# Airy stress and symmetry

• posted

Hi,

I am trying to calculate the airy stress function for a cantilever beam with a uniform load acting downward on its upper surface.

I am attempting a solution using an Airy function of polynomial degree

1. The beam has a rectangular cross section.

I know that if either simgaxx, sigmayy or sigmaxy are symetrical about any axis, the stress function derivative along the axis of symmetry can be reduced to an even function, and odd polynomial coefficients along the perpendicular axis can be eliminated.

I'm just a little unsure which stresses are symmetrical about which axes.

The only one I can imagine would be sigmaxx is symetrical about the x- axis, so that would mean odd polynomial coefficients of y can be eliminated.

Am I right, and are there any other symmetrical distributions that I have missed?

Any help would be greatly appreciated.

• posted

I'm not sure I fully understand what you are asking (not your fault).

The normal stresses, sigma-xx (assuming x is the long axis of the beam) should be symmetrical about the center of the beam. That is linear with sigma-xx going to zero at the middle.

The shear stress in the beam is also symmetrical about the center of the beam, reaching a maximum at the center and zero on the upper and lower surfaces.

By center I mean the plane running horizontally through the half- height of the beam. That is the plane that would contain the neutral axis of the beam at a given cross section.

• posted

Thanks very much, that is exactly what I needed

Many Thanks

• posted

To clarify, the normal stresses vary linearly with distance from the axis. Sigma-xx isn't symmetrical about the center. Sigma-xx should still be zero on the neutral-axis. But it sounds like you found what you needed. Glad to be helpful.

If anyone else is interested here is the answer I obtained:

phi = -( Wo / 4)(x^2) + (Wo / 8h^3)(x^2)(y^3) - (3Wo / 8h)(x^2)(y)

I assumed the x-axis to be the long axis (perpendicular to the wall) and the y-axis to be parallel to the wall (that is the y-axis is along the beam's height). I denoted the loading as "Wo" and assumed, as you said, it to be uniform. The beam is 2h high.

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