# Should the stiffness of this cantilever beam be different for these two conditions?

So, I was just doing some Finite Element Analysis (FEA) to figure out what would be the effect on the stiffness of a cantilever beam if another beam is attached to it on its free end.

Below shows the two cases I am using as examples.

I am only conducting geometric linear analysis i.e. change of stiffness of the green beam is not tracked by the solver during loading. Here are the deformation results for this beam.

So apparently, the deformation along Z axis only changes by a magnitude of 0.018 mm of the green cantilever beam. But my question is that is this change in deformation because of the addition of the vertical gray beam on the free end of it, or its just a result of the numerical errors coming from the FEA solver's nature?

I mean if the deformation is changing for the green body, then it means that its stiffness has changed since the solver is linear and same load is applied. So attaching another body, like the gray body in this example, somewhere along the length of the green body should change its stiffness (and therefore its resulting deformation) or not? And how would you anticipate the affect of increasing the height or area of connecting face of the gray body to green body would affect the green body's stiffness and its deformations?

If I talk about purely Statics, then the bodies are assumed to be Rigid. If I talk about Mechanics of Materials, then bodies are assumed to have very small deflections and rotations. Plus, the beam thoeries like Bernoulli and Timoshenko assume that the cross section is constant throughout the length of the beam. But what is the reality for this example I have mentioned here in my post?

The integration equation for the deflection of a cantilever beam provides the answer for the small difference between the two models.

$$\delta_y = \iint \frac{M}{EI}dx + Ax +B$$

Note, For beams with varying cross-sections, you need to break up the integration accordingly. And, the result will change if you add member self-weight in the analysis.

Comment:

The strain is greatest at the fixed end is an unchangeable fact. We know well that for a cantilever beam, the maximum moment and normal stresses occur at the support, thus the displacements (lengthen/shorten) of the extreme fibers. The moment and stresses then decrease from the maximum at the support to zero at the free end, as well as the strain.

• Can you just elaborate how exactly is the integration equation an answer for the small difference between the models? Feb 24 at 18:57
• Pay attention to I, which is not constant over the beam span of case 2.
– r13
Feb 24 at 19:40

Most of the strain occurs at the fixed end of a cantilever. Stiffening the free end has very little affect on the overall displacement. The FEA results seem reasonable.

This is why a tapered cantilever beam is thick at the fixed end and tapers to a smaller thickness at the loaded end.

• I don't think most strain occurs at fixed end since strain's basic definition is displacement over length. When the disp. are the lowest near the fixed end, so there is no reason to claim that strains should be largest there. Stresses, yes they should be since moments are highest, but I cannot say anything about strains (although stresses are supposed to be directly propotional to strains through Hooke's law). The purpose of this question was to understand if attaching any vertical beam (like gray) to the green body at any location should affect its stiffness and hence displacement or not. Feb 24 at 11:45
• @RameezUlHaq the moment is highest at the fixed end - it's a lever. Feb 24 at 14:43
• @TigerGuy, didn't I mention that already in my comment? lol. High moments at the fixed end are gonna generate the high stresses, but I cannot say anything about strains there. Feb 24 at 14:53
• @RameezUlHaq, stress & strain are related things Feb 24 at 15:15
• The hands of an analog clock have large displacement but the strain is zero. Don't confuse a total displacement, incremental displacement, and strain. Feb 25 at 3:27