u4 )يعيبط تبصتخم( x يطخ يعبرم نبملا x3 y3 2h y, v يعيبط

دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 1

x, u

y, v

1 (x1, y1)

(u1, v1)

2 (x2, y2)

(u2, v2)

3 (x3, y3)

(u3, v3)

2b

4 (x4, y4)

(u4, v4)

2h

1 (1, 1)

(u1, v1)

2 (1, 1)

(u2, v2)

3 (1, +1)

(u3, v3) 4 (1, +1)

(u4, v4)

( , ) ( , )x y x yu N d 31 2 4

31 2 4

Node 2 Node 3Node 1 Node 4

00 0 0

00 0 0

NN N N

NN N N

Nwhere

(مختصبت طبيعي) المبن مربعي خطي

توابع شكل المبن مربعي در دستگبه مختصبت طبيعي


)1)(1(

)1)(1(

)1)(1(

)1)(1(

41

4

41

3

41

2

41

1

N

N

N

N

113 4at node 11

113 4at node 2

1

113 4at node 31

113 4at node 4

1

(1 )(1 ) 0

(1 )(1 ) 0

(1 )(1 ) 1

(1 )(1 ) 0

N

N

N

N

Delta function

property

4

1 2 3 4

1

14

14

[(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )]

[2(1 ) 2(1 )] 1

i

i

N N N N N

Partition of unity

1 (1, 1)

(u1, v1)

2 (1, 1)

(u2, v2)

3 (1, +1)

(u3, v3) 4 (1, +1)

(u4, v4)

( )( )1j j j4

N 1 1


توابع شكل المبن مربعي در دستگبه مختصبت طبيعي


Rectangular elements have limited application

Quadrilateral elements with unparallel edges are more useful

Irregular shape requires coordinate mapping before using

Gauss integration


به دستگبه مختصبت طبيعي (x,y)انتقبل از دستگبه مختصبت


Coordinate mapping

2 (x2, y2)

y

x 1 (1, 1) 2 (1, 1)

3 (1, +1) 4 (1, +1)

3 (x3, y3) 4 (x4, y4)

1 (x1, y1)

Physical coordinates Natural coordinates

( , ) ( , ) u N d (Interpolation of displacements)

( , ) ( , ) e X N x (Interpolation of coordinates)




( , ) ( , ) e X N x

where x

y

X ,

1

1

2

2

3

3

4

4

coordinate at node 1




e

x

y

x

y

x

y

x

y

x

)1)(1(

)1)(1(

)1)(1(

)1)(1(

41

4

41

3

41

2

41

1

N

N

N

N

ii

i

xNx ),(4

1

ii

i

yNy ),(4

1




Substitute x = 1 into ii

i

xNx ),(4

1

2 (x2, y2)

y

x 1 (1, 1) 2 (1, 1)

3 (1, +1) 4 (1, +1)

3 (x3, y3) 4 (x4, y4)

1 (x1, y1)

1 12 32 2

1 12 32 2

(1 ) (1 )

(1 ) (1 )

x x x

y y y

or

)()(

)()(

2321

3221

2321

3221

yyyyy

xxxxx

Eliminating , 3 2 1 12 3 2 32 2

3 2

( ){ ( )} ( )

( )

y yy x x x y y

x x




Remarks

Shape functions used for interpolating the coordinates are

the same as the shape functions used for interpolation of

the displacement field. Therefore, the element is called an

isoparametric element.

Note that the shape functions for coordinate interpolation

and displacement interpolation do not have to be the same.

Using the different shape functions for coordinate

interpolation and displacement interpolation, respectively,

will lead to the development of so-called subparametric or

superparametric elements.


دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود

,

, centroid

c

c

c c

x x b ξ

y y h η

Nodes are at 1 1 in ξ - η space

x y

8

1 2

3 4

x

h

x = +1 x = -1

h = +1

h= -1

Parent Element

Natural Coordinates x-h

استخراج مبتريس سختي المبن ايسوپبرامتريك


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

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

1 2 3 4

5 6 7 8

1 2 3 4

1 2 3 4

x = a +a ξ +a η+a ξη

y = a +a ξ +a η+a ξη

1x = 1- ξ 1- η x + 1+ξ 1- η x + 1+ξ 1+η x + 1- ξ 1+η x

4

1y = 1- ξ 1- η y + 1+ξ 1- η y + 1+ξ 1+η y + 1- ξ 1+η y

4

1

1

2

1 2 3 4 2

1 2 3 4 3

3

4

4

x

y

x

N 0 N 0 N 0 N 0 yx

0 N 0 N 0 N 0 N xy

y

x

y

( )( )

( )( )

( )( )

( )( )

1

2

3

4

1- ξ 1- ηN

4

1+ξ 1- ηN

4

1+ξ 1+ηN

4

1- ξ 1+ηN

4



1

1

2

1 2 3 4 2

1 2 3 4 3

3

4

4

u

v

u

N 0 N 0 N 0 N 0 vu

0 N 0 N 0 N 0 N uv

v

u

v

x

y

xy

u

x

v

y

u v

y x

Bd

x

y

xy

0x

u0

vy

y x



Coordinate transformation is unique and invertible.

( , ) ( , )

( , ) ( , )

x x ξ η ξ ξ x y

y y ξ η η η x y

Chain Rule:

f f x f y

ξ x ξ y ξ

f f x f y

η x η y η

Jacobian matrix: x y

ξ ξ

x y

J

y

f

x

f

yx

yx

f

f



y

f

x

f

Jf

f

f

f

J

y

f

x

f

1

2221

1211][

JJ

JJ

yx

yx

J

1 y y

x J

1 x x

y J

x

y

xy

y y0

u1 x x0

vJ

x x y y



y y0

1 x x0

J

x x y y

DNd

D

ε

3 8 3 2 2 8

B D N

( )( ) ( )( )

+( )( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 2

3 4

1 2 3 4

1 2 3 4

1- ξ 1- η x + 1+ξ 1- η x1x =

1+ξ 1+η x + 1- ξ 1+η x4

x 11- η x 1- η x 1+η x 1+η x

ξ 4

x 11- ξ x 1+ξ x 1+ξ x 1- ξ x

η 4

( )( ) ( )( )

( )( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 2

3 4

1 2 3 4

1 2 3 4

1- ξ 1- η y + 1+ξ 1- η y1y =

+ 1+ξ 1+η y + 1- ξ 1+η y4

y 11- η y 1- η y 1+η y 1+η y

ξ 4

y 11- ξ y 1+ξ y 1+ξ y 1- ξ y

η 4



,T

c c

0 1 η η 1

η 1 0 1 t1X Y

η 1 0 η 18

1 η η 1 0

J

1 1

2 2

c c

3 3

4 4

x y

x yX Y

x y

x y

, ,

, ,

, , , ,

( , ) 1 2 3 4

i i η

i i η i

i η i i i η

1B η B B B B

a N b N 0

B 0 c N d N

c N d N a N b N

J



, ,

, ,

, ,

, ,

1 11 1 η

2 22 2 η

3 33 3 η

4 44 4 η

1 1 η η 1 1 1 1N NN N

4 4 η 4 4

1 1 η 1 η 1 1 1N NN N

4 4 η 4 4

1 1 η 1 η 1 1 1N NN N

4 4 η 4 4

1 1 η 1 η 1 1 1N NN N

4 4 η 4 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

a 1 4 y 1 y 1 y 1 y 1

b 1 4 y η 1 y 1 η y η 1 y 1 η

c 1 4 x η 1 x 1 η x η 1 x 1 η

d 1 4 x 1 x 1 x 1 x 1



1 1

T

b b

-1 -1

T

sL

= t d dη

= T t dL

f

f

N X J

N J

T

A

A A

1 1

T

-1 -1

=

=

=

t dxdy

f(x, y)dxdy f(ξ,η) dξdη

t dξdη

k B EB

J

k B EB J



Gauss integration

For evaluation of integrals in k (in practice)

In 1 direction: )()d(1

1

1jj

m

j

fwfI

m gauss points gives exact solution of polynomial

integrand of n = 2m - 1

1 1

1 11 1

( , )d d ( , )yx

nn

i j i j

i j

I f w w f

In 2 directions:



المبن مربعي درجه دو

Quadratic Quadrilateral Element (Q8)

This is the most widely used element for 2-D problems due

to its high accuracy in analysis and flexibility in modeling.


In the natural coordinate system (x,h), the eight shape functions are,

),(),( iii GFN

),( iF Give a value of zero along the sides of the element

That the given node does not contact

),( iGSelect such that when multiply by Fi, it will produce A value of

unity at node i and a value of zero at other neighboring nodes.

Example: Consider N3

)1)(1(),(3 F

3213 ),( cccG

1)1,1()1,1()1,1(; 0)1,0(; 0)0,1( 33333 GFNGG



1)(41)1,1(

00)1,0(

00)0,1(

3213

313

213

cccN

ccG

ccG

4/1

4/1

4/1

3

2

1

c

c

c

)1)(1)(1(4

13 N



In the natural coordinate system (x,h),

the eight shape functions are,



at any point inside the element:

The displacement field is given by:

which are quadratic functions over the element. Strains and

stresses over a quadratic quadrilateral element are linear

functions, which are better representations.



Q4 and T3 are usually used together in a mesh with linear

elements.

Notes:

Q8 and T6 are usually applied in a mesh composed of quadratic

elements.

Quadratic elements are preferred for stress analysis, because of

their high accuracy and the flexibility in modeling complex

geometry, such as curved boundaries.



محبسبه تنش در المبن

The stress in an element is determined by the following relation,

where B is the strain-nodal displacement matrix and d is the nodal

displacement vector which is known for each element once the

global FE equation has been solved.

Stresses can be evaluated at any point inside the element (such as the

center) or at the nodes. Contour plots are usually used in FEA

software packages (during post-process) for users to visually inspect

the stress results.


The von Mises stress is the effective or equivalent stress for 2-D

and 3-D stress analysis. For a ductile material, the stress level is

considered to be safe, if

where is the von Mises stress and the yield stress of the

material. This is a generalization of the 1-D (experimental) result

to 2-D and 3-D situations.

The von Mises stress is defined by

in which and are the three principle stresses at the

considered point in a structure.



For 2-D problems, the two principle stresses in the plane are

determined by

Thus, we can also express the von Mises stress in terms of the

stress components in the xy coordinate system. For plane stress

conditions, we have,

Averaged Stresses:

Stresses are usually averaged at nodes in FEA software packages to

provide more accurate stress values. This option should be turned

off at nodes between two materials or other geometry discontinuity

locations where stress discontinuity does exist.



.يك صفحه مربعي بب سوراخ دايره اي كه تحت تنش فشبري قرار گرفته است: مثبل

The dimension of the plate is 10 in. x 10 in., thickness is 0.1 in. and radius of the hole is 1 in. Assume E = 10x106 psi, n= 0.3

and p = 100 psi. Find the maximum stress in the plate.

ايمحبسبه تنش در يك مسئله تنش صفحه


.يك صفحه مربعي بب سوراخ دايره اي كه تحت تنش فشبري قرار گرفته است: مثبلFrom the knowledge of stress concentrations, we should expect the

maximum stresses occur at points A and B on the edge of the hole.

Value of this stress should be around 3p (= 300 psi) which is the

exact solution for an infinitely large plate with a hole.



.يك صفحه مربعي بب سوراخ دايره اي كه تحت تنش فشبري قرار گرفته است: مثبل

FEA Mesh (Q8, 493 elements) FEA Stress Plot (Q8, 493 elements)



Check the deformed shape of the plate

Check convergence (use a finer mesh, if possible)

Less elements (~ 100) should be enough to achieve the

same accuracy with a better or “smarter” mesh

Discussions:


u4 )يعيبط تبصتخم( x يطخ يعبرم نبملا x3 y3 2h y, v يعيبط

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