1. Finite Element Method CHAPTER 4: FEM FOR TRUSSES
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 4:
FEM FOR TRUSSES

2. CONTENTS INTRODUCTION FEM EQUATIONS Shape functions construction
Strain matrix
Element matrices in local coordinate system
Element matrices in global coordinate system
Boundary conditions
Recovering stress and strain
EXAMPLE
Remarks
HIGHER ORDER ELEMENTS

3. INTRODUCTION
Truss members are for the analysis of skeletal type systems – planar trusses and space trusses.
A truss element is a straight bar of an arbitrary cross-section, which can deform only in its axis direction when it is subjected to axial forces.
Truss elements are also termed as bar elements.
In planar trusses, there are two components in the x and y directions for the displacement as well as forces at a node.
For space trusses, there will be three components in the x, y and z directions for both displacement and forces at a node.

4. INTRODUCTION
In trusses, the truss or bar members are joined together by pins or hinges (not by welding), so that there are only forces (not moments) transmitted between bars.
It is assumed that the element has a uniform cross-section.

5. Example of a truss structure

6. FEM EQUATIONS Shape functions construction Strain matrix
Element matrices in local coordinate system
Element matrices in global coordinate system
Boundary conditions
Recovering stress and strain

7. Shape functions construction
Consider a truss element

8. Shape functions construction
Let
Note: Number of terms of basis function, xn determined by n = nd - 1
At x = 0, u(x=0) = u1
At x = le, u(x=le) = u2 

9. Shape functions construction
(Linear element)

10. Strain matrix or
where

11. Element Matrices in the Local Coordinate System
Note: ke is symmetrical
Proof:

12. Element Matrices in the Local Coordinate System
Note: me is symmetrical too

13. Element matrices in global coordinate system
Perform coordinate transformation
Truss in space (spatial truss) and truss in plane (planar truss)

14. Element matrices in global coordinate system
Spatial truss
(Relationship between local DOFs and global DOFs)
(2x1)
where ,
(6x1)
Direction cosines

15. Element matrices in global coordinate system
Spatial truss (Cont’d)
Transformation applies to force vector as well:
where

16. Element matrices in global coordinate system
Spatial truss (Cont’d)

17. Element matrices in global coordinate system
Spatial truss (Cont’d)

18. Element matrices in global coordinate system
Spatial truss (Cont’d)

19. Element matrices in global coordinate system
Spatial truss (Cont’d)
Note:

20. Element matrices in global coordinate system
Planar truss
where ,
Similarly
(4x1)

21. Element matrices in global coordinate system
Planar truss (Cont’d)

22. Element matrices in global coordinate system
Planar truss (Cont’d)

23. Singular K matrix  rigid body movement Constrained by supports
Boundary conditions
Singular K matrix  rigid body movement
Constrained by supports
Impose boundary conditions  cancellation of rows and columns in stiffness matrix, hence K becomes SPD
Recovering stress and strain
(Hooke’s law) x

24. EXAMPLE
Consider a bar of uniform cross-sectional area shown in the figure. The bar is fixed at one end and is subjected to a horizontal load of P at the free end. The dimensions of the bar are shown in the figure and the beam is made of an isotropic material with Young’s modulus E. P l

25. EXAMPLE
Exact solution of :
, stress:
FEM:
(1 truss element) 

26. Remarks FE approximation = exact solution in example
Exact solution for axial deformation is a first order polynomial (same as shape functions used)
Hamilton’s principle – best possible solution
Reproduction property

27. HIGHER ORDER ELEMENTS
Quadratic element
Cubic element