Understanding Terzaghi's Bearing Capacity Factors

Terzaghi's bearing capacity theory is a fundamental concept in geotechnical engineering used to determine the ultimate bearing capacity of shallow foundations. The theory provides an equation to calculate the maximum pressure a soil can withstand before shear failure occurs.

The ultimate bearing capacity ($q_u$) for a strip footing, according to Terzaghi, is given by the equation:

\(q_u = c N_c + q N_q + 0.5 \gamma B N_{\gamma}\)

Where:

• \(c\) is the cohesion of the soil
• \(q\) is the surcharge load at the foundation level (typically \(q = \gamma' D_f\), where \(\gamma'\) is the effective unit weight of soil above the foundation base and \(D_f\) is the depth of the foundation)
• \(\gamma\) is the unit weight of the soil below the foundation level
• \(B\) is the width of the foundation
• \(N_c\), \(N_q\), and \(N_{\gamma}\) are Terzaghi's bearing capacity factors

These bearing capacity factors, \(N_c\), \(N_q\), and \(N_{\gamma}\), are dimensionless quantities that account for the contribution of cohesion, surcharge, and soil weight, respectively, to the ultimate bearing capacity. A key aspect of Terzaghi's original theory is how these factors are determined.

Dependency of Terzaghi's Factors

Terzaghi derived the values for \(N_c\), \(N_q\), and \(N_{\gamma}\) based on the assumption of a specific failure surface in the soil beneath the foundation. These factors are solely dependent on one fundamental shear strength parameter of the soil: the angle of internal friction (\(\phi\)).

The formulas for \(N_q\) and \(N_c\) are explicitly given in terms of \(\phi\):

• \(N_q = e^{\pi \tan \phi} \tan^2 \left(45^\circ + \frac{\phi}{2}\right)\)
• \(N_c = (N_q - 1) \cot \phi\)

The factor \(N_{\gamma}\) is also a function of \(\phi\), although its derivation is more complex and its values are often presented in charts or tables based on experimental fits.

Therefore, changing the angle of internal friction of the soil will change the values of \(N_c\), \(N_q\), and \(N_{\gamma}\). Other soil properties or foundation dimensions do not directly influence the values of these specific factors in Terzaghi's original formulation.

Analyzing the Given Options

Let's examine the provided options in light of this understanding:

• Option 1: Uniformity coefficient of soil and dry density of soil
  The uniformity coefficient and dry density are soil classification and compaction parameters. While they are important soil properties, Terzaghi's bearing capacity factors \(N_c\), \(N_q\), \(N_{\gamma}\) are not calculated directly using these values.
• Option 2: Coefficient of curvature of soil and bulk density of soil
  Similar to Option 1, the coefficient of curvature is a grain size distribution parameter, and bulk density is a measure of soil mass per unit volume. Neither of these directly determines the values of Terzaghi's bearing capacity factors. Bulk density (or unit weight) is used in the bearing capacity equation itself (in the \(q\) and \(\gamma B N_{\gamma}\) terms), but it does not influence the factors \(N_c\), \(N_q\), \(N_{\gamma}\).
• Option 3: Angle of internal friction of soil only
  As discussed, Terzaghi's bearing capacity factors \(N_c\), \(N_q\), and \(N_{\gamma}\) are solely functions of the angle of internal friction (\(\phi\)) of the soil. This aligns perfectly with the theoretical basis of Terzaghi's method.
• Option 4: Angle of internal friction of soil and depth of foundation
  The angle of internal friction is indeed a determinant of the factors. However, the depth of the foundation (\(D_f\)) affects the surcharge term \(q\) in the bearing capacity equation (\(q = \gamma' D_f\)), but it does not directly affect the values of the bearing capacity factors \(N_c\), \(N_q\), \(N_{\gamma}\) themselves in Terzaghi's original formulation. Later theories by researchers like Meyerhof did introduce depth factors that modify \(N\) values, but the question specifically asks about Terzaghi's factors.

Based on the derivation and definition of Terzaghi's bearing capacity factors, they depend exclusively on the angle of internal friction of the soil.

Revision Table: Terzaghi's Bearing Capacity

Additional Information: Beyond Terzaghi

While Terzaghi's theory is foundational, it has limitations (e.g., assumes a rough base, simplified failure surface, no consideration for foundation shape or depth effects on factors). Later researchers refined the bearing capacity theory by introducing additional factors:

• Meyerhof's Theory: Introduced shape factors, depth factors, and inclination factors that modify Terzaghi's equation and \(N\) values to account for foundation geometry, depth, and load inclination. In Meyerhof's approach, some factors might depend on depth or shape in addition to the angle of internal friction.
• Hansen's and Vesic's Theories: Further refined bearing capacity calculations by introducing a wider range of correction factors for shape, depth, inclination, ground surface tilt, and base tilt.

However, for the specific question about Terzaghi's original bearing capacity factors (\(N_c\), \(N_q\), \(N_{\gamma}\)), their dependency is solely on the soil's angle of internal friction (\(\phi\)).