DYNAMIC OF STRUCTURES

CHAPTER 12INTRODUCTION TO FINITE ELEMENT

METHOD

Department of civil engineering, University of North Sumatera

Ir. DANIEL RUMBI TERUNA, MT;IP-U

INTRODUCTION TO FINITE LEMENT METHOD

To ilustrate this concept, consider the cantilever beam shown in Fig. 1

q General overview

Fig.1 Degrees of freedom of cantilever beam

6u

3u

2u

1u

4u

5u1−Nu

Nu

12 =u

11 =u )(1 xψ

)(2 xψ

INTRODUCTION TO FINITE LEMENT METHOD

6u

3u

2u

1u

4u

5u1−Nu

Nu

13 =u)(3 xψ

14 =u)(4 xψ

11 =−Nu)(1 xN−ψ

1=Nu)(xNψ

The frame structures is subdivided into a number of segment, called

finite element

The element are interconected only at nodes or nodal point, in this simple

case the nodal points are the ends of the elements

In the finite element method nodal displacement are selected as the generalized

coordinate

The deflection of the beam is expressed in terms of nodal displacement

through trial/shape fungtion

Coresponding to each DOF, a trial function is selected with the following

properties: unit value at the DOF; zero value at all other DOFs; continous function

with continous first derivative

These trial function satisfy the requirement of admissibility, because they are linerly

independent, continous and consistent with the geometric boundary contidition

)(xiψ

INTRODUCTION TO FINITE LEMENT METHOD

)(xmLConsider a straight beam of length , mass per unit length ,andflexural rigidity as shown Fig.2,

INTRODUCTION TO FINITE LEMENT METHOD

)(xEI

1ux2u

3u

4u

11 =u

12 =u

13 =u

14 =u

)(xm)(xEI

)(1 xψ

)(2 xψ

)(3 xψ

)(4 xψ

( ) ( )xtutxu ii

i ψ)(,4

1∑

=

= (1)

Where is the nodal displacement in the DOF and is

the associated trial function (interpolation function).

)(tui

The deflection of the beam can be expressed in nodal DOFs as

INTRODUCTION TO FINITE LEMENT METHOD

thi ( )xiψ

The interpolation function can be any arbitrary shapes stasfyingthe boundary condition. Thus satisfies the followingboundary condition

( )xiψ

INTRODUCTION TO FINITE LEMENT METHOD

(2)

The interpolation function can be any arbitrary shapes stasfyingthe boundary condition. The deflection of beam is selected acubic polynomial

( )3

4

2

321

+

+

+=

L

xa

L

xa

L

xaaxu (3)

(3)

iaThe constant can be determined for each of the four sets of boundaryconditions Eq.(1), to obtain

INTRODUCTION TO FINITE LEMENT METHOD

Ø ELEMENT STIFFNESS MATRIX

INTRODUCTION TO FINITE LEMENT METHOD

(5)

Using the principle of virtual displacement, the stiffness matrix can be derived :

(4)

Ø ELEMENT MASS MATRIX

INTRODUCTION TO FINITE LEMENT METHOD

(6)

Using the principle of virtual displacement, the mass matrix can be derived :

(5)

It is noted that the consistent mass matrix is not diagonal, where as the lump mass approximation lead to diagonal matrix

Ø ELEMENT MASS MATRIX

INTRODUCTION TO FINITE LEMENT METHOD

(7)

The mass matrix of a finite element can be simplified by assuming that thedistibuted mass of the element can be lumped as point masses alongtranslational DOF 31 uandu

For, example if the mass of a uniform element is m per unit length, a point mass of will be assigned to each end, leading to 2/mL

Ø ELEMENT (APPLIED) LOAD VECTOR

INTRODUCTION TO FINITE LEMENT METHOD

(8)

When members of a structures are oriented in different directions, itbecomes necesssary to transform the stiffness , mass, force relations foreach member from its local coordinate system to a single globalcoordinate system selected for the entire structures.

Since both local element DOF and structures DOF are defined along the same set of cartesian coordinate, no transformation of coordinate is required. Therefore, 2211 , kkkk ==

If local element DOF and structures DOF do not have the same set of cartesian coordinate, transformation of coordinate is required. Therefore,

eT

e akak =

The following example is given in derivation of transformation matrix.

(a)(b)

(c)

Assemble the element stiffness matrices

The element mass matrices in their local DOF; and as for the stiffnessMatrix, . Thus2211 , mmmm ==

21 , mm

[ ]{ } [ ]{ } 0=+ uKuM &&

Assemble the element mass matrices

Diffrential equation of free vibration can be written as:

The Natural frequencies are determined by solving the eigen problem

[ ] [ ] 02 =− MK nω

Repeat example using lump mass aproximation

Bcause the mass associated with the rotational DOF iszero, they can be eliminated from the stiffness matrix by staticcondensation. The resulting 2x2 stiffness matrix in term of thetranslational DOF 31 uandu

42 uandu

Solve the eigenvalue problem to obatain natural frequencies:

Comparison of Finite Element and Exact Solutions