pyflange.flangesegments

This module contains FlangeSegment classes, which model the mechanical behavior of a flange sector containig one bolt only.

Currently, the only two type of FlangeSegment available are an L-Flange segment and a T-Flange segment, implementing a polinomial relation between tower shell force force and bolt force / bolt moment. Nonetheless, this module has been structured to be easily extensible with other types of FlangeSegment model, such as Multilinear (Petersen) L-Flanges, Multilinear T-Flanges, etc.

The models implemented in this module are based on the following references:

[1]: Marc Seidel, SGRE TE TF PST: IEC 61400-6 AMD1 Background Document Fatigue design for ring flange connsections in wind turbine support structures.

[2]: IEC 61400-6:2020/AMD1 - wIND ENERGY GENERATION SYSTEMS - Part 6: Tower and foundation design requirements - AMENDMENT 1 Draft version 2024-10-01

[3]: Petersen, C.: Nachweis der Betriebsfestigkeit exzentrisch beanspruchter Ringflansch-verbindungen (Fatigue assessment of eccentrically loaded ring flange connections). Stahlbau 67 (1998), S. 191-203. https://onlinelibrary.wiley.com/doi/abs/10.1002/stab.199800690

[4]: Petersen, C.: Stahlbau (Steel construction), 4. Auflage Braunschweig: Wiesbaden: Springer Vieweg 2012. https://link.springer.com/book/10.1007%2F978-3-8348-8610-1

[9]: Tobinaga, I.; Ishihara, T.: A study of action point correction factor for L‐type flanges of wind turbine towers. Wind Energy 21 (2018), p. 801-806. https://doi.org/10.1002/we.2193

FlangeSegment

Bases: ABC

Abstract FlangeSegment class.

This class is meant to be extended and not to be instatiated directly.

Each FlangeSegment child class must implement the two transfer-function methods:

  • .bolt_axial_force(Z), returning the bolt axial force generated by a shell pull force Z
  • .bolt_bending_moment(Z), returning the bolt bending moment generted by a shekk pull force Z

PolynomialFlangeSegment

Bases: FlangeSegment

Generic FlangeSegment implementing polynomial transfer functions.

This is a generic FlangeSegment that implements a polynomial relation between tower shell force Z and bolt axial force Fs or bolt bending moment Ms.

It is not meant to be instantiated directly, but to be subclassed instead.

The polynomial transfer functions .bolt_axial_force(Z) and .bolt_bending_moment(Z) are defined based on 4 points, through which the polynomials pass. Those points are implementation specific.

The 4 reference points for the Fs(Z) polynomial are:

  • P1 = (Z1, Fs1) representing the flange segment state at rest (no loads applied, other than the self-weight). Each implementatio of this class should define Z1 as shell_force_at_rest property and Fs1 as bolt_force_at_rest property.

  • P2 = (Z2, Fs2) representing the flange segment ultimate tensile limit state (failure state B). Each implementatio of this class should define Z2 as shell_force_at_tensile_ULS property and Fs2 as bolt_force_at_ultimate_ULS property.

  • P3 = (Z3, Fs3) representing the flange segment in small tensile deformation condition. This point is meant to define the initial slope of the polynomial. Each implementatio of this class should define Z3 as shell_force_at_small_displacement property and Fs3 as bolt_force_at_small_displacement property.

  • P4 = (Z4, Fs4) representing the gap closure state. Each implementatio of this class should define Z4 as shell_force_at_closed_gap property, while Fs4 is automatically defined.

The 4 reference points for the Ms(Z) polynomial are:

  • Q1 = (Z1, Ms1) corresponding to P1 as defined above. Each implementation of this class should define Ms1 as bolt_moment_at_rest property.

  • Q2 = (Z2, Ms2) corresponding to P2 as defined above. Each implementation of this class should define Ms2 as bolt_moment_at_tensile_ULS property.

  • Q3 = (Z3, Ms3) corresponding to P3 as defined above. Each implementation of this class should define Ms3 as bolt_moment_at_small_displacement property.

  • Q4 = (Z4, Ms4) corresponding to P4 as defined above. Each implementatio of this class should define Z4 as shell_force_at_closed_gap property, while Ms4 is automatically defined.

bolt_axial_force(shell_pull)

Bolt axial force due to a given tower shell force force Z.

The relation between tower shell force Z and bolt axial force Fs, is a polynomial function, as defined in ref.[1], section 8.2 and in ref.[2], section G.4.2.

The passed shell_pull parameter must be either a real number or a numpy.array. If a numpy.array is passed, then the corresponding array of Fs values will be returned.

bolt_bending_moment(shell_pull)

Bolt bending moment due to a given tower shell force force Z.

The relation between tower shell force force Z and bolt bending moment Ms, is a polynomial function, as defined in ref.[1], section 8.3 and in Ref.[2], section G.4.2.

The passed shell_pull parameter must be either a real number or a numpy.array. If a numpy.array is passed, then the corresponding array of Ms values will be returned.

PolynomialLFlangeSegment dataclass

Bases: PolynomialFlangeSegment

A PolynomialFlangeSegment implementation for L-Flanges.

The ploynomial model implemented in this class is based on ref. [1] and ref. [2].

For this particular case of flange, this class defines the polynomial reference points P1, P2, P3, P4, Q1, Q2, Q3, Q4 and inherits the polynomial functions .bolt_axial_force(Z) and .bolt_bending_moment(Z) from the parent class.

Parameters:
  • a (float) –

    Distance between inner face of the flange and center of the bolt hole.

  • b (float) –

    Distance between center of the bolt hole and center-line of the shell.

  • s (float) –

    Shell thickness.

  • t (float) –

    Flange thickness.

  • R (float) –

    Shell outer curvature radius.

  • central_angle (float) –

    Angle subtended by the flange segment arc.

  • Zg (float) –

    Load applied to the flange segment shell at rest (normally dead weight of tower + RNA, divided by the number of bolts). Negative if compression.

  • bolt (Bolt) –

    Bolt object representing the flange segment bolt.

  • Fv (float) –

    Applied bolt preload, after preload losses.

  • Do (float) –

    Bolt hole diameter.

  • washer (Washer) –

    Washer object representing the fastener wahsers. If this parameter is None, then no washer is considered to be present.

  • nut (Nut) –

    Nut object representing the fastenrr nut.

  • gap_height (float) –

    Maximum longitudinal gap height.

  • gap_angle (float) –

    Angle subtended by the gap arc from the flange center.

  • gap_shape_factor (float, default: 1.0 ) –

    A correction factor that applies to Fs2-Fs1 and to Fs3-Fs1 to account for gap shape different that the default sinusoidal shape. If omitted, it defaults to 1.0 (sinusoida shape).

  • tilt_angle (float, default: 0.0 ) –

    Flange tilt angle, in radians.

  • E (float, default: 210000000000.0 ) –

    Young modulus of the flange. If omitted, it will be taken equal to 210 GPa.

  • G (float, default: 80770000000.0 ) –

    Shear modulus of the flange. If omitted, it will be taken equal to 80.77 GPa.

  • s_ratio (float, default: 1.0 ) –

    Ratio of bottom shell thickness over s. If omitted, it will be taken equal to 1.0, threfore, by default, s_botom = s.

  • r (float, default: 0.01 ) –

    Radius of the rouding between the shell and the flange.

  • k_shell (float, default: None ) –

    Optional individual initial shell stiffness value. This value can be calculated in a separate FE analysis. The unit must be [N/m/m]. If omitted, the interpolated formula from [1] will be used.

The given parameters are also available as attributes (e.g. fseg.a, fseg.Fv, etc.).

This class is designed to be immutable, therefore modifying the attributes after instantiation is not a good idea. If you need a segment with different attributes, just create a new one.

bolt_force_at_rest cached property

Bolt axial force when no external loads are applied.

The bolt force at rest is just the bolt pretension.

bolt_force_at_small_displacement cached property

Intermediate bolt pull, between rest and tensile failuse.

This is an auxiliary point that gives the polynomial the right value of initial slope. It is evaluated according to ref. [1], sec.8.2.2.3 and to ref. [2], section G.4.2.

bolt_force_at_tensile_ULS cached property

Bolt axial force at tensile failure.

This is the bolt tensile force corresponding to the ultimate tower shell force (.shell_force_at_rensile_ULS).

Assuming the failure mode B, in the ULS, the bolt is subjected to its maximum tensile capacity.

In refs. [1] and [2], this value is colled Fs2.

bolt_moment_at_rest cached property

Bolt bending moment when no external loads are applied.

bolt_moment_at_small_displacement cached property

Intermediate bolt moment, between rest and tensile failuse.

This is the bolt bending moment corresponding to the tower shell force Z3 (.shell_force_at_small_displacement).

bolt_moment_at_tensile_ULS cached property

Bolt bending moment at tensile failure.

This is the bolt bending moment corresponding to the ultimate tower shell force (.shell_force_at_rensile_ULS).

In refs. [1] and [2], this value is colled Ms2.

shell_force_at_closed_gap cached property

Shell force necessary to completely close the imperfection gap.

shell_force_at_rest cached property

Shell force when no external loads are applied

The shell loads at rest are normally the self-weights of the structure supported by the flange.

shell_force_at_small_displacement cached property

Intermediate tower shell force, between rest and tensile failure.

This is an auxiliary point that gives the polynomial the right value of initial slope. It is evaluated according to ref. [1], sec.8.2.2.3 and to ref. [2], section G.4.2.

shell_force_at_tensile_ULS cached property

Ultimate Limit State tower shell force.

This is tower shell force that brings the flange segment system in its tensile ultimate limit state. It is evaluated according to ref. [1], sec.8.2.2.2 and to ref. [2], section G.4.2, where it is referred to as Z2.

failure_mode(fy_sh, fy_fl, gamma_0=1.1)

Determine the failure mode of this flange.

Parameters:
  • fy_sh (float) –

    Yield stress of the shell material.

  • fy_fl (float) –

    Yield stress of the flange material.

  • gamma_0 (float, default: 1.1 ) –

    Material factor.
Returns:
  • mode( str ) –

    failure mode string (eiter "A", "B", "D" or "E").

  • limits( list ) –

    The list [Zu_A, Zu_B, Zu_D, Zu_E] of the tower shell force values at which each failure mode occurs.

validate(fy_sh, fy_fl, gamma_0=1.1)

Verifies the applicability of the polynomial model.

Check if this L-Flange Segment matches the polynomial model assumptions, that is, if it fails according to failure mode B. If not, it will throw an exceptions.

Parameters:
  • fy_sh (float) –

    Yield stress of the shell material.

  • fy_fl (float) –

    Yield stress of the flange material.

  • gamma_0 (float, default: 1.1 ) –

    Material factor.
Returns:
  • None –

    If successfull.
Raises:
  • ValueError –

    if the failure mode is not B.

PolynomialTFlangeSegment dataclass

Bases: PolynomialFlangeSegment

A PolynomialFlangeSegment implementation for T-Flanges.

The ploynomial model implemented in this class is based on ref. [1] and ref. [2].

For this particular case of flange, this class defines the polynomial reference points P1, P2, P3, P4, Q1, Q2, Q3, Q4 and inherits the polynomial functions .bolt_axial_force(Z) and .bolt_bending_moment(Z) from the parent class.

Parameters:
  • a (float) –

    Distance between inner face of the flange and center of the bolt hole.

  • b (float) –

    Distance between center of the bolt hole and center-line of the shell.

  • s (float) –

    Shell thickness.

  • t (float) –

    Flange thickness.

  • R (float) –

    Shell outer curvature radius.

  • central_angle (float) –

    Angle subtended by the flange segment arc.

  • Zg (float) –

    Load applied to the flange segment shell at rest (normally dead weight of tower + RNA, divided by the number of bolts). Negative if compression.

  • bolt (Bolt) –

    Bolt object representing the flange segment bolt.

  • Fv (float) –

    Applied bolt preload, after preload losses.

  • Do (float) –

    Bolt hole diameter.

  • washer (Washer) –

    Washer object representing the fastener wahsers. If this parameter is None, then no washer is considered to be present.

  • nut (Nut) –

    Nut object representing the fastenrr nut.

  • gap_height (float) –

    Maximum longitudinal gap height.

  • gap_angle (float) –

    Angle subtended by the gap arc from the flange center.

  • gap_shape_factor (float, default: 1.0 ) –

    A correction factor that applies to Fs2-Fs1 and to Fs3-Fs1 to account for gap shape different that the default sinusoidal shape. If omitted, it defaults to 1.0 (sinusoida shape).

  • tilt_angle (float, default: 0.0 ) –

    Flange tilt angle, in radians.

  • E (float, default: 210000000000.0 ) –

    Young modulus of the flange. If omitted, it will be taken equal to 210 GPa.

  • G (float, default: 80770000000.0 ) –

    Shear modulus of the flange. If omitted, it will be taken equal to 80.77 GPa.

  • s_ratio (float, default: 1.0 ) –

    Ratio of bottom shell thickness over s. If omitted, it will be taken equal to 1.0, threfore, by default, s_botom = s.

  • r (float, default: 0.01 ) –

    Radius of the rouding between the shell and the flange.

  • k_shell (float, default: None ) –

    Optional individual initial shell stiffness value. This value can be calculated in a separate FE analysis. The unit must be [N/m/m]. If omitted, the interpolated formula from [1] will be used.

The given parameters are also available as attributes (e.g. fseg.a, fseg.Fv, etc.).

This class is designed to be immutable, therefore modifying the attributes after instantiation is not a good idea. If you need a segment with different attributes, just create a new one.

bolt_force_at_rest cached property

Bolt axial force when no external loads are applied.

The bolt force at rest is just the bolt pretension.

bolt_force_at_small_displacement cached property

Intermediate bolt pull, between rest and tensile failuse.

This is an auxiliary point that gives the polynomial the right value of initial slope. It is evaluated according to ref. [1], sec.8.2.2.3 and to ref. [2], section G.4.2.

bolt_force_at_tensile_ULS cached property

Bolt axial force at tensile failure.

This is the bolt tensile force corresponding to the ultimate tower shell force (.shell_force_at_rensile_ULS).

Assuming the failure mode B, in the ULS, the bolt is subjected to its maximum tensile capacity.

In refs. [1] and [2], this value is colled Fs2.

bolt_moment_at_rest cached property

Bolt bending moment when no external loads are applied.

bolt_moment_at_small_displacement cached property

Intermediate bolt moment, between rest and tensile failuse.

This is the bolt bending moment corresponding to the tower shell force Z3 (.shell_force_at_small_displacement).

bolt_moment_at_tensile_ULS cached property

Bolt bending moment at tensile failure.

This is the bolt bending moment corresponding to the ultimate tower shell force (.shell_force_at_rensile_ULS).

In refs. [1] and [2], this value is colled Ms2.

shell_force_at_closed_gap cached property

Shell force necessary to completely close the imperfection gap.

shell_force_at_rest cached property

Shell force when no external loads are applied.

The shell loads at rest are normally the self-weights of the structure supported by the flange.

shell_force_at_small_displacement cached property

Intermediate tower shell force, between rest and tensile failure.

This is an auxiliary point that gives the polynomial the right value of initial slope. It is evaluated according to ref. [1], sec.8.2.2.3 and to ref. [2], section G.4.2.

shell_force_at_tensile_ULS cached property

Ultimate Limit State tower shell force.

This is the tower shell force that brings the flange segment system in its tensile ultimate limit state. It is evaluated according to ref. [1], sec.8.2.2.2 and to ref. [2], section G.4.2, where it is referred to as Z2.

failure_mode(fy_sh, fy_fl, gamma_0=1.1)

Determine the failure mode of this flange.

Parameters:
  • fy_sh (float) –

    Yield stress of the shell material.

  • fy_fl (float) –

    Yield stress of the flange material.

  • gamma_0 (float, default: 1.1 ) –

    Material factor.
Returns:
  • mode( str ) –

    failure mode string (eiter "A", "B", "D" or "E").

  • limits( list ) –

    The list [Zu_A, Zu_B, Zu_D, Zu_E] of the tower shell force values at which each failure mode occurs.

validate(fy_sh, fy_fl, gamma_0=1.1)

Verifies the applicability of the polynomial model.

Check if this L-Flange Segment matches the polynomial model assumptions, that is, if it fails according to failure mode B. If not, it will throw an exceptions.

Parameters:
  • fy_sh (float) –

    Yield stress of the shell material.

  • fy_fl (float) –

    Yield stress of the flange material.

  • gamma_0 (float, default: 1.1 ) –

    Material factor.
Returns:
  • None –

    If successfull.
Raises:
  • ValueError –

    if the failure mode is not B.

bolt_markov_matrix(fseg, flange_markov_matrix, bending_factor=0.0, macro_geometric_factor=1.0, mean_factor=1.0, range_factor=1.0)

Evaluates the bolt Markov matrix, given the shell Markov matix.

Parameters:
  • fseg (PolynomialFlangeSegment) –

    The flange segment object that contains the force and moment transfer function that convert shell forces to bolt forces.

    WARNING: Notice that currently this function works only with PolynomialFlangeSegments, while it should be extended to work with any FlangeSegment object.

  • flange_markov_matrix (DataFrame) –

    The flanges bending moments markov matrix, containg the colums 'Cycles', 'Range', 'Mean'.

  • bending_factor (float, default: 0.0 ) –

    The factor that considers the bending portion of the total stress range.

  • macro_geometric_factor (float, default: 1.0 ) –

    The factor that considers macro geometric influences. The factor affects the deadweigt of the tower, the mean values of the markov matrix and the range values of the markov matrix.

  • mean_factor (float, default: 1.0 ) –

    The factor that multiplies the mean values of the bending moments of the tower.

  • range_factor (float, default: 1.0 ) –

    The factor that multiplies the range of the bending moments of the tower.
Returns:
  • bolt_markov_matrix( DataFrame ) –

    The bolt markov matrix representing the stress history in the bolt due to the applied flange markov matrix. The returned pandas DataFrame contains the columns 'Cycles', 'Range' and 'Mean'.

shell_stiffness(shell_radius, shell_thickness, gap_angle)

Axial stiffness of a flange-segment shell.

This function will return the linear interpolation of the flange stiffnesses measured in 2160 different FEA analyses with several values of shell radius, shell thickness and gap angle.

Parameters:
  • shell_radius (float) –

    outer radius of the shell.

  • shell_thickness (float) –

    thickness of the shell.

  • gap_angle (float) –

    angular length of the flange gap.
Returns:
  • K_sh( float ) –

    shell stiffness.