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Method of Finite Elements I Chapter 2 The Direct Stiffness Method Direct Stiffness Method (DSM) Computational method for structural analysis Matrix method for computing the member forces and displacements in structures DSM implementation is the basis of most commercial and open-source finite element software

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The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration.It is common to use the finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable.

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where {F} is the applied force vector, {R} is the external reaction vector, [K] is the global stiffness matrix, and {U} is the nodal displacement vector.. In the equation above, {R} and {U} are unknowns. To simplify the solution of this equation, it is desired to solve for the reactions and displacements independently of one another.

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Definitions. Given two metric spaces (X, d X) and (Y, d Y), where d X denotes the metric on the set X and d Y is the metric on set Y, a function f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x 1 and x 2 in X, ((), ()) (,).Any such K is referred to as a Lipschitz constant for the function f and f may also be referred to as K-Lipschitz.

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16.810 (16.682) 2 Plan for Today FEM Lecture (ca. 50 min) FEM fundamental concepts, analysis procedure Errors, Mistakes, and Accuracy Cosmos Introduction (ca. 30 min) Follow along step-by-step Conduct FEA of your part (ca. 90 min) Work in teams of two First conduct an analysis of your CAD design You are free to make modifications to your original model

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We all remember the equation, {F} = [K] {u}, where F is the nodal force vector, K is the stiffness matrix and u is the nodal displacement vector. Well in a typical (linear) FEA model, this stiffness matrix, K, is defined at the beginning and it is assumed not to change throughout the entire analysis.

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The following approach will be used: 1. Look at force equilibrium of the element - assume that the sum of the nodal internal forces acting on the element is equal to zero. 2. Apply Hooke's law, which states that the stress \sigma σ is a linear function of the strain \epsilon ϵ. \begin {equation} \sigma = E \epsilon \end {equation} σ = E ϵ.