Element Loads Strain and Stress 2D Analyses

1 / 13

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

PowerPoint Slideshow about ' Element Loads Strain and Stress 2D Analyses' - keaton

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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
• Solution:
• Solve system of equations to determine unknown nodal displacements
• Determine strains and stresses from the nodal displacements

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 …

• 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 = ?

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?
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 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

+

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
• 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

sx

x

x

These are not always flat …

2D/3D elements extend this behavior dimensionally …

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