Infinite Beam on Elastic Foundation

Consider an infinite beam placed on an elastic foundation. The beam cross-section is a rectangle of width b and height h.

Beam on Elastic Foundation,

The force P is applied at the middle of the beam.

Sought quantity is the maximum beam deflection.
Let us use the following initial data: P= 500 N, b = 0.05 m, h= 0.005 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 =P/(8β3·EJ),

where J = bh3 / 12 = 5.208333 - the moment of inertia, k is modulus of subgrade reaction (k= 3e+06 N/m3), β = ( k·b/4E·J)1/4 = 4.303069.

Thus, w0max = 500/(8*4.303069^3*2.1e11*5.208333) = 7.171782 mm.

Before calculating, determine the following input values: area of loading face A=b·L=0.25 m2; distributed stiffness will be: k= 3e+06 N/m3; total stiffness of the base k1=k·A=3*106*0.25=7.5e+05 N/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
linear triangle	1255	2000

Table 2. Result "Displacement"

Numerical Solution Displacement w*, mm	Analytical Solution Displacement w , mm	Error δ =100%* \|w* - w \| / \|w\|
7.1675	7.171782	0.06

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

The numerical solution

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.5% for linear finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.