07-Str-B1 · May 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Examinations — May 2013 — 07-Str-B1 Geotechnical Design. Three-hour, OPEN-BOOK exam; any non-communicating calculator permitted (the candidate must record its make and model). Format: Section A carries five short-answer questions of 7 marks each, of which any FOUR are to be answered; Section B carries the long design questions at 24 marks each, of which any THREE are to be answered. The paper instructs candidates to state any interpretive assumptions and to identify the source of every design chart or assumed value used. Every question in both sections is worked below, because the set is intended as a study resource.
Reference texts: Das, B.M., Principles of Foundation Engineering (9th ed., Cengage) — shallow foundations, consolidation settlement, sheet-pile walls, retaining walls and drilled shafts; Das, B.M., Principles of Geotechnical Engineering (9th ed., Cengage) — method of slices, lateral earth pressure, consolidation theory; Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM, 4th ed., 2006) — Canadian practice for SPT interpretation, pile design, limit-states design and tolerable settlement; Craig, R.F. / Knappett, J.A., Craig's Soil Mechanics (8th ed., CRC Press) — effective stress, shear strength and slope stability; Duncan, J.M., Wright, S.G. & Brandon, T.L., Soil Strength and Slope Stability (2nd ed., Wiley) — choice of strength parameters and factors of safety for short- and long-term analyses.
NOTE 1 — question numbering in the source. The printed paper labels the retaining-wall problem (Figure 4) and the drilled-pier problem (Figure 5) both as "Question 9", while the Section B heading reads "answer any THREE of the following FOUR questions". Section B therefore contains five printed problems under four numbers. They are set out below as Question 9 (retaining wall) and Question 10 (drilled pier) in printed order, so that each can be referred to unambiguously; the marks shown are those printed against each problem.
NOTE 2 — dimensions scaled from Figure 1. Figure 1 is a hand-drawn slope on a 1 m × 1 m grid with no written dimensions other than $R=10$ m. The geometry used in Question 6 was scaled from that grid: slope height 7 m over a 9 m horizontal run, a 3 m thick lower layer, and the centre of the trial circle 1.4 m horizontally beyond the toe and 8.0 m above it. Every one of these values reproduces the drawing to within about 0.2 m (one fifth of a grid square). Check against the original if the paper is used for marking rather than study.
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.
The allowable bearing pressure of a footing is the smaller of two quite different quantities: the ultimate bearing capacity divided by a factor of safety, and the pressure that produces a settlement the structure can tolerate. Writing this as $$q_{\text{all}}=\min\left(\frac{q_u}{FS},\;q_{s}\right)$$ makes the point of the statement clear — the two criteria are checked independently, and for the great majority of real foundations the second one is the smaller.
The two criteria carry very different margins. Bearing capacity is a collapse limit state guarded by a factor of 2.5 to 3, or in limit-states form by a resistance factor near 0.5. Settlement is a serviceability limit state assessed at unfactored working load, with an effective factor of safety of 1.0 against the tolerable movement. A criterion applied with no margin will bind before a criterion applied with a margin of three, unless the ultimate capacity is genuinely small.
In sands the ultimate capacity is enormous and the tolerable settlement is tiny. A medium-dense sand with $\phi'=35^\circ$ gives $N_q\approx33$ and $N_\gamma\approx45$, so a 2 m square footing at 1.5 m depth has an ultimate capacity of several thousand kilopascals. No ordinary building applies anything approaching one third of that. What does bind is the settlement limit — conventionally 25 mm of total settlement and an angular distortion of about 1/500 for a framed building — which on sand is reached at a bearing pressure of a few hundred kilopascals. This is why the classical design methods for footings on sand, from Terzaghi and Peck through Meyerhof to Burland and Burbidge, are settlement methods from the outset: they return an allowable pressure for 25 mm of settlement and treat bearing capacity as a check made afterwards, not as the design equation.
In clays the mechanism differs but the conclusion is the same. A stiff clay has an ultimate capacity of $q_u\approx5.14 s_u+\gamma D_f$ which for $s_u=100$ kPa is well above 500 kPa, while the consolidation settlement of the clay stratum beneath is time-dependent, may continue for decades, and is frequently of the order of tens of millimetres for quite modest net pressures. Since $S_c\propto \log\!\big[(\sigma'_o+\Delta\sigma)/\sigma'_o\big]$, the settlement grows with the logarithm of the pressure increase while the capacity grows in proportion to $s_u$; the serviceability check is the one that runs out first. In soft normally consolidated clay both criteria become severe together, and it is only there that bearing capacity commonly governs a narrow footing.
Increasing the footing size sharpens, rather than relieves, the settlement criterion. Widening a footing on sand increases $q_u$ through the $\tfrac12\gamma B N_\gamma$ term, so the capacity criterion becomes easier. But a wider footing pushes its pressure bulb deeper — significant stress increase extends to roughly $2B$ below the base — so it stresses a greater depth of compressible soil and, at the same bearing pressure, settles more. For rafts and large mat foundations the bearing-capacity check is almost never the one that sizes the foundation.
Finally, structures are damaged by distortion, not by collapse. Cladding cracks, doors bind, services shear and floors go out of level at differential movements of 10 to 25 mm, long before any soil element approaches failure. Differential settlement, which is what actually damages a structure, arises from variability of the ground and of column loads and is conventionally taken as up to 75% of the total for footings on sand; keeping it inside tolerable limits is what fixes the design.
The proper qualification is that the statement is a generalisation, not a law. Bearing capacity does govern for footings on soft or organic clay and peat, for foundations carrying large inclined or eccentric loads where the effective area is small, for footings near a slope crest, and for any case where a punching or local shear mechanism can develop through a thin firm crust into soft material below. Canadian limit-states practice makes the discipline explicit: NBCC and CFEM require both a ULS check on geotechnical bearing resistance and an SLS check on settlement, and the design is not complete until both have been shown to be satisfied.
