Stage 6 / bearing capacity

Stage 6 bearing capacity.

The bearing-capacity route is a shallow-foundation ULS screen for vertical loading on the interpreted CPT section. It evaluates drained and undrained resistance separately, applies effective-dimension logic for eccentricity, builds Brinch-Hansen shape / depth / inclination factors from the effective dimensions, averages the unit weight across the water table for the Nγ term, and reports Belgian DA1-style design envelopes.

Scope and intent

1. Problem class and engineering purpose

The bearing route is a shallow-foundation ULS screen. It evaluates the resistance side of the bearing problem from CPT-derived soil parameters and reports drained and undrained resistance envelopes versus founding depth.

Model class. Brinch Hansen / EC7 Annex D screening for vertical loading. Not a full general-footing check with sliding, uplift, ground slope, or base tilt. Those live in dedicated structural-geotechnical tools.
Primary quantities
qult,d
Ultimate drained bearing resistance [kPa].
qult,u
Ultimate undrained bearing resistance [kPa].
B′, L′
Effective footing dimensions after eccentricity reduction [m].
Nc, Nq, Nγ
Bearing-capacity factors [-].

Theory

2. Prandtl–Reissner slip-line derivation

The drained bearing capacity is rooted in the Prandtl–Reissner plasticity solution for a weightless, cohesive, frictional half-space under a rigid strip footing. Slip lines partition the soil under the footing into three zones: a triangular active Rankine zone under the footing, a logarithmic-spiral Prandtl transition zone, and a triangular passive Rankine zone on each side. Equating the moments of the active and passive zones about the spiral centre gives the bearing expression with two of the three factors.

qult,weightless = c′ Nc + q′ Nq
Active zone angle: π/4 + φ′/2 (Rankine limit)
Passive zone angle: π/4 − φ′/2
Prandtl spiral: r(θ) = r0 exp(θ tan φ′)
Nq = exp(π tan φ′) · tan²(π/4 + φ′/2)
Nc = (Nq − 1) / tan φ′

The third factor, Nγ, accounts for the self-weight of the soil inside the failure mechanism. Prandtl–Reissner cannot express it in closed form because adding self-weight destroys the kinematic separability that makes the spiral solution tractable. Several empirical closed forms are in circulation (Meyerhof, Vesić, Brinch Hansen); the app uses the Meyerhof-style expression below and flags the choice in the output.

Nγ,Meyerhof = (Nq − 1) tan(1.4 φ′)
Nγ,Vesić = 2(Nq + 1) tan φ′ (alternative, reported as comparator)
Nγ,EC7-Annex-D (rough base) = 2(Nq − 1) tan φ′
  • For φ′ → 0, Nq → 1 and Nγ → 0; Nc takes the limit by L'Hôpital and approaches the undrained Prandtl value 5.14.
  • The drained form is exact for the cohesion and surcharge terms; the self-weight term is an empirical closure.

Factors

3. Undrained bearing factor and Prandtl's 5.14

For saturated clays loaded faster than pore-pressure dissipation, the strength is taken as an undrained shear strength cu. The same slip-line mechanism with φ′ = 0 collapses to Prandtl's original 1921 result: a kinematically admissible punching mechanism with spiral angle 2π and radius 2 r0, yielding the classical Nc,u = π + 2 ≈ 5.14.

qult,u = q + (π + 2) cu scu dcu icu
(π + 2) ≈ 5.14 (Prandtl 1921 result for φ′ = 0)

This is the limiting case and the reason drained and undrained routes are reported separately: a φ′ → 0 drained run approaches 5.14 cu + q only if c′ → cu and the depth / shape / inclination factors coincide. In layered CPT profiles, drained and undrained envelopes can cross over at different depths, so the engineer interprets them side-by-side rather than as a single mixed envelope.

Model

4. Complete drained and undrained expressions

qult,d = c′ Nc sc dc ic + q′ Nq sq dq iq + 0.5 γ′B B′ Nγ sγ dγ iγ
qult,u = q + (π + 2) cu scu dcu icu
Notation
c′
Effective cohesion [kPa].
φ′
Effective friction angle [°].
q′
Effective surcharge at foundation level [kPa].
γ′B
Effective unit weight used in the Nγ term [kN/m³]. Averaged across the water table where present — see §8.
cu
Undrained shear strength [kPa].
s*, d*, i*
Shape, depth, and inclination factors for each resistance term.

Eccentricity

5. Effective dimensions after load eccentricity

An eccentric vertical load is equivalent to a centred load on a reduced footing. The "effective dimensions" (Meyerhof 1953) are the plan sides of the largest centred rectangle such that the centred load on that rectangle gives the same maximum contact pressure as the original eccentric load.

eB = MB/V, eL = ML/V
B′ = B − 2 eB, L′ = L − 2 eL
r = B′/L′ with B′ ≤ L′
  • The app requires |eB| < B/6 and |eL| < L/6 to avoid tensile reactions under the footing corner; larger eccentricity is flagged.
  • Shape, depth, and inclination factors are evaluated with B′ and L′, not with the nominal plan.

Shape and depth factors

6. Brinch Hansen factors on effective dimensions

sq = 1 + r sin φ′
sc = (sq Nq − 1) / (Nq − 1)
sγ = max(0.6, 1 − 0.3 r)
scu = 1 + 0.2 r
η = Df / B′, k = η for η ≤ 1, k = atan(η) for η > 1
dq = 1 + 2 tan φ′ (1 − sin φ′)² k
dc = dq − (1 − dq)/(Nq tan φ′) for φ′ > 0
dγ = 1.0
dcu = 1 + 0.4 k
Notation
Df
Foundation embedment depth below ground level [m].
η
Embedment ratio Df/B′ [-].
k
Auxiliary depth-factor parameter [-], switching at η = 1 to cap k growth.
Transparency note. The public route can also be switched to a conservative shape-factor mode with all shape factors set to 1.0. That is a deliberate screening choice, not a second theory — it returns the strip-centreline envelope on rectangular geometry.

Inclination

7. Inclination factors for horizontal load on the base

Horizontal load on the footing base reduces the bearing capacity of each term. The Brinch Hansen form expresses the reduction in closed-form in terms of the horizontal / vertical load ratio and a geometric exponent. The app uses it only when the engineer enters a non-zero horizontal action on the footing.

iq = (1 − H/(V + A′ c′ cot φ′))m
iγ = (1 − H/(V + A′ c′ cot φ′))m+1
ic = iq − (1 − iq)/(Nq − 1) for φ′ > 0
icu = 0.5 (1 + √(1 − H/(A′ cu)))
Exponent m for load in B-direction: mB = (2 + B′/L′) / (1 + B′/L′)
Exponent m for load in L-direction: mL = (2 + L′/B′) / (1 + L′/B′)
General bi-axial load: m = mL cos²θ + mB sin²θ
Notation
H, V
Horizontal and vertical components of the footing load [kN].
A′
Effective plan area B′L′ [m²].
θ
Azimuth of the horizontal load component, from the B-direction [rad].
  • The undrained form icu is only valid while H ≤ A′ cu (the sliding limit on the footing base).
  • If H / V exceeds the angle-of-friction limit tan φ′, no bearing equation applies and the engineer must switch to a combined bearing + sliding check.

Unit-weight averaging

8. Three-case water-table averaging for the Nγ term

The Nγ term contains the soil unit weight inside the failure wedge, which extends about one B below the footing. When the water table intersects that wedge, the appropriate "effective" γ is a spatial average of dry and submerged unit weights over the wedge depth. The app resolves this with three canonical cases depending on where the water table sits relative to the footing and the wedge bottom.

zw ≤ Df (WT above footing): γ′B = γsat − γw
Df < zw ≤ Df + B (WT within wedge): γ′B = γ (zw − Df)/B + (γsat − γw)(Df + B − zw)/B
zw > Df + B (WT deep): γ′B = γ
Notation
zw
Water-table depth below ground level [m].
γ, γsat
Dry and saturated unit weight of the soil inside the wedge [kN/m³].
γw
Unit weight of water, 9.81 kN/m³.
  • The surcharge q′ in the Nq term uses the effective vertical stress at the foundation base, consistent with the standard EC7 convention.
  • Transient drawdown during excavation (if modelled in the dewatering route) raises the effective surcharge and can increase qult,d. The app exposes a coupled run for that sensitivity.

Design route

9. Belgian DA1/1 and DA1/2 handling

For Belgian EC7 practice, the app distinguishes DA1/1 and DA1/2 through resistance-side strength reduction. DA1/1 uses the unfactored M1 soil-strength set; DA1/2 reduces strength through the M2 soil-strength set. A resistance-side factor γRd scales the computed qult to a design resistance.

tan φ′d = tan φ′k / γφ, γφ = 1.00 (M1), 1.25 (M2)
c′d = c′k / γc, γc = 1.00 (M1), 1.25 (M2)
cu,d = cu,k / γcu, γcu = 1.00 (M1), 1.40 (M2)
qd = qult(φ′d, c′d, cu,d) / γRd
  • The Belgian ANB γRd uses 1.40 for drained bearing on shallow footings; see NBN EN 1997-1 ANB Table A.NB.5.
  • DA1/1 and DA1/2 envelopes are reported separately; the governing one is the minimum.
  • Action-side partial factors (γG, γQ) are not applied here — they belong in the structural load combination producing qgross.

Deformation cross-check

10. Relation to the Stage 2 deformation safety analysis

The bearing route gives a collapse-load estimate from slip-line theory. The Stage 2 deformation solver gives a strength-reduction factor from a finite-element c-phi analysis on the same CPT section. The two should be consistent: for the same geometry and strengths, the deformation solver's factor of safety times the applied load should equal approximately qult,d.

  • When the two envelopes disagree by more than ~20%, the most common explanation is strength or stiffness layering the slip-line form ignores but the finite element captures.
  • The engineer should run both routes on critical founding depths and compare.
  • See the Stage 2 MC chapter for the finite-element theory.

Outputs and limitations

11. Deliverables and boundaries of validity

  • Depth-wise drained and undrained resistance envelopes; DA1/1 and DA1/2 design envelopes.
  • Effective-dimension summary, shape / depth / inclination factor breakdown.
  • Nγ term's water-table case (above footing / in wedge / below wedge).
  • Ground slope, base tilt, dynamic or seismic inclination, uplift, and sliding are not in the current route.
  • Piled or combined-pile-raft foundations are outside the shallow-foundation screen.
  • For two-layer soils where a stiff stratum is within 2B of the footing base, the slip-line solution can over-estimate capacity; the app flags the configuration for the engineer to reconsider.
  • The deeper audit trail remains in the full technical specification.

References

References

  • Prandtl, L. (1921). Über die Eindringungsfestigkeit plastischer Baustoffe und die Festigkeit von Schneiden. Z. Angew. Math. Mech. 1(1), 15–20. Origin of the Nc = π + 2 result.
  • Reissner, H. (1924). Zum Erddruckproblem. Proc. 1st Int. Congr. Appl. Mech., Delft, 295–311. Extension of Prandtl to weightless frictional soil.
  • Meyerhof, G. G. (1953). The bearing capacity of foundations under eccentric and inclined loads. Proc. 3rd ICSMFE, Zurich, Vol. 1, 440–445.
  • Brinch Hansen, J. (1970). A revised and extended formula for bearing capacity. Bull. Danish Geotech. Inst. 28, 5–11. Source of the shape/depth/inclination factors.
  • Vesić, A. S. (1973). Analysis of ultimate loads of shallow foundations. J. Soil Mech. Found. Div. ASCE 99(SM1), 45–73.
  • EN 1997-1:2004+A1:2013 — Eurocode 7, Part 1. Annex D.
  • NBN EN 1997-1 ANB:2022 — Belgian National Annex to Eurocode 7.
  • EN 1990:2002+A1:2005 — Basis of structural design.