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G. Buckling Analysis - Iterative Method

In STAAD.Pro, a simple procedure has been adopted to incorporate the calculation of the Buckling Factor for any number of primary load cases. The buckling factor is the amount by which all of the loadings in a load case must be factored to cause global buckling of the structure. 

  1. First, the primary deflections are calculated by linear static analysis based on the provided external loading.
  2. Primary deflections are used to calculate member axial forces and plate center membrane stresses. These forces and stresses are used to calculate geometric stiffness terms. Both the large delta effects and the small delta effects are calculated. These terms are the terms of the Kg matrix which are multiplied by the estimated BF (buckling factor) and then added to the global stiffness matrix K.   

    Buckling Kg matrix effects are calculated for frame members and plate elements only. They are not calculated for solid elements. So buckling analysis is restricted to structures where members and plate elements carry the vertical load from one structure level to the next.

  3. For compressive cases, the Kg matrix is negative definite. If the buckling factor is large enough, then [ [K]+BF×[Kg] ] will also be negative definite which indicates that BF times the applied loads is greater than the loading necessary to cause buckling. 
  4. STAAD.Pro starts an iterative procedure with a BF estimate of 1.0. If that BF causes buckling, then a new, lower BF estimate is used in the next trial. If the BF does not cause buckling, then a higher BF estimate is used. On the first iteration, if the determinant of the K matrix is positive and lower than the determinant of the K+Kg matrix, then the loads are in the wrong direction to cause buckling; and STAAD.Pro will stop the buckling calculation for that case.   
  5. After a few iterations, STAAD.Pro will have the largest BF that did not cause buckling (lower bound) and the lowest BF that did cause buckling (upper bound). Then each trial will use a BF estimate that is halfway between the current upper and lower bounds for BF. 
  6. After the default iteration limit is reached or the user specified iteration limit, MAXSTEPS, is reached or when two consecutive BF estimates are within 0.1% of each other; then the iteration is terminated.
  7. Results for this load case are based on the last lower bound BF calculated.  
    • Only primary load cases may be solved
    • Any number of buckling cases may be solved.
    • Only the first buckling mode (lowest BF) is calculated.
    • The buckling shape may not be as expected even though the buckling factor is OK. To enhance the mode shape result, apply small loads in the locations and directions where you expect the large displacements. 
Note: During the buckling analysis using the iterative method, if the determinant of the matrix changes sign in one or more steps, then the resulting values of displacements and forces for the last successful step may not be accurate. If using STAAD.Pro Advanced, then it is recommended to check the buckling factor using the eigen method.