Cantilever beam: cubic stiffness question

[JolileChat]

Hello.

Supose that we have a cantilever beam.
For a small force P applied at the free side of the beam, we can find an expression for the maximum deflection:

$$\delta=\frac{P l^3}{3 E I}$$

If we want to use this beam as a string, we can find its equivalent stiffnes noting that $$P=K_{eq} \delta$$, so

$$K_{eq} = \frac{3 E I}{l^3}$$

In the case of large forces (and large deflections), it is known that the equivalent stiffness will have cubic powers of the deflection. Does anyone know a good reference on how to find the expression for this new relation force versus deflection with the cubic terms?

 

[Studiot]

Do you mean the equations for the vibration of a cantilever?

$$EI{\left( {\frac{{{\partial ^3}y}}{{\partial {x^3}}}} \right)_{x = L}} = - P\left( {\frac{{{\partial ^2}y}}{{\partial {t^2}}}} \right)$$

$$EI{\left( {\frac{{{\partial ^2}y}}{{\partial {x^2}}}} \right)_{x = L}} = 0$$

 

[JolileChat]

No, it is a static case where the displacement is large enough to invalidade the linear theory, which permits us to find the traditional deflection and equivalent stiffness expressions.