Sheet pile and embedded walls

Gravity, cantilever and MSE walls are built from the bottom up: the ground is excavated, the wall is built, and the backfill is placed behind it. Embedded walls are built from the top down. Steel sheet piles, soldier piles with lagging, secant or contiguous bored piles and diaphragm walls are installed into the ground first, and the soil in front of them is then excavated. They stand because the part of the wall below the excavation level is pushed against the soil in front, mobilizing passive resistance, and, if the wall is tall, because anchors or struts hold it near the top. This lesson covers cantilever and anchored sheet pile walls, using the outfall of the new storm sewer on Kestrel Creek.

The storm sewer from the operations centre discharges to the creek downstream of the culvert. Its outfall structure needs a headwall, and the utility chamber beside it needs a deeper excavation against the bank. Borehole BH-9 at the outfall found loose to compact alluvial sand, \(\phi' = 32^\circ\), to 10 m, with the water table 1.5 m below the bank top, a thicker sand than elsewhere on the site. Both walls will be steel sheet piles: a cantilever wall retaining 3.0 m for the headwall, and an anchored wall retaining 5.0 m for the chamber.

Cantilever embedded walls

A cantilever sheet pile wall rotates about a point a little above its toe. Above that point, active pressure acts behind and passive pressure in front; below it, the pressures reverse, with passive pressure behind the wall and active in front. Blum's simplified method replaces the reversed pressures below the point of rotation by a single concentrated force \(R\) at a depth \(D_0\), where the toe would be. Taking moments about that point eliminates \(R\) and gives one equation for \(D_0\): the moment of the active pressure on the full height \(h + D_0\) equals the moment of the passive pressure on \(D_0\). The wall is then extended by about 20 % to provide the length over which \(R\) develops:

Safety is introduced either by dividing the passive coefficient by a factor, typically 1.5 to 2.0, or by increasing the embedment by a further 20 to 40 % with the full passive coefficient. The two approaches should not be confused, and should not both be omitted. The maximum bending moment occurs below the excavation level, where the shear force is zero. Cantilever sheet pile walls are economical only for retained heights up to about 4 to 5 m, because the moment grows with the cube of the height and the deflection with its fifth power.

Anchored walls and free earth support

A wall anchored near its top can retain much greater heights. If the embedment is just sufficient for the passive resistance in front to balance the thrust not carried by the anchor, the toe is free to rotate and the wall behaves as a beam simply supported by the anchor and the soil in front. This is the free earth support method. Taking moments about the anchor gives the embedment directly:

where \(\bar{z}_a\) and \(\bar{z}_p\) are the lever arms of the active and passive resultants about the anchor. Horizontal equilibrium then gives the anchor force, \(T = P_a - P_p\). The maximum moment is found at the point of zero shear between the anchor and the dredge line. Deeper embedment produces fixity at the toe, the fixed earth support condition, which reduces the moment and the anchor force but needs a longer pile.

Flexible walls redistribute pressure. Rowe's model tests in the 1950s showed that a flexible anchored wall in sand deflects between the anchor and the excavation level, arching transfers load to the stiffer anchor and toe zones, and the actual maximum moment is less than the free earth support value. Design manuals give moment reduction charts based on Rowe's flexibility number. Modern practice often uses beam-on-springs or finite element analyses instead, especially when deflections must be controlled near structures.

The script solves both walls by numerical integration of the pressure diagrams, finds the embedments by bisection and locates the maximum moments.

# Sheet pile walls at the Kestrel Creek outfall: cantilever and anchored (free earth support)
from math import tan, radians
UNITS = "SI"
SI = UNITS == "SI"
L, F, P, MU = ("m", "kN/m", "kPa", "kN.m/m") if SI else ("ft", "lb/ft", "psf", "lb.ft/ft")
phi = 32.0
g, gsub, zw = (18.0, 10.0, 1.5) if SI else (115.0, 63.6, 5.0)       # moist and submerged unit weights; water table depth on the retained side
Ka = tan(radians(45 - phi / 2)) ** 2; Kp = tan(radians(45 + phi / 2)) ** 2; Kpd = Kp / 1.5
def sv(z):                       # vertical effective stress behind the wall (water table at zw)
    return g * z if z <= zw else g * zw + gsub * (z - zw)
def solve(h, D, anchor=None, n=600):
    """Net force and moment on a wall of retained height h embedded D, water level the same both sides below zw."""
    Fa = Ma = Fp = Mp = 0.0
    dz = (h + D) / n
    for i in range(n):
        z = (i + 0.5) * dz
        pa = Ka * sv(z)
        Fa += pa * dz; Ma += pa * dz * z
        if z > h:
            pp = Kpd * gsub * (z - h) if h >= zw else Kpd * (g * min(z - h, zw - h) + gsub * max(0, z - zw))
            Fp += pp * dz; Mp += pp * dz * z
    return Fa, Ma, Fp, Mp
print(f"Ka = {Ka:.3f}, Kp = {Kp:.2f}, Kp/1.5 = {Kpd:.2f} used for design")
# 1) cantilever wall: moment about the toe of the simplified (fixed-toe) method, then add 20 %
h1 = 3.0 if SI else 10.0
lo, hi = 0.1, 30.0
for _ in range(60):
    D = (lo + hi) / 2; Fa, Ma, Fp, Mp = solve(h1, D)
    zt = h1 + D
    if (Fa * zt - Ma) > (Fp * zt - Mp): lo = D       # active moment about the toe exceeds passive: embed deeper
    else: hi = D
D0 = D
print(f"\nCantilever, retained height {h1:g} {L}: D0 = {D0:.2f} {L}; design embedment 1.2 D0 = {1.2*D0:.2f} {L}; pile length {h1 + 1.2*D0:.2f} {L}")
# maximum moment at zero shear below dredge level
best = 0
for k in range(1, 600):
    z = h1 + D0 * k / 600; Fa, Ma, Fp, Mp = solve(h1, z - h1)
    if Fa - Fp <= 0: break
    best = max(best, Fa * z - Ma - (Fp * z - Mp))
print(f"  maximum bending moment about {best:.1f} {MU}")
# 2) anchored wall: free earth support, anchor at depth a
h2, a = (5.0, 1.0) if SI else (16.0, 3.0)
lo, hi = 0.1, 30.0
for _ in range(60):
    D = (lo + hi) / 2; Fa, Ma, Fp, Mp = solve(h2, D)
    if (Ma - Fa * a) > (Mp - Fp * a): lo = D          # moments about the anchor
    else: hi = D
Fa, Ma, Fp, Mp = solve(h2, D); T = Fa - Fp
print(f"\nAnchored, retained height {h2:g} {L}, anchor {a:g} {L} below the top: D = {D:.2f} {L}; specify {1.2*D:.2f} {L} ({h2 + 1.2*D:.2f} {L} long)")
print(f"  anchor force T = {T:.1f} {F}; with anchors at {2.5 if SI else 8.0} {L} spacing, {T*(2.5 if SI else 8.0):.0f} {'kN' if SI else 'lb'} per anchor")
Mmax = 0; zm = 0
for k in range(1, 2000):
    z = (h2 + D) * k / 2000; Fa_, Ma_, Fp_, Mp_ = solve(h2, max(z - h2, 0), n=400) if z > h2 else solve(z, 0, n=400)
    M = (Fa_ * z - Ma_) - (Fp_ * z - Mp_) - (T * (z - a) if z > a else 0)
    if abs(M) > abs(Mmax): Mmax, zm = M, z
print(f"  maximum moment {abs(Mmax):.1f} {MU} at {zm:.2f} {L} depth")
fy = 345e3 if SI else 50000 * 144                  # kPa | psf
S = abs(Mmax) / (0.65 * fy)
print(f"  required section modulus at 0.65 fy: {S*1e6:.0f} cm3/m".replace("cm3/m", "cm3/m") if SI else f"  required section modulus at 0.65 fy: {S*1728:.1f} in3/ft")

Ka = 0.307, Kp = 3.25, Kp/1.5 = 2.17 used for design

Cantilever, retained height 3 m: D0 = 4.20 m; design embedment 1.2 D0 = 5.04 m; pile length 8.04 m
  maximum bending moment about 70.9 kN.m/m

Anchored, retained height 5 m, anchor 1 m below the top: D = 2.70 m; specify 3.24 m (8.24 m long)
  anchor force T = 37.6 kN/m; with anchors at 2.5 m spacing, 94 kN per anchor
  maximum moment 60.3 kN.m/m at 4.07 m depth
  required section modulus at 0.65 fy: 269 cm3/m

Reading the results and choosing a section

With an allowable bending stress of about 0.65 of the yield strength of 345 MPa steel, the anchored wall needs a section modulus of about 270 cm³/m, and the cantilever headwall a little more. These are light sections, but sheet piles in practice are selected also for driveability, for corrosion allowance and for stiffness, and the lightest sections can be damaged when driven through dense layers or cobbles. The team selects a medium-weight Z-section for both walls, with a corrosion allowance for a design life of 75 years in fresh water with road-salt runoff, and specifies vibratory installation with a check on vibrations at the culvert.

Deflection matters next to the culvert and the road. A cantilever wall of this height will deflect a few tens of millimetres at its top; the anchored wall much less. The team estimates both with a beam-on-springs model and sets a monitoring limit for the top of the headwall.

Anchors, deadmen and tiebacks

The anchor of a sheet pile wall can be a tie rod to a deadman, a continuous concrete beam or sheet pile wall buried behind the active wedge, or a grouted ground anchor drilled through the wall. A deadman works by passive resistance, so it must lie outside the active wedge of the wall and its own passive wedge must not overlap that of the wall: the usual rule places it behind a line drawn from the dredge level at the angle of the passive plane. Ground anchors, the subject of lesson 6, are drilled at an inclination and bonded in soil or rock beyond the failure wedge; their vertical component adds axial load to the wall.

Waling beams spread the anchor force along the wall. The tie rod, the waling and the connections are designed as structural elements, usually with a margin above the calculated force to allow for the uncertainty of the earth pressure distribution and for the loss of one anchor.

Practical points

• Check the effect of scour and over-excavation on the embedment, and add length for it.
• Introduce safety once: either a factor on \(K_p\) or extra embedment, as the method requires.
• Locate deadmen outside the active wedge and with their own passive wedge clear of the wall's.
• Choose sections for driveability, corrosion and stiffness as well as moment.
• Estimate and monitor deflection where the wall is near structures or roads.