DT EC 03: Buckling in and out of the plane

Description

Geometry	Cross-section:	70x221mm
Geometry	Material:	C24
Load	In ULS FC:	P = 15kN q = 4.69kN/m
Standard		EN 1995-1-1 [- -]
Design parameters	k_mod: L_y and L_z:	0.8 3m

Independent reference results

Open handcalculations

$i_y=\sqrt{\frac{I_y}{A}}=\sqrt{\frac{62964189 mm^4}{15470 mm^2}}=63.8mm$

$i_z=\sqrt{\frac{I_z}{A}}=\sqrt{\frac{6316917 mm^4}{15470 mm^2}}=20.2mm$

$\lambda_y=\frac{l_{cr.y}}{i_y}=\frac{3m}{63.8mm}=47.02$

$\lambda_z=\frac{l_{cr.z}}{i_z}=\frac{3m}{20.2mm}=148.46$

$\[\]$

(6.21) $\begin{align*} \overline{\lambda_y}&=\frac{\lambda_y}{\pi}\sqrt{\frac{f_{c.0.k}}{E_{0.05}}} \\ &=\frac{47.02}{\pi}\sqrt{\frac{21 \text{MPa}}{7370 \text{MPa}}}\\ &=0.8 \end{align*}$

(6.29) $\begin{align*} \beta_c &=0.2 \\ \end{align*}$

(6.27) $\begin{align*} k_y&= 0.5 \cdot \left( 1+ \beta_c \cdot \left( \overline{\lambda_y} -0.3 \right) + \overline{\lambda_y}^2\right) \\ &=0.5 \cdot \left( 1+ 0.2 \cdot \left( 0.8 -0.3 \right) + 0.8^2\right) \\ &= 0.87\\ \end{align*}$

(6.28) $\begin{align*} k_z&= 0.5 \cdot \left( 1+ \beta_c \cdot \left( \overline{\lambda_z} -0.3 \right) + \overline{\lambda_z}^2\right) \\ &=0.5 \cdot \left( 1 + 0.2 \cdot \left( 2.52 -0.3 \right) + 2.52 ^2\right) \\ &=3.9 \end{align*}$

(6.25) $\begin{align*} k_{c.y}&= \frac{1}{k_y+\sqrt{k_y^2- \overline{\lambda_y^2}}} \\ &= \frac{1}{0.87+\sqrt{0.87^2- 0.8^2}} \\ &= 0.826 \end{align*}$

(6.26) $\begin{align*} k_{c.z}&= \frac{1}{k_z+\sqrt{k_z^2- \overline{\lambda_z^2}}} \\ &= \frac{1}{3.9+\sqrt{3.9^2- 2.52^2}} \\ &= 0.145 \end{align*}$

From EN 1995-1-1 §6.1.6 (2)

$k_m=0.7$

Because $\lambda_z$ > $0.3$ (EN 1995-1-1 §6.3.2) the unity check is performed using equations 6.23 and 6.24.

(6.23) $\begin{align*} N_{c.Rd}&=A \cdot f_{c.0.d}\\ &=15470 \text{mm²} \cdot \frac{k_{mod} \cdot f_{c.0.d}}{\gamma_m}\\ &=569812 \text{mm³} \cdot \frac{0.8 \cdot 21 \text{MPa}}{1.3}\\ &=199.9 \text{kN}\\ M_{y.Rd}&=W_y \cdot f_{m.d}\\ &=W_y \cdot \frac{k_{mod} \cdot f_{m.k}}{\gamma_m}\\ &=569812 \text{mm³} \cdot \frac{0.8 \cdot 24 \text{MPa}}{1.3}\\ &=8.4 \text{kNm}\\ UC_1&= \frac{N_{c.Ed}}{k_{c.y} \cdot N_{c.Rd}}+\frac{M_{y.Ed}}{M_{y.Rd}} + k_m \frac{M_{z.Ed}}{M_{z.Rd}} \\ &= \frac{15 \text{kN}}{0.826 \cdot 199.9 \text{kN}}+\frac{5.37 \text{kNm}}{8.4 \text{kNm}} + 0 \\ &= 72.9 \% \\ UC_2&= \frac{N_{c.Ed}}{k_{c.z} \cdot N_{c.Rd}}+k_m \frac{M_{y.Ed}}{M_{y.Rd}} + \frac{M_{z.Ed}}{M_{z.Rd}} \tag{6.24} \\ &= \frac{15 \text{kN}}{0.145 \cdot 199.9 \text{kN}}+ 0.7 \frac{5.37 \text{kNm}}{8.4 \text{kNm}} + 0 \\ &= 96.3 \% \\ UC&= max\left( UC_1;UC_2 \right) = 96.3 \% \end{align*}$

Diamonds results and comparison

Stability verification of the beam according to EN 1995-1-1

Intermediate results in Diamonds for buckling and lateral torsional buckling

Results	Independent reference	Diamonds	Difference
$\lambda_y$	47.02	47.02	0,00%
$\lambda_z$	148.46	148.46	0,00%
$\overline{\lambda_y}$	0.80	0.80	0,00%
$\overline{\lambda_z}$	2.52	2.52	0,00%
$k_y$	0.87	0.87	0,00%
$k_z$	3.90	3.90	0,00%
$k_{c.y}$	0.826	0.826	0,00%
$k_{c.z}$	0.145	0.145	0,00%
Unity check	96.3%	95.5%	-0,83%