OpenSpecEngineering

Finds how much a saturated clay layer settles as water squeezes out of it under a new load. It uses Terzaghi’s one-dimensional consolidation theory [1] with the compression indices from an oedometer test. Geotechnical engineers use it to estimate the long-term settlement of a footing or fill on clay.

Equations

The settlement depends on where the stress starts and ends relative to the preconsolidation pressure σ′_p, the highest stress the clay has carried before [2], [3]. With σ′_f = σ′₀ + Δσ:

σ′_f ≤ σ′_p:   S_c = C_r·H/(1 + e₀)·log(σ′_f/σ′₀)

σ′₀ ≥ σ′_p:   S_c = C_c·H/(1 + e₀)·log(σ′_f/σ′₀)

otherwise:   S_c = H/(1 + e₀)·[C_r·log(σ′_p/σ′₀) + C_c·log(σ′_f/σ′_p)]

H is the thickness of the clay layer and e₀ its void ratio before loading. C_c is the slope of the virgin compression line on a plot of e against log stress, and C_r the much flatter slope of reloading; C_r is often a fifth to a tenth of C_c. σ′₀ is the effective stress now at mid-depth of the layer and Δσ the stress increase there from the new load. The overconsolidation ratio OCR = σ′_p/σ′₀ tells you how much the clay can be reloaded before it starts compressing steeply. The logs are base 10.

Limits and common mistakes

• Use effective stresses: subtract the pore water pressure below the water table.
• Stresses are taken at the middle of the layer. For a layer thicker than a few meters, or a load that spreads with depth, split it into sublayers and add their settlements.
• σ′_p must not be less than σ′₀; a clay can’t be under more stress now than it has ever carried.
• This is primary consolidation only. Immediate settlement and secondary compression (creep) come on top, and the result says nothing about how long settlement takes.

Worked example

A footing adds 40 kPa at the middle of a 4 m clay layer. An oedometer test on that clay gave e₀ = 0.95, C_c = 0.32, C_r = 0.06 and σ′_p = 110 kPa. The effective stress there now is 85 kPa. How far will the footing settle?

1. Enter Clay Layer Thickness 4 m, Initial Void Ratio 0.95, Compression Index 0.32 and Recompression Index 0.06.
2. Enter Initial Effective Stress 85, Stress Increase 40 and Preconsolidation Pressure 110, and press Solve. The Solution shows: OCR = σ'p / σ'0: 1.29412
   Final effective stress σ'f: 125 kPa
   Case: Crossing σ'p (recompression + virgin compression)
   Settlement Sc: 50.2235 mm

Check by hand. H/(1 + e₀) = 4 / 1.95 = 2.051 m. Reloading from 85 to 110 kPa gives 2.051 × 0.06 × log(1.294) = 0.0138 m. Virgin compression from 110 to 125 kPa gives 2.051 × 0.32 × log(1.136) = 0.0364 m. Together they make 50.2 mm. Most of it comes from the last 15 kPa, above σ′_p.