Skip to content

formula_8_33

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_33

Formula 8.33 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_33.Form8Dot33ShearStressResistanceWithoutAxialForce 

Form8Dot33ShearStressResistanceWithoutAxialForce(
    gamma_v: DIMENSIONLESS, rho_l: DIMENSIONLESS, f_ck: MPA, d_dg: MM, d: MM
)

Bases: Formula

Class representing formula 8.33 for the calculation of the design value of the shear stress resistance without the effect of compressive normal forces.

[\(\tau_{Rdc,0}\)] Design value of the shear stress resistance without the effect of compressive normal forces [\(MPa\)].

FprEN 1992-1-1:2023 (E) art 8.2.2 (5) - Formula (8.33)

This is the expression of Formula (8.27) without its lower bound. It feeds Formula (8.32) and Formula (8.35).

For the factor [\(k_1\)] according to Formula (8.34), the effective depth [\(d\)] may be replaced by [\(a_{v,0}\)], determined according to Formulas (8.29) and (8.30) without considering in [\(M_{Ed}\)] and [\(V_{Ed}\)] the effect of prestressing or external load that produces the compressive axial force. That substitution is left to the caller.

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 [\(mm\)].

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_33.py 
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
def __init__(
    self,
    gamma_v: DIMENSIONLESS,
    rho_l: DIMENSIONLESS,
    f_ck: MPA,
    d_dg: MM,
    d: MM,
) -> None:
    r"""[$\tau_{Rdc,0}$] Design value of the shear stress resistance without the effect of compressive
    normal forces [$MPa$].

    FprEN 1992-1-1:2023 (E) art 8.2.2 (5) - Formula (8.33)

    This is the expression of Formula (8.27) without its lower bound. It feeds Formula (8.32) and
    Formula (8.35).

    For the factor [$k_1$] according to Formula (8.34), the effective depth [$d$] may be replaced by
    [$a_{v,0}$], determined according to Formulas (8.29) and (8.30) without considering in [$M_{Ed}$] and
    [$V_{Ed}$] the effect of prestressing or external load that produces the compressive axial force. That
    substitution is left to the caller.

    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 [$mm$].
    """
    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

codes.eurocode.fpr_en_1992_1_1_2023.chapter_8_ultimate_limit_states.formula_8_33.Form8Dot33ShearStressResistanceWithoutAxialForce.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 8.33.

Source code in blueprints/codes/eurocode/fpr_en_1992_1_1_2023/chapter_8_ultimate_limit_states/formula_8_33.py 
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 8.33."""
    _equation: str = r"\frac{0.66}{\gamma_V} \cdot \left(100 \cdot \rho_l \cdot f_{ck} \cdot \frac{d_{dg}}{d}\right)^{\frac{1}{3}}"
    _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}" + "}",
        },
        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" + "}",
        },
        unique_symbol_check=False,
    )
    return LatexFormula(
        return_symbol=r"\tau_{Rdc,0}",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="MPa",
    )