Nonlinear Finite Element Analysis of a Laminated Composite Plate

with Nonuniform Transient Thermal Loading

by

Thomas Harris Fronk

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

~faster of Science

APPROVED:

Dr. C. E. Kni'g t j~matka

March 25, 1988

Blacksburg, Virginia

LD SloSS V6~ 19BB (1,,5 C'~

Nonlinear Finite Element Analysis of a Laminated Composite Plate

with Nonuniform Transient Thermal Loading

by

Thomas Harris Fronk

Dr. J. R. Mahan, Chairman

(ABSTRACT)

Metal plates are being replaced by lighter but equally strong laminated composite

plates in order to improve efficiency and increase performance of aerospace vehicles.

But because of the complex construction of laminated plates they are very difficult to

analyze. Conventional thin plate theories prove to be inadequate in predicting laminated

composite plate behavior. Therefore, a finite element model which incorporates a first ..

order shear .. deformation theory and nonlinear von Karman strains is described. The

model is shown to accurately predict deflections in laminated composite plates due to

nonuniform transient heat fluxes and transverse mechanical loads.

Acknowledgements

I would like to thank my adviser Dr. 1.R. Mahan for his constant help and advice.

Also I appreciate the advice and time given to me by the members of my committee,

Dr. C.E. Knight, and Dr. J.B. Kosmatka. A special thanks goes to NASA Langley and

Dr. lohn S. Mixson for their funding of this project under contract NASl-18471, Task

3. I am indebted to my parents, brothers, and sisters for their example and encourage­

ment. And most of all I thank my wife Monica and two sons Alexander and Aaron.

Without their support it would not have been possible.

Acknowledgements iii

Table of Contents

1. Introouction .....,................................«................... t

2. Laminated Plate Analysis •••..••..••.•.•••.••.•.•..••.••••••.•.••••.•.••.. 6

2.1. I..a.mi.rtate Defulition ................................................. 6

2.2. Classical Lamination Theory (CL T) ..................................... 7

2.3. Shear-Defonnation Theory (SDT) ...................................... 9

3. Lamina Stresses, Strains, and Displacements •..••...•....•.••.•.•....•.••.•.• 12

3.1. I..a.mi.rta Stress-Strain Relations ........................................ 12

3.2. Strain-Displacement Relations ........................................ 20

4. Governing Equations for Laminated Plates .•.•••••.•••••••••••••••••.•••.••. 24

4.1. Laminate Forces and Moments ........................................ 24

4.2. Thennal Forces and Moments ........................................ 27

4.3. Equations of Motion ............................................... 33

5. .Finite Element Mooel ..................................... ,............ 37

Table of Contents iv

5.1. Variational Fonnulation ............................................. 37

5.2. Interpolation and Approximate Functions ................................ 45

5.3. Gauss-Legendre Quadrature and Reduced Integration ....................... 54

5.4. The Transient Solution (Newmark lVlethod) .............................. 58

5.5. Newton-Raphson Method ........................................... 60

6. Verification of Model and Results •.••.. 'O •••••••••••••••••• 'O............... 63

6.1. Static Mechanical Load ............................................. 64

6.2. Linear Temperature Variation Through an Isotropic Plate .................... 66

6.3. Transient Mechanical Load .......................................... 67

6.4. Transient ThenIlal Load ............................................. 68

7. Conclusions and Recommendations •..• •.••••..••.•..•••... 'O ••••••• 'O'O...... 70

Referen~ .............................................................. 88

Appendix A. COMMEC2 PROGRA't LISTING .•....•..•.....•..•••.......... 92

Vita ................................................................. 109

Table of Contents v

List of Illustrations

Fig. 1. Typical Laminated Composite Plate. . ........................... 71

Fig. 2. Plate Element with Seven Degrees of FreedoT, three Translations and Four Rotations. ................................. .... :......... 72

Fig. 3. (a) Undeformed Laminated Composite Plate (b) Deformed Laminated Plate with CLT Displacement Field (c) Deformed Laminated Plate with First-Order SDT Displacement Field. ..................... '. . . . . . . . . . 73

Fig. 4. (a) Forces on the Laminated Plate (b) Moments on the Laminated Plate .. 74

Fig. 5. Laminate Coordinate System and Nomenclature .................... 75

Fig. 6. Typical Infmitesimal Cube of Laminated Composite Plate. . ........... 76

Fig. 7. Master Element for (a) 4-Node Rectangular (b) 8-Node Rectangular, and (c) 9-Node Rectangular Elements. . ............................. 77

Fig. 8. Finite Element Mesh used in the Plate Analysis, Four 9-Node Quadrilateral Elements. . ............................................... 78

Fig. 9. Geometry and Boundary Conditions of the First Quadrant of a Simply-Supported Square Plate. . .................................... 79

Fig. 10. Normalized Center Deflection of a Simply-Supported Isotropic Plate Under a Uniformly-Distributed Transverse Load ......................... 80

Fig. 11. Geometry and Boundary Conditions of the First Quadrant of a Clamped Symmetric Cross-Ply [0/90/90/0] Laminated Composite Plate. . ........ 81

Fig. 12. Center Deflection of a Clamped Symmetric Cross-Ply [0/90/90/0] Laminate Subjected to a Uniform Transverse Load (a=4.72 cm (12 in.), h=0.038cm (0.096 in.». . ............................................. 82

Fig. 13. Normalized Deflection of a Simply-Supported Isotropic Plate Subjected to a Linear Temperature Distribution. . ............................ 83

List of Illustrations vi

Fig. 14. Evolution of Center Deflection with Time for a Simply-Supported Isotropic Square Plate Under a a Uniform Load of 100 kPa (a= 25.0 em, h= 5.0 em). 84

Fig. 15. Evolution of Center Deflection with Time for a [0/90/0] Simply-Supported Laminated Composite Plate Under a Uniform Load of 100 kPa (a = 25.0 em, h = 5.0 em). .............................................. 85

Fig. 16. Geometry and Boundary Conditions of the First Quadrant of a Simply­Supported Cross-Ply [0/90/90/0j Laminated Composite Plate (a= 2.0 em, h = 0.5 em). .............................................. 86

Fig. 17. Evolution of Center Deflection with Time for a Simply-Supported [0/90/90/0] Laminated Composite Plate Subjected to a Nonuniform Heat Flux (a = 2.0 em, h = 0.5 em). ................................. 87

List or Illustrations vii

1. Introduction

The need for a strong yet light·weight structural material has evolved with the evo­

lution of aerospace vehicles. Many missiles, aircraft, and even the space shuttle now use

parts made of composite materials which have large strength-to-weight and stiffness-to­

weight ratios. One of the most popular forms of composite material is the laminated

plate. Several layers of unidirectional aligned fibers bound with resin, called laminae, are

bonded together to form a solid plate, called a laminate. The individual laminae are ori­

ented in certain directions giving the laminate the necessary strength and stiffness in the

directions of the anticipated loads. A typical laminated plate is shown in Fig. 1. A

laminated composite plate is an orthotrop;c, nonhomogenous material. Orthotropic

means the material properties are directional dependent with three orthogonal planes of

material property symmetry and nonhomogenous means the material properties are lo­

cation dependent. Because of their orthotropy and nonhomogenity, many complications

and complexities arise in the analysis and description of laminated composites that do

not exist with conventional isotropic, homogenous materials. These include coupling

between bending and extension, bending and twisting, and shear and extension.

1. Introduction

The National Aeronautics and Space Administration (NASA) is currently studying

the interaction of thermal stress and sonic fatigue in laminated composite structures of

the proposed U.S ... U.K. advanced short take-off and vertical landing (ASTOVL) air­

craft [1). If the aft fuselage and thrust redirection and lifting surfaces of the ASTOVL

can be made of laminated composites, a substantial weight savings will be gained ..

However, the engine exhaust causes high levels of acoustic radiation and heat flux to

impinge on these structures. Although composite materials are said to have an almost

infinite fatigue life [2], the extent of damage caused by the coupling of acoustically­

induced vibration with thermally-induced stress is still unknown. Given the complex

nature of composite materials, it is unlikely that their behavior can be predicted only on

the basis of experience with isotropic materials. A finite element model of a laminated

composite plate, developed in this thesis, is shown to accurately and economically pre ..

dict the deflections caused by transient thermally-induced loads. This is the necessary

first step in the development of a finite element model capable of describing the

thermoacoustic response of fiber-reinforced composite materials.

The science of analyzing and designing laminated composite components has re­

ceived significant attention only in the last twenty to thirty years. In the 1950's and 60's

an increased interest arose in developing elasticity theories for plates that are nonho­

mogeneous and anisotropic. Stavsky and Reissner (3) established the elastostatic bend ..

ing and stretching theory for nonhomogenous plates based on the Euler-Bernoulli

hypothesis and the classical Kirchoff hypothesis. This evolved into what is now called

the classical plate theory (CPT). Classical lamination theory (CL T) is an extension of

CPT to laminated composite plates. Classical lamination theory is generally accepted

and widely used to predict laminated composite plate response and in designing compo­

nents made of laminated composites. Classical lamination theory is relatively easy to

use because it employs several simplifying assumptions discussed in Chapter 2.

1. Introduction 2

However, many researchers have recognized the inadequacies and limitations of

classical lamination theory. In 1969, Pagano published a series of papers [4,5,6] in which

the deficiencies of classical lamination theory are exposed and explained. One of the

most limiting consequences of the CL T assumptions is that accurate predictions are

obtained only when the plate has a high span-to-depth ratio and deflections are small.

Therefore, CL T is really only a thin plate approximation of laminated composite plates.

The varying material properties of laminated plates make it impossible to set a

specific thickness where CL T yields valid results. Hence, many have attempted to ac­

count for thickness effects by including shear-deformation terms. The Mindlin plate

theory [7] includes tranverse shear effects and, therefore, has served as the basis for de­

veloping theories which account for deformation caused by shear. Stravsky [8] was the

first to propose shear deformation theory as a solution to analyze laminated isotropic

plates. Later, Yang, Norris, and Stavsky [9] generalized t~e shear deformation theory

to include laminated anisotropic heterogenous plates. Whitney and Pagano [10] applied

the theory of Yang, Norris, and Stavsky (YNS) to bending of anti symmetric cross-ply

and angle-ply plate strips under -sinusoidal load distribution and free vibration of anti­

symmetric angle-ply strips and reported good results. Reissner [11], Lo, Christensen and

WU [12J, and Librescu [13] have all developed higher-order shear-deformation theories

to predict the response of laminated composites. Each one of these has included shear

deformation effects and consequently led to improved accuracy of their predictions over

that afforded by the traditional CL T results (e.g. Dong, Pister, and Taylor [.14], and Bert

and Mayberry [15]). Of great importance to the research of this thesis is the work done

by Reddy and N su [16]. They used the YNS theory to evaluate the effects of shear de­

formation and anisotropy on the thermal bending of layered composite plates. Recently,

Mei and Prasad [17] investigated the influence of large deflection and transverse shear

I. Introduction 3

on rectangular symmetric laminated composite plates due to acoustic loads. They They

found that for thick plates the effects due to shear deformation are considerable.

The finite element method has been used extensively to analyze isotropic plates, but

less effort has gone into the investigation of laminated composite plates. Thick lami­

nated plates have been analyzed by Pryor and Barker [18J, and Barker, Lin, and Dara

[19] using the conventional displacement finite element method. A plate element with

seven degrees of freedom per node (three displacements, two rotations, and two shear

rotations as shown in Fig. 2) was used. Implementing an element with 80 degrees of

freedom, Noor and Mathers [20] developed a finite element model based on Reissner;s

plate theory. They were able to study the effects of shear deformation on the accuracy

of several models. While these models are very accurate they are also computationally

very demanding. Models of plate elelnents with five degrees of freedom (three displace­

ments, two rotations) have been formulated by Yang, Norris, and Stravsky [9] and

Reddy [21,22]. They show that in most cases the use of only five degrees of freedom

proves sufficiently accurate. Reddy [23J has improved the accuracy of the predictions

of the fmite element model incorporating the method of reduced integration on the shear

deformation terms as proposed by Zienkiewicz, Taylor, and Too [241.

The problem of thermally-induced stress in nonhomogenous materials was first ad­

dressed by Pell [25], Sharma [26], and Nowacki [27]. Later Sugano [28] studied the effects

of transient thermal stresses in transverse isotropic cylinders. The thermal stress prob­

lem was expanded to include orthotropic laminated plates by Tauchert and Akoz [29].

However, this was only a steady-state solution. Most recently Wang, and Chou [30]

present a solution of transient thermal stresses in thermally and elastically orthotropic

two-dimensional materials based on a displacement-potential approach.

The previous work cited above has provided the basis for the nonlinear transient

finite element model developed in this thesis. The model is nonlinear in the sense that

1. Introduction 4

the in-plane strain-displacement relationships include the square of the transverse de­

flection; i.e. the von Karman strains are implemented. The model is capable of accu­

rately and economically predicting the dynamic behavior of the laminated composite

components of the type proposed for use in the ASTOVL aircraft, and is capable of ex­

amining the effects of imposed three-dimensional, transient 'temperature fields while ac­

counting for the nonlinear shear deformation terms with the use of reduced integration.

The temperature field is restricted to the semicoupled case in which the heat generated

by the kinetic behavior of the plate is assumed to be negligible. This assumption has

been extensively validated by Jia [31]. To the author's knowledge, the finite element

model in this thesis is unique in its accuracy, versatility and low computational demand.

The goals of the work described in this thesis are, first, to formulate the finite element

model just described and, second, to demonstrate its capabilities by considering the

transient, three-dimensional thermoelastic response of a laminated pl~te.

I. Introduction 5

2. Laminated Plate Analysis

The purpose, of this chapter is to describe the components of a laminated composite

plate and to develop the theories used in laminated composite plate analysis. One of the

most prevalent lamination theories, classical lamination theory (CL T), is described and

shown to be inadequate for the application envisioned in this thesis. Other theories

which assume higher-order displacement fields and include shear deformation effects are

discussed. Finally, a first-order theory which does not rely on Kirchoffs assumption is

shown to be the most suitable for the application of this thesis.

2.1. Laminate Definition

Consider a composite laminated plate of thickness h and width a as shown in Fig.

1. Throughout this thesis the laminate orientation will be referred to a global x, y, z

system of Cartesian coordinates. The x-y plane of the coordinate system is the midplane

of the plate and the z-axis is normal to the midplane. The top of the plate, therefore,

2. Laminated Plate Analysis 6

will be located at z = + h/2 and the bottom at z = - h/2. The principal material axes of

an individual lamina are aligned parallel to the direction of the unidirectional fibers in

each case. The angle between· the laminate x-axis and the principal material axis is re­

ferred to as the angle of rotation and is denoted by 0 in Fig. 1. The angle of rotation is

positive in the clockwise direction.

Each layer is assumed to be orthotropic and homogeneous. Orthotropic means the

lamina has three planes of geometric and material property symmetry. The orthotropic

assumption is made with little or no error. There are many methods and theories for

mathematically deriving orthotropic material properties of a heterogenous lamina.

However, these methods are tedious and rarely more accurate than empirical character­

izations of the material properties [3}. Therefore, for this analysis no micro mechanical

calculations are performed. The lamina properties are assumed to be known and

homogenous throughout each lamina.

2.2. Classical Lamination Theory (CL T)

Classical lamination theory (CL T) is the prevalent analysis and design tool of lami­

nated composite plates because of its relative simplicity compared to three-dimensional

theory. The assumptions made in classical lamination theory are listed below along with

the direct implication of each assumption.

1. Assumption: Plane sections originally plane and perpendicular to the middle surface

remain plane and perpendicular to the middle surface after extension and bending.

Implication: The deflection of the plate is associated only with the bending strains.

The shear strains ea and eyz are neglected.

2. Laminated Plate Analysis 7

2. Assumption: The displacements of the middle surface are small compared with the

thickness of the plate.

Impli~ation: The slope of the deflected surface is small compared to the in-plane

strains and the square of the slope is negligible; i.e., linear strain-displacement re­

lations are valid, and the influence of in-plane forces upon the transverse deflection

may be ignored.

3. Assumption: The thickness of the plate is small compared with other dimensions

of the plate.

Implication: The plate is in a state of plane stress. (1t = 1"xz = 1"yz = O.

4. Assumption: The laminate consists of perfectly bonded laminae and the bonds are

infmitesimally thin as well as nonshear-deformable.

Implication: There is no possibility of delamination since laminae cannot slip with

respect to each other.

S. Assumption: The plate is constructed of linearly-elastic material with properties

that are independent of temperature.

Implication: Hooke's law is applicable.

Although these assumptions simplify the analysis and design of composite materials, the

implications often cause erroneous results. Pagano and his associates in a series of pa­

pers [4,S,6] show that CL T predictions are reasonable only in the case of relatively thin

plates. This is because of:

1. the neglect of shear deformation terms, implied by the Kirchoff hypothesis (normals

remain normal)

2. Laminated Plate Analysis 8

2. the assumption of linear in· plane displacements through the thickness

3. the presence of only two boundary conditions per edge in the bending theory

4. the assumption of a state of plane stress which eliminates the possibility of

interlaminar stress calculations.

Research has been done to eliminate some of the restricting assumptions of CL T and

still maintain the computational simplicity of plate theory. Shear-deformation theories

(SDT) are a result of this research.

2.3. Shear-Deformation Theory (SDT)

The Kirchoff restriction is unnecessary in shear-deformation theories and, therefore,

they are a compromise between the accuracy of three-dimensional theory and the sim­

plicity of plate theory. Shear-deformation theories can be grouped into either stress­

based or displacement-based theories. Reissner [32] is responsible for the first

stress-based SDT and Basset [33] is one of the frrst to develop a shear deformation the­

ory based on displacements. The theory developed in this thesis is also a displacement­

based theory. Basset assumes that displacements can be expanded in power series of the

z-coordinate. For example, the displacement in the x-coordinate is written

N

u(x, y, z) = UO(x, Y) + L zn I/I~(x, y) , (2.1) n==l

where UO is the middle surface displacement in the x-direction and "'~) means

2. Laminated Plate Analysis 9

n = 1,2, ... (2.2)

A similar expression can be written for the displacement in the y-coordinate. The

quantities t/lz and t/I y have a physical significance when n = I as the slope of the deflected

surface. A special case of Basset's displacement field is the first-order shear-deformation

theory. The first-order SDT assumes the following displacement field

u (x, y, z) = UO(x, y) + zt/lix, y),

(2.3)

and w(x, y, z) = WO(x, y) .

The displacement fields assumed by the first-order SOT and CL T are similar in that

they both assume a linear through-the-thickness displacement field. However, in the

first-order SOT transverse displacements are allowed to rotate and are not necessarily

perpendicular to the middle surface after bending or extension. Figure 3 shows a

graphical comparison of the displacement field assumed by the first-order SOT and the

displacement field assumed in CLT because of the Kirchoff-Love hypothesis. Figure

3(a) shows an undeformed laminate, and the deformed laminate with the displacement

field assumed by CLT is shown in Fig. 3(b). Notice in Fig. 3(b) that the solid line de-

picting the displacement field is still normal to the middle surface. The solid line shown

in Fig. 3(c) is a typical f«st-order SDT displacement field and is not necessarily normal

to the middle surface.

2. Laminated Plate Analysis 10

Another observation about the first-order SOT is that the shear strains are assumed

uniform through the thickness of the plate. In general this assumption is incorrect and

for this reason Mindlin [7] introduced a shear correction factor, kl. The shear correction

factor is evaluated by comparing solutions with an exact elasticity solution. For the

materials most commonly used in laminated composite plates kl = ~ . This factor 'is

multiplied by the shear moduli G23 , and G31•

In 1969 Yang, Norris, and Stravsky [9J assumed the displacement field of the first-

order SOT in order to investigate elastic wave propagation in heterogenous plates and

found good agreement with the exact solution. Later Gol'denveizer [34], Schmidt [35],

Whitney and Pagano [10), Librescu [13], and Lo, Christensen, and Wu [12] independently

included higher order terms in Basset's power series assumption to describe the dis-~

placement fields. Reddy and his associates [22] have refined a higher-order theory that

is extremely accurate ~nd yet somewhat less cumbersome and computationally less de ..

manding than the three-dimensional elastic theory. However, even this refined higher­

order theory is substantially more cumbersome than the first-order SOT. In chapter 3

the nonlinear deflections are introduced. By including the nonlinear deflections it be­

comes even more computationally cumbersome to include the higher-order shear defor­

mation theories. Therefore, it is assumed in the present work that the first-order theory

is sufficiently accurate and the displacement field of Eq. (2.3) is used as the basis for

defining the strains. Thus only.five degrees of freedom are considered: u, v, W" t/lx, and

t/I y'

2. Laminated Plate Analysis 11

3. Lamina Stresses, Strains, and Displacements

Stress-strain relations for an orthotropic lamina are reviewed in this chapter. Starting

with the generalized Hooke's law for an elastic solid, the stress-strain relations are re­

duced to the three-dimensional orthotropic form using contracted notation. The

stiffness matrix is transformed so that the on-axis stresses and strains can be found for

a lamina with arbitrary orientation of the principal material direction. The stiffness

matrix is further reduced to exclude (I" and £z. Finally, the nonlinear strain­

displacement relations are derived.

3.1. Lamina Stress-Strain Relations

The generalized Hooke's law for an elastic solid in indicial notation is expressed

(3.1)

where alj is the stress tensor Cjjtl is the stiffness tensor and ttl is the strain tensor. It is

assumed in this relationship that there are no residual stresses from previous defor-

3. Lamina Stresses, Strains, and Displacements 12

mations. With this assumption the generalized Hooke's law is the most general form

of a linear relationship between the stress and strain components. In this general form

CUt! represents 81 independent material constants. From equilibrium conditions it can

be shown that, if there are no body moments, the O'lj tensor is symmetric (0'11 = 0'11)' And

by defmition of the strain tensor from the Eulerian viewpoint (see Eq. (3.18»), it can be

concluded that the strain tensor ttl is symmetric. Since O'ij and tlJ are symmetric, it follows

that C11t! is symmetric. The independent constants are now reduced to 36 and so con-

tracted notation can be employed to simplify the expression for Hooke's law to

(3.2)

Contracted notation is a simplification of indicial notation that reduces the number of

indices by one-half. The indices are replaced as follows:

11 -+ 1 t 22 -+ 2, 33 -+ 3 ,

23 -+ 4 , 31 -+ 5, 12 -+ 6 .

For example, O'u -+ 0'1' t» -+ t 4 , and Cllll -+ CS6 ' The number of independent elastic

constants is further reduced to 21 by recognizing that there exists a quadratic strain en­

ergy function W such that

(3.3)

Therefore Cil is symmetric.

In Chapter 2 the lamina was assumed to be orthotropic. This implies that there are

three planes of geometric and material symmetry. By systematically equating terms from

symmetric coordinate systems it can be shown that the number of independent material

3. Lamina Stresses, Strains, and Displacements 13

constants can be reduced to nine and that the generalized Hooke's law in matrix form

for an individual lamina can now be written [3)

0'1 Ctl Cn Cl3 0 0 0 £1

0'2 Cl2 C22 C23 0 0 0 £2

0'3 C13 C2l C33 0 0 0 £3 - (3.4)

0'4 0 0 0 C44 0 0 £4

O's 0 0 0 0 Css 0 £s

0'6 0 0 0 0 0 C66 £6

By inverting the stiffness matrix [CII]' the compliance matrix [Sij], can be found along

with the strain-stress relationship

£1 Sl1 S12 S13 0 0 0 0'1

£2 S12 S22 523 0 0 0 0'2

£3 S13 S23 533 0 0 0 0'3 - (3.5)

£4 0 0 0 544 0 0 0'4

£5 0 0 0 0 Sss 0 O's

£6 0 0 0 0 0 S66 0'6

From the strain-stress relationship, the compliance terms can be logically deduced in

terms of engineering constants [3] :

3. Lamina Stresses, Strains, and Displacements 14

1 - V21 - V31 0 0 0

El E2 E3

- v12 1 - v32 0 0 '0

El E2 E3

-v13 - V23 1 0 0 0 El E2 EJ

[5J= (3.6)

0 0 0 1 0 0 G23

0 0 0 0 _1_ 0 G31

0 0 0 0 0 _1_ G12

where E, are the Young's moduli in the i-direction, Vi) are the Poisson's ratios for strain

in the j direction resulting from stress in the i direction, and Glj are the Shear moduli in

the i-j plane. Recall from Chapter 2 that the shear moduli G23 and G13 can now be re-

duced by the shear correction factor, kl . Although twelve material constants appear in

Eqs. (3.6), there are only nine independent constants because

Vij vji -E = -E ,i = 1,2,3 , j = 1,2,3 ·

i j (3.7)

The stress-strain relations of Eqs. (3.4) are true only when the stresses and strains

are in the principal material direction of the lamina (direction in which the fibers are

aligned). When the various principal axes of the laminae are in different directions it

becomes difficult to analyze the laminate. This necessitates a method of transforming

3. Lamina Stresses, Strains, and Displacements IS

the stress-strain relations from one coordinate system to another. Consider a lamina

whose principal material direction is at some angle 0 with respect to the laminate x-axis.

In tensor notation the stresses in the x-y coordinate system (all)' are transformed to the

1-2 coordinate system (0"11)' by [36]

where

This yields

at

0'2

0'3

-1'23

1:'31

1'12

where now

cosO -sinO 0

[ajj] = sinO cosO 0

001

c2 c2 0 0 0 2cs

S2 S2 0 0 0 -2cs

0 0 I 0 0 0

0 0 o C -8 0

0 0 0 s c 0

-CS CS 0 0 0 c2 _ S2

c = cosO and s = sinO.

(3.8)

ax

O'y

O'z (3.9)

1'yz

't'zx

1:'xy

In this thesis the transformation matrix of Eq. (3.9) is represented by the symbol [T1].

Tensorial strains are transformed in the same way; however, tensorial strains and engi-

3. Lamina Stresses, Strains, and Displacements 16

neering strains are not equal. Tensorial shear strains are one-half of engineering strains.

Thus,

t1 1 0 0 0 0 0 t1

t2 .0 1 0 0 0 0 t2

t3 0 0 1 0 0 0 t3 - (3.10)

Y23 0 0 0 2 0 0 &4

Y31 0 0 0 0 2 0 t5

Yl2 0 0 0 0 0 2 t6

where the conversion matrix is conunonly called the Reuter matrix [RJ. Engineering

strains can now be transformed from the x-y coordinate system {t}~ to the 1-2 coordinate

system {e h by

{t} 1 = [R][TtJ[R]-l {e}x ,

or, carrying out the indicated matrix mUltiplications,

el c2 c2 0 0 0 es ex

t2 S2 S2 0 0 0 -cs ey

e3 0 0 1 0 0 0 ez - (3.11)

Y2l 0 0 0 e -e 0 Yyz

Yll 0 0 0 s s 0 Yzx

Y12 -2es 2es 0 0 0 e2 _ S2 Yxy

This transformation matrix is represented by the symbol [T.J. Expressing the on-axis

stresses and strains ({ah and {e}l) of Eqs. (3.4) in terms of the off-axis stresses and

strains (Eqs. (3.9) and (3.11» yields

3. Lamina Stresses, Strains, and Displacements 17

[Tl]{O}x = [C][T2] {e}x . (3.12)

Rearranging Eqs. (3.12) yields

or

{O}x = [C]{e}x' (3.13)

where

[C] = [T1] -t[C ][T2]'

Expanding Eqs. (3.13), the off-axis stress-strain relationship for the given lamina is

t1x Cll Cl2 C l3 0 0 C16 ex

t1y e12 C22 e23 0 0 C26 ey

t1z en C2l e33 0 0 C36 ez - (3.14)

Tyz 0 0 0 C44 C4S 0 Yyz

Tzx 0 0 0 C45 Css 0 Yzx

Txy Ct6 C26 C36 0 0 C66 Yxy

where

- 2 2 e u = cell + s e 2l ,

3. Lamina Stresses, Strains, and Displacements 18

- 2 2 C44 = C C44 + s C SS t

C4S = CS(CSS - C44) ,

- 2 2 C SS = S C44 + c C SS t

and

The transformed stiffness matrix is reduced by recognizing that a z = 0 from the plane

stress assumption. When zero is substituted for (lz in Eqs. (3.14) there results

(3.15)

Solving Eq. (3.15) for t z ,

(3.16)

3. Lamina Stresses, Strains, and Displacements 19

By eliminating O'z and substituting Eq. (3.16) for tu the reduced stiffness matrix and

stress-strain relationship become

O'x QI1 Q12 Q13 0 0 tx

O'y Q12 Q22 Q23 0 0 ty

1'xy - Q13 Q 23 Q33 0 0 Yxy (3.17)

1'yz 0 0 0 Q44 Q4S Yyz

1'xz 0 0 0 Q45 Qss "'Iv:

where [Qij] is the transformed reduced stiffness matrix. In terms of the transformed

stiffness matrix its elements are

Q4S = C4S t and Qs" = Css •

3.2. Strain-Displacement Relations

In the Eulerian viewpoint the strains are described in indicial notation as [36]

1 aUi aUj aUk aUk tij = 2" [ ax- + ax- + -ax.- ax- ]; i = 1,2,3 , j = 1,2,3 , (3.18)

1 I 1 1

3. Lamina Stresses, Strains, and Displacements 20

where Ult u2, and u] are the displacements in the x, y, and z directions and are referred

to as u, v, and w, respectively, in the analysis that follows; and the quantities Xl' Xl' and

X3 refer to the orthogonal coordinate directions redesignated x, y, and z, respectively, in

the analysis that follows. After substituting the first-order SDT displacement field into

the Eulerian strain description, the strains become

a·~ 2 a·~ atjl + z (_Y'_X ) + .L [ (_Y'_X )2 + ( __ y )2] ax 2 ax ax ' (3.19)

(3.20)

(3.21)

8t/1x atjly 2 a",x 8t/1x 8tj1y 8t/1y ] +z(-+-)+z (--+--) ay ax ax By ax ay (3.22)

(3.23)

3. Lamina Stresses, Strains, and Displacements 21

and

(3.24)

Many of the higher-order terms in the strain expressions are negligible in comparison

with the flrst-order terms. Using von Karman's criterion only the squares and products

of ow/ox and ow/oy are retained. Removing all other higher-order terms and replacing

tensorial strains with engineering strains, the strain-displacement relations become

(3.25)

(3.26)

t z = 0 t (3.27)

ouG ovo owo owo 0 VI 0 Vly + + __ +z( __ x + __ ) Yxy=ay ax ax oy oy ox (3.28)

(3.29)

and

(3.30)

For convenience in notation let

3. Lamina Stresses, Strains, and Displacements 22

The total strains can then be written

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

3. Lamina Stresses, Strains, and Displacements 23

4. Governing Equations for Laminated Plates

In Chap~er 3 the stress-strain relationships for an arbitrary lamina, Eq. (3.14), were

found. In this chapter those relationships are extended to an arbitrary laminate resulting

in force-strain relationships. Thermal forces are also included in the force-strain re­

lationship. Finally, the equations of motion are expanded into five equilibrium

equations which fully describe the five degrees of freedom of the laminate: u, v, w, tPx,

and t/ly.

4.1. Laminate Forces and lVloments

The resultant forces and moments per unit width acting on a laminate can be found

by integrating the laminae stress-strain relationship of Eq. (3.14) over the laminate

thickness, yielding

(4.1)

4. Governing Equations for Laminated Plates 24

(4.2)

and

(4.3)

where 1'xy = 1'yx, 1'zx = 1'xzt 1')'1 = 1'ty' Nxy = Nyx, and Mxy = Myx. These forces and mo ..

ments are labeled on the plate in Fig. 4. Figure 5 depicts a laminate of N laminae and

gives the lamina numbering system that is used in the analysis. The z-axis is. orthogonal

to the midplane, positive being up. The laminae are numbered sequentially from 1 to

N starting with the top lamina. The laminae interfaces are numbered corresponding to

lamina directly below the interface with the laminate top surface being numbered 1 and

the laminate bottom surface numbered N + 1. The z-coordinate of the midplane of the

kth lamina is denoted by Zto Equations (4.1), (4.2) and (4.3) can now be rewritten using

the notation of Fig. 5:

(4.4)

(4.5)

and

4. Governing Equations for Laminated Plates 25

N Zit

(Mx• My. Mxy) = If (ux• U'l' TXY) Z dz · (4.6) k==l ZIt+l

The off-axis stresses can now be replaced with the equivalent stiffness matrix and

middle surface strains, Eqs. (3.31) through (3.36). Neither the middle surface strains nor

the stiffness matrix are dependent on z so they can be brought outside the integral. After

these changes are made Eqs. (4.4), (4.5), and (4.6) can now be written in matrix form as

N x All Al2 A16 0 0 B11 B12 B16 0 ex

Ny Al2 A22 A 26 0 0 B12 B22 B26 0 ey

Nxy A 16 A 26 A66 0 0 B16 B26 B36 0

Yxy

Nyz 0 0 0 A44 A4s 0 0 0 0 Yyz

= (4;7) Nxz 0 0 0 A4S Ass 0 0 0 0

Yxz

Mx Bll B12 B16 0 0 011 0 12 0 16 Kx

My Btl B22 B26 0 0 0 12 0 22 0 26 Ky

Mxy B16 B26 B66 0 0 0 16 0 26 0 36 Kxy

where

N

Aij = I Qij(Zk - Zk+l) t (4.8) k=l

(4.9)

and

4. Governing Equations for Laminated Plates 26

(4.10)

4.2. Thermal Forces and Moments

The three-dimensional thermal strains are induced in the laminate when the temper-

ature of the laminate changes from its stress-free state. The stress free state occurs at a

temperature very close to that at which the composite plate is cured. If the cure tem-

perature and coefficients of thermal expansion are known, the thermally-induced strains

can be found from

t· = (X,AT 1 lL.1 , 1 = 1, 2, 3 , (4.11)

where (XI are the coefficients of thermal expansion in the ith material direction and &T is

the change in temperature from the stress-free state to the present temperature.

The lamina thermally-induced stresses can now be found in the principal material

directions from Hooke's law as [3]

0'1 ell e l2 ell 0 0 0 (Xl

0'2 e l2 e 22 e 23 0 0 0 (X2

(1) ell e 2l e 3l 0 0 0 (X3 - AT. (4.12)

<14 0 0 0 C44 0 0 0

as 0 0 0 0 ess 0 0

0'6 0 0 0 0 0 e 66 0 T

4. Governing Equations for Laminated Plates 27

As before, we are more concerned with stresses and strains on the laminate coordi-

nate axes than along the principal material directions. Therefore, the coefficients of

thermal expansion must be transformed to the on-axis directions by [T J-l :

(Xx c2 S2 0 0 0 -cs (Xl

(Xy S2 c2 0 0 0 cs (X2

(Xz 0 0 1 0 0 0 (Xl

= (4.13) 0 0 0 0 c s 0 0

0 0 0 0 -s c 0 0

(Xxy 2cs -2cs 0 0 0 c2 _ S2 0

or t written out,

(4.14)

(4.15)

(4.16)

and

(4.17)

Upon substituting the on-axis thermally-induced strains into Eq. (4.12), the corre-

sponding stresses can be expressed

4. Governing Equations for Laminated Plates 28

O'x Cll Cl2 C l3 0 0 C16 IXx

O'y Cl2 C22 C23 ·0 0 C26 IXy

O'z Cl3 C23 C33 0 0 C36 IXz. = AT • (4.18)

1'yz 0 0 0 C44 C45 0 0

1'xz 0 0 0 C4S CS5 0 0

1'xy C16 C26 C36 0 0 C66 IXxy T

As was done in Section 3.1, O'z is assumed to be zero and therefore, can be eliminated

from Eqs. (4.18). Now, IXz can be rewritten in terms of lXx, IXy, and IXxy as

(4.19)

Substituting Eq. (4.19) into Eq. (4.18) yields

O'x Q11 Q12 Q13 0 0 IXxdT

O'y Q12 Q22 Q23 0 0 IXyAT

1'xy - Q13 Q23 Q33 0 0 IXxyAT (4.20)

Tyz 0 0 0 Q44 Q45 0

Txz 0 0 0 Q45 Qss 0 T

where [Q] is the same stiffness matrix as in Eq. (3.17). At this point the strain caused

by mechanical loads can be added to the thermally-induced strains to give the total

strain. The purely mechanically induced stresses can now be expressed as

4. Governing Equations for Laminated Plates 29

(1x QI1 Q 12 Q13 0 0 tx - €XxAT

(1y Q12 Q22 Q23 0 0 ty - €XyAT

't'xy = Q13 Q23 Q33 0 0 Yxy - €XxyL\T (4.21)

't'yz 0 0 0 Q44 Q4S Yyz

't'xz 0 0 0 Q4S Qss Yxz M

where t x, ey, Yxyt YYI' and Yxz now represent the total strains, mechanical plus free ther-

mal.

With the use of the expressions developed at the beginning of this chapter, Eqs. (4.4)

through (4.10), Eqs. (4.21) can be integrated over the laminate thickness to yield an ex-

pression for the laminate mechanical forces and moments:

Nx Au A12 Al6 0 0 Bll B12 B 16 0 NT £x x

Ny Al2 A22 A 26 0 0 Bl2 B22 B26 0

£y NT Y

N xy A16 A26 A66 0 0 B16 B26 B36 0 T

Yxy N xy

Nyz 0 0 0 A44 A4S 0 0 0 0 0 Yyz = , (4.22)

N xz 0 0 0 A45 Ass 0 0 0 0 0 Yxz

Mx Bll B12 B16 0 0 0 11 0 12 0 16 ICx MT x

My Bt2 B22 B26 0 0 0 12 0 22 0 26 ICy MT y

Mxy B16 B26 B66 0 0 0 16 0 26 0 36 "'"y T

Mxy

where the superscript T on the right-hand side indicates thermal forces and moments.

The total forces and moments can be expressed by adding the thermal forces and mo-

ments to both sides of Eqs. (4.22) to give

4. Governing Equations for Laminated Plates 30

N" A11 Al2 AJ6 0 0 Bll B12 B16 -0 e"

Ny Al2 A22 A 26 0 0 B12 B22 B26 -0 ey

Nxy A 16 A 26 A66 0 0 B 16 B26 B36 -0 Yxy

Nyz 0 0 0 A44 A45 0 0 0 -0 Yyz

- (4.23) Nxz 0 0 0 A4S Ass 0 0 0 -0

Yxz

M" Btl B12 B16 0 0 Dtl D12 D 16 K"

My B12 B22 B26 0 0 D12 D22 D 26 Ky

Mxy Bt6 B26 B66 0 0 D 16 D 26 D36 Kxy

where

- - T My = My, and M"y = Mxy + M"y .

In the paragraphs which follow, expressions for the thermally induced forces and

moments (NxT, NyT, NJ, MxT, Ml, and MJ) are derived on the basis of a piecewise­

linear temperature distribution in the z-direction. A typical graphite lamina thickness is

on the order of 0.15 mm (.006 in.) and so it seems reasonable to assume that the tem-

perature distribution through the lamina thickness is linear and, therefore, piecewise

linear through the laminate thickness. The temperature field can then be described as

(4.24)

4. Governing Equations for Laminated Plates 31

where Tie is the temperature at the lamina interface above the kth lamina, T HI is the

temperature at the lamina interface above the ktb + 1 lamina, Zt is the z-coordinate of the

lamina interface above the kth lamina, and tie is the thickness of the kth lamina. If To re-

fers to the stress-free temperature then

AT = T(x,y,z) - To . (4.25)

The thermally-induced forces can now be found in the same manner as the mechan­

ical forces were found in Chapter 3. The thermal forces and moments are therefore

written

=Ir k=l Zt+l

and

Q11 Q 12 Q13

Qt2 Q22 Q23

Q13 Q23 Q33

Qt. Q12 Q13

Q 12 Q22 Q23

Q13 Q23 Q33

k

k

ATdz (4.26)

k

ATzdz . (4.27)

(Xxy k

The stiffness matrix [QIj] and the coefficients of thermal expansion (xx, (Xy, and (Xxy do

not depend on z and can be taken outside of the integral. The terms remaining in the

integrand are z and L\ T. Replacing AT with Eqs. (4.24) through (4.25) and inte-

grating over the lamina thickness leads to

f" ~Tdz = tk [(Tk + Tk+1)/2 - TJ • (4.28)

'k+l

4. Governing Equations for Laminated Plates 32

and

(4.29)

where Zt is the z-coordinate of the middle of the kth lamina. The thermal forces and

moments can now be written as

NT N Qtl Q12 Q13 txx x

=L NT Q12 Q22 Q23 a y tk [(Tk + Tk+1)/2 - To] (4.30) y

T N xy k-I QlJ Q23 Q33 txxy

k k

and

MT x N Q 11 Q12 QlJ ax

=L MT Q12 Q22 Q2J Zk tk zk tk _

y a y tk[Tk( 2" + 12) + Tk+1( 2" -12) - TozkJ(4.31) T

Mxy k-l Q13 Q23 Q33 tx xy k k

4.3. Equations of Motion

Up to this point the strains have been defined in terms of the five degrees offreedom:

u, v, w, l/Ix, and l/Iy. The stresses have been found from Hooke's law in terms of the

strains, and the laminate forces and moments have been defmed in terms of the me ..

chanical and thermal strains. In order to completely describe the dynamic behavior of

4. Governing Equations for Laminated Plates 33

the three-dimensional laminate, the equations of equilibrium must also be satisfied.

Consider the infinitesimal cube with dimensions dx, dy, and dz shown in Fig. 6. All of

the nonzero stresses considered in this thesis are assumed to be acting on the cube. The

net force in the x- and y-directions can be written

0(1 orxy F x = 0; dxdydz + --ay dydxdz (4.32)

and

oay orxy F y = ay dydxdz + a;:- dxdydz. (4.33)

Because of the plate geometry, the net force in the z-direction also contains in-plane

stress terms which cause nonzero z-directed forces when the plate becomes nonplanar

[37]. The net force in the z-direction is therefore

oryz OT F z = --ay dydxdz + o~z dxdydz +

o ow ow 0 ow ow ox (ax ox + TXYiiY )dxdydz + oy (ay oy + Txy ax )dxdydz. (4.34)

The equations of eqUilibrium can now be found by substituting the net forces into

Newton's second law of motion. By assuming that only symmetric laminates will be

analyzed we can ignore rotary inertia terms and therefore the acceleration terms in the

three directions are

4. Governing Equations for Laminated Plates . 34

and w,

and the mass is denoted by pdxdydz, where p is the mass density. Newton's second law

then becomes

.. [ oa x OT xy ] . p(u + zt/lJdxdydz = --ax + ay dxdydz,

.. [OTXy oay ] p(v + zt/ly)dxdydz = ax + ---ay dxdydz,

and

.. [OTyz OTxz a ow ow 0 aw aw ] pwdxdydz = ay + ax + ox (ax ax + Txy oy ) + oy (TXY ax + ay ay ) dxdydz.

Similarly, moment equilibrium equations can also be derived. The moment equilibrium

equations based on the net moments about the x- and y-axes are

(4.35)

and

.. aa OTxy pz(u + zt/lJdxdydz = (z oxx + z ay - T;u)dxdydz . (4.36)

4. Governing Equations for Laminated Plates 3S

Equations (4.32) through (4.36) can now be divided through by dxdydz. Recalling Eqs.

(4.4) through (4.6), the equilibrium equations can be integrated through the laminate

thickness yielding

a2 u a Nx 0 Nxy ph--2 =-a-+ a ' at x y

(4.37)

(4.38)

a2 w 8Nxz 8Nyz 8 ow ow 0 ow ow ph -2- = -a-+ -a-+ -0 (Nx -8 + Nxy -8 ) + -"1- (Nxy -8 + Ny -8 ) ,(4.39) a t x y X X Y oy x Y

ph3 a2 t/lx a Mx 0 Mxy = 8 x + 8 Y - Nxz t 12 8 t2 (4.40)

and

ph3 82

t/ly o Mxy a My 12 a t 2 - a x + oy - Nyz · (4.41)

4. Governing Equations for Laminated Plates 36

5. Finite Element Model

The salient features of the fmite element model are explained in this chapter. First,

the equations of dynamic equilibrium are expanded and recast into a variational formu-

lation. Interpolation functions are derived and the equilibrium equations are rewritten

in matrix form. The Newmark method is employed to integrate the equations with re-

spect to time. The nonlinearity of the dynamic equilibrium equations is treated using the

Newton-Raphson method. Gauss-Legendre quadrature is used to evaluate the integrals.

Finally, reduced integration is explained with respect to the shear terms of the dynamic

equilibrium equations.

5.1. Variational Formulation

The equations of dynamic equilibrium, Eqs. (4.37) through (4.41) are more easily

treated under the following substitutions:

5. Finite Element Model 37

When the total forces and moments are expressed in terms of the corresponding strains

and stiffness's, Eqs. (4.21), the equations of dynamic equilibrium, Eqs. (4.37) through

( 4.41), become

hft_.L [A (k+..L(k)2)+A (l!..+_1 (aw )2) p 8t2 OX 11 8x 2 ox 12 oy 2 oy

-.L[A (k+..L( ow )2)+A (l!..+_l (k)2)+A (k+ av + oy 16 ox 2 ox 26 oy 2 oy 66 oy ax

- - +B --+B -+B --+- -------ow ow) at/lx ot/ly (o"'x o"'y) ] oN~ oN;y ax oy 16 ax 26 oy 66 oy ox - 8x 8y' (5.1)

S. Finite Element Model 38

oNT oNT xy y =- ---ox oy , (5.2)

+ ow [A (1.!!.. + ..L ( ow )2) + A ( 1.Y.. + _I (l:!L Y) oy 16 OX 2 ox 26 oy 2 oy)

o OW A au + 1 ow + A OV + I Ow [ [ ( - 2) ( ()2) - oy ox 16 ox T ( ox ) 26 oy T oy

+ ow [A (1.!!..+..L ( ow )2) +A (1.Y..+_l ( iJw \2) oy 12 ox 2 ox 22 oy 2 oy)

S. Finite Element Model 39

A ( .'1 + ow ) ] = p +.L [NT ow + NT ow ] +.L [NT ow + NT OW] (5 3) 44 Y' Y oy ox ! x ox xy oy ay xy ox y oy , .

P Y' x 0 B OU + 1 ow + B ov + 1 ow h J a2

./. [( ()2 ) ( ()2 ) 12 7 - ax . 11 ox 2' ox 12 oy 2" oy

-.L[B (1!!.+_1 (.2:!!..)2)+B (k+_l (.2:!!..)2) ay 16 ox 2 ox 26 oy 2 oy

(5.4)

and

phJ

a2

VJy a [B (au + 1 ( ow )2) + B (OV + I ( ow )2) 12 7--ax- 16 -ax- 2" ax 26 Ty 2' ay

s. Finite Element Model 40

-.L[B (1.Y..+_l (aw )2)+B (k+_l (E!..)2) ay 12 ax 2 ax 22 ay 2 ay

(5.5)

Variational formulation in the context of this thesis means the recasting of differen-

tial equations, Eqs. (5.1) through (5.5), into an equivalent integral form by distributing

the differentiation between a test , A, and the dependent variables. The reason for this

manipulation is to minimize the order of differentiation of the dependent variables and

to specify the secondary variables of the problem. For example, Eq. (5.l) is first multi­

plied by a test'function, A, and then integrated over the element area, a, yielding

J [A h~-A.L [A (1.Y..+_l (E!..)2)+A (k+..L(E!..)2) Q p at2 ax 11 ax 2 ax 12 ay 2 ay

a [ (au 1 ( aw )2) (av 1 ( aw )2) -1 By A16 ax +2" ax +A26 By +2" ay (5.6)

5. Finite Element Model 41

Wherever there are second derivatives in Eq. (5.6) the differentiation can be equally

distributed between 1 and the dependent variable by noting that for any two

differentiable functions f and g

t g ~~ dxdy = - t f !! dxdy + t :x (fg)dxdy

and

J g ~f dxdy = -J f :g dxdy + J : (fg)dxdy. Q Y .Q Y .Q Y

But by the divergence theorem

t :x (fg)dxdy = Jr

(fg)Ilxds

and

J : (fg)dxdy = J (fg)nyds · .Q Y r

Therefore,

S. Finite Element Model 42

f or f og J , Q g ox dxdy = - Q f ox dxdy + r (fg)nxds

and

f or f og J g T dxdy = - f T dxdy + (fg)nyds , Q Y n Y r

where Ilx and n,. are the x and y components, respectively, of the unit normal to the ele­

ment boundary, r, and ds is an infinitesimal arc length along the boundary. With these

considerations Eq. (5.6) becomes

J [1 h~+.M. [A (k+_l (k)2)+A (1!..+_1 (k)2) n p ot2 ox 11 OX 2 ox 12 oy 2 oy

+.M.[A (k+_l (ow )2)+A (1!..+1..( OW)2)+ oy 16 ox 2 ox 26 oy 2 oy

f l[A (k+_l ( ow )2) +A (1!..+1..( ow)2) r t 1 ox 2 ox 12 oy 2 oy (5.7)

S. Finite Element Model 43

l[A (l.1!..+_1 (k)2) +A (l!..+_l ( OW )2) +A (l.1!..+l!..+ ow OW) 16 OX 2 ox . 26 oy . 2 oy 66 oy OX OX oy

The primary variables are defined as the dependent variables; u, v, w, t/lx, and t/ly. The

secondary variables are defined from the variational form ~s the coefficients of A. in the

boundary integrals. In Eq. (5.7) the secondary variables are

[A ( l.1!.. + ..L ( ow )2) + A ( l!.. + _1 (k)2) + A (l.1!.. + l!.. + ow k) 11 ax 2 ax 12 ay 2 oy 16 ay ax ax ay

(5.8)

and

[A ( l.1!.., + ..L ( k )2) + A ( l!.. + ..L ( k )2) + A ( l.1!.. + l!.. + kk ) 16 ax 2 ax 26 oy 2 ay 66 oy ox ax ay

(5.9)

which are recognized from Eqs. (4.7) as (Nx + Nxy)nx and (Nxy + Ny)n,.. The variational

form of Eq. (5.1) can fmally be written as

s. Finite Element Model 44

+R[A (1!!.+..l. (k)2) +A (k+_l (k)2) + ay 16 ax 2 ax 26 ay 2 ay

The variational forms of Eqs. (5.2) through (5.5) can be derived in a similar manner.

5.2. Interpolation and Approximate Functions

In the fmite element method the region being examined is discretized into small fmite

areas called elements. The element boundaries are established by discrete points called

nodes. Values of the dependent variables (u, v, w, t/lx, t/ly) are directly obtained only at

the nodes. The values between the nodes are found by interpolation. In the Ritz

method [37] the dependent variables for a single element are approximated by

s. Finite Element Model 4S

(5.11)

(5.12)

(5.12)

(5.14)

i=l

and (5.15)

i=l

where u(1), y><I), W<I), "'~), and "'~)' are the nodal values at the point (X;, YI), the cP, are the

interpolation junctions, and the symbol P represents the number of nodes the element

contains. Equations (5.11) through (5.15) are referred to as approximate junctions. The

interpolation functions of Eqs. (5.11) through (5.15) are not unique but must obey to the

following criteria:

S. Finite Element Model 46

1. The interpolation functions must be differentiable to the same order as the depend­

ent variables.

2. Any set of interpolation functions {tP,} must be linearly independent.

3. The set of interpolation functions must be complete; i.e., if the interpolation func­

tions contain a term of the nth degree, they must also contain terms of the

n - lit, n - 2nd ••• , and Oth degree.

4. Interpolation functions are nonzero only on the element to which they are assigned.

5. An interpolation function l/JI is equal to unity at the coordinate Xt, y, and equal to

zero at any other nodal locat~on.

6. At any point Xt, y. on an element the value of the interpolation functions at that

point must sum to unity.

If these criteria are satisfied the element is said to be conforming which means that:

1. There are no interelement gaps or overlaps and no sudden changes in slope between

elements.

2. Rigid body modes are present; i.e. when the body is in rigid motion the element must

exhibit zero strain.

3. The elemental interpolation functions are complete ..

s. Finite Element Model 47

Figure 7 shows three master rectangular elements in local Cartesian coordinates which

are all conforming elements. The following three sets of interpolation functions corre-

spond to these three elements.

Linear Rectangular Element (4 Nodes)

1 cP 1 = 4" (1 - ')( 1 - rr) ,

and

Quadratic Rectangular Element (8 Nodes)

1 2 2 cPs = 2" (1 -, )(" - rr) ,

5. Finite Element Model 48

and

Quadratic Rectangular Element (9 Nodes)

These are the same as the interpolation functions for the 8-noded element with the ad-

dition of

The interpolation functions therefore depend on the element type. The computer code

COMMEC2 contained in the appendix is capable of utilizing all of these elements how-

ever only results of the analysis using the 9-noded quadratic rectangular element are re ..

ported in this thesis.

The test function l must be capable of satisfying the boundary conditions but other

than that it is arbitrary. It is convenient and usual to let the test function assume the

same form as the interpolation functions. After this substitution is made along with the

substitution of the approximate functions, the variational formulation can be written in

matrix form as

5. Finne Element Model 49

M,l,l IJ 0 0 0 0 Ui

0 M~2 lJ 0 0 0 vi

0 0 M~3 I) 0 0 Wi +

0 0 0 44 Mij 0 "xi

0 0 0 0 M~S IJ t/lyi

K~l 1)

K!,2 IJ

K·1,) lJ

K,l.4 1J

K~·5 IJ ~

Fl 1

K~l IJ

K~2 I]

K~3 11

K~4 IJ

K~S 1J Vi F~

1

K~l 1]

K~2 1]

K~,3 IJ

K~4 IJ

K~S IJ W· 1 - F~

1 (5.16)

K~l K~2 K~3 44 K~5 l/Ixi ~ IJ IJ IJ Kij 11 1

K~l lJ

K~2 IJ

K~3 lJ

K~4 IJ

K~S IJ t/lyi F~

1

where

Mi/ = J [ph.pj.pj]dxdy , Q

s. Finite Element Model 50

5. Finite Element Model 51

5. Finite Element Model 52

43 34 Kij = Kji /2 ,

p

FI = - II t/>i[N~t/>j,Jl + N~yt/>J.yJdxdy • j-l Q

5. Finite Element Mode1 53

p

F~ = - If 4>i [N;4>J,Y + N;y4>j,x]dxdy • j=l Q

p

11 = - If 4>i[M;4>i,x + M;y4>j,y]dxdy • j=l Q

and

p

F~ = - II 4>;[M;4>i,y + M;y4>i,x]dxdy · j=l Q

The matrices and vectors of Eq. (5.16) are referred to as the mass matrix [M], the direct

stiffness matrix [K), the force vector {F} and the displacement vector {U}, so that Eq.

(5.16) can be written

[M]{U} + [K]{U} = {F} . (5.17)

5.3. Gauss-Legendre Quadrature and Reduced Integration

In order to incorporate the approximate and interpolation functions into the vari-

ational form, all the functions and areas must be described in the same coordinate sys-

S. Finite Element Model 54

tern. The transformation from the local coordinate system to the global coordinate

system is done with the Jacobian matrix. For example,

acPi acPi a, ax

= [JJ acPi acPi a" ay

and

dxdy = IJld,d11 ,

where

[J] =

The elements used in this thesis are isoparametric, meaning that the x- and y-

coordinates are approximated using the same nodes and interpolation functions as were

used to approximate the dependent variables. Thus

5. Finite Element Model

p

x = LXi<Pi i-I

55

P

and Y= LYicPi' i=l

For computational convenience the Jacobian matrix can be rewritten as

[J] =

A wide variety of numerical integration methods is available to evaluate the

integrands of the previous equations. Gauss-Legendre quadrature is one of the most

widely used and has proven to be reliable. For any function G(x) the integral can be

found by

b d 1 t GP GP J J G(x,y)dxdy = f f G(~.1/)IJld1/d{ = LLG(~.1/)WiWPI. (5.18) a c -1 - J i= I j= 1

where C and" are the so-called Gauss points in the local system (roots of the Legendre

polynomial), WI and wt are weight factors, and IJI is the determinant of the Jacobian

matrix. The integral of a polynomial of the nth degree can be found if the number of

Gauss points is greater than or equal to (n + 1)/2 [38].

Although the integral can be evaluated using the Gauss-Legendre quadrature, the

terms in Eq. (5.16) involving Aw, A..s, and Ass are still erroneous. This is because the

problem under consideration is based on two-dimensional plate theory, but the trans-

s. Finite Element Model 56

verse shear terms are included. The result is that these terms are too stiff in the finite

element model. In other words, the solution to the governing equations is dominated

by the presence of the shear deformation terms regardless of their value. Zienkiewicz,

Taylor, and Too [24] addressed this problem and developed a solution, called

reduced integration, which is very simple to employ. The first step in reduced inte-

gration is to separate the transverse shear terms from the tangent stiffness matrix. These

terms are

S. Finite Element Model S7

and

Instead of using (n + 1)/2 Gauss points, as is done when finding the exact solution, the

transverse shear terms are integrated using one less Gauss point. This relaxes the

stiffness and gives a better weighting to those terms.

5.4. The Transient Solution (Newmark Method)

There are many choices of nur.nerical solutions to the transient problem represented

by Eq. (5.12). However, the Newmark method is one of the most reliable and is un-

conditionally stable. In the Ne'wmark method the essential boundary conditions and

their derivatives with respect to time at the n + 1 It time step are given by

(5.19)

and

S. Finite Element Model 58

(5.20)

where now U represents anyone of the dependent variables. Solving for {u}n+l,

(5.21)

Now mUltiplying both sides of Eq. (5.21) by the mass matrix [M]n+l yields

However,

(5.23)

so that Eq. (5.22) can now be written

4 4 where Clo = (L1t)l and at = L1t' For simplification in notation let

and

Then

S. Finite Element Model S9

5.5. Newton-Raphson Method

The Newt on-Raphson method of iteration is a very fast and accurate scheme for

solving nonlinear problems. The theory is as follows. Consider a residual vector {R}

such that at the rth iteration

(5.24)

When the solution fmally converges the residual matrix ideally would be zero. The res-

idual matrix on the iteration after the rtb iteration will be

(5.25)

If {R}r+l is set to zero, Eq. (5.25) can be written as

8{R}r [Ur+1 _ UrJ = _ {R}r 8{Ur '

(5.26)

where

(5.27)

and is commonly called the tangent stiffness matrix. Unlike the direct stiffness matrix

[Kl, the tangent stiffness matrix, [KT] is symmetric. The individual terms of the tangent

stiffness matrix can be written in terms of the direct stiffness matrix and mass matrix:

K~l1 = K911 + M~.l 1] IJ lJ'

K!13 = 2K~13 1) lJ'

S. Finite Element Model 60

K!14 = K~14 IJ 1)' K!IS = K~lS KT22 KD22 M22

Ij IJ' ij = ij + 1 ij ,

K!23 = 2K~23 1J IJ'

K!34 _ K~34 I) - 1] ,

T44 D44 44 T45 D4S K·· = K·· + M·· K·· = K .. lJ I) 1) , I) 1)'

and

K!55 _ K~SS + M~5 IJ - Ij IJ'

where

J.. ow aW OtPi OtPj + _I ( aW )2 ~tPi atP j ] + A [au atPi atPj + . 2 oy ax oy ox 2 ox oy oy 16 oy ox oy

- - --+ - - -- +A - - --+ (OW )2 atPj atPj (ow)2 atPi atPj] [ou Ocf>i acf>j ax ox oy ax oy ax 26 oy oy oy

S. Finite Element Model 61

+ Bw 8w .Bq;,i 8q;,j ] + A (a.w)2 8cPi aq;,j + B 8t/1x aq;,i aq;,j + ax ay ay ax 22 By 8y ay 11 8x ax ax

Equation (5.26) can now be written as

(5.28)

Given the initial displacements, velocities, and accelerations, Eq. (5.28) can be numer­

ically solved for any other displacements, velocities, or accelerations in time.

S. Finite Element Model 62

6. Verification of Model and Results

The results predicted by the finite element model are presented and compared to

known results of plate problems. In order to verify the finite element model, discrete

boundary conditions are applied to plates of various geometries and the known results

are compared to those predicted by the model. Each condition verifies an aspect of the

model and demonstrates its capabilities. The conditions are:

• Simply-supported isotropic plate subjected to a unifonnly-distributed transverse

mechanical load at steady state.

• Clamped composite plate subjected to a unifonnly-distributed transverse mechanical

load at steady state.

• Simply-supported isotropic plate subjected to a linear temperature variation through

the thickness of the plate (steady state).

• Simply .. supported isotropic plate subjected to a sudden uniformly .. distributed trans­

verse mechanical load (transient).

6. Verification of Model and Results 63

• Simply-supported laminated composite plate subjected to a sudden uniformly­

distributed transverse mechanical load ( transient).

• Simply-supported composite plate subjected to a sudden heat flux on the upper

surface (transient).

Each case is analyzed with a finite element model of four quadrilateral 9-node elements

as shown in Fig. 8.

6.1. Static Mechanical Load

Numerical results are presented in this section for a simply-supported isotropic plate

under uniformly distributed static loads and for a clamped laminated plate under the

same loading.

The geometry and boundary conditions of the rectangular isotropic plate are shown

in Fig. 9. Because of the biaxial symmetry only the first quadrant of the plate must be

modeled. The isotropic material properties are E = 21.0 GPa, and v = 0.3.

An analytical solution for a simply supported rectangular isotropic plate with uni­

form transverse loading is available for comparison. However, the analytical theory is

based on a linear relationship between the deflection in the z-direction and the transverse

load and therefore it is limited to the case of a thin plate with small deflections. For the

case of a square plate with v = 0.3 the analytical theory gives a maximum deflection at

the center of the plate of

6. Verification of Model and Results

4 poa wmax = 0.0443 --3- .

Eh

64

Figure 10 shows the normalized maximum deflections versus the plate width-to ..

depth (a/h) ratio of three theories. The continuous line represents the thin plate theory

[39J and the dashed line represents the first-order SOT fmite element solution of this

thesis. The results of a higher-order shear deformation theory of Reddy [22] are indi-

cated with the triangles. The deflections in Fig. 10 are normalized as

The finite element solutions agree with the linear theory until the width-to-depth ratio

becomes less than about 10. At this point thin plate theory breaks down because it as­

sumes that the in"plane loads are negligible. However, as the 'width-to-depth ratio be-

comes small more of the load is being carried in-plane and less transversely. Because the

finite element solutions have incorporated this coupling between the deflections in the

u-, y-, and z-directions they diverge from the linear solution. The two finite element

solutions agree within 0.1 percent difference.

The geometry and boundary conditions of the laminated composite plate are shown

in Fig. 11. The plate is symmetric about the x-y plane with a O· and a 90· layer of equal

thickness above and below the midplane; i.e. a [0/90/90/0] layup. It is clamped on all

edges and subjected to uniformly-distributed transverse load. The orthotropic properties

of the laminae are

Ell = 12.605 GPa, E22 = 12.628 GPa, En = 12.628 GPa,

V12 = 0.23949,

G 12 = 2.155 GPa, G31 = 2.155 GPa, . and G23 = 2.155 GPa.

6. Verification of Model and Results 6S

The fmite element model results are compared to the linear solution of Zaghloul

and Kennedy [401 in Fig. 12. Experimental results are also available for this case [401

as w~ll as results from a higher-order finite element model [22]. The dot-dashed line in

Fig. 12 represents the linear solution, the dashed line is the experimental data, and the

solid line is the SDT fmite element solution from this thesis. The triangles represent the

values from the higher-order finite element model [22]. Again the nonlinear and linear

solutions converge for small loads. But now it can also be seen that the experimental

solution agrees very well with the finite element solutions. The small discrepancies that

occur between the two finite element solutions and the experimental results may be due

to a number of things, including errors in modeling experimental boundary conditions.

Another important result is the agreement between the higher-order finite element model

and the first-order model of this thesis. The deflections differ by less than 0.01 percent.

6.2. Linear Temperature Variation Through an Isotropic Plate

The nonlinearity of the plate deflection due to a thermal load is demonstrated using

a simply-supported isotropic plate. The boundary conditions and geometry of the plate

are shown in Fig. 9 (a = 25.4 em), and the material properties are

E == 68.9 GPa, v == 0.3, and ex = 0.000124 m/mrC. The midplane of the plate is kept

at the stress-free temperature and the temperature varies linearly through the thickness

of the plate and is uniform on any x-y plane. Figure 13. shows the nondimensional de­

flections versus the width-to-depth ratio (a/h). The deflection in nondimensional form

is

6. Verification or Model and Results

W= lO.4wh exa2 aT '

66

where ~ T is the temperature difference between the top and bottom surfaces, (X is the

coefficient of thermal expansion, and w is the actual deflection. The solid line in Fig.

13 depicts the thin-plate small-deflection solution (411. 'Yhen the temperature variation

is small producing small deflections the finite element predictions agree with the thin­

plate small-deflection theory. The dashed line in Fig. 13 shows the nondimensional fmite

element solution when the temperature difference between the top surface and bottom

surface is 5000 C. Again the effects of the nonlinear strains are shown in Fig. 13. As

the width-to-depth ratio becomes small the nonlinear and linear solutions converge.

6.3. Transient Mechanical Load

The ability to accurately predict deflections as a function of time due to a

uniformly-distributed transverse load is demonstrated for a simply-supported isotropic

square plate and laminated composite square plate. The isotropic material properties

are E = 2.1 GPa, v = 0.3, and p = 800.0 kg/ml. The plate is 5.0 cm thick and 25.0 cm

wide and suddenly subjected to a 100 kPa uniform transverse load. The deflections

through half of a cycle are shown in Fig. 14. The solid line represents the solution using

the present fIrst-order SDT model and the triangles represent the results using another

first-order SDT finite element model [421. There is no appreciable difference in the two

finite element solutions.

Figure 15 shows the center deflection versus time of a [O/90j01Iaminated composite

plate under the same 100 kPa load. The laminated plate is 5.0 cm thick and 25.0 cm

square and the lamina material properties are

El1 = 52.1 GPa, E22 = 2.1 GPa, E33 = 2.1 GPa,

6. Verification or Model and Results 67

V l 2 = 0.3,

G12 = 1.05 GPa, G31 - 1.05 GPa, G23 = 1.05 GPa.

It can be seen in Fig. 15 that the SnT finite element solution from the literature [42]

again agrees well with the results obtained using the first-order SnT model of this thesis.

6.4. Transient Thermal Load

The first-order SDT finite element model is shown to accurately predict the time­

dependent deflections caused by a nonuniform heat flux. The predictions are compared

to the soluti~n of a three-dimensional finite element model [31]. Since the finite element

model of this thesis does not have the capability to perform heat conduction calcu­

lations, the temperature field calculated by the three-dimensional model is read into the

model at time steps of L1t = 0.075seconds. The comparison is made on a simply­

supported square [0/90/90/0] laminated composite plate of thickness 0.005 m and 0.020

m width, as shown in Fig. 16. The lamina material properties of the plate are

El I = 132.0 GPa, E22 = 10.8 GPa, E33 = 10.8 GPa,

V12 = .24,

G12 = 5.7 GPa, G31 = 5.7 GPa, G23 = 3.6 GPa.

The plate is subjected to a nonuniform heat flux on the upper surface described by

6. Verification of Model and Results 68

Figure 17 shows the center deflections of the plate caused by the time-dependant thermal

load. The solid line shows the first-order SDT finite element solution and the dashed line

represents the three-dimensional finite element solution. Because the inertia effects. are

extremely small the plate response is dominated by the thermal load. The two finite el­

ement solutions differ by about 5.0 percent. There are probably two reasons for this

difference:

1. The differences that arise in simulating the boundary conditions of a two­

dimensional plate and a three-dimensional plate.

2. The three-dimensional finite element theory permits strains in the z-direction; in

other words the plate can become thinner. The two-dimensional theory does not

permit transverse strains.

6. Verification of Model and Results 69

7. Conclusions and Recommendations

The fust-order SOT fmite element model predicts the deflections of the isotropic and

laminated composite plates very accurately under steady-state and transient thermal and

mechanical loads. In all cases presented in this thesis the first-order SDT solution agre~s

with the other two-dimensional finite element solutions and is very close to the three­

dimensional finite element solution for transient thermal problems.

The convergence of the model is very fast using the Newton-Raphson method. No

more than 3 iterations were ever needed; thus, CPU time requirements are very small.

The fmite element model should be improved by including the capability to perform heat

conduction calculations. Also, to correctly predict the dynamic response of .nonsym­

metric laminates, rotary inertia terms must also be included. In order to fully predict

laminate behavior a postprocessor must be added in which the laminate stresses are

calculated. With these modifications the model will be a very valuable tool in predicting

laminated composite plate behavior.

7. Conclusions and Recommendations 70

z

x

Fig. t. Typical Laminated Composite Plate.

71

z

w

Fig. 2. Plate Element with Seven Degrees of Freedom, Three Tr~ations, and Four Rotations.

y

72

\ j \

) ~ ~----------~~----------~(

(a)

. (b)

(c)

ow ox

Fig. 3 <a) Undeformed Laminated Plate, (b) Deformed Laminated Plate witb CLT Displacement Field, (c) Deformed Laminated Plate with First-Order SDT Displacement Eield.

73

z

~--------------~~--~~--~y

-~-"Nxy

x (a)

z

y

x (b)

Fig. 4. (a) Forces and (b) l\'loments on the Laminated Plate.

74

• •

-ht2

z. A~

1

1

n

n+r

N-l

N

/'

Fig. 5. Laminate Coordinate System and Nomenclature.

,

7S

z

~--------dy --------~.I

dz y

~ II. + OX

x

Fig. 6. Typical Infinitesimal Cube of Laminated Composite Plate.

/

76

(a)

41

(b)

.... -

•

-(c)

Fig. 7. Master Elements for (a) 4-Node Rectangular (b) 8-Node Rectangular, and (c) 9-Node Rectangular Elements.

77

y

~~

21

It

16

11

~ . 6

lit

1

/

.-- 13 -22 14 25

• U • ~ . 17 18 19 20

.- -- ~

12 13 14 15

~

~

• u • u 7 8 9 10

- -- ~

2 3 4 5

Fig. 8. Finite Element l\'lesh used in the Plate Analysis, Four 9-Node Quadrilateral Elements.

.... - x

78

z

t/I. = u = o .

t/ly = v =0. ...-.---atl----1

h

'" y = v = w = o.

Fig. 9. Geometry and Boundary Conditions of the First Quadrant of a Simply-Supported Square Plate.

y

79

1.20-

1.15 ..

1.10-

W

I.OS-

1.00-

0.95-

I

o

\ - - First-Order SOT

\ • Higher-Order SDT (22)

\ Linear Theory [39]

\ \ '" ~- ..

I I • I • I , I I

10 20 JO 40 50

alb

Fig. 10. Normalized Center Deflection of a Simply-Supported Isotropic Plate Under a Uniformly-Distributed Transverse Load.

80

z

"'x = u = o .

.-...---- all ------..1 .", = v =0.

h

u = v = w = "'1 = "''I = o.

Fig. 11. Geometry and BouDdary Conditions of the First Quadrant

of a Oamped Symmetric Cross-Ply (0/90/90/0) Laminated

Composite Plate.

y

81

w (em)

/

1.1 First-Order SDT

1.0 Higher-Order SDT (22)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

.0.2

0.1

0.0

0 1.7 5.4 8.1 10.8 13.5

Load (kPa)

Fig. 12. Center Deflection of a Oamped Symmetric Cross-Ply

(0/90/90/0) Laminate Subjected to a Unifonn Transverse Load (a=4.72 cm (12. in.), b=O.038cm (0.096 in.».

82

1.00-

0.98 ..

0.96-

, 0.94-

,

First-Order SDT

0.90 .. --- Linear Theory (41)

, , , , " , , , ,

" a·-~------~------~~------~I--------r-I-------r, o 20 .w 60 80 100

alb

Fig. 13. Normalized Deflection of a Simply-Supported Isotropic Plate Subjected to a Linear Temperature Distribution.

83

1.0

1.5

1.0

w (em) 0.5

0.0

-0.5 --- First-Order SOT

First-Order SDT (42)

o 40 80 120 160 lOO

Time (.us)

Fig. 14. Evolution of Center Deflection with Time for a Simply-Supported Isotropic Square Plate Under a UDifonn Load of too kPa (a = 25.0 cm, h = 5.0 em).

84

0.7 0.6 0.5 0.4 0.3

w (em) 0.2

0.1

0.0 -G.l -G.2

-6.3 -6.4

/

--- First-Order SDT

First-Order SDT (42)

o 20 40 60 80 100 120 140 160 180 200 220

Time (ps)

Fig. 15. Evolution of Center Deflection with Time for a [0/90/0) Simply-Supported Laminated Composite Plate Under a Uniform Load of 100 kPa (a = 25.0 em, h = 5.0 em).

8S

"'s = u =0 .

.;, = , = o. ........... --- a/2 ---...... 1

y

h .;, = , = w = o.

x ';s = U = W = o.

Fig. 16. Geometry and Boundary Conditions of the First Quadrant of a Simply-Supported Cross-Ply (0/90/90/0) Laminated Composite Plate (a=1.0 em, h=O.S em).

86

0.12

0.10

0.08 .

w (em) 0.06

0.04

--- First-Order SDT 0.02

- - 3-D FElVl (31)

0.00 ~--~--~-~--~~--~--T---~--~---T o .125 .250 .375 .500 .625 .750 .875 1.00 1.125

Time (s)

Fig. 17. Evolution of Center Deflection with Time for a Simply-Supported (0/90/90/0) Laminated Composite Plate Subjected to a Nonuniform Heat Flux (a = 2.0 em, h = 0.5 em).

87

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40. Zaghloul, S. A., and J. B. Kennedy, Nonlinear Behavior of Symmetrically Laminated Plates, Journal of Applied Mechanics, Vol. 42, 1975, pp. 234-236.

41. Timoshenko 5., and S. Woinowsky, Theory of Plates and Shells, 2nd Ed. McGraw-Hill, New York, 1959, pp. 176-180.

42. Reddy, J. N., Dynamic Transient Analysis of Layered Anisotropic Composite Material Plates, Intemationl Journal for Numerical Methods in Engineering, Vol. 19, 1983, pp. 237-255.

91

Appendix A. COMlVIEC2 PROGRAM LISTING

C

IMPLICIT REAL*8(A-H,O-Z) DIMENSION IBDY(8S),YBDYC8SJ,XC2S),Y(2S),TITLE(70) DIMENSION IBFC1),YBFll),GSTIFC12S,6S),ELXYI',2),Ul12S),GRt12S) DIMENSION TNX( 2S), THY, 2S), TNXY( 2S), Tl1X( ZS), THY ( 2S), THXY( 2S) DIMENSION TEM(S,2S),T(S),TH(S),DX(S),DY(S),LMT(S) DIMENSION El(2),E2(2),E3f2),PIZI2),P31(2},P23(2) DIMENSION G12(Z),G31(2),G23(2),ALPl(2),ALPZC2),ALPlZ(2),RHO(2) DIMENSION C(2,6),Z(21),ZBC20),CSHC20,3),QB(20,6) DIMENSION ALPX(20),ALPY(20),ALPXY(20) DIMENSION GFOtl25),GFll12S),GF2(125} COHMONIMSH/NOD(4,') COMHONISTFlIElKT(4S,4S),RC4S),A(6),BC6),DI6),AS(3) COMHONISTF2/SAO,SAl,SA2,SA3,SA4,XINl,XINZ C~~TF3/ElUUt4S),ElU(4S),ElUIC4S),ElU2(4S)

COHMON/STF4/TTNX(,),TTNYe'1,TTNXYC'1)TTMXe,),TTHY"),TTHXY(,) DATA NRMAX,NCHAXl12S,6S/ READ(S,31TITlE READIS,*)NPE,NDF,IHESH IFCNPE.EQ.4)THEN

IElEM=2 ELSE

IElEH=3 ENDIF NN=NPE*NDF IFCIHESH.EQ.l)GOTO SOO READ(S,*)NEH,NNH DO SIO N=l,NEH

510 READ(S,*)(NODtN,I),I=l,NPE) READ(S,*)(X(I),YCI),I=l,NNH) GaTO 520

500 READ(S,*)NX,NY READCS,*)(DXtI),I=l,NX) READ(S,*)(DY(I),I=l,NYl CALL HESHCIELEH,NX,NY,NPE,X,Y,NNH,NEH,DX,DY)

520 CONTINJE READ(S,*)ITHAX,TOLER,SEED READ(S,*)HT,NL,SK,TEO DO S I=l,HT READlS,*)EIII),E2(I),E3(I),P12(I),P31lI),P23(I)

S READ(S,*)G12(I),G31CI),G23(I),ALPIlI),ALP2(I),ALP12(I),RHOfI) TT=O. DO 7 I=l,NL

READ(S,*)THtI),T(I),LMTII) 7 TT=TT+TC I)

READCS,*lNLOAD,PINIT,PDELT

COHOOOIO COH00020 C0I100030 COM00040 COHOOOSO COM00060 COH00070 COH00080 COH00090 COMOOIOO COMOOIIO COH00120 COH00130 COM00140 COH001SO COM00160 COH00170 COHO 0 180 COH00190 COM00200 COH00210 COH00220 COH00230 COM00240 COH002S0 COH00260 COHOOZ70 COH00280 COM00290 COM00300 COM00310 COH00320 COH00330 COH00340 C0t1003S0 COH00360 COH00370 COH00380 COH00390 COH00400 COM00410 COH00420 COH00430 COH00440 COM004S0 COH00460

C*-----*-*-------------------------*-*-*-*-*-*-*-------*---------*---*-*COH00470 C COM00480 C TRANSIENT INPUT COM00490 C COHOOSOO

READ(S,*)ITIH COHOOSIO IFIITIH.EQ.l)THEN COHOOSZO

READ(S,*)NT,THAX COMOOS30 DE L T=TMAX/NT COH00540

92

C

SAO=4.DO/(DELT**2) SA1=SAO*OELT SA2=1.DO SA3=DELT/2.DO SA4=SA3

ELSE SAO=O.DO SA1=0.DO SA2=O.DO SA3=O.DO SA4=O.DO

ENDIF

COHOOSSO COMOOS60 COMOOS70 COHOOS80 COMOOS90 COH00600 C0t100610 COM00620 COH00630 COM00640 COM006S0 COM00660 C0f100670

C*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*---*-*-*-*-*-*-*-*-*-*---*-*-*-*-*-*-*COH00680 C COM00690 C WRITE INPUT DATA TO OUTPUT FILE C0f100700 C COM00710

HRITE(7,3)TITLE COM00720 WRITE. 7,6399) COM00730 HRITE ( 7,938 )NPE COM00740 WRITE( 7,940 )NEH COM007S0 WRITE' 7,942 lNMt COMO 0 760 WRlTEl7,6399) COM00770 HRITE(7,8999) COH00780 DO 8998 I=l,NEM COM00790

8998 HRITE(7,2376}I,(NODtI,J),J=l,NPE) COH00800 HRITE17,6399) COM00810 HRITE(7,2999) COH00820 DO 3998 I=l,tH1 C0f100830

3998 HRITE(7,2876)I,XII),Y'I) COH00840 HRITEf7,6399} COM008S0 HRITE' 7,8210 lHT COM00860 WRITE ( 7,8211 )NL COM00870 WRITE(7,8212)TEO COH00880 HRITE( 7,8213 lSI{ COH00890 HRITE(7,8214)TOLER C0f100900 HRITE(7,821S)SEED COH00910 HRITE(7,8216)PINIT COM00920 HRITE(7,3416)PDELT COM00930 WRITE( 7,6399) COM00940 DO S164 I=l,HT COH009S0 HRITEf7,9922)HT COM00960 HRITE(7,3S221 COM00970 HRITE(7,3S23JE1(I),E2lI),E3tI),P12(I),P31lI),P23(I) COM00980 HRlTE(7,6382) COM00990 HRITE(7,6384)G12(I),G31(I),G23(I),ALP1(Il,ALP2(I),ALP12(I) COMOIOOO

Sl64 HRITE(7,3473)RHOlI) COMOIOIO WRITEf7,6399) COMOI020 HRITE(7,7444) COH01030 DO 7445 I=l,NL COMOI040

7445 HRITE(7,7446)THfIJ,T(IJ,LHT(I) COMOI0S0 HRITEf1,6399) COHOI060

1444 FORf1ATC' ','ANGLE',4X,'THICkNESS',4X,'MAT. TYPE') COMO 10 70 1446 FORMAT" ',F5.2,5X,F7.4,9X,I2) C0t101080 8210 FORf1AT' , ·,'NO. OF MATERIAL TYPES :',11) COMO 10 90 8211 FORMAT(' ','NO. OF LAYERS IN PLATE =',12) COMOllOO 8212 FORMATe' ','STRESS-FREE TEMPERATURE =',F8.4) COM01IIO 8213 FORf1AT(' ','SHEAR CORRECTION FACTOR =',F8.6) COHOl120 8214 FORHAT" ','TOLERANCE ON LS ERROR =',F8.4) COHOl130 8215 FORMAT.' ','INITIAL DISPLACEMENTS =',F8.6) COHOl140 8216 FORMAT(' ','INITIAL LOAD IN THE Z-DIRECTION=',F13.6) COMOllSO 3416 FORMAT" ','DELTA LOAD IN THE Z-DIRECTION=',F13.6) COMOl160 3 FORHAT" ',70Al) COHOl170 938 FORHAT(' ·,'NO. OF NODES PER ELEMENT= ',II) COMOl180 940 FORHATI' ','NO. OF ELEMENTS IN HESH= ',12) COMOl190 942 FORHAT.' ','NO. OF NODES IN MESH: ',12) COM01200 2876 FORMAT.' ',12,3X,2(F1.4,2X)) COM01210 2376 FORMAT" ',3X,I2,1X,9(I2,2X)) COM01220 8999 FORf1AT(' ','ELEHENT',3X,'NODES') COH01230 2999 FORf1ATC' ','NODE',SX,'X',8X,'Y') COHO 1 240 6399 FORMAT" ',' ',/) COMOl2S0

93

9922 FORMAT( I

3522 FORHA T( , 3523 FORMAT(' 6382 FORMAT( I

6384 FaRHAT(' 3473 FORMAT( I

','FOR MATERIAL TYPE ',11,':') COM01260 • ,4X,' El' ,lOX, 'E2' J lOX, tE3' ,ax, 'P12' ,7)(,' P31' ,7)(, 'P23 t) COM01270 t ,3( 010 .4,2X )'3( F8.6 ,2X) COH01280 , ,3X, 'G12' ,9X, 'G31' ,9X, 'G23' ,9X, 'TEl t ,9X, 'TE2' ,8)(, 'TE12' )COM01290 , ,6(010.4,2X)) I, 'RHO=' ,013.6)

COMOl300 COMOl3l0 COMOl320 COM01330 COMO 1340 COMO 1350 COM01360 COM01370 COMOl380 COMOl390 COMO 1400 COMOl410 COM01420 COM01430 COMO 1440 COM01450 COH01460 COMO 1470 COMOI480 COHOl490 COMOl500 COM01510 COM01520 COM01530 COMO 1540

C C C C

30

33

35

INPUT NONZERO BOUNDARY FORCES AND DISPLACEMENTS

READ(5,*JNBDY,NBF IFCNBDY.EQ.O)THEN

IBDYCl)=O VBOY(1)=0.OO

ELSE READIS,*)(IBDYCI1,I=1,NBDY) READ(S,*)(VBDYII),I=l,NBDY)

ENDIF IF(NBF.EQ.01THEN

IBF(l)=O VBF(1)=0.DO

ELSE DO 30 I=l,NBF REAOI5,*)IBFII) DO 33 I=l,NBF READ(S,*)VBF(I) HRlTE(7,35)'VBF(I),I=1,NBF) FORMATe' ','SECONDARY KNOHNS=',4(FIO.4,2)(),//)

ENOIF C* ........................ -* ............... -* ... -*-*-* ............................................. -*COM01550 C C C C C C C

27 C C C C C

CALCULATION OF MATERIAL PROPERTIES

CALCULATION OF STIFFNESS MATRIX

COM01560 COMO 1570 COMO 1580

- - - - - - - - - - - - - - COMOl590 COM01600 COM01610 COM01620

DO 27 I=l,HT COH01630 P32=E3(I)*PZ3(I)/EZ(I) COM01640 P21=E2(I)*PIZ(I)/El(I) COM01650 P13=El(I)*P31II}/E3(I) COM01660 COEl=(1.00-Pl2,CI)*P21-P23(I)*P3Z-P31fI)*PI3-Z.DO*PZl*P32*PI3)/ COM01670

& (El( I )*E2C I )*E31 I) ) COM01680 C(I,I)='I.DO-P23(I)*P3Z)/(COEl*E2CI)*E3(I)) COM01690 CII,Z)=(PIZII)+P32*P13)/(COEl*ElfI)*E3CI)) COM01700 C( I ,.3 )=( P31( I l+P21*P3Z )/(COEl*E2( I )*E3( I)) COMO 1710 CCI,4)=(1.00-P13*P3l(I))/(COEL*ElfIJ*E3CI)) COM01720 CII,5)=(P23II)+P21*P13)/(CDEL*EIII)*EZ(I)) COM01730 CCI,6)=Cl.DO-P12CI)*P21)/tCDEl*ElII)*EZ(I)) COMO 1740

TRANSFORMED REDUCED STIFFNESS MATRIX (Qa)

• • • • • • • • • • • COM01750 COM01760 CQM01770 COMO 1780

• • • • • • • • • • • • • • • • • • • • • • • • • • COH01790 Z(IJ=TT/Z.DO DO 15 I=l,NL

11=1+1 ZlII)=ZtIJ-TIIJ ZBCI)=(Z(I)+Z(II))/Z.DO TTH=TH(I)*3.14159Z653589793DO/180.00 CS=COS(TTH) SN=SIN( TTH ) CS4=CSH4 CS2=CS*CS SN4=SN**4 SN2=SN*SN K=LHTII) CCl=CfK,Z)+2.DO*GlZ(K) CC2=CtK,1)+C(K,4)-4.DO*GIZ(K) CC3=CIK,lJ-ctK,Z)-2.00*G12(K) CC4=CIK,IJ-2.DO*C(K,2)+C(K,4)

COM01800 COH01810 COH01820 COH01830 COMO 1840 COt101850 COM01860 COM01870 COMO 1880 COMO 1890 COM01900 COM01910 COMO 1 920 COM01930 COMO 1 940 COHO 1950 COMO 1960

94

15 C •• C C C C

18 17

C C C C C

21

19 C. • C C C C. •

c C C

23

CCS=C(K,2)-CCK,4)+2.DO*GI2(K) CB11=CS4*C(K,I)+2.DO*CS2*SN2*CC1+SN4*ClK,4) CBI2=CS2*SN2*CC2+(SN4+CS41*C(K,2) CBI3=CS2*CtK,3)+SN2*CIK,S) CBI6=5N*CS*tCS2*CC3+SN2*CCS) CB22=SN4*C(K,I)+2.DO*CS2*SN2*CC1+CS4*C(K,4) CB23=SN2*CCK,3)+CS2*C(K,S) CB26=SN*CS*(SN2*CC3+CS2*CCS) CB33=CtK,6) CB36=CS*SN*CCIK,3)-CIK,S» CB66=CS2*SN2*CC4+GIZ(KJ*(CSZ-SNZ)**Z) CSH(I,I)=CS2*G23(K)+SNZ*G31CK) CSHCI,21=CS*SN*(G31CK)-G23IK) CSHlI,3)=SN2*G23CK)+CS2*G31(K) ALPXII)=ALPICK)*CS2+AlP2(K)*SNZ ALPY(I)=ALPl(K)*SNZ+ALPZ(K)*CS2 ALPXYII)=Z.DO*(AlPI(KJ-ALPZ(K))*CS*SN QaCI,I)=CBI1-CCBI3**2)/CB33 Qa(I,2)=CBI2-fCBI3*CBZ3)/CB33 Qa(I,3)=CBI6-CCBI3*CB361/C833 QaII,4J=CB2Z-lCB23**21/CB33 Qa(I,S)=CB26-(CB23*CB26)/CB33 Qa(I,6)=CB66-CCB36**2)/CB33

ASSEt'BLE THE A, B, D, ttl TRICES

DO 17 J=I,6 A'J)=O. B(J)=O. DIJ)=O. DO 18 I=I,NL

11=1+1 At J :1=Qa( I ,J )*( Z( I )-Z( II) )+AC J) BtJ)=QB(I,Jl*(Z(I)**Z-Z(II)**2)/2.DO+B(J)

DlJ)=QaCI,J)*(ZtI)**3-Z(II)**3)/3.DO+DtJ) CONTINJE

SHEAR STIFFNESS TERMS

DO 19 J=I,3 AS(J)=O.DO DO 21 I=l,NL

11=1+1 AS(J)=CSH(I,J)*(ZII1-Z(II»+AS(J) AS(J)=AStJ)*SK

CONTINJE

AVERAGE DENSITY

RH=O.DO DO 23 I=I,NL RH=RHO'LHTCI»)*TtIl+RH RH=RHITT IF(ITIM.EQ.I1THEN

XIN1=RH*TT XIN2=RH*CTT**3)/12.DO

ELSE XINl=O. XIN2=0.

ENDIF

HRlTE LAMINATE PROPERTIES

HRITE(7,4572)(A(I),I=I,3) HRlTE(7,4S73),ACI},I=4,S)

COHO 1970 COM01980 COMO 1 990 COH02000 COM02010 COH02020 COH02030 COM02040 COH020S0 COH02060 COM02070 COHO 2080 COH02090 COM02100 CONOZ110 COMOZ120 COH02130 COM02140 COM021S0 COH02160 C(J102170 COHO 2180 COHO 2190

.COM02200 COH02210 COM02220 COH02230 COHO 2240 C0t1022S0 C0t102260 COH02270 COM02Z80 COH02290 COH02300 COH02310 COM02320 COH02330 COM02340 COMO 23S0 C(J102360 COH02370 COHOZ380 COH02390 COHO 2400 COH02410 COMO 24Z0 COH02430 COHOZ440 COMO 2450 COH02460

.COH02470 COMO 2480 COH02490 COM02S00

.C(J102510 COtl02520 COHO 2530 COHO 2540 COHOZ550 C0t102.560 COHOZ.570 COMO 2580 COHO 2590 COH02600 COHOZ610 COM02620 COH02630 COHOZ640 COHOZ650 COH02660 COH02670

95

HRITE(7,4574)A(6) HRITE(7,6399) HRITE'7,4572)'B(I),I=1,3) HRITE(7,5573)(B(I),I=4,5) HRITE(7,4574}B(6) HRITE(7,6399) HRITEI7,4572)IDCI),I=1,3) HRITE(7,7573)(D(I),I=4,5) HRITE(7,4574)D(6) HRITE(7,6399J HRITE(7,7571)(ASCI),I=1,2) HRITE(7,7572)AS(3) HRITE(7,6399)

4572 FORHATe' ',10x,'I',tx,3(D13.6,tx),'I')

COM026S0 COM02690 COM02700 COH02710 COM02720 COM02730 COHO 2740 COM02750 COM02760 COM02770 COMO 27S0 COH02790 COM02800 COH02810 COMO 2820 COMO 2830 COMO 2840 COMO 2850 COM02860 COMO 2870 COMO 2880 COM02S90 COH02900

4573 FORHATt' ·,5X,'Aij=',lX,'I',17X,2(D13.6,ZK),'I',· i,j=1,2,6 1)

4574 FaRHAT" ',10x,'I',32X,D13.6,2X,'I') 5573 FORHAT(' ',5X,'Bij=',lX,'I',17X,2(D13.6,2X),'I',' i,j=1,2,6') 7573 FORMAT" ',5X,'Dij=',lX,'I',17X,2CD13.6,2X),'I',' i.j=1.2,6') 7571 FORHAT(' ·,lX,'A44,A45=',lX,'J',2X,2(D13.6,2X),'I') 7572 FORHAT(' ',5X,'ASS=',lX,'J',17X,D13.6,tx,'(')

HRITE(7,3076)RH 3076 FORHAT.' ','AVERAGE DENSITY=',013.6)

C C*-*---*-*-*-*-*-*-----*-*---*-*-*-*-*-*-*-*-*---*---*-*-*-*-*-*-*---*-*COH02910 C COH02920 c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COH02.930 C COM02MO C CDHPUTAnCN OF HALF BAND HIDTH COHO 2950 C COH02960 c. . . . . __ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMO 2970

3100 NH8H=0 COH02980 DO 3110 N=l,HEH COH02990 DO 3110 I=l,NPE COHO 3000 DO 3110 J=I,NPE COM03010 NH=CIABSCNOOCN,IJ-NOOCN,J))+l)*NDF COH03020

3110 IF (NHBH.LT.NU NH8H=NH C0t103030 teH=NHBH*2 C0t103040 HRITE(7,34001NHBM,NBH COM03050

3400 FORMAT(' ','HALF BAND "IDTH=',I2,' FULL BAND HIDTH=',I3) COM03060 NEQ=NNH*NOF COH03070 DO 288 I=I,NEQ COH03080 GFO(Il=O.DO COH03090 GF1(I)=O.DO COH03100 GF2(I)=0.DO COH03110

288 UtI)=SEEO COH03120 . DO 3458 I=l,teDY COM03130

U( IBDY( I) )=WOY( I J COH03140 3458 WDY'I)=O.DO COH03150

READfS,*)ITEHP COH03160 C*-*-*-*-*-*-*-*-*-*-*-*-*-*-----*-*---*-*-*---*-*-*-*-*-*-*-*-*-*-*-*-*COH03170 C C C BEGIN TIME DO LOOP C

C- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -C

nHE=DELT IFCITIH.EQ.01GOTO 179

C NTl=NT+l DO 1999 NTIH=l,NT IF(NnM.GT.801GOTO 1001

c- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -C C TEMPERATURE INDUCED FORCES C

C- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -179 DO 5820 ILOAD=l,NLOAD

~=PINIT+PDELT*(ILOAD-l)

IF(ITEMP.EQ.l)THEN

COH03180 COH03190 COH03Z00 COH03210

-COH03220 COH03Z30 C0M03240 COH032S0 COH03260 COH03270 COH03Z80

-COH03290 COH03300 COH03310 COH03320

-COH03330 C0t103340 COH033S0 COH03360

96

C C

NNN=Nl+1 READ(5,*)(tTEMCI,J),I=1,NNN),J=1,NNH)

DO 9333 I=1,NNN 9333 READt.5 ,*)( TEM( I,J) ,J=l,tH1)

C DO 9444 J=l,NNH

COH03370 COH03380 COH03390 COM03400 COM03410 COH03420 COM03430 COH03440 COH034S0

C9444 HRITE(7,8368)(TEH(I,J),I=1,NNN) C8368 FORMAT(' ',2(D13.6,3X)) C

DO 39 J=1.t-H1 COH03460 TNX(J)=O.DO COH03470 TNY(J)=O.DO COH03480 TNXY(J)=O.DO COH03490 THXfJ)=O.DO COM03S00 THY(J)=O.DO COM03S10 Tt1XYl J )=0. DO COH03S20 DO 39 I=1.NL COH03S30

11=1+1 COH03S40 TEHT1=TII)*(.5DO*(TEHCI.JJ +TEHfII,J))-TEO) COH03550 TEHT2=TfI)*(TEMfI,J)*(ZB(I)/2.DO+TCIJ/12.DO)+ COH03S60

& TEM(II,J)*fZB(Il/2.DO-T(IJ/12.DO)-TEO*Z8(I)) COH03570 TNXr.J)=(Qa(I,1)*ALPXCI)+Q8(I.2)*AlPXtI)+Q8(I,3)*AlPXY(I))*TEHT1 COH03S80

& + TNX( J I COH03590 TNY(J)=(QBCI,21*AlPX(I)+QB(I,4)*AlPV(I)+Q8(I,5)*AlPXYIIJ)*TEHT1 COH03600

& +TNV( J) COH03610 TNXY(J)=(QBII,3)*AlPX(I)+QBCI,S)*AlPVfI)+Q8(I,6)*ALPXY.I))*TEHTICOH03620

& +TNXYf J ) COH03630 THXfJ)=(Q8(I,1)*ALPXII)+Q8CI,2)*ALPV(I)+QB(I,3)*AlPXYII))*TEHT2 COH03640

& + 111)e( J ) COH03650 THY.J)=(Q8eI,2)*AlPXeI)+Q8(I,4)*AlPV(I)+Q8fI,5)*ALPXYfI))*TEHT2 COH03660

& +THV( J) COH03670 THXYeJ)=(QBlI,3)*AlPX(I)+Qa(I,S)*AlPV(I)+Q8(I,6)*AlPXYtI))*TEHT2 COH03680

& +TI1XY( J J C0t103690 39 CONTINUE COH03700

c C C C C

C

ELSE COH03710 DO 1139 J=l,tN1 COM03720

TNX(J)=O.DO COH03730 TNYeJ)=O.DO COM03740 TNXYtJ}=O.DO COH03750 THX( J )=0. DO COH03760 THY ( J ) =0. DO COH03770

1139 THXY(J )=0.00 COH03780

80

ENDIF COH03790

LOOP ON NONLINEARITY

DO 999 lTER=l,ITHAX IFtITHAX.GT.30)GOTQ 1001 DO 80 I=l,NEQ

GRtI)=O.DO DO 80 J=l,NHBH GSTIF(I,J)=O.DO DO 100 N=l,HEM

COH03800 COH03810 COM03820 COH03830 Cot103840

C JlN(

COH03850 COH03860 COH03870 COHO 3880 COHO 3a 90 COH03900 COH03910 COH03920 COH03930 COH03940 COH03950 COH03960

C HRlTE(7,2087)N C2087 FORHAT(' ','ELEMENT. ~',Il) C C

91

DO 91 I=1,NPE NET=NDF*(I-l)

NAT=NDF*(NODIN,IJ-1) DO 91 J=l,NDF

NIT=J+NAT IJ=J+NET ELUU(IJ)=GFO(NITJ ELU1(IJ)=GF1(NIT) ELU2(IJ)=GF2(NITJ ELU( IJ J=U( NIT)

COH03970 COH03980 COH03990 COH04000 COH04010 COH04020 COH04030 COH04040 COH04050 COH04060 COH04070

97

C C

C C

90

DO 90 I=l,NPE TTNX(I)=TNXfNOD(N,I») TTNYII)=TNY(NODCN,I)) TTNXY(I)=TNXY(NOD(N,I)) TTMXII)=THXtNOO(N,I) TTHYCI)=THY(NOD(N,I}} TTHXY(I)=THXY(NODCN,I» ELXY(I,2)=YfNOO(N.I) ELXYlI,11=XlNODCN,I)

CALL STIFFCIELEM,NN,NPE,NDF,ElXY,QJ

ASSEMBLE ELEMENT STIFFNESS MATRICES TO GET GLOBAL STIFFNESS DO 280 I=l,NPE NR=(NODfN,Il-1)*NOF DO 280 II=l,NDF

NR=NR+l L=(I-1)*NOF+II GRfNR)=GRINRJ+R(Ll DO 260 J=l,NPE

NCL=(NODCN,J)-lJ*NDF 00 260 JJ=l,NDF f1=(J-l)*NDF+J.J NC=NCL+JJ+1-NR IFfNC.LE.O)GOTO 260 GSTIF(NR,NC)=GSTIF(NR,NC)+ELKTIL,H)

260 CONTINUE 280 CONTINUE

100 CONTINUE

COH04080 COM04090 COM04100 COM04110 COM04120 COH04130 COM04140 COM04150 COH04160 COM04110 COH04180 COH04190 COM04200 COM04210 COM04220 COH04230 COH04240 COM04250 COM04260 COM04210 COH04280 COM04290 COM04300 COH04310 COM04320 COM04330 COM04340 COM04350 COM04360 COM04310

111 IF(NBF.Eq.O)GOTO 110 COM04380 00 120 NF=l,teF COH04390

NB=IBFINF) COH04400 120 GR(NB)=GR(NB)-V8F(NF)' COM04410

C-----------------------------------------------------------------------COM04420 C ASSEHBLEO I1ATRIX EQUATIONS ARE NOH READY FOR IHPLEt1£NTATICI\I OF COH0443 0 C THE BOUNDARY CONDITIONS CI\I PRII1ARY AND SECONDARY VARIABLES COM04440 C-----------------------------------------------------------------------COM04450

110 CALL BNDYtNRt1AX,NCt1A)(,NEQ,NHBH,SSTIF ,GR,NBDY ,IBDY ,WDY) COM04460 IRES=O COM04410 CALL SOLVE (NRttAX ,NCHAX ,NEQ ,NHBH ,SSTI F ,GR, IRES) COM04480

C-----------------------------------------------------------------------COH04490 C CALL SUBROUTINE • SOLVE • TO SOLVE THE SYSTEH OF EQUATIONS FOR THE COM04500 C PRII1ARY DEGREES OF FREEDOI1 (THE SOLUTION IS RETURNED IN ARRAY GR COM04510 C-----------------------------------------------------------------------COH04520 C COM04530 C COM04540 C LEAST-SQUARES ERROR t1£ASUREHENT COM04550 C COM04560

ERR1=0.DO COH04510 ERR2=0.00 COM04580 00 43 I=l,NEli COM04590

C HRITE(1,768)I,UfI),GRII) COH04600 C768 FORHAT(' ',I3,2(013.6,2)()J COH04610

ERR1=ERR1+GR(I)**2 COH04620 43 ERR2=ERR2+(GRtI)+UII»)**2 COH04630

IFIERR2.EQ.O. )9OTO 1000 COH04640 ERROR=SQRTCERR1IERR2) COH04650 00 44 I=l,NEQ COH04660

44 U(I)=UII)+GR(I) COH04610 IF(ERROR.LT.TOLER1GOTO 1000 COH04680

999 CONTINUE COH04690 HRITE(1,998) COM04100

998 FORMAT(' ','SORRY, IT DID NOT CONVERGE') COM04110 GOTO 1001 COH04120

1000 HRITE'7,6399) COH04130 HRITE(7,6l2) COM04140 HRITE.7,6399) COM04150 HRITEI7,944)li,TIHE COH04760

944 FORMAT(' ','FOR Q=',F12.3,5X,'AT TIt1£=',F12.') COM04770 HRITE(7,933)ITER COM04780

98

C

933 FORMAn' I 71 IT CONVERGED AFTER • ,12 7' ITERATIONS') HRITE(7,7458)

6lZ FORMAn' I,' SOLUTION ' ,/) 7458 FORMATe' I, 'NODE' ,5)(7 IU' 7llX, ·V I ,11X, 'H' ,lOX, 'PHIX' ,9X, 'PHIV')

DO 3348 I=l,NNH II=l+NOF*CI-l) III=II+NDF-l

3348 HRITE(7,610)I,(UIJ),J=II,III) 610 FORMATe' ·713,2X75(DlO.4,2X)) 5820 CONTINUE

IFCITIM.EQ.01GOTO 1001 TlME=TIHE+DELT DO 180 I=17NEQ GFO(IJ=SAO*(UIIJ-GFOlI»)-SAl*GFlCI1-SAZ*GFZ(I) GFlCI)=GF1(I)+SA3*GF2(I)+GFOfI)*SA3 GFZ(Xl=GFOCI)

180 GFO(I)=U(I) 1999 CONTItlJE 1001 CONTINUE

STOP END SUBROUTINE STIFFCIELEH,NN,NPE 7NDF,ELXY,Ql IMPLICIT REAL*8CA-H,0-Z) COHMON/STFl/ELKT(45,45),R(45),A(6),8C6),D(6),AS(3) CotIU»LtSTFZ/SAO,SAl,SAZ,SA3,SA4,XIN1,XINZ COHMONVSTF3/ELUU(45),ELUC45),ELUl(45),ELUZ(45) COMHON/STF4/TTNX(9),TTNY(9),TTNXYt9),TTMX(9),TTMY(9),TTHXYf9) COHHONVSHP/GOSF(Z,91,SF(9) DIMENSION GAUSS(4 74),HTI4,4),ELXVe9,2J,TU(45),ELF3(9),ELFI19) DIMENSION ELFZ(9),ELF4(9),ELF5(9),TUUl45),TU1(45),TUZ(45) OIMENSION TKll(9.9),TKlZC9,9),TK13(9,9),TK1419,9),TK15(9,9) DIMENSION TKZl(9,9),TKZ2(9,9),TK23(9,9),TK24(9,9),TKZ5(9,9) DIMENSION TK31C9,9),TK3Z(9,9),TK33(9,9),TK34(9,9),TK35C9,9J DIMENSION TK4119,9),TK42(9,9},TK43C9,9),TK44(9,9),TK45(9,9) DIMENSION TK51(9,9),TK5Z(9,9),TK53t9,9),TK54(9,9),TK55(9,9) DIMENSION XK33(9,9),G33(9,9),G34e9,9),G3519,9),Glt3(9,9) DIMENSION G44(9,9),Glt5(9,9),G5319,9),G54(9,9),G55(9,9),GHASS(9,91 DATA GAuss/4*0.ODO,-.577350Z7DO,.577350Z7DO,2*0.ODO,-.77459667D0,

*0.000,.7745966700,0.ODO,-.8611363100, *-.3399810400,.33998104DO,.86113631DO/

DATA HT/Z.ODO,3*O.ODO,Z*1.ODO,2*0.ODO,.55555555DO,.88888888DO, *.5555555500,0.000,.3478548500,Z*.65Z1451500,.3478548500/

NGP=IELEH

C REARRANGE U C

NCl=l NCZ=2 Nt3=3 1«:4=4 1«:5=5 DO 89Z I=l,NPE IZ=I+NPE I3=IZ+NPE I4=I3+NPE I5=I4+NPE TUU(I)=ELUUfNCl) TUUI IZ )=ELUU( I«:Z ) TUUII3)=ELUU(NC3) TUUII4)=ELUUCNC4) TUUCI5)=ELUUI1«:5) TU(I)=ELUINCl) TUIIZ)=ELUINCZ) TUII3)=ELUtNC3) TUII4)=ELUlNC4) TU( IS )=ELUC 1«:5) TUlII)=ELU1(NC1) TUlII2)=ELUl(NCZl TUl(I3)=ELU1(NC3) TUl(I4)=ELU1(NC4) TUl(I5)=ELUIINC5)

COH04790 COH04800 COH04810 COH04820 COH04830 COH04840 COH04850 COH04860 COH04870 COH04880 COM04890 COH04900 COM04910 COH049Z0 COH04930 COtt04940 COH04950 COH04960 COH04970 COM04980 COH04990 COH05000 COH05010 COH05020 COH05030 COM05040 CDt1050S0 COM05060 COHOS070 COM05080 COM05090 COH05100 COH05110 COHOS120 COH05130 COH05140 COMOS150 COH05160 COH05170 COH05180 COH05190 COH05200 COH05210 COH05220 COH05230 COH05240 COH05250 COHOS260 COH05270 COH05Z80 COHOS290 COHOS300 COH05310 COH05320 COH05330 COH05340 COH05350 COH05360 COH05370 COH05380 COH05390 COH05400 COH05410 COH054Z0 COtt05430 COH05440 COH05450 COH0546 0 COH05470 COH05480 COH054CJO

99

892 C C C

415

416 C C C C C C

C

TU2fI)=ELU2INCl) TUZ(IZ)=ELU2(NC2) TU2(I3)=ELUZ(NC3) TU2(I4)=ELUZ(NC4) TUZe IS )=ELU2( NC5 1 NCl=NC1+NOF NC2=NC2+NOF NC3=NC3+NOF NC4=NC4+NOF NC5=Ne5+NOF

INITIALIZE MATRICES

DO 415 1=1,'" RCI)=O.OO DO 415 J=1,'"

ELKT( I,J )=0.00 DO 416 1=l,NPE

ELFU 1)=0.00 ELFZ(I)=O.DO ELF3(I)=0.DO ELF4(I)=0.DO ELFsII)=O.DO DO 416 J=1,NPE GHASSCI,J)=O.DO TK1U I,J )=0.00 TK12(1,J)=O.DO TK13(I,J)=O.DO TK14fI,J)=0.DO TK15(I,J)=O.DO TK2U I ,J )=0.00 TK22(I,J)=0.DO TK23fI,.J)=O.00 TKZ4f·I ,J )=0. DO TKZStI,J)=O.DO TK3lt I ,J )=0. DO TK32(1,J)=0.DO TK33lI,J)=0.DO TK34CI,J)=0.00 TK3S(I,.J)=0.DO TK4U I ,J )=0. DO TK42:(I,J)=O.00 TK43(I,J)=0.DO TK44(I,.J)=O.OO TK4Sf1,.J)=0.DO TK5U I ,J ) =0. DO TKS2fI,J)=O.DO TKs3(I,J)=O.DO TK54(I,J)=O.DO TK55(I,J)=0.DO )(I(33(1,J)=0.00 G33(I,J)=0.DO G34(1,J)=0.00 G35'I,J)=0.00 G43(I,J)=O.DO G44(I,J)=O.DO G4S(I,J)=O.DO Gs3(1,.J)=0.DO GS4(I,J)=O.OO G55(I,J)=0.00

THIS LOOP REPRESENTS THE Slft1ATION OF THE GAUSS POINTS CONTRIBUTION TO KT & Q.

DO 400 IGP=l,NGP XI=GAUSS( IGP ,NGP ) DO 400 JGP=l,NGP

ETA=GAUSS(JGP,NGP)

COMOS500 COHOS510 COH05520 COH05530 COMOS540 COMOSS50 COH05S60 COH05570 COMO 5580 COM05S90 COH05600 COH05610 COH05620 COHOS630 COMOS640 COHOS650 COH05660 COHOS670 COM05680 COHOS690 COHOS700 COMOS710 COH05720 COMOS730 COHOS740 Cot10S750 Cot10S760 COH05770 COHOS780 COMOS790 COMOSSOO COHOS810 COHOSS20 COHOSS30 COH0S840 COHOSS50 COH05860 COH05870 COH05880 Cot105890 COHOS900 COHOS 910 COHOS920 COMOs930 COHOS 940 COHOs9S0 COH05960 COHOS970 COHOS 980 COHOS990 COH06000 COH06010 COH060Z0 COH06030 COH06040 COH060s0 C0H06060 COH06070 COH06080 COH06090 COH06100 COH06110 COH06120 COH06130 COH0614O COH061s0 COH06160 COH06170 COM06180 COH06190 COH06200

100

C C C C

C C C C C C C

C C C C C C C C C615 C616 C617 C618 C

150

"GAUSS" CONTAINS THE GAUSS POINTS IN AN ARRAY WITH EACH COLUMN CONTAINING THE XI FOR 1, 2, 3, AND 4 GAUSS POINTS RESPECTIVELY

CALL SHAPEIXX,ETA,NPE,ELXY,IELEH,DET)

"SHAPE" CALCULATES THE ..JACOBIAN (G..J), THE LOCAL INTERPOLATION FUNCTIONS (SF)' AND THEIR DERIVATIVES (DSF). AND PUTS THE LOCAL DERIVATIVES INTO GLOBAL TERMS. GDSF=DSF/G..J

DHX=O.DO DHY=O.OO DUX=O.DO OUY=O.OO DVX=O.DO OVY=O.DO DFXX=O.OO DFXY=O.OO DFYX=O.DO DFYY=O.DO TNX=O.DO TNY=O.DO TNXY=O.DO Tt1X=O.DO THY=O.DO T11XY=O.DO DO 150 I=I,NPE

II=I+NPE III=II+NPE 14=III+NPE IS=I4+NPE TNX=TNX+TTNX(I)*SF(I) TNY=TNY+TTNYCI1*SFII) TNXY=TNXY+TTNXYCI)*SFtI) TMX=Tt1X+TTt1)((I)*SFfI) THY=THY+TTHYtI}*SF(I) T11XY=T11XY+TT11XY(I)*SF(I)

.JlN(

HRITE(7,615) HRlTE(7,61S)TTNXtI),TNX,TTNYCI),TNY "RITE(7,616) HRIlc(7,61S}TTt1)(fI),Tt1)(,TTHYCI),THY "RITE(7,617) "RI,'E( 7,618 )TTI1)(Y( I), T11XY, TTNXY( I), 'TNXY FORHAT( I • ,3X, 'TTNX( I) t ,lOX, 'TNX' ,lOX, 'TTNY( I)' ,lOX, 'THY' ) FORHAT(' I ,3X, 'TTt1)(C I)' ,lOX, 'TMX' ,lOX, 'TTHY( I)' ,lOX, 'THY I )

FORHAT(' I ,2X, 'TTI1)(Y( I)' ,9)(, 'T11XY' ,9X, 'TTNXY( I)' ,9)(, 'TNXY' ) FORMAT(' ·,4(DI3.6,2X»

DFYX=DFYX+GDSFll,I)*TUfIS) DFYY=DFYY+GDSF(2,Il*TUCIS) DFXX=DFXX+GOSF(I,I)*TUCI4) DFXY=OFXY+GOSF(2,I)*TUCI4) DHX=OHX+GDSF(I,I)*TUIIII) DHY=DHY+GDSF(2,I)*TU(III) DVX=DVX+GDSF(l,I)*TUIII) DVY=DVY+GDSF(2,I)*TU(II) OUX=OUX+GDSF(l,I)*TUII) DUY=DUY+GDSF(2,I)*TUtI) G=TNX*DHX+'TNXY*DHY H=TNXY*DHX+TNY*OHY CONST=HT(IGP,NGP)*HT(..JGP,NGP)*OET CONAll=A(I)*HT(IGP,NGP)*HT(..JGP,NGP)*DET CONAIZ=A(Z)*HT(IGP,NGP)*HTt..JGP,NGP)*DET CONAI6=A(3)*HTCIGP,NGP)*HT(..JGP,NGP)*DET CONAZZ=A(4)*HTtIGP,NGP)*HT(..JGP,NGP)*OET CONAZ6=A(S)*HT(IGP,NGP)*HT(..JGP,NGP)*OET C0NA66=A(6)*HTCIGP,NGP)*HTC..JGP,NGP)*OET

COM06Z10 COM06220 COM06230 COM06240 COM06250 COM06260 COH06Z70 COH06280 COM06290 COH06300 COM06310 COM06320 COM06330 COH06340 COHO 63S0 COH06360 COM06370 COM06380 COM06390 COM06400 COM06410 COM06420 COM06430 COM06440 COM064S0 COM06460 Cott06470 COH06480 COM06490 COM06500 COM06510 COM06520 COH06S30 COH06S40 COM06550 COM06560 COH06570 COM06580 COH06590 COH06600 COM06610 COM06620 COH06630 COMO 6640 COM06650 COH06660 COM06670 COHO 6680 COM06690 COM06700 COM06710 COM06720 COH06730 COH06740 COM06750 COH06760 COM06770 CDt106780 Cot106790 COH06800 COM06810 COH06820 COM06830 COHO 6840 COM06850 COH06860 COM06870 COH06880 COH06890 COH06900 COH06910

101

CONBll=8(1)*HTfIGP,NGP)*HTIJGP,NGP)*DET COH069Z0 CONB12=8(2)*HT(IGP,NGP)*HTfJGP,NGP)*DET COM06930 CONB16=8(3)*HT(IGP,NGP)*HT(JGP,NGP)*DET COM06940 CONBZ2=B(4)*HT(IGP,NGP)*HT(JGP,NGP)*DET COM06950 C0N826=8(5)*HT(IGP,NGP)*HTfJGP,NGP)*DET COM06960 C0N866=8(6)*HTtIGP,NGP)*HT(JGP,NGP)*DET COH06970 CONOll=OCI)*HTCIGP,NGP)*HT(JGP,NGP)*OET COMO 6980 CON012=O(2)*HTfIGP,NGP)*HT(JGP,NGP)*DET COH06990 CON016=0(3)*HT(IGP,NGP)*HT(JGP,NGP)*OET COH07000 CON022=0(4)*HTIIGP,NGP}*HT(JGP,NGP)*DET COMO 7010 C0N026=0(S)*HT(IGP,NGP1*HT(JGP,NGP)*OET CCH07020 C0N066=0(6)*HT(IGP,NGP)*HT(JGP,NGP)*DET COH07030 CONQ=~HT(IGP,NGP)*HT(JGP,NGP)*OET COH07040 00 160 I=l,NPE COMO 7050

DO 159 J=I,NPE COHO 7060 XX=GOSFCl,J)*GDSF(l,I) COMO 70 70 YY=GDSFt2,J)*GDSF(2,I) COH07080 XIYJ=GDSF(2,J)*GDSFI1,I) COH07090 XJYI=GDSF(1,J)*GOSF(2,I) COH07100 GHASS(I,J)=GHASS(I,J)+CONST*SFfIJ*SF(J) COMO 71 10 TKllfI,J)=CONAll*XX+C0NA66*YY+TKll(I,J)+CONA16*(XIYJ+XJYI) COM07120 TK12(I,J)=CONAI2*XIYJ+C0NA66*XJYI+CONA16*XX+C0NA26*YV COM07130

& +TK12(I,J) COHO 7140 TK13II,JJ=CONAII*XX*DHXlZ.OO+CONA12*XIYJ*DHY/2.DO+ COM071S0

& C0NA66*GOSF(Z,I) COH07160 & *tDHX*GDSF(2,J)+DHY*GDSF(I,J))/2.DO+C0NA26*YY*OHY/2.DO COH07170 & +C0NA16*(XIYJ*DHX+XX*DHY+XJYI*DHX)/2.DO+TK13(I,J) COM07180

TK14(I,J)=CONBll*XX+C0NB16*(XJYI+XIYJ)+C0NB66*YV+TK14(I,JJ COM07190 TKlSfI,J)=C0NB16*XX+C0NB66*XJYI+C0N812*XIYJ+C0NB26*YY+ COMO 7200

& TKlS( I ,J ) COM07210 TK21(I,J)=C0NA66*XIYJ+C0NA12*XJYI+CONA16*XX+C0NA26*YY+ COM07220

& TK2lt I,J) COH07230 TK22II,J)=C0NA66*XX+C0NA22*YY+C0NA26*fXIYJ+XJYI)+rK22(I,J) CCH07240 TK23(I,J)=(C0NA66*tXX*DHY+XIYJ*DHX)+CONA12*XJYI*DHX+C0NA22 COMO 7250

& *DHY*YV+CONA16*XX*OHX+C0NA26*(XIYJ*DHY+YV*DHX+XJYI*DHY)) COH07260 & l2.DO+TK23(I,J) COH07270

TK24(I,J)=CONB16*XX+CONB66*XIYJ+CONB12*XJYI+CONB26*YV+ COH07Z80 & TK24( I,J ) COHO 7290

TK25(I,J)=CONB66*XX+C0N826*fXJYI+XIYJ)+CONB22*YY+TK25(I,J) COM07300 TK31(I,J)=CONAll*XX*DHX+C0NA66*(OHY*XIYJ+DHX*YV)+CONAl2*DHY* COH07310

& XJYI+CONAl6*CXIYJ*OHX+OHY*XX+XJYI*DHX)+C0NA26*YV*OHY+ COH07320 & TK31fI,J) COH07330 TK32fI,J)=C0NA12*OHX*XIYJ+C0NA66*(DHY*XX+DHX*XJYI)+C0NA22*OHY*YV+ COH07340

&C0NA26*CXIYJ*DHY+YV*DHX+XJYI*DHY)+CONAl6*XX*DHX+TK32(I,J) COHO 7350 TK33(I,J)=CONAl1*XX*0HX*DHX/2.DO+CONA22*YY*DHY*DHY/2.DO+ COHO 7360

&C0NA66*DHY*DHX*(XIYJ+XJYI)+CONAl2*(XX*DHY*DHY+YV*DHX*DHX)/2.DO+ COM07370 &C0NA16*(DHX*DHX*CXIYJ+XJYI)/Z.DO+DHX*DHY*XX)+CONAZ6* COMO 7380 &(DHY*DHY*tXJYI+XIYJ)/2.DO+OHX*OHY*YY)+TK33(I,J) COH07390 TK34fI,J)=CONBll*XX*DHX+CONBl6*CXIYJ*DHX+XX*DHY+XJYI*DHX1+CONB26* COH07400 &YV*DHY+CONB12*XJYI*DHY+CONB66*(XIYJ*DHY+YY*DHX)+TK34(I,~) COH07410 TK3S(I,Jl=C0NB16*XX*DHX+C0N8Z6*CXIYJ*DHY+YY*DHX+XJYI*DHY)+C0NB22* COH07420

&YY*OHY+CONBlZ*XIYJ*DHX+C0N866*CXJYI*DHX+XX*DHY)+TK35(I,J) COHO 7430 TK4lfI,J)=TK41(I,J)+CONB11*XX+C0N866*VY+C0N816*CXIYJ+XJYI) COH07440 TK42tI,J)=TK42(I,J)+CONB12*XIYJ+C0N866*XJYI+CONB16*XX+CONB26*YV COMO 7450 TK43(I,Jl=TK43CI,J)+(CONBlZ*XIYJ*DHY+C0N866*(XJYI*DHY+YY*OHX)+ COHO 7460

& CONB26*YY*DHY+CONB16*(XIYJ*DHX+XX*DHY+XJYI*DHX)+ COHO 7470 & CONBl1*XX*DHX)/2.DO COH07480 TK44(I,J)=TK44(I,J)+CON011*XX+C0ND66*YY+CONDl6*CXIYJ+XJYI) COHO 7490 TK45(I,~)=TK45'I,J)+CONDIZ*XIYJ+COND66*XJYI+COND16*XX+CON026*YV COHO 7500 TKSlII,J)=TK5lII,J)+CONB1Z*XJVI+C0N866*XIYJ+C0N816*XX+CONaZ6*YV COHO 7510 TK52fI,J)=TK5ZCI,Jl+C0NB22*YY+C0NB66*XX+CONB26*(XIYJ+XJYI) COH075Z0 TK53CI,J}=TK53(I t J)+(CONBl2*XJYI*DHX+CONB66*(XIYJ*DHX+XX*DHY)+ COHO 7530

& CONBZ2*YV*DHY+CONB26*(XIYJ*DHY+YY*OHX+XJYI*OHY)+ COHO 7540 & CONB16*XX*DHX)/2.DO COHO 7550 TK54(I,Jl=TK54(I,J)+CONDl2*XJYI+COND66*XIYJ+COND16*XX+CONDZ6*YV COH07S60 TK5StI,J)=TK55(I,J)+COND2Z*VY+COND66*XX+CONDZ6*(XIYJ+XJYI) COHO 75 70 XK33(I,J)=XK33fI,J)+CONA11*(DHX*DHX+DUXl*XX+CONAlZ*(YY*OUX+XX*OVY COH07580

& +DHY*DHX*(XIYJ+XJYI))+C0NA16*CXX*CDUY+ COH07590 & DYX+2.DO*DHX*OHY)+(DUX+OHX*OHX)*(XIYJ+XJYI)) COHO 7600 & +CONAZ6*CYV*CDUY+DVX+2.00* COMO 7610 & DHX*DHY)+(DHY*DHY+DVY)*(XIYJ+XJYI))+C0NA66*(DHY*DHY*XX+YV* COM076Z0

102

& DHX*DHX+(XIY4+XJYI)*(DUY+DVX+DHX*DHY))+CONA22*YY*IDVY+DHY*DHY) COH07630 XK33II,4)=XK33II,4)+CONB11*XX*OFXX+CONB16*IXX*DFXY+DFXX*eXIY4+ COMO 7640

&X4YI)+DFYX*XX)+CONB66*((DFXY+OFYX)*(XIY4+XJYI)) COM07650 &+CONB12*(YY*DFXX+XX*DFYY)+C0N826*eYY*(OFXY+DFYX)+OFYY*(XIY4+XJYI)COM07660 &+CONB22*YY*OFYY COM07670 &+CONST*GDSF(1,I)*(TNX*GOSF(1,4)+TNXY*GDSF(2,4)) COMO 7680 &+CONST*GDSFI2,I)*(TNY*GDSF(2,4)+TNXY*GDSF(1,4)) COM07690

159 CONTINUE COM07700 ELF1(I)=ELF1(I)+(TNX*GDSF(1,I)+TNXY*GDSF(2,I))*CONST COM07710 ELF2(I)=ELF2(I)+(TNY*GDSF(2,I)+TNXY*GDSFC1,I)*CONST COM07720 ELF4e I )=ELF4( I )+e THX*GDSFI·1,I J+nlXY*GDSFI 2,1) )*CONST COM07730 ELF5eI)=ELF5{I)+(TMY*GDSF(2,I)+THXY*GDSF(1,I))*CONST COMO 7740

160 ELF3(I)=ELF3(I)+C~SFeI)+eG*GDSFe1,I)+H*GDSF(2,I))*CONST COMO 77S0 400 CONTINUE COH07760

C COHO 7770 C COM07780 C BEGIN REDUCED INTERGRATION COM07790 C COHO 7800 C COM07810

NGP=NGP-1 COM07820 DO 401 IGP=l,NGP COMO 7830

XI=GAUSSe IGP ,NGP ) Cot107840 DO 401 4GP=1,NGP COMO 78S0

ETA=GAUSS(4GP,NGP) COM07860 CALL SHAPE(XI,ETA,NPE,ELXY,IELEH,DET) COHO 7870 CONA44=AS(1)*HT(IGP,NGP)*HT14GP,NGP1*DET COMO 7880 C0NA45=AS(2)*HT(IGP,NGP)*HTe4GP,NGP)*DET COMO 7890 CONA55=AS(3)*HT(IGP,NGP)*HT(4GP,NGP)*OET COH07900 DO 402 I=l,NPE COH07910 DO 402 4=1,NPE COM07920

XX=GDSF(1,4)*GOSF(1,I) COH07930 YY=GOSF(2,4)*GDSF(2,I) COMO 7940 G33(I,4)=G33(I,4)+C0NA55*XX+C0NA44*YY+C0NA45*(GOSF(2,4) COM07950

~ *GDSF(1,I)+GOSFe1,J)*GOSFe2,I)) Cot107960 G34(I,J)=G34(I,J)+C0NAS5*GOSF(1,I)*SF(J)+C0NA45*GDSF(2,I)*SF(J) COM07970 G3SII,J)=G35(I,J)+CONA44*GDSF(2,I)*SF(J)+C0NA45*GDSF(1,I)*SF(J) COM07980 G43(I,J)=G43CI,J)+CONA55*GDSFe1,J)*SFeI)+C0NA45*GDSF(Z,J)*SF(I) COH07990 G44(I,J)=G44CI,J)+C0NAS5*SFCI)*SFeJ) COH08000 G45(I,J)=G45(I,J)+C0NA4S*SFII)*SFeJ) COH08010 G53(I,J)=G53(I,J)+C0NA44*GDSFe2,J}*SF(I)+C0NA45*GDSF(1,J)*SF(I) COM08020 GS4(I,J)=G54(I,J)+CONA45*SF(I)*SFeJ) COM08030

402 G55(I,J)=G55(I,4)+CONA44*SFtI)*SFtJ) COH08040 401 CONTINUE COM080S0

C COM08060 C COM08070 C ASSEMBLE ELEMENT KT & R MATRICES COM08080 C COH08090 C COH08100

11=1 COH08110 DO 170 I=1,NPE COH08120

12=11+1 COH08130 13=11+2 COH08140 14=11+3 COM08150 15=11+4 COH08160 JJ=1 COM08170

DO 180 J=1,NPE COH08180 J2=JJ+1 COM08190 J3=JJ+Z COM08200 44=JJ+3 COM08210 J5=4J+4 COM08220 JNPE=J+NPE COM08230 J2NPE=4+2*NPE COM08240 J3NPE=J+3*NPE COM08250 J4NPE=4+4*NPE COH08260 GH1=GHASSCI,J)*XIN1*SAO COM08270 GH2=GHASSCI,J)*XIN2*SAO COM08280 ELKT(II,JJ)=TKll(I,J)+GH1 COM08290 ELK.Tl II ,J2 )=TK12( I,J) COH08300 ELKTtII,43)=TK13(I,4)*2.DO COH08310 ELKT( II ,J4 )=TK14( I ,J) COH08320 ELKT(II,45)=TK15(I,J) Cot108330

103

ELKTCI2,JU}=TK2lfI,J) COH08340 ELKT(I2,J2)=TK22(I,J)+GMl COH083S0 ELKTfI2,J3)=TK23(I,Jl*2.00 COH08360 ELKTII2,J4)=TK24CI,J) COH08370 ELKT(I2,JS1=TK2S(I,J) COH08380 ELKT(I3,JJJ=TK3l(I,J) COM08390

'ELKTII3,J21=TK32tI,J) COM08400 ELKT(I3,J3)=TK33(I,J)+XK33(I,J)+G33(I,Jl+GHl COH08410 ELKTtI3,J4)=G34CI,J)+TK34(I,J) COM08420 ELKTII3,JS)=G3S(I,J)+TK35(I,J) COH08430 ELKT(I4,JJ)=TK41CI,J) COM08440 ELKTfI4,J2)=TK4ZII,J) COM084S0 ELKTII4,J3)=G43CI,J)+2.00*TK43fI,J) COH08460 ELKT(I4,J4)=TK44CI,J)+G44(I,J)+GM2 COH08470 ELKT(I4,JS)=TK4SII,J)+G4SCI,J) COM08480 ELf(,Tf IS,JJ )=TKS1( I,J) COH08490 ELf(,T( IS,J2 )=TKS2( I,J) COH08S00 ELKTCIS,J3)=GS3(I,J)+2.DO*TKS3(I,JJ COM08Sl0 ELKT(I5,J4)=TKS4(I,Jl+GS4fI,J) COM08520 ELt<T( IS,JS l=TKSS{ I,J HGSS( I,J J+GM2 COH08S30

ReII)=-(TKllCI,Jl+GM1)*TUfJ)-TK12{I,J)*TUfJNPE1-TK13(I,J) COH08S40 &*TUIJ2NPE1-TK14(I,JJ*TU(J3NPEJ-TK1S(I,J)*TUeJ4NPE1+R(II1+ COH08550 &GMASSCI,J)*(SAO*TUU(Jl+SAl*TUltJ)+SA2*TU2(J))*XlNl COH08560

R(I2)=-TK21lI,J)*TU(J)-CTK22(I,J)+GM1)*TUIJNPE1-TK23(I,J) COH08S70 &*TU(J2NPEJ-TK24(I,J)*TUIJ3NPE)-TK2S(I,J)*TUIJ4NPE)+R(I2)+ COM08580 &GHASS(I,J)*(SAO*TUU(JNPE)+SAl*TUlfJNPE)+SA2*TU2(JNPE))*XINl COH08590

RCI3)=-TK3lII,J)*TU(J)-TK3ZCI,J)*TU(JNPE1-CTK33(I,J)+G33(I,J)+GHllCOM08600 &*TU(J2NPEJ-CTK34(I,J)+G34CI,J)}*TUIJ3NPE1-CG3S(I,J)+TK3S(I,J})* COH08610 &TUfJ4NPE)+ReI3)+ COM08620 &GMASSCI,J)*CSAO*TUUfJZNPE)+SA1*TUlfJZNPE)+SA2*TUZ(J2NPE))*XINl COH08630

RCI4)=-TK41CI,J)*TU(JJ-TK4Z(I,J)*TUIJNPE) COM08640 &-(G43(I,J)+TK43fI,J))*TUCJ2NPE)-(TK44(I,J)+G44CI,J)+GHZ) COH086S0 &*TUeJ3NPE)-(TK4SII,J)+G45(I,J))*TUfJ4NPE)+RCI4)+ COM08660 &GHAS$(I,J)*CSAO*TUUIJ3NPE)+SAl*TUlIJ3NPE)+SA2*TUZ(J3NPE)1*XINZ COM08670 R(IS)=-TKSIII,J)*TUeJ)-TK52II,J)*TUIJNPE)-ITKS3(I,J)+G53(I,J))* COM08680

&TUeJ2NPE1-(G54II,J)+TKS4(I,J))*TUIJ3NPE) COH08690 &-CTK5S(I,J)+GSS(I,J)+GHZJ*TUlJ4NPE)+RlIS)+ COH08700 &GMASS(I,JJ*(SAO*TUU(J4NPE)+SA1*TUl(J4NPE)+SAZ*TU2(J4NPE)*XIN2 COH08710

ISO JJ=NOF*J+l COM087Z0 R(II)=R(II)+ELFl(I) COH08730 RfI2)=R(IZ)+ELF2(I) COH08740 RCI3)=RCI3)+ELF3(IJ COH08750 RII4)=R(I4)+ELF4(I) COH08760 RfI5)=R(IS)+ElF5'I) COH08770

170 II=NDF*I+l COH08780 C DO 8972 I=l,NPE COH08790 CS972 HRITE(7,2777)ElFltI),ELF2(I),ELF3(I),ELF4(I),ELFS(I) COH08800

C C C

2777 FORMAT" ·,S(012.S,2X) COH08810

410

RETURN COH08820 END COH08830 SUBROUTINE SHAPE(XI,ETA,NPE,ELXY,IELEH,OET) COH08840 IMPLICIT REAL*8(A-H,O-Z) COH08850 COHHON/SHP/GDSF(2,'),SF(') COH08860 OIHENSION DSF(2,'),ELXYI9,2),GJ(2,2),GJINV(2,2),XSIGN(9,2),NP(') COH08870 DATA XSIGNI-l.ODO,2*1.ODO,-1.0DO,O.000,1.ODO,O.ODO,-1.000,O.000, COH0888 0

&2*-1.000,Z*1.000,-1.0DO,0.000,1.0DO,2*0.000/ COH08890 DATA NP/1,2,3,4,S,7,6,8,9/ COH08900 IF(NPE.EQ.S)GOTO SOO COH08910 IF(NPE.EQ.')GOTO 900 COH08920

LINEAR INTERPOLATION FlN:TIONS

DO 410 1=1,4 XP=XSIGN(I,l) VP=XSIGNCI,2) XlO=I.00+XI*XP ETAO=l.DO+ETA*VP SFCI)=.ZSOO*XIO*ETAO OSFtl,I)=.2S00*XP*ETAO OSF(2,I)=.2S00*VP*XlO

COH08930 COH08940 COH08950 COH08960 COH08970 COH08980 COH08990 COH09000 COMO 9010 COH09020 COH09030

104

GOTO 100 C C QUADRATIC INTERPOLATION FUNCTIONS (EIGHT-NODE ELEMENTS) C

800 DO 810 1=1,8 NI=NP'I) XP=XSIGN(NI,l) VP=XSIGN(NI,Z) XIO=l.DO+XI*XP ETAO=l.DO+ETA*VP XI1=1.DO-XI*XI ETA1=1.DO-ETA*ETA IF(I.GT.4)GOTO 8Z0 SF(NI1=.Z5DO*XIO*ETAO*(XI*XP+ETA*VP-1.DO) DSF(l,NI)=.2SDO*XP*ETAO*(Z.DO*XI*XP+ETA*VP) DSF(Z,NI1=.Z5DO*VP*XIO*(Z.DO*ETA*YP+XI*XP) GOTO 810

8Z0 IF(I.GT.6)GOTO 830 SF(NIJ=.500*Xl1*ETAO OSF(l,NI)=-XI*ETAO DSFlZ,NI1=.500*VP*XIl GOTO 810

830 SFtNIJ=.5DO*ETA1*XIO DSFll,NI)=.500*XP*ETA1 DSF(Z,NI)=-ETA*XIO

810 CONTINUE GOTO 100

C C QUADRATIC INTERPOLATION FUNCTIONS (NINE -NODE ~LEMENTS) C

900 DO 910 1=1,9 N!=NP(I) XP=XSIGN(NI,l) YP=XSIGNCNI,Z) XIO=l.DO+XI*XP ETAO=1.00+ETA*VP XI1=1.DO-XI*XI ETA1=1.DO-ETA*ETA XIZ=XP*XI ETAZ=YP*ETA IF(I.GT.4)GOTO 9Z0 SF(NI)=.2S00*XIO*ETAD*XIZ*ETAZ DSFf1,N!)=.Z5DO*XP*ETAO*ETAZ*(1.DO+Z.DO*XIZ) DSF(Z,NI)=.2SDO*VP*XIO*XIZ*C1.00+Z.DO*ETAZ) GOTO 910

9Z0 IF(I.GT.6)GQTO 930 SFfN!)=.500*XI1*ETAO*ETAZ OSF(l,NI)=-XI*ETAO*ETAZ DSF(Z,NI1=.5DO*YP*XI1*{1.DO+Z.DO*ETAZ) GOTO 910

930 IFlI.GT.8lGOTO 940 SFINI)=.5DO*ETA1*XIO*XIZ DSF(l,NI)=.5DO*XP*ETA1*(1.DO+Z.DO*XIZ) DSF(Z,NI)=-ETA*XIO*XIZ GOTO 910

940 SF(NI)=ETA1*XI1 DSF(1,NI)=-Z.DO*XI*ETA1 OSF(Z,NI)=-ETA*XIl*Z.DO

910 CONTINJE 100 DO 560 I=l,Z

DO 560 J=l,Z GJ(I,J)=O.DO

DO 560 K=l,NPE 560 GJ(I,J)=GJ(I,J)+OSF(I,K)*ELXY(K,J)

OET=GJl1,1)*GJ(2,2)-GJ(I,2)*GJ(Z,1) GJINV(I,I)=GJ(Z,Z)/DET GJINV(l,Z)=-GJ(I,Z)/DET GJINV(Z,lJ=-GJ(Z,I)/DET GJINV(2,2)=GJ(I,l)/DET DO 570 1=1,2

DO 570 J=l,NPE

C0H09040 COH09050 COH09060 COHO 90 70 COHO 9080 COHO 9090 COM09100 COM09110 COM09120 COMO 9130 COM09140 COM09150 COM09160 COH09170 COHO 9180 COM091CJO COM09Z00 COMOCJZIO C0I10CJZ20 COH09230 COHO 9240 COMOCJ250 COHOCJZ60 COM09270 COH09280 COH09290 COMO 9300 COMO 93 10 COH09320 COHOCJ330 COMOCJ340 COH09350 COM09360 COMOCJ370 COHO 9380 COHOCJ3CJO Cot109400 COMOO/dO COH09420 COM09430 COHO 9440 COMO 9450 COMO 9460 COHO 9470 COHO 9480 COH094CJO COH09500 COH09510 COHOCJ5Z0 COH09530 COMO 9540 COHOCJ550 COHOCJ560 COHO 9570 COHO 9580 COHO CJ5 CJO COHOCJ600 COHOCJ610 COM096Z0 COM09630 COMOCJ640 COHOCJ650 COH09660 COMOCJ670 COHO 9680 COM096CJO COMOCJ700 COMOc)710 C0I10CJ720 COMOc)730 COMOC)740

105

GDSFfI,J'=O.DO DO 570 K=1,2

570 GDSF(I,J)=GDSF(I,J)+GJINV(I,K}*DSF(K,J) RETURN

C C C C C C C C

END SUBROUTINE BNDY(NRHAX,NCHAX,NEq,NHBH,S,SL,NBDY,IBDY,VBDY)

THIS PROGRAM IMPOSES THE PRESCRIBED BOUNDARY CONDITIONS ON THE BANDED SYMMETRIC MATRIX AND MODIFIES THE RIGHT-HAND-SIDE (SL). S IS THE COEFFICIENT MATRIX (STIFFNESS MATRIX) SL IS THE RIGHT-HAND-SIDE COLUMN (FORCE) VECTOR

IMPLICIT REAL*8(A-H,0-Z) DIMENSION SINRHAX,NCHAX),SLINRHAX),IBDY(NBDY),VBDY(NBDY) COMHON/IO/IN , ITT

C HRlTE(7,980)SI1,1),S(1,2),SC1,3),SC1.4) C "RITE(7,980)S(2,1),S(2,2),S(2,3),S(2,4) C HRITE(7,980)S(3,1),S(3,2},SC3,3),S(3,4) C HRITE(7,980)SC4,1),S(4,2),S(4,3),S(4,4) C HRITE(7,980)SC5,1),S(5,2),S(S,3),S(S,4) C "RITE(7,980)S(6,1),S(6,2),S(6,3),S(6,4) C HRITE{7,980)S(7,1),SC7,2),SC7,3),SC7,4) C980 FORMATe' ·,4(D13.6,ZX))

C C C C C C C C C

C

DO 30 NB=l,NBDY IE=IBDYC NB) SVAL=VBDY(NB) IT=NHBH-1 I=IE-NHBH DO 10 II=l,IT 1=1+1 IF (I.LT.1) GO TO 10 J=IE-I+1 SL(I)=SL(I)-SII,J)*SVAL S(I,J)=O.O

10 CONTINUE S(IE,1)=1.0 SLC IE )=SVAL I=IE DO 20 II=2,NHBH 1=1+1 IF e I.GT .NEq) GO TO 20 SLCI)=SL(I)-S(IE,II)*SVAL S(IE,II)=O.O

20 CONTINUE 30 CONTINJE

RETURN END SUBROUTINE SOLVE (NRtt,N:H,NEqNS,teH,8AND,RHS,IRES)

THIS PROGRAM SOLVES A BANDED SYMMETRIC SYSTEH OF EQUATIONS. THE BANDED HATRIX IS INPUT THROUGH BANDfNEQNS,teHh AND RHS IS THE RIGHT HAND SIDE (FORCE VECTOR) OF THE EQUATION. NEqNS IS THE NO. OF EQUATIONS AND NBH IS THE HALF BAND HIDTH. IN RESOLVItc, IRES .ST. 0, LHS ELIMINATION IS SKIPPED.

IMPLICIT REAL*8(A-H,0-Z) DIMENSION BAND(NRH,NCH),RHS(NRH) COI't'D-VIOIIN ,ITT ttEliNS=NEQNS-1 IF (IRES.ST.D) GO TO 40 DO 30 NPIV=l,HEQNS NPIVOT=NPIV+1 LSTSUB=NPIV+NBH-1 IF (LSTSlB.GT .NEqNS), LSTSU8=NEQNS DO 20 NROH=NPlVOT,LSTSUB INVERT RONS AND COLUMNS FOR ROM FACTOR NCOL=NROH-NPIV+1

coHo 97S0 COM09760 COM09770 COM09780 C0t109790 COM09800 COMO 9810 COM09820 COM09830 COMO 9840 COMO 9850 COHO 9860 COMO 9870 COHO 9880 COHO 9890 COMO 9900 COH09910 COH09920 COM09930 COHO 9940 COHO 99S0 C0H09960 COM09970 COMO 9980 COH09990 COH10000 COM10010 COM10020 COH10030 COH10040 COM100SO COH10060 COH10070 COHI0080 COM10090 COMI0100 COHIOIIO Ca110120 COM10130 Ca110140 COM101SO COMI0160 Ca110170. COH10180 COHI0190 COHI0200 COMI0210 COHI0220 COM10230 COH10240 COH102S0 CQt110260 COH10270 COH10280 COH10290 COM10300 COH10310 COH10320 COH10330 COH10340 COH10350 COH10360 COH10370 COHI0380 COHI0390 COH10400 COH10410 Cot110420 COH10430 COHI0440 COH104S0

106

FACTOR=BANDlNPIV,NCOLJ/BANDfNPIV,lJ DO 10 NCOL=NROH,LSTSUB ICOL=NCOL-NROH+l JCOL=NCOL-NPIV+l

10 BANDfNROH,ICOL)=BANDfNROH,ICOLJ-FACTOR*BAND(NPIV,JCOLJ 20 RHSCNROH)=RHSCNROH1-FACTOR*RHS(NPIV) 30 CONTINUE

GO TO 70 40 DO 60 NPIV=l,HEQNS

NPIVOT=NPIV+l lSTSU8=NPIV+NBH-l IF (LSTSUB. GT • NEQNS) LSTSlB=NEQNS DO 50 NROH=NPIVOT,LSTSUB NCOL=NROH-NPIV+l FACTOR=BANDtNPIV,NCOL)/BANDfNPIV,l)

50 RHSINROH)=RHS(NROH1-FACTOR*RHS(NPIV) 60 CONTINUE

C BACK SUBSTITUTION 70 DO 90 IJK=2,NEQNS

NPIV=NEQNS-IJK+2 RHSfNPIV)=RHSCNPIV)/BANDCNPIV,l) lSTSUB=NPIV-NBH+l IF (LSTSUB.LT.l) LSTSU8=l NPIVOT=NPIV-l DO 80 JKI=LSTSUB,NPIVOT NROH=NPIVOT-JKI+LSTSUB NCOL=NPIV-NROH+l FACTOR=BANDCNROH,NCOL)

80 RHS(NROH)=RHS(NROH)-FACTOR*RHS(NPIV) 90 CONTINUE

RHSCl)=RHS(l)/BANDfl,ll RETlJRN END SUBROUTINE HESH(IElEH,NX,NY,NPE,X,Y,NNH,NEH,DX,DY) IMPLICIT REAl*8(A-H,O-Z) COMHONIHSHINOD(4,,) DIMENSION X(9),Y(9J,DXI5),DYfS) IEL=IELEH-l

C GUADRILATERAL ELEMENTS NEXl=NX+l NEYl=NY+l NXX=NX*IEL NVY=NY*IEl NXXl =NXX+ 1 NVYl=NYY+l NEH=NX*NY NNH=NXX1*NYYl-(IEl-l)*NX*NY IFfNPE.EQ.9)NNH=NXXl*NYYl KO=O IFCNPE.EQ.9)KO=l NODIl,l)=l NOD(1,2)=IEl+l NDDCl,3)=NXXl+CIEl-l)*NEXl+IEL+l IFCNPE.EQ.9)NODCl,3)=4*NX+S NODfl,4)=NODCl,3)-IEl IFfNPE.EQ.4)GOTO 200 NODfl,S)=2 NODCl,61=NXXl+(NPE-6) NODfl,7)=NOD(l,3)-1 NOD(l,81=NXX1+l IF(NPE.EQ.9)NOOfl,9)=NXX1+2

200 IF(NY.EQ.l)GOTO 230 M=1 DO 220 N=2,NY L=(N-l)*NX+l DO 210 I=l,NPE

210 NOD(L,I)=NOD(H,I)+NXX1+(IEL-l}*NEXl+KO*NX 220 H=L 230 IFfNX.EQ.l)GOTO 270

00 260 NI=2,NX 00 240 I=l,NPE

COHI0460 COMI0470 COHI0480 COHI0490 COHIOSOO COHIOSIO COHI0520 COHI0530 COHI0540 COMI0550 COHI0560 COHI0570 COHIOS80 COHI0590 COHI0600 COHI0610 COHI0620 COHI0630 COHI0640 COHI0650 COHI0660 COHI0670 COHI0680 COHI0690 COHI0700 COHI0710 COHI0720 COHI0730 COHI0740 COHI0750 COHI0760 COHI0770 COHI0780 COHI0790 COHI0800 COHI08l0 COHI0820 COHI0830 COHI0840 COHI08S0 COHI0860 COHI0870 COHI0880 COHI0890 COHI0900 COHI0910 COHI0920 COHI0930 COHI0940 COHI09S0 COHI0960 COHl0970 COHI0980 Cct110990 Cct111000 COHIIOIO COHII020 COHII030 Cct1110ltO COHIIOSO COHII060 COHII070 Cct111080 Cct111090 COHIIIOO COHIIIIO COHll120 COHlll30 COHlll40 COHll150 COHl1160

107

K1=IEl IFII.EQ.6.0R.I.EQ.8)K1=1+KO

240 NOD(NI,I)=NOD(NI-1,Il+K1 M=NI DO 260 N..I=2,NY l=(N..I-1)*NX+NI DO 250 J=1,NPE

250 NODCl,J)=NODIM,J)+NXX1+(IEl-l)*NEXl+KO*NX 260 M=L 270 YC=O.DO

IF(NPE.EQ.9)GOTO 310 00 300 NI=1,NEY1 I=(NXX1+(IEL-l)*NEX11*(NI-l)+1 J=( NI-l UifIEl+1 X(I)=O.DO Y(Il=YC 00 280 N..I=l,NXX 1=1+1 X(I)=XCI-1)+DX(N..I)

280 Y(I)=YC IF(NI.GT.NY.OR.IEL.EQ.1)GOTO 300 J=J+1 YC=YC+DY(J-1J I=I+l X(I)=O.DO Y(I)=YC 00 290 II=1,N)(

K=2*II-1 1=1+1 X(I)=X(I-1)+DX(K)+DX(K+1)

290 Y(I)=YC 300 YC=YC+DY(J)

REnJRN 310 00 330 NI=1,NYV1

I=NXX1*(NI-l) XC=O.DO 00 320 N..I=1,NXX1 1=1+1 X(I)=XC Yfll=YC

320 XC=XC+DX(N..I) 330 YC=YC+DY(NIJ

REnJRN END

C0I111170 COH11180 COM11190 C0t111200 COM11210 COM11ZZ0 C0t111230 C0I111240 COM11Z50 COH11Z60 C0t111270 C0I111Z80 COH11Z90 COHl1300 COHl1310 COHl13Z0 C0I111330 COH11340 COH11350 COH11360 COH11370 C0I111380 COH11390 COtt11400 C0t111410 COH114Z0 COH11430 C0I111440 COH11450 COH11460 COH11470 COH11480 COH11490 C0I111500 C0t111510 COH115Z0 C0I111530 COH11540 C(x"Il1550 COH11560 COH11570 COH11580 COH11590 COH11600

108

Vita

Thomas Harris Fronk was born February 17, 1961 in Tremonton, Utah, the fifth of six children. He was educated in Tremonton until his graduation from Bear River High School in June 1979. He served a lay mission in South America for the Church of Jesus Christ of Latter-Day Saints from 1980 to 1982. He received his B.S. degree in mechanical engineering from Utah State University in June 1985 while working for Lockheed Missile and Space Co. his senior year. From June 1985 to September 1986 he was employed in the composite structures design group at Morton Thiokol Inc .. He began his graduate studies at Virginia Polytechnic and State University in September 1986. He married Monica Layne Sntith in June 1984 and they are the parents of two boys ages four months and two and one-half years.

109