Torsion Constant of a Rectangle

Plot the variation of the torsion constant with aspect ratio of a rectangle section.

In this example, the aspect ratio of a rectangular section is varied whilst keeping a constant cross-sectional area and the torsion constant calculated. The variation of the torsion constant with the aspect ratio is then plotted.

# sphinx_gallery_thumbnail_number = 1

import numpy as np
import matplotlib.pyplot as plt
import sectionproperties.pre.library.primitive_sections as sections
from sectionproperties.analysis.section import Section

Rectangle dimensions

d_list = []
b_list = np.linspace(0.2, 1, 20)
j_list = []  # list holding torsion constant results

Number of elements for each analysis

n = 100

Loop through all the widths

for b in b_list:
    # calculate d assuming area = 1
    d = 1 / b
    d_list.append(d)

    # compute mesh size
    ms = d * b / n

    # perform a warping analysis on rectangle
    geometry = sections.rectangular_section(d=d, b=b)
    geometry.create_mesh(mesh_sizes=[ms])
    section = Section(geometry)
    section.calculate_geometric_properties()
    section.calculate_warping_properties()

    # get the torsion constant
    j = section.get_j()
    print("d/b = {0:.3f}; J = {1:.5e}".format(d / b, j))
    j_list.append(j)

d/b = 25.000; J = 1.30315e-02
d/b = 17.060; J = 1.88338e-02
d/b = 12.380; J = 2.55717e-02
d/b = 9.391; J = 3.31309e-02
d/b = 7.367; J = 4.14010e-02
d/b = 5.934; J = 5.02289e-02
d/b = 4.881; J = 5.95117e-02
d/b = 4.086; J = 6.90261e-02
d/b = 3.470; J = 7.86512e-02
d/b = 2.983; J = 8.81645e-02
d/b = 2.593; J = 9.73762e-02
d/b = 2.274; J = 1.06053e-01
d/b = 2.010; J = 1.14042e-01
d/b = 1.790; J = 1.21141e-01
d/b = 1.604; J = 1.27224e-01
d/b = 1.446; J = 1.32194e-01
d/b = 1.310; J = 1.35982e-01
d/b = 1.192; J = 1.38616e-01
d/b = 1.090; J = 1.40125e-01
d/b = 1.000; J = 1.40617e-01

Plot the torsion constant as a function of the aspect ratio

(fig, ax) = plt.subplots()
ax.plot(np.array(d_list) / b_list, j_list, "kx-")
ax.set_xlabel("Aspect Ratio [d/b]")
ax.set_ylabel("Torsion Constant [J]")
ax.set_title("Rectangular Section Torsion Constant")
plt.show()