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formula_8_27

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_27

Formula 8.27 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_27.Form8Dot27DesignShearStressResistance 

Form8Dot27DesignShearStressResistance(
    gamma_v: DIMENSIONLESS,
    rho_l: DIMENSIONLESS,
    f_ck: MPA,
    d_dg: MM,
    d: MM,
    tau_rdc_min: MPA,
)

Bases: Formula

Class representing formula 8.27 for the calculation of the design value of the shear stress resistance of members without shear reinforcement.

[\(\tau_{Rd,c}\)] Design value of the shear stress resistance of members without shear reinforcement [\(MPa\)].

FprEN 1992-1-1:2023 (E) art 8.2.2 (2) - Formula (8.27)

The standard writes this as an expression bounded from below by [\(\tau_{Rdc,min}\)], which is implemented as the maximum of the two.

Parameters:

  • gamma_v (DIMENSIONLESS) –

    [\(\gamma_V\)] Partial factor for shear design according to Table 4.3 (NDP) or Tables A.1 (NDP) and A.2 (NDP) [\(-\)].

  • rho_l (DIMENSIONLESS) –

    [\(\rho_l\)] Longitudinal tensile reinforcement ratio according to Formula (8.28), see Form8Dot28LongitudinalReinforcementRatio [\(-\)].

  • f_ck (MPA) –

    [\(f_{ck}\)] Characteristic compressive strength of concrete [\(MPa\)].

  • d_dg (MM) –

    [\(d_{dg}\)] Size parameter describing the failure zone roughness, which depends on the concrete type and its aggregate properties. The standard gives it as [\(16 + D_{lower} \leq 40\)] for concrete with [\(f_{ck} \leq 60\)] MPa, and as [\(16 + D_{lower} \cdot \left(60/f_{ck}\right)^2 \leq 40\)] for concrete with [\(f_{ck} > 60\)] MPa, both in millimetres. [\(D_{lower}\)] is the smallest value of the upper sieve size [\(D\)] in an aggregate for the coarsest fraction of aggregates in the concrete permitted by the specification of concrete according to EN 206; where [\(D_{max}\)] is known it may replace [\(D_{lower}\)], see the NOTE 2 to 8.2.1(4) [\(mm\)].

  • d (MM) –

    [\(d\)] Effective depth [\(d_{nom}\)]. The value [\(d\)] may be refined according to 8.2.2(3) and 8.2.2(4) for non-slender members and members with axial force, by replacing it with the mechanical shear span [\(a_v\)] or by multiplying it with [\(k_{vp}\)]. That refinement is the responsibility of the caller, which passes the already adjusted value [\(mm\)].

  • tau_rdc_min (MPA) –

    [\(\tau_{Rdc,min}\)] Minimum shear stress resistance according to Formula (8.20), see Form8Dot20MinimumShearStressResistance [\(MPa\)].

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_27.py 
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def __init__(
    self,
    gamma_v: DIMENSIONLESS,
    rho_l: DIMENSIONLESS,
    f_ck: MPA,
    d_dg: MM,
    d: MM,
    tau_rdc_min: MPA,
) -> None:
    r"""[$\tau_{Rd,c}$] Design value of the shear stress resistance of members without shear reinforcement [$MPa$].

    FprEN 1992-1-1:2023 (E) art 8.2.2 (2) - Formula (8.27)

    The standard writes this as an expression bounded from below by [$\tau_{Rdc,min}$], which is implemented
    as the maximum of the two.

    Parameters
    ----------
    gamma_v : DIMENSIONLESS
        [$\gamma_V$] Partial factor for shear design according to Table 4.3 (NDP) or Tables A.1 (NDP)
        and A.2 (NDP) [$-$].
    rho_l : DIMENSIONLESS
        [$\rho_l$] Longitudinal tensile reinforcement ratio according to Formula (8.28), see
        Form8Dot28LongitudinalReinforcementRatio [$-$].
    f_ck : MPA
        [$f_{ck}$] Characteristic compressive strength of concrete [$MPa$].
    d_dg : MM
        [$d_{dg}$] Size parameter describing the failure zone roughness, which depends on the concrete type
        and its aggregate properties. The standard gives it as [$16 + D_{lower} \leq 40$] for concrete with
        [$f_{ck} \leq 60$] MPa, and as [$16 + D_{lower} \cdot \left(60/f_{ck}\right)^2 \leq 40$] for concrete
        with [$f_{ck} > 60$] MPa, both in millimetres. [$D_{lower}$] is the smallest value of the upper sieve
        size [$D$] in an aggregate for the coarsest fraction of aggregates in the concrete permitted by the
        specification of concrete according to EN 206; where [$D_{max}$] is known it may replace
        [$D_{lower}$], see the NOTE 2 to 8.2.1(4) [$mm$].
    d : MM
        [$d$] Effective depth [$d_{nom}$]. The value [$d$] may be refined according to 8.2.2(3) and 8.2.2(4) for
        non-slender members and members with axial force, by replacing it with the mechanical shear span
        [$a_v$] or by multiplying it with [$k_{vp}$]. That refinement is the responsibility of the caller,
        which passes the already adjusted value [$mm$].
    tau_rdc_min : MPA
        [$\tau_{Rdc,min}$] Minimum shear stress resistance according to Formula (8.20), see
        Form8Dot20MinimumShearStressResistance [$MPa$].
    """
    super().__init__()
    self.gamma_v = gamma_v
    self.rho_l = rho_l
    self.f_ck = f_ck
    self.d_dg = d_dg
    self.d = d
    self.tau_rdc_min = tau_rdc_min

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_27.Form8Dot27DesignShearStressResistance.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 8.27.

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_27.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 8.27."""
    _equation: str = (
        r"\max\left(\frac{0.66}{\gamma_V} \cdot \left(100 \cdot \rho_l \cdot f_{ck} \cdot "
        r"\frac{d_{dg}}{d}\right)^{\frac{1}{3}}, \tau_{Rdc,min}\right)"
    )
    _numeric_equation: str = latex_replace_symbols(
        template=_equation,
        replacements={
            r"\gamma_V": f"{self.gamma_v:.{n}f}",
            r"\rho_l": f"{self.rho_l:.{n}f}",
            r"f_{ck}": f"{self.f_ck:.{n}f}",
            r"d_{dg}": f"{self.d_dg:.{n}f}",
            r"{d}": "{" + f"{self.d:.{n}f}" + "}",
            r"\tau_{Rdc,min}": f"{self.tau_rdc_min:.{n}f}",
        },
        unique_symbol_check=False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        template=_equation,
        replacements={
            r"\gamma_V": f"{self.gamma_v:.{n}f}",
            r"\rho_l": f"{self.rho_l:.{n}f}",
            r"f_{ck}": rf"{self.f_ck:.{n}f} \ MPa",
            r"d_{dg}": rf"{self.d_dg:.{n}f} \ mm",
            r"{d}": "{" + rf"{self.d:.{n}f} \ mm" + "}",
            r"\tau_{Rdc,min}": rf"{self.tau_rdc_min:.{n}f} \ MPa",
        },
        unique_symbol_check=False,
    )
    return LatexFormula(
        return_symbol=r"\tau_{Rd,c}",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="MPa",
    )