Advanced Package Training Scaffolding 2011.1 - Scia-Software GbR

11. Stability
Linear Stability
During a linear stability calculation, the following assumptions are used:
- Physical Linearity.
- The elements are taken as ideally straight and have no imperfections.
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- The loads are guided to the mesh nodes, it is thus mandatory to refine the finite element mesh in
order to obtain precise results.
- The loading is static.
- The critical load coefficient is, per mode, the same for the entire structure.
- Between the mesh nodes, the axial forces and moments are taken as constant.
The equilibrium equation can be written as follows:
[ − K ] ⋅u
= F
KE G
The symbol u depicts the displacements and F is the force matrix.
As specified in the theory of the Timoshenko method, the stiffness K is divided in the elastic stiffness
KE and the geometrical stiffness KG. The geometrical stiffness reflects the effect of axial forces in
beams and slabs.
The basic assumption is that the elements of the matrix KG are linear functions of the axial forces in the
members. This means that the matrix KG corresponding to a λ th multiple of axial forces in the structure
is the λ th multiple of the original matrix KG.
The aim of the buckling calculation is to find such a multiple λ for which the structure loses stability.
Such a state happens when the following equation has a non-zero solution:
[ − ⋅ K ] ⋅u
= 0
K G
E λ
In other words, such a value for λ should be found for which the determinant of the total stiffness matrix
is equal to zero:
KE − λ ⋅KG
= 0
Similar to the natural vibration analysis, the subspace iteration method is used to solve this eigenmode
problem. As for a dynamic analysis, the result is a series of critical load coefficients λ with
corresponding eigenmodes.
To perform a Stability calculation, the functionality Stability must be activated.