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

These four terms represent a sequence from theoretical maximum to practical design value. Gross ultimate bearing capacity (qu) is the total contact pressure at the foundation base at which shear failure occurs; it is calculated directly from Terzaghi, Meyerhof, Hansen, or Vesic equations. Net ultimate bearing capacity (qnu) subtracts the pre-existing overburden stress at foundation depth (gamma x Df), representing only the additional stress beyond what was already in the ground before construction. Safe bearing capacity (qs) divides gross qu by a factor of safety (typically 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.

Nc, Nq, and Ngamma are dimensionless bearing capacity factors representing the contributions of soil cohesion, overburden pressure, and soil self-weight respectively. They are purely functions of friction angle phi and come from the theoretical failure mechanism. Nq represents the surcharge resistance; it equals 1.0 for phi = 0 (no frictional contribution). Nc represents cohesive resistance; its exact value for phi = 0 is pi + 2 = 5.14 (Prandtl 1920 exact solution). Ngamma represents the 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 (Hansen vs Vesic). The exponential sensitivity to phi means accurate determination of friction angle is the most critical input to bearing capacity analysis.

Bearing capacity depends on effective stress in the soil skeleton. Below the groundwater table, soil is buoyant; its effective unit weight drops from approximately 18 to 20 kN/m3 to approximately 8 to 11 kN/m3 (roughly halved). This reduction affects two terms in the bearing capacity equation: the overburden term (q = gamma x Df) and the soil self-weight term (0.5 x gamma x B x Ngamma). For a typical medium-dense sand, raising the groundwater table from deep below the footing to the footing base level reduces ultimate bearing capacity by approximately 30 to 50 percent. For a further rise to ground surface, reduction can reach 55 percent. Three correction cases apply: Case 1 with GWT at or above ground surface uses buoyant unit weight throughout; Case 2 with GWT at footing level uses full unit weight for the q term only; Case 3 with GWT within depth B below the footing uses interpolated unit weight for the self-weight term. If GWT is deeper than B below the footing base, no correction is needed.

IS 6403 recommends a factor of safety of 2.5 to 3.0 applied to the net ultimate bearing capacity. The choice within this range depends on: quality of soil investigation (more tests = lower FoS needed); soil variability (uniform deposit allows lower FoS); structure importance (critical structures use FoS 3.0); construction quality control; and load reliability (well-defined loads allow lower FoS). A FoS of 3.0 is used for most routine building foundations in India. AASHTO LRFD bridge design uses resistance factors (phi = 0.45 to 0.50 for bearing), equivalent to FoS approximately 2.0 to 2.5. Eurocode 7 uses partial factors on material properties rather than a single overall FoS. Important: the FoS addresses shear failure only; settlement must be checked independently with its own permissible limits.

Terzaghi (1943) gave separate equations for each shape. Strip footing: qu = c x Nc + q x Nq + 0.5 x gamma x B x Ngamma. Square footing: qu = 1.3 x c x Nc + q x Nq + 0.4 x gamma x B x Ngamma. Circular footing of diameter B: qu = 1.3 x c x Nc + q x Nq + 0.3 x gamma x B x Ngamma. Rectangular footing: 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. The empirical shape multipliers 1.3 for Nc and 0.3 to 0.4 for Ngamma capture the 3D effect of confined failure zones under non-strip footings. Meyerhof, Hansen, and Vesic later provided analytical shape factors that are more rigorous and are universally preferred for current design.

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) requiring inclination factors; the footing is deeply embedded (Df/B greater than 1) requiring proper depth factors; the footing base is tilted (Hansen base inclination factors); the ground surface is sloped (Hansen ground inclination factors); the load is eccentric requiring the effective width method; or a highly accurate shape factor formulation is needed for rectangular footings. In general professional practice, Meyerhof or Hansen is always preferred over Terzaghi because they handle all loading and geometry conditions analytically with published correction factors.

Increasing Df improves bearing capacity through two mechanisms. First, it increases the overburden term q = gamma x Df, which multiplies with Nq in the second term of the equation; 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) formally account for the additional confinement and the shearing resistance of the soil above the footing level. For purely cohesive clay (phi = 0), only the depth factor dc contributes (dc = 1 + 0.4 x Df/B in Hansen), giving a modest increase. For sandy soils, both the Nq term and depth factors contribute significantly. Practical minimum foundation depths per IS 1080: 0.5 m below ground surface; 0.5 to 1.0 m below the frost penetration depth; minimum 0.6 m below any soft topsoil or fill.

General shear failure occurs in dense or stiff soils (dense sand with Dr greater than 67%, stiff clay, SPT N greater than 30). A well-defined continuous shear surface extends from the footing edge to the ground surface with pronounced heaving on both sides. The load-settlement curve shows a clear peak followed by sudden collapse. Terzaghi and Meyerhof equations apply directly. Local shear failure occurs in medium-density soils (Dr 35 to 67%, SPT N = 10 to 30). The failure zone is partial; the shear surface 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) and for deeply embedded footings. The footing punches straight down with vertical shear planes at its edges; no surface heave; continuous settlement without a distinct failure load. Apply Vesic compressibility correction or Meyerhof depth factors to account for this mode.

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 is estimated using Peck correlations: phi from 25 to 28 degrees for N60 = 2 to 4, up to 38 to 43 degrees for N60 = 31 to 50. Undrained cohesion of clay is estimated as cu = 6 x N60 kPa (Terzaghi and Peck) or cu = 7.5 x N60 (Bowles). Meyerhof (1956) gave direct allowable bearing pressure 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. These SPT correlations are approximate empirical estimates and should be supplemented with laboratory tests and formal bearing capacity analysis for important structures.

When the resultant vertical force does not pass through the centroid of the footing, the contact pressure distribution is non-uniform. Meyerhof effective width method handles this by replacing the actual dimensions B and L with reduced effective dimensions: B prime = B - 2eB and L prime = L - 2eL, where eB = moment about B-axis divided by vertical load, and eL = moment about L-axis divided by vertical load. Bearing capacity is then 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. If e exceeds B/6, a tension zone develops which soil cannot resist, and the footing must be enlarged or the eccentricity reduced by repositioning the column.

IS 1904:1986 specifies the following permissible settlements for building foundations in India. For isolated footings: sand and hard clay 25 mm total, 18 mm differential between adjacent columns; soft clay 40 mm total (up to 65 mm in exceptional cases), 25 mm differential. For raft foundations: sand 50 mm total, 25 mm differential; clay 65 to 100 mm total, 40 to 65 mm differential. The angular distortion (differential settlement divided by column spacing) should not exceed 1/500 for frame buildings and 1/150 for flexible structures that can tolerate more movement. Differential settlement is more damaging than total settlement because it induces bending moments and cracking in structural elements. For structures sensitive to settlement (turbine foundations, precision equipment, historical buildings), project-specific limits much tighter than IS 1904 values must be set.

Drained analysis uses effective strength parameters c prime and phi prime from CD or CU triaxial tests with pore pressure measurement. It represents long-term conditions after full pore pressure dissipation. Undrained analysis uses total stress parameters cu and phi_u = 0 from UU triaxial or vane shear tests. It represents the critical short-term condition in saturated clay immediately after rapid loading when excess pore pressures have not dissipated. For saturated clay, both must be checked: short-term undrained (using cu, phi_u = 0) and long-term drained (using c prime, phi prime); the more critical governs. For normally consolidated soft clay, undrained strength is typically lower and governs the short-term design. As consolidation proceeds over time, undrained strength increases toward the drained value. For sandy soils, drainage is so rapid that the drained analysis always applies; undrained analysis is not used for clean sands.

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; Meyerhof gives sc = 1 + 0.2 x Nphi for B/L = 1 vs 1.0 for strip. This means a square footing has a higher unit bearing capacity in the cohesion term than a strip. For the self-weight term (Ngamma), shape factors for square or circular footings are less than 1.0 in Hansen (sgamma = 1 - 0.4 x B/L = 0.6 for square) and greater than 1.0 in Meyerhof, reflecting different theoretical treatments of the 3D self-weight contribution. For purely cohesionless soil (c = 0) with phi greater than 0, a square footing typically has 20 to 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.

Several empirical correlations relate SPT N60 to drained friction angle phi prime for sands and gravels. Peck, Hanson and Thornburn (1974): N60 = 0 to 4 corresponds to phi = 25 to 28 deg; 5 to 10: 28 to 30 deg; 11 to 30: 30 to 36 deg; 31 to 50: 36 to 41 deg; over 50: over 41 deg. Schmertmann (1975): phi prime = 27.1 + 0.3 x (N1)60 - 0.00054 x (N1)60 squared. Hatanaka and Uchida (1996): phi prime = sqrt(20 x (N1)60) + 20 degrees. These correlations are approximate because N-value depends on many factors beyond friction angle, including particle size distribution, stress history, and test procedure. For important projects, supplement SPT with laboratory angle-of-friction tests (direct shear or triaxial) on undisturbed or reconstituted samples. CPT provides a more reliable phi estimate through the Robertson-Campanella correlation.