Look before you leap! 1.Check the model that you have developed: Boundary conditions Loadings Symmetry? Element aspect ratios/shapes Mesh gradation 2.Check the results Eyeball Anything funny (nonzero displacements where they should be zero?) Are stress concentrations in places that you expect? Comparison with known analytical solution/literature 3.If you remesh the same problem and analyze, do the solutions converge? (specifically check for convergence in strain energy)
Stress equilibrium in FEM analysis Example: 80cm 1 2 Consider a linear elastic bar with varying cross section P=3E/80 x The governing differential (equilibrium) equation Boundary conditions Analytical solution Eq(1) E: Young’s modulus
Lets us discretize the bar using a 2-noded (linear) bar element. The finite element approximation within the bar is where the shape functions If we incorporate the boundary condition at x=0 Does this solution satisfy the equilibrium equation (Eq 1)?
Conclusion: The FEM displacement field does NOT satisfy the equilibrium equations at every point inside the elements. However, the solution gets better as the mesh is refined.
Stress equilibrium in FEM analysis To obtain exact solution of the mathematical model in solid mechanics we need to satisfy 1. Compatibility 2. Stress-strain law 3. Stress-equilibrium at every point in the computational domain. In a FE model one satisfies the first 2 conditions exactly. But stress-equilibrium is NOT satisfied point wise. Question: Then what is satisfied?
Let us compute the FEM solution using a bar element The stiffness matrix is The system equations to solve are With u 1x =0; we solve for (Note that the exact solution for the displacement at node 2 is 1cm!!)
Let us now compute the nodal forces due to element stresses using the formula
1. Element equilibrium 2. Nodal equilibrium P=3E/80 Two observations
The following two properties are ALWAYS satisfied by the FEM solution using a coarse or a fine mesh Property 1: Nodal point equilibrium Property 2. Element equilibrium El #4El #3 El #1 El #2 P PROPERTY 1: (Nodal point equilibrium) At any node the sum of the element nodal point forces is in equilibrium with the externally applied loads (including all effects due to body forces, surface tractions, initial stresses, concentrated loads, inertia, damping and reaction)
How to compute the nodal reaction forces for a given finite element? Once we have computed the element stress, we may obtain the nodal reaction forces as
El #4El #3 El #1 El #2 P This is equal in magnitude and in the same direction as P Sum of forces equal externally applied load (=0 at this node) Nodal point equilibrium implies:
PROPERTY 2: (Element equilibrium) Each element is in equilibrium under its forces f i.e., each element is under force and moment equilibrium e.g., F 3x F 2x F 1x F 4x F 1y F 2y F 3y F 4y Define But since this is a rigid body displacement, the strains are zero Hence as a rigid body displacement in x-direction
Example (Finite Element Procedures, Bathe 1996)
Hence a finite element analysis can be interpreted as a process in which 1. The structure or continuum is idealized as an assemblage of elements connected at nodes pertaining to the elements. 2. The externally applied forces are lumped to these nodes to obtain the equivalent nodal load vectors 3. The equivalent nodal loads are equilibriated by the nodal point forces that are equivalent to the element internal stresses. 4. Compatibility and stress-strain relationships are exactly satisfied, but instead of force equilibrium at the differential level, only global equilibrium for the complete structure, of the nodal points and of each element under its nodal point forces is satisfied.