Cosine interaction
While reviewing the changes introduced in the new ISO 19902:2020 standard, this one jumped at me:
tubular member strength formulae for combined axial and bending loading now of cosine interaction form instead of previously adopted linear interaction;
In ISO 19902:2020, the combined unity check for axial (tension  compression) + bending takes the following general expression:
$$ U_m = 1  \cos\left( \frac{\pi}{2} \frac{\gamma_{R,tc} \, \sigma_{tc}}{f_{tyc}} \right) + \frac{\gamma_{R,b} \sqrt{\sigma_{b,y}^2 + \sigma_{b,z}^2}}{f_b} $$
This form of unity check has existed since 1993 in API RP2A LRFD, 1st edition, and whose introduction into ISO 19902:2020 is briefly described in §A13.3.2 and §A13.3.3.^{1} This form makes its presence felt throughout §13 Strength of tubular members.^{2}
Previously, Um in ISO 19902:2007 was expressed as:
$$ U_m = \frac{\gamma_{R,tc} \, \sigma_{tc}}{f_{tyc}} + \frac{\gamma_{R,b} \sqrt{\sigma_{b,y}^2 + \sigma_{b,z}^2}}{f_b} $$
The reduction of Um in the first equation is notable, see Figure below. For example, if the axial unity check value (x) is, say, 0.2, then its contribution is reduced to 0.05 (\(= 1  \cos\left( \frac{\pi}{2} x\right)\)).^{3}
#!/usr/bin/env python3
# * coding: utf8 *
"""
cosplay.py  Effect of cosine interaction form on axial utilisation
component in the combined axial + bending utilisation expressions
2022 ckunte
"""
import numpy as np
import matplotlib.pyplot as plt
def cosinefunc(uc_t):
return list(
map(
lambda x: 1  np.cos(x * np.pi / 2.0), uc_t
)
)
def plot_tuc_under_cosinteraction(uc_t):
cfunc = cosinefunc(uc_t)
plt.xlabel('$\\frac{\\gamma_{R,tc}\\,\\sigma_{tc}}{f_{tyc}}$')
plt.ylabel('$1  \\cos\\left(\\frac{\\pi}{2}\\frac{\\gamma_{R,tc}\\,\\sigma_{tc}}{f_{tyc}}\\right)$')
plt.plot(uc_t, cosinefunc(uc_t))
x = np.arange(0.01, 1.0, 0.01)
plt.savefig("tuc_under_cosint.svg")
plt.close()
pass
if __name__ == '__main__':
# Increasing tension utilisation from 0.0 to 1.0
t1 = np.arange(0.01, 1.0, 0.01)
plot_tuc_under_cosinteraction(t1)
pass

I am still processing the explanation in §A13.3.2, and in the paper. ↩

This form, i.e., 1  cos(x) occurs in as many as eleven equations, viz., Eq. 13.31, 13.32, 13.34, 13.38, 13.318, 13.319, 13.321, 13.323, 13.47, 13.413, and 13.419 in ISO 19902:2020. Curiously, this is not applied to dented tubes in ยง13.7.3, whose combined UC expression(s) remains like before. ↩

Remember that cos() is in radians. ↩