Beam 3-point bending (Presentation)

This example considers a beam in three points bending. Linear elasticity and small strain are considered.

In three-point bending, the simply supported beam is supported on two outer points, and deformed by driving the third point downwards,which causes it to bend.

Geometry

The geometry of the beam is shown in the figure below. The displacements are blocked in all the direction at one of the extremity of the beam (however, rotation is possible). The other extremity can move freely in the $x$-direction. A point force is applied at the point $x=\alpha$.

The beam is assumed to have a rectangular cross section. The length L of the beam is 100mm, $\alpha$  = 25 mm. The quantity of interest is the displacement (in the vertical direction) at the middle of the beam.

Input parameters

Four parameters are considered in the study using COSSAN-X:

• The point force (denoted P)
• The Young modulus of the material (denoted E)
• The width of (the cross section of) the beam (denoted b)
• The height of (the cross section of) the beam (denoted h)

Approximate solution

The displacement of the mid-span of the beam can be approximated as: $u_y = -11 \cdot P \cdot 100^3/(768 EI)$ where uy  denotes the vertical displacement, I is the inertia moment of the cross section given as $I = \frac{bh^{3}}{12}$

Aims

This example is prepared to analyse the displacement of the mid-span of the beam. To start with, uncertainties will be considered and the variability of the displacement of the mid-point of the beam will be evaluated. Next, a threshold value for the mid-span displacement will be defined and reliability analysis is performed.