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# Element Loads Strain and Stress 2D Analyses - PowerPoint PPT Presentation

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 LoadsStrain and Stress2D Analyses

Structural Mechanics

Displacement-based Formulations

• 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

• Solution:

• Solve system of equations to determine unknown nodal displacements

• 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

E2

E1

No b.c.’s

N2

N1

E3

- or -

1000

This is it! Solve for the nodal displacements …

• 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.

• 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 = ?

• You might guess F = qL/2, but why?

• Setting dconc = ddist:

• 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?

sx

x

sx

x

• 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

• Add analytically-derived fixed-displacement strain and stress

sx

x

sx

x

+

• 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

• Strain and stress prediction improve

• Results converge toward the analytical solution even without inclusion of “fixed-displacement analytical stress”

• 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 …

• 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