# Determination critical load coefficient

The critical load coefficient is determined by solving an eigenvalue problem via the following equation:

With:

- K
_{E}= Elastic stiffness matrix EI - K
_{G}= Geometrical stiffness matrix of the normal forces T - Lambda = eigenvalue = critical load coefficient

The elastic stiffness matrix is known because it represents the stiffness of the structure elements (columns, beams,…).

The geometrical stiffness matrix is a correction matrix to get a more realistic stiffness of the structure. I will explain the purpose of the geometrical stiffness matrix with a simple example. Consider a beam on two supports which is loaded as follows:

The beam will deform according to the green line if it is only loaded by the vertical load F_{V}. The value of the deformation is related to the elastic stiffness (Stiffness matrix K_{E}) of that element.

This beam will deform less (blue line) when there is besides the vertical load F_{V} a horizontal load F_{H}. This horizontal load F_{H} has a stiffening effect on the vertical deformation of the beam. This stiffening effect is taken into account in the geometrical stiffness matrix K_{G}. K_{G} is in fact a correction on K_{E} in order to get realistic results of the structure. SCIA Engineer searches during the stability calculation for a lambda value that fulfills the following equation

K_{E} = Lambda x K_{G}.

This is done by solving the stiffness matrices of the above equation. Below you can find an example of such stiffness matrices (K_{E} & K_{G}).

__Extra information:__

Once the critical load coefficient is determined via the stability calculation the user can determine the critical normal force N_{Cr}. that causes instability (=buckling) of the structure. Because N_{Cr}. = alpha_critical x N_{ed} (Formula 5.1 of EN1993-1-1; Lambda = alpha_critical).

The user can also determine the buckling factors k_{y} & k_{z} when he knows N_{Cr}. Because the buckling factors are the only unknowns in the formula of Euler:

With:

- L=bucklinglength = k
_{y}x systemlength yy or k_{z}x systemlenght zz