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formula_8_4

codes.eurocode.en_1992_1_1_2004.chapter_8_detailing_of_reinforcement_and_prestressing_tendons.formula_8_4

Formula 8.4 from EN 1992-1-1:2004: Chapter 8: Detailing of reinforcement and prestressing tendons.

Classes:

codes.eurocode.en_1992_1_1_2004.chapter_8_detailing_of_reinforcement_and_prestressing_tendons.formula_8_4.Form8Dot4DesignAnchorageLength 

Form8Dot4DesignAnchorageLength(
    alpha_1: RATIO,
    alpha_2: RATIO,
    alpha_3: RATIO,
    alpha_4: RATIO,
    alpha_5: RATIO,
    l_b_rqd: MM,
    l_b_min: MM,
    min_product_alpha_2_3_5: RATIO | None = None,
)

Bases: Formula

Class representing formula 8.4 for the calculation of the design anchorage length [\(l_{bd}\)] [\(mm\)].

[\(l_{bd}\)] Design anchorage length [\(mm\)].

EN 1992-1-1:2004 art.8.4.4(1) - Formula (8.4)

Parameters:

  • alpha_1 (RATIO) –

    [\(α_{1}\)] Coefficient for the effect of the form of the bars assuming adequate cover (see figure 8.1) [\(-\)]. [\(= 1.0\)] for bars in compression. [\(= 1.0\)] for straight bars in tension. [\(= 1.0 \text{ if } c_{d} \leq 3 ⋅ Ø\)] for bars other than straight in tension (see figure 8.1 (b), (c) and (d)). [\(= 0.7 \text{ if } c_{d} > 3 ⋅ Ø\)] for bars other than straight in tension (see figure 8.1 (b), (c) and (d)). Note: see figure 8.3 for values of [\(c_{d}\)].

  • alpha_2 (RATIO) –

    [\(α_{2}\)] Coefficient for the effect of minimum concrete cover (see figure 8.3) [\(-\)]. [\(= 1.0\)] for bars in compression. [\(= 1 - 0.15 ⋅ (c_{d} - Ø) / Ø \leq 1\)] with a minimum of 0.7 for straight bars in tension. [\(= 1 - 0.15 ⋅ (c_{d} - 3 ⋅ Ø) / Ø \leq 1\)] with a minimum of 0.7 for bars other than straight in tension (see figure 8.1 (b), (c) and (d)). Note: see figure 8.3 for values of [\(c_{d}\)].

  • alpha_3 (RATIO) –

    [\(α_{3}\)] Coefficient for the effect of confinement by transverse reinforcement [\(-\)]. [\(= 1.0\)] for bars in compression. [\(= 1 - K ⋅ λ \leq 1\)] with a minimum of 0.7 for bars in tension. Where: [\(λ = (\Sigma A_{st} - \Sigma A_{st,min}) / A_{s}\)]. Where: [\(\Sigma A_{st,min}\)] = cross-sectional area of the minimum transverse reinforcement [\(= 0,25 ⋅ A_{s}\)] for beams and 0 for slabs. Note: see figure 8.4 for values of [\(K, A_{s}\)] and [\(A_{st}\)].

  • alpha_4 (RATIO) –

    [\(α_{4}\)] Coefficient for the influence of one or more welded transverse bars [\(Ø_{t} > 0,6 Ø\)] along the design anchorage length [\(l_{bd}\)] (see 8.6) [\(-\)]. [\(= 0.7\)] for all types, position and size as specified in figure 8.6 (e) in both tension and compression.

  • alpha_5 (RATIO) –

    [\(α_{5}\)] Coefficient for the effect of the pressure transverse to the plane of splitting along the design anchorage length [\(l_{bd}\)] (see 8.6) [\(-\)]. [\(= 1 - 0.04 ⋅ p \leq 1\)] with a minimum of 0.7 for all types of anchorage in compression. Where: p = transverse pressure at ultimate limit state along [\(l_{bd}\)] [\(MPa\)].

  • l_b_rqd (MM) –

    [\(l_{b,rqd}\)] Basic required anchorage length, for anchoring the force [\(A_{s}⋅σ_{sd}\)] in a straight bar assuming constant bond stress (formula 8.3) [\(mm\)]. Use your own implementation for this value or use the :class:Form8Dot3RequiredAnchorageLength class.

  • l_b_min (MM) –

    [\(l_{b,min}\)] Minimum anchorage length if no other limitation is applied [\(mm\)]. [\(= \max(0.3 ⋅ l_{b,rqd}, 10 ⋅ Ø, 100)\)] for tension anchorage (formula 8.6). [\(= \max(0.6 ⋅ l_{b,rqd}, 10 ⋅ Ø, 100)\)] for compression anchorage (formula 8.7). Use your own implementation of this formula or use the :class:Form8Dot6MinimumTensionAnchorage class for tension or :class:Form8Dot7MinimumCompressionAnchorage for compression.

  • min_product_alpha_2_3_5 (RATIO | None, default: None ) –

    Minimum value of the product of factors [\(α_{2}\)], [\(α_{3}\)] and [\(α_{5}\)]. When this argument is None, :class: Form8Dot5ProductAlphas235 is used for this condition. When this argument is given, the condition [\(\max(α_{2}⋅α_{3}⋅α_{5}) \geq\)] min_product_alpha_2_3_5 is used.

Notes

EN 1992-1-1:2004 art.8.4.4(1) - Formula (8.5) prescribes that [\(α_{2} ⋅ α_{3} ⋅ α_{5} \geq 0.7\)].

Source code in blueprints/codes/eurocode/en_1992_1_1_2004/chapter_8_detailing_of_reinforcement_and_prestressing_tendons/formula_8_4.py 
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def __init__(
    self,
    alpha_1: RATIO,
    alpha_2: RATIO,
    alpha_3: RATIO,
    alpha_4: RATIO,
    alpha_5: RATIO,
    l_b_rqd: MM,
    l_b_min: MM,
    min_product_alpha_2_3_5: RATIO | None = None,
) -> None:
    r"""[$l_{bd}$] Design anchorage length [$mm$].

    EN 1992-1-1:2004 art.8.4.4(1) - Formula (8.4)

    Parameters
    ----------
    alpha_1 : RATIO
        [$α_{1}$] Coefficient for the effect of the form of the bars assuming adequate cover (see figure 8.1) [$-$].
        [$= 1.0$] for bars in compression.
        [$= 1.0$] for straight bars in tension.
        [$= 1.0 \text{ if } c_{d} \leq 3 ⋅ Ø$] for bars other than straight in tension (see figure 8.1 (b), (c) and (d)).
        [$= 0.7 \text{ if } c_{d} > 3 ⋅ Ø$] for bars other than straight in tension (see figure 8.1 (b), (c) and (d)).
        Note: see figure 8.3 for values of [$c_{d}$].
    alpha_2 : RATIO
        [$α_{2}$] Coefficient for the effect of minimum concrete cover (see figure 8.3) [$-$].
        [$= 1.0$] for bars in compression.
        [$= 1 - 0.15 ⋅ (c_{d} - Ø) / Ø \leq 1$] with a minimum of 0.7 for straight bars in tension.
        [$= 1 - 0.15 ⋅ (c_{d} - 3 ⋅ Ø) / Ø \leq 1$] with a minimum of 0.7 for bars other than
        straight in tension (see figure 8.1 (b), (c) and (d)).
        Note: see figure 8.3 for values of [$c_{d}$].
    alpha_3 : RATIO
        [$α_{3}$] Coefficient for the effect of confinement by transverse reinforcement [$-$].
        [$= 1.0$] for bars in compression.
        [$= 1 - K ⋅ λ \leq 1$] with a minimum of 0.7 for bars in tension.
        Where: [$λ = (\Sigma A_{st} - \Sigma A_{st,min}) / A_{s}$].
        Where: [$\Sigma A_{st,min}$] = cross-sectional area of the minimum transverse
        reinforcement [$= 0,25 ⋅ A_{s}$] for beams and 0 for slabs.
        Note: see figure 8.4 for values of [$K, A_{s}$] and [$A_{st}$].
    alpha_4 : RATIO
        [$α_{4}$] Coefficient for the influence of one or more welded transverse bars [$Ø_{t} > 0,6 Ø$] along the design anchorage
        length [$l_{bd}$] (see 8.6) [$-$].
        [$= 0.7$] for all types, position and size as specified in figure 8.6 (e) in both tension and compression.
    alpha_5 : RATIO
        [$α_{5}$] Coefficient for the effect of the pressure transverse to the plane of splitting
        along the design anchorage length [$l_{bd}$] (see 8.6) [$-$].
        [$= 1 - 0.04 ⋅ p \leq 1$] with a minimum of 0.7 for all types of anchorage in compression.
        Where: p = transverse pressure at ultimate limit state along [$l_{bd}$] [$MPa$].
    l_b_rqd: MM
        [$l_{b,rqd}$] Basic required anchorage length, for anchoring the force [$A_{s}⋅σ_{sd}$] in a straight bar assuming constant
        bond stress (formula 8.3) [$mm$].
        Use your own implementation for this value or use the :class:`Form8Dot3RequiredAnchorageLength` class.
    l_b_min : MM
        [$l_{b,min}$] Minimum anchorage length if no other limitation is applied [$mm$].
        [$= \max(0.3 ⋅ l_{b,rqd}, 10 ⋅ Ø, 100)$] for tension anchorage (formula 8.6).
        [$= \max(0.6 ⋅ l_{b,rqd}, 10 ⋅ Ø, 100)$] for compression anchorage (formula 8.7).
        Use your own implementation of this formula or use the :class:`Form8Dot6MinimumTensionAnchorage` class for tension or
        :class:`Form8Dot7MinimumCompressionAnchorage` for compression.
    min_product_alpha_2_3_5: RATIO | None
        Minimum value of the product of factors [$α_{2}$], [$α_{3}$] and [$α_{5}$].
        When this argument is None, :class: `Form8Dot5ProductAlphas235` is used for this condition.
        When this argument is given, the condition [$\max(α_{2}⋅α_{3}⋅α_{5}) \geq$] min_product_alpha_2_3_5 is used.

    Notes
    -----
    EN 1992-1-1:2004 art.8.4.4(1) - Formula (8.5) prescribes that [$α_{2} ⋅ α_{3} ⋅ α_{5} \geq 0.7$].
    """
    super().__init__()
    self.alpha_1 = alpha_1
    self.alpha_2 = alpha_2
    self.alpha_3 = alpha_3
    self.alpha_4 = alpha_4
    self.alpha_5 = alpha_5
    self.l_b_rqd = l_b_rqd
    self.l_b_min = l_b_min
    self.min_product_alpha_2_3_5: RATIO | None = min_product_alpha_2_3_5

codes.eurocode.en_1992_1_1_2004.chapter_8_detailing_of_reinforcement_and_prestressing_tendons.formula_8_4.Form8Dot4DesignAnchorageLength.latex 

latex(n: int = 2) -> LatexFormula

Returns a LatexFormula representation of the formula.

Source code in blueprints/codes/eurocode/en_1992_1_1_2004/chapter_8_detailing_of_reinforcement_and_prestressing_tendons/formula_8_4.py 
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def latex(self, n: int = 2) -> LatexFormula:
    """Returns a LatexFormula representation of the formula."""
    _equation: str = latex_max_curly_brackets(
        r"\alpha_1 \cdot \alpha_2 \cdot \alpha_3 \cdot \alpha_4 \cdot \alpha_5 \cdot l_{b,rqd}",
        r"l_{b,min}",
    )
    _numeric_equation: str = latex_replace_symbols(
        _equation,
        {
            r"\alpha_1": f"{self.alpha_1:.{n}f}",
            r"\alpha_2": f"{self.alpha_2:.{n}f}",
            r"\alpha_3": f"{self.alpha_3:.{n}f}",
            r"\alpha_4": f"{self.alpha_4:.{n}f}",
            r"\alpha_5": f"{self.alpha_5:.{n}f}",
            r"l_{b,rqd}": f"{self.l_b_rqd:.{n}f}",
            r"l_{b,min}": f"{self.l_b_min:.{n}f}",
        },
        False,
    )
    _numeric_equation_with_units: str = latex_replace_symbols(
        _equation,
        {
            r"\alpha_1": f"{self.alpha_1:.{n}f}",
            r"\alpha_2": f"{self.alpha_2:.{n}f}",
            r"\alpha_3": f"{self.alpha_3:.{n}f}",
            r"\alpha_4": f"{self.alpha_4:.{n}f}",
            r"\alpha_5": f"{self.alpha_5:.{n}f}",
            r"l_{b,rqd}": rf"{self.l_b_rqd:.{n}f} \ mm",
            r"l_{b,min}": rf"{self.l_b_min:.{n}f} \ mm",
        },
        False,
    )
    return LatexFormula(
        return_symbol=r"l_{bd}",
        result=f"{self:.{n}f}",
        equation=_equation,
        numeric_equation=_numeric_equation,
        numeric_equation_with_units=_numeric_equation_with_units,
        comparison_operator_label="=",
        unit="mm",
    )