ME 475 Computer Aided Design of Structures Finite Element Analysis of Trusses – Part 1

1. ME 475Computer Aided Design of StructuresFinite Element Analysis of Trusses – Part 1 Ron Averill Michigan State University

2. Learning Objectives • Describe the differences between truss and frame systems • Recall the 2D bar finite element equations and assumptions • Define the orientation angle for a 2D plane truss element

3. Trusses A truss is a structure made from slender members that are joined together at their ends. The type of connection used to join the members is important in deciding how to represent the members in a finite element model.

4. Trusses versus Frames Pin 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. θ θ

5. Truss Assumptions • Members are joined at their ends by frictionless pins • Loads are applied at the joints These assumptions ensure that each F truss member acts as a two-force member: Tension Compression

6. Review of 1D Bar Finite Elements A 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

7. 1D Bar Finite Element Approximations Within a 2-noded linear bar element, we assume that the axial displacement u varies linearly between nodes 1 and 2: where x 1 2 h Element solution approximation Interpolation functions

8. 1D Bar Finite Element Equations For a 2-noded linear bar element, the final form of the local finite element equations is: where Stiffness matrix Nodal displacement vector Internal force vector

9. 2D Plane Truss Elements The 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. θ

10. Element Orientations The 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 θ

11. Exercise Determine the orientation angle for each of the truss elements: 1 1 Y 2 2 3 3 X The boolean array is: The orientations are: 45° 45°

12. Solution Recall: θis measured counter clockwise (CCW) from X to x. 1 1 Y 2 2 3 3 X The boolean array is: The orientations are: 45° 45°

13. Solution Recall: θis measured counter clockwise (CCW) from X to x. 1 1 Y 2 2 3 3 X The boolean array is: The orientations are: 45° θ1 45°

14. Solution Recall: θis measured counter clockwise (CCW) from X to x. 1 1 Y 2 2 3 3 X The boolean array is: The orientations are: θ2 45° 45°

15. Solution Recall: θis measured counter clockwise (CCW) from X to x. 1 1 Y 2 2 3 3 X The boolean array is: The orientations are: 45° θ3 45°