By prex @ http://www.xcalcs.com
You start by having the six stress components specified at every point along the segment representing the section thickness.
To get the membrane stress:
- Take the average of each stress component. In formula:
σim=( ∫σi dx )/t
where t is the thickness (length of segment), x=0 to t is the abscissa along the thickness and i=1 to 6.
Now you have a single set of six (membrane) stress components. - Determine the principal stresses of that set
- The membrane stress intensity is the absolute value of the largest algebraic difference of the principal stresses taken 2 by 2.
To get the membrane+bending stress the procedure is:
- Take the static moment of each stress component and obtain the average bending stress by dividing it by the static moment of a unit bending stress distribution (t2/6). In formula:
σib=( ∫σi (x-t/2) dx )/(t2/6)
Now you have a single set of six (bending) stress components. - Calculate two sets of m+b stress components at each segment end as:
σi(m+b)0=σim+σib
σi(m+b)1=σim-σib - Determine the principal stresses for each of the two sets
- Calculate the stress intensity as above for each of the two sets of principal stresses and take the larger value as the M+B stress intensity.
[Note: the formulae above are not exact for axisymmetric sections, as the stresses should also be averaged onto the radius. However, if the thickness to radius ratio is small, the impact of this error will also be small.]
The formula for the membrane stress becomes:
σim=( ∫σir dr )/trm with rm=(r1+r0)/2 and t=r1-r0.
[The formula for bending stress is more complex, but may derived in a similar way.]