SlipSurface IQ Pile Wall Analysis — theory

What SlipSurface IQ Pile Wall Analysis (SPWA) computes for a cantilever or anchored wall of steel sheet piles or of reinforced concrete bored piles, and where the methods stop being valid. Depth z is measured down from the top of the wall at the retained ground level; H is the retained height (the excavation depth), D the embedment below the dredge line. Pressures are in kPa, forces in kN per metre of wall, moments in kNm per metre of wall.

1. Design soil

The partial factors are applied to the strength before anything else, on the tangent of the friction angle and on the cohesion, layer by layer:

φd = atan( tan φ / FSφ )         cd = c / FSc

The pressures, the embedment and the internal forces below all use the design values. The unit weights are not factored. Below the water table the weight is γsat and the effective stress uses γ′ = γsat − γw.

2. Earth pressures

2.1 Coulomb

For a vertical wall, wall friction δ, a backfill sloping at β (retained side) and a dredge line sloping at α (excavation side):

Ka = cos² φ / ( cos δ · [ 1 + √( sin(φ+δ) sin(φ−β) / (cos δ cos β) ) ]² )
Kp = cos² φ / ( cos δ · [ 1 − √( sin(φ+δ) sin(φ+α) / (cos δ cos α) ) ]² )

The passive coefficient is undefined if the square root reaches 1 (a steep dredge line with a high δ); the program stops and says so. Coulomb overestimates the passive resistance for a large δ with a curved failure surface, so δ ≤ 2/3 φ is the usual choice.

2.2 Pressure at depth z

On the retained side

p_a(z) = max( 0,  Ka · (σ′v + q) − 2 c √Ka )  +  u_a(z)  +  p_hd(z)

with σ′v the vertical effective stress, q the uniform surcharge, u_a the pore pressure from the water level on that side, and p_hd the hydrodynamic pressure (§5). The cut-off at zero stops the cohesion term from producing a tension.

On the excavation side, below the dredge line

p_p(z) = Kp · σ′v,exc + 2 c √Kp  +  u_p(z)

where σ′v,exc counts the soil from the dredge line down, and the free water in front of the wall acts at every depth below the water level on that side, also above the dredge line.

The net pressure is p_a − p_p.

3. Limit equilibrium: embedment, anchor force, internal forces

3.1 The embedment

The wall is in limit equilibrium when the moments of the net pressure about a reference point vanish:

Σ M = ∫₀^(H+D)  ( p_a(z) − p_p(z) ) · arm(z)  dz = 0

• Cantilever (simplified method): the reference is the toe, arm = H + D − z. The wall rotates about a point near the toe; the simplified method takes the moment about the toe itself.
• Anchored, free earth support: the reference is the lowest anchor, arm = z − z_anchor. The anchor is a support at its depth and the toe is free to rotate (no fixity), so the wall is statically determinate with one anchor.

The root D_req is found by a bracketed search (Brent). It is the theoretical embedment; the design embedment is

D_design = roundup( embedment factor · D_req / rounding increment ) · rounding increment

with the factor 1.2 by default.

3.2 The anchor force

For one anchor, the horizontal force per metre follows from horizontal equilibrium at D_req:

T_h = ∫₀^(H+D_req) p_a dz − ∫₀^(H+D_req) p_p dz

and the axial force of the anchor, at the angle θ below the horizontal, is T_h / cos θ (the vertical component T_h tan θ goes to the wall). Free earth support is statically determinate for one anchor only; with several, the program distributes T_h by the tributary area of the net pressure, which does not satisfy moment equilibrium exactly: use the beam-spring analysis of §4 for a multi-anchor wall.

3.3 Internal forces

Shear and moment are the integrals of the net pressure from the top, with the anchor force as a point load at its depth; the moment is zero at the top and at the toe for the free-earth case, and the maximum bending moment M_max is the largest absolute value. Rotation and deflection come from twice integrating the curvature M / EI, with the boundary conditions of the support: rotation and deflection zero at the toe for the cantilever, deflection zero at the lowest anchor and at the toe of an anchored wall.

4. Beam on springs (Winkler)

For staged construction and for walls with several anchors the wall is modelled as an Euler–Bernoulli beam (Hermite cubic elements, horizontal displacement w positive towards the excavation and rotation at each node) on elastic–perfectly plastic soil springs.

• Springs. The pressure at a node is linear in the displacement between the active and the passive limit (the Coulomb or Mononobe–Okabe values of §2) and constant once a limit is reached:

  retained side:    p = clip( p_ref − k_s (w − w_ref),  p_a,  p_p )
  excavation side:  p = clip( p_ref + k_s (w − w_ref),  p_a,  p_p )
  

  The springs start from the at-rest pressure K₀ σ′v, with K₀ = 1 − sin φ (Jaky), and keep their state from one construction stage to the next. The retained side extends over the whole length of the wall; the excavation side only acts below the dredge level of the stage.

• Subgrade modulus k_s (kN/m³) of a layer: entered, or from the pressuremeter modulus E_M and the rheological coefficient α (sand 1/3, silt 1/2, clay 2/3):

  Ménard–Bourdon (1964):   k_s = E_M / [ α a / 2 + 0.133 (9 a)^α ],      a = max( 2D/3, 0.6 m )
  Schmitt (1995):          k_s = 2.1 (E_M / α)^(4/3) / (EI)^(1/3)          (E_M in kPa, EI in kNm²/m)
  

  with D the embedment and EI the stiffness of the wall per metre; k_s is a single value per layer.

• Anchors are tension-only springs of stiffness k_h = EA / (L_free · s) · cos² θ per metre of wall (s the horizontal spacing, θ the angle below the horizontal, L_free the free length), installed at a stage with a lock-off load P₀:

  T = max( 0,  P₀ cos θ / s + k_h (w − w_install) )
  

• Stages are generated in order: excavate to the anchor depth plus an overdig, install the anchor, carry on to the final level, where the non-staged variant instead installs every anchor in place and excavates in one step.

• Water pressures are fixed loads, with no seepage correction. They can be taken as given at every stage or, for a dry excavation, with the water at the current dredge level until the final stage.

• The equilibrium of the whole system is solved by Newton–Raphson, and the internal forces are recovered by integrating the converged nodal loads from the top, so shear and moment close at the toe to round-off.

The result is a set of moments, shears, anchor forces and displacements for each stage; the envelope of the stages is what the checks use. The model is deliberately simple: one k_s per layer with no dependence on depth or stress, horizontal anchor forces, no creep of the wall or the soil and no hysteresis on reloading. It is a good model of the wall's behaviour in service, but the limit pressures and k_s are only as good as the soil data.

5. Seismic action

• Earth pressure: Mononobe–Okabe, for a vertical wall,

  K_AE = cos²(φ − θ) / ( cos θ · cos(δ + θ) · [ 1 + √( sin(φ+δ) sin(φ−θ−β) / (cos(δ+θ) cos β) ) ]² )
  K_PE = cos²(φ − θ) / ( cos θ · cos(δ + θ) · [ 1 − √( sin(φ+δ) sin(φ−θ+α) / (cos(δ+θ) cos α) ) ]² )
  

  with the inertia angle θ = atan( kh / (1 − kv) ). Below the water table, with the pore water restrained to move with the soil, θ is computed with γsat / γ′ instead of 1 (an option). If θ ≥ φ − β the active coefficient is undefined, and the program caps it and warns. The vertical pressure is multiplied by (1 − kv).

• Hydrodynamic pressure of the free water in front of the wall, Westergaard (1933), as a driving load,

  p_hd(y) = 7/8 · kh · γw · √( H_w · y )
  

  with H_w the depth of the free water and y the depth below its surface.

6. Checks

6.1 Steel sheet pile

The section's elastic modulus W and moment of inertia I come from the sheet pile table. The bending stress is σ = M_max / W, and the check is

σ  ≤  f_y / FS_bending

The deflection is compared to the limit chosen: H/120, H/100 or H/240 (FHWA).

6.2 Reinforced concrete bored piles

The wall is analysed per metre of wall, and a pile carries the moment and shear of its spacing s.

• Stiffness: E_c = 3250 √f_ck + 14000 (MPa), and I = η · π D⁴ / 64 for one pile, spread over the spacing, EI per metre = E_c · I / s. The factor η (default 0.5) stands for the cracked section; EI feeds the beam-spring analysis.

• Bending, TS 500: the moment resistance M_Rd of the circular section with the bars on a ring is found from a rectangular stress block of 0.85 f_cd over the depth k₁·c (c the neutral axis depth), ultimate concrete strain 0.003 and elastic–perfectly plastic bars, the neutral axis from axial equilibrium for bending only. The weaker of two bar orientations governs. The check is

  γ_load · M_max · s  ≤  M_Rd
  

• Shear and links, TS 500: the section is taken as a rectangle of width D and effective depth d = 0.8 D (the usual rule for circular sections, as in ACI 318). With f_ctd = 0.35 √f_ck / γ_c:

  V_cr  = 0.65 f_ctd b d
  V_c   = 0.8 V_cr
  V_max = 0.22 f_cd b d
  V_w   = ( A_sw / s_link ) · f_ywd · d        (two legs)
  

  The check reports section too small (V_d > V_max), links below the minimum (A_sw / s_link < 0.3 f_ctd b / f_ywd), links too widely spaced (more than d/2 or 200 mm) or links insufficient (V_d > V_c + V_w), and gives the link spacing required. The shear is the largest of the analysis, without a reduction near the support.

6.3 Vertical equilibrium (indicative)

The vertical components of the anchors, ΣT_h tan θ, are compared with the skin friction of the embedded length, ∫ ( p_a + p_p ) tan δ dz; end bearing is neglected. It is an indication, not a bearing capacity check.

7. What is not covered

• Axial load, buckling and second-order effects of the wall; the capping beam and the arching of soil between the piles; the opening between piles and the water that passes through them.
• Crack width, durability, and the seismic detailing of the links (TBDY). Check the clause details of the governing code before using the numbers for a design.
• Overall (global) stability of the wall and the soil mass, base heave and piping: use SlipSurface IQ LE and a seepage analysis.
• The moment resistance of the circular section is bending only: with an axial force it changes.
• The soil is a stack of horizontal layers with Coulomb/Mononobe–Okabe pressures on a vertical wall; for a wall that is not vertical, or with a complicated surface, the coefficients do not apply.

References

• Coulomb, C.A. (1776). Essai sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs à l'architecture. Mém. Math. Phys. Acad. Roy. Sci. 7, 343–382.
• Okabe, S. (1926). General theory of earth pressure. J. Japan Society of Civil Engineers 12 (1).
• Mononobe, N. & Matsuo, H. (1929). On the determination of earth pressure during earthquakes. Proc. World Engineering Congress, Tokyo, 9, 177–185.
• Westergaard, H.M. (1933). Water pressures on dams during earthquakes. Trans. ASCE 98, 418–433.
• Ménard, L. & Bourdon, G. (1964) and Schmitt, G. (1995): the subgrade modulus of a wall from the pressuremeter modulus, as in the formulae of §4.
• FHWA (1999). Geotechnical Engineering Circular No. 4: Ground Anchors and Anchored Systems.
• Bowles, J.E. (1996). Foundation Analysis and Design, 5th ed. McGraw-Hill.
• TS 500 (2000). Requirements for Design and Construction of Reinforced Concrete Structures. Turkish Standards Institution.