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, to automatically analyze the geometry of a whole structure and to assign an appropriate value of the buckling length to individual structure columns considering:
- member supports
- parameters of adjoining members
- supports on the other ends of adjoining members.
Both end nodes of a column are analyzed separately and their stiffness is calculated according to the code regulations. To use code formulas, you must know the stiffness of the analyzed column (known from the definition), stiffness values of transverse beams adjoining the node as well as the stiffness of the adjoining column. These last two stiffnesses, which will be referred to here as the 'beam' stiffness and the 'column' stiffness, are evaluated as follows:
- A member adjoining the node is analyzed together with its further connections (i.e. with the entire member chain): the stiffness is calculated for the entire member chain which can affect either the beam stiffness or the column stiffness of the node depending on the member chain direction
- The first member of the member chain determines its direction:
- column direction (the direction included in the range of ±15° from the direction determined by the original analyzed column).
- beam direction (the direction included in the range of ±15° from the direction perpendicular to the original analyzed column).
- intermediate direction (all members that are not included in the above-presented classification belong to the group of 'intermediate' members).
- The stiffness of an 'intermediate' member chain (equal to J/L) is substituted by the equivalent column stiffness J c (J/L c ) and beam stiffness J b (J/L b ) assuming for a fictitious column and beam the same moment of inertia J as for an inclined member chain, and the modified lengths L c = k*L*cosα, L b = k*L*sinα (k is a multiplier, whereas a is an angle between the column and the direction of the vector connecting the beginning and end of the member chain). From the condition J = J c + J b , we obtain 1/L = 1/L c + 1/L b and that allows calculating the multiplier k = (sin*cos)/(sin+cos).
- The end of a member chain is determined by:
- branching of several members (the node at which at least 3 members meet)
- support
- nodal release or element (hinge) release
- change of the direction by an angle greater by ±30° from the original one
- too great number of changes in the member stiffness (more than 10). A change in the stiffness of the order of 1.0e-12 is considered insignificant and is disregarded in calculations. The equivalent stiffness is calculated according to the formula (J1*L1+J2*L2)/(L1+L2).
- A member chain ending with an unsupported end is not considered in the stiffness calculations, similarly as a member chain starting with a pinned support (an element release at the beginning of the member chain).
- The support (ending) method of beam chains is included in the calculation (rotational release, fixed support, elastic fixed support).
- Effect of the longitudinal force on the stiffnesses is ignored; this is a purely geometrical analysis.
Column and beam stiffnesses (calculated as a ratio of the moment of inertia to the length) for individual member chains are added together, once all the members meeting at the column node have been analyzed, to determine the final beam stiffness and column stiffness of the node. These values are substituted in the relevant code formulas.
If a support occurs at the node, analysis of a member chain is not performed and the appropriate equivalent stiffness of the node is implied by the support scheme. If both nodes are supported, buckling length coefficients corresponding to those known from the theory of material strength are adopted.