Application of improved element-free Galerkin combining with finite strip method for buckling analysis of channel-section beams with openings

Abstract

In this paper, a coupling of improved element-free Galerkin with the finite strip (IEFG-FS) is used to investigate the buckling analysis of cold-formed steel (CFS) channel-section beam with web holes. For this purpose, three sections of the CFS channel are considered under bending loading. These sections are divided into two sub-domains in which the IEFG method is used in sub-domain with openings, and the finite strip (FS) method is applied for another sub-domain. In the IEFG domain, the boundary conditions are enforced using the Lagrange multiplier method. In the following, hole size effects are investigated on the moment buckling load of sections. The results show that the IEFG-FS is an efficient method for buckling analysis of channel-section beams. Moreover, the presence of holes in CFS channel section reduces the moment buckling load, so that the decrease in the local buckling is more than the distortional and global buckling loads.

Appendix

The polynomial interpolation functions are defined as

$$\begin{gathered} f_{11} = \frac{1}{2}(1 - \eta ), \, f_{12} = \frac{1}{2}(1 + \eta ), \hfill \\ f_{21} = (1 - 3\eta^{2} + 2\eta^{2} ), \, f_{22} = b(\eta - 2\eta^{2} + \eta^{3} ), \hfill \\ f_{31} = 3\eta^{2} - 2\eta^{3} , \, f_{32} = b( - \eta^{2} + \eta^{3} ), \hfill \\ \end{gathered}$$

(32)

n which $\eta = \frac{y}{b}$.

${\mathbf{K}}_{{_{{\rm{(IEFG)}}} }}$ is

$$\left[ {{\mathbf{K}}_{{_{{\rm{(IEFG}}} )}} } \right]_{{ij}} = \int\limits_{{{\Omega }_{1} }} {\left( {{\mathbf{B}}_{{_{{\rm{(IEFG)}}} }}^{{}} } \right)_{i}^{T} {\mathbf{D}}\left( {{\mathbf{B}}_{{_{{\rm{(IEFG)}}} }}^{{}} } \right)_{j}^{{}} {{\rm d}\Omega }}$$

(33)

where

$$\left( {{\mathbf{B}}_{{_{{\rm{(IEFG}}} }}^{{}} } \right)_{i} = \left[ {\begin{array}{*{20}c} {\frac{{\partial \phi_{i} }}{\partial x}} & 0 & { - z\frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }}} \\ 0 & {\frac{{\partial \phi_{i} }}{\partial y}} & { - z\frac{{\partial^{2} \phi_{i} }}{{\partial y^{2} }}} \\ {\frac{{\partial \phi_{i} }}{\partial y}} & {\frac{{\partial \phi_{i} }}{\partial x}} & { - 2z\frac{{\partial^{2} \phi_{i} }}{\partial x\partial y}} \\ \end{array} } \right]$$

(34)

and

$${\mathbf{D = }}\frac{E}{{\rm{(1 - }\nu^{2} \rm{)}}}\left[ {\begin{array}{*{20}c} 1 & \nu & \rm{0} \\ \nu & 1 & \rm{0} \\ \rm{0} & \rm{0} & {{{(\rm{1 - }\nu )} \mathord{\left/ {\vphantom {{(\rm{1 - }\nu )} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right],$$

(35)

in which E and ν are Young's modulus and Poisson's ratio, respectively.

$$\left[ {{\mathbf{G}}_{{\rm{(IEFG)}}} } \right]_{ij} = - \int\limits_{{\Gamma _{u}^{(1)} }} {\rm{(}{\mathbf{L}}_{b} {{\varvec{\Phi}}}_{{_{i} }}^{{}} \rm{)}_{{_{{}} }}^{T} {\mathbf{N}}_{{_{j} }}^{{}} {\rm{d}\Gamma }} ,$$

(36)

$$\left[ {{\mathbf{H}}_{{\rm{(IEFG)}}}^{{}} } \right]_{ij} = \int\limits_{{\Gamma _{{_{I} }}^{{}} }} {\rm{(}{\mathbf{L}}_{I} {{\varvec{\Phi}}}_{{_{i} }}^{{}} \rm{)}_{{_{{}} }}^{T} {\mathbf{N}}_{{_{j} }}^{{}} {\rm{d}\Gamma }} ,$$

(37)

where

$${{\varvec{\Phi}}}_{{_{i} }}^{{}} = \left[ {\begin{array}{*{20}c} {\phi_{i} } & 0 & 0 \\ 0 & {\phi_{i} } & 0 \\ 0 & 0 & {\phi_{i} } \\ \end{array} } \right],$$

(38)

and the interpolation function ${\mathbf{N}}_{{_{j} }}^{{}}$ in the jth node is expressed as:

$${\mathbf{N}}_{{_{j} }}^{{}} = \left[ {\begin{array}{*{20}c} {{\rm N}_{{_{j} }}^{{}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\rm N}_{{_{j} }}^{{}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\rm N}_{{_{j} }}^{{}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {{\rm N}_{{_{j} }}^{{}} } \\ \end{array} } \right].$$

(39)

${\mathbf{K}}_{{\rm{(FS)}}}^{{}}$ for the strip e (Fig. 3a) is calculated as

$${\mathbf{K}}_{{\rm{(FS)}}}^{e} = \left[ {\begin{array}{*{20}c} {\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{e} } \right]_{11} } & \cdots & {\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{e} } \right]_{1r} } \\ \vdots & \ddots & \vdots \\ {\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{e} } \right]_{r1} } & \cdots & {\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{e} } \right]_{rr} } \\ \end{array} } \right],$$

(40)

where

$$\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{e} } \right]_{pq} = \int\limits_{{\Omega _{e} }} {\left( {{\mathbf{RB}}_{{p_{{\rm{(FS)}}} }}^{e} } \right)^{T} {\mathbf{D}}\left( {{\mathbf{RB}}_{{q_{{\rm{(FS)}}} }}^{e} } \right){\rm{d}\Omega }} ,$$

(41)

in which ${\mathbf{B}}_{{n_{{\rm{(FS)}}} }}^{e}$ is

$${\mathbf{B}}_{{n_{{\rm{(FS)}}} }}^{e} = {\mathbf{L}}_{{\mathbf{d}}}^{{}} {\mathbf{N}}_{{n_{{\rm{(FS)}}} }}^{e} , \, n\rm{ = }p\rm{,}q$$

(42)

where

$${\mathbf{L}}_{{\mathbf{d}}}^{{}} = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}} & 0 & { - z\frac{{\partial^{2} }}{{\partial x^{2} }}} \\ 0 & {\frac{\partial }{\partial y}} & { - z\frac{{\partial^{2} }}{{\partial y^{2} }}} \\ {\frac{\partial }{\partial y}} & {\frac{\partial }{\partial x}} & { - 2z\frac{{\partial^{2} }}{\partial x\partial y}} \\ \end{array} } \right]$$

(43)

and

$${\mathbf{N}}_{{n_{{\rm{(FS)}}} }}^{e} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{11} S_{n} } \\ \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ {f_{11} S_{n} } \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ {f_{21} S_{n} } \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ {f_{31} S_{n} } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{12} S_{n} } \\ \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ {f_{12} S_{n} } \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ {f_{22} S_{n} } \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ {f_{32} S_{n} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right].$$

(44)

${\mathbf{H}}_{{\rm{(FS)}}}^{{}}$ is calculated as

$${\mathbf{H}}_{{\rm{(FS)}}}^{{}} = \left[ {\begin{array}{*{20}c} {\left[ {{\mathbf{H}}_{{\rm{(FS)}}}^{{}} } \right]_{11} } & \cdots & {\left[ {{\mathbf{H}}_{{\rm{(FS)}}}^{{}} } \right]_{{1n_{\gamma } }} } \\ \vdots & \ddots & \vdots \\ {\left[ {{\mathbf{H}}_{{\rm{(FS)}}}^{{}} } \right]_{r1} } & \cdots & {\left[ {{\mathbf{H}}_{{\rm{(FS)}}}^{{}} } \right]_{{rn_{\gamma } }} } \\ \end{array} } \right],$$

(45)

in which $\left[ {{\mathbf{H}}_{{\rm{(FS)}}}^{{}} } \right]_{nj}$ is obtained from

$$\left[ {{\mathbf{H}}_{{\rm{(FS)}}}^{{}} } \right]_{nj} = \int\limits_{{\Gamma _{I} }} {\rm{(}{\mathbf{N}}_{{I_{{(\rm{FS})}} }}^{{}} \rm{)}_{n} {\mathbf{N}}_{{_{j} }}^{{}} {\rm{d}\Gamma }} ,$$

(46)

where

$$\rm{(}{\mathbf{N}}_{{I_{{(\rm{FS})}} }}^{{}} \rm{)}_{n} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{11} S_{n} } \\ \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ {f_{22} S_{n} } \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ {f_{31} S_{n} } \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ {f_{32} S_{n} } \\ \end{array} } \\ \end{array} } \right].$$

(47)

Similar to ${\mathbf{K}}_{{\rm{(FS)}}}^{e}$ (A.9), ${\mathbf{K}}_{\rm{0}}^{e}$ and ${\mathbf{K}}_{a}^{e}$ can be defined for the strip e in which their components $\left[ {{\mathbf{K}}_{\rm{0}}^{e} } \right]_{pq}$ and $\left[ {{\mathbf{K}}_{a}^{e} } \right]_{pq}$ are, respectively, obtained as

$$\left[ {{\mathbf{K}}_{\rm{0}}^{e} } \right]_{pq} = \int\limits_{{\Gamma _{{\rm{FS}_{\rm{1}} }} }} {\left( {{\mathbf{R}}\rm{(}{\mathbf{L}}_{k}^{{}} {\mathbf{N}}_{{p_{{\rm{(FS)}}} }}^{e} \rm{)}^{T} } \right){\mathbf{k}}_{{\mathbf{0}}} \left( {{\mathbf{R}}\rm{(}{\mathbf{L}}_{k}^{{}} {\mathbf{N}}_{{q_{{\rm{(FS)}}} }}^{e} \rm{)}} \right){\rm{d}\Gamma }} ,$$

(48)

$$\left[ {{\mathbf{K}}_{a}^{e} } \right]_{pq} = \int\limits_{{\Gamma _{{\rm{FS}_{\rm{2}} }} }} {\left( {{\mathbf{R}}\rm{(}{\mathbf{L}}_{k}^{{}} {\mathbf{N}}_{{p_{{\rm{(FS)}}} }}^{e} \rm{)}} \right)^{T} {\mathbf{k}}_{a} \left( {{\mathbf{R}}\rm{(}{\mathbf{L}}_{k}^{{}} {\mathbf{N}}_{{q_{{\rm{(FS)}}} }}^{e} \rm{)}} \right){\rm{d}\Gamma }} ,$$

(49)

where

$${\mathbf{L}}_{k} = \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{L}}}_{k} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\tilde{\mathbf{L}}}_{k} } \\ \end{array} } \right], \, {\tilde{\mathbf{L}}}_{k} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} \rm{1} \\ \rm{0} \\ \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{1} \\ \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \rm{1} \\ {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} \\ \end{array} } \\ \end{array} } \right] \, {.}$$

(50)

${\mathbf{K}}_{{\rm{(IEFG)}}}^{g}$ is

$${\mathbf{K}}_{{\rm{(IEFG)}}}^{g} = \int\limits_{{\Omega _{1} }} {{\mathbf{L}}_{{N_{{\rm{(IEFG)}}} }}^{T} S_{N} {\mathbf{L}}_{{N_{{\rm{(IEFG)}}} }}^{{}} {\rm{d}\Omega }} ,$$

(51)

where

$${\mathbf{L}}_{{N_{{\rm{(IEFG)}}} }}^{T} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial \phi_{i} }}{\partial x}} & {\frac{{\partial \phi_{i} }}{\partial y}} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \end{array} } \\ {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{\partial \phi_{i} }}{\partial x}} & {\frac{{\partial \phi_{i} }}{\partial y}} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \end{array} } \\ {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{\partial \phi_{i} }}{\partial x}} & {\frac{{\partial \phi_{i} }}{\partial y}} \\ \end{array} } \\ \end{array} } \right],$$

(52)

and

$$S_{N} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {s_{x} } & {s_{xy} } \\ {s_{xy} } & {s_{y} } \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \rm{0} & \rm{0} \\ \end{array} } \\ {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {s_{x} } & {s_{xy} } \\ {s_{xy} } & {s_{y} } \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \rm{0} & \rm{0} \\ \end{array} } \\ {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} & \rm{0} \\ \rm{0} & \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {s_{x} } & {s_{xy} } \\ {s_{xy} } & {s_{y} } \\ \end{array} } \\ \end{array} } \right].$$

(53)

${\mathbf{K}}_{{\rm{(FS)}}}^{ge}$ is the stability stiffness matrix for the strip e, and is defined similar to ${\mathbf{K}}_{{\rm{(FS)}}}^{e}$ (40). Therefore, $\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{ge} } \right]_{pq}$ is

$$\left[ {{\mathbf{K}}_{{\rm{(FS)}}}^{ge} } \right]_{pq} = \int\limits_{{\Omega _{e} }} {\left( {{\mathbf{L}}_{{N_{{\rm{(FS)}}} }}^{T} } \right)_{p} S_{N} \left( {{\mathbf{L}}_{{N_{{\rm{(FS)}}} }}^{{}} } \right)_{q} {{\rm{d}}\Omega }} ,$$

(54)

where

$$\begin{gathered} \left( {{\mathbf{L}}_{{N_{{\rm{(FS)}}} }}^{{}} } \right)_{n} = \left[ {\begin{array}{*{20}c} {f_{11} \frac{{\partial S_{n} (x)}}{\partial x}} & \rm{0} & \rm{0} & \rm{0} \\ {\frac{{\partial f_{11} }}{\partial y}S_{n} (x)} & \rm{0} & \rm{0} & \rm{0} \\ {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {f_{11} \frac{{\partial S_{n} (x)}}{\partial x}} \\ {\frac{{\partial f_{11} }}{\partial y}S_{n} (x)} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } \\ {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {f_{21} \frac{{\partial S_{n} (x)}}{\partial x}} \\ {\frac{{\partial f_{21} }}{\partial y}S_{n} (x)} \\ \end{array} } & {\begin{array}{*{20}c} {f_{31} \frac{{\partial S_{n} (x)}}{\partial x}} \\ {\frac{{\partial f_{31} }}{\partial y}S_{n} (x)} \\ \end{array} } \\ \end{array} } \right. \hfill \\ \, \left. {\begin{array}{*{20}c} {f_{12} \frac{{\partial S_{n} (x)}}{\partial x}} & \rm{0} & \rm{0} & \rm{0} \\ {\frac{{\partial f_{12} }}{\partial y}S_{n} (x)} & \rm{0} & \rm{0} & \rm{0} \\ {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {f_{12} \frac{{\partial S_{n} (x)}}{\partial x}} \\ {\frac{{\partial f_{12} }}{\partial y}S_{n} (x)} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } \\ {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} \rm{0} \\ \rm{0} \\ \end{array} } & {\begin{array}{*{20}c} {f_{22} \frac{{\partial S_{n} (x)}}{\partial x}} \\ {\frac{{\partial \partial f_{22} }}{\partial y}S_{n} (x)} \\ \end{array} } & {\begin{array}{*{20}c} {f_{32} \frac{{\partial S_{n} (x)}}{\partial x}} \\ {\frac{{\partial f_{32} }}{\partial y}S_{n} (x)} \\ \end{array} } \\ \end{array} } \right] n\rm{ = }p\rm{,}q. \hfill \\ \end{gathered}$$

(55)