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formula_6_41

codes.eurocode.en_1993_1_1_2005.chapter_6_ultimate_limit_state.formula_6_41

formula 6.41 from EN 1993-1-1:2005: chapter 6 - ultimate limit state.

Classes:

codes.eurocode.en_1993_1_1_2005.chapter_6_ultimate_limit_state.formula_6_41.Form6Dot41BiaxialBendingCheck 

Form6Dot41BiaxialBendingCheck(
    my_ed: NMM,
    m_n_y_rd: NMM,
    mz_ed: NMM,
    m_n_z_rd: NMM,
    alpha: DIMENSIONLESS,
    beta: DIMENSIONLESS,
)

Bases: Formula

class representing formula 6.41 for bi-axial bending.

Check the bi-axial bending criterion.

EN 1993-1-1:2005 art.6.2.9.1(6) - formula (6.41)

Parameters:

  • my_ed (NMM) –

    [\(m_{y,ed}\)] design bending moment about the y-axis [\(Nmm\)].

  • m_n_y_rd (NMM) –

    [\(m_{n,y,rd}\)] design resistance moment about the y-axis [\(Nmm\)].

  • mz_ed (NMM) –

    [\(m_{z,ed}\)] design bending moment about the z-axis [\(Nmm\)].

  • m_n_z_rd (NMM) –

    [\(m_{n,z,rd}\)] design resistance moment about the z-axis [\(Nmm\)].

  • alpha (DIMENSIONLESS) –

    [\(\alpha\)] exponent for the y-axis term.

  • beta (DIMENSIONLESS) –

    [\(\beta\)] exponent for the z-axis term.

    In which \(\alpha\) and \(\beta\) are constants, which may conservatively be taken as unity, otherwise as follows: - I and H sections: \(\alpha = 2\) and \(\beta = 5 \cdot n\), but \(\beta \geq 1\). - Circular hollow sections: \(\alpha = 2\) and \(\beta = 2\). - Rectangular hollow sections: \(\alpha = \beta = \frac{1.66}{1-1.13 \cdot n^2}\), but \(\alpha = \beta \leq 6\).

    Where \(n = N_{Ed} / N_{pl,Rd}\), see equation 6.38n.

Source code in blueprints/codes/eurocode/en_1993_1_1_2005/chapter_6_ultimate_limit_state/formula_6_41.py 
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def __init__(
    self,
    my_ed: NMM,
    m_n_y_rd: NMM,
    mz_ed: NMM,
    m_n_z_rd: NMM,
    alpha: DIMENSIONLESS,
    beta: DIMENSIONLESS,
) -> None:
    r"""Check the bi-axial bending criterion.

    EN 1993-1-1:2005 art.6.2.9.1(6) - formula (6.41)

    Parameters
    ----------
    my_ed : NMM
        [$m_{y,ed}$] design bending moment about the y-axis [$Nmm$].
    m_n_y_rd : NMM
        [$m_{n,y,rd}$] design resistance moment about the y-axis [$Nmm$].
    mz_ed : NMM
        [$m_{z,ed}$] design bending moment about the z-axis [$Nmm$].
    m_n_z_rd : NMM
        [$m_{n,z,rd}$] design resistance moment about the z-axis [$Nmm$].
    alpha : DIMENSIONLESS
        [$\alpha$] exponent for the y-axis term.
    beta : DIMENSIONLESS
        [$\beta$] exponent for the z-axis term.

        In which $\alpha$ and $\beta$ are constants, which may conservatively be taken as unity, otherwise as follows:
        - I and H sections:
            $\alpha = 2$ and $\beta = 5 \cdot n$, but $\beta \geq 1$.
        - Circular hollow sections:
            $\alpha = 2$ and $\beta = 2$.
        - Rectangular hollow sections:
            $\alpha = \beta = \frac{1.66}{1-1.13 \cdot n^2}$, but $\alpha = \beta \leq 6$.

        Where $n = N_{Ed} / N_{pl,Rd}$, see equation 6.38n.
    """
    super().__init__()
    self.my_ed = my_ed
    self.m_n_y_rd = m_n_y_rd
    self.mz_ed = mz_ed
    self.m_n_z_rd = m_n_z_rd
    self.alpha = alpha
    self.beta = beta

codes.eurocode.en_1993_1_1_2005.chapter_6_ultimate_limit_state.formula_6_41.Form6Dot41BiaxialBendingCheck.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 6.41.

Source code in blueprints/codes/eurocode/en_1993_1_1_2005/chapter_6_ultimate_limit_state/formula_6_41.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 6.41."""
    _equation: str = r"\left[ \frac{M_{y,Ed}}{M_{N,y,Rd}} \right]^{\alpha} + \left[ \frac{M_{z,Ed}}{M_{N,z,Rd}} \right]^{\beta} \leq 1"
    _numeric_equation: str = latex_replace_symbols(
        _equation,
        {
            "M_{y,Ed}": f"{self.my_ed:.{n}f}",
            "M_{N,y,Rd}": f"{self.m_n_y_rd:.{n}f}",
            "M_{z,Ed}": f"{self.mz_ed:.{n}f}",
            "M_{N,z,Rd}": f"{self.m_n_z_rd:.{n}f}",
            r"\alpha": f"{self.alpha:.{n}f}",
            r"\beta": f"{self.beta:.{n}f}",
        },
        False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        _equation,
        {
            "M_{y,Ed}": rf"{self.my_ed:.{n}f} \ Nmm",
            "M_{N,y,Rd}": rf"{self.m_n_y_rd:.{n}f} \ Nmm",
            "M_{z,Ed}": rf"{self.mz_ed:.{n}f} \ Nmm",
            "M_{N,z,Rd}": rf"{self.m_n_z_rd:.{n}f} \ Nmm",
            r"\alpha": rf"{self.alpha:.{n}f}",
            r"\beta": rf"{self.beta:.{n}f}",
        },
        False,
    )
    return LatexFormula(
        return_symbol=r"CHECK",
        result="OK" if self.__bool__() else "\\text{Not OK}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="\\to",
        unit="",
    )