BE Civil Engineering (IOE, TU) Foundation Engineering (IOE, CE 701) Question Paper 2077 Nepal

Terzaghi's equation for a strip footing (general shear):

$$q_u = c\,N_c + \gamma D_f\,N_q + 0.5\,\gamma B\,N_\gamma$$

Step 1 — Surcharge at founding level.

$$q = \gamma D_f = 18 \times 1.5 = 27\text{ kPa}$$

Step 2 — Substitute the three terms.

$$q_u = 212.28 + 200.88 + 80.51 = \mathbf{493.67\ kPa}$$

Step 3 — Net ultimate bearing capacity.

$$q_{nu} = q_u - \gamma D_f = 493.67 - 27 = \mathbf{466.67\ kPa}$$

Step 4 — Net safe bearing capacity (FoS = 3).

$$q_{ns} = \frac{q_{nu}}{F} = \frac{466.67}{3} = \mathbf{155.56\ kPa}$$

Step 5 — Safe load per metre run. For a 1 m length of strip footing, the loaded area $= B \times 1 = 1.8\text{ m}^2$.

$$Q_{safe} = q_{ns} \times B \times 1 = 155.56 \times 1.8 = \mathbf{280.0\ kN \ per\ metre\ run}$$

Note: The net safe value is used because the soil already carries the overburden $\gamma D_f$ before construction; the structure only adds net pressure.

Terzaghi's equation for a square footing:

$$q_u = 1.3\,c\,N_c + \gamma D_f\,N_q + 0.4\,\gamma B\,N_\gamma$$

Since the sand is cohesionless, $c = 0$ and the first term vanishes.

Step 1 — Surcharge.

$$q = \gamma D_f = 18 \times 1.2 = 21.6\text{ kPa}$$

Step 2 — Two contributing terms.

$$q_u = 616.03 + 386.93 = \mathbf{1002.96\ kPa}$$

Step 3 — Net ultimate.

$$q_{nu} = q_u - \gamma D_f = 1002.96 - 21.6 = 981.36\text{ kPa}$$

Step 4 — Net safe bearing capacity (FoS = 3).

$$q_{ns} = \frac{981.36}{3} = 327.12\text{ kPa}$$

Step 5 — Net safe column load.

$$Q_{safe} = q_{ns} \times A = 327.12 \times (2.0 \times 2.0) = \mathbf{1308.5\ kN}$$

(c) Effect of rising water table: If the water table rises to the footing base, the effective unit weight in the self-weight ($0.4\,\gamma B N_\gamma$) term reduces to the submerged value $\gamma' \approx 18 - 9.81 = 8.19\text{ kN/m}^3$, roughly halving that term and lowering $q_u$; the bearing capacity therefore decreases (the surcharge term, lying above the base, is unaffected for a table exactly at the base).

Step 1 — Locate the compressible layer and its mid-depth. Clay layer: $5.0\text{ m}$ to $7.0\text{ m}$ → thickness $H = 2.0\text{ m}$, mid-depth at $6.0\text{ m}$ below ground. Depth of the mid-plane below the footing base ($D_f = 2.0\text{ m}$):

$$z = 6.0 - 2.0 = 4.0\text{ m}$$

Step 2 — Stress increment by the 2:1 method (square footing).

$$\Delta\sigma = \frac{q\,B^2}{(B+z)^2} = \frac{90 \times 2.5^2}{(2.5+4.0)^2} = \frac{90 \times 6.25}{42.25} = \mathbf{13.31\ kPa}$$

Step 3 — Initial effective overburden $\sigma_0'$ at mid-clay (6.0 m).

$$\sigma_0' = 36.0 + 34.76 = \mathbf{70.76\ kPa}$$

Step 4 — Primary consolidation settlement (normally consolidated clay).

$$S_c = \frac{C_c\,H}{1+e_0}\,\log_{10}\!\left(\frac{\sigma_0' + \Delta\sigma}{\sigma_0'}\right)$$

With $H = 2000\text{ mm}$:

$$S_c = \frac{0.25 \times 2000}{1+0.72}\,\log_{10}\!\left(\frac{70.76 + 13.31}{70.76}\right)$$ $$= \frac{500}{1.72}\,\log_{10}(1.1881) = 290.7 \times 0.07486$$ $$S_c = \mathbf{21.8\ mm}$$

Conclusion: The estimated primary consolidation settlement is about 22 mm, which is within the usual permissible limit (≈ 25 mm for isolated footings on clay), so the foundation is acceptable on settlement grounds.

Part (a) — Single pile capacity ($\alpha$-method).

Base area: $A_b = \dfrac{\pi}{4}D^2 = \dfrac{\pi}{4}(0.5)^2 = 0.1963\text{ m}^2$

Shaft area: $A_s = \pi D L = \pi \times 0.5 \times 12 = 18.850\text{ m}^2$

End-bearing resistance:

$$Q_b = c_u N_c A_b = 60 \times 9 \times 0.1963 = 106.0\text{ kN}$$

Skin-friction resistance:

$$Q_s = \alpha\,c_u\,A_s = 0.45 \times 60 \times 18.850 = 508.9\text{ kN}$$

Ultimate single-pile capacity:

$$Q_u = Q_b + Q_s = 106.0 + 508.9 = \mathbf{614.9\ kN}$$

Allowable (FoS = 2.5):

$$Q_{all} = \frac{614.9}{2.5} = \mathbf{246.0\ kN}$$

Part (b) — Group efficiency (Converse–Labarre).

$$\theta = \tan^{-1}\!\left(\frac{D}{s}\right) = \tan^{-1}\!\left(\frac{0.5}{1.5}\right) = 18.43^\circ$$

With $m = 3$ rows, $n = 3$ columns:

$$E_g = 1 - \frac{\theta}{90}\cdot\frac{(n-1)m + (m-1)n}{m\,n}$$ $$= 1 - \frac{18.43}{90}\cdot\frac{(2)(3) + (2)(3)}{9} = 1 - 0.2048\times\frac{12}{9}$$ $$E_g = 1 - 0.2731 = \mathbf{0.727\ (72.7\%)}$$

Group capacity (individual-pile action):

$$Q_{g} = E_g \times (m n)\times Q_u = 0.727 \times 9 \times 614.9 = \mathbf{4023\ kN\ (ultimate)}$$

Block-failure check (for governance): Group plan $B_g = L_g = 2s + D = 2(1.5)+0.5 = 3.5\text{ m}$.

$$Q_{block} = c_u N_c B_g L_g + c_u(4 B_g)L = 60\times9\times3.5\times3.5 + 60\times(4\times3.5)\times12$$ $$= 6615 + 10080 = 16{,}695\text{ kN}$$

Since the block-failure capacity ($\approx 16{,}700\text{ kN}$) is far larger than the individual-pile-action capacity ($\approx 4023\text{ kN}$), individual-pile action governs. The design group capacity is therefore $\approx 4023\text{ kN}$ ultimate, or $\approx 1609\text{ kN}$ allowable at FoS 2.5.

Step 1 — Active earth pressure (Rankine, $\phi = 30^\circ$).

$$K_a = \frac{1-\sin\phi}{1+\sin\phi} = \frac{1-0.5}{1+0.5} = 0.333$$

Thrust from backfill (triangular), acting at $H/3 = 2.0\text{ m}$:

$$P_{a1} = \tfrac12 K_a \gamma H^2 = 0.5\times0.333\times18\times6^2 = 108.0\text{ kN}$$

Thrust from surcharge (rectangular), acting at $H/2 = 3.0\text{ m}$:

$$P_{a2} = K_a\, q\, H = 0.333\times10\times6 = 20.0\text{ kN}$$

Total horizontal thrust: $P_h = 108.0 + 20.0 = 128.0\text{ kN}$

Step 2 — Overturning moment about the toe.

$$M_o = P_{a1}\!\times\!2.0 + P_{a2}\!\times\!3.0 = 216.0 + 60.0 = 276.0\text{ kN·m}$$

Step 3 — Resisting (stabilising) moments — weights about the toe. Stem height $= 6.0 - 0.6 = 5.4\text{ m}$.

(a) Factor of safety against overturning:

$$FoS_{ot} = \frac{M_r}{M_o} = \frac{1083.5}{276.0} = \mathbf{3.93} \ \ge 2 \ \checkmark$$

(b) Factor of safety against sliding. Base friction: $F_r = W\tan\delta = 401.76\times\tan 20^\circ = 146.2\text{ kN}$. Passive resistance in front of toe (to $1.5\text{ m}$):

$$P_p = \tfrac12 K_p \gamma D_f^2 = 0.5\times3\times18\times1.5^2 = 60.75\text{ kN}$$ $$FoS_{sl} = \frac{F_r + P_p}{P_h} = \frac{146.2 + 60.75}{128.0} = \mathbf{1.62} \ \ge 1.5 \ \checkmark$$

(c) Location of resultant on the base. Distance of resultant from toe:

$$\bar{x} = \frac{M_r - M_o}{W} = \frac{1083.5 - 276.0}{401.76} = 2.01\text{ m}$$

Eccentricity: $e = \dfrac{B}{2} - \bar{x} = 2.25 - 2.01 = 0.24\text{ m} < \dfrac{B}{6} = 0.75\text{ m}$.

The resultant lies within the middle third → no tension at the heel. Base pressures:

$$q_{max,min} = \frac{W}{B}\left(1 \pm \frac{6e}{B}\right) = \frac{401.76}{4.5}\left(1 \pm \frac{6\times0.24}{4.5}\right)$$ $$q_{max} = 89.28(1.32) = \mathbf{117.9\ kPa}, \quad q_{min} = 89.28(0.68) = \mathbf{60.7\ kPa}$$

Conclusion: The wall is safe in overturning (3.93) and sliding (1.62), the resultant is within the middle third, and base pressures are compressive throughout — the section is satisfactory.

(a) Purpose of subsurface investigation. A site investigation is carried out to characterise the soil/rock beneath a proposed structure so that a safe and economical foundation can be designed. It establishes ground conditions, identifies hazards, and supplies design parameters.

Four pieces of information it should provide:

1. Soil profile / stratification — sequence, thickness and lateral extent of layers.
2. Groundwater table location and its seasonal variation.
3. Engineering properties — shear strength ($c$, $\phi$, $c_u$), compressibility ($C_c$, $m_v$), and unit weights.
4. Bearing capacity / allowable pressure and settlement characteristics, plus aggressivity of soil/water (e.g. sulphates) and seismic considerations.

(Acceptable alternatives: depth to bedrock, density/consistency, SPT/CPT profiles.)

(b) SPT overburden correction.

$$C_N = \sqrt{\frac{100}{\sigma_v'}} = \sqrt{\frac{100}{90}} = \sqrt{1.111} = 1.054$$ $$N_1 = C_N \times N = 1.054 \times 18 = 18.97 \approx \mathbf{19}$$

The corrected blow count is about 19. Because the in-situ overburden (90 kPa) is below the reference 100 kPa, $C_N > 1$, so the field value is increased slightly. (If a dilatancy correction for fine saturated sands with $N>15$ were also required, $N' = 15 + 0.5(N_1-15) = 15 + 0.5(4) = 17$.)

(a) Settlement scaling for footings on sand. For cohesionless soils the settlement of the prototype footing relative to the test plate is:

$$S_f = S_p\left[\frac{B_f\,(B_p + 0.3)}{B_p\,(B_f + 0.3)}\right]^2$$

where all widths are in metres.

Given $S_p = 5\text{ mm}$, $B_p = 0.3\text{ m}$, $B_f = 2.0\text{ m}$:

$$S_f = 5\left[\frac{2.0\,(0.3+0.3)}{0.3\,(2.0+0.3)}\right]^2 = 5\left[\frac{2.0\times0.6}{0.3\times2.3}\right]^2 = 5\left[\frac{1.2}{0.69}\right]^2$$ $$S_f = 5\times(1.739)^2 = 5\times3.025 = \mathbf{15.1\ mm}$$

The larger footing settles about 15 mm, roughly three times the plate settlement, because a wider footing stresses a much deeper zone of soil.

(b) Limitations of the plate load test.

1. Shallow influence zone — the plate stresses soil only to about twice its own width; weak strata below the (much deeper) zone influenced by a full-size footing are not detected (the size effect).
2. Short duration — the test reflects mainly immediate settlement; in clays it cannot capture long-term consolidation settlement, so results on clay are unreliable.

(Other acceptable points: results sensitive to water table and disturbance; gives a local value only.)

(a) Cantilever sheet pile — principle and pressure diagram. A cantilever sheet pile derives all its stability from passive resistance of the soil over its embedded (driven) depth; it behaves like a vertical cantilever fixed in the soil below dredge level. Under the active thrust from the retained side, the wall rotates about a point near its toe. Above that pivot, active pressure acts behind and passive in front; below the pivot the pressures reverse.

  Retained side            Dredge line
  (active, behind) |======================  ground surface
                   |\ active                 
                   | \  (Ka γ z)             
                   |  \                       
  ----- dredge ----+---\--------------------- 
  level            |   /  passive (front)    
                   |  / over embedment       
            pivot ~+ X  <- net pressure       
                   | \  reverses below pivot 
                   |__\  toe                  

Why limited to small heights: the required embedment and the bending moment grow rapidly (roughly with the cube of the retained height), so for large heights the section becomes uneconomically heavy and deflections excessive. Hence cantilever sheet piles are normally used only for heights up to about $4$–$5\text{ m}$; beyond this, anchored or braced walls are preferred.

(b) Active thrust over the exposed height (above dredge level).

$$K_a = \frac{1-\sin 32^\circ}{1+\sin 32^\circ} = \frac{1-0.5299}{1+0.5299} = \frac{0.4701}{1.5299} = 0.307$$

Active thrust on the $3.0\text{ m}$ exposed height (triangular distribution):

$$P_a = \tfrac12 K_a \gamma H^2 = 0.5\times0.307\times17\times3.0^2 = 0.5\times0.307\times17\times9$$ $$P_a = \mathbf{23.5\ kN \ per\ metre\ run}$$

It acts horizontally at the centroid of the triangle, i.e. at $H/3 = 3.0/3 = \mathbf{1.0\ m}$ above the dredge line. (Additional net pressures over the embedded depth must be added for the full design.)

(a) Components of a well foundation.

        Pier / abutment
   ============================
   |        Well cap          |   <- transfers load to steining
   |==========================|
   |  ||                  ||  |
   |  ||  Steining        ||  |   <- thick masonry/RCC wall (main body)
   |  ||  (dredge hole)   ||  |
   |  ||                  ||  |
   |__||__________________||__|
      \   bottom plug      /      <- concrete seal at base
       \__________________/
         cutting edge (sharp)     <- knife edge that eases sinking
   ----- top plug & sand filling inside dredge hole -----

Main components and their functions:

• Cutting edge: the sharp lower edge of the curb that cuts into and penetrates the soil under the well's self-weight, easing sinking.
• Well curb: transition ring above the cutting edge carrying the steining load to the edge.
• Steining: the thick wall (the body of the well) that resists vertical and lateral loads and provides weight for sinking.
• Bottom plug: a concrete seal cast at the founding level inside the curb; it transmits the pier load to the bearing stratum and prevents soil from entering or water from seeping up.
• Sand filling and top plug: stabilise the well and seal the top; well cap distributes the pier load to the steining.

(b) Sinking difficulties and remedies.

(Other acceptable points: sand blowing/quick condition — control dredging rate and maintain water head inside; tilting near a sloping rock — provide differential dredging.)

(a) Common ground-improvement techniques and suitable soils.

(Other acceptable answers: grouting — fissured rock/coarse soils; lime or cement stabilisation — clayey soils; geotextile/geogrid reinforcement — soft subgrades.)

(b) Mechanism of vibro-compaction. A vibrating poker (vibroflot) is lowered into the loose sand. Its horizontal vibrations, aided by water jetting, momentarily liquefy and break down the loose grain structure, allowing the sand particles to rearrange into a denser packing under gravity. As the probe is withdrawn in stages, granular backfill is added to compensate for the volume reduction. The result is increased relative density, higher friction angle and stiffness, and reduced liquefaction potential.

(a) Shallow vs. deep foundation.

In short, a shallow foundation spreads the load onto competent soil near the surface, whereas a deep foundation transmits it through weak upper layers to a stronger stratum further down.

(b) When a raft (mat) is preferred. A raft foundation is preferred when:

1. The soil is weak/compressible and the allowable bearing pressure is low, so isolated footings would need very large areas — typically when footings would cover more than about 50% of the plan area.
2. The structure carries heavy or unevenly distributed column loads, and a continuous mat helps to reduce differential settlement by averaging the contact pressure.
3. There is a need to bridge over local soft pockets or to provide a watertight base below the water table.

Floating (compensated) raft — advantage: It is particularly advantageous on deep soft clay, where the soil excavated for a basement is made (nearly) equal in weight to the structure. The net increase in stress on the clay is then small or zero, so consolidation settlement is minimised — ideal for heavy buildings on soft compressible ground.