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Modified three-dimensional finite element for general and composite shells
Compufers & Smntures Vol. 51. No. 3. pp. 289-298. 1994 Copyright 0 1994 Elwier Science Ltd Printed in Great Brimin. All rightsnswved
Pergamon
MODIFIED
@34s-7949/94 57.00 + 0.00
THREE-DIMENSIONAL FINITE ELEMENT GENERAL AND COMPOSITE SHELLS D. N.
BURAC~HAIN
and P. K.
FOR
RAVICHANDRAN
Department of Civil Engineering, Indian Institute of Technology, Bombay, India (Received 8 September
1992)
Abstract-A new shear deformable finite element formulation has been developed for the linear static analysis of general, homogeneous and laminated composite shells. It is based on the reduction of a three-dimensional element to a two-dimensional element by the use of substitute degrees of freedom. The proposed formulation incorporates a realistic cubic variation of the warping of the cross-section and also accounts for the transverse normal strain energy in the derivation of the total potential energy of the structure. Analytical integration along the shell thickness direction is carried out by taking quadratic variation of the determinant of the Jacobian matrix. The elements considered are a seven-node triangle and eight- and nine-node quadrilaterals. Reduced Gauss integration is used to evaluate the various element property matrices. A number of problems from thin isotropic shells to thick laminated composite shells have been solved to check the accuracy of the proposed formulation and to study the effect of transverse shear deformation on the response of thick sandwich and composite shells.
1. INTRODUCTION
Structural units such as plates and shells made of composite materials are finding increasing applications in advanced technical fields such as the aerospace and automotive industries where the main requirements are a high strength and/or stiffness coupled with low weight and other optimum properties. The analysis and design of such composite structural systems are, however, complicated because of their highly anisotropic behaviour. Also, owing to rile low ratio of the transverse shear modulus to the in-plane modulus in most of these advanced composite materials, shear deformation theories are necessary for the reliable analysis of laminated composite plate/shell structures. The first-order shear deformation theory in which constant transverse shear deformation is incorporated along the total thickness of the shell was used earlier by many investigators [l-3]. Since such analytical methods are limited to simple geometry and boundaries, many numerical methods, and in particular the finite element method, have been widely employed [4-91. The theories used in these references are either based on an assumed stress field or an assumed displacement field or on a mixed formulation. For thicker composite shells, or when the properties of the layers vary from layer to layer, an improved theory known as the layerwise constant shear angle theory (LCST) was adopted in [l&13]. However, this theory introduced two additional degrees of freedom with each additional layer and therefore was cumbersome. Both these theories also necessitated the use of arbitrary shear correction coefficients in their formulation. CAS Sl,3--E
Many higher order theories have, therefore, been proposed as an improvement over the first-order theory models by expanding the displacements in powers of the thickness coordinate. They have been successfully applied for the analysis of composite plates, for example [ 14, 151, but such theories for composite shells have been limited [ 16-201. Buragohain [19] presented a refined higher order theory for laminated general shells and proposed a modified three-dimensional (3-D) formulation for shear deformable shell analysis, both using economical explicit thickness integration schemes. The theory of [20] is based on hierarchical polynomial approximation of the displacements along the thickness direction and uses a numerical Gauss integration scheme to evaluate the element properties. Such a scheme becomes uneconomical as the order p of the polynomial is increased since p + 1 integration points are necessary along the thickness in each layer. Three-dimensional analysis of composite plates and shells has also been scarce [21,22] due to the inherent numerical complexity. In the present development, a simplified 3-D model based on the modified 3-D element concept of [19] is described in detail for the linear static analysis of layered composite shells in which many of the above-mentioned difficulties have been overcome. 2. THE MODIFIED 3-D (MOD 3-D) ELEMENT
2.1. Parent solid element A typical quadrilateral brick element with four nodes along the thickness at each of its corners and mid-sides is shown in Fig. I (a) (a total of 32 nodes for
289
D. N. BURAGOHAIN and P. K. ~AV~CHANDRA~
Nir(i,
Q, C)=
Lr(iWz(5aQ)
(W
N,,(L Q, i) = L&)N,(5, Q)
(24
in which N,(,(r,4) are the standard shape functions for Co continuity for the eight-noded quadrilateral shell element and L,(i)=&1 of
ORIGINAL
NODAL
POSIlIONS
+5 +912-95’)
(3a)
L,(~)=~(l
-31
L,(i) = $1
+ 35 - iz - 31’)
L,(i)=h(-1
-i?+31’)
-i
(3b)
(3c)
+9<*+91g.
(34
The global translations can now be written as (e.g. U) u = y N~(S, rl) t&p utp + Liq
uiq
,=I
b)
MODIFIEO DEGREES
ELEMENT WITH OF FREEDOM
SUBSTITUTE +
Fig. 1. Modified three-dimensional element. (a) Original nodal. positions (b) modified element with substitute degrees of freedom.
the solid element). Using natural coordinates (5, q, <), with (4, q) along the mid-surface and 5 along the thickness direction, the translations U, V and W along global X, Y and Z directions at any point in the solid element may be obtained using a 3-D isoparametric element formulation as (for example, displacement U)
u=zNf(5,~,W,,
j=1,2
,..., 32
(1)
in which NT denotes the shape functions of the solid element at node j. Although it is possible to evaluate the structure properties such as the stiffness matrix with three translational degrees of freedom at each node using standard procedures [23,24], various numerical difficulties are often encountered for the solution of the assembled structure equilibrium equations: ill-conditioned equations, large storage and computational efforts [23]. Hence, the following procedure is adopted to obtain the equivalent modified 3-D (MOD-3D) element. 2.2, D~sp~u&emen~fieid for the MOD-31)
element
Denoting four points, p, y, Y and s, at equal intervals along the thickness at any node point i (passing through the solid element nodes), the shape functions for global translations Ii, V and W at these levels may be written as
N&
Q,
i) = Lq(iWi(t, Q)
(2b)
L
uir
f
hs
ursl. t4)
Substituting eqns (3) into (4) and re-arranging terms according to powers of < leads to U = c N,(L n)]h(-
the
U,,,+9U,, + 9U,, - U,,)
+ ([/2)$(U, - 27U, + 27U, - U,) + (~~2)2~(uj~-
uiq
-
iir
-
ut,~)
+(i/2)'~(-Vi~+3U,,-3Ui,+Uj,)]
(5)
with similar expressions for V and W. The 12 transiationa~ degrees of freedom at the four points p, 4, r and s along the thickness are replaced by 12 substitute degrees of freedom at any nodal position i of the reduced model (Fig. lb) as U =~N,(5,~)iUr+(r/2)t,Ui + (;/2):r; U:’ -t (l/2)%; U:“] (6) in which iJ:+u,,+9u,+9(/,-
Cl>,)
(7a)
U: = $ (U,P- 27u, + 270;, - U,,)
(7b)
U~=~(u,~-u,,,-ui,+Li,)
(7c)
U~=~(-U,~+3u‘,-3u,+Ui,~
(7d)
and t, is the total thickness of the shell at node i. Similar expressions are obtained for V and W. The
291
Modi~ed thr~-djmensional finite elen tent for general and composite shells displacement field for the MOD-3D model can thus be written as ii=
=~N,(S,v)~~~+(~,Z)r,~ + (i/2)%5@ +
(~/2)3t$iy]
in which 8 = Wy, with 8 as the transformation matrix at the point. The global coordinates (X, Y, Z) at any point (t, r,i,[) on the shell element is obtained as
(8)
in which
p=
1~:”
v:’
WY}‘.
(9)
In a similar manner, seven-node triangular and nine-node rectangular elements with nodes at the comers, mid-sides and centre are also developed. 2.3. Strain-displacement relation
At each node i, a local Cartesian coordinate system (x, y, z) is set up uniquely with (x, y) in the tangent plane and z along the normal to the shell midsurface. The displacements (u, u, w) along (x, y, z) axes can be obtained by suitable transformation of the global displacements. The displacement components along global directions (A’, Y, 2) in eqn (9) may be transformed to similar displacement components along (x, y, z) axes by using the nodal transformation matrix 0, leading to
In order to show the necessary modi~~tions to be made on the strain-displacement matrix of the solid element formulation from which the present method is derived, the approach described here is followed. For the solid element, the strains along local Cartesian coordinates (x, y, z) at any point on the midsurface of the shell element may be obtained using standard procedure as -a/ax
0
0 -
;i = y N,(& q)@;[d; + ([/2)&d; i=t
+ ([/2)%fdj f ({/2)3ri’dy]
(IOa)
in which OOb)
is the nodal transformation matrix, the three rows of which are given by the unit vectors along the local axes (x, y, z); dp, dj, d; and d:,, are linear combinations of displacement components as in eqn (7) and are given by
(14)
or c =
L&i.
(15)
Using eqn (lOa) in eqn (14) t = (L@ii@d,= Lf? F &di
(16a)
1-I
=5&d = “c”Bidi i= I
The displacement components d = (u, u, w) along local axes (x, y, z) at any point on the mid-surface of the shell element is obtained again by transformation, d=t?;l as d = F N,@d; f (
(1W
in which mi is the coefficient matrix of the nodal degrees of freedom vector obtained from eqn (lOa). Thus, the three translational degrees of freedom at each of the four nodes along the thickness of the solid element are modified to equivalent substitute degrees of freedom at the nodes of the simplified 2-D model. The Cartesian derivatives in L of eqn (14) are obtained in terms of the derivatives in the natural coordinates using Jacobian matrix .J as {a/at
a/all
aiayy=qaiax
alay
a/aq
(17)
292
D. N. BURAGOHAINand P. K. RAVICHANDRAN
in which
be expressed in terms of the strains L in (x, y, z) axes using the strain transformation matrix T, as [25] c’=T,t.
In eqn (I@, V:, V, and V; are the tangent vectors along r, q and < directions and are obtained as
(22)
The stresses a in the (x, y, r) system are related to the strains c as u = Dc
(23)
in which D = TJCT,
(24a)
and d = {a, 2.5.
The strain-displacement now arranged as
matrix
Bi of eqn (l6b)
Bi = BO,+ [B,, + i*B,, + i’B3,. 2.4.
Stress-strain
are
T,.,
7,;
7wi
IT
Wb)
Element stiffness matrix
K,=
relation
in which C is the elasticity matrix of the layer material. The following approach is used to locate the orientation of the material axes in the case of an arbitrarily laminated general shell. At any point on the shell mid-surface, the z axis is fixed along the normal direction while the (x,v) axes are conveniently chosen in the tangent plane. Now, at one of the element modes, a system of base axes (x0, y”, z”) is defined (with z” coinciding with the nodal z axis) and the fibre orientation can be input with reference to one of these base axes. Based on this data, it is possible to obtain the orientation of the material principal axes (I, 2) at any point with respect to the laminate axes (x, y, z). Knowing the material principal axes (I, 2, 3) with respect to the laminate axes (x, _r. z), the strains t ’ can
(r.
Applying the principle of minimum potential energy, the element stiffness matrix, K,, for the shell element is obtained as
(20)
The stress-strain relation along the material axes (I, 2,3) of an orthotropic lamina (Fig. 2) is defined as [25] a’=Cc’ (21)
6,
=
BrDBdV sY
(25a)
+I
t,
+I
-I
-I
-I
ISI
BrDB(J] d< drj d<
(25b)
in which D is the transformed elasticity matrix of the shell layers along the (x. 13,z) axes and is given in eqn (24a). In eqn (25b), the integration along (5, ‘I) surface is done by Gauss quadrature. Since the elasticity matrix is different from layer to layer and not a continuous function in <, the integration along [ direction is to be carried out by splitting the limits through each layer. The integration along [ directrion may be carried out either by numerical quadrature or analytically. If numerical quadrature is to be used, the limits of integration should be from - I to + I in each layer in order to apply the known coefficients of the Gauss integration rule. This is achieved by suitably modifying the variable [ to ia in the k th layer such that ii vary from -I to + I in that layer. The change of variable is obtained from the transformation [4, 201 r, - t,
(I - ;r)1
and
where tA is the layer thickness at the point. written as
and t is the total eqn (25b) can be re-
Thus, +I
B’D,B]J] -1 Fig. 2. Lamina principal directions (I, 2,3) and laminate axes (x, y, z) for a typical layer of the composite shell.
x
-:” d< dtj di,. 0
(27)
Modified three-dimensional finite element for general and composite shells The numerical Gauss integration along the [ direction necessitates a four-point Gauss rule in each layer
for the complete integration of the stiffness matrix given by eqn (27). This becomes uneconomical for shells with a large number of layers. In such cases, analytical thickness integration can be used with advantage by assuming that the constituents of the strain-displacement matrix, B,,B, ,B2and B,do not vary across the thickness of the shell. Thus the constituents of strain-displacement matrix B need be evaluated only once at the Gauss points located along the midsurface of the shell element. In order to evaluate the element volume accurately, the determinant of the Jacobian matrix J is assumed to vary quadratically across the shell thickness as follows. The determinant of the Jacobian at the top, middle and bottom surfaces of the shell element is evaluated as IJr], IJM] and ]Js] at each Gauss point located along the shell mid-surface and its value at any other point along the thickness is interpolated as IJI = Jo + t;J, + r2J2
(28)
in which
Jo= IJ,L J, = (IJA- IJdP (IJA- 2lJ~l+ IJd.
(29)
The element stiffness matrix given in eqn (25b) can thus be rewritten as R,=
c” +’ +’ +1W+ge,+12B2+53B,lr k=lSIS -I -I -I
x W%
+ CB,+
12B2
+
53B311Jl d5 drt d6. (30)
Denoting ik +I DjZ k“c”, i* W”-“(J,+lJ, _ s
+12Jz)dC, j=l,2,...,7
(31)
in which the limits of c for the layer k are obtained from eqn (26a). The element stiffness matrix can be obtained as
X
and a three.-point scheme for the seven-node triangular elements. Consistent load vectors are derived for the various load cases for the modified 3-D formulation. It can be seen that load terms will be present not only for the mid-surface translations, but also for terms corresponding to the other degrees of freedom depending on the depth 4 at which these loads act. Details of the load vectors are omitted for brevity and are available in [26]. 3. BOUNDARYCONDITIONS
The displacement finite element formulation developed in the previous sections using the modified 3-D formulation is based entirely on assumed displacement functions, and hence only displacement boundary conditions can be specified. Since, in the present formulation, transformation is applied to the translation terms only, the same boundary conditions used for the mid-surface translations are used for the corresponding substitute degrees of freedom at any node. 4. NUMERICALRESULTSAND DISCUSSION
and J2 =
293
de dtt. (32)
The element stiffness matrix given by eqn (32) is computed using reduced integration, namely, a 2 x 2 Gauss rule for eight- and nine-node quadrilaterals
The modified three-dimensional formulation developed above is first used to solve thin homogeneous and isotropic shells which are standard tests for new formulations. The effect of transverse shear deformation on the displacements and stresses in layered composite shells are then investigated. Results from a first-order shear deformation theory model, FSDTS, from [26] are also given wherever comparative results are not available. All the computations have been done on CDC Cyber 180/84OA computer with single precision arithmetic. A nine-node Lagrangian quadrilateral has been used in all the analyses except when otherwise specified. Example 1. Pinched hemisphere
This is a NAFEMS benchmark problem given by Hitchings et al. [27] and consists of a hemisphere clamped at the top and free at the bottom subjected to two diametrically opposite concentrated loads at the base. Only a quarter of the shell is modelled with seven-node triangles as shown in Fig. 3. Convergence of the radial deflection under the loads is shown in Fig. 4. It is found that the results from the modified 3-D element formulation show smooth convergence to the exact solution for this sensitive problem. Example 2. Clamped under pressure load
laminated cylindrical shells
TO evaluate the present refined formulation for different fibre angles including angle-ply and unsymmetrical lamination scheme, two clamped cylindrical shells under internal pressure are considered for which comparative solutions are given by Hass and Lee [81 based on an assumed strain degenerated solid
294
D. N. BURAGOHAINand P. K. RAVICHANDRAN
FIXED
Y
Fig. 3. Pinched hemisphere (quarter only shown) (R = 10m, I = 0.04 m, E = 68.25 x lo6 Pa, Y = 0.3). element. The geometrical details of the shells; namely, one circular cylindrical shell and the other, a 90” segment of the same cylindrical shell are shown in Fig. 5. For the full cylinder, the lamination schemes considered are the symmetric (45’/ - 45”/ -45”/45”) and (O”/900/900/Oo) lay ups and hence only an octant of the shell is analysed. But, for the 90” cylindrical shell segment with lamination schemes (45”/ - 45”) and (45”/-45”/-45”/45”), the full shell is analysed since the former scheme is unsymmetrical. The layers are of equal thickness and the material properties assumed for the layers are
displacements in the shells for the different lamination schemes are shown in Tables 1 and 2. From the results, it is seen that the MOD-3D model performs well for both symmetric and unsymmetric angle-ply schemes showing convergence to the reference solutions. Shear deformation in this thin shell is not appreciable for the material properties and loading considered so that the first-order model FSDTS is able to predict satisfactorily the response of the shell. The present solutions converge to the reference solutions in all cases except for (45”/ -45”i -45”/45’) scheme for the full cylinder. However, comparison with another finite element solution by Witt et al. [28] indicates that thge present results are satisfactory. Example 3. Simply supported cross-ply (oc’/9oL/O’) cylindrical shell under sinusoidally distributed load The cylindrical shell panel in this example has a mean radius of 10 in. and sector angle of (n/3) radians and is infinitely long in the longitudinal direction (Fig. 6). It is simply supported along the two straight edges and is subjected to sinusoidal load of maximum intensity q0 at the centre. The material properties of the layers of equal thickness are taken as E, =25
E2= E,=
x IO’psi,
G,z = G,, = 0.5 x lO’psi,
106psi,
G,, = 0.2 x 106psi,
v,> = vI1 = v,, = 0.25. E, = 7.5 x IO6 psi,
E2 = 2 x lo6 psi,
G,, = 1.25 x 106psi G,, = G,, = 0.625 x lo6 psi,
vu = 0.25.
The Poisson ratios vu and v2, are assumed to be equal to I’,~. Both the shells are clamped all along their boundaries and subjected to uniform internal pressure q of value (6.41/x) psi. A comparison of the radial
This problem has an exact elasticity solution given by Ren [29]. Only one half of the shell with unit width along the longitudinal axis is analysed by taking a 1 x 12 mesh with the assumption of plane strain conditions along the length of the shell. A comparison of the non-dimensional central deflections for different (R/r) ratios are given in Table 3 along with the results of a finite element formulation from a higher order theory by Dennis and Palazotto [30].
1.20
0
0~~000 0
MOD-3D
40
20 ND.
OF
ELEMENTS
60
80 ( 7 NOOED
100 TRIANGLES
)
Fig. 4. Convergence of maximum radial displacement at the base of the pinched hemisphere.
Modified th~~imensional
finite ekment
for general and composite
295
shells
2 &
z t ttt?ttttT
L R
4
V
L
i, (a)
4 CLAMPED
’
f b)
CYLINDER
SO’ CLAMPED CYLINDRICAL SHELL
Fig. 5. Cylindrical shells under internal pressure [R = 20 in., L = 20 in., f = 1 in, 4 = (6.41/n) psi]. shows that the MOD3D model gives stresses closer to the exact solutions while the classical layer theory is in serious error. For the case of the transverse shear stress variation along the thickness of the shell, the present model is found to give parabolic variation. However, this is found to be inaccurate since it gives non-zero shear stresses at the free surfaces and also di~ontinuity at the junction between adjacent layers. Thus, even though the surface parallel stress prediction by the MOD-3D model is satisfactory for practical purposes, alternative methods are necessary to estimate the variation of the transverse shear stresses.
The variation of the in-plane displacement, U, along the ~rcumferential direction for a (R/f) ratio of 4 is shown in Fig. 7. The variation of the corresponding stress, uY, at the centre of the shell and the transverse shear stress, zvZ, at the support are shown in Figs 8 and 9 for (R/t) values 10 and 4, respectively. From the results it can be seen that the effect of transverse shear and transverse normal strain are significant in thick laminated shells. The classical lamination theory is found to have serious errors for the moderately thick and thick laminated shells. The first-order model, FSDTS, is also in error for thicker shells (R/t < 10). The present MODJD model is found to give solutions closer to the exact ones. Also, it is found to give better results than the higher order theory of Dennis and Palazotto [30] although they used a finer 1 x 20 mesh for one half of the shell. This is obviously due to the inclusion of the transverse normal strain in the MOD-3D model which is neglected in the higher order theory of [30]. However, the difference between the predictions by the present method as well as those of[30] from the exact solutions show an increasing trend as thickness is increased. Comparison of the circumferential stress au variation along the thickness at the centre of the shell
Example 4. Mulch-layered spherical ~~~~~~~1 load
sheik
In order to study the effect of shear deformation on thick multi-layered composite shells for various lamination schemes and thickness shear effects, a spherical shell panel of square planform with side a is considered. It is simply supported on all the four edges and subjected to a sinusoidal load of maximum intensity q,, at the centre. The different lamination schemes considered are (45”/-45”/45”/-45”), (0”/45”/90”/ -45°)sym and (0’/30”/ - 30”/60”/ - 60”/ 90”),, and a parametric study for various (a/t) ratios of 4, 10 and 20 was conducted for R/a = 3. The
Table 1. Maximum radial displacement ( x 10V4in.) for the clamped full cylindrical shell of Example 2 00/900/W/0 Method FsDT5 MOD-3D Ref. [8] Ref. [28]
2x2
3x3
1.794 I.782 1.792
1.787 1.807 (4 x 4) (8 x 8)
Table 2. Radial displacement
4x4
I.785 1.788 1.783 1.787
45”/-45”/ - 45”/45” 5X5
1.787 (6 x 6) (16 x 16)
2x2
3x3
4x4
2.354 2.398 2.337
2.353 2.368
2.3% 2.359 2.402 2.331
45”/-45”
Method
3x3 3.028 3.067 2.914 (6 x 6)
Ref. [8]
5x5 2.359 (6 x 6) (16 x 16)
( x 10e4 in.) at the centre for the clamped 90” cylindrical segment of Example 2
FSDTS MOD-3D
4x4
under
45”/ -45”/ -4s0/450
5x5
2.917 2.916 2.913 2.911 2.916 (10 x 10)
3x3 3.011 3.048 2.908 (6 x 6)
4x4
5x5
2.914 2.918 2.914 2.916 2.909 (10 x 10)
296
D. N. BURAGOHAIN and P. K. RAWCHANDRAN
v
100
= q.
-
E2 v
,R/i=b
(R/113
t
EXACT
---
CLT 0
t/2-
MOD-30
\ \ \ \ \ \ I
Fig. 6. Cross-ply (0’/90’/0’) cylindrical shell under soidal load.
t/6--
sinu-
‘, \ \ 1 I \ 20 \
I O-
following assumed
material
properties
for
the
layers
I 80 \
are
\
-t/6-
\ \ \ \
E, = 25E,,
G,? = G,, = 0.5E2,
\
G,, = 0.2Ez
viz = v2) = v,~ = 0.25.
Fig. 7. Variation of circumferential displacement c at the support for the thick cross-ply (0°/900/Oo) cylindrical shell (R/r = 4).
For the unsymmetric scheme, (45”/ -45’/45”/ -45’-), the complete shell is modelled by a 5 x 5 mesh while for the other symmetric cases only a quarter of the shell is modelled with a 3 x 3 mesh division, A comparison of the non-dimensional centre deflections for the various cases are presented in Table 4. From the table, it is again found that the effect of shear deformation on the central deflection for this thick multi-layered shell mainly depends on the (a/t) ratio. For the (45”/-45”/45-/-45”) scheme, for example, the increase in the central deflection obtained using the MOD-3D model over FSDTS are respectively 10.7%, 7.9% and 2.4% for (u/r) values of 4, 10 and 20. A similar trend is observed for the other lamination schemes. It can be noted that shear deformation is slightly larger in the unsymmetric laminate in comparison with that of the symmetrically laminated cases. For the lamination schemes considered, the unsymmetric schemes (45’/ -45”/ 45”/-45”) is found to be the stiffest.
a central concentrated load P. The problem involves severe warping of the shell cross-section under the load. The details of the sandwich shell are as follows: u = 32 in.,
Method
transverse
R/I
FSDTS MOD-3D Ref. [30] Classical layer theory [29] Exact (291
Facings:
E, = 10’ psi.
Core:
E, = E,/lO, I’, = 0.3,
V, = 0.3 E,/25
and
and
t, = 0.1 t E,/50,
t, = 0.8 t.
Keeping the mid-surface radius, R, the same, the total thickness t is varied to obtain different radius to total thickness (R/t) ratios of 25, 50 and 100. The material properties of the core are also varied for each (R/t) ratio. The radial deflection at the centre for the various cases obtained using the MOD-3D formulation is given in Table 5 along with the first-order theory FSDTS [26] results. Only one quadrant of the shell is modelled with a 3 x 3 mesh. From the results it is seen that the effect of shear deformation is very severe in this sandwich shell as a
In this problem, the spherical shell of the previous example is considered as a sandwich shell made of a weak core and strong facings. It is simply supported on its boundary of square planform and subjected to 3. Non-dimensional
P=lOOlb.
R/t = 25. 50, 100.
Example 5. Sandwich spherical shell under central concentrated load
Table
R=96in.,
w* at
displacement
the
centre
for
the
cylindrical
shell
[MS*= 10E,w/q,r(R/r)4] 2
4
10
50
100
I.1223 1.2481 I.1410 0.0799 I .436
0.3410 0.3970 0.3820 0.0781 0.457
0.1 I98 0.1311 0.1280 0.0777 0.144
0.0793 0.0797 0.0796 0.0776 0.0808
0.0780 0.078 I 0.078 I 0.0776 0.0787
Fig, 8. Circumferential stress Q?distribution at ths rr?ntreof the thick cross-ply (W/W’/W’) cylindrical shell,
-0
Rli
f
-
$0
n
EXdCf Rlt
a
4
Mm-30
Fig. 9. Transverse shear stre8s rv: distribution at the support of the thick cross-pIy (W/90*/0’)cylindrical shell. result of the weak core and the co~~n~c~ luadb Increase in the cxmtraI de&&ion obtained from the present refined method over FSDT5 suggests that shear deformation increases as the core is made weaker for any given (R/t) ratio. As in the previous example, shear deformation effects reduce as the shell is made thinner. The first-order theory is in serious error for this sandwich shelI when the (R(r) ratio is small and/or when the core is very weak; hence it is unsatisfactory, Again, the inclusion of transverse normal strain energy in the MOD-3D element leads to improved results for the thicker shells.
Table 4. ~~n~irn~~ona~
vertical de&&on
A new formulation for the analysis ofshear flexible homogeneous and layered composite shell structures has heen presented which incorporates warping of the cross-section and transverse narmal straining. The elements ~~~o~ hased on this me&l have been shown to be equaly effective for both thin and thick homogeneous and layered composite shells. This model is found to he suitable for the analysis af layered composite shells for which the first-order theory is inadequate. From the studies, it is found
w* at the centre for the spherical shell under sinusoidal !oad (Iv+ = 10%,Pw/qoa4)
alr
Method
(45”/-45”/4P/-45”)
4
FSDTS MUD-30
10.5234 If.6489 (10.7)$
Il.8940 12.8369(7-9)
I1,9590 f3.1821 (10.2)
If)
FSWE MOD-3D
1.7626 2.9820 (7*9)
3.37f9 3.5434 (5. If
3.3303 3.4747 (4.3)
(OO”/4Y/90”/ -45”/90”),,
(0”/30”/- 30~/60”/-60Q/90Q)*Ym
FSDTS i.1471 1A688 MOD9D 1.1748(2”4) 1.5076(2.6) t Values in parentheses denote prcentage deviation given by 100 [MOD-3D/FSDTS-I].
20
I ,4$43 1.4638(0.6)
298
D. N.
~URA~HAIN
and P. K.
RAVI~HANDRAN
Table 5. Comparison of radial displacement (X 10e4 in.) at the centre of the sandwich shell of Example 5 Method
E,/E, = 10
E&E, = 25
E&E, = SO
25
FSDTS MOD-3D
0.6575 1.0090 (54)?
0.7420 1.5954 (I 15)
0.7768 2.1851 (181)
50
FSDTS MOD-3D
2.9810 3.5220 (18)
3.3280 4.7030 (41)
3.4650 5.8680 (69)
100
FSDTS MOD-3D
12.5500 13.4780 (7.4)
14.2460 16.8780 (19)
14.9290 19.7930 (33)
R/l
t Values .n parentheses denote percentage deviation given by lOO[MOD-3D/FSDTS-I]. that neglect of transverse shear and normal strain energies of the shell affect the response of thick
layered composite shells significantly and these must be included for a reliable analysis. It is seen that although the surface parallel stress predictions by the proposed formulation are reasonably satisfactory, alternative methods are necessary for better prediction of transverse shear stress variation in thick layered composite shells. The analytical thickness integration adopted in the MOD-3D model was found to be satisfactory for thick layered shells up to (R/t) > 5. For thicker shells with (R/t) -=z5, the numerical four-point Gauss rule is recommended along the shell thickness as the proposed explicit integration scheme is inadequate in such cases. This is thought to be due to the simplifying assumptions made in the explicit integration scheme.
12. B. Dasgupta. Finite element analysis of layered composite shells. MTech dissertation, Department of Civil Engineering, I.I.T.. Bombay (1980). 13. P. Seide and R. A. Chaudhuri, Triangular finite element for analysis of thick laminated shells. In/. J. Numer. Meth. Engng 24, 1563- 1579 (1987). 14. N. D. Phan and J. N. Reddy, Analysis of laminated composite plates using a higher order shear deformation theory, In!. J. .?%rrer. Meth. Engng 21, 2201-2219 (1985). 1.5. T. Kant, D. R. J. Owen and 0. C. Zienkiewicz, A refined higher order Co plate bending element. Compu/. Struct. 15, 177-183 (1982). 16. T. Kant, On finite element discretisation of a higher order shell theory. In MAFELAP 1981 (Edited by J. R. Whiteman). Academic Press, London (1982). 17. A. Bhimaraddi, A. J. Carr and P. J. Moss, Finite element analysis of laminated shells of revolution with laminated stiffeners. Cornput. Strurt. 33, 2955305 (1989).
18. S. T. Dennis and A. N. Palazotto, Transverse shear deformation in orthotropic cylindrical pressure vessels. Am. fnst. Aeronaut. Astronaut.
REFERENCES
I. J. A. Zukas and J. R. Vinson, Laminated transversely isotropic cylindrical shells. J. Appl. Mech. ASME 38,
4oo-407 (1971). 2. J. M. Whitney and C. T. Sun, A refined theory for laminated anisotropic cylindrical shells. J. Appf. Mesh., ASME
41,471476
(1974).
3. J. N. Reddy, Exact solutions of moderately thick laminated shells. J. Engng Mech., AXE 110, 794809 (1984). 4. A. K. Noor and C. M. Anderson, Mixed isoparametric finite elements models of laminated composite shells. Comput.
Meth. Appf. Mech. Engng II, 255-280 (1977).
5, S. C. Panda and R. Natarajan, Finite element analysis of laminated composite plates. Int. J. Numet. Me&. Ertgng 14, 69-79 (1989). 6. C. S. Panda and R. Natarajan, Analysis of laminated composite shell structures by finite element method. Cornput. Struct. 14, 255-230 (1981). I. J. N. Reddy, Bending of laminated anisotropic shells by a shear deformable finite element. Fibre Sci. Tech. 17, 9-24 (1982). 8 D. J. Hass and S. W. Lee, A nine-node assumed-strain finite element for composite plate and shells. Compur. Struct. 26, 445452 (1987). 9 T. Y. Chang and K. Sawamiphakdi, Large deformation analysis of laminated shells by finite element method. Comput. Struct. 13, 331-340 (1981). IO A. S. Mawenya and J. D. Davis, Finite element bending analysis of multilayered plates. Inr. J. Nzuner. 1Weth. Engng 8, 215-225 (1974). 11. S.-T. Mau. T. H. H. Pian and P. Tong, Vibration analysis of laminated plates and shells by ahybrid stress element. Am. Inst. Aeronaut. Astronaut. Jnl 11, 1450-1452 (1973).
Jnl27,
1441-1447 (1989).
19. D. N. Buragohain, Use of higher order deformation theory in finite element analysis of shear deformable shells. Presented at Inr. Cotzl: on Engq Software ICENSOFT, I.I.T.. Delhi (1989). 20. K. S. Surana and R. M. Sorem. Curved shell elements based on hierarchical p-approximation in the thickness direction for linear static analysis of laminated composites. Int. J. Nrimcr. Meth. Engng 29, 1393-1420 (1990). 21. R. M. Barker, Fu-Tien Lin and J. R. Dana, Three dimensional finite element analysis of laminated plates. Comput.
Strucr. 2, 1013-- 1029 (1972).
22. Y. H. Kim and S. W. Lee, A solid element formulation for large deflection analysis of composite shell structures. Comput. Struct. 30, 269. 274 (1988). 23. 0. C. Zienkiewicz, The Finite Element method, 3rd Edn. McGraw-Hill, London (t 977). 24. K. J. Bathe, Finite Element Proredures in Engineering Analysis. Prentice-Hall of India, New Delhi (1990). 25. S. G. Lekhnitskii, Theory of Elasticity of an Anisotropir Elastic Body. Holden-Day, San Francisco (1963). 26. P. K. Ravichandran, Finite element analysis of layered composite and stiffened shells based on refined theories. Ph.D. thesis submitted to Department of Civil Engineering, I.I.T., Bombay (1992). 27. D. Hitchings, A. Kamoulkos and G. A. 0. Davies, Linear Static Benchmarks. NAFEMS Vol. I. New York (1987). 28. T. E. Witt, A. F. Saleeb and T. Y. Chang, A mixed element for laminated plates and shells. Comput. Struct. 37, 597611 (1990). 29 J. G. Ren, Exact solutions for laminated cylindrical shells in cylindrical bending. Camp. Sci. Tech. 29, 169-187 (1987). 30 S. T. Dennis and A. N. Palazotto, Laminated shell in cylindrical bending, two-dimensional approach vs exact. Am. Inst. Aeronaut. Astronaut. Jnl29, 647-650 (1991).
Pergamon
MODIFIED
@34s-7949/94 57.00 + 0.00
THREE-DIMENSIONAL FINITE ELEMENT GENERAL AND COMPOSITE SHELLS D. N.
BURAC~HAIN
and P. K.
FOR
RAVICHANDRAN
Department of Civil Engineering, Indian Institute of Technology, Bombay, India (Received 8 September
1992)
Abstract-A new shear deformable finite element formulation has been developed for the linear static analysis of general, homogeneous and laminated composite shells. It is based on the reduction of a three-dimensional element to a two-dimensional element by the use of substitute degrees of freedom. The proposed formulation incorporates a realistic cubic variation of the warping of the cross-section and also accounts for the transverse normal strain energy in the derivation of the total potential energy of the structure. Analytical integration along the shell thickness direction is carried out by taking quadratic variation of the determinant of the Jacobian matrix. The elements considered are a seven-node triangle and eight- and nine-node quadrilaterals. Reduced Gauss integration is used to evaluate the various element property matrices. A number of problems from thin isotropic shells to thick laminated composite shells have been solved to check the accuracy of the proposed formulation and to study the effect of transverse shear deformation on the response of thick sandwich and composite shells.
1. INTRODUCTION
Structural units such as plates and shells made of composite materials are finding increasing applications in advanced technical fields such as the aerospace and automotive industries where the main requirements are a high strength and/or stiffness coupled with low weight and other optimum properties. The analysis and design of such composite structural systems are, however, complicated because of their highly anisotropic behaviour. Also, owing to rile low ratio of the transverse shear modulus to the in-plane modulus in most of these advanced composite materials, shear deformation theories are necessary for the reliable analysis of laminated composite plate/shell structures. The first-order shear deformation theory in which constant transverse shear deformation is incorporated along the total thickness of the shell was used earlier by many investigators [l-3]. Since such analytical methods are limited to simple geometry and boundaries, many numerical methods, and in particular the finite element method, have been widely employed [4-91. The theories used in these references are either based on an assumed stress field or an assumed displacement field or on a mixed formulation. For thicker composite shells, or when the properties of the layers vary from layer to layer, an improved theory known as the layerwise constant shear angle theory (LCST) was adopted in [l&13]. However, this theory introduced two additional degrees of freedom with each additional layer and therefore was cumbersome. Both these theories also necessitated the use of arbitrary shear correction coefficients in their formulation. CAS Sl,3--E
Many higher order theories have, therefore, been proposed as an improvement over the first-order theory models by expanding the displacements in powers of the thickness coordinate. They have been successfully applied for the analysis of composite plates, for example [ 14, 151, but such theories for composite shells have been limited [ 16-201. Buragohain [19] presented a refined higher order theory for laminated general shells and proposed a modified three-dimensional (3-D) formulation for shear deformable shell analysis, both using economical explicit thickness integration schemes. The theory of [20] is based on hierarchical polynomial approximation of the displacements along the thickness direction and uses a numerical Gauss integration scheme to evaluate the element properties. Such a scheme becomes uneconomical as the order p of the polynomial is increased since p + 1 integration points are necessary along the thickness in each layer. Three-dimensional analysis of composite plates and shells has also been scarce [21,22] due to the inherent numerical complexity. In the present development, a simplified 3-D model based on the modified 3-D element concept of [19] is described in detail for the linear static analysis of layered composite shells in which many of the above-mentioned difficulties have been overcome. 2. THE MODIFIED 3-D (MOD 3-D) ELEMENT
2.1. Parent solid element A typical quadrilateral brick element with four nodes along the thickness at each of its corners and mid-sides is shown in Fig. I (a) (a total of 32 nodes for
289
D. N. BURAGOHAIN and P. K. ~AV~CHANDRA~
Nir(i,
Q, C)=
Lr(iWz(5aQ)
(W
N,,(L Q, i) = L&)N,(5, Q)
(24
in which N,(,(r,4) are the standard shape functions for Co continuity for the eight-noded quadrilateral shell element and L,(i)=&1 of
ORIGINAL
NODAL
POSIlIONS
+5 +912-95’)
(3a)
L,(~)=~(l
-31
L,(i) = $1
+ 35 - iz - 31’)
L,(i)=h(-1
-i?+31’)
-i
(3b)
(3c)
+9<*+91g.
(34
The global translations can now be written as (e.g. U) u = y N~(S, rl) t&p utp + Liq
uiq
,=I
b)
MODIFIEO DEGREES
ELEMENT WITH OF FREEDOM
SUBSTITUTE +
Fig. 1. Modified three-dimensional element. (a) Original nodal. positions (b) modified element with substitute degrees of freedom.
the solid element). Using natural coordinates (5, q, <), with (4, q) along the mid-surface and 5 along the thickness direction, the translations U, V and W along global X, Y and Z directions at any point in the solid element may be obtained using a 3-D isoparametric element formulation as (for example, displacement U)
u=zNf(5,~,W,,
j=1,2
,..., 32
(1)
in which NT denotes the shape functions of the solid element at node j. Although it is possible to evaluate the structure properties such as the stiffness matrix with three translational degrees of freedom at each node using standard procedures [23,24], various numerical difficulties are often encountered for the solution of the assembled structure equilibrium equations: ill-conditioned equations, large storage and computational efforts [23]. Hence, the following procedure is adopted to obtain the equivalent modified 3-D (MOD-3D) element. 2.2, D~sp~u&emen~fieid for the MOD-31)
element
Denoting four points, p, y, Y and s, at equal intervals along the thickness at any node point i (passing through the solid element nodes), the shape functions for global translations Ii, V and W at these levels may be written as
N&
Q,
i) = Lq(iWi(t, Q)
(2b)
L
uir
f
hs
ursl. t4)
Substituting eqns (3) into (4) and re-arranging terms according to powers of < leads to U = c N,(L n)]h(-
the
U,,,+9U,, + 9U,, - U,,)
+ ([/2)$(U, - 27U, + 27U, - U,) + (~~2)2~(uj~-
uiq
-
iir
-
ut,~)
+(i/2)'~(-Vi~+3U,,-3Ui,+Uj,)]
(5)
with similar expressions for V and W. The 12 transiationa~ degrees of freedom at the four points p, 4, r and s along the thickness are replaced by 12 substitute degrees of freedom at any nodal position i of the reduced model (Fig. lb) as U =~N,(5,~)iUr+(r/2)t,Ui + (;/2):r; U:’ -t (l/2)%; U:“] (6) in which iJ:+u,,+9u,+9(/,-
Cl>,)
(7a)
U: = $ (U,P- 27u, + 270;, - U,,)
(7b)
U~=~(u,~-u,,,-ui,+Li,)
(7c)
U~=~(-U,~+3u‘,-3u,+Ui,~
(7d)
and t, is the total thickness of the shell at node i. Similar expressions are obtained for V and W. The
291
Modi~ed thr~-djmensional finite elen tent for general and composite shells displacement field for the MOD-3D model can thus be written as ii=
=~N,(S,v)~~~+(~,Z)r,~ + (i/2)%5@ +
(~/2)3t$iy]
in which 8 = Wy, with 8 as the transformation matrix at the point. The global coordinates (X, Y, Z) at any point (t, r,i,[) on the shell element is obtained as
(8)
in which
p=
1~:”
v:’
WY}‘.
(9)
In a similar manner, seven-node triangular and nine-node rectangular elements with nodes at the comers, mid-sides and centre are also developed. 2.3. Strain-displacement relation
At each node i, a local Cartesian coordinate system (x, y, z) is set up uniquely with (x, y) in the tangent plane and z along the normal to the shell midsurface. The displacements (u, u, w) along (x, y, z) axes can be obtained by suitable transformation of the global displacements. The displacement components along global directions (A’, Y, 2) in eqn (9) may be transformed to similar displacement components along (x, y, z) axes by using the nodal transformation matrix 0, leading to
In order to show the necessary modi~~tions to be made on the strain-displacement matrix of the solid element formulation from which the present method is derived, the approach described here is followed. For the solid element, the strains along local Cartesian coordinates (x, y, z) at any point on the midsurface of the shell element may be obtained using standard procedure as -a/ax
0
0 -
;i = y N,(& q)@;[d; + ([/2)&d; i=t
+ ([/2)%fdj f ({/2)3ri’dy]
(IOa)
in which OOb)
is the nodal transformation matrix, the three rows of which are given by the unit vectors along the local axes (x, y, z); dp, dj, d; and d:,, are linear combinations of displacement components as in eqn (7) and are given by
(14)
or c =
L&i.
(15)
Using eqn (lOa) in eqn (14) t = (L@ii@d,= Lf? F &di
(16a)
1-I
=5&d = “c”Bidi i= I
The displacement components d = (u, u, w) along local axes (x, y, z) at any point on the mid-surface of the shell element is obtained again by transformation, d=t?;l as d = F N,@d; f (
(1W
in which mi is the coefficient matrix of the nodal degrees of freedom vector obtained from eqn (lOa). Thus, the three translational degrees of freedom at each of the four nodes along the thickness of the solid element are modified to equivalent substitute degrees of freedom at the nodes of the simplified 2-D model. The Cartesian derivatives in L of eqn (14) are obtained in terms of the derivatives in the natural coordinates using Jacobian matrix .J as {a/at
a/all
aiayy=qaiax
alay
a/aq
(17)
292
D. N. BURAGOHAINand P. K. RAVICHANDRAN
in which
be expressed in terms of the strains L in (x, y, z) axes using the strain transformation matrix T, as [25] c’=T,t.
In eqn (I@, V:, V, and V; are the tangent vectors along r, q and < directions and are obtained as
(22)
The stresses a in the (x, y, r) system are related to the strains c as u = Dc
(23)
in which D = TJCT,
(24a)
and d = {a, 2.5.
The strain-displacement now arranged as
matrix
Bi of eqn (l6b)
Bi = BO,+ [B,, + i*B,, + i’B3,. 2.4.
Stress-strain
are
T,.,
7,;
7wi
IT
Wb)
Element stiffness matrix
K,=
relation
in which C is the elasticity matrix of the layer material. The following approach is used to locate the orientation of the material axes in the case of an arbitrarily laminated general shell. At any point on the shell mid-surface, the z axis is fixed along the normal direction while the (x,v) axes are conveniently chosen in the tangent plane. Now, at one of the element modes, a system of base axes (x0, y”, z”) is defined (with z” coinciding with the nodal z axis) and the fibre orientation can be input with reference to one of these base axes. Based on this data, it is possible to obtain the orientation of the material principal axes (I, 2) at any point with respect to the laminate axes (x, y, z). Knowing the material principal axes (I, 2, 3) with respect to the laminate axes (x, _r. z), the strains t ’ can
(r.
Applying the principle of minimum potential energy, the element stiffness matrix, K,, for the shell element is obtained as
(20)
The stress-strain relation along the material axes (I, 2,3) of an orthotropic lamina (Fig. 2) is defined as [25] a’=Cc’ (21)
6,
=
BrDBdV sY
(25a)
+I
t,
+I
-I
-I
-I
ISI
BrDB(J] d< drj d<
(25b)
in which D is the transformed elasticity matrix of the shell layers along the (x. 13,z) axes and is given in eqn (24a). In eqn (25b), the integration along (5, ‘I) surface is done by Gauss quadrature. Since the elasticity matrix is different from layer to layer and not a continuous function in <, the integration along [ direction is to be carried out by splitting the limits through each layer. The integration along [ directrion may be carried out either by numerical quadrature or analytically. If numerical quadrature is to be used, the limits of integration should be from - I to + I in each layer in order to apply the known coefficients of the Gauss integration rule. This is achieved by suitably modifying the variable [ to ia in the k th layer such that ii vary from -I to + I in that layer. The change of variable is obtained from the transformation [4, 201 r, - t,
(I - ;r)1
and
where tA is the layer thickness at the point. written as
and t is the total eqn (25b) can be re-
Thus, +I
B’D,B]J] -1 Fig. 2. Lamina principal directions (I, 2,3) and laminate axes (x, y, z) for a typical layer of the composite shell.
x
-:” d< dtj di,. 0
(27)
Modified three-dimensional finite element for general and composite shells The numerical Gauss integration along the [ direction necessitates a four-point Gauss rule in each layer
for the complete integration of the stiffness matrix given by eqn (27). This becomes uneconomical for shells with a large number of layers. In such cases, analytical thickness integration can be used with advantage by assuming that the constituents of the strain-displacement matrix, B,,B, ,B2and B,do not vary across the thickness of the shell. Thus the constituents of strain-displacement matrix B need be evaluated only once at the Gauss points located along the midsurface of the shell element. In order to evaluate the element volume accurately, the determinant of the Jacobian matrix J is assumed to vary quadratically across the shell thickness as follows. The determinant of the Jacobian at the top, middle and bottom surfaces of the shell element is evaluated as IJr], IJM] and ]Js] at each Gauss point located along the shell mid-surface and its value at any other point along the thickness is interpolated as IJI = Jo + t;J, + r2J2
(28)
in which
Jo= IJ,L J, = (IJA- IJdP (IJA- 2lJ~l+ IJd.
(29)
The element stiffness matrix given in eqn (25b) can thus be rewritten as R,=
c” +’ +’ +1W+ge,+12B2+53B,lr k=lSIS -I -I -I
x W%
+ CB,+
12B2
+
53B311Jl d5 drt d6. (30)
Denoting ik +I DjZ k“c”, i* W”-“(J,+lJ, _ s
+12Jz)dC, j=l,2,...,7
(31)
in which the limits of c for the layer k are obtained from eqn (26a). The element stiffness matrix can be obtained as
X
and a three.-point scheme for the seven-node triangular elements. Consistent load vectors are derived for the various load cases for the modified 3-D formulation. It can be seen that load terms will be present not only for the mid-surface translations, but also for terms corresponding to the other degrees of freedom depending on the depth 4 at which these loads act. Details of the load vectors are omitted for brevity and are available in [26]. 3. BOUNDARYCONDITIONS
The displacement finite element formulation developed in the previous sections using the modified 3-D formulation is based entirely on assumed displacement functions, and hence only displacement boundary conditions can be specified. Since, in the present formulation, transformation is applied to the translation terms only, the same boundary conditions used for the mid-surface translations are used for the corresponding substitute degrees of freedom at any node. 4. NUMERICALRESULTSAND DISCUSSION
and J2 =
293
de dtt. (32)
The element stiffness matrix given by eqn (32) is computed using reduced integration, namely, a 2 x 2 Gauss rule for eight- and nine-node quadrilaterals
The modified three-dimensional formulation developed above is first used to solve thin homogeneous and isotropic shells which are standard tests for new formulations. The effect of transverse shear deformation on the displacements and stresses in layered composite shells are then investigated. Results from a first-order shear deformation theory model, FSDTS, from [26] are also given wherever comparative results are not available. All the computations have been done on CDC Cyber 180/84OA computer with single precision arithmetic. A nine-node Lagrangian quadrilateral has been used in all the analyses except when otherwise specified. Example 1. Pinched hemisphere
This is a NAFEMS benchmark problem given by Hitchings et al. [27] and consists of a hemisphere clamped at the top and free at the bottom subjected to two diametrically opposite concentrated loads at the base. Only a quarter of the shell is modelled with seven-node triangles as shown in Fig. 3. Convergence of the radial deflection under the loads is shown in Fig. 4. It is found that the results from the modified 3-D element formulation show smooth convergence to the exact solution for this sensitive problem. Example 2. Clamped under pressure load
laminated cylindrical shells
TO evaluate the present refined formulation for different fibre angles including angle-ply and unsymmetrical lamination scheme, two clamped cylindrical shells under internal pressure are considered for which comparative solutions are given by Hass and Lee [81 based on an assumed strain degenerated solid
294
D. N. BURAGOHAINand P. K. RAVICHANDRAN
FIXED
Y
Fig. 3. Pinched hemisphere (quarter only shown) (R = 10m, I = 0.04 m, E = 68.25 x lo6 Pa, Y = 0.3). element. The geometrical details of the shells; namely, one circular cylindrical shell and the other, a 90” segment of the same cylindrical shell are shown in Fig. 5. For the full cylinder, the lamination schemes considered are the symmetric (45’/ - 45”/ -45”/45”) and (O”/900/900/Oo) lay ups and hence only an octant of the shell is analysed. But, for the 90” cylindrical shell segment with lamination schemes (45”/ - 45”) and (45”/-45”/-45”/45”), the full shell is analysed since the former scheme is unsymmetrical. The layers are of equal thickness and the material properties assumed for the layers are
displacements in the shells for the different lamination schemes are shown in Tables 1 and 2. From the results, it is seen that the MOD-3D model performs well for both symmetric and unsymmetric angle-ply schemes showing convergence to the reference solutions. Shear deformation in this thin shell is not appreciable for the material properties and loading considered so that the first-order model FSDTS is able to predict satisfactorily the response of the shell. The present solutions converge to the reference solutions in all cases except for (45”/ -45”i -45”/45’) scheme for the full cylinder. However, comparison with another finite element solution by Witt et al. [28] indicates that thge present results are satisfactory. Example 3. Simply supported cross-ply (oc’/9oL/O’) cylindrical shell under sinusoidally distributed load The cylindrical shell panel in this example has a mean radius of 10 in. and sector angle of (n/3) radians and is infinitely long in the longitudinal direction (Fig. 6). It is simply supported along the two straight edges and is subjected to sinusoidal load of maximum intensity q0 at the centre. The material properties of the layers of equal thickness are taken as E, =25
E2= E,=
x IO’psi,
G,z = G,, = 0.5 x lO’psi,
106psi,
G,, = 0.2 x 106psi,
v,> = vI1 = v,, = 0.25. E, = 7.5 x IO6 psi,
E2 = 2 x lo6 psi,
G,, = 1.25 x 106psi G,, = G,, = 0.625 x lo6 psi,
vu = 0.25.
The Poisson ratios vu and v2, are assumed to be equal to I’,~. Both the shells are clamped all along their boundaries and subjected to uniform internal pressure q of value (6.41/x) psi. A comparison of the radial
This problem has an exact elasticity solution given by Ren [29]. Only one half of the shell with unit width along the longitudinal axis is analysed by taking a 1 x 12 mesh with the assumption of plane strain conditions along the length of the shell. A comparison of the non-dimensional central deflections for different (R/r) ratios are given in Table 3 along with the results of a finite element formulation from a higher order theory by Dennis and Palazotto [30].
1.20
0
0~~000 0
MOD-3D
40
20 ND.
OF
ELEMENTS
60
80 ( 7 NOOED
100 TRIANGLES
)
Fig. 4. Convergence of maximum radial displacement at the base of the pinched hemisphere.
Modified th~~imensional
finite ekment
for general and composite
295
shells
2 &
z t ttt?ttttT
L R
4
V
L
i, (a)
4 CLAMPED
’
f b)
CYLINDER
SO’ CLAMPED CYLINDRICAL SHELL
Fig. 5. Cylindrical shells under internal pressure [R = 20 in., L = 20 in., f = 1 in, 4 = (6.41/n) psi]. shows that the MOD3D model gives stresses closer to the exact solutions while the classical layer theory is in serious error. For the case of the transverse shear stress variation along the thickness of the shell, the present model is found to give parabolic variation. However, this is found to be inaccurate since it gives non-zero shear stresses at the free surfaces and also di~ontinuity at the junction between adjacent layers. Thus, even though the surface parallel stress prediction by the MOD-3D model is satisfactory for practical purposes, alternative methods are necessary to estimate the variation of the transverse shear stresses.
The variation of the in-plane displacement, U, along the ~rcumferential direction for a (R/f) ratio of 4 is shown in Fig. 7. The variation of the corresponding stress, uY, at the centre of the shell and the transverse shear stress, zvZ, at the support are shown in Figs 8 and 9 for (R/t) values 10 and 4, respectively. From the results it can be seen that the effect of transverse shear and transverse normal strain are significant in thick laminated shells. The classical lamination theory is found to have serious errors for the moderately thick and thick laminated shells. The first-order model, FSDTS, is also in error for thicker shells (R/t < 10). The present MODJD model is found to give solutions closer to the exact ones. Also, it is found to give better results than the higher order theory of Dennis and Palazotto [30] although they used a finer 1 x 20 mesh for one half of the shell. This is obviously due to the inclusion of the transverse normal strain in the MOD-3D model which is neglected in the higher order theory of [30]. However, the difference between the predictions by the present method as well as those of[30] from the exact solutions show an increasing trend as thickness is increased. Comparison of the circumferential stress au variation along the thickness at the centre of the shell
Example 4. Mulch-layered spherical ~~~~~~~1 load
sheik
In order to study the effect of shear deformation on thick multi-layered composite shells for various lamination schemes and thickness shear effects, a spherical shell panel of square planform with side a is considered. It is simply supported on all the four edges and subjected to a sinusoidal load of maximum intensity q,, at the centre. The different lamination schemes considered are (45”/-45”/45”/-45”), (0”/45”/90”/ -45°)sym and (0’/30”/ - 30”/60”/ - 60”/ 90”),, and a parametric study for various (a/t) ratios of 4, 10 and 20 was conducted for R/a = 3. The
Table 1. Maximum radial displacement ( x 10V4in.) for the clamped full cylindrical shell of Example 2 00/900/W/0 Method FsDT5 MOD-3D Ref. [8] Ref. [28]
2x2
3x3
1.794 I.782 1.792
1.787 1.807 (4 x 4) (8 x 8)
Table 2. Radial displacement
4x4
I.785 1.788 1.783 1.787
45”/-45”/ - 45”/45” 5X5
1.787 (6 x 6) (16 x 16)
2x2
3x3
4x4
2.354 2.398 2.337
2.353 2.368
2.3% 2.359 2.402 2.331
45”/-45”
Method
3x3 3.028 3.067 2.914 (6 x 6)
Ref. [8]
5x5 2.359 (6 x 6) (16 x 16)
( x 10e4 in.) at the centre for the clamped 90” cylindrical segment of Example 2
FSDTS MOD-3D
4x4
under
45”/ -45”/ -4s0/450
5x5
2.917 2.916 2.913 2.911 2.916 (10 x 10)
3x3 3.011 3.048 2.908 (6 x 6)
4x4
5x5
2.914 2.918 2.914 2.916 2.909 (10 x 10)
296
D. N. BURAGOHAIN and P. K. RAWCHANDRAN
v
100
= q.
-
E2 v
,R/i=b
(R/113
t
EXACT
---
CLT 0
t/2-
MOD-30
\ \ \ \ \ \ I
Fig. 6. Cross-ply (0’/90’/0’) cylindrical shell under soidal load.
t/6--
sinu-
‘, \ \ 1 I \ 20 \
I O-
following assumed
material
properties
for
the
layers
I 80 \
are
\
-t/6-
\ \ \ \
E, = 25E,,
G,? = G,, = 0.5E2,
\
G,, = 0.2Ez
viz = v2) = v,~ = 0.25.
Fig. 7. Variation of circumferential displacement c at the support for the thick cross-ply (0°/900/Oo) cylindrical shell (R/r = 4).
For the unsymmetric scheme, (45”/ -45’/45”/ -45’-), the complete shell is modelled by a 5 x 5 mesh while for the other symmetric cases only a quarter of the shell is modelled with a 3 x 3 mesh division, A comparison of the non-dimensional centre deflections for the various cases are presented in Table 4. From the table, it is again found that the effect of shear deformation on the central deflection for this thick multi-layered shell mainly depends on the (a/t) ratio. For the (45”/-45”/45-/-45”) scheme, for example, the increase in the central deflection obtained using the MOD-3D model over FSDTS are respectively 10.7%, 7.9% and 2.4% for (u/r) values of 4, 10 and 20. A similar trend is observed for the other lamination schemes. It can be noted that shear deformation is slightly larger in the unsymmetric laminate in comparison with that of the symmetrically laminated cases. For the lamination schemes considered, the unsymmetric schemes (45’/ -45”/ 45”/-45”) is found to be the stiffest.
a central concentrated load P. The problem involves severe warping of the shell cross-section under the load. The details of the sandwich shell are as follows: u = 32 in.,
Method
transverse
R/I
FSDTS MOD-3D Ref. [30] Classical layer theory [29] Exact (291
Facings:
E, = 10’ psi.
Core:
E, = E,/lO, I’, = 0.3,
V, = 0.3 E,/25
and
and
t, = 0.1 t E,/50,
t, = 0.8 t.
Keeping the mid-surface radius, R, the same, the total thickness t is varied to obtain different radius to total thickness (R/t) ratios of 25, 50 and 100. The material properties of the core are also varied for each (R/t) ratio. The radial deflection at the centre for the various cases obtained using the MOD-3D formulation is given in Table 5 along with the first-order theory FSDTS [26] results. Only one quadrant of the shell is modelled with a 3 x 3 mesh. From the results it is seen that the effect of shear deformation is very severe in this sandwich shell as a
In this problem, the spherical shell of the previous example is considered as a sandwich shell made of a weak core and strong facings. It is simply supported on its boundary of square planform and subjected to 3. Non-dimensional
P=lOOlb.
R/t = 25. 50, 100.
Example 5. Sandwich spherical shell under central concentrated load
Table
R=96in.,
w* at
displacement
the
centre
for
the
cylindrical
shell
[MS*= 10E,w/q,r(R/r)4] 2
4
10
50
100
I.1223 1.2481 I.1410 0.0799 I .436
0.3410 0.3970 0.3820 0.0781 0.457
0.1 I98 0.1311 0.1280 0.0777 0.144
0.0793 0.0797 0.0796 0.0776 0.0808
0.0780 0.078 I 0.078 I 0.0776 0.0787
Fig, 8. Circumferential stress Q?distribution at ths rr?ntreof the thick cross-ply (W/W’/W’) cylindrical shell,
-0
Rli
f
-
$0
n
EXdCf Rlt
a
4
Mm-30
Fig. 9. Transverse shear stre8s rv: distribution at the support of the thick cross-pIy (W/90*/0’)cylindrical shell. result of the weak core and the co~~n~c~ luadb Increase in the cxmtraI de&&ion obtained from the present refined method over FSDT5 suggests that shear deformation increases as the core is made weaker for any given (R/t) ratio. As in the previous example, shear deformation effects reduce as the shell is made thinner. The first-order theory is in serious error for this sandwich shelI when the (R(r) ratio is small and/or when the core is very weak; hence it is unsatisfactory, Again, the inclusion of transverse normal strain energy in the MOD-3D element leads to improved results for the thicker shells.
Table 4. ~~n~irn~~ona~
vertical de&&on
A new formulation for the analysis ofshear flexible homogeneous and layered composite shell structures has heen presented which incorporates warping of the cross-section and transverse narmal straining. The elements ~~~o~ hased on this me&l have been shown to be equaly effective for both thin and thick homogeneous and layered composite shells. This model is found to he suitable for the analysis af layered composite shells for which the first-order theory is inadequate. From the studies, it is found
w* at the centre for the spherical shell under sinusoidal !oad (Iv+ = 10%,Pw/qoa4)
alr
Method
(45”/-45”/4P/-45”)
4
FSDTS MUD-30
10.5234 If.6489 (10.7)$
Il.8940 12.8369(7-9)
I1,9590 f3.1821 (10.2)
If)
FSWE MOD-3D
1.7626 2.9820 (7*9)
3.37f9 3.5434 (5. If
3.3303 3.4747 (4.3)
(OO”/4Y/90”/ -45”/90”),,
(0”/30”/- 30~/60”/-60Q/90Q)*Ym
FSDTS i.1471 1A688 MOD9D 1.1748(2”4) 1.5076(2.6) t Values in parentheses denote prcentage deviation given by 100 [MOD-3D/FSDTS-I].
20
I ,4$43 1.4638(0.6)
298
D. N.
~URA~HAIN
and P. K.
RAVI~HANDRAN
Table 5. Comparison of radial displacement (X 10e4 in.) at the centre of the sandwich shell of Example 5 Method
E,/E, = 10
E&E, = 25
E&E, = SO
25
FSDTS MOD-3D
0.6575 1.0090 (54)?
0.7420 1.5954 (I 15)
0.7768 2.1851 (181)
50
FSDTS MOD-3D
2.9810 3.5220 (18)
3.3280 4.7030 (41)
3.4650 5.8680 (69)
100
FSDTS MOD-3D
12.5500 13.4780 (7.4)
14.2460 16.8780 (19)
14.9290 19.7930 (33)
R/l
t Values .n parentheses denote percentage deviation given by lOO[MOD-3D/FSDTS-I]. that neglect of transverse shear and normal strain energies of the shell affect the response of thick
layered composite shells significantly and these must be included for a reliable analysis. It is seen that although the surface parallel stress predictions by the proposed formulation are reasonably satisfactory, alternative methods are necessary for better prediction of transverse shear stress variation in thick layered composite shells. The analytical thickness integration adopted in the MOD-3D model was found to be satisfactory for thick layered shells up to (R/t) > 5. For thicker shells with (R/t) -=z5, the numerical four-point Gauss rule is recommended along the shell thickness as the proposed explicit integration scheme is inadequate in such cases. This is thought to be due to the simplifying assumptions made in the explicit integration scheme.
12. B. Dasgupta. Finite element analysis of layered composite shells. MTech dissertation, Department of Civil Engineering, I.I.T.. Bombay (1980). 13. P. Seide and R. A. Chaudhuri, Triangular finite element for analysis of thick laminated shells. In/. J. Numer. Meth. Engng 24, 1563- 1579 (1987). 14. N. D. Phan and J. N. Reddy, Analysis of laminated composite plates using a higher order shear deformation theory, In!. J. .?%rrer. Meth. Engng 21, 2201-2219 (1985). 1.5. T. Kant, D. R. J. Owen and 0. C. Zienkiewicz, A refined higher order Co plate bending element. Compu/. Struct. 15, 177-183 (1982). 16. T. Kant, On finite element discretisation of a higher order shell theory. In MAFELAP 1981 (Edited by J. R. Whiteman). Academic Press, London (1982). 17. A. Bhimaraddi, A. J. Carr and P. J. Moss, Finite element analysis of laminated shells of revolution with laminated stiffeners. Cornput. Strurt. 33, 2955305 (1989).
18. S. T. Dennis and A. N. Palazotto, Transverse shear deformation in orthotropic cylindrical pressure vessels. Am. fnst. Aeronaut. Astronaut.
REFERENCES
I. J. A. Zukas and J. R. Vinson, Laminated transversely isotropic cylindrical shells. J. Appl. Mech. ASME 38,
4oo-407 (1971). 2. J. M. Whitney and C. T. Sun, A refined theory for laminated anisotropic cylindrical shells. J. Appf. Mesh., ASME
41,471476
(1974).
3. J. N. Reddy, Exact solutions of moderately thick laminated shells. J. Engng Mech., AXE 110, 794809 (1984). 4. A. K. Noor and C. M. Anderson, Mixed isoparametric finite elements models of laminated composite shells. Comput.
Meth. Appf. Mech. Engng II, 255-280 (1977).
5, S. C. Panda and R. Natarajan, Finite element analysis of laminated composite plates. Int. J. Numet. Me&. Ertgng 14, 69-79 (1989). 6. C. S. Panda and R. Natarajan, Analysis of laminated composite shell structures by finite element method. Cornput. Struct. 14, 255-230 (1981). I. J. N. Reddy, Bending of laminated anisotropic shells by a shear deformable finite element. Fibre Sci. Tech. 17, 9-24 (1982). 8 D. J. Hass and S. W. Lee, A nine-node assumed-strain finite element for composite plate and shells. Compur. Struct. 26, 445452 (1987). 9 T. Y. Chang and K. Sawamiphakdi, Large deformation analysis of laminated shells by finite element method. Comput. Struct. 13, 331-340 (1981). IO A. S. Mawenya and J. D. Davis, Finite element bending analysis of multilayered plates. Inr. J. Nzuner. 1Weth. Engng 8, 215-225 (1974). 11. S.-T. Mau. T. H. H. Pian and P. Tong, Vibration analysis of laminated plates and shells by ahybrid stress element. Am. Inst. Aeronaut. Astronaut. Jnl 11, 1450-1452 (1973).
Jnl27,
1441-1447 (1989).
19. D. N. Buragohain, Use of higher order deformation theory in finite element analysis of shear deformable shells. Presented at Inr. Cotzl: on Engq Software ICENSOFT, I.I.T.. Delhi (1989). 20. K. S. Surana and R. M. Sorem. Curved shell elements based on hierarchical p-approximation in the thickness direction for linear static analysis of laminated composites. Int. J. Nrimcr. Meth. Engng 29, 1393-1420 (1990). 21. R. M. Barker, Fu-Tien Lin and J. R. Dana, Three dimensional finite element analysis of laminated plates. Comput.
Strucr. 2, 1013-- 1029 (1972).
22. Y. H. Kim and S. W. Lee, A solid element formulation for large deflection analysis of composite shell structures. Comput. Struct. 30, 269. 274 (1988). 23. 0. C. Zienkiewicz, The Finite Element method, 3rd Edn. McGraw-Hill, London (t 977). 24. K. J. Bathe, Finite Element Proredures in Engineering Analysis. Prentice-Hall of India, New Delhi (1990). 25. S. G. Lekhnitskii, Theory of Elasticity of an Anisotropir Elastic Body. Holden-Day, San Francisco (1963). 26. P. K. Ravichandran, Finite element analysis of layered composite and stiffened shells based on refined theories. Ph.D. thesis submitted to Department of Civil Engineering, I.I.T., Bombay (1992). 27. D. Hitchings, A. Kamoulkos and G. A. 0. Davies, Linear Static Benchmarks. NAFEMS Vol. I. New York (1987). 28. T. E. Witt, A. F. Saleeb and T. Y. Chang, A mixed element for laminated plates and shells. Comput. Struct. 37, 597611 (1990). 29 J. G. Ren, Exact solutions for laminated cylindrical shells in cylindrical bending. Camp. Sci. Tech. 29, 169-187 (1987). 30 S. T. Dennis and A. N. Palazotto, Laminated shell in cylindrical bending, two-dimensional approach vs exact. Am. Inst. Aeronaut. Astronaut. Jnl29, 647-650 (1991).