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

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 5 Example of a truss structure

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 7 Shape functions construction Consider a truss element

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Finite Element Method by G. R. Liu and S. S. Quek 8 Shape functions construction Let Note: Number of terms of basis function, x n determined by n = n d - 1 At x = 0, u(x=0) = u 1 At x = l e, u(x=l e ) = u 2

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Finite Element Method by G. R. Liu and S. S. Quek 9 Shape functions construction (Linear element)

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Finite Element Method by G. R. Liu and S. S. Quek 10 Strain matrix or where

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Finite Element Method by G. R. Liu and S. S. Quek 11 Element Matrices in the Local Coordinate System Note: k e is symmetrical Proof:

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Finite Element Method by G. R. Liu and S. S. Quek 12 Element Matrices in the Local Coordinate System Note: m e is symmetrical too

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Finite Element Method by G. R. Liu and S. S. Quek 13 Element matrices in global coordinate system Perform coordinate transformation Truss in space (spatial truss) and truss in plane (planar truss)

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Finite Element Method by G. R. Liu and S. S. Quek 14 Element matrices in global coordinate system Spatial truss (Relationship between local DOFs and global DOFs) where, Direction cosines (2 x1) ( 6x1)

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Finite Element Method by G. R. Liu and S. S. Quek 15 Element matrices in global coordinate system Spatial truss (Contd) Transformation applies to force vector as well: where

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Finite Element Method by G. R. Liu and S. S. Quek 16 Element matrices in global coordinate system Spatial truss (Contd)

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Finite Element Method by G. R. Liu and S. S. Quek 17 Element matrices in global coordinate system Spatial truss (Contd)

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Finite Element Method by G. R. Liu and S. S. Quek 18 Element matrices in global coordinate system Spatial truss (Contd)

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Finite Element Method by G. R. Liu and S. S. Quek 19 Element matrices in global coordinate system Spatial truss (Contd) Note:

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Finite Element Method by G. R. Liu and S. S. Quek 20 Element matrices in global coordinate system Planar truss where, Similarly (4 x 1)

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Finite Element Method by G. R. Liu and S. S. Quek 21 Element matrices in global coordinate system Planar truss (Contd)

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Finite Element Method by G. R. Liu and S. S. Quek 22 Element matrices in global coordinate system Planar truss (Contd)

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Finite Element Method by G. R. Liu and S. S. Quek 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 (Hookes law) x

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Finite Element Method by G. R. Liu and S. S. Quek 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 Youngs modulus E. P l

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Finite Element Method by G. R. Liu and S. S. Quek 25 EXAMPLE Exact solution of, stress: : FEM: (1 truss element)

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Finite Element Method by G. R. Liu and S. S. Quek 26 Remarks FE approximation = exact solution in example Exact solution for axial deformation is a first order polynomial (same as shape functions used) Hamiltons principle – best possible solution Reproduction property

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Finite Element Method by G. R. Liu and S. S. Quek 27 HIGHER ORDER ELEMENTS Quadratic elementCubic element

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