formula_8_22_23_24

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_22_23_24

Formula 8.22, 8.23 and 8.24 from FprEN 1992-1-1:2023: Chapter 8: Ultimate limit states (ULS).

Classes:

Form8Dot22To24EffectiveDepth –

Class representing formulas 8.22, 8.23 and 8.24 for the calculation of the effective depth of planar members,

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_22_23_24.Form8Dot22To24EffectiveDepth

Form8Dot22To24EffectiveDepth(v_ed_x: N_MM, v_ed_y: N_MM, d_x: MM, d_y: MM)

Bases: Formula

Class representing formulas 8.22, 8.23 and 8.24 for the calculation of the effective depth of planar members, taken as a function of the ratio of the shear forces.

[\(d\)] Effective depth of planar members, as a function of the ratio of the shear forces [\(v_{Ed,y}/v_{Ed,x}\)] [\(mm\)].

FprEN 1992-1-1:2023 (E) art 8.2.1 (5) - Formula (8.22), (8.23) and (8.24)

The boundaries of the ratio are one-sided as printed: a ratio of exactly 0.5 falls under Formula (8.22) and a ratio of exactly 2 falls under Formula (8.24). 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, and the standard does not settle that: it prints no absolute value bars and defines no sign convention. Read literally, the ratio is signed, so any pair with opposite signs is negative, always satisfies the first condition, and selects [\(d_x\)] even where [\(\left|v_{Ed,y}\right|\)] dominates. Flipping the positive direction of the y axis would then change the effective depth, which cannot be intended.

The alternative the standard prints on the same page settles it: Formula (8.26) gives [\(\alpha_v = \arctan(v_{Ed,y}/v_{Ed,x})\)] and Formula (8.25) gives [\(d = d_x \cdot \cos^2\alpha_v + d_y \cdot \sin^2\alpha_v\)], which is even in [\(\alpha_v\)] and therefore insensitive to the sign. Formulas (8.22) to (8.24) are the stepped approximation of that same continuous function, so their boundaries of 0,5 and 2 order a ratio of magnitudes. Formula (8.21) reaches the same place by squaring both components.

The LaTeX prints the absolute value bars that the standard leaves out, so that the substituted numbers match the branch that was taken.

The requirement that [\(d_x\)] and [\(d_y\)] be strictly positive is an addition made here: neither appears in a denominator in these three formulas, and the standard prints no such condition. An effective depth of zero is not a cross-section.

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 [\(N/mm\)].
d_x (MM) –

[\(d_x\)] Effective depth in the x direction [\(mm\)].
d_y (MM) –

[\(d_y\)] Effective depth in the y direction [\(mm\)].

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_22_23_24.py

def __init__(self, v_ed_x: N_MM, v_ed_y: N_MM, d_x: MM, d_y: MM) -> None:
    r"""[$d$] Effective depth of planar members, as a function of the ratio of the shear forces
    [$v_{Ed,y}/v_{Ed,x}$] [$mm$].

    FprEN 1992-1-1:2023 (E) art 8.2.1 (5) - Formula (8.22), (8.23) and (8.24)

    The boundaries of the ratio are one-sided as printed: a ratio of exactly 0.5 falls under Formula (8.22)
    and a ratio of exactly 2 falls under Formula (8.24). 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, and the standard does not settle that: it prints no absolute
    value bars and defines no sign convention. Read literally, the ratio is signed, so any pair with
    opposite signs is negative, always satisfies the first condition, and selects [$d_x$] even where
    [$\left|v_{Ed,y}\right|$] dominates. Flipping the positive direction of the y axis would then change
    the effective depth, which cannot be intended.

    The alternative the standard prints on the same page settles it: Formula (8.26) gives
    [$\alpha_v = \arctan(v_{Ed,y}/v_{Ed,x})$] and Formula (8.25) gives
    [$d = d_x \cdot \cos^2\alpha_v + d_y \cdot \sin^2\alpha_v$], which is even in [$\alpha_v$] and therefore
    insensitive to the sign. Formulas (8.22) to (8.24) are the stepped approximation of that same continuous
    function, so their boundaries of 0,5 and 2 order a ratio of magnitudes. Formula (8.21) reaches the same
    place by squaring both components.

    The LaTeX prints the absolute value bars that the standard leaves out, so that the substituted numbers
    match the branch that was taken.

    The requirement that [$d_x$] and [$d_y$] be strictly positive is an addition made here: neither appears
    in a denominator in these three formulas, and the standard prints no such condition. An effective depth
    of zero is not a cross-section.

    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 [$N/mm$].
    d_x : MM
        [$d_x$] Effective depth in the x direction [$mm$].
    d_y : MM
        [$d_y$] Effective depth in the y direction [$mm$].
    """
    super().__init__()
    self.v_ed_x = v_ed_x
    self.v_ed_y = v_ed_y
    self.d_x = d_x
    self.d_y = d_y

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_22_23_24.Form8Dot22To24EffectiveDepth.latex

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formulas 8.22, 8.23 and 8.24.

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_22_23_24.py

def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formulas 8.22, 8.23 and 8.24."""
    _equation: str = (
        r"\begin{cases} d_x & \text{if } \frac{\left|v_{Ed,y}\right|}{\left|v_{Ed,x}\right|} \leq 0.5 \\ "
        r"0.5 \cdot (d_x + d_y) & \text{if } 0.5 < \frac{\left|v_{Ed,y}\right|}{\left|v_{Ed,x}\right|} < 2 \\ "
        r"d_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"d_x": f"{self.d_x:.{n}f}",
            r"d_y": f"{self.d_y:.{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"d_x": rf"{self.d_x:.{n}f} \ mm",
            r"d_y": rf"{self.d_y:.{n}f} \ mm",
        },
        unique_symbol_check=False,
    )
    # Only the middle branch has a step between the selected expression and the result. LatexFormula skips
    # an empty intermediate result, so the other two branches show none.
    _intermediate: str = ""
    if 0.5 < abs(self.v_ed_y) / abs(self.v_ed_x) < 2:
        _intermediate = rf"0.5 \cdot ({self.d_x:.{n}f} + {self.d_y:.{n}f})"

    return LatexFormula(
        return_symbol=r"d",
        result=f"{self:.{n}f}",
        intermediate_result=_intermediate,
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="mm",
    )