Deflection of a Cantilevered Beam under a Weight at its End

Deflection of a Cantilever Beam under a Weight at its End

Let us consider the cantilever beam, the right end of which is under the weight.

Deflection of a Beam with a Weight

The length of the beam is L. The beam cross-section is a rectangle of width b and height h. The mass of the weight is M. The specific mass of the beam is m.
m = ρ F,
where F = b h, ρ is the density of the material of the beam.

Deflection of a Beam with a Weight, the finite element model with load and restraint

The finite element model with restraints

Let L equal 0.5 m, b equal 0.02 m, h equal 0.05 m.
The material properties are: the Young's modulus E = 2.1E+011 Pа, Poisson's ratio ν=0.28, the density ρ = 7800 kg / m3.
The mass of the weight M is equal to 20.m.L kg (i.e. 78 kg).
Analytical solution of this problem is given by the following formula:
,
.
Thus, |z|max= 4.6067E-003 m.

After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1. Parameters of finite element mesh

Finite Element Type	Number of nodes	Number of finite elements
quadratic tetrahedron	905	2377

Table 2. Result "Displacement"*

Numerical solution Displacement \|z\|*max, m	Analytical solution Displacement \|z\|max, m	Error δ = 100%\| \|z\|max-\|z\|max\| / \|z\|max
4.6065E-003	4.6067E-003	0.004

Deflection of a Beam with a Weight, Result "Displacement"

Conclusions:

The relative error of the numerical solution compared to the analytical solution for displacements is equal to 0.004% for quadratic finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.