D.5.B.14.3 Bending, Shear, and Axial Force Interaction

In addition to the 3 parts of the Cl: 6.2.10 of EC3, the Dutch National Annex specifies:

Interaction for Class 1 and 2 Sections

For Class 1 and 2 I section the interaction between bending, shear force, and axial force when bending about strong axis can be checked using:

\frac{M_{y, Ed}}{M_{y, V, Rd}} + \frac{\frac{N_{Ed}}{N_{Vz, Rd}} - \frac{a_{2}}{2}}{1 - \frac{a_{2}}{2}} \leq 1.0

where

$N_{Vz, Rd}$	=	$\frac{N_{pl, Rd} - (ρ A_{v} f_{y})}{γ_{M0}}$
$a_{2}$	=	$a_{1} (1 - ρ)$
$a_{1}$	=	${}_{min}{\| \begin{matrix} \frac{A - 2 b_{f} t_{f}}{A} \\ 0.5 \end{matrix}}$
$M_{y, Ed}$	=	the design bending moment about the major axis
$M_{y, V, Rd}$	=	the moment capacity calculated per Eqn. 6.30 of EN1993-1-1.
$N_{Ed}$	=	the design axial force
$N_{pl, Rd}$	=	the axial capacity (tension or compression) calculated per Eqn. 6.10 of EN1993-1-1.
$ρ$	=	the shear factor calculated per Cl. 6.2.8(3) of EN1993-1-1.
$A_{v}$	=	the shear area parallel to the web calculated per Cl. 6.2.6(3) of EN1993-1-1.
$b_{f}$	=	the width of the flange
$t_{f}$	=	the thickness of the flange
$A$	=	the cross-sectional area
$f_{y}$	=	yield strength

For Class 1 and 2 square or rectangular tube sections the interaction between bending, shear force, and axial force when bending about strong axis can be checked using:

\frac{M_{y, Ed}}{M_{y, V, Rd}} + \frac{\frac{N_{Ed}}{N_{Vz, Rd}} - \frac{a_{4z}}{2}}{1 - \frac{a_{4z}}{2}} \leq 1.0

where

$N_{Vz, Rd}$	=	$\frac{N_{pl, Rd} - [2 (1 - q_{z}) t \times h \times f_{y}]}{γ_{M0}}$
$a_{4z}$	=	$q_{z} a_{3}$
$a_{3}$	=	${}_{min}{\| \begin{matrix} \frac{A - 2 b t}{A} \\ 0.5 \end{matrix}}$
$M_{y, V, Rd}$	=	the moment capacity calculated per Eqn. NB 29 of the Dutch NA 2016.
$q_{z}$	=	factor calculated per Cl. 6.2.8(8) of EN1993-1-1.
$b$	=	the width of the member
$t$	=	the thickness of the member

For Class 1 and 2 I section the interaction between bending, shear force, and axial force when bending about weak axis can be checked using:

\frac{M_{z, Ed}}{M_{z, V, Rd}} + {(\frac{\frac{N_{Ed}}{N_{Vy, Rd}} - a_{1}}{1 - a_{1}})}^{2} \leq 1.0

where

$M_{z, V, Rd}$	=	the moment capacity calculated as: $= \frac{q_{y} \times W_{Pl, z, Rd}}{γ_{M0}}$
$N_{Vy, Rd}$	=	$\frac{N_{pl, Rd} - [2 (1 - q_{y}) b_{f} \times t_{f} \times f_{y}]}{γ_{M0}}$
$M_{z, Ed}$	=	the design bending moment about the minor axis
$q_{y}$	=	factor calculated per Cl. 6.2.8(7) of EN1993-1-1.
$A_{v}$	=	the shear area parallel to the flange calculated per Cl. 6.2.6(3) of EN1993-1-1.

For Class 1 and 2 square or rectangular tube sections the interaction between bending, shear force, and axial force when bending about weak axis can be checked using:

\frac{M_{z, Ed}}{M_{z, V, Rd}} + \frac{\frac{N_{Ed}}{N_{Vy, Rd}} - \frac{a_{4y}}{2}}{1 - \frac{a_{4y}}{2}} \leq 1.0

where

$N_{Vy, Rd}$	=	$\frac{N_{pl, Rd} - [2 (1 - q_{y}) t \times b \times f_{y}]}{γ_{M0}}$
$a_{4y}$	=	$q_{y} a_{3}$

For Class 1 and 2 round hollow section sections the interaction between bending, shear force, and axial force can be checked using:

\frac{M_{Ed}}{1.04 \times M_{V, Rd}} + {(\frac{N_{Ed}}{N_{V, Rd}})}^{1.7} \leq 1.0

\frac{M_{Ed}}{M_{V, Rd}} \leq 1.0

where

$M_{V, Rd}$	=	$\frac{q \times M_{pl, Rd}}{γ_{M0}}$
$N_{V, Rd}$	=	$\frac{q \times N_{pl, Rd}}{γ_{M0}}$

Biaxial Bending

Biaxial bending with axial and shear force can be determined using:

β_{0} \times {(\frac{M_{y, Ed}}{M_{y, N, V, Rd}})}^{α_{1}} + β_{1} \times {(\frac{M_{z, Ed}}{M_{z, N, V, Rd}})}^{α_{2}} \leq 1.0

Class 1 and 2 I section using:where

$M_{y, N, V, Rd}$	=	$M_{y, V, Rd} \times \frac{1 - \frac{N_{Ed}}{N_{Vz, Rd}}}{1 - \frac{a_{2}}{2}}$
$M_{z, N, V, Rd}$	=	$M_{z, V, Rd} [1 - {(\frac{\frac{N_{Ed}}{N_{Vy, Rd}} - a_{1}}{1 - a_{1}})}^{2}]$
$α_{1} = α_{2}$	=	$\begin{matrix} 1.6 - \| \frac{\frac{N_{Ed}}{N_{c, Rd}}}{2 ln (\frac{N_{Ed}}{N_{c, Rd}})} \| & when & b > 0.3 h \\ 1.0 & when & b < 0.3 h \end{matrix}$
$β_{0} = β_{1}$	=	1.0

Class 1 and 2 round tubing using:where

$M_{y, N, V, Rd}$	=	$1.04 M_{V, Rd} [1 - {(\frac{N_{Ed}}{N_{V, Rd}})}^{1.7}]$
$M_{z, N, V, Rd}$	=	$1.04 M_{V, Rd} [1 - {(\frac{N_{Ed}}{N_{V, Rd}})}^{1.7}]$
$α_{1} = α_{2}$	=	2.0
$β_{0} = β_{1}$	=	1.0

Class 1 and 2 square and rectangular tube NB.62:where

$M_{y, N, V, Rd}$	=	$M_{y, V, Rd} + \frac{1 - \frac{N_{Ed}}{N_{Vz, Rd}}}{1 - \frac{a_{4z}}{2}}$
$M_{z, N, V, Rd}$	=	$M_{z, V, Rd} + \frac{1 - \frac{N_{Ed}}{N_{Vy, Rd}}}{1 - \frac{a_{4y}}{2}}$
$α_{1}$	=	$\begin{matrix} 1.0 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} \leq \frac{2}{3} \\ 2.0 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} > \frac{2}{3} \end{matrix}$
$α_{2}$	=	$\begin{matrix} 2.0 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} \leq \frac{2}{3} \\ 1.0 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} > \frac{2}{3} \end{matrix}$
$β_{0}$	=	$\begin{matrix} 1.0 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} \leq \frac{2}{3} \\ 0.75 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} > \frac{2}{3} \end{matrix}$
$β_{1}$	=	$\begin{matrix} 0.75 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} \leq \frac{2}{3} \\ 1.0 & when & \frac{M_{Ed}}{M_{y, N, V, Rd}} > \frac{2}{3} \end{matrix}$

Note: The Dutch NA text compares the ratio of

\frac{M_{Ed}}{N_{N, V, Rd}}

\frac{2}{3}

. However, this appears to be an obvious printing error in the with respect to the ratios of moments. Thus the program uses the ratio of

\frac{M_{Ed}}{M_{y, N, V, Rd}}

Bending Stability for Class 4 Sections

when $\frac{M_{y, Ed}}{M_{y, N, f, Rd}} \leq 1.0$ the conditions of Cl. 6.2.6(6) must be met.
when $\frac{M_{y, Ed}}{M_{y, N, f, Rd}} > 1.0$ and $\frac{M_{y, Ed}}{M_{y, N, Rd}} \leq 1.0$ then the following condition must be met:
$\frac{M_{y, Ed}}{M_{y, f, N, Rd} + {(M_{y, N, Rd} - M_{y, f, N, Rd}) [1 - {(\frac{2 V_{z, Ed}}{V_{z, Rd}} - 1)}^{2}]}} \leq 1.0$

Additionally, the program check if Cl. 6.2.6(6) is met. This states that in addition to the shear buckling resistance for webs without intermediate stiffeners should satisfy Section 5:

\frac{h_{w}}{t_{w}} > 72 \frac{ε}{η}

where

$η$	=	may conservatively be taken = 1
$M_{y, f, N, Rd}$	=	the resistance to bending about the y-axis; calculated per Cl. 6.2.9(5)
$M_{y, N, Rd}$	=	the resistance to bending about the y-axis, taking into account the presence of the normal force and based on the effective cross section; calculated per Cl. 6.2.9(5)

For tube sections, both strong and weak axis should be considered.