# Lecture_FEM_Axially_Loaded_Structure - Class Workbook and...

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Class Workbook and Handouts - MME 412: Mechanics of Materials Prepared by: Kumar V. Singh -170- Introduction to Finite Element (Computational Framework: Rayleigh-Ritz Method) Powerful numerical technique that uses energy methods, calculus of variation and interpolation schemes in solving large boundary value problems in which a continuous structure is divided into well defined substructures (finite elements) having elemental material and geometrical parameters These elements are defined by nodes which connect the neighboring elements, possesses degree of freedom (translation and/or rotation) and at which the boundary conditions and loads are applied. Extremely useful for approximating the governing equations of complicated, large and complex geometry structures Provides a computational framework which can address wide range of static and dynamics problems FORMULATION FOR AXIALLY LOADED BAR Consider an axially loaded bar as shown in the figure here. The Potential energy of the axially loaded bar corresponding to the exact solution u(x) can be obtained as, dx u(x) x F dx dx du EA Π (u) L L = 0 0 2 ) ( 2 1 Similarly, the Potential energy of the axially loaded bar corresponding to the approximated solution w(x) can be obtained as, dx w(x) x F dx dx dw EA Π (w) L L = 0 0 2 ) ( 2 1 Steps for Finite Element analysis STEP 1: Divide the bar into small pieces - “finite elements” connected to each other through special points (“nodes”) x L F ) ( , x A E y ) ( x u x L F ) ( , x A E y ) ( x u y Elements Nodes

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Class Workbook and Handouts - MME 412: Mechanics of Materials Prepared by: Kumar V. Singh -171- STEP 2: Describe the behavior of each element TASK 1: Approximate the displacement within each element TASK 2: Approximate the strain and stress within each element TASK 3: Derive the stiffness matrix of each element using the Rayleigh-Ritz method TASK 1: Approximate the displacement within each element Assume displacement within the element varies linearly. x a a u(x) 1 0 + = Displacement of the nodes: j j j i i i x a a ) u(x u x a a ) u(x u 1 0 1 0 + = = + = = By solving them simultaneously for the coefficients we get, i j i j i j i j j i x x u u a x x x u x u a = = 1 0 ,. Hence, the displacement within the element can be expressed as: ( ) ( ) x L u u L x u x u u(x) e i j e i j j i + = This displacement form is commonly expressed in the terms of “ shape functions” or “interpolation functions” (x) N i and (x) N j as: j j i i j (x) N e i i (x) N e j (x)u N (x)u N u L x x u L x x u(x) j i + = + = ± ² ³ ± ² ³ In the matrix notation the displacement within the element can be expressed in terms of shape function and nodal displacements: e u(x) Nu = () = j i j i u u x N x N x u ) ( ) ( ) (
Class Workbook and Handouts - MME 412: Mechanics of Materials Prepared by: Kumar V. Singh -172- TASK 2: Approximate the strain and stress within each element The strain in the bar element can be approximated as, dx du ε = or equivalently in the matrix form: e e dx d ε Bu u N = = , where, () 1

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