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Element Loads Strain and Stress 2D Analyses Structural Mechanics Displacement-based Formulations

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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

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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 E1 E2 E3 N1 N2 N3 1000 32426694.11-7680008.00-26666666.670.00-5760027.447680008.00u1 = F1x -7680008.0010239972.560.00 7680008.00-10239972.56v1F1y -26666666.670.0026666666.670.00 u2F2x 0.00 20000000.000.00-20000000.00v2F2y -5760027.447680008.000.00 5760027.44-7680008.00u3F3x 7680008.00-10239972.560.00-20000000.00-7680008.0030239972.56v3F3y 32426694.11-7680008.000.00 7680008.00u1 = 0.00 -7680008.0010239972.560.00 -10239972.56v1-1000 0.00 1.000.00 u20.00 1.000.00 v20.00 1.000.00u30.00 7680008.00-10239972.560.00 30239972.56v30.00 This is it! Solve for the nodal displacements … 32426694.11-7680008.007680008.00u1 = 0 -7680008.0010239972.56-10239972.56v1-1000 7680008.00-10239972.5630239972.56v30 - or - No b.c.’s

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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.

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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) F = ? L

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You might guess F = qL/2, but why? Setting conc = dist :

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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?

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No xx x xx x

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Strain and Stress Calculation For bar/truss elements with just nodal boundary conditions: – Find axial elongation L from differences in node displacements – Find axial strain from the normal strain definition – Find axial stress from the stress-strain relationship Even when models become more complicated (higher order displacement/strain relationship, complex constitutive model) this is the general approach

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

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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”

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

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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

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Illustration of FE algorithm on the example of 1D problem Problem: Stress and displacement analysis of a one-dimensional bar, loaded only by its own weight,

Illustration of FE algorithm on the example of 1D problem Problem: Stress and displacement analysis of a one-dimensional bar, loaded only by its own weight,

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