Derivation of the Shell Element , Ahmed Element , Midlin Element in Finite Element Analysis- Hani Aziz Ameen

8/6/2019 Derivation of the Shell Element , Ahmed Element , Midlin Element in Finite Element Analysis- Hani Aziz Ameen

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Derivation of the shell element , Ahmed Element , Dr. Hani Aziz Ameen

Midlin Element in Finite Element Analysis

1

Derivation of the shell element ,

Ahmed Element , Midlin Element

in Finite Element Analysis

Asst. Prof. Dr. Hani Aziz Ameen

Technical College - BaghdadDies and Tools Eng. Dept.

E-mail:[email protected]

www.mediafire.com/haniazizameen

1- Definition of a Shell

Shell is defined as an object which, for the purpose of stress

analysis may be considered as the materialization of a curved surface [1].

This definition implies that the thickness of a shell must be small

compared with its other dimensions, but it does not require the smallness

be extreme. Most shells, of course, are made of a solid material, and

generally, it will be assumed that the material is isotropic and elastic.
http://www.mediafire.com/haniazizameenhttp://www.mediafire.com/haniazizameenhttp://www.mediafire.com/haniazizameen


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In most cases, a shell is bounded by two curved surfaces, the faces. The

thickness "ts" of the shell may be assumed the same everywhere or it may

vary from point to point. The middle surface of a shell is defined as the

surface which passes midway between the two faces. If the shape of the

middle surface and thickness are known, then the shell is geometrically

fully described.

2- Types of Shell Elements:

The analysis of shells with an arbitrarily defined shape presents an

intractable analytical problem. If, in addition, the shell is a thick one in

which the shear deformation is significant, the applicability of a classical

approach becomes a question. In many engineering structures, these

difficulties might be overcome, if satisfactory (and hopefully optimized)

designs are ever to be achieved. Over the years, much has been written on

the various attempts to produce efficient, accurate, and reliable shell

elements in (FEM). Three distinct classes of shell elements have emerged

[2]:

1. Flat, plate-like elements which are sometimes called facet elements

because they approximate the curved shell by a faceted surface.

2. Curved shell elements founded on some shell theory.

3. Degenerated shell elements based on the three-dimensional continuum

theory.

In the first approach, the shell is replaced by an assemblage of flat plate

elements which are either triangular or quadrilateral in shape, as the

triangle shown in figures (2) (a) and (b) [3]. Each plate element is

connected in some fashion to those surrounding it and undergoes both in-

plane (membrane) and bending (flexural) deformations. The method has


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3

the disadvantage that there is no coupling between bending and stretching

with each element. The coupling only appears indirectly through the

degrees of freedom at the nodal points linking the adjacent elements.

Consequently, a large number of elements must be used to achieve

satisfactory accuracy. Although of the certain shortcomings in the

approach, facet elements are very efficient for the approximate analysis

of many shell structures.

The second type includes curved shell elements based upon thin shell

theories of classical mechanics, consisting of the analysis of deep or

shallow shells. A commonly used theory of deep shells is based upon the

strain-displacement relationships ofNovozhilov [4]. On the other hand,

specialized theories of shallow shells follow the simplified strain-

displacement relationships of Vlasov [5]. The later method is more

approximate than the former, but accurate results have been obtained,

even when shallow-shell concepts were applied to deep shells, Cowper

et.al. [6]. Figure (3) shows the geometry for an arbitrary shallow shell

element. Although the above type of shell elements are quite popular, but

they also suffer from various limitations associated with the lack of

consistency in many shell theories and also with the difficulty in finding

appropriate deformation idealization which allows truly strain-free rigid

body movement.

Finally, the curved elements (third class) for shell analysis can be

devised by specializing three dimensional solid elements to be thin in one

direction while introducing constraint conditions on nodal displacements.

As examples, the hexahedron and the pentahedron in figures (4) (a) and

(b) can be specialized to become quadrilateral and triangular shell

elements that are curved in three dimensional spaces. The characteristicsand analysis of this type is illustrated in the following section.


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4

Figure (2) Flat-facet element: (a) Membrane components (b) Flexural

components.

Figure (3) Shallow shell element geometry and coordinate system [6].

3

1

3

1

w

yz

x

u i

2iq

1iq

yz

x

(a)

(b)

5iq

4iq

3iq

i

zy

x

1

b

a2

c

3

2

),(

1

3

wu

v


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5

3- Finite Element Formulation of Shell Element

Among all of the shell elements, the Ahmad [7] type

"degenerated" isoparametric shell element based on an independent

translational and rotational displacement interpolation, has become the

most popular in shell analyzation. In this element, the Mindlin [8] theory

is employed, where the "normal" to the middle surface of the element is

constrained to remain straight (but no longer normal) after deformation in

order to overcome the numerical difficulty associated with a large

stiffness ratio through the thickness direction. The strain energy

associated with the stress perpendicular to the middle surface is also

neglected. By adopting the isoperimetric geometric description, the

element can be used to represent thin and thick shell components with

arbitrary shapes.

Figure (5) (a), shows the original isoperimetric hexahedron element

which has a quadratic formula defining its geometry, where ui, vi, and wi

are translations in the global coordinates as demonstrated by Weaver and

Johnston [9]. In order to convert this hexahedron to a thin curved

quadrilateral element for the analysis of shell, one can first form a flat

rectangular solid by making the curvilinear coordinate's , , and

orthogonal and the dimension is small. The resulting element appears in

figure (5) (b) is the rectangular parent of shell element before constraints.

Note that groups of three nodes occur at the corners, while pairs of nodes

are at the mid-edge locations of the element. By invoking the former

constraints, each group and pair of nodes could be converted to a single

node on the middle surface, as shown in figure (5) (c), where i and i are

small rotations about two local tangential axes. The Relationships

between the nodal displacement at a corner and mid-edge with a node of


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6

shell elements can be seen more clearly in figures (6) (a), (b) and (c). So,

the nine nodal, translations in figure (6) (a), can

Figure (4) Specialization of solids [6]:

(a)Hexahedron (b) Pentahedron.

Figure (5) Specialization of Hexahedron[9]:

(a) Isoparametric hexahedron (b) Rectangular parent as shell elementbefore constraints(c) Constrained nodal displacements.

(a)(b)

2

8

1619

1511

7

13

1

5

4

1

3

12

1814

6

9

17

z

Yxi

iw

iuiv

2a2b

v,

u,st

w,

iw

i

iv

iiu

ki

j

(a)

(b)

(c)


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7

be related to the five nodal displacements in figure (6) (c) by the

following 95 constraint matrix:

00100

02

010

20001

00100

02

010

20001

00100

00010

00001

s

s

s

s

ai

t

t

t

t

G . (1)

Where, "ts" is the element thickness.

Similarly, the six nodal translations in figure (6) (b) are related to

the five nodal displacements in figure (6) (c) by the 65 constraint

matrix.

00100

0

2

010

20001

00100

02

010

20001

s

s

s

s

bi

t

t

t

t

G . (2)

If each of these constraint materials in four locations were applied, the

number of nodal displacements could be reduced from 606494 to

4058 . In the following articles, the direct formulations of shell

element in the manner described by Cook [10] will be pursued.


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8

Figure (6) Nodal displacement: (a) Corner of rectangular element (b)Mid-edge of rectangular element (c) Node of shell element, [9].

j

6

54

2/st

2/st

i

3

21

k 8

79

(a)

j

i

k

1

3

2

5

4

6

(b)

k

j

i

51

3

42

(c)

z

y

x


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9

4- Geometric Definition of the Element

Figure (7) shows the geometric layout of the shell element, in

which the global coordinates of any point take the form,

8

1

3

3

38

1 2ii

i

i

s

i

i

i

i

i

i

n

mt

N

z

y

x

N

z

y

x

. (3)

Where, ,iN represents the interpolation shape functions, and they are

illustrated and explained in appendix (A1).

In addition, the terms iii nm 333 ,, are the direction cosines of vector iV3 that

is normal to the middle surface and spans the thickness "t s" of the shell at

node i . Figure (7) (b) shows this vector which is obtained as:

s

i

i

i

kj

kj

kj

i t

n

m

zz

yy

xx

V

3

3

3

3

....... (4)

Point j and k in the figure are at the upper and lower surfaces of the shell,respectively. In a computer program, the direction cosines for iV3 must be

given as data.

Generic displacements at any point in the shell elements are taken

to be in the directions of global axes. Thus, the generic displacements

vector is:

w

vu

u .. (5)

On the other hand, nodal displacements consist of these same translations

(in global directions) as well as two small rotations i and i about two

local tangential axes x and y , as indicated in figure (7).

Hence,


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11

i

i

i

i

i

i w

v

u

. (6)

8,.......,3,2,1i

This represents the nodal displacements vector. Generic displacements in

terms of nodal displacements are:

i

i

i

s

i

i

i

i

i

i

i

tN

w

v

u

N

w

v

u

2

8

1

8

1

.... (7)

In this formula, the symbol i denotes the following matrix:

ii

ii

ii

i

nn

mm

12

12

12

... (8)

Column 1 in this matrix contains negative values of the direction cosines

of the second tangential vector iV2 , and column 2 has the direction cosines

for the first tangential vector iV1 (see figure (7)). These vectors are

orthogonal to the vector iV3 and to each other. As infinity of vector

directions, normal to a given direction can be generated, a particular

scheme has been devised to ensure a unique definition. This is given in

appendix (A2) [11].

Figure (7) shows the local generic translations u and v (in the

directions of iV1 and iV2 ) due to the nodal rotations i and i , respectively.

Their values are:

i

stu 2

& istv

2 .. (9)


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1

Contributions of these terms to the generic displacements at any point are

given by the second summation in equation (7).

The displacement shape functions in equation (7) may be cast into

the matrix form:

i

i

s

i

s

i

s

i

s

i

s

i

s

i N

nt

nt

mt

mt

tt

N

12

12

12

22100

22010

22001

.................................... (10)

8,......2,1i

In order to isolate terms in sub-matrix iN that multiplied by , let:

iAi NN

00100

00010

00001

. (11)

And,

is

ii

ii

ii

Bi Nt

nn

mmN2

000

000

000

12

12

12

..... (12)

Then,

BiAii NNN ... (13)

And, the shape function matrix becomes:

BA NNN . (14)

The last of these formulas will later be used to drive the consistent mass

matrix.

The 33 Jacobian matrix required for this element is given by:


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13

zyx

zyx

zyx

J .... (15)

The concept of Jacobian matrix is shown in appendix (A3). Derivatives in

the Jacobian matrix are found as follows:

8

1

3

8

1

3

8

1

8

1

2

2

i

isi

i

i

i

is

i

i

i

ii

tNxNx

tNx

Nx

8

1

32i

is

i lt

Nx

and so on.

The inverse of J becomes:

zzz

yyy

xxx

JJ

*1

..... (16)


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14

5- Strain Calculations

Certain derivatives of the generic displacements (equation (7)) with

respect to local coordinates are needed. These derivatives are listed in a

column vector of nine terms as follows:

Transformation of these derivatives to global coordinates requires that the

inverse of the Jacobian matrix be applied. Therefore,

w

u

u

J

J

J

z

w

y

ux

u

.....00

00

00

*

*

*

... (18)

Multiplying the terms in this equation yields,

17...........................

2

2

000

00

00

000

00

00

000

00

00

8

1

12

12

12

12

12

12

12

12

12

is

is

i

i

i

i

iiii

ii

iii

ii

iii

iiii

ii

iii

ii

iii

iiii

ii

iii

ii

iii

t

t

w

vu

nNnN

nN

nNN

nN

nNN

mNmN

mN

mNN

mN

mNN

NN

NNN

NNN

w

w

w

v

v

v

u

u

u


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15

19......................................................

00

00

00

00

00

00

00

00

00

8

1

12

12

12

12

12

12

12

12

12

i

i

i

i

i

i

iiiii

iiiii

iiiii

iiiii

iiiii

iiiii

iiiii

iiiii

iiiii

w

v

u

ngngc

neneb

ndnda

mgmgc

memeb

mdmda

lglgc

leleb

ldlda

z

w

y

wx

wz

v

y

vx

vz

u

y

ux

u

In which,

ii

i

NJ

NJa

*

12

*

11

ii

i

NJ

NJb

*

22

*

21

ii

i

NJ

NJc

*

32

*

31 . (20)

iisi NJat

d*

132

iisi NJbt

e*

232

iisi NJct

g*

332

The strain displacement vector may be written as:


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16

z

u

x

wy

w

z

vx

v

y

uz

wy

vx

u

xz

yz

xy

z

y

x

. (21)

Substituting equation (19) into equation (21) gives,

B . .. (22)

Where, is the nodal displacement vector (equation (6)), and B is the

strain-displacement matrix. The ith part of matrix B may be written as:

iiiiiiiiii

iiiiiiiiii

iiiiiiiiii

iiiii

iiiii

iiiii

gndgndac

nemgnemgbc

mdemdeab

ngngc

memeb

dda

B i

1122

1122

1122

12

12

12

0

0

0

00

00

00

. (23)

As with the sub-matrix iN , the terms in sub-matrix iB that multiply

could be isolated to get,

BiAii BBB ..... (24)

Sub-matrices AiB and BiB are composed from equations (20) and (23),

but the actual details are omitted. Altogether, one can have

BA BBB ... (25)

which will be convenient when determining the stiffness matrix for theshell element.


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17

6- Stress Calculations

Stress-strain relationships in the local (primed) axes for isotropic

material take the form:

D (26)

Where D is the stress-strain matrix or the elasticity matrix in local axes

[10] or,

xz

zy

yx

z

y

x

xz

zy

yx

z

y

x

k

k

2

100000

02

10000

002

1000

000000

00001

00001

12

(27)

In which, and are Young's modulus and Poisson's ratio, respectively.

The factor k included in the last two shear terms is taken as 1.2, and its

purpose is to improve the shear displacement approximation [12].

Coordinate transformation is applied to convert matrix D to the global

matrix D by using the 66 strain transformation matrix T .

Thus,

TDTDT .. (28)

Where,


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18

31131133113131313

23323322332323232

12212211221212121

333333

2

3

2

3

2

3

222222

2

2

2

2

2

2

111111

2

1

2

1

2

1

3

2

1

222

222

222

nnnmnmmmnnmm

nnnmnmmmnnmm

nnnmnmmmnnmmnnmmnm

nnmmnm

nnmmnm

T .(29)

Where etcm ..,.........,,, 1321 are directional cosines of the primed axes with

respect to the global axes. The concept of coordinate transformation is

illustrated in appendix (A4) [9]. To evaluate the matrix T at anintegration point, the directional cosines for vector 321 ,, VVV must be found

at that point. This may be done with the following sequence of

calculations:

.11 norm

Je , .213 norm

JJe , 132 eee .

In these expressions, the vector .1 normJ denotes the first row of the

Jacobian matrix normalized to a unit length, and so on.

Equation (28) would be more efficient if the third row and column of

matrix D (corresponding to z and z ) were deleted, along with the

third row of matrix T .

7- Stiffness Matrix [k]e for Shell Element

The stiffness matrix eK could be obtained from calculating the

strain energy U. Applying the principle of variational approach [13], the

strain energy for the element may be written as:

V

e

T

e dVU 2

1..... (30)


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2

through the thickness. While, the products aT

a BDB and bb BDB2

may be integrated with respect to at once. Thus, equation (35) is

reduced to

1

1

1

13

22 ddJBDBBDBK bba

T

ae (36)

Hence, the first part of matrix eK in equation (36) is due to the transverse

shearing deformations, whereas the second part is associated with flexural

deformations. To evaluate the integrals in equation (36) numerically,

Gauss-quadrature technique [15] is adopted using two integration points

in each of and coordinates. This method is explained in appendix

(A5).

8- Consistent Mass Matrix [M]e for Shell Element

The mass matrix eM might be obtained by calculation the kinetic

energy KE as follows:

For any body of infinitesimal mass dm and velocity vector eq , the kinetic

energy is:

dmqqKE eT

e 2

1...... (37)

Since, dVdm ..... (38)

Where, is the mass density. Then,

dVqqKE eV

T

e 2

1.. (39)

But, ee Nq . (40)


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1

Where,ee

dt

d ...... .. (41)

Substituting equation (40) gives

eV

TT

e dVNNKE

2

1.... (42)

Or,

eeTe MKE 21 .. (43)

Where, eM is the consistent mass matrix [16] and is defined as:

V

T

e dVNNM ... (44)

Where, N is the shape function matrix (equation (14)).

Substituting equations (14) and (34) into equation (44) yields,

1

1

1

1

1

1

dddJNNNNM BAT

BAe ... (45)

Employing the same techniques used in simplifying equation (35) gives,

1

1

1

1322 ddJNNNNM BTBATAe .... (46)

Hence, the first part of matrix eM consists of the translational inertias,

and the second part gives a rotational (or rotary) inertia. The integrals in

equation (46) are evaluated in the same manner as in integrals equation

(36).


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9- Equation of Motion for Finite Element

The equation of motion (or dynamic equation) can be derived, using the

energy balance principle which involves that "the summation of the

structure energies is stationary", i.e., the summation of kinetic energy,

dissipation energy, strain energy and potential energy is stationary, or

StationaryPEUDEKE (47)

If these energies are defined in terms of a nodal displacement vector ,

then,

0

PEUDEKE

.. (48)

The first and third terms of equation (47) are obtained by equations (43)

and (32), respectively. Now, the second and the fourth terms will be

created. The dissipation energy DE depends upon the nature of damping,

and for the case of viscous damping, a damping matrix ec can be defined

such that:

eeT

e CDE

2

1 . (49)

Finally, the potential energy PE (with the absence of body forces) can be

written as:

tFWPE eT

e (50)

Where, tFe is the nodal forces vector.

Substituting equations (32), (43), (49), and (50) in equation (48) gives,


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3

)(

2

1

2

1

2

1

2

1tFKCM e

T

e

T

ee

T

eee

T

eee

T

e

e

=0. (51)

The derivation of the first term of the upper equation is obtained as

follows:

eeeeeeTeee

eee

T

e

e

MMdt

dM

dt

dM

2

1

2

1

The other terms can be easily derived to get the final form of the dynamic

equation of finite element.

)(tFKCM eeeeeee ... (52)

Equation (52) with zero damping becomes,

)(tFKM eeeee (53)


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4

Appendices

Appendix (A1): The Shape Functions

Shell Element Shape Functions.

In the finite element analysis, the region of interest is subdivided

into a number of sub-regions known as elements, which are defined by

the locations of their nodal points. The main concept here is that the

geometry of the element is defined using the nodal coordinates and the

shape functions, which are used to interpolate the main unknowns (i.e.,

displacement) with an isoparametric formulations in terms of a non-

dimensional element coordinates ,, which varies from -1 to +1 over

the element called natural coordinates. This coordinate system is

particularly useful when the adoption of numerical integrations is

considered to evaluate any integrals which are required during the

stiffness matrix calculations for example. Figure (A1.1) shows the

rectangular parent element (a) of the isoparametric quadrilateral element

(b) which is geometrically similar to the shell element used. Since 8-node

elements have been employed, and according to Pascal's triangle, the 8

terms polynomials are assumed for the displacement function as follows

2

8

2

7

2

65

2

4321 ccccccccu (A.1.1)

(And similar polynomials for other displacements)

Several methods could be used in obtaining the displacement shape

function. Hence, a direct substituting method will be used by applying the

above equation to each node in the element. Thus,

2

1181

2

17

2

16115

2

14131211 ccccccccu

2

2282

2

27

2

26215

2

24232212 ccccccccu

And so on, substituting the values of ii , (where i the node number,

i 1,2,.8) which are listed in table (A.1.1) into the above equations


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5

and solve them simultaneously, the values of the constants1c ,

2c ,..etc.

can be calculated. Substituting these constants into equation (A.1.1), the

displacement shape functions are obtained as follows:

Figure (A1.1) (a) Rectangular parent element (b) isoparametric element.

74

8

1 25

6

3

1

1

1

1

(a)

(b)

4

8

1 5

2

6

3

7

y

x

y

x


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6

8877665544332211 NuNuNuNuNuNuNuNuu (A.1.2)

Or, it can be written in the form:

8

1i

iiuNu , and similar for other displacements.

Where, the shape functions could be written as follows:

1114

11

N

1114

12 N

1114

13 N

1114

14 N

112

1 25N

26 112

1 N

112

1 27N

28 112

1 N

These shape functions must satisfy two conditions:

1-

8

1

1),(i

iN

2-

jiif

jiifN jii

0

1),(

Table (A.1.1) Nodal coordinates for shell element

i 1 2 3 4 5 6 7 8

i -1 1 1 -1 0 1 0 -1

i -1 -1 1 1 -1 0 1 0


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7

The geometric interpolation functions are taken to be the same as the

displacement shape functions obtained. Physically, this means that the

natural coordinates ,, are curvilinear, and all sides of the element

become quadratic curves.

Thus,

8

1i

iixNx ,

8

1i

iiyNy ,

8

1i

iizNz

Fluid Element Shape Functions

Figure (5) (a) in chapter three shows the isoparametric hexahedron

element used in the fluid finite element formulation. Both types of shape

functions for the fluid element could be obtained in the same manner as

for the shell element. Thus, the velocity shape functions viN could be

written as :

8,......2,121118

1000000 iNvi

002 1114

1 viN 19,17,11,9i

002 1114

1

viN 20,18,12,10i

002 1114

1 viN 16,15,14,13i

Where,

i

0 , i0 , i0

The values of i , i , and i required in these formulas are given in table

(A1.2).


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8

Table (A1.2) Nodal coordinates for fluid element.

i i i i i i i i

1 -1 -1 -1 11 0 1 -1

2 1 -1 -1 12 -1 0 -1

3 1 1 -1 13 -1 -1 0

4 -1 1 -1 14 1 -1 0

5 -1 -1 1 15 1 1 0

6 1 -1 1 16 -1 1 0

7 1 1 1 17 0 -1 1

8 -1 1 1 18 1 0 1

9 0 -1 -1 19 0 1 1

10 1 0 -1 20 -1 0 1

Also, the pressure shape functions piN could be written as :

8,......2,11118

1000 iNpi

Where,

i

0 , i0 , i0

The values of i , i , and i required in these formulas are given in table

(A1.2).


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9

Appendix (A2): Unique Definition of Directions Normal to a

Reactor [56]

If a vector 3V is defined (by its three Cartesian components for

instance), it is possible to erect an infinity of mutually perpendicular

vectors orthogonal to it. Some scheme therefore has to be adopted to

eliminate this choice, and indeed quite arbitrary decisions can be made

here. A convenient scheme adopted in the present work related the choice

to the global x and y axis.

If i for instance is the unit vector along the x axis,

31 ViV

makes the vector 1V perpendicular to the plane defined by the direction

3V and the x axis. As 2V has to be orthogonal to both 1V and 3V , one can

have,

132

VVV

To obtain unit vectors in the three directions, 1V , 2V , and 3V are simply

divided by their scalar lengths, giving the unit vectors:

1v ,

2v , and 3v .


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32

Appendix (A3): The Jacobian Matrix [J]

In calculating the element strain, certain derivatives of the generic

displacement wvu ,, with respect to the global coordinates zyx ,, are

needed. But, since the shape functions are expressed in terms of the local

coordinates ,, , it is useful to use a convenient transformation as

follows:

The chain rule of partial differential calculus for differentiation of shape

functions ,,N with respect to, , and produces:

z

z

Ny

y

Nx

x

NN

z

z

Ny

y

Nx

x

NN

z

z

Ny

y

Nx

x

NN

In matrix form:

z

Ny

Nx

N

zyx

zyx

zyx

N

N

N

For this arrangement, the terms in the coefficient matrix are easily

obtained by differentiating equation (3) . This array is called the Jacobianmatrix [J] which contains the derivatives of the global coordinates with

respect to the local coordinates. Thus,

z

Ny

Nx

N

J

N

N

N

i

i

i

i

i

i


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31

Where,

Jacobian matrix [J] =

zyx

zyx

zyx

Finally, to find the global derivatives, [J] must be inverted as:

i

i

i

i

i

i

N

N

N

J

z

Ny

NxN

1][

Where, [J]-1

is the inverse of the Jacobian matrix.


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33

Any state of stress and strain may be expressed in either coordinate

system as and in x y z coordinates or as and in x y z

coordinates. Stresses and are arranged in the order:

zx

yz

xy

z

y

x

... (a)

xz

zy

yx

z

y

x

(b)

Also the strains and :

zx

yz

xy

z

y

x

..(c)

xz

zy

yx

z

y

x

.(d)

Stress-strain relationships may be written in either coordinate system, as

][D . (A4.1)

Or,

][D .. (A4.2)

Where, D and D are the stress-strain matrix (see sec.3.3.4) in the either

coordinate system, respectively. Now, the transformation of D to D

and vice versa can be implemented through the following approach [13].

For the convince in rotation of axes, the stress vector may be recast

into the form of a symmetric 33 matrix as follows:

zzyzx

yzyyx

xzxyx

. (A4.3)

Then, the rotation-of-axes transformation for stress can be stated as:


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34

TRR (A4.4)

Where, [R] is the rotation matrix and has the form

333

222

111

nm

nm

nm

R

.. (A4.5)

In this matrix, the terms1 , 1m and so on, are the directional cosines.

Similarly, the strain vector may be recast as the symmetric 33 matrix:

zzyzx

yzyyx

xzxyx

. (A4.6)

For which the rotation transformation is:

TRR . (A4.7)

Now, rewrite the expanded result of equation (A4.7) as:

T ..... (A4.8)

In this equation, the strains are in the forms of equation (A4.c) and (A4.d)

instead of equation (A4.6). The 66 strain transformation matrix T in

equation (A4.8) is as follows:

311313133113131313

233232322332323232

122112211211212121

333333

2

3

2

3

2

3

222222

2

2

2

2

2

2

111111

2

1

2

1

2

1

222222

222

nnmnnmmmnnmmnnmnnmmmnnmm

nnmnnmmmnnmm

nnmmnm

nnmmnm

nnmmnm

T (A4.9)

The form of the stress transformation matrix T is derived from the

argument that during any virtual displacement, the resulting increment in

strain energy density oU must be the same regardless of the coordinate

system in which it is computed. Thus,

TT

oU . (A4.10)


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35

Then, substituting the transposed incremental from of equation (A4.8)

into equation (A4.10) to obtain:

TTT

T . (A4.11)

Hence, one can conclude that,

T (A4.12)

Where,

TTT . (A4.13)

Thus, the stress transformation matrix T

is proven to be the transposedinverse of the strain transformation matrix T .

Now, to transform the stress-strain relationships from one set of

coordinates to another, substitute equation (A4.8) and equation (A4.12)

into equation (A4.1) to obtain:

TDT . (A4.14)

Then, premultiply equation (A4.13) by 1

T and use equation (A4.12) to

find:

TDTT ... (A4.15)

Or,

D ... (A4.16)

Where,

TDTD T .... (A4.17)

which represents the transformation of D to D .

The reverse transformation is:

TTDTD ... (A4.18)


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36

Appendix (A5): Gaussian Quadrature[6]

The process of computing the value of a definite integral (see

figure A5.1 (a)) from a set of numerical values of the integral is called

numerical integration.

2

1

)(

x

x

x dxxfI .. (A5.1)

The problem is solved by representing the integrand by an interpolation

formula and then integrating this formula between specified limits. When

applied to the integration of a function of a single variable, the method is

referred to as mechanical quadrature. The most accurate quadrature

formula in common usage is that of Gauss, which involves unequally

spaced points that are symmetrically placed. To apply Gauss's method,

the variable is changed from x to the dimensionless coordinate with its

origin at the center of the range of integration, as shown in figure (A5.1

(b)). The expression for x in term of is

21 112

1xxx . (A5.2)

Substitution of equation (A5.2) into the function in equation (A5.1) gives,

)()( xf ..... (A5.3)

Also,

dxxdx )(21 12 .... (A5.4)

Then, substituting equations (A5.3) and (A5.4) into equation (1) and

changing the limits of integration yields,

1

1

12 )()(2

1 dxxIx ... (A5.5)

Gauss's formula for determining the integral in equation (A5.5) consists

of summing the weighted values of )( at n specified points as follows:


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37

Figure (A5.1) Gaussian quadrature.

Figure (A5.2) Infinitesimal area in natural coordinates.

)(xf

)(xf

0 1x 2x x

)(

1 0 1

)(

(a)

(b)

d

rr

d

x

x

r

y

y

x

z

k

j

i

d

y

d

r

dA

dr

dx

dy

d

rr


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38

1

1 1

)()(n

j

jjRdI

Or,

)(...............)()(2211 nnRRRI .. (A5.6)

In this expression, j is the location of integration point j relative to the

center, jR is a weighting factor for point j , and n is the number of points

at which )( is to be calculated. The values of these parameters are listed

in table (A5.1).

For quadrilaterals in Cartesian coordinates, the type of integration to be

performed is:

2

1

2

1

),(

x

x

y

y

dxdyyxfI (A5.7)

However, this integral is more easily evaluated if it is first transformed to

the natural coordinates for a quadrilateral. One can accomplish this by

expressing the function f in terms of , and using the limits -1 to 1

for each of the integrals. In addition, the infinitesimal area dxdydA must

be replaced by an appropriate expression in terms of d and d . For this

purpose, figure (A5.2) shows an infinitesimal area dA in the natural

coordinates. Vector r locates a generic point in the Cartesian coordinates

x and y , as follows:

yixiyxr ..... (A5.8)

The rate of change of r with respect to is :

jy

ixr

... (A5.9)

Also, the rate of change of rwith respect to is:

jy

ixr

.... (A5.10)


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39

Table (A5.1) Coefficients for Gaussian quadrature.

n i iR

1 0.0 2.0

2 0.5773502692 1.0

3 0.7745966692

0.0

0.5555555556

0.8888888889

4 0.8611363116

0.3399810436

0.3478548451

0.6521451549

5 0.9061798459

0.53884693101

0.0

0.2369268851

0.4786286705

0.5688888889

6 0.9324695142

0.6612093865

0.2386191861

0.1713244924

0.3607615730

0.4679139346

7 0.9491079123

0.7415311856

0.4058451514

0.0

0.1294849662

0.2797053915

0.3818300505

0.4179591837

8 0.9602898565

0.7966664774

0.5255324099

0.1834346425

0.1012285363

0.2223810345

0.3137066459

0.3626837834


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42

When multiplied by d and d, the derivatives in equations (A5.9) and

(A5.10) form two adjacent sides of the infinitesimal parallelogram of area

dA in the figure. This area may be determined from the following vector

triple product:

kdr

dr

dA

. (A5.11)

Substitution of equations (A5.9) and (A5.10) into equation (A5.11)

produces,

ddyxyx

dA

(A5.12)

The expression in the parentheses of equation (A5.12) may be written as

a 22 determinate. That is,

ddJdd

yx

yx

dA

.. (A5.13)

In which J is the determinate of the 22 Jacobian matrix. Thus, the new

form of the integral in equation (A5.7) becomes,

1

1

1

1

),( ddJfI .. (A5.14)

Two successive applications of Gaussian quadrature result in,

n

k

n

j

kjkjkj JfRRI1 1

),(),( .. (A5.15)

Where, jR and kR are weighting factors for the function evaluated at the

point kj , . Integration points for ,3,2,1n and 4 (each way) on a

quadrilateral are illustrated in figure (A5.3).


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41

Figure (A5.3) Integration points for quadrilateral (a) 1n (b) 2n

(c) 3n (d) 4n (each way).

For hexahedral in Cartesian coordinates, the type of integral to be

evaluated has the form:

dxdydzzyxfI ),,( (A5.16)

Before integrating, one can rewrite the functions in terms of the natural

coordinates , , and and using the limits -1 to 1 for each of the

integrals. In addition, the infinitesimal volume dxdydzdV must be

(a) (b)

(c) (d)


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4

replaced by an appropriate expression of d , d , and d . By employing

the same procedure used for the quadrilaterals, dV can be written as:

dddJddd

zyx

zyx

zyx

dV

... (A5.17)

In which, J is the determinate of the 33 Jacobian matrix. Hence, the

revised form of the integral in equation (A5.16) becomes,

1

1

1

1

1

1

),,( dddJfI . (A5.18)

Three successive applications of Gaussian quadrature yield,

),,(),,(1 1 1lkjlkjlk

n

l

n

k

n

jj JfRRRI ... (A5.19)

Where jR , kR , lR are weighting factors for the function evaluated at the

point ),,( lkj . Integration points for 3,2,1n and 4 (each way) are:

1,8,27, and 64, respectively.


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43

References

[1] Flugg W., "Stress in shell", Springer-Verlag, 4th

ed., New York, 1967.

[2] Hou-Cheng H., "Static and Dynamic Analysis of Plate and Shells",

Springer-Verlag, Berlin, 1989.

[3] Zienkiewiz O.C. and Taylor R.L., "The Finite Element Method",

Butterworth-Heinemann, 5th

ed., Oxford, 2000.

[4] Novozhlov V.V., "The Theory of Thin Shells", 2nd

ed., Noordhoff

Ltd., Groningen, Netherlands, 1964.

[5] Vlasov V.Z., "General Theory of shells and its Applications in

Engineering", NASA TTF-99, 1964.

[6] Cowper G.R., Lindberg G.M., and Olson M.D., "A shallow Shell

Finite Element of Triangular Shape", International Journal of

Solution Structures, Vol., 6, No.8, pp. 1133-1156, 1970.

[7] Ahmed S., Itons B.M. and Zienkiewic O.C., "Analysis of Thick and

Thin Shell Structure by Curved Finite Elements", International

Journal of Numerical and Mechanical Engineering, Vol.2, No.3, pp.

419-451, 1970.

[8] Mindlin R.O., "Influence of Rotary Inertia and Shear in Flexural

Motion of Isotropic and Elastic Plates", Journal of Applied

Mechanics, Vol.73, pp.31-38, 1951.

[9] Weaver W.Jr. and Johnston P.R., "Finite Element for Structural

Analysis", Prentice-Hall, Englewood Cliffs, N.J. 1984.

[10] Cook R.D., "Concepts and Applications of Finite Element Analysis",

2nd

ed., Wiley, New York, 1981.

[11] Ahmad S., Irons B.M., and Zienkiewic O.C, "A Simple Matrix-

Vector Handling Scheme for Three-Dimensional and Shell

Analysis", IJNME, Vol.2, No.4, pp.509-522, 1970.


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Documents

Derivation of the Shell Element , Ahmed Element , Midlin Element in Finite Element Analysis- Hani Aziz Ameen