Bearing Capacity of Soil — Terzaghi's Equation & IS 6403

1. Introduction to Bearing Capacity

The bearing capacity of a soil is its ability to safely support the loads applied to it by a foundation without undergoing shear failure or excessive settlement. It is the most critical parameter in shallow foundation design and directly links the structural load from the superstructure to the geotechnical properties of the underlying soil.

Bearing capacity analysis answers two questions: (1) Will the soil shear and fail catastrophically under the applied load? (2) Will settlements be within acceptable limits? Shear failure governs at high load intensities; settlement limits often govern at moderate loads on compressible soils. Both must be checked.

2. Modes of Shear Failure

Terzaghi’s bearing capacity equation was derived for general shear failure. For local shear failure, the shear strength parameters are reduced (see Section 9).

3. Terzaghi’s Bearing Capacity Equation

Terzaghi (1943) derived the first complete bearing capacity equation by assuming a failure mechanism consisting of three zones: an active Rankine zone immediately below the footing, a radial shear (Prandtl) zone, and a passive Rankine zone at the edge.

3.1 General Equations

3.2 Terms Defined

4. Bearing Capacity Factors

4.1 Terzaghi’s Factors (General Shear)

4.2 Values of N_c, N_q, N_γ for Common φ Values

4.3 Special Cases

5. Net vs Gross Bearing Capacity

The soil at foundation level was already supporting an overburden pressure q = γ D_f before the foundation was constructed. The net bearing capacity is the additional pressure above this pre-existing overburden that the soil can carry.

In practice, q_ns (net safe bearing capacity) is the most useful quantity — it is the additional load per unit area beyond existing overburden that the foundation can safely carry. The total load on the footing divided by the footing area equals q_s (gross); subtracting the weight of soil excavated and replaced by the footing gives the net applied pressure which must be ≤ q_ns.

6. Safe and Allowable Bearing Capacity

6.1 Factor of Safety

6.2 Settlement Criterion

Even if the shear FOS is satisfactory, the footing must not settle more than allowable limits. IS 1904 specifies:

The allowable bearing capacity q_a is taken as the lesser of:

7. IS 6403:1981 — General Bearing Capacity Formula

IS 6403 uses the Meyerhof/Hansen general bearing capacity formula with correction factors for foundation shape, embedment depth, and load inclination:

7.1 Shape Factors (IS 6403)

7.2 Depth Factors (IS 6403 / Hansen)

7.3 Inclination Factors (IS 6403)

8. Effect of Water Table

The presence of the water table reduces the effective unit weight of soil, thereby reducing bearing capacity. Three cases arise based on the depth of the water table (D_w) relative to the foundation depth (D_f) and footing width (B):

8.1 Case I — Water Table at or Above Foundation Level (D_w ≤ D_f)

8.2 Case II — Water Table Below Foundation Level but Within B (D_f < D_w ≤ D_f + B)

8.3 Case III — Water Table Below D_f + B

8.4 Summary Table

9. Local Shear Failure — Reduced Parameters

For soils prone to local shear failure (loose sands, soft clays: relative density D_r < 65 %, or q_u < 100 kPa for clays), Terzaghi recommended using reduced shear strength parameters in the bearing capacity equation:

The FOS used for local shear is 2.5 (vs 3.0 for general shear) because the failure is more gradual and settlements provide warning before total collapse.

10. Worked Examples

Example 1 — Strip Footing on Sandy Clay

Problem: A 1.5 m wide strip footing is placed at a depth of 1.0 m below the ground surface. Soil properties: c = 30 kPa, φ = 20°, γ = 18 kN/m³. Water table is deep. Find the net safe bearing capacity (FOS = 3).

Bearing Capacity Factors (φ = 20°)

Gross Ultimate Bearing Capacity (Strip)

Net Ultimate and Net Safe Bearing Capacity

Example 2 — Square Footing on Pure Clay (φ = 0)

Problem: A 2 m × 2 m square footing is placed at D_f = 1.5 m in a saturated clay with undrained cohesion c_u = 60 kPa, φ_u = 0°, γ_sat = 19 kN/m³. Water table is at ground surface. Find net safe bearing capacity (FOS = 3).

For φ = 0: N_c = 5.14, N_q = 1.0, N_γ = 0

Example 3 — Effect of Water Table on Square Footing

Problem: A 1.8 m × 1.8 m square footing is at D_f = 1.2 m. Soil: c = 0, φ = 30°, γ_moist = 17 kN/m³, γ_sat = 19 kN/m³. Water table is at D_w = 1.5 m below ground (i.e., 0.3 m below the footing base). Find q_ns (FOS = 3).

Water Table Check

Bearing Capacity (φ = 30°, c = 0)

Example 4 — GATE-Style: Find Safe Load on Footing

Problem (GATE CE type): A 2.5 m × 2.5 m square footing is founded at 1.2 m depth. Soil: c = 25 kPa, φ = 25°, γ = 18 kN/m³. Water table is deep. FOS = 3. Find the safe load the footing can carry.

Bearing Capacity Factors (φ = 25°)

Square Footing

11. Common Mistakes

Mistake 1: Dividing Gross Ultimate Bearing Capacity by FOS

What happens: q_s = q_u/FOS is used instead of q_s = q_net,u/FOS + q. Dividing the gross capacity by FOS applies the safety factor to the pre-existing overburden — which has nothing to do with the structural load. This underestimates the safe bearing capacity and leads to over-designed, uneconomical foundations.

Fix: The factor of safety is applied only to the net ultimate bearing capacity: q_ns = q_net,u/FOS. The overburden q = γD_f is then added back to get the gross safe bearing capacity if needed.

Mistake 2: Using the Strip Footing Formula for Square/Circular Footings

What happens: q_u = cN_c + qN_q + 0.5γBN_γ is used for a square footing without the shape factors 1.3 (c-term) and 0.4 (N_γ-term). This underestimates the bearing capacity of square and circular footings by 20–30 %.

Fix: Square: multiply c-term by 1.3 and γ-term coefficient by 0.4. Circular: same 1.3 for c-term, 0.3 for γ-term. These shape factors account for 3D failure mechanism.

Mistake 3: Applying the Wrong Water Table Correction

What happens: When the water table is at the foundation level, some students replace γ only in the N_γ-term but forget that the q-term (γD_f) also needs correction if the water table is within the soil above the foundation. This is only relevant if the WT is above the foundation level.

Fix: Identify Case I, II, or III. Case I (WT above foundation level): correct γ in both q-term and N_γ-term. Case II (WT between D_f and D_f+B): q-term uses moist γ; N_γ-term uses interpolated γ_avg. Case III (WT below D_f+B): no correction needed.

Mistake 4: Ignoring the N_q−1 Correction in the Net Ultimate Bearing Capacity

What happens: Net bearing capacity is written as q_net,u = cN_c + qN_q + 0.5γBN_γ − q, forgetting that subtracting q from qN_q gives q(N_q−1), not cN_c + qN_q + 0.5γBN_γ − γD_f separately. Numerically identical but confusing if not tracked.

Fix: q_net,u = cN_c + q(N_q−1) + 0.5γBN_γ. This form clearly shows that the q-term contribution to net capacity is q(N_q−1), not qN_q.

Mistake 5: Confusing D_f and B in the Water Table Correction

What happens: The boundary between Case II and Case III is at D_w = D_f + B — beyond this depth, the water table has no effect. Students sometimes use D_f + D_f or 2B instead of D_f + B, leading to incorrect identification of the case.

Fix: The zone of influence of the foundation extends approximately one footing width B below the foundation base. The boundary is: if D_w > D_f + B → no correction needed (Case III).

12. Frequently Asked Questions

Q1. Why does bearing capacity increase with foundation depth D_f?

Terzaghi’s equation shows that bearing capacity increases with depth through two mechanisms. First, the overburden term q = γD_f increases directly with depth — deeper soil is subjected to a higher confining pressure, which increases its resistance to shear failure in the passive zone at the edge of the failure surface. Second, the depth factors (d_c, d_q in IS 6403) account for the shear strength of soil above the foundation level, which contributes additional resistance. Deeper foundations also encounter soil that has been compressed by the overburden and typically has higher density and strength. This is why, all else being equal, increasing foundation depth increases bearing capacity — a key design lever when the surface soil is weak.

Q2. What is the difference between the bearing capacity from Terzaghi’s formula and the Safe Bearing Capacity (SBC) used in practice?

Terzaghi’s formula gives the ultimate bearing capacity — the theoretical load at which the soil shears and the footing sinks catastrophically. The Safe Bearing Capacity (SBC) used in practice is the net ultimate bearing capacity divided by an appropriate factor of safety (typically 3), added back to the overburden. The SBC also incorporates the settlement criterion — if the settlement-limited pressure is lower than the shear-failure-limited net safe bearing capacity, the SBC is governed by settlement. In Indian practice, the SBC reported in a soil investigation report is the lesser of the two criteria (shear failure and settlement), making it the true allowable bearing capacity as per IS 6403 and IS 1904.

Q3. Why is the factor of safety 3 for general shear but 2.5 for local shear failure?

General shear failure is sudden and catastrophic — once the failure surface is fully developed, there is little warning. The higher FOS of 3 provides both a safety margin against uncertainty in soil parameters and serviceability margin against excessive settlement. Local shear failure, by contrast, is more gradual — settlements increase progressively and the failure is not sudden. The gradual nature provides some warning, justifying a slightly lower FOS of 2.5. However, local shear typically occurs in weaker, more compressible soils where settlements are large — the settlement criterion often governs before the shear FOS is reached.

Q4. How do the bearing capacity factors N_c, N_q, N_γ physically represent the three contributions to bearing capacity?

The three terms in the bearing capacity equation represent three independent contributions, each multiplied by a factor derived from the plasticity theory of the failure mechanism. cN_c represents the contribution of soil cohesion — how much the shear strength of the material itself (independent of normal stress) contributes to resisting failure. qN_q represents the contribution of overburden surcharge — the confining pressure from the soil above foundation level that strengthens the passive resistance zone in the failure mechanism. 0.5γBN_γ represents the contribution of soil weight within the failure wedge — the self-weight of the failing soil mass that must be overcome for the footing to punch through. For purely cohesive soil (φ = 0), only the first term applies. For cohesionless sand at the surface (no overburden), only the third term applies.