ME 475Computer Aided Design of Structures

Finite Element Analysis of Trusses – Part 1

Ron AverillMichigan State University

Learning Objectives1. Describe the differences between truss and frame systems

2. Recall the 2D bar finite element equations and assumptions

3. Define the orientation angle for a 2D plane truss element

2

Trusses

A truss is a structure made from slender members that are joined together at their ends.

The type of connection used to jointhe members is important indeciding how to represent themembers in a finite element model.

3

Trusses versus FramesPin joints in trusses can transmit forces, but not moments.

So members do not bend.

Angle θ is free to change

during loading.

Rigid joints in frames can transmit forces and moments.

So there is bending in members.

Angle θ remains fixed

during loading.

4

θ

θ

Truss Assumptions1. Members are joined at their ends by frictionless pins

2. Loads are applied at the joints

These assumptions ensure that each F

truss member acts as a two-force member:

Tension

Compression

5

Review of 1D Bar Finite ElementsA 2-noded linear bar element e is depicted as follows:

y

e

x

1 2

h Local coordinates x and y are associated with the element

Local nodes are always numbered “1” and “2” with x2 > x1

We use lower case letters for all local (element) quantities 6

1D Bar Finite Element ApproximationsWithin a 2-noded linear bar element, we assume that the axial displacement u varies linearly between nodes 1 and 2:

where

x

1 2

h7

Element solution approximation

Interpolation functions

1D Bar Finite Element EquationsFor a 2-noded linear bar element, the final form of the local finite element equations is:

where

8

Stiffness matrix

Nodal displacementvector

Internal force vector

2D Plane Truss ElementsThe members of a truss are really just bar elements that are oriented arbitrarily relative to the global XY coordinate system: Y x

y e 2

1 X** θ is measured counter clockwise (CCW) from X to x.** Local z and global Z coordinates are in the same direction.

9

θ

Element OrientationsThe orientation of an element is defined by the direction of the local x coordinate, which is from node 1 to node 2.

Note that

Y e 1

y x 2

X

10

θ

ExerciseDetermine the orientation angle for each of the truss elements:

1

1 Y 2 2 3

3 X

11

45°

45°

The boolean array is:

The orientations are:

Element θ

SolutionRecall: θ is measured counter clockwise (CCW) from X to x.

1

1 Y 2 2 3

3 X

12

The boolean array is:

The orientations are:

45°

45°

Element θ

SolutionRecall: θ is measured counter clockwise (CCW) from X to x.

1

1 Y 2 2 3

3 X

13

The boolean array is:

The orientations are:

45°

45°

Element θ

1 135°

θ1

SolutionRecall: θ is measured counter clockwise (CCW) from X to x.

1

1 Y 2 2 3

3 X

14

The boolean array is:

The orientations are:

45°

45°

Element θ

1 135°

2 270°

θ2

SolutionRecall: θ is measured counter clockwise (CCW) from X to x.

1

1 Y 2 2 3

3 X

15

The boolean array is:

The orientations are:

45°

45°

Element θ

1 135°

2 270°

3 225°

θ3