F inite Element Method

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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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### Finite Element Method

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.
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

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
• Planar truss

where

,

Similarly

(4x1)

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

, stress:

Exact solution of

:

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