Bending Moment and Axial Force from Stresses in Soild Elements

[bdbd]

Hi,
If you model a member with solid elements, you can get different stress results, such as Syy, Sxy, Sxx etc. depending on the notation used by the software. In a report that I read (by a reputable consultant), they have used these stresses to calculate the bending moment and axial forces, using very simple equations such as (Syy+Sxy)/2 = Axial stress and (Syy-Sxy)/2 = Bending stress.
Of course there might be million complicated things, but thinking simply, for a simple beam modelled as volume element, can you do this? Are you aware of any calculation (not necessarily the one I shared) to convert element stresses to axial / bending stresses?

 

[centondollar]

If you have several solid elements through-thickness (i.e., a rectangular shape divided into three or more parts in the direction of vertical deflection), it is possible to calculate bending moment using definitions, yes. If you have one or two elements in the through-thickness direction, you cannot find bending moments with any reasonable accuracy.

Assume "z" = into paper, "x" = longitudinal axis, "y" = vertical axis for a typical beam. Then:

Normal force = N = integral(sigma_xx) dz dy
Bending moment about principal axis z = Mz = integral (sigma_xx*y) dz dy (uniaxial bending, typical case)
Bending moment about principal axis y = My = integral (sigma_zz*z) dz dy (transverse bending, "into the paper" so to speak)

If the stresses very also in the "into paper" (z) direction, you need to integrate in both "y" and "z" directions and also find the principal axes. If the stress state is such that it depends mostly or only on the through-thickness coordinate "y", you can simplify the expressions by omitting integration in direction "y" and replacing it by multiplying each integration interval/panel with a width dependent on location in the through-thickness direction: "width_z_direction = function". For a typical beam in bending (not combined torsion and bending about several axes), the integration would be required in the through-thickness direction, but check to make sure that it is the case in your calculations.

The equations you showed are not correct, since the assumption of linear variation of normal stresses is an assumption of classical beam theory and not what actually happens if you model a beam with 3D volume elements. The stresses will rather be of a parabolic shape (similar to a 2D plane stress thin-slab solution of the beam bending problem), and the integration should be performed numerically using the basic definitions.

 

[centondollar]

I should add that you can cross-check the results yourself if you have access to output of the stresses of interest, the meshing and the geometry. No need to take the word of a consultant - however reputable - at face value. To me, what you described seems odd (calculating normal forces and bending moments by some type of averaging), and I would urge you to investigate it further.

 

[rb1957]

why do you want to separate the stress into bending and tension ? to take advantage of plastic bending (increased allowable) ?

I think that'll be a hard sell, unless the solid is clearly in bending ... so stress on one surface is tension, stress on the opposite face is compression ...

I'm pretty sure 3D solids are considered to work up to Ftu. If you want to go beyond that I think it'll take tetsing.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.