Foundation Engineering

Soil Bearing Capacity: Terzaghi, Meyerhof, Hansen & Vesic Theories, All Formulas & Design Guide

Soil bearing capacity overview – Terzaghi Meyerhof Hansen Vesic theories comparison
Fig. 1: Overview of the four bearing capacity theories showing their evolution, key assumptions, and applicable conditions for shallow foundation design.

The complete all-in-one reference: every bearing capacity formula with full derivation, all correction factors, $N_c$/$N_q$/$N_\gamma$ tables for all four theories, three failure modes, groundwater corrections, SPT/CPT/PLT field methods, net vs gross vs allowable pressure, IS 6403/Eurocode 7/AASHTO standards, settlement criteria, complete worked example, and live calculator.

All 4 Theories Full Correction Factors Worked Example IS 6403 / Eurocode 7
By Bimal Ghimire • Published July 20, 2024 • 30 min read
At a Glance

Key Numbers: Why Bearing Capacity Matters

Four quick visual snapshots before diving in — typical allowable pressures by soil type, the dramatic effect of groundwater, how phi drives all BC factors, and why settlement usually governs over shear.

Typical Allowable BC by Soil Type (kPa)
GWT Impact on qu — Medium Dense Sand (φ=30°)
Nq Growth with Friction Angle — All Theories
Shear Capacity vs. Settlement-Governed qa (kPa)
Core Concepts

Bearing Capacity: Every Term Precisely Defined

Soil bearing capacity is the ability of soil to safely carry the loads transferred to it from a foundation without shear failure of the soil mass or excessive settlement of the structure. It is expressed as pressure (kPa, kN/m², or tsf). Two criteria must always be checked independently: (1) shear failure, and (2) settlement. The more critical governs the final footing size.

1. Gross Ultimate Bearing Capacity (qu)
$$q_u = \text{Total contact pressure at foundation base at which the soil fails in shear}$$
Calculated directly from Terzaghi, Meyerhof, Hansen, or Vesic equations. Includes the weight of soil above the footing level (overburden) as part of the total stress.
2. Net Ultimate Bearing Capacity (qnu)
$$q_{nu} = q_u - \gamma D_f$$
Subtracts the pre-existing overburden stress $q_0 = \gamma D_f$ at the foundation level. Represents the net additional stress beyond what already existed before construction. $\gamma$ = unit weight of soil above foundation base (kN/m³). $D_f$ = depth to foundation base (m).
3. Safe Bearing Capacity and Net Safe Bearing Capacity
$$q_s = \frac{q_u}{F_s} \qquad q_{ns} = \frac{q_{nu}}{F_s} = \frac{q_u - \gamma D_f}{F_s}$$
$F_s$ = factor of safety against shear failure, typically 2.5 to 3.0 per IS 6403. Provides a margin against shear collapse only; does not account for settlement.
4. Allowable Bearing Pressure (qa) — the Final Design Value
$$q_a = \min\!\left(\frac{q_{nu}}{F_s} + \gamma D_f,\; q_{\text{settlement-limited}}\right)$$
The governing design pressure: smaller of (a) net safe BC plus original overburden (shear criterion), and (b) the contact pressure that limits total/differential settlement to the permissible value. For most foundations on clay or loose/medium sand, settlement governs. Permissible settlements per IS 1904: isolated footing on sand: 25 mm total, 18 mm differential; on clay: 40 mm total, 25 mm differential.
2.5–3.0
FoS for shear failure (IS 6403)
25 mm
Max. total settlement, sand (IS 1904)
3
Shear failure modes
4
Major theories: Terzaghi / Meyerhof / Hansen / Vesic
Mechanism

Three Shear Failure Modes: Theory, Diagrams & Criteria

Three shear failure modes in soil: general shear, local shear, and punching shear failure diagrams with load-settlement curves
Fig. 2: The three shear failure modes under shallow foundations. Left: General shear failure in dense/stiff soil — clear failure surface, sharp load-settlement peak. Centre: Local shear in medium soil — incomplete failure zone, no clear peak. Right: Punching shear in loose/soft soil — vertical shear planes, no surface heave. (After Vesic, 1975)
FOOTING Zone IIIIIZone III (Rankine passive)
General Shear Failure

Dense/stiff soils (Dr > 67%, stiff clay). Continuous failure surface to ground; sharp load-settlement peak; heave on both sides. Terzaghi formula applies directly.

FOOTING Compression zoneFailure surface incomplete
Local Shear Failure

Medium soils (Dr 35–67%). Failure zone does not reach surface. Large settlement before failure; slight heave. Use reduced c* = 0.67c, tan(φ*) = 0.67tanφ in Terzaghi.

FOOTING Footing punchesNo surface heave
Punching Shear Failure

Very loose/soft soils (Dr < 35%) or deep footings. No defined failure surface; footing punches down; no heave. Vesic/Meyerhof depth factors correct for this.

Failure ModeSoil ConditionSPT N (sand)Dr (sand)Load-Settlement CurveSurface HeaveFormula Applicability
General ShearDense sand, stiff clay>30>67%Sharp peak, sudden dropPronounced both sidesTerzaghi, Meyerhof, Hansen, Vesic directly
Local ShearMedium dense sand, medium clay10–3035–67%Gradual; no clear peakSlightTerzaghi: c* = 0.67c; φ* = arctan(0.67tanφ)
Punching ShearLoose sand, soft clay; deep footings<10<35%Continuous settlement; no peakNoneMeyerhof/Vesic depth and compressibility factors
Influencing Parameters

Factors Affecting Soil Bearing Capacity

FactorSymbolEffect on quQuantitative ImpactDesign Note
Cohesionc (kPa)Direct proportional: c·Nc termEach +1 kPa cohesion adds Nc kPa; e.g. at φ=30°, Nc=30: each +1 kPa c adds 30 kPa to quUndrained φ=0: qu = 5.14cu (strip). Measure cu from UU triaxial or vane shear
Friction angleφ (degrees)Exponential via Nq and Nγφ: 30° to 35° doubles Nq; nearly triples NγUse drained φ' for long-term; undrained φu=0 for short-term saturated clay
Unit weightγ (kN/m³)Affects overburden q and Nγ·B termAbove GWT: 18–21; below GWT: buoyant γ' ≈ 8–11 kN/m³Use γ' below groundwater table in both q and γ·B terms
Foundation depthDf (m)Increases overburden q = γ×Df; Nq term growsDoubling Df roughly doubles q term contribution to quMin. Df per IS 1080: 0.5 to 1.0 m; deeper = higher capacity but higher cost
Foundation widthB (m)Increases Nγ×B term; for clay (φ=0): no effectSand: wider footing has higher qu. Clay: qu independent of BClay: increasing B improves settlement capacity; does not change unit bearing capacity
Footing shapeB/LShape factors sc, sq, sγ modify each termSquare vs strip: sc increases from 1.0 to 1.3; sγ decreases from 1.0 to 0.8Always apply shape factors for non-strip footings
Groundwater tabledw (m)Reduces effective unit weight below GWTGWT at footing level: qu reduced 35–55% for typical sand vs deep GWTThree correction cases based on GWT position relative to Df and B
Load inclinationα (deg)Inclination factors ic, iq, iγ reduce quα=20°: capacity reduced 30–60% depending on soilApply for wind, seismic, earth pressure on basement walls
Ground slopeβ (deg)Ground inclination factors gc, gq, gγ (Hansen)10° slope: reduces qu 15–30%Footings on embankments or near excavations
Theory 1 (1943)

Terzaghi's Bearing Capacity Theory: Full Equations, Factors & Mechanism

Karl Terzaghi (1943) was the first to propose a rigorous analytical method for the ultimate bearing capacity of a shallow footing. His assumptions: (1) general shear failure; (2) soil above the foundation base acts as a surcharge only ($q = \gamma D_f$, not as soil with shear strength); (3) rough footing base; (4) failure mechanism has three zones: Zone I (elastic wedge at angle φ beneath footing), Zone II (radial shear zone, logarithmic spiral boundary), Zone III (Rankine passive zone). The condition Df ≤ B defines a shallow footing in Terzaghi's original framework.

Terzaghi bearing capacity failure mechanism showing Zone I elastic wedge Zone II radial shear and Zone III Rankine passive zone beneath a strip footing
Fig. 3: Terzaghi's (1943) three-zone failure mechanism for a strip footing on cohesive-frictional soil. Zone I (elastic wedge) moves with the footing; Zone II (radial fan) has a logarithmic spiral upper boundary; Zone III is a Rankine passive zone. (After Terzaghi, 1943)
Terzaghi Strip Footing (B/L → 0)
$$q_u = c\,N_c + q\,N_q + \tfrac{1}{2}\,\gamma B\,N_\gamma$$
$c$ = cohesion (kPa). $q = \gamma D_f$ = overburden at footing base (kPa). $\gamma$ = unit weight of soil below footing base (kN/m³). $B$ = footing width (m). $N_c$, $N_q$, $N_\gamma$ = dimensionless bearing capacity factors, functions of φ only.
Terzaghi Shape Factor Equations for Other Shapes
$$q_u(\text{square}) = 1.3\,c\,N_c + q\,N_q + 0.4\,\gamma B\,N_\gamma$$
$$q_u(\text{circular, diameter }B) = 1.3\,c\,N_c + q\,N_q + 0.3\,\gamma B\,N_\gamma$$
$$q_u(\text{rect.}) = \!\left(1+0.3\tfrac{B}{L}\right)\!c\,N_c + q\,N_q + \!\left(1-0.2\tfrac{B}{L}\right)\!\tfrac{1}{2}\gamma B\,N_\gamma$$
$B$ = short side, $L$ = long side. When $B/L = 1$ (square), rect. formula gives same result as square formula. The Nq term carries no shape factor in Terzaghi's original formulation.
Terzaghi's Bearing Capacity Factor Expressions
$$N_q = \frac{e^{2\pi(0.75 - \phi/360)\tan\phi}}{2\cos^2(45+\phi/2)}$$
$$N_c = (N_q - 1)\cot\phi \quad (\phi>0); \qquad N_c = 5.7 \text{ for } \phi = 0$$
$$N_\gamma \approx 2(N_q+1)\tan\phi \quad \text{(Meyerhof 1963 approximation)}$$
Terzaghi used Nc = 5.7 for φ = 0; the more accurate Prandtl value is 5.14 (π + 2), used by Meyerhof, Hansen, and Vesic. Note: Terzaghi's original Nγ was obtained graphically; the Meyerhof (1963) approximation is widely used.
Local Shear Modification (Terzaghi) — for Loose Sand and Soft Clay
$$c^* = \tfrac{2}{3}c; \qquad \phi^* = \arctan\!\left(\tfrac{2}{3}\tan\phi\right)$$
For loose sands and soft clays where local shear governs: replace c with c* and φ with φ* in all Terzaghi equations, then look up Nc*, Nq*, Nγ* from tables at φ*. Criterion: local shear occurs when Dr = 35–67% (sand) or SPT N = 10–30.
φ (°)Nc (Terzaghi)NqNγ (approx.)Soil Description
05.710Saturated clay, undrained (φ=0)
57.341.570.07Soft clay
109.612.470.37Firm clay
1512.863.941.1Medium clay
2017.696.42.87Stiff clay / very loose sand
2525.1310.666.77Loose to medium sand
2831.6114.7211.19Medium sand
3037.1618.415.1Medium-dense sand
3244.0423.1820.79Dense sand
3557.7533.333.92Very dense sand / dense gravel
3873.0348.9352.19Dense gravel
4095.6764.273.4Very dense gravel
45172.29134.87192Rock-like dense gravel
Theory 2 (1951, 1963)

Meyerhof's Theory: Shape, Depth & Inclination Factors

G.G. Meyerhof (1951, 1963) extended Terzaghi's formula by rigorously incorporating shape factors, depth factors, and inclination factors into a unified general equation. His $N_q$ and $N_c$ are derived analytically from the Prandtl logarithmic spiral mechanism and are considered more accurate than Terzaghi's. Meyerhof's method is the basis for most North American practice (FHWA GEC No.6, AASHTO). A key innovation: Meyerhof extended the failure surface above the footing base, accounting for the shear strength of soil above footing level — which Terzaghi ignored.

Meyerhof's General Bearing Capacity Equation
$$q_u = c\,N_c\,s_c\,d_c\,i_c \;+\; q\,N_q\,s_q\,d_q\,i_q \;+\; \tfrac{1}{2}\,\gamma B\,N_\gamma\,s_\gamma\,d_\gamma\,i_\gamma$$
Every term carries three correction factors: shape (s), depth (d), inclination (i). For a strip footing with vertical central load: all nine correction factors = 1.0 and the equation reduces to the same form as Terzaghi's strip equation.
Meyerhof's Bearing Capacity Factors (Prandtl-based)
$$N_q = e^{\pi\tan\phi}\,\tan^2\!\left(45 + \tfrac{\phi}{2}\right)$$
$$N_c = (N_q - 1)\cot\phi \quad (\phi>0); \qquad N_c = 5.14 \text{ for } \phi=0$$
$$N_\gamma = (N_q - 1)\tan(1.4\phi)$$
Meyerhof's Nc = 5.14 for φ = 0 is the exact Prandtl (1920) solution: Nc = π + 2 = 5.14. More accurate than Terzaghi's 5.7. The Nq and Nc expressions are now standard across all four theories; only Nγ differs between methods.
Shape Factors (Meyerhof)Depth Factors (k = Df/B)Inclination Factors
Factorφ = 0φ > 10°Factorφ = 0φ > 10°FactorExpressionNotes
sc1 + 0.2 B/L1 + 0.2(B/L)Nφdc1 + 0.2(Df/B)1 + 0.2k√Nφic(1 − α/90)²α = load inclination from vertical (deg)
sq1.01 + 0.1(B/L)Nφdq1.01 + 0.1k√Nφiq(1 − α/90)²Same as ic for φ > 0
sγ1.01 + 0.1(B/L)Nφdγ1.01 + 0.1k√Nφiγ(1 − α/φ)²iγ = 0 when α ≥ φ
Nφ = tan²(45 + φ/2). For square/circle: B/L = 1. For strip: B/L = 0 (all shape factors = 1.0).
Theory 3 (1970)

Hansen's Method: Full Factor Set Including Base & Ground Slope

J. Brinch Hansen (1970) published the most comprehensive bearing capacity formulation. Beyond Meyerhof's factors, he added base inclination factors ($b_c$, $b_q$, $b_\gamma$) for tilted footing bases, and ground inclination factors ($g_c$, $g_q$, $g_\gamma$) for footings on sloping ground. Hansen's depth factors differ from Meyerhof's and are more accurate for deep footings (Df/B > 1). Hansen's method is the basis of Eurocode 7 Annex D and British Standard BS 8004.

Hansen's Complete Equation
$$q_u = c\,N_c\,s_c\,d_c\,i_c\,b_c\,g_c + q\,N_q\,s_q\,d_q\,i_q\,b_q\,g_q + \tfrac{1}{2}\,\gamma B\,N_\gamma\,s_\gamma\,d_\gamma\,i_\gamma\,b_\gamma\,g_\gamma$$
Hansen's Nc and Nq are the same as Meyerhof's (Prandtl-based). His Nγ = 1.5(Nq − 1)tan(φ), slightly smaller than Vesic's. For simple vertical central load on level ground: all b, g, i correction factors = 1.0.
Factor GroupFactorExpressionNotes / Conditions
Shapesc1 + (Nq/Nc)(B/L) for φ>0;  1 + 0.2 B/L for φ=0Square: B/L=1; strip: B/L=0
sq1 + (B/L)tanφφ > 0 only
sγ1 − 0.4(B/L)Minimum 0.6; strip: 1.0
Depth (k = Df/B if ≤1; k = arctan(Df/B) rad if >1)dc1 + 0.4k for φ>0;  1 + 0.4arctan(Df/B) for φ=0More accurate than Meyerhof for deep footings
dq1 + 2tanφ(1−sinφ)² kSame k definition
dγ1.0 (always)Hansen sets depth factor for γ term to 1.0
Inclination (QH = horiz. load; m = (2+B/L)/(1+B/L))iciq − (1−iq)/(Nq−1) for φ>0;  1 − m·QH/(A·c·Nc) for φ=0Must satisfy QH < QVtanφ + cA
iq(1 − QH/(QV + Ac·cotφ))mUse mB or mL depending on load direction
iγ(1 − QH/(QV + Ac·cotφ))m+1= 0 when φ = 0
Base inclination (ψ = base tilt from horizontal, radians)bc1 − 2ψ/(2+π) for φ>0;  1 − ψ/147° for φ=0ψ in radians in formula
bq = bγ(1 − ψ·tanφ)²Positive ψ tilts toward direction of horizontal load
Ground slope (β = slope angle from horizontal, radians)gc1 − 2β/(2+π) for φ>0;  1 − β/147° for φ=0For footing on inclined ground surface
gq = gγ(1 − tanβ)²β in radians in formula
Theory 4 (1973, 1975)

Vesic's Method: Revised Nγ and Soil Compressibility Correction

A.S. Vesic (1973, 1975) refined Hansen's framework by revising $N_\gamma$ and introducing a soil compressibility correction factor ($I_{rc}$) for loose or soft soils where local/punching shear governs. Vesic's shape factors closely follow Hansen's. His method is widely used in US practice (AASHTO LRFD, FHWA).

Vesic's Nγ (the Key Difference from Hansen)
$$N_\gamma^{\text{Vesic}} = 2(N_q + 1)\tan\phi \qquad \text{vs} \qquad N_\gamma^{\text{Hansen}} = 1.5(N_q-1)\tan\phi$$
Vesic's Nγ is consistently 10–30% larger than Hansen's. At φ = 30°: Vesic Nγ = 2(18.40+1)tan30° = 22.4; Hansen Nγ = 1.5(18.40−1)tan30° = 15.1. This makes Vesic slightly less conservative for sands. Always state which is used in the report.
Vesic Compressibility (Rigidity Index) Correction
$$I_r = \frac{G_s}{c + q\tan\phi}; \qquad I_{r,crit} = \exp\!\!\left[\left(3.30 - 0.45\tfrac{B}{L}\right)\!\cot\!\left(45-\tfrac{\phi}{2}\right)\right]$$
$$\text{If } I_r \geq I_{r,crit}: \quad I_{rc} = 1.0 \quad\text{(rigid; no correction needed)}$$
$$\text{If } I_r < I_{r,crit}: \quad \text{apply compressibility correction (reduces }q_u\text{ by up to 30--40\%)}$$
Gs = shear modulus of soil (kPa); q = overburden at footing base. For stiff soils (dense sand, stiff clay), Ir typically exceeds Ir,crit and no correction is needed. For loose sand or soft clay, the correction effectively converts the theoretical general-shear capacity to a local/punching shear capacity.
φ (°)Nc (all theories)Nq (all)Nγ HansenNγ VesicNγ Meyerhof
05.141000
56.491.570.090.450.07
108.352.470.471.220.37
1510.983.941.422.491.1
2014.836.43.543.642.87
2520.7210.668.116.776.77
3030.1418.415.0722.415.67
3546.1233.333.9248.0333.92
4075.3164.279.54109.4173.4
45133.88134.87200.81271.76192
Critical Adjustment

Groundwater Table Correction: Three Cases Fully Explained

The groundwater table (GWT) reduces the effective unit weight of saturated soil below it to the buoyant value: $\gamma' = \gamma_{sat} - \gamma_w \approx 8$–$11\;\text{kN/m}^3$ (roughly half of moist unit weight). This affects two separate terms in the bearing capacity equation: the overburden term $q = \gamma D_f$ and the self-weight term $\frac{1}{2}\gamma B N_\gamma$. Three cases are defined by depth of GWT relative to $D_f$ and footing width $B$.

Groundwater table correction cases for bearing capacity Case 1 at surface Case 2 at footing level Case 3 within depth B Case 4 below B
Fig. 4: Four groundwater table (GWT) correction cases for bearing capacity calculation. The position of the water table relative to Df and B determines which unit weights to use in the overburden (q) and self-weight (½γBNγ) terms. (After IS 6403:1981, Cl. 5.4)
CaseGWT LocationOverburden term q = γDfSelf-weight term ½γBNγ
Case 1GWT at or above ground surface (dw = 0)Use γ' throughout: q = γ' × DfUse γ' in place of γ
Case 2GWT exactly at footing base (dw = Df)No change: q = γ × Df (γ above GWT is full)Use γ' in place of γ
Case 3GWT within B below footing base (Df < dw < Df + B)No change: q = γ × DfUse interpolated γ̄ (see formula below)
Case 4GWT deeper than Df + B below surfaceNo changeNo change; GWT has no effect on qu
Case 3: Interpolated Unit Weight for Self-Weight Term
$$\bar\gamma = \gamma' + \frac{d_w - D_f}{B}\,(\gamma - \gamma')$$
dw = depth of GWT below ground surface. Linear interpolation: when GWT is at foundation base (dw = Df), γ̄ = γ'; when GWT is at Df + B, γ̄ = γ (no effect). Use γ̄ only in the 0.5 × γ × B × Nγ term. The overburden term q is unaffected in Case 3.

Quantitative impact example: For a 2.0 m square footing at Df = 1.5 m in medium dense sand (φ = 32°, γ = 19 kN/m³, γ' = 9.5 kN/m³): GWT well below (Case 4): qu ≈ 1,380 kPa. GWT at foundation level (Case 2): qu ≈ 980 kPa. GWT at ground surface (Case 1): qu ≈ 620 kPa. GWT can reduce qu by 35–55%. Always determine GWT during site investigation and recheck seasonally if GWT fluctuates.

Special Conditions

Special Cases: Saturated Clay, Eccentric Load, Two-Layer Soil

Undrained (Short-Term) Bearing Capacity of Saturated Clay (φu = 0 Analysis)
$$q_u = c_u\,N_c\,s_c\,d_c + \gamma D_f; \qquad N_c = \pi + 2 = 5.14 \text{ (Prandtl exact)}$$
$$q_{nu} = 5.14\,c_u\,s_c\,d_c \quad \text{(net ultimate, \(\phi\)=0 analysis)}$$
For φ = 0: Nq = 1 and Nγ = 0, so only the c·Nc term contributes to net bearing capacity. Nc = π + 2 = 5.14 is the exact Prandtl solution. cu = undrained shear strength from UU triaxial test, unconfined compression, or field vane shear. The φ = 0 analysis is appropriate for short-term (rapid) loading of saturated clay; use drained parameters for long-term design.
Eccentric Loading: Meyerhof Effective Area Method
$$B' = B - 2e_B; \qquad L' = L - 2e_L$$
$$e_B = \frac{M_B}{Q_V}; \qquad e_L = \frac{M_L}{Q_V}; \qquad \text{Limit: } e \leq B/6 \text{ (resultant in middle third)}$$
Replace B and L with B' and L' in all bearing capacity formula terms (including shape and depth factor calculations). Total allowable load = qa × B' × L'. If e > B/6, the resultant falls outside the middle third; tension occurs in the soil (which cannot resist tension) and the footing must be enlarged or the eccentricity reduced. Apply effective dimensions to shape factor B/L ratio as well.
Two-Layer Soil: Strong over Weak (Meyerhof and Hanna 1978)
$$q_u = q_{u2} + \frac{(q_{u1} - q_{u2})\,H^2\,K_s\tan\phi_1}{B^2/4 + H^2 + H\sqrt{B^2/4 + H^2}} \leq q_{u1}$$
qu1 = ultimate BC of upper (stronger) layer. qu2 = ultimate BC of lower (weaker) layer. H = thickness of upper layer. Ks = punching shear coefficient (from Meyerhof-Hanna charts; depends on ratio qu1/qu2 and φ1). Simple conservative approach: project load at 2V:1H through upper layer to interface; check qu2 at that stress level. If H/B > 2–3, upper layer governs entirely.
In-Situ Testing

Field Determination: SPT, CPT and Plate Load Test

Standard Penetration Test (SPT)

The SPT (ASTM D1586 / IS 2131) is the most widely used in-situ test worldwide. A 51 mm split-spoon sampler is driven 450 mm into the borehole by a 63.5 kg hammer falling 762 mm; the blow count for the last 300 mm is recorded as the raw N-value. Raw N is corrected to $N_{60}$ (standardised to 60% energy) and then $(N_1)_{60}$ (overburden-corrected to 100 kPa reference) before use in correlations.

Standard Penetration Test SPT setup showing split-spoon sampler borehole hammer and correction factors N60 N160
Fig. 5: SPT test setup and the correction cascade from raw N to (N1)60. The four corrections (energy efficiency CEM, borehole diameter CB, rod length CR, sampler type CS) give N60; the overburden correction CN gives (N1)60. (After Skempton, 1986; ASTM D1586)
SPT Corrections (ASTM D1586 / IS 2131)
$$N_{60} = N_{raw} \times \frac{E_m}{60} \times C_B \times C_R \times C_S$$
$$(N_1)_{60} = N_{60} \times C_N; \qquad C_N = \sqrt{\frac{P_a}{\sigma'_v}} \leq 2.0$$
$E_m$ = hammer energy efficiency (%): donut 45%, safety 55–60%, automatic trip 75–78%. $C_B$: 1.0 for 65–115 mm; 1.05 for 150 mm; 1.15 for 200 mm. $C_R$: 0.75 (<3 m); 0.85 (3–4 m); 0.95 (4–6 m); 1.00 (>6 m). $C_S$: 1.0 with liner; 1.1–1.3 without. $P_a$ = 100 kPa. $C_N$ limit of 2.0 prevents over-correction at shallow depth.
N60Sand Classificationφ' estimateqa for 25 mm settlement (kPa)Clay Classificationcu estimate (kPa)
0–4Very loose25–28°0–50Very soft0–25
5–10Loose28–30°50–100Soft25–50
11–20Medium dense30–35°100–200Medium stiff50–100
21–30Dense35–40°200–300Stiff100–200
31–50Very dense38–43°300–500Very stiff200–400
>50Rock-like / cemented>43°>500Hard>400
Meyerhof (1956) Direct Allowable BC from SPT (Sand, 25 mm settlement)
$$q_a = \frac{N_{60}}{0.05} \text{ kPa} \quad (B \leq 1.2\text{ m})$$
$$q_a = \frac{N_{60}}{0.08}\left(\frac{B+0.3}{B}\right)^2 \text{ kPa} \quad (B > 1.2\text{ m})$$
$B$ in metres. These give qa corresponding to 25 mm settlement. For other target settlement s mm: multiply by s/25. Bowles (1996) revised upward by ≈50% for modern practice. Always treat as preliminary estimates; formal analysis required for important structures.

Cone Penetration Test (CPT)

The CPT (ASTM D3441 / IS 4968 Part III) pushes a 10 cm² cone at 20 mm/s and continuously measures tip resistance $q_c$ (MPa), sleeve friction $f_s$ (kPa), and pore pressure $u_2$ (kPa). The friction ratio $F_r = f_s/q_c \times 100\%$ is used for soil classification. CPT gives a continuous profile without sampling disturbance.

CPT Bearing Capacity Correlations
$$\phi' = \arctan\!\left[0.1 + 0.38\log\!\left(\frac{q_c}{\sigma'_v}\right)\right] \quad \text{(Robertson and Campanella 1983, sand)}$$
$$c_u = \frac{q_c - \sigma_v}{N_k}; \qquad N_k = 14\text{ to }20 \text{ (typically 16 for most clays)}$$
$q_c$ in kPa. $\sigma_v$ = total overburden (kPa). $\sigma'_v$ = effective overburden (kPa). $N_k$ = cone factor: lower (14) for sensitive clays; higher (20) for insensitive, stiff clays. For normalised CPT: use Qt and Fr on Robertson (1990) soil behaviour type chart.

Plate Load Test (PLT)

The PLT (IS 1888 / ASTM D1194) places a rigid steel plate (typically 300–600 mm square) at the proposed foundation level and applies incremental loads while measuring settlement. A load-settlement curve is plotted; $q_u$ from the plate is defined at the intersection of two tangents drawn to the bilinear curve.

PLT Scale Corrections to Foundation Size
$$q_u(\text{footing}) = q_u(\text{plate}) \quad \text{for clay } (\phi=0, \text{ c}_u\text{ independent of B)}$$
$$q_u(\text{footing}) = q_u(\text{plate}) \times \frac{B_f}{B_p} \quad \text{for sand (q}_u\text{ proportional to B)}$$
$$S_f = S_p \times \left[\frac{2B_f}{B_f+B_p}\right]^2 \quad \text{(settlement scaling for sand, Terzaghi)}$$
$B_f$ = footing width; $B_p$ = plate width (m). Limitation: PLT tests a small soil volume; cannot detect weak layers below depth ≈ $B_p$. Conduct test at the actual proposed foundation level. Minimum 3 tests per soil type recommended.
Codes of Practice

IS 6403, AASHTO, Eurocode 7: Requirements and FoS

StandardCountryRecommended TheoryFoS (Shear)Settlement Limit (isolated)Key Distinguishing Feature
IS 6403:1981IndiaTerzaghi for simple cases; Hansen for inclined/eccentric2.5–3.0 (net qu)25 mm sand; 40 mm clay (IS 1904)Specific GWT correction rules; presumptive BC table; widely used in Indian practice
IS 1904:1986IndiaSettlement criteria companion to IS 6403Referenced from IS 640325/40/65 mm (sand/clay/raft); diff.: 18–25 mmGoverns most Indian foundation design; differential settlement limits are critical
FHWA GEC No. 6 (2006)USAMeyerhof shape/depth; Vesic Nγ2.5–3.5 (net qu)25–50 mmStandard US reference for shallow foundations; detailed factor tables
AASHTO LRFD 9th Ed.USA (bridges)Meyerhof/Vesic; LRFD resistance factorsφbc = 0.50 (equiv. FoS ≈2.0–2.5)Structure-specific; typically 25 mmLoad and Resistance Factor Design; separate factors per limit state
Eurocode 7 (EN 1997-1:2004)EuropeHansen (Annex D); Design Approaches 1, 2, or 3Partial factors: γc = 1.25 (GEO)S4 performance class; structure-specificPartial material and load factors; three Design Approaches per National Annex
BS 8004:2015UKHansen; references EC7 DA1EC7 partial factors25–50 mm guidanceSupersedes BS 8004:1986 presumptive BC tables; aligned with EC7
NBC 105:2020NepalIS 6403 methods + NBC seismic load combinations2.5–3.0Per IS 1904Seismic zone factor reduces allowable BC; NBC seismic provisions mandatory
Comparison of Eurocode 7 partial factor design approach and IS 6403 global factor of safety approach for shallow foundation bearing capacity
Fig. 6: Design philosophy comparison: IS 6403 (global factor of safety, Fs = 2.5–3.0 on net qu) vs. Eurocode 7 Design Approach 1 (partial factors on loads and material properties). Both approaches target similar structural reliability but EC7 is more transparent about sources of uncertainty.

IS 6403 presumptive bearing capacity values (Table 1) for preliminary design only: Soft/medium clay 50–100 kPa. Stiff clay 100–200 kPa. Hard clay 200–400 kPa. Loose sand 100 kPa. Medium sand 100–200 kPa. Dense sand 200–400 kPa. Gravel 400–600 kPa. Weathered rock 600–2,000 kPa. These are gross allowable pressures. Never use for final design of any structure requiring a proper geotechnical investigation.

Full Numerical Example

Complete Worked Example: Square Footing by Meyerhof (IS 6403)

Problem: A 2.0 m × 2.0 m square footing is to be placed at $D_f = 1.5$ m in medium dense sandy soil. Soil: $c = 0$, $\phi = 30°$, $\gamma = 18$ kN/m³, $\gamma_{sat} = 19.5$ kN/m³. The groundwater table is at 3.0 m below ground surface (1.5 m below footing base; $B = 2.0$ m: Case 3 GWT correction). Vertical central load only. Required: (1) $q_u$ by Meyerhof; (2) $q_{nu}$; (3) $q_a$ with $F_s = 3.0$; (4) allowable column load.

Step-by-Step Solution

1
Meyerhof bearing capacity factors at φ = 30°:
$N_q = e^{\pi\tan 30°}\tan^2(60°) = e^{1.8138} \times 3.0 = 6.115 \times 3.0 = \mathbf{18.40}$
$N_c = (18.40 - 1)\cot 30° = 17.40 \times 1.732 = \mathbf{30.14}$
$N_\gamma = (N_q-1)\tan(1.4\phi) = 17.40 \times \tan 42° = 17.40 \times 0.9004 = \mathbf{15.67}$ (Meyerhof)
2
Groundwater correction (Case 3: GWT between Df and Df + B):
$d_w = 3.0$ m; $D_f = 1.5$ m; $d_w - D_f = 1.5$ m; $B = 2.0$ m; $1.5 < 2.0$ so Case 3 applies.
$\gamma' = 19.5 - 9.81 = 9.69$ kN/m³
Overburden: all soil above footing base is above GWT → $q = 18 \times 1.5 = \mathbf{27.0}$ kPa (no GWT correction on q term).
$\bar\gamma = 9.69 + \frac{1.5}{2.0}(18.0 - 9.69) = 9.69 + 6.23 = \mathbf{15.92}$ kN/m³
3
Shape factors (Meyerhof, square: B/L = 1.0):
$N_\phi = \tan^2(60°) = 3.0$
$s_c = 1 + 0.2 \times 1.0 \times 3.0 = \mathbf{1.60}$; $s_q = 1 + 0.1 \times 1.0 \times 3.0 = \mathbf{1.30}$; $s_\gamma = \mathbf{1.30}$
4
Depth factors (Meyerhof, Df/B = 1.5/2.0 = 0.75):
$d_c = 1 + 0.2 \times 0.75 \times \sqrt{3.0} = 1 + 0.260 = \mathbf{1.260}$
$d_q = 1 + 0.1 \times 0.75 \times 1.732 = 1 + 0.130 = \mathbf{1.130}$; $d_\gamma = \mathbf{1.130}$
5
Inclination factors: Vertical central load only → $i_c = i_q = i_\gamma = \mathbf{1.0}$
6
Ultimate bearing capacity ($c = 0$, first term vanishes):
$q_u = q\,N_q\,s_q\,d_q + \tfrac{1}{2}\,\bar\gamma B\,N_\gamma\,s_\gamma\,d_\gamma$
$= 27.0 \times 18.40 \times 1.30 \times 1.130 + 0.5 \times 15.92 \times 2.0 \times 15.67 \times 1.30 \times 1.130$
$= 729.8 + 367.4 = \mathbf{1{,}097}$ kPa
7
Net ultimate bearing capacity:
$q_{nu} = q_u - \gamma D_f = 1{,}097 - 27.0 = \mathbf{1{,}070}$ kPa
8
Allowable bearing pressure ($F_s = 3.0$ per IS 6403):
$q_a = q_{nu}/F_s + \gamma D_f = 1{,}070/3.0 + 27.0 = 356.7 + 27.0 = \mathbf{384}$ kPa
9
Total allowable column load on the 2.0 m × 2.0 m footing:
$Q_{allow} = q_a \times B \times L = 384 \times 2.0 \times 2.0 = 1{,}536$ kN (total, including footing self-weight).
Footing self-weight + backfill: $W_f \approx 2.0 \times 2.0 \times 1.5 \times 24 = 144$ kN.
Net structural column load: $\approx 1{,}536 - 144 = \mathbf{1{,}392}$ kN.
Always check settlement separately; if settlement exceeds IS 1904 permissible limit, qa must be reduced to the settlement-limited value.
Settlement Analysis

Settlement Criteria and Calculation Methods

Total foundation settlement = Immediate (elastic) settlement $S_i$ + Primary consolidation $S_c$ + Secondary compression $S_s$. For sands and gravels, $S_i$ dominates (drainage is fast; $S_c$ and $S_s$ negligible). For soft to medium clays, $S_c$ dominates and may continue for years to decades.

Immediate (Elastic) Settlement
$$S_i = \frac{q_a\,B\,(1-\nu^2)}{E_s}\,I_w\,\mu_0\,\mu_1$$
$q_a$ = net applied contact pressure (kPa). $B$ = footing width (m). $\nu$ = Poisson's ratio: sand 0.25–0.35; clay 0.40–0.50. $E_s$ = soil elastic modulus (kPa): estimate from SPT as $E_s \approx 0.4 N_{60}$ MPa (sand) or $E_s \approx 500\,c_u$ (clay). $I_w$ = influence factor for shape and rigidity (flexible square 0.82; flexible circle 0.85; rigid circle 0.79; flexible rectangle L/B = 5: 1.22). $\mu_0$, $\mu_1$ = depth and shape correction factors (Janbu et al. charts).
Primary Consolidation Settlement of Clay (Oedometer Method)
$$S_c = \frac{C_c\,H}{1+e_0}\log\frac{\sigma'_{v0}+\Delta\sigma'_v}{\sigma'_{v0}} \quad \text{(normally consolidated: } \sigma'_{v0} \leq \sigma'_p\text{)}$$
$$S_c = \frac{C_s\,H}{1+e_0}\log\frac{\sigma'_{v0}+\Delta\sigma'_v}{\sigma'_{v0}} \quad \text{(overconsolidated: } \sigma'_{v0}+\Delta\sigma'_v \leq \sigma'_p\text{)}$$
$C_c$ = compression index (from oedometer). $C_s$ = swelling/recompression index ($\approx 0.1$–$0.2\,C_c$). $e_0$ = initial void ratio. $H$ = compressible clay layer thickness (m). $\sigma'_{v0}$ = initial effective overburden at mid-layer (kPa). $\Delta\sigma'_v$ = stress increase at mid-layer from footing load (Boussinesq or 2:1 distribution). $\sigma'_p$ = preconsolidation pressure.
Foundation TypeSoilMax. Total SettlementMax. Differential SettlementStandard
Isolated footingSand and gravel25 mm18 mmIS 1904; IS 6403; FHWA
Isolated footingClay40 mm (up to 65 mm with caution)25 mmIS 1904
Raft foundationSand50 mm25 mmIS 1904
Raft foundationClay65–100 mm40–65 mmIS 1904
Frame structureAny25–50 mmL/500 (angular distortion)Eurocode 7; AASHTO
Sensitive machineryAny5–10 mm2–5 mmProject-specific

Why settlement often governs over shear: For most practical footings on loose sand or clay, the allowable bearing pressure derived from settlement limits (25 mm on sand, 40 mm on clay) is typically 100–200 kPa, which is well below the safe bearing capacity from shear analysis (often 300–600 kPa). This means the footing must be sized to keep contact pressure below 100–200 kPa, not the 300–600 kPa the soil could resist in shear. Always check both criteria and use the lower governing value.

Visual Reference

Charts: Bearing Capacity Factors, GWT Impact & SPT Correlations

Chart 1 — Nc, Nq, Nγ vs. Friction Angle φ (log scale)

Comparison of all bearing capacity factors on a logarithmic scale. Note the rapid exponential growth above φ = 30° and the divergence between Vesic, Meyerhof, and Hansen Nγ expressions.

Chart 2 — qu vs. φ for Different GWT Positions (2.0 m square footing, Df = 1.5 m)

Meyerhof method with shape and depth factors; c = 0, γ = 19 kN/m³, γ' = 9.5 kN/m³. Shows how GWT position dramatically reduces qu — up to 55% in extreme cases.

Chart 3 — Meyerhof Allowable Bearing Pressure from SPT vs. Footing Width (Sand, 25 mm settlement)

Direct preliminary design chart. Allowable pressure decreases with increasing width beyond 1.2 m due to deeper influence zone. Use as first estimate only; confirm with formal analysis.

Chart 4 — Bearing Capacity Reduction Due to Load Inclination (Meyerhof iq, iγ)

Relative capacity (% of vertical-load capacity) vs. inclination angle α from vertical. Horizontal loads from wind, seismic, or lateral earth pressure can reduce capacity by 50–80%.

Bar chart comparing Ngamma bearing capacity factor for Meyerhof Hansen and Vesic theories at phi 20 25 30 35 40 degrees
Fig. 7: Nγ compared across Meyerhof, Hansen, and Vesic theories. Vesic's expression gives consistently higher values. The difference increases significantly above φ = 30°; at φ = 40° Vesic Nγ is 38% higher than Hansen's.
Load-settlement curves for three shear failure modes general shear local shear and punching shear showing different curve shapes
Fig. 8: Typical load-settlement (q–s) curves for the three failure modes. General shear shows a clear abrupt peak. Local shear has a gradual knee. Punching shear curves continuously without a distinct failure point. (After Vesic, 1973)
Interactive Tool

Ultimate Bearing Capacity Calculator (Meyerhof / Terzaghi)

Bearing Capacity Calculator

Computes $q_u$, $q_{nu}$, and $q_a$ for vertical central load. GWT assumed below $D_f + B$ (Case 4 — no groundwater correction). Adjust results manually using Case 1/2/3 formulas if GWT is shallower.

Cohesion c (kPa)
Friction angle φ (°)
Unit weight γ (kN/m³)
Width B (m)
Length L (m)
Foundation depth Df (m)
Factor of safety
Footing shape
Method
Help

Frequently Asked Questions

1. What is the difference between gross, net, safe, and allowable bearing capacity?

These four terms represent a sequence from theoretical maximum to practical design value. Gross ultimate bearing capacity (qu) is the total contact pressure at which shear failure occurs, calculated from Terzaghi, Meyerhof, Hansen, or Vesic equations. Net ultimate bearing capacity (qnu) subtracts the pre-existing overburden stress (gamma x Df), representing only the additional stress beyond what already existed. Safe bearing capacity (qs) divides qu by a factor of safety (2.5 to 3.0) against shear failure. Allowable bearing pressure (qa) is the final design value: the smaller of the net safe bearing capacity plus overburden (shear criterion) and the contact pressure that limits settlement to permissible values. For most foundations on clay or loose-medium sand, settlement governs and qa is the settlement-limited pressure.

2. What are Nc, Nq, and Ngamma in bearing capacity equations?

Nc, Nq, and Ngamma are dimensionless bearing capacity factors representing contributions of soil cohesion, overburden pressure, and soil self-weight. They are purely functions of friction angle phi. Nq represents surcharge resistance; it equals 1.0 for phi = 0. Nc represents cohesive resistance; its exact value for phi = 0 is 5.14 (pi + 2, Prandtl 1920). Ngamma represents resistance from soil self-weight below the footing; it is 0 for phi = 0. All three increase rapidly with phi: at phi = 30 degrees, Nq is approximately 18, Nc approximately 30, and Ngamma approximately 15 to 22 depending on the source. The exponential sensitivity to phi means accurate determination of friction angle is the most critical input.

3. Why does the groundwater table reduce bearing capacity and by how much?

Bearing capacity depends on effective stress. Below the groundwater table, effective unit weight drops from approximately 18-20 kN/m3 to approximately 8-11 kN/m3 (roughly halved). This affects two terms: the overburden term (q = gamma x Df) and the soil self-weight term (0.5 x gamma x B x Ngamma). For typical medium-dense sand, raising GWT from deep below the footing to the footing base level reduces ultimate bearing capacity by approximately 30-50 percent. A further rise to ground surface can cause reduction up to 55 percent. Three correction cases apply based on GWT position relative to Df and depth B below the footing base.

4. What factor of safety is appropriate for bearing capacity design?

IS 6403 recommends FoS = 2.5 to 3.0 applied to the net ultimate bearing capacity. The choice depends on: quality of soil investigation; soil variability; structure importance (critical structures use 3.0); construction quality control; and load reliability. AASHTO LRFD uses resistance factors (phi = 0.45 to 0.50), equivalent to FoS approximately 2.0 to 2.5. Eurocode 7 uses partial factors on material properties rather than a single FoS. Important: the FoS addresses shear failure only; settlement must be checked independently.

5. What is the Terzaghi formula for strip, square, circular, and rectangular footings?

Terzaghi (1943) gave separate equations for each shape. Strip: qu = c x Nc + q x Nq + 0.5 x gamma x B x Ngamma. Square: qu = 1.3 x c x Nc + q x Nq + 0.4 x gamma x B x Ngamma. Circular (diameter B): qu = 1.3 x c x Nc + q x Nq + 0.3 x gamma x B x Ngamma. Rectangular: qu = (1 + 0.3B/L) x c x Nc + q x Nq + (1 - 0.2B/L) x 0.5 x gamma x B x Ngamma. Meyerhof, Hansen, and Vesic later provided analytical shape factors that are more rigorous and are universally preferred for current design.

6. When should Meyerhof or Hansen be used instead of Terzaghi?

Terzaghi is adequate only for simple cases: shallow strip, square, or circular footings with vertical central load on level ground. Meyerhof, Hansen, or Vesic must be used whenever: the load is inclined (horizontal component from wind, seismic, lateral earth pressure); the footing is deeply embedded (Df/B greater than 1); the footing base is tilted (Hansen base inclination factors); the ground surface is sloped (Hansen ground inclination factors); the load is eccentric; or a rigorous shape factor formulation is needed. In general professional practice, Meyerhof or Hansen is always preferred over Terzaghi.

7. How does foundation depth Df affect bearing capacity?

Increasing Df improves bearing capacity through two mechanisms. First, it increases the overburden term q = gamma x Df, which multiplies with Nq; for sandy soils with large Nq this effect is substantial. Second, depth factors dc, dq, dgamma (all greater than 1.0 in Meyerhof and Hansen) account for the additional confinement and shearing resistance of the soil above the footing level. For purely cohesive clay (phi = 0), only dc contributes (dc = 1 + 0.4 x Df/B in Hansen), giving a modest increase. Practical minimum foundation depths per IS 1080: 0.5 to 1.0 m below any soft topsoil or fill.

8. What is the difference between general shear, local shear, and punching shear failure?

General shear failure occurs in dense or stiff soils (Dr greater than 67%, SPT N greater than 30). A well-defined continuous shear surface extends to the ground surface with pronounced heaving on both sides. The load-settlement curve shows a clear peak followed by sudden collapse. Local shear failure occurs in medium-density soils (Dr 35-67%, SPT N = 10-30). The failure zone does not reach the ground surface. Settlement is large before collapse and the curve has no sharp peak. Use Terzaghi with reduced parameters: c* = 0.67c and phi* = arctan(0.67tan phi). Punching shear failure occurs in very loose or soft soils (Dr less than 35%, SPT N less than 10). The footing punches straight down with no surface heave; continuous settlement without a distinct failure load.

9. How is bearing capacity estimated from SPT N-values?

SPT N-values are first corrected to N60 (standardising to 60 percent hammer energy) accounting for hammer type, borehole diameter, rod length, and sampler liner. Then corrected to (N1)60 (normalised to 100 kPa reference overburden) using CN = sqrt(100/sigma_v prime) limited to 2.0. Friction angle estimated from Peck correlations: phi from 25-28 degrees for N60 = 2-4, up to 38-43 degrees for N60 = 31-50. Undrained cohesion of clay: cu = 6 x N60 kPa (Terzaghi and Peck). Meyerhof (1956) direct allowable BC for sand: qa = N60/0.05 kPa for B less than 1.2 m; qa = N60/0.08 x ((B+0.3)/B) squared for B greater than 1.2 m, both for 25 mm settlement.

10. What is the effective width method for eccentric loading?

When the resultant vertical force does not pass through the centroid, Meyerhof effective width method replaces actual dimensions B and L with reduced effective dimensions: B prime = B minus 2eB and L prime = L minus 2eL, where eB = moment about B-axis divided by vertical load. Bearing capacity is computed using B prime and L prime for all terms including shape and depth factors. Total load capacity = qa x B prime x L prime. The eccentricity limit e less than or equal to B/6 ensures the resultant stays within the middle third and contact pressure remains compressive throughout.

11. What are the permissible settlements for shallow foundations per IS 1904?

IS 1904:1986 specifies: for isolated footings: sand and hard clay 25 mm total, 18 mm differential; soft clay 40 mm total (up to 65 mm), 25 mm differential. For raft foundations: sand 50 mm total, 25 mm differential; clay 65-100 mm total, 40-65 mm differential. The angular distortion should not exceed 1/500 for frame buildings. Differential settlement is more damaging than total settlement because it induces bending moments and cracking in structural elements.

12. What is the drained versus undrained analysis for bearing capacity?

Drained analysis uses effective strength parameters c prime and phi prime from CD or CU triaxial tests and represents long-term conditions. Undrained analysis uses total stress parameters cu and phi_u = 0 from UU triaxial or vane shear tests and represents the critical short-term condition in saturated clay immediately after rapid loading. For saturated clay, both must be checked; the more critical governs. For normally consolidated soft clay, undrained strength is typically lower and governs short-term design. For sandy soils, drainage is so rapid that the drained analysis always applies.

13. How does foundation shape (square, circular, strip) affect bearing capacity?

Shape affects bearing capacity through shape factors sc, sq, and sgamma. For the cohesion term (Nc), a square or circular footing has a higher factor than a strip: Terzaghi gives sc = 1.3 for square/circular vs 1.0 for strip. For the self-weight term (Ngamma), Hansen gives sgamma = 1 - 0.4 x B/L = 0.6 for square, reflecting 3D geometry. For purely cohesionless soil (c = 0) with phi greater than 0, a square footing typically has 20-30 percent higher qu per unit area than a strip footing of the same width, due to depth factor and shape factor advantages in the Nq term.

14. What is the relationship between SPT N-value and soil friction angle?

Several empirical correlations relate SPT N60 to drained friction angle phi prime for sands. Peck, Hanson and Thornburn (1974): N60 = 0-4 corresponds to phi = 25-28 degrees; 5-10: 28-30 degrees; 11-30: 30-36 degrees; 31-50: 36-41 degrees; over 50: over 41 degrees. Schmertmann (1975): phi prime = 27.1 + 0.3 x (N1)60 minus 0.00054 x (N1)60 squared. Hatanaka and Uchida (1996): phi prime = sqrt(20 x (N1)60) + 20 degrees. These correlations are approximate; for important projects, supplement SPT with laboratory tests.

15. What are the Terzaghi bearing capacity factors for phi = 0 (saturated clay, undrained)?

For phi = 0 (saturated clay, undrained analysis): Nc = 5.7 (Terzaghi original; Prandtl exact = 5.14), Nq = 1.0, and Ngamma = 0. The Nq = 1 term means the overburden term exactly cancels in the net bearing capacity calculation. The Ngamma = 0 term means soil self-weight below the footing contributes nothing. The entire net ultimate bearing capacity is therefore: qnu = Nc x cu = 5.14 x cu (strip, Prandtl) or 5.7 x cu (Terzaghi). For a square footing: qnu = 1.3 x 5.14 x cu = 6.68 x cu. This simplicity makes the phi = 0 analysis very practical for preliminary design on saturated clays.

16. What are the Meyerhof bearing capacity factors for phi = 36 degrees?

At phi = 36 degrees using the Meyerhof/Prandtl expressions: Nq = e^(pi x tan36) x tan^2(63) = approximately 39.15; Nc = (Nq minus 1) x cot36 = 38.15 x 1.376 = approximately 52.49; Ngamma (Meyerhof) = (Nq minus 1) x tan(50.4 degrees) = approximately 46.40; Ngamma (Hansen) = 1.5 x (Nq minus 1) x tan36 = approximately 41.59; Ngamma (Vesic) = 2 x (Nq plus 1) x tan36 = approximately 58.36. These values apply to very dense sands and compacted gravels. The large Nq of approximately 39 means even a modest Df of 1.5 m in dense sand contributes approximately 1,110 kPa to qu from the overburden term alone.

17. What is the Meyerhof shape factor formula sq = 1 + 0.1 B/L and the Hansen sgamma = 1 minus 0.4 B/L?

Meyerhof shape factors for phi greater than 10 degrees: sc = 1 + 0.2 x Nphi x (B/L); sq = 1 + 0.1 x Nphi x (B/L); sgamma = 1 + 0.1 x Nphi x (B/L); where Nphi = tan^2(45 + phi/2). For square at phi = 30 degrees (Nphi = 3.0): sq = 1.30, sgamma = 1.30. Hansen shape factors: sc = 1 + (Nq/Nc)(B/L); sq = 1 + (B/L)tan phi; sgamma = 1 minus 0.4(B/L). For square at phi = 30 degrees: sq = 1 + tan30 = 1.577; sgamma = 1 minus 0.4 = 0.60. The Hansen sgamma = 0.60 for square footings is significantly lower than Meyerhof s 1.30, reflecting different treatments of the 3D self-weight contribution. Meyerhof is more common in Indian textbooks; Hansen is standard in Eurocode 7.

18. What is the Meyerhof depth factor dq = 1 + 0.1 Df/B formula and how does it work?

The Meyerhof depth factor dq = 1 + 0.1 x k x sqrt(Nphi) for phi greater than 10 degrees, where k = Df/B. This accounts for the shear resistance of the soil above the footing base along the failure surface. For phi = 0, dq = 1.0. For phi = 30 degrees, sqrt(Nphi) = sqrt(3.0) = 1.732, so dq = 1 + 0.1 x (Df/B) x 1.732. For typical Df/B = 0.75: dq = 1.130, meaning a 13 percent increase in the Nq term from properly accounting for footing depth. Hansen depth factors are more conservative for Df/B less than 1 and more accurate for Df/B greater than 1, where Hansen uses arctan(Df/B) in radians instead of the ratio directly.

19. What is the Eurocode 7 Annex D undrained bearing capacity formula for square foundations?

For a square foundation on saturated clay under undrained conditions (phi_u = 0), Eurocode 7 Annex D gives: Rd/Af = (pi + 2) x cu x sc x dc + q, where pi + 2 = 5.14 (Prandtl exact). For a square footing with B = L: sc (Hansen) = 1 + (Nq/Nc)(B/L) = 1 + (1/5.14) x 1 = 1.195. The depth factor dc = 1 + 0.4 x arctan(Df/B) in radians. For Df/B = 0.5: dc = 1 + 0.4 x arctan(0.5) = 1.185. So qnu = 5.14 x 1.195 x 1.185 x cu = approximately 7.28 x cu. The simplified version (1 + 0.2 x B/L) x 5.14 = 1.2 x 5.14 = 6.17 x cu for shape factor only. This formula also appears in the formula (1 + 0.2 B/L) x 5.14 x cu that is sometimes written as (1+0.2(B/L)) x 5.14 x cu in examination notation.

20. What is the typical allowable bearing capacity of loose sandy soil in kPa?

The typical allowable bearing capacity of loose sandy soil is 50-100 kPa for a 25 mm settlement criterion, corresponding to SPT N60 values of approximately 5-10. The Meyerhof (1956) formula gives: for a 2.0 m wide footing with N60 = 8: qa = 8/0.08 x ((2.0+0.3)/2.0) squared = 100 x 1.32 = 132 kPa. IS 6403 Table 1 gives a presumptive allowable bearing capacity of 100 kPa for loose sand as a conservative preliminary value. From formal Meyerhof analysis at phi = 28 degrees with c = 0, B = 2.0 m, Df = 1.5 m, the allowable bearing pressure from shear failure (Fs = 3.0) is approximately 130-180 kPa, but settlement typically governs at 75-100 kPa for loose sand. BS 8004 historical presumptive value was also approximately 100 kPa for loose sand.

21. What is the typical allowable bearing capacity of firm clay in kPa?

Firm clay is defined by undrained shear strength cu approximately 50-75 kPa (SPT N60 approximately 7-11 in clay). Net ultimate bearing capacity for a strip footing: qnu = 5.14 x cu = 257 to 386 kPa. Applying Fs = 3.0: net safe BC = 86 to 129 kPa. Adding overburden (gamma x Df = 18 x 1.0 = 18 kPa): qa approximately 104 to 147 kPa. For a square footing with shape and depth factors this increases by about 20-30 percent. IS 6403 Table 1 gives a presumptive value of 100-200 kPa for medium stiff clay. BS 8004 (1986) quoted 100-200 kPa for firm to stiff clay. In practice 75-150 kPa is used accounting for the IS 1904 settlement criterion of 40 mm maximum for isolated footings on clay.

22. What is the soil bearing capacity symbol and standard notation?

In standard geotechnical notation: qu or qult = gross ultimate bearing capacity (kPa). qnu or qnet,ult = net ultimate bearing capacity = qu minus gamma x Df. qs or qsafe = safe bearing capacity = qu divided by Fs. qns = net safe bearing capacity = qnu divided by Fs. qa or qall or qallow = allowable bearing pressure, the final design value. qnet = net contact pressure from the structure. In IS 6403 these are specifically designated: qu (gross ultimate), qnet,ult (net ultimate), qs,gross (gross safe), qs,net (net safe), qa (allowable). In AASHTO LRFD the nominal bearing resistance qn replaces qu, and the factored resistance phi x qn is compared to the factored applied stress. In Eurocode 7 the design bearing resistance Rd/Af (kPa) is compared to the design bearing pressure sigma_v,d.

23. What is the soil bearing capacity chart showing typical values by soil type?

A soil bearing capacity chart organises allowable bearing pressures by soil type for preliminary design. Typical allowable gross bearing pressures (Fs = 3.0): Massive crystalline igneous rock 3,000-10,000 kPa; foliated metamorphic rock 1,500-3,000 kPa; weathered rock 300-800 kPa; compact well-graded gravel 600-900 kPa; loose gravel 200-400 kPa; dense coarse sand (N greater than 30) 300-500 kPa; medium dense sand (N = 15-30) 150-300 kPa; loose sand (N less than 10) 50-100 kPa; hard clay (cu greater than 200 kPa) 300-500 kPa; very stiff clay 150-300 kPa; stiff clay 100-150 kPa; firm clay 50-100 kPa; soft clay 25-50 kPa (settlement usually governs). These values are from IS 6403 Table 1, BS 8004:1986, and NAVFAC DM-7.02. Formal analysis is always required for final design.

24. What are the Terzaghi bearing capacity factors for phi = 32 degrees?

At phi = 32 degrees using the Meyerhof/Prandtl expressions (now universally standard): Nq = e^(pi x tan32) x tan^2(61) = approximately 23.18; Nc = (23.18 minus 1) x cot32 = 22.18 x 1.600 = approximately 35.49; Ngamma (Meyerhof) = (23.18 minus 1) x tan(44.8 degrees) = approximately 22.02; Ngamma (Hansen) = 1.5 x 22.18 x tan32 = approximately 20.79; Ngamma (Vesic) = 2 x (23.18 + 1) x tan32 = approximately 30.22. For local shear at phi = 32 degrees: phi* = arctan(2/3 x tan32) = arctan(0.4166) = 22.6 degrees; use the bearing capacity factors at 22.6 degrees.

25. What are the Terzaghi bearing capacity factors for phi = 35 degrees?

At phi = 35 degrees using Meyerhof/Prandtl expressions: Nq = e^(pi x tan35) x tan^2(62.5) = approximately 33.30; Nc = (33.30 minus 1) x cot35 = 32.30 x 1.428 = approximately 46.12; Ngamma (Meyerhof) = (33.30 minus 1) x tan(49 degrees) = 32.30 x 1.1504 = approximately 37.16; Ngamma (Hansen) = 1.5 x 32.30 x tan35 = approximately 33.93; Ngamma (Vesic) = 2 x (33.30 + 1) x tan35 = approximately 48.05. Note: Terzaghi s original tabulated values at phi = 35 degrees were: Nc = 57.75, Nq = 33.30, Ngamma = 34.00, which differ because Terzaghi used a slightly different Nq formula. For IS 6403 practice in India, Meyerhof/Hansen values are used.

Authoritative Sources

References & Official Standards

  1. Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley & Sons, New York.
  2. Meyerhof, G.G. (1951). The ultimate bearing capacity of foundations. Géotechnique, 2(4), 301–332.
  3. Meyerhof, G.G. (1963). Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal, 1(1), 16–26.
  4. Hansen, J.B. (1970). A Revised and Extended Formula for Bearing Capacity. Danish Geotechnical Institute Bulletin No. 28, Copenhagen.
  5. Vesic, A.S. (1973). Analysis of ultimate loads of shallow foundations. Journal of the Soil Mechanics and Foundations Division, ASCE, 99(1), 45–73.
  6. Vesic, A.S. (1975). Bearing capacity of shallow foundations. In H.F. Winterkorn & H.-Y. Fang (Eds.), Foundation Engineering Handbook (Ch. 3). Van Nostrand Reinhold, New York.
  7. Bureau of Indian Standards. (1981). IS 6403: Code of Practice for Determination of Bearing Capacity of Shallow Foundations. BIS, New Delhi. www.bis.gov.in
  8. Bureau of Indian Standards. (1986). IS 1904: Code of Practice for Design and Construction of Foundations in Soils — General Requirements. BIS, New Delhi.
  9. Bureau of Indian Standards. (2002). IS 2131: Method for Standard Penetration Test for Soils. BIS, New Delhi.
  10. Bureau of Indian Standards. (1992). IS 1888: Method of Load Test on Soils. BIS, New Delhi.
  11. European Committee for Standardization. (2004). EN 1997-1: Eurocode 7 — Geotechnical Design, Part 1: General Rules. CEN, Brussels. Annex D: Sample procedures to determine bearing resistance from analytical calculations.
  12. Bowles, J.E. (1996). Foundation Analysis and Design (5th ed.). McGraw-Hill, New York.
  13. Das, B.M. (2021). Principles of Foundation Engineering (9th ed.). Cengage Learning.
  14. Federal Highway Administration. (2006). FHWA NHI-06-089: Shallow Foundations — Reference Manual. US Department of Transportation. www.fhwa.dot.gov
  15. Robertson, P.K. & Campanella, R.G. (1983). Interpretation of cone penetration tests. Part I: Sand. Canadian Geotechnical Journal, 20(4), 718–733.
  16. Skempton, A.W. (1986). Standard penetration test procedures and the effects in sands of overburden pressure, relative density, particle size, ageing and overconsolidation. Géotechnique, 36(3), 425–447.
  17. Meyerhof, G.G. & Hanna, A.M. (1978). Ultimate bearing capacity of foundations on layered soils under inclined load. Canadian Geotechnical Journal, 15(4), 565–572.
  18. AASHTO. (2020). AASHTO LRFD Bridge Design Specifications (9th ed.). American Association of State Highway and Transportation Officials, Washington, D.C.

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