Apply a load and watch the beam settle into a spring bed; the deflection wave decays over the characteristic length 1/β. Deflection y(x), bending moment, and soil pressure p=k·y are visualized live.
Parameters
Point load P
kN
Representative point load on the foundation beam.
Flexural rigidity EI
MN m2
Beam flexural rigidity.
Subgrade modulus k
kN/m2
Winkler foundation reaction coefficient.
Allowable reaction
kPa
Allowable foundation reaction for screening.
Load type
Choose the type of load applied to the beam.
Results (live)
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Characteristic β (1/m)
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Characteristic length 1/β (m)
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Max deflection (mm)
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Max moment M (kN·m)
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Max soil pressure (kN/m)
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Reaction utilization (%)
Beam settling into the spring bed (real time)
The blue beam settles under load and the soil springs compress. After loading, deflection decays over the characteristic length 1/β, and a slight uplift (negative reaction) appears far away.
Classical Hetényi solution for an infinite Winkler beam under a central point load P. β is the characteristic coefficient and 1/β is the characteristic length, the decay scale of the deflection wave. Max deflection y₀=Pβ/2k, max moment M₀=P/4β, soil pressure p=k·y. The Winkler model treats the soil as independent springs; verify continuous soil, piles and nonlinear bearing with detailed analysis.
What is a beam on a Winkler foundation
The Winkler model is the most basic elastic-foundation model: it treats the soil as countless independent springs. When the beam settles a distance y, the soil at that point pushes back with a reaction p = k·y. Here k is the subgrade modulus, larger for stiffer soils. It is widely used for first-pass evaluation of structures in continuous contact with the ground, such as strip footings, mat foundations, rails, and pavement slabs.
The governing equation is the fourth-order ODE EI·y'''' + k·y = q(x): deflection and soil reaction are coupled, which sets it apart from a simply supported beam. For an infinite beam under a central point load P the classical Hetényi solution is y(x) = (Pβ/2k)·e^(−βx)(cosβx + sinβx) and M(x) = (P/4β)·e^(−βx)(cosβx − sinβx).
Physical model and key equations
The key quantities are the characteristic coefficient β = (k/4EI)^(1/4) and its inverse, the characteristic length 1/β. Because deflection decays as e^(−βx) away from the load, 1/β represents the distance over which the load has appreciable influence. A stiffer beam (large EI) gives a smaller β, so the influence spreads out and deflection is shallow. A stiffer soil (large k) gives a larger β, so deflection concentrates near the load.
Max deflection y₀ = Pβ/2k, max moment M₀ = P/4β. In the animation the beam sinks while the deflection wave decays in the cosβx + sinβx form; beyond about one characteristic length the sign flips and a small uplift (negative reaction) appears. Since real soil cannot carry tension, this marks where contact would lift off.
How to read this simulator
First check the live values: characteristic β, characteristic length 1/β, max deflection, max moment, and max soil pressure. Then use the "beam settling into the spring bed" animation to feel how far the load reaches (1/β), and read the distribution chart to relate deflection, moment, and reaction.
Raising the subgrade modulus k increases β, concentrating the deflection wave at the load and sharpening the reaction peak. Raising EI decreases β, spreading the influence and softening the peak. Sweep the sliders back and forth to see which one dominates.
For early design, focus less on absolute values and more on how much margin the peak reaction keeps against the allowable, and how that margin moves with scatter in k and EI.
Real-world applications
First-pass contact pressure and bending of continuous footings and mat foundations, sleeper reactions for ballasted railway track, stresses in rigid road and airport pavement slabs, and local settlement of tank ring foundations. Broadly, any structure that spreads load through continuous contact with the ground.
Common misconceptions and cautions
Because the springs are assumed independent, the model cannot reproduce the bowl-shaped settlement of real soil with shear continuity (continuum solutions like Boussinesq or the Pasternak model are needed). Also, k is not a soil constant but an apparent quantity that depends on loaded width, so plate-load-test values used directly on a large foundation overestimate stiffness. Regions of negative reaction (uplift) correspond to lift-off in real soil and should be confirmed with a no-tension nonlinear analysis.
Learn Beam Foundation Winkler by dialogue
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When reading Beam Foundation Winkler, where should I look first? Moving Point load P changes both the plots and the result cards.
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Start with the characteristic length 1/β, but do not treat the number as the whole answer. Watch the spring-bed animation to see how far the load reaches, then read the distribution chart for deflection, moment, and soil pressure together. The main view shows the controlling trend, including the uplift far from the load that a single result card can hide.
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I can see why Point load P changes the deflection. How should I judge the influence of Flexural rigidity EI?
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Raise EI in small steps and watch how 1/β grows and the deflection wave flattens out. A stiffer beam spreads the load over a longer span, so the reaction peak drops. The Winkler model treats the soil as independent springs; continuous soil, piles, and nonlinear bearing need detailed modeling. Sweep the realistic scatter range rather than trusting a single point.
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Why does a little uplift appear far from the load? Is that physically real?
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It comes from the cosβx − sinβx form of the solution: beyond about one characteristic length the wave swings negative. In the math the spring is pulled up, but real soil cannot take tension, so it just lifts off. For example under a stiff mat the corners can rise slightly off soft ground. That is exactly where a no-tension check matters.
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So if the peak reaction is within the allowable, can I accept the condition?
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Treat this as a first-pass review. It helps narrow controlling factors and worst-side conditions before detailed analysis, and to teach the equation, numbers, and visualization under the same inputs, but final decisions still need standards, measured data, detailed analysis, and vendor limits. For early design, focus on which input controls the margin before trusting the absolute value.
Practical use
First-pass comparison of design options before review.
Narrowing controlling factors and worst-side conditions before detailed analysis.
Teaching or explaining the equation, numbers, and visualization under the same inputs.
FAQ
Start with Characteristic beta and Center deflection. Then use Deflection and reaction profile to confirm the assumed state and Load and foundation reaction to read distribution or bias. Use the main plot to read the controlling trend, including break points that a single result card can hide
Move Point load P alone, then move Flexural rigidity EI by a comparable amount and compare the change in Characteristic beta. Load-foundation margin map shows combinations where margin or performance changes quickly.
Use it for First-pass comparison of design options before review. Instead of trusting a single point, widen the input range and check whether Characteristic beta keeps enough margin before moving to detailed analysis.
The Winkler model treats the soil as independent springs. Continuous soil behavior, piles, and nonlinear bearing require detailed modeling. Final decisions still require standards, measured data, detailed analysis, and vendor limits.