Development of higher order triangular element for accurate stress resultants in plated and shell structures 6

CHAPTER 5 Nonlinear Continuum Spectral Shell Element In Chapter 4, we presented a detailed derivation of HT-CS and HT-CS-X elements, the associated linear finite element formulation an

CHAPTER 5

Nonlinear Continuum

Spectral Shell Element

In Chapter 4, we presented a detailed derivation of HT-CS and HT-CS-X elements, the associated linear finite element formulation and several challenging benchmark problems to assess the performances of higher order elements Based on the performances of HT-CS and HT-CS-X elements in

‘Discriminating and Revealing Test Cases’, it was found that HT-CS-X elements are more robust in handling a wide range of shell problems Further, the element was also tested for its capability to handle stress resultants in challenging linear plate bending problems (Morley’s skew plate and corner supported square plate) Having assessed the performance of HT-CS-X element in linear plate\shell analysis, in this chapter, we shall extend the linear finite element formulation of HT-CS-X elements to a nonlinear formulation that caters for large deflection problems The performance of the developed nonlinear continuum shell element will be assessed in several geometric nonlinear shell problems Moreover, its superiority over lower order elements

in handling stresses in the nonlinear regime will be discussed

The large deflection analysis of shells has drawn the attention of many researchers due to its importance in engineering practice There have been numerous research studies on the geometric nonlinear analysis of shells that

al.,1989; Saleeb et al, 1990; Sze et al., 1999; Balah and Al-Ghamedy, 2002;

Arciniega and Reddy, 2007) Most of the researchers presented the nonlinear load versus deflection response of shell structures computed from various kinds of shell elements that were developed to tackle nonlinear behaviour The ability of finite elements in handling stresses in nonlinear regime has received relatively less attention The accurate prediction of stress distributions in the nonlinear region is very crucial from a design point of view Furthermore, the correct estimation of peak stresses and their localization in nonlinear region provides a sound basis to perform reliability and failure studies which decide the safety of a structural component Achieving good and reliable levels of accuracy in the highly nonlinear range with lesser computational resources is certainly not possible with lower order finite elements Although the nonlinear load versus deflection response of the structure may be traced accurately with coarser mesh designs of lower order finite elements, one may require very fine mesh designs in order to achieve good accuracy of stress values Furthermore, the accuracy of stresses predicted by lower order elements may be highly erroneous in problems which involve steep stress gradients Hence, we use higher order finite elements that have enriched shape functions and render many advantages over conventional lower order finite elements such as the accommodation of high aspect ratio elements, better prediction of stresses with coarser meshes, ability to handle steep stress gradients, less sensitivity to input data (locking mechanisms) and lesser computational resources as compared to lower order finite elements

In this chapter, we shall present the nonlinear finite element formulation of

a continuum shell element that accounts for large deflections and moderate

rotations of shell structures The accuracy of the proposed element will be verified via several nonlinear benchmark problems and their superior performance over conventional lower order shell finite elements will be illustrated in the nonlinear stress analysis of the shell problems

This chapter has been organized into four main sections The first section deals with the development of nonlinear finite element model for shells under the framework of Total Lagrangian approach followed by a description on nonlinear solution algorithms adopted in the present work In the second section, we verify the performance of HT-CS-X elements in selected nonlinear shell benchmark problems Following this, we will present the performance of HT-CS-X elements in handling stresses in nonlinear region much efficiently as compared to ABAQ US S8R lower order shell elements The performances of HT-CS-X element and ABAQUS S8R element will be compared in the context of accuracy and distribution of stresses, relative ease of mesh designs and number of degrees of freedom to achieve a smooth stress variation In the last section of this chapter, we will present a detailed nonlinear analysis of laminated composite hyperboloid shells which are challenging due to their negative value of Gaussian curvature and complex behaviour

5.1 Description of motion

Consider a deformable body of known geometry, constitution and loading that occupies an initial configuration Â0in which a particle X occupies the position

X having Cartesian coordinates (X, Y, Z) After the application of loads, the

body assumes a new position x in the deformed configuration  having

coordinates (x, y, z) The objective is to determine the final configuration of a

the final configuration  from a known initial configuration Â0 is to assume

that the total load Pmax is applied in increments so that the body occupies several intermediate configurations Âi (i =1,2,……) prior to attaining the final configuration The magnitude of load increments should be such that the computational procedure that is employed to trace the response of the body (such as the Newton Raphson method and the Arc- length method) is capable

of predicting the deformed configuration at the end of each load step In the determination of an intermediate configuration Âi, one may use any of the previously known configurations Â0, Â1, … ,Âi-1 as the reference configuration ÂR If the initial configuration is used as the reference configuration with respect to which all quantities are measured, it is called the Total Lagrangian description

We consider three equilibrium configurations of the body namely, Â0, Â1 and Â2 which correspond to three different loads Â0 denotes the initial undeformed configuration, Â1 denotes the last known deformed configuration and Â2 denotes the current deformed configuration to be determined It is assumed that all variables such as displacements, strains and stresses are known up to configuration Â1 The objective is to develop a formulation to determine the displacements and stresses of the body in the deformed configuration Â2

In the next section, we present the strain and stress measures employed in the Total Lagrangian formulation A detailed derivation of relevant stress and strain measures for a Total Lagrangian approach can be seen in standard textbooks on nonlinear finite element formulation (Reddy, 2004) Hence we

present the final equations that are necessary for the development of nonlinear finite element model

5.1.1 Green strain tensor

We adopt the Green-Lagrange strain tensor or simply referred to as the Green strain tensor to measure the deformation of a body The Green strain tensor is symmetric and is expressed as follows:

E= (F T ×F-I)= (C-I)

2

12

1

(5.1)

where C = F T ×F is called the right Cauchy-Green deformation tensor and F

is the deformation gradient tensor defined as

÷ =

ø

öçè

û

ù

êêêêêêê

z x z

z

y y

y x y

z

x y

x x x

0 1 0 1 0 1

0 1 0 1 0 1

0 1 0 1 0 1

¶

¶+

¶

¶

=

J K I K I

J J

I J

I

X

u X

u X

u X

u E

2

1

(5.3)

where u I denotes the component of displacement The subscripts I,J,K take

the values of 1,2,3 (u1 =u,u2 =v and u3 =w ) u, v denote the in-plane displacements and w denotes the transverse displacements Likewise, X I

denotes the components of Cartesian coordinates (X1 = X ,X2 =Y

andX3 =Z)

The Green-strain components can be written as

ú

úû

ùê

êë

é

÷ø

öçè

æ

¶

¶+

÷ø

öçè

æ

¶

¶+

÷ø

öçè

æ

¶

¶+

¶

¶

=

2 2

v X

u X

ùê

êë

é

÷ø

öçè

æ

¶

¶+

÷ø

öçè

æ

¶

¶+

÷ø

öçè

æ

¶

¶+

¶

¶

=

2 2

v Y

u Y

ùê

êë

é

÷ø

öçè

æ

¶

¶+

÷ø

öçè

æ

¶

¶+

÷ø

öçè

æ

¶

¶+

¶

¶

=

2 2

v Z

u Z

¶

¶

¶

¶+

¶

¶

¶

¶+

¶

¶+

w Y

v X

v Y

u X

u X

v Y

u

21

ø

öç

¶

¶

¶

¶+

¶

¶

¶

¶+

¶

¶+

w Z

v X

v Z

u X

u X

w Z

u

E x z

21

ø

öç

¶

¶

¶

¶+

¶

¶

¶

¶+

¶

¶+

w Z

v Y

v Z

u Y

u Y

w Z

The equation of equilibrium has to be derived for the deformed configuration

of the body, i.e at configuration Â2 Since the geometry of the deformed configuration is unknown, the equations are written in terms of the known reference configuration Â0 In doing so, it becomes necessary to introduce various measures of stress These stress measures emerge when the elemental volumes and areas are transformed from the deformed configuration to the undeformed configuration In the Total Lagrangian approach we use the

second Piola-Kirchhoff stress tensor denoted as S The second Piola-Kirchhoff stress S can be expressed in terms of Cauchy stress tensor s (which is defined

to be the current force per unit deformed area) by the following transformation

S =JF-1×s×F-T (5.5)

where, J denotes the determinant of the deformation gradient tensor F The

second Piola-Kirchhoff stress tensor S, gives the transformed current force per unit undeformed area The stress tensor S is symmetric whenever the Cauchy stress tensor s is symmetric For details on the transformation of various

measures one may refer to books by Bathe (1996) and Reddy (2004)

Having mentioned about the strain and stress measures, it can be shown that the rate of internal work done in a continuous medium in the current configuration can be expressed as (Reddy 2004):

Thus the second Piola-Kirchhoff tensor S is the work conjugate to the rate of

the Green-Lagrange strain tensor E& The following notations are used in this

chapter A left superscript on a quantity denotes the configuration in which the quantity occurs and a left subscript denotes the configuration with respect to which the quantity is measured For example, j i H refers to a quantity H (say

displacements, stresses) that occurs in configuration Âi but is measured in configuration Âj When the quantity is measured in the same configuration, the left subscript is omitted The left superscript will be omitted for all incremental quantities that occur between configurations Â1 and Â2 The right subscript refers to the components of Cartesian coordinate system

When the body deforms under the action of externally applied loads, a

particle X occupying position (X, Y, Z) in configuration Â0 moves to a new

position x having coordinates (x, y, z) in configuration Â2 The components of

particle X can be written as 0x=(0x,0y,0z) and that of x can be written

as2x=(2x,2y,2z) The total displacements of a particle X in the two

configurations Â1 and Â2 can be written as:

01u i =1x i-0x i , (i =1,2,3) (5.7 a)

02u i = 2x i-0x i , (i =1,2,3) (5.7 b)

The displacement increment of a point from configuration Â1 to Â2 is

u i = 02u i-01u i , (i =1,2,3) (5.8)

5.1.3 Green Strain tensor and stress tensor for various configurations

The components of Green strain tensor in configurations Â1 and Â2 are given

in terms of displacements as:

÷

÷ø

öç

çè

¶

¶+

¶

¶

=

j k i k i

j j

i ij

x

u x

u x

u x

u

1 0 0

1 0 0

1 0 0

1 0 1

öç

çè

¶

¶+

¶

¶

=

j k i k i

j j

i ij

x

u x

u x

u x

u

2 0 0

2 0 0

2 0 0

2 0 2

0

2

1

(5.9 b)

The incremental Green-Lagrange strain components 0e ij which are obtained

in moving from configuration Â1 to Â2 are given as

0e j = 0e j + 0h j (5.10) where, 0e j are linear components of strain increment tensor expressed as

÷

÷ø

öç

çè

¶

¶

¶

¶+

¶

¶+

¶

¶

=

j k i k j k i k i

j j

i j

x

u x

u x

u x

u x

u x

u

1 0 0 0

0

1 0 0

0 0

k j

x

u x

u

0 0 0

For geometrically linear analysis, only two configurations Â1 = Â0 and Â2 are involved Thus 1u i =0and 2u i =u i The terms involving products of

(¶u k ¶0x i) and (¶u k ¶0x j) are small and hence are neglected Consequently,

the linear components of strain increment tensor 0e jbecome the same as the components of the Green- Lagrange strain tensor 02E ij and both reduce to infinitesimal strain components

÷

÷ø

öç

çè

æ

¶

¶+

¶

¶

=

i j j

i j

x

u x

The second Piola-Kirchhoff stress tensor components in configurations Â1 and

Â2 are denoted by 01S j and 02S j respectively They are related by the following equation

02S j = 01S j+0S j (5.14) where, 0S j are the components of the K irchhoff stress increment tensor and are given by:

0S j=0C ijkl 0e k l (5.15)

ijkl

C

0 denotes the incremental constitutive tensor with respect to configuration

Â0 In the present work, since we deal with geometric nonlinearity, the components of the constitutive matrix are the same as that obtained for a linear analysis

5.1.4 Total Lagrangian Formulation

Having defined the necessary terms involved in the Total Lagrangian formulation, we now present the final equations of equilibrium The equations

finite element model considered herein are derived from the principle of virtual displacements The detailed derivation of the equations of equilibrium can be found in the book by Reddy (2004)

The weak form of the equilibrium equation that is suited for the development of displacement finite element model based on the Total Lagrangian formulation is given to be:

0 0

1 0 1

i i

0 2

0 0 2

(sum of body force and traction force) The total stress components 01S j are evaluated using the following constitutive relation

01S j = 0C j k l 01E k l (5.17) where, 01E k l are the Green-Lagrange strain components described in Eq (5.4)

5.2 Finite Ele ment Model Continuum Shell Element

The equilibrium equation that is required for the development of nonlinear displacement based degenerated shell finite element model for a solid continuum is given in Eq (5.16) In order to derive the finite element equations for a shell element, the first step is to select appropriate interpolation (shape) functions for the displacement field and geometry The coordinates and displacements are interpolated using the isoparametric concept which involves the same interpolation functions This is done to ensure the displacement compatibility across element boundaries is preserved at all configurations

The degenerated shell element is deduced from the 3D continuum element

by imposing two kinematic constraints, i.e (i) the straight line normals to the midsurface before deformation remain straight but not necessarily normal after deformation, allowing for the effect of transverse shear deformation and (ii) transverse normal strain/stress components are neglected, allowing for the conversion of the 3D shell model into a 2D model Moreover, the strains are assumed to be small

The layout of HT-CS-X is shown in Fig 5.1 with Lobatto nodal

distribution in the element In Fig 5.1, r and s denote the curvilinear coordinates of the element and t denotes the coordinate in the thickness

ùê

êêë

é

ïþ

ïýü

ïî

ïí

ì

÷ø

öçè

æ +ïþ

-ïýü

ïî

ïí

ì

÷ø

öçè

æ +Q

top i i i

i

i

z y

x t z

y

x t s

r z

y

x

2

12

1,

i y

x , and z denote the coordinates in the x, y and z direction at node i In Fig i

5.1, E , ˆ1 E and ˆ2 Eˆ3 denote the unit vectors defined along the global (x, y, z)

coordinate system Let V3i be a vector connecting the upper and lower points

of the shell’s normal at node i, i.e

bottom i i i

top i i

i i

z y x

z y

x V

ïþ

ïýü

ïî

ïí

ì-ïþ

ïýü

ïî

ïí

eˆ3 = 3 3 Equation (5.18) can now be written as

úúúû

ùê

êêë

é

+ï

þ

ïýü

ïî

ïí

ìQ

=úúúû

ùê

êêë

é

+ï

þ

ïýü

ïî

ïí

ìQ

i i

i i i

midsurface i

i i

i

z y

x s r V

t z

y

x s r z

y

x

3 45

1 3

45

(5.20) where h i =h= V3i is the thickness of the shell at node i The curvature of the

shell is described in terms of the shell director vector V3i When the director

vector has no components in the x and y directions, the corresponding unit

vector e ˆ becomes a unit vector in the global z direction and the resulting 3i

structure represents a plate

The displacements and incremental displacements are given by:

=

)(

2)

,

45 1 1 0 1

i k i k k i k

k i

2)

,

45 1

1

i k i k k i k

k i

i

(01[K L]+01[K NL]){D e}=2{ }R -01{F} (5.23) where, {De}is the vector of nodal incremental displacements from time t to time t + Dt in an element, and 10[K L]{De},01[K NL]{De} and 01{F are obtained }

by evaluating the following integrals

K B C B L d V

V

T L L

0 1

0 0 1 0 1

K B S B NL d V

V

T NL NL

0 1

0 1 0 1

0 1

0[ ]=ò0 [ ] [ ] [ ] (5.24b)

F B S d V

V

T L

0 1 0 1 0 1

0{ }=ò0 [ ] {ˆ} (5.24c)

in which 01[B L] and 10[B NL] are the linear and nonlinear strain–displacement transformation matrices, 0[C is the incremental stress-strain material property ]matrix, 01[S is a matrix of 2] nd Piola-Kirchhoff stress components, 01{Sˆ}is a vector of these stresses and 2{ }R is the vector of applied loads All matrix elements are defined with respect to the configuration Â0 and the solution at

Â2 is sought Equation (5.23) represents the nonlinear equilibrium equation and has to be iterated for each time step until it satisfies a specified tolerance

Hence, Eq (5.22) can be expressed as

2)

ùê

é

+

-Q

=

i e

e h

t u s r

e e e e

E

e E

3 1 2 3 1 2 1 1

ˆˆˆ,ˆˆ

ˆˆ

u u

where { } { k k k }T

i e

u q1 q2

=

D ,(i = 1, 2, 3, k = 1, 2,…, 45) is the vector of nodal

incremental displacements (five per node).1[ ]H is the incremental displacement interpolation matrix given by

úúúúúú

û

ù

êêêêêê

ë

é

Q-Q

Q

Q-Q

Q

Q-Q

KK

KK

k k k k

k k k

k k k k

k k k

k k k k

k k k

e h t e

h t

e h t e

h t

e h t e

h t H

23 1 13

1

22 1 12

1

21 1 11

10

0

2

12

100

2

12

100

(5.30)

z y z x y z

w x

w z

v y

v x

v z

u y

u x

u u

[ ]

ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú

¶

¶ +

¶

¶ +

¶

¶

¶

¶ +

w y

v z

v y

u z

u

x

w z

w x

v z

v x

u z

u

x

w y

w x

v y

v x

u y

u

z

w z

v z

u

y

w y

v y

u

x

w x

v x

u

A

1 0 1 0 1

0 1

0 1

0 1

0

1 0 1

0 1

0 1

0 1 0 1

0

1 0 1

0 1

0 1 0 1

0 1

0

1 0 1

0 1

0

1 0 1

0 1

0

1 0 1

0 1

0

9

1

1 0 1

0 0

0 1

0 1

0

0 0

1 0 1

1 0 0

0 0

0 0

0 0

0 1

0 0

0

0 0

0 0

0 0

1

(5.32) The vectors { }0u and { }0e are related to the displacement increments at nodes

by the following equations

{ } [ ]{ } [ ] [ ] { }e

H u

{ } [ ]{ } [ ] [ ] [ ] { }e [ ] { }e

B H

A u

z y x

z y x

T

0 0 0

0 0 0

0 0 0

000000

0000

00

000000

[ ]

úúúúúúú

û

ù

êêêêêêê

û

ù

êêêêêêê

v t

u

s

w s

v s

u

r

w r

v r

u

J

z

w z

v z u

y

w y

v y u

x

w x

v x u

u j

1 1 1

1 1 1

1 1 1

1 0

0 1 0 1 0 1

0 1 0 1 0 1

0 1 0 1 0 1

û

ù

êêêêêêê

y t x

s

z s

y s x

r

z r

y r x

J

0 0 0

0 0 0

0 0 0

0

(5.39)

[ ]J

0

is computed from the coordinate definition of Eq (5.20) The derivatives

of displacement 1u iwith respect to the coordinates r , s,and t can be computed

from Eq (5.27) In the evaluations of element matrices in Eq (5.24), the integrands of 10[ ]B L , 0[C , ] 01[B NL], 01[S , ] 1[H and ] 01{Sˆ}should be expressed

in the same coordinate system(0x,0y,0z)which is the global coordinate system

or a local coordinate system that is aligned in the shell element’s direction(x¢,y¢,z¢)

The number of stress and strain components is reduced to five since we neglect the transverse normal components of stress and strain Hence, the global derivatives of displacements, 01u,jwhich are obtained in Eq 5.38, are transformed to the local derivatives of the local displacements along the orthogonal coordinates (shell element aligned by the following relation

3 3 01 , 3 3

1 1

1

1 1

1

1 1

1

][][]

=

úúúúúúú

û

ù

êêêêêêê

v z

u

y

w y

v y

u

x

w x

v x

u

(5.40)

where [q]Tis the transformation matrix between the local coordinate system(x¢,y¢,z¢)and the global coordinate system(0x,0y,0z) at the integration point The transformation matrix [ ]q is obtained by interpolating the three orthogonal unit vectors (1eˆ1,1eˆ2,1eˆ3)at each node:

úúúú

û

ù

êêêê

ë

é

QQ

Q

QQ

Q

QQ

Q

=

å å

å

å å

å

å å

1 231 45

1 131

45

1 321 45

1 221 45

1 121

45

1 311 45

1 211 45

1 111

]

[

i

i i i

i i i

i i

i

i i i

i i i

i i

i

i i i

i i i

i i

e e

e

e e

e

e e

e

Since the element matrices are evaluated using numerical integration, the transformation must be performed at each integration point during the numerical integration In order to obtain the strain-displacement matrix,01[ ]B L , the vector of derivative of incremental displacements {0u}needs to be evaluated Equations (5.38) can be used again except that 1u iare replaced by

i

u and the interpolation equation for u (Eq 5.24) is applied i

Next, the development of the matrix of material stiffness 0[C¢] will be discussed The material stiffness matrix for shell element composed of orthotropic material layers with the principal material coordinates (x,y,z)oriented arbitrarily with respect to local coordinate system (x¢,y¢,z¢) (with

z

z = ¢) is formulated For a kth lamina of a laminated composite shell the matrix of material stiffness is given by:

úúúúúú

û

ù

êêêêêê

45 44

66 26 16

26 22 12

16 12 11

) ( 0

000

000

00

00

00

][

C C

C C

C C C

C C C

C C C

where

) ( )

(

44 2 55 2 55

44 55 45

55 2 44 2 44

66 2 2 2 12

22 11 2 2 66

66 12

2 2 22

2 11 2 26

22 4 66 12

2 2 11 4 22

66 12

2 2 22 2 11 2 16

12 4 4 66

22 12 2 2 12

22 4 66 12

2 2 11 4 11

sin,

cos

)(

,

)(

)2(

)]

2)(

(][

)2(2

)]

2)(

([

)(

)4(

)2(2

k

m

Q n Q m C

Q Q mn C

Q n Q m C

Q n m Q

Q Q n m C

Q Q n m Q

m Q n mn C

Q m Q Q n m Q n C

Q Q n m Q n Q m mn C

Q n m Q

Q Q n m C

Q n Q Q

n m Q m C

=

¢

+-

-+

=

¢

+-

+-

=

¢

++

+

=

¢

+-

-

-=

¢

++-

+

=

¢

++

2 12 12

1 v v

E v Q

21 12

2 22

1 v v

E Q

where K is the shear correction factor (assumed to be 5/6), E is the modulus i

in the x direction, i G , (i j) are the shear moduli in the j x - i x plane and j n ij

are the associated Poisson’s ratios (Reddy, 2004)

To evaluate the element matrices defined in Eqs (5.24), we employ the Gauss quadrature Since we are dealing with laminated composite structures, integration through the thickness involves individual lamina O ne way is to use a 1D Gauss quadrature through the thickness direction Since the

constitutive relation 0[ ]C is different from layer to layer and is not a continuous function in the thickness direction, the integration should be performed separately for each layer The integration of stiffness matrix in the in-plane direction follows the procedure used for integrating the stiffness matrix in triangular plate elements

Element external load vector

In this section, we present the element load vector due to gravity loads, uniform normal surface pressure and uniform vertical loading

(a) Gravity loads

Gravity loads are computed from uniform weight density r throughout the

element From Eq (5.21), the vertical displacement is given by

=

))ˆˆ

(2)

,

45 1

k k k k k k k

e h t P

k k k

k k k

k

k ò ò ò

-ïïïï

þ

ïïï

ýü

ïïïï

î

ïïï

íì

Q-

Q

Q

=

1 0 1 0 1 1

23 2 13 2

ˆ2

1

ˆ2

1

00

By integrating the above equation analytically with respect to t we get,

{ } ò ò

ïïïþ

ïïýü

ïïïî

ïïí

ìQ

=

1 0 1 0

00

00

ds dr J

P k r k (5.45)

(b) Uniform normal surface pressure

In order to evaluate the nodal loads for surface pressure, we require the displacements normal to the surface to of the shell This is given by

ïþ

ïýü

ïî

ïí

ì

=

3 3 3

n m

l w v u

u n (5.46)

Let q be the normal pressure applied at the top surface of the shell Hence t n

=1 By substituting for the displacements u,v,w from Eq (5.21), we get,

þ

ïïï

ýü

ïïïï

î

ïïï

íì

++

Q-

++

Q

QQQ

=

1 0 1 0

23 2 3 22 2 3 21 2 3

13 2 3 12 2 3 11 2 3 3 3 3

ˆˆ

ˆ2

1

ˆˆ

ˆ2

e n e m e l h

e n e m e l h

n m l

q P

k k

k k k

k k

k k k

k k k

n

33 2

* 23 2

*

J J

ùê

êêë

é

=

-* 33

* 23

* 13

* 32

* 22

* 12

* 31

* 21

* 11 1

0

J J J

J J J

J J J

(c) Uniform vertical load

Let q denote the vertical pressure applied at the top surface t = 1 By making v

use of Eq (5.43) for the vertical displacement w, the nodal load vector is given

by

{ } ò ò

ïïïï

þ

ïïï

ýü

ïïïï

î

ïïï

íì

Q-Q

Q

=

1 0 1 0

23 2 13 2

ˆ2

1

ˆ2

1

00

dA

e h

e h q

P

k k k

k k k

k v

k (5.49)

5.3 Nonlinear solution procedure

In the present work, the nonlinear equilibrium equation given in Eq (5.23) is solved using two techniques, namely, (a) the Newton-Raphson method and (b) the arc length method The latter method is used when the load deflection path

to be traversed contains snap-through, snap-back points or bifurcation points

In order to trace the nonlinear response of the structure up to a load say Pmax,

we divide the total load Pmax into several load steps The nonlinear equilibrium equation is solved using an appropriate nonlinear solution procedure at a

particular load step say q Having obtained the response of the structure at load step q, we proceed to the next load step (q+1) and we iterate to obtain the solution of the equilibrium equation at this step The two nonlinear solution procedures adopted in the present work will be discussed in the following sections

5.3.1 Newton-Raphson method

The finite element equation can be expressed in the following manner

[ e( { }e ) ] { } { }e e

F u

u

K = (5.50) where [ ]e

K is the element stiffness matrix which depends on the solution vector { }e

F is the vector of element nodal forces Equation (5.50) can

be written as K( )u ×u= F or alternatively R( )u =0 Hence the Eq (5.50) can

be expressed as:

R( )u =K( )u ×u-F (5.51)

We assume that we know the solution of Eq (5.51) at the iteration index

(m-1) The solution for unknown displacement variables have to be obtained for

the next iteration which is m Therefore, R( )u given in Eq (5.51) is expanded about the known solution (m- 1)

u by using Taylor’s series

1 1

=+

×

÷÷

ø

öççè

æ

¶

¶+

×

÷ø

öçè

æ

¶

¶+

=

-

-HOT u

u

R u

u

R u

R

u

R

m m

u u

=

×-

m

u u

K F u

K u

R u

K u

(01[K L]+01[K NL]){De}=2{ }R -01{F} (5.56)

The tangent stiffness matrix [ ]K T is given by [K T]= 01[K L]+01[K NL] and the

residual or the imbalance force vector is given by{ } { } 1{ }

0 2

F R

(5.54) gives the increment of displacement vector uat the m th iteration and hence the total solution is given by:

u(m) =u(m-1)+d u(m) (5.57) The iteration is continued until the following convergence criterion is reached, i.e

m

m m

1 Evaluation of element stiffness matrix [ ]e

F

2 Computation of element tangent stiffness matrix [ ]e

T

K and residual force vector (imbalance force vector) { }e

3 Assembly of element tangent stiffness matrix and residual force vector

to obtain global tangent stiffness matrix [ ]K T and residual force vector { }R

4 Application of boundary conditions on the assembled set of equations

5 Solution of the assembled set of equations using standard linear solvers

6 Updating of the solution vectors for use in the subsequent iterations and load steps

7 Checking for convergence

8 If the convergence criterion is met, the load is increased to the next load step value and steps 1 to 6 are repeated If the convergence criterion is not met, we check for the maximum number of iterations set If the maximum number of iterations allowed is exceeded, the computation terminates Otherwise, the computation begins with the next iteration (i.e step 1)

5.3.2 Arc-length method

The Newton-Raphson method works efficiently for most of the nonlinear system of solutions But when the nonlinear equilibrium path contains limit points, the method fails This is due to the reason that in the vicinity of the limit point, the tangent stiffness matrix becomes singular and the iteration procedure diverges Wempner (1971) and Riks (1972) presented a procedure called as the arc-length method to predict the nonlinear equilibrium path through limit points The method introduces a modification to the Newton-Raphson method to control progress along the equilibrium path In the arc-length method, the load increment for each load step is considered to be an unknown and is solved as a part of the solution A detailed explanation of the Riks method is given by Reddy (2004) The basic idea of the Riks method is to introduce a load multiplier that increases or decreases the intensity of applied load Hence the load is assumed to vary proportionally during the response calculation

{ }F =l{Pmax} (5.58a) The assembled equations associated with Eq (5.51) are given to be

{R( )u ,l}=[ ]K { }u -l{Pmax}=0 (5.58b)

The residual vector { }R is considered to be a function to both the unknown

displacement vector and the load factorl We assume that the solution

u l at the (m-1) iteration and q load step is known The th

residual force vector { }R is expanded by invoking the Taylor series

,

1 1

1 1

= +

×

÷÷

ø

ö çç è

æ

¶

¶ +

×

÷ ø

ö ç è

æ

¶

¶ +

=

-

-

-HOT u

u

R R

u R u

m m

n

m m

q m q m

q

m

l l

(5.59)

HOT denotes the higher order terms involving increments of load factor and

displacements which we neglect in the derivation due to their smaller

magnitude Equation (5.59) can be written as:

q T

m q m

q T

d , { }d uˆ q =[K T]-1{P max} and d l( )q m is the load

increment which is to be determined at every iteration For the first iteration of

any load step, d l( )q0 is given by the following expression,

( ) ( { } { }q)

T q q

d d

l

d = ±D (5.62) where, Ds qis the length of an arc whose center is at the current equilibrium

computed using the formula

Ds q = ( { }Du T q- 1 { }Du q- 1) (5.63)

where, { }Du q-1 is the converged solution increment of the previous load step

In order to control the number of iterations taken to converge in the subsequent load steps, the arc length is scaled using the formula,

Ds q =Ds q-1×(I d I0) (5.64)

1

-Ds q is the arc length used in the last iteration of ( )th

q 1- load step I is the d

number of desired iterations (usually taken to be < 5) and I is the number of 0

iterations required for convergence in the previous step Thus, Eq (5.64) will automatically give small arc lengths in regions having severe nonlinear behaviour and longer arc lengths when the response is nearly linear The maximum arc length is usually specified within the program in order to avoid convergence of the solution at higher equilibrium paths For the first iteration

of the first load step, we use

(usually { } { }u0 = 0 )

The incremental load parameter dl at the th

m iteration, d l( )q m is computed from the following quadratic equation

a1( )dl 2 + 2a2dl +a3 =0 (5.66) The coefficients a1, a2and a are given as follows: 3

u

T m q m

u

a = d + D - × d + D - - D The solving of Eq (5.66) yields two solutions namely, dl and1 dl Thus we 2

obtain two vectors { }( )m

D -1 is selected If both the roots of dl and 1 dl give 2

positive values of the product, we choose the one giving the smallest value

of(-a3 2a2)

The solution increment is updated by

{ } { }( ) { }( )m

q m

q m

D -1 (5.67) The total solution at the current load step is given by

{ } { } { }( )m

q q

u = -1 + D (5.68) Further details regarding the stepwise implementation of the arc-length method is discussed by Reddy (2004)

Having discussed about the nonlinear finite element formulation of a degenerated continuum shell element and two efficient nonlinear solution procedures that are adopted in the present work, the next section begins with the assessment of HT-CS-X elements in several nonlinear shell benchmark problems

5.4 Nonlinear Benchmark Shell Problems

The accuracy of HT-CS-X elements in geometrically nonlinear analysis will

be verified by using the following three well-known nonlinear benchmark example problems

1 Cylindrical shell panel clamped along its four edges and subjected to

2 Cylindrical shell supported by rigid diaphragms along the curved edges

5.4.1 Clamped Cylindrical Shell Panel

Consider a cylindrical shell panel of length a = 508 mm, radius R = 2540 mm, thickness h = 3.175 mm and the half- angle subtended by the cylindrical shell

panel f =0.1 radians (see Fig 5.2) The material properties assumed are: the Young’s modulus E =3.103´103 N mm2and Poisson’s ratio n = 0.3 The cylindrical shell is clamped along its four edges and is subjected to a uniform

surface pressure of intensity q Owing to symmetry of the problem, we

consider one quadrant of the cylindrical shell as the computational domain The problem at hand is to trace the nonlinear load versus deflection response

of the shell panel at point A up to a maximum load of intensityqmax =3´10-3N mm2 In order to achieve this, the total load qmaxis divided into 16 equal load steps and the Newton-Raphson method discussed in Section 5.3.1 is employed to determine the load versus deflection response of the shell

The results of transverse deflection at point O obtained using a 3´3 mesh of HT-CS-X elements will be compared with the following three types of finite elements:

1 HMSH5 which is a geometric nonlinear five noded hybrid strain

element developed by Saleeb et al (1990)

2 CSH9 which is a geometric nonlinear nine noded quasi conforming shell element developed by Guan and Tang (1995)

3 NSQ9 which is a geometric nonlinear nine noded hybrid stabilized degenerated shell element developed by Sze (1994)

The aforementioned shell elements are specially tailored to overcome shear and membrane locking mechanisms in shells whose thickness-to-length ratio

h/a is small

Figure 5.3 presents the nonlinear plot of load versus transverse deflection that is monitored at point O of the cylindrical shell panel It can be observed that the results of transverse deflection obtained by using HT-CS-X elements are closer to NSQ9 results and show minor deviations with respect to CSH9 and HMSH5 results The nonlinear load versus deflection response predicted

by CSH9 and HMSH9 elements are stiffer at certain points as compared to NSQ9 and HT-CS-X elements However HMSH5 elements seem to conform well to the results of HT-CS-X and NSQ9 elements in the later part of the load versus deflection response The value of maximum transverse deflection at the end of final load step qmaxobtained using HT-CS-X elements differ by 2% with respect to NSQ9 and HMSH5 elements Thus the present higher order triangular element HT-CS-X works on par with some of the well-known shell finite elements that are claimed to be free of shear and membrane lockings which manifest when the thickness of the shell is small The deformed configuration of the cylindrical shell panel at the initial and final load steps are shown in Fig 5.4

Fig 5.2 Geometry of clamped cylindrical shell panel

Fig 5.3 Nonlinear load versus deflection response at point O of a clamped cylindrical shell panel subjected to uniform surface pressure

Initial undeformed configuration Deformed final configuration

Fig 5.4 Deformation of a clamped cylindrical shell panel subjected to uniform surface pressure

5.4.2 Cylindrical Shell Supported on Rigid Diaphragms

This example problem is concerned with a cylindrical shell supported on rigid diaphragms and subjected to two point loads that are located at diametrically opposite points as shown in Fig 5.5 The convergence behaviour of HT-CS-X elements was assessed in a similar problem (see Section 4.3.1 of Chapter 4) Herein we assess the performance of HT-CS-X elements in predicting the nonlinear behaviour of the aforesaid shell under the action of point loads The

cylindrical shell has a length a = 200 mm, radius R = 100 mm and thickness h

= 1 mm It should be noted that ratio of thickness-to- length (h/a) considered in

this problem is less as compared to the previous example problem The material properties assumed are: the Young’s modulus 4 2

to a maximum load valuePmax =730.4N The Newton-Raphson method is employed to trace the nonlinear response in 16 equal load steps

Fig 5.5 Geometry of a cylindrical shell supported on rigid diaphragms and

Figure 5.6 shows a comparison of nonlinear load versus transverse deflection

at point O obtained using a 3´3 mesh of HT-CS-X elements, 4´4 mesh of NSQ9 and 16´16 mesh of HMSH5 elements The results of transverse deflection at point O obtained using HT-CS-X elements are seen to match the nonlinear deflection results of HMSH5 elements exactly The nonlinear load versus deflection response predicted by NSQ9 elements are seen to be slightly stiffer with respect to HT-CS-X and HMSH5 results This minor difference can be attributed to the use of a coarser 4´4 mesh of NSQ9 elements as compared to a finer 16´16 mesh of HMSH5 elements The maximum value of transverse deflection (at point O) computed at the end of the last load stepPmaxusing HT-CS-X and HMSH5 are exactly the same The maximum

transverse deflection w obtained at Pmaxusing HT-CS-X and HMSH5 elements differ by 6% with respect to the results obtained from NSQ9 elements Figure 5.6 shows the undeformed and deformed configuration of the cylinder at the initial and final load steps

Fig 5.6 Nonlinear load versus deflection response at point O of a cylindrical shell supported by rigid diaphragms and subjected to two diametrically opposite point loads

Initial undeformed configuration Deformed final configuration Fig 5.7 Deformation of a cylindrical shell subjected to two diametrically

opposite point loads

5.4.3 Hinged cylindrical shell panel

In this example problem, we study the behaviour of cylindrical shell panels

subjected to a central point load (Fig 5.8) The cylindrical shell panel has a

length a = 508 mm, radius R = 2540 mm, the half-angle subtended by the shell

panel is f =0.1 radians and thickness h = 25.4, 12.7 and 6.35 mm The nature

of nonlinear load versus deflection response at point O will be studied for the

following two problems namely,

· An isotropic cylindrical shell panel having h/a = 0.05, 0.025 and

0.0125

090

900

cylindrical shell panels having h/a = 0.0125

The layers are assumed to have equal thicknesses For isotropic cylindrical

shell panels, the material properties assumed are: the Young’s modulus

2

75

E = and Poisson’s ratio n = 0.3 For laminated composite

cylindrical shell panels, the material properties assumed are: the Young’s

modulus along circumferential direction E1 =1100 N mm2 , the Young’s

given to beG13 =440 N mm2 , G12 =G23 =660 N mm2 and Poisson’s ratios

25.0

Fig 5.8 Geometry of a hinged cylindrical shell panel subjected to point load

This example is a well-known benchmark problem for nonlinear analysis of cylindrical shells and is particularly significant due to its snapping behaviour This problem has been considered by various researchers (Sabir and Lock,

1972; Crisfield, 1981; Ramm, 1981; Simo et al., 1990; Brank et al., 1995; Sze

et al., 2004 and Arciniega and Reddy, 2007) Since the nonlinear response of

the structure contains snapback and snap through points, the arc-length method discussed in Section 5.3.2 will be used to follow the nonlinear path The arc length method is a very efficient method to tackle problems that involve multiple limit points

Isotropic cylindrical shell panel

Figures 5.9,5.10 and 5.11 present the nonlinear load versus deflection response

of an isotropic cylindrical shell panel having thicknesses h = 25.4 mm, 12.5

mm and 6.35 mm respectively The results of maximum transverse deflection

deflection is recorded up to a maximum load value Pmax =22kN for h = 25.4

mm, Pmax =3kN for h = 12.5 mm and 6.35 mm The results of transverse

deflections are compared with a 4×4 mesh of Q25 elements (which is a quadrilateral higher order element having degree of polynomial shape functionsp=4) developed by Arciniega and Reddy (2007) It can be observed that the transverse deflection results obtained by using HT-CS-X elements show excellent agreement with the results reported by Arciniega and Reddy

It can be seen from Fig 5.9 that the nonlinear load versus deflection

response of an isotropic cylindrical shell panel having thickness h = 25.4 mm contains no limit points As the thickness h of the cylindrical shell panel

decreases, the nonlinear response changes dramatically Figures 5.9 and 5.10 show the nonlinear load versus deflection behaviour of the cylindrical shell

panel under point load with thickness h =12.7 and 6.35 mm, respectively