22 May Calculate stress concentration using FEA
The Saint-Venant theory, first developed in XIX century, became possible the design of big steel structures, bridges and buildings. Thanks to this, a lot of simple analytical formula are used by engineers to calculate the link between external loads (forces and torques) and stress state. The link between loads and stress is generally given by geometrical properties (section, area moment of inertia, area, …) and material properties (Young modulus, Poisson modulus, …). In a simple case, for example, we get a simple relation between bending moment and stress in a simple beam (right image). In this formula, σ is the stress due to external moment, M is the bending moment modulus, J is the moment of inertia (in mm2) and y is the distance from the neutral axis. This formula, such as all those obtained under the Saint-Venant hypothesis, has a major defect: it is valid only when considered from a certain distance of the location where external loads are applied. Unluckily, this means that geometry irregularities, loads applications and similar critical points are not correctly predicted by simple analytical formula. In order to increase the results accuracy in these particular situations,  introduced some correction coefficient, known as stress intensification factors, very popular in mechanical design. These stress, named as Kt, vary according to the geometry and represent an intensification between the real stress in the material and the equivalent stress obtained from the Saint-Venant formulation.
Thanks to modern FEM – Finite Element Method – analysis available in CONSELF – https://conself.com/my-account/), the calculation of these coefficients is possible in real time. Furthermore, a comparison with values given by  can actually give a validation of the analysis performed. Keep in mind that CONSELF provides you a graphical interface to Code Aster, one of the greatest opensource FEM software. A simple is here used to show how FEM analysis compares against other type of formulation. The image here shows the geometry we are going to simulate: a simple plate, with two round semi-circular cavities on the two sides. Obviously, according to the radius of the circular cavities r and the high of the plate H the concentration factor varies, and the following graph shows this variation obtained from  (blue curve) and FEM analysis in CONSELF (red dots)
Thanks to this we clearly demonstrated two points:
- FEM analysis in CONSELF (so with Code Aster) is highly reliable and reproduces the analytical results present in 
- FEM analysis actually are the easiest way to analyse stresses in a general shaped geometry. Thanks to this approach any user just need to correctly define the geometry and apply the constraints and loads rather than apply a certain number of formula and graphs.
 Pilkey, W. D., & Peterson, R. E. (1997). Peterson’s stress concentration factors. New York: Wiley.
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