# What is the optimal shape of a beam subject to three point bending?

Assume we have a beam with density $$\rho$$ that is subject to three-point bending according to the image above. The beam cross section is rectangular with the variable height $$H(x)$$ and constant width $$B$$.

What function describing the non-constant height, $$H(x)$$ will yield a minimal total mass of the beam assuming a maximum allowed deflection of $$\delta$$?

The mass of the beam can easily be described as:

$$m(H) = \int_0^L \rho BH(x)\text{d}x$$

I started off trying to solve this using Euler-Bernoulli beam theory but later realized that this formula is only valid for constant second moments of area. Any ideas how to solve this problem?

• The problem is that if you do not include other constraints then your solution will include such things as infinitely thinn slices. Anyway i would test a catenary. – joojaa Feb 4 '19 at 7:09
• For the record, Euler-Bernoulli works just fine with variable cross-sections. You just need to use the general form $$q(x) = \dfrac{\partial^2}{\partial x^2}\left(E(x)I(x)\dfrac{\partial^2 w(x)}{\partial x^2}\right)$$ So, the shear force is the integral of the load and the bending moment is the integral of the shear force, exactly the same as you always do it. However, the tangent angle now needs to include $EI$ in the integral: $$\theta(x) = \int\dfrac{M(x)}{EI(x)}\text{d}x$$ The deflection is just $w = \int \theta \text{d}x$ (but $\theta(x)$ will be really ugly). – Wasabi Feb 4 '19 at 21:10
• Shape=cross section of beam? – Rhodie Aug 19 '19 at 19:58
• @Rhodie In this case Shape = profile depth as defined by H – RasmusN Aug 21 '19 at 9:59
• A T beam performs differently to say an I beam – Rhodie Aug 22 '19 at 14:33