Deflection of the Beam under the Action of Three Forces

Deflection of the beam under the action of three forces

Let us consider a beam of length L, loaded with the force P at the middle and reaction forces R1, R2 at the ends. The cross-section of the beam is a rectangle of width b and height h.

Deflection of the beam under the action of three forces

Sought quantity is the maximum deflection of the beam .
Assume: P = 1000 N, R1=500 N, R2=500 N, L = 0.5 m, b = 0.05 m, h = 0.02 m.
Characteristics of material have values: Young's modulus E = 2.1E+011 Pa, Poisson's ratio ν = 0.28.
Both ends of the beam are assumed free and subjected to the loads R1, R2 directed vertically. The force P is applied at the middle of the beam.
To solve this study in AutoFEM Analysis, it is necessary to turn on the option "Stabilize the unfixed model" with additional stiffness equal 1.
You should check this box at the Properties dialog of Static Analysis on the page "Solving".

Deflection of the beam under the action of three forces, 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 ) / ( 48 . E . J ) = 3.720E-004 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:

(beam deflection is equal to (2.2485E-004)-(-1.4819E-004)=3.7304E-004 m)

Table 1.Parameters of the finite element mesh

Finite Element Type	Number of Nodes	Number of Finite Elements
quadratic tetrahedron	912	2401

Table 2. Result "Displacement, 0Z"*

Numerical Solution Displacement 0Z*, m	Analytical Solution Displacement 0Z, m	Error δ =100%* \|0Z* - 0Z\| / \|0Z\|
3.704E-004	3.720E-004	0.28

Deflection of the beam under the action of three forces, Result "Displacement, 0Z

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.28% 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.