## Simulation Mechanical and Multiphysics

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# Shear area, shear delfection, and bending stresses for linear beam elements

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As I discovered when comparing hand deflection calculations to the simulation, and as specified in the help section, the FEA takes into account the shear deflection when reporting displacements. I have a couple of different questions regarding this.

1) How does the software compute the effective shear area for different shapes in the cross-section library for user-defined shapes? For example, for a pipe, the area for SA1 and SA2 is 25% of the total cross-sectional area (CSA). According to most sources, the shear area for a thin walled pipe is 50% of the CSA. Is this a mistake? Shouldn't there be some documenation on how the software is calculating these? For other shapes, it appears like the effective shear area is in line with other sources. For rectangular shapes, the effective shear area is 85% of the CSA and in line with other sources which specify (5/6)CSA.

2) When a user specifies a shear area (SA2), the software adds the shear deflection to bending deflection and reports a total displacement. Does the shear area effect the bending stresses in a beam? It appears that the bending stresses increase very slightly when defining a shear area in comparison to when not entering a shear area. The reason I ask is because I was comparing hand calculations to the software for a very simple grillage. I have two beams which intersect at the mid-point and all have fixed ends and uniform loading. Using hand calculations, I can match almost excatly the maximum bending stress at the end of a beam by adding the bending moment when considering just one of the beams under uniform loading to the bending moment caused by the relative displacement between the intersection of the two beams to the fixed end. When I consider the displacement from only bending, my hand calculations match the software; however, when I consider the displacement from both shear and bending, the stresses in the software remain almost the same as when I didn't imput a SA2 (or SA3) and my hand calculations show greater bending stresses. I'm not sure if I am wrong in assuming that shear deflections would cause a bending moment, or if the software just doesn't account for bending stresses from shear defelction.

# Re: Shear area, shear delfection, and bending stresses for linear beam elements

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1) I agree that it looks like a bug.

2) I do not know for certain, but it looks to me that the displacement due to the shear area will change the bending moment and bending stress in the beam in general.Take a look at the attached image (which shows the elements with "Results Options > View > Shrink Elements" turned on). The displacements and bending moments are clearly different between the two structures: one with the shear areas and one without the shear areas.

In my test model, the structure is (purposely) not symmetric. Because the beams are connected, the short member deflects more than it would if it were not connected, and the long member deflects less than it would if it were not connected. Another way to say it is that there is a transfer of force between the two beams, and this changes the moment diagram.

If all of the members of the grill are the same length and intersect at the midpoints and at a 90 degree angle, then the displacement of each individual member is the same regardless of whether the shear area is included or not. So there is no extra force transmitted; hence, the bending moments would be the same.

Of course, the above discussion ignores the effects of the moment at the joint which may change things, too.

Mechanical Engineer

Pittsburgh, PA

16.9 years experience with Simulation Mechanical