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.

Bending of a T-shaped Beam, the finite element model with applied loads and restraints

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	2671	7920

Table 2. Result "Displacement OZ"

Numerical Solution Displacement OZ \|w*\|, m	Analytical Solution Displacement w, m	Error δ =100%* \|w* - w\| / \|w\|
5.7246E-004	5.6989E-004	0.45

Bending of a T-shaped Beam, result "Displacement" of finite element analysis

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.45% 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.