Bearing capacity¶

Four classical methods in one function. The general bearing-capacity equation (Meyerhof / Hansen / Vesic):

\[ \boxed{\;q_u = c\,N_c\,s_c\,d_c\,i_c + q\,N_q\,s_q\,d_q\,i_q + \tfrac{1}{2}\gamma B\,N_\gamma\,s_\gamma\,d_\gamma\,i_\gamma\;} \]

Terzaghi (1943) uses the same first/second/third-term form but without shape, depth, or inclination corrections.

Quick example¶

Python

import geoeq as ge

# Square footing on dense sand
bc = ge.bearing_capacity(
    c=0, gamma=18, Df=1, B=2, L=2,
    phi=35, method="meyerhof",
)
print(bc["q_u"])                    # ultimate bearing capacity, kPa

# Allowable with FS = 3 (standard for static loading)
ge.bearing_allowable(bc["q_u"], FS=3)

Method comparison¶

The four methods diverge most for \(N_\gamma\) in granular soils:

Python

for m in ("terzaghi", "meyerhof", "hansen", "vesic"):
    f = ge.bearing_factors(phi=30, method=m)
    print(f"{m:9s} -> Nc={f['Nc']:6.2f} Nq={f['Nq']:6.2f} Ng={f['Ngamma']:6.2f}")

φ°	Nc	Nq	\(N_\gamma\) Meyerhof	\(N_\gamma\) Hansen	\(N_\gamma\) Vesic
0	5.14	1.00	0.00	0.00	0.00
20	14.83	6.40	2.87	2.95	5.39
30	30.14	18.40	15.67	14.39	22.40
40	75.31	64.20	93.69	79.54	109.41

A design chart showing \(q_u\) vs \(B\) for all four methods:

Bearing capacity chart — Ultimate bearing capacity vs footing width for c = 0, φ = 30°, D_f = 1 m. Generated by `ge.bearing_capacity_plot()`.

Shape, depth, inclination factors¶

The corrections are independently callable if you want to inspect them:

Python

ge.bearing_shape_factors(B=2, L=3, phi=30, method="meyerhof")
# {'sc': 1.40, 'sq': 1.20, 's_gamma': 1.20}

ge.bearing_depth_factors(Df=1, B=2, phi=30, method="meyerhof")
# {'dc': 1.17, 'dq': 1.09, 'd_gamma': 1.09}

ge.bearing_inclination_factors(beta=10, phi=30)
# {'ic': 0.79, 'iq': 0.79, 'i_gamma': 0.45}

The unified bearing_capacity() function applies them automatically unless you pass shape=False, depth=False, or inclination=0.

Water-table effects¶

Pass water_table="at_base" to use the effective (buoyant) unit weight under the footing. This is the most common case for shallow footings on saturated sand.

API reference¶

bearing_factors ¶

Python

bearing_factors(phi: float, method: str = 'meyerhof') -> dict

Bearing-capacity factors Nc, Nq, N_gamma for a given friction angle.

PARAMETER	DESCRIPTION
`phi`	Effective friction angle (degrees). TYPE: `float`
`method`	`'terzaghi'` \| `'meyerhof'` \| `'hansen'` \| `'vesic'`. TYPE: `str` DEFAULT: `'meyerhof'`

RETURNS	DESCRIPTION
`dict`	`{'Nc': ..., 'Nq': ..., 'Ngamma': ...}`.

Reference

Das (2014) Tables 4.1-4.3.

Source code in geoeq/design/bearing.py

Python
def bearing_factors(phi: float, method: str = "meyerhof") -> dict:
    """Bearing-capacity factors Nc, Nq, N_gamma for a given friction angle.

    Parameters
    ----------
    phi : float
        Effective friction angle (degrees).
    method : str
        ``'terzaghi'`` | ``'meyerhof'`` | ``'hansen'`` | ``'vesic'``.

    Returns
    -------
    dict
        ``{'Nc': ..., 'Nq': ..., 'Ngamma': ...}``.

    Reference
    ---------
    Das (2014) Tables 4.1-4.3.
    """
    check_non_negative(phi, "phi")
    check_range(phi, "phi", 0, 50)
    method = method.lower()
    phi_rad = np.radians(phi)

    # Nq is identical across Meyerhof/Hansen/Vesic (Prandtl-Reissner).
    if phi == 0:
        Nq = 1.0
    else:
        Nq = np.exp(np.pi * np.tan(phi_rad)) * \
            np.tan(np.pi / 4 + phi_rad / 2) ** 2

    # Nc via classical relation Nc = (Nq - 1) / tan(phi); Nc = 5.14 at phi=0.
    if phi == 0:
        Nc = 5.14  # = pi + 2
    else:
        Nc = (Nq - 1) / np.tan(phi_rad)

    if method == "terzaghi":
        # Terzaghi's original Nq, Nc differ slightly (he used local shear);
        # use the Bowles-tabulated forms for general shear.
        Nq = np.exp((1.5 * np.pi - phi_rad) * np.tan(phi_rad)) / \
            (2 * np.cos(np.pi / 4 + phi_rad / 2) ** 2)
        if phi == 0:
            Nc = 5.7  # Terzaghi original
        else:
            Nc = (Nq - 1) / np.tan(phi_rad)
        # Terzaghi N_gamma (empirical fit to chart, Bowles 1996):
        Ngamma = (Nq - 1) * np.tan(1.4 * phi_rad) if phi > 0 else 0.0
    elif method == "meyerhof":
        Ngamma = (Nq - 1) * np.tan(1.4 * phi_rad)
    elif method == "hansen":
        Ngamma = 1.5 * (Nq - 1) * np.tan(phi_rad)
    elif method == "vesic":
        Ngamma = 2 * (Nq + 1) * np.tan(phi_rad)
    else:
        raise ValueError(
            "method must be 'terzaghi', 'meyerhof', 'hansen', or 'vesic'.")
    return {"Nc": float(Nc), "Nq": float(Nq), "Ngamma": float(Ngamma)}