Winkler foundations

The Winkler model represents soil as a bed of independent linear springs. It is the oldest soil–structure idealisation still in general use, it is what "elastic foundation" means in almost every structural package, and its limitations are as well known as its convenience.

For the practical side (setting up a raft, choosing values, bracketing) see Elastic foundations.

The model

Contact pressure at a point is proportional to settlement at that point, and at that point only:

$k_s$ is the modulus of subgrade reaction, in force per unit volume (kN/m³). For a beam of bearing width $b$, the line stiffness is $k=k_{s}b$, in force per unit length per unit deflection. That is exactly what a line support takes, and it is the form available today.

Combined with plate bending, the same idea gives the governing equation for a plate on an elastic foundation. This is the area case, which is not switched on yet:

where $D$ is the flexural rigidity of the plate:

The characteristic length

The single most useful quantity in the model. For a beam:

It sets the distance over which a local load spreads:

The same idea applies to plates with an equivalent radius of relative stiffness. It tells you how far apart columns must be before a raft stops behaving as one unit, and how much of a strip footing a single column load actually mobilises.

The central assumption, and why it is wrong

The springs are independent. Settlement at one point produces no settlement at its neighbour.

Real soil is a continuum. It shears, so load at one point settles the ground around it. Three consequences follow, all visible in real structures:

1. A uniformly loaded flexible raft on uniform Winkler bedding settles uniformly. Real soil dishes, with more settlement in the middle than at the edges, because the middle receives overlapping influence from the whole loaded area.
2. Winkler produces no settlement outside the loaded area. Real ground settles beyond the footprint, which is how neighbouring structures get affected.
3. Edge pressure concentrations under a stiff raft on clay are not reproduced. The real contact pressure rises sharply at the edge; Winkler gives a smooth distribution.

Why it is still used

It is cheap, stable and, for the purpose it is used for, usually adequate. The structural engineer's question is generally "what are the bending moments in my raft", not "what is the settlement profile". Moments are much less sensitive to the soil model than settlements are, and bracketing $k_s$ covers the residual uncertainty.

k_s is not a soil property

The modulus of subgrade reaction depends on the size and shape of the loaded area as well as on the soil. The same ground gives a much higher $k_s$ under a 1 m pad than under a 20 m raft, because the larger foundation mobilises deeper soil.

Consequences:

• A value from a plate loading test cannot be used directly for a raft without correction for size.
• A value quoted without the foundation dimensions it applies to is close to meaningless.
• The geotechnical engineer should supply $k_s$ for the foundation you are designing, ideally as a range.

Where linearity fails

Winkler springs in a linear analysis resist tension. Where a combination lifts part of the foundation, the model holds it down, overestimating restraint and misdistributing moments.

Compression-only support requires contact nonlinearity, which is on the roadmap and not in this release. Today the options are to interpret the linear result with judgement, check the affected region by hand, or re-run with the uplifting region unsupported and compare.

When to use something better

Reporting

State $k_s$, its source, the foundation dimensions it applies to, the bounds you ran, and the fact that the support is tension-capable in the linear model.