SlipSurface IQ Wall — theory

What the program computes, with the expressions it uses and where they come from. Units are kN, m and kPa throughout (mm, mm², kNm and MPa in the concrete design); every force is per metre run of wall.

References: Das, Principles of Foundation Engineering; Bowles, Foundation Analysis and Design; EN 1997-1 (Eurocode 7) §9 and Annex D; EN 1998-5 §7 and Annex E; TBDY 2018, Türkiye Bina Deprem Yönetmeliği, §2 (spectra), §16.7 (shallow foundations) and §16.12 (retaining structures); TS 500 (2000), Betonarme Yapıların Tasarım ve Yapım Kuralları; Mononobe & Matsuo (1929), Okabe (1926); Seed & Whitman (1970); Wood (1973); Bishop (1955); Fellenius (1936); Vesić (1973), Meyerhof (1963), Brinch Hansen (1970), Terzaghi (1943) for the bearing capacity factors (as in SlipSurface IQ Bearing).

1. Geometry

The front edge of the footing, at its underside, is the origin; x runs towards the backfill, y up. H is the height from the underside of the footing to the top of the stem. The footing is B = toe + t_bot + heel wide and t_f thick; the stem rises Hs = H − t_f from it, t_bot thick at its base and t_top at its top, one face vertical and the other inclined. A shear key of depth k_d and width k_w hangs below the base. The ground in front stands D above the top of the footing (the embedment is D + t_f); the backfill starts at the top of the stem and slopes at β.

2. Earth pressure

The earth pressure acts on the virtual back, the vertical plane through the end of the heel, of height h = H + (B − x_back,top)·tan β. The soil between the stem and that plane moves with the wall and counts as its weight.

method	coefficient	the thrust acts at
Rankine	Ka = cos β·(cos β − √(cos²β − cos²φ)) / (cos β + √(cos²β − cos²φ))	β (parallel to the surface)
Coulomb	Ka = sin²(θ + φ) / [Γ·sin²θ·sin(θ − δ)], Γ = [1 + √(sin(φ+δ)·sin(φ−β) / (sin(θ−δ)·sin(θ+β)))]², θ = 90°	δ
at rest	K0 = (1 − sin φ)·√OCR·(1 + sin β) (EN 1997-1 eq. 9.2)	β
trial wedge	the largest P over planes through the heel at α ∈ (β, 90°): W + Q on the wedge, R at φ from the plane's normal, P at δ	δ

The trial wedge is solved numerically: 720 planes, then a golden-section refinement. It reproduces Coulomb's coefficient (the tests hold it to 2·10⁻⁴) and returns the critical plane.

The pressure at a height y is p = K·σ′v(y) (+ K·q for a surcharge), with σ′v the effective vertical stress under the ground at the virtual back: γ above the water table behind the wall, γsat − γw below it. Rankine's pressure takes the cohesion of the backfill, p = K·σ′v − 2c·√K, never below zero (a tension crack is not counted on); the other methods leave the cohesion out, and the program says so. The water pushes hydrostatically, u = γw·(hw − y). The resultants and their heights are integrated numerically over 241 points.

Passive resistance in front, from the ground (D + t_f above the base) down to the base, or to the bottom of the key:

pp = Kp·σ′v + 2c·√Kp,   Kp = tan²(45 + φ/2) (Rankine)  or  Coulomb's Kp with δ ≤ φ/2
                         and the ground in front at −β_toe

Only a share of it (entered) is counted, since it needs movement to develop.

3. Forces and stability

Vertical forces: the stem, the footing and the key (γc), the soil on the heel and on the toe (γ above the water, γsat below), the permanent surcharge over the heel, a line load on the stem, the vertical components of the thrusts at x = B, the uplift

U = γw·(hf + hw)/2·B   at   x = B·(hf + 2hw) / (3·(hf + hw))

Horizontal forces: the thrusts of the soil and of the surcharges, the water behind, a load at the top of the stem; the water in front resists. The live surcharge over the heel counts as a weight only for the bearing capacity (it never helps sliding or overturning).

sliding:      FS = [V·tan(k_b·φ) + k_a·c·B + share·Pp] / ΣH
overturning:  FS = M_R / M_O       (about the toe; uplift and upward inertia overturn)
eccentricity: e = B/2 − (M_R − M_O)/V ≤ B/6 (static), B/3 (seismic)

k_b = δb/φ (2/3 usual; EN 1997-1 6.5.3(10) allows 1 for concrete cast against the soil) and k_a = ca/c are entered. The base pressure is

q = V/B·(1 ± 6e/B)              e ≤ B/6
q_max = 2V / (3·(B/2 − |e|))    beyond, over a length 3·(B/2 − |e|)

4. Bearing capacity

A strip of Meyerhof's effective width B′ = B − 2|e| under σv = ΣV/B′:

q_ult = c·Nc·dc·ic·gc + q·Nq·dq·iq·gq + ½·γ′·B′·Nγ·dγ·iγ·gγ

with the factor sets of Terzaghi, Meyerhof, Brinch Hansen, Vesić and EN 1997-1 (SlipSurface Bearing's, unchanged), q = σ′v at the base from the ground in front (when the embedment is counted), the load inclination from ΣH/ΣV, the depth factors when asked, the ground factors of a slope in front, and γ′ reduced for a water table within B′ below the base. FS = q_ult/σv. An allowable soil pressure from the geotechnical report, if given, is compared with q_max.

5. Global stability

Circles through the soil passing under the footing and the key, coming out in front of the wall and in the backfill. The mass above a circle is cut into 60 vertical slices; the weight of a slice is everything above its base (foundation soil, concrete, backfill, the soil in front, saturated below the water table, the surcharge). The pore pressure at the base comes from a water table that runs from the water in front to the water behind (or lies water_depth below the base).

Bishop:     F = Σ [c·b + (W − u·b)·tan φ] / mα  /  Σ [W·sin α + kh·W·(yc − yg)/R]
            mα = cos α + sin α·tan φ / F   (not below 0.2)
Fellenius:  F = Σ [c·l + (W·cos α − kh·W·sin α − u·l)·tan φ]  /  the same sum

The centres are searched on a 13 × 11 grid with 10 tangent depths below the base, refined three times around the lowest Bishop factor. Fellenius' factor is reported on Bishop's critical circle and on its own; the verdict is Bishop's.

6. Earthquake — TBDY 2018

Fs, F1   from Tables 2.1 and 2.2 by the site class, linear between the tabulated Ss, S1
SDS = Ss·Fs,  SD1 = S1·F1,  TA = 0.2·SD1/SDS,  TB = SD1/SDS,  TL = 6 s
Sae(T) = (0.4 + 0.6·T/TA)·SDS  |  SDS  |  SD1/T  |  SD1·TL/T²
kh = βr·0.4·SDS,   kv = ratio·kh   (up and down)

βr is 1 for a wall that cannot move (EN 1998-5 Table 7.1 puts flexural reinforced concrete walls there), 0.67 or 0.5 when a displacement of 200 or 300·(0.4·SDS) mm is acceptable; a kh can also be given directly. Ss and S1 come from the AFAD hazard map for the site.

The dynamic increment of the thrust on the virtual back:

Mononobe–Okabe:  ΔP = (1 − kv)·KAE·(S + q·h) − K·(S + q·h),   S = ∫σ′v dy (½γh² when dry)
                 KAE = cos²(φ − ψ) / {cos ψ·cos(δ + ψ)·[1 + √(sin(φ+δ)·sin(φ−ψ−β) / (cos(δ+ψ)·cos β))]²}
                 ψ = atan(kh / (1 − kv))
Seed–Whitman:    ΔKAE = ¾·kh
Wood:            ΔP = kh·γ·h²   (an unyielding wall, EN 1998-5 E.9)

acting at the chosen share of h (½ by EN 1998-5 7.3.2.3; 0.6 by Seed & Whitman), as a linear pressure whose resultant sits there. Where φ − ψ − β < 0 Mononobe–Okabe has no real solution; its limiting value is used and a warning says the backfill itself is at failure. The wall, the soil on its heel, the permanent surcharge and the line load carry kh·W horizontally at their centroid and kv·W vertically; the passive resistance becomes Mononobe–Okabe's KPE; the live load is kept with its share ψ. Every check is made with kv up and with kv down, and the worse governs. The required factors are those of TBDY 2018 §16.7 for shallow foundations — sliding γRh = 1.1, bearing γRv = 1.4 — and the others entered.

7. Reinforced concrete — TS 500

Materials: fcd = fck/1.5, fyd = fyk/1.15, fctk = 0.35·√fck, fctd = fctk/1.5, k1 = 0.85 up to C25, less 0.006 per MPa above, not below 0.70; ρb = 0.85·k1·(fcd/fyd)·600/(600 + fyd).

Combinations (TS 500 6.2.6 and TBDY 2018 4.4.4): C1 = 1.4G + 1.6Q + 1.6H, C2 = 0.9G + 1.6H, C3 = G + Q + H + E and C4 = 0.9G + H + E, the last two with kv up (+) and down (−). G are the weights, Q the live loads (the live surcharge's thrust included, ψ·Q in an earthquake), H the earth pressure, the water at 1.4 (1.0 with the earthquake), E the earthquake.

Stem — a cantilever from the footing under the pressure on a vertical back of its own height Hs (the design method's coefficient, horizontal component), the water, the surcharges, a load at its top, and in an earthquake the dynamic increment over Hs and its own inertia kh·γc·t(y). V(y) and M(y) are integrated from the top down and enveloped.

Toe and heel — cantilevers from the faces of the stem. For each combination the factored forces give V and e, the contact pressure under the base follows (trapezoid or triangle), and

M_toe  = ∫₀^toe (q + u − w)·(toe − x) dx                      (bottom tension +)
M_heel = ∫_heel (w − q − u)·(x − x_stem) dx + P_v·heel        (top tension +)

w being the slab, the soil and the surcharge on it and P_v the vertical components of the thrusts at the end of the heel. A combination whose resultant leaves the base (less than a tenth of it in contact) is not one the wall survives; the stability checks report it, and the footing is designed for the others.

Shear key — a cantilever under the full passive pressure in front of it, ×1.6.

Sections, one metre wide:

a  = d − √(d² − 2·Md / (0.85·fcd·b)),   As = 0.85·fcd·b·a / fyd
As ≥ ρmin·b·h,   As/(b·d) ≤ min(0.85·ρb, 0.02)
Vd ≤ Vcr = 0.65·fctd·b·d,   Vd ≤ 0.22·fcd·b·d        (no stirrups)

Vd is taken at d from the face of the stem for the stem and the toe, at the face for the heel (a support in tension). Every diameter from Ø12 is tried with its own d and the spacings 100…300 mm; the layout with the least steel that suffices and keeps the spacing within min(1.5·h, 200 mm) is chosen (or the diameter given). The distribution steel is at least a fifth of the main steel and ρdist·b·h over both faces, no further apart than 300 mm; the front face of the stem has at least ρfront·b·h.

Curtailment — half the bars of the back face of the stem stop where the capacity Mr of the other half (with the depth d(y) of the tapering stem) covers the envelope for good, extended by max(d, 12φ), and never below lb above the footing. It is done only if the remaining bars are still within the spacing limit and above ρmin there; if the chosen spacing does not allow it, the bars at half the widest spacing are tried.

Anchorage — lb = 0.12·(fyd/fctd)·φ ≥ 20·φ (TS 500 eq. 9.2), ×1.4 for top bars with more than 300 mm of concrete below; the straight length available beyond each critical section is reported, and the ends are hooked.

8. Bars, drawing and schedule

Positions: (1) the stem's back face, full height, and (2) curtailed, each with a foot on the bottom mesh towards the heel; the stem's front face with a foot towards the toe; horizontal bars on both faces; the bottom and top transverse bars of the footing with hooks at both ends; the longitudinal bars of the footing; the U-bars and the corner bars of the key. Hooks are 12·φ, at least 0.20 m, within the section. The schedule gives, per metre of wall, the diameter, the spacing, the shape with its segment lengths, the length of one bar, the number, the total length and the mass (7850 kg/m³); the quantities give the concrete, the steel and their ratio.

The drawing is one list of primitives (polylines, circles, texts on layers) drawn by Matplotlib and written as an AutoCAD R12 DXF in millimetres, the diameter sign as %%c and the texts in the ANSI 1254 (Turkish) code page.

9. The heel the wall needs, and the first dimensions

The heel is found by bisection, from 0 to 3·H, as the shortest one for which every stability and bearing check holds, static and seismic. The first dimensions (Suggest dimensions) are the textbook proportions: stem 0.25–0.30 m at the top and H/10 at the base, footing H/10 thick (≥ 0.40 m), base 0.6·H wide (more under a slope), a third of it toe, all rounded to 5 cm.

10. Not included

Settlement; the drainage of the backfill and the pressure of water that cannot drain; the hydrodynamic pressure of free water in an earthquake; the vertical component of the seismic thrust in the slip circles (kv); joints, lap splices and the reinforcement along the wall beyond the distribution steel; crack widths. Check them separately.