Principle of superposition with examples

The principle of superposition states that for a linearly elastic structure, the total effect (reaction, internal force, stress or displacement) caused by several loads acting together is equal to the algebraic sum of the effects caused by each load acting separately.

Conditions for application

1. The material obeys Hooke's law (stress is proportional to strain), i.e. the material is linearly elastic.
2. Displacements are small, so the geometry of the structure does not change appreciably under load (equilibrium equations are written on the undeformed geometry).
3. The support conditions do not change when loads are applied (no settlement-dependent or yielding supports).
4. Loads are applied statically and do not change the stiffness of the structure.

Examples

1. Beam deflection. A simply supported beam carries a central point load $P$ and a UDL $w$ over the full span. The mid-span deflection is the sum of the two separate deflections:

$$\delta = \frac{PL^3}{48EI} + \frac{5wL^4}{384EI}$$

2. Reactions and bending moment. For a beam with several point loads, the reaction at a support is the sum of the reactions due to each load alone. The bending moment at a section is the sum of the moments due to each load alone.

3. Truss member force. The force in a member under combined loading is the sum of the forces due to each joint load applied separately.

Uses

• Complicated loading is split into simple standard cases whose results are tabulated.
• Basis of influence lines, the force method (compatibility) and the method of consistent deformations for indeterminate structures.

Limitations

• It is not valid if the material is non-linear (stress beyond the proportional limit, cracked concrete, plastic hinges).
• It is not valid if deformations are large, so the geometry changes (cables with large sag, slender columns, $P$-$\Delta$ effects).
• It fails when the effect of one load changes the effect of another, e.g. a beam-column where axial load changes the bending stiffness: the deflection under $P_1+P_2$ is not the sum of the separate deflections.
• It cannot be used when supports change their nature under load (e.g. a gap closing or a support lifting off).
• It does not apply to quantities that are not linear in load, such as strain energy: $U \ne U_1 + U_2$, since $U$ depends on the square of the load.