# Principle of Superposition of Forces

By Aina Parasher|Updated : May 5th, 2022

The principle of superposition is one of the most important concepts in the Mechanics of Solids. The principle of superposition can be defined as "If two equal and opposite collinear forces are added to removed from the system, there will be no change in the system as well as the position of the body".

The principle of superposition can also be defined as "If a force system is acting on a body is superimposed with another force system that itself is in equilibrium, then, the effect of both the force systems remains same as that of initial force system". Let us discuss further the laws of superposition of forces.

## What is the Superposition of Forces?

When a number of charges interact, the total force on a particular charge is the vector sum of the forces exerted on it by all other charges, according to the principle of superposition of forces. The existence of other charges has no effect on the force between two charges.

In other words, it is assumed that the resultant force of an ensemble of forces is the vectorial sum of these forces, implying that the individual forces do not interact with one another except through the entire system.

## Law of Superposition of Forces

The law of superposition of forces states that the combined effect of a force system acting on a particle or a rigid body is the sum of the effects of individual forces. Consider one body on which three forces P1, P2, and P3 are acting simultaneously. These forces create a deformation of the triangle in that body.

Deformation in the body due to P1, P2 & P3 is Δ.

Now, apply each force P1, P2 & P3 separately and calculate the deformation in the body because of the individual force.

Deformation due to P1 = Δ1

Deformation due to P2 = Δ2

Deformation due to P3 = Δ3

After applying the Principle of Superposition

Δ = Δ1 + Δ2 + Δ3

Δ= PL/AE

Overall deformation of the body is equal to the summation of deformation of the body when subjected to a single force system.

Let us understand the concept of superposition of forces with one more example.

Consider a body with two points A and B as shown in the figure. Apply a force of magnitude P at A.

To differ this force system and superimpose this with another force system such that the effect created by force P at location A remains the same.

There will be no additional effect on the body because of the new force system. Therefore, whatever effect that has been created in the first case remains the same in the second case also.

Now, consider the distance between A and B as ‘d’.

The effect of this force system is similar to that of the first case. All three force systems create the same effect on the body.

All three force systems create the same effect on the body.

## Application and Limitations of Principle of Superposition

The Principle of Superposition of forces is mainly used in all types of problems but is effectively used in changing the type of force system without changing the effect of the forces on the body. This principle is widely used in the problem of non-concurrent force systems. In Engineering Mechanics, it is applicable to all types of problems. In Strength of Materials, the only limitation is body should behave as an elastic body with small deformations.

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## FAQs

• (Length) LAB = LBC = LCD = 200mm

(Diameter) dAB = dBC = dCD = 30mm

(Area) AAB = ABC = ACD = π/4 x 302 = 706.87 mm2 ≈ 707 mm2

E = 200 x 103 N/mm2

Analyzing the sections AB, BC, and CD separately.

Force in section AB = 60kN (T)

Force in section BC = 80kN (T)

Force in section CD = 40kN (T)

(δL)AD = (δL)AB + (δL)BC + (δL)CD

= (PL/AE)AB+ (PL/AE)BC+ (PL/AE)CD

= L/AE (P)AB+ (P)BC+(P)CD

= 2.55 mm (Ans)

• Principle of Superposition can also be defined as “ If a force system is acting on a body is superimposed with another force system that itself is in equilibrium, then, the effect of both the force systems remains same as that of initial force system.

• In Strength of Materials, principle of superposition is applicable only for small deformation and the bodies should behave as elastic body but this principle is applicable to all types of problems in Engineering Mechanics.

• A = 400 mm2

E = 84 x 103

LAB = 500mm

LBC = 1400mm

LCD =1000mm

Analyzing the sections AB,BC & CD individually.

Force on BC = 50 kN

Force on BC = 30kN (Compressive)

Force on CD = 10 kN (Compressive)

Total elongation

(δL)AD = (δL)AB + (δL)BC + (δL)CD

= (PL/AE)AB+ (PL/AE)BC+ (PL/AE)CD

= -0.8035mm

Negative value indicates contraction.

• Length : (L)AB = 1000mm

(L)BC = 2000mm

(L)CD = 1000mm

Area:

ACD = AAB = π ×1024=78.5 mm2

ABC = π ×2024=314 mm2

Analyzing the section AB, BC, CD separately.

Force on AB = 30kN

Force on BC = 20kN

Force on CD = 35kN