Bending a T-shaped Beam

Let us consider a T-shaped beam.

A length of the beam is L. The beam is clamped on the left end and loaded with the force P on the right end.

Let us use the following initial data: length L of the beam is 1 m, the lengths of the sides of a cross-section b1,h1,b2,h2 are 0.01m, 0.1m, 0.006 m, 0.05 m, respectively, the magnitude of the applied force P is 100 N.
Material characteristics: E = 2.1E+011 Pa, ν = 0.28.

The finite element model with applied loads and restraints

The analytical solution is calculated with the expression:

where
- moment of inertia with respect to central axis of inertia;
, ;
, ;
z01,z02 are distance between axes Y1 and Y, Y2 and Y, respectively.

Thus, w= 5.6989E-004 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

2463

7427

Table 2. Result "Displacement"

Numerical Solution
Displacement w*, m

Analytical Solution
Displacement w, m

Error δ =100%* |w* - w| / |w|

5.7232E-004

5.6989E-004

0.59

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.59% 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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