Bar Loaded with Sinusoidal Force on the Free End

Let us consider longitudinal vibrations of a steel beam. Its left end is fixed. Longitudinal force is applied to the right end starting from time t = 0. The force changes under the law P=P0∙sin(ωt).

Bending of a Cantilever Beam under a Concentrated Load

It is necessary to define the maximum displacement of the beam end.

Beam length l=0.5 m. Cross-section is rectangular with height h=50 mm and width b= 20 mm. Modulus of elasticity E=200 GPa, Poisson ratio ν=0.29, density ρ=7900 kg/m3. Amplitude and frequency of external force are correspondingly equal P0=10 kN and ν0=ω/2π=2 kHz.

Displacements of the beam are determined by the formula:

where

Limited by six terms of series (n=0…5) we create a graph of sections displacements from time t on ten intervals of external force. The sections have the following coordinates x=l (right beam end), x=0.75l, x=0.5l and x=0.25l .

Displacements of beam sections

Maximum displacements appear on the right end of the beam and are under tension 1.02E-004 m, under compression -1.031E-004 m.

Let us calculate AutoFEM Analysis study: We create two studies, one Transitional processes, another Mode superposition with the same loads and restraints.

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

We create a full restraint on the left butt end of the beam. The right end will stay free. We apply distributed force to the free end and specify its value using graph:

For both of the studies, we set the finite modeling time 0.005 s, the time step of integration 5∙10-6 s. The method of time integration: Newmark. We set a number of the lower natural frequencies in the Mode superposition study: 5.

Let us find the step on which the maximum deviation by graph appears after calculation using AutoFEM Analysis:

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

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	809	2645

Table 2.Parameters of temporal discretization

Total calculation time (s)	Time step (s)	Number of time layers
5E-003	5E-006	1001

Table 3. Transitional processes, Result "Displacement, OX"*

Numerical Solution Displacement OZ w*, mm	Analytical Solution Displacement OZ w, mm	Error δ =100%\|w* - w\| / \|w\|
1.0180E-04	1.02E-004	0.19

Table 4. Mode superposition, Result "Displacement, OX"*

Numerical Solution Displacement OZ w*, mm	Analytical Solution Displacement OZ w, mm	Error δ =100%\|w* - w\| / \|w\|
1.0123E-04	1.02E-004	0.78

Conclusions:

The maximum displacement of the free end of the beam, found using AutoFEM Analysis is: for the Transitional process 1.0180E-04 (relative error 0.19%), for the Mode superposition 1.0120E-04 (relative error 0.78%).

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