Finite Element Method-The Direct Stiffness Method

8/10/2019 Finite Element Method-The Direct Stiffness Method

1/24

Finite

Element

MethodBy the Direct Stiffness Method (DSM)Engr Y. K. Galadima


2/24

General Procedure Pre-processing

1. Idealisation

, ,

, ,

,

,

,

,

2

31 4

Idealisation simply

means creating a

mathematical

model of the

physical systemby making

necessary

assumptions

1 2 3

NOTE

The subscript notations adopted in this

presentation are slightly different from those

used in the lecture notes

This presentation may contain error


3/24

General Procedure Pre-processing

2. Discretisation (decomposition) (2.4.2.2)

a) Disconnection (2.4.2.2.1)

2

3

1 4

1 2 3

y

The local or element

axes are denoted by

and

The local coordinate

system is selected

such that the

axis aligned with the

longitudinal axis ofthe element

The global

coordinate axes are

denoted by and


4/24

General Procedure

Pre-processing2. Discretisation (decomposition) (2.4.2.2)

b) Localisation/isolation (2.4.2.2.2)


5/24


b) Localisation/isolation (2.4.2.2.2)

We will use this

element as a genericelement to derive the

stiffness equations for

the truss elements


6/24

General Procedure


c) Derivation of Member Stiffness Equations (2.4.2.2.3)

=

= cos 90


7/24


8/24

General ProcedurePre-processing

2. Discretisation (decomposition) (2.4.2.2)c) Derivation of Member Stiffness Equations (2.4.2.2.3)

No shear for truss elements, hence

=

00 0

00 0

0

0 0

0

0 0


9/24

General Procedure

Pre-processing2. Discretisation (decomposition) (2.4.2.2)c) Derivation of Member Stiffness Equations (2.4.2.2.3)

Now the task is to find expressionsfor the INFLUENCECOEFFICIENTS , ,

and


10/24


2. Discretisation (decomposition) (2.4.2.2)

c) Derivation of Member Stiffness Equations (2.4.2.2.3)

=

, =

=

, =

Hence


11/24

General Procedure

Pre-processing2. Discretisation (decomposition) (2.4.2.2)c) Derivation of Member Stiffness Equations (2.4.2.2.3)

=

1 0

0 0

1 0

0 01 0

0 0

1 0

0 0


12/24

General Procedure

Pre-processing3. Globalisation (2.4.2.3)

,

,

,

,


13/24

General Procedure

Pre-processing3. Globalisation: Force fieldexpressing the joint forces wrt their components in theglobal coordinate gives

= cos + sin

= s i n + cos

And

= cos + sin

= s i n + cos


14/24


3. Globalisation: Force field

=

0 0

0 00 0

0 0

Or in short notation

=


15/24

General Procedure

Pre-processing3. Globalisation: Displacement field

=

0 0

0 0

0 00 0

Or in short notation

=


16/24

General Procedure

Pre-processing

3. Globalisation: substituting theexpressions for and will give

=

Or

=


17/24

General Procedure

Pre-processing

3. Globalisation: if we write therelationship btw the nodal forces in theglobal coordinate system and their

corresponding displacement as =

Then


18/24

General Procedure

Pre-processing

3. Globalisation: comparing the last twoexpression shows that

= Thus


19/24

General Procedure

Pre-processing3. Globalisation

=

0 0

0 00 0

0 0

1 0

0 0

1 0

0 01 0

0 0

1 0

0 0

0 0

0 00 0

0 0


20/24

General Procedure

Pre-processing3. Globalisation

=

Is the global stiffness matrix


21/24

General Procedure

Pre-processing3. Globalisation: thus the global stiffness matrix forthe element becomes

=


22/24

General Procedure

Pre-processing3. Globalisation: therefore, for the

element, the global stiffness equation is

=


23/24

I think we should take a break here


24/24

Questions

???

Documents

Finite Element Method-The Direct Stiffness Method