Deflection of a Beam under a Uniformly Distributed Load

Let us consider a beam under a uniformly distributed load q. A length of the beam is L. The beam cross-section is a square. The length of the side of the square is a.

Sought quantity is the maximum deflection of the beam.
Let us use the following initial data: q = 3000 Pa, L = 0.5 m, a = 0.02 m.
Material characteristics: the Young's modulus E = 2.1E+011 Pa, Poisson's ratio ν = 0.28.

The finite element model with applied loads and restraints

The analytical solution is calculated by the formula:
, .

The maximal deflection of the beam obtains at x = L / 2 :
,
where J = a4 / 12 - the moment of inertia.

Thus, | w | = 1.7439E-005 m.

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

249

596

Table 2. Result "Displacement"

Numerical Solution
Displacement | w* |, m

Analytical Solution
Displacement | w |, m

Error δ =100%* |w* - w | / | w |

1.7512E-005

1.7439E-005

0.42

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.42% 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.

 

 

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