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

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

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.

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.

Example of a truss structure

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

Shape functions construction Consider a truss element

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 

Shape functions construction (Linear element)

Strain matrix or where

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

HIGHER ORDER ELEMENTS Quadratic element Cubic element