Assignment#02

Finite Element Methods

Statement:

1.9 For the tapered bar shown in Figure 1.20. The area of cross section changes along

the length as . where A0 is the cross-sectional area at x=0, and is

the length of the bar. By expressing the strain and kinetic energies of the bar in

matrix forms, identify the stiffness and mass matrices of a typical element. Assume

a linear model for the axial displacement of the bar element.

Figure:

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Solution:

Step 1: Idealization

As shown in the above figure the bar is distributed into three equal and same elements.

We take a typical element of length 𝑙 (e) with nodal unknowns

1

e

and

2

e

at nodes 1

and 2.we take linear displacement model of bar element.

12

.......... 1xx

Step 2: B.C’s

At x=0

1

e

x

And at x=

2

e

x

Putting these conditions in equation.1 and finding them values of constants

gives

11

e

and

21

2

ee

e

Putting these values in equation 1 we get

21

1

ee

e

e

xx

Element strain:

21

ee

e

e

d

dx

Element stress:

21

ee

ee

e

E

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Step 3: Element stiffness Matrix

First we calculate the potential energy of the bar

12

p

w

Where

0

1

2

e e e e

A dx

Or

2

0

2

ee

ee

AE dx

2

21

0

2

ee

ee

e

e

AE dx

22

2 1 1 2

2

0

2

2

e e e e

ee

e

e

AE dx

22

2 1 1 2

2

2

ee

e e e e e

e

AE

1

2

e

e e T e

K

1

2

e

e

e

And

11

11

ee

e

e

AE

K

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Now

12

2

eAA

A

Also

xl

o

A x A e

Therefore

1

20.3678

0.3678 1.3678 0.684

22

o

o

eo o o o

AA

AA

A A A

AA

And

so

11

2.052 ....... 1

11

e

eo

AE

K Ans

Step 4: Mass Matrix

The element mass matrix can be found from kinetic energy equation of the beam

2

0

1..........2

2

T m dx

t

Now

21

1

ee

e

ex

t

Putting this value in equation 2

3

e

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##### Document information

Uploaded by:
ramu

Views: 4180

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3

University:
Aligarh Muslim University

Subject:
Finite Element Method

Upload date:
08/07/2012