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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

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 …

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)

L

F = ?

slide6
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?
slide8
No

sx

x

sx

x

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

sx

x

sx

x

+

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 …

sx

sx

x

x

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