„§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.

REFERENCES

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36. Gulati, S. T. and Essenberg, F., "Effects of anisotropy inaxisymmetric cylindrical shells," J. Appl. Mech., Vol. 34, p. 650,1967.

37. Zukas, J. A. and Vinson, J. R., "Laminated transversely isotropiccylindrical shells," J. Appl. Mech., Vol. 38, p. 400, 1971.

38. Dong, S. B. and Tso, F. K. W., "Dn a laminated orthotropic shelltheory including transverse shear deformation," J. Appl. Mech.,Vol. 39, p. 1091, 1972.

39. whitney, J. M. and Sun, C. T., "A higher order theory forextensional motion of laminated anisotropic shells and plates," J;Sound Vib., Vol. 30, p. 85, 1973.

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82

41. Dong, S. B., "Analysis of laminated shells of revolution," J. Eng.Mech. Div., Vol. 92(EM6), p. 135, 1966.

42. Dong, S. B. and Selna, L. G., "Natural vibrations of laminatedorthotropic shells of revolution," J. Composite Mat., Vol. 4(I), p.2, 1970.

V43. Nilson, E. A. and Parsons, B., "The Finite element analysis oFFilament—reinForced axisymmetric bodies," Fibre Science & Tech.,Vol; 2, p. 155, 1969.

44. Schmit, L. A. and Monforton, G. R., "Finite element analysis ofsandwich plate and laminate shells with laminated Faces," AIAA J.,Vol. 8, p. 1454, 1970.

L/45. Panda, S. C. and Natarajan, R., "Finite element analysis oFlaminated shells of revolution," Computers & Structures, Vol. 6, p.61, 1976.

46. Shivakumar, K. N. and Krishna Murty, A. V. Krishna, "A highprecision ring element For vibrations of laminated shells," J;Sound & Vib., Vol. 58, No. 3, p. 311, 1978.

47. Rao, K. P., "A rectangular anisotropic shallow thin shell Finiteelement," Comp. Meth. Appl. Mech. Eng., Vol. 15, p. 13, 1978.

48. Seide, P. and Chang, P. H. H., "Finite element analysis oflaminated plates and shells," NASA CR-157106, 1978.

49. Venkatesh, A. and Rao, K. P., "A doubly curved quadrilateral Finiteelement For the analysis of laminated anisotropic thin shell oFrevolution," Computers & Structures, Vol. 12, p. 825, 1980.

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52. Chao, N. C. and Reddy, J. N., "Geometrically nonlinear analysis oflayered composite plates and shells," Report No. VPI—E-83-10,Virginia Polytechnic Institute & State Univ., Blacksburg, VA.

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83

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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84

68. Hu, H., "On some variational principles in the theory of elasticityand the theory of plasticity," Scientia Sinic, Vol. 4, pp. 33-54,1955.

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· 1962.

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85

82. Chia, C. Y. and Prabhakara, M. K., "A general mode approach tononlinear Flexural vibrations of laminated rectangular plates,"ASME, J. Agpl. Mech., Vol. 45, No. 3, p. 623, 1978.

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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