numerical modeling and analysis of composite beam structures

Numerical Modeling and Analysis of Composite Beam

Structures Subjected to Torsional Loading

Kunlin Hsieh

Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science in

Engineering Mechanics

John C. Duke (Chair) Raymond H. Plaut

Scott W. Case

May 3, 2007 Blacksburg, Virginia

Keywords: Homogenization, Volume Fraction, Laminated Composites, Torsional Rigidity

Copyright ©2007, Kun-Lin Hsieh

Numerical Modeling and Analysis of Composite Beam Structures Subjected to Torsional Loading

Kunlin Hsieh

ABSTRACT

Torsion of cylindrical shafts has long been a basic subject in the classical theory of elasticity.

In 1998 Swanson proposed a theoretical solution for the torsion problem of laminated

composites. He adopted the traditional formulation of the torsion problem based on Saint

Venant’s torsion theory. The eigenfunction expansion method was employed to solve the

formulated problem. The analytical method is proposed in this study enabling one to solve the

torsion problem of laminated composite beams. Instead of following the classical Saint Venant

theory formulation, the notion of effective elastic constant is utilized. This approach uses the

concept of elastic constants, and in this context the three-dimensional non-homogeneous

orthotropic laminate is replaced by an equivalent homogeneous orthotropic material. By adopting

the assumptions of constant stress and constant strain, the effective shear moduli of the

composite laminates are then derived. Upon obtaining the shear moduli of the equivalent

homogeneous material, the effective torsional rigidity of the laminated composite rods can be

determined by employing the theory developed by Lekhnitskii in 1963. Finally, the predicted

results based on the present analytical approach are compared with those by the finite element,

the finite difference method and Swanson’s results.

iii

Acknowledgements

I would like to sincerely thank my committee chair and advisor Dr. John C. Duke for his

generosity, patience, and guidance throughout this thesis and my graduation program. His

thoughtfulness and willingness to help are deeply appreciated. It has been a pleasure to have him

as an advisor. My gratitude and appreciation also go to Dr. Raymond H. Plaut and Dr. Scott W.

Case for serving as members of my committee.

I would like to thank my friend Mr. Haekyu Hur for taking time with me to discuss my

research problem. Your long-term support and encouragement are truly appreciated.

Most importantly, I would like to thank my parents, Mr. Joneming Hsieh and Mrs. Wei Jui,

my older sister Huiya Hsieh, and my younger sister Menghua Hsieh for their love, support, and

encouragement throughout my entire graduate study. This work is dedicated to all of you.

iv

Table of Contents .

Abstract……………………………………………………………………………………………ii

Acknowledgements………………………………………………………………………………iii

List of Figures…………………………………………………………………………………….vi

List of Tables…………………………………………………………………………………….vii

Chapter 1: Introduction and Literature Review…………………………………………………...1

1.1 Introduction…………………………………………………………………………….1

1.2 Literature Review………………………………………………………………………3

1.2.1 Torsion of Circular Cross Section……………………………………………..3

1.2.2 Torsion of Non-Circular Cross Section……………………………………….4

1.2.3 Solutions using Prandtl’s Stress Function…………………………………….7

1.2.4 Analytical Solutions for Rectangular Cross Section…………………………..9

Chapter 2: Effective Elastic Constants for Laminated Composite………………………………11

2.1 Introduction…………………………………………………………………………..11

2.2 Theoretical Background……………………………………………………………...13

2.3 Hooke’s Law for Monoclinic Materials……………………………………………...14

2.4 Effective Constants…………………………………………………………………..18

2.5 Verification Examples……………………………………………………………….27

Chapter 3: Torsional Response of Laminated Composite Beam………………………………...32

3.1 Introduction…………………………………………………………………………..32

3.2 General Formulation…………………………………………………………………33

3.3 Formulation for Orthotropic Composite Beams………………………………..……35

3.4 Analytical Model for Rectangular Laminated Composites…………..……………...37

Chapter 4: Finite Element Analysis……………………………………………………………...45

4.1 Introduction…………………………………………………………………………..45

4.2 Finite Element Procedure…………………………………………………………….52

v


Chapter 5: Finite Difference Method…………………………………………………………….69

5.1 Introduction…………………………………………………………………………..69

5.2 Finite Difference Equations…..……………………………………………………...70


Chapter 6: Results and Comparisons…………………………………………………………….84

6.1 Material Properties…………………………………………………………………...84

6.2 Analysis Models……………………………………………………………………...87

6.2.1 Convergence Study…………………………………………………………….88

6.2.2 Results and Comparisons………………………………………………………96

Chapter 7: Summary and Conclusions...………………………………………………………..100

References………………………………………………………………………………………102

Vita……………………………………………………………………………………………...105

vi

List of Figures

Figure 1.1: Circular Prismatic Bar under Torsional Loading……………………………………4

Figure 1.2: Elliptical Cross Section……………………………………………………………...8

Figure 1.3: Equilateral Triangle Cross Section…………………………………………………..8

Figure 1.4: Rectangular Cross Section…………………………………………………………...9

Figure 2.1: The Representative Element for Monoclinic Materials……………………………14

Figure 2.2: The Representative Element of Layered Medium………………………………….19

Figure 3.1: General Cross Section Composed of Orthotropic Media…………………………..34

Figure 3.2: Cross Section of Laminated Composites…………………………………………...39

Figure 4.1: 2D Rectangular Beam Cross Section……………………………………………....49

Figure 4.2: PLANE77 Geometry……………………………………………………………….53

Figure 4.3: Rectangular Cross Section………………………………………………………….54

Figure 4.4: Rectangular Cross Section Composed of Two Different Materials………………..57

Figure 4.5: Cross Section of Orthogonal Beam………………………………………………...64

Figure 5.1: Finite Difference Approximation of Function )(xg ……………………………….71

Figure 5.2: Rectangular Mesh…………………………………………………………………..74

Figure 5.3: Finite Difference Grids……………………………………………………………..76

Figure 6.1: Transformation Relations between 1-2-3 and x-y-z Coordinate Systems………….86

Figure 6.2: The Analysis Model………………………………………………………………..87

Figure 6.3: Convergence Study for Finite Element Method…………………………………....89

Figure 6.4: Convergence Study for Finite Difference Method………………………………....90

Figure 6.5: Contour Plot for Prandtl’s Stress Function

(Graphite-polymer Laminates s]0/15[ )…………………………………………….91

Figure 6.6: Mesh Plot for Analysis Model (element size = 0.8 mm)…………………………..92

Figure 6.7: Mesh Plot for Analysis Model (element size = 0.5 mm)…………………………..93

Figure 6.8: Mesh Plot for Analysis Model (element size = 0.25 mm)………………………..94

Figure 6.9: Mesh Plot for Analysis Model (element size = 0.2 mm)…………………………95

vii

List of Tables

Table 2.1: Direction Cosine Table……………………………………………………………...15

Table 2.2: Material Properties of Laminas in the Sublaminate……………………………...…28

Table 2.3: Comparison of Effective Moduli for Boron/Epoxy Angle-Plied Laminates……..…29

Table 2.4: Comparison of Effective Moduli for Hybrid Laminates…………………………....31

Table 4.1: Analogy between Torsion Problem and Heat Transfer Problem…………………....51

Table 4.2: Comparison of 1k Value between Analytical and ANSYS Results…………….…...54

Table 4.3: Comparison of 2k Value between Analytical and ANSYS Results……………...….56

Table 4.4: Comparison of Torsional Rigidity between Analytical and ANSYS Solutions

with the Change of Ratio of Shear Moduli 21 / GG ……............................................60


with the Change of Ratio of Shear Moduli ba /1 ………...……………………...….61


with the Change of Ratio of Shear Moduli ba /2 …………...……………...……….62

Table 4.7: Comparison of β Value between Analytical and ANSYS Results…........................67

Table 4.8: Comparison of 3k Value between Analytical and ANSYS Results………………....67

Table 4.9: Comparison of 4k Value between Analytical and ANSYS Results…………………68

Table 5.1: Comparison of 1k Value between Analytical and FDM Results…………………….77

Table 5.2: Comparison of 2k Value between Analytical and FDM Results……………………77

Table 5.3: Comparison of Torsional Rigidity between Analytical and FDM Solutions

with the Change of Ratio of Shear Moduli 21 / GG ……………...………………….78


with the Change of Ratio of Shear Moduli ba /1 ………………...…………………79


with the Change of Ratio of Shear Moduli ba /2 …………………………………...80

Table 5.6: Comparison of β Value between Analytical and FDM Results…………………….81

viii



Table 6.1: Material Properties of Graphite-polymer Composite…………………………….…84

Table 6.2: Material Properties of Glass-polymer Composite…………………………………..85

Table 6.3: Shear Moduli for Different Ply-angles (Graphite-polymer Composite)…………....85

Table 6.4: Shear Moduli for Different Ply-angles (Glass-polymer Composite)……………….85

Table 6.5: Convergence Study for Finite Element Method…………………………………….88

Table 6.6: Convergence Study for Finite Difference Method………………………………….89

Table 6.7: Comparisons of Torsional Rigidity between Present Analytical Method

and ANSYS Results (Graphite-polymer Composite)……………………….………97

Table 6.8: Comparisons of Torsional Rigidity between Present Analytical Method

and ANSYS Results (Glass-polymer Composite)……………………….………....98

１

Chapter 1 Introduction and Literature Review 1.1 Introduction

Composite materials have been widely used to improve the performance of various types of

structures. Compared to conventional materials, the main advantages of composites are their

superior stiffness to mass ratio as well as high strength to weight ratio. Because of these

advantages, composites have been increasingly incorporated in structural components in various

industrial fields. Some examples are helicopter rotor blades, aircraft wings in aerospace

engineering, and bridge structures in civil engineering applications.

Torsion of cylindrical shafts has long been a basic subject in the classical theory of elasticity

(Timoshenko and Goodier, 1970). The stiffness of a cylindrical shaft under torsional loading is

often of interest in the study of torsion problems. The axial displacement field of the cross-

section is assumed to not vary along the axial direction away from the ends of the shaft. Under

this assumption, the torsional rigidity is only dependent on the shape of the cross-section. The

governing equations of this boundary value problem can be formulated in terms of a Laplace or

Poisson Equation. Within the former one, one uses the warping function as a dependent variable,

while in the latter the Prandtl’s stress function is used. The solutions for the warping and

Prandtl’s stress function have been obtained exactly for simple cross-sectional shapes such as a

circle, annulus, ellipse, rectangle, and triangle. For more complicated shapes, numerical methods

are usually employed, such as, for example, the finite difference method (Ely and Zienkiewicz,

1960), finite element method (Herrmann, 1965; Karayannis, 1995; Li at al., 2000), and boundary

element method (Jawson and Ponter, 1963; Friedman and Kosmatka, 2000; Sapountzakis, 2001;

Sapountzakis and Mokos, 2001, 2003). Some authors use the approach of combination of

experimental and analytical methods to predict the effective in-plane and out-of-plane shear

moduli of structural composite laminates (Davalos, 2002).

Due to the extensive use of composite materials, the study of compound bars under torsion

becomes a very important topic. Compared to homogeneous cylindrical shafts, the torsional

behavior of composite shafts is considerably more complicated. The torsional rigidity not only

depends on the global cross-sectional geometry, but also on the properties and configurations of

each constituent. The analytical solution of compound bars under torsion was first obtained by

Muskhelishvili (1963), where the solution was expressed in terms of eigenfunctions. Packham

２

and Shail (1978) used linear combinations of solutions of a homogeneous shaft to solve the

problem in which the cross section is symmetric with respect to the common boundary.

The elastic properties of non-homogeneous anisotropic beams are usually of engineering

interest. Torsional rigidities of multilayered composite beams are especially needed when

structures are under torsional loading. Savoia and Tullini (1993) analyzed the torsional response

of composite beams of arbitrary cross section. The boundary value problem was formulated in

terms of both warping and Prandtl’s stress function. Using the eigenfunction expansion method,

the exact solution of rectangular multilayered orthotropic beams under uniform torsion was

derived. Swanson (1998) extended the existing solutions of torsion of orthotropic laminated

rectangular beams to the high aspect ratio case. Based on the membrane analogy, an approximate

solution of general, thin, laminated, open cross sections was derived.

In this study, one analytical approach will be proposed to solve the torsion problem of

laminated composite beams that consist of orthotropic sublaminates. The present approach uses

the concept of elastic constants (Chou, et al., 1972), in which the three-dimensional non-

homogeneous orthotropic laminate is replaced by an equivalent homogeneous orthotropic

material. By considering a small element consisting of n layers from the composite material, this

small element is assumed to represent the behavior of the overall composite laminate. We will

consider that this element is under a uniform state of stress when the composite laminate is under

arbitrary loading. Two assumptions have to be implemented: first, the normal strains and shear

strains parallel to the plane of layers are uniform and the same for each constituent and the

corresponding stresses are averaged. Second, the normal stresses and shear stresses

perpendicular to the plane of layers are uniform and equal for each constituent and the

corresponding strains are averaged. Under these two assumptions, the equilibrium at each sub-

laminate interface and compatibility conditions of materials are satisfied automatically. As the

thickness of each layer approaches zero, the overall effective elastic constants are developed.

The effective shear moduli of the composite laminates are used to calculate the overall torsional

rigidity of the orthotropic laminated shaft.

３

1.2 Literature Review

1.2.1 Torsion of Circular Cross Section

Figure 1.1 shows a prismatic bar under torsional loading (Sadda, 1993). The prismatic bar

with radius r is fixed at one end (x-y plane) and subjected to a torque T at the other end. Based

on three assumptions, the torsional response of the circular cross-sectional beam can be derived.

The three assumptions are stated as follows:

(1) The plane normal to the OZ axis remains plane after deformation.

(2) The rotation angle of each cross section is proportional to the distance from the fixed

end (z=0).

(3) The twist angle of each cross section is assumed to be small, so the torsion problem can

be treated as a linear elasticity problem.

Satisfying the equations of equilibrium and boundary conditions, the explicit expressions

of twisting moment and torsional rigidity can be derived. Saint-Venant’s principle states that the

solutions are excellent as long as the plane of solutions is one or two diameters away from the

plane of torque application. The torsional rigidity of an isotropic, circular cross section rod is

defined by the applied torque divided by the twist angle per unit length.

θTGJ = (1.1)

where

=G Shear modulus

== 24rJ π Polar moment of inertia (r = radius of circular cross section)

=T Applied torque

=θ Angle of twist per unit length

４

Figure 1.1 Circular Prismatic Bar under Torsional Loading

1.2.2 Torsion of Non-Circular Cross Section

For a non-circular bar, the torsion problem is solved by the semi-inverse method, which

was developed by Saint-Venant. To use this method, the following two assumptions have to be

made:

(1) The angle of twist of each cross section is proportional to the distance from the fixed

end without any inplane distortion.

(2) All the cross sections will warp in the same way, which means the warping function is

the function of the in-plane coordinates (x, y) only, and independent of the longitudinal

coordinate (z).

The torsion problem of non-circular prismatic bars can be formulated by two approaches.

The first approach is using the warping function, and the second one uses the stress function. The

governing equation, i.e., Laplace’s equation, and associated equations of the torsion problem by

using warping function are displayed below:

X

Y

Z T

O

５

02

2

2

22 =

∂∂+

∂∂=∇

yxψψψ on beam cross section (1.2)

0)()( =+∂∂−−

∂∂

dsdxxyds

dyyxψψ on the boundary surface (1.3)

∫∫ ∂∂−

∂∂++=

R

dxdyxyyxyxGT )( 22 ψψθ (1.4)

∫∫ ∂∂−

∂∂++==

R

dxdyxyyxyxGTGJ )( 22 ψψθ

(1.5)

where

=ψ Warping function

x, y = Coordinates in the cross section

s = Arc length along the bar

=T Applied torque

=G Shear modulus of the prismatic bar

=R Beam cross section

=GJ Torsional rigidity


By employing the relations between the warping function and Prandtl’s stress function, the

torsion problem of non-circular cross section can be formulated by the stress function approach.

The relations are shown below:

)( yxGy −∂∂=

∂∂ ψθφ (1.6)

)( xyGx +∂∂−=

∂∂ ψθφ (1.7)

where

６

=φ Prandtl’s stress function

=ψ Warping function



The governing equation, i.e., Poisson equation, and associated equations of the torsion

problem by using Prandtl’s stress function are displayed below:

θφφφ Gyx

22

2

2

22 −=

∂∂+

∂∂=∇ on beam cross section (1.8)

0=φ on the boundary surface (1.9)

∫∫=R

dxdyT φ2 (1.10)

θTGJ = (1.11)

where

=φ Prandtl’s stress function



=T Applied torque

=R Beam cross section

=GJ Torsional rigidity

７

1.2.3 Solutions Using Prandtl’s Stress Function

For some simple geometry cases, the Prandtl’s stress function can easily be used to express

the torsion problem. If the boundary equation of the beam cross section is a simple function of x

and y, the stress function can be chosen so as to satisfy the governing equation. The constant

included in the boundary function is then derived by solving the Poisson equation. This approach

is only useful for some simple geometrical cross section. The following shows solutions of two

simple cases by employing this procedure. The first case is an elliptical cross section, and the

other one shows the solutions of an equilateral triangle cross section (Boresi and Schmidt, 2002).

Elliptical Cross Section (Figure 1.2):

)1( 2

2

2

2

22

22−+

+−=

by

ax

baGba θφ (1.12)

22

33

baGbaT

+= θπ (1.13)

22

33

baGbaGJ

+= π (1.14)

Equilateral Triangle Cross Section (Figure 1.3):

)3)(323)(3

23(2axayxayxa

G +−+−−= θφ (1.15)

315

4θGaT = (1.16)

315

4GaGJ = (1.17)

８

Figure 1.2 Elliptical Cross Section

Figure 1.3 Equilateral Triangle Cross Section

Y

X

b

b

aa

3aY

X

32a

９

1.2.4 Analytical Solutions for Rectangular Cross Section

Figure 1.4 Rectangular Cross Section

The analytical solutions of rectangular cross section beams can be derived by employing

membrane analogy and Fourier series for both isotropic (Boresi and Schmidt, 2002) and

orthotropic (Lekhnitskii, 1981) cases. For the isotropic case, the governing equation, boundary

conditions, and solutions can be expressed as:

Governing Equation:

θφ Gyx 2),(2 −=∇ over the cross section (1.18)

Boundary Conditions:

0),( =± yaφ and 0),( =±bxφ (1.19)

Torsional Rigidity:

]2tanh1)(1921[3)2()2(

,...5,3,155

3

∑∞

=

−=k a

bkkb

abaGGJ ππ

(1.20)

Y

X

a a

b

b

１０

For the orthotropic case, the governing equation, boundary conditions, and analytical

solution can be expressed as:

Governing Equation:

θφφ 2),(1),(12

2

2

2

−=∂

∂+∂

∂y

yxGx

yxG zxzy

(1.21)

Boundary Conditions:

0),( =± yaφ and 0),( =±bxφ (1.22)

Torsional Rigidity:

])2

tanh(11921[3

16,...5,3,1

55

3

∑∞

=

−=k zy

zx

zx

zyzy G

Ga

kbkG

GbaGbaGJ π

π (1.23)

１１

Chapter 2 Effective Elastic Constants for Laminated Composite

2.1 Introduction

The overall elastic constants for composite materials are of interest in both the academic and

industry field. For thin laminated composites, classical plate theory is employed to analyze the

behavior. To solve the thick laminate problem, high order theory provides adequate accuracy.

The shortages of this theory are that it usually involves more mathematical derivations and is

only suitable for 2-dimensional behavior predictions. Hashin and Shtrikman (1963) proposed

upper and lower bounds for arbitrary geometrical composites. For layered composite materials,

many papers have displayed formulas for elastic moduli.

Three approaches are employed to solve the laminated composite problems. The common

point of these methods is assuming that the laminate is a homogeneous material. These three

approaches are the rule of mixtures, Voigt’s hypothesis, and Reuss’s hypothesis. The basic idea

of the rule of mixtures is averaging the elastic constants by volume. This approach is only

suitable for overall Young’s moduli prediction when the specimen is under axial tension in one

direction. But when applied to other elastic constants, the theory produces poor results. Voigt’s

hypothesis states that the strain components are the same throughout the layered media. The

results from this approach violate classical elasticity theory. The stresses between layered

interfaces do not satisfy the equilibrium equation. Similar to Voigt’s hypothesis, Reuss’s

hypothesis assumes that throughout the composite layers the stresses are uniform. This approach

also does not agree with the theory of elasticity. The strain and displacement predictions from

Reuss’s hypothesis violate the compatibility condition.

Many researchers have proposed methods to predict the effective elastic moduli of laminated

composites. Postma (1955) displayed a longwave approach, which treats a laminated medium as

an alternating layered medium consisting of two isotropic materials. The overall elastic constants

show that the medium behavior is like a transversely isotropic material. The application of this

approach to thick laminates is found to be more accurate than the classical laminated theory. To

avoid warping behavior, the layers’ stacking sequence has to be maintained periodically. The

non-homogeneous properties of a layered medium can be smeared out if the deformation’s

characteristic length is much larger than the periodicity. The non-homogeneous anisotropic

１２

laminates can then be treated as a homogeneous anisotropic material. Pagano (1974) proposed a

modified longwave approach to derive the effective moduli formula of the anisotropic laminates.

This approach uses a representative volume consisting of the overall thickness of the laminates to

analyze the overall elastic constants. Since the representative mdium has finite thickness, the

effect of bending moments has to be encountered in the stress-strain relations. Other similar

approaches were also proposed by many investigators. A dispersion method was used by

Behrens (1967) to examine the wave propagations of an alternating laminated medium. The

result shows that the averaged elastic moduli can be derived by calculating the phase speed in

different propagation directions while the long wavelength approaches a limit. White and

Angona (1955) proposed a static approach which assumes the stresses and strains in laminated

medium would not disappear as the wave propagates through the layered medium to determine

the elastic constants. A static approach was used by Salamon (1968) to study the elastic constants

of a stratified rock. A stratified rock was modeled by a representative n-layered medium. Instead

of deriving the effective stiffness of a layered medium, Salamon calculated the compliance term.

By proper algebraic manipulations, the elastic moduli derived by the aforementioned authors can

be shown to be equivalent.

Composite materials can be modeled as equivalent anisotropic materials under most loading

conditions. The results from the above model are fairly satisfactory. Sun, et al. (1968), however,

showed that under certain loading condition, especially under dynamic loading in which the

characteristic length of deformation is small, the predicted results are not accurate. A

microstructure continuum theory was then proposed to replace the equivalent anisotropic model

to predict the mechanical behavior of layered composites. Chou and Wang (1970) investigated a

one dimensional elastic wave front in a layered medium by employing a control volume

approach. By relating the in-plane averaged normal stress and strain, the stiffness terms for the

layered medium can be derived. The results were shown to be identical with those from the

equivalent material approach.

１３

2.2 Theoretical Background

In present approach (Chou et al, 1971), the laminated composite is treated as an equivalent

homogeneous material. The relations between overall stiffness and compliance with the layered

stiffness and compliance will be derived. The present approach can be applied in either isotropic

or anisotropic layered media. A solution procedure for a boundary value problem that is similar

to classical laminated plate theory is explained. The classical plate theory is limited to the two

dimensional case: the third normal stress and transverse shear stress are ignored. The present

method can be considered as an extension of classical laminated plate theory to the three

dimensional problem. By using the plate theory assumptions and considering the overall forces

throughout the layers, a continuous solution can be derived. The stresses of each layer can then

be obtained by employing the constitutive law.

The present approach combines the assumptions of Voigt’s and Reuss’s hypotheses. The

method first isolates an n layer element from the overall composite material. This element is

considered as a small element, which is usually used in classical elasticity theory. Hence, the

element is assumed under a uniform state of stress in arbitrary loading conditions. Two

assumptions have to be made. First, the normal strains and shear strains parallel to the layering

are assumed to be uniform and the same values for each layer and the corresponding stresses are

then averaged. Second, the normal stresses and shear stresses perpendicular to layers are

assumed to be uniform and the same, and then the corresponding strains are averaged. Under

these two assumptions, the equilibrium and compatibility conditions at the interfaces of layers

are both satisfied. Hence, the layered composite materials can be smeared out to equivalent

homogeneous anisotropic materials to determine the effective elastic moduli. The present theory

will approach exact solutions when the thickness of each laminated layer approaches zero.

The present approach treats the laminated composite as an n-layered repeating sub-laminate.

The thickness of each layer is assumed to be small compared with the total laminate thickness.

The laminated composite is modeled as a three dimensional homogeneous anisotropic material.

The anisotropic degree of each layer is monoclinic when the plane of symmetry is parallel to the

layering. Under the assumptions of constant stress and constant strain, the effective elastic

moduli can be derived. The results show that the mechanical behavior of the equivalent

homogeneous material is also monoclinic.

１４

2.3 Hooke’s Law for Monoclinic Materials

Figure 2.1 The representative element for monoclinic materials

The generalized Hooke’s Law for anisotropic materials can be expressed as

[ ]C

xy

xz

yz

zz

yy

xx

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

σσ

σσ

σσ

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

xy

xz

yz

zz

yy

xx

γγ

γε

εε

(2.1)

Z

X

Y

１５

where

[ ]

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

16

15

14

13

12

11

CCCCCC

C

26

25

24

23

22

12

CCCCCC

36

35

34

33

23

13

CCCCCC

46

45

44

34

24

14

CCCCCC

56

55

45

35

25

15

CCCCCC

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

66

56

46

36

26

16

CCCCCC

(2.2)

Monoclinic materials have planes of symmetry parallel to layers. Let us assume that the x-y

plane is the plane of symmetry in this case. Then under the coordinate transformation, i.e.,

xx → , yy → ,and zz −→ , the coordinates reflect with respect to the x-y plane, and the

stiffness matrix will remain the same.

Table 2.1 Direction Cosines Table (Boresi and Schmidt, 2003)

x y z

X 1l 1m 1n

Y 2l 2m 2n

Z 3l 3m 3n

By applying the direction cosines table to this case, the direction cosines under the

transformation are displayed as

121 == ml , 13 −=n , 0213132 ====== nnmmll (2.3)

The stresses and strains after transformation can be expressed as

xyxzyzzzyyxxXX mllnnmnml σσσσσσσ 1111112

12

12

1 222 +++++= (2.4)

xyxzyzzzyyxxYY mllnnmnml σσσσσσσ 2222222

22

22

2 222 +++++= (2.5)

xyxzyzzzyyxxZZ mllnnmnml σσσσσσσ 3333332

32

32

3 222 +++++= (2.6)

yzzzyyxxXY nmnmnnmmll σσσσσ )( 1221212121 ++++= (2.7)

xyxz mlmlnlnl σσ )()( 12211221 ++++

１６

yzzzyyxxXZ nmnmnnmmll σσσσσ )( 1331313131 ++++= (2.8)


yzzzyyxxYZ nmnmnnmmll σσσσσ )( 2332323232 ++++= (2.9)


xyxzyzzzyyxxXX mllnnmnml εεεεεεε 1111112

12

12

1 222 +++++= (2.10)

xyxzyzzzyyxxYY mllnnmnml εεεεεεε 2222222

22

22

2 222 +++++= (2.11)

xyxzyzzzyyxxZZ mllnnmnml εεεεεεε 3333332

32

32

3 222 +++++= (2.12)

yzzzyyxxXYXY nmnmnnmmll εεεεεγ )(21

1221212121 ++++== (2.13)

xyxz mlmlnlnl εε )()( 12211221 ++++

yzzzyyxxXZXZ nmnmnnmmll εεεεεγ )(21

1331313131 ++++== (2.14)


yzzzyyxxYZYZ nmnmnnmmll εεεεεγ )(21

2332323232 ++++== (2.15)


Substituting equation 2.3 into equations 2.4 to 2.15, the stresses and strains after coordinate

transformation are as follows:

xxXX σσ = , yyYY σσ = , zzZZ σσ = , xyXY σσ = , xzXZ σσ −= , yzYZ σσ −= (2.16)

xxXX εε = , yyYY εε = , zzZZ εε = , xyXY εε = , xzXZ εε −= , yzYZ εε −= (2.17)

The generalized Hooke’s Law after coordinate transformations can be shown to be

[ ]C

XY

XZ

YZ

ZZ

YY

XX

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

σσσσσσ

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

XY

XZ

YZ

ZZ

YY

XX

γγγεεε

(2.18)

１７

Substituting equations 2.16 and 2.17 into equation 2.18, the stress-strain relations can be

expressed as

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−−

xy

xz

yz

zz

yy

xx

σσ

σσ

σσ

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

= [ ]C

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−−

xy

xz

yz

zz

yy

xx

εε

εε

εε

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

(2.19)

By merging a negative sign into stiffness matrix [ ]C in equation 2.2, the stress-strain relation

can be obtained as

[ ]

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

C

γγ

γε

εε

σσ

σσ

σσ

(2.20)

where

[ ] =C

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

16

15

14

13

12

11

CCCCCC

−−

26

25

24

23

22

12

CCCCCC

−−

36

35

34

33

23

13

CCCCCC

−

−−−

46

45

44

34

24

14

CCCCCC

−

−−−

56

55

45

35

25

15

CCCCCC

−−

66

56

46

36

26

16

CCCCCC

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

(2.21)

As mentioned above, under coordinate transformation, the stiffness constants will not change.

Hence, by equating equation 2.21 with equation 2.2, the following results can be obtained:

04636352515342414 ======== CCCCCCCC (2.22)

１８

By substituting equation 2.22 into equation 2.21, the stiffness matrix for monoclinic

materials can be obtained:

[ ]

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

16

13

12

11

00

C

CCC

C

26

23

22

12

00

C

CCC

36

33

23

13

00

C

CCC

0

000

45

44

CC

0

000

55

45

CC

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

66

36

26

16

00

C

CCC

(2.23)

2.4 Effective Constants

Consider a thick laminated medium which is formed by stacking n-layered plates (Chou and

Carleone, 1971). The material property of each layer is considered as anisotropic. The degree of

anisotropy here is restricted to monoclinic in which the plane of symmetry of each layer is

parallel to the layering. This monoclinic material property includes orthotropic, transversely

isotropic, and isotropic layering. Figure 2.1 shows a representative laminated element. The

element is assumed to be small compared with the overall composite. An equivalent

homogeneous medium will represent this representative element. To derive the constitutive

formula of the equivalent medium in terms of the material properties of each constituent layer,

certain assumptions between stresses and strains of each layer have to be made.

１９

Figure 2.2 The representative element of layered medium

The summation notation will be used when employing subscript notation. Each layer will be

assigned by superscript notation, which does not follow the summation convention rule. The

quantities of the equivalent homogeneous medium do not use the superscript notation. The

coordinate system used here is the plane formed by x and y axes parallel to each layer. Stresses

and strains in this representative element are assumed to satisfy compatibility and equilibrium

X

Y

Z

Layer 1

Layer 2

Layer K

Layer N

.

.

.

.

.

.

.

２０

conditions. To satisfy the displacement continuity between each layer, certain assumptions about

strains have to be made. The normal strains in the x and y directions and shear strain in the x-y

plane are assumed to be uniform and equal for each individual layer. The values of these strains

are assumed to be the same with those of the equivalent homogeneous element:

kii εε = ),...,2,1;6,2,1( nki == (2.24)

In order to satisfy the continuity condition of stress in the interface of each layer, certain

assumptions about stresses have to be employed. The normal stress in the z direction and shear

stresses in z directions are assumed to be uniform and equal in each layer of the representative

element. The values of these stresses are assumed to be the same as for the equivalent

homogeneous element:

kii σσ = ),...,2,1;5,4,3( nki == (2.25)

The values of the rest of the stresses and strains of the equivalent homogeneous element are

assumed to be averaged for each constituent layer of the laminated element:

∑=

=n

k

ki

ki v

1

εε )5,4,3( =i (2.26)

∑=

=n

k

ki

ki v

1σσ )6,2,1( =i (2.27)

where

elementcompositeofvolumekmaterialofvolumevk =

２１

Hooke’s Law for a monoclinic material can be given as

kj

kij

ki C εσ = ),...,1;6,...,1( nki == (2.28)

kj

kij

ki S σε = ),...,1;6,...,1( nki == (2.29)

where

kijC =

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

k

k

k

k

C

C

C

C

61

31

21

11

00

k

k

k

k

C

C

C

C

62

32

22

12

00

k

k

k

k

C

C

C

C

63

33

23

13

00

0

000

54

44k

k

C

C

0

000

55

45k

k

C

C

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

k

k

k

k

C

C

C

C

66

36

26

16

00

kijS =

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

k

k

k

k

S

S

S

S

61

31

21

11

00

k

k

k

k

S

S

S

S

62

32

22

12

00

k

k

k

k

S

S

S

S

63

33

23

13

00

0

000

54

44k

k

S

S

0

000

55

45k

k

S

S

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

k

k

k

k

S

S

S

S

66

36

26

16

00

kji

kij CC = , ji ≠

kji

kij SS = , ji ≠

Equations 2.24 to 2.28 represent 12n+6 linear equations for 12n+12 variables. Hence, the

stresses of the equivalent element in terms of strains of the equivalent element can be derived.

The homogeneous equivalent elastic constants can then be in terms of constants of each

individual layer. The effective elastic constants are obtained as

２２

jiji C εσ = )6,...,1,( =ji

where

∑∑

∑=

=

=+−=n

kn

ll

lk

n

ll

lj

lki

k

kj

kik

ijk

ij

CvC

CCv

C

CCC

CvC1

1 3333

1 33

33

33

33 )( )6,3,2,1,( =ji (2.30)

0== jiij CC )5,4;6,3,2,1( == ji (2.31)

∑∑

∑

= =

=

−ΔΔ

Δ= n

k

n

l

lklk

lk

lk

n

kij

k

k

ij

CCCCvv

Cv

C

1 154455544''

1'

)( )5,4,( =ji (2.32)

where

k

k

k

C

C

54

44' =Δ

k

k

C

C

55

45

From the above stiffness formulas, the equivalent homogeneous material is shown to be

monoclinic and the plane of symmetry is the x-y plane.

The compliance terms of the equivalent homogeneous element can also be obtaied by

following the similar procedure of deriving the stiffness terms. Equations 2.24 to 2.27 and

equation 2.29 yield 12n+6 linear equations and 12n+12 variables. Hence, the strains can be

obtained in terms of stresses of the equivalent homogeneous material:

jiji S σε = )6,...,1,( =ji (2.33)

The compliance matrix can be shown in terms of the combination of constituent compliances

in two different formats. The first form uses a reference layer, while the second one does not

employ any reference layer. The first form is shown below:

２３

)(13'62'21'1 j

mij

mij

miij SSSS Δ+Δ+Δ

Δ= (2.34)

;6,2,1, =ji ⎪⎩

⎪⎨

⎧=

3'

jj

if

if

6

2,1

=

=

j

j

∑∑= =

Δ+Δ+ΔΔΔ

Δ−=

n

k ll

mil

mil

mi

k

lk

kmii SSSvSS

1

3

136221133 )(1 (2.35)

;6,2,1, =ji

∑∑= =

Δ+Δ+ΔΔΔ

=n

k lli

lk

klk

klk

k

k

k

i SSSvS1

3

1'

336

232

1313 )(1 σ (2.36)

;6,2,1, =ji ⎪⎩

⎪⎨

⎧=

3'

ii

if

if

6

2,1

=

=

i

i

∑=

Δ+Δ+ΔΔ

+=n

kk

kk

kk

k

k

kkk SSSvSvS

1

336

232

1313333 )([

∑∑∑= = =

ΔΔ+Δ+ΔΔΔ

ΔΔ−

n

p l ili

ik

kik

kik

k

p

lp

p

kSSS

v

1

3

1

3

1

336

232

131 ])(1 (2.37)

∑=

=n

k

kij

kij SvS

1 )5,4( =j (2.38)

0== jiij SS )5,4;6,3,2,1( == ji (2.39)

The superscript m in the above formula is referred to as any convenient reference layer. The

symbols shown in equations 2.33 to 2.37 are defined below:

kk Ddet=Δ (2.40)

２４

where

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

k

k

k

k

S

S

S

D

61

21

11

k

k

k

S

S

S

62

22

12

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

k

k

k

S

S

S

66

26

16

ijk

ijk Ddet=Δ (2.41)

where

=ijkD replace column i of kD by column j of mD

ik

ik Ddet=Δ (2.42)

where

=ikD replace column i of kD by column vector S

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−

−

−

=

km

km

km

SS

SS

SS

S

6363

2323

1313

Edet=Δ (2.43)

where

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΔΔ

ΔΔ

ΔΔ

=

∑

∑

∑

=

=

=

n

k k

kk

n

k k

kk

n

k k

kk

v

v

v

E

1

31

1

21

1

11

∑

∑

∑

=

=

=

ΔΔ

ΔΔ

ΔΔ

n

k k

kk

n

k k

kk

n

k k

kk

v

v

v

1

32

1

22

1

12

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

ΔΔ

ΔΔ

ΔΔ

∑

∑

∑

=

=

=

n

k k

kk

n

k k

kk

n

k k

kk

v

v

v

1

33

1

23

1

13

２５

=Δ ij the thij cofactor of matrix E (2.44)

The convenient reference layer is used while deriving equations 2.33 to 2.37. If the

mechanical behavior of the reference layer is isotropic, the calculations of the compliance matrix

will be largely simplified. The forms shown in equations 2.33 to 2.37, however, do not display

the symmetry of the compliance matrix.

The second expression of the compliance matrix is the general form of compliance terms

which does not depend on any reference layer. These two forms of the compliance matrix can be

proved to be equivalent. The second form of compliance terms is the following:

∑∑= =

ΔΔ−ΔΔΔΔΔ

=n

k

n

l

lqs

kps

lqs

kpr

lk

lk

ijvvS

1 1* )(1 (2.45)

where

)2,1,6(),1,3,2(),3,2,1(),,(),,,( =srjqpi

∑∑∑= = =

ΔΔΔΔΔ

==n

k

n

k

n

ki

kkk

kkk

iivvvSS

1 1 1'*33

1 2 3 321

321 ~1 (2.46)

where

;6,2,1=i ⎩⎨⎧

=3

'i

j ifif

6

2,1==

ii

∑=

+Δ−Δ

+=n

k

kk

k

kk SSSvS1

113133333 )[(1{

]})()( 3636322323kkkk SSSS Δ−+Δ− (2.48)

where

=Δrij

thij cofactor of matrix rD (2.49)

２６

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

r

r

r

r

S

S

S

D

61

21

11

r

r

r

S

S

S

62

22

12

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

r

r

r

S

S

S

66

26

16

∑∑∑= = =

ΔΔΔΔ

=ΔΔ=Δ

n

k

n

k

n

k kkk

kkk

m

vvv1 1 1

*

1 2 3 321

321 ~ (2.50)

Fdet~ =Δ

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Δ

Δ

Δ

=

1

1

1

31

21

11

k

k

k

F

2

2

2

32

22

12

k

k

k

Δ

Δ

Δ

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

Δ

Δ

Δ

3

3

3

33

23

13

k

k

k

'det~' Fi =Δ (2.51)

where

='F replace thi' column of matrix F by column vector 'ikjΔ

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

Δ+Δ+Δ

Δ+Δ+Δ

Δ+Δ+Δ

=Δ

''''''

''''''

''''''

'

333632323131

233622322131

133612321131

iiiiii

iiiiii

iiiiii

i

kkkkkk

kkkkkk

kkkkkk

kj

SSS

SSS

SSS

The elastic constants shown above are obtained by using Cartesian coordinate systems, but

any other orthogonal coordinates can also be applied to the layered medium. The orthogonal

coordinate system is restricted in that one direction is perpendicular to the layering, and the other

two are parallel to the layering. By using the transformation relations between coordinate

systems, after the effective elastic constants are derived for one coordinate system, the others can

also be obtained.

２７

2.5 Verification Examples

The effective elastic moduli of laminated composites will be calculated in this section to

show the accuracy of the present approach. After obtaining the effective elastic compliance

matrix by the formulas in the previous section, the effective engineering moduli can be obtained

through the following relations:

11

1S

E x = 22

1S

E y = 33

1S

E z = (2.52)

22

23

SS

yz −=ν 11

31

SS

xz −=ν 11

21

SS

xy −=ν (2.53)

44

1S

G yz = 55

1S

G xz = 66

1S

G xy = (2.54)

where

[ ]

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

16

13

12

11

00

S

S

S

S

S

26

23

22

12

00

S

S

S

S

36

33

23

13

00

S

S

S

S

0

000

45

44

S

S

0

000

55

45

S

S

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

66

36

26

16

00

S

S

S

S

The material data used in the examples shown in this section are from Sun and Li (1988).

Table 2.2 shows the material properties used in the following examples.

２８

Table 2.2 Material properties of laminas in the sublaminate

Boron/Epoxy

(Br)

Glass/Epoxy

(Gl)

Graphite/Epoxy

(Gr)

Al O3

(Al)

1E (msi) 33.30 7.80 26.00 10.20

2E (msi) 3.30 2.60 1.45 10.20

3E (msi) 3.30 2.60 1.45 10.20

23ν 0.32 0.25 0.49 0.33

13ν 0.26 0.25 0.23 0.33

12ν 0.26 0.25 0.26 0.33

23G (msi) 0.50 1.10 1.04 3.84

13G (msi) 0.80 1.30 1.04 3.84

12G (msi) 0.80 1.30 1.04 3.84

Example 1: Angle-plied Boron/Epoxy Laminates

The effective moduli of three boron/epoxy laminates with sub-laminates ]30[± , ]45,90,0[ ±

and ]45,30[ are calculated by the formulas mentioned in the previous section and compared with

the results from Sun and Li (1988) and Enie and Rizzo (1970). The comparisons of the effective

moduli are shown in table 2.3.

２９

Table 2.3 Comparison of effective moduli for boron/epoxy angle-plied laminates

Effective Moduli ]30[± ]45,90,0[ ± ]45,30[

Chou et al. (1972) 10.4 12.8 4.48

Sun and Li (1988) 10.4 12.8 4.60 xE (msi)

Enie and Rizzo

(1970) 10.34 12.8 4.48

Chou et al. (1972) 2.54 12.8 3.39

Sun and Li (1988) 2.54 12.8 3.37 yE (msi)

Enie and Rizzo

(1970) 2.54 12.8 3.39

Chou et al. (1972) 3.42 3.57 3.30

Sun and Li (1988) 3.42 3.57 3.31 zE (msi)

Enie and Rizzo

(1970) 3.33 3.57 3.31

Chou et al. (1972) 0.216 0.224 0.215

Sun and Li (1988) 0.216 0.224 0.215 yzν

Enie and Rizzo

(1970) 0.183 0.212 0.159

Chou et al. (1972) -0.125 0.224 0.179

Sun and Li (1988) -0.125 0.224 0.175 xzν

Enie and Rizzo

(1970) 0.112 0.212 0.123

３０

Table 2.3 Comparison of effective moduli for boron/epoxy angle-plied laminates (Continued)

Effective Moduli ]30[± ]45,90,0[ ± ]45,30[

Chou et al.

(1972) 1.40 0.336 0.459

Sun and Li

(1988) 1.40 0.336 0.472 xyν

Enie and Rizzo

(1970) 1.39 0.336 0.255

Chou et al.

(1972) 0.552 0.615 0.582

Sun and Li

(1988) 0.552 0.615 0.582 yzG (msi)

Enie and Rizzo

(1970) 0.555 0.615 0.582

Chou et al.

(1972) 0.696 0.615 0.653

Sun and Li

(1988) 0.696 0.615 0.653 xzG (msi)

Enie and Rizzo

(1970) 0.695 0.615 0.653

Chou et al.

(1972) 6.78 4.79 2.55

Sun and Li

(1988) 6.79 4.79 2.62 xyG (msi)

Enie and Rizzo

(1970) 6.72 4.79 2.55

３１

Example 2: Hybrid Laminates

The effective moduli of two hybrid laminates are calculated in this example. The results

obtained from the formulas in the previous section are compared with those from Sun and Li

(1988). The comparisons are shown in table 2.4. The stacking sequences of these two materials

are displayed below:

Laminate A: ]30,30,30,30[ BrGlGrAl −−

Laminate B: ]45,45,90,0[ BrGlGrAl −

Table 2.4 Comparisons of effective moduli for hybrid laminates

Effective Moduli Laminate A Laminate B

Chou et al. (1972) 7.94 5.55 xE (msi)

Sun and Li (1988) 8.47 5.61

Chou et al. (1972) 4.42 10.97 yE (msi)

Sun and Li (1988) 4.42 11.00

Chou et al. (1972) 2.99 3.04 zE (msi)

Sun and Li (1988) 3.04 3.06

Chou et al. (1972) 0.27 0.17 yzν

Sun and Li (1988) 0.27 0.15

Chou et al. (1972) 0.18 0.32 xzν

Sun and Li (1988) 0.11 0.29

Chou et al. (1972) 0.56 0.25 xyν

Sun and Li (1988) 0.59 0.25

Chou et al. (1972) 1.02 1.09 yzG (msi)

Sun and Li (1988) 1.02 1.09

Chou et al. (1972) 1.15 1.09 xzG (msi)

Sun and Li (1988) 1.15 1.09

Chou et al. (1972) 3.26 3.60 xyG (msi)

Sun and Li (1988) 3.38 3.53

３２

Chapter 3 Torsional Response of Laminated Composite Beam

3.1 Introduction

The elastic response of inhomogeneous beams is often of interest in many engineering

fields (Savoia and Tullini, 1993). In this chapter, the elastic response of an arbitrarily shaped

composite beam will be analyzed. By employing Prandtl’s stress function, the expressions for

shear stress distribution, cross-sectional warping, and torsional rigidity can be determined. The

warping behavior is the beginning of the analyses of composite beams under a variety of loading

and boundary conditions. This chapter also presents the theoretical solution of the response of a

multilayered orthotropic beam under uniform torsion loading. The solution is in the form of a

series, which is a similar form with the classical elasticity solution of an orthotropic rectangular

beam. Previous works about torsional response of composite beams include a two-layer bonded

isotropic beam (Muskhelishvili, 1963), isotropic symmetric sandwich beam (Cheng et al., 1989),

homogeneous anisotropic beam (Lekhnitskii, 1963) and orthotropic laminated beams (Tsai et al.,

1990). The last paper mentioned above proposes a Reissner-Mindlin theory approach to predict

the torsional behavior of orthotropic laminates. The result was compared with the exact solutions,

and it shows that the Reissner-Mindlin approach doesn’t approximate the torsional behavior well.

The shortcoming of this approach is that it underestimates the torsional stiffness of thick

laminates with a small number of plies and very distinct elastic properties.

Because of the high stiffness and strength to weight ratio, fiber composite materials have

become more and more important for industrial applications. For classical elasticity and

mechanics of materials problems of orthotropic laminated composites, the solutions are readily

available. The bending and axial loading behavior can be easily determined (Hyer, 1998). The

solution of torsional response of a rectangular isotropic homogeneous beam can be found in a

classical elasticity book (Timoshenko and Goodier, 1970). The solution of a rectangular

homogeneous orthotropic beam can also be determined (Lekhnitskii, 1981). The derivation of the

isotropic solution is based on the membrane analogy. This approach can be applied to the

solution of open cross sections, such as U shape, I shape, and T shape. A similar approach can be

employed in the orthotropic laminated composite beams.

３３

3.2 General Formulation

A general cross section of a prismatic bar composed of orthotropic material is considered

(Figure 3.1). The right hand orthogonal coordinate system (X, Y, Z) is used to analyze the beam.

The X-Y plane lies in the plane of Z=0, and the Z axis is along the centroidal axis. The cross

section is composed of many regions and holes, the area of the region is denoted as rA , and the

corresponding external boundary and interface boundary as )(rsΓ and )(r

fΓ respectively. The

orthotropic axis of each region is assumed to coincide with the reference axis. The area and

boundary of the n-th hole are denoted as )(nhA and )(n

hΓ respectively. Two assumptions have to

made, that the boundary )(rΓ of r-th region is a piecewise curve, and that the same interface is

not shared by more than two regions. The St. Venant torsion problem is considered here. The

problem states that the resultants of two ends of the prismatic bar are produced by the tangential

shear stress distributions, and can be considered as the problem under a twisting torque T. By the

following elasticity conditions, the governing differential equations can be derived.

From the equilibrium equation:

0)()(

=∂∂

+∂∂

yx

ryz

rxz ττ

in rA (3.1)

From the traction-free equations for the lateral boundaries:

0)()( =+ yr

yzxr

xz nn ττ on )(rsΓ (3.2)

From the traction-free equations for the boundaries of holes:

0)()( =+ yr

yzxr

xz nn ττ on )(rsΓ (3.3)

From the stress continuity condition at the layer interfaces:

0)( )()( =+Δ yr

zyxr

zx nn ττ (3.4)

where Δ stands for the jump of the values between layer interfaces.

From stress balance at the ends of the beam:

∫∫ =A

xz dA 0τ (3.5)

∫∫ =A

yz dA 0τ (3.6)

TdAyxA

xzyz =−∫∫ )( ττ (3.7)

３４

From the stress-strain and strain-displacement relations:

)()()(

)()(

zu

xu

Gr

xr

zrxz

rxz ∂

∂+

∂∂

=τ in rA (3.8)

)()()(

)()(

zu

yuG

ry

rzr

yzr

yz ∂∂

+∂∂

=τ in rA (3.9)

From the displacement compatibility conditions at the interfacial boundaries:

0=Δ=Δ=Δ zyx uuu on )(rfΓ (3.10)

where the symbol Δ means the jump of the displacements between boundaries. For multi-

connected regions, the conditions of single value displacement have to be supplied in equations

3.1 to 3.10. By imposing the conditions of setting the line integrals around the boundaries of

holes to zero, the single-value requirement can be satisfied:

∫ ∫ ∫Γ Γ Γ

===h h h

zyx dududu 0 for h=1, 2, …, H (3.11)

Figure 3.1 General Cross Section Composed of Orthotropic Media

Y

X

rA

)(rfΓ

)(rsΓ

)(nhΓ

)(nhA

hΓ

３５

3.3 Formulation for Orthotropic Composite Beams

The problem of equations 3.1 to 3.10 can be solved by either use of the warping function or

in terms of the Prandtl’s stress function. Two assumptions have to be made: the first one is that

the plane of the cross section remains plane, and the second one is that the plane has warping

deformation which is assumed to be a constant value along the axial direction of beam. From the

above assumptions, the displacements can be expressed as

θyzux −=

θxzu y = (3.12)

θ),()( yxwu rr

z =

where θ is the angle of twist per unit beam length and rw is the warping function of the r-th

layer. Introducing equation 3.11 in equations 3.7 and 3.8, the stress components can be shown in

the form

)()()( yx

wG rr

zxr

zx −∂∂

= θτ (3.13)

)()()( xy

wG rrzy

rzy +

∂∂

= θτ (3.14)

Substituting equations 3.12 and 3.13 into equation 3.1, the generalized Laplace equation for

the r-th region can be determined:

02

2)(

2

2)( =

∂∂

+∂∂

ywG

xwG rr

zyrr

zx in rA (3.15)

Equation 3.14 can be reduced to a harmonic function if the material property is transversely

isotropic, where zyzx GG = . Substituting equations 3.12 to 3.14 into equations 3.2, 3.3, 3.4, 3.10,

3.11, the traction-free equations for the lateral and the hole boundary conditions become

yr

zyxr

zxyrr

zyxrr

zx xnGynGny

wGnx

wG )()()()( −=∂∂

+∂∂ on )(r

sΓ (3.16)

yr

zyxr

zxyrr

zyxrr

zx xnGynGny

wGnx

wG )()()()( −=∂∂

+∂∂ on )(n

hΓ (3.17)

The stress continuity condition at the interfaces of layers becomes

][][ )()()()(y

rzyx

rzxy

rrzyx

rrzx xnGynGn

ywGn

xwG −Δ=

∂∂

+∂∂

Δ on )(rfΓ (3.18)

３６

The condition of displacement compatibility at the interfaces becomes

0=Δ rw on )(rfΓ (3.19)

The condition of vanishing of the line integral along the closed loop hΓ surrounding a hole

of the cross section becomes

∫ ∑ ∫Γ Γ

=∂∂

+∂∂

=h

rh

s

rr dyy

wdxx

wdw)(

0][ for h=1,…,H (3.20)

The uniform torsion problem can also be formulated by the Prandtl’s stress function.

Assume rφ is the Prandtl stress function of the r-th layer. In order to satisfy the equilibrium

equation, the relations between shear stress and the Prandtl’s stress function for r-th layer can be

defined as

yrr

zx ∂∂

=φ

θτ )( (3.21)

xrr

zy ∂∂

−=φ

θτ )( (3.22)

By equating equations 3.21 and 3.22 with equations 3.13 and 3.14, the relation between the

warping function and Prandtl stress function for layer r can be obtained:

yyGx

w rr

zx

r +∂∂

=∂∂ φ

)(1 (3.23)

xxGy

w rr

zy

r −∂∂

−=∂∂ φ

)(1 (3.24)

By employing the following equality, the governing equations for the r-th layer can be

writtern in terms of the Prandtl’s stress function:

xyw

yxw rr

∂∂∂

=∂∂

∂ 22

(3.25)

2112

2

)(2

2

)( −=∂∂

+∂∂

yGxGr

rzx

rr

zy

φφ in rA (3.26)

３７

3.4 Analytical Model for Rectangular Laminated Composites

Figure 3.1 shows the cross-section of a rectangular laminated composite beam with

dimensions 2b and 2h in the x and y directions respectively (Swanson, 1998). The layering is

parallel to the x axis, and the x-axis is also the symmetric axis of the composite. The material

characteristic of each layer is considered as orthotropic material behavior and is dominated by

two shear moduli, in the in-plane ( xzpG ) and through-thickness ( yxpG ) directions. The subscript p

is the p-th layer and the total number of layers is n. The classical lamination theory will be used

to obtain the shear moduli for both in-plane and through-thickness properties.

While obtaining the in-plane shear modulus xzpG , only the x and z coordinates will be

considered. The stress-strain relations for plane stress can be defined as (Hyer, 1998)

{ } [ ]{ }εσ Q= (3.27)

or

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

12

2

1

τσσ

=⎢⎢⎢

⎣

⎡

012

11

QQ

022

12

QQ

⎥⎥⎥

⎦

⎤

66

00

Q

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

12

2

1

γεε

(3.28)

where

2112

1111 1 νν−=

EQ

2112

121

2112

21212 11 νν

ννν

ν−

=−

=EEQ

2112

222 1 νν−=

EQ

1266 GQ =

The transformation matrix for a fiber-oriented composite can be given as

[ ] [ ][ ][ ][ ][ ]11 −−= RTRQTQ (3.29)

where

[ ]R =⎢⎢⎢

⎣

⎡

001

010

200

⎥⎥⎥

⎦

⎤ (3.30)

３８

⎢⎢⎢

⎣

⎡=][T

mnnm

−

2

2

mnmn

2

2

22

22

nmmn

mn

−

− ⎥⎥⎥

⎦

⎤ (3.31)

θcos=m

θsin=n

Matrix R is used for transforming tensor shear strain to engineering shear strain. The in-

plane shear modulus xzpG can be obtained by transformation matrix Q.

Through-thickness shear moduli 13G and 23G can also be employed to obtain the through-

thickness shear properties of laminated composites. Again, the classical lamination theory will be

used. The shear stress-strain relation can be defined as

⎩⎨⎧

13

23

ττ

⎭⎬⎫

= ⎢⎣

⎡

023G

13

0G ⎥

⎦

⎤ ⎩⎨⎧

13

23

γγ

⎭⎬⎫

(3.32)

The through-thickness shear modulus yzG can then be obtained by

θθ 223

213 sincos GGGyz += (3.33)

The material property of each layer of the laminated composite beam is considered as

orthotropic. In order to maintain this material behavior, each layer has consisted of well

dispersed θ± angle fibers.

３９

Figure 3.2 Cross-section of laminated composites

The displacements of each layer are assumed to be the same and can be expressed as

θyzux −=

θxzuy = (3.34)

θψ ),( yxu pz −=

where xu , yu , and zu are the displacements in x, y, and z directions, respectively, θ is the

angle of twist per unit length, and ),( yxpψ is the warping function of layer p. Substituting

equation 3.34 into the strain-displacement relations, the strain of layer p can be shown to be

h

h

X

Y

b b

p=1

p=n

nb

４０

)( yxp

xz −∂∂

=ψ

θγ (3.35)

)( xyp

yz +∂∂

=ψ

θγ (3.36)

Substituting equation 3.35 and equation 3.36 into the stress-strain relations, the stress of

layer p can be obtained:

)( yxG pxzpxz −

∂∂

=ψ

θτ (3.37)

)( xyG pyzpyz +

∂∂

=ψ

θτ (3.38)

The relation between the stress and the Prandtl’s stress function for layer p can be defined

as

yp

xz ∂∂

=φ

τ (3.39)

xp

yz ∂∂

−=φ

τ (3.40)

Substituting equation 3.39 and equation 3.40 into equation 3.37 and equation 3.38,

respectively, the governing equation for layer p can be found:

θφφ

2112

2

2

2

−=∂

∂+

∂

∂

xGyGp

yzp

p

xzp (3.41)

In order to solve the governing equation, the continuity condition between each layer has to

be employed. The first condition is that the displacement zu must be continuous at the interfaces

of each layer. Hence, the warping function and the warping function derivative with respect to x

are also continuous at the boundary between each layer. By substituting equation 3.39 into

equation 3.37, the first continuity condition can be obtained:

11

1

1

11

−− =

−

−=∂

∂=

∂∂

pp by

p

xzpby

p

xzp yGyGφφ

(3.42)

The second continuity condition is that the shear stress yzτ has to be continuous at the

interface of each layer. From equation 3.40, the second continuity condition can be obtained:

４１

11

1

−− =

−

=∂

∂=

∂∂

pp by

p

by

p

xxφφ

(3.43)

Since the stress function derivative with respect to x has to be continuous at the interface of

each layer, the stress function must also be continuous between the boundaries of each layer.

This condition leads us to the following equation:

111

−− =−==

pp bypbyp φφ (3.44)

The stress boundary condition has to be employed also, i.e., the stress at the outside of the

boundary will vanish. Hence, the shear stress xzτ must equal zero at the boundaries bx ±= , and

the shear stress yzτ must vanish at the edges hy ±= . From equations 3.39 and 3.40, the stress

function for each layer has a constant value at the edges of bx ±= , and the stress function for the

first and last layer must be constant at the boundaries hy ±= .

In order to solve the governing equation with the associated boundary conditions, a similar

solving procedure to that in Lekhnitskii (1981) can be used. The assumed solution is taken as a

series form

)2cos(∑=n

pnp xbnF πφ (3.45)

The constant term in the governing equation can also be expanded into a series form:

∑−=−n

n xbnS )2cos(2 πθθ (3.46)

Substituting equations 3.45 and 3.46 into the governing equation, the solution can be

obtained:

πnSn

n8)1( 2

1−−= , n=1, 3, 5, … (3.47)

θπ

αα yzp

n

pnpnpnpnpn Gn

byDyDyF 33

22

1

2132)1(sinhcosh)(

−

−++= (3.48)

where

bn

GG

yzp

xzppn 2

πα = (3.49)

The total solution can be obtained by introducing equation 3.48 into equation 3.45:

４２

∑∞

=

−

−−−=,...5,3,1

32

1

3

2)2cos()sinhcosh1(1)1(32

npnpnpnpn

n

yzpp xbnyByA

nGb παα

πθφ (3.50)

The resultant torque acting on the end of the beam due to the shear stress distribution has to

be identical with the applied torsional moment. This leads us to the following equation:

xdxdyydxdydT yzxz ττ +−= (3.51)

Substituting the relations between shear stresses and stress function into equation 3.51, and

integrating over the whole cross-section of the beam, the total torque can be obtained:

dydxxx

yy

T ppb

b

h

h⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−= ∫∫−−

φφ (3.52)

The equation mentioned above is satisfied within each individual layer. Integration in

equation 3.52 can be set equal to integration over each layer and summed up. By using the

technique of integration by parts,

∫ ∫∑− ⎥

⎥⎦

⎤

⎢⎢⎣

⎡−−=

−−

b

b

b

bp

b

bpp

dxdyyTp

p

p

p1

1)( φφ dydxx

b

bp

b

bp

b

bp

p

p

⎥⎦

⎤⎢⎣

⎡−− ∫∫∑

−−

−

φφ )(1

(3.53)

where pb and 1−pb are the locations of layers as shown in figure 3.1. The first b value is

equal to –b. From the boundary conditions, the stress function at the exterior boundary must be

constant value. In the laminated composite case, we assume it is equal to zero as employed in the

classical elasticity textbooks. Using the continuity condition that the stress function is continuous

at the interface in equation 3.53, the resulting equation is obtained as

∫∑∫∫∫−−−−

==p

p

b

bp

p

b

b

b

b

h

h

dxdydydxT1

22 φφ (3.54)

Applying equation 3.50 and condition 0=φ at the exterior boundary in equations 3.42 and

3.44, two equations can be obtained:

)coshsinh( 11 −− − ppnpnppnpnpnxzp

yzp bBbAGG

ααα

)coshsinh( 1,1,11,1,1,11

1−−−−−−−

−

− −= pnpnppnpnpnpxzp

yzp bBbAGG

ααα (3.55)

)sinhcosh1( 11 −− −− ppnpnppnpnyzp bBbAG αα

)sinhcosh1( 1,1,11,1,11 −−−−−−− −−= pnpnppnpnpyzp bBbAG αα (3.56)

４３

If the layering of the laminated composite is symmetric with respect to the center line, the

constant pnB will vanish, and the simple form of the constants nA1 and pnA can be expressed as

hAn

n1

1 cosh1α

= (3.57)

⎥⎦

⎤⎢⎣

⎡−−= −−

−−

)cosh1(1cosh1

1,111

ppnnpyzp

yzp

ppnpn bAG

GbA α

α (3.58)

Substituting equations 3.57 and 3.58 into equation 3.50, the explicit solution for the stress

function can be obtained. Then performing the integration in equation 3.54, the total torque can

be determined:

∑ ∑∞

=

=p n

yzp nGahbT

,...5,3,144

31)()2(32

πθ

⎩⎨⎧

−−−

−− )sinh(sinh22 11

ppnppnpn

pnpp bbhA

hbb

ααα

⎭⎬⎫

−− − )cosh(cosh2 1ppnppnpn

pn bbhB

ααα

(3.59)

The effective torsional rigidity can be obtained as

∑ ∑∞

=

=p n

yzp nGahbGJ

,...5,3,144

31)()2(32

π

⎩⎨⎧

−−−

−− )sinh(sinh22 11

ppnppnpn

pnpp bbhA

hbb

ααα

⎭⎬⎫

−− − )cosh(cosh2 1ppnppnpn

pn bbhB

ααα

(3.60)

By introducing equation 3.50 in equation 3.39 and 3.40, and employing the relationGJT=θ ,

the shear stresses can be determined as

yzpxz TGGJbyx 2

16),(π

τ −=

[ ]∑∞

=

−

+−

,...5,3,12

21

2coscoshsinh)1(n

pnpnpnpn

n

xbnyByA

nπαα (3.61)

４４

yzpyz TGGJbyx 2

16),(π

τ =

[ ] xbnyByA

nnpnpnpnpn

n

2sinsinhcosh1)1(,...5,3,1

2

21

παα∑∞

=

−

−−− (3.62)

Shear stresses at the mid-point of the boundary can also be obtained as

( )hBhAn

GGTGJbh pnpnpnpn

n

n

xzpyzpxz ααπ

τ coshsinh)1(16),0(,...5,3,1

2

21

2 +−

= ∑∞

=

−

(3.63)

⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑

∞

= ,...5,3,122

812)0,(n

pnyzpyz n

AG

GJbTb

πτ (3.64)

４５

Chapter 4 Finite Element Analysis

4.1 Introduction

By employing the laws of physics, physical phenomena appearing in nature can be

formulated in mathematical forms which include algebraic equations, differential equations, and

integral equations (Reddy, 2006). The physical phenomena mentioned here may include the

range of biology, geology, and mechanics. Examples can be shown in the following real-world

problems. For example, the search of pollutants in the river, seawater, or the atmosphere, the

determination of stress distribution in a pressure vessel that is subjected to thermal, mechanical,

and aerodynamic loading, the formulation of tornados and thunderstorms. Two major stages are

involved in the study of physical phenomena: the first one is the mathematical formulation, and

the second is analyzing the mathematical model gained from the first step by employing

numerical methods.

Mathematical methods and laws of physics are required to formulate a physical problem

into a mathematical form. The resulting mathematical forms usually appear as differential

equations. These kinds of equations often contain physical quantities which researchers are

interested in to solve the physical problem or apply into design work. Certain assumptions have

to be made in order to simplify the complex physical phenomena and develop suitable

mathematical models. By employing numerical methods and computers, researchers can simulate

the mathematical models and understand the characteristics of the physical process.

The derived governing differential equations are often unable to be solved by traditional

analytical methods. When encountering these situations, approximate or numerical methods

become the only route to solve the governing equations. Two methods are usually considered as

effective methods. These are the finite difference method and variational methods such as the

Galerkin method and the method of Rayleigh-Ritz. The finite difference formulations of the

differential equations are developed by replacing the differential equations with the finite

difference quotients which contain the function values at each mesh point of the domain. The

function values at each mesh point of the domain can be determined by solving the algebraic

equations with associated boundary conditions. By employing the variational method in the

governing differential equations, the equations are in terms of equivalent weighted-integral forms

and the approximate solutions of the governing equations are assumed to a be linear combination

４６

of approximation functions, with unknown coefficients. These unknown coefficients are

determined so that the integral form is equivalent with the governing differential equation.

Depending on different choices of integral form and approximation functions, the variational

method can be divided into many sub-division methods which include Rayleigh-Ritz method,

Galerkin method, and least squares method. These traditional variational methods will become

ineffective for the problems involving an arbitrary domain. The shortcoming is that the

approximation functions will be difficult to formulate in these types of problems.

The advantage of the finite element method over the traditional variational methods is that

it overcomes the disadvantage of constructing the approximation functions over the complex

domain. A systematic procedure of formulating the weighting function or approximation

function in the sub-domain, which is usually a simple domain, is employed in the finite element

analysis. There are three main reasons that make the finite element method more suitable for

complex problem than other traditional numerical methods. First, instead of formulating the

whole complex domain at once, the complex domain is formulated with an assembly of simple

geometrical sub-domains, called finite elements. Second, the approximating functions are

derived by employing functions that can be represented as a combination of algebraic

polynomials. Third, for each element, the unknown coefficients can be determined by equating

the algebraic relations with the governing equations which are in the weighted-integral form.

From the perspective mentioned above, the finite element method can also be considered as an

application of element combinations of the Rayleigh-Ritz method or weighted-residual method.

The approximating functions are usually in the form of polynomials, and the unknown

coefficients are the values at nodal points. The nodal points exist in the interior and boundary of

the elements. The approximating functions are sometimes called interpolation functions because

the derivation of the approximation functions uses the theory of interpolation. The choices of the

number of nodes and the orders of the differential equations will influence the degree of

interpolation functions.

Although the term finite element method is used in the present, the idea behind it is not

new. The ancient mathematicians used a similar concept to approximate the value of π by

inscribing a polygon in a circle. By increasing the number of the sides of the polygon inscribed

in a circle, the value of π can be approximated to the very high degree of accuracy. In the present

ages, the first application of the finite element method was found in the analysis of aircraft

４７

structures. The analysis of wings of fuselages can be in terms of the analysis of combinations of

stringers, skins, and shear panels. A similar method, the framework method, was proposed by

Hrenikoff in 1941. In this paper, a plastic plane is treated as a combination of beams and bars.

Courant (1943) proposed to approximate an unknown function as an assembly of piecewise

continuous functions over simpler sub-domains. The St. Venant torsion problem is solved by

employing the minimum total potential energy principle and representing the whole domain by a

collection of triangular elements. Although similar concepts were proposed by the two authors

mentioned above, the formal literature contributed to Argyris and Kelsey (1960) and Turner et al.

(1956). The term finite element method was first used in the paper of Clough (1960). Since then,

the development, research, and applications of the finite element method have progressed very

fast. The historical review of the advancement of the finite element method can be found in

numerous books and journal papers. The application of the finite element method has expanded

to almost every science field.

The finite element method is a method that divides the whole complex domain into many

simple sub-domains, which are called finite elements, and constructs the approximating

functions over each element. There are two advantages of dividing the domain into piece-wise

connected domains. First, the complex domain can be accurately represented and the problem

involving dissimilar materials in the whole domain can be solved. Second, the solution over each

element can be accurately represented and problems involving local effects can be studied. Three

major finite element procedures can be stated. First, in order to represent the geometry and the

solution of the problem, the whole domain is divided into piece-wise connected simple domains.

Second, the approximated solution over each element is assumed to be a linear combination of

approximating functions and solution values at nodes. Third, through the algebraic relations, the

nodal solutions are determined, and then the solution over the whole domain can be obtained by

assembling the elements.

Despite the three major procedures, there are some detail features of the finite element

method. First, more than one type of element can be used when the dividing work is needed. For

example, a complex domain can be divided into a combination of triangular and rectangular

elements. Second, if more than one type of element is employed over the whole domain, the

development of approximating functions for each type of element should be executed separately.

Third, the governing equations are usually in the form of complex differential equations. The

４８

equations may not be able to be solved for the following reasons. The exact solution cannot be

found; hence the traditional variational method should be used. The algebraic equations obtained

by a variational method cannot be solved without utilizing the appropriate continuity, boundary,

and initial conditions. Fourth, two major differences of the form of approximating functions can

be found between the use of the finite element method and traditional variational methods. For

example, the approximating function in variational methods is applied over the whole domain.

The first difference appears in the choice of unknown parameters in the linear combination of

approximating functions and undetermined coefficients. The traditional variational method uses

arbitrary parameters while the finite element method employs the solution value or the derivative

value at each nodal point. The second difference is that the approximating function in the finite

element method is often in the form of a polynomial that is derived from interpolation theory.

The approximate solution for the finite element method is usually in terms of a linear

combination of interpolation functions and nodal point values. The approximation function can

also be in terms of nodeless variables instead of nodal values.

The location and number of nodal points in one element depends on the following three

impact factors: first, the degree of the approximating polynomial; second, the form of the integral

in the governing equation; and third, the elemental geometry. The approximate solution can be

determined by representing the solution in terms of values at the nodal points. The idea of

assembling each element into the whole domain is used because the solution is assumed to be

continuous at the inter-element boundaries. Sometimes the condition of continuity of solution

uses solution derivatives instead. After assembling all finite elements, the boundary conditions

and initial conditions can then be applied to the problem. The equation for each finite element

can be solved by imposing the associated boundary conditions and initial conditions.

Errors in numerical methods are unavoidable. Knowing the source of errors makes it easier

to understand and decrease numerical errors. There are three major sources of numerical errors:

first, the error caused by approximating the domain; second, the error due to the approximation

of the problem solution; third, the error caused by numerical computation, including round-off

and numerical integration errors. The evaluation of error is not a easy task. For certain problems,

the error can be evaluated. Depending on the governing differential equations, the forms of

integrals, and the element employed, the accuracy and convergence of the solution in the finite

element method can be determined. The term accuracy means the difference between the

４９

analytical solution and the solution obtained from the finite element method. The term

convergence means that increasing the number of elements will increase the accuracy of the

finite element solution and approach the exact solution.

Two major stages are involved when solving a time-dependent problem. In the first one,

the governing differential equation is approximated by the finite element method in order to gain

ordinary differential equations in time. In the second one, the obtained ordinary differential

equation is solved exactly by an analytical method or approximately by the finite difference

method or variational methods. The approximated equations are then in the form of algebraic

equations. The obtained algebraic equations can then be solved for the values at nodal points. A

similar numerical method is called the discrete element method, which uses contact conditions

instead of continuity conditions in the inter-element boundary. In this method, finite motions,

such as displacements and rotations, are permitted. The application of this method can be found

in rock mechanics, such as tunneling and mining engineering, and in ice mechanics.

Figure 4.1 2D rectangular beam cross section

Y

X

５０

This chapter describes the details of the finite element analysis used in this study. The

analogy between torsion problems and heat transfer problems is employed to develop the finite

element models for beams under torsional loading. The finite element commercial package

ANSYS 9.0 is used for the finite element analysis. Consider the rectangular beam shown in

figure 4.1. The governing equations for the torsion problem and steady heat transfer problem for

this rectangular beam are shown in the following:

Governing equation for torsion problem:

θφφ 2112

2

2

2

−=∂∂+

∂∂

yGxG zxzy (4.1)

where

=φ Prandtl’s Stress Function

=zyG Shear Modulus in z-y plane

=zxG Shear Modulus in z-x plane

=θ Twist Angle per Unit Length

Governing equation for steady heat transfer problem:

fy

kx

k yx −=∂∂+

∂∂

2

2

2

2 φφ (4.2)

where

=φ Temperature

=xk Thermal Conductivity in x Direction

=yk Thermal Conductivity in y Direction

=f Heat Generation Rate

By comparing equations 4.1 and 4.2, the similarities between these two equations can be

observed. Hence, an analogy between the torsion problem and steady heat transfer problem can

be summarized in table 4.1. By employing this analogy, the torsion problem of a rectangular

cross-sectional beam can be transformed to the steady heat transfer problem for the same cross-

sectional beam.

５１

Table 4.1 Analogy between Torsion Problem and Heat Transfer Problem

Torsion Problem Heat Transfer Problem

φ

(Stress Function)

φ

(Temperature)

zyG1

(The Reciprocal of Shear Modulus in z-y

Plane)

xk

(Thermal Conductivity in x Direction)

zxG1

(The Reciprocal of Shear Modulus in z-x

Plane)

yk

(Thermal Conductivity in y Direction)

θ2

(Twice the Angle of Twist Per Unit Length)

f

(Heat Generation)

yzτ−

(Minus the Shear Stress in y-z Plane) x∂

∂φ

(Temperature Gradient in x Direction)

xzτ

(Shear Stress in x-z Plane) y∂

∂φ

(Temperature Gradient in y Direction)

５２

4.2 Finite Element Procedure

The ANSYS PLANE77 (2D 8-node Thermal Solid) element will be used in this study. This

element has one degree of freedom, temperature, at each node (ANSYS, 2004). Figure 4.2 shows

the geometry of the element PLANE77. This eight-node element not only can be used to model a

straight boundary but is suitable for modeling a curved boundary problem also. The steady-state

and transient thermal analyses are problems to which this 8-node thermal element can be applied.

Instead of using free meshing, a mapped meshing technique in ANSYS is used to mesh the plane

domain model into small elements. The advantage of mapped meshing is that it restricts the

element shape and the pattern of the meshing. Only rectangular or triangular elements can be

chosen since the mapped meshing technique is employed. In this study, the rectangular elements

will be used, since the analysis model is a rectangular shape domain and the selection of

rectangular elements usually results in better engineering results. The material property of the

analysis model can be specified by thermal conductivity in x and y directions for PLANE77

elements. The values of thermal conductivity in x and y direction are the reciprocal of shear

modulus in the z-y plane and z-x plane, respectively. The torsion problem states that along the

boundary of the beam, the stress function will be set to zero. By the analogy mentioned in table

4.1, the analysis model is set to be isolated. Hence, the temperature along the boundary of the

analysis model is set to be zero. The boundary conditions can be defined at the keypoints or

nodal points in ANSYS. In this study, since the model has been built and meshed, there is no

keypoint left in the modeling domain. The node points became the only choice and one can set

all the nodal points along the analysis model boundary to be zero. For the loading condition in

the heat transfer problem, the heat generation rate is specified to be twice the value of the angle

of twist per unit length. After solving the problem, the temperature at each nodal point can be

obtained. The torsional rigidity can be decided by summing the temperature values at each nodal

point.

５３

Figure 4.2 PLANE77 Geometry


Example 1: Torsional Parameters for Rectangular Cross Sections

The verification example shown in figure 4.3 is from a textbook (Boresi and Schmidt,

2003). The value of the torsional parameter from the textbook is compared with the value

obtained from the ANSYS finite element analysis. The analytical solution of this rectangular

cross section beam subjected to uniform torsional loading is shown below:

The torsional constant is expressed as

]2

tanh11921[3

)2()2(,...5,3,1

55

3

∑∞

=

−=n h

bnnb

hbhJ ππ

(4.3)

The other form of the torsional constant can be written as

13 )2()2( kbhJ = (4.4)

where

x x

x x

x

xx x K

N

J

L O

P

I

MX

Y

５４

]2

tanh11921[31

,...5,3,1551 ∑

∞

=

−=n h

bnnb

hk ππ

(4.5)

1k = torsional parameter for rectangular cross section

Figure 4.3 Rectangular Cross Section

Table 4.2 Comparison of 1k value between analytical and ANSYS results

b/h 1.0 1.5 2.0 2.5 3.0 4.0 6.0 10

1k

(Boresi

and

Schmidt)

0.141 0.196 0.229 0.249 0.263 0.281 0.299 0.312

1k

(ANSYS) 0.1406 0.1957 0.2287 0.2493 0.2633 0.2808 0.2983 0.3123

hh

b

b

x

y

５５

The expression of shear stress in two orthogonal directions can be shown as

∑∞

=

−−−=

,...5,3,1 2

2/)1(

2

2cosh

2sinh

2cos)1(

16n

n

zx

hbnn

yh

nxh

nhG

π

ππ

πθτ (4.6)

∑∞

=

−−−=

,...5,3,1 2

2/)1(

2

2cosh

2cosh

2sin)1(

162n

n

zy

hbnn

yh

nxh

nhGxG

π

ππ

πθθτ (4.7)

In order to determine the value of maximum shear stress in the torsional rectangular

member shown in figure 4.3, the case where b is greater than h is considered here. From the

membrane analogy, the maximum slope of the stress function occurs in the middle point of the

longer side. This means the maximum shear stress will appear at the same position. By

substituting the conditions hx = and 0=y into equations 4.6 and 4.7, the value of maximum

shear stress can be expressed as

0)0,( === yhxzxτ (4.8)

...]

23cosh9

1

2cos

1[162)0,( 2max ++−====

hb

hb

GhGhyhxzy πππθθττ (4.9)

In terms of the k value, the maximum shear stress can be expressed as

kGh ]2[max θτ = (4.10)

where

∑=

−=,...5,3,1 22

2cosh

181n

hbnn

kππ

The relation of applied torque and 1k value can be expressed as

)2()2( 31 bhGkT θ= (4.11)

By comparing equations 4.11 and 4.10, the relation of maximum shear stress and the

applied torque can be expressed with a new torsional parameter 2k :

)2()2( 22

max bhkT=τ (4.12)

where

５６

kk

k 12 =

In order to compare the value of the torsional parameter 2k , re-write equation 4.12 in a

more convenient form:

2max

2 )2)(2( hbTk

τ= (4.13)


b/h 1.0 1.5 2.0 2.5 3.0 4.0 6.0 10

2k

(Boresi

and

Schmidt)

0.208 0.231 0.246 0.256 0.267 0.282 0.299 0.312

2k

(ANSYS) 0.2081 0.2309 0.2459 0.2576 0.2672 0.2816 0.2983 0.3117

Example 2: Torsional Rigidity of Rectangular Cross Section Composed of Two Different

Materials

Torsional rigidity of a rectangular beam cross section consisting of two different materials

shown in figure 4.4 will be calculated by ANSYS and compared with the analytical solution

obtained from an analytical method (Muskhelishvili, 1963).

Considering a rectangular cross section composed of two parts, each part is made of

different material and with a common boundary. The side of the first part has 1a as length and

b2 as width, and the second part has 2a as length and b2 as width. The shear modulus of each

part is 1G and 2G , respectively. The coordinate is chosen such that the x axis bisects the width of

５７

the rectangular cross section and the y axis is along the boundary interface between these two

different materials. Assuming the warping functions of each material are 1ϕ and 2ϕ , respectively,

the harmonic function for each part can be defined as

xy+= 11 ϕφ (4.14)

xy+= 22 ϕφ (4.15)

Figure 4.4 Rectangular Cross Section Composed of Two Different Materials

From the above definition of harmonic functions, the associated boundary conditions can

be found as follows:

yx

21 =∂∂φ

at 1ax −= and byb ≤≤− (4.16)

yx

22 =∂∂φ

at 2ax = and byb ≤≤− (4.17)

X

Y

b

b

1a 2a

1A 2A

５８

)(2 212

21

1 GGyx

Gx

G −=∂∂

−∂∂ φφ

at 0=x and byb ≤≤− (4.18)

21 φφ = at 0=x and byb ≤≤− (4.19)

01 =∂∂

yφ at by ±= and 01 ≤≤− xa (4.20)

02 =∂∂

yφ at by ±= and 20 ax ≤≤ (4.21)

Satisfying all the boundary conditions mentioned above, the harmonic function of each part

can be determined as

myBA

nb

n

n

sin)12(

)1()2()2(1

1

03

321 ∑

∞

= +−

=π

φ (4.22)

myBA

nb

n

n

sin)12(

)1()2()2(2

2

03

322 ∑

∞

= +−

=π

φ (4.23)

where

)(cosh]cosh)([ 122121 axmmaGGGA +−+=

mxmaGmxmaG coshcoshsinhsinh 2122 −+

2121211 sinhcoshsinhcosh mamaGmamaGB +=

)(cosh]cosh)([ 212112 axmmaGGGA −−+−=

mxmaGmxmaG coshcoshsinhsinh 1211 ++

2121212 sinhcoshsinhcosh mamaGmamaGB +=

πb

nm2

12 +=

The warping functions of these two different parts can be determined from equations 4.14

and 4.15 as

xy−= 11 φϕ in 1A (4.24)

xy−= 22 φϕ in 2A (4.25)

Since the forms of both harmonic functions are known, the explicit forms of the warping

function for these two different parts can be shown as

５９

xymyBA

nb

n

n

−+

−= ∑

∞

=

sin)12(

)1()2()2(1

1

03

321 π

ϕ (4.26)

xymyBA

nb

n

n

−+

−= ∑

∞

=

sin)12(

)1()2()2(2

2

03

322 π

ϕ (4.27)

The torsional rigidity of bars consisting of two different materials can be defined as

∫∫ ∂∂

−∂∂

++=1

)( 11221

A

dxdyx

yy

xyxGGJϕϕ

∫∫ ∂∂

−∂∂

+++2

)( 22222

A

dxdyx

yy

xyxG ϕϕ (4.28)

The explicit form of torsional rigidity of rectangular beam composed of two different

materials can then be determined:

∑ ∑∞

=

∞

=

−++=0 0 2

221

1

14532211 ][)4()(

38

n n DCGG

DCbbaGaGGJ

π (4.29)

where

212

22

112

222

11 coshcosh)(coshcosh mamaGGmaGmaGC +−+=

]sinhcoshsinhcosh[)12( 2121215

1 mamaGmamaGnD ++=

1)(coshcoshcosh 21212 −−−+= aammamaC

]sinhcoshsinhcosh[)12( 2121215

2 mamaGmamaGnD ++=

６０

Case 1: Torsional rigidity with the change of the ratio of shear modulus 21 / GG

Parameters used in this case are as follows:

ma 41 = , ma 52 = , mb 5.1= , 22 /1 mNG =

The unit of torsional rigidity obtained here is 2mN −

Table 4.4 Comparison of torsional rigidity between analytical and ANSYS solutions

with the change of ratio of shear modulus 21 / GG

21 / GG 1.0 1.5 2.0 2.5

GJ

(Muskhelishvili) 63.9861 76.9984 88.9217 100.2226

GJ

(ANSYS) 63.9838 76.9955 88.9183 100.2186


with the change of ratio of shear modulus 21 / GG (Continued)

21 / GG 3.0 3.5 4.0 4.5

GJ

(Muskhelishvili) 111.1345 121.7871 132.2581 142.5970

GJ

(ANSYS) 111.1300 121.7820 132.2525 142.5909

６１

Case 2: Torsional rigidity with the change of the ratio of ba /1


ma 12 = , mb 1= , 21 /12 mNG = , 2

2 /24 mNG =



with the change of ratio of ba /1

ba /1 1.0 1.5 2.0 2.5

GJ

(Muskhelishvili) 37.8176 54.0636 70.2765 86.3944

GJ

(ANSYS) 37.8112 54.0562 70.2681 86.3850


with the change of ratio of ba /1 (Continued)

ba /1 3.0 3.5 4.0 4.5

GJ

(Muskhelishvili) 102.4525 118.4798 134.4925 150.4983

GJ

(ANSYS) 102.4421 118.4684 134.4801 150.4849

６２



ma 11 = , mb 1= , 21 /12 mNG = , 2

2 /24 mNG =




ba /2 1.0 1.5 2.0 2.5

GJ

(Muskhelishvili) 37.8176 62.5551 90.9069 121.1658

GJ

(ANSYS) 37.8112 62.5467 90.8965 121.153



ba /2 3.0 3.5 4.0 4.5

GJ

(Muskhelishvili) 152.3550 183.9818 215.8108 247.7327

GJ

(ANSYS) 152.3406 183.9654 215.7924 247.7123

６３

Example 3: Torsional coefficients for orthogonal rectangular rod

Torsional coefficients of a rectangular beam cross section with two orthogonal shear

properties shown in figure 4.5 will be calculated by ANSYS and compared with the analytical

solution obtained from an analytical method (Lekhnitskii, 1963).

Consider a rectangular cross section beam with orthogonal material elasticity. One end of

the rectangular rod is fixed while the other end is subjected to a torsional moment T. The side of

the beam has lengths a and b, respectively. The solution of this problem can be solved by St.

Venant theory. The referring coordinates will change from the x-y system to the x’-y’ system.

From the theory, the shear stresses can be expressed as

yzx ′∂∂=′ ϕτ (4.30)

xzy ′∂∂−=′ ϕτ (4.31)

where

=ϕ Prandtl’s Stress Function

The governing equation for Prandtl’s stress function is

θϕϕ 2112

2

2

2

−=′∂

∂+′∂

∂yGxG zxzy

(4.32)

６４

Figure 4.5 Cross Section of Orthogonal Beam

In order to satisfy the boundary conditions on two sides ( 0=′x and ax =′ ) of the

rectangular beam, the solution of the stress function is sought in the form

∑∞

=

′′=,...5,3,1

sin)(k

k xa

kyY πϕ (4.33)

Expanding the right hand side of equation 4.32 in a Fourier sine series in the interval between 0

and a, the governing equation can be rewritten as

∑∞

=

′−=′∂

∂+′∂

∂

,...5,3,12

2

2

2

sin1811kzxzy

xa

kkyGxG

πθπ

ϕϕ (4.34)

By substituting the expression for the stress function into equation 4.34, the function kY can then

be determined:

X

Y

a

b X ′

Y ′

６５

kY

ak

GY

G kzy

kzx

18)(11 2 θπ

π −=−′′ (4.35)

The equation above is a second order ordinary differential equation in the function kY , hence the

function can be solved as

23)(

8coshsinh ak

ya

kBya

kAY kkk πθπμπμ +′+′= (4.36)

where

zy

zx

GG

=μ

The arbitrary constants kA and kB will be determined from the boundary conditions on the other

two sides ( 2/by =′ 2/by −=′ ) of the rectangular rod. The final expression for the stress

function can be shown as

xa

kBA

kaG

k

k

zyπ

πϕ cos)1()1(8

,...5,3,13

2/)1(2

3 −−

= ∑∞

=

−

(4.37)

where

ya

kA πμcosh=

abkB

2cosh πμ=

The torsional rigidity of the orthogonal rectangular rod can be obtained by the following

definition:

∫ ∫− −

=2/

2/

2/

2/

2a

a

b

b

GJ ϕ dxdy (4.38)

β3abGGJ zx= (4.39)

The applied torque can be shown as

θβzxGabT 3= (4.40)

where

∑∞

=

−=,...5,3,1

42

4 )2

tanh21(132k c

kckk

c πππ

β (4.41)

６６

zx

zy

GG

bac = (4.42)

The shear stresses can be determined by employing the stress function in equations 4.30 and 4.31:

xa

kBA

kc

abT

k

k

zxπ

πτ cos)1(8

,...5,3,12

2/)1(

2 ∑∞

=

−−−= (4.43)

xa

kDC

kc

GG

abT

k

k

zx

zyzy

πβπ

τ sin)1()1(8,...5,3,1

2

2/)1(

22 −−

= ∑∞

=

−

(4.44)

where

ya

kA πμsinh=

abkB

2cosh πμ=

ya

kC πμcosh=

abkD

2cosh πμ=

Depending on the ratio of shear moduli and the ratio of the sides of the rectangular cross section

beam, the maximum shear stress may occur at the middle point of either side, i.e.

0=x and2by ±= , or

2ax ±= and 0=y . The maximum shear stress can be expressed as

32max kabT=τ at 0=x and

2by ±= (4.45)

or

42max kGG

abT

zx

zy=τ at 2ax ±= and 0=y (4.46)

where

∑∞

=

−

=,...5,3,1

2

2/)1(

23 2tanh)1(8

k

k

ck

kck π

βπ (4.47)

]

2cosh

181[,...5,3,1 224 ∑

∞

=

−=k

ckk

ckππβ

(4.48)

６７

The analytical solution for torsional coefficients (β , 1k , 2k ) with the change of parameter c

obtained by equations 4.41, 4.47 and 4.48 will be compared with ANSYS results to show the

accuracy and effectiveness of the finite element method.

Table 4.7 Comparison of β value between analytical and ANSYS results

c 1 1.25 1.5 1.75 2 2.5

β

(Lekhnitskii) 0.141 0.172 0.196 0.214 0.229 0.249

β

(ANSYS) 0.1406 0.1717 0.1957 0.2142 0.2286 0.2493

Table 4.7 Comparison of β value between analytical and ANSYS results (continued)

c 3 4 5 10 20

β

(Lekhnitskii) 0.263 0.281 0.291 0.312 0.323

β

(ANSYS) 0.2633 0.2807 0.2912 0.3122 0.3226


c 1 1.25 1.5 1.75 2 2.5

3k

(Lekhnitskii) 4.804 4.521 4.330 4.186 4.068 3.882

3k

(ANSYS) 4.8044 4.5213 4.330 4.1861 4.0678 3.8829

６８

Table 4.8 Comparison of 3k value between analytical and ANSYS results (continued)

c 3 4 5 10 20

3k

(Lekhnitskii) 3.742 3.550 3.430 3.202 3.098

3k

(ANSYS) 3.7423 3.5506 3.4301 3.2023 3.0982


c 1 1.25 1.5 1.75 2 2.5

4k

(Lekhnitskii) 4.804 4.141 3.767 3.434 3.234 2.975

4k

(ANSYS) 4.8044 4.1413 3.7671 3.4345 3.2342 2.9746

Table 4.9 Comparison of 4k value between analytical and ANSYS results (continued)

c 3 4 5 10 20

4k

(Lekhnitskii) 2.538 2.644 2.548 2.379 2.274

4k

(ANSYS) 2.5374 2.6437 2.5474 2.3784 2.2741

６９

Chapter 5 Finite Difference Method

5.1 Introduction

The basic idea of the finite difference method is to represent the governing differential

equations and the associated boundary conditions with finite difference equations (Boresi et al.,

2003). The finite difference quotients are employed to approximate the derivatives in the

differential equations. Combinations of the values of unknown functions at the specified

locations of the independent variable form the finite difference quotients. The finite difference

equations with the specified values of the independent variables then lead to a system of

simultaneous algebraic equations that can be solved by computers.

To show how the difference quotients represent approximately the derivatives of the

differential equations, consider one simple case, one independent variable. Let y(x) be a function

of an independent variable x, where x is defined at ],[ 1 nbb . By dividing the domain into n-1 parts

(either equally or unequally), we get sub-dividing points nbbb ,...,, 21 called pivotal or nodal

points. By employing the interpolation polynomial theory, the n-th degree polynomial is assumed

to satisfy the conditions )()( iin bfbP = , i=1, 2, 3, …, n. The function y(x) can also be expressed

in terms of the sum of interpolation polynomial and remainder, )()()( xRxPxy nn += . Obviously

the derivative kn

kkn

kkk dxxRddxxPddxxyd /)(/)(/)( += , hence the derivative of the polynomial

is used to represent the derivative of the interpolation function y(x).

If the error in the derivative approximations can be observed, the accuracy of the

approximations is then easily found. By removing the error terms, the value of the derivative can

be expressed as kn

kkk dxxPddxxyd /)(/)( ≈ , where )(xPn is a linear combination of the

approximate values of )( iby , which can be denoted as )( ibY . Through the same procedure, all the

derivatives of the differential equations and boundary conditions can be replaced by the

corresponding finite difference equations. The formulation of the finite difference equations at

the specified nodal points will lead to a system of simultaneous algebraic equations in )( ibY . For

a linear boundary-value problem, the algebraic equations are a set of linear simultaneous

equations. For the nonlinear problem, a system of nonlinear equations will be obtained.

７０

5.2 Finite Difference Equations

The basic expression of finite difference equation follows the rule of fundamental calculus

(Ugural and Fenster, 2003). The definition of the first derivative of function g(x) with respect to

the independent variable x can be expressed as (Figure 5.1)

xgg

xxgxxg

dxdg nn

x

nn

xn Δ−

=Δ

−Δ+= +

→Δ→Δ

1

00lim)()(lim)( (5.1)

where

n = any point on the curve

Instead of using infinitely small increments in the limit calculations, the finite difference

formulation assumes that the increment is a finite amount, denoted xhx =Δ . The expression of

equation 5.1 can then be expressed in an alternative form in the finite difference analysis:

x

n

x

nnn h

gh

ggdxdg Δ

=−

≈ +1)( (5.2)

By mathematical manipulations, the first forward difference at point nx can be defined as

nxnnn dxdghggg )(1 ≈−=Δ + (5.3)

Similarly, the first backward difference at point nx can be expressed as

nxnnn dxdghggg )(1 ≈−=∇ − (5.4)

The forward and backward difference analysis is usually employed for the nodal points on

the boundary when the central difference can not be used. The central difference method needs

three pivot points to formulate, which makes it more accurate than the forward and backward

difference formulations. By using pivot points forward and backward at the point nx , the first

derivative of function g(x) can be approximated by

)(21

2)()()( 11 −+ −=

−−+≈ nn

xx

xnxnn gghh

hxghxgdxdg (5.5)

By mathematical manipulations of equation 5.5, the first central difference can be

expressed as

nxnn dxdghggg )()(2

111 ≈−= −+δ (5.6)

７１

Figure 5.1 Finite difference approximation of function g(x)

Similarly, by following the same procedure as the first difference formulation, the higher

order difference formulation can be derived. By employing equation 5.2 and equation 5.3, the

second-order derivative of function g(x) at pivot point nx can be expressed as

)(1)(1)()( 1222

2

nnx

nx

nn ggh

ghdx

dgdxd

dxgd −Δ=ΔΔ≈= + (5.7)

)(1)(1)(11212212 nn

xnn

xnn

x

ggh

ggh

ggh

−−−=Δ−Δ= ++++ (5.8)

)2(1122 nnn

x

gggh

+−= ++ (5.9)

By equating equation 5.7 and equation 5.9, the second forward difference at nodal point nx

can be derived as

nxnnnn dxgdhgggg )(2 2

22

122 ≈+−=Δ ++ (5.10)

1−nx

x

g(x)

xΔ xΔ

nx 1+nx

1−ng ng

1+ng

７２

By employing equation 5.2 and equation 5.4, the second-order derivative of function g(x)

at the pivot point nx can be expressed as

)(1)(1)()( 1222

2

−−∇=∇∇≈= nnx

nx

nn ggh

ghdx

dgdxd

dxgd (5.11)

)(1)(1)(12121212 −−−− −−−=∇−∇= nn

xnn

xnn

x

ggh

ggh

ggh

(5.12)

)2(1212 −− +−= nnn

x

gggh

(5.13)

By equating equation 5.11 and equation 5.13, the second backward difference at nodal

point nx can be found as

nxnnnn dxgdhgggg )(2 2

22

212 ≈+−=∇ −− (5.14)

The coefficients of the nodal function values in the kth forward and backward differences

are the same as the coefficients of the binomial expansion kba )( − . Following the rules of

binomial expansion, the higher-order forward and backward differences can be easily obtained.

By employing the rule of equation 5.6, the second central difference at nodal point nx can

be obtained

)(21)(2

1)( 11112

−+−+ −=−== nnnnnn gggggg δδδδδδ

)](21)(2

1[21

22 −+ −−−= nnnn gggg

)2(41

22 −+ +−= nnn ggg (5.15)

The central difference value in equation 5.15 depends on the pivot function values at

second forward and backward nodal points. In order to improve the accuracy of the

approximations, the second central difference has to be evaluated by the first forward and

backward nodal point function values. By the following mathematical manipulations, the

transformation mentioned above can be done.

The following equality will be employed:

)(2nn gg ∇Δ=δ or )(2

nn gg Δ∇=δ (5.16)

７３

The second central difference at pivot point nx can then be obtained by employing the

equality above.

112 )()( −− Δ−Δ=−Δ=∇Δ= nnnnnn ggggggδ

)()( 11 −+ −−−= nnnn gggg

11 2 −+ +−= nnn ggg

nx dxgdh )( 2

22≈ (5.17)

By following the same procedure, the third central difference can be displayed as

111123 2)2()( −+−+ +−=+−== nnnnnnnn gggggggg δδδδδδδ

)(21)()(

21

2112 −−++ −+−−−= nnnnnn gggggg

)22(21

2112 −−++ −+−= nnnn gggg

nx dxgdh )( 3

33≈ (5.18)

Similarly, the fourth central difference at the pivot point nx can be expressed as

122

12

112224 2)2()( −+−+ +−=+−== nnnnnnnn gggggggg δδδδδδδ

)2()2(2)2( 211112 −−−+++ +−++−−+−= nnnnnnnnn ggggggggg

2112 464 −−++ +−+−= nnnnn ggggg

nx dxgdh )( 4

44≈ (5.19)

It is easy to verify that the coefficients of the pivotal function values for even-order (kth-

order) central differences are exactly the same as the coefficients of the binomial

expansion kba )( − .

The extension of the finite difference method to two independent variables can be readily

derived. The partial differential equations are put in terms of finite difference equations by

employing the same technique used in the one independent variable case. In order to illustrate the

finite difference formulation for two independent variables, figure 5.2 will be used. By taking

xhx =Δ and yhy =Δ , a rectangular mesh is formed by the horizontal and vertical lines. The nodal

or pivotal points are the intersections of mesh lines.

７４

Figure 5.2 Rectangular Mesh

The finite difference formulation of a continuous function ),( yxv will be discussed here.

Only the central difference analysis will be determined in this section. Two new symbols

( xδ and yδ ) will be defined: xδ means the first central difference with the pivotal points in the x-

axis direction and yδ means the first central difference considering the nodal points along the y-

axis direction only. Following the same procedure developed for the one independent variable

case, the first and second derivatives can be approximated by central difference equations:

vhxv

xxδ1≈

∂∂ (5.20)

vhyv

yyδ1≈

∂∂ (5.21)

vhx

vx

x

222

2 1 δ≈∂∂ (5.22)

ab c

d

e

f g

h

i

j k

l m

X

Y

７５

vhy

vy

y

222

2 1 δ≈∂∂ (5.23)

)(12

yv

hyxv

xx ∂

∂≈∂∂

∂ δ (5.24)

By associating with figure 5.2, equations 5.20 to 5.24 for the nodal point a can be re-

written as

)(21)],(),([2

1bc

xxx

xvvhyhxvyhxvhx

v −=−−+≈∂∂ (5.25)

)(21)],(),([2

1ed

yyy

yvvhhyxvhyxvhy

v −=−−+≈∂∂ (5.26)

)],(),(2),([122

2yhxvyxvyhxv

hxv

xxx

−+−+≈∂∂

)2(12 bac

x

vvvh

+−= (5.27)

)],(),(2),([122

2

yyy

hyxvyxvhyxvhy

v −+−+≈∂∂

)2(12 ead

y

vvvh

+−= (5.28)

)](21[12

edy

xx

vvhhyxv −≈∂∂

∂ δ

)(21

exdxyx

vvhh δδ −=

)](21)(2

1[21

lmjkyx

vvvvhh −−−=

)(41

lmfkyx

vvvvhh +−−= (5.29)

７６

Figure 5.3 Finite Difference Grids

Domain of the problem

yh

xh

Boundary of the problem

Boundary Node

Inner Domain Node

７７


Three verification examples will be presented in this chapter. Instead of using the finite

element commercial package to produce the results, a numerical program employing the finite

difference method will be used. In order to provide the accuracy and efficiency of the finite

difference method, the results obtained from the numerical program will be compared with those

from the analytical method which is available from previous work.

Example 1: Torsional Parameters for Rectangular Cross Sections

Table 5.1 Comparison of 1k value between analytical and FDM results

b/h 1.0 1.5 2.0 2.5 3.0 4.0 6.0 10

1k

(Boresi

and

Schmidt)

0.141 0.196 0.229 0.249 0.263 0.281 0.299 0.312

1k

(FDM) 0.1404 0.1963 0.2289 0.2483 0.2627 0.2811 0.2983 0.3121


b/h 1.0 1.5 2.0 2.5 3.0 4.0 6.0 10

2k

(Boresi

and

Schmit)

0.208 0.231 0.246 0.256 0.267 0.282 0.299 0.312

2k

(FDM) 0.2076 0.2312 0.2464 0.2567 0.2667 0.2826 0.2993 0.3116

７８

Example 2: Torsional rigidity of rectangular cross section composed of two different materials

Case 1: Torsional rigidity with the change of the ratio of shear modulus 21 / GG


ma 41 = , ma 52 = , mb 5.1= , 22 /1 mNG =


Table 5.3 Comparison of torsional rigidity between analytical and FDM solutions

with the change of ratio of shear modulus 21 / GG

21 / GG 1.0 1.5 2.0 2.5

GJ

(Muskhelishvili) 63.9861 76.9984 88.9217 100.2226

GJ

(FDM) 64.3677 75.8554 87.5213 101.5136


with the change of ratio of shear modulus 21 / GG (Continued)

21 / GG 3.0 3.5 4.0 4.5

GJ

(Muskhelishvili) 111.1345 121.7871 132.2581 142.5970

GJ

(FDM) 112.3252 120.2129 134.2354 143.3424

７９



ma 12 = , mb 1= , 21 /12 mNG = , 2

2 /24 mNG =




ba /1 1.0 1.5 2.0 2.5

GJ

(Muskhelishvili) 37.8176 54.0636 70.2765 86.3944

GJ

(FDM) 37.1325 55.2135 70.5162 87.5621



ba /1 3.0 3.5 4.0 4.5

GJ

(Muskhelishvili) 102.4525 118.4798 134.4925 150.4983

GJ

(FDM) 103.1532 119.1526 135.5264 152.1151

８０



ma 11 = , mb 1= , 21 /12 mNG = , 2

2 /24 mNG =




ba /2 1.0 1.5 2.0 2.5

GJ

(Muskhelishvili) 37.8176 62.5551 90.9069 121.1658

GJ

(FDM) 37.4685 63.1535 92.0135 122.1652



ba /2 3.0 3.5 4.0 4.5

GJ

(Muskhelishvili) 152.3550 183.9818 215.8108 247.7327

GJ

(FDM) 153.7648 184.4685 217.0561 248.2545

８１

Example 3: Torsional coefficients for orthogonal rectangular rod

Table 5.6 Comparison of β value between analytical and FDM results

c 1 1.25 1.5 1.75 2 2.5

β

(Lekhnitskii) 0.141 0.172 0.196 0.214 0.229 0.249

β

(FDM) 0.1403 0.1715 0.1954 0.2138 0.2285 0.2492

Table 5.6 Comparison of β value between analytical and FDM results (Continued)

c 3 4 5 10 20

β

(Lekhnitskii) 0.263 0.281 0.291 0.312 0.323

β

(FDM) 0.2636 0.2806 0.2917 0.3124 0.3225

８２


c 1 1.25 1.5 1.75 2 2.5

3k

(Lekhnitskii) 4.804 4.521 4.330 4.186 4.068 3.882

3k

(FDM) 4.8042 4.5214 4.3304 4.1856 4.0674 3.8826

Table 5.7 Comparison of 3k value between analytical and FDM results (Continued)

c 3 4 5 10 20

3k

(Lekhnitskii) 3.742 3.550 3.430 3.202 3.098

3k

(FDM) 3.7425 3.5508 3.4303 3.2012 3.0984

８３


c 1 1.25 1.5 1.75 2 2.5

4k

(Lekhnitskii) 4.804 4.141 3.767 3.434 3.234 2.975

4k

(FDM) 4.8041 4.1408 3.7677 3.4335 3.2347 2.9743

Table 5.8 Comparison of 4k value between analytical and FDM results (continued)

c 3 4 5 10 20

4k

(Lekhnitskii) 2.538 2.644 2.548 2.379 2.274

4k

(FDM) 2.5378 2.6435 2.5483 2.3795 2.2742

８４

Chapter 6 Results and Comparisons

6.1 Material Properties

The material properties used in this chapter are shown in tables 6.1 and 6.2 (Hyer, 1998) and

the equivalent shear moduli of lamina with different ply angles are shown in tables 6.3 and 6.4.

Figure 6.1 shows the relations between the 1-2-3 coordinates and x-y-z coordinate system.

Several laminated composite models will be presented here and the results obtained from the

proposed analytical method will be compared with numerical results.

Table 6.1 Material properties of graphite-polymer composite

1E )(GPa 155.0

2E )(GPa 12.10

3E )(GPa 12.10

23ν 0.458

13ν 0.248

12ν 0.248

23G )(GPa 3.20

13G )(GPa 4.40

12G )(GPa 4.40

８５

Table 6.2 Material properties of glass-polymer composite

1E )(GPa 50.0

2E )(GPa 15.20

3E )(GPa 15.20

23ν 0.428

13ν 0.254

12ν 0.254

23G )(GPa 3.28

13G )(GPa 4.70

12G )(GPa 4.70

Table 6.3 Shear moduli for different ply-angles (graphite-polymer composite)

θ )(deg ree 0 15 30 45 60 75 90

zxG )(GPa 4.4000 5.1672 7.9340 10.8347 7.9340 5.1672 4.4000

zyG )(GPa 4.4000 4.2922 4.0229 3.7053 3.4341 3.2595 3.2000

Table 6.4 Shear moduli for different ply-angles (glass-polymer composite)

θ )(deg ree 0 15 30 45 60 75 90

zxG )(GPa 4.7000 5.4478 7.9902 10.4222 7.9902 5.4478 4.7000

zyG )(GPa 4.7000 4.5675 4.2410 3.8637 3.5480 3.3478 3.2800

８６

Figure 6.1 Transformation relations between 1-2-3 and x-y-z coordinate systems

Z

1

X

2Y3

θ

８７

6.2 Analysis Models

The analysis model is shown in figure 6.2. Several models are analyzed to predict the

torsional rigidity of the laminated composite beam. The parameters (W=50mm, t=1.3208mm)

employed in this chapter are from an example of a previous paper (Swanson, 1998). The results

of the analysis model are shown in tables 6.5 and 6.6 for graphite-polymer and glass-polymer

composites, respectively. The results obtained by the present analytical method are compared

with those from ANSYS, finite difference method, and the formula in a previous paper (Swanson,

1998). The comparisons of results obtained by present analytical method and ANSYS are shown

in figures 6.7 and 6.8 for graphite-polymer and glass-polymer composites.

Figure 6.2 The analysis model

X

Y

W

t

８８

6.2.1 Convergence Study

In order to determine the element size for finite element analysis and the nodal point

distances along x and y directions for finite difference method, the convergence study is

performed for both numerical methods. The study case used in this section is s]0/15[ for

graphite-polymer composite laminate. The test element sizes for finite element analysis are

0.8mm, 0.4mm, 0.2mm, 0.1mm, and 0.05mm. The associated torsional rigidity can then be

obtained through the finite element method. The test distances between each nodal point in x

direction for finite difference analysis are 5mm, 2.5mm, 1.25mm. The test distances between

each nodal point in y direction for finite difference method are 0.13208mm, 0.013208mm, and

0.0052832mm. The associated torsional rigidity can also be obtained for each specific mesh. The

results of convergence study for both numerical methods are shown in Table 6.5 and 6.6 and

Figure 6.3 and 6.4. From table 6.5, the element size 0.1mm will be used in the following models

for finite element analysis. From table 6.6, the nodal distances mmhx 5.2= and

mmhy 0052832.0= will be utilized in the following models in finite difference analysis.

Table 6.5 Convergence study for finite element method

Element size (mm) )( 2mNGJ −

0.8 6.5511

0.5 9.8902

0.25 11.0490

0.2 11.1376

0.1 11.4561

0.05 11.5726

８９

Table 6.6 Convergence study for finite difference method

)(mmhx )(mmhy )( 2mNGJ −

5 0.13208 8.3650

5 0.013208 10.6456

2.5 0.013208 11.0181

1.25 0.013208 11.1328

2.5 0.0052832 11.2316

Figure 6.3 Convergence study for finite element method

９０

Figure 6.4 Convergence study for finite difference method

９１

Figure 6.5 Contour plot for Prandtl’s stress function

(Graphite-polymer laminates s]0/15[ )

９２

Figure 6.6 Mesh plot for analysis model (element size = 0.8 mm)

９３


９４


９５


９６

6.2.2 Results and Comparisons

Table 6.5 Comparisons of torsional rigidity of graphite-polymer composite beam

Present ANSYS Finite Difference

Method

Swanson

(1998)

s]0/15[ 15.7015 11.5708 11.2316 11.3498

s]15/30[ 18.3443 16.9941 16.5335 16.8983

s]30/45[ 22.0441 22.9560 22.3845 22.9342

s]60/45[ 21.9371 22.7832 22.2282 22.8769

s]75/60[ 18.1272 16.7771 16.3440 16.7689

s]90/75[ 15.4700 11.4223 11.1083 11.3267

Table 6.6 Comparisons of torsional rigidity of glass-polymer composite beam

Present ANSYS Finite Difference

Method

Swanson

(1998)

s]0/15[ 12.6238 12.2214 11.8625 12.1221

s]15/30[ 15.7472 17.2221 16.7516 17.2311

s]30/45[ 20.9357 22.2895 21.7266 22.3283

s]60/45[ 20.8274 22.1078 21.5631 22.2892

s]75/60[ 15.5597 16.9782 16.5393 16.7829

s]90/75[ 12.4429 12.0434 11.7145 12.1298

９７

Table 6.7 Comparisons of torsional rigidity between present analytical method

and ANSYS results

(Graphite-polymer composite)

Present ANSYS Present/ANSYS

s]15/0[ 15.7015 10.2935 1.5254

s]0/15[ 15.7015 11.5708 1.3570

s]30/15[ 18.3443 12.4845 1.4694

s]15/30[ 18.3443 16.9941 1.0795

s]45/30[ 22.0441 18.3550 1.2010

s]30/45[ 22.0441 22.9560 0.9603

s]60/45[ 21.9371 22.7832 0.9629

s]45/60[ 21.9371 18.3223 1.1973

s]75/60[ 18.1272 16.7771 1.0805

s]60/75[ 18.1272 12.3840 1.4638

s]90/75[ 15.4700 11.4223 1.3544

s]75/90[ 15.4700 10.1784 1.5199

９８

Table 6.8 Comparisons of torsional rigidity between present analytical method

and ANSYS results

(Glass-polymer composite)

Present ANSYS Present/ANSYS

s]15/0[ 12.6238 10.9747 1.1503

s]0/15[ 12.6238 12.2214 1.0329

s]30/15[ 15.7472 13.0636 1.2054

s]15/30[ 15.7472 17.2221 0.9144

s]45/30[ 20.9357 18.3972 1.1380

s]30/45[ 20.9357 22.2895 0.9393

s]60/45[ 20.8274 22.1078 0.9421

s]45/60[ 20.8274 18.3606 1.1344

s]75/60[ 15.5597 16.9782 0.9165

s]60/75[ 15.5597 12.9450 1.2020

s]90/75[ 12.4429 12.0434 1.0332

s]75/90[ 12.4429 10.8347 1.1484

９９

Tables 6.7 and 6.8 show the comparisons of torsional rigidity of laminated composite

between the present analytical approach and finite element results. The results obtained from the

present analytical method seem somewhat strange in that different lay-up sequences of the same

lamina component have the same torsional rigidity. This phenomenon is due to the effective

elastic constants of laminated composites being independent of lay-up sequence. The effective

elastic constants are solely dependent on the components of each lamina of the laminated

composites. From the results obtained from tables 6.7 and 6.8 and the associated shear moduli of

each different angle-ply lamina, one can observe that for the same lamina component but

different lay-up sequence, if the lamina with the higher zxG and lower zyG value lay-up first, the

prediction of torsional rigidity of the laminated composite beam will be more accurate. The

greatest difference between the finite element results and the present analytical results are 52.5%

and 20.5% for graphite-polymer composite and glass-polymer composites, respectively. If the

lamina lay-up order uses the lamina with higher zxG and lower zyG value places in the outer layer,

the greatest difference between the finite element results and present analytical method will

decrease to 35.7% and 8.56% for graphite-polymer and glass-polymer composites, respectively.

１００

Chapter 7 Summary and Conclusions

The primary objective of this study is to analytically investigate the torsional rigidity of

laminated composite beams and compare the results with those from the finite element method,

finite difference method, and a previous paper (Swanson, 1998). Torsion of cylindrical shafts has

long been a basic subject in classical theory of elasticity. In the first chapter, many previous

works about the torsional problem were being discussed. All of the previous works formulated

the torsion problem by following the classical approach, which was proposed by Saint Venant.

For some simple geometrical shapes, analytical solutions are readily available. For other more

complex geometry, numerical methods, such as the finite element or finite difference method, are

usually utilized to obtain the solution.

The analytical method proposed here utilized the concept of the effective elastic constant of

laminated composite materials. The formulation of the effective elastic constant of laminated

composites has been proposed by many previous researchers. This study primarily adopts the

formulation from Chou et al. (1971). From that approach, a layered medium with monoclinic

anisotropy is considered. Within these laminated composites, the conditions of uniform strain in

the layering direction and constant stress in the direction normal to layering are assumed. After a

long mathematical derivation, the non-homogeneous anisotropic laminated composite is

transformed into an equivalent homogeneous anisotropic material. The obtained effective shear

moduli of the homogenized material can then be used to determine the effective torsional rigidity

of the laminated composite rods by the theory developed by Lekhnitskii (1963).

The classical formulation of the torsion problem of composite laminates is presented in

chapter 3. A theoretical solution for the torsion problem of laminated composites is presented.

Two continuity conditions have to be satisfied between the interfaces of each layer. The partial

derivative of the stress function with respect to x must be equal between two neighboring layers,

and the partial derivative of the stress function with respect to y times the reciprocal of the shear

modulus in the x-z direction must be constant within each layer. The outer boundary conditions

are set to be stress free as in the classical formulation. Finally, the eigenfunction expansion

method is employed to solve the formulated problem.

Two numerical methods are employed in this study, the finite element method and the

finite difference method. In the finite element method, the finite element commercial package

１０１

ANSYS 9.0 is used to model the laminated composite beam problem. In the finite difference

method, the finite difference codes are developed to analyze the torsional problem of laminated

composite rods. In order to show the effectiveness and accuracy of both numerical methods,

three verification examples with different material properties and geometrical dimensions are

used in the validation stage.

A total of twenty-four analysis models composed of different ply-angle layers are created

and analyzed in this study. Two commonly used composite materials (Graphite-polymer and

Glass-polymer composites) are used to build the analysis model. The dimensional parameters of

the analysis models are used from previous work (Swanson, 1998), and the material properties of

each lamina are from a composites book (Hyer, 1998).

The finite element models and finite difference models developed in this study are capable

of predicting the torsional rigidity of laminated composite rods. Good agreement was found

between the finite element method, finite difference method, and previous results (Swanson,

1998). From the results in chapter 6, one can see that the present analytical method doesn’t

predict very well for the torsional rigidity compared with the previous work and both numerical

methods. The reason behind the difference is that the first stage of the present analytical method

employed the idea of effective elastic constants instead of the classical Saint Venant approach.

The formulation of effective elastic constant considers a three dimensional layered medium. The

deformation mode of this typical laminate is assumed as constant stress in the direction normal to

layering and uniform strain in the layering direction. Both these assumptions will be violated

when the laminated composite is subjected to torsional loading. Hence, the concept of effective

elastic constants is not suitable in the prediction of the torsional rigidity of laminated composite

rods.

１０２

REFERENCES

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Enie, R. B. and Rizzo, R. R., “Three-Dimensional Laminate Moduli,” Journal of Composite Materials, Vol. 4, 1970, pp. 150-154. Friedman, Z. and Kosmatka, J. B., “Torsion and Flexure of A Prismatic Isotropic Beam Using the Boundary Element Method,” Computers and Structures, Vol. 74, 2000, pp. 479-494. Hashin, Z. and Shtrikman, S., “A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials,” Journal of the Mechanics and Physics of Solids, Vol. 11, 1963, pp. 127-140. Herrmann, R. L., “Elastic Torsional Analysis of Irregular Shapes,” Journal of Engineering Mechanics Division, ASCE , Vol. 9, 1965, pp. 11-19. Hrenikoff, A., “Solution of Problems in Elasticity by the Framework Method,” Journal of Applied Mechanics, Vol. 8, 1941, pp. 169-175. Hyer, M. W., “Stress Analysis of Fiber-Reinforced Composite Materials,” McGraw-Hill, 1998. Jawson, M. A. and Ponter, R. A., “An Integral Equation Solution of the Torsion Problem,” Proceedings of the Royal Society London, Vol. 275, 1963, pp. 237-246. Karayannis, C. G., “A Numerical Approach to Steel Fiber Reinforced Concrete under Torsion,” Structural Engineering Review, Vol. 7, 1995, pp. 83-91. Lekhnitskii, S. G., “Theory of Elasticity of an Anisotropic Elastic Body,” MIR Publishers, Moscow, 1963. Li, Z., Ko, J. M. and Ni, Y. Q., “Torsional Rigidity of Reinforced Concrete Bars with Arbitrary Sectional Shape,” Finite Elements in Analysis and Design, Vol. 39, 2000, pp. 349-361. Muskhelishvili, N. I., “Some Basic Problems of the Mathematical Theory of Elasticity,” Noordhoff, Groningen, 1963. Packham, B. A. and Shail, R., “St. Venant Torsion of Composite Cylinders,” Journal of Elasticity, Vol. 8, 1978, pp. 393-407. Pagano, N. J., “Exact Moduli of Anisotropic Laminates,” in Composite Materials, Broutman, L. J. and Krock, R. H., eds., Vol. 2, Mechanics of Composite Materials, Sendeckyj, G. P., ed., Academic Press, 1974, pp. 23-45. Postma, G. W., “Wave Propagation in a Stratified Medium,” Geophysics, Vol. 20, 1955, pp. 780-806. Reddy, J. N., “An Introduction to the Finite Element Method,” McGraw-Hill, 2006. Saada, A. S., “Elasticity: Theory and Applications,” Krieger Publishing Company, 1993.

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Vita Kunlin Hsieh was born in Kaohsiung, Taiwan on December 3, 1978 to Wei Jui and Joneming

Hsieh. He is the middle of three children. In 1999 Kunlin Hsieh graduated from National

Kaohsiung University of Applied Sciences, where he obtained his Associate Degree in Civil

Engineering. He graduated from Tamkang University at Taipei, where he obtained his Bachelor

of Science Degree in Construction Engineering in June 2001. In the fall of 2004, he joined

Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State

University to continue an academic career towards a Master’s Degree in Engineering Mechanics.

Kunlin Hsieh proudly received the degree of Master of Science in Engineering Mechanics in

May 3, 2007.

numerical modeling and analysis of composite beam structures

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