Closed-form solution for the buckling analysis of Euler-Bernoulli columns and beam-columns with jump discontinuities

Abstract

Structural elements under compression can fail due to instability. Such type of failure may present itself as flexural buckling, which is the sudden change of configuration of one state to another at a critical compressive load. Actual structure characteristics, such as cracks, real joints, sudden changes of flexural stiffness, bracing, and actual boundary conditions, make current analytical buckling methods inefficient. For this reason, most structures are evaluated using Finite Element (FE) software based on the Finite Element Method (FEM). However, such a numerical method may have a high computational cost when parametric studies need to be performed. The motivation for this thesis was to develop an analytical alternative for the buckling analysis of columns and beam-columns.
The thesis analyses jointed Euler-Bernoulli columns and beam-columns with N discontinuities due to step-changes of flexural stiffness, and influence of open edge cracks, real joints, bracing, and actual support conditions. The approach considered utilizes the Heaviside function to obtain a single piecewise expression for the deflection, slope, moment, and shear of columns and beam-columns. The use of the Heaviside function, along with proposed closed-form expressions, reduce the number of unknown integration constants to only four.
Main findings are listed as following: (i) Linear buckling analysis of a column only depends on solving the determinant of the 4x4 matrix of unknown coefficients, obtained by imposing four boundary conditions. (ii) Solving the four unknown constants of integration results in performing a geometrical non-linear static analysis of the beam-column, which perfectly agrees with results obtained through FE software. (iii) Closed-form expressions reduce the number of unknown constants of integrations to only four, regardless of the number of sections composing a column or beam-column.
Concluding remarks and recommendations include: (i) Closed-form expressions accurately provide linear buckling loads for columns and geometrical non-linear expressions for beam-columns. (ii) Equations solved algebraically can be stored and used for future analysis, saving the computation time of rerunning the analysis for specific columns or beam-columns. Having the equations expressed algebraically allows for the ease of performing parametric analysis and exploring the performance of new designs. (iii) Computational cost (time needed for the computer to obtain the results) may depend on the mathematical software used for the calculations. Other software may result in having lower computational time for the same calculations.