F inite Element Method

1. Finite Element Method 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. 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 , stress: Exact solution of : 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