Understanding Soil Bearing Capacity Factors

Bearing capacity factors ($N_c$, $N_q$, and $N_r$) are crucial dimensionless parameters used in geotechnical engineering to determine the ultimate bearing capacity of soil beneath a foundation. The ultimate bearing capacity is the maximum pressure that a soil can support before shear failure occurs.

These factors are derived from theoretical analyses of soil failure mechanisms and depend primarily on the internal shear strength properties of the soil.

What are Bearing Capacity Factors a Function of?

The question asks what the bearing capacity factors $N_c$, $N_q$, and $N_r$ are a function of. Let's look at the options provided and their relationship with these factors and the overall bearing capacity calculation.

• Width and depth of footing: The width (B) and depth (D) of the footing are used in the bearing capacity equation itself (e.g., Terzaghi's or Meyerhof's equations), often appearing in terms like $q N_q$ (depth effect from overburden) and $\frac{1}{2} \gamma B N_r$ (width effect related to the wedge failure zone). However, the factors $N_c$, $N_q$, and $N_r$ themselves are not direct functions of B or D. Shape, depth, and load inclination factors are applied to the basic bearing capacity factors to account for these effects.
• Angle of internal friction of soil: The angle of internal friction ($\phi$) is a fundamental shear strength parameter of soil, particularly for granular soils. Theoretical solutions for bearing capacity factors, such as those by Terzaghi, Meyerhof, Hansen, and others, show that $N_c$, $N_q$, and $N_r$ are calculated directly using formulas that involve the angle of internal friction ($\phi$). For example, Terzaghi's original factors are complex functions of $\phi$.
• Density of soil: The density (or unit weight, $\gamma$) of the soil is used in the bearing capacity equation, specifically in the terms related to overburden pressure ($q = \gamma D$) and the self-weight of the soil within the failure wedge ($\frac{1}{2} \gamma B N_r$). While density is essential for calculating bearing capacity, the factors $N_c$, $N_q$, and $N_r$ themselves are not functions of soil density.
• Cohesion of soil: The cohesion (c) is another fundamental shear strength parameter, particularly for cohesive soils. Cohesion appears as a separate term in the bearing capacity equation ($c N_c$). The factor $N_c$ is related to the contribution of cohesion to bearing capacity. However, similar to the angle of internal friction, the factors $N_c$, $N_q$, and $N_r$ are derived based on the soil's shear strength parameters, primarily the angle of internal friction ($\phi$). The factor $N_c$ is a function of $\phi$, and for cohesive soils with $\phi=0$, $N_c$ has a specific value (e.g., 5.7 for Terzaghi).

Detailed Analysis of Bearing Capacity Factors Dependency

The bearing capacity factors $N_c$, $N_q$, and $N_r$ are dimensionless values that quantify the influence of soil properties on bearing capacity. These factors are derived from analyzing the soil failure mechanisms under the foundation. The most significant soil property governing these factors is the soil's ability to resist shear stress through friction, which is quantified by the angle of internal friction ($\phi$).

Different theories provide slightly different formulas for $N_c$, $N_q$, and $N_r$, but all show a direct mathematical relationship with $\phi$. For instance, a common set of formulas (derived from Prandtl and Reissner for $N_c$ and $N_q$, and various authors for $N_r$) clearly shows this dependency:

$$N_q = e^{\pi \tan\phi} \tan^2\left(45^\circ + \frac{\phi}{2}\right)$$

$$N_c = (N_q - 1) \cot\phi \quad (\text{for } \phi > 0)$$

$$N_c = 5.14 \quad (\text{for } \phi = 0)$$

$$N_r \text{ is a more complex function of } \phi \text{ and varies between theories}$$

These formulas explicitly show that $N_c$, $N_q$, and $N_r$ change as the angle of internal friction ($\phi$) changes.

While cohesion (c) and unit weight ($\gamma$) are critical soil parameters and are used in the full bearing capacity equation, they are multiplied by the factors $N_c$, $N_q$, and $N_r$, respectively, to calculate the bearing capacity. The factors themselves are primarily functions of $\phi$.

Therefore, the bearing capacity factors $N_c$, $N_q$, and $N_r$ are fundamentally functions of the angle of internal friction of the soil.

Conclusion on Bearing Capacity Factors

Based on the analysis of the relationship between soil properties and bearing capacity factors, it is clear that $N_c$, $N_q$, and $N_r$ are calculated values that depend significantly on the soil's angle of internal friction ($\phi$). Other parameters like footing dimensions, soil density, and cohesion are used alongside these factors in the ultimate bearing capacity equation, but they do not determine the values of $N_c$, $N_q$, and $N_r$ themselves.

The correct option is the one stating that the bearing capacity factors are a function of the angle of internal friction of soil.

Revision Table: Bearing Capacity Factors

Additional Information: Ultimate Bearing Capacity Equation

The general form of the ultimate bearing capacity ($q_{ult}$) equation for a strip footing (by Terzaghi) is:

$$q_{ult} = c N_c + q N_q + \frac{1}{2} \gamma B N_r$$

Where:

• $q_{ult}$ = Ultimate bearing capacity
• $c$ = Soil cohesion
• $q$ = Surcharge pressure ($\gamma D$)
• $\gamma$ = Unit weight of soil
• $B$ = Width of footing
• $N_c, N_q, N_r$ = Bearing capacity factors (functions of $\phi$)

This equation clearly shows that while $c$, $q$, $\gamma$, and $B$ are part of the equation, $N_c$, $N_q$, and $N_r$ are coefficients whose values depend on the angle of internal friction ($\phi$). For other footing shapes (square, circular), shape factors are applied to this basic equation, and for inclined loads or deep foundations, depth and load inclination factors are also used. However, the core dependency of $N_c$, $N_q$, and $N_r$ on $\phi$ remains.