Location of the load generated due to gravity is at the Center of Gravity of the bar which is at the middle of a uniform bar. Load due to gravity experienced by the bar at the top-most is full weight of bar, and that at the bottom-most point is equal to zero. Hence, the average load along the length of the bar = weight/2. Hence, displacement at the tip = PL/AE = wL/(2AE)

Deformation of a bar under its own weight vs direct load

Definition of deformation: Deformation refers to the change in the shape or size of a body due to the application of external forces.

Deformation under own weight:
When a bar is subjected to its own weight, it experiences a force called the self-weight force. This force is equal to the weight of the bar and acts vertically downwards. Due to this force, the bar deforms and bends downwards.

Direct load equal to weight:
When the same bar is subjected to a direct load equal to the weight of the bar, the force acting on the bar is the same as in the previous case, i.e. the weight of the bar. However, this force is applied directly to the top of the bar, as opposed to being applied through the bar's own weight. This causes a different type of deformation in the bar.

Comparison of the two types of deformation:
The deformation of a bar under its own weight is greater than the deformation when the same body is subjected to a direct load equal to the weight of the body. This is because, in the former case, the force causing the deformation is distributed along the length of the bar, whereas in the latter case, the force is applied directly to the top of the bar.

Mathematical explanation:
The amount of deformation in a body is proportional to the magnitude of the force acting on it. When a bar is subjected to its own weight, the force causing the deformation is equal to the weight of the bar, which is distributed along the length of the bar. However, when the same bar is subjected to a direct load equal to its weight, the force causing the deformation is applied directly to the top of the bar. This means that the force acting on any point along the length of the bar is greater in the latter case. Therefore, the deformation in the latter case is less than the deformation under the self-weight of the bar.

Conclusion:
The deformation of a bar under its own weight is greater than the deformation when the same body is subjected to a direct load equal to the weight of the body. This is because, in the former case, the force causing the deformation is distributed along the length of the bar, whereas in the latter case, the force is applied directly to the top of the bar.

B