Effective stress and capillarity

Karl Terzaghi, the founder of soil mechanics, called the principle of effective stress the most important idea in the subject. It says that what deforms soil and gives it strength is not the total stress applied to it, but the part of that stress carried by the grains: the total stress minus the pressure of the water in the pores. Every calculation of settlement, every check of stability and every interpretation of a strength test rests on it. This lesson computes the effective stress profile under the operations centre, examines the artesian water in the till and the capillary water above the water table, and shows why lowering the groundwater for the pump station will settle the clay.

Total stress

At a depth \(z\) below a level ground surface, the total vertical stress \(\sigma_v\) is the weight of everything above a unit area: soil grains, the water in the pores and any surface load. For layered soil it is the sum of unit weight times thickness:

using the moist unit weight above the water table and the saturated unit weight below it. If a lake or river covers the ground, the weight of the free water above the surface is added too.

Pore water pressure

Water in the pores is connected through the soil, so under static conditions its pressure is hydrostatic: it increases linearly with depth below the water table, the level at which the pore pressure is zero (atmospheric).

where \(h_w\) is the depth below the water table, or more generally the height of water that would rise in a standpipe above the point, the pressure head. When water is flowing, as it is upward through the site clay, the pore pressure departs from hydrostatic, and the piezometers of lesson 1 measure it directly.

The principle of effective stress

Terzaghi defined the effective stress as

The idea is easiest to see with a thought experiment. A box of saturated sand is submerged in a tank and the water level is raised by a metre. The total stress on the sand rises, because there is more water above it, but the pore pressure rises by exactly the same amount, so the effective stress does not change, and the sand does not compress or get stronger. If instead a weight is laid on the sand, the total stress rises and, once the water has drained, the pore pressure is unchanged, so the effective stress rises and the sand compresses. Only changes in effective stress cause soil to deform or to change its strength.

Physically, the effective stress is the average contact force between grains per unit of total area. It is not the true stress at the tiny grain contacts, which is very much larger, and it is not a stress that can be measured with a gauge. It is a bookkeeping quantity that turns out to control behaviour. Soil cannot carry tension through its grains, so the effective stress of an uncemented soil cannot be negative; when the pore pressure rises to equal the total stress, the grains float apart and the soil loses its strength. That is the condition behind quicksand, piping and liquefaction.

The stress profile at the garage

The BH-3 profile gives the layers: compact sand to 4.0 m, weighing 18.0 kN/m³ above the water table at 2.5 m and 19.8 kN/m³ below; clay to 11.0 m at 18.0 kN/m³; and till to bedrock at 15.5 m, at 21.5 kN/m³. A piezometer sealed in the till shows a water level 1.5 m below ground, a metre above the water table in the sand. The till is under artesian pressure, fed from higher ground to the north.

Sofia's script computes the whole profile, the upward gradient through the clay, the height of capillary rise in the sand and the effect of dewatering.

# Total stress, pore pressure and effective stress at BH-3
gw = 9.81
# (top, bottom, unit weight above water table, unit weight below water table) in m and kN/m3
layers = [(0.0, 4.0, 18.0, 19.8, "sand"), (4.0, 11.0, 18.0, 18.0, "clay"), (11.0, 15.5, 21.5, 21.5, "till")]
wt = 2.5                 # water table in the sand (m)
h_till = 1.5             # piezometric level of the till, as a depth below ground (m)

def total_stress(z, wt):
    s = 0.0
    for top, bot, g_moist, g_sat, _ in layers:
        if z <= top:
            break
        dz_top, dz_bot = top, min(z, bot)
        above = max(0.0, min(dz_bot, wt) - dz_top)
        below = max(0.0, dz_bot - max(dz_top, wt))
        s += g_moist * above + g_sat * below
    return s

def pore_pressure(z, wt, h_till):
    if z <= 4.0:                                    # sand: hydrostatic below the water table
        return gw * max(0.0, z - wt)
    u_top = gw * (4.0 - wt)                         # top of clay, hydrostatic from the sand
    u_bot = gw * (11.0 - h_till)                    # bottom of clay, from the till piezometer
    if z <= 11.0:                                   # steady seepage through the clay: linear
        return u_top + (u_bot - u_top) * (z - 4.0) / 7.0
    return gw * (z - h_till)                        # till: hydrostatic about its own head

print("  z (m)   sigma   u      sigma'   (kPa)")
for z in (0.0, 2.5, 4.0, 7.5, 11.0, 15.5):
    s, u = total_stress(z, wt), pore_pressure(z, wt, h_till)
    print(f"{z:6.1f} {s:7.1f} {u:6.1f} {s - u:8.1f}")

# Upward gradient through the clay from the artesian head in the till
i = ((11.0 - h_till) - (11.0 - wt) ) / 7.0
print(f"\nExcess head at the base of the clay = {wt - h_till:.1f} m, upward gradient i = {i:.3f}")

# Capillary rise in the sand: h_c = C / (e D10), with C about 0.1 to 0.5 cm2
e, D10_cm = 0.62, 0.0097
for C in (0.1, 0.3, 0.5):
    print(f"C = {C:.1f} cm2: capillary rise h_c = {C / (e * D10_cm) / 100:.2f} m")
hc = 0.5
z = wt - hc
print(f"Top of the capillary zone at {z:.1f} m: u = {-gw*hc:.1f} kPa, sigma' rises by {gw*hc:.1f} kPa above the dry estimate")

# Dewatering for the pump station lowers the water table in the sand to the top of the clay
s0, u0 = total_stress(4.0, wt), pore_pressure(4.0, wt, h_till)
s1 = total_stress(4.0, 4.0)             # sand between 2.5 and 4.0 m now drained, at its moist unit weight
u1 = 0.0
print(f"\nTop of clay before: sigma' = {s0 - u0:.1f} kPa; after dewatering: sigma' = {s1 - u1:.1f} kPa; "
      f"increase {s1 - u1 - (s0 - u0):.1f} kPa")

  z (m)   sigma   u      sigma'   (kPa)
   0.0     0.0    0.0      0.0
   2.5    45.0    0.0     45.0
   4.0    74.7   14.7     60.0
   7.5   137.7   54.0     83.7
  11.0   200.7   93.2    107.5
  15.5   297.4  137.3    160.1

Excess head at the base of the clay = 1.0 m, upward gradient i = 0.143
C = 0.1 cm2: capillary rise h_c = 0.17 m
C = 0.3 cm2: capillary rise h_c = 0.50 m
C = 0.5 cm2: capillary rise h_c = 0.83 m
Top of the capillary zone at 2.0 m: u = -4.9 kPa, sigma' rises by 4.9 kPa above the dry estimate

Top of clay before: sigma' = 60.0 kPa; after dewatering: sigma' = 72.0 kPa; increase 12.0 kPa

Seepage and effective stress

Flowing water drags on the grains it passes. Where water flows downward, the drag adds to the weight of the grains and the effective stress is higher than hydrostatic conditions would give; where it flows upward, the drag lifts the grains and the effective stress is lower. The excess head across the clay here is small, and the upward hydraulic gradient, the head lost per unit length of flow, is

Lesson 7 shows that upward flow reduces the effective stress by \(i\gamma_w\) per unit depth, and that the effective stress falls to zero when the gradient reaches about 1.0. The site is far from that, but the pump station excavation, which will cut down toward the clay, changes the picture: removing the sand above the clay removes the weight that holds the clay down against the artesian pressure in the till. Lesson 8 checks the base of that excavation against heave.

Capillarity

Above the water table, water is drawn up into the narrow pores by surface tension, just as it rises in a thin glass tube. In a tube of diameter \(d\), water rises to a height

where \(T\) is the surface tension of water, about 0.073 N/m at room temperature, and \(\alpha\) is the contact angle, near zero for clean glass and mineral surfaces. The water in the tube hangs from the meniscus, so its pressure is below atmospheric: negative pore pressure, or suction, equal to \(-\gamma_w h\) at a height \(h\) above the water table. Soil pores are not tubes, but the same physics applies, and an empirical form due to Hazen relates the height of capillary rise to the effective size:

with \(C\) between about 0.1 and 0.5 cm² and \(D_{10}\) in centimetres. Rise is a few centimetres in gravel, tens of centimetres in sand, metres in silt and many metres in clay, although in clay it takes a very long time to develop.

For the site sand, \(D_{10} = 0.097\) mm and \(e = 0.62\) give a capillary rise between about 0.2 and 0.8 m; Owen takes 0.5 m. In that zone, from 2.0 to 2.5 m, the sand is close to saturated and the pore pressure is negative, as low as \(-4.9\) kPa at the top. The suction adds to the effective stress: it is what lets a child build a sandcastle from damp sand, and it disappears when the sand dries out or is submerged.

Engineers rarely count on suction in design, because it can vanish after heavy rain, a burst pipe or a rise in the water table. Its loss is the reason a trench cut in damp sand can stand vertically for a few hours and then collapse as it dries. Capillarity also feeds the ice lenses of frost heave described in lesson 3, which is why a layer of clean coarse gravel, a capillary break, is placed under floor slabs and road bases in cold regions.

Partly saturated soils

In soils that are partly saturated, such as compacted fills and the sand above the capillary zone, the pores contain both air and water, and the simple form of the effective stress principle no longer holds exactly. Bishop proposed a modified expression that weights the suction by a factor between 0 and 1 depending on the degree of saturation, and modern unsaturated soil mechanics treats the net stress and the suction as separate variables. For most foundation work, engineers assume either zero pore pressure above the water table or a conservative capillary zone, as Owen has done.

Changing the water table

Because effective stress depends on the pore pressure, lowering the groundwater increases the effective stress in every layer below. In sand the effect is immediate and causes little settlement, because sand is stiff. In clay it causes consolidation settlement that develops over months or years, the subject of lessons 10 and 11. Groundwater pumping has caused large regional subsidence in many cities built on clay, including Mexico City and Bangkok.

Owen points out that the excavation itself will remove soil and reduce the total stress inside the pump station, so the clay there will unload. Outside the excavation, however, the dewatering effect spreads as far as the drawdown cone reaches, and that is where the garage footings sit. Dr. Kowalczyk adds a requirement for settlement monitoring points on the garage during the pump station work and suggests recharge wells if the measured drawdown spreads too far.

Horizontal stress

The effective stress principle applies in every direction, but the horizontal effective stress is not simply equal to the vertical one. For a soil that was deposited and loaded only vertically, the ratio of horizontal to vertical effective stress at rest is the coefficient \(K_0\), about \(1 - \sin\phi'\) for normally consolidated soils, roughly 0.4 to 0.6. In overconsolidated soils, from which overburden has been removed by erosion or melting ice, \(K_0\) is higher and can exceed 1 near the surface. The horizontal stress matters for retaining walls, piles and the stress paths of lesson 12, and it is covered in the foundation and geotechnical design courses.

Practical points

• Always draw the profile: layers, unit weights, water table and any piezometric levels in deeper layers.
• Use measured piezometric levels, not an assumed hydrostatic profile, when layers are separated by clay.
• Do not rely on capillary suction for long-term stability.
• Check the effect of any dewatering on nearby structures founded over compressible soil.