Overview of Finite Element Methods Group Members: Daniel Braley Heriberto Cortes Adam Hollrith

Outline Fundamental Concept of FEM Reasons for Using FEM Various Steps in FEM Analysis Examples Brief Outline of Algor Explanation of Homework Problem

The Fundamental Concept of FEM A continuous field  of a domain  and an infinite number of degrees of freedom is broken up and approximated by a set of piecewise continuous functions of a finite number of degrees of freedom. The piecewise functions are then defined over a set of subdomains called elements. The unknown ’s are defined at nodes and are evaluated using the equation, [K]{} = {F}

Concept of FEM Discrete Element Node Continuous function Finite Degrees of Freedom Infinite Degrees of Freedom

Reasons For Using FEM Serves as a tool for: - Stress and vibration analysis - Fluid flow analysis - Electrostatic analysis - Displacement analysis Allows for an approximation of otherwise impossible calculations

Various Steps in FEM Analysis 1) Discretize the Structure a) identify and label nodes i) must be at points where loads act ii) must be at points where geometry changes b) identify and label elements c) identify symmetry conditions

Steps in FEM Analysis Cont. 2) Select a displacement function that is defined within the element, using the nodal values of the element. 1 1 2 6 2 6 3 5 3 4 5 4

Steps in FEM Analysis Cont. 3) Define the strain and stress displacement relationships In the case of one-dimensional deformation, strain in the x-direction Now apply Hooke’s Law for the stress analysis x is the stress in the x direction E is the modulus of elasticity

Steps in FEM Analysis Cont. 4) Derive the element stiffness matrix by the work or energy methods, or by methods of weighted residuals {f}=[k]{d} Utilizes the method of weighted residuals {f} is the vector of the element nodal forces [k] is the element stiffness matrix {d} is the vector of unknown element nodal degrees of freedom or generalized displacements

Steps in FEM Analysis Cont. 5) Assemble the element equations to obtain the global or total equations, and also introduce boundary conditions {F}=[K]{d} {F} is the vector global nodal forces [K] is the structure global or total stiffness matrix {d} is now the vector of known and unknown structure nodal degrees of freedom or generalized displacements

Steps in FEM Analysis Cont. 6) Solve for unknown degrees of freedom or generalized displacements by such methods as the Gauss-Elimination Method 7) Solve for the element strains and stresses 8) Interpret the results and analyze them for use in the design/analysis process. Let’s Look at an Example!!

Example Find the nodal displacements at points 1,2, and 3, and find the stress in each element 3 1 1 2 2 Where P is a load applied at node 2 to the center of the bar, the bar has a constant area A, and an elastic modulus of E

Example Continued 1) Discretize the function 1 3 1 2 2 1 2 2 3 1 2 u1

Example Continued 2) Select a displacement function for bar 1 2 2 3 1 Note that this a one-dimensional problem and the displacement is only in the x-direction

Example Continued 3) Define the stress and strain relationships Now apply Hooke’s Law for the stress analysis x is the stress in the x direction E is the modulus of elasticity

Example Continued 4) Derive the element stiffness matrix for each element Model Using a 1-D bar of the following dimensions Given: σx = Eεx. σx = P/A εx = du/dx du/dx = (d2x – d1x) By substitution: Eεx = P/A , P = EA εx , P = f -f1 = EA (d2x – d1x) f1 = EA (d1x – d2x) L f2 = EA (d2x – d1x) L A f2 f1 d1x d2x L L The stiffness matrix [k] can now be found by [k] = EA -1 -1 1 L

Example Continued {f} = [k]{d} For element 1: For element 2: 1 3 1 2 2 EA 1 -1 -1 1 u1 u2 u1 u2 = L 1 2 2 3 1 2 u1 u2 u2 u3 u2 u3 f22 f3 EA 1 -1 -1 1 u2 u3 = L

Example Continued 5) Construct the global matrix and introduce the boundary conditions and known values -1 0 -1 2 -1 0 -1 1 f1 f21+f22 f3 EA u1 u2 u3 = L u1=0 , u3=0 , f2 = f21+f22 = P -1 0 -1 2 -1 0 -1 1 f1 P f3 EA u2 = L

Example Continued 6) Solve for the Unknowns -1 0 -1 2 -1 0 -1 1 f1 P -1 0 -1 2 -1 0 -1 1 f1 P f3 EA u2 = L Reaction 1= f1 = (-EAu2)/L P = (2EAu2)/L, so u2 = (PL)/(2EA) Reaction 3 = f3 = (-EAu2)/L

Example Continued 7) Solve for the element strains and stresses u1 u2 u2 u2 x1 = 1/L [-1 1] x1 = E/L [-1 1] u2 u2 x2 = E/L [-1 1] x2 = 1/L [-1 1]

End of Example 8) Analyze the results found - Do they make sense? - In being a static problem, do all of the forces add up to 0?

Algor A method for computer aided analysis (i.e. can import Pro/E files into Algor) Saves time and money for companies by not having to calculate everything by hand

References Chandrakath, Shet. “FEM Class Notes”. http://www.eng.fsu.edu/~chandra/courses/eml4536 2003. Logan, Daryl L. A First Course in the Finite Element Method: Third Edition. Brooks/Cole. 2002. Fancher, Darren, et.al., IAS2 Spring Report: Integrated Advanced Surveillance System Final Report. 2003.