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

Determine the resultant force and specify where it acts on the beam measured from A. Assume p = 230 lb/ft.

1.Determine the magnitude of the resultant force.

2.Determine the distance between A and the resultant's line of action.

## General guidance

Concepts and reason
The concepts required to solve this problem are Resultantforce, Moment, Uniformly distributed load, uniformly varying load and Free body diagram.

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.

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

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.

Fundamentals

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.

## Step-by-step

### Step 1 of 2

(1)

Free body diagram of the beam,

Here, is the distance between point and point , is the distance between point and point and is load per unit length acting on the beam.

From the above figure, the beam is carrying linearly varying load from point to point and the beam is carrying uniformly distributed load from point to point .

Calculate the magnitude of the resultant force.

Add equation (1) and equation (2) to calculate total resultant of force acting on the beam.

Substitute for, for andfor.

The magnitude of the resultant force is .

Part 1

The magnitude of the resultant force is .

Determine the expression for total resultant force by adding the expression of resultant force due to uniformly distributed load and uniformly varying load. Substitute the values of , and .Then calculate the total resultant force on the beam.

### Step 2 of 2

(2)

Calculate the distance between point A and the resultant’s line of action.

Here, is the resultant force acting due to uniformly varying load on member of the beam, is the resultant force acting due to uniformly distributed load on member of the beam, is the net resultant force acting at from point . is the distance between point and point and is the distance between point and point .

Distance between and line of action of

For uniformly varying load at member, acts at from point .

Distance between and line of action of

For uniformly distributed load at member, acts at from point .

Calculation for resultant force acting on member.

Here, is load acting per unit length and is the length of member.

Substitute for andfor

Calculate resultant force acting on member.

Here, is load acting per unit length and is the length of member.

Substitute for andfor

Use moment equilibrium equation at point.

Thus,

Substitute for, for, for, for and for

Part 2

The distance between point A and the resultant line of action is .

Calculate the values of and . Determine the point of application of and . Apply moment equilibrium equation at point. Substitute the values of and in moment equilibrium equation. Solve for the distance between point and resultant’s line of action.

Part 1

The magnitude of the resultant force is .

Part 2

The distance between point A and the resultant line of action is .

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