Determination of Critical Loads by Successive Approximation
Introduction
The method of determining critical buckling loads by successive approximation is to derive two approximate deflected shapes of the beam based upon the expected buckled shape and the conjugate-beam or double-integration methods. The process is then iterated until a satisfactory solution is reached.
The following example problem was solved symbolically in Section 2.15 of Theory of Elastic Stability by Timoshenko (1961).
Problem Description
A beam with three segments which is geometrically symmetrical about its midpoint is loaded axially. Segment 1 and 3 have a moment of inertia of I1 while the middle segment has a moment of inertia of I2 .The ratio of I1 / I2 is 0.4. The deflected shape is assumed initially to be sinusoidal and the conjugate beam method was then used to successively approximate the deflected shape until the solution converged. In this example, we will add values to check the validity of SACS critical buckling load calculation against the theoretical symbolic values that were calculated.
Input Parameters
Geometric Properties:
Material Properties:
Load Properties:
Section Properties:
Comparison:
Calculated Values:
SACS Values:
Percent Error:
Conclusion
The SACS solution very closely matches the theoretical solution by Timoshenko. The maximum error of 0.1% in this example is acceptable for an iterative solution. This error could be due to a few factors like round-off error due to the use of large numbers like E and I in the calculations, different values used for conversion of units, different number of sections used to calculate the average deflected shape of the beam as well as different convergence tolerances.