Elastic Foundation Beam Calculator and Equations Loading Case 2

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Elastic Foundation Beam Calculator and Equations Loading Case 2

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Beam on Elastic Foundation Calculator and Equations Loading Case 2

Beam with a Single moment loading at center

Elastic Beam with Single Moment Loading

When a beam rests on one or more elastic foundations the final beam deflection and internal shears and bending moment depends on the foundation stiffness as well as the beam stiffness. To make the equation convenient the foundation constant β is available from Figure 1 and the damping functions (γ, λ, ψ, η,) are obtained from Figure 2.

Preview: Beam on Elastic Foundation Calculator and Equations Loading Case 2

Figure. 1. Foundation Constant β

Figure 2 Damping Constants

Eq. 1
M = M_o ( λ ) / 2

Eq. 2
V = -M_o β ( γ ) / 2

Eq. 3
y = M_o β² ( ψ ) / K

Eq. 4
K = p / y

Eq. 5
β = [ K / ( 4 E I ) ] ^(1/4)

Eq. 6
γ = e^-βx ( cos βx + sin βx )

Eq. 7
ψ = e^-βx sin βx

Eq. 8
η = e^-βx ( sinβx - cosβx )

Eq. 9
λ = e^-βx cos βx

Declarations

P = Applied Load (lbs)
p = Load per unit length (lbs / in)
M = Moment (in-lbs)
V = Shear (lbs)
E = Modulus of Elasticity (psi)
I = Area moment of inertia (in⁴)
K = p / y = spring constant of the foundation
p = load per unit length lbs/in
y = deflection (in)
x = distance from applied force (in)
Note: The beam must extend a minimum distance x = 5 / β beyond the applied loads of moments
β = Foundation constant
ψ = damping constant (Figure 2)
η = damping constant (Figure 2)
γ = damping constant (Figure 2)
λ = damping constant (Figure 2)

Credits:

M. Hete’nyi, “Beams on Elastic Foundation”