NASA Contractor Report 4056

A Higher-Order Theory for Geometrically Nonlinear Analysis of Composite Laminates

J. N. Reddy and C. F. i i u

https://ntrs.nasa.gov/search.jsp?R=19870008689 2020-01-19T13:54:24+00:00Z

NASA Contractor Report 4056

A Higher-Order Theory for Geometrically Nonlinear Analysis of Composite Laminates

J. N. Reddy and C . F. Liu Virginia Polytechnic Institute altd State University BZacksburg, Virginia

Prepared for Langley Research Center under Grant NAG1-459

National Aeronautics and Space Administration

Scientific and Technical Information Branch

1987

Table o f Contents

Page

ABSTRACT ......................................................... 1 . INTRODUCTION .................................................

1.1 Background .............................................. 1.2 Review o f L i t e r a t u r e .................................... 1.3 Present Study ...........................................

2 . FORMULATION OF THE NEW THEORY ................................ 2.1 Kinematics .............................................. 2.2 Displacement F i e l d ...................................... 2.3 Strain-Displacement Relat ions ........................... 2.4 Cons t i t u t i ve Relat ions .................................. 2.5 Equations o f Motion .....................................

3 . THE N A V I E R SOLUTIONS ......................................... 3.1 In t roduc t i on ............................................ 3.2 The Navier Solut ions ....................................

4 . M I X E D VARIATIONAL PRINCIPLES ................................. 4.1 In t roduc t i on ............................................ 4.2 Var ia t i ona l P r i n c i p l e s ..................................

5 . F I N I T E ELEMENT MODEL ......................................... 5.1 In t roduc t i on ............................................ 5.2 Fini te-Element Model .................................... 5.3 So lu t i on Procedure ......................................

6 . SAMPLE APPLICATIONS .......................................... 6.1 I n t r o d u c t i o n ............................................ 6.2 Exact Solut ions ......................................... 6.3 Approximate (Fini te-Element) Solut ions ..................

6.3.1 Bending Analys is ................................. 6.3.2 V i b r a t i o n Analysis ...............................

7 . SUMMARY AND RECOMMENDATIONS .................................. 7.2 Some Comments on Mixed Models ........................... 7.3 Recommendations ......................................... 7.1 Summary and Conclusions .................................

REFERENCES ....................................................... APPENDIX A: Coe f f i c i en ts o f the Navier So lu t i on ................. APPENDIX B: S t i f f n e s s Coe f f i c i en ts o f the Mixed Model ...........

i v

10 10 12 16 17 19

24 24 26

28 28 29

34 34 34 40

46 46 46 57 57 68

73 73 74 74

75

82

85

iii

A HIGHER-ORDER THEORY FOR GEOMETRICALLY NONLINEAR ANALYSIS OF

COMPOSITE LAMINATES

J. N. Reddy and C. F. Liu Department of Engineering Science and Mechanics

B1 acksburg, Virginia Virginia Polytechnic Institute and State University

ABSTRACT

A refined, third-order laminate theory that accounts for the

transverse shear strains is developed, the Navier solutions are derived,

and its finite element models are developed. 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 also accounts for small

strains but moderately large displacements (i.e., von Karman strains).

Closed-form solutions o f the linear theory for certain cross-ply and

angle-ply plates and cross-ply shells are derived. 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 i nterelement

boundaries). The finite element is used to analyze cross-ply and angle-

ply laminated plates and shells for bending and natural vibration. Many

of the numerical results presented here for laminated shells should

serve as references for future investigations.

V

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', the laminate is assumed to be in a

state of plane stress, the individual laminae are assumed to be elastic,

and perfect bonding between layers is assumed. 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.

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 significantly smaller than the effective elastic moduli

The classical

1

2

dsn

along fiber directions.

render the classical laminated plate theory inadequate for the analysis

of thick composite plates.

These high ratios of elastic to shear moduli

The first, stress-based, shear deformation plate theory is due to

Reissner [l-31. The theory is based on a linear distribution of the

, n = 0,1,2, ... s=O

inplane normal and shear stresses through the thickness,

where (a l ,u ) and a6 are the normal and shear

are the associated bending moments (which are 2 stresses, (M1,M2) and M6

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

and u ) is determined from the equilibrium shear stresses (a3, a4

equations of the three-dimensional elasticity theory. 5

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 attributed to Basset

[ 4 ] . 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 u 1 along the c1 coordinate in the surface of the shell can be written in the form

(1.2a)

where c1 and c2 are the curvilinear coordinates in the middle surface of the shell, and uin) have the meaning

(1.2b)

3

Basset’s work did not receive as much attention as it deserves. In a

1949 NACA technical note, Hi ldebrand, Rei ssner and Thomas [ 51 presented

a displacement-based shear deformation theory for shells (also see

Hencky [6]). They assumed the following displacement field,

principle of minimum total potential energy. This approach results in

t

five differential equations in the five displacement functions, u, v,

Y. w, bX, and 4

The shear deformation theory based on the displacement field in Eq.

(1.3) for plates is often referred to as the Mindlin plate theory.

Mindlin [7] presented a complete dynamic theory of 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 [lo] to shear

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 [171. In

4

these higher-order theor ies, w i t h each a d d i t i o n a l power o f t h e th ickness

coordinate an a d d i t i o n a l dependent unknown i s in t roduced i n t o t h e

theory. Levinson [ 181 and Murthy [ 191 presented t h i r d - o r d e r t h e o r i e s

t h a t assume transverse i n e x t e n s i b i l i t y .

were reduced t o f i v e by r e q u i r i n g t h a t the t ransverse shear s t resses

vanish on the bounding planes o f t h e p la te .

the equ i l ib r ium equations o f the f i r s t - o r d e r theory i n t h e i r analys is .

As a r e s u l t , the higher-order terms o f the displacement f i e l d a re

accounted f o r i n the c a l c u l a t i o n o f the s t r a i n s b u t no t i n the governing

d i f f e r e n t i a l equations o r i n t h e boundary condi t ions. These t h e o r i e s

can be shown (see Librescu and Reddy [ZO]) t o be the same as those

described by Reissner 11-31. Recently, Reddy [21-231 developed a new

th i rd -order p l a t e theory, which i s extended i n t h e present study t o

laminated she l l s .

The n ine displacement f u n c t i o n s

However, both authors used

1.2 Review o f L i t e r a t u r e

Shel l s t ruc tu res are abundant on t h e e a r t h and i n space. Use o f

s h e l l s t ruc tu res dates back t o ancient Rome, where t h e r o o f s o f t h e

Pantheon can be c l a s s i f i e d today as t h i c k she l l s . She l l s t r u c t u r e s f o r

c a l and

t h e

a long t ime have been b u i l t by experience and i n t u i t i o n . No l o g

s c i e n t i f i c study had been conducted on the design o f s h e l l s u n t i

eighteenth century.

The e a r l i e s t need f o r design c r i t e r i o n f o r s h e l l s t r u c t u r e s

probably came w i t h the development of t h e steam engine and i t s a t tendant

accessories. However, i t was n o t u n t i l 1888, by Love [ 2 4 ] , t h a t t h e

f i r s t general theory was presented. Subsequent t h e o r e t i c a l e f f o r t s have

5

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 requirements and

derive 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):

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 [25] and Bert [26 ] . These theories, known as the Love's first

approximation theories (see Love 1241) are expected to yield

sufficiently accurate results when (i) the lateral dimension-to-

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

(1) plane sections normal to the

6

shells could lead to as much as 30% or more errors in deflections,

stresses, and frequencies.

As noted by Koiter [ 2 7 ] , refinements to Love's first approximation theory o f 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 to analyze

laminates that incorporated the bending-stretching coupling due to

material anisotropy. The laminates that Ambartsumyan analyzed are 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.

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 was derived by Widera and his colleagues [33,34] by

means of asymptotic integration of the elasticity equation.

homogeneous, isotropic material, the theory reduced to the Donnell's

Cheng and Ho [31]

For a

7

equation. An exposition of various shell theories can be found in the

article by Bert [261 and monograph by Librescu [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 [361 and Zukas and Vinson [37].

[38] presented a theory applicable to layered, orthotropic cylindrical

shells.

theory.

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 o f the Sanders shell

theory for laminated composite shells.

constant transverse shear stresses through thickness, and therefore

require a correction to the transverse shear stiffness.

Dong and Tso

Whitney and Sun [391 developed a higher-order shear deformation

This theory is based on a displacement field in which the

Such theories account for

As far as the finite element analysis of shells i s concerned, the

early works can be attributed to those of Dong [41], Dong and Selna

[ 42) , Wilson and Parsons [43] , and Schmit and Monforton [ 441. The

studies in [41-441 are confined to the analysis of orthotropic shells of

revolution.

composite shells include the works of Panda and Natarajan [45],

Shivakumar and Krishna Murty [46], Rao [471, Seide and Chang [481,

Venkatesh and Rao [49], Reddy and his colleagues [15,50-511, and Noor

and his col leagues [ 52-541.

Other finite element analyses of laminated anisotropic

8

1.3 Present Study

While the three-dimensional theories [ 55-60] give more accurate

results than the lamination (classical or shear deformation) theories,

they are intractable. For example, the 'local' theory of Pagano [ 60 ]

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.

631, the refined plate theory provides improved global response

estimates for deflections, vibration frequencies and buckling loads for

laminated composite plates.

findings, deals with the extention of the third-order plate theory of

Reddy [21-231 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 study. The significant and novel contributions of the

research conducted (in addition to those reported in the first year's

report [ 2 3 ] ) are:

As demonstrated by Reddy [21-231 and his colleagues [61-

The present study, motivated by the above

1. The formulation of a new third-order theory of laminated shells

that accounts for a parabolic distribution of the transverse

shear stresses and the von Karman strains.

2. The derivation of the exact solutions of the new theory for

certain simply supported laminated composite shells.

9

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, Co-finite-element and its

application to the bending and vibration analysis of laminated

composite shells.

2. FORMULATION OF THE NEW THEORY

2.1 Kinematics

Let (c1,&2,s) denote the orthogonal curvilinear coordinates (or

shell coordinates) such that the

curvature on the midsurface s=O,

perpendicular to the surface s=O

the lines of principal curvature

e l - and c2-curves are lines of

and 5-curves are straight lines

. For cylindrical and spherical shells

coincide with the coordinate lines.

The values of the principal radii of curvature of the middle surface are

denoted by R1 and R2.

The position vector of a point on the middle surface is denoted

by r, and the position of a point at distance 5 from the middle surface

is denoted by R. .., The distance ds between points ( E , ~ , ~ ~ , O ) and (cl+dcl,

c2+dc2,0) is determined by (see Fig. 2.1)

..,

(ds)' = dr I dr ..,

2 2 = + a2(dc2)

ar I

where dr = sldcl + g2dc2, the vectors g1 and g2 (gi = -) are tangent I . . , I aci

to the t1 and c2 coordinate lines, and a1 and a2 are the surface metrics

.., .., I I

The distance dS between points ( c1,c2,c) and ((l+dc1,c2+dc2,s+dr) is

given by

(dS)2 = dR dR I

11

Figure 2 .1 Geometry and stress resultants o f a shell

12

aR aR - ac1 at2 2 as

- a R - where dR = - de1 + - dg +

coefficients

ds, and L1, L2 and L3 are the Lame’

aR aR ... It should be noted that the vectors s1 E and G E are parallel -2 ac2

to the vectors g1 and g2, 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 s ~ - ~ be the top and bottom

5-coordinates of the k-th layer. Before we proceed, a set of

simplifying assumptions that provides 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 displacements 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:

where t is time, (U,V,.) are the displacements along the (g1,c2,c)

13

f

U a, >

0 7 0 -0

ii

14

coordinates, (u,v,w) are the displacements o f a p o i n t on the middle

surface and $1 and $2 are the r o t a t i o n s a t 5 = 0 o f normals t o the

midsurface w i t h respect t o the c2- and cl-axes, respec t i ve l y . A l l o f

(u,v,w,~l,~2,bl,b2,el,e2) are funct ions of el and 6 only. The 2 p a r t i c u l a r choice o f the displacement f i e l d i n Eq. (2.5) i s d i c t a t e d by

the des i re t o represent the t ransverse shear s t r a i n s by quadra t ic

funct ions o f t h e th ickness coordinate, 5, and by the requirement t h a t

the transverse normal s t r a i n be zero. The kinematics o f deformat ion o f

a t ransverse normal i n var ious theo r ies i s shown i n F ig. 2.3.

The func t ions bi and ei w i l l be determined us ing the c o n d i t i o n t h a t

the transverse shear stresses, ul3 =

and bottom surfaces o f t he she l l :

and aZ3 = vanish on the top u4

These condi t ions are equ iva len t to , f o r s h e l l s laminated of o r t h o t r o p i c

layers, t he requirement t h a t t he corresponding s t r a i n s be zero on these

surfaces.

r a d i i of curvature are g iven by

The transverse shear s t r a i n s o f a s h e l l w i t h two p r i n c i p a l

a i 1 a i u = - + - - - - ‘5 as a1 a t 1 R~

- - U 1 aw u - + 41 + 2sal + 3s2e l + - - - - R1 OL1 a % R1

(2.7)

1 aw v V - - - + b2 + 2cb2 + 3s2e2 + -- - - R2 O2 a % R2

Se t t i ng E5(c1,c2,k 7 h ,t) and ~ q ( t l r S 2 9 k 7 h ,t) t o zero, we o b t a i n

15

1 UNDEFORMED

DEFORMED IN CLASSICAL

(KIRCHHOFF) THEORY

DEFORMED IN THE THIRD ORDER (PRESENT) THEORY

F igure 2.3 Assumed deformat ion pa t te rns o f the t ransverse normals i n var ious displacement-based theor ies.

16

- 4 1 aw

- 4 1 aw

e 1 - - 2 ( 4 1 + - -)

e2 - - 2 (@2 + (r -)

3h O 1 a c l

3h 2 ac2

Substituting Eq. (2.8) into Eq. (2.5), we obtain

5 - v = (1 + -)v R, c

This displacement fie

3 4 1 aw

3 4 1 aw

3h

(2.9)

d 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-Displacement Relations

Substituting Eq. (2.9) into the strai n-di spl acement re1 at ions

referred to an orthogonal curvilinear coordinate system, we obtain

where

(2.10)

17

0 - aw 1 - 4 aw h a x2

K4 - - 2 (Cb2 + -) €4 - @2 +

0 - aw 1 4 aw h axl

K5 = - 2 (e1 + -) €5 - @1 + q

a @ 2 a@l + - O a v + - + - - au aw aw 0 €6 = a x2 axl ax2 ’ K6 = ax2

(2.11)

Here xi denote the Cartesian coordinates (dxi = aidcis i = 1,Z).

2.4 Constitutive Relations

The stress-strain relations f o r the k-th lamina in the lamina

coordinates are given by

0

0

Q66 - ( k )

0

0

0

- (k) Q44

€1

€ 6

€4

€ 5 (k)

(2.12)

where vi ( k ) are the plane stress reduced stiffnesses o f the k-th lamina

18

in the lamina coordinate system.

in terms of the engineering constants of a lamina:

The coefficients qij can be expressed

(2.13)

- - - ‘44 = ‘23 Q55 = ‘13 Q66 = ‘12

To determine the laminate constitutive equations, Eq. (2.12) should be

transformed to the laminate coordinates. We obtain

r4) = 1;: r4} (2.14) ‘5 ( k ) 55 ( k ) €5 (k )

where 4 2 2 4 -

Q,, = Qll cos e + 2(Q12 + 2066)~in e cos e + gz2 sin e

- 4066)sin 2 e cos 2 e + Q12(sin 4 e + cos 4 e ) Q12 = (Q11 + Q22

- 4 2 2 4 Q,, = Qll sin e + 2(Q1, + 2Q66)~in e cos e + Q,, cos e

Q,, = (Qll + qz2 - 2Q12 - ~ T j ~ ~ ) s i n 2 e cos 2 e + Q66(~in 4 e + cos 4 e)

2 2 - Q4, = Q4, cos e + V,, sin e

19

2 2 - = Q,, cos e t q44 s i n e Q55 (2.15)

2.5 Equations o f Mot ion

The dynamic vers ion of the p r i n c i p l e o f v i r t u a l work f o r t h e

present case y i e l d s

dxldx2}dc - q6wdxldx2 - 6(J h/2 { $ k ) ~ [ (t)2+(t)Z+(~)2]dxldx2}dc) ] d t

D -h/2 Q

1617 ( a 2" 2" - - - - + 9) + IlW)6W av

2 9h4 axl ax2 + ('3 % + '5

(2.16)

where q i s t he d i s t r i b u t e d transverse load, Niy Miy e tc . a re t h e

resu 1 tants ,

20

The inert?

and

(2.17)

s Ti and Ti (i = 1,2,3,4,5) are defined by th, equations,

T1 = I 1 + li;; 2 I 2 ,

4 4 - - 1 - I = I + - 2 2 R 1 '3 3h

4 4 2 I5

- - 1 - 5 = I2 + 5 I3 3h2 I4 -

l6 I 8 3h

- I 4 = I3 - - Z 1 5 + g h 4 7

l6 I 4 I = - -

3h 2 1 5 - 2 7 '

l6 I 4 1' = - - 3h 2 1 5 - 2 7 (2.18a)

I

21

The governing equations o f motion can be derived from Eq. (2.16) by

integrating the displacement gradients by parts and setting the

coefficients o f 6u,6v,6w and 6'i (i = 1,2) to zero separately:

a N 1 + - = aN6 aw axl ax2 6u: -

6v: -

2 2 a '6 )

a p 2 + 2 axlax2

a91 aQ2 aK1 aK2 4 a2P1 +2 2 ( 2

ax2 3h axl ax2 axl ax2 h2 axl ( + - ) + - - + - - - - 6w: -

- - - - N1 N2 + N(w) = - I3 .; au - - a 4 ~ av + T i 3 a42 + I1w I5 + % 3 R1 R2

1617 ,aZ; + a2; 7) - q - - -

9h4 axl 2 ax2

aw aP1 aP6 - - 2' + '4'1 - '5 + - - + - K 4 - - - ( + - )=I " Q1 h2 1 3h2 axl ax2 641: - aM1 aM6

axl ax2

(2.19)

where

aw =) (2.20) a aw aw a (N1 + N6 + - (N6 + N2 ax2 N(w) = -

axl ax2

The essential (i .e., geometric) and natural boundary conditions o f the

theory are given by:

22

Un’ Uns’ W ’ an’ aw as’ aw @n’ @ns (essent ia l )

_ _ _ - I

Nn’ Nns* Qn, Pn’ Ps’ Mn’ Mns (na tu ra l ) (2.21)

where 2 2 = n N + n N + 2 n n N

‘n x 1 y 2 X Y 6

Nns = (N2 - N ) n n + N6(nx 2 2 - ny) 1 X Y

= nxP1 2 + n 2 P ‘n Y 2 X Y 6

+ 2n n P

= (P2 - P ) n n + P6(nx 2 2 - ny)

= (M2 - M ) n n + M6(nx 2 2 - ny)

‘ns 1 X Y

= nxM1 2 + n 2 M Mn Y 2 X Y 6

Mns 1 X Y

+ 2n n M

aPn aPns 4 K - - aw aw 4 Qn - Nn an + Nns + 3 (K + -) + Qn - -

h2 as

= n K + n K Kn x 1 y 2

P n - - - - 4 P ..,

3h2 - - - - - 4 P

‘ns 3h2 “’ - Mn = Mn - - 4 P

3h2 - - - 4 P

“ns = Mns 3h2 ns (2.22)

23

and nx and ny are the d i r e c t i o n cosines o f the u n i t normal on the

boundary o f t he laminate.

The r e s u l t a n t s can be expressed i n terms o f t he s t r a i n components

I us ing Eqs. (2.10) and (2.12) i n Eq. (2.14). We ge t

Ni - - Ai j ~ g + 6. .IC. 0 + E . .K 2 1 J J 1 J j

Mi - - B i j ~ j 0 + D

O + F i j'j P. = E 1

2 + F..K jKj 1~ j

jKj 1~ j 2 O + H..K

(i,j = 1,2,6) (2.23)

0 1 5 j j

K1 = D 5 c j + F K

where Aij, Elij, etc. are the laminate s t i f fnesses ,

( A . B. . ,Di ,Ei j,Fi j,H. .) ij' ij 'J

f o r i,j = 1,2,4,5,6.

(2.25)

3. THE NAVIER SOLUTIONS

3.1 Introduction

Exact solutions of the partial differential equations (2.19) on

arbitrary domains and for general conditions is not possible. However,

for simply supported shells whose projection in the xlx2-plane is a

rectangle, the linear version of Eq. (2.19) can be solved exactly,

provided the lamination scheme is of antisymmetric cross-ply [Oo/900/

0°/90 O . . . ] or symmetric cross-ply [0°/900...]s type.

solution exists if the following stiffness coefficients are zero 1211:

The Navier

Fi6 = Hi6 = 0 , (i = 1,2) (3 .1 ) Ai6 = Bi6 - - Di6 - - Ei6 - -

= o A45 = O45 = F45 The boundary conditions are assumed to be of the form,

u(xl,O) = u(xl,b) = v(0,x2) = v(a,x2) = 0

w(xl,O) = w(xl,b) = w(0,x2) = w(a,x2) = 0

aw (0,x2) = - (a,x2) = 0 aw aw aw axl axl ax2 ax2 - ( ~ 1 ~ 0 ) = - (xl,b) = -

N2(x1,0) = N (x ,b) = Nl(0,x2) = Nl(a,x2) = 0 2 1

where a and b denote the lengths along the xl- and x2-directions,

respectively (see Fig. 3.1).

24

25

/

Figure3.1 The geometry and the coordinate system f o r a Projected area of -she1 1 element

26

3.2 The Navier Solution

Following the Naviw solution procedure (see Reddy [21]), we assume

the following solution form that satisfies the boundary conditions in

Eq. (3.2):

m

m

where

f (x ,x ) = cosax1sin~x2, f (x ,x ) = sinax C O S B X ~

f3(x1,x2) = sinaxlsin~x2, a = mn/a, B = nn/b

1 1 2 2 1 2 1 (3.4)

Substituting Eq. (3.3) into Eq. (2.19), we obtain

where Qmn are the coefficients in the double Fourier expansion o f the

transverse load,

27

and the coefficients Mi j and Ci j(i,j = 1,Z ,... ,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).

are evaluated using a finite number o f terms in the series.

vibration analysis, Eq. (3.5) can be expressed as an eigenvalue

equation,

The series in Eq, (3.3)

For free i

I ( [ C I - UZIMl){A} = (0) (3.7)

where {A} = {Umn,Vmn,Wmn,m,!,n,m~n}T, and w is the frequency of natural

vibration. For static bending analysis, Eq. (3.5) becomes

4. MIXED VARIATIONAL PRINCIPLES

4 .1 Introduction

The variational formulations used for developing plate and shell

elements can be classified into three major categories:

(i) formulations based on the principle o f 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).

formulations are called, respectively, the displacement models,

equilibrium models and mixed models.

The finite-element models based on these

In the principle of virtual displacements, one assumes that the

kinematic relations (i .e. , strain-displacement relations and geometric 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

a U ( E . .)

ij ij --J-J-=.

a E

where E . . and uij are the components of strain and stress tensors,

respectively. 1 J

When the principle of total potential energy is used to

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

only in a variational sense.

There are a number of mixed variational principles in elasticity

(see [64-711). The phrase 'mixed' is used t o 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

in which both primal and dual

y in the interior and on the

two kinds of mixed models: (i) models

variables are interpolated independent

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 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 } = [ A * ~ { E ~ } + [B*I{M} + IE*I{P}

30

t o -{K’} = [B*1 { E } - [D*l{M} - [F*l{P}

-{K} = [E*] { E } - [F*lt{M} - [H*]{P} t o

where

{KC} = - w 1 a w 2 ay R2 2 ay a v + - + - (-)

au av aw aw 1 ay ax ax ay - + - + - -

A c1 = 2 and c = - 4 3h h2

[B*] = [B][D*] + [E][F*]

[E*] = [B][F*] + [E][H*]

(not symmetric)

(not symmetric)

-

-

[Dl [FI

[F*I [H*l [FI [HI (4.2)

Note that in this part the notation x = x1 and y = x2 is used, and [ I t denotes the transpose o f a vector or matrix.

To develop a mixed variational statement o f the third-order

laminate theory, the generalized displacements (u, v, w, 41,

generalized moments (MI, M2, Me, PI, P2, P6) are treated as the

dependent variables.

and

31

1 o t = s a [y { E 1 + { E O } ~ ( [ B * ] { M } + [E* ] {p } )

A A A

- J- (Nnun + Nsus + Qnw + Mn+,, + M,$,)ds r

(4.3)

where the q u a n t i t i e s w i t h a ha t over them denote s p e c i f i e d values, and

Qn = Qlnx + Q2ny

un = unx + vny , us = -uny + vnx

a - n - - n - a an x ax y ay ' as x ay y ax

a a _ - a - n - a + n - - -

2 2 2 2 Nn = N n + 2N n n + N2ny , NS = (N2 - N )n n + N6(nx - ny) l x 6 X Y 1 X Y

w i t h

(4.4)

(4.5)

32

Note that the bending moments (MI, M2, M6) do not enter the set of

essential boundary conditions, and the resultants (PI, 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.

laminate can assume, the one that renders the functional nH1 stational-y

also satisfies the equations of equilibrium (2.19) and the kinematic-

constitutive equations [i.e., the last two equations of Eq. (4.3)].

Of all possible configurations a

We note that the variational statement in Eq. (4.3) contains the

second-order derivatives of the transverse deflection w (through {K}).

To relax the continuity requirements on w, we integrate the

term {P}t{~} by parts to trade the differentiation to {P}. We obtain

A A A A

aw en an 9 e~ as

- - - aw = -

33

I

We note that the stress resultants Pn and Ps (hence PI, 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.

5. FINITE-ELEMENT MODEL

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 [72,73] and Hellan

[741

Since the bending moments do not enter the set of essential

boundary conditions, they are not required to be continuous across

interelement boundaries.

discontinuous (between elements) bi 1 inear or higher-order approximation

of the bending moments. The present section deals with the development

o f a mixed model based on nH2.

Thus, one can develop mixed models with

5.2 Finite-Element Model

Let n,

represented

where = n

the domain (i.e., the midplane) of the laminate, be

by a collection of finite elements

(5.1) - n = u n , ne n af = empty for e # f

e= 1

u r is the closure of the open domain a and r is its

boundary. Over a typical element ne, each of the variables u, v, w,

etc. are interpolated by expressions of the form

N N N u = c U.$ , v = c V.$ , w = W.$ (5.2) j=l J j j= 1 J j j=l ~j

34

35

etc. , where {s, .} denote the set of admissible interpolation functions

and j denotes the number of 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, (Q Q ),

(M1,M2,M6) and (Pl,PZ,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.

resulting element is called a Co-element, because no derivative of the

dependent variable is required to be continuous across interelement

boundaries.

J

x y Y

The

To develop the finite-element equations of a typical element, we

first compute the strains, rotations, and resultants in terms of their

finite element approximations. We have

{ E O } = {EOL} + { E O N } (5-3)

= [HL]{nl} + [H"]{A*} (5.4)

36

aw aw 2 - - a x a y

N 2 ,: $ [ H ] { A } (5.5)

{&.ON} = [ HN] {SA'}

37

{Q} =

4 4 [ H ]{A } (5.11)

(5.12)

(5.13)

aw , f = - and aw Y aY Here, fx = - ax

{$i} = $2 0 . . $ N I 9 E$i,x} = {$1,x $2,x * e * $N,xl, etc .( 5.14)

and {u}, for example, denotes the column o f the nodal values of u.

The finite-element model for t h e refined shear deformation theory

is derived from the variational statement in Eq. (4.6) over an

element.

element Q~ is given by

The first variation o f the functional in (4.6) for a typical

0 = 5 [{~E~}~([A*~{C'} + [ B * l { M } + [E*l{P})

+- {sM}t({~s} + [B*] {E } - [ D * l { M } - [F*l{P}) ne

t o

38

+ { 6 P } t ( - C1{Ks} + [E*]{E'} - [F*l{M} - [H*l{P})

- (Nn6un + N 6u + Qn6w + Mn6$,, + Ms6$, s s re

+ c e SP + cles6Ps)ds I n n

Subs t i t u t i on o f Eqs. (5.2)-(5.14)

Eq. (5.15), we ob ta in

[Kl1] [K12] [K13] [K14] [K15]

[K21] [kZ2] [K23] [K24] [KZ5]

[K31] [K3'] [K33] [K34] [K35]

[K41] [K4'] [K43] [K44] [K45]

[K51] [K52] [K53] [K54] [K55]

where

(5.15)

n t o the v a r i a t i o n a l statement i n

[K"] = [HLIt[A*][HL]dA , IF1} = 8 [ H 2 I t ds ae re

[K l2 1 ' 7 8 [HLIt[A*]([HN] + 2[Hol)dA

ne

(5.16)

[K14] = [HLlt[B*I[H3]dA = [K 41 1 t ne

39

[K15] = [ [HLIt[E*][H3]dA = [K 51 ] t ae

[iZ21 = s (Hol + [H N t 1 ) [A*l([Hol + 7 1 N [H ])dA + s [

ae ae

1

[KZ41 = (([HOI + [H N t I) [B*][H3])dA ne

= .f ($ [H31t[B*lt([HN] + 2[Ho]))dA ne

N t I t L t ([Hol + [H I) [E*l[H31 + cl[H I [H I )dA [KZ5] =

ne

[K5'1 = ([H3lt[E*l([Ho1+ 7 1 N [H I) + cl[HL1[H1J)dA ae

[K -3 3 ] = .f [H 2 t - ] [A][H2]dA, [icZ3l = S [ H I t A ] [A][H2]dA = [K -32 ] t ae ne

L t 3 43 t = S [ H I [ H IdA = [K ] ae

[K441 = - [H3lt[D*1[H31dA, Qe

[K45] = - J [H3It[F*][H 3 ]dA = [K 54 ] t ae

\

40

[ K 55 1 = - I [H3It[H*1[H3]dA ne

IF5} = 1 [H3It[Tlt ds ne

(5.17)

Clearly, the element stiffness matrix is pJ symmetric.

case (i.e., [HN] = [O]), the element stiffness matrix is symmetric.

For the linear

The

theories can be

form of the

finite-element models for the classical and first-order

obtained as special cases from Eq. (5.16). An explicit

coefficients of the stiffness matrix in Eq. (5.16) is g

B.

ven in Appendix

5.3 Solution Procedure

The assembled finite-element equations are of the form

tKO(A)l{A) = {F) (5.18)

where [ K O ] is the assembled (direct) stiffness matrix, unsymmetric in

general, and {A} is the global solution vector.

matrix i s a nonlinear function o f the unknown solution, Eq. (5.18)

should be solved iteratively.

quite commonly used in the finite-element analysis o f nonlinear

problems:

iteration methods.

nonlinear equations.

Because the stiffness

Two iterative methods of analysis are

the Picard-type direction iteration and the Newton-Raphson

Here the Newton-Raphson method is used to solve the

41

Suppose that the solution is required for a load of P. The load is

divided into a sequence of load steps AP1, A P ~ , ..., A P ~ such that P

= c A P ~ . At any load step i, Eq. (5.18) is solved iteratively to

obtain the solution. At the end of the r-th iteration the solution for

n

i=l

the next iteration is obtained solving the following equation

(5.19) [K T r (A )]{6Ar+'} = - {R} 3 -[K D r (A )]{Ar} + {F}

for the increment of the solution {sar+'}, and the total solution is

computed from {Ar+'} = {Ar} + {&A r+l

where [KT] is the tangent stiffness matrix,

(5.20)

(5.21)

A geometrical explanation o f the Newton-Raphson iteration is given in

Fig. 5.1.

For the finite-element models developed here, the tangent stiffness

matrix is symmetric.

model in Eq. (5.16) for plates (1/R1 = 1/R2 = 0) is given by

For example, the tangent stiffness matrix for the

[K21T] [KZzT] symm.

[ K~~~ I. 5 1T [K ] .... ........ where

[K1lT] = [K"

[KZzT] = [kz2

(5.22)

42

F

Load, F

0 Displacement, U

Figure 5.1 Geometric i n t e r p r e t a t i o n o f t he Newton-Raphson i t e r a t i o n f o r the s o l u t i o n o f one-parameter p rob l ems.

43

[K31T] = [O],

ae

44 [K44T] = [K

t[A*IIHNl + ["?[A*] [$l)dA{A2}

t[B*][H3]dA){A4} ae

32 [K32T] = [K 1 ,

42T 24 , [K 1 = [K

53 5 4T 54 [K53T] = [K 1 , [K 1 = [K

-N [H 1 =

[K33T] = [K33]

K25] ,

K55]

(5.23)

For a1 1 coupled (i .e. bending-stretching coupling) problems, the

mixed models of classical, first-order and third-order theories have

respectively. If the

, the element degrees six, eight and eleven degrees of freedom per node,

bending moments are eliminated at the element leve

of freedom can be reduced by 3 (see Fig. 5.2).

The finite-element model (5.16) for the dynam

form,

[ K I M + [ M I m = {F}

c case i s of the

(5.24)

where [MI is the mass matrix (see Appendix 6).

(5.24) can be written as

For free vibration, Eq.

44

w. / //

2

u1

(a ) Displacement model

/

J t *

'a 1 1 "1 M1 ' p l

( b ) Hixed model

Figure5.2 The displacement and mixed f i n i t e elements f o r the t h i r d - o r d e r shear deformation theory

45

[ K ] { A } = A [ M l { A } (5 .25)

where ;x i s t h e square of the frequency.

The eva lua t ion o f the element matrices requ i res numerical

in tegra t ion .

c o e f f i c i e n t s associated w i t h t h e shear energy terms. More s p e c i f i c a l l y ,

the 2 x 2 Gauss r u l e i s used f o r shear terms and t h e standard 3 x 3

Gauss r u l e i s used f o r the bending terms when the nine node quadrat ic

isoparametr ic element i s considered.

Reduced i n t e g r a t i o n i s used t o evaluate the s t i f f n e s s

6. SAMPLE APPLICATIONS

6.1 Introduction

A number of representative problems are analyzed using the higher-

order theory developed in the present study.

included here illustrate the accuracy of the present theory. These

problems are solved using the closed-form solutions presented in Section

3.

the mixed finite-element model described in Section 5.

The first few problems

Then problems that do not allow closed-form solutions are solved by

The geometries of typical cylindrical and spherical shell panels

are shown in Fig. 6.1. Of course, plates are derived as special cases

from cylindrical or spherical shell panels.

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.

distribution of the transverse load, the series reduces to a single

term.

For a sinusoidal

1. Four-layer, cross-ply (0/90/90/0) square laminated flat plate under

This example demonstrates the relative accuracy of the present

sinusoidal load

higher-order theory when compared to three-dimensional elasticity theory

and to the first-order theory. Square, cross-ply laminates under

simply-supported boundary conditions [see Eq. (3.2)] and a sinusoidal

distribution of the transverse load are studied for deflections. The

lamina properties are assumed to be

46

47

Figure 6.1 Geometry of a typical cylindrical and spherical shell

48

Plots o f the nondimensional center deflection [w = wE2h3)/qoa4] versus

the side to thickness ratio (a/h) obtained using various theories are

shown in Figure 6.2. The present third-order theory gives the closest

solution to the three-dimensional elasticity solution [58] than either

the first-order theory or the classical theory.

2. An isotropic spherical shell segment under point load at the center.

The problem data are (see Fig. 6.1 for the geometry)

R1 = R2 = 96.0 in., a = b = 32.0 in., h = 0.1 in., 7 El = E2 = 10 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 f irst-order shear deformation theory

(FSDT) and classical shell theory (CST) for various terms in the series

is presented in Table 1 (for simply supported boundary conditions). It

should be noted that Vlasov [76] did not consider transverse shearing

strains in his study. The difference between the values predicted by

HSDT and FSOT is not significant for the thin isotropic shell problem

considered here.

3. Cross-ply spherical shell segments under sinusoidal, uniform, and point loads.

The geometric parameters used are the same as those used in Problem

2, and the material parameters used are the same as those used in

Problem 1. The shell segments are assumed to be simply supported.

Nondimensionalized center deflection of various cross-ply shells under

sinusoidal, uniform, and point loads are presented in Tables 2 through

49

z t;; 4

T

u U Y

w

M

2 0 w I I-

0

si

9

0 M

0 m

0 rl

0

0

50

Table 1. Center d e f l e c t i o n ( -w x 103) of a simply supported spher ica l s h e l l segment under p o i n t load a t the center (see Fig. 6.1)

Number o f terms i n the ser ies h Theory N = 9 N = 49 N = 99 N = 149 N = 199

CST Vlasov [79 ] - -

0.1 FSDT [40 ] 32.594 39.469 HSDT 32.584 39.458

0.32 FSDT 3.664 3.902 HSDT 3.661 3.899

1.6 FSDT 0.165 0.171 HSDT 0.164 0.170

3.2 FSDT 0.035 0.038 HSDT 0.035 0.037

6.4 FSDT 0.007 0.008 HSDT 0.007 0.007

39.591 39.560 39.724 39.714

3.919 3.916 0.174 0.172 0.039 0.037 0.009 0.007

- 39.786 39.775

3.927 3.923 0.175 0.172 0.039 0.037 0.009 0.007

39.647

39.814 39.803

3.932 3.927 0.176 0.172 0.040 0.037 0.009 0.007

-

51

4, respectively.

higher-order theory and the first-order theory increases with increasing

values of R/a. For a/h = 10, the difference between the deflections

given by FSDT and HSDT is larger than those for a/h = 100.

unsymmetric laminates (0/90) , FSDT yields higher deflections than HSDT,

whereas for symmetric laminates HSDT yields higher deflections than FSDT

for values of a/h = 10.

difference between the solution predicted by the first-order theory and

the higher-order theory is more significant, with FSDT results higher

than HSDT, especially for antisymmetric cross-ply laminates.

The difference between the solutions predicted by the

For

Note that for point-loaded shells the

4. Natural vibration of cross-ply cylindrical shell segments

Nondimensionalized fundamental frequencies of cross-ply cylindrical

[Oo/9O0], shells are presented in Table 5 for three lamination schemes:

[O0/9Oo/O0] , and [0°/900/900/00]. 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 shell segments

Nondimensionalized natural frequencies obtained using the first-

and higher-order theories are presented in Table 6 for various cross-ply

spherical shell segments. Analogous to cylindrical she Is, the first-

order theory underpredicts fundamental natural frequenc es of

antisymmetric cross-ply shells; for symmetric thick she 1s and symmetric

shallow thin shells the trend reverses.

52

Table 2. Nondimensionalized center deflections, w = (-wh3E /q a4)103, of cross-ply laminated spherical she1 1 segments uGdeP sinusoidally distributed load (a/b = 1, R 1 = R2 = R, q, = 100)

- W

oo/900 0 O /goo /oo oo/900/900/oo

R/a Theory a/h=100 a/h=lO a/h=100 a/h=lO a/h=100 a/h=lO

5 FSDT 1.1948 HSDT 1.1937

10 FSDT 3.5760 HSDT 3.5733

20 FSDT 7.1270 HSDT 7.1236

50 FSDT 9.8717 FSDT 9.8692

100 FSDT 10.4460 HSDT 10.4440

Plate FSDT 10.6530 R/a=m FSDT 10.6510

11.429 11.166 12.123 11.896 12.309 12.094 12.362 12.150 12.370 12.158 12.373 12.161

1.0337 1.0321 2.4109 2.4099 3.6150 3.6170 4.2027 4.2071 4.3026 4.3074 4.3370 4.3420

6.4253 6.7688 6.6247 7.0325 6.6756 7.1016 6.6902 7.1212 6.6923 7.1240 6.6939 7.1250

1.0279 1.0264 2.4030 2.4024 3.6104 3.6133 4.2015 4.2071 4.3021 4.3082 4.3368 4.3430

6.3623 6.7865 6.5595 7.0536 6.6099 7.1237 6.6244 7.1436 6.6264 7.1464 6.6280 7.1474

53

Table 3. Nondimensionalized center de f l ec t i ons , w = (-WE h3/q a4) x 103, o f c ross-p ly laminated spher ica l s h e l l s2gmen?s under un i fo rm ly d i s t r i b u t e d load

0"/90" 0" /90 O /oo oo/900/900/oo R - Theory a/h=100 a/h=lO a/h=100 a/h=lO a/h=100 a/h=lO a

5 FSDT HSDT

10 FSDT HSDT

20 FSDT HSDT

50 FSDT HSDT

100 FSDT HSDT

P la te FSDT m

1.7535 1.7519 5.5428 5.5388

11.273 11.268 15.714 15.711 16.645 16.642 16.980 16.977

19.944 17.566 19.065 18.744 19.365 19.064 19.452 19.155 19.464 19.168 19.469 19.172

1.5118 1.5092 3.6445 3.6426 5.5473 5.5503 6.4827 6.4895 6.6421 6.6496 6.6970 6.7047

9.794 10.332 10.110 10.752 10.191 10.862 10.214 10.893 10.218 10.898 10.220 10.899

-

1.5358 1.5332 3.7208 3.7195 5.6618 5.666 6.6148 6.6234 6.7772 6.7866 6.8331 6.8427

9.825 10.476 10.141 10.904 10.222 11.017 10.245 11.049 10.294 11.053 10.251 11.055

54

Table 4. Nondimensionalized center de f l ec t i on o f cross-p ly spher ica l s h e l l segments under p o i n t load a t t he center

- w h 3 E 2 w = - (- )102, a/b = 1, a/h = 10 Pan

Center de f l ec t i on , w R/a Theory 0"/90" 0" /90 " / O " oo/900/900/oo

5 FSDT HSDT

10 FSDT HSDT

20 FSDT HSDT

50 FSDT HSDT

100 FSDT HSDT

P1 a t e FSDT m HSDT

7.1015 5.8953 7.3836 6.1913 7.4692 6.2714 7.4909 6.2943 7.4940 6.2976 7.4853 6.2987

5.1410 4.4340 5.2273 4.5470 5.2594 4.5765 5.2657 4.5849 5.2666 4.5861 5.2572 4.5865

4.9360 4.3574 5.0186 4.4690 5.0496 4.4982 5.0557 4.5065 5.0565 4.5077 5.0472 4.5081

55

Table 5 Nondimensionalized fundamental frequencies o f cross-ply c y l i n d r i c a l s h e l l panels (see Fig. 6.1 f o r geometry).

0"/9O0 oo/900/oo oo/900/900/oo

R/a Theory a/h=100 a/h=lO a/h=100 a/h=lO a/h=100 a/h=lO

5 FSDT HSDT

10 FSDT HSDT

20 FSDT HSDT

50 FSDT HSDT

100 FSDT HSDT

P la te FSDT m HSDT

16.668 16.690 11.831 11.840 10.265 10.270 9.7816 9.7830 9.7108 9.7120 9.6873 9.6880

8.9082 9.0230 8.8879 8.9790 8.8900 8.9720 8.8951 8.9730 8.8974 8.9750 8.8998 8.9760

20.332 20.330 16.625 16.620 15.556 15.550 15.244 15.240 15.198 15.190 15.183 15.170

12.207 11.850 12.173 11.800 12.166 11.791 12.163 11.790 12.163 11.790 12.162 11.790

20.361 20.360 16.634 16.630 15.559 15.550 15.245 15.230 15.199 15.190 15.184 15.170

12.267 11.830 12.236 11.790 12.230 11.780 12.228 11.780 12.227 11.780 12.226 11.780

56

Table 6. Nondimensionalized fundamental f requencies o f cross-p ly laminated spher ica l s h e l l segments

a2 - h

- w = w - Jp/E2

0"/90" 0" /goo /oo oo/900/900/oo

R/a Theory a/h=100 a/h=lO a/h=100 a/h=lO a/h=100 a/h=lO

5 FSDT HSDT

10 FSDT HSDT

20 FSDT HSDT

50 FSDT HSDT

100 FSDT HSDT

P la te FSDT HSDT

28.825 28.840 16.706 16.710 11.841 11.84 10.063 10.06 9.7826 9.784 9.6873 9.6880

9.2309 9.3370 8.9841 9.0680 8.9212 8.999 8.9034 8.980 8.9009 8.977 8.8998 8.9760

30.993 31.020 20.347 20.350 16.627 16.62 15.424 15.42 15.244 15.24 15.183 15.170

~~

12.372 12.060 12.215 11.860 12.176 11.81 12.165 11.79 12.163 11.79 12.162 11.790

31.079 31.100 20.380 20.380 16.638 16.63 15.426 15.42 15.245 15.23 15.184 15.170

12.437 12.040 12.280 11.840 12.240 11.79 12.229 11.78 12.228 11.78 12.226 11.780

57

6.3 Approximate (Finite-Element) Solutions

6.3.1 Bending Analysis

1. Linear analysis of a rectangular plate under uniformly distributed 1 oad . 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 lo6 psi

and v = 0.3). The problem was also solved by Timoshenko [77], Herrmann

[72] and Prato [78].

displacement and bending moments obtained by various investigators

(using the linear theory).

and others is very good, verifying the accuracy of the theory and the

finite element formulation.

Figures 6.3-6.5 contain plots of the transverse

The agreement between the present solution

2. Nonlinear analysis of rectangular laminates under uniformly distributed load

Figure 6.6 contains load-deflection curves for a simply supported

square orthotropic plate under uniformly distributed load [23]. The

following geometric and material properties are used:

a = b = 12 in., h = 0.138 in. 6 0.37 x 10 psi El = 3 x 10 psi, E2 = 1.28 x 10 psi, GI2 = 613 = 623 = 6 6

The experimental results and classical solutions are taken from the

paper by Zaghloul and Kennedy [79]. The agreement between the present

solution and the experimental solution is extremely good. It is clear

that, even for thin plates, the shear deformation effect is significant

in the nonlinear range.

58

4, ~~

0 L

- PRESENT SOLuTIOr4 (LINEAR ANALYSIS)

HERRMA”

a PRATO

A TIWSHENKO 0 /

/ C W E D

/ SS = SIfVLY SUPPORTED

THICKNESS = 0,O~‘’ 0 I I I

0 0 2 o14 0,6 018 DISTANCE ALONG X i (IN.)

Figure 6.3 Comparison o f the transverse deflection o f a clamped- simply-supported-free isotropic rectangular p la te under uniformly d is t r ibu ted transverse load,,

59

0

b

PRESENT SOLUTION

H E R R W "

-

0 PRATO

A TIMOSHENKO

002 0,4 0,6 008 DISTANCE ALONG X2 (in.)

Figure 6.4 Comparison o f the bending moment along the line x1=0.4" for the problem of Figure 6.3.

60

6

8 4 X

4

2 W z c.

f

0

DISTANCE ALONG X i ( I N . )

Figure 6.5 Comparison o f the bending moment M6 along the line x2=0.4" f o r the problem i n Figure 6.3.

61

EXPERIMENTAL - - PRESEM' --- 0,4

n

z -

v" 0,3 2 0 - G k 012

4 * 0 8 1

111 oc W

w

0 0 0,4 018 182 1 '6 280

LOAD INTENSITY (PSI)

Figure 6.6 Center deflection versus load in tens i ty fo r a simply 'supported square orthotropic p l a t e under uniformly distributed transverse load. The following Simply supported boundary condi- t ions were used :

V = W = $y = Idx = Px = 0 on s ide x = .a

U = W = $ = M = P = O o n s i d e y = b X Y Y

62

Figure 6.7 contains load-deflection curves for a clamped

bidirectional 1 aminate [ 0/90/90/0 I under uniformly distributed load. The geometric parameters and layer properties are given by

a = b = 12 in., h = 0.096 in., 6 6 El = 1.8282 x 10 psi, E2 = 1.8315 x 10 psi,

6 G12 = 613 = G23 = 3.125 x 10 psi, v12 = v13 = '23 = 0.2395

.The experimental results and the classical laminate solutions are taken

from [79]. 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.

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 = - 6 6 El = 2 x 10 psi, E2 = 7.5 x 10 psi

1.25 x 10 psi 6 G12 =

6 613 = 623 = 0.625 x 10 psi , h = 1 in.

= 0.25 - '12 = '13 - "23

a = 10 in. , P = 6.41/~ psi

This problem has an analytical solution (see 1771) for the linear

case, and Rao [47] used the finite-element method to solve the same

problem. Both solutions are based on the classical theory. The center

deflections from [771 and [471 are 0.000367 in. and 0.000366 in.,

respectively. Chao and Reddy 1501 obtained 0.0003764 in. and 0.0003739

63

INTENSITY OF TRANSVERSE LOAD (PSI)

Figure6.7 Center def lect ion versus load intensi ty fo r a clamped (CC-1) under uniform transverse l o a d ( v o n Karman theory).

square laminate [0 /90 /90 /O ]

cc-1: lJ = \I = \J = $ - - qY ’= 0 on a l l

f o u r cld;?oed edges

64

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 [501,

as expected.

For the nonlinear analysis of the same problem, the present results

are compared to those of Chang and Sawamiphakdi [80 ] and Chao and Reddy

[50] in Fig. 6.9, which contains plots of the center deflection versus

the load obtained by various investigators. The agreement between the

various results is very good.

4. Nine-layer cross-ply spherical shell segment subjected to uniform

1 oadi ng . Consider a nine-layer [Oo/900/Oo.. . / O o ] cross-ply laminated

spherical shell segment with the following material and geometric data:

, a = b = 100 in. R1 = R2 = 1000 in. 6 h = 1 in. , El = 40 x 10 psi

6 0.5 x 10 psi E2 = 10 psi , G12 = 0.6 x 10 psi , 613 = 623 = 6 6

= 0.25 '12 = '13 = '23 The present results are compared with those obtained by Noor and

Hartley [52 ] and Chao and Reddy [50 ] in Fig. 6.10. Noor and Hartley

used mixed i soparametric elements with 13 degree-of-freedom per node

which is based on a shear deformation shell theory.

agree with both investigations.

The present results

65

Figure 6.8 Geometry and boundary conditions for the octant o f the clamped cylindrical shell.

66

10

8

6 '

4

2

- 0

A

PRESENT SOLUTION

CHANG AND SAWAMIPHAKDI

CHAO AND REDDY

&

A / VERTICAL DEFLECTION ( IN, )

Figure 6.9 Center deflection versus l o a d f o r the clamped cylinder with internal pressure.

67

0-

PRESENT SOLUTION

0 NOOR AND HARTLEY

A CHAOAND REDDY

-

1 I 1

1 2 3 4 " D I E N S I W L I Z E D CENTER DEFLECTION, W/h

Figure 6.10 Nonlinear bending of a nine-layer cross-ply spherical shell (0"/90"/0"/90"/ ... /o").

68

6.3.2 V ib ra t ion Analysis

Equation (5.24) can be expressed i n the a l t e r n a t i v e form as

where

For f r e e v i b r a t i o n analysis, we wish t o e l im ina te { h 2 } as fo l lows.

Eq. (6.1) we have

From

[ K l l I { A l } + IK121EA21 = - I M 1 1 1 { q

[ K 2 l l { A l 1 + [K221{"21 = 0

(6.3)

(6.4)

From (6.4) we have,

{A21 = -[K221-1[K211{A11 (6.5)

S u b s t i t u t i n g Eq. (6.5) i n t o Eq. (6.3), we obtain,

([K111 - [Kl2I [ K 2 2 1 - 1 1 K z ~ 1 ) { A 1 1 = - [MI11 {A11

(K1 - u 2 1 M 1 ) I A 1 1 = 0

(6.6)

For t h e f r e e v i b r a t i o n case, Eq. (6.6) reduces t o

(6.7)

where w i s the na tura l f requencies o f the system.

1. Natural frequencies of a two-layer [Oo/900] laminated p l a t e .

The geometric and mater ia l p r o p e r t i e s used are

a = b = 100 in. , h = 0.1 in.

El = 40 x 10 p s i , E2 = 10 p s i . 6 6

Vi3 = V23 = 0.25 G12 = G13 = GZ3 = 0.5 x 10 p s i , u12 - - 6

2 4 p = 1 lb-sec / i n

69

The boundary conditions are shown in Fig. 6.11, which also contains

the plots of the ratios wNL/wL versus wo/h.

nonlinear 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 [81], and Reddy and Chao [82]. The present

results are slightly higher compared to those obtained by the classical

Here wNL and wL denote the

and first-order shear deformation theories.

2. Natural frequencies of two-layer [45"/-45'1 angle-ply square plate.

The material and geometric parameters used are

El = 10 x 10 psi , E2 = 10 psi 6

p = 1 lb-sec. /in , a = b = 100 in. , h = 0.1 in.

6 6

= 0.3 G12 = 613 = 623 = 0.3333 x 10 psi , w12 = w13 = ~ 2 3 2 4

The boundary conditions used are shown in Fig. 6.12, which also

contains plots of uNL/uL versus wo/h.

available in the literature for comparison.

For this case, no results are

3. Natural vibration of two-layer [Oo/9O0] cross-ply plates.

Consider a two-layer cross-ply plate with the following geometric

and material properties: 6 6 El = 7.07 x 10 psi , E2 = 3.58 x 10 psi

- ~ 2 3 = 0.3 6 G12 = 623 = 613 = 1.41 x 10 psi , w12 = vi3 - 2

p = 1 lb-sec. /in.4 , a/h = 1000 , a = 100 in.

The results of uNL/uL versus wo/h are shown in Fig. 6.13. Compared

to the results of Reddy [82] and Chandra and Raju (831, the results of

the present study are in general a little higher,

present results for natural frequencies are higher than those predicted

by the first-order theory indicates that the additional inertia terms

The fact that the

contribute to the increase of natural frequencies.

70

- PRESENT SOLUTION

0 CHIA AND Pw\f3HAw\RA

REDDY

f X l w = + / = O

u = o +,= 0 W = @2=

- 0

Figure 6.11 Fundamental frequencies of a two-layer cross-ply (0 '190") square laminate under simply-supported boundary conditions.

71

0 086 182 1,'s 2 8 - 4 3 8 0 AFIPLITUDE TO THICKNESS RATIO, W o / h

Figure 6.12 Ratio o f nonlinear to linear frequency versus amplitude to thickness ratio for two-layer 'angle-ply square plate ( 4 5 O / - 4 5 O ) .

72

BC 1 boundary cond i t i ons :

u = w = $ l = o ; M Z = P2 = O a long x2 = b/2

v = w = ~ ~ ' 0 ; M1 = P 1 = O a l o n g x = a / 2 u = o 1 = 0 ; 1'1 = P = O a long x = 0

v = 02=0; M = P = O a long x = 0

1 6 6 1

6 6 2

Figure 6.13 Ratio of the nonlinear t o l inear frequency versus the amplitude t o thickness r a t io of a two-layer cross-ply (oo/900) square laminate.

7. SUMMARY AND RECOMMENDATIONS

7.1 Summary and Conclusions

The present study dealt with the following major topics:

The development of a variationally-consistent, 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.

coefficients.

The development of the closed-form solutions (for the linear

theory) for the simply supported cross-ply laminates. These

solutions are used as a check for the numerical analysis of

she1 1s.

The construction of a mixed variational principle for the

It does not require the use of the shear correction

third-order theory that includes the c

first-order theory as special cases.

The development and application o f the

of the third-order theory for laminate

assical theory and the

finite-element model

composite shells,

accounting for the geometric nonlinearity in the sense of von

Karman (moderate rotations).

The increased accuracy of the present third-order theory (for thin

as well as thick laminates) 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 and vibration analysis included here can

serve as references for future investigations.

73

74

7.2 Some Comments on Mixed Models

The displacement model of the classical laminate theory requires 1 the use of C elements, which are algebraically complex and

computational ly expensive.

simple and allows the direct computation of the bending moments at the

nodes. The inclusion of the bending moments as nodal degrees of freedom

not only results in increased accuracy of the average stress compared to

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.

that the bending moments are not required to be continuous across

interelement boundaries, as was shown in Chapter 4.

based on the shear deformation theories also have the same advantages,

except that the displacement model of the first-order shear deformation

theory is a so a Co element.

programming efforts are less with the mixed elements.

The Co mixed elements are algebraically

It should be noted

The mixed models

In general , the formulative and

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 o f the present theory t o include nonlinear

material models is awaiting. Of course, the inclusion of thermal loads

and damping in the present theory is straight forward.

AcknowZedgements The authors are grateful to Dr. Norman Knight, Jr. for

his support, encouragement and constructive criticism of the results

reported here. Our sincere thanks to Mrs. Vanessa McCoy for typing.

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Solids & Structures, Vol. 14, pp. 385-400, 1978.

isotropic, orthotropic-and laminated plates using a higher-order shear deformation theory," J. Sound & Vibration, Vol. 98, No. 2, pp. 157-1703 1985.

62. Putcha, N. S.; and Reddy, J. N.: " A refined mixed shear flexible finite element for the nonlinear analysis of laminated plates," Computers & Structures, Vol. 22, No. 4, pp. 529-538, 1986.

63. Reddy, J. N.; and Putcha, N. S.: "Stability and vibration analysis o f laminated plates by using a mixed element based on a refined plate 1986.

J. Sound & Vibration, Vol. 104, No. 2, pp. 285-300,

64. Hellinger, E.: "Die allegemeinen Ansatze der Mechanik der Kontinua," Art 30, in Encyklopadie der Mathematischen Wissenchaften, Vol. IV/4, 654-655, F. Klein and C. Muller (eds.), Leipzig, Teubner, 1914.

65. Reissner, E.: "On a variational theorem in elasticity," J. Math.

66. Hu. H.: "On some variational principles in the theory of elasticity

PhyS., Vol. 29, pp. 90-95, 1950.

and the theory of plasticity," Scientia Sinic, Vol. 4, pp. 33-54, 1955.

80

67. Reissner, E.: "On variational principles of elasticity,Il Proc.

68. Washizu, K.: "Variational principles in continuum mechanics,'' Dept.

Symp. Appl. Math., Vol. 8, pp. 1-6, 1958.

of Aeronautical Engineering, Report 62-2, Univ. of Washington, 1962.

.

69. Tonti, E.: "Variational principles of elastostatics," Mechanica, Vol. 2, pp. 201-208, 1967.

70. Oden, J. T.; and Reddy, J. N.: "On dual complementary variational principles in mathematical physics," Intl. J. Engng.- Sci. , Vol. 12, pp. 1-29, 1974.

71. Oden, J. T.; and Reddy, J . N.: Variational Methods in Theoretical Mechanics, Second Edition, Springer-Verlag, Berlin, 1983.

72. Herrmann, L. R.: "A bending analysis of plates," Proc. 1st Conf. Matrix Mech. Struct. Mech., Wright-Patterson AFB, AFFDL-TR-66-80, Dayton, OH, pp. 577-604, 1966.

73. Herrmann, L. R.: "Finite element bending analysis for plates," J. Enqng. Mech. Div., ASCE, Vol. 93, EM5, pp. 13-26, 1967.

74. Hellan, K.: "Analysis of elastic plates in flexure by a simplified finite element method," Acta Polytech. Scand. Civil Engng., Series No. 46, Trondheim, 1967.

75. Reddy, J. N.: An Introduction to the Finite Element Method, McGraw- Hill, New York, 1984.

76. Vlasov, V. Z.: General Theory of Shells & Its Applications in Engineering, (Translation of Obshchaya teoriya obolocheck i yeye prilozheniya v tekhnike), NASA TT F-99, NASA, 1964.

77. Timoshenko, S.; and Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, 1959.

78. Prato, C. A.: "Shell finite element method via Reissner's principle," Intl. J. Solids Struct., Vol. 5, p. 11-19, 1969.

79. Zaqhloul, S. A.; and Kennedy, J. B.: "Nonlinear behavior of symmetrically laminated plates," J. Appl. Mech., Vol. 42, pp. 234- 236, 1975.

80. Chang, T. Y.; and Sawamiphakdi, K.: "Large deflection analysis o f laminated shells by finite element method," Comp. & Struct., Vol. 13, pp. 331-340, 1981.

81

81.

82.

83.

Chia, C. Y.; and Prabhakara, M. K.: "A general mode approach to nonlinear flexural vibrations of laminated rectangular plates," ASME, J. Appl. Mech., Vol. 45, No. 3, pp. 623-628, 1978.

Reddy, J. N.; and Chao, W . C.: "Nonlinear oscillations o f laminated anisotropic rectangular plates," Trans. o f ASME, Vol. 49, pp. 396- 402, 1982.

Chandra, R.; and Raju, B. Basava: "Larqe arnDlitude flexural vibration o f cross-ply lamianted compo;ite plates," Fib. Sci. Tech., Vol. 8, pp. 243-263, 1975.

APPENDIX A

COEFFICIENTS OF THE N A V I E R SOLUTION

4E66 + -1 4E 12 c(1,5) = a0(-B12 - 866 + -

3h2 3h2

C(2,3) = 0(- A12 + R) A22 + a 2 "(-7- 4E12

R1 2 3h

2 2 4 16H22 32a B - 9h4 (H12 + 2H66)

c(3,3) = a 4 (- z) 16Hll + 5 (-

8E12

8E22 + 2)

-) 2 8D55 16F55 8Ell

2 8D44 16F44 8E 12

- - - - - 2 2 + a (-A55 + -

h2 h4 3h R1 3h R2

+ 6 (-A44 + 7 - - - - 2 h4 3h R1 3h R2

82

83

- - All 2A12 A22 2

R1 -m-G- 2 16aB

(H12 + 2F66) - - 3 4a

3h 9h4 C(3,4) = -

B1l B12 4 E12 + 2ti66) + a [ r + - - - + T I 1 - A55a 1 R2 3h2 (T 2

8aD55 16aF55

h2 h4 +- -

8BD44 168Fqq

h2 h4 +- -

86' 16 2 2 2 2 + H66B ) - Dlla - 0668

2 + - 8a

3h 2 F66 - 2 (H1la C(4,4) = - 2 F1l 3h

2 8F66 2 8F22 16H -

:6) + 0 (-DZ2 + - - +3h2 9h 3h2 c(595) = a (-066

84

M(1,2) = 0

4 4 M(1,3) = (7 I4 + - 3h 2 R1 I 5 l a 3h

4 - - '3 4 2 I 5 1 M(1,4) = - ( I 2 + - - -

R1 3h2 I4 3h R1

M(1,5) = 0

M(292) = -(I1 + -) 212

R2

4 4 M(2,3) = (7 I4 + - 3h 3h R2 2 ' 5 ) B

M(2,4) = 0

I 3 4 4 M(295) = - ( I 2 + - - 7 I4 - - 2 I 5 ) R2 3h 3h R2

16 2 16 2 M(3,3) = - 7 17a - 7 1 7 ~ - I1 9h 9h

16 3h

l6 I ) 8

3h M(4,4) = -(I3 - 7 I5 + 2 M(4,5) = 0

8 16 M(5Y5) = 4 1 3 - 2 I 5 + gh4 17)

3h

APPENDIX B

STIFFNESS COEFFICIENTS FOR THE MIXED MODEL

The element equation (5.16) can be written in the form 12

22 [K I ...... [K ] ......

[ K(ll)l] [ K(ll)'] . . .

1 aw 1 aw 1 aw + - A 2 12 [axp '12' + 2 A13 '5 '12' + 2 A13 [$ 'ill

+ 7 1 A32 '3 aw '22' + 7 'A 33 [% axl '22' + 7 A 33 [x ax2 SZ1]

14 15 [K l = [ K l = O

a5

86

23 A3 1 1 aw A32 [ K 1 3 - R1 [slO1 + 2 A 3 1 ' T '11' + ['lo1

1 aw 1 aw 1 aw + 2 A32 '3 '12' + 2 A33 '12' + 2 A33 [axp '11'

1 aw 2 22 ax2 + - A [- SZ2

24 25 [ K ] = [ K 1 - 0

A22 s211 + RE [s201

87

A12 A13 aw aw ['0Z1 + R + A 1 2 [ q '12' + A 1 3 [ q '11'

= - R1 1

A 1 13 [aw '02' + 2 7 5 '01'

A 1 13 [ a w A 1 12 [aw

2 R1 + - - 3 '02' + 7

aw 1 aw 2 A 1 ~ [ a w - s l o l + - 1- '101 + 7 A l l I(-) S I 1 1 + -

R1 aX1 axl R2 axl

+ - A 1 [-- aw aw aw 2 2 12 ax1 ax2 '12' ' $ A13 '(q) '12'

s ] + - A21 + - A 1 [-- aw aw 2 13 axl ax2 11 R I R Z

A A22 1 A22 aw 1 23 [aw + 2 [Soo1 + - - [- SO2] + - - - 2 R2 ax2 2 R 2 axl '02' R2

1 2 3 [ ~ A21 - S20] + - A 1 [-- a w a w [ a w

A22 [aw 1 aw 2 1 aw a w

1 aw 2 A31 [ a w 1 a w 2 2 23 ax2

A

+ - - 2 R2 ax2 '01' + 2 21 axl ax2 '211 ax2

+ - - s201 + 2 A22[($ s221 + - 2 A 23 [-- a x l - ax2 '22' R2 ax2

+ - A [(-) S211 + - - '201 + 2 A31[($) 5211 R1 axl

a w 2 A I(-) S221 1 aw aw 2 32 ax1 ax2 '22' + 7 33 axl A32 [E S20] + - A [-- + -

R2 axl

1 aw aw A31 aw 1 aw aw 2 33 axl ax2 '21' + [- ax2 '10' + 7 A 3 1 [ q 3 '11' + - A [--

1 a w 2 1 aw a w '10' + 7 A 3 2 ' ( q ) '12' + 2 A 3 3 ' q 3 '12' + -

+ x45

36 5 1 a w B2 1 R1 R2

[ K I = - [So , ] + B l l [ ~ Slo] + -

aw + B 3 1 ' 3 '20' + '31[% '10'

'001 + 8 2 4 % s20

89

+ B [aw S ~ ~ I + ~32[$ aw slOl 32 axl

[K3'] =

[ K3 11]

41 [K I = 42 [K ] = O

[ K ~ ~ I = R ~ ~ [ s ~ ~ I + R ~ ~ [ s ~ ~ I

90

IK4’1 = CPISol] , [K4,lo] = 0 , [K4,11] = CP[So2]

[K ] = [ K ] = 0 51 52

63 Bll aw aw [K ] = - R1

91

67 [K I = -D12[Sool

[K681 = -Dl3[Soo1

69 [K 1 = -F11[SooI

[K6’lo1 = -Fl2[So0]

[K6’11] = -F [S ] 13 00

[K7lI = Bl2[So1I + B32[So21

[K7‘I = B22[S021 + B32[Soll

aw 1 aw + - B [- so11 1

+ - 2 32 [- axl ’02’ 2 32 a x 2

74 75 [K 1 = 0 [K 1 = [So21

76 [K 1 = -D21[Sool

[K I = -D22[Soo1

[K 1 = -D23[Sool

[ K 1 = -F21[So0l

77

78

79

[K7’l0] = -F [S ]

[K7’11] = -Fz3[So0]

[K

22 00

[ S 1 I = Bl3[soll + B33 o2 81

92

83 '13 1 aw '23 1 aw [ K ] = - R1 [sOO1 + 2 B13[$ '01' + [sOO1 + 7 B23[$ '02'

93

+ - 1 E [- aw So21 + - 1 E [- aw Sol] + CPISll] 2 31 axl 2 31 ax2

94 95 [K I = CP[Slo1 [K I = 0

[ K ~ ~ I = - F ~ ~ [ s ~ ~ I

[Kg71 = - F21[Sool

[Kg8I = - F31[Sool

99 [K I = - H1l[Sool

[K9’10] = - H [S ]

[K9y111 = - H13[Soo]

[K1O’ll = El2[Sol1 + E32[So21

12 00

[K1Oy2I = E22[So21 + E32 [S 01 1

+ - 1 E [- aw So2] + - 1 E [- aw S o l ] + CP[S22] 2 32 axl 2 32 ax2

[K1Os41 = 0 , [K10*5] = CP[Sz0]

[K1096] = - F [ S ] 12 00

[K10971 = - F22[Soo]

94

[K1OY8] = - F32[Soo]

[K109’] = - HZ1[Soo]

[ KIO 9 l o ] = - H22[Sool

[K1o”ll = - H23[Soo]

[ K 1 1 9 1 ] = E 13 [S 01 .] + E33 [ I d2

[K11921 = E23[So21 + E33 [S o1 I

[K1194] = C P [ S z 0 ]

[ K 1 l y 5 1 = CP[Sl0]

[ K 1 l Y 8 ] = - F33[Soo]

[K1l’lo] = - H32[Soo]

[K11911] = - H33[Soo]

M a s s M a t r i x [ M I f o r the M i x e d M o d e l

95

.... ..........................

- - - - I [S ] [M31] 3hZ 4 01

= I [S ] - - u s I 2 00 3h2 4 00

- - - u s I 3h2 4 10

41

-

1 3hZ 4 00 [ M 1 = '2~sool - -

u s 1 - - - - I [ S ] [ M ] = - - 3hZ 4 02 3hZ 4 20

32

= (I2 - - 42 I ~ ) [ s ~ ~ I , [M5'] = [MZ5] 3h

44 = [ M 1

al l others are zero, and Ii are defined i n Eq. (2.18b)

96

- 4 2 A45 = A45 - 2E45 h2 + G45(7)

- 4 4 2 = A55 - 2E55 7 + G55 $7)

4 CP = - 3h2

aJli

a xO with - - - 6i

For the tangent stiffness matrix, the coefficients are given by

[(KT) 13 ] = 2[K13] = [K31]

[(KT)23] = 2[KZ3] = [K 32 ]

[ (KT)331 = [ K 3 3 1

au au au + A 1 3 [ 3 '11l + A 2 3 [ 3 '22' + A 3 3 [ 3 ('21 + '12)'

97

a v av a v + A 1 3 [ q '11' + A 2 3 [ q '22' + A 3 3 [ q ('21 + '12)'

aw a w a w 2 s211 + [(-I '2211 axl

1 2 21 axl ax2 + - A [[--

1 aw aw a w 2 2 23 axl ax2

1 a w a w a w 2 2 31 axl ax2

+ - A [[-- '221 + [(-) '211

[[-- 'ill + [(-) '121

ax2

axl + - A

1 a w 2 a w aw + 7 A33[[(-) '111 + [ax ax '1211

+ I(-) '2111

a x2 1 2

aw aw a w 2 1 + - A 2 32 [[-- axl ax2 '22l ax2

1 a w aw + 7 q iA1l ["X1 '011 + A12 1% '021

98

99

[ ( K T ) ~ ~ ~ = [K371

[ ( K T ) ~ ~ ~ = [K381

[ (KT)931 = IK3’1

[ (K,)10’3] = [K3”’]

[ ( K T ) l l y 3 ] = [K3’11]

A l l others are same as those i n [ K ] .

Standard Bibliographic Page

17. Key Words (Suggested by Authors(s)) klgl e-ply , composite laminates, cross-ply, closed- form solution, f i n i t e element analysis,

Report No. NASA CR-4056

18. Distribution Statement

Unclassified - Unlimited

, Titie and Subtitle

A Higher-Order Theory f o r Geometr ical ly Nonlinear Analysis o f Composite Laminates

-

. Author(s)

J . N . Reddy and C. F. L i u

- ,

Uncl ass i f i ed

. Performing .Organization Name and Address Virginia Polytechnic I n s t i t u t e and State University Department of Engineering Science and Mechanics Blacksburg, VA 24061

Uncl assi f ied I 105 I A06

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, DC 20546

3. Recipient's Catalog No.

5 . Report Date

MARCH 1987 6 . Performing Organization Code

8. Performing Organization Report No. VPI- E-86.21

10. Work Unit No.

11. Contract or Grant No.

NAG 1 - 459 13. Type of Report and Period Covered

Contractor Report _ _ 14. Sponsoring Agency Code

505-63- 31-02 5. Supplementary Notes

NASA Langley Research Center Technical Monitor: Dr. Norman K n i g h t , J r . Final Report

h i gher-order theory , pl a t e s and she1 1 s , nonlinear analysis, shear deformation theory, vibration

Subject Category 39

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19. Securitv Classif.(of this report) 1 20. Security Classif.(of this page) I 21. No. of Pages I 22. Price

For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Langley, 1987