Following a post in the Eng-Tips Forum about finding the buckling load of a stepped strut I have modified the Frame4 spreadsheet to carry out a buckling analysis of any straight member subject to axial load, including stepped or tapered cross sections. The spreadsheet, including full open source code, may be downloaded from: Frame4Buckle.zip
The buckling load is found using the method of Vianello, described in a paper from the University of Ljubljana (Note 20 Nov 14: The link to the University no longer works, but the iterative procedure used is described in Theory of Elastic Stability by Timoshenko and Gere). This is an iterative method using successive approximations of the deflected shape:
- Assume an approximation of the deflected shape. In the spreadsheet a linear approximation is used from the fixed point(s) to the point of maximum deflection, which is assumed to be at the top of the column if the top is unrestrained in the X direction, or at mid height if the top is restrained.
- Find the bending moments due to a unit axial load applied at the top in a column with this shape .
- Find the deflections of a column subject to this bending moment distribution.
- Normalise the deflections based on the maximum absolute deflection.
- Update the column shape using the normalised deflections, and repeat the analysis.
- Repeat until the difference in maximum deflection between successive iterations is negligible.
- The buckling load is the inverse of the maximum deflection from the last iteration (before normalisation).
The Frame4 spreadsheet has been modified to generate the required input data and carry out the analysis simply and quickly for a straight member with axial load applied at one end. The input required is:
- End conditions: either 1 (fixed) or 0 (free) for base rotation, top X deflection and top rotation. X and Y deflection at the base must be fixed, and Y deflection at the top must be free.
- Cross section details: for a member of constant cross section specify the cross section area, second moment of area, and E (Young’s Modulus), and member length and number of sections in the analysis. For a varying cross section enter the section details for every cross section, with the height to the top of the section.
- If the number of sections is entered in the constant cross section data the constant cross section is used. If this cell is blank or zero the varying cross section data table is used.
Typical input and results are shown in the screen shot below:
The buckling shape may be plotted to a chosen scaled deflection:
Calculated deflections and normalised deflections are listed on the “Output” sheet: