Beam on Elastic Foundation Alpha · 2026-09-02

Winkler foundation — shear, moment, slope and deflection for a uniform beam resting on a continuously reacting elastic medium; finite-length and semi-infinite formulas selected automatically

What this calculator does

A beam resting on a continuous elastic medium is not carried at a few points but reacted along its whole length. The foundation pushes back in proportion to the local deflection (q = b_o k_o y, the Winkler assumption), which adds one term to the governing equation:

EI\,\dfrac{d^4y}{dx^4} + b_o k_o\,y = w(x)

That single extra term changes the character of the solution completely: the load's influence becomes local, decaying as e^{-\beta x}, and the deflected shape crosses zero — there are upward-curving regions with reversed bending moment on either side of the load.

The characteristic length

\beta = \left(\dfrac{b_o k_o}{4EI}\right)^{1/4} has units of 1/length and is the reciprocal of the decay length. Influence falls to about 37% at 1/\beta, 5% at 3/\beta and 0.25% at 6/\beta. The half wavelength l_1 = \pi/\beta is the yardstick for whether discrete supports (sleepers, piers) may be treated as continuous: at least three reaction points are needed within one half wavelength.

Why two sets of formulas

Once \beta l exceeds 6, the finite-length expressions subtract two nearly equal large numbers (for example \sinh^2\beta l - \sin^2\beta l, which at \beta l = 10 pits 1.2\times10^8 against a quantity below 1) and lose significant digits. The semi-infinite formulas exist for exactly this reason. The split is a numerical-stability criterion, not a change of physical model — at \beta l = 6 the two agree to about 1 part in 10,000, so switching costs no accuracy. The calculator reports which set it used.

Scope of this version

• Semi-infinite path (\beta l > 6): six load types — concentrated load, distributed load, concentrated moment, imposed angular displacement, imposed lateral displacement, top-to-bottom temperature differential — with four left-end restraints.
• Finite-length path (\beta l \le 6): concentrated and distributed load, with all sixteen combinations of end restraints.
• Note the distributed load is defined differently in the two: the finite-length formulation runs it from a to the right-hand end, the semi-infinite one from a to b.

Model boundaries

• Winkler foundation: the reaction depends only on the local deflection and carries no shear.
• Linear elastic material.
• Small deflection.
• Uniform section along the beam.
• The foundation carries compression only — if the beam lifts off, that is a contact non-linearity outside this solution, and the calculator says so.

Engineering data

• Material Query

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