thesis3.2 minor change

LOSS OF ACCURACY USING SMEARED PROPERTIES IN

COMPOSITE BEAM MODELING

A Thesis

Submitted to the Faculty

of

Purdue University

by

Ning Liu

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Aeronautics and Astronautics

May 2015

Purdue University

West Lafayette, Indiana

ii

I dedicate my thesis work to my loving parents G. Liu and X. Yang who have been

encouraging me during my darkest period of time; to my girlfriend J. Li who has

been staying by my side and accepting my occasional idiosyncrasies; to my

roommates S. Jia, Z. Gao and Y. Chen who have been my nearest neighbors; to my

teammates in Defense of the Ancients, Ximen, Lanshao, Xiangshen, Biaoge,

Guangguang and etc with whom I have been fighting side by side with.

iii

ACKNOWLEDGMENTS

I would like to express the deepest appreciation to my advisor, the committee

chair, Professor Wenbin Yu, who has continually and convincingly conveyed a spirit

of adventure in regard to research. I would like to thank him for his understanding,

wisdom, patience, enthusiasm, and encouragement and for pushing me further than

I thought I could go. I would also like to thank my committee members, Professor

Vikas Tomar and Professor Arun Prakash. I am extremely grateful for their assistance

and suggestions throughout my project. To my lab mates, I would like to thank you

for your persistent help.

iv

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Smeared Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Transformation Law . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Constitutive relation for orthotropic materials . . . . . . . . . . . . 72.3 Smeared stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Benchmark examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 A rectangular Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 A multi-layer composite pipe . . . . . . . . . . . . . . . . . . . . . . 213.3 A thin-walled box-beam . . . . . . . . . . . . . . . . . . . . . . . . 273.4 A realistic wind turbine blade . . . . . . . . . . . . . . . . . . . . . 353.5 A fiber reinforced laminate . . . . . . . . . . . . . . . . . . . . . . . 47

4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

v

LIST OF TABLES

Table Page

3.1 Rectangular beam’s material properties and lay-up sequences . . . . . . 14

3.2 Rectangular beam: relative errors of classical stiffness matrices . . . . . 15

3.3 Rectangular beam: classical stiffness matrices, using original properties(top), smeared properties (bottom) . . . . . . . . . . . . . . . . . . . . 15

3.4 Multi-layer composite pipe: relative errors . . . . . . . . . . . . . . . . 22

3.5 Multi-layer composite pipe: classical stiffness matrices, using original prop-erties (top), smeared properties (bottom) . . . . . . . . . . . . . . . . . 22

3.6 Thin-walled box-beam lay-up sequence . . . . . . . . . . . . . . . . . . 28

3.7 Thin-walled box beam: relative errors . . . . . . . . . . . . . . . . . . . 28

3.8 Thin-walled box beam: classical stiffness matrices, using original proper-ties (top), smeared properties (bottom) . . . . . . . . . . . . . . . . . 29

3.9 Cross sectional configuration of the wind turbine blade . . . . . . . . . 37

3.10 Material properties of different material types that are used in the windturbine blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.11 Relative errors of classical stiffness matrix for the wind turbine blade . 39

3.12 Classical stiffness matrices for the wind turbine blade, using original ma-terial properties (top) and smeared material properties (bottom) . . . . 39

3.13 Fiber reinforced laminate’s material properties . . . . . . . . . . . . . . 48

3.14 Relative errors of classical stiffness matrix for the fiber reinforced laminate 49

3.15 Classical stiffness matrices for the fiber reinforced laminate, using originalmaterial properties (top) and smeared material properties (bottom) . . 49

vi

LIST OF FIGURES

Figure Page

1.1 A composite wind turbine blade, adopted from Gurit [1] . . . . . . . . 1

1.2 Efficient high-fidelity modeling of VABS . . . . . . . . . . . . . . . . . 2

1.3 Smeared properties at constituent level . . . . . . . . . . . . . . . . . 3

1.4 Smeared properties at lamina level . . . . . . . . . . . . . . . . . . . . 3

2.1 Coordinate system of structure, ply and fiber, adopted and reproducedfrom Yu [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Sketch of rectangular beam geometry, unit: mm . . . . . . . . . . . . . 14

3.2 Screenshot of rectangular beam mesh from ABAQUS . . . . . . . . . . 16

3.3 Displacement counter plots of original material properties (left) and smearedproperties (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Rectangular beam: vertical deflection . . . . . . . . . . . . . . . . . . 18

3.5 Rectangular beam: stress component σ11 . . . . . . . . . . . . . . . . 18



3.8 Cross sectional sketch of a multi-layer composite pipe, adopted from Jianget al. [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.9 A screenshot of mesh generating from ABAQUS for multi-layer compositepipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.10 Displacement contour plots of original material properties (left), and smearedmaterial properties (right) . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.11 Multi-layer composite pipe: deflection . . . . . . . . . . . . . . . . . . 24

3.12 Multi-layer composite pipe: stress σ11 . . . . . . . . . . . . . . . . . . 25

3.13 Multi-layer composite pipe: stress σ22 . . . . . . . . . . . . . . . . . . 26

3.14 Cross sectional sketch of thin-walled box-beam, adopted from Jiang etal. [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.15 A screenshot of thin-walled box beam mesh . . . . . . . . . . . . . . . 29

vii

Figure Page

3.16 A screenshot of thin-walled box beam mesh . . . . . . . . . . . . . . . 30

3.17 Thin-walled box beam deflection . . . . . . . . . . . . . . . . . . . . . 31

3.18 Thin-walled box beam stress σ11 . . . . . . . . . . . . . . . . . . . . . 32



3.21 A typical cross sectional sketch of a wind turbine blade, adopted fromChen et al. [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.22 Wind turbine blade: airfoil cross sectional key positions, adopted fromJiang et al. [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.23 Wind turbine blade: segment and web numbering details, adopted fromJiang et al. [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.24 A screenshot of mesh of wind turbine blade . . . . . . . . . . . . . . . 40

3.25 Contour plots of displacement of the wind turbine blade, using originalmaterial properties (top) and using smeared material properties (bottom) 41

3.26 Extension of wind turbine blade . . . . . . . . . . . . . . . . . . . . . 42

3.27 Stress σ11 of wind turbine blade . . . . . . . . . . . . . . . . . . . . . 44

3.28 Zoom in at the end of stress curve σ11 of wind turbine blade . . . . . . 44





3.33 Geometry of the fiber reinforced laminate . . . . . . . . . . . . . . . . 47

3.34 Geometry of the fiber reinforced laminate . . . . . . . . . . . . . . . . 48

3.35 Mesh of the fiber reinforced laminate . . . . . . . . . . . . . . . . . . . 50

3.36 Contour plots of displacement of the fiber reinforced laminate, using orig-inal material properties (top) and using smeared material properties (bot-tom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.37 Deflection of the fiber reinforced laminate . . . . . . . . . . . . . . . . 52

3.38 Stress σ11 of the fiber reinforced laminate . . . . . . . . . . . . . . . . 53


viii


ix

ABSTRACT

Liu, Ning MSAA, Purdue University, May 2015. Loss of Accuracy Using SmearedProperties in Composite Beam Modeling. Major Professor: Wenbin Yu.

Advanced composite materials have broad, proven applications in many engineer-

ing systems ranging from sports equipment sectors to components on the space shuttle

because of their lightweight characteristics and significantly high stiffness. Together

with this merit of composite materials is the challenge of improving computational

simulation process for composites analysis. Composite structures, particularly com-

posite laminates, usually consist of many layers with different lay-up angles. The

anisotropic and heterogeneous features render 3D finite element analysis (FEA) com-

putationally expensive in terms of the computational time and the computing power.

At the constituent level, composite materials are heterogeneous. But quite often

one homogenizes each layer of composites, i.e. lamina, and uses the homogenized

material properties as averaged (smeared) values of those constituent materials for

analysis. This is an approach extensively used in design and analysis of composite

laminates. Furthermore, many industries tempted to use smeared properties at the

laminate level to further reduce the model of composite structures. At this scale,

smeared properties are averaged material properties that are weighted by the layer

thickness. Although this approach has the advantage of saving computational time

and cost of modeling significantly, the prediction of the structural responses may not

be accurate, particularly the pointwise stress distribution. Therefore, it is important

to quantify the loss of accuracy when one uses smeared properties. In this paper,

several different benchmark problems are carefully investigated in order to exem-

plify the effect of the smeared properties on the global behavior and pointwise stress

distribution of the composite beam.

x

In the classical beam theory, both Newtonian method and variational method in-

clude several ad hoc assumptions to construct the model, however, these assumptions

are avoided if one uses variational asymptotic method. VABS (Variational Asymp-

totical Beam Sectional Analysis) is a code implementing the theory of classical beam

modeling based on the variational asymptotic method. We will also show in this

thesis that using VABS with the same set of benchmark examples enables efficient

and high fidelity analysis of composite beams by comparing to the detailed 3D FEA

using a commercial finite element software.

1

1. Introduction

In these days there is an increasing need for using composite materials in the industries

because of their better performance than isotropic homogenous materials. Composite

structures consist of the layers which could have different material properties and

different lay-up angles. There are many advantages of using composite materials,

such as high strength and stiffness and low thermal expansions. On the other hand,

there will be a disadvantage of using composite materials, i.e. longer computation

time which comes from the characteristics of the composite materials. Many com-

posite structures, especially helicopter rotor blades and wind turbine blades, consist

of hundreds of layers with different lay-up angles in their cross-sections, as shown in

Figure 1.1. The complex geometries and varying material features make 3D analysis

computationally expensive in terms of the limitation on the computational time and

the computer power.

Figure 1.1. A composite wind turbine blade, adopted from Gurit [1]

2

There were not many tools that have a capability of analyzing the complex com-

posite beams in the past. In these days, however, there is an efficient tool, called

VABS (Variational Asymptotical Beam Sectional Analysis) that can carry out the

detailed layer by layer modeling. The efficient high-fidelity modeling approach that

VABS uses is illustrated in the flowchart in Figure 1.2. It carries out asymptotic

analysis using the variational asymptotic method (VAM) to mathematically split the

original 3D problem into a constitutive modeling over the 2D cross section and a

macroscopic 1D beam analysis. The constitutive modeling will provide constitutive

relations for the beam analysis to obtain global structural behavior as well as recovery

relations to obtain the local fields. This approach has two unique advantages: one is

that it uses VAM as the mathematical foundation to avoid ad hoc assumptions; the

other is the macroscopic structural analysis can be formulated exactly as a general

continuum and all the approximations are confined in the constitutive modeling over

the cross section, the accuracy of which is guaranteed to be the best by VAM.

Figure 1.2. Efficient high-fidelity modeling of VABS

Even though there is a general-purpose tool that can analyzes the complex com-

posite beams, the advantage of using smeared properties cannot be overlooked. There

3

are two levels of smeared properties, one level is at the constituents: one uses the ho-

mogenized material properties of different components in a lamina, for example fibers

and matrices, as the averaged (smeared) values of that lamina for analysis, as Figure

1.3. Then at the lamina level, one takes the averaged material properties of multi-

ple layers as the material properties of one laminate, as Figure 1.4 described. The

computational time and the cost of the modeling would be decreased by using the

smeared material properties. For these reasons, many industries are in favor of using

the smeared material properties in order to simplify design and analysis of composite

beams.

Figure 1.3. Smeared properties at constituent level

Figure 1.4. Smeared properties at lamina level

In engineering design, the feasibility and trade-offs among different design pro-

posals have to be assessed by engineers. Each model has not necessarily to be tested

precisely, instead, the technique of smeared properties is usually adopted to narrow

down the field of configurations. Pitarresi et al. [2] appreciated this advantage of us-

ing smeared properties in the early stage of engineering design, but in his case study

he mentioned high errors occurred to several frequency predictions and suggested a

4

further refinement of the smeared material properties. Freels [3] mentioned that the

smeared property used in his analysis is viable to approximate the mode 1 failure

response but results from his model did not present a good agreement with exper-

imental results. Lall et al. [4] praised the value of using smeared properties in his

research for providing a reasonable computational efficiency, but he also mentioned

that errors in his research might come from missing the progressive delamination in

layers that would not be captured by the smeared properties model. In the analysis

of Mollineaux et al. [5], they limited their analysis on the isotropic material property.

However, in their case, the structural response due to the lamination sequence could

be disregarded. In general, smeared material properties result in the loss of accuracy

for orthotropic and anisotropic material. Unfortunately the loss remains vague since

no researcher before has explicitly show how much loss of accuracy smeared properties

will introduce.

Hence, in the present work, the objective is to prove the loss of accuracy introduced

by using smeared properties. Several different composite beams are investigated in

order to exemplify the effect of the smeared material properties on the behavior of the

composite beams. Finite element analysis (FEA) is carried out using original material

properties and smeared properties, respectively. Meanwhile, VABS is used to provide

a counterpart which presents efficient and accurate analysis. Errors are introduced by

using smeared properties, however, good agreement between VABS and FEA results

using the original material properties is observed.

5

2. Smeared Properties

2.1 Transformation Law

Coordinate transformation is frequently addressed when one deals with anisotropic

materials. To smear material properties, elastic constants of different layers of com-

posites has to be presented under one coordinate system, and usually we choose

global coordinate or problem coordinate. Therefore it is necessary to first elaborate

the transformation law that connects tensors under different coordinate systems.

The quantities which have more than one direction, such as stresses and strains,

can be described using tensors. The tensor field of order two is a system which has

nine components eij in the variables x1, x2, x3 and nine components e′ij in the variables

x′1, x′2, x′3, and these components are related through the characteristic transformation

law

e′ij = βimβjnemn (2.1)

written it in a matrix form, it becomes

[e′] = [β] [e] [β]T (2.2)

where β is the direction cosine matrix describing the transformation from xi coordi-

nate system to x′i coordinate system.

6

Now suppose eij is a stress tensor. Since only six stress components are indepen-

dent, we can write the stress tensor in a column matrix

[e] =

e11

e22

e33

e23

e13

e12

=

e1

e2

e3

e4

e5

e6

(2.3)

then the transformation law can be further expressed in the contracted form as

[e′] = [Te][e] (2.4)

where

[Te] =

β211 β2

12 β213 2β12β13 2β11β13 2β11β12

β221 β2

22 β223 2β22β23 2β21β23 2β21β22

β231 β2

32 β233 2β32β33 2β31β33 2β31β32

β21β31 β22β32 β23β33 (β22β33 + β23β32) (β23β31 + β21β33) (β31β22 + β21β32)



(2.5)

To understand the coordinate transformation particularly for fiber reinforced com-

posites, we need to first find the relationships among three coordinate systems: the

global coordinate system (x1, x2, x3), the ply coordinate system (y1, y2, y3) and the

material coordinate system (e1, e2, e3).

As shown in Figure 2.1, the ply coordinate system (y1, y2, y3) is formed by rotating

the global coordinate system (x1, x2, x3) in the right-hand sense about x1 by the

amount 0 ≤ θ1 ≤ 360◦. Then to form the material coordinate system (e1, e2, e3) we

rotate ply coordinate system (y1, y2, y3) about y3 in the right-hand sense by −90◦ ≤

θ3 ≤ 90◦.

Consider that we use primed coordinate system to denote global coordinate system

and unprimed to denote material coordinate system, then to transform a tensor from

7

Figure 2.1. Coordinate system of structure, ply and fiber, adoptedand reproduced from Yu [6]

the material coordinate system to the global coordinate system, we need to rotate θ3

about negative e3 axis first then rotate θ1 about negative y1 axis, which results in the

following transformation matrix

β =

1 0 0

0 cos θ1 − sin θ1

0 sin θ1 cos θ1

cos θ3 − sin θ3 0

sin θ3 cos θ3 0

0 0 1

(2.6)

2.2 Constitutive relation for orthotropic materials

In linear elasticity, the relation between the stress and strain is called the consti-

tutive relation and obeys the generalized Hookes law as

σij = Cijklεkl (2.7)

where Cijkl is a 6 by 6 stiffness matrix whose elements are the elastic constants of

the material. The matrix of elastic constants, [C], is positive definite and symmetric

from the strain energy consideration.

8

For orthotropic materials, the constitutive relation is expressed as

σ11

σ22

σ33

σ23

σ13

σ12

=

C11 C12 C13 0 0 0

C12 C22 C23 0 0 0

C13 C23 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

0 0 0 0 0 C66

ε11

ε22

ε33

2ε23

2ε13

2ε12

(2.8)

Note that it has been a common practice to use engineering notations, i.e. shear

strains are arranged as 2ε23, 2ε13, 2ε12.

In terms of the engineering elastic moduli, the stress and strain relation for an

orthotropic material can be expressed using compliance matrix [S] as ε = Sσ, i.e.

ε11

ε22

ε33

2ε23

2ε13

2ε12

=

1E1

−ν21E2−ν31

E30 0 0

−ν12E1

1E2

−ν32E3

0 0 0

−ν13E1−ν23

E2

1E3

0 0 0

0 0 0 1G23

0 0

0 0 0 0 1G13

0

0 0 0 0 0 1G12

σ11

σ22

σ33

σ23

σ13

σ12

(2.9)

9

Since stiffness matrix is the inverse of compliance matrix, we can formulate the

elastic constants of stiffness matrix from compliance matrix where we have explicit

expression in terms of engineering constants

C11 =1− ν23ν32E2E3∆′

C22 =1− ν13ν31E1E3∆′

C33 =1− ν12ν21E1E2∆′

C12 =ν21 + ν31ν23E2E3∆′

(2.10)

C13 =ν31 + ν21ν32E2E3∆′

C23 =ν32 + ν12ν31E1E3∆′

C44 = G23, C55 = G13, C66 = G12

where

∆′ =1− ν12ν21 − ν23ν32 − ν31ν13 − 2ν21ν32ν13

E1E2E3

(2.11)

2.3 Smeared stiffness

For composite structures that have multiple layers each having a fiber orientation,

the constitutive relation is valid only under the material coordinates. Because each

layer could orientate differently with respect to each other and differently from beam

coordinate system, so to undertake structural analysis, one has to transform all the

constitutive relations to the beam coordinate system.

For a single layer in a new coordinate system the constitutive relation becomes

σ′ij = C ′ijklε′kl (2.12)

where σ′, C ′ and ε′ are the stress tensor, elastic constants and strain tensor, respec-

tively in the new coordinate system.

10

In terms of stress tensor, if we arrange the stress components in a new coordinate

system, the relation between the stress in an old and a new coordinate system can be

obtained by

{σ}new = [Rσ]{σ}old (2.13)

where the transformation matrix is the same as [Te] in Eq. (2.5) and expressed as

[Rσ] =

β211 β2

12 β213 2β12β13 2β11β13 2β11β12

β221 β2

22 β223 2β22β23 2β21β23 2β21β22

β231 β2

32 β233 2β32β33 2β31β33 2β31β32




(2.14)

As for strain, the engineering strain components do not form a second order ten-

sor, thus do not follow the coordinate transformation law. However, since they are

related to the components of the strain tensor, their coordinate transformation can

be performed via the components of the strain tensor, i.e.

[ε] =

ε1

ε2

ε3

ε4

ε5

ε6

=

ε11

ε22

ε33

2ε23

2ε13

2ε12

(2.15)

By using the transformation law Eq. (2.4) and Eq. (2.5), the coordinate transfor-

mation for the engineering strain components is obtained as

{ε}new = [Rε]{ε}old (2.16)

11

where

[Rε] =

β211 β2

12 β213 β12β13 β11β13 β11β12

β221 β2

22 β223 β22β23 β21β23 β21β22

β231 β2

32 β233 β32β33 β31β33 β31β32

2β21β31 2β22β32 2β23β33 (β22β33 + β23β32) (β23β31 + β21β33) (β31β22 + β21β32)



(2.17)

Note that [Rε] 6= [Rσ].

The transformation of the stiffness matrix from material coordinate system (un-

primed) to global coordinate system (primed) can be derived as

σ′ = Rσσ = RσCε = RσCR−1ε ε′ = C ′ε′ (2.18)

which derives

C ′ = RσCR−1ε (2.19)

Next we are going to show that R−1ε = RTσ .

It is assumed that there exists a strain energy function W (εi) such that

σi =∂W

∂εii = 1, 2, ..., 6 (2.20)

The strain energy function for linearly elastic materials can be expressed in terms

of the strain components as

W =1

2Cijεiεj =

1

2{ε}T [C]{ε} =

1

2{ε}T{σ} (2.21)

where the superscript T denotes the transposed matrix.

Alternatively, the strain energy function can be expressed in terms of stress com-

ponents and the compliances Sij as

W =1

2Sijσiσj =

1

2{σ}T [S]{σ} =

1

2{σ}T{ε} (2.22)

12

Since the strain energy function is a scalar and is invariant with respect to coor-

dinate transformation, we can write

W =1

2{ε′}T{σ′} (2.23)

or

W =1

2{σ′}T{ε′} (2.24)

Using coordinate transformations for stress and strain, Eq.(2.22) and Eq.(2.24)

become

W =1

2{σ}T [Rε]

−1{ε′} (2.25)

and

W =1

2{σ}T [Rσ]T{ε′} (2.26)

Compare Eq. (2.25) and Eq. (2.26), it is obvious that R−1ε = RTσ .

Since R−1ε = RTσ , Eq. (2.19) can be written as

C ′ = RσCRTσ (2.27)

Now all stiffness matrices of different materials are presented in global coordinate,

then we are able to smear those material properties by

C∗ =1

T

n∑i=1

Citi (2.28)

where Ci and ti are the stiffness matrices and the thickness of the ith layer, respectively,

T is the total thickness of a composite laminate. Thus the smeared material properties

are the averaged material properties that is weighted by the layer thickness.

13

3. Benchmark examples

In this chapter, we are going to use several benchmark examples to demonstrate the

loss of accuracy using smeared properties. In each example, we provide solutions from

finite element analysis, i.e. ABAQUS using original material properties and smeared

properties, and from VABS (Variational Asymptotic Beam Sectional Analysis).

VABS is designed to model structures for which one dimension is much larger

than the other two (i.e. a beam-like body), even if the structures are made of com-

posite materials and have a complex internal structure. VABS implements a rigorous

dimensional reduction: from a 3D elasticity description to a 1D continuum model.

All the details of the cross-sectional geometry and material properties are included

as inputs to calculate both structural and inertial coefficients. These properties can

be directly imported into 1D beam analyses to predict the global behavior, which is

necessary for predicting pointwise 3D distributions of displacement, strain and stress

over the cross section by VABS.

Exactness of solutions from VABS is already proven to be very close to the elas-

ticity solutions [8] [9] [10] [11] [12]. First, the 4 × 4 classical stiffness matrices using

original material properties and smeared properties are obtained from VABS and

compared. The relative error is defined as∣∣∣Csmr−Cori

Cori

∣∣∣ ∗ 100%, where Csmr is the stiff-

ness property evaluated using the smeared properties and Cori is the corresponding

stiffness property evaluated using the original material properties. FEA is carried

out using the original material properties and the smeared properties, respectively.

On the other hand, GEBT (Geometrically Exact Beam Theory) uses the beam cross-

sectional properties, i.e. classical stiffness from VABS to carry out 1D beam analysis.

VABS then uses 1D displacement results from GEBT to recover 3D displacements and

stresses. Eventually, displacements, such as deflections and extensions are compared

as well as local stress distributions.

14

3.1 A rectangular Beam

2

4

40

Figure 3.1. Sketch of rectangular beam geometry, unit: mm

The first case is a composite beam with rectangular cross-section. The geometry

of the beam is given in Figure 3.1. Each ply has the thickness of 0.25 mm, a total

of 8 plies contributes to a height of 2 mm. The width is 4 mm. Since by definition

beams are slender structures, so then we set the span to be 10 times the largest

cross sectional dimension, which results in 40 mm. Material properties and lay-up

sequences are given in Table 3.1. The Young’s modulus and the shear modulus are

represented with the unit of GPa.

Table 3.1. Rectangular beam’s material properties and lay-up sequences

Lay-up Sequence E11 E22 E33 G12 G13 G23 ν12 ν13 ν23

[252/50/0/50/0/252] 41.5 7.83 7.83 3.15 3.01 3.01 0.3 0.3 0.3

The relative errors between two classical stiffness matrices are shown in Table

3.2. Both classical stiffness matrices are calculated using VABS. The procedure is

to create a cross sectional mesh containing the geometry and material properties as

an input file to VABS. VABS implements cross sectional analysis and returns the 1D

15

Table 3.2. Rectangular beam: relative errors of classical stiffness matrices

2.18% 100.00% 100.00% 0%

6.50% 34.97% 0%

10.24% 0%

Symm 4.59%

classical stiffness matrices. The only difference is one uses original lay-up sequence

and corresponding material properties and the other uses the smeared properties. As

it can be seen in Table 3.3, the smeared material properties introduce errors to all

entries in the stiffness matrix. The errors are relatively low on diagonal terms except

bending stiffness about x2 direction, which is 10.24%. However, higher loss occurs in

off diagonal terms, especially two coupling terms related with extension: the relative

errors of these two are 100%, which means, using the smeared properties would totally

ignore these two coupling effects. Note that we only calculate the relative errors of

terms that have values greater than 0.001.

Table 3.3. Rectangular beam: classical stiffness matrices, using orig-inal properties (top), smeared properties (bottom)1.72510× 1011 −2.88873× 108 8.26416× 109 2.93858× 10−4

4.02001× 1010 −2.22269× 1010 −2.32365× 10−5

5.83441× 1010 −1.38595× 10−4

Symm 2.24713× 1011

1.76277× 1011 8.08888× 10−7 −2.60799× 10−4 2.45839× 10−4

3.75883× 1010 −1.44535× 1010 −5.67488× 10−5

6.43168× 1010 −9.69945× 10−5

Symm 2.35036× 1011

16

Since the diagonal terms play a major role in structural behavior, we then carry

out the study on bending behavior about x2 direction to further quantify the loss of

accuracy.

We first use ABAQUS to carry out the finite element analysis. Coordinate origin

is placed at geometrical center of root surface. Geometry is built exactly the same

as described in Figure 3.1 and materials are put in as described in Table 3.1. The

boundary conditions are set to be one end fixed and one end free, subject to a moment

of −1×102 N ·mm about x2 direction. For mesh, use total 3520 quadratic hexahedral

elements of type C3D20. And to be comparable, smeared case has the same mesh

number and mesh type with the original one. Mesh are presented in Figure 3.2. As

for VABS, use 88 quadratic quadrilateral elements for cross section meshing, and then

use 40 beam elements in GEBT analysis, so that results form three are as much as

comparable.

Figure 3.2. Screenshot of rectangular beam mesh from ABAQUS

17

Figure 3.3. Displacement counter plots of original material properties(left) and smeared properties (right)

Contour plots of displacement using original material properties (left) and smeared

properties (right) are presented in Figure 3.3, note that they are set under same

deformation scale factor 3000. Even so, it is hard to tell the difference between these

two sets of results.

So then we plot the vertical displacement versus distance on beam axis, at a

path where x2 = x3 = 0 and x1 goes from root to tip, i.e., 0 to 40 mm. As we

can see from Figure 3.4, for the vertical deflection, smeared properties introduce

a significant accuracy loss, up to 21.41% loss at the tip; displacement from VABS

correlates with FEA using original material properties very well. The inaccurate

displacement prediction using smeared properties can not be overlooked as engineers

may overestimate the structural stiffness, and hence it misleads the assessment of

the structure. Note that in the legend of displacement and stress plots throughout

this chapter, “FEA” denotes ABAQUS results using the original material properties;

“SMP” denotes ABAQUS results using the smeared properties; “VABS” denotes

VABS results using the original material properties.

18

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x1 / L

u3 (

10

−6 m

m)

VABS

FEA

SMP

Figure 3.4. Rectangular beam: vertical deflection

0 0.5 1 1.5 2−50

−40

−30

−20

−10

0

10

20

30

40

x3 (mm)

σ11 (

MP

a)

FEA

VABS

SMP

Figure 3.5. Rectangular beam: stress component σ11

19

Figures 3.5, 3.6 and 3.7 are local stress distributions along the path where x1 = 20

mm, x2 = 0 mm and x3 goes from bottom surface to top surface, i.e., -1 mm to 1 mm.

In Figure 3.5 we can see, even though using smeared properties gives us a correct trend

of stress distribution for σ11, it cannot capture the rise and fall, peak and bottom

of stress curve as the laminate has one single smeared orthotropic material, which

leads to the continuous stress distribution along the stacking direction. For general

composites, stresses are discontinuous across the interfaces due to the different fiber

orientations. For σ12 and σ22, smeared properties present poor stress predictions as

seemingly zeros compared to the true stress distributions. Even though σ22 and σ12 are

relatively small compared to σ11, the loss of prediction on these stress components

may affect future analysis such as failure analysis, for example, Hill-Tsai criterion

includes both σ22 and σ12 into critical load calculation. So the loss of σ22 and σ12

can not be overlooked. On the other hand, the stress analysis using VABS is carried

out. VABS gives accurate predictions of local stress distributions. All the other

stress components’ absolute values are less than 0.01 MPa, with magnitude 100 times

smaller than three major stresses σ11 σ22 σ12, so all the other stress components are

not included into the study.

20

0 0.5 1 1.5 2−10

−8

−6

−4

−2

0

2

4

6

x3 (mm)

σ22 (

MP

a)

FEA

VABS

SMP


0 0.5 1 1.5 2−10

−8

−6

−4

−2

0

2

4

6

x3 (mm)

σ12 (

MP

a)

FEA

VABS

SMP


21

3.2 A multi-layer composite pipe

Figure 3.8. Cross sectional sketch of a multi-layer composite pipe,adopted from Jiang et al. [13]

The second case is a multi-layer composite pipe, the exactness of the VABS so-

lution is validated in Chen et al [14]. The geometry of the pipe and lamination

information are given in Figure 3.8, span is equal to 30 in. It is a thin-walled

cross-section with the thickness of wall to the chord length ratio being less than

0.1. Each layer is made of a composite material that has the material properties of

E11 = 20.59× 106 psi, E22 = E33 = 1.42× 106 psi, G12 = G13 = G23 = 0.87× 106 psi,

and ν12 = ν13 = ν23 = 0.42.

Relative errors are only calculated for the values which are greater than 1 and are

given in Table 3.4, classical stiffness matrices are shown in Table 3.5

22

Table 3.4. Multi-layer composite pipe: relative errors

3.11% 100.00% 0% 0%

29.56% 0% 0%

19.06% 0%

Symm 1.04%

Table 3.5. Multi-layer composite pipe: classical stiffness matrices,using original properties (top), smeared properties (bottom)1.03890× 107 9.83566× 104 −4.21294× 10−2 −2.84977× 10−2

6.87060× 105 −7.74104× 10−2 −1.29056× 10−1

1.88227× 106 −6.47980× 10−2

Symm 5.38148× 106

1.00658× 107 −7.36471× 10−3 9.38870× 10−4 5.51866× 10−3

8.90142× 105 −3.40423× 10−3 −8.58568× 10−3

1.52347× 106 −1.43136× 10−2

Symm 5.32552× 106

The differences shown in Table 3.5 are also high. For those terms on diagonal,

torsion stiffness and bending stiffness about x2 direction include high errors, and

for coupling between torsion and extension, it has even higher loss. The decrease

in bending stiffness will result in greater deflection, and this will be verified later

comparing the global beam behavior.

In ABAQUS, coordinate center is placed at the geometric center. Materials are

put in as the same with mentioned above. Total 21600 quadratic hexahedral elements

of type C3D20 are generated, and a screenshot of mesh is presented in Figure 3.9.

Boundary conditions are root fixed and tip subject to a -1000 lb·in bending moment

23

about x2 direction. Again, case for smeared properties has the same number of mesh

element and mesh type as case for original properties. VABS uses the same number

of mesh as ABAQUS generates for the cross section.

Figure 3.9. A screenshot of mesh generating from ABAQUS for multi-layer composite pipe

Contour plots of displacement are presented in Figure 3.10. Deformation scale

factors are set to be 20 for both cases. As we can see, there is a noticeable difference

in the vertical deflection between original case and smeared case, obviously smeared

case has higher deformation.

Then set a path at x2 = 0, x3 = 0.5 in, x1 goes from root to tip, which essentially

is a path along the top surface, we plot the vertical deflection versus x1. From

the displacement plot in Figure 3.11, we see that case of smeared properties has

higher vertical deflection, and this is due to the decrease of bending stiffness about x2

direction. The error in this plot agrees with the expectation from previous stiffness

analysis. The error reaches its maximum of 18.83% at the tip.

24

Figure 3.10. Displacement contour plots of original material proper-ties (left), and smeared material properties (right)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x1 / L

u3 (

in)

VABS

FEA

SMP

Figure 3.11. Multi-layer composite pipe: deflection

25

Not only displacement has loss of accuracy, also the loss will be introduced to

local stress distributions. Then in local stress analysis, set the path at x1 = 15 in,

x2 = 0 and x3 goes from 0.8 to 1 in, which goes across the laminate on the top. For

the normal stresses σ11 and σ22, shown in Figure 3.12 and Figure 3.13, we see that

stresses are discontinuous and experience a drop across the interface, whereas stresses

of smeared properties vary linearly along the path in an averaged sense. Note that

we did not compare shear stress and other stress components here in this case, the

reason is that except σ11 and σ22 all the other stress components, i.e. σ33 σ12 σ13 σ23

have values less than 10−3, which means their magnitudes are 105 times smaller than

σ11 and σ22, so these stress components can be considered as negligible and thus their

results are not presented here.

0 0.05 0.1 0.15 0.2−6

−5

−4

−3

−2

−1

0

x3 (in)

σ11 (

ksi)

FEA

VABS

SMP

Figure 3.12. Multi-layer composite pipe: stress σ11

26

0 0.05 0.1 0.15 0.2−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x3 (in)

σ22 (

ksi)

FEA

VABS

SMP

Figure 3.13. Multi-layer composite pipe: stress σ22

27

3.3 A thin-walled box-beam

Figure 3.14. Cross sectional sketch of thin-walled box-beam, adoptedfrom Jiang et al. [15]

Another benchmark case is a thin-walled box beam with span equal to 10 in, and

the cross section is shown in Figure 3.14. The exactness of VABS solution has been

verified by Yu et al. [16]. Each wall of the box-beam is made up of different lay-up

angles and their material properties are given the same as the previous case (a multi-

layer composite pipe) except G23 = 0.696 × 106 psi. The lamination information of

this beam is given in Table 3.6.

Relative errors are only calculated for the values which are greater than 10−5. As

is shown in Table 3.8, for thin-walled box-beam, diagonal terms have relative low

percentage errors. High errors appear on two coupling terms, i.e. the coupling term

between two bending directions as well as the coupling term between bending about

x3 and torsion. Even though their magnitudes (top) are small compared to diagonal

terms, but using smeared properties totally disregards these coupling effects. In the

28

Table 3.6. Thin-walled box-beam lay-up sequence

Lay-up Sequence

Right Wall [15◦/− 15◦]3

Left Wall [−15◦/15◦]3

Top Wall [−15◦]6

Bottom Wall [15◦]6

following finite element analysis, we will demonstrate that this difference will result

in distinct outcomes.

Table 3.7. Thin-walled box beam: relative errors

0.14% 0% 0% 0%

0.15% 0.14% 100.00%

0.19% 100.00%

Symm 0.21%

Compared to previous examples, now in a thin-walled box beam example, smeared

properties introduce less errors on diagonal terms, all being less than 1%. However,

coupling terms that appear off diagonally have noticeably high error and structure

could behave differently due to loss of accuracy of these coupling terms introduced

by smeared properties.

In ABAQUS, total 72000 quadratic hexahedral elements of type C3D20 are created

in mesh, shown in Figure 3.15. Boundary conditions are root fixed and tip subject to

a -1000 lb·in bending moment about x3 direction.

Contour plots of displacement using original material properties (left) and smeared

properties (right) are present in Figure 3.16. Deformation scale factors are set to be

29

Table 3.8. Thin-walled box beam: classical stiffness matrices, usingoriginal properties (top), smeared properties (bottom)

9.84050× 105 1.04120× 10−6 3.30297× 10−6 −5.44303× 10−6

1.69336× 104 1.76045× 104 −3.54096× 102

5.90734× 104 −3.70816× 102

Symm 1.41252× 105

9.85409× 105 −8.52174× 10−7 3.47044× 10−6 −8.98934× 10−6

1.69590× 104 1.76300× 104 4.33835× 10−7

5.91851× 104 −1.78467× 10−6

Symm 1.41547× 105

Figure 3.15. A screenshot of thin-walled box beam mesh

10 for both cases. As we can see, there is barely no difference in the deformation

between original case and smeared case. This is because as previously we see, errors

of diagonal terms are very low.

Then to confirm the high errors at off diagonal terms, set the path at x2 = 0.4765

in, x3 = 0 in and x1 goes from root to tip, which is along the wall on the right, plot

the displacement along x3 direction versus distance along x1 direction. As we can

30

Figure 3.16. A screenshot of thin-walled box beam mesh

see from displacements in Figure 3.17, case of smeared properties has little response

along x3 direction which means using smeared properties essentially cannot present

that coupling effect, and this corresponds to previous stiffness analysis. Using original

material properties has noticeable deflection along that direction. Although the value

of deflections are small, with magnitude being 10−4, VABS matches with the FEA

result quite well.

31

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

x1 / L

u3 (

10

−4 in

)

VABS

FEA

SMP

Figure 3.17. Thin-walled box beam deflection

Set the path at x1 = 5 in, x3 = 0 in and x2 goes from -0.4765 in to -0.4465 in,

which essentially is a path normal to the left wall and goes along positive x2 direction.

Stresses are presented in Figures 3.18, 3.19 and 3.20.

And as loss of accuracy is introduced to stress analysis, in both σ11 and σ33

we see that even though smeared properties could possibly present an overall stress

distribution across the thickness but cannot capture the rise and fall at interfaces. One

thing worth notice is at the interface between the first and second layer, i.e. x2 = 0.005

in, stress σ11 from FEA reaches its maximum value 56040.8 psi through the thickness,

higher than stress value 56017.2 psi at the beginning i.e. x2 = 0. Same thing happens

to stress σ33 and it is becoming more obvious. Using smeared properties we may come

to a misleading conclusion that stress reaches its maximum at the lateral surface,

i.e. x2 = 0 whereas the truth is that the maximum stress appears at a different

place. In terms of the shear stress σ13, the true stress varies back and forth, and

with black dots indicating results from VABS, VABS matches with FEA results quite

32

0 0.005 0.01 0.015 0.02 0.025 0.0352

52.5

53

53.5

54

54.5

55

55.5

56

56.5

x2 (in)

σ11 (

ksi)

FEA

VABS

SMP

Figure 3.18. Thin-walled box beam stress σ11

well; smeared properties, on the other hand, cannot present the oscillating stress

distribution precisely. All the other stress components have maximum values less

than 5 psi, considering the fact that we have made the point, so they are negligible

in comparison to σ11, σ33 and σ13 distributions.

33

0 0.005 0.01 0.015 0.02 0.025 0.03−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x2 (in)

σ33 (

ksi)

FEA

VABS

SMP


34

0 0.005 0.01 0.015 0.02 0.025 0.03−15

−10

−5

0

5

10

15

x2 (in)

σ13 (

ksi)

FEA

VABS

SMP


35

3.4 A realistic wind turbine blade

Figure 3.21. A typical cross sectional sketch of a wind turbine blade,adopted from Chen et al. [14]

So far benchmark examples are limited to simple cross-section geometry, to im-

prove the demonstration, we include a complex example. We analyze a realistic wind

turbine blade. A typical wind turbine blade cross-section configuration is shown in

Figure 3.21. It has five varying skin segments and two webs. The detail of cross

sectional data, such as airfoil, chord length, lay-up sequence, material properties are

described in Figures 3.22 and 3.23, Tables 3.9 and 3.10, where the layer thickness

has the unit of inch, the elastic and shear moduli have the units of lb/in2 and the

material density has the unit of lb·sec2/in4. Span is set to be 800 in.

36

Figure 3.22. Wind turbine blade: airfoil cross sectional key positions,adopted from Jiang et al. [17]

Figure 3.23. Wind turbine blade: segment and web numbering details,adopted from Jiang et al. [17]

37

Table 3.9. Cross sectional configuration of the wind turbine blade

Components No. of plies Layer thickness Fiber orientations Matl. ID

Segments 1 1 0.015 0 3

&2 1 0.02007874 0 4

18 0.020866142 20 2

Segment 3 1 0.015 0 3

1 0.02007874 0 4

33 0.020866142 20 2

Segment 4 1 0.015 0 3

1 0.02007874 0 4

17 0.020866142 20 2

38 0.020866142 30 1

1 0.123031496 0 5

37 0.020866142 30 1

16 0.020866142 20 2

Segment 5 1 0.015 0 3

1 0.02007874 0 4

17 0.020866142 20 2

1 0.123031496 0 5

16 0.020866142 0 2

Webs 1 38 0.020866142 0 1

&2 2 0.061515748 0 5

38 0.020866142 0 1

38

Tab

le3.

10.

Mat

eria

lpro

per

ties

ofdiff

eren

tm

ater

ial

typ

esth

atar

euse

din

the

win

dtu

rbin

ebla

de

Mat

eria

lN

ame

Mat

eria

lID

E11

E22

=E

33

G12

=G

13

=G

23

ν 12

=ν 1

3=ν 2

3ρ

uni-

dir

ecti

onal

FR

P1

5.36

64E

+06

1.30

53E

+06

5.80

15E

+05

0.28

1.74

0449

E-0

4

dou

ble

-bia

sF

RP

21.

4939

E+

061.

4939

E+

061.

1603

E+

060.

31.

7123

78E

-04

Gel

coat

31.

4504

E-0

31.

4504

E-0

31.

4504

E-0

40.

31.

7123

78E

-04

Nex

us

41.

4939

E+

061.

4939

E+

061.

1603

E+

060.

31.

5570

47E

-04

Bal

sa5

1.45

04E

+03

1.45

04E

+03

2.90

08E

+01

0.3

1.19

7729

E-0

5

39

From relative errors in Table 3.11 and the classical stiffness matrices shown in

Table 3.12, we see that a realistic wind turbine blade has full elastic coupling between

extension, twist and bendings. For a composite wind turbine blade, the effects of

these couplings are significant that they are necessarily accounted for an accurate

prediction. However, using smeared properties introduces errors to all terms to various

degrees.

Table 3.11. Relative errors of classical stiffness matrix for the wind turbine blade

1.94% 31.55% 17.49% 50.55%

13.18% 14.95% 147.70%

0.80% 26.56%

Symm 16.28%

Table 3.12. Classical stiffness matrices for the wind turbine blade, us-ing original material properties (top) and smeared material properties(bottom)

5.37615× 108 −2.97501× 108 −2.39815× 108 −4.34302× 109

7.51335× 109 −1.72957× 107 5.06916× 108

6.68380× 109 4.40740× 109

Symm 1.57591× 1011

5.48023× 108 −3.91348× 108 −1.97875× 108 −2.14773× 109

8.50387× 109 −1.47098× 107 −2.41817× 108

6.73732× 109 3.23685× 109

Symm 1.31928× 1011

40

In the finite element analysis, build the geometry as exactly the same as we de-

scribed before. Then use same amount and same type of mesh for both the original

case and smeared properties case, i.e. 81480 quadratic hexahedral elements of type

C3D20. Mesh is presented in Figure 3.24. Boundary conditions are root fixed and

tip subject to a 1× 106 lb·in twisting moment about x1 direction.

Figure 3.24. A screenshot of mesh of wind turbine blade

Contour plots of displacement are presented in Figure 3.25. As we can see, there is

a slight difference in displacement distribution at tip surface between original material

properties and smeared properties, but it is not very obvious.

41

Figure 3.25. Contour plots of displacement of the wind turbine blade,using original material properties (top) and using smeared materialproperties (bottom)

42

Then we set a path along the blade to compare extensions, i.e., x2 = 2.3725 in,

x3 = 3.9060 in and x1 goes from root to tip.

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

8

9

x1 / L

u1 (

10

−2 in

.)

VABS

FEA

SMP

Figure 3.26. Extension of wind turbine blade

As we can see from Figure 3.26, using smeared properties leads to a higher exten-

sion meanwhile VABS gives an accurate prediction which aligns with FEA solutions

using original material quite well. Then in stress analysis, shown in Figures 3.27,

3.29 and 3.31, path is set along x3 direction across the LPS (low pressure surface)

at x1 = 400 in, x2 = 2.3725 in. In a wind turbine blade, each lamina has the differ-

ent material properties and different fiber orientation angles from place to place. So

as expected, there is a more varying and irregular stress distribution than previous

examples because of this complex lay-up configuration across the thickness. Again,

what smeared properties simply present is an overall trend on how stresses vary across

43

the thickness, and it does not capture the variations. VABS, on the other hand, could

capture all the details, the peak and bottom, the rise and fall and each corner of the

stress curve. All the other stress components have absolute values less than 1, so they

are not included here.

From Figures 3.28, 3.30 and 3.32, it can be observed that at the end of each stress

curve, there is a sharp decrease. This is because a ply which is made of Gelcoat is

paved at the lateral surfaces of all segments of airfoil, which can be observed in Tables

3.9 and 3.10. Due to the low strength of Gelcoat, that ply has little capability to

withstand stresses. Thus all stress curves experience drops at the end. Besides, at

the end of stress curve σ22 in Figure 3.29, a leap of stress appears before the drop;

this is because the second outermost ply, which is made of Nexus, has higher strength

in x2 direction than adjacent plies. To be more specific, if we look at the cross

sectional configuration of Segment 4 in Table 3.9, we find that next to the second

ply (of material ID: 4) is 17 plies that are made of double-bias FRP (Fiberglass

Reinforced Plastic) and have 20 degree fiber orientation angle. Double-bias FRP

has same strength in x1 and x2 directions, but the 20 degree fiber orientation angle

weakens its strength in x2 direction. Nexus has the same material properties as

double-bias FRP, but ply of Nexus has no fiber orientation, so the ply of Nexus has

higher strength in x2 direction than adjacent plies thus experiences higher stress in

x2 direction. This explains why there is an unexpected jump in stress distribution in

Figure 3.29.

44

0 0.5 1 1.5 2 2.5−800

−600

−400

−200

0

200

400

600

x3 (in)

σ11 (

psi)

FEA

VABS

SMP

Figure 3.27. Stress σ11 of wind turbine blade

2.1 2.2 2.3 2.4 2.5 2.6 2.7

0

50

100

150

200

250

300

x3 (in)

σ11 (

psi)

FEA

VABS

SMP

Figure 3.28. Zoom in at the end of stress curve σ11 of wind turbine blade

45

0 0.5 1 1.5 2 2.5−150

−100

−50

0

50

100

150

200

250

x3 (in)

σ22 (

psi)

FEA

VABS

SMP


2 2.1 2.2 2.3 2.4 2.5

−30

−20

−10

0

10

20

30

40

50

60

x3 (in)

σ22 (

psi)

FEA

VABS

SMP

(a) bottom

2.1 2.2 2.3 2.4 2.5 2.6 2.7

180

190

200

210

220

230

240

250

260

270

x3 (in)

σ22 (

psi)

FEA

VABS

SMP

(b) top


46

0 0.5 1 1.5 2 2.5−1200

−1000

−800

−600

−400

−200

0

200

x3 (in)

σ12 (

psi)

FEA

VABS

SMP


2 2.1 2.2 2.3 2.4 2.5

−950

−900

−850

−800

−750

−700

−650

x3 (in)

σ12 (

psi)

FEA

VABS

SMP

(a) bottom

2.3 2.35 2.4 2.45 2.5 2.55

−100

−80

−60

−40

−20

0

20

40

60

x3 (in)

σ12 (

psi)

FEA

VABS

SMP

(b) top


47

3.5 A fiber reinforced laminate

0.25 in

0.25 in

4 in

10 in

Figure 3.33. Geometry of the fiber reinforced laminate

At the most fundamental level, composites are mixtures of fibers and matrix. At

the constituent level, material is heterogeneous, therefore its behavior varies signifi-

cantly among different constituent materials. But engineers directly use the elastic

constants from experimental results for a lamina in their analysis or give a lamina the

smeared material properties, the assumption under which is that laminae are treated

as homogeneous. This is a direct violation of the truth. Secondly, material proper-

ties are supposed to be continuous across the interface between two laminae if two

laminae are bonded perfectly. But an discontinuity of material is artificially imposed

across the interfaces when laminae have different fiber orientations and engineers as

mentioned above, attribute the smeared material properties to those laminae. This is

another violation of the truth. In this section, we will study the smeared properties

in the constituent level and demonstrate the different behavior of laminae between

48

using original material properties and using the smeared properties. The benchmark

example we present here is a fiber reinforced laminate with two 0◦ laminae. Both

laminae have 40% fiber volume ratio. The geometry is described in the Figure 3.33.

Materials properties are described in Figure 3.34 and presented in Table 3.13.

Table 3.13. Fiber reinforced laminate’s material properties

Material E (GPa) ν

Upper fiber 379.3 0.1

Lower fiber 250.0 0.1

Matrix fiber 68.3 0.3

Figure 3.34. Geometry of the fiber reinforced laminate

49

Classical stiffness matrices using original properties and smeared properties are

presented in Table 3.15 and their relative errors are calculated in Table 3.14. Using

smeared properties again introduces various errors to different places in classical stiff-

ness matrix, and overestimates the twisting stiffness and bending stiffness about x2

direction to a large extent.

Table 3.14. Relative errors of classical stiffness matrix for the fiberreinforced laminate

1.96% 0% 0.96% 100.00%

63.74% 0% 0%

12.17% 100.00%

Symm 2.11%

Table 3.15. Classical stiffness matrices for the fiber reinforced lam-inate, using original material properties (top) and smeared materialproperties (bottom)

3.35167× 1011 0 6.47837× 109 −4.20974× 107

6.71092× 109 0 0

6.34689× 109 5.47390× 106

Symm 4.46251× 1011

3.41737× 1011 0 6.54037× 109 −1.49996× 101

1.09887× 1010 0 0

7.11952× 109 3.41712

Symm 4.55649× 1011

50

In ABAQUS, total 337750 quadratic hexahedral elements of type C3D20 are cre-

ated in mesh, shown in Figure 3.35. Boundary conditions are root fixed and tip

subject to a 1× 104 lb extension force in x1 direction.

Figure 3.35. Mesh of the fiber reinforced laminate

Contour plots of displacement using original material properties (top) and smeared

properties (bottom) are presented in Figure 3.36, note that they are set under same

deformation scale factor. It is hard to tell the difference between two results.

Then set a path at x2 = 0, x3 = 0, x1 goes from root to tip, we plot the vertical

deflection u3 versus x1. From the displacement plot in Figure 3.37, we see that using

smeared properties predicts lower vertical deflection, and this is a consequence of

the overestimating stiffness. The error in this plot agrees with the expectation from

previous stiffness analysis.

Not only displacement has loss of accuracy, also the loss will be introduced to

local stress distributions. Then in local stress analysis, set the path at x1 = 5 in,

x2 = 0.125 in and x3 goes from -0.25 to 0.25 in, which goes across the center of a fiber

from the bottom of the laminate to the top of the laminate. For the stresses σ11 and

σ22, shown in Figures 3.38 and 3.39, we see that FEA using the original material prop-

erties has stresses that are continuous across the interface, i.e. x3 = 0.25 in, whereas

51

Figure 3.36. Contour plots of displacement of the fiber reinforcedlaminate, using original material properties (top) and using smearedmaterial properties (bottom)

stresses of using smeared properties experience a jump at the interface; this demon-

strates the artificial material discontinuity introduced by using smeared properties.

Then within each lamina, using original material properties gives stresses that varies

significant among different constituents but using smeared properties gives linearly

varied stress distributions. VABS, on the other hand, gives accurate predictions on

both macroscopic displacement and local stress distributions. Note that we did not

compare other stress components here in this case, the reason is that except σ11, σ22

and σ13, all the other stress components, i.e. σ23, σ13 and σ12 have values less than 5

52

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

x1 / L

u3 (

10

−7 in)

VABS

FEA

SMP

Figure 3.37. Deflection of the fiber reinforced laminate

psi, so these stress components can be considered as negligible and thus their results

are not presented here.

53

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

x3 (in)

σ11 (

ksi)

FEA

VABS

SMP

Figure 3.38. Stress σ11 of the fiber reinforced laminate

54

0 0.1 0.2 0.3 0.4 0.5−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x3 (in)

σ22 (

ksi)

FEA

VABS

SMP


55

0 0.1 0.2 0.3 0.4 0.5−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

x3 (in)

σ33 (

ksi)

FEA

VABS

SMP


56

4. Summary

Smeared properties have their advantages of saving computational cost. At the early

stage of composite beam design and analysis, smeared properties could be a viable

tool that could possibly help engineer to make a prediction about which design will

function the best. Using original material properties requires the structure to be com-

putationally built as exactly the same as physical problem described; however using

smeared properties, geometric model is greatly reduced because multiple laminas can

be combined into one lamina with a homogenized material. But if engineers only con-

sider the speed of analysis but overlook the accuracy of analysis, no matter how fast

they can get the analysis done, inaccurate result could cause a unwise decision. So if

we have a tool that can carry out the analysis rapidly meanwhile retain the accuracy,

then engineers can analyze composite beam problems efficiently and effectively.

To provide a detailed analysis about loss of accuracy using smeared properties, this

work presents several benchmark examples to test the viability of smeared properties

in composite beam analysis. First we presented the concept of smeared properties.

Smeared properties in the lamina level are the averaged material properties weighted

by the layer thickness; smeared properties in the constituent level are the homogenized

material properties of different constituents. Finite element analysis was carried out

using original material properties to provide the exact solution, then FEA was carried

out again using smeared properties. VABS was also carried out as a counterpart to

finite element analysis.

Through comparison, we found that using smeared properties in composite beam

analysis could introduce errors not only into the macro analysis for example displace-

ments but also the local stress recoveries. In our studies, in terms of the displace-

ment, smeared properties have higher predictions in some cases and lower predictions

in other cases, showing no universal trend of prediction. Then in terms of stresses,

57

smeared properties provide a very rough stress distribution, and details of stress vari-

ation along the thickness are missing. However, to provide a accurate analysis as

finite element analysis using original material properties does, VABS only requires

the cross section mesh and this can be done rapidly using any mature softwares, both

commercial and open-source softwares. Unfortunately, the present work is limited to

static linear analysis. For a more complex composite beam analysis, for example non-

linear analysis, multiple iterations are required. VABS mathematically splits the 3D

problem into a 2D constitutive modeling over the cross section and a 1D beam anal-

ysis; as long as the cross sectional properties do not change throughout the analysis,

constitutive models will remain the same, so only the 1D beam analysis needs to be

iterated to have results converged without repeating constitutive modeling over the

cross section. Apparently, this is computationally efficient than analysis on original

3D structure.

In conclusion, present work showed that using smeared material properties in-

troduced significant error to the analysis. Solutions of VABS, on the other hand,

agreed with exact solutions very well, and particularly in stress distributions, VABS

have captured all the rise and fall, peak and bottom of stress curves. For not only

engineers but also researchers to carry out an accurate yet efficient analysis in their

studies, this paper suggests them to use VABS to carry out their analysis.

REFERENCES

58

REFERENCES

[1] Breakdown of a wind turbine blade. http://gurit.fangle.co.uk/breakdown-of-a-turbine-blade.aspx.

[2] JM Pitarresi, DV Caletka, R Caldwell, and DE Smith. The “smeared” propertytechnique for the FE vibration analysis of printed circuit cards. Journal ofElectronic Packaging, 113(3):250–257, 1991.

[3] Jason K Freels. Modeling fracture in z-pinned composite co-cured laminatesusing smeared properties and cohesive elements in DYNA3D. Technical report,DTIC Document, 2006.

[4] Pradeep Lall, Dhananjay Panchagade, Yueli Liu, Wayne Johnson, and Jeff Suh-ling. Smeared-property models for shock-impact reliability of area-array pack-ages. Journal of Electronic Packaging, 129(4):373–381, 2007.

[5] Mark G Mollineaux, Kendra L Van Buren, Francois M Hemez, and Sezer Atam-turktur. Simulating the dynamics of wind turbine blades: part I, model devel-opment and verification. Wind Energy, 16(5):694–710, 2013.

[6] Wenbin Yu. VABS manual for users. Department of Mechanical and AerospaceEngineering, Utah State University, Logan, Utah, 2011.

[7] Wenbin Yu. Beam Models. Department of Mechanical and Aerospace Engineer-ing, Utah State University, Logan, Utah, 2012.

[8] Wenbin Yu and Dewey H Hodges. Elasticity solutions versus asymptotic sec-tional analysis of homogeneous, isotropic, prismatic beams. Journal of AppliedMechanics, 71(1):15–23, 2004.

[9] Wenbin Yu and Dewey H Hodges. Generalized Timoshenko theory of the varia-tional asymptotic beam sectional analysis. Journal of the American HelicopterSociety, 50(1):46–55, 2005.

[10] Wenbin Yu, Dewey H Hodges, and Jimmy C Ho. Variational asymptotic beamsectional analysis–an updated version. International Journal of Engineering Sci-ence, 59:40–64, 2012.

[11] Wenbin Yu, Dewey H Hodges, Vitali Volovoi, and Carlos ES Cesnik. OnTimoshenko-like modeling of initially curved and twisted composite beams. In-ternational Journal of Solids and Structures, 39(19):5101–5121, 2002.

[12] Wenbin Yu, Dewey H Hodges, Vitali V Volovoi, and Eduardo D Fuchs. A gener-alized Vlasov theory for composite beams. Thin-Walled Structures, 43(9):1493–1511, 2005.

59

[13] Fang Jiang and Wenbin Yu. Validation of VABS-ANSYS GUI: multi-layer com-posite pipe example. School of Aeronautics and Astronautics, Purdue University,West Lafayette, Indiana, 2014.

[14] Hui Chen, Wenbin Yu, and Mark Capellaro. A critical assessment of computertools for calculating composite wind turbine blade properties. Wind Energy,13(6):497–516, 2010.

[15] Fang Jiang and Wenbin Yu. Validation of VABS-ANSYS GUI: thin-walled box-beam example. School of Aeronautics and Astronautics, Purdue University, WestLafayette, Indiana, 2014.

[16] Wenbin Yu, Vitali V Volovoi, Dewey H Hodges, and Xianyu Hong. Validation ofthe variational asymptotic beam sectional analysis. AIAA Journal, 40(10):2105–2112, 2002.

[17] Fang Jiang and Wenbin Yu. Validation of VABS-ANSYS GUI: airfoil exam-ple. School of Aeronautics and Astronautics, Purdue University, West Lafayette,Indiana, 2014.

thesis3.2 minor change

Documents