# How to calculate the force of a cantilever beam at a given point of deflection?

I have put together a simple cantilever beam using finite difference modeling according to the following standard wave equation

$$ρAy_{tt} = -EIy_{xxxx}$$

where $$ρ$$ is density, $$A$$ is cross sectional area, $$E$$ is Young's modulus, and $$I$$ is the moment of inertia.

I have run this simulating a simple deflection where this beam (clamped to horizontal at points 1 & 2) is moving upwards in the positive y-axis until it strikes an immovable object at point 12.

The deflection looks reasonable.

I am now wondering how I would calculate the force exerted by the beam at point 12 based on my finite difference model.

In the finite difference wave equation, local force at any given node would be given by:

$$(ΔxρA)y_{tt} = -ΔxEIy_{xxxx}$$

$$F = -ΔxEIy_{xxxx}$$

Because $$F = ma$$ and $$m = ΔxρA$$ and $$a = y_{tt}$$.

The problem is I think this is just the local force for that node relative to the neighbors. It's not giving a reasonable/useful total force output just by calculating the force at that one point.

This makes sense as the force it calculates is based on the fourth spatial derivative, and at the collision point it's almost perfectly straight so this will be near zero. Most of the restoring beam force is coming from the strong bend closer to nodes 1 and 2.

If I want the total force exerted on the "immovable object" striking the beam at point 12, how would I get that?

I know that in a general approximation, this is given by:

$$F = \frac{6EI(y)}{x^2(3x-a)}$$

Where $$y$$ is y deflection of a given point, $$a$$ is the distance to point of collision contact, and $$x$$ is the position of the point you're assessing. But I actually want to figure this out from my finite difference model so it can account for funnier shapes/motions.

I wonder if this is given by bending moment and stress force:

$$M = -EIy_{xx}$$

$$F = \frac{M (y)}{IA}$$

Where $$y$$ is the deflection from neutral and $$A$$ is cross sectional area.

However, I'm getting nothing useful from that. For a simple small plastic lever which should only generate 12 N or so of force from the general estimate equation I'm getting 10s of thousands of newtons using this approach.

Any help? Thanks.