# Solving linear elasticity equations for a cantilever problem

Has anyone successfully used the linear elasticity equations to model a cantilever problem?

The equation I am solving is simply: $$\nabla \cdot \boldsymbol{\sigma} = 0$$

subject to boundary conditions at the fixed end, free surfaces, as well as where the load is applied. I apply the load as a shear stress/traction boundary condition on the end face.

I am having no luck getting my results to match the analytical solution. I plotted the lateral displacement along the length of the beam and the shape of my curve is approximately linear, whereas the analytical solution is a cubic. When comparing max displacement in my numerical result vs the analytical result, I am off by orders of magnitude.

I've studied my code and numerics carefully and I do not see any mistakes, but I could be wrong. Is the cantilever problem able to be modeled with the equation above?

Editing this post based on the comments: In addition to the above equation, I am also using Hooke's law and the stress strain displacement relationship:

$$\boldsymbol{\sigma} = 2\mu \boldsymbol{\epsilon} + \lambda tr(\boldsymbol{\epsilon}) I$$ $$\boldsymbol{\epsilon} =0.5[\boldsymbol{\nabla u} + \boldsymbol{(\nabla u)}^T]$$

Editing the post to include the analytical solution and dimensions. The analytical solution that I am using is (http://ruina.mae.cornell.edu/Courses/ME4735-2012/Rand4770Vibrations/BeamFormulas.pdf): $$v = \frac{Fx^2}{6EI}(3L-x)$$ Where v is the lateral displacement or the y-displacement in the link. The beam I am currently modeling has L = 10m with a square cross section of 1m x 1m. So the moment of inertia, $I=\frac{1}{12}$ for this problem. I apply a force of $10^6$N at the free end, which is equivalent to a $10^6$Pa shear stress at the free end. I prescribe this as a traction boundary condition so that the boundary condition at the free surface is: $$\sigma = \begin{bmatrix} 0 & -10^6 & 0 \\ -10^6 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

or in terms of tractions: $$T = \sigma n = \begin{bmatrix} 0 \\ -10^6 \\ 0 \end{bmatrix}$$

Attached is the result I am getting for lateral displacement for $E=2*10^{11} Pa$.

• Yes, people have used linear elasticity to model that problem successfully. But that equation alone does not represent linear elasticity. Feb 20, 2018 at 12:02
• To expand on the comment of @nicoguaro, the equation you show is actually three equations but with six unknowns-- the stress components. So, clearly, that equation by itself is not sufficient. Essentially all elasticity texts solve the cantilever beam problem with various loadings. See, for example, these notes: solidmechanics.org/Text/Chapter5_2/Chapter5_2.php#Sect5_2_4 Feb 20, 2018 at 14:45
• Sorry for the vagueness. Yes I do use the additional equations, which I have now edited into my OP. The other two equations are Hooke's law and the stress strain relationship. Are there additional equations I should be considering? Feb 20, 2018 at 18:16
• A solution found using Airy stress functions also give cubic response (see en.wikiversity.org/wiki/Introduction_to_Elasticity/…). Using Finite Elements I obtain the same results: nbviewer.jupyter.org/github/AppliedMechanics-EAFIT/SolidsPy/… Feb 20, 2018 at 19:32
• I can't think of any explanation for that displacement shape except some kind of software bug. Do you want to post some kind of image of the complete, deformed position of the beam to see if we can get some clues from that? Feb 21, 2018 at 13:26