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formula_8_38_39_40

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_38_39_40

Formula 8.38, 8.39 and 8.40 from FprEN 1992-1-1:2023: Chapter 8: Ultimate limit states (ULS).

Classes:

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_38_39_40.Form8Dot38To40ReinforcementRatioPlanarMembers 

Form8Dot38To40ReinforcementRatioPlanarMembers(
    v_ed_x: N_MM,
    v_ed_y: N_MM,
    rho_l_x: DIMENSIONLESS,
    rho_l_y: DIMENSIONLESS,
    alpha_v: DEG | None = None,
)

Bases: Formula

Class representing formulas 8.38, 8.39 and 8.40 for the calculation of the reinforcement ratio of planar members with different reinforcement ratios in both directions, taken as a function of the ratio of the shear forces.

[\(\rho_l\)] Reinforcement ratio of planar members, as a function of the ratio of the shear forces [\(v_{Ed,y}/v_{Ed,x}\)] [\(-\)].

FprEN 1992-1-1:2023 (E) art 8.2.2 (7) - Formula (8.38), (8.39) and (8.40)

The boundaries of the ratio are one-sided as printed: a ratio of exactly 0.5 falls under Formula (8.38) and a ratio of exactly 2 falls under Formula (8.40). The standard gives no rule for [\(v_{Ed,x} = 0\)], for which the ratio is undefined, so that input is rejected.

Both shear forces are taken as magnitudes. They are components of a shear force vector and the standard places no restriction on their sign, but the printed boundaries only order a ratio of magnitudes: a signed ratio turns negative as soon as one component does, and every negative value would fall under Formula (8.38) regardless of which direction carries the shear. This matches Formulas (8.22) to (8.24), which the standard writes with the same ratio and the same two boundaries.

Parameters:

  • v_ed_x (N_MM) –

    [\(v_{Ed,x}\)] Out-of-plane design shear force per unit width acting on the cross-section perpendicular to the x direction [\(N/mm\)].

  • v_ed_y (N_MM) –

    [\(v_{Ed,y}\)] Out-of-plane design shear force per unit width acting on the cross-section perpendicular to the y direction [\(N/mm\)].

  • rho_l_x (DIMENSIONLESS) –

    [\(\rho_{l,x}\)] Reinforcement ratio in the x direction [\(-\)].

  • rho_l_y (DIMENSIONLESS) –

    [\(\rho_{l,y}\)] Reinforcement ratio in the y direction [\(-\)].

  • alpha_v (DEG, default: None ) –

    [\(\alpha_v\)] Angle between the principal shear force and the x-axis, which the standard defines in 8.2.1(5). When it is left out it is computed from the two shear forces with Formula (8.26), [\(\arctan\left(\left|v_{Ed,y}\right| / \left|v_{Ed,x}\right|\right)\)], the value that clause allows it to be taken as and the only one consistent with the forces passed here. That clause prints Formula (8.26) as what the angle "may be taken as", so a directly determined direction of the principal shear force is admissible as well and can be passed here instead; it is then the caller's responsibility to keep it consistent with the two forces. The angle only reaches Formula (8.39). It lies between 0 and 90 degrees, as Formula (8.26) can produce nothing else [\(degrees\)].

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_38_39_40.py 
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def __init__(
    self,
    v_ed_x: N_MM,
    v_ed_y: N_MM,
    rho_l_x: DIMENSIONLESS,
    rho_l_y: DIMENSIONLESS,
    alpha_v: DEG | None = None,
) -> None:
    r"""[$\rho_l$] Reinforcement ratio of planar members, as a function of the ratio of the shear forces
    [$v_{Ed,y}/v_{Ed,x}$] [$-$].

    FprEN 1992-1-1:2023 (E) art 8.2.2 (7) - Formula (8.38), (8.39) and (8.40)

    The boundaries of the ratio are one-sided as printed: a ratio of exactly 0.5 falls under Formula (8.38)
    and a ratio of exactly 2 falls under Formula (8.40). The standard gives no rule for [$v_{Ed,x} = 0$],
    for which the ratio is undefined, so that input is rejected.

    Both shear forces are taken as magnitudes. They are components of a shear force vector and the standard
    places no restriction on their sign, but the printed boundaries only order a ratio of magnitudes: a
    signed ratio turns negative as soon as one component does, and every negative value would fall under
    Formula (8.38) regardless of which direction carries the shear. This matches Formulas (8.22) to (8.24),
    which the standard writes with the same ratio and the same two boundaries.

    Parameters
    ----------
    v_ed_x : N_MM
        [$v_{Ed,x}$] Out-of-plane design shear force per unit width acting on the cross-section perpendicular
        to the x direction [$N/mm$].
    v_ed_y : N_MM
        [$v_{Ed,y}$] Out-of-plane design shear force per unit width acting on the cross-section perpendicular
        to the y direction [$N/mm$].
    rho_l_x : DIMENSIONLESS
        [$\rho_{l,x}$] Reinforcement ratio in the x direction [$-$].
    rho_l_y : DIMENSIONLESS
        [$\rho_{l,y}$] Reinforcement ratio in the y direction [$-$].
    alpha_v : DEG, optional
        [$\alpha_v$] Angle between the principal shear force and the x-axis, which the standard defines in
        8.2.1(5). When it is left out it is computed from the two shear forces with Formula (8.26),
        [$\arctan\left(\left|v_{Ed,y}\right| / \left|v_{Ed,x}\right|\right)$], the value that clause allows
        it to be taken as and the only one consistent with the forces passed here. That clause prints
        Formula (8.26) as what the angle "may be taken as", so a directly determined direction of the
        principal shear force is admissible as well and can be passed here instead; it is then the caller's
        responsibility to keep it consistent with the two forces. The angle only reaches Formula (8.39). It
        lies between 0 and 90 degrees, as Formula (8.26) can produce nothing else [$degrees$].
    """
    super().__init__()
    self.v_ed_x = v_ed_x
    self.v_ed_y = v_ed_y
    self.rho_l_x = rho_l_x
    self.rho_l_y = rho_l_y
    self.alpha_v = self._angle(v_ed_x, v_ed_y) if alpha_v is None else alpha_v

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_38_39_40.Form8Dot38To40ReinforcementRatioPlanarMembers.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formulas 8.38, 8.39 and 8.40.

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_38_39_40.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formulas 8.38, 8.39 and 8.40."""
    _equation: str = (
        r"\begin{cases} \rho_{l,x} & \text{if } \frac{\left|v_{Ed,y}\right|}{\left|v_{Ed,x}\right|} \leq 0.5 \\ "
        r"\rho_{l,x} \cdot \cos^4(\alpha_v) + \rho_{l,y} \cdot \sin^4(\alpha_v) "
        r"& \text{if } 0.5 < \frac{\left|v_{Ed,y}\right|}{\left|v_{Ed,x}\right|} < 2 \\ "
        r"\rho_{l,y} & \text{if } \frac{\left|v_{Ed,y}\right|}{\left|v_{Ed,x}\right|} \geq 2 \end{cases}"
    )
    _numeric_equation: str = latex_replace_symbols(
        template=_equation,
        replacements={
            r"v_{Ed,x}": f"{self.v_ed_x:.{n}f}",
            r"v_{Ed,y}": f"{self.v_ed_y:.{n}f}",
            r"\rho_{l,x}": f"{self.rho_l_x:.{n}f}",
            r"\rho_{l,y}": f"{self.rho_l_y:.{n}f}",
            r"\alpha_v": f"{self.alpha_v:.{n}f}",
        },
        unique_symbol_check=False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        template=_equation,
        replacements={
            r"v_{Ed,x}": rf"{self.v_ed_x:.{n}f} \ N/mm",
            r"v_{Ed,y}": rf"{self.v_ed_y:.{n}f} \ N/mm",
            r"\rho_{l,x}": f"{self.rho_l_x:.{n}f}",
            r"\rho_{l,y}": f"{self.rho_l_y:.{n}f}",
            r"\alpha_v": rf"{self.alpha_v:.{n}f} \ degrees",
        },
        unique_symbol_check=False,
    )
    return LatexFormula(
        return_symbol=r"\rho_l",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="-",
    )