Stiffness of a Cantilever Beam

Question

I was wondering that for a Cantilever beam subjected to point load at extreme end. Does the stiffness goes on changing as the deflection of free end changes?

Logically it should provide more resistance as deflection goes on increasing i.e., stiffness should increase with deflection.

But the stiffness formula of cantilever beam with point load is 3EI / (L^3)

E = youngs modulus I = second moment of area L = length/span of beam

All these terms are constant, so stiffness should be constant.

So does it actually change or remain constant? & If changes what would be the approach to calculate that changing stiffness.

Would love to have your comments on these. Thanks

Edit 22-Oct:

I have gone through the references suggested in the answer.

I was trying to find two things,

1. The deflection at each point on the length of the beam, plot it in excel and simulate for varying point loads.

2. Calculate stiffness from the maximum deflection i.e. the tip deflection.

To achieve this i tried solving equations 8, 11 & 12 of the paper "large and small deflections of a Cantilever beam". Wolfram alpha and other calculators failed so i tried solving in manually using Runge kutta fourth order. But RK method requires trial and error and it fails too because i tried reaching the solution using excel goalseek, but it did not gave any result.

Also i feel like this is not the right approach to solve it. Can you suggest me an approach to reach the solution? ****

Answer 1

Stiffness is resistance per unit deformation, which is why it has SI units of Newtons per metre. So it will be constant as long as resistance is a linear function of deformation.

For large loads/deflections the linear approximation will break down and the simple stiffness formula for a point load at the end of the beam will no longer apply. The deflection for a given load is then likely to be a much more complex function of the beam geometry and material properties.

Answer 2

As others have said, the simple beam deflection equation assumes small deflections.

You say that...

Logically it should provide more resistance as deflection goes on increasing

It's true that you would expect the load to keep increasing, but it is not always the case that the stiffness (load/deflection) will keep increasing.

A real-world example where the stiffness decreases is a metal bar where the metal is failing at the surface near the support. The stiffness decrease in this case is due to material non-linearity.

A real-world example where the stiffness increases is a fiberglass fishing rod. These rods can be so flexible that the end of the rod bends to the point of being almost parallel to the line. This effectively causes the portion of the rod undergoing bending to become shorter, thus increasing the stiffness. The stiffness increase in this case is due to geometric non-linearity.