Consolidation Settlement

Theory: Settlement

How the consolidation settlement of a clay is worked out from its e–log σ′ curve, why the preconsolidation stress matters, and how long the settlement takes.

Settlement and the change in void ratio

Under a wide load the soil strains only vertically, as in the oedometer. A layer of thickness H going from void ratio e₀ to e settles in proportion to the change in the volume of its voids.

s = \frac{H\, \Delta e}{1 + e_0}

H: layer thickness · Δe: fall in void ratio · e₀: initial void ratio

In a saturated clay, Δe doesn't happen at once: the water has to leave the pores. The load is first carried by the pore pressure, which dissipates with time, handing it to the soil skeleton. That's primary consolidation.

The e–log σ′ curve

In the consolidation test the void ratio falls almost linearly with the logarithm of effective stress along two branches: recompression, of slope C_s, up to the preconsolidation stress σ′_p, and the virgin line, of slope C_c, beyond it.

Normally consolidated (σ′_v0 = σ′_p): the whole load follows the virgin line.
Overconsolidated with σ′_vf ≤ σ′_p: the load stays on recompression, and the settlement is small.
Overconsolidated with σ′_vf > σ′_p: recompression up to σ′_p and the virgin line from there on.

s = \frac{H}{1 + e_0} \left[C_s \log\frac{\sigma'_p}{\sigma'_{v0}} + C_c \log\frac{\sigma'_{vf}}{\sigma'_p}\right]

σ′_v0: initial effective stress at mid-layer · σ′_vf = σ′_v0 + Δσ · each term only enters on the branch the stress path crosses

The overconsolidation ratio OCR = σ′_p/σ′_v0 tells how far the clay has been loaded in the past. Even a modest OCR cuts the settlement a lot, since C_s is usually 5 to 10 times smaller than C_c.

Sublayers

Since σ′_v0 grows with depth and the settlement depends on its logarithm, working out a whole layer at its midpoint is an approximation. Cutting the clay into sublayers and adding up their settlements converges on the exact value; for thick, shallow layers the difference can be marked.

Progress with time

By Terzaghi's theory, a layer's average degree of consolidation U depends only on the time factor T_v, which brings together the coefficient of consolidation, the time and the distance the water travels to a draining face.

T_v = \frac{c_v\, t}{H_{dr}^2}

c_v: coefficient of consolidation · H_dr: drainage path, H/2 draining at top and bottom, H through one face · s_c: final settlement

U = 1 - \sum_{m=0}^{\infty} \frac{2}{M^2}\, e^{-M^2 T_v}

Solution for a uniform initial excess pore pressure; for U < 60%, U ≈ √(4 T_v/π)

U	T_v
50%	0.197
90%	0.848
95%	1.129

Since t grows with H_dr², a layer draining through one face takes four times as long to consolidate as the same layer draining through both. Sands above and below a clay usually act as draining faces.

Limitations

The load is infinite: under footings and fills of finite width, Δσ falls with depth and calls for solutions such as Boussinesq's.
C_c, C_s, e₀, σ′_p and c_v are constant through each layer. In practice c_v varies with stress, and in the field it's often larger than in the laboratory.
The load is applied at once; building a fill up gradually delays the start of the settlement.
Secondary compression (creep), immediate settlement and large strains are left out.

References

Terzaghi, K. Erdbaumechanik auf bodenphysikalischer Grundlage. Vienna: Deuticke, 1925.
Das, B. M.; Sobhan, K. Principles of Geotechnical Engineering. 9th ed. Boston: Cengage, 2018.
Holtz, R. D.; Kovacs, W. D.; Sheahan, T. C. An Introduction to Geotechnical Engineering. 2nd ed. Upper Saddle River: Pearson, 2011.
Pinto, C. S. Curso Básico de Mecânica dos Solos. 3. ed. São Paulo: Oficina de Textos, 2006.

Enter the soil layers, the water table and the stress added by a wide fill to get the primary consolidation settlement of the clay layers — from the indices C_c and C_s and the preconsolidation stress — and how it progresses with time by Terzaghi's theory.

Final settlement, s_c: 27.2 cm
t₅₀: 0.615 years
t₉₀: 2.65 years

Ground and load

Added stress, ΔσkPaSublayers per clayWater table, z_wmγ_wkN/m³

Δσ is the fill's weight per area (γ_fill × height). With 1 sublayer, each clay is worked out at its middle, as in textbooks.

Layers

Thickness, hmγ (above WT)kN/m³γ_sat (below WT)kN/m³
Thickness, hmγ (above WT)kN/m³γ_sat (below WT)kN/m³
e₀C_cC_sσ′_pkPac_vm²/yr
Drainage
Thickness, hmγ (above WT)kN/m³γ_sat (below WT)kN/m³

One-dimensional consolidation under a load of infinite extent: the same Δσ at every depth, applied at once.
Each clay layer is cut into equal sublayers, worked out at the middle of each.
e₀ and σ′_p are constant through each layer; where σ′_v0 exceeds σ′_p, the sublayer is taken as normally consolidated.
Each clay consolidates on its own, with its own drainage.
Secondary compression and immediate settlement are left out.