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

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Presentation Transcript

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

- 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

- 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

- 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 = ?

F = ?

- You might guess F = qL/2, but why?
- Setting dconc = ddist:

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

- 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

- 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

- 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

- 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

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