bar elements finite element methods, Lecture notes of Metallurgy

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Bar Elements in 2-DSpace By S. Ziaei Rad

Distributed Load

Uniformly distributed axial load

q

(N/mm, N/m, lb/in) can

be converted to two equivalent nodal forces of magnitude

qL/

We verify this by considering the work done by the load

q,

Distributed Load

The new nodal force vector is^ In an assembly of bars,

Bar Elements in 2-D Note:

Lateral displacement

d

oes not contribute to the stretch of the bar, within thelinear theory.

Transformation

For the two nodes of the bar element, we have

or,

The nodal forces are transformed in the same way,

Stiffness Matrix in the 2-DSpace

In the local coordinate system, we have

Augmenting this equation, we write

or ,

Stiffness Matrix in the 2-DSpace Explicit form

Calculation of the

directional cosines l

and

m

Note:

The structure stiffness

matrix is assembled by using the Element stiffness matrices in the usual way a in the 1-D case.

Assembly Rules

Compatibility

The joint displacements of all

members meeting at a joint

must be the same.

Equilibrium

The sum of forces exerted by all

members that meet at a joint

must balance

the

external force applied to that joint.

The Direct Stiffness Method (DSM)Steps

Example 2.

Problem:

A simple plane truss is made

of two identical bars (with

E, A,

and

L

and loaded as shown in the figure. Find1) displacement of node 2;2) stress in each bar.

Example 2.

(Globalization)

Element 1

Using formula (*), we obtain the stiffness matrix in theglobal system

Example 2.

(Globalization)

Element 2

Using formula (*), we obtain the stiffness matrix in theglobal system

Example 2.

(Application of BCs and

Solution)^ Load and boundary conditions (BC):

Condensed FE equation,

Solving this, we obtain the displacement of node 2,

Example 2.

(Recovery)

Using formula for stress, we calculate the stresses in the two bars