Reddy 1985
-
Upload
kamineni-jagath -
Category
Documents
-
view
233 -
download
0
Transcript of Reddy 1985
-
8/18/2019 Reddy 1985
1/4
440
AIAA
J O U R N A L
V O L .
23, NO. 3
Nonlinear Analysis of Laminated Shells
Including Transverse Shear Strains
J . N. R eddy* and K. Chandrashekharat
Virginia
Polytechnic
Institute
a nd State University,
Blacksburg, Virginia
Abstract
N
UMERICAL results obta ined us ing a doubly curved,
shear deformable shell element are presented for
geometr ical ly nonlinear anlaysis of laminated composite
shells. The element is based on an extension of Sanders ' shell
theory and accounts for the von Karman strains and trans-
verse
shear strains. The sample numerical results presented
here
for the
geometr ica lly nonlinear analys is
o f
laminated
composite shells should serve
as
references
fo r
fu tu re
in -
vestigations.
Contents
Laminated shells are f inding increased applications in
aerospace, automobile, and petrochemical industries. This is
due
primarily to the high stiffness-to-weight
ratio,
high-
strength-to-weight
ra t io , and lower machining and main-
tenance costs associated with composite structures. However,
the analysis of com posite structures is more com plicated when
compared to meta ll ic s tructures , because laminated composite
structures
are anisotropic and characterized
by
bending-
stretching coup ling. Fu rther, the classical shell theories, whic h
ar e based on the
K i rc h h of f -Love
kinematic hypothesis, are
k n o w n to y ield deflec tions and stresses in laminated shells that
are as much as
3 0%
in error. This error is due to the neglect of
t ransverse shear strains in the classical shell theories.
T he effects o f
t ransverse shear deformat ion
an d
thermal
expansion through
th e
th ickness
fo r
laminated, t ransversely
isotropic, cylindrical shells were considered by Zukas and
Vinson.
1
Dong an d
T so
2
constructed a laminated o rthotropic
shell
theory that includes transverse shear deformation. Other
shear deformation theories, specialized to anisotropic cylin-
drical shells, were presented by Whitney and
Sun .
3
'
4
Recent ly ,
Reddy
5
extended
Sanders '
theory to account for the trans-
verse
shear strains, and presented exact solutions for simply
supported cross-ply laminated shells. All of these studies are
limited to
small displacement theories
a nd
static analyses.
In the present paper , the extended Sanders ' shell theory that
accounts for the shear deformation and the von Karman
strains is used to develop a displacement
finite
element model
fo r
the bending analysis of laminated composite shells.
Numer ica l
results for cylindrical and doubly curved shells are
presented, showing the effect of geometry and material or-
thotropy
on the
def lect ions
an d
stresses.
Consider a laminated shell constructed of a finite n u m b e r
of
uniform -thickness orthotropic layers, or iented arbit rari ly
with
respect
to the
shell coordinates ( ?
7
, b * f ) .
The o r -
thogonal cu rviline ar coordin ate system £ / , £ 2 > D is chosen
such
that the £ / and £
2
curves are lines of curvature on the
midsurface f = 0 , and f curves are straight lines perpendicular
to the
surface
f = 0 .
Received N ov.
29 ,
1983; revision received
M a r c h 1 5,
1984.
Copyr igh t
© Amer i can
Inst i tute
of Aeronaut i cs and As tronaut i cs ,
Inc., 1984.
All
rights
reserved. Ful l paper to be avai lable
from
National Technical Information
Service,
Springfield, Va. ,
at the
s tandard
pr ice.
*Professor ,
D e p a r t m e n t o f
Engineering Science
an d
Mechanics.
jGraduate
Research
A ssistant ,
D epar tment
of Engineer ing Science
an d
M echanics.
The principle of virtual displacements can be used to derive
th e
governing equat ions
5
'
6
:
dN,
i
x
dN
2
—l
dx
2
.dx,
d x
2
d
X ]
where
c
0
=
0.5 \/R
2
- 1
/ / ? / ) ,
and
du
3
dx?
(1 )
T—
dx
2
du
3
du
3
—t
dx
2
(2)
The rel ationship between the stress resultants T V / , M /, Q ,) are
generalized displacements (u , v,
w ,
< / > / , 4 >
2
) can be found in
R e f .
6.
The finite elem ent model of the equation s is of the form
where [A) = ( \ u
l
) , \u
2
} ,
(u
3
J ,
{ < / > / ,
4 >
2
} )
7
, [A] is the
element
stiffness
matrix, and
( F )
is the force vector. The
coefficients
of the stiffness matr ices ar e included in R e f . 6.
Note that th e stiffness matr ix [K ] is a
funct ion
of the
u n k n o w n solution vector ( A ) ; therefore, an iterative solution
procedure is required for each
load
step. In the present study,
the authors have used the direct (i.e., Picard-type) iteration
technique. Sample numerical results for laminated shell
problems are presented in Figs. 1-3. A discussion of these
results
i s
presented
in the
fo llowing paragraphs.
The following types of boundary conditions for a quarter-
panel were used (along the symmetry lines, normal in-plane
displacement , an d normal rotation are set to zero):
= w=(f>
J
=0 at y —
= w = l >
2
= 0 at x=a
4)
Here
u, v,
an d
w
denote th e displacements along th e three
coordinate axes, an d
< >
7
an d < />
2
denote the rotations of the
transverse normals
abou t
the x
2
an d x
I
axes, respectively. T he
fol lowing material properties
ar e
used:
Material 1:
E, =25 x
10
6
psi, E
2
= 10
6
psi,
G
]2
=
G
]3
=0.5 x
10
6
psi
G
23
=
0.2x
10
6
psi, v,
2
= 0.25 (5a)
Materia l
2:
Ej =
40 x
10
6
psi,
E
2
=
10
6
psi,
G
]2
= G
}3
=
0.6 x
10
6
psi
G
23
=0.5 x 10
6
psi,
V] 2
=0 .25 (5b)
-
8/18/2019 Reddy 1985
2/4
MARCH 1985
NONLINEAR
A N A L Y S I S
OF LAMINATED SHELLS
441
Load
P
= F
3 .
0.0 0.2 0.4 0.6 0.
Deflection, -w/h)
Fig.
1 Bending of a nine layer cross ply [0°/90°/0 ..] s imply
supported
spherical shell
subjected t o a uniformly distributed load.
cross-ply
I
Deflection, -w
10
in)
• — angle-ply
0.0 0.1 0.2 0.3 0.4 0.5
3.0 i———i———i———i———i———i———i———i———i———i———|
angle-oly
Load, 2.0
q
n
psi)
0.4
Load ,
ci
nsi)
0.1 0. 2 0.3 0.4
Deflection, w
in in.)
0.0
.5 -»-cross-ply
Fig. 2
Bending
of a two layer cross ply and angle ply s imply sup
ported spherical shells under uniform
loading.
T he
first
problem is concerned with the bending of a nine-
layer [0
0
/90°/0°/90%..] cross-ply spherical
panel
under
external pressure load. T he following geometrical data ar e
used
in the
analysis: RI
=R
2
=R= 1000 in., a = b =5 Q in.,
h =
1 in. Individual layers are assumed to b e of equal thickness
h j = h/9). Results using
tw o
sets
of
orthotropic-material
constants, typical
of
high-modulus graphite-epoxy material
(the
ratios are more pertinent here) for in d iv idua l layers, are
presented
i n Fig . 1.
Figure 2 contains the pertinent data an d results (with
different
scales)
fo r
two-layer cross-ply
an d
angle-ply
simply
supported spherical panels (of material 2). It is interesting to
note that the
type
o f
nonlinearity
exhibited by the two
shells
i s
quite different;
th e
cross-ply
shell
gets
softer,
whereas
th e
arigle-ply
shell gets stiffer with an increase in the applied load.
While both
shells have bending-stretching coupling due to the
laminat ion
scheme (B
22
=
— B
11
nonzero for the cross-ply
shell an d
B
} 6
an d B
2
6 ar e nonzero for the angle-ply shell), the
Fig.
3
Bending
of a
c lamped
cross ply
[ 0 ° / 9 0 ° ] cylindrical shell
under
a
uniform loading.
angle-ply
experiences shear coupling that stiffens the spherical
shell
relatively more than the normal coupling (note that, in
general, shells get
softer
under
externally applied pressure
loads).
Lastly,
Fig.
3 contains
results (i.e.,
w ,
o
y
, o
xz
vs
load)
for a
cross-ply [0°/90°] clamped cylindrica l
panel
under un i form
load. Clearly, the
load
deflection curve indicates
that
th e
nonlinearity exhibited by the cross-ply shell is of the har-
dening
type.
The shear-flexible finite
element
presented here is com-
putat ional ly more efficient than those based on fully o r
degenerated three-dimensional elasticity theory. Since only
the von Karman-type nonlinearity is accounted for here, the
element is not suitable for
applicat ions involving
severe
dis tort ions (i.e., large strains). F rom the numerica l com-
putat ions it is
observed
that
boundary condi t ions
on the in-
plane d isplacements
have
a significant effect on the shell
deflections and stresses. It is
also
observed
that
the
form
of
nonlinearities exhibited by laminated shells depends on the
lamination schemes
(i.e., cross-ply
a nd
angle-ply).
It
wou ld
be
of considerable interest
to
verify these
f indings by ex-
periments .
Acknowledgments
T he
results reported
here were obtained dur ing in -
ves t igations supported by Structura l Mechanics Divisions of
AFOSR and NASA
Lewis .
The authors are
grateful
to Dr.
Anthony A mo s and Dr. Chris Chamis for the support .
References
Zukas , J . A. and Vinson, J. R., "Laminated
Transversely
Isotropic
Cylindrical
Shells, Journa l
of
Applied Mechanics, Vol .
38 ,
J u n e
1971,
pp .
400-407.
2
D o n g , S. B. and Tso,
F .K .W . ,
On a Laminated
Or tho t rop ic
Shell Theory Including
Transverse Shear Deformation,"
Journal
o f
Applied Mechanics, Vol .
39, Dec.
1972,
pp. 1091-1097.
*Whitney,
J. M. and Sun, C. T., A
Higher Order Theory
for
Extensional
Mot ion
of Laminated
Composi tes , "
Journal
o f Sound
a n d Vibrat ion, Vol .
3 0,
Sept.
1973, p. 85.
4
W h i t n e y ,
J . M. and
Sun,
C. T., A Refined Theory fo r
Laminated
Anisotropic,
Cyl indr ical Shells,
Journal
o f Applied
Mechanics, Vol .
41, June 1974, pp.
471-476.
5
R e d d y ,
J . N.,
'Exact
Solut ions of
Moderate ly Th ick Laminated
Shells,
Journal of Engineering Mechanics,
ASCE,
Vol.
110,
M ay
1984, pp. 794-809.
6
R e d d y ,
J.
N. , "Nonl inear Analysis
o f
Layered
Composite Plates
an d Shells, Virginia Poytechnic Inst i tute and
State
Univers i ty,
Blacksburg, Va.,
Final Report to
A F O S R , Rept. VPI-E-84-5,
Jan.
1984.
-
8/18/2019 Reddy 1985
3/4
This article has been cited by:
1. Achchhe Lal, B.N. Singh, Soham Anand. 2011. Nonlinear bending response of laminated composite spherical shell
panel with system randomness subjected to hygro-thermo-mechanical loading. International Journal of Mechanical
Sciences 53:10, 855-866. [CrossRef ]
2. Ireneusz Kreja. 2011. A literature review on computational models for laminated composite and sandwich panels.
Central European Journal of Engineering 1:1, 59-80. [CrossRef ]
3. M.X. Shi, Q.M. Li. 2011. Instability in the transformation between extensional and flexural modes in thin-walled
cylindrical shells. European Journal of Mechanics - A/Solids 30:1, 33-45. [CrossRef ]
4. Surendra Kumar. 2008. Analysis of impact response and damage in laminated composite shell involving large
deformation and material degradation. Journal of Mechanics of Materials and Structures 3:9, 1741-1756. [CrossRef ]
5. Chih-Ping Wu, Yen-Wei Chi. 2005. Three-dimensional nonlinear analysis of laminated cylindrical shells under
cylindrical bending. European Journal of Mechanics - A/Solids 24:5, 837-856. [CrossRef ]
6. M.H. Toorani. 2003. Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells.
International Journal of Non-Linear Mechanics 38:9, 1315-1335. [CrossRef ]
7. Jianwu Zhang, Yanhai Xu, Jinsong Wu. 2003. Buckling and Postbuckling of Antisymmetrically Laminated Cross-ply
Shear-Deformable Cylindrical Shells under Axial Compression. Journal of Engineering Mechanics 129:1, 107-116.
[CrossRef ]8. K. Y. Sze, W. K. Chan, T. H. H. Pian. 2002. An eight-node hybrid-stress solid-shell element for geometric non-linear
analysis of elastic shells. International Journal for Numerical Methods in Engineering 55:7, 853-878. [CrossRef ]
9. A.M. Zenkour, M.E. Fares. 2001. Bending, buckling and free vibration of non-homogeneous composite laminated
cylindrical shells using a refined first-order theory. Composites Part B: Engineering 32:3, 237-247. [CrossRef ]
10. C.-P. Wu, C.-C. Liu. 2001. An Asymptotic Theory for the Nonlinear Analysis of Laminated Cylindrical Shells.
International Journal of Nonlinear Sciences and Numerical Simulation 2:4, 329-342. [CrossRef ]
11. Il-Kwon Oh, In Lee. 2001. Thermal snapping and vibration characteristics of cylindrical composite panels using
layerwise theory. Composite Structures 51:1, 49-61. [CrossRef ]
12. M.H. TOORANI, A.A. LAKIS. 2000. GENERAL EQUATIONS OF ANISOTROPIC PLATES AND SHELLS
INCLUDING TRANSVERSE SHEAR DEFORMATIONS, ROTARY INERTIA AND INITIAL CURVATURE
EFFECTS. Journal of Sound and Vibration 237:4, 561-615. [CrossRef ]13. K. Sandeep, Y. Nath. 2000. Nonlinear Dynamic Response of Axisymmetric Thick Laminated Shallow Spherical
Shells. International Journal of Nonlinear Sciences and Numerical Simulation 1:3. . [CrossRef ]
14. C.W.S. To, B. Wang. 1999. Hybrid strain based geometrically nonlinear laminated composite triangular shell finite
elements. Finite Elements in Analysis and Design 33:2, 83-124. [CrossRef ]
15. S. Ganapathy, K.P. Rao. 1998. Failure analysis of laminated composite cylindrical/spherical shell panels sub jected
to low-velocity impact. Computers & Structures 68:6, 627-641. [CrossRef ]
16. Vijaya Kumar, Anand V. Singh. 1997. Geometrically non-linear dynamic analysis of laminated shells using Bézier
functions. International Journal of Non-Linear Mechanics 32:3, 425-442. [CrossRef ]
17. C.P. Chaplin, A.N. Palazotto. 1996. The collapse of composite cylindrical panels with various thickness using finite
element analysis. Computers & Structures 60:5, 797-815. [CrossRef ]
18. C.S. Xu, Z.Q. Xia, C.Y. Chia. 1996. Non-linear theory and vibration analysis of laminated truncated, thick, conical
shells. International Journal of Non-Linear Mechanics 31:2, 139-154. [CrossRef ]
19. F. Gruttmann, W. Wagner. 1996. Coupling of two- and three-dimensional composite shell elements in linear and non-
linear applications. Computer Methods in Applied Mechanics and Engineering 129:3, 271-287. [CrossRef ]
20. Deokjoo KimReaz A. Chaudhuri. 1995. Full and von Karman geometrically nonlinear analyses of laminated
cylindrical panels. AIAA Journal 33:11, 2173-2181. [Citation] [PDF] [PDF Plus]
21. J.R. Kommineni, T. Kant. 1995. Pseudo-transient large deflection analysis of composite and sandwich shells with a
refined theory. Computer Methods in Applied Mechanics and Engineering 123:1-4, 1-13. [CrossRef ]
22. F. Gruttmann, W. Wagner. 1994. On the numerical analysis of local effects in composite structures. Composite
Structures 29:1, 1-12. [CrossRef ]
23. Mohamad S. Qatu. 1994. On the validity of nonlinear shear deformation theories for laminated plates and shells.
Composite Structures 27:4, 395-401. [CrossRef ]
http://dx.doi.org/10.1016/0263-8223(94)90266-6http://dx.doi.org/10.1016/0263-8223(94)90266-6http://dx.doi.org/10.1016/0263-8223(94)90032-9http://dx.doi.org/10.1016/0045-7825(94)00771-Ehttp://dx.doi.org/10.1016/0045-7825(94)00771-Ehttp://dx.doi.org/10.2514/3.12963http://arc.aiaa.org/doi/pdf/10.2514/3.12963http://arc.aiaa.org/doi/pdf/10.2514/3.12963http://arc.aiaa.org/doi/pdfplus/10.2514/3.12963http://arc.aiaa.org/doi/pdfplus/10.2514/3.12963http://dx.doi.org/10.1016/0020-7462(95)00051-8http://dx.doi.org/10.1016/0045-7949(95)00344-4http://dx.doi.org/10.1016/S0020-7462(96)00068-6http://dx.doi.org/10.1016/S0045-7949(98)00080-7http://dx.doi.org/10.1016/S0168-874X(99)00028-1http://dx.doi.org/10.1515/IJNSNS.2000.1.3.225http://dx.doi.org/10.1016/S0263-8223(00)00123-9http://dx.doi.org/10.1515/IJNSNS.2001.2.4.329http://dx.doi.org/10.1016/S1359-8368(00)00060-3http://dx.doi.org/10.1061/(ASCE)0733-9399(2003)129:1(107)http://dx.doi.org/10.1016/S0020-7462(02)00073-2http://dx.doi.org/10.1016/S0020-7462(02)00073-2http://dx.doi.org/10.1016/0263-8223(94)90266-6http://dx.doi.org/10.1016/0263-8223(94)90032-9http://dx.doi.org/10.1016/0045-7825(94)00771-Ehttp://arc.aiaa.org/doi/pdfplus/10.2514/3.12963http://arc.aiaa.org/doi/pdf/10.2514/3.12963http://dx.doi.org/10.2514/3.12963http://dx.doi.org/10.1016/0045-7825(95)00897-7http://dx.doi.org/10.1016/0020-7462(95)00051-8http://dx.doi.org/10.1016/0045-7949(95)00344-4http://dx.doi.org/10.1016/S0020-7462(96)00068-6http://dx.doi.org/10.1016/S0045-7949(98)00080-7http://dx.doi.org/10.1016/S0168-874X(99)00028-1http://dx.doi.org/10.1515/IJNSNS.2000.1.3.225http://dx.doi.org/10.1006/jsvi.2000.3073http://dx.doi.org/10.1016/S0263-8223(00)00123-9http://dx.doi.org/10.1515/IJNSNS.2001.2.4.329http://dx.doi.org/10.1016/S1359-8368(00)00060-3http://dx.doi.org/10.1002/nme.535http://dx.doi.org/10.1061/(ASCE)0733-9399(2003)129:1(107)http://dx.doi.org/10.1016/S0020-7462(02)00073-2http://dx.doi.org/10.1016/j.euromechsol.2005.04.006http://dx.doi.org/10.2140/jomms.2008.3.1741http://dx.doi.org/10.1016/j.euromechsol.2010.09.002http://dx.doi.org/10.2478/s13531-011-0005-xhttp://dx.doi.org/10.1016/j.ijmecsci.2011.07.008
-
8/18/2019 Reddy 1985
4/4
24. T Kant, J.R Kommineni. 1994. Geometrically non-linear analysis of symmetrically laminated composite and sandwich
shells with a higher-order theory and C0 finite elements. Composite Structures 27:4, 403-418. [CrossRef ]
25. R.S. Alwar, M.C. Narasimhan. 1993. Nonlinear analysis of laminated axisymmetric spherical shells. International
Journal of Solids and Structures 30:6, 857-872. [CrossRef ]
26. Y. Basar, U. Montag, Y. Ding. 1993. On an isoparametric finite-element for composite laminates with finite rotations.
Computational Mechanics 12:6, 329-348. [CrossRef ]
27. K. Chandrashekhara, Kiran M. Bangera. 1993. Linear and geometrically non-linear analysis of composite beamsunder transverse loading. Composites Science and Technology 47:4, 339-347. [CrossRef ]
28. R.S. Alwar, M.C. Narasimhan. 1992. Axisymmetric non-linear analysis of laminated orthotropic annular spherical
shells. International Journal of Non-Linear Mechanics 27:4, 611-622. [CrossRef ]
29. Changshi Xu. 1992. Multi-mode nonlinear vibration and large deflection of symmetrically laminated imperfect
spherical caps on elastic foundations. International Journal of Mechanical Sciences 34:6, 459-474. [CrossRef ]
30. Wojciech Gilewski, Maria Radwa#ska. 1991. A survey of finite element models for the analysis of moderately thick
shells. Finite Elements in Analysis and Design 9:1, 1-21. [CrossRef ]
31. E. Graff, G.S. Springer. 1991. Stress analysis of thick, curved composite laminates. Computers & Structures 38:1,
41-55. [CrossRef ]
32. S.T. Dennis, A.N. Palazotto. 1990. Large displacement and rotational formulation for laminated shells including
parabolic transverse shear. International Journal of Non-Linear Mechanics 25:1, 67-85. [CrossRef ]
33. H.T.Y. Yang, S. Saigal, D.G. Liaw. 1990. Advances of thin shell finite elements and some applications—version I.
Computers & Structures 35:4, 481-504. [CrossRef ]
34. Y.H. Kim, S.W. Lee. 1988. A solid element formulation for large deflection analysis of composite shell structures.
Computers & Structures 30:1-2, 269-274. [CrossRef ]
35. V.P. Iu, C.Y. Chia. 1988. Effect of transverse shear on nonlinear vibration and postbuckling of anti-symmetric cross-
ply imperfect cylindrical shells. International Journal of Mechanical Sciences 30:10, 705-718. [CrossRef ]
36. RAKESH K. KAPANIAT. Y. YANG. 1987. Buckling, postbuckling, and nonlinear vibrations of imperfect plates.
AIAA Journal 25:10, 1338-1346. [Citation] [PDF] [PDF Plus]
37. Paul Seide, Reaz A. Chaudhuri. 1987. Triangular finite element for analysis of thick laminated shells. International
Journal for Numerical Methods in Engineering 24:8, 1563-1579. [CrossRef ]
http://dx.doi.org/10.1002/nme.1620240812http://arc.aiaa.org/doi/pdfplus/10.2514/3.9788http://arc.aiaa.org/doi/pdf/10.2514/3.9788http://dx.doi.org/10.2514/3.9788http://dx.doi.org/10.1016/0020-7403(88)90036-7http://dx.doi.org/10.1016/0045-7949(88)90232-5http://dx.doi.org/10.1016/0045-7949(90)90071-9http://dx.doi.org/10.1016/0020-7462(90)90039-Chttp://dx.doi.org/10.1016/0045-7949(91)90122-3http://dx.doi.org/10.1016/0168-874X(91)90016-Rhttp://dx.doi.org/10.1016/0020-7403(92)90012-6http://dx.doi.org/10.1016/0020-7462(92)90066-Ghttp://dx.doi.org/10.1016/0266-3538(93)90003-Yhttp://dx.doi.org/10.1007/BF00364242http://dx.doi.org/10.1016/0020-7683(93)90044-8http://dx.doi.org/10.1016/0263-8223(94)90267-4