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formula_5_9

codes.eurocode.en_1993_5_2007.chapter_5_ultimate_limit_states.formula_5_9

Formula 5.9 from EN 1993-5:2007 Chapter 5 - Ultimate limit state.

Classes:

codes.eurocode.en_1993_5_2007.chapter_5_ultimate_limit_states.formula_5_9.Form5Dot9ReducedBendingMomentResistance 

Form5Dot9ReducedBendingMomentResistance(
    beta_b: DIMENSIONLESS,
    w_pl: MM3,
    rho: DIMENSIONLESS,
    a_v: MM2,
    t_w: MM,
    alpha: DEG,
    f_y: MPA,
    gamma_m_0: DIMENSIONLESS,
    mc_rd: KNM,
)

Bases: Formula

Class representing formula 5.9 for reduced design bending moment resistance of the cross-section.

[\(M_{V,Rd}\)] Calculate reduced design bending moment resistance of the cross-section allowing for the shear force in [\(kNm\)].

This calculation is specifically for sheet pile cross-sections, particularly U-profiles and Z-profiles.

EN 1993-5:2007(E) art.5.2.2(9) - Formula (5.9)

Parameters:

  • beta_b (DIMENSIONLESS) –

    [\(β_{b}\)] Reduction factor for the bending resistance of the cross-section, which takes account of possible lack of shear force transmission in the interlocks [\(-\)]. Defined in EN 1993-5:2007(E) art. 5.2.2(2) or CUR166, part 2, par. 3.3.2.

  • w_pl (MM3) –

    [\(W_{pl}\)] Plastic section modulus in [\(mm^3\)].

  • rho (DIMENSIONLESS) –

    [\(ρ\)] Reduction factor for shear resistance of the cross-section, according EN 1993-5:2007(E) art. 5.2.2(9) formula 5.10 [\(-\)].

  • a_v (MM2) –

    [\(A_{V}\)] Projected shear area for each web, acting in the same direction as VEd in [\(mm^2\)].

  • t_w (MM) –

    [\(t_{w}\)] Thickness of the web in [\(mm\)].

  • alpha (DEG) –

    [\(α\)] the inclination of the web according to EN 1993-5:2007(E) Figure 5-1 in [\(degrees\)].

  • f_y (MPA) –

    [\(f_{y}\)] Yield strength in [\(MPa\)].

  • gamma_m_0 (DIMENSIONLESS) –

    [\(γ_{M0}\)] Partial factor for material properties in [\(-\)].

  • mc_rd (KNM) –

    [\(M_{c,Rd}\)] Design moment resistance of the cross-section in [\(kNm\)].

    The mc_rd parameter represents the design moment resistance of the cross-section. In the context of the formula for reduced bending moment resistance, it serves as an upper bound. The formula calculates the reduced design bending moment resistance (m_v_rd) and then returns the minimum of m_v_rd and mc_rd. This means that the result of the formula will never exceed mc_rd, making mc_rd an upper bound for this formula.

    [\(M_{v,Rd} \leq M_{c,Rd}\)]

Source code in blueprints/codes/eurocode/en_1993_5_2007/chapter_5_ultimate_limit_states/formula_5_9.py 
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def __init__(
    self,
    beta_b: DIMENSIONLESS,
    w_pl: MM3,
    rho: DIMENSIONLESS,
    a_v: MM2,
    t_w: MM,
    alpha: DEG,
    f_y: MPA,
    gamma_m_0: DIMENSIONLESS,
    mc_rd: KNM,
) -> None:
    r"""[$M_{V,Rd}$] Calculate reduced design bending moment resistance of the cross-section allowing for the shear force in [$kNm$].

    This calculation is specifically for sheet pile cross-sections, particularly U-profiles and Z-profiles.

    EN 1993-5:2007(E) art.5.2.2(9) - Formula (5.9)

    Parameters
    ----------
    beta_b : DIMENSIONLESS
        [$β_{b}$] Reduction factor for the bending resistance of the cross-section, which takes account of
        possible lack of shear force transmission in the interlocks [$-$].
        Defined in EN 1993-5:2007(E) art. 5.2.2(2) or CUR166, part 2, par. 3.3.2.
    w_pl : MM3
        [$W_{pl}$] Plastic section modulus in [$mm^3$].
    rho : DIMENSIONLESS
        [$ρ$] Reduction factor for shear resistance of the cross-section, according EN 1993-5:2007(E) art. 5.2.2(9) formula 5.10 [$-$].
    a_v : MM2
        [$A_{V}$] Projected shear area for each web, acting in the same direction as VEd in [$mm^2$].
    t_w : MM
        [$t_{w}$] Thickness of the web in [$mm$].
    alpha : DEG
        [$α$] the inclination of the web according to EN 1993-5:2007(E) Figure 5-1 in [$degrees$].
    f_y : MPA
        [$f_{y}$] Yield strength in [$MPa$].
    gamma_m_0 : DIMENSIONLESS
        [$γ_{M0}$] Partial factor for material properties in [$-$].
    mc_rd : KNM
        [$M_{c,Rd}$] Design moment resistance of the cross-section in [$kNm$].

        The `mc_rd` parameter represents the design moment resistance of the cross-section.
        In the context of the formula for reduced bending moment resistance, it serves as an upper bound.
        The formula calculates the reduced design bending moment resistance (`m_v_rd`) and then returns the minimum of `m_v_rd` and `mc_rd`.
        This means that the result of the formula will never exceed `mc_rd`, making `mc_rd` an upper bound for this formula.

        [$M_{v,Rd} \leq M_{c,Rd}$]
    """
    super().__init__()
    self.beta_b: DIMENSIONLESS = beta_b
    self.w_pl: MM3 = w_pl
    self.rho: DIMENSIONLESS = rho
    self.a_v: MM2 = a_v
    self.t_w: MM = t_w
    self.alpha: DEG = alpha
    self.f_y: MPA = f_y
    self.gamma_m_0: DIMENSIONLESS = gamma_m_0
    self.mc_rd: KNM = mc_rd

codes.eurocode.en_1993_5_2007.chapter_5_ultimate_limit_states.formula_5_9.Form5Dot9ReducedBendingMomentResistance.latex 

latex(n: int = 2) -> LatexFormula

Returns LatexFormula object for formula 5.9.

Source code in blueprints/codes/eurocode/en_1993_5_2007/chapter_5_ultimate_limit_states/formula_5_9.py 
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def latex(self, n: int = 2) -> LatexFormula:
    """Returns LatexFormula object for formula 5.9."""
    latex_equation = latex_min_curly_brackets(
        r"\left(\beta_b \cdot W_{pl} - \frac{\rho \cdot A_v^2}{4 \cdot t_w \cdot \sin(\alpha)}\right) \cdot \frac{f_y}{\gamma_{M0}}, M_{c,Rd}"
    )
    fraction = latex_fraction(numerator=f"{self.f_y:.{n}f}", denominator=f"{self.gamma_m_0:.{n}f}")
    return LatexFormula(
        return_symbol=r"M_{V,Rd}",
        result=f"{self:.{n}f}",
        equation=latex_equation,
        numeric_equation=(
            latex_min_curly_brackets(
                rf"\left({self.beta_b:.{n}f} \cdot {self.w_pl:.{n}f} - \frac{{{self.rho:.{n}f} "
                rf"\cdot {self.a_v:.{n}f}^2}}{{4 \cdot {self.t_w:.{n}f} \cdot "
                rf"\sin({self.alpha:.{n}f})}}\right) \cdot "
                rf"{fraction} \cdot 10^{{-6}}, {self.mc_rd:.{n}f}"
            )
        ),
        comparison_operator_label="=",
    )