Shallow Foundation and Bearing Capacity

Question 1
A square footing is to be designed to carry a column load of 500 \mathrm{kN} which is resting on a soil stratum having the following average properties: bulk unit weight =19 \mathrm{kN} / \mathrm{m}^{3}; angle of internal friction =0^{\circ}, and cohesion =25 \mathrm{kPa}. Considering the depth of the footing as 1 \mathrm{~m} and adopting Meyerhof's bearing capacity theory with a factor of safety of 3 , the width of the footing (in \mathrm{m} ) is ____ (round off to one decimal place)
[Assume the applicable shape and depth factor values as unity; ground water level at greater depth.]
GATE CE 2023 SET-2   Geotechnical Engineering
Question 1 Explanation: 
As per Meyerhoff's approach
q_{u}=C N_{c}\left[i_{c} S_{c} d_{c}\right]+q N_{q}\left[{ }_{q} S_{q} d_{r}\right]+\frac{1}{2} \beta \gamma N_{r}\left[i_{r} S_{r} d_{r}\right]
Given; shape factor S_{c}, S_{q}, S_{r}=1
depth factor d_{c}, d_{v}, d_{r}=1

Assuming no inclination of loading hence
i_{c} i_{q} i_{r}=1
For \phi=0^{\circ}
\begin{aligned} & \mathrm{N}_{\mathrm{c}}=5.14 \\ & N_{\mathrm{q}}=1 \\ & N_{r}=0 \\ & \therefore \quad \mathrm{q}_{\mathrm{u}}=25 \times 5.14 \times 1 \times 1 \times 1+19 \times 1 \times \\ & 1 \times 1 \times 1 \\ & =147.5 \mathrm{kPa} \\ & \mathrm{q}_{\mathrm{nu}}=\mathrm{q}_{\mathrm{u}}-\mathrm{rD}_{\mathrm{f}} \\ & =147.5-19 \times 1 \\ & =128.5 \mathrm{kPa} \end{aligned}
q_{n s}=\frac{q_{n u}}{F O S}=\frac{128.5}{3}=42.83 \mathrm{kPa}
Hence, \frac{500}{B^{2}} \leq 42.83
or B \geq 3.416 m
Hence B=3.4 m
Question 2
The reason(s) of the nonuniform elastic settlement profile below a flexible footing, resting on a cohesionless soil while subjected to uniform loading, is/are:
Variation of friction angle along the width of the footing
Variation of soil stiffness along the width of the footing
Variation of friction angle along the depth of the footing
Variation of soil stiffness along the depth of the footing
GATE CE 2023 SET-2   Geotechnical Engineering
Question 2 Explanation: 
The modulus of elasticity varies with width of footing so there is a variation in the stiffness along the width of footing.

Question 3
A square footing of size 2.5 \mathrm{~m} \times 2.5 \mathrm{~m} is placed 1.0 \mathrm{~m} below the ground surface on a cohesion homogeneous soil stratum. Considering that the groundwater table is located at the base of the footing, the unit weights of soil above and below the groundwater table are 18 \mathrm{kN} / \mathrm{m}^{3} and 20 \mathrm{kN} / \mathrm{m}^{3}, respectively, and the bearing capacity factor N_{q} is 58 , the net ultimate baring capacity of the soil is estimated as 1706 \mathrm{kPa} (unit weight of water 10 \mathrm{kN} / \mathrm{m}^{3} ).
Earlier, a plate load test was carried out with a circular place of 30 \mathrm{~cm} diameter in the same foundation pit during a dry season, when the water table was located beyond the plate influence zone. Using Terzaghi's bearing capacity formulation, what is the ultimate bearing capacity (in \mathrm{kPa}_{-}of the plate ?
GATE CE 2023 SET-1   Geotechnical Engineering
Question 3 Explanation: 

\mathrm{q}_{\mathrm{nu}}=1706 \mathrm{kPa}, \quad \gamma_{\mathrm{w}}=10 \mathrm{kN} / \mathrm{m}^{3}
Plate load test data:
dia. (d) =30 \mathrm{~cm} with no water able effect.
\mathrm{q}_{\mathrm{u} \text { plate }}=\text { ? }

For footing :
\begin{aligned} & \mathrm{q}_{\mathrm{nu}}=1706 \mathrm{kPa} \\ & \mathrm{q}_{\mathrm{nu}}=0+\mathrm{q}\left(\mathrm{N}_{\mathrm{q}}-1\right)+0.4 \mathrm{~B} \gamma \mathrm{N}_{\gamma} \\ & \mathrm{q}_{\mathrm{nu}}=18 \times 1 \times(58-1 \times)+0.4 \times 2.5 \times(20-10 \times) \times \mathrm{N}_{\gamma} \\ & \mathrm{N}_{\gamma}=\frac{(1706-18 \times 57)}{(0.4 \times 2.5 \times 10)} \Rightarrow \mathrm{N}_{\gamma}=68 \\ & \mathrm{q}_{\text {yplage }}=0.3 \times \mathrm{d} \times \gamma_{\mathrm{t}} \times \mathrm{N}_{\gamma} \\ & \mathrm{q}_{\text {uplate }}=0.3 \times 0.3 \times 18 \times 68 \\ &=110.16 \mathrm{kPa} \end{aligned}
Question 4
Read the following statements:

(P) While designing a shallow footing in sandy soil, monsoon season is considered for critical design in terms of bearing capacity.
(Q) For slope stability of an earthen dam, sudden drawdown is never a critical condition.
(R) In a sandy sea beach, quicksand condition can arise only if the critical hydraulic gradient exceeds the existing hydraulic gradient.
(S) The active earth thrust on a rigid retaining wall supporting homogeneous cohesionless backfill will reduce with the lowering of water table in the backfill.

Which one of the following combinations is correct?
(P)-True, (Q)-False, (R)-False, (S)-False
(P)-False, (Q)-True, (R)-True, (S)-True
(P)-True, (Q)-False, (R)-True, (S)-True
(P)-False, (Q)-True, (R)-False, (S)-False
GATE CE 2022 SET-2   Geotechnical Engineering
Question 4 Explanation: 
In monsoon season sand will be fully saturated hence this will be critical condition in designing of shallow foundation.
In case of sudden drawdown flow direction reverses hence for slope stability, it will be critical condition.
In sandy sea beach, quicksand condition can arise only if existing hydraulic gradient exceeds the critical hydraulic gradient.
Question 5
A square concrete pile of 10 m length is driven into a deep layer of uniform homogeneous clay. Average unconfined compressive strength of the clay, determined through laboratory tests on undisturbed samples extracted from the clay layer, is 100 kPa. If the ultimate compressive load capacity of the driven pile is 632 kN, the required width of the pile is _______ mm. (in integer)
(Bearing capacity factor N_c 9; adhesion factor \alpha =0.7 )
GATE CE 2022 SET-1   Geotechnical Engineering
Question 5 Explanation: 
\begin{aligned} Q_{up}&=q_bA_b+q_sA_s\\ C_u&=\frac{q_u}{2}=\frac{100}{2}=50kN/m^2\\ &=9 \times CB^2+\alpha \bar{C}(4BL)\\ 632kN&=9 \times 50 \times B^2 +0.7 \times 50(4B \times 10)\\ B&=0.4m\\ B&=400mm \end{aligned}

There are 5 questions to complete.

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