For beams of uniform strength, if depth is constant, A. Width $${\text{b}} \propto {\text{M}}$$ B. Width $${\text{b}} \propto \sqrt {\text{M}} $$ C. Width $${\text{b}} \propto 3\sqrt {\text{M}} $$ D. Width $${\text{b}} \propto \frac{1}{{\text{M}}}$$

The correct answer is: B. Width $${\text{b}} \propto \sqrt {\text{M}} $$

The width of a beam is proportional to the square root of the bending moment, M. This is because the bending moment is a measure of the force that is trying to bend the beam, and the width of the beam is a measure of its resistance to bending. The greater the bending moment, the greater the force trying to bend the beam, and the greater the width of the beam needed to resist the bending.

The other options are incorrect because they do not take into account the relationship between the bending moment and the width of the beam. Option A states that the width is proportional to the bending moment, but this is not the case. The width is proportional to the square root of the bending moment. Option C states that the width is proportional to three times the square root of the bending moment, but this is also not the case. The width is only proportional to the square root of the bending moment. Option D states that the width is inversely proportional to the bending moment, but this is also not the case. The width is proportional to the square root of the bending moment, not inversely proportional to it.