Bending a Cantilevered Beam under a Concentrated Load

Bending a Cantilever Beam under a Concentrated Load

Let us consider a cantilevered beam of length L, loaded with the force P at the right-hand end. The beam cross-section is a rectangle of width b and height h.

Bending of a Cantilever Beam under a Concentrated Load

Sought is the maximum beam deflection.
Assume P = 825 N, L = 0.5 m, b = 0.05 m, h = 0.02 m.
Material characteristics assume default values: the Young's modulus E = 2.1E+011 Pa, Poisson's ratio ν = 0.28.
The left-hand end of the beam is fixed, and the right-hand end is subjected to the load amount P, directed vertically downward.

Bending of a Cantilever Beam under a Concentrated Load, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

The analytical solution appears as:

w = ( P . L3 ) / ( 3 . E . J ) = 4.9107E-003 m

where P – is the force, L – the beam length, E – the material Young's modulus, J = b . h3 / 12 - the moment of inertia.

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

Table 1.Parameters of the finite element mesh

Finite Element Type	Number of Nodes	Number of Finite Elements
quadratic tetrahedron	1017	3593

Table 2. Result "Displacement, magnitude"*

Numerical Solution Displacement w*, m	Analytical Solution Displacement w, m	Error δ =100%* \|w* - w\| / \|w\|
4.8792E-003	4.9107E-003	0.64

Bending of a Cantilever Beam under a Concentrated Load, Result "Displacement, magnitude" of finite element analysis

Dependence of the relative error on the number of finite elements

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.64% 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.