Method Of Finite Elements I: Shape Functions
Method of Finite Elements I:Shape FunctionsAdrian Egger
Why shape functions? Discretization leads to solution in the nodes, but no information concerning thespace in betweenShape 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 transformation3/24/2015Adrian Egger FEM I FS 20152
What can shape functions be used for?1. Used to interpolate between nodesi.e. discrete nodal quantities continuous across element𝑢 𝑥 𝐻𝑖 (𝑥)𝑢𝑖 𝐻1 (𝑥)𝑢1 𝐻2 (𝑥)𝑢23/24/2015Adrian Egger FEM I FS 20153
What can shape functions be used for?2. Used to discretize continuous quantities to nodal DOFi.e. continuous across element discrete nodal 𝑒𝑛𝑡3/24/2015𝐿 𝑻𝐿𝑯 𝑞(𝑥) 𝑑𝑥 𝑥 0𝑥 01𝐻011𝐻111𝐻021𝐻12Adrian Egger FEM I FS 2015𝑞𝐿2𝑞𝐿2𝑞(𝑥) 𝑑𝑥 12𝑞𝐿2𝑞𝐿2 12𝑉1𝑀1𝑉2𝑀24
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 developin the nodal DOF we are interested in! Equivalent to solving differential equation𝐸𝐼𝑤 𝐼𝑉 𝑞3/24/2015Adrian Egger FEM I FS 20155
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 optimizednumerical schemes, and thus natural coordinates (r,s,t) areintroduced ranging from -1 r,s,t 1.3/24/2015Adrian Egger FEM I FS 20156
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 polynomial3. Evaluate A at each DOF by substituting values of “r”3/24/2015Adrian Egger FEM I FS 20157
Derivation of shape functions:Bar element (II)4. Reorder the previous equation5. Substitute into previous equation6. Extract shape functions (as a function of “r”)3/24/2015Adrian Egger FEM I FS 20158
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 polynomial3/24/2015Adrian Egger FEM I FS 20159
Derivation of shape functions:Beam element (II)3. Find an expression linking displacements and rotations4. Evaluate A at each DOF by substituting values of “r”3/24/2015Adrian Egger FEM I FS 201510
Derivation of shape functions:Beam element (III)4. Reorder the previous equation5. Substitute into previous equation6. Extract shape functions (as a function of “r”)3/24/2015Adrian Egger FEM I FS 201511
Questions3/24/2015Adrian Egger FEM I FS 201512
Method of Finite Elements I: Shape Functions. 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 . i.e. continuous across element discrete nodal quantities
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