Probabilistic buckling analysis of beam-column elements with geometric imperfections and various boundary conditions

Abstract

Although the manufacturing process of structural members has been improved over the last decades, inherent geometric imperfections in slender beam-columns cannot be avoided altogether. Such imperfections largely affect the buckling behavior of the beam by modifying (reducing) the actual critical load in comparison with the theoretically perfect beam idealization. Whereas the deterministic characterization of such imperfections is a rather difficult task that should be performed from case to case, a more general approach would require dealing with imperfections within a probabilistic framework, i.e., treating them as random fields. The aim of this paper is to provide a simple but effective method to characterize probabilistically the response of imperfect beam-column elements based on the stochastic description of the imperfections. The proposed method assumes that the imperfections are expressed through series expansion wherein the critical shapes of the corresponding perfect beam represent the series polynomials. Once the continuum deflection field and the random imperfection field are discretized, some simple closed-form relationships between the imperfection parameters and the main response quantities (e.g. beam deflection and critical load) are established. In contrast to other methods, any type of boundary conditions can be incorporated without computational effort. A few boundary conditions are numerically investigated that clarify how the proposed method can be usefully employed to characterize the buckling response of imperfect beam-column elements probabilistically.

Appendix

The elements of the ${\mathbf{K}}$ and $\varLambda$ matrices defined in (10) and (16) are here reported in closed form for a variety of b.c. In the below expressions $\delta_{nm}$ is the Kronecker delta, and the following positions are made

$$\begin{aligned} f_{n} = 1 + \alpha_{n}^{2} \lambda^{2} ; \, f_{m} = 1 + \alpha_{m}^{2} \lambda^{2} ; \, \eta_{nm} = \frac{l\lambda }{{f_{m} }}\delta_{nm} ; \, \hfill \\ \gamma { = }e^{ - l/\lambda } ; \, \beta_{mn} = \alpha_{m} \alpha_{n} . \hfill \\ \end{aligned}$$

(32)

$$\begin{aligned} {\text{Simply supported beam}} \hfill \\ {\mathbf{K}}_{nm} = 0.5l\delta_{nm} ; \, \hfill \\ {\varvec{\Lambda}}_{nm} = \lambda^{4} \beta_{mn} \frac{{( - ( - 1)^{m} - ( - 1)^{n} )\gamma + 1 + ( - 1)^{m + n} }}{{f_{n} f_{m} }}{\kern 1pt} + \eta_{nm} \hfill \\ \end{aligned}$$

(33)

$$\begin{aligned} {\text{Fixed beam (clamped}}{-}{\text{clamped)}} \hfill \\ {\mathbf{K}}_{nm} = l\left( {1 + 0.5\delta_{nm} } \right); \, \hfill \\ {\varvec{\Lambda}}_{nm} = 2\lambda \frac{{lf_{n} f_{m} + \beta_{mn}^{2} \lambda^{5} (\gamma - 1)}}{{f_{n} f_{m} }} + \eta_{nm} \hfill \\ \end{aligned}$$

(34)

$$\begin{aligned} {\text{Cantilever beam (clamped-free)}} \hfill \\ {\mathbf{K}}_{nm} = \frac{{( - 1)^{m} }}{{\alpha_{m} }} + \frac{{( - 1)^{n} }}{{\alpha_{n} }} + l\left( {1 + 0.5\delta_{nm} } \right); \, \hfill \\ {\varvec{\Lambda}}_{nm} = \frac{{\lambda (2l - \lambda )(f_{n} + f_{m} - 1) + 2\beta_{mn}^{2} \lambda^{5} (l - \lambda )}}{{f_{n} f_{m} }} + \hfill \\ \frac{{( - 1)^{m} \lambda (1 + f_{m} )}}{{\alpha_{m} f_{m} }} + + \frac{{( - 1)^{n} \lambda (1 + f_{n} )}}{{\alpha_{n} f_{n} }} + \frac{{( - 1)^{m + n} \beta_{mn} \lambda^{4} }}{{f_{n} f_{m} }} + \hfill \\ \gamma {\kern 1pt} \lambda^{4} \frac{{\alpha_{m}^{2} f_{n} + \alpha_{n}^{2} f_{m} - \lambda \beta_{mn} (( - 1)^{n} \alpha_{m} + ( - 1)^{m} \alpha_{n} )}}{{f_{n} f_{m} }} + \eta_{nm} \hfill \\ \end{aligned}$$

(35)

$$\begin{aligned} {\text{Guided-hinged beam}} \hfill \\ {\mathbf{K}}_{nm} = 0.5l\delta_{nm} ; \, \hfill \\ {\varvec{\Lambda}}_{nm} = \,\gamma {\kern 1pt} \lambda^{3} \frac{{( - 1)^{n} \alpha_{n} + ( - 1)^{m} \alpha_{m} }}{{f_{n} f_{m} }} + \lambda^{2} \frac{{( - 1)^{m + n} \beta_{mn} \lambda^{2} - 1}}{{f_{n} f_{m} }} + \eta_{nm} \hfill \\ \end{aligned}$$

(36)

$$\begin{aligned} {\text{Guided-clamped beam}} \hfill \\ {\mathbf{K}}_{nm} = ( - 1)^{m + n} l + 0.5l\delta_{nm} ; \, \hfill \\ {\varvec{\Lambda}}_{nm} = \lambda ( - 1)^{m + n} \frac{{2f_{n} f_{m} (l - \lambda ) + \lambda (f_{n} + f_{m} - 1)}}{{f_{n} f_{m} }} - \hfill \\ \lambda^{2} \frac{{1 - ( - 1)^{n} f_{n} - ( - 1)^{m} f_{m} }}{{f_{n} f_{m} }} - \gamma {\kern 1pt} \lambda^{4} \frac{{( - 1)^{m} \alpha_{m}^{2} + ( - 1)^{n} \alpha_{n}^{2} }}{{f_{n} f_{m} }} + \hfill \\ ( - 1)^{m + n} \gamma {\kern 1pt} \lambda^{4} \frac{{\alpha_{m}^{2} f_{n} + \alpha_{n}^{2} f_{m} }}{{f_{n} f_{m} }} + \eta_{nm} \hfill \\ \end{aligned}$$

(37)