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Method of initial functions for thick shells
hr. J. Solids Stnrcrttra Vol. 21. No. 8. pp. 851463. Printed in Great Britain.
METHOD
Civil engineering
0020-76831x5 Pqamon
1985
OF INITIAL FUNCTIONS SHELLS
%3.W t .# Press Ltd.
FOR THICK
SUSAN FARAJI and ROBERT R. ARCHER Department, University of Massachusetts, Amherst, MA 01003
Abstract-In
the present work for circular cylindrical shells, three-dimensional elasticity equations are solved by assuming Taylor series expansions, in the radial direction, for the stresses and displacements. Depending upon the number of terms retained in the expansion, different order shell theories are derived,, classical theories (referred to as eighth-order), the shear deformation-transverse normal stress theories (referred to as tenth-order), and higher order theories (referred to as twelfth-order). In each case. by carrying out the symbolic algebra using the digital computer, partial differential equations are derived. The procedure was carried out in detail for the case of a circular cylindrical shell with no loading on the interior surface and a given pressure distribution on the exterior surface. Then, numerical comparisons are made between the current theories and various shell theories, as well as the exact (three dimensional) theory. Thus, using this method with its associated computer programs, one can realize a spectrum of approximate shell theories ranging from the classical thin shell, through all current thick shell theories, right up to the three-dimensjonal elastic theories.
NOTATION
Cylindrical coordinates. Normalized displacement components. Middle surface radius of cylinder. Inner radius of cylinder. Outer radius of cylinder. Normalized radial and axial coordinate. Thickness of shell. Normalized thickness. Normalized shear stress components. Normalized normal stress components.
Exponential
coefficient.
INTRODUCTION
In the present work, the method of initial functions has been used to solve the elasticity equations for isotropic materials by assuming Taylor series expansions for stresses and displacements in the radial direction at interior or exterior edges, for circular cylindrical shells. Comparison is then made with classical she11 theories of different orders, as well as three-dimensional shell theory as a special case of elasticity theory. The method of initial functions, in which the elasticity equations are solved by assuming Taylor series expansions for stresses and displacements was introduced by Vlasov[ll for elasto-static problems. Das and Setlur[2] applied the method to twodimensional elasto-dynamic problems both in plane stress and plane strain in which Maclaurin series were expanded in the thickness direction. Haydll31 used Vlasov’s method of initial parameters for symmetric bending of 851
852
SUSAN FARAJIAND ROBERTR. ARCHER
cylindrical shells, and by an example he showed his method is applicable for the solution of shells with many supports or concentrated loading. Iyengar et. ~11[4]used this method for investigating thick rectangular plates, for the case of two opposite edges simply supported and the other two edges clamped. Das and Rao[.5] applied this method to thick plates subjected to antisymmetric and symmetric lateral loads. This method was extended by Iyengar and Chandrashekharalh] for the case of axisymmetric cylindrical shells. In the present work, the earlier work is extended in two important ways. First, the more general case of an unsymmetric shell is allowed; secondly. the computer algorithm to be developed will allow the derivation of thick shell theories of arbitrarily high order. The present work addresses itself to isotropic circular cylindrical shells, although it will be seen that the more general case of anisotropic materials would follow along similar lines, but with more involved algebra. For circular cylindrical shells, three-dimensional elasticity equations are solved by assuming Taylor series expansions, in the radial direction for stresses and displacements, making use of a convenient point within or at the edge of the shell. The formulation becomes exact when an infinite number of terms are used in the Taylor series expansions. In practice. only a finite number of terms can be used, and the number of terms to be retained depends on the accuracy desired for a given ratio of thickness to radius (Ma). For thick shells, it is necessary to retain more terms. Depending upon the number of terms retained, one recovers the classical theories (referred to later as eighth-order), the shear deformation-transverse normal stress theories (referred to as tenth-order), and higher order theories (referred to as twelfthorder, etc.). In each case, by carrying out the symbolic algebra using the digital computer, it is possible to derive partial differential equations for the different shell theories of arbitrarily high order (only limited by the storage capability of the computer). One can recover the associated boundary conditions for any given problem to the same order of accuracy as used in the derivation of the differential equation. Thus, using this method with its associated computer program, one can realize a spectrum of approximate shell theories ranging from the classical thin shell, through all current thick shell theories, right up to three-dimensional elastic theories. In many problems (i.e. involving stress concentrations and other problems where analytic solutions are preferred to numerical solutions found by finite element or finite difference methods), this method is of particular value for deriving suitable differential equations and boundary conditions for a shell with a given h/a. In order to demonstrate the comparative accuracy of the various theories, numerical comparisons are made between the current theories and various shell theories, as well as the exact (three-dimensional) theory.
Anulysis
Starting from three-dimensional elasticity equations, particular relations among stresses and displacements and their derivatives are found in matrix form. Then, by successive differentiations of these equations, relations between stresses and displacements and higher derivatives are found. Using a matrix notation and substituting these derivatives into Taylor series expansions at a given point along the radius (exterior or interior edges or mid-point) yields a formal series solution of the elasticity equations. The differential equations of equilibrium and strain components of an isotropic elastic body in cylindrical coordinates r, 0, z with corresponding displacement components u, ~1,w in the general case are:
853
Method of initial functions for thick shells
(1)
y,=!C+l!E. z
r at3’
az
and, according
to the generalized
aTi; E, = az
Hooke’s law for an isotropic material,?
(3)
For cylindrical layer studies, we can reduce eqns (l), (2), and (3) from a system of 15 equations to a system of six first-order equations with respect to the r derivatives regarding ,!A%, d/az as formal symbolic operators. In matrix form, the six equations become
[ul’ = [Sl [Yl?
(4)
where
-
( )
--(Y
- CivP
-- 1 r
0
-vP- P r
0
-- 2 r
0
0
-- VP r
0
1
1
0
VP
1
r
( --v 21
) P
2uPcX
2pp -131r
T _ z$z
-,PP
(l)
-
(,,,,‘+
-2fy
-(u2
+ y.q
-
@p
$1
(!+
-- VPP r
_ vPa
-- P
-1
0
-a
0
0
r
t This method can also be used for anisotropic
r
materials with the same algebraic procedure.
854
SUSAN FARAJIAND ROBERTR. ARCHER
and I
ur I
Trz
[yl’ =
r:t3 u’ 71’ W'
with ( )’ = a 5 ( ) remaining stresses can be obtained from 2vP 2vP u + pv + 2PakV + v P CT, 52 = Y
r
2P oe =-u
2PP + -71+
r
T,e
=
a71 +
2Paw
Y
+ VPU,
P M’.
-
r
Higher derivatives
of [u]’ can be obtained from eqn (4) by successive (1) tions. (Note: It is convenient to write Si,i = Si,i) then
(1)’ _YY=
(5)
C
(Sij
differentia-
(1) Yj
+
Sij
(1)’ (l)(l) YY= 2 (Si/, + SijSjk)
yj’)
(6)
yk,
or (2) Y; =
C
Sil,
Yk
(7)
where (2) (1)’ (l)(l) Six = Si/, + Sij S,jk, etc. Now for the nth derivative
n (n) [Yl = [Sl [Yl
(8)
In eqn (8) [S ]is a matrix of differential operators 01, l3 for any order of derivation. For any given Poisson’s ratio v and any radius r in radial direction ranging from (n) 1 to 1 + p, computer program& give elements of [S ] up to n = 7 as function of (Y, P* $ Copies of these programs are given in 171.
855
Method of initial functions for thick shells HIGHER ORDER THEORIES
FOR THICK CYLINDRICAL
SHELLS
Now, by making use of shell boundary conditions in the radial direction, we derive partial differential equations for cylindrical shells ranging from classical through current thick elastic shell theories, as well as new higher-order shell theories. In order to compare the various theories with the exact theory, we are going to study, as a special case, a circular cylinder with no loading on the interior surface and a given pressure on the exterior surfaceD, i.e. at r=
1;
r=l+p:
(Jr =
7,
=
Tr%
or = - q,7,
=
0
= T,t) = 0.
It may be noted that other classes of practically important problems can be studied along very similar lines, e.g. a cylindrical elastic layer with a stress-free outer surface and bonded to a rigid cylindrical shaft at Y = 1 would have boundary conditions: at u=~=w=O
1;
r=
#+=1+&L;
or = T& = I,
= 0
In order to derive partial differential equations governing the various theories, we use Taylor series expansions at the interior (r = 1) or exterior edge (Y = 1 + IL), by using the notation previously developed. Expanding about the interior edge:
(1) LYWI = [Y(l)1 + (r - 1) IS(l)1 [Y(l)1 + -@ ,, L!
l)*$I)][Y(l)]
+..*.
or
{
Expanding
Mfr-m! l)” ;?(l,,[Y(l)].
=
)
I
ml=
(9)
about the exterior edge:
Derivation of equations for shells with un inner surface freeof loading, and an outer surface under pressure For a cyfinder with no loading on the inner surface, and pressure acting on the outer surface, we have: r=
I;
r=l+p; Now, by applying the above boundary
(Jr = 7, = Tr%= 0
conditions
[YtbWl = [Y(l)1 +
to eqn (9), we find:
M pm(m) 2 -g [S (111 IYU)l m=l *
8 The more general case of three stress components line.
(11)
0, = - q, 7, = Tr@= 0.
(12)
presented on both edges would follow along similar
856
FARAJIAND ROBERT R. ARCHER
SUSAN
or b
0
ur -
b
0 0
Ti-9
u
-
a
b
-
a
b
0
TII
T12
T,3 tTI4 L
T,5
T,h;
0
0
(ml
(ml
(m);(m)
Cm) (m)]
T2,
T22
T23 ; T24
7-25 7-26;
0
0
(ml
(ml
(m):(m)
Cm) kd;
T~I
TX
TV 1%--.-_-;-_J T,s
7’36’
(un> (ml
(ml
(ml
(ml
(02)
T41
T42
T43
T44
T45
7-46
(m>
(m)
(ml
Cm) (ml
(ml
T51
Ts2
7;3
T1(4 Ts,
7-5~
(ml
(ml
(ml
(ml
(ml
(ml
T~I
Th2 Tm
TM
TM
T,,
=
dl) b
2)
w
‘(mlCm)(m,yi, c;r;,cmJ
0
-4
u
0
+
rt(l)
-
a
-
Ml)
a
0 0
0
411
(13)
41) 41
where ftn)
1,1
.
Ttj = C
,,)
5
Cm)
S,i (I)
or, because the present problem involves stress boundary conditions at b/a, we write: 0
1: 14(l)
Cm)Cm)(ml
which can be rewritten
T24
T25
T2h
(ml
(ml
(m)
T14
T,5
TM
>
(14)
-q
M’( 1)
in the form:
0
=
z,(l)
Let , Cm)
Cm)
ai,j =
,1
SXI (1)
where k = 4 - i;
I-j+3,
therefore ,,1
A;.;
= C k”’
(m) ai,i ,
(16)
or with [U(1)17’ = [u(l), o(l), w(l)] and [Q]’ = [O, 0, -q]
[Al [U(l)1 = LQI.
(17)
Method of initial functions for thick shells
857
We note that u, 11,w are functions of r, 8, z, while u(l), v(l), w(l) are function of 8 and z only, and [A] is a matrix of differential operators. Equation (17) represents a set of three partial differential equations. Nontrivial homogeneous solutions of (17) involve an equation of the form:
(18)
lAlF=O,
where F is a convenient auxiliary function. In practice, the number of terms to be retained in eqns (9) or (10) in the Taylor series expansions depends on the ratio of thickness to radius (h/a) and the accuracy desired. In the expansions, if one retains terms up to third derivative, then the result 1A 1F = 0 will be called the eighth-order shell theory; then, by adding fourth derivative terms, the result will be called the improved eighth-order shell theory; and then by adding fifth up to eighth derivative terms, the results will be called tenth-, improved tenth-, twelfth-, and improved twelfth-order shell theories. By comparing the results of 1A 1F = 0 for each case, one notices that by adding the fourth derivative terms, the coefficients of what we call the eighth-order shell theory have been stabilized, which means adding the fifth, sixth, seventh, and eighth higher derivatives does not change the coefficients of terms in the eighth-order theory. Also by adding the sixth derivative terms, the coefficients of the tenth-order theory have been stabilized, and by adding the eighth derivatives, the coefficients of twelfthorder theory have been stabilized. The following table shows the different order shell theories corresponding to the number of derivative terms retained in the Taylor series expansion. Computer programs which produce, respectively, elements of [A] and the coefficients of 1A 1as functions of (Y, B, and k for expansion at interior edge (r = 1) for a given v for different orders eighth- up to improved twelfth-order, are given in [7]. As a check on the computer algorithms up to the fourth derivative, elements of [A] have been derived by hand and are given in Table 2. They are given as a function of v, whereas the computer programs are written to accept a given numerical value for v. For the eighth- or improved eighth-order, expanding the determinant after dividing by ps P2/3 gives: 1A 1 = V8 + 6(1 - v*) o? + 2B6 + 8a2B4 + 6v2a4B2 + 63
1
+ B” + 4aZB2 + G, = 0.
Where G, represents terms in (19) of k and higher order. As mentioned before, for a given v, computer programs give coefficients terms in I A ( which include G,.
Table I. Relation of the number of derivative terms to the order of the shell theory No. of Derivative Terms = IX 3 4 5 6 7 8
Order of Shell Theory = IO Eighth-order Improved eighth-order Tenth-order Improved tenth-order Twelfth-order Improved twelfth-order
(19)
of all
SUSAN FARAJIANDRWERT R. ARCHER
8.58
Table 2. Elements of [A] A,,
up to fourth
derivatives as functions of a, p, p, v
= /L{ - 2Pp} + k2/2{2Pp3 + 2Pa73 + BPP} + w3/6{ -4P&p( I + )*?24{ -4Pp-’ - 4Pd’P - 8Pa’p’ + 3oJq33 -t 2a$3P(l2v
+ VI .- 8Pp’ - 38Pp} t II) + ZIJPp}
AI2 = w{ -_(y’ - 2P(3’} + ~‘/2{& + 6Pp”] f ~‘161a’ + 4Pp4 + a’@‘P(.s +. p4/24{ - 2i4 - 40Pp4 + 6a’fl”P(5 - VI + 30~1’ t 14OPp’}
VI .- ha’ - 26Pp’j
A17 = p{ -c+P(l
+ Y,} + ~%japP(Sv + 3) + p%{a’pP(v + 3) + c@“P(iJ + 3) - 2af3P(6 + 13v)} i p4/24{ -2aj3’P( I5 + 7~) - Za’PP(9 + 5~) + 2afiVY30 + 77vl}
&I = p{ - 2vPctf + (1’!2{2Pu’ + Zf’cwj3”-I- ?avP} + p316{ - 2a’ - ho@’ - hvPa} + @‘/24{ - 4Pu5 - 4PcqJ
- 8Pczf3
+ Za~P(5 - 2v) + 2+3~P(l3
-
l&f
+ 24vPcYj
+ v,} + p2/2{af3/2} + p’/6{&P(v + 3) + ap’P(v A22 = +{-c@P(I + p4124{2Pa’f3(v - 3) - 2$“P(9 -t v) + 2Pc$(6 - v,} A 23 - /.L{- 2Pct? - p’} + p’/2{3@’ t p4/24{ - 8Pa4 - lop4 t
t 3) + @P(u
+ 2f’a’} + y3/6{4Pa4 + f3” t a’p’P(S 6cx”p2P(v- 5) + 24Pa’ + SOP’}
- 3)}
- v) - 6Pa’
-
I I p’)
A ‘I, = p,{ZP} + )1’/2{ -6P} + ~*‘/6{ - 2Pd - 2Pp4 - 4Pu’p’ + 4vPa’ - XPP’ + 24P) + p4/24{4Pa4 + 20Pp4 + 24Poi’p” - 24vPa’ t SOPP’ - l2OP} A32 = J,L{ZP@} + p2i2{2Pf&
+ 2Ppz - 4P@} + p5’6{- 8P&” - 4Pa4@ + I ICIPp’
+ p4/24{-4Pf3’ As
16Pp’ + 4a’PP(v - 2) + l4P@) - 12~) - 66Pp)
t 2&P(23
= p{2vPa}
+ pL’!2{2Paz + 2Pa@ .- BvPa} + ~?‘6{-2Pab + ‘1 - 6a@P~(v t 2) + 24vPaj - XPa’f3’ + hct’P(3 + 3~) + 2Pa@(37 + 30~) - l?Ov/‘n} + *4124{ -4PaC - 4P&
A class of homogeneous solutions In order to compare the present higher-order shell theories with some of the previously-derived shell theories and with the elasticity theory, one could solve a specific boundary value problem and compare the predicted stresses and displacements. In the interest of comparing the theories for a whole class of practical shell problems, we take a particular class of homogeneous solutions which are used in books on shell theory to study bending effects in shells[8]. For closed cylindrical shells, solutions that are periodic in 8 are useful, since the various cylindrical shell theories have constant coefficients, as seen in (19). A useful class of solutions is of the form: ~(1)
=
2
Ah,,e” cos r&I
v(I) = 2 Bh,,eh’ sin n0
(20)
w(1) = 2 CX,jeATcos ne, where X is to be determined. Now, if we substitute the above expressions into eqn ( 15) for different cases (eighthorder up to improved twelfth-order), we will get, in each case, a set of three simultaneous equations for Ax,, , Bh,,, Cxn which will be satisfied, provided that:
~~~
:,i
~~1~~~1
(21)
= 0.
For nontrivial solutions, 1L 1must vanish, which results in an equation for A and n for given p. and u values. For the eighth-order case: P* = 12 X4(1 + u) + J.L*p[(v2 - ?lq4 - 2nh - 6(1 - V’)Xh - 6v2h4n2 + 8X”n4 + 63 A” (1 - v’) + -
4X2nT] - 30 h” (1 + V) i* = 0.
d -
122)
Method of initial functions for thick shells
8.59
Terms of higher order than p,* have been neglected in above equation, because k is small for thin shells studied with the eighth-order theory. For a given u., the corresponding equation from the Flugge[8] and Sanders[9] theories are P R,Fliigge
=
12h4(1 + u) + /.LZP[(A2 - n2)4 - 2n6 + 2vA’ - 6A4n2 - 2X2n4 (v - 4) + X4 (4 - 3~‘) + n4 - 2 X*n* (2 - v)] = 0
PWanders= 12A4(1 + v) + p*P
[(A* -
(23) n2)4 - 2n6
- -6A4n2 + -8A2n4 + -n4 - 4A2n2] = 0,
(24)
where the underlined terms in eqns (22-24) indicate terms that are in common to pairs of the above theories. In comparing the three theories, we see that the zeroth-order p” term, 12 A4 (1 + v), appears in all three theories, as well as the eighth-order terms in A and n. The lower-order terms in A and n, i.e. sixth- and fourth-order, differ for the three theories. This means that, for larger A or n, they would all asymptotically give the same roots. Computer programs” calculate elements of [L] and coefficients of 1L 1 = 0 as a function of A, n, k for a given v for eighth-order up to improved twelfth-order theories. SOME NUMERICAL
RESULTS FOR “END SOLUTIONS”
OF HIGHER ORDER
SHELL THEORIES
We are going to calculate roots of polynominals which provide solutions for different-order shell theories. These solutions will be compared with solutions from classical shell theories and three-dimensional elasticity solutions. Roots of characteristic polynominals Eighth-order theory. From eqn (21) we see that eighth- or improved eighth-order theory leads to fourth degree equations for A*. We will see that its four roots are two pairs of conjugate complex roots, and therefore will have four pairs of conjugate complex numbers for A-roots. The eight roots may be written in the following form
AI = -
XI
+
iy,,
As = x,
+ iy,
A2
XI
-
iy,,
A6 = x,
-
=
-xz+iyz,
Ax= A4
-
=
-
~2
-
iy,
(25)
A7=x2+iy2 iy2,
As = x2 - iy2,
where the &, yk are real. Each of the eight values A; yields in independent solution of (15). Tenth-order theory. In the case of tenth- or improved tenth-order theory, we must have at least one real root for A*, since we have a fifth-order polynomial in A* with real coefficients; therefore, since it turns out that the real X*-root is positive, we get As = x3; where x3 is real. ’ See [7].
Alo = - x3,
(26)
860 Table
SUSANFARAJI END RCIBER~ R. ARCHER 3. Exponential
coefficients
for eighth-order and improved and 11/r-,~= .Ol. I Eighth-order
n hiro
.Ol
.I
Table
.33 1273 .315894 13.2272 12.4340
.990640 .860523 13.8865 I I .88Y6
.33SY4 .320096 13.1977 12.3962
xI ?‘I _Y? .vz
.7538 .52lO
2.0543 .X486 6.6279 2.6185
I.1052 .73 17 5.145x 2.8910
4. Exponential
coefficients
for tenth-order and improved Ill,0 = .Ol, .I. .2
tenth-order
.Ol
.I
.2
5. Exponential
x
.I
.2
5
I .00497 .8700998 13.8355 I I .877Y 182.050 0 2.7723 I.1970 6.7928 2.559Y 18.225 0
.335898 .320054 13.1666 12.4286 1X1.684 0 I .0x40 ,716s 5.063 I 3.0170 17.556 0.
I .0049x .87 I()(34 13.8353 I I .87X0 1X1.728 0 7.8092 I.201 I 6.7X37 2.5322 17.9x3 0
I.1748 .6388 4.1164 I.8210 8.8143 0
2.352 I .5298 5.7731 I.277 8.98’336 0
I.4103 .7345 4. IO9 I .658X 8.6874 0
3.2523 I .0X0 I 5.8X72 I .3022 9.5X1 I 0
coefficients
for twelfth-order and improved and hirl, = .Ol, .I. .2 Twelfth-order
3
twelfth-order
theories
Improved 5
3 .33589X 320054 13.1668 12.4287 207.990 78.7330
Y2 -Y) Yx
I .08222 .7157 5.0697 3.0216 20. I48 7.4184
2.8026 I.1995 6.7916 2.5373 20.443 7.3465
I .0X25 .715x 5.0695 3.0215 20.026 7.459
XI VI X2 Y2 .Y? vj
1.3441 .7l I2 4. I265 I .7089 9.8563 3.5541
XI ?‘I x2
tenth-order
3
I .0049X .871004 13.8355 I I.8781 208.094 78.6830
1’1 x2 ?? x7 ‘3
(V = .3). II = 3. 5 and
Improved
.335898 .320054 13.1668 12.4287 208.061 78.6954
.K!
.Ol
2.84X9 I .23Yl 6.X65’) 2.4002
.335896 .320052 13.1668 12.4285 182.007 0 I .0727 .7l IX 5.0731 3.0285 17.980 0
n hlro
I .OOJYX .87lO50 13.8663 I I .X457
I
theories
5
3
h/r,)
eighth-order 5
.YI ?‘I .Y? Y2
5.1083 3.1586
(V = .3). II = 3. 5
3
Tenth-order
Table
theories
Improved 5
3
h
eighth-order
3.0260 I .0232 5.883156 I .39250 IO. I91 I 3.591 I
I .3666 .71X5 4.1232 I .69X2 9.7259 3.4505
(V = .3). II = 3, 5
twelfth-order 5 I .004YX .x7 IO04 13.8355 I I .87X I 208.023 7X.7204 2.8038 I.1996 6.7912 2.5371 20.341 7.345 3.12YO I .0424 5.X82.55 I .363X 10.3-W 3.2678
Method of initial functions for thick shells
861
aweigh-order theory. In the case of twelfth- or improved twelfth-order theory, it turns out that we have three pairs of conjugate complex roots A*, which give four more complex roots, eight of which are of the form given in eqn (29, along with
hs = x3 + iy3,
A,, = - x3 - iy3
h,o = - x3 + iy3,
(27)
h 12 = x3 - iy3*
Corresponding to each root, we have a homogeneous solution for the corresponding shell theory. These can be superimposed to solve end problems where shells are subjected to edge forces and moments. Computer programsf71 give $ for eighth- up to improved twelfth-order, for a given u, by inputting IE, CL. Numerical
comparisons
The following tables show roots of polynominals for n = 3, 5 and h/r0 = .Ol, .1, .2 (h/a = h/rOl(l - .5 hlro)) for present eighth- up to improved twelfth-order theories (Tables 3, 4, and 5). Results for Love’s first approximation type theory, such as the theory given by Fliiggejg], are given in Table 6. Table 6. Exponential
coefficients
(V = .3), n = 3, 5 and
h/r, = .Ol, .I
hir0
.Ol
Xn
XI Yl x2
Y2
.I
Xl 4’1 x2
Y2
Exponential Fliigge(81.
3
5
.337750 .321524 13.20597 12.5183
I .01058 .875004 13.8784 Il.9652
1.14818 .748438 5.25693 3.27008
2.97213 7.08014 7.08014 2.77926
coefficients
Table 7. Exponential
have
coefficients
been
calculated
from
(v = .3), n = 3, 5 and
h/r0 = .Ol, .I, .2
n hlro
.Ol
h
3
5
Xl Yl
.337725 .32 1503 13.2363 12.4880 316.247 0
1.01037 .874888 13.9086 I I .9350 316.272 0
1.1407 .7484 5.3454 3.1744 31.818 0
2.95078 I .25401 7.1546 2.6694 32.071 0
1.5159 .787l 4.5864 1A836 16.2083 0
3.4604 1.1423 6.52985
x2
Y2 x3
Y’1 xi Yl .I
A-2
Y2 x3
Y-c
XI YI .2
x2
Yr x3
Y3
Exponential coefficients Naghdi and Cooper[ IO].
have
1.52634 16.7152 0 been
calculated
from
SUSAN FARAJI AND ROBERTR. ARCHER
862
Table 8. Exponential coefficients for the elasticity solution (V = .3), n = 3. 5 and /I/Q, = .Ol. _I, .2 n
i?lU()
3
h
.338353 .322 I73 13.1151 12.3741 310.539 0
.Y ,
.Ol
.I
.2
.vl X3 ??? .Y? ? .q
3
-
s, ?‘I _s: ?‘3 X? \‘i
.716860 5.05255 2.99760 29.7129 0
.yI .vl x2 .k I? I3
I .3738 .7157 4.108 I .6703 14.3856 0
I .OI2656 .875Y?4 Q 13.7825 I I.Xlh8 Gz 310.564 0 0
I .08868
@ @ @ 0 0 0
2.8162 I.1959 6.7690 2.5056 29.9575 0 3. I200 I .0365 5.86889 I .35463 14.8506 0
Numbers in circles indicate number of steps in numerical method.
Table 9. Comparison of exponential coefficients between current theories and classical shell theories as well as the three-dimensional theory for v = .3. n = 3. h/r,)= .Ol Theory Eighth Fltigge’ Improved eighth Tenth Naghdi’ Improved tenth Twelfth Improved twelfth Elasticity’
X2
Au2 x IO_ 7
13.2272 13.20597 13.1977 13.1668 13.2363 13.1666 13.1668 13.1668 13.1151
-II2 -91 -82 -52 - I21 -52 52 -52 0
c/c Error in s2 .8S .69 .63 .39 .Y? .3Y .39 .3Y 0
,I!:
12.4340 l2.Sl83 12.3962 12.4285 12.4880 12.4286 12.4287 12.4287 12.3741
x IO 1
-60 - 144 _ 22 -54 - I I4 -55 -55 -55 0
% Error in ~2 .4x I.15 IX 144 .Y2 .44 .44 .44 0
’ Exponential coefficient has been calculated ’ Exponential coefftcient has been calculated ’ Calculated using a numerical solution of the in a program available from the Civil Engineering
from Flugge[R]. from Naghdi and Cooperl IO]. elasticity equation by a Runge-Kutta method as contained Department, Univ. of Mass. (TSHELLRI).
Table IO. Comparison of exponential coefficients well as the three-dimensional
between current theories and classical shell theories. as theory for v = .3. n = 5, idrg = .2
Theory Tenth Naghdi Improved tenth Twelfth Improved twelfth Elasticity
x2
5.7731 6.52985 5.8872 5.88356 5.88255 5.86889
Ax2 x
IO j
96 - 660 - I8 - I5 - I4 0
% Error in s2 1.6 II .3 .25 .23 0
?‘z 1.277 I .52634 I .3022
1.39250 I .36381 I .35463
A?‘? x lo-? 78 - 172 52 -38 -9 0
% Error in ~2 6 I3 4 2.8 .68 0
Method of initial functions for thick shells
863
Results for a shear deformation-transverse normal stress type theory, such as has been given by Naghdi and Cooper[lO], are given in Table 7, and results of the threedimensional elasticity theory” in Table 8. For purpose of comparison, in Tables 9-10 the various higher-order theories are compared in terms of the real and imaginary parts of predicted exponential roots for As we notice in Table 9 for thin shells (h/r0 = .01), tenthto improved twelfth-order theory are about the same and eighth-order theory gives good results compared to higher order theories, and as we expected, eighth-order theory gives good results for thin shells. But in Table 10 for a thick shell (h/r0 = .2) by going from tenth- to improved twelfth-order theory, we get noticeable corrections with respect to the elasticity solution. In Tables 3-5, by going from the present eighth- to improved twelfth-order theory, we generally get improvement with respect to the elasticity solution, but as we notice, the energy-type-based Naghdi and Cooper[lO] theory gives more accurate results for the highest mode compared to elasticity solution, because the example which is chosen is in favor of Naghdi and Cooper’s theory. If we would take other examples with loads in radial direction, we would expect the present theories results to improve considerably, as Iyengar and Chandrashekhara[6] showed in their study. They used the method of initial functions for the case of axisymmetric circular cylindrical shell, subjected to periodically spaced band loads. Based on their numerical comparison with the Naghdi[ 1I] and Reissner[l2] theories as well as elasticity solutions, they concluded that their higher order theories (equivalent to present eighth-, improved eighth-, and tenth-order theory for the axisymmetric case) are more accurate for problems with radial loads. So, in our case, which is more general (i.e. variation with 0) and allows for higher-order equations compared to Iyengar and Chandrashekhara’s theory, we would also expect very accurate results. Even for the end problem, we could, of course, go to higher-order (e.g. fourteenth-, sixteenth-, etc.) theories to get more accurate results. REFERENCES I. V. Z. Vlasov, The method of initial functions in problems of theory of thick plates and shells, Proc~. 9th Intl. Gong. Applied Mech. 6, 321-330. Brussels, Belgium (1957).
2. Y. C. Das and A. V. Setlur, Method of initial functions in two-dimensional elastodynamic problems. (J. Applied Mech. 37, 137-140 (1970). 3. H. M. Haydl, Bending of cylindrical shells by initial parameter method (J. Engngfor Industry), Truns. ASME Series B, 93(3), 845-850 (1971). 4. K. T. lyengar, K. Chandrashekhara, and V. K. Sebastian, On the analysis of thick rectangular plates, Ing. Arch. 43, 317-330 (1974). 5. Y. C. Das and N. S. V. Rao Kameswara. A Mixed Method in Elasticity. Indian Institute of Technology, Kanpur, India (1976). 6. K. T. lyengar and K. Chandrashekhara. On thick circular cylindrical shells under axisymmetric loading. Ac,ta Mechunicu
23 (1975).
7. S. Fardji. Method of initial functions for thick shells. dissertation, University of Massachusetts, Amherst, MA (1983). 8. W. Fltigge, Stresses in Shells, Springer-Verlag, Berlin (1960). 9. J. L. Sanders. Jr., An Improved Firsf Approximation Theory ,for Thin Shell, NASA Technical Report R-24 (1959). IO. P. M. Naghdi and R. M. Cooper, Progagation of elastic waves in cylindrical shells, including the effects of transverse shear and rotatory inertia, J. Acoustical Sot. America, 28(I), 56-63 (1955). II. P. M. Naghdi, On the theory of thin elastic shells, Quurt. Appl. Math. 14, 369-380 (1957). 12. E. Reissner, Stress strain relation in the theory of thin elastic shells, .I. Math. and Physics 31, 109-I I9 (1952).
” Calculated using a numerical solution of the elasticity equation by a Runge-Kutta method, as contained in a program available from the Civil Engineering Department, Univ. of Mass. (TSHELL8I).
METHOD
Civil engineering
0020-76831x5 Pqamon
1985
OF INITIAL FUNCTIONS SHELLS
%3.W t .# Press Ltd.
FOR THICK
SUSAN FARAJI and ROBERT R. ARCHER Department, University of Massachusetts, Amherst, MA 01003
Abstract-In
the present work for circular cylindrical shells, three-dimensional elasticity equations are solved by assuming Taylor series expansions, in the radial direction, for the stresses and displacements. Depending upon the number of terms retained in the expansion, different order shell theories are derived,, classical theories (referred to as eighth-order), the shear deformation-transverse normal stress theories (referred to as tenth-order), and higher order theories (referred to as twelfth-order). In each case. by carrying out the symbolic algebra using the digital computer, partial differential equations are derived. The procedure was carried out in detail for the case of a circular cylindrical shell with no loading on the interior surface and a given pressure distribution on the exterior surface. Then, numerical comparisons are made between the current theories and various shell theories, as well as the exact (three dimensional) theory. Thus, using this method with its associated computer programs, one can realize a spectrum of approximate shell theories ranging from the classical thin shell, through all current thick shell theories, right up to the three-dimensjonal elastic theories.
NOTATION
Cylindrical coordinates. Normalized displacement components. Middle surface radius of cylinder. Inner radius of cylinder. Outer radius of cylinder. Normalized radial and axial coordinate. Thickness of shell. Normalized thickness. Normalized shear stress components. Normalized normal stress components.
Exponential
coefficient.
INTRODUCTION
In the present work, the method of initial functions has been used to solve the elasticity equations for isotropic materials by assuming Taylor series expansions for stresses and displacements in the radial direction at interior or exterior edges, for circular cylindrical shells. Comparison is then made with classical she11 theories of different orders, as well as three-dimensional shell theory as a special case of elasticity theory. The method of initial functions, in which the elasticity equations are solved by assuming Taylor series expansions for stresses and displacements was introduced by Vlasov[ll for elasto-static problems. Das and Setlur[2] applied the method to twodimensional elasto-dynamic problems both in plane stress and plane strain in which Maclaurin series were expanded in the thickness direction. Haydll31 used Vlasov’s method of initial parameters for symmetric bending of 851
852
SUSAN FARAJIAND ROBERTR. ARCHER
cylindrical shells, and by an example he showed his method is applicable for the solution of shells with many supports or concentrated loading. Iyengar et. ~11[4]used this method for investigating thick rectangular plates, for the case of two opposite edges simply supported and the other two edges clamped. Das and Rao[.5] applied this method to thick plates subjected to antisymmetric and symmetric lateral loads. This method was extended by Iyengar and Chandrashekharalh] for the case of axisymmetric cylindrical shells. In the present work, the earlier work is extended in two important ways. First, the more general case of an unsymmetric shell is allowed; secondly. the computer algorithm to be developed will allow the derivation of thick shell theories of arbitrarily high order. The present work addresses itself to isotropic circular cylindrical shells, although it will be seen that the more general case of anisotropic materials would follow along similar lines, but with more involved algebra. For circular cylindrical shells, three-dimensional elasticity equations are solved by assuming Taylor series expansions, in the radial direction for stresses and displacements, making use of a convenient point within or at the edge of the shell. The formulation becomes exact when an infinite number of terms are used in the Taylor series expansions. In practice. only a finite number of terms can be used, and the number of terms to be retained depends on the accuracy desired for a given ratio of thickness to radius (Ma). For thick shells, it is necessary to retain more terms. Depending upon the number of terms retained, one recovers the classical theories (referred to later as eighth-order), the shear deformation-transverse normal stress theories (referred to as tenth-order), and higher order theories (referred to as twelfthorder, etc.). In each case, by carrying out the symbolic algebra using the digital computer, it is possible to derive partial differential equations for the different shell theories of arbitrarily high order (only limited by the storage capability of the computer). One can recover the associated boundary conditions for any given problem to the same order of accuracy as used in the derivation of the differential equation. Thus, using this method with its associated computer program, one can realize a spectrum of approximate shell theories ranging from the classical thin shell, through all current thick shell theories, right up to three-dimensional elastic theories. In many problems (i.e. involving stress concentrations and other problems where analytic solutions are preferred to numerical solutions found by finite element or finite difference methods), this method is of particular value for deriving suitable differential equations and boundary conditions for a shell with a given h/a. In order to demonstrate the comparative accuracy of the various theories, numerical comparisons are made between the current theories and various shell theories, as well as the exact (three-dimensional) theory.
Anulysis
Starting from three-dimensional elasticity equations, particular relations among stresses and displacements and their derivatives are found in matrix form. Then, by successive differentiations of these equations, relations between stresses and displacements and higher derivatives are found. Using a matrix notation and substituting these derivatives into Taylor series expansions at a given point along the radius (exterior or interior edges or mid-point) yields a formal series solution of the elasticity equations. The differential equations of equilibrium and strain components of an isotropic elastic body in cylindrical coordinates r, 0, z with corresponding displacement components u, ~1,w in the general case are:
853
Method of initial functions for thick shells
(1)
y,=!C+l!E. z
r at3’
az
and, according
to the generalized
aTi; E, = az
Hooke’s law for an isotropic material,?
(3)
For cylindrical layer studies, we can reduce eqns (l), (2), and (3) from a system of 15 equations to a system of six first-order equations with respect to the r derivatives regarding ,!A%, d/az as formal symbolic operators. In matrix form, the six equations become
[ul’ = [Sl [Yl?
(4)
where
-
( )
--(Y
- CivP
-- 1 r
0
-vP- P r
0
-- 2 r
0
0
-- VP r
0
1
1
0
VP
1
r
( --v 21
) P
2uPcX
2pp -131r
T _ z$z
-,PP
(l)
-
(,,,,‘+
-2fy
-(u2
+ y.q
-
@p
$1
(!+
-- VPP r
_ vPa
-- P
-1
0
-a
0
0
r
t This method can also be used for anisotropic
r
materials with the same algebraic procedure.
854
SUSAN FARAJIAND ROBERTR. ARCHER
and I
ur I
Trz
[yl’ =
r:t3 u’ 71’ W'
with ( )’ = a 5 ( ) remaining stresses can be obtained from 2vP 2vP u + pv + 2PakV + v P CT, 52 = Y
r
2P oe =-u
2PP + -71+
r
T,e
=
a71 +
2Paw
Y
+ VPU,
P M’.
-
r
Higher derivatives
of [u]’ can be obtained from eqn (4) by successive (1) tions. (Note: It is convenient to write Si,i = Si,i) then
(1)’ _YY=
(5)
C
(Sij
differentia-
(1) Yj
+
Sij
(1)’ (l)(l) YY= 2 (Si/, + SijSjk)
yj’)
(6)
yk,
or (2) Y; =
C
Sil,
Yk
(7)
where (2) (1)’ (l)(l) Six = Si/, + Sij S,jk, etc. Now for the nth derivative
n (n) [Yl = [Sl [Yl
(8)
In eqn (8) [S ]is a matrix of differential operators 01, l3 for any order of derivation. For any given Poisson’s ratio v and any radius r in radial direction ranging from (n) 1 to 1 + p, computer program& give elements of [S ] up to n = 7 as function of (Y, P* $ Copies of these programs are given in 171.
855
Method of initial functions for thick shells HIGHER ORDER THEORIES
FOR THICK CYLINDRICAL
SHELLS
Now, by making use of shell boundary conditions in the radial direction, we derive partial differential equations for cylindrical shells ranging from classical through current thick elastic shell theories, as well as new higher-order shell theories. In order to compare the various theories with the exact theory, we are going to study, as a special case, a circular cylinder with no loading on the interior surface and a given pressure on the exterior surfaceD, i.e. at r=
1;
r=l+p:
(Jr =
7,
=
Tr%
or = - q,7,
=
0
= T,t) = 0.
It may be noted that other classes of practically important problems can be studied along very similar lines, e.g. a cylindrical elastic layer with a stress-free outer surface and bonded to a rigid cylindrical shaft at Y = 1 would have boundary conditions: at u=~=w=O
1;
r=
#+=1+&L;
or = T& = I,
= 0
In order to derive partial differential equations governing the various theories, we use Taylor series expansions at the interior (r = 1) or exterior edge (Y = 1 + IL), by using the notation previously developed. Expanding about the interior edge:
(1) LYWI = [Y(l)1 + (r - 1) IS(l)1 [Y(l)1 + -@ ,, L!
l)*$I)][Y(l)]
+..*.
or
{
Expanding
Mfr-m! l)” ;?(l,,[Y(l)].
=
)
I
ml=
(9)
about the exterior edge:
Derivation of equations for shells with un inner surface freeof loading, and an outer surface under pressure For a cyfinder with no loading on the inner surface, and pressure acting on the outer surface, we have: r=
I;
r=l+p; Now, by applying the above boundary
(Jr = 7, = Tr%= 0
conditions
[YtbWl = [Y(l)1 +
to eqn (9), we find:
M pm(m) 2 -g [S (111 IYU)l m=l *
8 The more general case of three stress components line.
(11)
0, = - q, 7, = Tr@= 0.
(12)
presented on both edges would follow along similar
856
FARAJIAND ROBERT R. ARCHER
SUSAN
or b
0
ur -
b
0 0
Ti-9
u
-
a
b
-
a
b
0
TII
T12
T,3 tTI4 L
T,5
T,h;
0
0
(ml
(ml
(m);(m)
Cm) (m)]
T2,
T22
T23 ; T24
7-25 7-26;
0
0
(ml
(ml
(m):(m)
Cm) kd;
T~I
TX
TV 1%--.-_-;-_J T,s
7’36’
(un> (ml
(ml
(ml
(ml
(02)
T41
T42
T43
T44
T45
7-46
(m>
(m)
(ml
Cm) (ml
(ml
T51
Ts2
7;3
T1(4 Ts,
7-5~
(ml
(ml
(ml
(ml
(ml
(ml
T~I
Th2 Tm
TM
TM
T,,
=
dl) b
2)
w
‘(mlCm)(m,yi, c;r;,cmJ
0
-4
u
0
+
rt(l)
-
a
-
Ml)
a
0 0
0
411
(13)
41) 41
where ftn)
1,1
.
Ttj = C
,,)
5
Cm)
S,i (I)
or, because the present problem involves stress boundary conditions at b/a, we write: 0
1: 14(l)
Cm)Cm)(ml
which can be rewritten
T24
T25
T2h
(ml
(ml
(m)
T14
T,5
TM
>
(14)
-q
M’( 1)
in the form:
0
=
z,(l)
Let , Cm)
Cm)
ai,j =
,1
SXI (1)
where k = 4 - i;
I-j+3,
therefore ,,1
A;.;
= C k”’
(m) ai,i ,
(16)
or with [U(1)17’ = [u(l), o(l), w(l)] and [Q]’ = [O, 0, -q]
[Al [U(l)1 = LQI.
(17)
Method of initial functions for thick shells
857
We note that u, 11,w are functions of r, 8, z, while u(l), v(l), w(l) are function of 8 and z only, and [A] is a matrix of differential operators. Equation (17) represents a set of three partial differential equations. Nontrivial homogeneous solutions of (17) involve an equation of the form:
(18)
lAlF=O,
where F is a convenient auxiliary function. In practice, the number of terms to be retained in eqns (9) or (10) in the Taylor series expansions depends on the ratio of thickness to radius (h/a) and the accuracy desired. In the expansions, if one retains terms up to third derivative, then the result 1A 1F = 0 will be called the eighth-order shell theory; then, by adding fourth derivative terms, the result will be called the improved eighth-order shell theory; and then by adding fifth up to eighth derivative terms, the results will be called tenth-, improved tenth-, twelfth-, and improved twelfth-order shell theories. By comparing the results of 1A 1F = 0 for each case, one notices that by adding the fourth derivative terms, the coefficients of what we call the eighth-order shell theory have been stabilized, which means adding the fifth, sixth, seventh, and eighth higher derivatives does not change the coefficients of terms in the eighth-order theory. Also by adding the sixth derivative terms, the coefficients of the tenth-order theory have been stabilized, and by adding the eighth derivatives, the coefficients of twelfthorder theory have been stabilized. The following table shows the different order shell theories corresponding to the number of derivative terms retained in the Taylor series expansion. Computer programs which produce, respectively, elements of [A] and the coefficients of 1A 1as functions of (Y, B, and k for expansion at interior edge (r = 1) for a given v for different orders eighth- up to improved twelfth-order, are given in [7]. As a check on the computer algorithms up to the fourth derivative, elements of [A] have been derived by hand and are given in Table 2. They are given as a function of v, whereas the computer programs are written to accept a given numerical value for v. For the eighth- or improved eighth-order, expanding the determinant after dividing by ps P2/3 gives: 1A 1 = V8 + 6(1 - v*) o? + 2B6 + 8a2B4 + 6v2a4B2 + 63
1
+ B” + 4aZB2 + G, = 0.
Where G, represents terms in (19) of k and higher order. As mentioned before, for a given v, computer programs give coefficients terms in I A ( which include G,.
Table I. Relation of the number of derivative terms to the order of the shell theory No. of Derivative Terms = IX 3 4 5 6 7 8
Order of Shell Theory = IO Eighth-order Improved eighth-order Tenth-order Improved tenth-order Twelfth-order Improved twelfth-order
(19)
of all
SUSAN FARAJIANDRWERT R. ARCHER
8.58
Table 2. Elements of [A] A,,
up to fourth
derivatives as functions of a, p, p, v
= /L{ - 2Pp} + k2/2{2Pp3 + 2Pa73 + BPP} + w3/6{ -4P&p( I + )*?24{ -4Pp-’ - 4Pd’P - 8Pa’p’ + 3oJq33 -t 2a$3P(l2v
+ VI .- 8Pp’ - 38Pp} t II) + ZIJPp}
AI2 = w{ -_(y’ - 2P(3’} + ~‘/2{& + 6Pp”] f ~‘161a’ + 4Pp4 + a’@‘P(.s +. p4/24{ - 2i4 - 40Pp4 + 6a’fl”P(5 - VI + 30~1’ t 14OPp’}
VI .- ha’ - 26Pp’j
A17 = p{ -c+P(l
+ Y,} + ~%japP(Sv + 3) + p%{a’pP(v + 3) + c@“P(iJ + 3) - 2af3P(6 + 13v)} i p4/24{ -2aj3’P( I5 + 7~) - Za’PP(9 + 5~) + 2afiVY30 + 77vl}
&I = p{ - 2vPctf + (1’!2{2Pu’ + Zf’cwj3”-I- ?avP} + p316{ - 2a’ - ho@’ - hvPa} + @‘/24{ - 4Pu5 - 4PcqJ
- 8Pczf3
+ Za~P(5 - 2v) + 2+3~P(l3
-
l&f
+ 24vPcYj
+ v,} + p2/2{af3/2} + p’/6{&P(v + 3) + ap’P(v A22 = +{-c@P(I + p4124{2Pa’f3(v - 3) - 2$“P(9 -t v) + 2Pc$(6 - v,} A 23 - /.L{- 2Pct? - p’} + p’/2{3@’ t p4/24{ - 8Pa4 - lop4 t
t 3) + @P(u
+ 2f’a’} + y3/6{4Pa4 + f3” t a’p’P(S 6cx”p2P(v- 5) + 24Pa’ + SOP’}
- 3)}
- v) - 6Pa’
-
I I p’)
A ‘I, = p,{ZP} + )1’/2{ -6P} + ~*‘/6{ - 2Pd - 2Pp4 - 4Pu’p’ + 4vPa’ - XPP’ + 24P) + p4/24{4Pa4 + 20Pp4 + 24Poi’p” - 24vPa’ t SOPP’ - l2OP} A32 = J,L{ZP@} + p2i2{2Pf&
+ 2Ppz - 4P@} + p5’6{- 8P&” - 4Pa4@ + I ICIPp’
+ p4/24{-4Pf3’ As
16Pp’ + 4a’PP(v - 2) + l4P@) - 12~) - 66Pp)
t 2&P(23
= p{2vPa}
+ pL’!2{2Paz + 2Pa@ .- BvPa} + ~?‘6{-2Pab + ‘1 - 6a@P~(v t 2) + 24vPaj - XPa’f3’ + hct’P(3 + 3~) + 2Pa@(37 + 30~) - l?Ov/‘n} + *4124{ -4PaC - 4P&
A class of homogeneous solutions In order to compare the present higher-order shell theories with some of the previously-derived shell theories and with the elasticity theory, one could solve a specific boundary value problem and compare the predicted stresses and displacements. In the interest of comparing the theories for a whole class of practical shell problems, we take a particular class of homogeneous solutions which are used in books on shell theory to study bending effects in shells[8]. For closed cylindrical shells, solutions that are periodic in 8 are useful, since the various cylindrical shell theories have constant coefficients, as seen in (19). A useful class of solutions is of the form: ~(1)
=
2
Ah,,e” cos r&I
v(I) = 2 Bh,,eh’ sin n0
(20)
w(1) = 2 CX,jeATcos ne, where X is to be determined. Now, if we substitute the above expressions into eqn ( 15) for different cases (eighthorder up to improved twelfth-order), we will get, in each case, a set of three simultaneous equations for Ax,, , Bh,,, Cxn which will be satisfied, provided that:
~~~
:,i
~~1~~~1
(21)
= 0.
For nontrivial solutions, 1L 1must vanish, which results in an equation for A and n for given p. and u values. For the eighth-order case: P* = 12 X4(1 + u) + J.L*p[(v2 - ?lq4 - 2nh - 6(1 - V’)Xh - 6v2h4n2 + 8X”n4 + 63 A” (1 - v’) + -
4X2nT] - 30 h” (1 + V) i* = 0.
d -
122)
Method of initial functions for thick shells
8.59
Terms of higher order than p,* have been neglected in above equation, because k is small for thin shells studied with the eighth-order theory. For a given u., the corresponding equation from the Flugge[8] and Sanders[9] theories are P R,Fliigge
=
12h4(1 + u) + /.LZP[(A2 - n2)4 - 2n6 + 2vA’ - 6A4n2 - 2X2n4 (v - 4) + X4 (4 - 3~‘) + n4 - 2 X*n* (2 - v)] = 0
PWanders= 12A4(1 + v) + p*P
[(A* -
(23) n2)4 - 2n6
- -6A4n2 + -8A2n4 + -n4 - 4A2n2] = 0,
(24)
where the underlined terms in eqns (22-24) indicate terms that are in common to pairs of the above theories. In comparing the three theories, we see that the zeroth-order p” term, 12 A4 (1 + v), appears in all three theories, as well as the eighth-order terms in A and n. The lower-order terms in A and n, i.e. sixth- and fourth-order, differ for the three theories. This means that, for larger A or n, they would all asymptotically give the same roots. Computer programs” calculate elements of [L] and coefficients of 1L 1 = 0 as a function of A, n, k for a given v for eighth-order up to improved twelfth-order theories. SOME NUMERICAL
RESULTS FOR “END SOLUTIONS”
OF HIGHER ORDER
SHELL THEORIES
We are going to calculate roots of polynominals which provide solutions for different-order shell theories. These solutions will be compared with solutions from classical shell theories and three-dimensional elasticity solutions. Roots of characteristic polynominals Eighth-order theory. From eqn (21) we see that eighth- or improved eighth-order theory leads to fourth degree equations for A*. We will see that its four roots are two pairs of conjugate complex roots, and therefore will have four pairs of conjugate complex numbers for A-roots. The eight roots may be written in the following form
AI = -
XI
+
iy,,
As = x,
+ iy,
A2
XI
-
iy,,
A6 = x,
-
=
-xz+iyz,
Ax= A4
-
=
-
~2
-
iy,
(25)
A7=x2+iy2 iy2,
As = x2 - iy2,
where the &, yk are real. Each of the eight values A; yields in independent solution of (15). Tenth-order theory. In the case of tenth- or improved tenth-order theory, we must have at least one real root for A*, since we have a fifth-order polynomial in A* with real coefficients; therefore, since it turns out that the real X*-root is positive, we get As = x3; where x3 is real. ’ See [7].
Alo = - x3,
(26)
860 Table
SUSANFARAJI END RCIBER~ R. ARCHER 3. Exponential
coefficients
for eighth-order and improved and 11/r-,~= .Ol. I Eighth-order
n hiro
.Ol
.I
Table
.33 1273 .315894 13.2272 12.4340
.990640 .860523 13.8865 I I .88Y6
.33SY4 .320096 13.1977 12.3962
xI ?‘I _Y? .vz
.7538 .52lO
2.0543 .X486 6.6279 2.6185
I.1052 .73 17 5.145x 2.8910
4. Exponential
coefficients
for tenth-order and improved Ill,0 = .Ol, .I. .2
tenth-order
.Ol
.I
.2
5. Exponential
x
.I
.2
5
I .00497 .8700998 13.8355 I I .877Y 182.050 0 2.7723 I.1970 6.7928 2.559Y 18.225 0
.335898 .320054 13.1666 12.4286 1X1.684 0 I .0x40 ,716s 5.063 I 3.0170 17.556 0.
I .0049x .87 I()(34 13.8353 I I .87X0 1X1.728 0 7.8092 I.201 I 6.7X37 2.5322 17.9x3 0
I.1748 .6388 4.1164 I.8210 8.8143 0
2.352 I .5298 5.7731 I.277 8.98’336 0
I.4103 .7345 4. IO9 I .658X 8.6874 0
3.2523 I .0X0 I 5.8X72 I .3022 9.5X1 I 0
coefficients
for twelfth-order and improved and hirl, = .Ol, .I. .2 Twelfth-order
3
twelfth-order
theories
Improved 5
3 .33589X 320054 13.1668 12.4287 207.990 78.7330
Y2 -Y) Yx
I .08222 .7157 5.0697 3.0216 20. I48 7.4184
2.8026 I.1995 6.7916 2.5373 20.443 7.3465
I .0X25 .715x 5.0695 3.0215 20.026 7.459
XI VI X2 Y2 .Y? vj
1.3441 .7l I2 4. I265 I .7089 9.8563 3.5541
XI ?‘I x2
tenth-order
3
I .0049X .871004 13.8355 I I.8781 208.094 78.6830
1’1 x2 ?? x7 ‘3
(V = .3). II = 3. 5 and
Improved
.335898 .320054 13.1668 12.4287 208.061 78.6954
.K!
.Ol
2.84X9 I .23Yl 6.X65’) 2.4002
.335896 .320052 13.1668 12.4285 182.007 0 I .0727 .7l IX 5.0731 3.0285 17.980 0
n hlro
I .OOJYX .87lO50 13.8663 I I .X457
I
theories
5
3
h/r,)
eighth-order 5
.YI ?‘I .Y? Y2
5.1083 3.1586
(V = .3). II = 3. 5
3
Tenth-order
Table
theories
Improved 5
3
h
eighth-order
3.0260 I .0232 5.883156 I .39250 IO. I91 I 3.591 I
I .3666 .71X5 4.1232 I .69X2 9.7259 3.4505
(V = .3). II = 3, 5
twelfth-order 5 I .004YX .x7 IO04 13.8355 I I .87X I 208.023 7X.7204 2.8038 I.1996 6.7912 2.5371 20.341 7.345 3.12YO I .0424 5.X82.55 I .363X 10.3-W 3.2678
Method of initial functions for thick shells
861
aweigh-order theory. In the case of twelfth- or improved twelfth-order theory, it turns out that we have three pairs of conjugate complex roots A*, which give four more complex roots, eight of which are of the form given in eqn (29, along with
hs = x3 + iy3,
A,, = - x3 - iy3
h,o = - x3 + iy3,
(27)
h 12 = x3 - iy3*
Corresponding to each root, we have a homogeneous solution for the corresponding shell theory. These can be superimposed to solve end problems where shells are subjected to edge forces and moments. Computer programsf71 give $ for eighth- up to improved twelfth-order, for a given u, by inputting IE, CL. Numerical
comparisons
The following tables show roots of polynominals for n = 3, 5 and h/r0 = .Ol, .1, .2 (h/a = h/rOl(l - .5 hlro)) for present eighth- up to improved twelfth-order theories (Tables 3, 4, and 5). Results for Love’s first approximation type theory, such as the theory given by Fliiggejg], are given in Table 6. Table 6. Exponential
coefficients
(V = .3), n = 3, 5 and
h/r, = .Ol, .I
hir0
.Ol
Xn
XI Yl x2
Y2
.I
Xl 4’1 x2
Y2
Exponential Fliigge(81.
3
5
.337750 .321524 13.20597 12.5183
I .01058 .875004 13.8784 Il.9652
1.14818 .748438 5.25693 3.27008
2.97213 7.08014 7.08014 2.77926
coefficients
Table 7. Exponential
have
coefficients
been
calculated
from
(v = .3), n = 3, 5 and
h/r0 = .Ol, .I, .2
n hlro
.Ol
h
3
5
Xl Yl
.337725 .32 1503 13.2363 12.4880 316.247 0
1.01037 .874888 13.9086 I I .9350 316.272 0
1.1407 .7484 5.3454 3.1744 31.818 0
2.95078 I .25401 7.1546 2.6694 32.071 0
1.5159 .787l 4.5864 1A836 16.2083 0
3.4604 1.1423 6.52985
x2
Y2 x3
Y’1 xi Yl .I
A-2
Y2 x3
Y-c
XI YI .2
x2
Yr x3
Y3
Exponential coefficients Naghdi and Cooper[ IO].
have
1.52634 16.7152 0 been
calculated
from
SUSAN FARAJI AND ROBERTR. ARCHER
862
Table 8. Exponential coefficients for the elasticity solution (V = .3), n = 3. 5 and /I/Q, = .Ol. _I, .2 n
i?lU()
3
h
.338353 .322 I73 13.1151 12.3741 310.539 0
.Y ,
.Ol
.I
.2
.vl X3 ??? .Y? ? .q
3
-
s, ?‘I _s: ?‘3 X? \‘i
.716860 5.05255 2.99760 29.7129 0
.yI .vl x2 .k I? I3
I .3738 .7157 4.108 I .6703 14.3856 0
I .OI2656 .875Y?4 Q 13.7825 I I.Xlh8 Gz 310.564 0 0
I .08868
@ @ @ 0 0 0
2.8162 I.1959 6.7690 2.5056 29.9575 0 3. I200 I .0365 5.86889 I .35463 14.8506 0
Numbers in circles indicate number of steps in numerical method.
Table 9. Comparison of exponential coefficients between current theories and classical shell theories as well as the three-dimensional theory for v = .3. n = 3. h/r,)= .Ol Theory Eighth Fltigge’ Improved eighth Tenth Naghdi’ Improved tenth Twelfth Improved twelfth Elasticity’
X2
Au2 x IO_ 7
13.2272 13.20597 13.1977 13.1668 13.2363 13.1666 13.1668 13.1668 13.1151
-II2 -91 -82 -52 - I21 -52 52 -52 0
c/c Error in s2 .8S .69 .63 .39 .Y? .3Y .39 .3Y 0
,I!:
12.4340 l2.Sl83 12.3962 12.4285 12.4880 12.4286 12.4287 12.4287 12.3741
x IO 1
-60 - 144 _ 22 -54 - I I4 -55 -55 -55 0
% Error in ~2 .4x I.15 IX 144 .Y2 .44 .44 .44 0
’ Exponential coefficient has been calculated ’ Exponential coefftcient has been calculated ’ Calculated using a numerical solution of the in a program available from the Civil Engineering
from Flugge[R]. from Naghdi and Cooperl IO]. elasticity equation by a Runge-Kutta method as contained Department, Univ. of Mass. (TSHELLRI).
Table IO. Comparison of exponential coefficients well as the three-dimensional
between current theories and classical shell theories. as theory for v = .3. n = 5, idrg = .2
Theory Tenth Naghdi Improved tenth Twelfth Improved twelfth Elasticity
x2
5.7731 6.52985 5.8872 5.88356 5.88255 5.86889
Ax2 x
IO j
96 - 660 - I8 - I5 - I4 0
% Error in s2 1.6 II .3 .25 .23 0
?‘z 1.277 I .52634 I .3022
1.39250 I .36381 I .35463
A?‘? x lo-? 78 - 172 52 -38 -9 0
% Error in ~2 6 I3 4 2.8 .68 0
Method of initial functions for thick shells
863
Results for a shear deformation-transverse normal stress type theory, such as has been given by Naghdi and Cooper[lO], are given in Table 7, and results of the threedimensional elasticity theory” in Table 8. For purpose of comparison, in Tables 9-10 the various higher-order theories are compared in terms of the real and imaginary parts of predicted exponential roots for As we notice in Table 9 for thin shells (h/r0 = .01), tenthto improved twelfth-order theory are about the same and eighth-order theory gives good results compared to higher order theories, and as we expected, eighth-order theory gives good results for thin shells. But in Table 10 for a thick shell (h/r0 = .2) by going from tenth- to improved twelfth-order theory, we get noticeable corrections with respect to the elasticity solution. In Tables 3-5, by going from the present eighth- to improved twelfth-order theory, we generally get improvement with respect to the elasticity solution, but as we notice, the energy-type-based Naghdi and Cooper[lO] theory gives more accurate results for the highest mode compared to elasticity solution, because the example which is chosen is in favor of Naghdi and Cooper’s theory. If we would take other examples with loads in radial direction, we would expect the present theories results to improve considerably, as Iyengar and Chandrashekhara[6] showed in their study. They used the method of initial functions for the case of axisymmetric circular cylindrical shell, subjected to periodically spaced band loads. Based on their numerical comparison with the Naghdi[ 1I] and Reissner[l2] theories as well as elasticity solutions, they concluded that their higher order theories (equivalent to present eighth-, improved eighth-, and tenth-order theory for the axisymmetric case) are more accurate for problems with radial loads. So, in our case, which is more general (i.e. variation with 0) and allows for higher-order equations compared to Iyengar and Chandrashekhara’s theory, we would also expect very accurate results. Even for the end problem, we could, of course, go to higher-order (e.g. fourteenth-, sixteenth-, etc.) theories to get more accurate results. REFERENCES I. V. Z. Vlasov, The method of initial functions in problems of theory of thick plates and shells, Proc~. 9th Intl. Gong. Applied Mech. 6, 321-330. Brussels, Belgium (1957).
2. Y. C. Das and A. V. Setlur, Method of initial functions in two-dimensional elastodynamic problems. (J. Applied Mech. 37, 137-140 (1970). 3. H. M. Haydl, Bending of cylindrical shells by initial parameter method (J. Engngfor Industry), Truns. ASME Series B, 93(3), 845-850 (1971). 4. K. T. lyengar, K. Chandrashekhara, and V. K. Sebastian, On the analysis of thick rectangular plates, Ing. Arch. 43, 317-330 (1974). 5. Y. C. Das and N. S. V. Rao Kameswara. A Mixed Method in Elasticity. Indian Institute of Technology, Kanpur, India (1976). 6. K. T. lyengar and K. Chandrashekhara. On thick circular cylindrical shells under axisymmetric loading. Ac,ta Mechunicu
23 (1975).
7. S. Fardji. Method of initial functions for thick shells. dissertation, University of Massachusetts, Amherst, MA (1983). 8. W. Fltigge, Stresses in Shells, Springer-Verlag, Berlin (1960). 9. J. L. Sanders. Jr., An Improved Firsf Approximation Theory ,for Thin Shell, NASA Technical Report R-24 (1959). IO. P. M. Naghdi and R. M. Cooper, Progagation of elastic waves in cylindrical shells, including the effects of transverse shear and rotatory inertia, J. Acoustical Sot. America, 28(I), 56-63 (1955). II. P. M. Naghdi, On the theory of thin elastic shells, Quurt. Appl. Math. 14, 369-380 (1957). 12. E. Reissner, Stress strain relation in the theory of thin elastic shells, .I. Math. and Physics 31, 109-I I9 (1952).
” Calculated using a numerical solution of the elasticity equation by a Runge-Kutta method, as contained in a program available from the Civil Engineering Department, Univ. of Mass. (TSHELL8I).