Elastic foundations

Grade beams, strip footings and walls on ground as line supports on springs — choosing a subgrade modulus, and where the model stops being true.

Modeling 4 min read

Elementarium models a Winkler bed as a line support: a stiffness per unit length along a whole member, or along one edge of a slab. That covers a grade beam, a strip footing and a wall bearing on ground, without meshing the soil.

The theory is in Winkler foundations; this page is about using it.

The parameter

The bedding modulus $k_{s}$ has units of force per unit volume, i.e. pressure per unit settlement:

p = k_{s} w

where $p$ is contact pressure and $w$ is settlement.

A line support takes a stiffness per unit length, $k = k_{s} b$ , where $b$ is the bearing width. A 600 mm wide strip footing on ground of $k_{s}$ = 30 000 kN/m³ is a line support of:

k = k_{s} b = 30000 \times 0.6 = 18000 kN/m per metre = 18 MN/m^{2}

Each of the six directions is set independently, and the vertical one is usually all you want.

The stiffness is lumped onto the analysis nodes along the line by tributary length, so the total is exactly $k \times$ length however finely the line is meshed.

Choosing a value

Soil	Indicative $k_{s}$ (kN/m³)
Soft clay	5 000 – 15 000
Firm to stiff clay	15 000 – 50 000
Loose sand	10 000 – 25 000
Medium dense sand	25 000 – 60 000
Dense sand and gravel	60 000 – 150 000
Weak rock	150 000+

Bracketing, because you must

The bedding modulus is the least certain number in the model and it changes the answer. The standard defence is to run upper and lower bounds (commonly half and double the nominal value) and design for the envelope:

Bound	Effect on the raft
Low $k_{s}$ (soft)	More settlement and more uniform pressure; the raft spans between columns, so its moments are larger
High $k_{s}$ (stiff)	Less settlement and smaller moments (they scale with $k_{s}^{- 1/4}$ ), but pressure concentrated under the columns

Neither bound is conservative for everything, which is why you run both.

Modeling a grade beam or strip footing

Model the footing as a member with its real section.
Add a line support on that member and set the vertical stiffness to $k_{s} b$ .
Leave the other directions free unless something really restrains them. A footing still needs restraint in plan, from a horizontal bedding stiffness or a nominal support. Check the reaction at any nominal support: if it carries real load, the model is leaning on it.
Subdivide the member. A line support acts only where there are analysis nodes, and the bending along a beam on bedding is a wave, not a parabola. See the characteristic length below.
Apply loads, including self-weight.
Solve, then look at the reactions along the line before reading any moment.

A slab bearing on a wall is the same construct on a plate edge instead of a member.

A raft, today

A bed under the face of a raft is not available yet. Until it is, the closest models are line supports along the raft's supported edges, or node springs at the mesh nodes with each spring's stiffness $k_{s}$ times its tributary area. Neither reproduces a continuous bed, so treat raft moments from them as indicative.

Where the model breaks

Springs are independent. Real soil shears, so settlement at one point influences its neighbours. Winkler ignores this, which is why a uniformly loaded raft on uniform bedding settles uniformly instead of dishing.
Edge effects. Real contact pressure concentrates at the edges of a stiff raft on clay. A Winkler model does not reproduce that.

Characteristic length

The characteristic length tells you how far a point load's effect spreads:

L_{e} = (\frac{4 E I}{k})^{1/4}

A point load on a long beam on a Winkler bed. Settlement dies away over a few characteristic lengths and crosses zero at 2.36 L_e; beyond that the beam lifts slightly, and a linear spring pulls it back down, which a real soil cannot do.

Footing length	Behaviour
Shorter than about $L_{e}$	Effectively rigid: the pressure distribution is close to linear
About $L_{e}$ to $3 L_{e}$	Intermediate, the usual case: the bed stiffness and the beam's own stiffness both matter
Longer than about $3 L_{e}$	An infinitely long beam: end conditions barely matter

The limits are the same as in Winkler foundations.

Reporting

State the bedding modulus, its source, the bearing width you multiplied it by, the bounds you ran, and that the support is tension-capable in the linear model. A reviewer's first question about any foundation analysis is which $k_{s}$ you used; the second is what happens if it is wrong.

Something wrong or unclear on this page? Tell us, and it gets corrected.