A raft foundation of 6 m × 9 m is placed at a depth of 3 m in a cohesive soil having c = 120 kN/m ². The net ultimate bearing capacity of the soil using Terzaghi's theory will be.

Question

A raft foundation of 6 m &times; 9 m is placed at a depth of 3 m in a cohesive soil having c = 120 kN/m 2. The net ultimate bearing capacity of the soil using Terzaghi's theory will be.

Prepp Team · Accepted Answer

Raft Foundation Net Ultimate Bearing Capacity Calculation

This problem requires calculating the net ultimate bearing capacity of a raft foundation placed in cohesive soil using Terzaghi's bearing capacity theory.

Given Parameters:

Foundation dimensions (Rectangular): Breadth (B) = 6 m, Length (L) = 9 m
Depth of foundation ($D_f$) = 3 m
Soil type: Cohesive soil (Clay)
Cohesion (c) = 120 kN/m²

Applying Terzaghi's Bearing Capacity Theory for Cohesive Soil (ϕ = 0)

According to Terzaghi's theory, the ultimate bearing capacity ($q_u$) for a foundation is given by:

$q_u = c N_c + q N_q + \frac{1}{2} \gamma B N_{\gamma}$

For purely cohesive soil (when the angle of internal friction, $\phi = 0$), Terzaghi's bearing capacity factors are:

$N_c = 5.7$
$N_q = 1$
$N_{\gamma} = 0$

Substituting these values into the ultimate bearing capacity equation, we get:

$q_u = c(5.7) + q(1) + \frac{1}{2} \gamma B (0)$

$q_u = 5.7c + q$

Where $q = \gamma D_f$ is the overburden pressure at the foundation level. The term $\frac{1}{2} \gamma B N_{\gamma}$ becomes zero because $N_{\gamma} = 0$.

Calculating Net Ultimate Bearing Capacity

The net ultimate bearing capacity ($q_{nu}$) is the ultimate bearing capacity minus the overburden pressure $q$.

$q_{nu} = q_u - q$

Substituting the simplified $q_u$ for $\phi=0$:

$q_{nu} = (5.7c + q) - q$

$q_{nu} = 5.7c$

Considering Shape Factors in Terzaghi's Theory

Terzaghi's theory includes shape factors to account for the geometry of the foundation (square, rectangle, circle) compared to a strip footing. For a rectangular foundation on cohesive soil ($\phi=0$), the shape factor is applied to the $N_c$ term. The modified $N_c$ value ($N_c'$) is given by:

$N_c' = N_c \left(1 + 0.3 \frac{B}{L}\right)$

Where $B$ is the breadth and $L$ is the length of the foundation. For a rectangular foundation, $B < L$.

In this problem, $B = 6$ m and $L = 9$ m.

$\frac{B}{L} = \frac{6}{9} = \frac{2}{3} \approx 0.667$

The modified $N_c'$ is:

$N_c' = 5.7 \left(1 + 0.3 \times \frac{2}{3}\right)$

$N_c' = 5.7 (1 + 0.2)$

$N_c' = 5.7 \times 1.2$

$N_c' = 6.84$

Now, the net ultimate bearing capacity using the shape-modified $N_c'$ is:

$q_{nu} = c \times N_c'$

Given $c = 120$ kN/m² and $N_c' = 6.84$:

$q_{nu} = 120 \, \text{kN/m}^2 \times 6.84$

$q_{nu} = 820.8 \, \text{kN/m}^2$

Rounding to the nearest value among the options, the net ultimate bearing capacity is approximately 820 kN/m².

Conclusion

Using Terzaghi's theory, considering the shape factor for a rectangular foundation in cohesive soil, the net ultimate bearing capacity is found to be 820.8 kN/m², which closely matches option 1.

Revision Table: Key Concepts for Foundation Bearing Capacity

Concept	Description	Relevance to Problem
Ultimate Bearing Capacity ($q_u$)	The maximum gross pressure at the base of the foundation that the soil can support before shear failure occurs.	Starting point for calculation.
Net Ultimate Bearing Capacity ($q_{nu}$)	The ultimate bearing capacity minus the overburden pressure ($q$) at foundation level. It's the maximum extra pressure the soil can support.	The value requested in the question.
Cohesive Soil (Clay)	Soil where strength is primarily due to cohesion (c) rather than internal friction ($\phi$). Often characterized by $\phi = 0$.	Determines which bearing capacity factors and formulas apply.
Terzaghi's Theory	One of the early and fundamental theories for calculating bearing capacity based on shear failure zones below the foundation. Provides bearing capacity factors $N_c, N_q, N_{\gamma}$ and shape factors.	The specified method for this problem.
Shape Factors	Multipliers used to adjust bearing capacity factors for different foundation shapes (square, rectangle, circle) from the basic strip footing assumption.	Essential for accurate calculation for a rectangular raft.

Additional Information: Factors Affecting Raft Foundation Bearing Capacity

The bearing capacity of a raft foundation is influenced by several factors beyond just soil cohesion and foundation dimensions. Understanding these helps in practical foundation design:

Soil Type and Properties: Cohesion (c), angle of internal friction ($\phi$), unit weight ($\gamma$), and compressibility (affecting settlement, which is often critical for rafts).
Foundation Size and Shape: Larger foundations distribute load over a wider area but can influence the depth of the critical shear surface. Shape factors account for the difference between strip, square, rectangular, and circular footings.
Depth of Foundation ($D_f$): Placing the foundation deeper generally increases bearing capacity due to increased overburden pressure and confinement, although in pure clay ($\phi=0$) using net capacity, the effect is through surcharge unless considering upper layers.
Groundwater Table: A high water table can reduce the effective stress and soil shear strength, significantly lowering bearing capacity. It requires using submerged unit weight below the water table.
Method of Analysis: Different theories (Terzaghi, Meyerhof, Hansen, Vesic, etc.) use different assumptions, bearing capacity factors, and shape/depth/inclination factors, leading to potentially different results. IS codes often provide guidelines based on these theories.
Layered Soils: If soil properties vary significantly with depth, the analysis becomes more complex, potentially requiring consideration of punching shear failure through a strong layer into a weak layer.

Prepp Team · Answer

1020 kN/m2

Prepp Team · Answer

1220 kN/m2

Prepp Team · Answer

1420 kN/m2