Principle of superposition, example and limitations

Principle of superposition: the total effect (reaction, force, moment, stress, strain, deflection) of a system of loads on a structure is the algebraic sum of the effects of the loads applied one at a time. It is valid only when the response is linear.

Example

A cantilever of length $L$ carries a point load $P$ at the free end and a UDL $w$ over the whole length. Free-end deflection:

$$\delta = \delta_P + \delta_w = \frac{PL^3}{3EI} + \frac{wL^4}{8EI}$$

and the fixed-end moment is $M = PL + \dfrac{wL^2}{2}$, the sum of the two separate cases.

Limitations

1. The material must obey Hooke's law (stress proportional to strain); it fails once the material yields or for non-linear materials.
2. Deformations must be small, so the geometry (lever arms of the loads) is the same before and after loading. If deflection changes the geometry appreciably (large deflection), the effects are no longer additive.
3. Deflection must not change the action of the loads: for a beam-column (axial load together with lateral load) the axial load produces extra bending $P\delta$, so the effects of axial and lateral loads cannot simply be added.
4. The supports and the structure must not change during loading (no slip, no support yielding, no contact that starts or stops as load increases).
5. It applies to quantities that are linear in load, e.g. deflection and bending moment of a linear structure, but not to strain energy, which depends on the square of the load.