The principle of superposition is a central concept in the analysis of structures. This is applicable when there exists a linear relationship between external forces and corresponding structural displacements. The principle of superposition may be stated as the deflection at a given point in a structure produced by several loads acting simultaneously on the structure can be found by superposing deflections at the same point produced by loads acting individually. This is illustrated with the help of a simple beam problem. Now consider a cantilever beam of length L and having constant flexural rigidity EI subjected to two externally applied forces P

_{1}and P_{2}as shown in Fig.1. From moment-area theorem we can evaluate deflection below , which states that the tangential deviation of point from the tangent at point A is equal to the first moment of the area of the M/EI diagram between A and C about . Hence, the deflection below due to loads P_{1}and P_{2}acting simultaneously is (by moment-area theorem),
u=A

_{1}x_{1}+A_{2}x_{2}+A_{3}x_{3}
where u is the tangential deviation of point C with respect to a tangent at A. Since, in this case the tangent at A is horizontal, the tangential deviation of point C is nothing but the vertical deflection at C. x

_{1},x_{2}and x_{3}are the distances from point C to the cancroids of respective areas respectively.Hence

After simplification one can write,

Now consider the forces being applied separately and evaluate deflection at in each of the case.

where u

_{22}is deflection at C(2) when load P

_{1}is applied at (2) itself. And,

where u

_{21}is the deflection at C(2) when load is applied at B(1). Now the total deflection at C when both the loads are applied simultaneously is obtained by adding u

_{21}and u

_{22}

Hence it is seen from equations (2.3) and (2.6) that when the structure behaves linearly, the total deflection caused by forces P

The method of superposition is not valid when the material stress-strain relationship is non-linear. Also, it is not valid in cases where the geometry of structure changes on application of load. For example, consider a hinged-hinged beam-column subjected to only compressive force as shown in Fig. 2.3(a). Let the compressive force P be less than the Euler’s buckling load of the structure. Then deflection at an arbitrary point C u

_{1},P_{2},P_{3},....P_{n}at any point in the structure is the sum of deflection caused by forces acting P_{1},P_{2},P_{3},....P_{n}independently on the structure at the same point. This is known as the Principle of Superposition.The method of superposition is not valid when the material stress-strain relationship is non-linear. Also, it is not valid in cases where the geometry of structure changes on application of load. For example, consider a hinged-hinged beam-column subjected to only compressive force as shown in Fig. 2.3(a). Let the compressive force P be less than the Euler’s buckling load of the structure. Then deflection at an arbitrary point C u

_{c}^{1}is zero. Next, the same beam-column be subjected to lateral load Q with no axial load as shown in Fig. 2.3(b). Let the deflection of the beam-column at C u_{c}^{2}be . Now consider the case when the same beam-column is subjected to both axial load and lateral load. As per the principle of superposition, the deflection at the centre u_{c}^{3 }must be the sum of deflections caused by P and Q when applied individually. However this is not so in the present case. Because of lateral deflection caused by Q, there will be additional bending moment due to at C.Hence, the net deflection u_{c}^{3}will be more than the sum of deflections u_{c}^{1}and u_{c}^{2}.