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1 Finite Element Method FEM FOR FRAMES for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 6:

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Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS INTRODUCTION FEM EQUATIONS FOR PLANAR FRAMES – Equations in local coordinate system – Equations in global coordinate system FEM EQUATIONS FOR SPATIAL FRAMES – Equations in local coordinate system – Equations in global coordinate system REMARKS

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Finite Element Method by G. R. Liu and S. S. Quek 3 INTRODUCTION Deform axially and transversely. It is capable of carrying both axial and transverse forces, as well as moments. Hence combination of truss and beam elements. Frame elements are applicable for the analysis of skeletal type systems of both planar frames (2D frames) and space frames (3D frames). Known generally as the beam element or general beam element in most commercial software.

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Finite Element Method by G. R. Liu and S. S. Quek 4 FEM EQUATIONS FOR PLANAR FRAMES Consider a planar frame element

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Finite Element Method by G. R. Liu and S. S. Quek 5 Equations in local coordinate system Combination of the element matrices of truss and beam elements Truss Beam From the truss element, (Expand to 6 x 6)

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Finite Element Method by G. R. Liu and S. S. Quek 6 Equations in local coordinate system From the beam element, (Expand to 6 x 6)

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Finite Element Method by G. R. Liu and S. S. Quek 7 Equations in local coordinate system +

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Finite Element Method by G. R. Liu and S. S. Quek 8 Equations in local coordinate system Similarly so for the mass matrix and we get And for the force vector,

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Finite Element Method by G. R. Liu and S. S. Quek 9 Equations in global coordinate system Coordinate transformation where,

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Finite Element Method by G. R. Liu and S. S. Quek 10 Equations in global coordinate system Direction cosines in T: (Length of element)

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Finite Element Method by G. R. Liu and S. S. Quek 11 Equations in global coordinate system Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 12 FEM EQUATIONS FOR SPATIAL FRAMES Consider a spatial frame element Displacement components at node 1 Displacement components at node 2

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Finite Element Method by G. R. Liu and S. S. Quek 13 Equations in local coordinate system Combination of the element matrices of truss and beam elements

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Finite Element Method by G. R. Liu and S. S. Quek 14 Equations in local coordinate system where

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Finite Element Method by G. R. Liu and S. S. Quek 15 Equations in global coordinate system

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Finite Element Method by G. R. Liu and S. S. Quek 16 Equations in global coordinate system Coordinate transformation where,

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Finite Element Method by G. R. Liu and S. S. Quek 17 Equations in global coordinate system Direction cosines in T 3

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Finite Element Method by G. R. Liu and S. S. Quek 18 Equations in global coordinate system Vectors for defining location and orientation of frame element in space k, l = 1, 2, 3

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Finite Element Method by G. R. Liu and S. S. Quek 19 Equations in global coordinate system Vectors for defining location and orientation of frame element in space (contd)

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Finite Element Method by G. R. Liu and S. S. Quek 20 Equations in global coordinate system Vectors for defining location and orientation of frame element in space (contd)

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Finite Element Method by G. R. Liu and S. S. Quek 21 Equations in global coordinate system Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 22 REMARKS In practical structures, it is very rare to have beam structure subjected only to transversal loading. Most skeletal structures are either trusses or frames that carry both axial and transversal loads. A beam element is actually a very special case of a frame element. The frame element is often conveniently called the beam element.

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Finite Element Method by G. R. Liu and S. S. Quek 23 CASE STUDY Finite element analysis of bicycle frame

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Finite Element Method by G. R. Liu and S. S. Quek 24 CASE STUDY Youngs modulus, E GPa Poissons ratio, elements (71 nodes) Ensure connectivity

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Finite Element Method by G. R. Liu and S. S. Quek 25 CASE STUDY Constraints in all directions Horizontal load

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Finite Element Method by G. R. Liu and S. S. Quek 26 CASE STUDY M = 20X

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Finite Element Method by G. R. Liu and S. S. Quek 27 CASE STUDY x 10 5 Pa x 10 6 Pa x 10 5 Pa x 10 5 Pa x 10 5 Pa x 10 5 Pa x 10 5 Pa Axial stress

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