Let's suppose I have a two-sided fixed beam:
...and I want to find the equivalent spring constant... can I do the following:
$$\delta =\frac{FL^3}{192EI}$$
Then I know that force = spring_constant * displacement : $$f =k x$$ hence also: $$f_{max} = F = k x_{max} = k \delta$$
Finally: To get the equivalent spring constant, I do: $$ k =\frac{F}{\delta}= \frac{129 EI}{L^3}$$
Is that correct ?
I'm assuming your equation for deflection is correct. Notice that the force is proportional to the deflection divided by the cube of the length between the supports, which for very small deflections, is approximately the same as the length of the beam. Similarly, the distance L/2 from the end of the beam to the application of the force is approximately the same as the length of the beam from the end to the force for very small deflections. However, for large deflections, the the beam stretches so that the distance between the supports (L) and between the supports and the force (L/2) is different than the beam lengths.
Bottom line, I assume that the first equation is based on small deflections. If that is the case, the value of k would be correct as a function of the distance between the supports, but only for very small deflections.
Hope this helps.
