Combined stress (normal and shear)

Input

Normal stress in x

\sigma_x

Normal stress in y

\sigma_y

Shear stress

\tau_{xy}

Allowable tensile stress

f_t

Combined stress

\sigma

Combined stress check

f_{cs}

Notes

Based on Eq. (5.24) in 5.5 "Allowable stress where combined stresses occur" of the AIJ Standard for Allowable Stress Design of Steel Structures (2019).
The Standard states it as f_t² ≧ σ_x² + σ_y² − σ_xσ_y + 3τ_xy². This page takes the square root of the right-hand side and reports it as a combined stress, back in units of stress. Both sides are non-negative, so the inequality still reads the same way.
σ_x and σ_y are direct stresses at right angles to each other; τ_xy is the shear stress in the plane the two act in.
If direct stress acts in one direction only, enter 0 for σ_y. The commentary to the same Standard derives this relation with σ_y = 0.
Work out the allowable tensile stress separately and enter it here. f_t comes from 5.1(1) of the same Standard. For short-term loading, add 50 % (5.6 of the same Standard).
The check for axial force acting together with bending is on a separate page. That one follows Chapter 6 of the same Standard, which is a different route from the 5.5 used here.
Where the Standard writes "stress intensity", this page writes simply "stress".