Pile groups, settlement and downdrag

Piles rarely work alone. Under each fleet building column, four piles will share the load through a cap, and the yard around the building will be raised by a fill that will settle the clay through which the piles pass. This lesson examines how piles behave in groups, how much a pile group settles, and what happens when the soil around a pile settles more than the pile itself, dragging it down. That last phenomenon, downdrag, has caused more misunderstanding than almost any other topic in foundation engineering, and the modern treatment, the unified method developed by Fellenius, is presented in detail.

Group efficiency

The capacity of a group of piles is not necessarily the sum of the capacities of its individual piles. The ratio of the two is the group efficiency \(\eta\). Several empirical formulas exist; the best known, the Converse-Labarre formula, depends only on geometry:

for a group of \(m\) rows of \(n\) piles of diameter \(D\) at spacing \(s\). It predicts efficiencies of 0.7 to 0.8 at typical spacings, but it has no basis in mechanics and does not agree with load tests in sand. Driven piles in sand compact the soil between them, and groups of driven piles in loose to medium dense sand commonly have efficiencies greater than one. In practice, the capacity of a group of driven piles in sand or till is taken as the sum of the individual capacities, and the formula is mainly of historical interest.

Block failure in clay

In clay, closely spaced piles may fail together as a block, the soil between them moving with the piles. The block capacity is the shear resistance on the perimeter of the block plus the bearing capacity of its base:

with \(N_c\) up to 9 for a deep block. The group capacity is the lesser of the block capacity and the sum of the individual capacities. Block failure governs only for closely spaced piles in clay, typically at spacings below about two to three diameters, and for groups whose toes are also in clay. For the fleet building, the piles end in dense till, and the block capacity is about twice the sum of the individual capacities.

Settlement of pile groups

Even when the capacity of a group is the sum of its parts, its settlement is not. A single pile under its working load settles a few millimetres. A group settles more, because the stresses from all the piles overlap in the soil below the toes, stressing a much larger and deeper volume. Terzaghi and Peck proposed a simple and still widely used model, the equivalent footing. For piles that pass through soft soil into a firm bearing layer, the group load is assumed to act on an imaginary footing the size of the group, located at two thirds of the depth of penetration into the bearing layer, and to spread below it at 2:1. The settlement of the equivalent footing is calculated by the methods of lessons 5 and 6, and the elastic shortening of the piles above it is added:

For friction piles in a uniform clay, the equivalent footing is placed at two thirds of the pile length. More refined methods based on elastic interaction factors, developed by Poulos and Davis, and numerical analyses are used for large groups and piled rafts.

Negative skin friction and downdrag

When the soil around a pile settles relative to the pile, the shear stress on the pile shaft reverses: instead of supporting the pile, the soil drags it down. This is negative skin friction. It arises when a fill is placed on soft ground around piles, when the water table is lowered, when a soft clay is still consolidating under its own weight or a recent fill, or when a loose fill is wetted or vibrated. The load it adds to the pile is called the drag force, and the settlement it causes is called downdrag.

For many years, designers treated the drag force as an additional load to be added to the structural load and carried by the positive resistance below the settling layer, with a factor of safety on both. Load tests and instrumented piles showed that this view is wrong. Fellenius, Bozozuk and others measured the forces in piles over many years and found that negative skin friction develops along the upper part of a pile and positive resistance along the lower part, and that the two meet at a depth called the neutral plane. At the neutral plane, there is no relative movement between the pile and the soil, the load in the pile is greatest, and the pile and soil settle by the same amount.

The unified method

The unified method, now reflected in the CFEM and in many other design guides, separates three questions.

• Geotechnical capacity. The drag force is not a load that can cause the pile to plunge, because it disappears as soon as the pile moves down relative to the soil. The ultimate limit state is checked with the structural loads only, against the full positive shaft and toe resistance, as in lesson 9.
• Structural strength. The maximum load in the pile, at the neutral plane, is the sustained load plus the drag force. The pile material must carry it; transient live loads are not added, because they temporarily reduce the drag.
• Settlement. The pile head settles as much as the soil at the neutral plane, plus the elastic shortening of the pile. If the neutral plane lies in the bearing layer, where the soil settles little, downdrag is negligible even if the drag force is large. If it lies in a settling clay, the pile will follow the clay down.

The neutral plane is found by plotting two curves: the load in the pile from the head downward, starting with the sustained load and adding the negative skin friction, and the resistance from the toe upward, starting with the mobilized toe resistance and adding the positive shaft resistance. They intersect at the neutral plane. Its depth depends on how much toe resistance is mobilized, which depends on the toe movement; the more the toe moves, the more resistance it develops and the lower the neutral plane.

Sofia's script checks the group efficiency and block capacity, estimates the group settlement with the equivalent footing, and finds the neutral plane for full and half mobilization of the toe resistance under the yard fill.

# Four-pile group: efficiency, block check, group settlement and downdrag from the yard fill
from math import pi, sin, tan, radians, sqrt, atan, degrees, log10
D, tip, s, n, m = 0.406, 15.0, 1.2, 2, 2          # pile diameter, tip depth, spacing 3D (m), 2 x 2 group
theta = degrees(atan(D/s))
E = 1 - theta*((n - 1)*m + (m - 1)*n)/(90*m*n)
print(f"Converse-Labarre efficiency (for comparison only): theta = {theta:.1f} deg, E = {E:.2f}")
# block failure through the clay: perimeter shear on su plus bearing on the till at the block base
Bb = s + D
R_block = 4*Bb*7.0*55.0 + 4*Bb*4.0*120.0 + Bb**2*9.0*400.0     # clay su, till average shaft, till base (kPa)
print(f"Block width {Bb:.2f} m: block resistance about {R_block:.0f} kN vs 4 single piles 4 x 1781 = {4*1781:.0f} kN")

# group settlement: equivalent footing at 2/3 of the penetration into the till, load spread 2V:1H
z_eq = 11.0 + 2/3*(tip - 11.0); Q_sus = 1900.0 + 0.5*1100.0 + 166.0     # sustained load incl. cap (kN)
sett = 0.0; z = z_eq
while z < z_eq + 20.0:
    dz = 0.1; Es = 100e3 if z < 15.5 else 1000e3                        # till 100 MPa, shale 1,000 MPa
    sett += Q_sus/(Bb + (z - z_eq + dz/2))**2*dz/Es; z += dz
EA = 200e6*0.0157 + 25e6*0.1138                # steel pipe plus concrete fill (kN)
elastic = (Q_sus/4)*(tip*0.6)/EA               # pile shortening, effective length about 0.6 L
print(f"Equivalent footing at {z_eq:.2f} m: settlement below it {1000*sett:.1f} mm, pile shortening {1000*elastic:.1f} mm")

# downdrag: 1.2 m of yard fill (24 kPa) consolidates the clay; find the neutral plane (unified method)
def eff(z, fill=24.0):
    s_ = 18.0*min(z, 2.5) + 19.8*max(0.0, min(z, 4.0) - 2.5) + 18.0*max(0.0, min(z, 11.0) - 4.0) + 21.5*max(0.0, z - 11.0)
    u = 9.81*max(0.0, z - 2.5) if z <= 4.0 else 14.7 + 78.5*(z - 4.0)/7.0 if z <= 11.0 else 9.81*(z - 1.5)
    return s_ - u + fill
def fs(z):
    se = eff(z)
    if z < 4.0: return tan(radians(0.8*34))*se
    if z < 11.0: return min(1.0, 0.5*(55/se)**-0.5 if 55/se <= 1 else 0.5*(55/se)**-0.25)*55
    return tan(radians(0.8*38))*se
per, dz = pi*D, 0.05
zs = [k*dz for k in range(int(tip/dz) + 1)]
Qd = (1900.0 + 166.0)/4                         # dead load per pile (kN); transient live load does not add to drag
Rt_full = 50*(eff(tip) - 24.0)*pi*D**2/4
for frac in (1.0, 0.5):
    Rt = frac*Rt_full
    down = [Qd]; up = [0.0]*len(zs); up[-1] = Rt
    for z in zs[1:]: down.append(down[-1] + fs(z - dz/2)*per*dz)
    for k in range(len(zs) - 2, -1, -1): up[k] = up[k + 1] + fs(zs[k] + dz/2)*per*dz
    k = min(range(len(zs)), key=lambda i: abs(down[i] - up[i]))
    print(f"Toe mobilized {100*frac:.0f} %: neutral plane at {zs[k]:.1f} m, drag force {down[k] - Qd:.0f} kN, "
          f"maximum load in the pile {down[k]:.0f} kN")
# settlement of the soil at the neutral plane under the fill (clay below the plane only)
def clay_settlement_below(zn):
    t = 0.0
    for k in range(70):
        z = 4.05 + 0.1*k
        if z < zn: continue
        s0 = eff(z, 0.0); sp = s0 + 45.0; sf = s0 + 24.0
        t += 0.1*(0.047/2.06*log10(sf/s0) if sf <= sp else (0.047*log10(sp/s0) + 0.316*log10(sf/sp))/2.06)
    return 1000*t
print(f"Fill settlement of the ground surface {clay_settlement_below(0):.0f} mm; of the soil at 11.0 m and below: "
      f"{clay_settlement_below(11.0):.0f} mm; below 10.6 m: {clay_settlement_below(10.6):.1f} mm")

Converse-Labarre efficiency (for comparison only): theta = 18.7 deg, E = 0.79
Block width 1.61 m: block resistance about 14842 kN vs 4 single piles 4 x 1781 = 7124 kN
Equivalent footing at 13.67 m: settlement below it 9.4 mm, pile shortening 1.0 mm
Toe mobilized 100 %: neutral plane at 13.1 m, drag force 720 kN, maximum load in the pile 1237 kN
Toe mobilized 50 %: neutral plane at 10.6 m, drag force 470 kN, maximum load in the pile 987 kN
Fill settlement of the ground surface 18 mm; of the soil at 11.0 m and below: 0 mm; below 10.6 m: 0.8 mm

Reading the results

The Converse-Labarre formula gives an efficiency of 0.79, which Dr. Kowalczyk notes but does not use, because the piles are driven displacement piles ending in dense till. The block capacity is about twice the sum of the four single piles, so block failure does not govern. The equivalent footing at 13.67 m, two thirds of the way into the till, settles 9.4 mm under the sustained load of 2,616 kN, and the piles shorten by about 1 mm: the group settles about 10 mm, an order of magnitude less than spread footings would.

With the full toe resistance mobilized, the neutral plane is at 13.1 m in the till and the drag force is 720 kN, so the maximum load in the pile is 1,237 kN. With only half the toe resistance mobilized, the neutral plane rises to 10.6 m, near the base of the clay, and the drag force is 470 kN. In either case the soil at the neutral plane settles by less than 1 mm under the fill, although the ground surface settles 18 mm. The piles will therefore not suffer downdrag, and the maximum load is far below the structural resistance of about 6,800 kN. The ground around the building, however, will settle relative to the building, and the floor slab, the aprons and the service connections must be detailed for it.

Owen asks what would have happened if the piles had been friction piles ending in the clay. The neutral plane would then lie in the clay, the piles would settle with the clay, and the building would settle almost as much as the yard. That is the classic downdrag failure: piles that have adequate capacity on paper but follow the consolidating soil down. Ending the piles in a stratum that does not settle is the most reliable cure.

Reducing drag force

Where the drag force would overload a pile structurally, or where the neutral plane would lie in settling soil, several measures help. Bitumen coatings on the pile shaft through the settling layers reduce negative skin friction substantially, as long as the coating survives driving. Placing and consolidating the fill before the piles are installed, with vertical drains to speed the process, removes the cause. Lightweight fills, such as expanded polystyrene blocks, lightweight aggregate or foamed concrete, reduce the stress increase. Increasing the toe resistance, by driving the piles deeper into the bearing layer, lowers the neutral plane. And a pile can be sleeved through the settling soil, though this also gives up the positive resistance of that soil.

Uplift and lateral effects of groups

A group resisting uplift can fail by pulling out a block of soil if the piles are close together, and its capacity is the lesser of the sum of the individual tension capacities and the weight of the block plus the shear on its perimeter. Under lateral loads, piles in a group shadow one another: a pile in a trailing row pushes against soil already loaded by the leading row, and carries less load at a given deflection. Group reduction factors, called p-multipliers, are applied in lateral analyses, as lesson 11 notes.

Practical points

• Take the capacity of driven pile groups in sand and till as the sum of the individual capacities; check block failure for close groups in clay.
• Estimate group settlement with an equivalent footing in the bearing layer, plus elastic shortening.
• Treat the drag force as a structural load and a settlement issue, not as a reduction of geotechnical capacity.
• Find the neutral plane by the unified method and check the settlement of the soil there.
• End piles in a stratum that does not settle wherever fills, dewatering or consolidation will occur.