 # Cosine interaction

While reviewing the changes introduced in the new ISO 19902:2020 standard, this one jumped at me:

In ISO 19902:2020, the combined unity check for axial (tension | compression) + bending takes the following general expression:

$$U_m = 1 - \cos\left( \frac{\pi}{2} \frac{\gamma_{R,t|c} \, \sigma_{t|c}}{f_{t|yc}} \right) + \frac{\gamma_{R,b} \sqrt{\sigma_{b,y}^2 + \sigma_{b,z}^2}}{f_b}$$

This form of unity check has existed since 1993 in API RP-2A 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:

$$U_m = \frac{\gamma_{R,t|c} \, \sigma_{t|c}}{f_{t|yc}} + \frac{\gamma_{R,b} \sqrt{\sigma_{b,y}^2 + \sigma_{b,z}^2}}{f_b}$$

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 Figure: Axial utilisation versus axial component under cosine interaction in the combined utilisation expression.
#!/usr/bin/env python3
# -*- coding: utf-8 -*-

"""
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('$\\frac{\\gamma_{R,t|c}\\,\\sigma_{t|c}}{f_{t|yc}}$')
plt.ylabel('$1 - \\cos\\left(\\frac{\\pi}{2}\\frac{\\gamma_{R,t|c}\\,\\sigma_{t|c}}{f_{t|yc}}\\right)$')
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


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

2. This form, i.e., 1 - cos(x) occurs in as many as eleven equations, viz., Eq. 13.3-1, 13.3-2, 13.3-4, 13.3-8, 13.3-18, 13.3-19, 13.3-21, 13.3-23, 13.4-7, 13.4-13, and 13.4-19 in ISO 19902:2020. Curiously, this is not applied to dented tubes in §13.7.3, whose combined UC expression(s) remains like before.

3. Remember that cos() is in radians.