16-Civ-B3 Geotechnical Design · December 2017

Question 3 of 9: Terzaghi's bearing-capacity equation versus the general equation

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examinations, December 2017 — 16-Civ-B3 Geotechnical Design; three hours, open book, any non-communicating calculator. Section A holds five discussion questions worth 7 marks each of which four are marked; Section B holds four design questions worth 24 marks each of which three are marked, so the examinable total is 4 × 7 + 3 × 24 = 100 marks. All nine questions are worked below, because the set is a study resource rather than a marked script.

Reference texts. B. M. Das, Principles of Foundation Engineering, 8th–9th ed. (Cengage); B. M. Das, Principles of Geotechnical Engineering, 9th ed.; R. F. Craig / J. Knappett, Craig's Soil Mechanics, 9th ed.; Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed.; D. P. Coduto, Foundation Design: Principles and Practices; J. E. Bowles, Foundation Analysis and Design; ASTM D1586 (SPT), D5778 (CPTu), D2573 (field vane).

Source of design charts and assumed values (page 1, Note 6). Rankine active and passive coefficients from Das, Principles of Foundation Engineering, Ch. 7 (Eqs. 7.11 and 7.30); cantilever-wall stability procedure from Das Ch. 8 (Eqs. 8.11–8.14). Drilled-shaft adhesion factor alpha* = 0.55 and the 1.5 m surface exclusion from Reese & O'Neill (1989), tabulated in Das Ch. 12; bearing factor Nc* = 9 from Skempton (1951); block-failure check from Das Ch. 11 (Eq. 11.55). Compression index from Skempton's Cc = 0.009(LL − 10). Strain-influence factors and the C1, C2 corrections from Schmertmann, Hartman & Brown (1978) as presented in Das Ch. 5; Es = 500(N60 + 15) kPa from Das Table 5.7 (Bowles). Friction angle from the SPT via Wolff (1989), phi' = 27.1 + 0.3(N1)60 − 0.00054[(N1)60]², with the Liao & Whitman (1986) overburden correction; Vesic bearing-capacity, shape and depth factors from Das Tables 4.2 and 4.3. Every assumed value (unit weights of concrete, specific gravity for the void ratio, pile spacing, factors of safety) is stated in a callout beside the step that uses it.

Question 3: Terzaghi's bearing-capacity equation versus the general equation (7 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Terzaghi's 1943 equation, q_u = c'N_c + qN_q + 0.5 gamma B N_gamma, was derived for one idealised case: a rigid, rough, continuous (strip) footing bearing on the surface of a homogeneous, isotropic, semi-infinite soil mass, carrying a vertical concentric load, with the soil above founding level replaced by an equivalent surcharge q = gamma D_f. Every limitation follows from one of those idealisations, and the general equation of Meyerhof (1963), extended by Hansen and Vesic, removes them one at a time by attaching dimensionless correction factors to each of the three terms: q_u = c'N_cF_csF_cdF_ci + qN_qF_qsF_qdF_qi + 0.5 gamma B N_gammaF_{gamma s}F_{gamma d}F_{gamma i}.

Shape. Terzaghi solved the plane-strain problem and then handled square and circular footings with two fixed empirical multipliers (1.3 c'N_c and 0.4 gamma B N_gamma for a square, 1.3 and 0.3 for a circle). There is nothing for a rectangle of arbitrary B/L, so a 2 m by 6 m footing has to be forced into one of the three cases. The general equation gives continuous shape factors, F_qs = 1 + (B/L)tan phi', F_{gamma s} = 1 − 0.4(B/L), F_cs = 1 + (B/L)(N_q/N_c), which reduce correctly to unity for a strip.

Depth. Terzaghi treats the soil between the ground surface and founding level as a weightless surcharge and ignores its shear strength completely. For a footing at any appreciable embedment this is significantly conservative, typically by 10 to 30 per cent. The general equation restores that resistance through depth factors F_qd = 1 + 2 tan phi'(1 − sin phi')²(D_f/B) and F_cd, with separate expressions for D_f/B greater than one.

Load inclination and eccentricity. Terzaghi admits only a vertical, centrally applied load. A footing under a retaining wall, a bracing pile cap or a portal frame column carries a horizontal component and a moment, and inclination is severe: F_{gamma i} = (1 − beta/phi')² collapses towards zero as the load inclination beta approaches the friction angle. The general equation supplies F_ci, F_qi and F_{gamma i}, and eccentricity is handled by Meyerhof's effective-area rule, B' = B − 2e, which Terzaghi's formulation has no way of expressing.

Failure mode, geometry and soil idealisation. Terzaghi assumes general shear failure with a fully developed log-spiral mechanism; local or punching shear in loose sand or soft clay is handled only by the crude expedient of substituting c'* = (2/3)c' and tan phi'* = (2/3)tan phi'. He assumes the failure wedge beneath the base rises at the friction angle rather than at 45 + phi'/2, which is why his N_gamma differs numerically from Meyerhof's and Vesic's. He assumes a horizontal ground surface and a horizontal footing base, whereas Hansen's and Vesic's forms carry ground-slope and base-tilt factors for footings on or near a slope. And both formulations share the assumption of a single homogeneous stratum, so neither handles layering, a stiff crust over soft clay or a soil whose compressibility limits the mobilised capacity — for that, Vesic's rigidity-index correction or a layered-soil solution is needed.

What Terzaghi is still good for. Because it errs on the safe side and needs only phi', c', gamma, B and D_f, it remains a sound first estimate for a shallow strip or square footing under a vertical load, and it is the version most exam questions and many codes still quote. The general equation should be used whenever the footing is rectangular, embedded more than about half its width, or loaded eccentrically or at an angle — which in practice is most real footings.

Topic. Scope and correction factors of the classical versus the general bearing-capacity formulations.

Key relations. Terzaghi strip: q_u = c'N_c + qN_q + 0.5 gamma B N_gamma, with N_q = e^{(1.5π − phi')tan phi'}/[2cos²(45 + phi'/2)]. General: the same three terms each multiplied by shape, depth and inclination factors; N_q = e^{π tan phi'}tan²(45 + phi'/2) (Reissner), N_c = (N_q − 1)cot phi' (Prandtl), N_gamma = 2(N_q + 1)tan phi' (Vesic).

Why this works. All bearing-capacity formulae are upper- or lower-bound plasticity solutions for the same log-spiral mechanism; they differ in what is assumed about the wedge angle, the base roughness and the boundaries. Because superposition of the three terms is itself an approximation (the true solution is not the sum of the three separate problems), attaching empirical factors calibrated against model and full-scale tests is a legitimate and, in practice, more accurate way to extend the solution than re-deriving it for every geometry.

Common pitfall. Mixing N factors from one author with correction factors calibrated for another — Terzaghi's N_gamma with Vesic's shape factors, for instance — and applying depth factors to a footing whose surrounding soil may later be excavated for services, which removes exactly the resistance the factor credits.

Source. Das, Principles of Foundation Engineering, Ch. 4, Sections 4.3 (Terzaghi), 4.6 (general equation), Tables 4.1–4.3; Canadian Foundation Engineering Manual, 4th ed., Ch. 10.

← Question 2 Next: Question 4 →