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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

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Element loads strain and stress 2d analyses

Element LoadsStrain and Stress2D Analyses

Structural Mechanics

Displacement-based Formulations

Computational procedure
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


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




- or -


This is it! Solve for the nodal displacements …

Other loading conditions
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
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 = ?

F = ?

  • You might guess F = qL/2, but why?

  • Setting dconc = ddist:

Consistent nodal loads
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?






Strain and stress calculation
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

Adjusting strain and stress
Adjusting Strain and Stress

  • Add analytically-derived fixed-displacement strain and stress

  • This must be done for thermally-induced distributed loading






Note the added constraint …

Mesh refinement
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
Piece-wise Interpolation

  • If you remember nothing else about FEA, remember this …





These are not always flat …

2D/3D elements extend this behavior dimensionally …

To refine or not to refine
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