--- title: "Terzaghi's bearing capacity theory" subject: "Foundation Engineering" chapter: 6 key: "semester_5/foundation_engineering:terzaghi-theory-assumptions" source: https://www.bctengineer.com/q/semester_5/foundation_engineering/terzaghi-theory-assumptions site: BCT Engineering --- # Terzaghi's bearing capacity theory Foundation Engineering · Chapter 6: Bearing Capacity and Settlement of Shallow Foundations - Asked in: 2082 Bhadra, 4 marks - Video: [Terzaghi's bearing capacity theory](https://www.youtube.com/watch?v=lQ2-ouCE2KQ) — Civil Engineer - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/terzaghi-theory-assumptions - Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-6 ## Question Discuss Terzaghi's bearing capacity theory. Also state the assumptions. ## Short answer Terzaghi's theory gives ultimate bearing capacity for a shallow strip footing with a rough base, considering equilibrium of a failure wedge. Assumptions: strip footing (plane strain), Df ≤ B, rough base, homogeneous isotropic soil obeying Mohr-Coulomb, shear strength above base ignored (replaced by surcharge q = γ Df), general shear failure, vertical central load. The equation is qu = cNc + γ Df Nq + 0.5 γ B Nγ. For local shear, c and tanφ are reduced to two-thirds. ## Answer **Terzaghi's theory (1943)** gives the ultimate bearing capacity of a shallow footing by considering the equilibrium of a failure wedge. It is the most widely used classical theory and extends Prandtl's and Reissner's work by including soil weight and a rough base. #### Assumptions 1. The footing is a long **strip** (plane strain), and $D_f \le B$ (shallow foundation). 2. The base is **rough**, so the soil wedge directly below it stays elastic and moves down with the footing. 3. The soil is **homogeneous and isotropic**, and obeys Mohr–Coulomb's law $\tau = c + \sigma\tan\phi$. 4. The shear strength of the soil above the base level is ignored. It is replaced by a uniform **surcharge** $q = \gamma D_f$. 5. Failure occurs in **general shear** (for local shear, strength is reduced to two-thirds). 6. The load is vertical and central. #### Failure mechanism ```text q_u ____________ | I | II /\ /\ II III / \ / \ III ``` Under the footing, **zone I** is an elastic (triangular) wedge with base angles $\phi$. **Zone II** is a radial shear zone bounded by a logarithmic spiral. **Zone III** is a passive Rankine zone. #### Derivation (outline) Considering vertical equilibrium of the wedge: $q_u B + W = 2P_p + 2C\sin\phi$, where $C = \dfrac{cB}{2\cos\phi}$ is the cohesive force on each inclined face, $W$ the weight of the wedge, and $P_p$ the passive resistance from zones II and III (made of three parts, due to $c$, $q$ and $\gamma$). This gives $$ q_u = cN_c + \gamma D_f N_q + 0.5\gamma B N_\gamma \quad (\text{strip}) $$ For other shapes: square $q_u = 1.3cN_c + \gamma D_fN_q + 0.4\gamma BN_\gamma$; circular $q_u = 1.3cN_c + \gamma D_fN_q + 0.3\gamma BN_\gamma$. $N_c, N_q, N_\gamma$ depend only on $\phi$ (e.g. for $\phi = 35^\circ$: $57.8, 41.4, 42.4$). For local shear use $c' = \tfrac{2}{3}c$, $\tan\phi' = \tfrac{2}{3}\tan\phi$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why does Terzaghi assume a rough base? A. So the soil wedge below the footing stays elastic and moves down with it B. So the soil above the base can carry shear C. So the footing can be circular D. So the soil weight can be ignored Answer: A. Base friction keeps zone I as an elastic wedge with base angles phi moving with the footing. (Key point: Rough base, elastic wedge) 2. How does Terzaghi treat the soil above the footing base level? A. Its shear strength is added in full B. Its shear strength is ignored and it is replaced by a surcharge q = gamma*Df C. It is assumed weightless D. It is treated as part of the failure wedge Answer: B. The overlying soil only gives a uniform surcharge load. (Key point: Surcharge) 3. Which zone is bounded by a logarithmic spiral? A. Zone I, the elastic wedge B. Zone III, the passive zone C. Zone II, the radial shear zone D. The surcharge zone Answer: C. Zone I is elastic, II is radial shear (log spiral), III is Rankine passive. (Key point: Failure zones) 4. Which is Terzaghi's strip footing equation? A. qu = 1.3cNc + gamma*Df*Nq + 0.4*gamma*B*Ngamma B. qu = cNc + gamma*Df*Nq + 0.3*gamma*B*Ngamma C. qu = cNc + 0.5*gamma*B*Nq D. qu = cNc + gamma*Df*Nq + 0.5*gamma*B*Ngamma Answer: D. The strip form uses 0.5 for the gamma term and no shape factor on c. 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