Figure 81: Geometry of the box

The Box section (contributed by O. Bernhardi) is simulated using a 'parent' beam element of type B32R.

The outer cross sections are defined by $ a$ and $ b$, the wall thicknesses are $ t_1$, $ t_2$, $ t_3$ and $ t_4$ and are to be given by the user (Figure 81).

The cross-section integration is done using Simpson's method with 5 integration points for each of the four wall segments. Line integration is performed; therefore, the stress gradient through an individual wall is neglected. Each wall segment can be assigned its own wall thickness.

The integration in the beam's longitudinal direction $ \xi$ is done using the usual Gauss integration method with two stations; therefore, the element has a total of 32 integration points.

From the figure, we define, for example, the local coordinates of the first integration point

$\displaystyle \xi_1=-\frac{1}{\sqrt{3}}; \hspace{1cm} \eta_1=1-\frac{t_4}{b}; \hspace{1cm}\zeta_1=1-\frac{t_1}{a}$ (19)

The other three corner points are defined correspondingly. The remaining points are evenly distributed along the center lines of the wall segments. The length $ p$ and $ q$ of the line segments, as given w.r.t. the element intrinsic coordinates $ \eta$ and $ \zeta$, can now be calculated as

$\displaystyle p=2-\frac{t_1}{a} - \frac{t_3}{a}; \hspace{1cm}q=2-\frac{t_2}{b} - \frac{t_4}{b};$ (20)

An integral of a function $ f(\eta, \zeta$), over the area $ \Omega$ of the hollow cross section and evaluated w.r.t the natural coordinates $ \eta$, $ \zeta$, can be approximated by four line integrals, as long as the line segments $ \Gamma_1$, $ \Gamma_2$, $ \Gamma_3$ and $ \Gamma_4$ are narrow enough:

$\displaystyle \intop_{\Omega} f(\eta, \zeta) d\Omega \;$ $\displaystyle \approx$    
$\displaystyle \frac{2t_1}{a} \intop f\left(\eta(\Gamma_1), \zeta\right) d\Gamma_1 \;$ $\displaystyle +$ $\displaystyle \;
\frac{2t_2}{b} \intop f\left(\eta, \zeta(\Gamma_2)\right) d\Gamma_2 \;+ \;$  
$\displaystyle \frac{2t_3}{a} \intop f\left(\eta(\Gamma_3), \zeta\right) d\Gamma_3 \;$ $\displaystyle +$ $\displaystyle \;
\frac{2t_4}{b} \intop f\left(\eta),\zeta(\Gamma_4)\right) d\Gamma_4$ (21)

According to Simpson's rule, the integration points are spaced evenly along each segment. For the integration weights we get, for example, in case of the first wall segment

$\displaystyle w_k = \{1,4,2,4,1\}\frac{q}{12}$ (22)

Therefore, we get, for example, for corner Point 1

$\displaystyle w_1=\frac{1}{6} \frac{t_1}{a} q + \frac{1}{6} \frac{t_4}{b} p$ (23)

and for Point 2

$\displaystyle w_2=\frac{4}{6} \frac{t_1}{a} q$ (24)

The resulting element data (stresses and strains) are extrapolated from the eight corner integration points (points 1,5,9 and 13) from the two Gauss integration stations using the shape functions of the linear 8-node hexahedral element.