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Element Loads Strain and Stress 2D Analyses 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: – 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 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 No b.c.’s N3 32426694.11 -7680008.00 -26666666.67 0.00 -7680008.00 10239972.56 0.00 0.00 -26666666.67 0.00 26666666.67 0.00 0.00 0.00 0.00 20000000.00 -5760027.44 7680008.00 0.00 0.00 7680008.00 -10239972.56 0.00 -20000000.00 32426694.11 -7680008.00 -7680008.00 10239972.56 0.00 0.00 0.00 0.00 0.00 0.00 7680008.00 -10239972.56 E2 E1 0.00 0.00 1.00 0.00 0.00 0.00 -5760027.44 7680008.00 7680008.00 -10239972.56 0.00 0.00 0.00 -20000000.00 5760027.44 -7680008.00 -7680008.00 30239972.56 0.00 0.00 0.00 1.00 0.00 0.00 u1 v1 u2 v2 u3 v3 0.00 7680008.00 0.00 -10239972.56 0.00 0.00 0.00 0.00 1.00 0.00 0.00 30239972.56 u1 v1 u2 v2 u3 v3 = = - or - N2 N1 E3 32426694.11 -7680008.00 7680008.00 -7680008.00 10239972.56 -10239972.56 7680008.00 -10239972.56 30239972.56 u1 v1 v3 = 0 -1000 0 1000 This is it! Solve for the nodal displacements … F1x F1y F2x F2y F3x F3y 0.00 -1000 0.00 0.00 0.00 0.00 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. 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=? F=? • You might guess F = qL/2, but why? ddist L 0 dconc P (x)dx 1 L qL2 qxdx EA EA 0 2EA PL FL EA EA • Setting dconc = ddist: F qL 2 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? No sx x sx x 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 • Even when models become more complicated (higher order displacement/strain relationship, complex constitutive model) this is the general approach DL from nodal displacements DL e L s Ee Adjusting Strain and Stress • Add analytically-derived fixed-displacement strain and stress • This must be done for thermally-induced distributed loading sx x sx + Note the added constraint … x 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” Piece-wise Interpolation • If you remember nothing else about FEA, remember this … sx x sx These are not always flat … 2D/3D elements extend this behavior dimensionally … x 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