Question 9 of 9: Schmertmann settlement, and ten more storeys

Question 9: Schmertmann settlement, and ten more storeys (24 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.

Given. Figure 5 shows a footing 2.5 m × 3 m in plan founded 2 m below ground in sand, carrying a net pressure of 50 kPa, with a three-band modulus profile. The zero of the depth axis is drawn at the underside of the footing — the horizontal E_s axis is the continuation of the line at the base of the footing — so all three bands are measured from founding level, not from the ground surface. Getting that datum wrong is the single largest error available in this question, and this subjectʹs papers use both conventions, so the drawing has to be read rather than assumed.

Find. The elastic settlement of the existing foundation by Schmertmannʹs strain-influence-factor method, then the settlement after ten more storeys, and a recommendation on whether the extension is safe.

Approach. Build the strain-influence diagram appropriate to the footingʹs aspect ratio, integrate I_zΔz/E_s band by band down to the depth of influence, and scale the result by the embedment and creep correction factors. Then repeat with the increased pressure and a longer creep period.

1. Settle the interpretation of the 50 kPa. With γ = 20 kN/m³ and D_f = 2.0 m the overburden removed at founding level is q = γD_f = 40 kPa. Reading the figureʹs 50 kPa as a gross pressure would make the net increase only 10 kPa and the embedment factor $$C_1 = 1 - 0.5\left[\frac{q}{\Delta q}\right] = 1 - 0.5\left[\frac{40}{10}\right] = -1$$ which is physically impossible. The 50 kPa is therefore the net pressure increase at founding level (a gross bearing pressure of 90 kPa), and that reading is adopted throughout. It is also the reading that makes the second half of the question sensible, since a five-storey building at 10 kPa per storey is exactly 50 kPa.
2. Choose the influence diagram. The aspect ratio is L/B = 3.0/2.5 = 1.2, nowhere near the L/B ≥ 10 that would justify the plane-strain form, so the axisymmetric (square/circular) diagram applies: $$\begin{aligned} I_z &= 0.1 \text{ at } z = 0 \\ I_z &= I_{zp} \text{ at } z = \tfrac{B}{2} = 1.25\ \text{m} \\ I_z &= 0 \text{ at } z = 2B = 5.0\ \text{m} \end{aligned}$$ The influence zone therefore closes 5.0 m below the base, which is above the 8 m boundary — so the 25 500 kPa band never enters the calculation at all. That third band is a distractor.
3. Evaluate the peak influence factor. The peak sits B/2 below the base, i.e. at 2.0 + 1.25 = 3.25 m below ground, where the effective overburden is $$\sigma^{\prime}_{zp} = \gamma(D_f + \tfrac{B}{2}) = 20(3.25) = 65.0\ \text{kPa}$$ so that $$\boxed{I_{zp} = 0.5 + 0.1\sqrt{\frac{\Delta q}{\sigma^{\prime}_{zp}}} = 0.5 + 0.1\sqrt{\frac{50}{65.0}} = 0.5877}$$ Note that σʹ_zp belongs at the depth of the peak, not at the base and not at mid-influence.
4. Sum the influence over the modulus bands. Splitting the influence zone at the peak (1.25 m) and at the modulus boundary (2.0 m), and taking I_z at the middle of each sub-layer:

5. Apply the two correction factors. The embedment factor credits the wall of the excavation with taking part of the load, and the creep factor accounts for the time-dependent strain of sand: $$\begin{aligned} C_1 &= 1 - 0.5\left[\frac{q}{\Delta q}\right] = 1 - 0.5\left[\frac{40}{50}\right] = 0.60 \\ C_2 &= 1 + 0.2\log_{10}\!\left(\frac{t}{0.1}\right) = 1 + 0.2\log_{10}(50) = 1.3398 \end{aligned}$$
6. Compute the settlement of the existing foundation. Assembling the parts, $$S_e = C_1 C_2 \Delta q \sum \frac{I_z}{E_s}\Delta z = 0.60(1.3398)(50)(6.4834\times 10^{-5})$$ $$\boxed{S_e = 2.606\ \text{mm}}$$ a very small settlement, which is what a lightly loaded footing on a medium dense to dense sand should give.
7. Recompute for ten additional storeys. At approximately 10 kPa per storey the extension adds 100 kPa, taking the net pressure to 50 + 100 = 150 kPa (a gross pressure of 190 kPa). Both the peak influence factor and the embedment factor change, because both depend on Δq: $$\begin{aligned} I_{zp} &= 0.5 + 0.1\sqrt{\frac{150}{65.0}} = 0.6519 \\ C_1 &= 1 - 0.5\left[\frac{40}{150}\right] = 0.8667 \end{aligned}$$ and, following the hint, the creep period is extended to 10 years so that C₂ = 1 + 0.2 log₁₀(100) = 1.4. Re-summing the three bands with the new I_zp gives ∑I_zΔz/E_s = 7.1576 × 10⁻⁵ m³/kN, so that $$\boxed{S_{e,\text{extended}} = 0.8667(1.4)(150)(7.1576\times 10^{-5}) = 13.03\ \text{mm}}$$ The settlement rises by a factor of 5.0 — note that this is less than the threefold increase in pressure, because C₁ improves as the net load grows relative to the overburden relief.

Recommendations, and is the extension safe?

On settlement, yes. A total elastic settlement of about 13 mm is well inside the 25 mm that is conventionally tolerable for a building on sand, and the margin is large enough to absorb a good deal of uncertainty in E_s. Extending the creep period further does not change that conclusion: at 25 years the estimate is 13.8 mm and at 50 years 14.3 mm, because C₂ grows only logarithmically. Sand also settles essentially as it is loaded, so most of the movement will occur during construction of the new storeys rather than after it, which is favourable for the finishes.

But settlement is not the question that decides it. Four checks matter more than the one the question asks for, and the recommendation has to name them:

1. Bearing capacity at the new pressure. The gross pressure rises from 90 to 190 kPa, and Figure 5 gives no strength parameters at all — only moduli. A bearing check is mandatory before the extension is approved, and it needs φʹ from an SPT or CPT programme. As an order of magnitude, inverting the usual E_s ≈ 500(N₆₀ + 15) correlation on the 20 000 kPa upper band implies N₆₀ ≈ 25 and hence φʹ of the order of 36° to 38°, which on a 2.5 m footing at 2 m depth would give an ultimate capacity of well over 1000 kPa and an ample margin. That is an indication, not a design: the correlation runs the wrong way and must be replaced by real data.
2. Structural capacity of the existing foundation and frame. The footing, its reinforcement and the columns above were designed for a five-storey building. Trebling the load is far more likely to exhaust the structural capacity of the existing concrete than the geotechnical capacity of the sand, and this is the check most likely to stop the project.
3. Differential settlement across the building. The calculation above is for one footing on an assumed uniform profile. If the sand varies across the footprint, or if the additional load is not distributed in proportion to the original column loads, the extra 10 mm of settlement will not be uniform, and it is differential movement that damages a structure. Additional boreholes across the footprint should be required.
4. Settlement already realised. The existing building has been in place for some years, so most of its 2.6 mm has already occurred. The additional movement the extension imposes is therefore about 10 mm, not 13 mm — a distinction that matters when assessing the effect on existing finishes, cladding and services.

Recommendation. The foundation is not settlement-critical and the ten additional storeys are geotechnically feasible on the evidence given. Approval should nonetheless be conditional on a site investigation that supplies strength parameters and confirms the modulus profile across the whole footprint, on a bearing-capacity check at the full 190 kPa gross pressure, and above all on a structural appraisal of the existing footings, columns and connections. Monitoring points should be installed on the existing structure and read through construction, so that the predicted 10 mm of additional movement is verified as it occurs rather than assumed.