Bending Moment at any Point

The bending moment at any point along a beam is a measure of the internal forces that are causing the beam to bend. It is defined as the algebraic sum of the moments of all the forces acting on one side of the point.

Algebraic Sum of Forces
The bending moment at any point can be calculated by considering the algebraic sum of the forces on either side of the point. This includes both the vertical and horizontal forces acting on the beam.

Moments of Forces
The bending moment at a point is equal to the sum of the moments of all the forces on either side of the point. The moment of a force is calculated by multiplying the magnitude of the force by the perpendicular distance from the point to the line of action of the force.

Explanation of Option D
Option D states that the bending moment at any point is equal to the moments of forces on either side of the point. This is the correct answer because the bending moment is determined by the forces that are causing the beam to bend. These forces can be vertical or horizontal, and their moment arms (perpendicular distances) from the point of interest are taken into account.

By calculating the moments of the forces on either side of the point, we can determine the net bending moment at that point. The algebraic sum of these moments will give us the bending moment, which is an important parameter in analyzing the structural behavior of beams.

Example
Let's consider a simple example to illustrate this concept. Suppose we have a beam with a vertical force of 10 kN acting downward at a distance of 2 meters from a point. On the other side of the point, there is a horizontal force of 5 kN acting to the left at a distance of 3 meters. The bending moment at this point can be calculated as follows:

Moment due to the vertical force = 10 kN * 2 m = 20 kNm (clockwise)
Moment due to the horizontal force = 5 kN * 3 m = 15 kNm (counterclockwise)

Bending moment at the point = 20 kNm - 15 kNm = 5 kNm (clockwise)

This example demonstrates how the moments of forces on either side of a point are used to calculate the bending moment at that point. Option D correctly represents this concept, making it the correct answer.

D