Beam on Elastic Foundation

Consider a beam on elastic foundation.The length of the beam is L. The beam cross-section is a rectangle of width b and height h.

Beam on Elastic Foundation,

The beam is under a uniformly distributed load q . The force P is applied at the left end of the beam.

Sought quantity is the maximum beam deflection.
Let us use the following initial data: qL = 200 N/m, P= 1000 N, L = 5 m, b = 0.05 m, h= 0.02 m.
Material characteristics: the Young's modulus E = 2.1E+011 Pa, Poisson's ratio ν = 0.28.

Beam on Elastic Foundation, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

The analytical solution is calculated by the formula:
w0max =(2β·P - qL) / (k·b),

where J = bh3 / 12 - the moment of inertia, k is modulus of subgrade reaction (k= 3e+06 N/m3), β = ( k·b/4E·J)1/4 = 1.52136.

Thus, w0max = (2*1.52136 *1000-200)/(3*106*0.05) = 18.95147 mm.

- qL / k·b = -1.333 mm.

Before calculating determine the following input values: area of loading face A=b·L=0.25 m2; pressure on the face q=qL·L/A=4000 Pa; total stiffness of the base k1=k·A=3*106*0.25=7.5e+05 N/m; or distributed one will be: k= 3e+06 N/m3.

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	1699	3412

Table 2. Result "Displacement"

Numerical Solution Displacement w*, m	Analytical Solution Displacement w , m	Error δ =100%* \|w* - w \| / \|w\|
1.8953E-002	1.8951E-002	0.01


Numerical Solution: Bending the beam	Analytical Solution: Bending the beam

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.01% 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.