Element Loads Strain and Stress 2D Analyses

1. Element LoadsStrain and Stress2D Analyses 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. N3 Example B.C.’s • Displacements are handled by 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 No b.c.’s N2 N1 E3 - or - 1000 This is it! Solve for the nodal displacements …

4. Other Loading Conditions • Consider the assembled equation system [K]{D} = {F} • The only things we can manipulate are: • Terms of the stiffness matrix (element stiffness, connectivity) • The unknown or specified nodal displacement components • The applied nodal force components • How do we manage “element” loads? • Self-weight, structural systems where gravity loads are significant • Distributed applied loads, axial, torsional, bending, pressure, etc.

5. Conversion to Nodal Loads • All loads must be converted to nodal loads • This is more difficult than it appears • It is a place where FEA can go wrong and give you bad results • It has consequences for strain and stress calculation q (N/m) L F = ?

6. F = ? • You might guess F = qL/2, but why? • Setting dconc = ddist:

7. Consistent Nodal Loads • Consistent nodal loading: • Utilizes the same shape (interpolation) functions (more later) as displacement shape functions for the element • The bar (truss) shape functions specify linear displacement variation between the nodes • We choose a concentrated nodal force that results in an equivalent nodal displacement to the distributed force • Question: Are element strain and stress equivalent?

8. No sx x sx x

9. 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 s from the stress-strain relationship • Even when models become more complicated (higher order displacement/strain relationship, complex constitutive model) this is the general approach

10. Adjusting Strain and Stress • Add analytically-derived fixed-displacement strain and stress • This must be done for thermally-induced distributed loading sx x sx x + Note the added constraint …

11. Mesh Refinement • What if we model a bar (truss) or beam element not as a single element, but as many elements? • No gain is made in displacement prediction • Holds true for node and element loading • Strain and stress prediction improve • Results converge toward the analytical solution even without inclusion of “fixed-displacement analytical stress”

12. Piece-wise Interpolation • If you remember nothing else about FEA, remember this … sx sx x x These are not always flat … 2D/3D elements extend this behavior dimensionally …

13. To Refine, or Not To Refine … • It depends on the purpose of the analysis, the types of elements involved, and what your FEA code does • For bar (truss) and beam elements: • Am I after displacements, or strain/stress? • Does my FEA code include analytical strain/stress? • What results does my FEA code produce? • Can I just do my own post-processing? • Always refine other element types