Cosine interaction

eng Its effect in the combined axial + bending utilisation checks.

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:

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

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 (\(= 1 - \cos\left( \frac{\pi}{2} x\right)\)). Remember cos() is in radians.

#!/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 1 - np.cos(uc_t * np.pi / 2.0)


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, cfunc)
    plt.savefig("tuc_under_cosint.svg")
    plt.show()


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)