1D OF FINITE ELEMENT METHOD Session 4 – 6 Course : S0912 - Introduction to Finite Element Method Year : 2010 1D OF FINITE ELEMENT METHOD Session 4 – 6

COURSE 4 Content: 1D Element Types 1D Element Modelling 1D Solution Example/Case Study Bina Nusantara

1D ELEMENT TYPES Bina Nusantara

Forces and Moments on 1D Element 1D ELEMENT MODELLING Forces and Moments on 1D Element Bina Nusantara

APPLICATION TO FINITE ELEMENT Bina Nusantara

1D SOLUTION Global and Local Coordinate System Bina Nusantara

1D SOLUTION Bina Nusantara

1D SOLUTION Bina Nusantara

1D ELEMENT EXAMPLE u1 u2 Deformed shape f2 f1 x Node (a hinge) Element Bina Nusantara

1D ELEMENT EXAMPLE Conjecture a displacement function u(x) x Bina Nusantara

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

1D ELEMENT EXAMPLE Sub (2) into (1) Displacement polynomial that satisfies boundary conditions Bina Nusantara

Bar Element example Derive strain-displacement relationship by using mechanics theory Axial Strain Bina Nusantara

1D ELEMENT EXAMPLE Derive stress-displacement relationship by using elasticity theory Axial Stress Elastic Modulus Bina Nusantara

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

1D ELEMENT EXAMPLE Equate internal and external work Stiffness matrix Bina Nusantara

1D ELEMENT EXAMPLE Resultant stiffness matrix Bina Nusantara

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

EXAMPLE Model the following shaft using two beam finite elements neglecting axial deformation, given the following data: Bina Nusantara

EXAMPLE Bina Nusantara

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

EXAMPLE Simple Form Bina Nusantara

EXAMPLE Bina Nusantara

EXAMPLE Bina Nusantara

EXAMPLE Bina Nusantara

EXAMPLE Bina Nusantara