A rectangular beam of uniform strength and subjected to a bending moment ‘M’ has a constant width. The variation in depth will be proportional to

Concept:

We have the bending equation for a beam, as

\(\frac{\text{ }\!\!\sigma\!\!\text{ }}{\text{y}}=\frac{\text{M}}{\text{I}}=\frac{\text{E}}{\text{R}}\)

Where,

σ = Bending stress, y = distance from the neutral axis, M = Bending moment of any section, I = Moment of inertia, E = Modulus of Elasticity of material, and R = Radius of curvature.

When a beam is designed such that the extreme fibers are loaded to the maximum permissible stress ρ_max by varying c/s, it will be known as beam of uniform strength.

\(\therefore \frac{\sigma }{y}=\frac{M}{I}\)

\(\text{ }\!\!\sigma\!\!\text{ }=\frac{\text{M}\times \text{y}}{\text{b}{{\text{d}}^{3}}/12}\)

\(\text{ }\!\!\sigma\!\!\text{ }=\frac{\text{M}\times {{\text{d}}_{\text{x}}}\times 12}{\text{bd}_{\text{x}}^{2}\times 2}\therefore \text{y}={{\text{d}}_{\text{x}}}/2\)

\(\text{ }\!\!\sigma\!\!\text{ }=\frac{6\text{M}}{\text{bd}_{\text{x}}^{2}}\)

\(\text{d}_{\text{x}}^{2}=\frac{6\text{M}}{\text{ }\!\!\sigma\!\!\text{ b}}\Rightarrow {{\text{d}}_{\text{x}}}=\sqrt{\frac{6\text{M}}{\text{ }\!\!\sigma\!\!\text{ b}}}\)

\(\therefore {{\text{d}}_{\text{x}}}\propto \sqrt{\text{M}}\)