D9.C.10 Von Mises Stresses Check

The von Mises stress equation shown below, which is modified for beam elements based on the corresponding equation in AIJ steel design code (both 2002 and 2005 editions of AIJ), indicates that the left-hand side in the equation should be less than unity. These checks are performed at locations indicated by the BEAM parameter.

Note: As with other design checks, the unity check value can be modified by use of the RATIO parameter.

The von Mises stresses are evaluated and checked per AIJ clause 5.16 as follows:

\frac{\sqrt{σ_{x}^{2} + 3 τ_{x y}^{2}}}{k \times f_{t}} < 1.0

where

σ_x	=	Longitudinal stress in beam element. The following equation is used when the `MISES` parameter is set to 1 or 2. This is performed multiple times, once in each corner with the appropriate sign of the moment and value of elastic modulus. The largest stress is then used. $= \frac{F_{x}}{A_{x}} + \frac{M_{y}}{Z_{y}} + \frac{M_{z}}{Z_{z}}$ When the `MISES` parameter is set to 3 or 4, then the longitudinal stress is calculated once using the smalles elastic modulus for each axis as follows. $= \| \frac{F_{x}}{A_{x}} \| + \| \frac{M_{y}}{Z_{y}} \| + \| \frac{M_{z}}{Z_{z}} \|$
F_x	=	axial force
M_y	=	bending moment about y-axis
M_z	=	bending moment about z-axis
A_x	=	cross-sectional area
Z_y	=	section modulus about y-axis
Z_z	=	section modulus about z-axis
τ_xy	=	shear stress in the beam. When the `MISES` parameter is set to 1 or 3, this includes torsion stresses: $= \| \frac{M_{x}}{Z_{x}} \| + \sqrt{{\| \frac{F_{y}}{A_{y}} \|}^{2} + {\| \frac{F_{z}}{A_{z}} \|}^{2}}$ When the `MISES` parameter is set to 2 or 4, the torsion stresses are excluded: $= \sqrt{{\| \frac{F_{y}}{A_{y}} \|}^{2} + {\| \frac{F_{z}}{A_{z}} \|}^{2}}$
M_x	=	torsional moment.
F_y	=	shear stress in the y direction
F_z	=	shear stress in the z direction
Z_x	=	torsional section modulus
D_x	=	depth of the member
I_x	=	torsional constant
A_y	=	effective shear area in the y direction
A_z	=	effective shear area in the z direction
f_t	=	allowable tensile stress
k	=	loading duration factor as specified by the `TMP` parameter: 1.0 for permanent 1.5 for temporary

In the STRESSES output category, stress value of (numerator of the von Mises stress equation) is output as the value of fm. Along with slenderness ratios, stresses, and deflections, von Mises stress equation is checked. When its left-hand side yields the maximum ratio value, it is printed as RATIO and "VON MISES" is printed as CRITICAL COND.