# Does the Superposition principle actually make sense in reality for a rod subjected to bending and axial force togehter?

Consider a rectangular rod, for example. I fix it from one end and apply a transverse force on the other (to cause bending) and also a axial force (lets just assume its compressive). Now, according to theory and what we have learned at the University level is that I can directly use the superposition principle and calculate the maximum compressive stress in the rod, by adding the max compressive stress coming from bending to the compressive stress in the rod arising from the compressive axial force, at any cross section of the rod. Now what I am confused about is does this supperposition principle makes sense in reality, and is parallel to what we observe in reality or not?

In my understanding, if solely bending exists in the rod (with no axial force applied) then at any cross section of the rod, there won't be a net axial force, no matter what shape of the cross section is used. The part above the neutral axis will have a net force equal and opposite to the part below the neutral axis, and basically when this value is multiplied by the distance between them will give me bending moment at that cross section. But now, if I also include an axial force like compression (as mentioned in first paragraph), so still would the bending law hold or not? I mean now the net force in the part above the neutral axis is not equal to the net force in the part below the neutral axis (if we add the bending forces to the axial forces). So it doesn't actually satisfy the basic requirement for bending to occur (i.e. these upper net force and lower net force must be equal and opposite). So will bending still occur or not?

(I know I can actually observe it by experimentally using a flexible beam, but I want to learn if the theory makes sense or not).

• Theory? Why not see what Timoshenko has to say? Commented Sep 15, 2021 at 18:45

The superposition principle is valid if the assumptions it makes are valid.

A commonly used set of valid assumptions are small displacements and strains and linear elastic material behavior.

For large displacements, large strains, nonlinear elasticity, plastic deformation, etc it is usually not valid.

• I mean you are mentioning the assumptions as made by the beam theories. Should the same assumptions apply to superposition principle as well? Commented Sep 16, 2021 at 16:17

In setting small-angle deformations as a limit we set the limit for strains in all configurations to be elastic and totally reversible.

Meaning if the beam deflects under the load or twists under the torque, etc, it will recover its original shape fully after removing the load.

As long as the strains and stresses are within this envelope, you can add or subtract stresses and vice versa strains. It has a beneficial side effect that one can assume the functions, sin, and tan of an angle equal to the angle. and it has made engineering calculations much easier.