Watch deflection, bending moment, and shear decay along the beam from the load, and see how a stiffer subgrade localizes the response.
Parameters
Load P / q
kN
Point load in kN, or uniform load treated as kN/m.
Span L
m
Beam length being evaluated.
Flexural rigidity EI
MN·m²
Beam flexural rigidity. Larger values spread the response.
Subgrade modulus k
kN/m²
Foundation reaction coefficient per area. Larger values localize the response.
Load type
Switch point / uniform / moving to compare response shapes.
Results
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Characteristic beta (1/m)
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Max deflection (mm)
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Max moment (kN·m)
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Max shear (kN)
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Characteristic length 1/beta (m)
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Max contact pressure (kPa)
Beam response animation
Deflection y(x)Bending moment M(x)Shear force V(x)Foundation reaction p=k·y (springs)
Subgrade stiffness and response localization
Deflection curves for the same load and rigidity with only the subgrade modulus k changed. A larger k localizes the response near the load and shortens the characteristic length 1/beta.
$\beta$ is the characteristic factor that sets the decay rate, and $1/\beta$ is the response length scale. Because this is a linear Winkler model, boundary conditions, soil coupling, cracking, and yielding need separate checks.
What is a beam on an elastic foundation (Winkler model)?
A beam on an elastic foundation models members that rest on a continuous support — rails, foundation beams, pipelines, slabs. The soil is treated as a row of countless independent springs, so each point pushes back with a reaction $p(x)=k\,y(x)$ proportional to the local deflection $y(x)$. This is the Winkler foundation model, and the governing equation is the fourth-order ODE $EI\,y'''' + k\,y = q(x)$.
Unlike its sibling tool (choosing the subgrade modulus), this tool focuses on the beam response itself — drawing the deflection curve, bending-moment diagram, and shear diagram together along the beam. You can watch the peak rise under the load and decay in a damped oscillation that changes sign on either side, alongside the spring compression (contact pressure).
Characteristic length beta sets how far the response reaches
The key quantity is the characteristic factor $\beta=(k/4EI)^{1/4}$. Its reciprocal $1/\beta$ is the response length scale: roughly $1/\beta$ away from the load, the deflection has decayed strongly. A stiffer subgrade $k$ or a lower flexural rigidity $EI$ makes $\beta$ larger, so the response localizes near the load. Soft soils or rigid beams spread the response over a wider zone.
Beyond about $\pi/\beta$ from a point load, the deflection reverses and the beam lifts off the foundation. You can spot this where the animated deflection curve crosses the neutral (ground) line.
Learn the beam on an elastic foundation by dialogue
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Where should I look first in this simulator? Deflection, moment, and shear all move at once, so it is a little confusing.
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Roughly speaking, look right under the load first. That is where the blue deflection curve dips the most and the orange bending moment peaks. Moving out to either side, you see the wave decay while changing sign. That is the signature of a beam on an elastic foundation — a completely different response from an ordinary simply supported beam.
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When I raise the subgrade k, the response packs tightly near the load. Why is that?
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It is the characteristic factor beta=(k/4EI)^(1/4) at work. Raising k makes beta larger, so its reciprocal 1/beta (the response length) gets shorter. A stiffer soil settles only just under the load and barely moves a short distance away. A railway rail, for example, sits on sleepers and ballast with high k, so only right under the wheel settles locally.
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What do the green springs along the bottom represent?
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Those are the Winkler foundation reactions. The reaction at each point is p(x)=k*y(x), proportional to the deflection. So the spring compression is literally the contact-pressure distribution — most compressed under the load means the contact pressure is highest there. Where the deflection lifts up, the spring stretches, and you need to watch whether contact is being lost.
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How does the shape differ between a point load and a uniform load?
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A point load gives a sharp peak under the load, with deflection y0=P*beta/2k and maximum moment M0=P/4*beta. Switch to a uniform load and the response averages out: the mid-span deflection approaches roughly q/k, and only the ends vary on the characteristic-length scale. Toggle the presets and the difference in peak sharpness and decay range is obvious.
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So if the characteristic length and contact pressure are within limits, can I just accept this condition?
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It helps a lot for early study, but it is too soon for a final call. The Winkler model treats the soil as independent springs, so shear coupling between springs, nonlinearity, and adhesion during uplift are not captured. Check boundary conditions, soil coupling, cracking, and yielding separately, and reconcile them with standards, measurements, and detailed analysis — that is the practical workflow.
Real-world applications
Used for wheel-load response and characteristic length of railway rails (continuously supported beams), contact-pressure and bending-moment estimates for foundation beams and strip/raft footings, local settlement response of buried pipelines, and deflection/reaction checks of pavement slabs under wheel loads.
In particular it gives a quick estimate of "how far the load reaches" through the characteristic length 1/beta, useful for a first cut at the effective support length and reinforcement zone.
Common misconceptions and caveats
The subgrade modulus k is not a material constant; it depends on the loaded width and the soil layering, so it is a "system coefficient." Using a plate-load-test value directly on a beam tends to overestimate stiffness and needs a size correction.
When the characteristic length is shorter than one third of the beam length, the infinite-beam approximation errors grow. For short beams or loads near the ends, the boundary conditions (free / fixed) cannot be ignored, and a finite-beam solution or FEM comparison is recommended.
In uplift regions (y<0) the soil carries no tension, so it is really a nonlinear contact problem. Treat the linear Winkler solution as an approximation valid only while uplift is small.
FAQ
Start with the characteristic factor beta and the maximum deflection. The animation draws the deflection curve, bending-moment diagram, and shear diagram together along the beam, so you can watch the maximum deflection and maximum moment rise under the load and decay in a damped oscillation toward the ends. The stiffer the subgrade, the more the response localizes near the load and the shorter the characteristic length 1/beta.
beta=(k/4EI)^(1/4) sets how quickly the beam response decays in space. Its reciprocal 1/beta is the response length scale, and beyond about pi/beta from a point load a reversed (uplift) deflection appears. A larger subgrade modulus k or a smaller flexural rigidity EI makes beta larger, so the response localizes near the load.
A point load creates a sharp peak under the load, with deflection y0=P*beta/2k and maximum moment M0=P/4*beta. A uniform load averages the response: the mid-span deflection approaches q/k, and only the regions near the ends vary on the characteristic-length scale. Switching the load type in the animation makes the difference in peak sharpness and decay range obvious.
In the Winkler model the foundation reaction at each point is p(x)=k*y(x), so the deflection curve is also the shape of the contact-pressure distribution. The animation shows the reaction as the compression of the spring row beneath the beam, with the maximum contact pressure under the load. Check that it stays below the allowable bearing pressure and that contact is not lost in uplift regions where y<0.
This is a linear Winkler-foundation model. Because the soil is treated as a row of independent springs, shear coupling between springs, soil nonlinearity, and tension (adhesion during uplift) are not represented. Boundary conditions, soil continuity, cracking, and yielding need separate checks, and final decisions still require standards, measured data, detailed analysis, and vendor limits.
How to Use
Enter the load P/q (kN for a point load, kN/m for a uniform load), flexural rigidity EI (MN·m²), and subgrade modulus k (kN/m²)
Set the beam length L (m) and the load type (point / uniform / moving); the characteristic factor β=⁴√(k/4EI) is computed automatically
Watch the deflection curve, bending-moment diagram, shear diagram, and foundation reaction (springs) along the beam, and read the max deflection, max moment, max contact pressure, and characteristic length
Worked Example
Consider an RC continuous beam (EI=900 MN·m²) on sandy soil (k=22000 kN/m²) with a central point load P=100 kN. The characteristic factor is β=⁴√(22000/(4×900×1000))≈0.280 m⁻¹, giving characteristic length 1/β≈3.58 m, maximum deflection y0=Pβ/2k≈0.64 mm, and peak moment under the load M0=P/4β≈89.4 kN·m. The maximum contact pressure is p0=k·y0≈14.0 kPa. About π/β≈11.2 m from the load, the deflection reverses into an uplift region.
Practical Notes
For sandy soils k = 30,000–80,000 kN/m³ and for clays k = 10,000–30,000 kN/m³ are typical measured ranges; reflect site investigation results at the design stage
When the characteristic length 1/β is less than one third of the beam length, infinite-beam approximation errors grow; a finite-beam solution or FEM comparison is recommended
If settlement exceeds the serviceability limit (typically 15–25 mm), consider ground improvement or a larger EI (deeper section)
In uplift regions the soil carries no tension, so the linear Winkler solution is approximate; large uplift requires a contact-nonlinear detailed analysis