Shape functions
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This document discusses shape functions in finite element analysis. Shape functions are used to approximate quantities like displacements, strains and stresses between nodes in an element. They are used to interpolate between discrete nodal values and discretize continuous quantities into nodal degrees of freedom. Shape functions are derived for specific element types by choosing interpolation polynomials and natural coordinates that relate the physical coordinates to a standard coordinate system. The document outlines the derivation process for shape functions of bar and beam elements.
Shape functions
- 1. Adrian Egger Method of Finite Elements I: Shape Functions
- 2. Why shape functions? Discretization leads to solution in the nodes, but no information concerning the space in between Shape functions required to approximate quantities between nodes Underlying assumption of how quantities are distributed in an element (stiffness, mass, element loads; displacements, strains, stress, internal forces, etc.) Geometry transformation 3/24/2015 2Adrian Egger | FEM I | FS 2015 ?
- 3. What can shape functions be used for? 1. Used to interpolate between nodes i.e. discrete nodal quantities continuous across element = 諮() = 1()1+2()2 3/24/2015 3Adrian Egger | FEM I | FS 2015
- 4. What can shape functions be used for? 2. Used to discretize continuous quantities to nodal DOF i.e. continuous across element discrete nodal quantities 3/24/2015 4 = =0 () = =0 01 1 11 1 02 1 12 1 () = 2 2 12 2 2 12 1 1 2 2 Adrian Egger | FEM I | FS 2015
- 5. Alternative way to derive loading vector Recap: We calculate the solution in the nodes What is the influence of element loading in the nodes We must fix the element such that reaction forces develop in the nodal DOF we are interested in! Equivalent to solving differential equation 乞腫 腫 = 3/24/2015 5Adrian Egger | FEM I | FS 2015
- 6. How to derive shape functions Interpolation functions are generally assumed! (within certain parameters and restrictions) Minimal amount of continuity / differentiability Etc. Wish to implement this repetitive task as easily as possible, i.e. computer implementation using highly optimized numerical schemes, and thus natural coordinates (r,s,t) are introduced ranging from -1 < r,s,t < 1. 3/24/2015 6Adrian Egger | FEM I | FS 2015
- 7. Derivation of shape functions: Bar element (I) 1. Find a relationship for r(x). We choose -1 < r < 1. 2. Choose an appropriate shape function polynomial 3. Evaluate A at each DOF by substituting values of r 3/24/2015 7Adrian Egger | FEM I | FS 2015
- 8. Derivation of shape functions: Bar element (II) 4. Reorder the previous equation 5. Substitute into previous equation 6. Extract shape functions (as a function of r) 3/24/2015 8Adrian Egger | FEM I | FS 2015
- 9. Derivation of shape functions: Beam element (I) 1. Find a relationship for r(x). We choose 0 < r < 1. 2. Choose an appropriate shape function polynomial 3/24/2015 9Adrian Egger | FEM I | FS 2015
- 10. Derivation of shape functions: Beam element (II) 3. Find an expression linking displacements and rotations 4. Evaluate A at each DOF by substituting values of r 3/24/2015 10Adrian Egger | FEM I | FS 2015
- 11. Derivation of shape functions: Beam element (III) 4. Reorder the previous equation 5. Substitute into previous equation 6. Extract shape functions (as a function of r) 3/24/2015 11Adrian Egger | FEM I | FS 2015
- 12. Questions 3/24/2015 12Adrian Egger | FEM I | FS 2015