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1D OF FINITE ELEMENT METHOD Session 4 – 6
Course : S Introduction to Finite Element Method Year : 2010 1D OF FINITE ELEMENT METHOD Session 4 – 6
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COURSE 4 Content: 1D Element Types 1D Element Modelling 1D Solution
Example/Case Study Bina Nusantara
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1D ELEMENT TYPES Bina Nusantara
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Forces and Moments on 1D Element
1D ELEMENT MODELLING Forces and Moments on 1D Element Bina Nusantara
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APPLICATION TO FINITE ELEMENT
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1D SOLUTION Global and Local Coordinate System Bina Nusantara
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1D SOLUTION Bina Nusantara
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1D SOLUTION Bina Nusantara
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1D ELEMENT EXAMPLE u1 u2 Deformed shape f2 f1 x Node (a hinge) Element
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1D ELEMENT EXAMPLE Conjecture a displacement function u(x) x
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1D ELEMENT EXAMPLE Express u(x) in terms of nodal displacements by using boundary conditions. Deformed shape u(0) = u1 u(L) = u2 Bina Nusantara
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1D ELEMENT EXAMPLE Sub (2) into (1)
Displacement polynomial that satisfies boundary conditions Bina Nusantara
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Bar Element example Derive strain-displacement relationship by using mechanics theory Axial Strain Bina Nusantara
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1D ELEMENT EXAMPLE Derive stress-displacement relationship by using elasticity theory Axial Stress Elastic Modulus Bina Nusantara
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1D ELEMENT EXAMPLE Use principle of Virtual Work
Work = Stress x Strain x Volume Bar cross-sectional area A Internal work External work Bina Nusantara
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1D ELEMENT EXAMPLE Equate internal and external work Stiffness matrix
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1D ELEMENT EXAMPLE Resultant stiffness matrix Bina Nusantara
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EXAMPLE Axial deformation of a bar subjected to a uniform load
(1-D Poisson equation) u = axial displacement E=Young’s modulus = 1 A=Cross-sectional area = 1 Bina Nusantara
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EXAMPLE Model the following shaft using two beam finite elements neglecting axial deformation, given the following data: Bina Nusantara
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EXAMPLE Bina Nusantara
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EXAMPLE Global and element coordinates are parallel. The Global nodal coordinates are then defined as ui, i =1, 2, , 6 . Now assign a set of generalized coordinates qi along same directions. Bina Nusantara
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EXAMPLE Simple Form Bina Nusantara
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EXAMPLE Bina Nusantara
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EXAMPLE Bina Nusantara
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EXAMPLE Bina Nusantara
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EXAMPLE Bina Nusantara
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