An arbitrary section can only be checked for strength according to the formulas given in the strength of materials manual. However, when you have to consider the elastoplastic stage of work, check for the web buckling, check for the out-of-plane buckling, or to make some other checks according to design codes, it appears that all the codes are oriented only toward certain types of cross-sections. Engineers usually use the following approach – the strength is checked for a real cross-section and all the other checks are carried out for a "similar" section, the geometrical properties of which are selected according to the consideration of equivalence.
The equivalence is understood as the proximity of the cross-sectional geometrical properties (an area, moments of inertia, moments of resistance, etc.). Sometimes in the process of reduction some additional considerations are used which can help to specify the very concept of equivalence. For example, only the equality of the moments of inertia has to be achieved, if it is only the buckling that has to be checked.
The Sezam program is intended for finding such a section (in this version only a hollow section, a channel, a Tee section, or an I-beam), which approximates an arbitrary section set by the user according to its geometrical properties the best. An initial section can be set:
At any method of a section setting only geometrical properties are used for the calculation in the program. The following properties are approximated for a section:
Apart from the parameters mentioned above, it is necessary to set weight coefficients for each of the properties (all the weights are equal to 1 by default).
The task is to select geometrical dimensions of a hollow section, a channel, a Tee section, or an I-beam at which the functional is minimized.
\[ \begin{array}{l} k_{1} \left( {1-\frac{A}{A^{0}}} \right)^{2}+k_{2} \left( {1-\frac{I_{u} }{I_{u}^{0} }} \right)^{2}+k_{3} \left( {1-\frac{I_{v} }{I_{v}^{0} }} \right)^{2}+k_{4} \left( {1-\frac{W_{u+} }{W_{u+}^{0} }} \right)^{2}+k_{5} \left( {1-\frac{W_{u-} }{W_{u-}^{0} }} \right)^{2}+ \\ +k_{6} \left( {1-\frac{W_{v+} }{W_{v+}^{0} }} \right)^{2}+k_{7} \left( {1-\frac{W_{v-} }{W_{v-}^{0} }} \right)^{2}\quad , \\ \end{array} \]
where A0, Iu0, Iv0, W0u+, W0u-, W0v+, W0v- are respective geometrical properties of the equivalent section (a hollow section, a channel, a Tee section, or an I-beam).
Coefficients ki (i=1…7) indicate the importance assigned to each geometrical property; in particular, by setting one of the coefficients equal to zero it is possible to reject the approximation of a corresponding geometrical property.