2D element elasticity theory

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  • Last Post 29 March 2020
jfm418 posted this 27 March 2020

Hello,

I am trying to reproduce compression, tension, and shear on a 2D element/surface (see picture below) by applying normal or tangential force but I am not sure where I should be placing the fixed support. If I do not put any support I get pivot errors because the system is under-constrained. I would really appreciate any tip on this.

Thanks a lot.

Jad

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peteroznewman posted this 27 March 2020

Jad,

The diagrams and matrices refer to y and z, but in ANSYS, 2D is in X and Y directions.

Are you trying to solve 2D problems assuming Plane Stress or Plane Strain?

For applying tension or compression forces along the X axis, assume Quarter Symmetry. That means consider the lower left corner of the square to be the center of a square twice as large. Apply a Displacement of X = 0, Y = Free to the left edge.  Apply an X component of force to the right edge. Apply a Displacement of X = Free, Y = 0 to the bottom edge.  This will work for a single element or a mesh of many elements, however it is not useful for shear strain since that is not a symmetric load.

Note that applying forces to an element will not result in the strain matrix you show.  You are showing a strain matrix where the strain in the Y direction is zero.  That is an unusual constraint.  If there are no forces acting in the Y direction, but only in the X direction, there will be a strain in the X direction of Stress/E but the strain in the Y direction will occur due to the Poisson's Ratio.

If you want zero strain in the Y direction as shown in your matrix, then you need to apply Y=0 Displacement constraint to the top edge. If you want to apply strain in the X direction instead of force, you do that by using a Displacement on the right edge that is nonzero in X and leave Y free. For a strain of 0.01 use a Displacement that is 1% of the X dimension of the surface.

We can discuss Shear Strain in a later post.

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jfm418 posted this 29 March 2020

Actually you are right. I am trying to assume plane strain which makes everything a lot easier.

For tension and compression, I simply set displacements of Y=0 on the top and bottom ledge and opposite nonzero displacements in the X direction on the right and left edges with Y = 0.

For shear, I only applied opposite nonzero displacements on the top and bottom edges with Y=0.

For the last example (twisting), I tried to combine the displacements I applied for tension and shear but I get an error due to conflicting DOF constraints at one or more nodes. Not sure how else to simulate it. Would like to hear your thoughts on this.

Below are images of the deformation for:

-Tension

-Compression

-Shear

 

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