Analysis#

Section Object#

The first step in running a cross-section analysis in sectionproperties involves the creation of a Section object. This object stores the cross-section geometry and finite element mesh, providing methods to perform various types of cross-sectional analyses.

Section.__init__(geometry: Geometry | CompoundGeometry, time_info: bool = False) → None[source]

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

Raises:

  • ValueError – If geometry does not contain a mesh

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

Checking Mesh Quality#

Before carrying out a section analysis it is a good idea to check the quality of the finite element mesh. Some useful methods are provided to display mesh statistics and to plot the finite element mesh:

Section.display_mesh_info() → None[source]

Prints mesh statistics to the command line.

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

Plots the finite element mesh.

Parameters:

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

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

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

  • title (str) – Plot title

  • 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)

../_images/analysis-1.png

Composite section mesh#

Geometric Analysis#

A geometric analysis calculates the area properties of the section.

Section.calculate_geometric_properties() → None[source]

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

Plastic Analysis#

A plastic analysis calculates the plastic properties of the section.

Warning

The plastic analysis in sectionproperties assumes all materials are able to reach their yield stress defined in the material properties. Care should be taken if analysing materials or cross-sections exhibiting non-linear behaviour, e.g. reinforced concrete or non-compact steel sections.

Section.calculate_plastic_properties(verbose: bool = False) → None[source]

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.

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 not calculated.

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, non-composite only (centroidal and principal axe)

Warping Analysis#

A warping analysis calculates the torsion and shear properties of the section.

Warning

There must be connectivity between all elements of the mesh to perform a valid warping analysis. This is a limitation of the elastic theory that this implementation is based on, as there is no way to quantify the transfer of shear and warping between two unconnected regions.

Section.calculate_warping_properties(solver_type: str = 'direct') → None[source]

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

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

Frame Analysis#

Calculates the section properties required for a 2D or 3D frame analysis.

Note

This method is significantly faster than performing a geometric and a warping analysis and has no prerequisites.

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

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

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

Stress Analysis#

A stress analysis calculates the section stresses arising from a set of forces and moments. Executing this method returns a StressPost object, which stores the section stresses and provides stress plotting methods.

Warning

The stress analysis in sectionproperties is linear-elastic and does not account for the yielding of materials or any non-linearities.

Section.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[source]

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

  • vx (float) – Shear force acting in the x-direction

  • vy (float) – Shear force acting in the y-direction

  • mxx (float) – Bending moment about the centroidal xx-axis

  • myy (float) – Bending moment about the centroidal yy-axis

  • m11 (float) – Bending moment about the centroidal 11-axis

  • m22 (float) – Bending moment about the centroidal 22-axis

  • mzz (float) – Torsion moment about the centroidal zz-axis

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.