Deflection of simply supported beam

The deflection of a simply supported rectangular beam is influenced by various factors such as length, depth, and width of the beam. Let's analyze each option to understand their impact on the deflection of the beam.

a) Directly proportional to the cube of its length:
When the length of a beam increases, the deflection also increases. This relationship is not linear but cubic. The deflection is directly proportional to the cube of the length. This means that if the length of the beam is doubled, the deflection will increase by a factor of 8 (2^3). Similarly, if the length is tripled, the deflection will increase by a factor of 27 (3^3).

b) Inversely proportional to the cube of its depth:
The depth of a beam refers to the dimension perpendicular to the length and width. When the depth of a beam increases, the deflection decreases. This relationship is also not linear but inverse cubic. The deflection is inversely proportional to the cube of the depth. If the depth is doubled, the deflection will decrease by a factor of 8 (2^3). Similarly, if the depth is tripled, the deflection will decrease by a factor of 27 (3^3).

c) Inversely proportional to its width:
The width of a beam refers to the dimension perpendicular to the length and depth. When the width of a beam increases, the deflection decreases. This relationship is inverse, but not cubic. The deflection is inversely proportional to the width. If the width is doubled, the deflection will decrease by a factor of 2. Similarly, if the width is tripled, the deflection will decrease by a factor of 3.

d) All of the above:
Considering the above explanations, we can conclude that all the given options are correct. The deflection of a simply supported rectangular beam is directly proportional to the cube of its length, inversely proportional to the cube of its depth, and inversely proportional to its width.

In summary, the deflection of a simply supported rectangular beam is influenced by its length, depth, and width. The relationships between these factors and the deflection are not linear, but cubic or inverse. These relationships are derived from the principles of structural mechanics and are widely used in engineering calculations and design.