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formula_8_55

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_55

Formula 8.55 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_55.Form8Dot55EnhancedShearStressResistance 

Form8Dot55EnhancedShearStressResistance(
    nu: DIMENSIONLESS,
    f_cd: MPA,
    theta: DEG,
    beta_incl: DEG,
    rho_w: DIMENSIONLESS,
    f_ywd: MPA,
)

Bases: Formula

Class representing formula 8.55 for the calculation of the enhanced shear stress resistance where a concentrated load is applied close to a support.

[\(\tau_{Rd}\)] Enhanced shear stress resistance for concentrated loads applied close to a support [\(MPa\)].

FprEN 1992-1-1:2023 (E) art 8.2.3 (12) - Formula (8.55)

The standard offers this enhancement in cases where concentrated loads are applied at a distance [\(a_v = z \cdot \cot\beta_{incl}\)] less than [\(z \cdot \cot\theta\)] from a support. That is a condition of application, not a bound on the result, so it is not enforced here.

Parameters:

  • nu (DIMENSIONLESS) –

    [\(\nu\)] Strength reduction factor for concrete cracked in shear. A value of 0,5 may be adopted when using the angles of the compression field given in 8.2.3(4), see 8.2.3(6). A higher value may be adopted under the conditions of 8.2.3(7), calculated according to Formula (8.45) [\(-\)].

  • f_cd (MPA) –

    [\(f_{cd}\)] Design value of the compressive strength of concrete [\(MPa\)].

  • theta (DEG) –

    [\(\theta\)] Inclination of the compression field in the web, selected within the range of Formula (8.41). A flatter field, [\(\cot\theta < 1\)], is allowed under 8.2.3(11), but only if [\(f_{ywd}\)] is replaced by [\(\sigma_{swd}\)] of Formula (8.56); making that substitution is left to the caller [\(degrees\)].

  • beta_incl (DEG) –

    [\(\beta_{incl}\)] Angle that follows from the distance of the concentrated load to the support through [\(a_v = z \cdot \cot\beta_{incl}\)], so [\(\beta_{incl} = \arctan(z/a_v)\)] [\(degrees\)].

  • rho_w (DIMENSIONLESS) –

    [\(\rho_w\)] Shear reinforcement ratio according to Formula (8.43) [\(-\)].

  • f_ywd (MPA) –

    [\(f_{ywd}\)] Design value of the yield strength of the shear reinforcement [\(MPa\)].

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_55.py 
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def __init__(
    self,
    nu: DIMENSIONLESS,
    f_cd: MPA,
    theta: DEG,
    beta_incl: DEG,
    rho_w: DIMENSIONLESS,
    f_ywd: MPA,
) -> None:
    r"""[$\tau_{Rd}$] Enhanced shear stress resistance for concentrated loads applied close to a support [$MPa$].

    FprEN 1992-1-1:2023 (E) art 8.2.3 (12) - Formula (8.55)

    The standard offers this enhancement in cases where concentrated loads are applied at a distance
    [$a_v = z \cdot \cot\beta_{incl}$] less than [$z \cdot \cot\theta$] from a support. That is a condition
    of application, not a bound on the result, so it is not enforced here.

    Parameters
    ----------
    nu : DIMENSIONLESS
        [$\nu$] Strength reduction factor for concrete cracked in shear. A value of 0,5 may be adopted when
        using the angles of the compression field given in 8.2.3(4), see 8.2.3(6). A higher value may be
        adopted under the conditions of 8.2.3(7), calculated according to Formula (8.45) [$-$].
    f_cd : MPA
        [$f_{cd}$] Design value of the compressive strength of concrete [$MPa$].
    theta : DEG
        [$\theta$] Inclination of the compression field in the web, selected within the range of Formula
        (8.41). A flatter field, [$\cot\theta < 1$], is allowed under 8.2.3(11), but only if [$f_{ywd}$] is
        replaced by [$\sigma_{swd}$] of Formula (8.56); making that substitution is left to the caller
        [$degrees$].
    beta_incl : DEG
        [$\beta_{incl}$] Angle that follows from the distance of the concentrated load to the support
        through [$a_v = z \cdot \cot\beta_{incl}$], so [$\beta_{incl} = \arctan(z/a_v)$] [$degrees$].
    rho_w : DIMENSIONLESS
        [$\rho_w$] Shear reinforcement ratio according to Formula (8.43) [$-$].
    f_ywd : MPA
        [$f_{ywd}$] Design value of the yield strength of the shear reinforcement [$MPa$].
    """
    super().__init__()
    self.nu = nu
    self.f_cd = f_cd
    self.theta = theta
    self.beta_incl = beta_incl
    self.rho_w = rho_w
    self.f_ywd = f_ywd

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_55.Form8Dot55EnhancedShearStressResistance.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 8.55.

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_55.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 8.55."""
    _equation: str = (
        r"\min\left(\nu \cdot f_{cd} \cdot \frac{\cot(\theta) - \cot(\beta_{incl})}"
        r"{1 + \left(\cot(\theta)\right)^2} + \rho_w \cdot f_{ywd} \cdot \cot(\beta_{incl}), "
        r"\nu \cdot f_{cd} \cdot \frac{\cot(\theta)}{1 + \left(\cot(\theta)\right)^2}\right)"
    )
    _numeric_equation: str = latex_replace_symbols(
        template=_equation,
        replacements={
            r"\nu": f"{self.nu:.{n}f}",
            r"f_{cd}": f"{self.f_cd:.{n}f}",
            r"\beta_{incl}": f"{self.beta_incl:.{n}f}",
            r"\theta": f"{self.theta:.{n}f}",
            r"\rho_w": f"{self.rho_w:.{n}f}",
            r"f_{ywd}": f"{self.f_ywd:.{n}f}",
        },
        unique_symbol_check=False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        template=_equation,
        replacements={
            r"\nu": f"{self.nu:.{n}f}",
            r"f_{cd}": rf"{self.f_cd:.{n}f} \ MPa",
            r"\beta_{incl}": rf"{self.beta_incl:.{n}f} ^\circ",
            r"\theta": rf"{self.theta:.{n}f} ^\circ",
            r"\rho_w": f"{self.rho_w:.{n}f}",
            r"f_{ywd}": rf"{self.f_ywd:.{n}f} \ MPa",
        },
        unique_symbol_check=False,
    )
    return LatexFormula(
        return_symbol=r"\tau_{Rd}",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="MPa",
    )