Convergence of the Rayleigh–Ritz Method for buckling analysis of arbitrarily configured I-section beam–columns


Design of the slender members requires calculation of buckling loads in addition to stress and deflection demand/capacity ratios. The Rayleigh–Ritz Method, which allows one to present approximate closed-form solutions for certain cases, is one of the simplest methods for this purpose. This study evaluates the buckling analysis of the I-section prismatic beam–columns with the Rayleigh–Ritz Method in detail. First, algebraic, trigonometric, and exponential trial functions for various restraint configurations are derived carefully in finite series form. Then, an iterative procedure to calculate buckling loads and modes is described. Finally, a software is developed with Mathematica and the sensitivity of the results and performance to trial function type and the number of terms is investigated over 1000 computer-generated numerical examples, which include doubly and singly symmetric sections, simply supported and cantilever members, intermediate torsional and lateral restraints, transversal concentrated and distributed loads acting above/below the shear center, and axial loads.

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This study is supported by the Eskisehir Osmangazi University’s Scientific Research Projects Department (Project ID: 2015-806).

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Correspondence to Hakan Ozbasaran.

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Appendix A: Derivation of the exponential trial function for restraint configuration 2

The boundary conditions for torsional rotation of the cantilever restraint configuration 2 are \(\phi \left( 0 \right) =0\) and \(\phi _{,z} \left( 0 \right) =0\) (see Table 1). Equations A.1 and A.2 can be obtained by applying these boundary conditions to the exponential series (Eq. 2.6).

$$\begin{aligned}&\phi \left( 0 \right) =\mathop \sum \limits _{i=1}^n k_i i^{\frac{0}{L}}=k_1 +k_2 +\cdots +k_n =\mathop \sum \limits _{i=1}^n k_i =0 \end{aligned}$$
$$\begin{aligned}&\phi _{,z} \left( 0 \right) =\mathop \sum \limits _{i=2}^n \frac{\ln i}{L}k_i i^{\frac{0}{L}}=\frac{\ln 2}{L}k_2 +\frac{\ln 3}{L}k_3 +\cdots +\frac{\ln n}{L}k_n =\mathop \sum \limits _{i=2}^n \frac{\ln i}{L}k_i =0 \end{aligned}$$

The coefficients of the last terms (\(k_n \)) can be solved from Eqs. A.1 and A.2 as given in Eqs. A.3 and A.4, respectively.

$$\begin{aligned}&k_n =-k_1 -k_2 -\cdots -k_{n-1} =-\mathop \sum \limits _{i=1}^{n-1} k_i \end{aligned}$$
$$\begin{aligned}&k_n =-\frac{1}{\ln n}\left[ {\ln 2\,k_2 +\ln 3\,k_3 +\cdots +\ln \left( {n-1} \right) k_{n-1} } \right] =-\frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-1} \ln ik_i \end{aligned}$$

Since the solutions for the \(k_n \) coefficients must be equal, it can be shown that the coefficient \(k_{n-1} \) can be obtained in terms of the other coefficients from \(k_1 \) to \(k_{n-2} \). Equation A.5 can be written by separating the terms with the coefficient \(k_{n-1} \) in Eqs. A.3 and A.4.

$$\begin{aligned}&-\mathop \sum \limits _{i=1}^{n-2} k_i -k_{n-1} =-\frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\frac{1}{\ln n}\ln \left( {n-1} \right) k_{n-1}\nonumber \\&\quad \rightarrow \frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\mathop \sum \limits _{i=1}^{n-2} k_i =\left( {1-\frac{\ln \left( {n-1} \right) }{\ln n}} \right) k_{n-1} \end{aligned}$$

Arranging the series and simplifying Eq. A.5 by using the properties of the logarithms leads to the solution for \(k_{n-1} \) (Eq. A.6).

$$\begin{aligned} k_{n-1} =\frac{1}{\ln \left( {\frac{n}{n-1}} \right) }\left( {\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\ln n\mathop \sum \limits _{i=2}^{n-2} k_i -\ln n\,k_1 } \right) =-\frac{1}{\ln \left( {\frac{n-1}{n}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n}} \right) k_i \end{aligned}$$
Fig. 13

Buckling analysis of a doubly symmetric cantilever beam with a torsional restraint at \(z=0.4L\) using trigonometric series

Fig. 14

Buckling analysis of a singly symmetric cantilever column with a lateral restraint at \(z=0.8L\) using exponential series

Then, the coefficient \(k_n \) can be written (Eq. A.7) by substituting \(k_{n-1} \) (Eq. A.6) into Eq. A.4.

$$\begin{aligned}&k_n =-\frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\frac{\ln \left( {n-1} \right) }{\ln n}k_{n-1}\nonumber \\&\quad \rightarrow k_n =-\frac{1}{\ln n}\left[ {\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i +\frac{\ln \left( {n-1} \right) }{\ln \left( {\frac{n}{n-1}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n}} \right) k_i } \right] \nonumber \\&\quad \rightarrow k_n =-\frac{1}{\ln \left( {\frac{n}{n-1}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n-1}} \right) k_i \end{aligned}$$

The last (\(k_n \)) and the one before the last (\(k_{n-1} \)) coefficients of the exponential series are obtained in terms of the other coefficients (\(k_1 ,k_2 \ldots ,k_{n-2} \)) to satisfy the boundary conditions. Substituting these new expressions for \(k_n \) and \(k_{n-1} \) into the exponential trial functions template (Eq. 2.6) gives Eq. A.8, which is a proper trial function for configuration 2.

$$\begin{aligned} \phi \left( z \right) =\mathop \sum \limits _{i=1}^{n-2} k_i i^{\frac{z}{L}}-\left( {n-1} \right) ^{\frac{z}{L}}\left[ {\frac{1}{\ln \left( {\frac{n-1}{n}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n}} \right) k_i } \right] -n^{\frac{z}{L}}\left[ {\frac{1}{\ln \left( {\frac{n}{n-1}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n-1}} \right) k_i } \right] \end{aligned}$$

Thus, it is shown that the last two coefficients (\(k_n \) and \(k_{n-1} \)) of the exponential series for the considered restraint configuration are dependent on the rest (\(k_1 ,k_2 \ldots ,k_{n-2} \)), no matter how many terms are considered. In other words, if it is desired to approximate the buckling eigenvalue and mode with n different coefficients, \(n+2\) terms must be considered. As the final touch, replacing n with \(n+2\) in Eq. A.8 gives the trial function presented in Table 4 for the configuration 2.

Appendix B: Selected cases

See Figs. 1314, and 15.

Fig. 15

Buckling analysis of a singly symmetric cantilever beam–column with torsional and lateral restraints at \(z=0.5L\) using power series

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Ozbasaran, H. Convergence of the Rayleigh–Ritz Method for buckling analysis of arbitrarily configured I-section beam–columns. Arch Appl Mech 89, 2397–2414 (2019).

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  • Buckling
  • I-section
  • Beam–column
  • Rayleigh–Ritz
  • Trial function
  • Intermediate restraint