Determine the resultant force and specify where it acts on the beam measured from a.

Resultant force:

A resultant force is the single force which is obtained by combining a system of forces acting on a rigid body. The effect of the resultant force is the same as the original system of forces.

Moment:

Moment is the turning effect of a force and is written as an expression that involves the product of the distance and the physical quantity.

Uniformly distributed load:

It is a type of distributed load spread across a region of beam. It has constant magnitude throughout its length of application.

Uniformly varying load:

It is a type of distributed load in which load varies uniformly over a region of beam from minimum at one end to maximum at other end.

Free body diagram (FBD): It is a diagram of the member, representing all external forces and moments acting on it, considering all the support reactions on that member. It helps us to analyze the forces and moments acting on the system.

Sketch the free body diagram of the beam. Determine the expression for total resultant force by adding the expression of resultant force due to uniformly distributed load and the expression of resultant force due to uniformly varying load. Use moment equilibrium equation at the point to calculate the distance between point and the resultant’s line of action.

Expression to find theresultant force on beam carrying uniformly distributed load

…… (1)

Here, is the resultant force on the beam acting in downward direction,is the load on beam per unit length of the beam andis the length of the portion of the beam on which uniformly distributed load is applied.

Expression to find theresultant force on beam carrying linearly varying load

…… (2)

Here, is the resultant force on the beam acting in downward direction, is the load acting on beam per unit length andis the length of the portion of the beam on which linearly varying load is applied.

General sign convection for moment:

The moment is considered positive in counter-clockwise direction and negative in clockwise direction

Expression for the equilibrium of the rigid body:

Net force acting on a body is equal to zero. Also, the sum of the moment of all the forces at any point is equal to zero.

In plane,

And

Here, is the force acting on the rigid body indirection, is the force acting on the rigid body indirection andis the moment of forces acting at point.

Point of application of resultant force for uniformly distributed load

Resultant force lies at distance from left hand side.

Point of application of resultant force for linearly varying load:

Resultant force lies at distance from left hand side or low end side of the triangle.