Section

class sectionproperties.analysis.section.Section(geometry: Geometry | CompoundGeometry, time_info: bool = False)

Bases: object

Class for structural cross-sections.

Stores the finite element geometry, mesh and material information and provides methods to compute the cross-section properties.

Variables:

• geometry (Geometry | CompoundGeometry) – Cross-section geometry

• materials (list[Material]) – List of materials in the geometry

• material_groups (list[MaterialGroup]) – List of class:~sectionproperties.pre.pre.MaterialGroup objects, which contain the finite elements and stress results (if applicable) for each defined material

• mesh (dict[str, Any]) – Finite element mesh generated by CyTriangle

• num_nodes (int) – Number of nodes in the finite element mesh

• elements (list[Tri6]) – List of finite element objects describing the cross-section mesh

• mesh_search_tree (STRtree) – STRtree to enable efficient lookup of elements in the mesh

• section_props (SectionProperties) – Class to store calculated section properties

Methods

assemble_torsion	Assembles the warping stiffness matrix.
calculate_frame_properties	Calculates section properties to be used for frame analysis.
calculate_geometric_properties	Calculates geometric (area) properties.
calculate_plastic_properties	Calculates plastic properties.
calculate_stress	Calculates cross-section stresses.
calculate_warping_properties	Calculates warping properties.
display_mesh_info	Prints mesh statistics to the command line.
display_results	Prints the results that have been calculated to the terminal.
display_transformed_results	Prints the transformed results that have been calculated to the terminal.
get_area	Returns the cross-section area.
get_as	Returns the cross-section centroidal axis shear area.
get_as_p	Returns the cross-section princicpal axis shear area.
get_beta	Returns the cross-section global monosymmetry constants.
get_beta_p	Returns the cross-section principal monosymmetry constants.
get_c	Returns the cross-section elastic centroid.
get_e_eff	Returns the cross-section effective elastic modulus.
get_e_ref	Extract transformed elastic modulus from e_ref (float or material).
get_ea	Returns the cross-section axial rigidity.
get_eas	Returns modulus-weighted the cross-section centroidal axis shear area.
get_eas_p	Returns the modulus-weighted cross-section princicpal axis shear area.
get_egamma	Returns the modulus-weighted cross-section warping constant.
get_eic	Returns the modulus-weighted cross-section centroidal second moments of area.
get_eig	Returns the modulus-weighted cross-section global second moments of area.
get_eip	Returns the modulus-weighted cross-section principal second moments of area.
get_ej	Returns the modulus-weighted cross-section St Venant torsion constant.
get_eq	Returns the modulus-weighted cross-section first moments of area.
get_ez	Returns the modulus-weighted cross-section centroidal elastic section moduli.
get_ezp	Returns the modulus-weighted cross-section principal elastic section moduli.
get_g_eff	Returns the cross-section effective shear modulus.
get_gamma	Returns the cross-section warping constant.
get_ic	Returns the cross-section centroidal second moments of area.
get_ig	Returns the cross-section global second moments of area.
get_ip	Returns the cross-section principal second moments of area.
get_j	Returns the cross-section St Venant torsion constant.
get_mass	Returns the cross-section mass.
get_mp	Returns the cross-section plastic moment about the centroidal axis.
get_mp_p	Returns the cross-section plastic moment about the principal axis.
get_my	Returns the yield moment for bending about the centroidal axis.
get_my_p	Returns the yield moment for bending about the principal axis.
get_nu_eff	Returns the cross-section effective Poisson's ratio.
get_pc	Returns the cross-section centroidal axis plastic centroid.
get_pc_p	Returns the cross-section principal axis plastic centroid.
get_perimeter	Returns the cross-section perimeter.
get_phi	Returns the cross-section principal bending angle.
get_q	Returns the cross-section first moments of area.
get_rc	Returns the cross-section centroidal radii of gyration.
get_rp	Returns the cross-section principal radii of gyration.
get_s	Returns the cross-section centroidal plastic section moduli.
get_sc	Returns the cross-section centroidal shear centre (elasticity).
get_sc_p	Returns the cross-section principal shear centre (elasticity).
get_sc_t	Returns the cross-section centroidal shear centre (Trefftz's method).
get_sf	Returns the cross-section centroidal axis shape factors.
get_sf_p	Returns the cross-section principal axis shape factors.
get_sp	Returns the cross-section principal axis plastic section moduli.
get_stress_at_points	Calculates the stress for a list of points.
get_z	Returns the cross-section centroidal elastic section moduli.
get_zp	Returns the cross-section principal elastic section moduli.
is_composite	Returns whether or not a composite section is being analysed.
plot_centroids	Plots the calculated centroids over the mesh.
plot_mesh	Plots the finite element mesh.
plot_warping_function	Plots the warping function over the mesh.

__init__(geometry: Geometry | CompoundGeometry, time_info: bool = False) → None

Inits the Section class.

The constructor extracts information from the provided mesh object and creates and stores the corresponding finite element objects.

Parameters:

• geometry (Geometry | CompoundGeometry) – Cross-section geometry object used to generate the mesh

• time_info (bool) – If set to True, a detailed description of the computation and the time cost is printed to the terminal for every computation performed. Defaults to False.

Raises:

• ValueError – If geometry does not contain a mesh

• ValueError – If the number of materials does not equal the number of regions

calculate_geometric_properties() → None

Calculates geometric (area) properties.

Calculates the geometric properties of the cross-section and stores them in the SectionProperties object contained in the section_props variable.

Note

If materials are specified for the cross-section, the moments of area and section moduli are elastic modulus weighted.

Geometric Properties

Below is a list of the section properties calculated by the calculate_geometric_properties() method:

• Cross-sectional area

• Cross-sectional perimeter

• Cross-sectional mass

• Area weighted material properties (composite only) \(E_{eff}\), \(G_{eff}\), \({\nu}_{eff}\)

• Modulus weighted area (axial rigidity)

• First moments of area

• Second moments of area about the global axis

• Second moments of area about the centroidal axis

• Elastic centroid

• Centroidal section moduli

• Radii of gyration

• Principal axis properties

• Yield moments (composite only)

calculate_warping_properties(solver_type: str = 'direct') → None

Calculates warping properties.

Calculates all the warping properties of the cross-section and stores them in the SectionProperties object contained in the section_props variable.

Parameters:

solver_type (str) – Solver used for solving systems of linear equations, either using the "direct" method or "cgs" iterative method. Defaults to "direct".

Raises:

RuntimeError – If the geometric properties have not been calculated prior to calling this method

Note

If materials are specified, the values calculated for the torsion constant, warping constant and shear areas are elastic modulus weighted.

Warning

The geometric properties must be calculated prior to the calculation of the warping properties.

Warping Properties

Below is a list of the section properties calculated by the calculate_warping_properties() method:

• Torsion constant

• Shear centre

• Shear area

• Warping constant

• Monosymmetry constant

calculate_frame_properties(solver_type: str = 'direct') → tuple[float, float, float, float, float, float]

Calculates section properties to be used for frame analysis.

Calculates and returns the properties required for a frame analysis. The properties are also stored in the SectionProperties object contained in the section_props variable.

This method is more efficient than running a geometric and warping analysis as unnecessary values are not calculated.

Parameters:

solver_type (str) – Solver used for solving systems of linear equations, either using the "direct" method or "cgs" iterative method. Defaults to "direct".

Returns:

Cross-section properties to be used for a frame analysis (area, ixx, iyy, ixy, j, phi)

Return type:

tuple[float, float, float, float, float, float]

Note

If materials are specified, the values calculated for the moments of area and the torsion constant are elastic modulus weighted.

Warping Properties

Below is a list of the section properties calculated by the calculate_frame_properties() method:

• Cross-sectional area

• Second moments of area about the centroidal axis

• Torsion constant

• Principal axis angle

calculate_plastic_properties(verbose: bool = False) → None

Calculates plastic properties.

Calculates the plastic properties of the cross-section and stores them in the SectionProperties object contained in the section_props variable.

Parameters:

verbose (bool) – If set to True, the number of iterations required for each plastic axis is printed to the terminal. Defaults to False.

Raises:

RuntimeError – If the geometric properties have not been calculated prior to calling this method

Note

If materials are specified, the values calculated for the plastic section moduli are displayed as plastic moments (i.e \(M_p = f_y S\)) and the shape factors are calculated as the ratio between the plastic moment and the yield moment.

Warning

The geometric properties must be calculated prior to the calculation of the plastic properties.

Plastic Properties

Below is a list of the section properties calculated by the calculate_plastic_properties() method:

• Plastic centroids (centroidal and principal axes)

• Plastic section moduli (centroidal and principal axes)

• Shape factors (centroidal and principal axes)

calculate_stress(n: float = 0.0, vx: float = 0.0, vy: float = 0.0, mxx: float = 0.0, myy: float = 0.0, m11: float = 0.0, m22: float = 0.0, mzz: float = 0.0) → StressPost

Calculates cross-section stresses.

Calculates the cross-section stress resulting from design actions and returns a StressPost object allowing the post-processing of the stress results.

Parameters:

• n (float) – Axial force. Defaults to 0.0.

• vx (float) – Shear force acting in the x-direction. Defaults to 0.0.

• vy (float) – Shear force acting in the y-direction. Defaults to 0.0.

• mxx (float) – Bending moment about the centroidal xx-axis. Defaults to 0.0.

• myy (float) – Bending moment about the centroidal yy-axis. Defaults to 0.0.

• m11 (float) – Bending moment about the centroidal 11-axis. Defaults to 0.0.

• m22 (float) – Bending moment about the centroidal 22-axis. Defaults to 0.0.

• mzz (float) – Torsion moment about the centroidal zz-axis. Defaults to 0.0.

Returns:

Object for post-processing cross-section stresses

Raises:

RuntimeError – If a geometric and warping analysis (if required) have not been performed prior to calling this method

Return type:

StressPost

Note

A geometric analysis must be performed prior to performing a stress analysis. Further, if the shear force or torsion is non-zero a warping analysis must also be performed.

assemble_torsion(progress: Progress | None = None, task: TaskID | None = None) → tuple[csc_matrix, npt.NDArray[np.float64]]

Assembles the warping stiffness matrix.

Assembles stiffness matrices to be used for the computation of warping properties and the torsion load vector (f_torsion). A Lagrangian multiplier (k_lg) stiffness matrix is returned. The stiffness matrix is assembled using the sparse COO format and returned in the sparse CSC format.

Parameters:

• progress (Progress | None) – Rich progress object

• task (TaskID | None) – Rich task object

Returns:

Lagrangian multiplier stiffness matrix and torsion load vector (k_lg, f_torsion)

Return type:

tuple[csc_matrix, npt.NDArray[np.float64]]

plot_mesh(alpha: float = 0.5, materials: bool = True, mask: list[bool] | None = None, title: str = 'Finite Element Mesh', **kwargs: Any) → matplotlib.axes.Axes

Plots the finite element mesh.

Parameters:

• alpha (float) – Transparency of the mesh outlines: \(0 \leq \alpha \leq 1\). Defaults to 0.5.

• materials (bool) – If set to True shades the elements with the specified material colors. Defaults to True.

• mask (list[bool] | None) – Mask array, of length num_nodes, to mask out triangles. Defaults to None.

• title (str) – Plot title. Defaults to "Finite Element Mesh".

• kwargs (Any) – Passed to plotting_context()

Returns:

Matplotlib axes object

Return type:

matplotlib.axes.Axes

Example

The following example plots the mesh for a rectanglular steel-timber composite section:

from sectionproperties.pre import Material
from sectionproperties.pre.library import rectangular_section
from sectionproperties.analysis import Section

steel = Material(
    name="Steel",
    elastic_modulus=200e3,
    poissons_ratio=0.3,
    density=7.85e-6,
    yield_strength=250,
    color="grey",
)
timber = Material(
    name="Timber",
    elastic_modulus=8e3,
    poissons_ratio=0.35,
    density=6.5e-7,
    yield_strength=20,
    color="burlywood",
)

geom_steel = rectangular_section(d=50, b=50, material=steel)
geom_timber = rectangular_section(d=50, b=50, material=timber)
geom = geom_timber.align_to(geom_steel, on="right") + geom_steel
geom.create_mesh(mesh_sizes=[10, 5])

Section(geometry=geom).plot_mesh()

(Source code, png, hires.png, pdf)

Composite section mesh

plot_centroids(alpha: float = 0.5, title: str = 'Centroids', **kwargs: Any) → matplotlib.axes.Axes

Plots the calculated centroids over the mesh.

Plots the elastic centroid, the shear centre, the plastic centroids and the principal axis, if they have been calculated, on top of the finite element mesh.

Parameters:

• alpha (float) – Transparency of the mesh outlines: \(0 \leq \alpha \leq 1\). Defaults to 0.5.

• title (str) – Plot title. Defaults to "Centroids".

• kwargs (Any) – Passed to plotting_context()

Returns:

Matplotlib axes object

Return type:

matplotlib.axes.Axes

Example

The following example analyses a 200 PFC section and displays a plot of the centroids:

from sectionproperties.pre.library import channel_section
from sectionproperties.analysis import Section

geom = channel_section(d=200, b=75, t_f=12, t_w=6, r=12, n_r=8)
geom.create_mesh(mesh_sizes=[20])

section = Section(geometry=geom)
section.calculate_geometric_properties()
section.calculate_warping_properties()
section.calculate_plastic_properties()

section.plot_centroids()

(Source code, png, hires.png, pdf)

200PFC centroids

plot_warping_function(title: str = 'Warping Function', level: int = 20, cmap: str = 'viridis', alpha: float = 0.2, with_lines: bool = True, **kwargs: Any) → matplotlib.axes.Axes

Plots the warping function over the mesh.

Parameters:

• title (str) – Plot title. Defaults to "Warping Function".

• level (int) – Number of contour levels. Defaults to 20.

• cmap (str) – Colormap. Defaults to "viridis".

• alpha (float) – Transparency of the mesh outlines: \(0 \leq \alpha \leq 1\). Defaults to 0.2.

• with_lines (bool) – If set to True, contour lines are displayed. Defaults to True.

• kwargs (Any) – Passed to plotting_context()

Raises:

RuntimeError – If a warping analysis has not been performed

Returns:

Matplotlib axes object

Return type:

matplotlib.axes.Axes

display_mesh_info() → None

Prints mesh statistics to the command line.

display_results(fmt: str = '8.6e') → None

Prints the results that have been calculated to the terminal.

Parameters:

fmt (str) – Number formatting string, see https://docs.python.org/3/library/string.html. Defaults to "8.6e".

display_transformed_results(e_ref: float | Material, fmt: str = '8.6e') → None

Prints the transformed results that have been calculated to the terminal.

All results that are scaled by the elastic modulus in display_results() (e.g. e.ixx_c) are divided by e_ref.

This is a composite only method, as such this can only be called if material properties have been applied to the cross-section.

Parameters:

• e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results.

• fmt (str) – Number formatting string, see https://docs.python.org/3/library/string.html. Defaults to "8.6e".

Raises:

RuntimeError – If material properties have not been applied

is_composite() → bool

Returns whether or not a composite section is being analysed.

If the only material is the default material, a regular analysis is being conducted. Otherwise, a composite analysis is being conducted.

Returns:

Whether or not a composite section is being analysed.

Return type:

bool

get_e_ref(e_ref: float | Material) → float

Extract transformed elastic modulus from e_ref (float or material).

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results

Returns:

Transformed elastic modulus

Return type:

float

get_area() → float

Returns the cross-section area.

Returns:

Cross-section area

Raises:

RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_perimeter() → float

Returns the cross-section perimeter.

Returns:

Cross-section perimeter

Raises:

RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_mass() → float

Returns the cross-section mass.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Cross-section mass

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_ea(e_ref: float | Material = 1.0) → float

Returns the cross-section axial rigidity.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted area (axial rigidity)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_q() → tuple[float, float]

Returns the cross-section first moments of area.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

First moments of area about the global axis (qx, qy)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_eq(e_ref: float | Material = 1.0) → tuple[float, float]

Returns the modulus-weighted cross-section first moments of area.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus-weighted first moments of area about the global axis (e.qx, e.qy)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_ig() → tuple[float, float, float]

Returns the cross-section global second moments of area.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Second moments of area about the global axis (ixx_g, iyy_g, ixy_g)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float]

get_eig(e_ref: float | Material = 1.0) → tuple[float, float, float]

Returns the modulus-weighted cross-section global second moments of area.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted second moments of area about the global axis (e.ixx_g, e.iyy_g, e.ixy_g)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float]

get_c() → tuple[float, float]

Returns the cross-section elastic centroid.

Returns:

Elastic centroid (cx, cy)

Raises:

RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_ic() → tuple[float, float, float]

Returns the cross-section centroidal second moments of area.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Second moments of area about the centroidal axis (ixx_c, iyy_c, ixy_c)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float]

get_eic(e_ref: float | Material = 1.0) → tuple[float, float, float]

Returns the modulus-weighted cross-section centroidal second moments of area.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted second moments of area about the centroidal axis (e.ixx_c, e.iyy_c, e.ixy_c)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float]

get_z() → tuple[float, float, float, float]

Returns the cross-section centroidal elastic section moduli.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Elastic section moduli about the centroidal axis with respect to the top and bottom fibres (zxx_plus, zxx_minus, zyy_plus, zyy_minus)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float, float]

get_ez(e_ref: float | Material = 1.0) → tuple[float, float, float, float]

Returns the modulus-weighted cross-section centroidal elastic section moduli.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted elastic section moduli about the centroidal axis with respect to the top and bottom fibres (e.zxx_plus, e.zxx_minus, e.zyy_plus, e.zyy_minus)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float, float]

get_my() → tuple[float, float]

Returns the yield moment for bending about the centroidal axis.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Yield moment for bending about the centroidal x and y axes (my_xx, my_yy)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_rc() → tuple[float, float]

Returns the cross-section centroidal radii of gyration.

Returns:

Radii of gyration about the centroidal axis (rx, ry)

Raises:

RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_ip() → tuple[float, float]

Returns the cross-section principal second moments of area.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Second moments of area about the principal axis (i11_c, i22_c)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_eip(e_ref: float | Material = 1.0) → tuple[float, float]

Returns the modulus-weighted cross-section principal second moments of area.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted second moments of area about the principal axis (e.i11_c, e.i22_c)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_phi() → float

Returns the cross-section principal bending angle.

Returns:

Principal bending axis angle

Raises:

RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_zp() → tuple[float, float, float, float]

Returns the cross-section principal elastic section moduli.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Elastic section moduli about the principal axis with respect to the top and bottom fibres (z11_plus, z11_minus, z22_plus, z22_minus)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float, float]

get_ezp(e_ref: float | Material = 1.0) → tuple[float, float, float, float]

Returns the modulus-weighted cross-section principal elastic section moduli.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted elastic section moduli about the principal axis with respect to the top and bottom fibres (e.z11_plus, e.z11_minus, e.z22_plus, e.z22_minus)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float, float, float]

get_my_p() → tuple[float, float]

Returns the yield moment for bending about the principal axis.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Yield moment for bending about the principal 11 and 22 axes (my_11, my_22)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_rp() → tuple[float, float]

Returns the cross-section principal radii of gyration.

Returns:

Radii of gyration about the principal axis (r11, r22)

Raises:

RuntimeError – If a geometric analysis has not been performed

Return type:

tuple[float, float]

get_nu_eff() → float

Returns the cross-section effective Poisson’s ratio.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Effective Poisson’s ratio

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_e_eff() → float

Returns the cross-section effective elastic modulus.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Effective elastic modulus based on area

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_g_eff() → float

Returns the cross-section effective shear modulus.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Effective shear modulus based on area

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a geometric analysis has not been performed

Return type:

float

get_j() → float

Returns the cross-section St Venant torsion constant.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

St. Venant torsion constant

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

float

get_ej(e_ref: float | Material = 1.0) → float

Returns the modulus-weighted cross-section St Venant torsion constant.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted St. Venant torsion constant

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

float

get_sc() → tuple[float, float]

Returns the cross-section centroidal shear centre (elasticity).

Returns:

Centroidal axis shear centre (elasticity approach) (x_se, y_se)

Raises:

RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_sc_p() → tuple[float, float]

Returns the cross-section principal shear centre (elasticity).

Returns:

Principal axis shear centre (elasticity approach) (x11_se, y22_se)

Raises:

RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_sc_t() → tuple[float, float]

Returns the cross-section centroidal shear centre (Trefftz’s method).

Returns:

Centroidal axis shear centre (Trefftz’s method) (x_st, y_st)

Raises:

RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_gamma() → float

Returns the cross-section warping constant.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Warping constant

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

float

get_egamma(e_ref: float | Material = 1.0) → float

Returns the modulus-weighted cross-section warping constant.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted warping constant

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

float

get_as() → tuple[float, float]

Returns the cross-section centroidal axis shear area.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Shear area for loading about the centroidal axis (a_sx, a_sy)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_eas(e_ref: float | Material = 1.0) → tuple[float, float]

Returns modulus-weighted the cross-section centroidal axis shear area.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted shear area for loading about the centroidal axis (e.a_sx, e.a_sy)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_as_p() → tuple[float, float]

Returns the cross-section princicpal axis shear area.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Shear area for loading about the princicpal bending axis (a_s11, a_s22)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_eas_p(e_ref: float | Material = 1.0) → tuple[float, float]

Returns the modulus-weighted cross-section princicpal axis shear area.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

This value can be transformed by providing a reference elastic modulus or a material property.

Parameters:

e_ref (float | Material) – Reference elastic modulus or material property by which to transform the results. Defaults to 1.0.

Returns:

Modulus weighted shear area for loading about the princicpal bending axis (e.a_s11, e.a_s22)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float]

get_beta() → tuple[float, float, float, float]

Returns the cross-section global monosymmetry constants.

Returns:

Monosymmetry constants for bending about both global axes (beta_x_plus, beta_x_minus, beta_y_plus, beta_y_minus).

Return type:

tuple[float, float, float, float]

Note

The plus value relates to the top flange in compression and the minus value relates to the bottom flange in compression.

Raises:

RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float, float, float]

get_beta_p() → tuple[float, float, float, float]

Returns the cross-section principal monosymmetry constants.

Returns:

Monosymmetry constant for bending about both principal axes (beta_11_plus, beta_11_minus, beta_22_plus, beta_22_minus)

Return type:

tuple[float, float, float, float]

Note

The plus value relates to the top flange in compression and the minus value relates to the bottom flange in compression.

Raises:

RuntimeError – If a warping analysis has not been performed

Return type:

tuple[float, float, float, float]

get_pc() → tuple[float, float]

Returns the cross-section centroidal axis plastic centroid.

Returns:

Centroidal axis plastic centroid (x_pc, y_pc)

Raises:

RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float]

get_pc_p() → tuple[float, float]

Returns the cross-section principal axis plastic centroid.

Returns:

Principal bending axis plastic centroid (x11_pc, y22_pc)

Raises:

RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float]

get_s() → tuple[float, float]

Returns the cross-section centroidal plastic section moduli.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Plastic section moduli about the centroidal axis (sxx, syy)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float]

get_mp() → tuple[float, float]

Returns the cross-section plastic moment about the centroidal axis.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Plastic moment (\(M_p = f_y S\)) about the centroidal axis (mp_xx, mp_yy)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float]

get_sp() → tuple[float, float]

Returns the cross-section principal axis plastic section moduli.

This is a geometric only property, as such this can only be returned if material properties have not been applied to the cross-section.

Returns:

Plastic section moduli about the principal bending axis (s11, s22)

Raises:

• RuntimeError – If material properties have been applied

• RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float]

get_mp_p() → tuple[float, float]

Returns the cross-section plastic moment about the principal axis.

This is a composite only property, as such this can only be returned if material properties have been applied to the cross-section.

Returns:

Plastic moment (\(M_p = f_y S\)) about the principal axis (mp_11, mp_22)

Raises:

• RuntimeError – If material properties have not been applied

• RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float]

get_sf() → tuple[float, float, float, float]

Returns the cross-section centroidal axis shape factors.

For a geometric-only analysis, the shape factor is defined as the ratio between the plastic section modulus and the elastic section modulus. For a composite analysis the shape factors is defined as the ratio between the plastic moment and the yield moment.

Note

For composite analyses, the plus values will always equal the minus values as the yield moment occurs when the first fibre reaches its yield strength. For geometric-only analyses, the elastic section moduli is calculated with respect to the top and bottom fibres (i.e. plus and minus).

Returns:

Centroidal axis shape factors with respect to the top and bottom fibres (sf_xx_plus, sf_xx_minus, sf_yy_plus, sf_yy_minus)

Raises:

RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float, float, float]

get_sf_p() → tuple[float, float, float, float]

Returns the cross-section principal axis shape factors.

For a geometric-only analysis, the shape factor is defined as the ratio between the plastic section modulus and the elastic section modulus. For a composite analysis the shape factors is defined as the ratio between the plastic moment and the yield moment.

Note

For composite analyses, the plus values will always equal the minus values as the yield moment occurs when the first fibre reaches its yield strength. For geometric-only analyses, the elastic section moduli is calculated with respect to the top and bottom fibres (i.e. plus and minus).

Returns:

Principal bending axis shape factors with respect to the top and bottom fibres (sf_11_plus, sf_11_minus, sf_22_plus, sf_22_minus)

Raises:

RuntimeError – If a plastic analysis has not been performed

Return type:

tuple[float, float, float, float]

get_stress_at_points(pts: list[tuple[float, float]], n: float = 0.0, mxx: float = 0.0, myy: float = 0.0, m11: float = 0.0, m22: float = 0.0, mzz: float = 0.0, vx: float = 0.0, vy: float = 0.0, agg_func: Callable[[list[float]], float] = <function average>) → list[tuple[float, float, float] | None]

Calculates the stress for a list of points.

Calculates the stress at a set of points within an element for given design actions and returns the global stress components for each point.

Parameters:

• pts (list[tuple[float, float]]) – A list of points [(x, y), ..., ]

• n (float) – Axial force

• mxx (float) – Bending moment about the centroidal xx-axis. Defaults to 0.0.

• myy (float) – Bending moment about the centroidal yy-axis. Defaults to 0.0.

• m11 (float) – Bending moment about the centroidal 11-axis. Defaults to 0.0.

• m22 (float) – Bending moment about the centroidal 22-axis. Defaults to 0.0.

• mzz (float) – Torsion moment about the centroidal zz-axis. Defaults to 0.0.

• vx (float) – Shear force acting in the x-direction. Defaults to 0.0.

• vy (float) – Shear force acting in the y-direction. Defaults to 0.0.

• agg_func (Callable[[list[float]], float]) – A function that aggregates the stresses if the point is shared by several elements. If the point, pt, is shared by several elements (e.g. if it is a node or on an edge), the stresses are retrieved from each element and combined according to this function. Defaults to np.average.

Raises:

RuntimeError – If a warping analysis has not been carried out and a shear force or a torsion moment is supplied

Returns:

Resultant normal and shear stresses (sigma_zz, tau_xz, tau_yz) for each point pt. If a point is not in the section then None is returned for that element in the list.

Return type:

list[tuple[float, float, float] | None]