I do a lot of beam-based analysis of structures with circular tube cross-sections. As I've been thinking about the definition of "worst stress" for beams, it seems that this may not be totally accurate for a circular or hollow circular cross-sections.
The help documentation indicates that mathematical worst-stress could be at a non-existent location for asymmetrical beams. For a round beam, despite it being symmetrical, it seems that the worst stress reported may not be exactly correct either. It would seem that in-between the '2' and '3' axes on a circular cross-section (say at a 45° angle) that the stress would be somewhere between the values of |P/A| + |M2/S2| and |P/A| + |M3/S3|.
Does this make sense, or am I thinking about this incorrectly?
I'd appreciate any feedback.
I agree with your thinking Adam. The highest stress on the surface of a circle or tube is really lower than |M2/S2| + |M3/S3|.
So, if you want to have some fun today , you can check my math. The stress at a point depends on the distance from the neutral axis, it seems reasonable (but perhaps not correct!) that the
worse stress = stress2*cosine + stress3*sine.
where the angle is measured from one of the axes. The maximum result occurs at the angle where the derivative equals zero, or
change in worse stress/change in angle = -stress2*sine + stress3*cosine = 0
So the angle depends on the values for the maximum stress about axes 2 and 3. If we look at the ratio stress3/stress2 = X, the equation simplifies to
-stress2*sine + (X*stress2)*cosine = 0
stress2*(X*cosine - sine) = 0
Of course, stress2 could be 0, but the more interesting result is when
X*cosine - sine = 0
X*cosine = sine
X = sine/cosine = tangent
So by calculating a few ratios of X, the angle where the maximum is located can be calculated, which can then be used to calculate the maximum worse stress of stress2*(cosine+X*sine). Here are a few examples:
stress3/stress2 Angle worse stress/stress2 software's (|M2/S2| + |M3/S3|)/stress2
0 0 1
0.578 30 1.16
1 45 1.41
1.5 56.3 1.80
2 63.4 2.24
If you do not want to use the conservative values from the software, you might be able to write a custom formula to do the above calculation. That would be neat to see if it works!
Simulation Mechanical user since Dec 1997