Computation of w , S_{w} and I_{w} for a double L-shaped section.
Question : Computation of ω, Sω and Iω for a double L-shaped section.
Let us consider the thin-walled section shown in the next page:
Distribution of w_{s}\left ( s \right )
DA segment: 0\leq s\leq b ; -b\leq y\leq 0 ; z=-\frac{h}{2}
d_{s}=d_{y} , c_{n}=-\frac{h}{2} , w_{s}\left ( s \right )=\int_{0}^{s}c_{n}d_{s}=-\frac{h}{2}s
AB segment: b\leq s\leq b+h ; y=0 ; -\frac{h}{2}\leq z\leq \frac{h}{2}
d_{s}= d_{z} , c_{n}=0 , w_{s}\left ( s=b \right )=-\frac{h}{2}b
BF segment: b+h\leq s\leq b+2h ; 0\leq y\leq b ; z=\frac{h}{2}
d_{s}=d_{y} , c_{n}=\frac{h}{2} , w_{s}\left ( s \right )=-\frac{h}{2}b+\int_{b+h}^{s}\frac{h}{2}d_{s}=-\frac{h}{2}b+\frac{h}{2}\left ( s-b-h \right )
Mean value: w_{m}=-w_{D}
w_{m}=\frac{1}{2b+h}\int_{0}^{2b+h} w_{s}d_{s}=-\frac{bh\left ( b+h \right )}{2L_{s}}
Sectorial area w and S_{w}
w\left ( s,\zeta \right )=g\left ( s \right )-c_{t}\left ( s \right )\zeta with g\left ( s \right ) w_{s}+w_{D}
Neglecting the thickness variation, w\left ( s \right )=g\left ( s \right ) and S_{w}\left ( s,0 \right )=\int_{0}^{s}g\left ( s \right )ds.
Sectorial inertia modulus: I_{w}=\int_{A}^{}w^{2}dA=\frac{th^{2}b^{3}}{12}\left ( \frac{b+2h}{2b+h} \right )
The contribution of the term -c_{t}\zeta of w\left ( s,\zeta \right ) in I_{w} is negligible. Its value is I_{w}=\frac{t^{3}}{12}\left ( \frac{2b^{3}}{3}+\frac{h^{3}}{12} \right ).
The figures below shows the distribution of w_{s}\left ( s \right ) , g\left ( s \right ) and S_{w}\left ( s \right ) in the double L-shaped section considered.
