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formula_5_46

codes.eurocode.en_1992_1_1_2004.chapter_5_structural_analysis.formula_5_46

Formula 5.46 from EN 1992-1-1:2004: Chapter 5 - Structural Analysis.

Classes:

codes.eurocode.en_1992_1_1_2004.chapter_5_structural_analysis.formula_5_46.Form5Dot46Part1TimeDependentForceLosses 

Form5Dot46Part1TimeDependentForceLosses(
    a_p: MM2,
    epsilon_cs: DIMENSIONLESS,
    e_p: MPA,
    e_cm: MPA,
    delta_sigma_pr: MPA,
    phi_t_t0: DIMENSIONLESS,
    sigma_c_qp: MPA,
    a_c: MM2,
    i_c: MM4,
    z_cp: MM,
)

Bases: Formula

Class representing formula 5.46 for the calculation of the time dependent pre- and post-tensioning losses at location x under the permanent loads, [\(\Delta P_{c+s+r}\)].

[\(\Delta P_{c+s+r}\)] Time dependent pre- and post-tensioning losses at location x under the permanent loads [\(N\)].

EN 1992-1-1:2004 art.5.10.6(2) - Formula (5.46)

Parameters:

  • a_p (MM2) –

    [\(A_p\)] Area of all the prestressing tendons at the location x [\(mm^2\)].

  • epsilon_cs (DIMENSIONLESS) –

    [\(\epsilon_{cs}\)] The estimated shrinkage strain according to 3.1.4(6) in absolute value [\(-\)].

  • e_p (MPA) –

    [\(E_p\)] Modulus of elasticity for the prestressing steel, see 3.3.3 (9) [\(MPa\)].

  • e_cm (MPA) –

    [\(E_{cm}\)] Modulus of elasticity for the concrete (Table 3.1) [\(MPa\)].

  • delta_sigma_pr (MPA) –

    [\(\Delta \sigma_{pr}\)] is the absolute value of the variation of stress in the tendons at location x, at time t, due to the relaxation of the prestressing steel. It is determined for a stress of [\(\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)\)]where [\(\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)\)] is the initial stress in the tendons due to initial prestress and quasi-permanent actions. [\(MPa\)].

  • phi_t_t0 (DIMENSIONLESS) –

    [\(\phi(t, t_0)\)] Creep coefficient at a time t and load application at time t0 [\(-\)].

  • sigma_c_qp (MPA) –

    [\(\sigma_{c,QP}\)] stress in the concrete adjacent to the tendons, due to self-weight and initial prestress and other quasi-permanent actions where relevant. [\(MPa\)].

  • a_c (MM2) –

    [\(A_c\)] Area of concrete section [\(mm^2\)].

  • i_c (MM4) –

    [\(I_c\)] Second moment of area of concrete section [\(mm^4\)].

  • z_cp (MM) –

    [\(z_{cp}\)] Distance between the centre of gravity of the concrete section and the tendons [\(mm\)].

Source code in blueprints/codes/eurocode/en_1992_1_1_2004/chapter_5_structural_analysis/formula_5_46.py 
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def __init__(
    self,
    a_p: MM2,
    epsilon_cs: DIMENSIONLESS,
    e_p: MPA,
    e_cm: MPA,
    delta_sigma_pr: MPA,
    phi_t_t0: DIMENSIONLESS,
    sigma_c_qp: MPA,
    a_c: MM2,
    i_c: MM4,
    z_cp: MM,
) -> None:
    r"""[$\Delta P_{c+s+r}$] Time dependent pre- and post-tensioning losses at location x under the permanent loads [$N$].

    EN 1992-1-1:2004 art.5.10.6(2) - Formula (5.46)

    Parameters
    ----------
    a_p : MM2
        [$A_p$] Area of all the prestressing tendons at the location x [$mm^2$].
    epsilon_cs : DIMENSIONLESS
        [$\epsilon_{cs}$] The estimated shrinkage strain according to 3.1.4(6) in absolute value [$-$].
    e_p : MPA
        [$E_p$] Modulus of elasticity for the prestressing steel, see 3.3.3 (9) [$MPa$].
    e_cm : MPA
        [$E_{cm}$] Modulus of elasticity for the concrete (Table 3.1) [$MPa$].
    delta_sigma_pr : MPA
        [$\Delta \sigma_{pr}$] is the absolute value of the variation of stress in the tendons at location x, at
        time t, due to the relaxation of the prestressing steel. It is determined for a stress of
        [$\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)$]where [$\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)$] is the initial
        stress in the tendons due to initial prestress and quasi-permanent actions. [$MPa$].
    phi_t_t0 : DIMENSIONLESS
        [$\phi(t, t_0)$] Creep coefficient at a time t and load application at time t0 [$-$].
    sigma_c_qp : MPA
        [$\sigma_{c,QP}$] stress in the concrete adjacent to the tendons, due to self-weight and
        initial prestress and other quasi-permanent actions where relevant. [$MPa$].
    a_c : MM2
        [$A_c$] Area of concrete section [$mm^2$].
    i_c : MM4
        [$I_c$] Second moment of area of concrete section [$mm^4$].
    z_cp : MM
        [$z_{cp}$] Distance between the centre of gravity of the concrete section and the tendons [$mm$].
    """
    super().__init__()
    self.a_p = a_p
    self.epsilon_cs = epsilon_cs
    self.e_p = e_p
    self.e_cm = e_cm
    self.delta_sigma_pr = delta_sigma_pr
    self.phi_t_t0 = phi_t_t0
    self.sigma_c_qp = sigma_c_qp
    self.a_c = a_c
    self.i_c = i_c
    self.z_cp = z_cp

codes.eurocode.en_1992_1_1_2004.chapter_5_structural_analysis.formula_5_46.Form5Dot46Part1TimeDependentForceLosses.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 5.46.

Source code in blueprints/codes/eurocode/en_1992_1_1_2004/chapter_5_structural_analysis/formula_5_46.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 5.46."""
    return LatexFormula(
        return_symbol=r"\Delta P_{c+s+r}",
        result=f"{self:.{n}f}",
        equation=r"A_p \cdot \frac{\epsilon_{cs} \cdot E_p + 0.8 \cdot \Delta \sigma_{pr} + \frac{E_p}{E_{cm}} \cdot "
        r"\phi(t, t_0) \cdot \sigma_{c,QP}}{1 + \frac{E_p}{E_{cm}} \cdot \frac{A_p}{A_c} \cdot \left(1 + \frac{A_c}{I_c} "
        r"\cdot z_{cp}^2\right) \cdot \left(1 + 0.8 \cdot \phi(t, t_0)\right)}",
        numeric_equation=rf"{self.a_p:.{n}f} \cdot \frac{{{self.epsilon_cs:.6f} \cdot {self.e_p:.{n}f} + 0.800"
        rf" \cdot {self.delta_sigma_pr:.{n}f} + \frac{{{self.e_p:.{n}f}}}{{{self.e_cm:.{n}f}}} \cdot {self.phi_t_t0:.{n}f} "
        rf"\cdot {self.sigma_c_qp:.{n}f}}}{{1 + \frac{{{self.e_p:.{n}f}}}{{{self.e_cm:.{n}f}}} "
        rf"\cdot \frac{{{self.a_p:.{n}f}}}{{{self.a_c:.{n}f}}} "
        rf"\cdot \left(1 + \frac{{{self.a_c:.{n}f}}}{{{self.i_c:.{n}f}}} \cdot {self.z_cp:.{n}f}^2\right) \cdot \left(1 + 0.800 "
        rf"\cdot {self.phi_t_t0:.{n}f}\right)}}",
        comparison_operator_label="=",
        unit="N",
    )

codes.eurocode.en_1992_1_1_2004.chapter_5_structural_analysis.formula_5_46.Form5Dot46Part2TimeDependentStressLosses 

Form5Dot46Part2TimeDependentStressLosses(
    a_p: MM2,
    epsilon_cs: DIMENSIONLESS,
    e_p: MPA,
    e_cm: MPA,
    delta_sigma_pr: MPA,
    phi_t_t0: DIMENSIONLESS,
    sigma_c_qp: MPA,
    a_c: MM2,
    i_c: MM4,
    z_cp: MM,
)

Bases: Formula

Class representing formula 5.46 for the calculation of the time dependent pre- and post-tensioning losses at location x under the permanent loads, [\(\Delta \sigma_{p,c+s+r}\)].

[\(\Delta \sigma_{p,c+s+r}\)] Time dependent pre- and post-tensioning stress losses at location x under the permanent loads [\(MPa\)].

EN 1992-1-1:2004 art.5.10.6(2) - Formula (5.46)

Parameters:

  • a_p (MM2) –

    [\(A_p\)] Area of all the prestressing tendons at the location x [\(mm^2\)].

  • epsilon_cs (DIMENSIONLESS) –

    [\(\epsilon_{cs}\)] The estimated shrinkage strain according to 3.1.4(6) in absolute value [\(-\)].

  • e_p (MPA) –

    [\(E_p\)] Modulus of elasticity for the prestressing steel, see 3.3.3 (9) [\(MPa\)].

  • e_cm (MPA) –

    [\(E_{cm}\)] Modulus of elasticity for the concrete (Table 3.1) [\(MPa\)].

  • delta_sigma_pr (MPA) –

    [\(\Delta \sigma_{pr}\)] is the absolute value of the variation of stress in the tendons at location x, at time t, due to the relaxation of the prestressing steel. It is determined for a stress of [\(\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)\)]where [\(\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)\)] is the initial stress in the tendons due to initial prestress and quasi-permanent actions. [\(MPa\)].

  • phi_t_t0 (DIMENSIONLESS) –

    [\(\phi(t, t_0)\)] Creep coefficient at a time t and load application at time t0 [\(-\)].

  • sigma_c_qp (MPA) –

    [\(\sigma_{c,QP}\)] stress in the concrete adjacent to the tendons, due to self-weight and initial prestress and other quasi-permanent actions where relevant. [\(MPa\)].

  • a_c (MM2) –

    [\(A_c\)] Area of concrete section [\(mm^2\)].

  • i_c (MM4) –

    [\(I_c\)] Second moment of area of concrete section [\(mm^4\)].

  • z_cp (MM) –

    [\(z_{cp}\)] Distance between the centre of gravity of the concrete section and the tendons [\(mm\)].

Source code in blueprints/codes/eurocode/en_1992_1_1_2004/chapter_5_structural_analysis/formula_5_46.py 
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def __init__(
    self,
    a_p: MM2,
    epsilon_cs: DIMENSIONLESS,
    e_p: MPA,
    e_cm: MPA,
    delta_sigma_pr: MPA,
    phi_t_t0: DIMENSIONLESS,
    sigma_c_qp: MPA,
    a_c: MM2,
    i_c: MM4,
    z_cp: MM,
) -> None:
    r"""[$\Delta \sigma_{p,c+s+r}$] Time dependent pre- and post-tensioning stress losses at
    location x under the permanent loads [$MPa$].

    EN 1992-1-1:2004 art.5.10.6(2) - Formula (5.46)

    Parameters
    ----------
    a_p : MM2
        [$A_p$] Area of all the prestressing tendons at the location x [$mm^2$].
    epsilon_cs : DIMENSIONLESS
        [$\epsilon_{cs}$] The estimated shrinkage strain according to 3.1.4(6) in absolute value [$-$].
    e_p : MPA
        [$E_p$] Modulus of elasticity for the prestressing steel, see 3.3.3 (9) [$MPa$].
    e_cm : MPA
        [$E_{cm}$] Modulus of elasticity for the concrete (Table 3.1) [$MPa$].
    delta_sigma_pr : MPA
        [$\Delta \sigma_{pr}$] is the absolute value of the variation of stress in the tendons at location x, at
        time t, due to the relaxation of the prestressing steel. It is determined for a stress of
        [$\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)$]where [$\sigma_p = \sigma_p(G+P_{m0}+\Psi_2 Q)$] is the initial
        stress in the tendons due to initial prestress and quasi-permanent actions. [$MPa$].
    phi_t_t0 : DIMENSIONLESS
        [$\phi(t, t_0)$] Creep coefficient at a time t and load application at time t0 [$-$].
    sigma_c_qp : MPA
        [$\sigma_{c,QP}$] stress in the concrete adjacent to the tendons, due to self-weight and
        initial prestress and other quasi-permanent actions where relevant. [$MPa$].
    a_c : MM2
        [$A_c$] Area of concrete section [$mm^2$].
    i_c : MM4
        [$I_c$] Second moment of area of concrete section [$mm^4$].
    z_cp : MM
        [$z_{cp}$] Distance between the centre of gravity of the concrete section and the tendons [$mm$].
    """
    super().__init__()
    self.a_p = a_p
    self.epsilon_cs = epsilon_cs
    self.e_p = e_p
    self.e_cm = e_cm
    self.delta_sigma_pr = delta_sigma_pr
    self.phi_t_t0 = phi_t_t0
    self.sigma_c_qp = sigma_c_qp
    self.a_c = a_c
    self.i_c = i_c
    self.z_cp = z_cp

codes.eurocode.en_1992_1_1_2004.chapter_5_structural_analysis.formula_5_46.Form5Dot46Part2TimeDependentStressLosses.latex 

latex(n: int = 3) -> LatexFormula

Returns LatexFormula object for formula 5.46 for stress losses.

Source code in blueprints/codes/eurocode/en_1992_1_1_2004/chapter_5_structural_analysis/formula_5_46.py 
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def latex(self, n: int = 3) -> LatexFormula:
    """Returns LatexFormula object for formula 5.46 for stress losses."""
    return LatexFormula(
        return_symbol=r"\Delta \sigma_{p,c+s+r}",
        result=f"{self:.{n}f}",
        equation=r"\frac{\epsilon_{cs} \cdot E_p + 0.8 \cdot \Delta \sigma_{pr} + \frac{E_p}{E_{cm}} \cdot "
        r"\phi(t, t_0) \cdot \sigma_{c,QP}}{1 + \frac{E_p}{E_{cm}} \cdot \frac{A_p}{A_c} \cdot \left(1 + \frac{A_c}{I_c} "
        r"\cdot z_{cp}^2\right) \cdot \left(1 + 0.8 \cdot \phi(t, t_0)\right)}",
        numeric_equation=rf"\frac{{{self.epsilon_cs:.6f} \cdot {self.e_p:.{n}f} + 0.800"
        rf" \cdot {self.delta_sigma_pr:.{n}f} + \frac{{{self.e_p:.{n}f}}}{{{self.e_cm:.{n}f}}} \cdot {self.phi_t_t0:.{n}f} "
        rf"\cdot {self.sigma_c_qp:.{n}f}}}{{1 + \frac{{{self.e_p:.{n}f}}}{{{self.e_cm:.{n}f}}} "
        rf"\cdot \frac{{{self.a_p:.{n}f}}}{{{self.a_c:.{n}f}}} "
        rf"\cdot \left(1 + \frac{{{self.a_c:.{n}f}}}{{{self.i_c:.{n}f}}} \cdot {self.z_cp:.{n}f}^2\right) \cdot \left(1 + 0.800 "
        rf"\cdot {self.phi_t_t0:.{n}f}\right)}}",
        comparison_operator_label="=",
        unit="MPa",
    )