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Basic FEA Procedures

Basic FEA Procedures. Structural Mechanics Displacement-based Formulations. Computational Procedure. Element Matrices : Generate characteristic matrices that describe element behavior Assembly : Generate the structure matrix by connecting elements together Boundary Conditions :

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Basic FEA Procedures

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  1. Basic FEA Procedures Structural Mechanics Displacement-based Formulations

  2. Computational Procedure • Element Matrices: • Generate characteristic matrices that describe element behavior • Assembly: • Generate the structure matrix by connecting elements together • Boundary Conditions: • Impose support conditions, nodes with known displacements • Impose loading conditions, nodes with known forces • Solution: • Solve system of equations to determine unknown nodal displacements • Gradients: • Determine strains and stresses from the nodal displacements

  3. v2 F2y y y Element Physics N2 N2 u2 F2x E E • Truss (bar) element, axial tension or compression, not bending or torsion • Arbitrarily oriented in the (x,y) plane • Uniform cross-sectional area and material properties • Two nodes per element, linear displacement variation along element N1 N1 b b F1x v1 x x u1 F1y

  4. Element Matrices [k] • An element is fundamentally a matrix that relates nodal displacements with nodal forces • It reflects equilibrium between concentrated forces applied at nodes and the spring-like response when the bar changes length • It embodies constitutive behavior in the forma of a modulus value • It maintains compatibility between displacements and strains

  5. Element Matrices • Complexity arises quickly: here is a 3D Timoshenko beam element • These are “simple” elements that can be formulated directly • They embody exact variations of the field quantities (displ’s, rotations)

  6. N3 Assembly • Establish geometry and connectivity, number nodes and elements • Calculate characteristic matrix for each element independently • Load into a structure matrix [sdof x sdof] in size (sdof = num nodes x dof per node) E2 E1 N2 N1 E3

  7. Structure Matrix [K] • Coefficients on the diagonal are positive • Symmetric if loads are linearly related to displacements (superimposable) • Sparse (most coefficients = 0) for problems with more than a few elements • Represents a set of equations for simultaneous solution, order can be changed • Can be solved more efficiently if non-zero entries cluster along the diagonal • Is singular (non-invertible, non-solvable) with inadequate support (free to move) • Once it is solvable, it is “just” a (generally) very big linear algebra problem

  8. Boundary Conditions • Specification of known values associated with nodal degrees of freedom • Two types, two different approaches to implementation • Displacements • This is the quantity we are solving for • They are the “essential” boundary conditions in that a set must be specified to create adequate support conditions for the structure • Non-support conditions can also be specified, e.g. a known nodal displacement • Forces • These are “right hand side” values for our set of equations • They are the “natural” boundary conditions in that they “load” the structure • They are mutually exclusive • You cannot specify displacement and force for the same degree of freedom • If one is specified (e.g. displacement = 0), the other develops (e.g. reaction force = ?)

  9. N3 Example B.C’s • Displacements are handled moving the reaction influences to the right hand side and creation of equations that directly reflect the condition • Forces are simply added into the right hand side E2 E1 N2 N1 E3 - or - 1000 This is it! Solve for the nodal displacements …

  10. Solving 1 • Rarely is {D} = [K]-1{R} actually carried out • [K]-1 is “full” even if [K] is “sparse”, therefore storage is an issue for large models • The structure stiffness inverse is not needed to find {D} • Solution by Gauss elimination, LU decomposition, Cholesky decomp, etc. • Solution can start as [K] is assembled (wavefront methods) • Element numbering does make a difference in storage and efficiency • Most codes offer renumbering schemes to minimize storage and/or maximize solution speed

  11. Solving 2 • To iterate, or not to iterate … • Direct solvers are most useful for modest-size problems with many load cases • Iterative solvers are often used even if direct solution is possible • A convergence criteria is required to judge when a “solution” is found • The convergence rate is related to the condition number of [K] • Condition number is related to the ratio of highest to lowest Eigenvalues • Many factors produce poor [K] conditioning (bad ele shape, mat prop extremes)

  12. Strain and Stress Calculation • For bar/truss elements with just nodal boundary conditions: • Find axial elongation DL from differences in node displacements • Find axial strain e from the normal strain definition • Find axial stress sfrom the stress-strain relationship

  13. Element Loads • This is a surprisingly complicated issue • Distributed axial loads, self-weight, pressure, etc. • It is a place where FEA can go wrong and give you bad results • Element loads must be converted to nodal loads • “Lost” loads and inaccurate stresses must be dealt with • What if one of the nodes below is constrained? • Is the element stress state the same for the two representations? • These loading states are “consistent”, but require special handling

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