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F inite Element Method. 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

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f inite element method

Finite Element Method

for readers of all backgrounds

G. R. Liu and S. S. Quek

CHAPTER 4:

FEM FOR TRUSSES

contents
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
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.
introduction1
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.
fem equations
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
Shape functions construction
  • Consider a truss element
shape functions construction1
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

element matrices in global coordinate system
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 system1
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 system2
Element matrices in global coordinate system
  • Spatial truss (Cont’d)

Transformation applies to force vector as well:

where

element matrices in global coordinate system7
Element matrices in global coordinate system
  • Planar truss

where

,

Similarly

(4x1)

boundary conditions
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
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

example1
EXAMPLE

, stress:

Exact solution of

:

FEM:

(1 truss element)

remarks
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
HIGHER ORDER ELEMENTS

Quadratic element

Cubic element