All problems I solved in MoM class regarding beam bending involved a beam with continuous cross section. But is the process the same for a beam with varying cross section? For 'same process' I mean applying the basic MoM equations learned and following the same steps to make the shear, moment and deflection diagrams. Of course, the only thing being different is the value of I when calculating stresses and deflections at different point along the beam's length. Are there any reference books that deal with this type of beams? Thank you, guys, in advance.
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$\begingroup$ It is usually dealt with the piecewise integration method. $\endgroup$– r13Commented Nov 13, 2022 at 16:41
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2 Answers
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If you have FEM at your disposal, you could of course use it for finding an answer.
Alternatively, you could also describe the beam by its pieces, finding the deflection for the pieces (that are then uniform) and find the answer by superposition.
One very simple test case would be a beam that is flexible for a short piece near the wall mount and rigid otherwise. Divide it in those two pieces, find the bending moment, compute deflection and curvature and put them back together. In this case you will find that the deflection at the end of the beam equals the curvature at the end of the short flexible piece near the wall times the length of the beam. The rigid part doesn't bend, but it does add to the total deflection due to its angle.
The principle of superposition is taught in many standard textbooks on stiffness and strength of materials.
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There are formulas for tapered or curved (on one or both sides) but not for something like your figure! If the openings are equally spaced there are solutions in handbooks for Vierendeel trusses.
Basically, the hollowed sections are going to experience stress concentration at the corners, and each act as a fix-fix beam deforming like a letter S with an inflection point in the middle.
The solid sections will have a complex mix of stresses that are hard to calculate except for allowing a large safety factor and assuming they are rigid plates! Unless one uses FEM.
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$\begingroup$ Thank you for the answer, kamran. The figure I added was only for reference. I have never worked with a beam like that. But then would it be better to use the FEM to analyze beams with non-continuous cross sections? $\endgroup$– VXR.Commented Nov 13, 2022 at 19:07