Settlement

Three components, each as its own function:

• Immediate (elastic) — Janbu / Steinbrenner closed form
• Primary consolidation — 1-D Terzaghi for NC, OC-below-pc, or OC-crossing-pc
• Secondary compression — Mesri & Godlewski \(C_\alpha\) log-time
• Schmertmann strain-influence — Schmertmann (1970, 1978) for sands

Plus the time-rate of consolidation: \(T_v \leftrightarrow U\) and \(t = T_v\,H_{dr}^2/c_v\).

Immediate (elastic)

\[ S_i = q B\,\frac{1 - \mu^2}{E_s}\,I_p \]

Python

ge.settlement_immediate(q=200, B=2, Es=10000, mu=0.3,
                         shape="rigid_square")          # 32 mm

shape picks the influence factor \(I_p\) automatically (0.88 for a rigid square, 0.79 for a rigid circle, 1.12 for a flexible centre, 0.56 for a flexible corner, ~2 for a strip).

Primary consolidation

Three regimes, selected automatically by kind="auto":

\[ \begin{aligned} \text{NC:}\;\;\; & S = \frac{C_c H}{1+e_0}\log_{10}\frac{\sigma'_f}{\sigma'_0}\\[4pt] \text{OC, }\sigma'_f \leq \sigma'_p:\;\;\; & S = \frac{C_r H}{1+e_0}\log_{10}\frac{\sigma'_f}{\sigma'_0}\\[4pt] \text{OC crossing }\sigma'_p:\;\;\; & S = \frac{C_r H}{1+e_0}\log_{10}\frac{\sigma'_p}{\sigma'_0} + \frac{C_c H}{1+e_0}\log_{10}\frac{\sigma'_f}{\sigma'_p} \end{aligned} \]

Python

S = ge.settlement_primary(Cc=0.27, e0=0.92, H=3.66,
                          sigma0=80, delta_sigma=15)
print(f"S = {S*1000:.1f} mm")               # 38.3 mm — matches Das Ex 5.1

Secondary compression

Python

ge.settlement_secondary(C_alpha=0.005, H=2,
                         t1=100, t2=1000)       # 10 mm

For the void-ratio form \(C_\alpha\) (slope of \(e\) vs \(\log t\)), pass the end-of-primary void ratio:

Python

ge.settlement_secondary(C_alpha=0.02, H=2,
                         t1=100, t2=1000, e_p=1.0)

Schmertmann (sands)

\[ S = C_1\,C_2\,q_{\text{net}}\,\sum_i \frac{I_{z,i}}{E_{s,i}}\,\Delta z_i \]

with embedment correction \(C_1 = 1 - 0.5(\sigma'_0/q_{\text{net}})\) and creep correction \(C_2 = 1 + 0.2\log_{10}(t/0.1)\) years. The strain-influence diagram \(I_z\) peaks at \(z = 0.5B\) (axisymmetric) or \(z = 1.0B\) (strip).

Python

ge.settlement_schmertmann(
    q_net=100, B=2,
    Es=[10000, 30000],
    layers=[(0, 2), (2, 4)],
    Df=1, t_years=10, shape="square",
)
# {'S': 0.0157, 'C1': 0.91, 'C2': 1.40, 'Iz_peak': 0.665, ...}

Time rate of consolidation

Python

ge.time_factor(U=0.5)                        # 0.196
ge.time_factor(U=0.9)                        # 0.848

ge.consolidation_degree(Tv=0.197)            # 0.501
ge.consolidation_time(U=0.9, Hdr=2, cv=1e-7) # 3.4e7 seconds (393 days)

Terzaghi 1-D consolidation curve: average degree of consolidation U vs time factor T_v. Generated by ge.settlement_time_plot().

API reference

settlement_immediate

Python

settlement_immediate(q: float, B: float, Es: float, mu: float = 0.3, Ip: float = None, shape: str = 'rigid_square') -> float

Elastic (immediate) settlement under a uniformly loaded footing.

Text Only

S_i = q * B * (1 - mu^2) / Es * Ip

PARAMETER	DESCRIPTION
q	Surface pressure (kPa). TYPE: float
B	Footing width (m) -- shorter dimension for rectangles. TYPE: float
Es	Soil elastic modulus (kPa). TYPE: float
mu	Poisson's ratio (-). Default 0.3. TYPE: float DEFAULT: 0.3
Ip	Influence factor. If None, picked from shape. TYPE: float DEFAULT: None
shape	'rigid_square' -> Ip = 0.88 'rigid_circle' -> Ip = 0.79 'flexible_centre' -> Ip = 1.12 'flexible_corner' -> Ip = 0.56 'rigid_strip' -> Ip = 2.0 (approx, depth-dependent) TYPE: str DEFAULT: 'rigid_square'

Reference

Janbu et al. (1956); Das (2014) Eq. 5.20, Table 5.1.

Source code in geoeq/design/settlement.py

def settlement_immediate(
    q: float, B: float, Es: float, mu: float = 0.3,
    Ip: float = None, shape: str = "rigid_square",
) -> float:
    """Elastic (immediate) settlement under a uniformly loaded footing.

        S_i = q * B * (1 - mu^2) / Es * Ip

    Parameters
    ----------
    q : float
        Surface pressure (kPa).
    B : float
        Footing width (m) -- shorter dimension for rectangles.
    Es : float
        Soil elastic modulus (kPa).
    mu : float
        Poisson's ratio (-). Default 0.3.
    Ip : float, optional
        Influence factor. If None, picked from ``shape``.
    shape : str
        'rigid_square'  -> Ip = 0.88
        'rigid_circle'  -> Ip = 0.79
        'flexible_centre' -> Ip = 1.12
        'flexible_corner' -> Ip = 0.56
        'rigid_strip'   -> Ip = 2.0 (approx, depth-dependent)

    Reference
    ---------
    Janbu et al. (1956); Das (2014) Eq. 5.20, Table 5.1.
    """
    check_positive(B, "B")
    check_positive(Es, "Es")
    if Ip is None:
        table = {
            "rigid_square": 0.88, "rigid_circle": 0.79,
            "flexible_centre": 1.12, "flexible_center": 1.12,
            "flexible_corner": 0.56, "rigid_strip": 2.0,
        }
        if shape not in table:
            raise ValueError(
                f"shape must be one of {list(table.keys())}, got {shape!r}.")
        Ip = table[shape]
    return float(q * B * (1 - mu ** 2) / Es * Ip)

settlement_primary

Python

settlement_primary(Cc: float, e0: float, H: float, sigma0: float, delta_sigma: float, Cr: float = None, sigma_pc: float = None, kind: str = 'auto') -> float

1-D primary consolidation settlement.

Three regimes:

1. NC (normally consolidated): S = (Cc * H / (1 + e0)) * log10((sigma0 + delta_sigma) / sigma0)

2. OC, final stress below preconsolidation: S = (Cr * H / (1 + e0)) * log10((sigma0 + delta_sigma) / sigma0)

3. OC, final stress crosses preconsolidation: S = (Cr*H/(1+e0)) * log10(sigma_pc / sigma0)

   • (Cc*H/(1+e0)) * log10((sigma0+delta_sigma)/sigma_pc)

PARAMETER	DESCRIPTION
Cc	Compression index (-). TYPE: float
e0	Initial void ratio (-). TYPE: float
H	Drainage path length (m) -- full layer thickness for one-way drainage or half-thickness for two-way drainage; pass actual layer thickness. TYPE: float
sigma0	Initial effective vertical stress (kPa) at mid-depth. TYPE: float
delta_sigma	Stress increase at mid-depth (kPa). TYPE: float
Cr	Recompression (swell) index (-). Required for OC cases. TYPE: float DEFAULT: None
sigma_pc	Preconsolidation pressure (kPa). Required for OC cases. TYPE: float DEFAULT: None
kind	'NC', 'OC', or 'auto' (decide from sigma_pc). TYPE: str DEFAULT: 'auto'

RETURNS	DESCRIPTION
S	Primary settlement (m). TYPE: float

Reference

Terzaghi (1925); Das (2014) Eq. 5.31-5.34.

Source code in geoeq/design/settlement.py

def settlement_primary(
    Cc: float, e0: float, H: float, sigma0: float, delta_sigma: float,
    Cr: float = None, sigma_pc: float = None, kind: str = "auto",
) -> float:
    """1-D primary consolidation settlement.

    Three regimes:

    1. **NC** (normally consolidated):
       S = (Cc * H / (1 + e0)) * log10((sigma0 + delta_sigma) / sigma0)

    2. **OC**, final stress below preconsolidation:
       S = (Cr * H / (1 + e0)) * log10((sigma0 + delta_sigma) / sigma0)

    3. **OC**, final stress crosses preconsolidation:
       S = (Cr*H/(1+e0)) * log10(sigma_pc / sigma0)
         + (Cc*H/(1+e0)) * log10((sigma0+delta_sigma)/sigma_pc)

    Parameters
    ----------
    Cc : float
        Compression index (-).
    e0 : float
        Initial void ratio (-).
    H : float
        Drainage path length (m) -- full layer thickness for one-way drainage
        or half-thickness for two-way drainage; pass actual layer thickness.
    sigma0 : float
        Initial effective vertical stress (kPa) at mid-depth.
    delta_sigma : float
        Stress increase at mid-depth (kPa).
    Cr : float
        Recompression (swell) index (-). Required for OC cases.
    sigma_pc : float
        Preconsolidation pressure (kPa). Required for OC cases.
    kind : str
        'NC', 'OC', or 'auto' (decide from sigma_pc).

    Returns
    -------
    S : float
        Primary settlement (m).

    Reference
    ---------
    Terzaghi (1925); Das (2014) Eq. 5.31-5.34.
    """
    check_positive(H, "H")
    check_positive(sigma0, "sigma0")
    check_positive(delta_sigma, "delta_sigma")
    if e0 <= 0:
        raise ValueError("e0 must be positive.")

    sigma_f = sigma0 + delta_sigma

    if kind.lower() == "auto":
        if sigma_pc is None or sigma_pc <= sigma0:
            kind = "NC"
        elif sigma_f <= sigma_pc:
            kind = "OC_below"
        else:
            kind = "OC_cross"
    else:
        kind = kind.upper()

    if kind == "NC":
        return float(Cc * H / (1 + e0) * np.log10(sigma_f / sigma0))

    if Cr is None:
        raise ValueError("Cr is required for an OC analysis.")
    if sigma_pc is None:
        raise ValueError("sigma_pc is required for an OC analysis.")

    if kind in ("OC_below", "OC"):
        return float(Cr * H / (1 + e0) * np.log10(sigma_f / sigma0))

    if kind == "OC_cross":
        S1 = Cr * H / (1 + e0) * np.log10(sigma_pc / sigma0)
        S2 = Cc * H / (1 + e0) * np.log10(sigma_f / sigma_pc)
        return float(S1 + S2)

    raise ValueError(
        f"kind must be 'NC', 'OC', 'OC_below', 'OC_cross', or 'auto', "
        f"got {kind!r}.")

settlement_secondary

Python

settlement_secondary(C_alpha: float, H: float, t1: float, t2: float, e_p: float = None) -> float

Secondary compression S_s = C_alpha_eps * H * log10(t2/t1).

PARAMETER	DESCRIPTION
C_alpha	Secondary compression index. If e_p is given this is the e-vs-log-t form C_alpha (slope of e vs log t after EOP); converted to strain form via C_alpha_eps = C_alpha / (1 + e_p). Otherwise treated directly as the strain form. TYPE: float
H	Layer thickness (m). TYPE: float
t1	Reference (end of primary) and target times (any consistent unit). TYPE: float
t2	Reference (end of primary) and target times (any consistent unit). TYPE: float
e_p	Void ratio at end-of-primary, for converting C_alpha -> C_alpha_eps. TYPE: float DEFAULT: None

Reference

Mesri & Godlewski (1977); Das (2014) Eq. 5.51.

Source code in geoeq/design/settlement.py

def settlement_secondary(
    C_alpha: float, H: float, t1: float, t2: float, e_p: float = None,
) -> float:
    """Secondary compression S_s = C_alpha_eps * H * log10(t2/t1).

    Parameters
    ----------
    C_alpha : float
        Secondary compression index. If ``e_p`` is given this is the
        e-vs-log-t form C_alpha (slope of e vs log t after EOP);
        converted to strain form via C_alpha_eps = C_alpha / (1 + e_p).
        Otherwise treated directly as the strain form.
    H : float
        Layer thickness (m).
    t1, t2 : float
        Reference (end of primary) and target times (any consistent unit).
    e_p : float, optional
        Void ratio at end-of-primary, for converting C_alpha -> C_alpha_eps.

    Reference
    ---------
    Mesri & Godlewski (1977); Das (2014) Eq. 5.51.
    """
    check_positive(H, "H")
    check_positive(t1, "t1")
    check_positive(t2, "t2")
    if t2 <= t1:
        raise ValueError("t2 must be > t1.")
    if e_p is not None:
        C_alpha_eps = C_alpha / (1 + e_p)
    else:
        C_alpha_eps = C_alpha
    return float(C_alpha_eps * H * np.log10(t2 / t1))

settlement_schmertmann

Python

settlement_schmertmann(q_net: float, B: float, Es: Union[float, Sequence[float]], layers: Sequence[tuple] = None, Df: float = 0.0, sigma_v_at_base: float = None, gamma: float = 18.0, t_years: float = 1.0, shape: str = 'square') -> dict

Schmertmann's strain-influence settlement for sandy soils.

Text Only

S = C1 * C2 * q_net * sum_i ( Iz_i / Es_i * dz_i )

where: C1 = 1 - 0.5 * (sigma_v' / q_net) -- embedment correction C2 = 1 + 0.2 * log10(t_years / 0.1) -- creep correction

The strain-influence diagram Iz is taken from Schmertmann et al. (1978):

• Square / circular (axisymmetric): Iz peaks at 0.5*B (Iz_peak), = 0.1 at z=0, 0 at z=2B.
• Strip (plane strain): Iz peaks at 1.0*B, = 0.2 at z=0, 0 at z=4B.

Iz_peak = 0.5 + 0.1 * sqrt(q_net / sigma_v_at_peak).

PARAMETER	DESCRIPTION
q_net	Net surface pressure (kPa) above sigma_v at footing base. TYPE: float
B	Footing width (m). TYPE: float
Es	Elastic modulus of subsoil (kPa). If a single number, applied to one layer of depth 0 .. (2B or 4B). If sequence, must match layers. TYPE: float or sequence
layers	Optional. If None and Es is scalar, the function builds a single-layer model spanning the full influence depth. TYPE: sequence of (top, bot) in metres below the footing. DEFAULT: None
Df	Footing depth (m). TYPE: float DEFAULT: 0.0
sigma_v_at_base	Effective vertical stress at the base of the footing (kPa). If None, estimated as gamma * Df. TYPE: float DEFAULT: None
gamma	Soil unit weight, only used if sigma_v_at_base is None. TYPE: float DEFAULT: 18.0
t_years	Time after construction for the creep correction (yrs). Default 1. TYPE: float DEFAULT: 1.0
shape	'square' / 'circle' (axisymmetric) or 'strip' (plane strain). TYPE: str DEFAULT: 'square'

RETURNS	DESCRIPTION
dict	{'S': settlement_metres, 'C1', 'C2', 'Iz_peak', 'layers': [...]}

Reference

Schmertmann (1970); Schmertmann et al. (1978); Das (2014) Ch. 5.10.

Source code in geoeq/design/settlement.py

def settlement_schmertmann(
    q_net: float, B: float, Es: Union[float, Sequence[float]],
    layers: Sequence[tuple] = None, Df: float = 0.0,
    sigma_v_at_base: float = None, gamma: float = 18.0,
    t_years: float = 1.0, shape: str = "square",
) -> dict:
    """Schmertmann's strain-influence settlement for sandy soils.

        S = C1 * C2 * q_net * sum_i ( Iz_i / Es_i * dz_i )

    where:
        C1 = 1 - 0.5 * (sigma_v' / q_net)            -- embedment correction
        C2 = 1 + 0.2 * log10(t_years / 0.1)          -- creep correction

    The strain-influence diagram Iz is taken from Schmertmann et al. (1978):

    * Square / circular (axisymmetric):  Iz peaks at 0.5*B (Iz_peak),
      = 0.1 at z=0, 0 at z=2B.
    * Strip (plane strain):             Iz peaks at 1.0*B,
      = 0.2 at z=0, 0 at z=4B.

    Iz_peak = 0.5 + 0.1 * sqrt(q_net / sigma_v_at_peak).

    Parameters
    ----------
    q_net : float
        Net surface pressure (kPa) above sigma_v at footing base.
    B : float
        Footing width (m).
    Es : float or sequence
        Elastic modulus of subsoil (kPa). If a single number, applied
        to one layer of depth 0 .. (2B or 4B). If sequence, must match ``layers``.
    layers : sequence of (top, bot) in metres below the footing.
        Optional. If None and ``Es`` is scalar, the function builds a
        single-layer model spanning the full influence depth.
    Df : float
        Footing depth (m).
    sigma_v_at_base : float, optional
        Effective vertical stress at the base of the footing (kPa).
        If None, estimated as ``gamma * Df``.
    gamma : float
        Soil unit weight, only used if sigma_v_at_base is None.
    t_years : float
        Time after construction for the creep correction (yrs). Default 1.
    shape : str
        'square' / 'circle' (axisymmetric) or 'strip' (plane strain).

    Returns
    -------
    dict
        {'S': settlement_metres, 'C1', 'C2', 'Iz_peak', 'layers': [...]}

    Reference
    ---------
    Schmertmann (1970); Schmertmann et al. (1978); Das (2014) Ch. 5.10.
    """
    check_positive(q_net, "q_net")
    check_positive(B, "B")
    shape = shape.lower()
    if shape in ("square", "circle", "circular", "axisymmetric"):
        z_peak = 0.5 * B
        z_max = 2.0 * B
        Iz_z0 = 0.1
    elif shape == "strip":
        z_peak = 1.0 * B
        z_max = 4.0 * B
        Iz_z0 = 0.2
    else:
        raise ValueError("shape must be 'square', 'circle', or 'strip'.")

    if sigma_v_at_base is None:
        sigma_v_at_base = gamma * Df

    # sigma_v at z_peak (initial effective stress).
    sigma_v_peak = sigma_v_at_base + gamma * z_peak
    Iz_peak = 0.5 + 0.1 * np.sqrt(q_net / max(sigma_v_peak, 1e-6))

    # Build the Iz(z) function: linear from (0, Iz_z0) to (z_peak, Iz_peak)
    # then linear to (z_max, 0).
    def Iz(z):
        if z < 0 or z > z_max:
            return 0.0
        if z <= z_peak:
            return Iz_z0 + (Iz_peak - Iz_z0) * (z / z_peak)
        return Iz_peak * (1 - (z - z_peak) / (z_max - z_peak))

    # Build layers if not provided.
    if layers is None:
        if np.isscalar(Es):
            layers = [(0.0, z_max)]
            Es = [float(Es)]
        else:
            raise ValueError(
                "If Es is a sequence, layers must be provided.")
    Es_arr = np.atleast_1d(np.asarray(Es, dtype=float))
    if len(Es_arr) != len(layers):
        raise ValueError("Es and layers must have the same length.")

    # Integrate sum(Iz / Es * dz) using midpoint of each layer.
    integ = 0.0
    layer_info = []
    for (top, bot), Esi in zip(layers, Es_arr):
        dz = bot - top
        z_mid = 0.5 * (top + bot)
        contribution = Iz(z_mid) / Esi * dz
        integ += contribution
        layer_info.append({"top": top, "bot": bot, "Es": Esi,
                           "Iz_mid": Iz(z_mid), "dS": contribution})

    C1 = max(0.5, 1 - 0.5 * sigma_v_at_base / q_net)
    C2 = max(1.0, 1 + 0.2 * np.log10(max(t_years, 0.1) / 0.1))
    S = C1 * C2 * q_net * integ

    return {
        "S": float(S), "C1": float(C1), "C2": float(C2),
        "Iz_peak": float(Iz_peak), "z_peak": float(z_peak),
        "z_max": float(z_max), "layers": layer_info,
    }

time_factor

Python

time_factor(U: float) -> float

Time factor Tv from average degree of consolidation U (0..1).

Approximate Terzaghi solution (Das Eq. 5.46): Tv = (pi/4) * U^2 for U < 0.6 Tv = 1.781 - 0.933 * log10(100*(1-U)) for U >= 0.6

Reference

Terzaghi (1925); Das (2014) Eq. 5.46.

Source code in geoeq/design/settlement.py

def time_factor(U: float) -> float:
    """Time factor Tv from average degree of consolidation U (0..1).

    Approximate Terzaghi solution (Das Eq. 5.46):
        Tv = (pi/4) * U^2                   for U < 0.6
        Tv = 1.781 - 0.933 * log10(100*(1-U))   for U >= 0.6

    Reference
    ---------
    Terzaghi (1925); Das (2014) Eq. 5.46.
    """
    if not 0 <= U < 1:
        raise ValueError("U must be in [0, 1).")
    if U < 0.6:
        return (np.pi / 4) * U ** 2
    return 1.781 - 0.933 * np.log10(100 * (1 - U))

consolidation_degree

Python

consolidation_degree(Tv: float) -> float

Average degree of consolidation U (0..1) from time factor Tv.

Inverse of time_factor (Das Eq. 5.46).

Source code in geoeq/design/settlement.py

def consolidation_degree(Tv: float) -> float:
    """Average degree of consolidation U (0..1) from time factor Tv.

    Inverse of ``time_factor`` (Das Eq. 5.46).
    """
    if Tv < 0:
        raise ValueError("Tv must be non-negative.")
    if Tv <= 0.286:  # boundary at U = 0.6
        return float(np.sqrt(4 * Tv / np.pi))
    return float(1 - 10 ** ((1.781 - Tv) / 0.933) / 100)

consolidation_time

Python

consolidation_time(U: float, Hdr: float, cv: float) -> float

Time to reach degree of consolidation U.

Text Only

t = Tv * Hdr^2 / cv

PARAMETER	DESCRIPTION
U	Target degree of consolidation (0..1). TYPE: float
Hdr	Drainage path length (m). Half the layer thickness for double drainage. TYPE: float
cv	Coefficient of consolidation (m^2/s, or any consistent unit). TYPE: float

RETURNS	DESCRIPTION
t	Time (same time units as cv). TYPE: float

Reference

Das (2014) Eq. 5.43.

Source code in geoeq/design/settlement.py

def consolidation_time(U: float, Hdr: float, cv: float) -> float:
    """Time to reach degree of consolidation U.

        t = Tv * Hdr^2 / cv

    Parameters
    ----------
    U : float
        Target degree of consolidation (0..1).
    Hdr : float
        Drainage path length (m). Half the layer thickness for double drainage.
    cv : float
        Coefficient of consolidation (m^2/s, or any consistent unit).

    Returns
    -------
    t : float
        Time (same time units as cv).

    Reference
    ---------
    Das (2014) Eq. 5.43.
    """
    Tv = time_factor(U)
    check_positive(Hdr, "Hdr")
    check_positive(cv, "cv")
    return float(Tv * Hdr ** 2 / cv)

settlement_time_plot

Python

settlement_time_plot(ax=None, save_as: str = None)

The classical Terzaghi consolidation curve: U vs Tv (log scale).

Plots the approximate piecewise formula used by time_factor: Tv = (pi/4) U^2 for U < 0.6, Tv = 1.781 - 0.933 log10(100(1-U)) for U >= 0.6.

Source code in geoeq/design/plots.py

def settlement_time_plot(ax=None, save_as: str = None):
    """The classical Terzaghi consolidation curve: U vs Tv (log scale).

    Plots the approximate piecewise formula used by ``time_factor``:
    Tv = (pi/4) U^2 for U < 0.6, Tv = 1.781 - 0.933 log10(100(1-U))
    for U >= 0.6.
    """
    import matplotlib.pyplot as plt
    U_arr = np.linspace(0.001, 0.99, 200)
    Tv_arr = np.array([time_factor(u) for u in U_arr])
    if ax is None:
        fig, ax = plt.subplots(figsize=(7, 5))
    else:
        fig = ax.figure
    ax.plot(Tv_arr, U_arr * 100, color="#1f3a93", linewidth=2.0)
    # Annotated reference points.
    for U in (0.5, 0.9):
        Tv = time_factor(U)
        ax.plot([Tv], [U * 100], "o", color="#c0392b", markersize=6)
        ax.annotate(f"$U={int(U*100)}\\%$\n$T_v={Tv:.3f}$",
                    xy=(Tv, U * 100), xytext=(10, -10),
                    textcoords="offset points", fontsize=9,
                    color="#c0392b")
    ax.set_xscale("log")
    ax.set_xlim(1e-3, 3)
    ax.set_ylim(0, 100)
    ax.set_xlabel(r"Time factor $T_v$")
    ax.set_ylabel(r"Average degree of consolidation $U$ (%)")
    ax.set_title("Terzaghi 1-D consolidation: $U$ vs $T_v$")
    ax.grid(True, which="both", alpha=0.3)
    if save_as:
        fig.savefig(save_as, dpi=300, bbox_inches="tight")
    return fig