + b12 Wxx
+ b 23[1 + WxO
+ 23 x X C
xt x c
12Ct ( uo
+' 2 W s
wo )
xc
xwxe we
c c
h/2
 2T z dz,
3 /2
MxoA  b3 [u{ + xtw.]
+ b23 s + I w( + ]
xs xs2
23+ 2 t( x0  
I x \ s xs
h /2
 /cT z (1  z/xt) dz,
h/2
Mo = 2 {br3 wxx + b23 [U  t
wee 12tC
sc txw h2
+ b33  _
S12Ct ( u
+ h \h
v Wxo woe
x C XC
xwxe we
c c
hA/2
 c1 T z dz.
Jh/2
(55)
With Eq. (53), Eqs. (26) through (31) become
aN. N. 1 8N0x
ONx + Ný + 1 aNox
ax x xs 0o
ONxo No0 + Nox
Ox + x
No
x
+ 1  No Q9 + Po = 0
xs &0 xt
OQx x a1 Qo + P 0
Ox x xs 00 xt
ý(56)
III. SPECIALIZATIONS OF THE GENERAL THEORY
aM, M 1 9M0x
x + x xs 6
9x x xs a0
Mo£
(56)
a8Mf M 1 aMM 1M M_
Mx + M + M + MexQo = 0
ax x xs 0 x
Nox  Nxe =  x
xt
In Eq. (56), the terms Rx and R, have been dis
carded. By Eqs. (32) and (53), the moment
equilibrium equation is
a2Mx 2 9OM 1 02 (MOe + Mxo)
+ +
9x2 x ax xs 9x96
1 aMo 1 9 (Mxo + Mo)
x Ox x2s 90
1 02M0 No
+ ++ + P =0.
x2s2 O02 xt
By Eqs. (34) and (54), the strains are
Ex = Ux  z Wxx
vo uw/t
xs xz/t
V US/
"x =V + ±
X X
z
xz/t w+
s z
z/xtxt 
c W co  xz/t W
wee oo
xs2
v (58)
x
The strain energy for the conical shell is obtained
by substitution of Eq. (54) into Eqs. (43) and
(44) as
U = ffh 2bi U+ 12 b22 (V +u
wY
W )2
t
S, ± O )2 2b12 ( V
+b33 v + s 2b2b( 2 
+ u + 2b13 u v   + 
+ 2b23 (U + VO t)( V
x x)}[
2uw ,  12C u
+ uxwx\ + b22 ( 2
xt I( b h Lx
 t (wx + 2 )]2 3  cy,
c w +e )2 + 2 2
 cV + Wxo   I   (uo  xtwxo
x X / l
(57)
+ twe), + 2b12 [w + Woo + V
2 x +xst
+ 2b1 (2wxx + )( 
xt xt
we wo)e + 2b2 1 1 (
+ w  6 + Vx
c xc x2 XSt2
V Wxro
+x
x c
Wx /vo +
xC C
12C( / w
+ xs t wx  ,u t
 xtw ) (ue  xtWxo + two) ]}xs dx dO
+ fflxs dx dO
where
(59)
1/2 z ye
 h/2 xt ) C X8
+ C3x  C3 + c3 + C C
C2 (Wx /
x x82
z(
C3  W.x0
X8
 z (1  z/xt) ciw + ca (V 
+ WxO Wo) 1jl}Tdz.
c xc xt
C. SPHERICAL SHELL
The middle surface of a spherical shell with
center at the origin is given by the equations
X = Rsinx cosy, Y = Rsinxsiny,
Z = R cos x
(61)
where R is the radius of the middle surface, x is
the colatitude, and y is the longitude (see Fig. 4).
By Eqs. (4), (9), (11), (13), and (61),
E = R2, G = R2sin2 x, f = 0,
(62)
A = R, B = Rsinx, ri = r = R.
With Eq. (62), Eqs. (35) and (38) become
P x P  1 w
Pl R 'P Rs' 1 = R
2 R ' , ~ R2 ,
(63)
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
Wyy WrC  ucC v  vc
R2 Rs2 + R2 ' J = '
uY vc  v WxY  Wyc
K Rs' L 2 R2s '
S= ±w cwY  u
M Rs C1 = C2 = 0,
h/2
M = 12 , bn (u.  w.x) + b12
+ Cu  CWx 
 c + 2cw, + u,  2w ,)
h /J/2
 ciTz ( + z/R) dz,
J h/2
My = F fb2 (ux  W,) + b22 (V
_12_R2I S
+ CU  CWx 
w ) + b2 (V
Figure 4
where s = sin x, c = etn x. Substituting Eq. (63)
into Eq. (37), we obtain tensions, shears, and
moments as follows:
Nx =  bi (ux + w) + b12 + cu
+ W +b1(u + v + cv)
 /h2cT (1 + z/R) dz,
_h/2
Ny h bl2 (Ux + W) + b22 ("V + cu
 h/ c2T (1 + z/R) dz,
Ny = Ny = h [b,3 (uý + w) + b23 (v
+ cu + w) +b3 (s + vx  cvy
2cw, + uy  2Wxy
//2
 / c,,Tz (1 + z/R) dz,
+ b23  + c  Cw,  
" b3 (vcvc 2cw +uy2w )s
I hc3Tz (1 + z/R) dz.
hI1
Substitution of Eq. (62) into Eqs. (26) to (31)
yields the following equilibrium equations:
aN, 1 aN.
N + 1  + c (Nx  Ny)
Ox s ay
+ Q. + RP. = 0,
ONly + 1 . ON+ ±2cN~,
Ox s Oy
+ Qy + RP, = 0,
aQý + aQ,
Q + _ . (Nx + N,,)
Ox s Oy
+ cQ. + RPP =0,
aMý + 1 OM + c (M  M)
Ox s Oy
 RQx + RRy = 0,
(64)
(65)
wY )+ b1 (vx
8")
(64)
t
III. SPECIALIZATIONS OF THE GENERAL THEORY
aMx+ 1 aM,
+  + 2cMx
 RQy,  RRx = 0.
(65)
J
By Eqs. (32) and (62), the moment equilibrium
equation is
a2M, 1 a2M, 2 92Mx, a9
+ 82 +  +c (2M2
9x2 s2 2 s axay ax
 M,) +2 c Mx + (My  Mx)
s OJy
 R (NA + N,) + R2P, = 0.
Equations (34) and (63) yield
Ux W
zx=k+ R+z
zwxx
R (R + z) '
= C ,y w
"= Rs R + z
(R +z) R (\ s2" C)
x  VC Uy1
sR (R + z) (w  ).
" 2b13 (ux  wx) ( + 2cw,  2wxy
+ vz  c)± + 2b23  + CU  x
wL, ) ( u,, + 2cws  2w,,
+ v,  cv )}sdxdy + ff I RS dx dy,
where
(66)
1 (67)
The strain energy for the spherical shell is ob
tained from Eqs. (43), (44), and (63) as
U = f W { +u (w u +u b22 (
+ w + cu) + b33 u + v  cy2
+2b2 (u + w) ( +w+cu)
+ 263 (Ux + w) u( + v  c)
+ 2b23 + w + cu  + ; x
 cv) sdx dy + ff 24R2 bii (uxw..)2
+ b22  + cu  cw  W
( ± s2 )
b33 Uy + 2cwy  2wxy + V v
s + x  c
(68)
Sh f (1+ z/R)2 v,
±h/2 I R
±+ (c 1R) [w + c2 (w + CM) + C3 Vx
 CV +  z (1 + z/R) 
 c2 + clwzx
+ c2 + CWx  CU + C3 C  x
+ 2wy  2cwy  Uy )}T dz.
8 J
(69)
For isotropic shells the results derived in Sections
A, B, and C reduce to those obtained by Langhaar
and Boresi.(')(9)
D. AXIALLY SYMMETRICAL ORTHOTROPIC
SHELL OF REVOLUTION
The shell of revolution is shown in Fig. 5. The
middle surface of the shell is described by r=r(x)
+ 2b2 (uxwx) ( +cucw, w2 )
\ s s1 /
Figure 5
(67)
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
and Z= Z(x). A point in the surface is located by
the coordinates x and 0 where x is arc length from
the origin measured along a generator to the point
and 0 is the angle between the XZ plane and the
plane containing the point and the Zaxis. From
Fig. 5,
X = r cos 0, Y = r sin 0, Z = Z. (70)
Equations (4), (9), (11), (13), and (70) yield
E= 1, F =0, G=r2, A =1, B =r,
(71)
1/ri = rZXX  rzZx, 1/r2 = Z//r.
Consider an orthotropic shell of revolution
which undergoes axially symmetrical deformation.
Then
a
v bs = b23 = C3 = N = N
= MxY = MA = Qy = 0.
With these restrictions, Eq. (35) becomes
Pt = uX, P2 = 0, Q1 = 
Q2 = Urx
r r2'
R = u a + w(,
ax ( ri )
R2 = 1 ur + rxws,
=r r
J =K =L =M = 0.
Substitution of Eq. (73) into Eq. (37) yields
Nx = h bniu  (1  C)
ri
+ riC u ( ) + wj)
I ax ri
+b12( ur w)}
h/2
  h/ ciT (1  z/r2) dz,
Jh/2
No = h b22 (1  C2) ( r
w)
r2
+ r2C2 ( ur rxwx + b+2 [u.
\ rri r
r  c2T (1  z/ri) dz,
M= h, 2r 12Ch
±1_2r a (1)_] i 12rr2C
+ri n ^ +wzz 1 ^J
+ b2r2 ur,+ r± )
\ rr r j
h12
 _ ciTz (1  z/r2) dz,
Jh/2
h { r12 U~2 r2 ( x
12rI [ Lh r
+r2/r (UrXw) (1 12r r2 C2)]
+ b12 [ r  (u +) u + rwzzX]}
 C2Tz (1  z/ri) dz.
11/2
(74)
With Eq. (71), the equilibrium equations (Eqs.
(26) to (31)) become
(72)
a (rNx)  rNo r Qx + rP, = 0
Ox ri
ox i rNo rP = 0
a rl r2
ax (rMr)  Morx  rQ. + rRo = 0.
ax
(73)
(75)
Substitution of Eq. (71) into Eq. (32) yields
2M (2M M
r ax + r (2M.  M6) + rzx (Mx  M6)
axax
rl r2
(76)
The strains, given by Eqs. (34) and (73), are
Ex =
w
Ux 
rl  z
r O a(x r( ) +  1  Zz/ri
urx W ) 1
0 r r 1  z/r2
1 ( ur z
I + rxw 0
r \ r 1 z/r2
YXo = 0.
(77)
By Eqs. (43) and (73), the strain energy of the
shell is
(74) U =ffh b[u_ b  + [
2 r w [r r2 w rcx
+&4u rJ 72Jjr
r2
^
III. SPECIALIZATIONS OF THE GENERAL THEORY
ux+riu a (1 1wxx
Sxi, b ±x wxxri
+ J 12 [ 2 rjr2
 u12C w a (1  2Wxx
h L  ur ax ri /
b22 [ri urx + rxw 2
S2 Lr rr1 r
12C2 (ur  w r2ur r2rx )2]
h r r2 rr1 r
( + nu+() + rwx]
+ b12 " ax1 r1, )
F ur + Jrw )ArdxdO +fflrdxdo (78)
where
I =  (1  z/rj) (1  z/r2) clu,
Jh/2 1
+ (1  z/r) c2 ur 
+ (1  z/r2) cl  z (1
ri rr
r( r
 z/r2) c 1 ( + wax } Tdz. (79)
Equations (74) through (79) may be expressed
in terms of displacement components V1 and V2
in the axial and radial directions, respectively, as
u = VI sin 4 + V2 cos4'
w = V1 cos 1  V2 sin 4'
(80)
where # (x) is the angle between the normal to the
middle surface and the Zaxis. In addition, the
radii of curvature may be expressed in the form
1 1 =
sin 4
(81)
It is convenient to introduce the quantities
< = (Vi) cos'  (V2)x sin 4
e = (Vi), sin + (V/2) cos j
(82)
where p is the rotation under deformation of the
normal to the middle surface and E is the strain
of a generator on the middle surface.
With Eqs. (80), (81), and (82), the tractions
become
Nx = h b [(1  Ci) Clx + b1V
fh/2
h/2 1
z sin dz,
r
I r tan J
1/2
 fcT (1  z4x) dz,
h/2
_ bit F r 12Cr / 1
MX 12r  L  2
+ px sin J/ + bi2 P cos 'P
'(83)
'h2 1
 fcTz(1 
Jh/2
z sin 4, dz,
r dz,
ha 12C2 / V
12 h2 sin4 \ tank
+ 4:ý + b12 b }
tan (
_h/2
_ /c2Tz (1  z) dz.
Jhf2
Figure 6
selected by Reissner.(10 Referring to Fig. 6, we
obtain the relations for u and w in terms of Vi and
V2 as follows:
Similarly, the equilibrium equations (Eqs. (75)
and (76)) become
a (rN.)  cos i No  ripxQx + rP. = 0, (84)
dx
0 K
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
aO (rQ.) + rnN.x + sin 4 No + rP, = 0,
a (rM,)  cos V Mo  rQ, + rRo = 0,
r a1x + cos J, a (2Mx  Mo)
 4x sin 4 (Mx  Mo)
+ 4xrNý + sin 4 No + rP 0.
(84)
The strains are obtained by substitution of
Eqs. (80), (81), and (82) into Eq. (77). The
results are
x  1 zt
V2  zI cos 4'
r  z sin '
0 = 0. J
(85)
The strain energy expression (Eqs. (78) and
(79)) may be written in terms of Vi and V2 in
the form
U = ir fh {bu 2 + b22 (2)2 + 2b12 0V rdx
P2C ( + L r2 tan P P
1202 V2 2
h \ r tan / J
+ 2b12 ýx COS o )4 rdx + 2f Irdx
where
S hl2 _
+ C2(1  zI) (V2 Z cos) Tdz.
(86)
(87)
Solutions to problems of axially symmetrical
shells of revolution may be obtained with either
system of equations; i.e., equations (74) through
(79) or equations (83) through (87). The choice
will depend on the type of loading and the bound
ary conditions that the particular problem presents.
1^
IV. SPECIAL PROBLEMS OF CYLINDRICAL SHELLS
A. AXIALLY SYMMETRICAL DEFORMATION OF
AN ORTHOTROPIC CIRCULAR CYLINDER
Consider a circular cylinder, with radius a,
oriented so that its axial coordinate x and its
circumferential coordinate y coincide with the
principal directions of orthotropy. Let the cylin
der undergo axially symmetrical deformation. Then
v  = b13 = b23 = ca = Nxy = Ny,
ay
= Mx" = MY = QY = 0.
(88)
The equilibrium equations, determined by Eqs.
(48), (49), and (88), are
aNx= 0,aM QX = 0,
a9x , ax
aQ +± NY + P 0,
a2M N
x +  + Pý = 0.
ax2 a
(89)
The strain energy for the axially symmetrical
deformation of an orthotropic circular cylinder is
obtained from Eqs. (51) and (52). It is
U = Trf ah [bilu2 + b22 ( )  2b,2 uzw dx
+ ah bll (Wx 2 + 2uxx )
f12 1 (a
12C ) W2
 b22 2 w2 dx + 27rf aldx
where
I = h/2 Ci (1  z/a) (u.x  zwxx)
 c2 ]T dz.
a
(91)
(92)
If the elastic constants byi are considered inde
pendent of x, the equilibrium equations may be
expressed conveniently in terms of displacement
components u and w. Substituting Eq. (90) into
Eq. (89), we obtain
By Eq. (47), the tractions are
r ( h2wxx "
Nx = h[bn u + h2w)
12a
b12 "
rh/2
NY = h b22 (C  1) + b12 MU]
 I chT dz,
MX = 12a (ux + awxx)
1.2a
h/2
= h (3 12aC b22w, b
My = 12a  h2 + bxhawxd
Jh/2
Wxxx + auxx
a ch/2 [ a :
hb11 ih/2 La 1j(
h2
12 wxxx
z/a) dz = 0,
hb2, h3
ab1 (C  1) w 1+ u
a bni 1 2a
hb12 P, +  h/2
  c2Tdz
abn bn abn Jh/2
h/2 2( 1
+ /2 (cT) Iz (1 z/a) dz = 0.
bn Jh/2 L x~
(90)
,(93)
With the preceding equations, analyses of
problems of axially symmetrical deformations of
orthotropic circular cylinders may be obtained
by two procedures. One may express the total
potential energy of the shell as a function of u
and w and then minimize this function. Alter
natively, one may solve the equilibrium equations
(Eq. (93)) subject to the boundary conditions of
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
the particular problem. In the following articles,
examples are presented which serve to illustrate
the two procedures.
B. CIRCULAR CYLINDER SUBJECTED TO
TEMPERATURE WHICH VARIES
THROUGH THE THICKNESS
Let the cylinder be constrained at its ends so
that there is no strain in the x direction (i.e.,
u=0). This implies that the material properties,
the radial displacement component w, and the
temperature distribution are independent of x.
Hence, the strain energy is
U = 2 w + W c2Tdz dx. (94)
f, L 2a 1Jh/2 J
Since the integrand of Eq. (94) is independent of x,
the integration is readily performed. The result is
b2h/2
U = 4l hb2(1 C) w2 + 2 , (95)
1 2a wh/2
where the length of the cylinder is 21.
If the cylinder is subjected to a pressure Pi
on the inside and Po on the outside, the potential
energy of the external loads is
0 = 4ral (Pi  Po) w.
ab12 h/2 [ c2 Tdz
x hb22( C) (a z) iLh/2
+ a (Pi  P)]  ciT,
a r f[/2 (100)
07/ = h(1  C) (a  z) h/2
+ a (Pi  Po)]  c2T,
Txy = 0.
Equations (99) and (100) express stress compo
nents ax, ra, 7x, and deflection w for arbitrary tem
perature distribution through the shell thickness.
C. SEMIINFINITE CYLINDER WITH END
MOMENTS AND SHEARS
Consider a semiinfinite cylinder as shown in
Fig. 7. From Eq. (89),
ON0
Ox 0.
(9x
(96)
The total potential energy V of the system is
the sum of the strain energy U and the potential
energy Q2 of the external loads. Hence, by Eqs.
(95) and (96),
V = 4rl hb22 (1 C)
2a
+ I 2ciTdz + a (Pi  Po)) w . (97)
lJh/2
By the principle of stationary potential energy,
a necessary condition for equilibrium to exist is
Figure 7
Therefore, Nx = constant =0 since there is no net
axial tension. Equation (90) then yields
b12 W
x  bil a
h2
12a wxx
1 h/2
+ c /T (1  z/a) dz.
hbil h/2
Therefore, by Eqs. (97) and (98), we obtain
W  C22 c [ 2Tdz+ a(PiPo). (99)
w = h(1C) _/2 1 (99)
Substitution of Eq. (99) in Eqs. (36) and (50)
yields the stress components as
Substitution of Eq. (101) into Eq. (93) yields the
equilibrium equation for w:
Here,
w.... + 4a2wx + 404w = f (x).
a2 = b12
2a2b,16'
(102)
(103)
dV
dw
(98)
(101)
IV. SPECIAL PROBLEMS OF CYLINDRICAL SHELLS
$4 = 3 b22 (1 C) _ b122 1
a'h25 [ bll bi12 J
12 1 f2/2 b ,2
f (x) h3b6 a fh/2 b, ci (1
 z/a)  c2] Tdz
 2 da2 (T)l
 A/a2 ( c2 ) (1 ,
5=1 
12a2 *
The general solution of Eq. (102) is
w = e V/ 2x [B, cos %/ 2 + a2 x
+ B2 sin / $2 + a2 x]
+ e0/2 2x [B, cos V/ 02 + a2
+ B4 sin V/ 2 + a2 x] + F (x)
where B1, B2, B,, B4 are arbitrary constan
pending on the boundary conditions and F
the particular solution corresponding to the
perature function f(x).
Since w must remain finite, Ba= B4=0.
constants B, and B2 are determined by the b
ary conditions at x = 0.
When x = 0, Mx = Mo,
dM
dx = Qo.
Since B, and B2 depend on F(x), consider thi
T = Toexia. Substitution of this function of
Eq. (103) yields
f (x) = ah2 (cl b2  C2 Toex/a.
Then, by Eq. (102),
F(x) = yex/a
where
12aTo (cl b2 _
x /
= bih2b (1 + 4a2a2 + 4a4 4)
Equation (104) then reduces to
w = e "/2x [Bi cos V/$52 + a2 x
Substituting Eq. (109) into Eq. (90), eliminating
Uz by Eq. (101), and applying the boundary condi
tions of Eq. (105), we obtain
 (103)
B [1M ', (6 + b) 28 (2
b i xh / , (2Lb
b12  28 ( 2 2 2)
][a2b+2a2)
12. 25 (2 a2) ,
a2bil I
(104) B2 = {[2a2 b2 (6 +
12Q [26 ($2 + 2a2)
ts de y ( b1
(x) is a2bij _ + b(
tem
12Mo
The + $ I
ound  2 26 (2  2a2) + 12
a2bl J L a2bn J
(105)  [2 ( + 22) _ b1 26 ($2
a2bn
e case
r into  a2) 2 + a2.
(106)
(107)
(108)
+ B2 sin V/2 + a2 x] + ye i/. (109)
Substitution of Eqs. (103), (108), and (110)
into Eq. (109) gives w as a function of x. Equa
tions (101) and (109) yield u.. Then Eqs. (36),
(50), (101), and (109) give the stress components
in the shell as functions of x and z.
D. NUMERICAL EXAMPLE
In this section, numerical values of deflections
and stresses are computed with the results of Sec
tion C for five combinations of elastic coefficients
bi" and thermal coefficients c, (see Table 1), for a
cylinder with mean diameter of 20 inches, and
with thickness of 1 inch. For isotropic material
(Case 1),
(110)
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
E
bi& = b22 = E
Ek
Cl = C2  
(111)
where E is Young's modulus, v is Poisson's ratio,
and k is the coefficient of thermal expansion. In
Case 2, the thermal coefficient in the longitudinal
direction has been reduced by a factor of onehalf
as compared to Case 1. Similarly, in Case 3, the
thermal coefficient in the circumferential direction
has been reduced by a factor of onehalf. In Case 4,
the ratio of circumferential stiffness to longitudinal
stiffness has been doubled; in Case 5, it has been
reduced by a factor of onehalf compared to Case 1.
For brevity in the following discussion, the ratio
of circumferential stiffness to longitudinal stiffness
will be designated simply as the "stiffness ratio."
Two types of loading with constant tempera
ture are considered: (a) a constant shear Qo is
applied at the free end of the cylinder (Fig. 7);
(b) a constant moment Mo is applied at the free
end of the cylinder (Fig. 7). Also, deflections and
stresses are computed for a temperature distribu
tion T = Toex1a, where To is a constant, x is the
distance measured from the free end, and a is the
mean radius of the cylinder.
The deflection w is computed by means of Eqs.
(103), (108), (109), and (110). Numerical results
are presented graphically in Figs. 8a, 8b, and 8c.
Temperature effects are not included in Cases 1,
2, and 3 of Figs. 8a and 8b. In all cases, for end
shear Qo, the deflection w is a maximum at the
free end (Fig. 8a). The stiffness ratio is doubled
in Case 4. Then the deflection w attains a value
of approximately 57% that attained in the iso
tropic case. When the stiffness ratio is reduced
by onehalf (Case 5), the deflection is 81% greater
than in the isotropic case.
With end moment Mo, the deflection w is again
largest at the free end (Fig. 8b). However, for end
moment, the deflection w is less sensitive to changes
in the stiffness ratio; in Case 4, w decreases to 69%
of its value in Case 1, and in Case 5, it is increased
by 48% of its value in the isotropic case.
Deflections due to temperature distribution
T = Toexla are shown in Fig. 8c. As with end
shear and end moment, the deflection w is a
maximum at the free end. When the thermal
coefficient in the longitudinal direction is decreased
by a factor of onehalf, compared to the isotropic
Table I
Values of the Ratios of Elastic and Thermal Coefficients
for the Five Cases of Article 14
Case b b e22 i F c per oF
No. b b prF t, bper° F
1 .3 1 105 105
2 .3 1 5X106 105
3 .3 1 105 5X106
4 .3 2 105 10
5 .3 .5 105 105
case, the deflection w increases by approximately
21% (Case 2, Fig. 8c); whereas, if the thermal
coefficient in the circumferential direction is de
creased by a factor of onehalf, w decreases by
approximately 71% (Case 3, Fig. 8c). Thus, for
the assumed temperature distribution, the deflec
tion w is more sensitive to changes in the thermal
coefficient in the circumferential direction than in
the longitudinal direction. Furthermore, a de
crease of the thermal coefficient in the circum
ferential direction results in a decrease in the
lateral deflection w; a decrease in the thermal
coefficient in the longitudinal direction results in
an increase in the lateral deflection. Hence, if
lateral deflection due to temperature is to be kept
small, a shell should be designed so that these two
opposing effects cancel each other. Cases 4 and
5 of Fig. 8c illustrate the effect of the stiffness
ratio. If the stiffness ratio is doubled, the deflec
tion w is decreased to 52% of its value in the
isotropic case. If the stiffness ratio is reduced by
a factor of two, w is increased to 216% of its value
in Case 1. Hence, for the type of temperature
distribution considered, the deflection w is very
sensitive to the stiffness ratio.
Numerical values of stress components cx and
ry may be computed by means of Eqs. (36), (50),
(101), and (109). In this example, o, and cr have
been computed numerically for the free end (x = 0)
for all three types of loading. The numerical
values of cax and ay are illustrated graphically in
Figs. 8d, 8e, and 8f.
For end shear Qo, the longitudinal stress ax is
zero at the free end (Fig. 8d). However, the
circumferential stress zy varies through the thick
ness as shown; it attains a maximum value on the
inside of the cylinder. When the stiffness ratio is
doubled (Case 4), the maximum value of a, is
increased by approximately 20% of its value in
the isotropic case. If the stiffness ratio is decreased
by a factor of two (Case 5), o,, decreases approxi
mately 18%. Consequently, a,, is somewhat less
sensitive to variations in the stiffness ratio than
is deflection.
IV. SPECIAL PROBLEMS OF CYLINDRICAL SHELLS
x in inches
0 4 8 /2
.25 0
z in inches
.25 .5
r 1 1
.25 0 .25
z in inches
25 0
z in inches
(d) Circumferential stress
oy due to shear vs thickness
coordinate z at x 0 0
4
I.Z,3
/23
(e) Axial stress ac and
circumferentoia stress cry
due to moment vs thickness
coordinate z at x 0
6 ____ .__/
4
.2,3 /
5 /
/
r __I_
/
/
I + I
/
/ I
I _____ _____ ± I J
/ ,.(b) Radial deflection w
/ 2 3 due to moment vs oxial
coordinote x

x in inches
4
§C6
(f) Circumferential stress
cy due to temperature vs
thickness coordInote z
at x  0
3
I .~fl I I
(c) Rodio/ deflection w
due to temperature vs
ox/ai coordinate x
8
x in inches

.5
Figure 8
25 .5
4
I
/6
BULLETIN 458. NONHOMOGENEOUS ANISOTROPIC ELASTIC SHELLS
For end moment Mo, the variations of stress
components o and ay are illustrated in Fig. 8e.
The axial stress component ax is the same in all
cases. The variation of the circumferential stress
component a, is similar to that obtained for end
shear Qo.
For temperature distribution T= Toexla, the
circumferential stress component oy is shown in
Fig. 8f, the axial stress component being zero
at the free end. The largest compressive value
of a, is attained in Case 2, its magnitude being
approximately 21% larger than the value for
the isotropic case. In Case 3, the maximum value
of the compressive stress is approximately 72%
less than for the isotropic case. If the stiffness
ratio is doubled (Case 4), the maximum com
pressive stress is decreased by about 10%. How
ever, the maximum tensile stress increases in
comparison to the isotropic case (Fig. 8e). If the
stiffness ratio is decreased by a factor of two
(Case 5), the maximum compressive stress is in
creased approximately 16%, whereas the maxi
mum tensile stress decreases.
In general, the maximum thermal stress is
small. For example, for aluminum, bn is approxi
mately 107 lb/in.2 Then, for the value To = 1000F,
the maximum compressive stress at the outer sur
face is approximately 540 lb/in.2
V. DISCUSSION OF RESULTS
The geometrical preliminaries in Section I A are
well known topics of the differential geometry of
surfaces. Section I B explains the shell coordinates
which are used throughout the remainder of the
paper. The equilibrium and straindisplacement
relations discussed in Sections II A and II B have
been established previously in the theory of shells.
The form in which they are presented here follows
Langhaar and Boresi.(j)
In Section II C, a stressstraintemperature rela
tion for a linearly elastic anisotropic material is
chosen. The equations for the tractions, shears,
bending moments, and twisting moments in terms
of the displacement components u, v, w of the
middle surface are developed.
In Section II D, the strain energy of the shell in
terms of the displacement components u, v, w is
reduced to a surface integral by integration of the
strain energy density through the thickness.
In Chapter III, Sections A, B, C, and D, the
general theory is specialized for cylindrical shells,
conical shells of circular cross section, spherical
shells, and axially symmetrical orthotropic shells
of revolution. The latter portion of Section D
expresses the results for the shell of revolution in
terms of axial and radial displacement components
as selected by Reissner.(10
As an illustration of the theory, two special
problems for axially symmetrical orthotropic circu
lar cylinders are analyzed in Chapter IV. Section
B treats an infinite cylinder with temperature
varying through the thickness. Section C treats
a semiinfinite cylinder subjected to end moments
and end shears and a temperature distribution
which varies in the axial direction. The problems
are solved by two different procedures. The first
problem is solved by expressing the potential
energy of the system in terms of the displacement
w, and then applying the principle of stationary
potential energy. The second problem is solved
by expressing the equilibrium relation in terms of
displacement components u and w. Then the re
sulting differential equations are solved for w.
For the second problem (Section IV D), nu
merical values for stresses and deflections are
computed. The results are illustrated graphically
in Figs. 8a, b, c, d, e, and f for the isotropic case
and for four types of orthotropy.
VI. REFERENCES CITED
1. D. Struik. Differential Geometry. Cambridge, Mass.:
AddisonWesley Press, 1950.
2. W. Graustein. Difierential Geometry. New York:
The MacMillan Co., 1935.
3. H. Langhaar and A. Boresi, "Strain Energy and
Equilibrium of a Shell Subjected to an Arbitrary Tempera
ture Distribution," Proceedings, Third U.S. National Con
gress of Applied Mechanics, Brown University, Providence,
R.I., 1958.
4. A. E. H. Love. The Mathematical Theory of Elas
ticity. (4th ed.) Cambridge: Cambridge University Press,
1934.
5. W. Fltigge. Static und Dynamic der Schalen. Ann
Arbor, Mich.: Edwards Brothers, Inc., 1943.
6. V. Novozhilov. Foundations of the Nonlinear
Theory of Elasticity. Rochester, N.Y.: Graylock Press,
1953.
7. F. S. Shaw. Linear Theories of Shells. (PIBAL Re
port No. 247. Contract No. N6onr26303, Project No.
NR 064167) Brooklyn, N.Y.: Polytechnic Institute of
Brooklyn, 1954.
8. R. Courant and D. Hilbert. Methods of Mathemati
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Inc., 1953.
9. H. Langhaar and A. Boresi. Thermal Stress Problems
of Shells. (T.&A.M. Report No. 131, Contract No. NR
1834(14), Project No. NR 064413) Urbana, Ill.: University
of Illinois, 1958.
10. E. Reissner. On the Theory of Thin Elastic Shells.
(Reissner Anniversary Volume) Ann Arbor, Mich.: J. W.
Edwards, 1949.
11. H. Langhaar, R. Miller, and A. Boresi. Deflections
of NonHomogeneous Anisotropic Elastic Plates Subjected
to Heating. (T.&A.M. Report No. 136, Contract No. NR
1834(14), Project No. NR 064413) Urbana, Ill.: University
of Illinois, 1958.