Consolidation and stress

This chapter covers the following:

• Consolidation
• Stress in soil mass

Consolidation of clay soils

Definitions

Consolidation is the time-dependent reduction in volume of saturated fine-grained soils caused by the dissipation of excess pore water pressure.

Settlement of fine-grained soils occurs in three stages:

• Immediate settlement occurs rapidly and is based on elasticity theory.
• Primary consolidation occurs due to the expulsion of water under pressure.
• Secondary compression (also known as “creep”) occurs as soil particles readjust and compress over time.

Consolidation refers to the primary consolidation phase in fine-grained soils (silts and clays). Because these soils have low permeability, excess pore water pressure ($\Delta u$) dissipates slowly, so settlement takes time.

Coarse-grained soils (sands and gravels) also settle, but they do so much faster because their high permeability allows pore water pressure to dissipate quickly.

Secondary compression begins after all excess pore pressure ($\Delta u$) has dissipated. At that point, settlement continues mainly because soil particles slowly rearrange into a denser structure.

Compaction vs. consolidation

Definitions

Compaction is the process of increasing soil density by reducing air voids in an unsaturated soil.

Compaction and consolidation both increase soil density, but they happen in different ways:

• Compaction increases the density of an unsaturated soil by reducing the volume of air.
• Consolidation increases the density of a saturated soil over time by squeezing water out.

Stress history of clay soils

Definitions

A clay is normally consolidated if the present effective overburden pressure is the maximum pressure the soil has experienced.

Definitions

A clay is overconsolidated if the present effective overburden pressure is less than a past maximum pressure, called the preconsolidation pressure.

• Normally consolidated (NC) clay: The present effective overburden pressure is the maximum pressure the soil has been subjected to in the recent past.

• Overconsolidated (OC) clay: The present effective overburden pressure is less than what the soil has experienced in the past. The past maximum effective overburden pressure is called the preconsolidation pressure.

The stress history is summarized as:

• For normally consolidated soils:

  $${\sigma }_0^{\prime }={\sigma }_c^{\prime }$$

• For overconsolidated soils:

  $${\sigma }_0^{\prime }<{\sigma }_c^{\prime }$$

Where:

• ${\sigma }_0^{\prime }$ = Initial (present) effective overburden pressure
• ${\sigma }_c^{\prime }$ = Preconsolidation pressure

The Overconsolidation Ratio (OCR) is defined as:

$$OCR=\frac{{\sigma }_c^{\prime }}{{\sigma }_0^{\prime }}$$

Consolidation of normally consolidated (NC) clays

For a normally consolidated (NC) soil, primary consolidation settlement is calculated using the Compression Index $C_c$:

$$S_c=\left(\frac{C_c}{1+e_0}\right)H\log \left(\frac{{\sigma }_f^{\prime }}{{\sigma }_0^{\prime }}\right)$$

Where:

• $S_c$ = Primary consolidation settlement
• ${\sigma }_0^{\prime }$ = Initial effective overburden pressure
• ${\sigma }_f^{\prime }$ = Final effective overburden pressure (${\sigma }_0^{\prime }+\Delta \sigma$)
• $H$ = Thickness of soil layer
• $C_c$ = Compression index
• $e_0$ = Initial void ratio

Consolidation of overconsolidated (OC) clay

Case I: ${\sigma }_f^{\prime }\le {\sigma }_c^{\prime }$

Use the Recompression Index $C_r$:

$$S_c=\left(\frac{C_r}{1+e_0}\right)H\log \left(\frac{{\sigma }_f^{\prime }}{{\sigma }_0^{\prime }}\right)$$

Case II: ${\sigma }_f^{\prime }>{\sigma }_c^{\prime }$

Use both $C_r$ and $C_c$:

$$S_c=\left(\frac{C_r}{1+e_0}\right)H\log \left(\frac{{\sigma }_c^{\prime }}{{\sigma }_0^{\prime }}\right)+\left(\frac{C_c}{1+e_0}\right)H\log \left(\frac{{\sigma }_f^{\prime }}{{\sigma }_c^{\prime }}\right)$$

Empirical relationships of $C_c$ and $C_r$

For all clays:

$$C_c=1.15(e_0-0.35),C_r=(0.05\text{ to }0.10)\cdot C_c$$

For undisturbed clays of low to moderate sensitivity:

$$C_c=0.009(LL-10)$$

Where:

• $LL$= Liquid Limit

Rate of consolidation

Degree of consolidation using settlement ratio:

$$U=\frac{S_t}{S_s}$$

Where:

• $S_t$ = Settlement at time $t$
• $S_s$ = Total settlement

Degree of consolidation using pore pressure:

$$U=\frac{u_i-u}{u_i}$$

Where:

• $u_i$ = Initial excess pore pressure
• $u$ = Excess pore pressure at time $t$

Average degree of consolidation as a function of time factor:

$$T_v=\frac{C_{v}t}{H_d^2}$$

Where:

• $T_v$ = time factor
• $C_v$ = coefficient of consolidation
• $t$ = time
• $H_d$ = length of the drainage path
• $H_d$ = $H/2$ for two-way drainage and $H$ for one-way drainage, where, $H$ is thickness of the layer.

Please refer to the FE Handbook for detailed variation of time factor with degree of consolidation.

Stress in soil mass

Stress definitions

Where:

• $s$ = mean normal stress
• $t$ = maximum shear stress
• ${\sigma }_1$ = major principal stress
• ${\sigma }_3$ = minor principal stress
• $\theta$ = orientation angle between the plane of existing normal stress and the plane of the major principal stress

Total normal stress

$${\sigma }_N=\frac{P}{A}$$

Where:

• $P$ = normal force
• $A$ = cross-sectional area over which force acts

Effective stress

Definitions

Effective stress governs soil strength and deformation.

$${\sigma }^{\prime }=\sigma -u$$

$$u=h_u{\gamma }_w$$

Where:

• $h_u$ = uplift or pressure head

Shear Stress

$$\tau =\frac{T}{A}$$

Where:

• $T$ = shearing force

Shear stress at failure

$${\tau }_F=c+{\sigma }_N\tan \varphi$$

Where:

• $c$ = cohesion
• $\varphi$ = angle of internal friction