„§EOMETRICALLY NONLINEAR ANALYSIS DFCOMPOSITE LAMINATES USING A REFINED SHEAR
_ DEFORMATIDN SHELL THEORYI
by
Chorng-Fun\LiuH
Dissertation Submitted to the Faculty of theVirginia Polytechnic Institute and State University
in partial fulfillment of the requirementfor the degree of
DOCTOR DF PHILOSOPHY
Ein
Engineering Mechanics
Approved:
llhää IDr. J. N. Re Chairman .; //
;, E / ·
Dr. D. Frederick Dr. L. L r
4. ¢„.—„„\^—m ß [ FE;Dr. L. Meirovitch Dr. D. Post
December 1985
Blacksburg, Virginia
Ü GEOMETRICALLY NONLINEAR ANALYSIS OFZ COMPOSITE LAMINATES USING A REFINED
SHEAR DEFORMATION SHELL THEORY
> by
Chorng—Fuh LiuF AßsiRAcT
The theory is based on an assumed displacement field, in which the
surface displacements are expanded in powers of the thickness coordinate
up to the third order. The theory allows parabolic description of the
transverse shear stresses, and therefore the shear correction factors of
the usual shear deformation theory are not required in the present
theory. The theory accounts for small strains but moderately large
displacements (i.e., von Karman strains). Exact solutions for certain
cross-ply shells and finite-element models of the theory are also
developed. The finite-element model is based on independent
approximations of the displacements and bending moments (i.e., mixed
formulation), and therefore only C°-approximations are required.
· Further, the mixed variational formulations developed herein suggest
that the bending moments can be interpolated using discontinuous
approximations (across interelement boundaries). The finite element is
used to analyze cross-ply and angle-ply laminated shells for bending,
vibration and transient response. Numerical results are presented to
show the effects of boundary conditions, lamination scheme (i.e.,
bending-stretching coupling and material anisotropy) shear deformation
and geometric nonlinearity on deflections and frequencies. Many of the
numerical results presented here for laminated shells should serve as
references for future investigations.
iii
Table of Contents
Page
ABSTRACT......................................................... ii
ACKNOWLEDGEMENTS................................................. iii
1. INTRODUCTION................................................. 1
1.1 Background.............................................. 1
1.2 Review of the Literature on Shells...................... 4
1.3 Present Study........................................... 8
2. FORMULATION OF THE NEW THEORY................................ 10
2.1 Kinematics.............................................. 10
2.2 Oisplacement Field...................................... 12
2.3 Strain-Oisplacement Relations........................... 16
2.4 Constitutive Relations.................................. 17
2.5 Equations of Motion..................................... 19
3. THE NAVIER SOLUTION FOR CROSS-PLY LAMINATEO SHELLS........... 24
3.1 Introduction............................................ 24
3.2 The Navier Solution..................................... 26
4. MIXED VARIATIONAL PRINCIPLES................................. 28
4.1 Introduction............................................ 28
4.2 Mixed Variational Principles............................ 29
5. FINITE ELEMENT MODELS........................................ 34
5.1 Introduction............................................ 34
5.2 Finite-Element Oiscretization........................... 34
5.3 Finite-Element Model.................................... 37
5.4 Solution Procedure...................................... 40
iv
Table of Contents QContinued)
_ Page
6. SAMPLE APPLICATIONS.......................................... 46
6.1 Introduction............................................ 46
6.2 Exact Solutions...................................._..... 46
6.3 Approximate (Finite-Element) Solutions.................. 54
6.3.1 Bending Analysis................................. 54
6.3.2 Vibration Analysis............................... 66
6.3.3 Transient Analysis............................... 71
7. SUMMARY AND RECOMMENDATIDNS.................................. 76
7.1 Summary and Conclusions................................. 76
7.2 Some Comments on Mixed Models........................... 77
7.3 Recommendations......................................... 77
REFERENCES....................................................... 79
APPENDIX A: Coefficients of the Navier Solution................. 86
APPENDIX B: Stiffness Coefficients of the Mixed Model........... 89
VITA............................................................. 104
V
CHAPTER 1
INTRODUCTION
1.1 Background
The analyses of composite laminates in the past have been based on
one of the following two classes of theories:
(i) Three-dimensional elasticity theory
(ii) Laminated plate theories
In three—dimensional elasticity theory each layer is treated as an
elastic continuum with possibly distinct material properties in adjacent
layers. Thus the number of governing differential equations will be 3N,
where N is the number of layers in the laminate. At the interface of
two layers the continuity of displacements and stresses give additional
relations. Solution of the equations becomes intractable as the number
of layers becomes large.
In a 'laminated plate theory', it is assumed that the laminate is
in a state of plane stress, the individual laminae are elastic, and
there is perfect bonding between layers. The laminate properties (i.e.
stiffnesses) are obtained by integrating the lamina properties through
the thickness. Thus, laminate plate theories are equivalent single-
layer theories. In the ’classical laminate plate theory' (CLPT), which
is an extension of the classical plate theory (CPT) to laminated plates,
the transverse stress components are ignored. The classical laminate
plate theory is adequate for many engineering problems. However,
laminated plates made of advanced filamentary composite materials, whose
'elastic to shear modulus ratios are very large, are susceptible to
thickness effects because their effective transverse shear moduli are
I
2
significantly smaller than the effective elastic moduli along fiber
directions. These high ratios of elastic to shear modulus render the
classical laminated plate theory inadequate for the analysis of thick
composite plates.
The first, stress-based, shear deformation plate theory is due to
Reissner [1-3]. The theory is based on a linear distribution of the
inplane normal and shear stresses through the thickness,
<¤>(h /6) (h /6) (h /6)
where (61,62) and 66 are the normal and shear stresses, (Ml,M2) and M6
are the associated bending moments (which are functions of the inplane
coordinates x and y), z is the thickness coordinate and h is the total
thickness of the plate. The distribution of the transverse normal and
shear stresses (63, 64 and 65) is determined from the equilibrium
equations of the three-dimensional elasticity theory. The differential
equations and the boundary conditions of the theory were obtained using
Castigliano‘s theorem of least work.
The origin of displacement-based theories is apparently attributed
to Basset [4l. Basset assumed that the displacement components in a
shell can be expanded in a series of powers of the thickness
coordinate 5. For example, the displacement component ul along
the 51 coordinate is written in the form
(1.2ö)
where 51 and 52 are the curvilinear coordinates in the middle surface of
the shell, and uän) have the meaning
3 .
n0§")(;1,;2) = , n = 0,1,2,... (1.2b)
d; ;=0
Basset's work did not receive as much attention as it deserves. In a
1949 NACA technical note, Hildebrand, Reissner and Thomas [5] presented
(also see Hencky [6]) a displacement-based shear deformation theory for
shells, which obviously can be specialized to flat plates. They assumed
the following displacement field,
¤l(a1.+;2.c) = ¤(6l,«:2) + c ¢X(-21,62)¤2(;1.+;2.c) = 1/(61,62) + c ¢y(»;1.-:2)
¤3(+;1.62.c) = w(a1.62) (1-3)The differential equations of the theory are then derived using the
principle of minimum total potential energy. This gives five
differential equations in the five displacement functions, u, v,
w, ox, and ay.
The shear deformation theory based on the displacement field in Eq.
(1.3) for plates is often referred to as the Mindlin's plate theory.
Mindlin [7] presented a complete theory for isotropic plates based on
the displacement field (1.3) taken from Hencky [6]. We shall refer to
the shear deformation theory based on the displacement field (1.3) as
the first-order shear deformation theory.
A generalization of the first-order shear deformation plate theory
for homogeneous isotropic plates to arbitrarily laminated anisotropic
plates is due to Yang, Norris, and Stavsky [8] and Whitney and Pagano
[9]. Extensions of the classical von Karman plate theory [10] to shear
4
deformation theories can be found in the works of Reissner [11,12]
Medwadowski [13], Schmidt [14] and Reddy [15].
Higher-order, displacement-based, shear deformation theories have
been investigated by Librescu [16] and Lo, Christensen and wu [17]. In
these higher-order theories, with each additional power of the thickness
coordinate an additional dependent unknawn is introduced into the
theory. Levinson [18] and Murthy [19] presented third-order theories
that assume transverse inextensibility. The nine displacement functions
were reduced to five by requiring that the transverse shear stresses
vanish on the bounding planes of the plate. However, both authors used
the equilibrium equations of the first—order theory in their analysis.
Stating in other words, the higher-order terms of the displacement field
are accounted for in the calculation of the strains but not in the
governing differential equations or in the boundary conditions. These
theories can be shown (see Librescu and Reddy [20]) to be the same as
those described by Reissner [1-3]. Recently, Reddy [21,22] developed a
new third—order plate theory, which is extended in the present study to
laminated shells.
1.2 Review of the Literature on Shells
Shell structures are abundent in structures around us on the earth
and in space. Use of shell structures dates back to ancient Rome, where
the roofs of the Pantheon can be classified today as thick shells.
Shell structures have been built by experience and intuition. No
‘1ogical and scientific study has been conducted on the design of shells
until the eighteenth century.
5
The earliest need for design criterion for shell structures
probably came with the development of the steam engine and its attendent
accessories. However, it was not until 1888, by Love, that the first
general theory was presented. Subsequent theoretical efforts have been
directed towards improvements of Love's formulation and the solutions of
the associated differential equations.
An ideal theory of shells require a realistic modeling of the
actual geometry and material properties, and an appropriate description
of the deformation. Even if we can take care of these requirments and
set up the governing equations, analytical solutions to most shell
problems are nevertheless limited in scope, and in general do not apply
to arbitrary shapes, load distributions, and boundary conditions.
Consequently, numerical approximation methods must be used to predict
the actual behavior.
Many of the classical theories were developed originally for thin
elastic shells, and are based on the Love—Kirchhoff assumptions (or the
first approximation theory): (1) plane sections normal to the
undeformed middle surface remain plane and normal to the deformed middle
surface, (2) the normal stresses perpendicular to the middle surface can
be neglected in the stress-strain relations, and (3) the transverse
displacement is independent of the thickness coordinate. The first
assumption leads to the neglect of the transverse shear strains.
Surveys of various classical shell theories can be found in the works of
Naghdi [23] and Bert [24,25l. These theories, known as the Love's first
approximation theories (see Love [26]) are expected to yield
sufficiently accurate results when (i) the lateral dimension-to-
6
thickness ratio (a/h) is large; (ii) the dynamic excitations are within
the low-frequency range; (iii) the material anisotropy is not severe. _
However, application of such theories to layered anisotropic composite
shells could lead to as much as 30% or more errors in deflections,
stresses, and frequencies.
As pointed out by Koiter [27], refinements to Love's first
approximation theory of thin elastic shells are meaningless, unless the
effects of transverse shear and normal stresses are taken into account
in the refined theory. The transverse normal stress is, in general, of
order h/R (thickness to radius ratio) times the bending stresses,
whereas the transverse shear stresses, obtained from equilibrium
conditions, are of order h/a (thickness to the length of long side of
the panel) times the bending stresses. Therefore, for a/R < 10, the
transverse normal stress is negligible compared to the transverse shear
stresses.
Ambartsumyan [28,29] was considered to be the first one to analyze
laminates that incorporated the bending—stretching coupling. The
laminates that Ambartsumyan analyzed were now known as laminated
orthotropic shells because the individual orthotropic layers were
oriented such that the principal axes of material symmetry coincided
with the principal coordinates of the shell reference surface. In 1962,
Dong, Pister and Taylor [30] formulated a theory of thin shells
laminated of anisotropic shells. Cheng and Ho [31] presented an
analysis of laminated anisotropic cylindrical shells using Flugge‘s
shell theory [32]. A first approximation theory for the unsymmetric
deformation of nonhomogeneous, anisotropic, elastic cylindrical shells
7
was derived by widera and his colleagues [33,34] by means of asymptotic
integration of the elasticity equation. For a homogeneous, isotropic
material, the theory reduced to the Donnell's equation. An exposition
of various shell theories can be found in the article by Bert [35].
The effect of transverse shear deformation and transverse isotropy
as well as thermal expansion through the shell thickness were considered
by Gulati and Essenberg [36] and Zukas and Vinson [37]. Dong and Tso
[38] presented a theory applicable to layered, orthotropic cylindrical
shells. whitney and Sun [39] developed a higher order shear deformation
theory. This theory is based on a displacement field in which the
displacements in the surface of the shell are expanded as linear
functions of the thickness coordinate and the transverse displacement is
expanded as a quadratic function of the thickness coordinate. Recently,
Reddy [40] presented a shear deformation version of the Sanders shell
theory for laminated composite shells. Such theories account for
constant transverse shear stresses through thickness, and therefore
require a correction to the transverse shear stiffness.
As far as the finite element of shells is concerned, the early
works can be attributed to those of Dong [41], Dong and Selna [42],
wilson and Parsons [43], and Schmit and Monforton [44]. The studies in
[41-44] are confined to the analysis of orthotropic shells of
revolution. Other finite element analyses of lamianted anisotropic
composite shells include the works of Panda and Natarajan [45],
Shivakumar and Krishna Murty [46], Rao [47], Seide and Chang [48],
Venkatesh and Rao [49], Reddy and his colleagues [50-53], and Noor and
his colleagues [54-56].
8
1.3 Present Study
while the three—dimensional theories give more accurate results
than the lamination (classical or (see Seide [57], Srinivas et al.
[58,59] and Pagano, et al. [60-61]) shear deformation) theories, they
are intractable. For example, the 'local' theory of Pagano [62] results °
in a mathematical model consisting of 23N partial differential equations
in the laminate's midplane coordinates and 7N edge boundary conditions,
where N is the number of layers in the laminate. The computational
costs, especially for geometrically nonlinear problems or transient
analysis using the finite element method, preclude the use of such a
theory. As demonstrated by Reddy [21,22] and his colleagues [63-65],
the refined plate theory provides improved global response estimates for
deflections, vibration frequencies and buckling loads for laminated
composite plates. The present study, motivated by the above findings,
deals with the extention of the third-order plate theory of Reddy
[21,22] to laminated composite shells. The theory also accounts for the
von Karman strains. The resulting theory contains, as special cases,
the classical and first—order theories of plates and shells. Mixed
variational formulations and associated finite—element models are
developed in this dissertation. The significant and novel contributions
of the research conducted for this dissertation are:
1. The Formulation of a new third-order theory of lamianted shells
that accounts for a parabolic distribution of the transverse
shear stresses and the von Karman strains.
9
2. The derivation of the exact solutions of the new theory for
certain simply supported laminated composite shells.
3. The development of a mixed variational principle for the new
theory of shells, which yields as special cases those of the
classical (e.g., Love—Kirchhoff) and the first order theory. _
4. The development of a mixed, C°—finite—element and its
application to the bending, vibration and transient analysis of
laminated composite shells.
The theoretical formulation of the new shell theory is presented in
Chapter 2, and the exact solutions for certain shells are presented in
Chapter 3. Chapter 4 contains the development of a mixed variational
principle, and the associated finite element model is described in
Chapter 5. Chapter 6 contains the discussion of numerical results of
the bending, vibration and transient analysis of laminated composite
plates and shells. Comparison of the present results with the
experimental results for certain unidirectional and bidirectional
' composites show excellent agreement. Many additional numerical results
of laminated composite shells are also included, which should serve as
references for future investigations. A summary, conclusions and
recommendations of the present study are presented in Chapter 7.
CHAPTER 2
FORMULATION OF THE NEW THEORY
2.1 Kinematics
Let (51,22,;] denote the orthogonal curvilinear coordinates (or
shell coordinates) such that the gl- and 52- curves are lines of
curvature on the midsurface ;=0, and ;- curves are straight lines
perpendicular to the surface ;=0 . For cylindrical and spherical shells
the lines of principal curvature coincide with the coordinate lines.
The values of the principal radii of curvature of the middle surface are
denoted by Rl and R2.
The position vector of a point on the middle surface is denoted
by r, and the position of a point at distance ; from the middle surface
is denoted by R. The distance ds between points [51,52,0) and [5l+dg1,
g2+d52,0) is determined by (see Fig. 2.1)
(ds)2 = dr - dr
= ¤§(¤2,)2 + «§(¤3212 (2.1)af
where dr = gldgl + gzdgz, the vectors gl and gz [gi = SE;) are tangent‘ to the 51 and 52 coordinate lines, and al and az are the surface metrics
“ä ‘gl ° gl · °§ ‘
Q2 ° Q2· (2·2)The distance dS between points (21,52,;) and [;1+dg1,;2+dg2,;+d;] is
given by
(dS)2 = dR - dR
(2.3)
10
ll
‘c
R1R2
(a)R ß
%·h_ A
Rlg 2
(b)
c°1d€1° d£2
00 N l{ X "15 N
N .111) N|]|( N6R IB I °¤N N2M5 M2
(c)
Figure 2.l Geometry and stress resultants of a shell
12
aß aß aßwhere dß = SE;
dgl + SEEdg? + Sg dg, and L1, L2 and L3 are the Lame'
coefficients
' Q-=
$1 :
‘
L1 - a1[1 + R1) , L2 a2(l + R2] , L3 1 (2.4)
_ aR aRIt should be noted that the vectors G ; —;— and G ; —l— are parallel~1 agl ~2 agz
to the vectors gl and gz, respectively.
2.2 Displacement Field
The shell under consideration is composed of a finite number of
orthotropic layers of uniform thickness (see Fig. 2.2). Let N denote
the number of layers in the shell, and ck and gk_l be the top and bottom
g-coordinates of the k—th layer. Before we proceed, a set of
simplifying assumptions that provide a reasonable description of the
behaviors are as follows:
1. thickness to radius and other dimensions of shell are small.
2. transverse normal stress is negligible
3. strains are small, yet displacement can be moderately large
compared to thickness
Following the procedure similar to that presented in [21] for flat
plates, we begin with the following displacement field:
V(sl„s2,c,t) = (l + %;]v + c¢2 + czwz + cßöz
W(6l,62,c„t) = w (2-5)
where t is time, (ü,v,wj are the displacements along the (g1,g2,;]
13
--Jvi
(U :o\ g/‘ E5
D ?Z•
äo1 EN oxu: E
.C‘
E#5- g
¤¤
In "— · >~„Q \.
*I"""”.2
oI I ÜI I GI| I If*_____I,.- 01
' SI 2I I :1 ä 9
N*-U
14
coordinates, (u,v,w) are the disp1acements of a point on the midd1e
surface and 61 and 62 are the rotations at 5 = 0 of norma1s to the
midsurface with respect to the 52 and 51 -axes, respective1y. A11 of
(u,v,w,6l,62,w1,w2,6l,62) are functions of 51 and 52 on1y. The
particu1ar choice of the disp1acement fie1d in Eq. (2.5) is dictated by
the desire to represent the transverse shear strains by quadratic
functions of the thickness coordinate, 5, and by the requirement that
the transverse norma1 strain be zero. The kinematics of deformation of
a transverse norma1 in various theories is shown in Fig. 2.3.
The functions wi and ai wi11 be determined using the condition that
the transverse shear stresses, 613 = 65 and 623 = 64 vanish on the top
and bottom surfaces of the she11:
5 1* 2*‘
2* * 4 1* 2*‘
2*‘6 (5 5 + Ü t] = 0 6 (5 5 + E t] = 0 (2 6)
These conditions are equiva1ent to, for she11s Taminated of orthotropic
Tayers, the requirement that the corresponding strains be zero on these
surfaces. The transverse shear strains of a she11 with two principa1
radii of curvature are given by
Gl agl R1
65 + 62 652 R2 (2°7)
-X_ 1 2 Lil'.,‘ R2 T 92 T 2992 T 39 92 T 62 652 R2
Setting 55(5l,52,1 g ,t} and 54(5l,52, 1 g ,t} to zero, we obtain
15 1 ·
—-• X1u11¤61=oRmEo
WU
Jk 1>e1=o1=11~11a¤ 111 CLASSICAL
(KIRCHHOFF) meoav
1Jß oeroameo IN THE msi
01211611 THEORY
¢>—\dDEFORMED 111 me THIRDowner: (PRESENT) 1111201211
Figure 2.3 Assumed deformation patterns of the transverse normals
in various dispiacement-based theories.
16
Lvlzlbzzo
- L LL91 -_
3h2(¢l + Gl aal]
- L LL62 - —3hz
(¢2 + az agz] (2.8)
Substituting Eq. (2.8) into Eq. (2.5), we obtain
R1 1 3hZ 1 al 651
V = (1 + $—]v + 56 + 53 -£— [—¢ — L- gw-} (2.9)R2 2 3hZ 2 az 652
This displacement field is used to compute the strains and
stresses, and then the equations of motion are obtained using the
dynamic analog of the principle of virtual work.
2.3 Strain—Disglacement Relations
Substituting Eq. (2.9) into the strain-displacement relations
referred to an orthogonal curvilinear coordinate system, we obtain
E1: gg + + .
= 62 + ;2<ä
€6=¤°§*¤2‘§
E6 = EE (2,10)
where
17
0 au w 1 aw 2 0 **1 2 4 (**1*2w]
6=—·———+—+—(—) ,1c =—,•c =-i—-+———1 axi R1 2 axl 1 axl 1 3h2 axl ax?
0 av w l aw 2 0 **2 2 4 **2azw
0 aw 1 4 aw~
6 = ¢ + ——— 6 = -—— [4 + ——·)
4 2 axz 4 h2 2 axz
¤ - 21. 1 - L EL*6 ’ *1‘“
axl **6 ‘ ‘ h2 (*1+ axl]
84 a¢
1 2 1 2 1 2
2 _ 4 **2 **1 azw
Here xi denote the cartesian coordinates (dxi = aidgi, i = 1,2).
2.4 Constitutive Relations
The stress—strain relations for the k—th lamina in the lamina
coordinates are given by
· (k) — (k)U1 *11 *12 * * * *1· (k)62 022 0 0 0 62
- · (k)*6 ‘ *66 * * *6*4
”"‘“‘· *440*) Eaéßq*4— (k)*6 (ki *66 *6 (k)
(2.12)
where Öij(k) are the plane stress reduced stiffnesses of the k—th lamina
18
in the lamina coordinate system. The coefficients 0ij can be expressed
in terms of the engineering constants of a lamina:
Ö =E1
Ö = °12E2 = V2lE111 1-612621 12 1-612621 1-612621
6 — E2 <213>22 1-612621 °
044 = G23 066 ‘ G13 066 ‘ 012To determine the laminate constitutive equations, Eq. (2.12) should be
transformed to the laminate coordinates. we obtain
°1 011 012 016 61 O Q Q E_ 4 _ 44 45 4°2 ° 012 022 026 02 {O} ‘[QQ { } (202)
Q Q Q € 5 (k) 45 55 (k) 05 (k)06 (k) 16 26 66 (k) 6 (k)’
where
011 = 011 COS49 + 2(0l2 + 2066)sin26 COS29 + 022 sin46
012 = (011 + 022 - 4066)sin26 COS26 + 012(sin46 + cos46)
022 = 011 sin46 + 2(0l2 + 2066)sin26 cos26 + 022 COS46
016 = (011 — 012 — 2066)sin6 COS39 + (012 - 022 + 2066)sin36 cos6
026 = (011 - 012 - 2066)sin26 cos6 + (012 — 022 + 2066)sin6 COS38_ - - - — . 2 2 - . 4 4
066 - (011 + 022 —2012
— 2066)s1n 6 cos 6 + 066(s1n 6 + cos 6)
64. = 644 @$29 9 066 $*9129045 = (055
— 044)cos6 sin6
19
0 = Ö cosza + 0 sin26 (2 15)55 55 44 '
2.5 Eguations of Motion
The dynamic version of the principle of virtual work for the
present case yields _
t h/2O 02 <Sc$k)+ 06 66ék)+ 04 66:51k)+ 05 öcék)]
o —h/2 Q
h/2 (k) L L L 2dx1dx2}d; - f Q6WdXldX2 - 6 f {f p[[u)2+(v]2+(w) |dx1dx2}d;]d1Q —h/2 Q
t o o 2 o o= fo{fQ[[N1661 + M16¤1 + P16«1 + N2662
q6w]
+ ((116 + 1261 - 13 + (11Q + 1162 - Tä gäpv
" _ 66 _ " _ 66 161 " " „BX1 1 2 2 9h“ 8Xi axé
_ „ _ „ _ Q1 _1„ _ „ _1 iQ- 1+ (12u + 1461 - 15 EYI)661 + (12v + 1Ä62 -15(2.16)
where q is the distributed transverse load, N1, M1 etc. are the
resultants,
n Ck(N1,M1,P1) = 1; f ¤$k)(1,;,;3)d; (1 = 1,2,6)‘<=1
;k—16n k »
(0 „K ) = 2 f ¤(k)(l.c2)dc1 1 k=1 C 5 _
k—1 -
20
n Ck(02,uz) = z ( ¤$k)(1,;2)d; (2.17)
k’I Ck-1The inertias Ii and 1% (1 = 1,2,3,4,5) are defined by the equations,
- _ Ä- ,I1 ‘ I1 4 ul I2 ·— 2I'=I+—-I2 1 R2 2
— 1 4 4I = I + —— I — ——— I — ————— I ,2 2 R1 3 3h2 4 3h2R1 5
— 1 4 4I‘=I+—I -;I ————I2 2 R2 3 3h2 4 3h2R2 5
- 4 4I=—I+-—I,3 anz 4 anzul 3— 4 4
II'=·—I+———I3 anz 4 anzuz 3
— 8 16I=I-—-I+—I4 3 3h2 5 9h4 7
— 8 16I' = I - ——— I + ——— I4 3 3h2 5 9h4 7
— 4 16I=-——I——I,5 3h2 5 9h4 7
— 4 16I' = ——- I -——— I (2.18a)5 3h2 5 9h4 7
and
21
"Ck (k) 2 2 4 6(11,12,17),14,15,17) = 2 f 6 (1,;,; ,; ,; ,; )d; (2.186)K’l Ck—1
The governing equations of motion can be derived from Eq. (2.16) by
integrating the displacement gradients by parts and setting the
coefficients of 6u,6v,6w and 661 (i = 1,2) to zero separately:
6N 6N "..1 ..2.—" —" _— L6U' axl + 6x2 Ilu + I2°1 I3 axl
6N 6N "
""· 611 T 6612 ‘ Iiv * ‘2°2 ‘ I2 6612
601 602 4 6Kl 6K2 462Pl 62P2 62P6
l 2 h 1 2 3h axl axz 1 2N N " 3111 " 64} ..12'Ä;
‘Ä; + NKW) ” I3 axl Is axl + Iä axz + Is 6x2 + Ilw
9h4 612 6..2“
1 26M 6M 6P 6P ".1.3 2. LL.£-—"—i‘ —Lö¢1· 6;; + 612 · °1 + h2 K1 · 3hz (611 * 6x,) · Izu * I4¢1 · I6 6116M 6M 6P 6P "
• 6 Ä L _L..Äl="¤”
"•”_“¤H'..Q2 T h2 K2 3h2 (6xl + 6612) I2" T I4¢2 I6 6612(2.19)
where
- L L L L L LNKW) —SX1 (N1 SX1 + N6 SX2) + SX2 (N6 SX1 + N2 SX2) (2°20)
The essential (i.e., geometric) and natural boundary conditions of the
theory are given by:
22
Specify:
un, uns, w, gg, gg, an, ans (essential)
Nn, Nns, an, En, BS, Mn, Mns (natural) (2.21)where _
Nn = niul + nöuz + znxnyuö
nns = (N2 -Nl)nxnypn
= nipl + näpz + znxnypn
png = (P2 -Pl)nXnyMn
= n§Ml + n§M2 + 2nXnyM6V
Mns = (M2 -Ml)nXnyQn
= nxql + nyQ2
Kn = nxKl + nyK2
An = Mn - gi? pn
pns (2.22)
23
and nx and ny are the direction cosines of the unit normal on theboundary of the laminate.
The resultants can be expressed in terms 0f the strain components
using Eqs. (2.10) and (2.12) in Eq. (2.14). we get
N. = A.. 9 6.. 9 E., 9J JJEJ + JJKJ + JJ‘J
M. = 6.. 9 6.. 9 F.. 9J ‘¤‘J + J¤"¤ + ‘J‘J <i.1=1.2.6>(2.23)_ 0 0 2Pi — Eijej + Fijxj + Hijncj
_ 0 1Q2 — Adjej + 04jxj
= A ..Q15JEJ + D5J"J (_J = 4,5) (2.24)
_ 0 1K2 — Dajaj + Fajxj_ 0 1K1- Ü5ej + F5j•<j
where Aij, Bij, etc. are the plate stiffnesses,
(A..,B..,D..,E..,F..,H..)_ JJ JJ JJ JJ JJ JJ
cn k= E I 001) (l„c.c2„c3„c4„c6)dc (2-25)
k’l Ck-11J
for i,j = 1,2,4,5,6.
CHAPTER 3
THE NAVIER SOLUTION FOR CROSS-PLY LAMINATED SHELLS
3.1 Introduction
Exact solutions of the partial differential equations (2.19) on
arbitrary domains and for general conditions is not possible. However,
Ifor simply supported shells whose projection in the xlxz-plane is a
rectangle, the linear version of Eq. (2.19) can be solved exactly,
provided the lamination scheme is of antisymmetric cross-ply [O°/90°/
0°/90°...] or symmetric cross-ply [O°/9O°...]S type. The Navier
solution exists if the following stiffness are zero [21]:
A16 = B16 = U16 = Eiö = Fiö = Hiö = O , (T = 1,2) (3.1)Aas ‘ Das ‘ Pas ‘ 0
The boundary conditions are assumed to be of the form,
U(xl!0) = = = = 0
w(xl,O) = w(xl,b) = w(O,x2) = w(a,x2) = O
aw - Ä!. = E!. = Ä!. =5;- (xlso) ' ax (xlsb) ax (osxz) ax (aexz) 01 1 2 2
N2(x1,O) = N2(xl,b) = N1(O,x2) = Nl(a,x2) = O
M2(x1,O) = M2(x1,b) = M1(O,x2) = Ml(a,x2) = O (3.2)
P2(x1,O) = P2(xl,b) = Pl(O,x2) = Pl(a,x2) = O
0
where a and b denote the lengths along the xl- and x2—directions,
respectively (see Fig. 3.1).
24
25 y
*2
X3\ F;
Figure 3.l The geometry and the coordinate system for a·projecti"on of a shell element
. 263
3.2 The Navier Solution
Following the Navier solution procedure (see Reddy [21]), we assume
the following solution form that satisfies the boundary conditions in
Eq. (3.2):
U =m ä=l
Umnf1(x1’X2)
v =m ä=l
Vmnf2(xl,x2)
w = z N f (x ,x ) (3.3)mn 3 1 2
*1m,n=1
*2wherefl(x1,x2) = cosaxlsinßxz, f2(xl,x2) = sinaxlcosßxz
(3.4)f3(xl,x2) = sinaxlsinßxz, ¤ = mn/a, 6 = nn/b
Substituting Eq. (3.3) into Eq. (2.19), we obtain
Umn Umn O
Vmn Vmn 0
[Ml wmn + [C] Vmn = Qmn , for any m,n"1 1°mn °mn 0
oän ¢än 0 (3.5)where Qmn are the coefficients in the double Fourier expansion of the
transverse load,
27
q(x,,x2) =m,E=l
Qmnf3(x,,x2) (3.6)
and the coefficients Mij and C,j(i,j = 1,2,...,5) are given in Appendix
A.
Equation (3.5) can be solved for Umn,Vmn, etc., for each m and n,
and then the solution is given by Eq. (3.3). The series in Eq. (3.3)
are evaluated using a finite number of terms in the series. For free
vibration analysis, Eq. (24) can be expressed as an eigenvalue equation,
(IC) - ·»2lMl){A} = {0) (3-7)where {A} = and u is the frequency of natural
vibration. For static bending analysis, Eq. (3.5) becomes
00{CMA} = $0,,,,,; (3-8)0
_ 0
CHAPTER 4
MIXED VARIATIONAL PRINCIPLES
4.1 Introduction
The variational formulations used for developing plate and shell
bending elements can be classified into three major categories: (i)
formulations based on the principle of virtual displacements (or the
principle of total potential energy), (ii) formulations based on the
principle of virtual forces (or the principle of complementary energy)
and (iii) formulations based on mixed variational principles(Hu—washizu-
Reissner's principles). The finite-element models based on these
formulations are called, respectively, the displacement models,
eguilibrium models and mixed models.
In the principle of virtual displacements, one assumes that the
kinematic relations (i.e., strain—displacement relations and geometry
boundary conditions) are satisfied exactly (i.e., point-wise) by the
displacement field, and the equilibrium equations and force boundary
conditions are derived as the Euler equations. No a—priori assumption
concerning the constitutive behavior (i.e., stress-strain relations) is
necessary in using the principle. The principle of (the minimum) total
potential energy is a special case of the principle of virtual
displacements applied to solid bodies that are characterized by the
strain energy function U such that
.1%".aeij ij
where aij and Gij are the components of strain and stress tensors,
' respectively, when the principle of total potential energy is used to
28
29
develop a finite-element model of a shell theory, the kinematic
relations are satisfied point-wise but the equilibrium equations are met
only in an integral (or variational) sense. The principle of
complementary potential energy, a special case of the principle of
virtual forces, can be used to develop equilibrium models that satisfy
the equilibrium equations point-wise but meet the strain compatibility
in a variational sense.
There are a number of mixed variational principles in elasticity
(see [66-73]). The phrase 'mixed' is used to imply the fact that both
the displacement (or primal) variables and (some of the) force (or dual)
variables are given equal importance in the variational formulations.
The associated finite—element models use independent approximations of
dependent variables appearing in the variational formulation. There are
two kinds of mixed models: (i) models in which both primal and dual
variables are interpolated independently in the interior and on the
boundary of an element, (ii) models in which the variables are
interpolated inside the element and their values on the boundary are
interpolated by the boundary values of the interpolation. The first
kind are often termed hybrid models, and the second kind are known
simply as the mixed models. In the present study the second kind (i.e.,
mixed model) will be discussed.
4.2 Mixed Variational Principles
To develop a mixed variational statement of the third-order
laminate theory, we rewrite Eq. (2.23) in matrix notation as follows:
{N} = l^*}{¤°} + {B*{{N} + lE*l{F’}
30
T—{·<-} = 18*1 {@0} - [D*l{M} — (F*l{P} (4-1)T T-{¤} = lE*l {¤°} · {F*l {M} - lH*l{P}
where
EBX
¤¢ . ·{-1 53431 8¢2WTS?
ä M 1;+ 1 (M)?BXZ ax R1 2 ax
2 1 aw 2{°T= M.T°l=‘W2 E ay R2 2 ay2a w au av aw aw2 axay Sy + Fx + Ex Sy
4 4c = ——— and c = ——1 3h2 _ 2 h2
{Bl T {D} {Fl -1 {Bl _[A*] = [A] - (symmetric)
[El {F1 {H} [El
[B*] = [B][D*] + [E][F*] (ngt symmetric)
{6*1 = {BllF*l + {EllH*l (gg symmetric)
{0*} {F*l {D} {FI -1s (symmetric) (4.2)[FTI lH*l {Fl {Hl
Note that in this part of the dissertation we are using the notation x =
xl and y = x2.
Now we proceed to develop a mixed variational statement of the
third-order laminate theory. In the present mixed formulation, we treat
31
the generalized displacements (u, v, w, ¢1, ¢2) and generalized moments
(M1, M2, M6, P1, P2, P6) as the dependent variables.
The mixed functional associated with the static version of Eqs.
(2.19), (4.1) is given by [A = (u, v, w, ¢1, ¢2, M1, M2, M6, P1, P2,
_ 1 o T*
o o T* *¤Hl(^) -f {5 {6 } {A l{6 } + {6 } (TB l{M} + {E l{P})
Q
T T 1 T+ {M} {KS} - {M} {F*l{P} · 5 {M} lD*l{M}
T 1 T+ {P} {•<} - 5 {P} {H*l{P}
+5 {?°}TlÄl{?°} · qW}dXdY
- f (Nnun + Nsus + Önw + Mn¢n + MS¢S)ds (4.3)P
where
Qn T Qlnx T Q2ny
un = unx + vny , us = -uny + vnx
2 - 2. 2.. 2. - 2. _ 2.sa
T nx ax T ny ay’
asT nx ay ny ax
_ 2 2 _ 2 2Nn — N1nX + 2N6nXny + Nzny , NS - (N2 - N1)nXny + N6(nX — ny)(4.4)
aw_ $1+ E[Po} T aw¢2 +
E
[Ä] = [Ä} - 2c2[Ü[ + cä[P[ (4.5)
32
with
[Ä] _ ^66 A45 [61 _ Das Das [51 _ Fss F45' A A ' D D " F F45 44 45 44 45 44
Note that the bending moments (M1, M2, M6) do not enter the set of
essential boundary conditions, and the resultants (P1, P2, P6) do not
enter the set of natural boundary conditions also.
The (mixed) variational principle corresponding to the functional
in Eq. (4.3) can be stated as follows. Of all gossible configurations a
laminate can assume, the one that renders the functional HH1 stationarg
also satisfies the eguations of eguilibrium (2.19) and the kinematic—
. constitutive eguations [i.e. the last two equations of Eq. (4.3)].
we note that the variational staement in Eq. (4.3) contains the
second-order derivatives of the transverse deflection (through {¤}). To
relax the continuity requirements on w, we integrate the term {P}T{¤} by
parts to trade the differentiation to {P]. Ne obtain
_ 1 o T A* o o T 8* M * P¤H5(^)-{ {5{¤}{ l{¤}+{¤}(l }{}+{E}{})Q
T 1 T— {M} {F*}{P} · 5 {M} {¤*l{M}
T 1 T+ {<S} <{M} - ¤1{P}> · 5 {P} lH*l{P}
1 -0 T ^ -0apl apö aw+ 5 {E } {^}{¤ } + C1{(gg- + gg-) gg
aP 6P6 2 aw+ (gg- + gg-) gg} — ¤w}d¤dy
- fr [Nnun + Nsus + Onw + Mn¢n + MS¢S
+ c1(ÄnPn + ÄSPS)]ds (4.6)
33
-M -ä!an - an’ es ' as
we note that the stress resultants Pn and PS (hence Pl, P2 and P6)
enter the set of essential boundary conditions in the case of the second
mixed formulation of the higer-order theory. As special cases, the
mixed variational statements for the classical and the first order
theories can be obtained from Eq. (4.6) by setting appropriate terms to
zero.
CHAPTER 5
FINITE—ELEMENT MODELS
5.1 Introduction
The mixed variational principles presented in Chapter 4 can be used
to develop mixed Finite-element models, which contain the bending
moments as the primary variables along with the generalized
displacements. Historically, at least with regard to bending of plates,
the first mixed formulation is attributed to Herrmann [74,75] and Hellan
[76].
Since the bending moments do not enter the set of essential
boundary conditions, they are not required to be continuous across
interelement boundaries. Thus, one can develop mixed models with
discontinuous (between elements) bilinear or higher-order approximation
of the bending moments. The present section deals with the development
of a mixed model based on nH2.
5.2 Finite—Element Discretization
Let Q, the domain (i.e., the midplane) of the laminate, be
represented by a collection of Finite elements
— E —e e FQ = U Q , Q fW.Q = empty for e z F (5.1)e=l
where Q = Q L;r is the closure of the open domain Q and r is its
boundary. Over a typical element QB, each of the variables u, v, w,
etc. are interpolated by expressions of the form
N N Nu =
ä ujwj , v = jil vjwj , w = j;l wjoj (5.2)
34
35
etc., where {pj} denote the set of admissible interpolation functions
(see Reddy and Putcha [64]). Although the same set of interpolation
functions is used, for simplicity, to interpolate each of the dependent
variables, in general, different sets of interpolation can be used for
(u,v), w, (¢X,¢y), (M1,M2,M6) and (Pl,P2,P6). From an examination of
the variational statement in Eq. (4.6), it is clear that the linear,
quadratic, etc., interpolation functions of the Lagrange type are
admissible. The resulting element is called a C°—element, because gg
derivative of the dependent variable is required to be continuous across
interelement boundaries.
To develop the finite—element equations of a typical element, we
first compute the strains, rotations, etc. in terms of the interpolation
functions wj and nodal values uj, vj, etc. we have
(6.4)
{w· } [0} Liv}ax R1 1,x R1 i
(5.4)
36
$2 {fXwi_X}
2 ä ä {fxwid + Fywpnx}
(M)
(91} [Ni} {9} 2 3wi ' {6} {wi} {wi}
‘" 'M} {5*6}
E8X6
{KS}= 3% E
[HLHA3} (SJ)
66i 662W”+F
BW{9} = {gw i= {w} E lHl}{02} (5.8)
{$*},3/}
{1 {{2}{€°‘={}+{ 12 g}} 9 ((2 UM { {M I] (A3} (5.9)
M1 {wi} {9} {0} {Mi}
{M} = M2 = [{9} {wi} {9}] {W2}; E {H3}{A“} (5.10)
M6 {9} {9} {wi} {M6}
37
aMl aM65;* + 5;* {P11;} {0} {P11;} {M1}
{0} = = {M2}aM6 aM25;* + 5;* {0} {P1_;} {P11;} {M6}
E {H“}{1“} (5.11)
{P1}{P} = {H3} {P2} E {n31{a5} (5.12)
{P6}
E . igax ayap BP g {H4}{a5} (5.13)i + ...äax ay
- id - B!Here, FX — ax , Fy — ay and
{wi} = {wl wz ... wN} , {wi X} etc.(5.14)and {u}, {v}, etc. denote the columns of the nodal values oF u, v, etc.,
respectively.
5.3 Finite—Element Model
The Finite-element model For the refined shear deformation theory
is derived From the variational statement in Eq. (4.6) over an
element. The First variation oF the Functional in (4.6) For a typical
element oe is given by
O = } {{ö¤O}T({^*}{¤O} + {B*{{M} + {E*l{P})SIE
~ T T+ 16M} ({<S} + {8*1 {6°} — {Ü*|{M} — {F*l{P})
38
Böpl a6P6 a6P6 86P2 aP1 aP6
P P —+ (% + gyääéäl — ¤<Sw}d><dy
— { (Nnsun + Nsaus + Qnsw + Mn6¢n + MsaasPB
+ Clünöpn + c16S6PS)ds (5.15)
Substitution of Eqs. (5.2)-(5.14) into the variationai statement in
Eq. (5.15), we obtain
[K11) [K12} {P1}IKZSI {A2} {F2}_ = _ (5.16)
[K55) (15} {P5}where
NH1Kl11 = 1 {HL}T{A*l{HLldA . 111} = 1 1H21T dsge re NS
1Kl21 =—§ 1QB
{K2ll = 1 (1H"1 + H°])T{A*I{HL}dAQB
1Kl“1 = 1QB
39
lKl5} = I [HLlT[E*llH3ldA = {K5llTS28 „
läzzl = I <H°1 + IH"1>TIA*1<1H°1 + é 1H"I>dA + I lHllTlÄl[HlldAS26 S26
lK24l = I <<IH°1 + lHN])TlB*l{H3l)dAQB
[K42l = I I; lH3lTlB*lT({HNl + 2lH°l))dAQB
IK251 = I (IH°1 + [HN])T[E*][H3] + c1[HllT[HL]T)dAQB
IK521 = I (lH3lT{E*l(lH°l+ § [HNI) + ¤lIHL1IH‘1>¤AQB
{F2} = I {wi}T¤dA + I {¤»i}T0„dsQB FB
1E331 = I IH21TIÄI1H21¤A, IR231 = I [HllT[Äl[H2ldA = I¥321TQB QB
[KIM] zf [HL]T[H3]dA = [K43]TQB
[KBS] = _j- cl[HL]T[H3]dA = [KSBITQB
2 2T TV"} [nx ny]IF}=_I [Hl[R| dS,[Rl=
FB MS —ny nx
lK44l = - I lH3lTl¤*llH3ldA,QB
lK45l = — I {H3lTIF*llH3ldA = IK5“1TQB
40
1¤<551 = -1 [H3}T[H*l[H3ldAQe
nz nz 2n n
—-nxny nxny6
{F5} =} [H3]T[T]T{ n}as(5.17)
oe °s
Clearly, the element stiffness matrix is ngt symmetric. For the linear
case (i.e. [HN] = [0]), the element stiffness matrix is symmetric. The
finite-element models for the classical and first—order theories can be
obtained as special cases from Eq. (5.16). An explicit form of the
coefficients of the stiffness matrix in Eq. (5.16) is given in Appendix
B.
5.4 Solution Procedure
The assembled finite-element equations are of the form
{KD(A)|{A} = {F} (5-18)
where [KD] is the assembled stiffness matrix, unsymmetric in general,
and {A} is the global solution vector. Because the stiffness matrix is
a nonlinear function of the unknown solution, Eq. (5.18) should be
solved iteratively. Two iterative methods of analysis are quite
commonly used in the finite-element analysis of nonlinear problems: the
Picard type direction iteration and the Newton—Raphson iteration
methods. Here the Newton—Raphson method is used to solve the nonlinear
equations.
41
Suppose that the solution is required for a load of P. The load is
divided into a sequence of load steps API, AP2, ..., APn such that P
= .2 AP,. At any load step, Eq. (5.18) is solved iteratively to obtain
the=;olution. At the end of the r—th iteration the solution for the
next iteration is obtained solving the following equation
(RT(A”))(6A”*1) = - (R) E -(•(°(A')1(1’) + (6) (5.19)for the increment of the solution {6Ar+l}, and the total solution is
computed from
(N1) = (J) + (61**1) (5.20)where [KT] is the tgggggt stiffness matrix,
T 0 9•<D{K ] = [K 1 + lüj}·l·€^1 (521)
A geometrical explanation of the Newton—Raphson iteration is given in
Fig. 5.1.
For the finite—element models developed here, the tangent stiffness
matrix is symmetric. For example, the tangent stiffness matrix for the
model in Eq. (5.16) for plates (1/R1 = 1/R2 = 0) is given by[K11T]
T [K21T] [K22T] Sym"'[K 1 = ; (5.22)
where
[K11T] :[K11]IKZZTI
= [12221 + (T E lgNlT1A*llHl}d^){4l}sz
42
nenmeu
R = KU ‘ F R = KU
0 ul 02 UCcouvsnaaucs
FIGURE 5.1 CONVERGENCE OF THE NEWTON"RAPHSON ITERATION
43
+1 (lFNlTlA*l[HNl + lHNlT[A*llWN})d^{A2}GE
—N T 4+ (I [H l {B*l[H3ld^){^ }eQ
+<I {FNlT[E*l[H3ld^){/A5}E I
•Q
1¤<3“1 = [Ol, lK32Tl = [K32l„[K33Tl[K44Tl
[KSSTI = [K55]
T{‘”1,x} {'*’k,x}—N _ T
l"For a11 coup1ed (i.e. bending-stretching coup1ing) prob1ems, the
mixed mode1s of c1assica1, first-order and third—order theories have
six, eight and e1even degrees of freedom per node, respective1y. If the
bending moments are e1iminated at the e1ement 1eve1, the e1ement degrees
of freedom can be reduced by 3 (see Fig. 5.2).
The finite—e1ement mode1 (5.16) for the dynamic case is of the
form,
lKl{A} + IMIÄÄ} = {F} (5-24)where [M] is the mass matrix (see Appendix B). For free vibration, Eq.
(5.24) can be written as
44
/*2
T, ?< 12W
@2·¢2 ]
(a) Disp1acement mode1
/’TZ P;17< gg;
1 1 "11U] M1 P1vl 1 1
(b) Mixed mode1
Figure5.2 The disp1acement and mixed finite e1ementsfor the third—0rder shear deformation theory
45
[K]{A} = X{M}{A} (5.25)
where A is the square of the frequency. For transient analysis, Eq.
(5.24) can be reduced, after using the temporal approximation (see Reddy
[77]), to the form
(5.26)where{A}n+1 denotes the solution at time t = (n+l)At,At being the time
step. In the present study the Newmark beta method is used. In this
case [kl and {f} are given by (see Reddy [77])
[I2} = [Kl + aO[Ml
ao = —-Ä-? , al = dOAt , az = gg — I (5.27)(8At)
Once the solution {A} is known at time t = (n + 1)At, the first and
second derivatives (velocity and accelerations) of {A} at t = (n
+ l)At can be computed from
(5.28)
where ä3 = (1 — ¤)At and a4 = ¤At.
The evaluation of the element matrices requires numerical
integration. Reduced integration is used to evaluate the stiffness
coefficients associated with the shear energy terms. More specifically,
the 2 x 2 Gauss rule is used for shear terms and the standard 3 x 3
Gauss rule is used for the bending terms when the nine node quadratic
isoparametric element is considered.
CHAPTER 6
SAMPLE APPLICATIONS
6.1 Introduction
A number of representative problems are analyzed using the higher-
order theory developed in the present study. First few problems are
included here to illustrate the accuracy of the present theory. TheseI
problems are solved using the closed—form solutions presented in Chapter
3. Then problems that do not allow closed—form solutions are solved by
the mixed finite—element model described in Chapter 5.
The geometries of typical cylindrical and spherical shells are
shown in Fig. 6.1. Of course, plates are derived as special cases from
cylindrical or spherical shells.”
6.2 Exact Solutions
It is well known that the series solution in Eq. (3.3) converges
faster for uniform load than for a point load. For a sinusoidal
distribution of the transverse load, the series reduces tova single
term.
1. Four—layer, cross—ply (O/90/90/O) sguare lamiante under sinusoidalßThis example is included here to demonstrate the relative accuracy
of the present higher—order theory when compared to three-dimensional
elasticity theory and the first—order theory. Square, cross-ply
laminates under simply-supported boundary conditions [see Eq. (3.2)] and
sinusoidal distribution of the transverse load are studied for
deflections. The lamina properties are assumed to be
A6
47
PJ
jf 8
.b - \
iFigure 6.1 Geometry of a typical cylindrical and spherical shell
48
E1 = 25E2, G12 = Gl3 = 0.5E2, G23 = 0.2E2, ul? = GI3 = 0.25
Figure 6.2 contains plots of the nondimensional center deflection [W =
wE2h3)/qOa4] versus the side to thickness ratio (a/h) obtained using
various theories. It is clear that the present third-order theory gives
the closest solution to the three-dimensional elasticity solution [60]
than either the first—order theory or the classical theory.l
2. An isotropic spherical shell under point load at the center.
The problem data is
R1 = R2 = 96.0 in., a = b = 32.0 int, h = 0.1 in.,
El = E2 =107 psi, v = 0.3, intensity of load = 100 lbs.
A comparison of the center transverse deflection of the present theory
(HSDT) with that obtained using the first—order shear deformation theory
(FSDT) and classical shell theory (CST) for various terms in the series
is presented in Table 1. It should be noted that Vlasov [79] did not
consider transverse shear strains in his study. The difference between
the values predicted by HSDT and FSDT is not significant for the thin
isotropic shell problem considered here.
3. Cross-ply spherical shells under sinusoidal, uniform, and pointlgggs.
The geometric and material parameters used here are the same as
those used in Problem 1. Table 2-4 contain the nondimensionalized
center deflection of various cross-ply shells under sinusoidal, uniform,
and point loads, respectively. The difference between the solutions
predicted by the higher—order theory and the first-order theory
increases with the increasing value of R/a. Also, the difference is the
49
.·¤
‘Oc.:
cu ··-+-I6
‘„:·== 9 I1..E
uN2%
Nr-
u.n\ >"°gw
Ü nÜ;
Zig L5 m••-g
·"
>Hgc-
UnIn ' "°
X(XI
II • _‘ U
.,.N ·
____O
In Q nn ·mc.,
O•.U
8 9 > 32n>;I
Ö N cs'- :rQ nn
.C
V Q 3 N "¤>•'f'!'*
¤' O ¤. I.|.I 2 I-I-ng;
LU \ ggUn
I IU•—
IUJU)
I- : 1.;.1 I
«—m
'I—6: E 6 ·¤I= 'E2nä 3
'cuG O s.‘P > r ~= — ·»·%-’
MI- C4-J ‘
UJ LUG}
I I Iog
LDI., I U) :
I(D__
1 |-IJ (Dv)0
I-IJ ' I! o I;.
wO ca 1:
I-IJ I I€>·•
5cn ,. ' öl I- c>«¤
m ää In I | 6 ‘*:cc
¤. ,., I I- cs.
I .“· }/ ! u.n gg
NN. . Q •,.q_
‘/’ ·—· ‘¤ -Bw ¢:s..„._,
. /
_ oo,_
1/-/3-/
9*-, 6I l EU
J: DI
_ u.@
|"* Q
I¤ c: LUNS N .9 3: : aa °°!Ü°,„8 8: ~*~ <>ä2
E
50
Table 1. Center deflection (-w x 103) of a spherical shell under pointload at the center
Number of terms in the seriesh Theory N = 9 N = 49 N = 99 N = 149 N = 199
CST 39.591 39.647Vlasov [79] — - 39.560 — -
0.1 FSDT [40] 32.594 39.469 39.724”
39.786 39.814HSDT 32.584 39.458 39.714 39.775 39.803
0.32 FSDT 3.664 3.9019 3.9194 3.9269 3.9318HSDT 3.661 3.8986 3.9158 3.9230 3.9274
1.6 FSDT 0.1646 0.1713 0.1735 0.1748 0.1757HSDT 0.1637 0.1701 0.1715 0.1720 0.1722
3.2 FSDT 0.03485 0.03756 0.03864 0.03927 0.03972HSDT 0.03450 0.03668 0.03704 0.03713 0.03716
6.4 FSDT 0.00671 0.00798 0.008523 0.008838 0.009061HSDT 0.00658 0.007275 0.007337 0.007350 0.007354
5l
Table 2. Nondimensionalized center deflections, W = (—wh3E /q a“)103,of cross-ply laminated spherical shells under sinösoqdallydistributed load (a/b = 1, R1 = R2 = R, qo = 100)
0°/90° 0°/90°/0° 0°/90°/90°/0°
R/a Theory a/h=l00 a/h=l0 a/h=l00 a/h=l0 a/h=l00 a/h=l0
5 FSDT 1.1948 11.429 1.0337 6.4253 1.0279 6.3623HSDT 1.1937 11.166 1.0321 6.7688 1.0264 6.7865
10 FSDT 3.5760 12.123 2.4109 _ 6.6247 2.4030 6.5595HSDT 3.5733 11.896 2.4099 7.0325 2.4024 7.0536
20 FSDT 7.1270 12.309 3.6150 6.6756 3.6104 6.6099HSDT 7.1236 12.094 3.6170 7.1016 3.6133 7.1237
50 FSDT 9.8717 12.362 4.2027 6.6902 4.2015 6.6244FSDT 9.8692 12.150 4.2071 7.1212 4.2071 7.1436
100 FSDT 10.446 12.370 4.3026 6.6923 4.3021 6.6264HSDT 10.444 12.158 4.3074 7.1240 4.3082 7.1464
Plate 10.653 12.373 4.3370 6.6939 4.3368 6.628010.651 12.161 4.3420 7.1250 4.3430 7.1474
52
Table 3. Nondimensionalized center deflections, W = (—wE h3/q a*•)
x 103, of cross—ply laminated spherical shells 0nder°uniformlydistributed load
— 0°/90° 0°/90°/0° 0°/90°/90°/0°RE Theory a/h=10O a/h=10 a/h=100 a/h=1O a/h=1OO a/h=1O
5 FSDT 1.7535 19.944 1.5118 °9.7937 1.5358 9.8249HSDT 1.7519 17.566 1.5092 10.332 1.5332 10.476
10 FSDT 5.5428 19.065 3.6445 10.110 3.7208 10.141HSDT 5.5388 18.744 3.6426 10.752 3.7195 10.904
20 FSDT 11.273 19.365 5.5473 10.191 5.6618 10.222HSDT 11.268 19.064 5.5503 10.862 5.666 11.017
50 FSDT 15.714 19.452 6.4827 10.214 6.6148 10.245HSDT 15.711 19.155 6.4895 10.893 6.6234 11.049
100 FSDT 16.645 19.464 6.6421 10.218 6.7772 10.294HSDT 16.642 19.168 6.6496 10.898 6.7866 11.053
Plate 16.980 19.469 6.6970 10.220 6.8331 10.25116.977 19.172 6.7047 10.899 6.8427 11.055
53
Table 4. Nondimensionalized center deflection of cross—ply sphericalshells under point load at the center (101 terms are used inthe series)
_ wh3E2W H/b :1,
6/hR/aTheory O°/90° _0°/90°/0° 0°/90°/90°/0°
5 FSDT 7.1015 5.1410 4.9360HSDT 5.8953 4.4340 4.3574
10 FSDT 7.3836 5.2273 5.0186HSDT 6.1913 4.5470 4.4690
20 FSDT 7.4692 5.2594 5.0496HSDT 6.2714 4.5765 4.4982
50 FSDT 7.4909 5.2657 5.0557HSDT 6.2943 4.5849 4.5065
100 FSDT 7.4940 5.2666 5.0565HSDT 6.2976 4.5861 4.5077
Plate FSDT 7.4853 5.2572 5.0472HSDT 6.2987 4.5865 4.5081
54
most for the symmetric laminates, especially for shallow shells and
plates. Note that for point-loaded shells the difference between the
solution predicted by the first-order theory and the higher-order theory
is more significant, especially for antisymmetric cross-ply laminates.
4. Natural vibration of cross-ply cylindrical shells „
Table 5 contains the nondimensionalized fundamental frequencies of
cross-ply cylindrical shells for three lamination schemes: [0°/90°],
[0°/90°/0°], and [0°/90°/90°/0°]. For thin antisymmetric cross-ply
shells, the first-order theory underpredicts the natural frequencies
when compared to the higher-order theory. However, for symmetric cross-
ply shells, the trend reverses.
5. Natural vibration of cross-ply spherical shells
Table 6 contains nondimensionalized natural frequencies obtained
using the first- and higher-order theories for various cross-ply
spherical shells. Analogous to cylindrical shells, the first-order
theory underpredicts fundamental natural frequencies of antisymmetric
cross-ply shells; for symmetric thick shells and symmetric shallow thin
shells the trend reverses.
6.3 Approximate (Finite-Element) Solutions
6.3.1 Bending Analysis
1. Rectangular plate under uniformly distributed load and mixedboundary conditions.
The geometry and boundary conditions for the problem are shown in
Fig. 6.3. The plate is assumed to be made of steel (E = 30 x 106
and u = 0.3). The problem was also solved by Timoshenko [78l, Herrmann
55
Tab1e 5 Nondimensionalized fundamental frequencies of cross-p1ycylindrical she11s
;=R/aTheory a/h=100 a/h=l0 a/h=100 a/h=l0 a/h=10O a/h=l0
5 FSDT 16.668 8.9082 20.332 12.207 20.36112.267HSDT 16.690 9.0230 20.330 11.850 20.36011.830
10 FSDT 11.831 8.8879 16.625 12.173 16.63412.236HSDT 11.840 8.9790 16.620 11.800 16.63011.790
20 FSDT 10.265 8.8900 15.556 12.166 15.55912.230HSDT 10.27 8.972 15.55 11.791 15.55 11.78
50 FSDT 9.7816 8.8951 15.244 12.163 15.24512.228HSDT 9.783 8.973 15.24 11.79 15.23 11.78
100 FSDT 9.7108 8.8974 15.198 12.163 15.19912.227HSDT 9.712 8.975 15.19 11.79 15.19 11.78
Plate FSDT 9.6873 8.8998 15.183 12.162 15.18412.226HSDT 9.6880 8.9760 15.170 11.790 15.17011.780
56
Table 6. Nondimensionalized fundamental frequencies of cross-plylaminated spherical shells
E = „. § /,762
0°/90° 0°/90°/0° 0°/90°/90°/0°
R/a Theory a/h=100 a/h=1O a/h=100 a/h=10 a/h=100 a/h=1O
5 FSDT 28.825 9.2309 30.993 12.372 31.07912.437HSDT 28.840 9.3370 31.020 12.060 31.10012.040
10 FSDT 16.706 8.9841 20.347 12.215 20.38012.280HSDT 16.710 9.0680 20.350 11.860 20.38011.840
20 FSDT 11.841 8.9212 16.627 12.176 16.63812.240HSDT 11.84 8.999 16.62 11.81 16.63 11.79
50 FSDT 10.063 8.9034 15.424 12.165 15.426l2.229HSDT 10.06 8.980 15.42 11.79 15.42 11.78° 100 FSDT 9.7826 8.9009 15.244 12.163 15.24512.228HSDT 9.784 8.977 15.24 11.79 15.23 11.78
Plate FSDT 9.6873 8.8998 15.183 12.162 15.18412.226HSDT 9.6880 8.9760 15.170 11.790 15.17011.780
57
LI*' PRESENT SOLUTION Q
Q 6°HERRMANN¤PRATO
A TIPOSHENKO OQ‘é
><z„ 26 TX2 FREE5 ° i
SSQ•8”
T SS
ä 1I —°XI
I; CLAIVPED ·
ä"' SS = SIPPLY SUPPORTEDIIm1cI<NEss = 0.01
00 0.2 0.LI 0.6 0.8
DISTANCE ALONG xl
FIGURE 6.3 CONPARISON OF THE TRANSVERSE DEFLECTIGV OBTAINED BYVARIOUS INVESTIGATORS (x2 = 0.LI)
58
[74] and Prato [80]. Figures 6.3-6.5 contain plots of the transverse
displacement and bending moments obtained by various investigators
(using the linear theory). The agreement between the present solution
and others is very good.
2. Rectangular laminates under uniformly distributed load
Figure 6.6 contains load—deflection curves for a simply supported
square unidirectional laminate (0°) under uniformly distributed load.
The following geometric and material properties are used:
a = b = 12 in., h = 0.138 in.
E1 = 3 x 106psi, E2 = 1.28 x 106psi, G12 = 613 = 623 = 0.37 x 106psi
v = 0.32
The experimental results and classical solutions are taken from the
paper by Zaghloul and Kennedy [86]. The agreement between the present
solution and the experimental solution is extremely good.
Figure 6.7 contains load—deflection curves for a clamped
bidirectional laminate [0/90/90/0] under uniformly distributed load.
The geometric parameters and layer properties are given by
a = b = 12 in., h = 0.096 in.,
E1 = 1.8282 x 106 psi, E2 = 1.8315 x 106 psi,
G12 = 613 = 623 = 3.125 x 106 psi, u = 0.2395
The experimental results and the classical laminate solutions are taken
from [81]. The present solution is in good agreement with the
experimental results, and the difference is attributed to possible
errors in the simulation of the material properties and boundary
conditions.
59
8¤— FREsEm SOLUTION A
6 ¤ HERRMANN °¤ PRATO O
LI A T1msHEN¤<o ~A§
><I
E 0 FREES ° Eä _2 SS ggEä .
CLAMPED—LI
-60 0.2 0.LI 0.6 0.8
DISTANCE ALONG X2
FIGLRE 6.LI COMPARISON OF THE BENDING IIUENT MI ALONG THE LINE XI =O.LI
60 _
6'“
PRESENT SOLUI' ION
° HERRMANN¤8
¤ PRATOceL, A TINOSHENK0§ 8 .X ¤ZU
é FREEä 2gg ss ss °ä“°
cuwzn
00 0.2 OA 0.6 0.8
DI STANCE ALLNG X,
FIGLRE 6.5 COMPARISON OF M6 ALONG THE LINE X2 =Ü.L|
6l
ÜI5
- —— EXPERIMENIA1.—°— PRESENV _0.Ll —·— cLAss1cA1.
-(
E 0 3‘ LINEAR
ZI
-9.1- / .
ä, NO 1.111 .
lg 0.2 Ia Ä?„, '/e . -/ä 0 1
//I0
0 0.LI 0.8 1.2 1.6 2.0LOAD INTENSITY (pg;)
Figure 6.6 Center deflection versus load intensity for asimply supported square orthotropic plateunder uniformly distributed transverse load,The following simply supported boundary condi-‘ClO|'lS WEPE used:
· v=w=¤py=MX=Px=Oonsidex=·a
u=w=¤pX=My=Py=Oonsidey=b
62
ÜIS
—-— EXPERIMENTAL'-°- PRESENT .
0.Ll —·— CLASSICAL/ «
é .
5 0-3 ' LINEAR ‘EZ .ä
NONLINEAR
1:. „ · 1’
¤ 0.2E 2
"
tu _·
E / ·UJ·
•L) ;
’ •
Ol]- '. _ ·,/Q
00 0.LI 0.8 1.2 1.6 2.0
INTENSITY OF TRANSVERSE LOAD (pg;) ·
Figureö.7 Center deflection versus load intensity for aclamped (cc-1) square laminate [0 /90 /90 /0 ]under uniform transverse load (von Karman theory).
CC-l: u=v=w=¢x=wy‘=0onall
four clamped edges
63
3. Orthotropic cylinder subjected to internal pressure.
Consider a clamped orthotropic cylinder with the following
geometric and material properties (see Fig. 6.8)
R1 = 20 in. R2 = ¤
El = 2 x 106 psi, E2 = 7.5 x106 psi
·v21 = 0.25 , G12 = 1.25 x 106 psi
013 = 023 = 0.625 x 106 psi , h = 1 in.
a = 10 in. , P = 6.41/« psi
This problem has an analytical solution (see [78]) for linear case,
and Rao [47] used the finite-element method to solve the same problem.
Both are based on the classical theory. The center deflections from
[78] and [47] are 0.000367 in. and 0.000366 in., respectively. Chao and
Reddy [52] obtained 0.0003764 in. and 0.0003739 in. using the finite-
element model based on the first-order shear deformation theory and 3-D
degenerate element, respectively. The current result is 0.0003761 in.,
which is closer to Chao and Reddy [52], as expected.
For the nonlinear analysis of the same problem, the prsent results
are compared to those of Chang and Sawamiphakdi [81] and Chao and Reddy
[52] in Fig. 6.9, which contains plots of the center deflection versus
the load obtained by various investigators. The agreement between
various results is very good.
4. Nine—layer cross-gly spherical shell subjected to uniform loading.
Consider a nine—layer [0°/90°/0°.../0°] cross-ply laminated
spherical shell with the following material and geometric data:
R1 = R2 = 1000 in. ., a = b = 100 in.
64
xx- -—
- -- x2/l
\¢,—v—0_
¢'=¢2:M=%=0CLAMPED"" I M6=B,=U CLAVPED
‘
I\‘[ j U=¢’\{=M6 =P6=/U
\ [ X ·/\~„«.Ö .. .......J(Nor
TO A SCALE) „RADIUS OF SHELL = ZU
FIGURE GEGVETRY AND BOLNDARY CGIDITIONS FOR THE OCTANTOF THE CLAIVPED CYLINDRICAL SHELL
65
10
8 —· PRESENT SOLUTION ^¤¤ CHANG AND SAWAMIPHAKDI
. A cHAo AND REDDYG 6 A¤O.M
...75U)U)
ä¤. Q M
ä „LIJE
2
00 0.25 0.5 0.75 1.0 1.25
VERTICAL uaaaacrxou (1N.)
FIGURE 6.9 CENTER DEFLECTION VERSUS LOAD FOR THE CLAMPED CYLINDERWITH INTERNAL PRESSURE (ORTHOTROPIC)
66
h = 1 in. , E1 = 40 x 106 psi
E2 = 106 psi , G12 = 0.6 x 106 psi , G13 = G23 = 0.5 x 106 psi
V12 = 0.75The present results are compared with those obtained by Noor and
Hartley [54] and Chao and Reddy [52] in Fig. 6.10. Noor and Hartley
used a 13 degree-of-freedom element based on a shear deformation shell
theory. The present results agree with both investigations.
6.3.2 Vibration Analysis
Equation (5.24) can be expressed in the alternative form as
(6.1)[K21l [K22l {A2} {Mzll {M22] {A2} {0}
where
(6.2)For free vibration analysis, we wish to eliminate {A2} as follows. From
Eq. (6.1) we have
(6-3)
From (6.4) we have,
{A2} = ‘{K22}—l{K2l}{^1} (6-5)
Substituting Eq. (6.5) into Eq. (6.3), we obtain,
(IKM] — [Kl2llK22l_llK2ll){A1} = — lM1ll{A-1} (6-6)For the free vibration case, Eq. (6.6) reduces to
(TE} 0 (6.7)
67
10 O
·· PRESENT SOLUTHN ·
8 ° NOOR A 0
A CHAO AND REDDY
~‘i.
6 A 0~
cx..
LUI-
§ L} A 0CL
E~
2
00 1 2 3 LI
NONDIIVENSIONALIZED CENTER DEFLECTICN; W/I1
FIGURE NONLINEAR BENDING OF A NINE"LAYER CROSS*PLY SPHERICALSHELL [0/90/0/90/. . ./0]
68
where u is the natural frequencies of the system.
1. Natural freguencies of a two—layer |0°g90°] laminated plate.
The geometric and material properties used areI
a = b = 100 in. , h = 0.1 in.
E1 = 40 x 106 psi , E2 = 106 psi.·
G12 = 613 = 623 = 0.5 x 106 psi , plz = 0.25p=].
The boundary conditions are shown in Fig. 6.11, which also contains
the plots of the ratios uNL/mL versus wo/h. Here uNL and mL denote thenonlinear and linear natural frequencies, and wo is the normalized
center deflection of the first node. The results are compared with
those of Chia and Prabhakara [82], and Reddy [83]. The prsent results
are slightly higher compared to those obtained by the classical and
first—order shear deformation theories.
2. Natural freguencies of two-layer [45°[—45°] angle-ply sguare plate.
The material and geometric parameters used are
E1 = 10 x 106 psi , E2 =106 psi
612 = 613 = 623 = 0.3333 x 106 psi , plz = 0.3a = b = 100 in. , h = 0.1 in.
The boundary conditions used are shown in Fig. 6.12, which also
contains plots of uNL/uL versus wo/h. For this case no results are
available in the literature for comparison.
69
1.83.1E " PRESENT SOLUTIONSZ
° CHIA AND PRABHAKARAL: A nenuvä <·
9 gigI - w=¢,=0 A
1- „ °
ä—’X1
.1 g§ 6L5
lI2
91-
A
l0 0I6 l|2 l•8 2•
3.0
w,,/h
FIGIRE 6.1.1 FLNDANENTAL FREOUENCIES OF A TWO‘LAYER CROSS'PLY(0/90) s¤uAne ¤.AMmATe umen smnev sunnonrem B0umAnvCONDITIONS
70
LIXZT v = w = ¢1= 0
V = 0 U : w :¢2: O‘
..1 :3 ¢1 O\
é 3 —-•• X]3 u = ¢2= 0
cz?E
äEIZLL
ä 2...|on-
; .;'5Z
1.0 0.6 1.2 1.8 2.Ll 3.0
ANPLITLIJE TO THICKNESS RATIO; wo/h
FIGLRE 6.12 RATIO OF NONLINEAR TO LINEAR FREOUENCY VERSUS AMPLITUDETO THICKNESS RATIO FOR TWO"LAYER ANGLE··PLY SQUARE
PALTE (LI5/-45).
71
3. Natural vibration of two-layer [0°g90°| cross-ply plates.
Consider a two-layer cross-ply plate with the following geometric
and material properties:
E1 = 7.07 x 106 psi , E2 = 3.58 x 106 psi
G12 = G23 = G13 = 1.41 x 106 psi , V12 = 0.3
a/h = 1000 , a = 100 in.
The results of NNL/wL versus wo/h are shown in Fig. 6.13. Compared
to the results of Reddy [83] and Chandra and Raju [84], the results of
the present study are in general a little higher.
6.3.3 Transient Analysis
The selection of a time step is a critical problem in transient
analysis. we may expect better results if smaller time steps are
used. In practice, one wishes to take as large a time step as possible
to save computational expense. In the prsent study, the selection of
time step follows that given by Tsui and Tong [85]. Tsui and Tong
presented an estimate formula of the time step for conditionally stable
integration schemes:
2 (A¤)2lP<l-(At)S "—*‘T—""*ü2 + (l - V) T? [1 + 1.5 (F-) ]
where AX is the minimum distance between the element node points.
1. Two layer cross-ply |0°g90°| laminated, clamped cylinder underinternal pressure load.
The geometric parameters and boundary conditions are the same as
those shown in Fig. 6.8. The material properties used are
El = 7.5 x 106 psi , E2 = 2.0 x 106 psi
72
BCl boundary conditions:
X2l,„,„=¢2=0;ml =l>l =0 along xl = a/2u=¢1=O; M6=P6 =0 along xl = 0
v=¢2=0; M6= P6 =0 along x2 = 0S.]$2] 3
X21TZ T _5 - PRESENT SOLUTION E AQ 2 Bcl 13 ¤ cHANDRA AND RAJU JL96 aA REDDY l·— ä —·lZ I3 2 A; 2Z
z Aä
lA/
E 0 0.6 1.2 1.8 2.LI 3.0wo/h
FIGURE 6.13 RAT10 ol= THE NONLINEAR TO LINEAR FREQUENCY VERSUSTHE ANPLITUDE TO TH1cl<NEss RATI0 0l= A TWO·LAYERcnoss-l>l.v (0/90) SQUARE LAMINATE
73
G12 = GI3 = G23 = 1.25 x 106 psi , r = 0.25„ = 1 ib-sec?/1n4 , q = sooo psi
Figure 6.14 contains the plot of the center deflection versus time of
the current solutions and compare to those by Reddy and Chandrashekhara
[53]. with the time step At = 0.005, the current solutions match very
well with those results by Reddy and Ghandrashekhara with a smaller
step At = 0.001. In other words, a lot of computation time has been
saved by the current formulation to get the same accuracy.
2. Two—layer angle—ply |-45°g45°] laminated, clamped cylinder withinternal pressure load.
The geometric and material properties are the same as those used in
Problem 1 of this section. The boundary conditions are also the same
except that the moment resultants Mi, Pi are not specified. A 2 x 2
mesh of nine-node element was used to simulate the transient response of
the cylinder. Only one octant was used. Figure 6.15 contains the plot
of the transverse deflection versus time for q = 5000 psi and At = 0.005
and compare to the solutions by Reddy and Chandrashekhara [53].
74
1.25
i At = 0.005 CURRENT SOLUTIONS
_ o At: = 0.001 REDDY ANDLOC A At = 0.005 CHANDRASHEKHARA
AA
0.73 5A
EiH ASg 0.6 ^rz.fä o
S1-• A~E 0 25
C A
0A
-.25O 0.025 0.06 0.076 0.1 1*25
Time (sec)
Figure 6.14 NONLINEAR TRANSIENT RESPONSE OF A TWO-LAYER CROSS-PLY(0/90) CYLINDRICAL SHELL UNDER INTERNAL PRESSURE
75
1.50
———- At: = 0.005 CURRENT SOLUTIONS0 At = 0.001 REDDY ANDA At = 0.005 CHANDRASHEKHARA
1.258 2
01.0 A
O
3 0.7 ‘
.. OZO•-•[—*
E5A 0.5 ·LuI:-1Q
M
E oä 0.26U
0
^ A
0.0 0A
-.25() 0.025 0.05 0.075 0.100 1.25
Time (sec) .
FIGURE 6.15 NONLINEAR TRANSIENT RESPONSE OF A ANCLE-PLY(-65/45) CYLINDRICAL SHELL UNDER INTERNALPRESSURE
CHAPTER 7
SUMMARY AND RECOMMENDATIONS e7.1 Summary and Conclusions
The present study dealt with the following major topics:
(i) The development of a third-order shear deformation theory of
laminated composite doubly—curved shells. The theory
accounts for (a) the parabolic variation of the transverse
shear strains, and (b) the von Karman strains, It does not
require the use of the shear correction coefficients.
(ii) The development of the closed—form solutions (for the linear
theory) for the simply supported cross-ply laminates. These
solutions can be used for a check in the numerical analysis
of shells.
(iii) The construction of a mixed variational principle for the
third-order theory that includes the classical theory and the
first order theory as special cases.
(iv) The development and application of the finite—element model
of the third-order theory for laminated composite shells,
accounting for the geometric nonlinearity in the sense of von”
Karman.
The increased accuracy of the present third-order theory over the
— classical or first—order shear deformation theory is demonstrated via
examples that have either the three—dimensional elasticity solution or
experimental results. Many of the other results on bending, vibration
and transient analysis included here can serve as references for future
investigations.A
76
77
7.2 Some Comments on Mixed Models
The displacement model of the classical laminate theory requires
the use of cl elements, which are algebraically complex and
computationally expensive. The CO mixed elements are algebraically
simple and allows the direct computation of the bending moments at the
nodes. The inclusion of the bending moments as the nodal degrees of
freedom not only results in increased accuracy of the average stress
over that determined from the displacement model but it allows us to
determine stresses at the nodes. This feature is quite attractive in
contact problems and singular problems in general. It should be noted
that the bending moments are not required to be continuous across
interelement boundaries, as was shown in Chapter 4. The mixed models
based on the shear deformation theories also have the same advantages,
except that the displacement model of the first-order shear deformation
theory is also a CO element. In general, the formulative and
programming efforts are less with the mixed elements.
7.3 Recommendations
The theory presented here can be extended to a more general theory;
for example, the development of the theory in general curvilinear
coordinates, and for more general shells (than the doubly—curved shells
considered here). Extension of the present theory to include nonlinear
material models is awaiting. Of course, the inclusion of thermal loads
and damping in the present theory is straight forward.
The displacement models require Cl-approximation of the transverse
deflection, CO—approximation of other displacement variables. Such a
78
displacement fin1te—element model was not developed in the present
study, and interested researchers could pursue the development.
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55. Noor, A. K. and Andersen, C. M., "Mixed isoparametric finiteelement models of laminated composite shells," Comp. Meth. Appl.Mech. Eng., Vol. 11, p. 255, 1977.
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85
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APPENDIX A
COEFFICIENTS OF THE NAVIER SOLUTION
C(1,1) = -All62 - A66B2
C(1,2) = - Alzaß - A66aB'
- L L L 2 2 2C(1·3) ‘ A11 R1 * A12 R2 * ghz (E11° * 5128 * 25668 14E 4E2 11 2 66C(l„4) = 6 <- B + -—-> + 6 (- B + ————>
ll gh2 66 gh2
4E 4E12 66C(l·5) ‘ °8(‘B12 ‘ B66 * ‘“z‘ * °‘z'”)3h 3h ~
c(z,z) = -62A66 - 62 A22A A 4E 8E 35% E221 2 3h 3h 3h
4E 4E12 668<2·"> = ¤B(_Bl2 · 866 + TIT + T13h 3h
4E 4EI
2 66 2 22C(2,5) = 6 (-B + —) + 6 (—B + ——)66 gh2 22 gh2
16H 16H 2 24 11 4 22 32C(3,3) = a B (H12 + 21166)
2 8055 16F55 8Ell 8El2+8hh 3h Rl 3h R2
80 16F 8E 8E* 82(‘^A4 5 ’”%5 ‘ "E55 ' ‘°%g' 5 °”§g‘)
h h 3h R 3h R1 2
86
B7
1 E 1 L2 1 ERä R1R2 Rg3 4H 2 2_ i 11 4aß 16018C(3·") ‘ 3hz (F11 ‘ * fz (F12 * ZF66) ‘ gha (H12
B B E E11 12 4 11 12+2H )+ [·—+—-—-——(—·-——+—-)]-A66 ** R1 R2 3hz R1 R2 66**
1 Eis;.1h2 h4
3 4H 2 2_ Ä 22 4a B ].6¤ BC(3·5) ‘3hz (F22 ‘ 3‘h"z" ) * 3"h2" (F12 * ZF66) ’ gha (H12
B B E E12 22 4 12 22+2H )+6[——+——-——(—+——)]-A 666 R1 R2 3hZ R1 R2 44
1 1 Elah2 „ ha
C(44)—£F+ä‘£F lw 2+6 62)-0 2-ÜBZ
·‘
3hz 11 3hz 66 ‘ gha 11** 66 11° 66
- A +_55h2 h4
C(45)=¤6[—D -D +l(F +F)-£(H +H)l’ 12 66 3h2 12 66 9h4 12 66
BF 16H BF 16H2 66 66 2 22 22C(5,5)=¤(—D +——-——)+6(—D +——i)-A66 3h2 9h4 22 3h2 9h4 44
1 1 Elli62 6**212
88
M(l,2) = O
4 4M(1„3) = (—·I +71 )¤3h2 4 3h2R1 5
M(1 4) = -(I + ig- -5- 1 - -5- I )
· 2 R1 3h2 4 3h2R1 5M(l,S) = 0
212M(2„2) = -(I1+ 7)
2
4 4M(2,3) = (-— I + -—-— 1 )63hz "
ghzhz 5M(2,4) = O
mas)2 R2 3h2 4 3h2R2 5
16 2 16 2M(3,3) = -
——— I ¤ --—— I 6 - I9h4 7 9h4 7 1
4 16M(3,4) = (—-— 1 — -—- I )¤3h2 5 ghd 7
4 16M(3,5) = (·—T — — I )63h2 5 9h4 7
8 16M(4,4) = -(I - -- 1 + —-— 1 )3 3h2 5 9h4 7
M(4,5) = O
8 16M(5,5) = -(I -
-—— I + ——- I )3 3h2 5 9h4 7
APPENDIX 6STIFFNESS COEFFICIENTS Fon THE MIXED M0¤61
The element equation (5.16) can be written in the form[K11} [6121 ...... [KA[ll)1 {21} {Fl}[Kzll [6221 ...... [K2[11)} {22} {F2}
}K(11)1] }K(11)2} ___ }K(11)(11)} {A11} {F11}
where {Al} = {U}, {A2} = {V}, etc., and[Kll[ Z A11[S11[ * A12[S12[ * A21[S12[ * A22[S22[[Kl2[ Z A12[S12[ * A12[S11[ * A22[S22[ + A22[S21[
13 _ A11 1 aw A12[K [S10] + 2 A11 [Q $11[ [S10]
1 aw 1 aw 1 aw+ 2 A12 [Q S12] + 2 A12 [Q $12[ + 2 A12 [Q S11[A A31 1 aw 32+ ET [$2¤[ + 2 A21 [Q $21[ + FQ [$211[
1 aw 1 aw 1 aw* 2 A22 [22; S22[ * 2 A22 [222 S22[ * 2 A22 [22; S21[[K14} 2 [KIS} 2 0[Kl6[ Z B11[S1o[ * B21[S20[
[Kl7[ Z B12[S10[ * B22[S20[[Kl8[ Z B13[S10A Z B33[S2OA
89
90
[K[9[ ‘ El1[S10[ + E21[S20[1,10 _[K [ ‘ E12[S1O[ * E22[S20[1,11 _[K [ ‘ E13[S10[ * E33[S20[21 _ °[K [ ‘ A31[S11[ * A22[S12[ * A21[S21[ * A22[S22[22 _[K [ ‘ A22[S12[ + A22[S11[ * A22[S22[ * A22[S21[
A A23 - _§1 1 22. .§2[K [ ‘ R1 [S10[ * 2 A21[2x1 Sl1[ * R2 [S10]
1 aw 1 aw 1 aw1 2 A32 [Q $12[ + 2 ^22 [Q S12[ + 2 A22 [Q S1l[A A21 1 aw 22+ 2““ [S20] * 2 A21 [2§“ S21[ * 2" [S20[
1 1 21 aw 1 aw 1 aw* 2 A22[22; S22[ * 2 A23[$YI S22[ * 2 A22 [22; S21[
[K2A] 2 [K25] 2 0
112**1 = 111111111 1 B21[S20[[K27[ ‘ B32[Sl0[ * B22[S2O[[K28[ ‘ B33[S10[ * B22[S20[11291 =111111111 1 E21[S20[
2,10 _[K [ ‘ E32[S1O[ * E22[S2O[
91
[Kul] ‘ E33[Sl0A * E22[$20][K3]! = Allläg $111
* A1a[% $12] + % [$01] * [$02] * A21[?t $21]
+ A23[% $22] * A31[% $21] * A33[%i $22]
* A31[% $11] + Aaää $12]
[A32] ‘% [$02] * ä [$01] * A12[% $12] * Al3[% $11]
* [$02] * [$01] * A22[% $22] + A22[% $21]
* A32[% $22] * A33]-äi $21] * A32[i;l;<[—g $12]
* Asüä $11]
1
$10] A11 S10]
A 2 A12 [?t ät $12] * 2 A13 [(%]2$12]
92
A A1 aw aw 21 1 21 aw+—A [———S ]+l—[S ]+——[—S]2 13 axl axz 11 R1R2 00 2 R2 axl 01A A A22 1 22 aw 1 23 aw+y[$¤¤‘+2V[3g$o2[*2V[q$¤2[
2
[ÄLELS]2 R2 axz O1 R1 axz 20 2 21 axl axz 21
A22 aw 1 aw 2 1 aw aw* §“‘ [32* S20] * 2 ^22[[S§‘) S22[ * 2 ^23[2§’ 2Y* S22[2 2 2 1 2
12 23 axz 21 R1 axl 20 2 31 axl 21
A 32 aw 1 .aw aw 1 aw 2* 2“’ [2§‘ S20] * 2 S22[ * 2 ^33[[3§‘[ S22[2 1 1 2 1
,,1,, ,,1,, [ÄÄLSI2 33 axl ax? 21 R1 axz 10 2 31 axl axz 11
A12 aw 1 aw 2 1 aw aw+ V [V $16* + 2 ^32[[V) $121 + 2 $12}2 2 2 1 2
1 aw 2 — —* 2 ^33[[3§;) S11[ + ^s6[S11[ * A45[KSl2 * S21)[
* Ä44[S22[34 _ — —[K [ ‘ ^6s[S10[ * A45[S2O[35 _ — —[K [ ‘ A45[S10[ * A44[S2O[
B B36 - ll 22. .11 22.[K [ ‘ 2;’ [S00[ * B11[sxl S10] * R2 [S00[ * B21[ax2 S20]
é
93
BW BW$20] S10]
[K37l=Eg[S 1+6 IÄS 1+2lS 1+6 IÄS 1R1 00 12 SX1 10 R2 00 22 SX2 20
Sw SwS20] S10]
[K38l·§[S 1+6 1gg—s 1+älS 1+6 IÄS 1— R1 00 13 SX1 10 R2 00 23 SX2 20
Sw Sw* Bss]S§I S20] * B22]3§; S10]
[x391 — ggg1s 1 + 6 [gg- s 1 + ggg 1s 1 + 6 [gg- s 1”
R 00 11 ax 10 R 00 21 ax 201 1 2 2
BW BW+ E31]SYI S20] S10] * CP]S11]
[K3·1°1 - ggg 1s 1 + 6 [gg- s 1 + ggg 1s 1 + 6 [gg- s 1_R1 00 12 SX1 10 R2 00 22 SX2 20
· Sw Sw* E32]3§; $20] * E32]€§; $10] * CP]$22]
[¤<3·]]1—$[s 1+6 1%:6 l+2[S 1+6 IÄS 1”R 00 13 ax 10 R 00 23 ax 201 l 2 2
Sw Sw* Es2]E§I S20] * E33]5§§ S10] * CP]]$12 * $21)]
[K41] = {K42] = 0
43 _— —]K ] ‘ ^66]$01] * A45[S02]
00
10661 = 00010001
10661 = 10101 10671 = 0 10661 = 1020110661 = CP[SO1] 11<6·661 = 0 11<6·661 = 0010021
6
[K51] = 1K621 = 0666536 ‘ Ä456S0l6 6 64066026
10661 = 0001000110661 = 0 10671 = 10001 10661 = 1010110661 = 0 11<6·661 = c1>1s021 11<6·661 = CP[S01]666616 ’ B1l6SOl6 6 B3l6S026
666626 6 B2l6S026 6 B316S0l6
10661 - 666 10 1 + 6 0 0 1 + 666 10 1 + 6 0 166- 0 1· 01 00 2 ll 001 01 R2 00 2 21 002 O2
1 16 2 621 $026 6 2 3316% 6016
10661 = [Sol] 10661 = 010661 = -01110001
95
[K6?] ’ ”D12[S0O][KGS] = ·°131$00*1K691 = -F221s001
l [K6°10] ’ ”Fl2[S00][K6’lll = -F221S001[K71] ’ Bl2[S01] * Bszlsozl[*721 ‘ B22[S02]
1S001 S 1 + ggg 1S 1 + g 6 1gg- S 11 axl 01 R2 00 2 22 axz 02
$02] Bszlääg $01]
[K74l = 0 lK75l = 1s021
1K7’1=-02216001lK7’lll
= —F221S001
96B
[K8l[ ‘ B12[S01[ * B22[S02[
[K82[ ‘ B23[S02[ * B22[S01[
83 B12 1 aw B22 1 aw[K [ ‘ 22’ [S00[ * 2 $01] * 2E" [S0o[ * 2 B23[$YZ $02]
$02] $01]
{KBA] = 1602116851 = 16021
16861 = -¤221S00116871 = °D32[S0O]
lK89l = _F3l[S0O]1KB·l°1 = - F32[S0O]{K8’llI = — 62216001[K91[ ‘ B11[S01[ * B21[S02[[K92[ ‘ B21[S02[ * B21[S01[
93 Ell l aw E2l l aw[K [ ‘ 2]’ [S00[ + 2 $01] * 22’ [S00[ * 2 $02]
97
$02] $01] + EE°]$11][KM} = cP[sl01 [K95} = 0IKQGI = - Fl01s0011¤°’1
= - F2l[S00]lK98l
E22]$02]]"l0°2] ‘ E22]$02] * E22]$01]
10 2 E12 1 aw E22 1 aw]K ° ] ’Y ]$00] * 2 E12]§>Y $01] * Y ]$00] * E $02]
$02] $01] * E]°]$22]
[KlO,4] = 0 [Kl0,5] = 00[S20]lKlO’6l = — F001s001
98
[K10,10] 2 _ Hzzlsool[Kl0°ll] “ ‘ Hzslsool[Kll°1‘ ’ E13[S01l * Easlsozl[Kll°2‘ = Ezalsozl * E33[S01]
[$00] $01] [$00] $02]
Esslgäi $02] $01] * CP[(S12 * $21)]
[Kll’4l = CPISZOI[Kll’5l = cpxslon[Kll’6l = — Fl3[$O0l[Kll’7l = - F23lS00l[Kll”8‘ ‘ ‘ Faalsool[Kll·91 = - Hgltsool[Kll,lO] 2 _ Hgzlsool[Kll,1l] 2 _ Hgglsool
99
Mass Matrix [M] for the Mixed Model
[MHSHI [24].].,2] ... [Mll,l].] e
111111 - Il[SOO]13 _ 4 31 _ 4
111141 ’ 121S001 121S001 141S0011M221 = IIISOOI
23 _ 4 32 _ 4
1M151 - 112 - 222 1511$551 1M511 = 1M151
lM33l = 151$551 + 121222121$55 + $2211M121 - 1- 222 15 + 1222>21,>1$5511M211
- 1- -22 15 + 1-22>212>1$5513h 3h1M351 - 1- -22 15 + 1-221112>1$2513h 3hlM53l = 1- 222 15 + 1222>21,>1$521[M44l - 115 - 222 15 + 1222>11,11$5511M551 - [M441
a11 others are zero, and Ii are defined in Eq. (2.18b)
100
· - L L 2Aaa ° Aaa ° 2Eaa h2 * G44(h2)
· - L L ZA4s ‘ Aas ‘ 2Eas h2 + G45(h2)
· ° - L L 2Ass ‘ Ass ‘ 2Ess h2 + Gss (h2)
4CP = 3 · ——h2
avi av.=
J[SIJ] 1 [axl ax.]dX1dX2]113 J
. 3*1with
Q= ‘
av. av.M. = BL
-
.1[ax? SIJA [Ina axz axl axJ dX1dx23
For the tangent stiffness matrix, the coefficients are given by
1<KT>331 = 2lK23l = lK32l1<KT>331 = lK33]
au au au* A11 [EI; S11] S22] * A31[axI (S21 * $12)]
au au au* A13[axE S11] * Azsläig S22] + Aaaläig (S21 * $1233
101 2
H. H., l* A12*2x1 $11* * A22*1x1 $22* * A22*2x1 ($21 “’ $12**
8V 3V BV““ (*12*% $11* + A22*@ $22* + A32*@ ($21 ‘“ $12**
1 aw aw aw 2* 2 217$12*12 2
1 aw 2 aw aw $11]]
1 aw aw aw 2* 2 A21*[§ sr $21* + *(s;r* $22**1 2 1
1 aw aw aw 2+ 2 A23*[Ü 212 $22* + *(w* $21**1 2 2
1 aw aw aw 2’“2 A21**3T 57 $11* * *(3T* $12**1 2 1
1 aw 2 aw aw+ 2 A33**(K§* $11* * *Q Q $12**
1 aw aw _ aw 2* 2 A32*[B—>T Q $22* + ((7* $21**1 2 2
1 aw 2 aw aw* 2 ^22**(Q* $22* * *Q Q $21**1 1 2
aw 2 aw aw“’ A11*(Q* $11* * A12 *Q Q $11*
1 aw aw* Q *A11 *Q $01* “’ A12 *Q $02]
102
BW BW1 $01* 1 $02***
1 aw aw aw1 EE- [A21 *3Y’ $01* 1 A22 *5§“ $02* 1 $01]2 1 2 2
aw l1 *5I] $02***1 EI *A11*“$11* 1 A21*A$22**
1 1+ äg [A12[wSlll + A22[NS22]] + EI [A3l[w(Sl2 + $21)}}
1 aw 21 äg *A22**“$12 1 $21*** 1 A22*($Y§* $22*
aw aw aw 2 aw aw* A23 *$>q FQ $22* + ^32**sg* $12* + **33*ä sg $12*
aw 2 aw aw1 A21**S§]* $21* 1 A33[$YI Si; $21*
1 M1*B11*$11* 1 B21*$22* 1 B21**$12 1 $21***1 M2*B12*$11* 1 $22*$22* 1 $22**$12 1 $21***1 M6*B12*$11* 1 B22*$22* 1 B331(S12 1 $21***1 P1*E11*$11* 1 E21*$22* 1 E21**$21 1 $12***1 P2*E12*$11* 1 $22*$22* 1 E32*(S2l 1 $12***1 P6*El3*S1l* 1 E22*$22* 1 E33*(S2l 1 $12***
[(KT)63l = {K36l
3103
[(KT)73I = IK37l
l(KT)831 = lK38l
[<KT>°31 = [K39l
[(KT)ll,3] = [K3,ll]
AVI others are Same öS those in
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