A cantilever whose cross section increases uniformly from 200 mm to 400 mm at fixed end carries a load of 30 kN at the free end over a span of 2m. At what distance will the bending stress be maximum?

A cantilever whose cross section increases uniformly from 200 mm to 400 mm at fixed end carries a load of 30 kN at the free end over a span of 2m. At what distance will the bending stress be maximum?

Answer

BM obviously increases as you travel from the free end to the fixed end

But the Moment of inertia and section modulus too increase due to increasing depth.

Both BM and bending stress will be maximum at the fixed support in this case where the starting depth is 200mm and it increases to 400 mm at the support.

But if the starting depth were less than 200mm, say 150 mm or less and it increased to 400 mm at the fixed end then the maximum stress would be somewhere between the free end and the fixed end.

The exact location where the stress would be maximum will depend on the depths at the free end, the depth at the fixed end and how the variation takes place (linearly or otherwise)

In a general case, there is no way you can predict it unless you consider different locations at close intervals and work out the BM, section modulus and the bending stress at each location.

I have done the calculations for this particular case and the results were:

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