Linear Non Linear FEM

8/2/2019 Linear Non Linear FEM

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Paper: ASAT-13-ST-11

13th

International Conference on

AEROSPACE SCIENCES & AVIATION TECHNOLOGY,

ASAT- 13, May 26 28, 2009, E-Mail: [email protected]

Military Technical College, Kobry Elkobbah, Cairo, Egypt

Tel : +(202) 24025292 24036138, Fax: +(202) 22621908

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Linear and Nonlinear Finite Element Modeling

of Advanced Isotropic and Anisotropic Beams

Part I: Euler Bernoulli Theory

M. K. Abass * M. A. Elshafei *

Abstract: In the present work, a linear and nonlinear finite element modeling for both

isotropic and anisotropic advanced Euler beams is presented. A hermit cubic and linear shape

functions are used to represent the beam deformations in the modeling. The principle of

virtual work is used to formulate the equilibrium equations for both models. A MATLABcode is developed to analyze both models linear and nonlinear of steel beam andCarbon/Epoxy laminated composite beam. A ready-made function is used to solve the

nonlinear system of equilibrium equations. The Steel and the composite beams are tested in

the laboratory to measure the deflections, which compared with the proposed finite element

model. For further verification, a COSMOS software program is used. This comparison

highlighted the need of including the shear strain in the geometric nonlinear analysis for the

case of large deformation.

Keywords: Nonlinear Finite Element Analysis, laminated composite structure, Euler beam

theory, static testing of beams.

1. IntroductionNonlinearity naturally arises in the true. Based on the assumptions of smallness of certainquantities of the formulation, the problem may be reduced to a linear problem. Linear

solutions may be obtained with considerable ease and less computational cost when compared

to nonlinear solutions. In many instances, assumptions of linearity lead to reasonable

idealization of the behavior of the system. However, in some cases assumption of linearity

may result in an unrealistic approximation of the response. The type of analysis, linear or

nonlinear, depends on the goal of the analysis and errors in the system's response that may be

tolerated. In some cases, nonlinear analysis is the only option left for the analyst as well as the

designer.

There are two common sources of nonlinearity: geometric nonlinearity and materialnonlinearity. The geometric nonlinearity arises purely from geometric consideration (e.g.

nonlinear strain-displacement relations), and the material nonlinearity is due to nonlinear

constitutive behavior of the material of the system. A third type of nonlinearity may arise due

to changing initial or boundary conditions.

* Egyptian Armed Forces


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A numerical model for layered composite structures based on a geometrical nonlinear shell

theory was developed on 1993 by F. Guttmann, et.al, [1]. In thin-walled open-section beams

made of fiber-reinforced laminates, at which the bending and torsion are coupled, a non-linear

finite-element (NLFE) analysis based on the updated lagrangian formulation is developed tosolve the problem numerically by B. Omidvar A. Ghorbanpoor on 1996, [2]. 0n 2006 R

Murali et.al investigated systematically the discretisation errors that appear uniquely in a non-linear beam formulation due to the presence of non-linear derivative terms in the membrane

strain term. These appear in the form of degraded performance (locking), [3]. On 2007 P.

Krawczyk, et.al developed a layer-wise beam model for geometric nonlinear finite element

analysis of laminated beams with partial layer interaction. The model is built assuming first

order shear deformation theory (FSDT) at layer level and moderate interlayer slips, [4]. On

2008 Li Jun, et.al developed the exact dynamic stiffness matrix of a uniform laminated

composite beam based on trigonometric shear deformation theory, [5]. Maxwell Blair, et.al

used Euler-Bernoulli beam mechanics to drive a Finite element beam formulations with

geometric non-linear mechanics including geometric bend-twist coupling in the context oflarge deformations and follower forces, [6]. On 2009 Yagci, Baris; Filiz Sinan, et.al presenteda spectral-Tchebychev technique for solving linear and nonlinear beam problems. The

technique uses Tchebychev polynomials as spatial basis functions, and applies Galerkin's

method to obtain the spatially discretized equations of motion, [7]. J. N. Reddy deduced a

nonlinear formulation of straight isotropic beam using Euler Bernoulli beam theory and

Timoshenko beam theory to formulate the kinematic behavior of the beam. The princible of

virtual displacement was used to formulate the equlibrium equations, [8].

Concerning the geometric nonlinearity of the classical beam theory, this paper presents asimple finite element model able to estimate the large deformations, axial, transverse, and

rotation angle due to distributed and concentrated general loading. The model is able to solvelaminated composite straight beams with different fiber orientation angles of arbitrary number

of layers. The modeled beam has variable cross-section and variable material property along

its longitudinal axis. Linear anisotropic model is also deduced. By applying simplified

assumptions on the derived equations, the linear and nonlinear isotropic models are obtained.

Two beams are introduced as an example of isotropic and anisotropic beams. The two beams

are statically loaded and the deflections are measured in the lab. A MATLAB code is

developed to formulate and solve the system of equilibrium equations to get the requireddeflections. The numerical results are compared with the measured data. For further

verification, a COSMOS software program is used.

2. Theoretical FormulationThere are four different theories to model the kinematics behavior of beams; Euler-Bernoulli

beam theory (EBT) that neglects the transverse shear strain, Timoshenko beam theory (TBT),

which accounts for the transverse shear strain in the simplest way, the second-order beam

theory (SOBT), and the third-order beam theory (HOBT), which add additional terms into the

assumed displacement field [9].

The general form of the assumed displacement field is expressed as, [9];


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3

20 00 0 1 2 3

0

( , ) ( ) ( ) ( ) ( )

( , ) 0

( , ) ( )

x x x

dw dwzu x z u x z C C x C z x C x

dx h dx

v x z

w x z w x

Where: , ,u v w are the displacements along coordinate directions, longitudinal, lateral, and

transverse, (x,y,z) respectively.0 0( ), ( )u x w x denote the displacement of a point (x, y, 0) on

the mid plane of an undeformed beam along the axial (x) and the transverse (z) directions,

respectively. ( )x x and ( )x x are functions of x. h is the beam thickness along z-axis.

In the First-Order Beam Theory, FOBT, which known as Timoshenko Beam theory, the

constants are; 0 0c , 1 1c , 2 0c , 3 0c , by substituting in Eq. (1), the assumed

displacement field is, [9];

0

0

( , ) ( ) ( )

( , ) 0

( , ) ( )

xu x z u x z x

v x z

w x z w x

In the Second-Order Beam Theory (SOBT), the constants are taken as; 0 0c , 1 1c , 2 1c ,

3 0c , from Eq. (1), the assumed displacement field takes the form, [9];

20

0

( , ) ( ) ( ) ( )

( , ) 0

( , )

x xu x z u x z x z x

v x z

w x z w x

In the Higher-Order Beam Theory (HOBT) ,third-order, the constants are taken as; 0 0c ,

1 1c , 2 0c , 34

( )3

c h . By substituting in Eq. (1), the assumed displacement field is,

[9];

3 00

0

4( , ) ( ) ( ) ( )

3

( , ) 0

( , ) ( )

x x

dwzu x z u x z h x

h dx

v x z

w x z w x

Our concern in this study is geometric nonlinearity of the classical beam theory which is

based on the Euler-Bernoulli hypothesis that plane sections perpendicular to the mid-plane of

the beam before deformation remain plane, rigid (not deform), and rotate such that they

remain perpendicular to the (deformed) mid-plane after deformation, as shown in Fig. 1. Theassumptions lead to neglect the Poisson effect and transverse shear strains.


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Omitting large strain terms except the square of 0dw

dx, the axial strain can be written as:

22

0 0 0

2

1

2xx

du d w dwzdx dx dx

where

2

01

2

dw

dx

is the nonlinear term. Eq. (8) can be written as;

0 1

xx xx xxz

where 0xx

, is strain of mid-plane, 1xx

is the mid-plane curvatures in the x direction.

2 20 10 0 0

2

1,

2 xx xx

du dw d w

dx dx dx

4. Stress-Strain RelationsFor a linear elastic behavior of isotropic materials, Hokes' law can be written as, [14]:

xxxx E

and for anisotropic materials this relation takes the form, [10]:

i, j = 1,2,3,...6i ij jQ

where i and j are the stress and strain components, ijQ are the components of the lamina

transformed stiffness matrix.

Thus for our case, xx is the only non zero stress, [10].

11 xx xx

Q

5. Variational FormulationThe principle of virtual work will be used to formulate the equilibrium equations, [13].

0I EW W W

WhereIW is the internal virtual work, i.e. virtual strain energy stored in the beam due to

actual stress, and EW is the work done by external applied load, [13].


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I ij ij

v

W dv

where v is beam volume.

0 0 0 0 0

0 0

( ) ( ) ( ) ( ) ( ) ( ) ( )

L L

E N P M w q x w x dx f x u x dx N u x P w x M x

where ( )q x is the distributed transverse load, ( )x is the distributed axial load, , , N P M

are concentrated axial, transverse and bending moment load respectively, as shown in Fig. 2.

0 0 0( ), ( ), ( )u x w x x are the virtual axial, transverse, and rotational displacements

respectively, and 00( )

( )d w x

x

dx

.

Fig. 2. Beam under general loading

By substituting the stress and strain values, Eq (13), Eq. (9), and performing integration of the

internal virtual work, Eq. (15);

0

L

I xx xx

A

W dAdx

0 10

L

I xx xx xx

A

W z dA dx

0 1

11

0

L

I xx xx xx

A

W z Q dAdx

N , 0 ( )Nu x

z

h

b

M , 0 ( )Mx q(x),

0 ( )w x

x1q

x

z

P , 0 ( )Pw x

L

xP

xM

x2qx2f

x1f

xN

f(x),0( )u x

y


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0 1 0 1110

L

I xx xx xx x x

A

W z Q z dA dx

0 0 0 1 1 0 2 1 1110

L

I xx xx xx xx xx xx xx xx

A

W Q z z dA dx

0 0 0 1 1 0 1 111 11 110

L

I xx xx xx xx xx xx xx xxW A B D dx

where 211 11 11 11( , , ) (1, , )kA

A B D Q z z dA

whereij

A are the components of extensional stiffness matrix,ij

D are the components of

bending stiffness matrix,ij

B are the components of coupling stiffness of the laminate, and

( )ij kQ are the components of the transformed stiffness matrix of the kth lamina, [10].

2 2

11 11 11 1

12 2

( ) ( ) ( )

bhN

k k k K

kh b

A Q dydz b Q z z

2 22 2

11 11 11 1

12 2

1( ) ( ) ( )

2

bhN

k k k k

kh b

B Q zdydz b Q z z

2 22 3 3

11 11 11 1

12 2

1( ) ( ) ( )

3

bhN

k k k k

kh b

D Q z dydz b Q z z

By taking the variation of the strain components, Eq. (10);

20 10 0 0 0

2, xx xx

d u dw d w d w

dx dx dx dx

Substituting in the internal virtual work statement, Eq. (22), we can get;

2 2

0 0 0 0 0 0 011

22 2 2

0 0 0 0 0 0 0 011 2 2 2

2 2

0 011 2

1 1

2 2

1

2I

d u du dw d w dw du dwA

dx dx dx dx dx dx dx

d u d w d w dw d w d w du dwW B

dx dx dx dx dx dx dx dx

d w d w

D dx dx

0

2

L

dx


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6. Finite Element FormulationThe displacement of a beam subjected to axial stretching is given by, [15]:

02

2

x

u

By solving the above governing equation and applying boundary conditions:

2

0 1 2

1

( ) j jj

u x u u

1 2 1 21 , ,Tx x

u u uL L

The governing equation for a beam under pure bending is, [16]:

4

40

w

x

By solving the above equation and imposing the nodal boundary conditions yields to, [17];

4

0 1 2 3 4

1

( ) J JJ

w x

where 1 2 3 4 1 1 2 2T

w w , and the shape functions take the form;

2 3 2 3

1 22 3 2

2 3 2 3

3 42 3 2

3 2 21 ,

3 2,

x x x xx

L L L L

x x x x

L L L L

The element nodal displacements vector is d .

1 1 2 2 3 4T

d u u

6.1. Nonlinear stiffness matrix of anisotropic materials

By taking the variation of Eq. (29) and Eq. (32), then substituting the shape functions into Eq.

(27), we can get the following;


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2 40

1 1

112 4

0 0

1 1

2 24 40

2 21 1

112

1.

2

1. . .2

.

j Jii j J

j J

j JI I j J

j J

J Ji Ii J I J

J J

I

II

d dw ddu u

dx dx dx dxA

ddw dw d d udx dx dx dx dx

d dw d d du

dx dx dx dx dxW B

d

dx

2 40 0

21 1

22 4

11 2 21

1. .

2

.

L

j Jj J

j J

JI

I J

J

dxd dw d

udx dx dx

ddD

dx dx

where i,j=1,2 and I,J=1,2,3,4 from now on.

2

11

1 0

240

11 11 21 0 0

220

11 11 21 0 0

2

011

. .

1. . . .

2

. . . .

1

2

L

jij

j

iL L

J Ji i

JJ

L L

j jI II j

j

i

dd A d x u

d x d xu

d d w d d d A d x B d x

d x dx d x d x dx

d ddw d d W A d x B d x u

dx d x d x d x d x

d wA

d x

2

11 240 0

22 21

011 112 2 2

0 0

. . .

1. . . .

2

L L

J JI I

JL LJ

J JI I

d dd dd x B d x

dx dx d x d x

d w d d d d B d x D dx

dx dx dx d x d x

The internal virtual work can be expressed in matrix form as:

2

6 6

T T

I X

uW u K

Where the stiffness matrix 11 12

2 2 2 4

21 22

4 2 4 4

K KK

K K

and its coefficients are,


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11

11

0

2

12 011 11 2

0 0

221 0

11 11 2

0 0

2 222 0

11 11

0

. .

1 . . . .2

. . . .

1. .

2

Lji

ij

L L

J Ji iiJ

L Lj jI I

Ij

L

JI IIJ

ddK A dx

dx dx

dw d d d dK A dx B dx

dx dx dx dx dx

d ddw d dK A dx B dx

dx dx dx dx dx

dw d d d dK A dx B

dx dx dx dx

20

22 2

011 112 2 2

0 0

.

1. . . .

2

L

J

L L

J JI I

dxdx

dw d d d d B dx D dx

dx dx dx dx dx

Its clear that if we substitute the value of0( )w x as a function of the shape functions, it will

arise a square term in ju and j , which leads to form a nonlinear, second order, system of

algebraic equations. Thus to solve this system we should make iteration by assuming an initial

value of 0 ( )w x , then calculate the corresponding stiffness matrix K , Eq.(39). Then, get

the corrected value of 0 ( )w x from the calculated element nodal displacements d , Eq.(32).

Repeat the iteration process and substitute the new value of 0 ( )w x in the stiffness matrix and

calculate the nodal displacement until the system of equations converge and reach the

required accuracy.

Note that the stiffness matrix is asymmetric matrix;12 21

T

K K . As marked above,

12K contain the factor

1

2, where

21K does not, Eq.(39). To make it symmetric

i.e.12 21

T

K K , split the linear strain0

du

dx, in Eq.(27), into two equal parts, as marked in

the following, [8];

2 2

0 0 0 0 0 0 0 011

22 2 2

0 0 0 0 0 0 0 011 2 2 2

2

11

1 1 1 1

2 2 2 2

1

2I

d u du dw d w dw du du dwA


d u d w d w dw d w d w du dwW B


d wD

0

2

0 0

2 2

L

dx

d w

dx dx


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2 2

0 0 0 0 0 0 0 0 0 011

22 2 20 0 0 0 0 0 0 0

11 2 2 2

1 1 1

2 2 2

12

I

d u du dw d w dw du d w dw du dwA

dx dx dx dx dx dx dx dx dx dx

d u d w d w dw d w d w du dwW Bdx dx dx dx dx dx dx dx

0

2 2

0 011 2 2

L

dx

d w d wD

dx dx

By substituting the shape functions in the internal virtual work:

2 4 20 0

1 1 111 24

0 0

1

24 40

21 1

11

1 1. .

2 21

.2

.

j jJi Ii j J I j

j J j

JII J

J

Ji Ii J I J

J J

I

d ddw d dwd du u u

dx dx dx dx dx dx dxAd du dwd

dx dx dx dx

d dwd du

dx dx dx dxW B

2

2

2 2 40 0

21 1

22 4

11 2 21

1. .

2

.

J

L

j JI I j J

j J

JI

I JJ

d

dxdx

d dw ddu

dx dx dx dx

dd

D dx dx

The internal virtual work can be expressed in matrix form as:

2

6 6I X

uW u K

and the stiffness matrix

11 12

2 2 2 4

21 22

4 2 4 4

K KK

K K

, which is symmetric.

The symmetric stiffness matrix coefficients are;


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11 11

11

0

2

12 12 011 11 2

0 0

221 0

11 11 2

0 0

2

22 0 011

. .

1 . . . .2

1. . . .

2

1

2

Lji

ij ij

L L

J Ji iiJ iJ

L Lj jI I

Ij

IJ

ddK K A dx

dx dx

dw d d d dK K A dx B dx

dx dx dx dx dx

d ddw d dK A dx B dx

dx dx dx dx dx

du dw dK A

dx dx

2

11 2

0 0

22 2

011 112 2 2

0 0

. . .

1. . . .

2

L L

J JI I

L L

J JI I

d dddx B dx

dx dx dx dx

dw d d d d B dx D dx

dx dx dx dx dx

It is clear that the term1

2exist in the two submatrices

12K and

21K , which give

symmetric stiffness matrix.

In the symmetric case 0 0( ), ( )u x w x should be known from a previous iteration to calculate

the stiffness matrix. Thus we can solve the equilibrium equations for the nodal displacements

d , and repeat the iteration process.

6.2. Linear stiffness matrix of anisotropic materials

Omit the nonlinear term

2

0dw

dx

from the longitudinal strain expression, Eq.(8). So, the linear

strain will be;

2

0 0

2xx

du d wz

dx dx

This results to omit all nonlinear terms in the stiffness matrix coefficients as shown:

11 11

11

0

212 12

11 2

0

221

11 2

0

2 2222

11 112 2 2

0 0

. .

.

.

. . .

Lji

ij ij

L

Ji

iJ iJ

LjI

Ij

L L

J JI IIJ

ddK K A dx

dx dx

ddK K B dx

dx dx

ddK B dx

dx dx

d dd dK B dx D dx

dx dx dx dx


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6.3. Nonlinear stiffness matrix of isotropic materials

For isotropic materials, 11 ( )Q E x . If the geometric x -axis is along the principle centroidal

axis, 11 0B , 11 ( ) ( )A E x A x , and 11 ( ) ( )yD E x I x , where ( )A x is the beam cross-

section area, ( )E x is the beam modulus of elasticity, and ( )yI x is the beam cross-section

second moment of area about y-axis.

Thus, Euler nonlinear symmetric stiffness matrix coefficients for isotropic materials are asfollows;

11 11

0

12 12 0

0

21 0

0

2

22 0 0

0

( ). ( ). . .

1( ). ( ). . . .

2

1( ). ( ). . . .

2

1( ). ( ). . .

2

Lji

ij ij

L

JiiJ iJ

LjI

Ij

L

JIIJ

ddK K E x A x dx

dx dx

dw ddK K E x A x dx

dx dx dx

ddw dK E x A x dx

dx dx dx

du dw d dK E x A x dx

dx dx dx dx

22

2 2

0

( ) ( ) . .

L

JIy

dd E x I x dx

dx dx

6.4. Linear stiffness matrix of isotropic materialsThe stiffness matrix coefficients are as follow;

11

0

12

21

2222

2 2

0

. .

0

0

. .

Lji

ij

iJ

Ij

L

JI IJ y

ddK EA dx

dx dx

K

K

ddK EI dx

dx dx

By substituting the shape functions into external virtual workE

W , Eq.(16), we can get the

following;

2

1 1 2 2 1 1 2 3

1 0

4

1 2 2 4

1 0

( ) ( )

( ) ( )

L

E i i

i

L

I I

I

W N u N u f x x u dx P P

q x x dx M M

T

EW d F

where the element nodal load vector is c bF F F


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where cF is the concentrated nodal load vector;

1 1 1 2 2 2TcF N P M N P M

and bF is the equivalent body load vector;

1 1 2 2 3 40

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .

LT

bF f x x q x x q x x f x x q x x q x x dx

Thus the equilibrium equation is;

.K d F

where K is the rearranged symmetric stiffness matrix, according to the displacement

vector d , Eq.(34), [8]. The linear symmetric stiffness matrices of isotropic and anisotropicmodels are presented in appendix (A).

11 12 12 11 12 12

11 11 12 12 13 14

21 22 22 21 22 22

11 11 12 12 13 14

21 22 22 21 22 22

21 21 22 22 23 24

11 12 12 11 12 12

21 21 22 22 23 24

21 22 22 21 12 22

31 31 32 32 33 34

11 22 22 21 22 22

41 41 42 42 43 44

K K K K K K

K K K K K K

K K K K K K KK K K K K K

K K K K K K

K K K K K K

7. Solution of Nonlinear System of EquationsAfter assembling the elements stiffness matrices and force vectors, a new system of nonlinear

algebraic equations will be achieved.

g g g gK d d F

Where g gK d is the global stiffness matrix, which is function of the, unknown, global

nodal displacement vector gd , and gF is the global force vector.This nonlinear system should be linearized to be solved and to get the nodal displacements

gd . Two linearization methods were used in this study; the direct iteration and Newton-Raphson iterative methods.

In the direct iteration procedure, the solution at the (rth) iteration is determined from the

following assembled set of equations, [8];


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11r r

g g gd K d F

where the global stiffness matrix is determined using the nodal displacement vector from the

previous iteration, 1r

d

.

In the Newton Raphson procedure, the linearized element equation is of the form, [8, 11];

1

1 ( 1) ( 1)r r r r

g g g gd d T d R d

where the residual

( 1) ( 1)r r

g g g g R d K d F

and the tangent stiffness matrix ( 1)r

gT d

element is calculated using the definition,

[8, 11];

1

( 1)

r

gr

g

g

R dT d

d

The tangent stiffness matrix associated with the Euler Bernoulli beam is as follows,[8];

11 11

12 12

21 21

2

22 22 0 011

0

ij ij

iJ iJ

Ij Ij

L

JI IJ IJ

T K

T K

T K

du dw d dT K A dx

dx dx dx dx

In both methods, direct and Newton Raphson, the first iteration can be calculated using linear

stiffness matrix, i.e. assume 1

0r

gd

, and calculate

r

gd using Eq.(56) or Eq.(57).

Then calculate the residual, and repeat iteration process till reach a sufficient residual. At the

exact solution, the residual equals zero. The iteration process is explained by a flow chart in

the appendix (B).

During solution, direct iteration and Newton Raphson iteration methods, sometimes are notable to converge to a certain solution. A MATLAB ready-made function "fsolve" is used for

iteration process to obtain the proposed results.


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8. Validation and DiscussionTwo examples of cantilever beams, steel and carbon fiber, are loaded, as shown in Fig.3, and

their deflections are measured in the laboratory. MATLAB code is used to analyze the

existing beams, for both linear and nonlinear models. After solving the obtained systems ofequations, the calculated deflections are compared with the measured ones. A relation

between nondimensional maximum deflections and the applied load is drawn for each beam.

Example (1)A cantilever beam made of steel has the following data:

Table 1. Steel cantilever beam dimensions and Properties

Length (L) 800 mm

Width (b) 20 mmHeight (h) 6 mm

Modulus of elasticity (E) 2.1e11 Pa

Shear modulus (G) 79.3 GPa

Poisson's ratio( ) 0.3

Load (P)Transverse concentrated load

Vary from 0.58 N to 60.58 N

Load location(xP) 600 mm from fixed end

Density 7850 kg/m3

Fig. 3. Cantilever beam dimensions

On the lab, the test rig shown in Fig. 4 was used for measurements. The beam was fixed onthe vertical stand, and at a specific distance, different weights were hanged. The maximum

traverse displacement of each weight was measured by displacement gage, for smalldisplacements, and a ruler, for larger ones. The different results were noted.

To get the relation between the maximum traverse displacement and the applied load

numerically, A MATLAB code is developed to formulate and solve the system of equilibrium

equations previously formulated in the theoretical part of the present study.

xP

L

P

b

h


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Fig. 4. Deflection measurement of steel cantilever beam

Fig. 5. Convergence of linear and nonlinear solutions of steel cantileverat load = 8.08 N

Studying the convergence of the model, different number of elements is used for both models;

linear and nonlinear, once at a light load (8.08 N), and another at a heavy load (76.58 N) Fig.5, 6. The results show that, at the light load a convergence occurs for the linear model at 12

elements, and for the nonlinear model at 20 elements, Fig.5. Accordingly this specific

numbers of elements is selected for the numerical calculations. However, for the relatively

heavy load linear solution is converging away from the measured data, and nonlinear solution

needs larger number of elements to converge to the solution. This takes a lot of time, and it is

some times the iteration process proves fruitless, Fig.6.


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Fig. 6. Convergence of linear and nonlinear solutions of steel cantilever

at load = 76.58 N

For further verification, a COSMOS software program is used. The beam is solved using the

Nonlinear 2D Beam Element (BEAM2D), which is 2-node uniaxial beam element for twodimensional nonlinear structural models. The element has three degrees of freedom, two

translations and one rotation, per node. The element include the shear strain effect.

In this step, the numerical results are compared with the measured data, Fig.7.

Fig. 7. Nondimensional maximum deflection for steel cantilever beam


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Example (2)The previous procedures are re-applied on a carbon fiber beam. The laminated Carbon

Fabric/Epoxy Prepreg cantilever beam has the following data:

Table 2. Laminated Carbon /Epoxy cantilever beam dimensions and Properties

Length (L) 500 mm

Width (b) 20 mm

Height (h) 2.35 mm

E1 39.8 GPa

E2 37.9 GPa

G12 1.9027

12 0.14

Number of plies 8 plies

Orientation angles [0]8

Load (P)Transverse concentrated load

Vary from 0.58 N to 8.58 N

Load location(xP) 400 mm from fixed end

Density 1227.565 kg/m3

Fig. 8. Deflection measurement of Carbon /Epoxy cantilever beam

Studying the convergence of the model, different number of elements is used for both models;

linear and nonlinear, once at a light load (1.08 N), and another at a heavy load (7.08 N) Fig. 9,

10. The results show that, at the light load a convergence occurs for the linear model at 15

elements, and for the nonlinear model at 20 elements, Fig.9. Accordingly this specific

numbers of elements is selected for the numerical calculations. However, for the relativelyheavy load linear solution is converging away from the measured data, and nonlinear solution

needs larger number of elements to converge to the solution. This takes a lot of time, and it issome times the iteration process proves fruitless, Fig.10.


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Fig. 9. Convergence of linear and nonlinear solutions of Carbon /Epoxy cantilever

at load = 1.08 N

Fig. 10. Convergence of linear and nonlinear solutions of Carbon /Epoxy cantileverat load = 7.08 N


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Fig. 11. Nondimensional maximum deflections for Carbon /Epoxy cantilever beam

In this step, the numerical results are compared with the measured data, Fig.7, 11. It is clear

that both linear and nonlinear solutions converge to the measured data and COSMOS

solutions at relative deflection less thanmax

0.1w

L . For relative deflection range of (0.1 - 0.3),

the linear model solutions are quite close to COSMOS linear and nonlinear solutions and the

measured data also. For large transverse displacements, the Euler linear model solutions and

COSMOS linear solutions are quite identical, while COSMOS nonlinear solutions are closer

to measured data. However, the transverse deflections of the Euler nonlinear model go far

from the measured data for the relative deflection more than 0.1.

9. Conclusion and Future WorkThis paper is investigating the geometric nonlinearity of advanced Euler Bernoulli beam to

estimate the large deformations due to multi applied loads. When applied loads on the beamare large, the linear load-deflection relationship is not valid, because the beam develops

internal forces that resist deformation, and the magnitude of internal forces increases with

loading as well as the deformation.

Although the nonlinear model is used, Euler Bernoulli theory falls short to estimate the larger

deflections, because it neglects the Poisson effect and transverse shear strains. So it is

prescribed to include shear strain in the geometric nonlinear analysis. Timoshenko beam andhigher order beam models provide a better hypothesis to enhance the estimation of large

deflections. For future work, part II provides a complementary study that focuses on findingbetter solutions with fewer limitations, saving time and achieving much satisfying results.


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References[1] F. Guttmann, W. Wagner, L. Meyer, and P. Wriggers, "A nonlinear composite shell

element with continuous interlaminar shear stresses," Computational Mechanics, Vol.

13, pp. 175-188, 1993.[2] B. Omidvar A. Ghorbanpoor, " Nonlinear FE Solution for Thin-Walled Open-SectionComposite Beams," Journal of Structural Engineering, Vol. 122, November 1996, pp.

1369-1378, 1996.[3] R Murali and G Prathap, Field-consistency Aspects of Locking in a Geometrically Non-

linear Beam Formulation, CSIR Centre for Mathematical Modelling and Computer

Simulation, RR CM 0601, 2006.

[4] P. Krawczyk, F. Frey, and A.P. Zielinski, " Large deflections of laminated beams withinterlayer slips: Part 1: model development," Engineering Computations, Vol. 24, pp. 17

32, 2007.

[5] 2008 Li Jun, and Hua Hongxing, " dynamic stiffness analysis of a laminated compositebeams using trigonometric shear deformation theory," Composite Structures, Vol. 89,Issue 3, pp. 433-442, July 2009.

[6] 2008,Maxwell Blair, and Alfred G. Striz, "Finite Element Beam Assemblies withGeometric Bend-Twist Coupling," AIAA, 2008.

[7] Yagci, Baris; Filiz, Sinan; Romero, Louis L.; Ozdoganlar, O. BurakThis, "A spectral-Tchebychev technique for solving linear and nonlinear beam equations," Journal of

Sound and Vibration, Vol. 321, Issue 1-2, pp. 375-404, 2009.

[8] J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford UniversityPress, Oxford.

[9] A.A. Khdeir and J. N. Reddy, An exact Solution for the Bending of Thin and ThickCross-ply Laminated Beams, Composite Structure, Vol. 37, pp. 195-203, 1997.

[10] Ronald F. Gibson, Principles of Composite Material Mechanics, McGraw Hill Inc.,New York, 1994.

[11] William H., Saul A. Teukolsky, William T. Vetterling, and Brain P. Flannery,Numerical Recipes in Fortran, 2nd ed., Cambridge University Press, Cambridge, 1992.

[12] Logan, D.L., A First Course in the Finite Element Method, PWS-KENT, 1992.[13] David H. Allen, and Walter E. Haisler, Introduction to Aerospace Structural Analysis,

John Wiley & Sons, New York, 1985, pp. 250-328

[14] Ferdlanand P. Beer, and E. Russell Johnston, Jr., Mechanics of Materials, McGraw- HillInternational Book Company, Auckland, 1981.

[15] Yildirm V., Sancaktar, E. and Kiral, E., Comparison Of The In-Plane NaturalFrequencies of Symmetric Cross-Ply Laminated Beams Based on The Bernoulli-Euler

and Timoshenko Beam Theories, Journal of Applied Mechanics, Vol. 66, pp. 410-417(1999).

[16] Robbins, D. H., and Reddy, J. N., Analysis of Piezoelectrically Actuated Beams Usinga Layer-Wise Displacement Theory, Computers & Structures, Vol. 41, No. 2, pp. 265-

279, 1991.

[17] Bendary L. M., El Shafei M. A. and Riad A. M., Finite Element Model of a SmartIsotropic Beam with Distributed Piezoelectric Actuators, Proceeding of the 13th

International AMME Conference,MTC, 27-29 May, 2008.


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Appendix (A)

Linear stiffness matrix of anisotropic materials:-

11 11 11 11

11 11 11 11

3 2 3 2

11 11 11 11

2

11 11

11 11

3 2

11

0 0

12 6 0 12 6

4 6 2

0

. 0 12 6

4

A B A B

L L L L

D D D D

L L L L

D B D D

L L L LK

A B

L L

D D

sym L L

D

L

Linear stiffness matrix of isotropic materials:-

3 2 3 2

2

3 2

0 0 0 0

12 6 0 12 6

4 0 6 2

0 0

. 12 6

4

EA EA

L L

EI EI EI EI L L L L

EI EI EI

L L LK

EA

L

EI EI sym

L L

EI

L


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Appendix (B)

Flow chart of solution of nonlinear system of equations by iteration

Linear Non Linear FEM

Documents

Transcript of Linear Non Linear FEM