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On a general theory of anisotropic shells
ON A GENERAL TREORY OF ANISOTROPICSHELLS (K
OBSBCBBI TBOUII AN~ZO~UOPNYKB OBOLOCSEK) PYY Yot.22, No.2, f958, pp.2!26-237 S. A. AYBARTSUYIAN (Erevan) (Received
2 July
1958)
1. We consider a thin anisotropic shell of constant thickness h. Assume that the material of the shell obeys the generalized Hooke’s and that
at each point
parallel
to the middle surface
used as surface linear
lines
coordinates of that
along
the normal,
point
(a,
(b)
of the shell.
/3, y)
s and p,
will
surface
will
be
to curvi-
with the principal
the distance,
measured
(a, /3) of the middle surface
and the
We assume that
of the shell, normal to the middle after deformation;
the normal stresses*
law
syuvaetry,
be referred
which coincide
Let y represent
between the point
of the shell.
of elastic
The latter
and the shell
surface.
(a) the line elements not change their lengths
ua,o
is only one plane
of coordinates,
orthogonal
curvature
there
o,, are small
surface,
do
as compared with the stresses
andr
P $3; (c) the shear
stresses
thickness of the shell bola f 13 I . king
more rigorous
can state
of the rayand'PY wrthvarythein lawthe ofdirection the quadratic para-
in accordance
in the formulation
here the assumptions
(a)
and (I)
of the hypotheses
[ 2,s
in the following
e 0 approximately; w= (b) the stresses ui do not exert any essential components eaa and they can be neglected and “ss equations of the generalized Hooke’s law.
1 , we
form:
(a)
2. By virtue and r~,,
%Y==
*
the shear
stresses
r
ay
x+-x2 y+-y-
+XV++X-)++!p-$)p(a,
2
+
Here and in the following theory of shells.
Reprint
(~1 concerning
on the strain
in the corresponding
we have
%vf-
the
of the assumption
influence
-$(Y’
+
u-)
we adopt
p, +
+(r’-q)$(.,
the well-known
Order No. PMM25. 305
B)
notations
(2-l)
used
in
306
&A.
Aubartsurian
where A” (a, p), Y+ (a, fl) and X’ (a, f3), Y- (a, fl) are the components along the axes of the moving trihedron (in the directions of the positive tangents to the lines /3 = const., a = const., respectively) of the intensity vectors of the surface loads, applied to the boundary surfaces andy=-%h, respectively, while $(a, @), t,!&z, f3) are unknown Y =%h functions. Substituting the values of the tangential stresses r o$sr13Y from (2.1) into the corresponding equations of the generalized % for the shear strain components e and eBY the law 16 I, we obtain aY formulas
Here we have introduced
the following
notations:
x = a [Ub6(x+ -
x-)
+ UA5(Y’ -
Y = f [a44(Y” -
Y-) + a45(X’ -
r-)] (2.3)
x-)1
X’=a,5(X++x-)+a*,(I-++Y-) Y’ =
where the quantities
lx,,
(Y’ -f- Y-j +
aik are elastic
lQ6
(X’ +
constants
From the equations of the three-dimensional have for the strain components [ 1 1
x-j
[6 I. theory
of elasticity
we
(2.7)
(2.9) H,=A(lfk,y), In these
formulas A = A(a,
1r2=B(l+k,y) f3) and 3 = BCa, j3) are the coefficients
(2.10) of
On a general
theory
of anisotropic
307
sheiis
the first quadratic form of the middle surface, k, = k (a, @> and f surface, k, = kZ(a, ,5?> are the principal curvatures of the midd e (a, /3,y) are the displace(a, f3, y) and ~,(a, /3, y), up = u % = p="/ ment components of an arg itrary point 0 the shell in the directions of the tangents to the coordinate lines, respectively. &
the basis of the assumption (a) we find from (2.7) &L
$ =
0,
lly =
+(a,
/3) = w(a,
(2.11)
f3)
Thus, like in all existing theories of thin shells, the displacement of any point of the shell is independent of the coordinate y. This "r displacement component has for all points of a line element of a normal to the shell a constant value, equal to the normal displacement component w = ~(a, 6) of the corresponding point of the middle surface of the shell. Substituting the expressions for e , ep,, HII H2 and u,,from (2.2), (2.10) and (2.11) into equations (2.9y we obtain differential equations for the displacement components ua and UP. Integrating these equations and taking into consideration that ua = ~(a, 8) and u = ~(a, /I) when a y = 0, we find
(2.12) up = (1 + k2Y) u -
-$- a+
-
Y(l
+y
$)$(D,+
where u = u(a, /!I), tr= ~(a, /!3> are the tangential displacement components of the corresponding point of the middle surface. In the process of deriving the formulas (2.12) the accuracy was being confined to consideration of quantities up to those of the order of magnitude of yki, i.e. whenever a sufficiently precise estimation was possible, terms of the order of magnitude of (yki)2 were being neglected in comparison with unity. Our formulas (2.12) show that, in contrast to known theories of thin shells [ 1,2,5,7 I, the tangential displacement components ua and u P .Of any point of the shell at a distance y from the middle surface are, in the case considered here, as in the publications [ 8,9 1, non-linear functions of the distance y. By virtue of (2.12) the strain components eaa, e PP' pressed by ~l~~ials in powers of y, namely
e4
can be ex-
308
S.A.
AMbartsurian
(2.13)
from (2.12) and (2.111, respectSubstituting the values of Us, u into the relations (2.6) an~~2~~~, and comparing the resulting ively, with the corresexpressions for the strain components eaa, e Pp * .“aP ponding expressions (2.131, we obtain the fol owing formulas for the coefficients of the expansions:
(2.14)
On (I general
h
=
---1 6
---
;
of anirotropic
nhelfs
309
1 aa B
aft
;B$G?2+
+x-
theory
f[+&$
(k,X’)-
-++$-
* k&$$X’]
+ +[+&$(kpY’)-
-
_j_ _.;__k
$kl.;.a$
1 2As
aB __ y’ aa I
+
(2.25)
310
Arbor tsuaian
S.A.
In formulas (2.17) to (2.19) we have, in conformity with the usual definition of curvature changes and torsion of the middle surface of the shell 12,s I,
1 a
--1 aw ..-. _ A aa
u- H, >
Xl0 = -
-,;*- -a;
x20 = -
1 a - iaw. -~~--_-- v _ R ag ( n ag Rz 1
.p=-_
2 - 8% _ ns ( da ap
a.4 -.I._.1 aw -- 1 --AB
a? ( B a?
1 al3 _ 1 __aw _ ______ AB aa i A aa
aA& - 1 .~. .-_ ._- _ A $3 da
ii2 > ii _-N1 >
(2.29) (2.30)
f ai3 __. _._. .awR da a,3 ) -I-
(2.31)
where R, = Rl(a, ,k?) and R, = RZ(a, ,d> are the principal radii of curvature of the middle surface. Considering the expansions (2.13) we note that they have some similarity with the analogous expansions used in ref. El 1 ; the similarity is, however, only a superficial one. In the determination of the strain components eaa, e ref. [l] actually uses expressions in terms of powers of y ke!Zj%gea8 at the same time the hypothesis of non-deformable normals C 1,2 I, while in the present paper, as in the publications [8,9 I, the relations (2.13) are being obtained on the basis of the basic assumptions of the theory offered here. On the basis of (2.13) and of the original assumption (b) we derive from the generalized Hooke's law the following expressions for the stress components iraa,agp, ~4 : =a = 4~ -I- B,,4
+ &+a
+ y2 P,,ril
+ %A
+ &A)
= Baasa+ &A .%a4
B,,rlz -i k-4
+ &d)
+ u” @WI + &b + B,sw + Y(&A
~a = &,a, + &A
+ y” (B&
+ &A
+ y” PWI
+ Y” (&zriz -t- B nri, + B,,4
+ &z%
+ua VMir
+ &ciw + Y (&xx,
+
+ &,E,
-t + (2.32)
+ &t&) + &A
+
+ r’ V&&z + + %I)
+ %,a + y(&exr + &A
f Bzsrlz + &,v) + y3 (&A C B&z -+ y" (R& + Be& + B&
+ &A
(2.33) +
+ B,,Q + (2.34)
In these formulas the constants Bik are given by the following expressions in terms of the elastic constants aiR [lo,111 :
On a general theory of anisotropic
T ‘$t
~tges~
shells
311
;1p4, CX~“, ‘4, rQy, rpr prodye internal_fmces (‘fit an moments (M1, M , HI, which must satisfy the
f~~loZ,ng2~at~L.icZ,1
conditions
51:
[ 1, 2,
& (BT,) - T, ;$ + A- (AS,) + S, $ -$ (ANTE) - I’, ;$
+ &- (BS,) + S, 7;
+ ABk,N,
= - ABX”
+ ABk,N,
= - ABY
- (k,T, + J&T,) + $&&-(BN1)+
$- (AN,)] = - 2’
&(BH)+H’&
+-&(AM,)-M+ABN,=O
ix (AH) + H ;+
+ -& (B&f,) -M,
;; -
(2.36)
ABNl = 0
S,-SS,+klH-kk,H=O In these
formulas
X’ = X’(a, represent load,
the symbols p),
f” = Y’(a,
the components of the intensity
referred
to the middle
surface
f3),
Z*=
vector
of the shell
2Y(a,
of the applied
P stands generally
‘Ihe stress manner [l, elasticity statics:
resultants
surface
[ 7 1 , namely
P=P+(i+~)(l+~)+P-(l--)(~-~) where
f3)
(2.37)
for X, Y, 2. appearing
in (2.36)
2, 10 I. Without going into details, formulas, which identically satisfy
are determined
in the usual
we give here the simplest the sixth equation of
312
L-t.
Aatsartsurian
(2.41)
N, = t
(X’ - X-) -
N, =+(Y+
$- cp(a, p)
-Y-)--$-+(a,B)
(2.45) (2.46)
In these relations we have the following formulas for the rigidity constants Ci, of compression and Di, of bending: Cik = hBi&, Dik = +Z-3ie
(2.47)
We state here that, in the process of substitution of the values of . . . . 6, all terms containing X and Y can be omitted, still maintainEl’ ing a sufficiently high degree of accuracy [7 I, in all elasticity relations. Using the formulas (2.29), (2.30), we can eliminate the displacement components u, v, I of the middle surface from the relations (2.14) to (2.16); this leads to
The equation (2.48) is the third continuity relation for the deformation of the middle surface of the shell. As it should be expected, the relation does not differ in any way from the corresponding relation of the classical theory of thin shells [2, 5 I. The remaining two conditions of continuity for the deformation of the middle surface will not be needed in the present paper. 'lheequations (2.14) to (2.31), (2.36), (2.38) to (2.46) taken together represent a complete system of equations of the theory of shells.
On a general
theory
of
anisotropic
313
shells
It is known [1,2,5 1 that such a complete system can be established in various ways. In view of its extreme complexity in the general case of a shell of arbitrary form, the complete system of equations will be considered here for one practically important type of shell only. In the process of solving actual boundary value problems the differential equations of the shell have to be completed in the usual manner by statement of the boundary conditions [1,2,3 I . 3. Avoiding discussion of details, special types of boundary conditions. Free edge. of the shell,
‘Ibis designation for which M 1 =O,
will
H =O,
we mention here some possible characterize
S, = 0,
T, =0,
Simply supported edge. This designation (a = const. 1 of the shell, for which M 1=O,
T,=O,
w=O,
Fixed edge with a hinge. (a = const. 1 of the shell,
Clamped the shell,
v = 0,
u=O,
edge. ‘Ihis for which
designation
u=o
,
will
v=o,
v=O,
be used for such an edge
refers
+ B,,h = 0
characterizes W=
(3.1)
iV,=O
II,,01 + B&.
This designation for which
M,=O,
such an edge (a = const.)
0,
such an edge
+=o
Of course,
other
w=o,
qJ=o
The boundary conditions analogous manner.
(3.41
6u
boundary conditions for
(3.3)
to such an edge (a = const. 1 of
-.1 aw _- -lck,u~+qD,=O A
(3.2)
are still
an edge 6 = const.
Concluding this Section we note that the subject ditions requires special investigations.
possible. can be stated
in an
of the boundary con-
A detailed study of the results presented in the first three Sections of this paper reveals the following fact: the special case, characterized by au4 = 0, as5 = 0, a,,5 = 0, leads to the basic relations and equations of the theory of anisotropic shells based upon the hypotheses of nondeformable normals. 4. Consider a shell in the form of a circular cylinder of radius R. We take the a and p coordinate lines to be directed along the generators and the parallel circles of the middle surface, respectively. Assume that the shell is being acted upon by normally applied loading only. For such a shell
S.A.
314
Ambartsumian
B = eonst,
A = const,
(4.1)
kz = -$
k, = 0,
The coefficients of the expansions (2.13) are 1
au
E1= A xi
Es=
’
1
%l=---__-.-.-
1 ~2
x2=:---
1
1av
B@-+
aaw
A2 ~3~2 i32w
ap
au
1
-B-w,
W==v- + h2 1 6 A
1 1 au .+_-__-_8_Ba3 R Bag
1 -A
av aa
(4.2)
a@, aa ha
1
(4.3)
L’@)z
(4.4)
(4.5)
(4.6)
l'heequations of equilibrium assume the form I A
.d$!%+
$.a$++N2=0,
aw da +
1 aM2
---
3
ag
+-a$+fa+-N1=O
N 2==0 (4.7)
Substituting the expressions for tl, .... (= from (4.2) to (4.61 into the formulas (2.38) to (2.46), we obtain the stress resultants in terms of the unknown functions u, v, W, gS,$. Substituting the obtained expressions of the stress resultants into the equations of equilibrium (4.7), we find a final system of the five differential equations for the five unknown functions u, u, m, 4, 9% namely
+
Q*
(Cik,
aik)
J, +
Q6 (Cikr ‘%k)‘9 = 0
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
316
S.A.
Anbartsunian
Thus, the problem of the anisotropic a system of five differential equations known functions. difficulty
Having obtained
the stress
the formulas
(2.32)
The system of equations cation that
in the case of
for
B,l
as well (2.38)
(4.8)
isotropic a*$ = 0,
to (2.46)
solid
find without by means of
and (4.2)
undergoes
isotropic
shell
to (4.6).
substantial [lo
1.
It
simplifiis known
we have
Qt5= a&j = 0,
= Bw = &?,
we will
as the stresses,
to (4.12)
a transversely
a transversely a I6 = 0,
the latter,
resultants, to (2.34),
cylindrical shell is reduced to (4.8) to (4.12) for the five un-
n4, = U&5= -$ E
B,, = pB,j,
Bw, = -2 (1 +
I*)
(4.13)
where E is the modulus of elasticity in the plane of isotropy, p is Poisson’s ratio, G’ is the shear modulus for planes normal to the plane of
isotropy. We assume the plane
each point
of the shell,
of
isotropy
of the material
to the middle
surface
to be parallel,
at
of the latter.
The coordinates a, /3 are to be chosen in such a manner that the coefficients of the first quadratic form assume the following values [ 1,2 3 :
A= By virtue of (4.13) and (4.14) simpler and assumes the following
1,
(4.14)
B=K
the final form:
system of equations
becomes
(4.15)
(4.16)
On a general
theory
of
anisotropic
317
shells
(4.18) I a2v I ---_ asw __ --aaag _-..-- xi aaap2 R”-
saw aps
As an example we shall treat here the problem of a horizontal tube, of simply supported at its ends. The tube transversely isotropic material, is entirely filled with a liquid of specific weight y. ‘Ihe weight of the tube material shall be neglected [ 7,12 1. Measuring the angle /3 from the lowest the tube, we use the expansions
point
m
w=zr,
C,, cos nfl sin mT ,
of
the cross
section
of
n
m n
The chosen functions fulfil the boundary conditions of simple support of periodicity along the edges a = 0, Q = 2, as well as the conditions with the period 2n for the argument /3. ‘Ihe acting load, the radial pressure of the fluid, is (4.21)
P=RY(f-t-cosFd
It
can be represented
by the double
series
Z= x;Tjq,,cosnj
sinmF
(4.22)
7n n
where the coefficients
q,,
are given [ 7,12 ] by
Qmn=O,
q,*=
In view of the good convergence subscript RI= 1, 3, 5, . . . we will to the first term. Substituting
the functions
4YR
- nzx ’
4Yfi 9 ml =- ?l,Z
of the expansions confine ourselves
with respect to the in the following
u, II, UJ,q5, $I from (4.201,
and the function
318
S.A.
from (4.22) into the (4.191, we obtain, for equations for the five the special case, when cations. z
Anbartsumian
corresponding equations of the system (4.15) to each pair of values of m and n, a system of five unknown coefficients Ancn, Rnn, C,,, Dm,, Emn. In n = 0, these systems undergo essential sirnplifi-
Let us consider the numerical example treated in [7,12 f ; take a = 50 cm, 1 = 25 cm, h = 7 cm, while p = 0.3. For the dimensions just given we shall examine three cases, for which the ratio E/C' equals 2.6; 5.0; 10.0, respectively. In the case E/G’ = 2.6 we have evidently to deal with an isotropic shell, while in the second and in the third case we have transversely isotropic shells. 'Ihevalues of the coefficients Cn,,of the normal displacement component of the shell are given in Table 1 in the form of the ratio C,,/N, where N = 24yR3Z2/Enh. In the last column of Table 1 are given the values of TABLE 1.
2.6 5.0 10.0 .~
0.7022 0.6708 0.8103 0.8004 0.9OOO 0.9138 1.0616 _- 1.0275
1.3730 1.6107 1.8138 2.0891
the coefficient of the maximum normal displacement, i.e. the values of the coefficient of w at the point /3= 0, /?= % 1. For comparison we give in the first line of Table 1 the values of the same coefficients CAJUN, where N = 24~R~l'/~~, calculated by means of the theory based upon the hypothesis of non-deformable normals [7,12 1. The comparison shows that the results obtained on the basis of the latter theory essentially differ from those derived from the theory offered in the present paper. We see that even in the case of an isotropic shell the error incurred in the classical theory (based upon the hypothesis of non-deformable normals) can amount to 15%. In the case of transversely isotropic shells the error can become quite substantial for the case of the example considered here, depending on the ratio E/G’. For instance, in the case of a ratio E/G* = 10 the error just mentioned rises to 35%.
BIBLIOGRAPHY 1.
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teoriia
obolochek
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Goldenveizer, Elastic
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She1
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izdat,
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ortotropnykh obolochek shells). Izvestiia
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Anizotropnyye
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i obolochki
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Ambartsumian,
tonkikh
(Theory
tsilindricheskikh
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tonkostennykh
Elastic
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Ambartsumian, &A., (On the analysis
nauk.
Arsl.
Teoriia
Lur’ye,A.I.,Statika
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13.
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i Tekh.
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1951.
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Mat.
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Translated
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1.M.
OBSBCBBI TBOUII AN~ZO~UOPNYKB OBOLOCSEK) PYY Yot.22, No.2, f958, pp.2!26-237 S. A. AYBARTSUYIAN (Erevan) (Received
2 July
1958)
1. We consider a thin anisotropic shell of constant thickness h. Assume that the material of the shell obeys the generalized Hooke’s and that
at each point
parallel
to the middle surface
used as surface linear
lines
coordinates of that
along
the normal,
point
(a,
(b)
of the shell.
/3, y)
s and p,
will
surface
will
be
to curvi-
with the principal
the distance,
measured
(a, /3) of the middle surface
and the
We assume that
of the shell, normal to the middle after deformation;
the normal stresses*
law
syuvaetry,
be referred
which coincide
Let y represent
between the point
of the shell.
of elastic
The latter
and the shell
surface.
(a) the line elements not change their lengths
ua,o
is only one plane
of coordinates,
orthogonal
curvature
there
o,, are small
surface,
do
as compared with the stresses
andr
P $3; (c) the shear
stresses
thickness of the shell bola f 13 I . king
more rigorous
can state
of the rayand'PY wrthvarythein lawthe ofdirection the quadratic para-
in accordance
in the formulation
here the assumptions
(a)
and (I)
of the hypotheses
[ 2,s
in the following
e 0 approximately; w= (b) the stresses ui do not exert any essential components eaa and they can be neglected and “ss equations of the generalized Hooke’s law.
1 , we
form:
(a)
2. By virtue and r~,,
%Y==
*
the shear
stresses
r
ay
x+-x2 y+-y-
+XV++X-)++!p-$)p(a,
2
+
Here and in the following theory of shells.
Reprint
(~1 concerning
on the strain
in the corresponding
we have
%vf-
the
of the assumption
influence
-$(Y’
+
u-)
we adopt
p, +
+(r’-q)$(.,
the well-known
Order No. PMM25. 305
B)
notations
(2-l)
used
in
306
&A.
Aubartsurian
where A” (a, p), Y+ (a, fl) and X’ (a, f3), Y- (a, fl) are the components along the axes of the moving trihedron (in the directions of the positive tangents to the lines /3 = const., a = const., respectively) of the intensity vectors of the surface loads, applied to the boundary surfaces andy=-%h, respectively, while $(a, @), t,!&z, f3) are unknown Y =%h functions. Substituting the values of the tangential stresses r o$sr13Y from (2.1) into the corresponding equations of the generalized % for the shear strain components e and eBY the law 16 I, we obtain aY formulas
Here we have introduced
the following
notations:
x = a [Ub6(x+ -
x-)
+ UA5(Y’ -
Y = f [a44(Y” -
Y-) + a45(X’ -
r-)] (2.3)
x-)1
X’=a,5(X++x-)+a*,(I-++Y-) Y’ =
where the quantities
lx,,
(Y’ -f- Y-j +
aik are elastic
lQ6
(X’ +
constants
From the equations of the three-dimensional have for the strain components [ 1 1
x-j
[6 I. theory
of elasticity
we
(2.7)
(2.9) H,=A(lfk,y), In these
formulas A = A(a,
1r2=B(l+k,y) f3) and 3 = BCa, j3) are the coefficients
(2.10) of
On a general
theory
of anisotropic
307
sheiis
the first quadratic form of the middle surface, k, = k (a, @> and f surface, k, = kZ(a, ,5?> are the principal curvatures of the midd e (a, /3,y) are the displace(a, f3, y) and ~,(a, /3, y), up = u % = p="/ ment components of an arg itrary point 0 the shell in the directions of the tangents to the coordinate lines, respectively. &
the basis of the assumption (a) we find from (2.7) &L
$ =
0,
lly =
+(a,
/3) = w(a,
(2.11)
f3)
Thus, like in all existing theories of thin shells, the displacement of any point of the shell is independent of the coordinate y. This "r displacement component has for all points of a line element of a normal to the shell a constant value, equal to the normal displacement component w = ~(a, 6) of the corresponding point of the middle surface of the shell. Substituting the expressions for e , ep,, HII H2 and u,,from (2.2), (2.10) and (2.11) into equations (2.9y we obtain differential equations for the displacement components ua and UP. Integrating these equations and taking into consideration that ua = ~(a, 8) and u = ~(a, /I) when a y = 0, we find
(2.12) up = (1 + k2Y) u -
-$- a+
-
Y(l
+y
$)$(D,+
where u = u(a, /!I), tr= ~(a, /!3> are the tangential displacement components of the corresponding point of the middle surface. In the process of deriving the formulas (2.12) the accuracy was being confined to consideration of quantities up to those of the order of magnitude of yki, i.e. whenever a sufficiently precise estimation was possible, terms of the order of magnitude of (yki)2 were being neglected in comparison with unity. Our formulas (2.12) show that, in contrast to known theories of thin shells [ 1,2,5,7 I, the tangential displacement components ua and u P .Of any point of the shell at a distance y from the middle surface are, in the case considered here, as in the publications [ 8,9 1, non-linear functions of the distance y. By virtue of (2.12) the strain components eaa, e PP' pressed by ~l~~ials in powers of y, namely
e4
can be ex-
308
S.A.
AMbartsurian
(2.13)
from (2.12) and (2.111, respectSubstituting the values of Us, u into the relations (2.6) an~~2~~~, and comparing the resulting ively, with the corresexpressions for the strain components eaa, e Pp * .“aP ponding expressions (2.131, we obtain the fol owing formulas for the coefficients of the expansions:
(2.14)
On (I general
h
=
---1 6
---
;
of anirotropic
nhelfs
309
1 aa B
aft
;B$G?2+
+x-
theory
f[+&$
(k,X’)-
-++$-
* k&$$X’]
+ +[+&$(kpY’)-
-
_j_ _.;__k
$kl.;.a$
1 2As
aB __ y’ aa I
+
(2.25)
310
Arbor tsuaian
S.A.
In formulas (2.17) to (2.19) we have, in conformity with the usual definition of curvature changes and torsion of the middle surface of the shell 12,s I,
1 a
--1 aw ..-. _ A aa
u- H, >
Xl0 = -
-,;*- -a;
x20 = -
1 a - iaw. -~~--_-- v _ R ag ( n ag Rz 1
.p=-_
2 - 8% _ ns ( da ap
a.4 -.I._.1 aw -- 1 --AB
a? ( B a?
1 al3 _ 1 __aw _ ______ AB aa i A aa
aA& - 1 .~. .-_ ._- _ A $3 da
ii2 > ii _-N1 >
(2.29) (2.30)
f ai3 __. _._. .awR da a,3 ) -I-
(2.31)
where R, = Rl(a, ,k?) and R, = RZ(a, ,d> are the principal radii of curvature of the middle surface. Considering the expansions (2.13) we note that they have some similarity with the analogous expansions used in ref. El 1 ; the similarity is, however, only a superficial one. In the determination of the strain components eaa, e ref. [l] actually uses expressions in terms of powers of y ke!Zj%gea8 at the same time the hypothesis of non-deformable normals C 1,2 I, while in the present paper, as in the publications [8,9 I, the relations (2.13) are being obtained on the basis of the basic assumptions of the theory offered here. On the basis of (2.13) and of the original assumption (b) we derive from the generalized Hooke's law the following expressions for the stress components iraa,agp, ~4 : =a = 4~ -I- B,,4
+ &+a
+ y2 P,,ril
+ %A
+ &A)
= Baasa+ &A .%a4
B,,rlz -i k-4
+ &d)
+ u” @WI + &b + B,sw + Y(&A
~a = &,a, + &A
+ y” (B&
+ &A
+ y” PWI
+ Y” (&zriz -t- B nri, + B,,4
+ &z%
+ua VMir
+ &ciw + Y (&xx,
+
+ &,E,
-t + (2.32)
+ &t&) + &A
+
+ r’ V&&z + + %I)
+ %,a + y(&exr + &A
f Bzsrlz + &,v) + y3 (&A C B&z -+ y" (R& + Be& + B&
+ &A
(2.33) +
+ B,,Q + (2.34)
In these formulas the constants Bik are given by the following expressions in terms of the elastic constants aiR [lo,111 :
On a general theory of anisotropic
T ‘$t
~tges~
shells
311
;1p4, CX~“, ‘4, rQy, rpr prodye internal_fmces (‘fit an moments (M1, M , HI, which must satisfy the
f~~loZ,ng2~at~L.icZ,1
conditions
51:
[ 1, 2,
& (BT,) - T, ;$ + A- (AS,) + S, $ -$ (ANTE) - I’, ;$
+ &- (BS,) + S, 7;
+ ABk,N,
= - ABX”
+ ABk,N,
= - ABY
- (k,T, + J&T,) + $&&-(BN1)+
$- (AN,)] = - 2’
&(BH)+H’&
+-&(AM,)-M+ABN,=O
ix (AH) + H ;+
+ -& (B&f,) -M,
;; -
(2.36)
ABNl = 0
S,-SS,+klH-kk,H=O In these
formulas
X’ = X’(a, represent load,
the symbols p),
f” = Y’(a,
the components of the intensity
referred
to the middle
surface
f3),
Z*=
vector
of the shell
2Y(a,
of the applied
P stands generally
‘Ihe stress manner [l, elasticity statics:
resultants
surface
[ 7 1 , namely
P=P+(i+~)(l+~)+P-(l--)(~-~) where
f3)
(2.37)
for X, Y, 2. appearing
in (2.36)
2, 10 I. Without going into details, formulas, which identically satisfy
are determined
in the usual
we give here the simplest the sixth equation of
312
L-t.
Aatsartsurian
(2.41)
N, = t
(X’ - X-) -
N, =+(Y+
$- cp(a, p)
-Y-)--$-+(a,B)
(2.45) (2.46)
In these relations we have the following formulas for the rigidity constants Ci, of compression and Di, of bending: Cik = hBi&, Dik = +Z-3ie
(2.47)
We state here that, in the process of substitution of the values of . . . . 6, all terms containing X and Y can be omitted, still maintainEl’ ing a sufficiently high degree of accuracy [7 I, in all elasticity relations. Using the formulas (2.29), (2.30), we can eliminate the displacement components u, v, I of the middle surface from the relations (2.14) to (2.16); this leads to
The equation (2.48) is the third continuity relation for the deformation of the middle surface of the shell. As it should be expected, the relation does not differ in any way from the corresponding relation of the classical theory of thin shells [2, 5 I. The remaining two conditions of continuity for the deformation of the middle surface will not be needed in the present paper. 'lheequations (2.14) to (2.31), (2.36), (2.38) to (2.46) taken together represent a complete system of equations of the theory of shells.
On a general
theory
of
anisotropic
313
shells
It is known [1,2,5 1 that such a complete system can be established in various ways. In view of its extreme complexity in the general case of a shell of arbitrary form, the complete system of equations will be considered here for one practically important type of shell only. In the process of solving actual boundary value problems the differential equations of the shell have to be completed in the usual manner by statement of the boundary conditions [1,2,3 I . 3. Avoiding discussion of details, special types of boundary conditions. Free edge. of the shell,
‘Ibis designation for which M 1 =O,
will
H =O,
we mention here some possible characterize
S, = 0,
T, =0,
Simply supported edge. This designation (a = const. 1 of the shell, for which M 1=O,
T,=O,
w=O,
Fixed edge with a hinge. (a = const. 1 of the shell,
Clamped the shell,
v = 0,
u=O,
edge. ‘Ihis for which
designation
u=o
,
will
v=o,
v=O,
be used for such an edge
refers
+ B,,h = 0
characterizes W=
(3.1)
iV,=O
II,,01 + B&.
This designation for which
M,=O,
such an edge (a = const.)
0,
such an edge
+=o
Of course,
other
w=o,
qJ=o
The boundary conditions analogous manner.
(3.41
6u
boundary conditions for
(3.3)
to such an edge (a = const. 1 of
-.1 aw _- -lck,u~+qD,=O A
(3.2)
are still
an edge 6 = const.
Concluding this Section we note that the subject ditions requires special investigations.
possible. can be stated
in an
of the boundary con-
A detailed study of the results presented in the first three Sections of this paper reveals the following fact: the special case, characterized by au4 = 0, as5 = 0, a,,5 = 0, leads to the basic relations and equations of the theory of anisotropic shells based upon the hypotheses of nondeformable normals. 4. Consider a shell in the form of a circular cylinder of radius R. We take the a and p coordinate lines to be directed along the generators and the parallel circles of the middle surface, respectively. Assume that the shell is being acted upon by normally applied loading only. For such a shell
S.A.
314
Ambartsumian
B = eonst,
A = const,
(4.1)
kz = -$
k, = 0,
The coefficients of the expansions (2.13) are 1
au
E1= A xi
Es=
’
1
%l=---__-.-.-
1 ~2
x2=:---
1
1av
B@-+
aaw
A2 ~3~2 i32w
ap
au
1
-B-w,
W==v- + h2 1 6 A
1 1 au .+_-__-_8_Ba3 R Bag
1 -A
av aa
(4.2)
a@, aa ha
1
(4.3)
L’@)z
(4.4)
(4.5)
(4.6)
l'heequations of equilibrium assume the form I A
.d$!%+
$.a$++N2=0,
aw da +
1 aM2
---
3
ag
+-a$+fa+-N1=O
N 2==0 (4.7)
Substituting the expressions for tl, .... (= from (4.2) to (4.61 into the formulas (2.38) to (2.46), we obtain the stress resultants in terms of the unknown functions u, v, W, gS,$. Substituting the obtained expressions of the stress resultants into the equations of equilibrium (4.7), we find a final system of the five differential equations for the five unknown functions u, u, m, 4, 9% namely
+
Q*
(Cik,
aik)
J, +
Q6 (Cikr ‘%k)‘9 = 0
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
316
S.A.
Anbartsunian
Thus, the problem of the anisotropic a system of five differential equations known functions. difficulty
Having obtained
the stress
the formulas
(2.32)
The system of equations cation that
in the case of
for
B,l
as well (2.38)
(4.8)
isotropic a*$ = 0,
to (2.46)
solid
find without by means of
and (4.2)
undergoes
isotropic
shell
to (4.6).
substantial [lo
1.
It
simplifiis known
we have
Qt5= a&j = 0,
= Bw = &?,
we will
as the stresses,
to (4.12)
a transversely
a transversely a I6 = 0,
the latter,
resultants, to (2.34),
cylindrical shell is reduced to (4.8) to (4.12) for the five un-
n4, = U&5= -$ E
B,, = pB,j,
Bw, = -2 (1 +
I*)
(4.13)
where E is the modulus of elasticity in the plane of isotropy, p is Poisson’s ratio, G’ is the shear modulus for planes normal to the plane of
isotropy. We assume the plane
each point
of the shell,
of
isotropy
of the material
to the middle
surface
to be parallel,
at
of the latter.
The coordinates a, /3 are to be chosen in such a manner that the coefficients of the first quadratic form assume the following values [ 1,2 3 :
A= By virtue of (4.13) and (4.14) simpler and assumes the following
1,
(4.14)
B=K
the final form:
system of equations
becomes
(4.15)
(4.16)
On a general
theory
of
anisotropic
317
shells
(4.18) I a2v I ---_ asw __ --aaag _-..-- xi aaap2 R”-
saw aps
As an example we shall treat here the problem of a horizontal tube, of simply supported at its ends. The tube transversely isotropic material, is entirely filled with a liquid of specific weight y. ‘Ihe weight of the tube material shall be neglected [ 7,12 1. Measuring the angle /3 from the lowest the tube, we use the expansions
point
m
w=zr,
C,, cos nfl sin mT ,
of
the cross
section
of
n
m n
The chosen functions fulfil the boundary conditions of simple support of periodicity along the edges a = 0, Q = 2, as well as the conditions with the period 2n for the argument /3. ‘Ihe acting load, the radial pressure of the fluid, is (4.21)
P=RY(f-t-cosFd
It
can be represented
by the double
series
Z= x;Tjq,,cosnj
sinmF
(4.22)
7n n
where the coefficients
q,,
are given [ 7,12 ] by
Qmn=O,
q,*=
In view of the good convergence subscript RI= 1, 3, 5, . . . we will to the first term. Substituting
the functions
4YR
- nzx ’
4Yfi 9 ml =- ?l,Z
of the expansions confine ourselves
with respect to the in the following
u, II, UJ,q5, $I from (4.201,
and the function
318
S.A.
from (4.22) into the (4.191, we obtain, for equations for the five the special case, when cations. z
Anbartsumian
corresponding equations of the system (4.15) to each pair of values of m and n, a system of five unknown coefficients Ancn, Rnn, C,,, Dm,, Emn. In n = 0, these systems undergo essential sirnplifi-
Let us consider the numerical example treated in [7,12 f ; take a = 50 cm, 1 = 25 cm, h = 7 cm, while p = 0.3. For the dimensions just given we shall examine three cases, for which the ratio E/C' equals 2.6; 5.0; 10.0, respectively. In the case E/G’ = 2.6 we have evidently to deal with an isotropic shell, while in the second and in the third case we have transversely isotropic shells. 'Ihevalues of the coefficients Cn,,of the normal displacement component of the shell are given in Table 1 in the form of the ratio C,,/N, where N = 24yR3Z2/Enh. In the last column of Table 1 are given the values of TABLE 1.
2.6 5.0 10.0 .~
0.7022 0.6708 0.8103 0.8004 0.9OOO 0.9138 1.0616 _- 1.0275
1.3730 1.6107 1.8138 2.0891
the coefficient of the maximum normal displacement, i.e. the values of the coefficient of w at the point /3= 0, /?= % 1. For comparison we give in the first line of Table 1 the values of the same coefficients CAJUN, where N = 24~R~l'/~~, calculated by means of the theory based upon the hypothesis of non-deformable normals [7,12 1. The comparison shows that the results obtained on the basis of the latter theory essentially differ from those derived from the theory offered in the present paper. We see that even in the case of an isotropic shell the error incurred in the classical theory (based upon the hypothesis of non-deformable normals) can amount to 15%. In the case of transversely isotropic shells the error can become quite substantial for the case of the example considered here, depending on the ratio E/G’. For instance, in the case of a ratio E/G* = 10 the error just mentioned rises to 35%.
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i Tekh.
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