Beam with Distributed Loading on Elastic Foundation Calculator and Equations

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Beam with Distributed Loading on Elastic Foundation Calculator and Equations Loading Case 4

Elastic Beam Distributed Even 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 4

Figure. 1. Foundation Constant β

Figure 2 Damping Constants

Eq. 1
M = w [ ψ ( x ) + ψ ( d - x ) ] / ( 4 β² )

Eq. 2
V = w [ - η (x) + η ( d - x ) ] / ( 4 β )

Eq. 3
y = w [ 2 - λ ( x ) - λ ( d - x ) ] / ( 2 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)
w = distributed load ( lbs / in )
M_o = 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