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formula_8_49_50

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50

Formulas 8.49 and 8.50 from EN 1993-1-1:2022: Chapter 8 - Ultimate Limit State.

Classes:

  • Form8Dot49And50MomentReduction –

    Class representing formulas 8.49 and 8.50 for the calculation of reduced bending moment.

  • Form8Dot50A –

    Class representing formula 8.50a for the calculation of [\(a\)].

  • Form8Dot50N –

    Class representing formula 8.50 for the calculation of [\(n\)].

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50.Form8Dot49And50MomentReduction 

Form8Dot49And50MomentReduction(
    mpl_z_rd: NMM, a: DIMENSIONLESS, n: DIMENSIONLESS
)

Bases: Formula

Class representing formulas 8.49 and 8.50 for the calculation of reduced bending moment.

[\(M_{N,z,Rd}\)] Reduced bending moment [\(Nmm\)].

EN 1993-1-1:2022 art.8.2.9.1(6) - Formulas (8.49) and (8.50)

Parameters:

  • mpl_z_rd (NMM) –

    [\(M_{pl,z,Rd}\)] Plastic bending moment about the z-axis [\(Nmm\)].

  • a (DIMENSIONLESS) –

    Reduction factor for cross-sectional area, see equation 8.50a (Form8Dot50A) [-].

  • n (DIMENSIONLESS) –

    Axial force ratio, see equation 8.50n (Form8Dot50N) [-].

Source code in blueprints/codes/eurocode/en_1993_1_1_2022/chapter_8_ultimate_limit_state/formula_8_49_50.py 
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def __init__(
    self,
    mpl_z_rd: NMM,
    a: DIMENSIONLESS,
    n: DIMENSIONLESS,
) -> None:
    r"""[$M_{N,z,Rd}$] Reduced bending moment [$Nmm$].

    EN 1993-1-1:2022 art.8.2.9.1(6) - Formulas (8.49) and (8.50)

    Parameters
    ----------
    mpl_z_rd : NMM
        [$M_{pl,z,Rd}$] Plastic bending moment about the z-axis [$Nmm$].
    a : DIMENSIONLESS
        Reduction factor for cross-sectional area, see equation 8.50a (`Form8Dot50A`) [-].
    n : DIMENSIONLESS
        Axial force ratio, see equation 8.50n (`Form8Dot50N`) [-].
    """
    super().__init__()
    self.mpl_z_rd = mpl_z_rd
    self.a = a
    self.n = n

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50.Form8Dot49And50MomentReduction.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formulas 8.49 and 8.50.

Source code in blueprints/codes/eurocode/en_1993_1_1_2022/chapter_8_ultimate_limit_state/formula_8_49_50.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formulas 8.49 and 8.50."""
    _equation: str = (
        r"\begin{cases} M_{pl,z,Rd} & \text{if } n \leq a \\ "
        r"M_{pl,z,Rd} \cdot \left[1 - \left(\frac{ n - a}{1 - a}\right)^2\right] & \text{if } n > a \end{cases}"
    )
    _numeric_equation: str = latex_replace_symbols(
        _equation,
        replacements={
            "M_{pl,z,Rd}": f"{self.mpl_z_rd:.{n}f}",
            " n": f" {self.n:.{n}f}",
            " a": f" {self.a:.{n}f}",
        },
        unique_symbol_check=False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        _equation,
        replacements={
            "M_{pl,z,Rd}": rf"{self.mpl_z_rd:.{n}f} \ Nmm",
            " n": rf" {self.n:.{n}f}",
            " a": rf" {self.a:.{n}f}",
        },
        unique_symbol_check=False,
    )
    return LatexFormula(
        return_symbol=r"M_{N,z,Rd}",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="Nmm",
    )

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50.Form8Dot50A 

Form8Dot50A(capital_a: MM2, b: MM, tf: MM)

Bases: Formula

Class representing formula 8.50a for the calculation of [\(a\)].

[\(a\)] Reduction factor for cross-sectional area [dimensionless].

EN 1993-1-1:2022 art.8.2.9.1(6) - Formula (8.50a)

Parameters:

  • capital_a (MM2) –

    [\(A\)] Cross-sectional area [\(mm^2\)].

  • b (MM) –

    [\(b\)] Width of the cross-section [\(mm\)].

  • tf (MM) –

    [\(t_f\)] Thickness of the flange [\(mm\)].

Source code in blueprints/codes/eurocode/en_1993_1_1_2022/chapter_8_ultimate_limit_state/formula_8_49_50.py 
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def __init__(self, capital_a: MM2, b: MM, tf: MM) -> None:
    r"""[$a$] Reduction factor for cross-sectional area [dimensionless].

    EN 1993-1-1:2022 art.8.2.9.1(6) - Formula (8.50a)

    Parameters
    ----------
    capital_a : MM2
        [$A$] Cross-sectional area [$mm^2$].
    b : MM
        [$b$] Width of the cross-section [$mm$].
    tf : MM
        [$t_f$] Thickness of the flange [$mm$].
    """
    super().__init__()
    self.capital_a = capital_a
    self.b = b
    self.tf = tf

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50.Form8Dot50A.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 8.50a.

Source code in blueprints/codes/eurocode/en_1993_1_1_2022/chapter_8_ultimate_limit_state/formula_8_49_50.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 8.50a."""
    _equation: str = r"\min\left(\frac{A - 2 \cdot b \cdot t_f}{A}, 0.5\right)"
    _numeric_equation: str = latex_replace_symbols(
        _equation,
        replacements={
            r"A": f"{self.capital_a:.{n}f}",
            r"b": f"{self.b:.{n}f}",
            r"t_f": f"{self.tf:.{n}f}",
        },
        unique_symbol_check=False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        _equation,
        replacements={
            r"A": rf"{self.capital_a:.{n}f} \ mm^2",
            r"b": rf"{self.b:.{n}f} \ mm",
            r"t_f": rf"{self.tf:.{n}f} \ mm",
        },
        unique_symbol_check=False,
    )
    return LatexFormula(
        return_symbol=r"a",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="-",
    )

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50.Form8Dot50N 

Form8Dot50N(n_ed: N, n_pl_rd: N)

Bases: Formula

Class representing formula 8.50 for the calculation of [\(n\)].

[\(n\)] Axial force ratio [dimensionless].

EN 1993-1-1:2022 art.8.2.9.1(6) - Formula (8.50)

Parameters:

  • n_ed (N) –

    [\(N_{Ed}\)] Design axial force [\(N\)].

  • n_pl_rd (N) –

    [\(N_{pl,Rd}\)] Plastic resistance of the cross-section [\(N\)].

Source code in blueprints/codes/eurocode/en_1993_1_1_2022/chapter_8_ultimate_limit_state/formula_8_49_50.py 
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def __init__(self, n_ed: N, n_pl_rd: N) -> None:
    r"""[$n$] Axial force ratio [dimensionless].

    EN 1993-1-1:2022 art.8.2.9.1(6) - Formula (8.50)

    Parameters
    ----------
    n_ed : N
        [$N_{Ed}$] Design axial force [$N$].
    n_pl_rd : N
        [$N_{pl,Rd}$] Plastic resistance of the cross-section [$N$].
    """
    super().__init__()
    self.n_ed = n_ed
    self.n_pl_rd = n_pl_rd

codes.eurocode.en_1993_1_1_2022.chapter_8_ultimate_limit_state.formula_8_49_50.Form8Dot50N.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 8.50n.

Source code in blueprints/codes/eurocode/en_1993_1_1_2022/chapter_8_ultimate_limit_state/formula_8_49_50.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 8.50n."""
    _equation: str = r"\frac{N_{Ed}}{N_{pl,Rd}}"
    _numeric_equation: str = latex_replace_symbols(
        _equation,
        replacements={
            r"N_{Ed}": f"{self.n_ed:.{n}f}",
            r"N_{pl,Rd}": f"{self.n_pl_rd:.{n}f}",
        },
        unique_symbol_check=False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        _equation,
        replacements={
            r"N_{Ed}": rf"{self.n_ed:.{n}f} \ N",
            r"N_{pl,Rd}": rf"{self.n_pl_rd:.{n}f} \ N",
        },
        unique_symbol_check=True,
    )
    return LatexFormula(
        return_symbol=r"n",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="-",
    )