Calculating deflections on an inclined beam

I'm applying a force to a stepped inclined beam (that's what I'm calling it; don't know what the correct technical term would be) in which it's essentially a cantilever with the first section angled at some angle $$\theta_1$$. After some distance the beam is angled, or bent, at some angle $$\theta_2$$. There is a point load, $$F$$, applied to the end of the beam. I would like to calculate the maximum deflection of the beam (I assume it's at the end), but I am not exactly sure how to approach this problem.

Of course I'd need to calculate the reactions at the support. Would I then need to split the beam at $$x_1$$, calculate deflections and then do the same at $$x_2$$? Any help will be greatly appreciated. Thanks guys

The flexibility method of virtual work is an alternate (and potentially easier) way solving such multiple-segment elasticity problems. Instead of applying cantilever deflection theory to each beam, one adds up the strain energies, performs a matrix inversion, and differentiates the result with respect to the end motion. An example for a similar geometry is here (starting with Eq. (15)).

• Thank you so much for this. It's also a very good approach
– Mark
Commented Apr 4, 2023 at 6:26

yes you are right. We find the deflection of the section from x2 to x1 by applying $$\delta_{x1-x2}=\frac {F1* (sin\theta_2(x1-x2))^3}{3EI}$$ And repeat for the 0-x1 section . assuming the F1 is applyed at X1.

$$\delta_{0-x1}=\frac {F1* (sin\theta_1(0-x1))^3}{3EI}$$ and we calculate the angle of slope at X1

$$\theta_{X1}= \frac{F1*(0-X1)(2(0-X2)-(0-X1))}{2EI}$$

So the end of cantilever beam is bent down by $$\delta{slope}=(X1-X2)*sin(\theta_{X1})$$

Then we add these three deflections to get the deflection of the entire beam.

This is a good approximation for small angles, for large angles we need to consider second order effects.

• Thank you very much for clarifying this!
– Mark
Commented Mar 28, 2023 at 6:43