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Static, stability and dynamic analyses of second strain gradient elastic Euler–Bernoulli beams

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Abstract

A simplified second strain gradient Euler–Bernoulli beam theory with two non-classical elastic coefficients in addition to the classical constants is presented. The governing equation and the associated classical and non-classical boundary conditions are derived with the aid of variational principles. The simplified second strain gradient theory is governed by an eighth-order differential equation with displacement, slope, curvature and triple derivative of displacement as degrees of freedom. This theory can be reduced to the first strain gradient and classical Euler–Bernoulli beam theories. Analytical solutions for static behaviour, free vibration and stability analyses are presented for different boundary conditions and length scale parameters. Using the numerical Laplace transform, a spectral element is developed for dynamic analysis of a cantilever beam subjected to a Gaussian pulse. Further, spectrum and dispersion relations are derived to study wave propagation characteristics. The gradient effects on the structural response are assessed and compared with the corresponding first strain gradient and classical beam theories. Observations show that the second strain gradient theory exhibiting stiffer behaviour in comparison to the first strain gradient and classical theories. The beam deflection decreases whereas frequencies and buckling load increase for increasing values of the gradient coefficient in comparison to the first strain gradient and classical theories. The forced response for a finite beam reveals a decrease in the amplitude and a shift to smaller time values with an increase in the value of length scale parameter. Additionally, the second strain gradient beam shows a dispersive behaviour, and for a given frequency the wavenumber decreases and the phase speed increases with an increase in the length scale parameter as compared to the first strain gradient beam theory.

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  1. Department of Aerospace Engineering, Indian Institute of Science, Bengaluru, 560012, India

    Md. Ishaquddin & S. Gopalakrishnan

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Appendix

Appendix

1.1 A. Stiffness matrix and force vector for static analysis of second strain gradient Euler–Bernoulli beam

The following is the list of stiffness matrices and load vectors for different boundary conditions:

(a) Simply supported beam:

$$\begin{aligned}{}[K]= \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1\\ 1 &{} L &{} L^2 &{} L^3 &{} e^{m_{1}L} &{} e^{m_{2}L} &{} e{n_{1}L} &{} e^{n_{2}L}\\ 0 &{} 0 &{} 2 &{} 0 &{} a_{11} &{} a_{12} &{} a_{13} &{} a_{14}\\ 0 &{} 0 &{} 2 &{} 6L &{} b_{11} &{} b_{12} &{} b_{13} &{} b_{14}\\ 0 &{} 0 &{} 2 &{} 0 &{} m_{1}^{2} &{} m_{2}^{2} &{} n_{1}^{2} &{} n_{2}^{2} \\ 0 &{} 0 &{} 2 &{} 6L &{} m_{1}^{2}e^{m_{1}L} &{} m_{2}^{2}e^{m_{2}L} &{} n_{1}^{2}e^{n_{1}L} &{} n_{2}^{2}e^{n_{2}L}\\ 0 &{} 0 &{} 0 &{} 6 &{} m_{1}^{3} &{} m_{2}^{3} &{} n_{1}^{3} &{} n_{2}^{3}\\ 0 &{} 0 &{} 0 &{} 6 &{} m_{1}^{3}e^{m_{1}L} &{} m_{2}^{3}e^{m_{2}L} &{} n_{1}^{3}e^{n_{1}L} &{} n_{2}^{3}e^{n_{2}L}\\ \end{bmatrix}, \quad \{f\}=\begin{Bmatrix} 0 \\ -qL^{4}/24EI \\ g_{1}^{2}q/EI\\ g_{1}^{2}q/EI-qL^{2}/2EI\\ 0 \\ -qL^{2}/2EI\\ 0\\ -qL/EI\\ \end{Bmatrix}, \end{aligned}$$
(A.1)

where

$$\begin{aligned}&a_{11}= m_{1}^{2}-g_{1}^{2}m_{1}^{4}+g_{2}^{4}m_{1}^{6},\quad\quad \quad a_{12}= m_{2}^{2}-g_{1}^{2}m_{2}^{4}+g_{2}^{4}m_{2}^{6},\\&a_{13}=n_{1}^{2}-g_{1}^{2}n_{1}^{4}+g_{2}^{4}n_{1}^{6}\, , \quad \quad \quad \, a_{14}= n_{2}^{2}-g_{1}^{2}n_{2}^{4}+g_{2}^{4}n_{2}^{6},\\&b_{11}= (m_{1}^{2}-g_{1}^{2}m_{1}^{4}+g_{2}^{4}m_{1}^{6})e^{m_{1}L},\quad b_{12}= (m_{2}^{2}-g_{1}^{2}m_{2}^{4}+g_{2}^{4}m_{2}^{6})e^{m_{2}L},\\&b_{13}= (n_{1}^{2}-g_{1}^{2}n_{1}^{4}+g_{2}^{4}n_{1}^{6})e^{n_{1}L},\quad \,b_{14}= (n_{2}^{2}-g_{1}^{2}n_{2}^{4}+g_{2}^{4}n_{2}^{6})e^{n_{2}L}. \end{aligned}$$

(b) Cantilever beam:

$$\begin{aligned}{}[K]= \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1\\ 0 &{} 0 &{} 0 &{} 6 &{} a_{21} &{}a_{22} &{} a_{23} &{}a_{24}\\ 0 &{} 1 &{} 0 &{} 0 &{} m_{1} &{} m_{2} &{} n_{1} &{} n_{2}\\ 0 &{} 0 &{} 2 &{} 6L &{} b_{11} &{} b_{12} &{} b_{13} &{} b_{14}\\ 0 &{} 0 &{} 2 &{} 0 &{} m_{1}^{2} &{} m_{2}^{2} &{} n_{1}^{2} &{} n_{2}^{2} \\ 0 &{} 0 &{} 0 &{} 6g_{1}^{2} &{} b_{21} &{} b_{22} &{} b_{23} &{} b_{24}\\ 0 &{} 0 &{} 0 &{} 6 &{} m_{1}^{3} &{} m_{2}^{3} &{} n_{1}^{3} &{} n_{2}^{3}\\ 0 &{} 0 &{} 0 &{} 0 &{} c_{11} &{} c_{12} &{} c_{13} &{} c_{14}\\ \end{bmatrix}, \quad \{f\}=\begin{Bmatrix} 0 \\ -qL/EI \\ 0\\ g_{1}^{2}q/EI-qL^{2}/2EI\\ 0\\ -g_{1}^{2}Lq/EI\\ 0\\ 0\\ \end{Bmatrix} ,\end{aligned}$$
(A.2)

where

$$\begin{aligned}&a_{21}=(m_{1}^{3}-g_{1}^{2}m_{1}^{5}+g_{2}^{4}m_{1}^{7})e^{m_{1}L},\quad a_{22}=(m_{2}^{3}-g_{1}^{2}m_{2}^{5}+g_{2}^{4}m_{2}^{7})e^{m_{2}L},\\&a_{23}=(n_{1}^{3}-g_{1}^{2}n_{1}^{5}+g_{2}^{4}n_{1}^{7})e^{n_{1}L},\quad \quad a_{24}=(n_{2}^{3}-g_{1}^{2}n_{2}^{5}+g_{2}^{4}n_{2}^{7})e^{n_{2}L},\\ \end{aligned}$$
$$\begin{aligned}&b_{21}=(g_{1}^{2}m_{1}^{3}-g_{2}^{4}m_{1}^{5})e^{m_{1}L},\quad b_{22}=(g_{1}^{2}m_{2}^{3}-g_{2}^{4}m_{2}^{5})e^{m_{2}L},\\&b_{23}=(g_{1}^{2}n_{1}^{3}-g_{2}^{4}n_{1}^{5})e^{n_{1}L},\quad \quad b_{24}=(g_{1}^{2}n_{2}^{3}-g_{2}^{4}n_{2}^{5})e^{n_{2}L}\\ \end{aligned}$$
$$\begin{aligned}&c_{11}=g_{2}^{4}m_{1}^{4}e^{m_{1}L},\quad c_{12}=g_{2}^{4}m_{2}^{4}e^{m_{2}L},\quad c_{13}=g_{2}^{4}n_{1}^{4}e^{n_{1}L},\quad c_{14}=g_{2}^{4}n_{2}^{4}e^{n_{2}L},\\&h_{21}=(m_{1}^{2}-g_{1}^{2}m_{1}^{4}+g_{2}^{4}m_{1}^{6})e^{m_{1}L},\quad h_{22}=(m_{2}^{2}-g_{1}^{2}m_{2}^{4}+g_{2}^{4}m_{2}^{6})e^{m_{2}L},\\&h_{23}=(n_{1}^{2}-g_{1}^{2}n_{1}^{4}+g_{2}^{4}n_{1}^{6})e^{n_{1}L},\quad \quad h_{24}=(n_{2}^{2}-g_{1}^{2}n_{2}^{4}+g_{2}^{4}n_{2}^{6})e^{n_{2}L}.\\ \end{aligned}$$

(c) Clamped beam:

$$\begin{aligned}{}[K]= \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1\\ 1 &{} L &{} L^{2} &{} L^{3} &{} e^{m_{1}L} &{} e^{m_{2}L} &{} e^{n_{1}L} &{}e^{n_{2}L}\\ 0 &{} 1 &{} 0 &{} 0 &{} m_{1} &{} m_{2} &{} n_{1} &{} n_{2} \\ 0 &{} 1 &{} 2L &{} 3L^{2} &{} m_{1}e^{m_{1}L} &{} m_{2}e^{m_{2}L} &{} n_{1}e^{n_{1}L} &{} n_{2}e^{n_{2}L}\\ 0 &{} 0 &{} 2 &{} 0 &{} m_{1}^{2} &{} m_{2}^{2} &{} n_{1}^{2} &{} n_{2}^{2} \\ 0 &{} 0 &{} 2 &{} 6L &{} m_{1}^{2}e^{m_{1}L} &{} m_{2}^{2}e^{m_{2}L} &{} n_{1}^{2}e^{n_{1}L} &{} n_{2}^{2}e^{n_{2}L}\\ 0 &{} 0 &{} 0 &{} 6 &{} m_{1}^{3} &{} m_{2}^{3} &{} n_{1}^{3} &{} n_{2}^{3} \\ 0 &{} 0 &{} 0 &{} 6 &{} m_{1}^{3}e^{m_{1}L} &{} m_{3}^{2}e^{m_{2}L} &{} n_{1}^{3}e^{n_{1}L} &{} n_{2}^{3}e^{n_{2}L}\\ \end{bmatrix}, \quad \{f\}=\begin{Bmatrix} 0 \\ -qL^{4}/24EI \\ 0\\ -qL^{3}/6EI \\ 0\\ -qL^{2}/2EI\\ 0\\ -qL/EI\\ \end{Bmatrix} \end{aligned}$$
(A.3)

1.2 B. Frequency equations for free vibration analysis of second strain gradient Euler–Bernoulli beam

The following are the frequency equations for different boundary conditions:

(a) Simply supported beam:

$$\begin{aligned}{}[F(\omega )]=\begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1&{} 1 &{} 1\\ {e}^{({\alpha }_{1}L)} &{} {e}^{({\alpha }_{2}L)} &{} {e}^{({\alpha }_{3}L)} &{} {e}^{({\alpha }_{4}L)} &{} {e}^{({\alpha }_{5}L)} &{} {e}^{({\alpha }_{6}L)}&{} {e}^{({\alpha }_{7}L)} &{} {e}^{({\alpha }_{8}L)} \\ {{\alpha }_{1}}^2 &{} {{\alpha }_{2}}^2 &{} {{\alpha }_{3}}^2 &{} {{\alpha }_{4}}^2 &{} {{\alpha }_{5}}^2 &{} {{\alpha }_{6}}^2&{} {{\alpha }_{7}}^2 &{} {{\alpha }_{8}}^2\\ t_{1} &{}t_{2}&{}t_{3}&{}t_{4}&{}t_{5}&{}t_{6}&{}t_{7}&{}t_{8}\\ t_{1}{e}^{({\alpha }_{1}L)} &{}t_{2}{e}^{({\alpha }_{2}L)}&{}t_{3}{e}^{({\alpha }_{3}L)}&{}t_{4}{e}^{({\alpha }_{4}L)}&{}t_{5}{e}^{({\alpha }_{5}L)}&{}t_{6}{e}^{({\alpha }_{6}L)}&{}t_{7}{e}^{({\alpha }_{7}L)}&{}t_{8}{e}^{({\alpha }_{8}L)}\\ {\alpha }_{1}^{3} &{} {\alpha }_{2}^{3} &{} {\alpha }_{3}^{3} &{} {\alpha }_{4}^{3} &{} {\alpha }_{5}^{3} &{} {\alpha }_{6}^{3}&{} {\alpha }_{7}^{3} &{} {\alpha }_{8}^{3} \\ {\alpha }_{1}^{3}{e}^{({\alpha }_{1}L)} &{} {\alpha }_{2}^{3}{e}^{({\alpha }_{2}L)} &{} {\alpha }_{3}^{3}{e}^{({\alpha }_{3}L)} &{} {\alpha }_{4}^{3}{e}^{({\alpha }_{4}L)} &{} {\alpha }_{5}^{3}{e}^{({\alpha }_{5}L)} &{} {\alpha }_{6}^{3}{e}^{({\alpha }_{6}L)} &{} {\alpha }_{7}^{3}{e}^{({\alpha }_{7}L)} &{} {\alpha }_{8}^{3}{e}^{({\alpha }_{8}L)}\\ \end{bmatrix} \end{aligned}$$
(B.1)

(b) Cantilever beam:

$$\begin{aligned}{}[F(\omega )]=\begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1&{} 1 &{} 1\\ {{\alpha }_{1}} &{} {{\alpha }_{2}} &{} {{\alpha }_{3}} &{} {{\alpha }_{4}} &{} {{\alpha }_{5}} &{} {{\alpha }_{6}}&{} {{\alpha }_{7}} &{} {{\alpha }_{8}}\\ {{\alpha }_{1}}^2 &{} {{\alpha }_{2}}^2 &{} {{\alpha }_{3}}^2 &{} {{\alpha }_{4}}^2 &{} {{\alpha }_{5}}^2 &{} {{\alpha }_{6}}^2&{} {{\alpha }_{7}}^2 &{} {{\alpha }_{8}}^2\\ {{\alpha }_{1}}^3 &{} {{\alpha }_{2}}^3 &{} {{\alpha }_{3}}^3 &{} {{\alpha }_{4}}^3 &{} {{\alpha }_{5}}^3 &{} {{\alpha }_{6}}^3&{} {{\alpha }_{7}}^3 &{} {{\alpha }_{8}}^3\\ p_{1}{e}^{({\alpha }_{1}L)} &{}p_{2}{e}^{({\alpha }_{2}L)}&{}p_{3}{e}^{({\alpha }_{3}L)}&{}p_{4}{e}^{({\alpha }_{4}L)}&{}p_{5}{e}^{({\alpha }_{5}L)}&{}p_{6}{e}^{({\alpha }_{6}L)}&{}p_{7}{e}^{({\alpha }_{7}L)}&{}p_{8}{e}^{({\alpha }_{8}L)}\\ r_{1}{e}^{({\alpha }_{1}L)} &{}r_{2}{e}^{({\alpha }_{2}L)}&{}r_{3}{e}^{({\alpha }_{3}L)}&{}r_{4}{e}^{({\alpha }_{4}L)}&{}r_{5}{e}^{({\alpha }_{5}L)}&{}r_{6}{e}^{({\alpha }_{6}L)}&{}r_{7}{e}^{({\alpha }_{7}L)}&{}r_{8}{e}^{({\alpha }_{8}L)}\\ q_{1}{e}^{({\alpha }_{1}L)} &{}q_{2}{e}^{({\alpha }_{2}L)}&{}q_{3}{e}^{({\alpha }_{3}L)}&{}q_{4}{e}^{({\alpha }_{4}L)}&{}q_{5}{e}^{({\alpha }_{5}L)}&{}q_{6}{e}^{({\alpha }_{6}L)}&{}q_{7}{e}^{({\alpha }_{7}L)}&{}q_{8}{e}^{({\alpha }_{8}L)}\\ {\alpha }_{1}^{4}{e}^{({\alpha }_{1}L)} &{} {\alpha }_{2}^{4}{e}^{({\alpha }_{2}L)} &{} {\alpha }_{3}^{4}{e}^{({\alpha }_{3}L)} &{} {\alpha }_{4}^{4}{e}^{({\alpha }_{4}L)} &{} {\alpha }_{5}^{4}{e}^{({\alpha }_{5}L)} &{} {\alpha }_{6}^{4}{e}^{({\alpha }_{6}L)} &{} {\alpha }_{7}^{4}{e}^{({\alpha }_{7}L)} &{} {\alpha }_{8}^{4}{e}^{({\alpha }_{8}L)}\\ \end{bmatrix} \end{aligned}$$
(B.2)

(c) Clamped beam:

$$\begin{aligned}{}[F(\omega )]=\begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1&{} 1 &{} 1\\ {e}^{({\alpha }_{1}L)} &{} {e}^{({\alpha }_{2}L)} &{} {e}^{({\alpha }_{3}L)} &{} {e}^{({\alpha }_{4}L)} &{} {e}^{({\alpha }_{5}L)} &{} {e}^{({\alpha }_{6}L)}&{} {e}^{({\alpha }_{7}L)} &{} {e}^{({\alpha }_{8}L)} \\ {{\alpha }_{1}} &{} {{\alpha }_{2}} &{} {{\alpha }_{3}} &{} {{\alpha }_{4}} &{} {{\alpha }_{5}} &{} {{\alpha }_{6}}&{} {{\alpha }_{7}} &{} {{\alpha }_{8}}\\ {{\alpha }_{1}}{e}^{({\alpha }_{1}L)} &{} {{\alpha }_{2}}{e}^{({\alpha }_{2}L)} &{} {{\alpha }_{3}}{e}^{({\alpha }_{3}L)} &{} {{\alpha }_{4}}{e}^{({\alpha }_{4}L)} &{} {{\alpha }_{5}}{e}^{({\alpha }_{5}L)} &{} {{\alpha }_{6}}{e}^{({\alpha }_{6}L)}&{} {{\alpha }_{7}}{e}^{({\alpha }_{7}L)} &{} {{\alpha }_{8}}{e}^{({\alpha }_{8}L)} \\ {{\alpha }_{1}}^2 &{} {{\alpha }_{2}}^2 &{} {{\alpha }_{3}}^2 &{} {{\alpha }_{4}}^2 &{} {{\alpha }_{5}}^2 &{} {{\alpha }_{6}}^2&{} {{\alpha }_{7}}^2 &{} {{\alpha }_{8}}^2\\ {\alpha }_{1}^{2}{e}^{({\alpha }_{1}L)} &{} {\alpha }_{2}^{2}{e}^{({\alpha }_{2}L)} &{} {\alpha }_{3}^{2}{e}^{({\alpha }_{3}L)} &{} {\alpha }_{4}^{2}{e}^{({\alpha }_{4}L)} &{} {\alpha }_{5}^{2}{e}^{({\alpha }_{5}L)} &{} {\alpha }_{6}^{2}{e}^{({\alpha }_{6}L)} &{} {\alpha }_{7}^{2}{e}^{({\alpha }_{7}L)} &{} {\alpha }_{8}^{2}{e}^{({\alpha }_{8}L)}\\ {{\alpha }_{1}}^3 &{} {{\alpha }_{2}}^3 &{} {{\alpha }_{3}}^3 &{} {{\alpha }_{4}}^3 &{} {{\alpha }_{5}}^3 &{} {{\alpha }_{6}}^3&{} {{\alpha }_{7}}^3 &{} {{\alpha }_{8}}^3\\ {\alpha }_{1}^{3}{e}^{({\alpha }_{1}L)} &{} {\alpha }_{2}^{3}{e}^{({\alpha }_{2}L)} &{} {\alpha }_{3}^{3}{e}^{({\alpha }_{3}L)} &{} {\alpha }_{4}^{3}{e}^{({\alpha }_{4}L)} &{} {\alpha }_{5}^{3}{e}^{({\alpha }_{5}L)} &{} {\alpha }_{6}^{3}{e}^{({\alpha }_{6}L)} &{} {\alpha }_{7}^{3}{e}^{({\alpha }_{7}L)} &{} {\alpha }_{8}^{3}{e}^{({\alpha }_{8}L)}\\ \end{bmatrix} ,\end{aligned}$$
(B.3)

where

$$\begin{aligned} t_{1}&= {} (-g_{1}^{2}{\alpha }_{1}^{4}+g_{2}^{4}{\alpha }_{1}^{6}),\quad t_{2}=(-g_{1}^{2}{\alpha }_{2}^{4}+g_{2}^{4}{\alpha }_{2}^{6}) ,\quad t_{3}=(-g_{1}^{2}{\alpha }_{3}^{4}+g_{2}^{4}{\alpha }_{3}^{6}),\\ t_{4}&= {} (-g_{1}^{2}{\alpha }_{4}^{4}+g_{2}^{4}{\alpha }_{4}^{6}) ,\quad t_{5}=(-g_{1}^{2}{\alpha }_{5}^{4}+g_{2}^{4}{\alpha }_{5}^{6}), \quad t_{6}=(-g_{1}^{2}{\alpha }_{6}^{4}+g_{2}^{4}{\alpha }_{6}^{6}),\\ t_{7}&= {} (-g_{1}^{2}{\alpha }_{7}^{4}+g_{2}^{4}{\alpha }_{7}^{6}) ,\quad t_{8}=(-g_{1}^{2}{\alpha }_{8}^{4}+g_{2}^{4}{\alpha }_{8}^{6}) ,\end{aligned}$$
$$\begin{aligned} p_{1}&= {} ({\alpha }_{1}^{3}-g_{1}^{2}{{\alpha }_{1}}^{5}+g_{2}^{4}{{\alpha }_{1}}^{7}), \quad p_{2}=({\alpha }_{2}^{3}-g_{1}^{2}{{\alpha }_{2}}^{5}+g_{2}^{4}{{\alpha }_{2}}^{7}),\\ p_{3}&= {} ({\alpha }_{3}^{3}-g_{1}^{2}{{\alpha }_{3}}^{5}+g_{2}^{4}{{\alpha }_{3}}^{7}), \quad p_{4}=({\alpha }_{4}^{3}-g_{1}^{2}{{\alpha }_{4}}^{5}+g_{2}^{4}{{\alpha }_{4}}^{7}),\\ p_{5}&= {} ({\alpha }_{5}^{3}-g_{1}^{2}{{\alpha }_{5}}^{5}+g_{2}^{4}{{\alpha }_{5}}^{7}), \quad p_{6}=({\alpha }_{6}^{3}-g_{1}^{2}{{\alpha }_{6}}^{5}+g_{2}^{4}{{\alpha }_{6}}^{7}),\\ p_{7}&= {} ({\alpha }_{7}^{3}-g_{1}^{2}{{\alpha }_{7}}^{5}+g_{2}^{4}{{\alpha }_{7}}^{7}), \quad p_{8}=({\alpha }_{8}^{3}-g_{1}^{2}{{\alpha }_{8}}^{5}+g_{2}^{4}{{\alpha }_{8}}^{7}).\\ \end{aligned}$$
$$\begin{aligned} r_{1}&= {} ({\alpha }_{1}^{2}-g_{1}^{2}{{\alpha }_{1}}^{4}+g_{2}^{4}{{\alpha }_{1}}^{6}) ,\quad r_{2}=({\alpha }_{2}^{2}-g_{1}^{2}{{\alpha }_{2}}^{4}+g_{2}^{4}{{\alpha }_{2}}^{6}),\\ r_{3}&= {} ({\alpha }_{3}^{2}-g_{1}^{2}{{\alpha }_{3}}^{4}+g_{2}^{4}{{\alpha }_{3}}^{6}) ,\quad r_{4}=({\alpha }_{4}^{2}-g_{1}^{2}{{\alpha }_{4}}^{4}+g_{2}^{4}{{\alpha }_{4}}^{6}), \\ r_{5}&= {} ({\alpha }_{5}^{2}-g_{1}^{2}{{\alpha }_{5}}^{4}+g_{2}^{4}{{\alpha }_{5}}^{6}), \quad r_{6}=(k_{6}^{2}-g_{1}^{2}{{\alpha }_{6}}^{4}+g_{2}^{4}{{\alpha }_{6}}^{6}), \\ r_{7}&= {} ({\alpha }_{7}^{2}-g_{1}^{2}{{\alpha }_{7}}^{4}+g_{2}^{4}{{\alpha }_{7}}^{6}), \quad r_{8}=({\alpha }_{8}^{2}-g_{1}^{2}{{\alpha }_{8}}^{4}+g_{2}^{4}{{\alpha }_{8}}^{6}), \end{aligned}$$
$$\begin{aligned} q_{1}&= {} (g_{1}^{2}{{\alpha }_{1}}^{3}-g_{2}^{4}{{\alpha }_{1}}^{5}) ,\quad q_{2}=(g_{1}^{2}{{\alpha }_{2}}^{3}-g_{2}^{4}{{\alpha }_{2}}^{5}), \\ q_{3}&= {} (g_{1}^{2}{{\alpha }_{3}}^{3}-g_{2}^{4}{{\alpha }_{3}}^{5}) ,\quad q_{4}=(g_{1}^{2}{{\alpha }_{4}}^{3}-g_{2}^{4}{{\alpha }_{4}}^{5}) ,\\ q_{5}&= {} (g_{1}^{2}{{\alpha }_{5}}^{3}-g_{2}^{4}{{\alpha }_{5}}^{5}), \quad q_{6}=(g_{1}^{2}{{\alpha }_{6}}^{3}-g_{2}^{4}{{\alpha }_{6}}^{5}) ,\\ q_{7}&= {} (g_{1}^{2}{{\alpha }_{7}}^{3}-g_{2}^{4}{{\alpha }_{7}}^{5}), \quad q_{8}=(g_{1}^{2}{{\alpha }_{8}}^{3}-g_{2}^{4}{{\alpha }_{8}}^{5}) ,\end{aligned}$$

1.3 C. Geometric stiffness matrix for buckling analysis of second strain gradient Euler–Bernoulli beam

The following are the geometric stiffness matrices for different boundary conditions:

(a) Simply supported beam:

$$\begin{aligned}{}[G(P)]= \begin{bmatrix} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1&{} 1 &{} 1\\ 1 &{} L &{} {e}^{(m_{1}L)} &{} {e}^{(m_{2}L)} &{} {e}^{(m_{3}L)} &{} {e}^{(n_{1}L)}&{} {e}^{(n_{2}L)} &{} {e}^{(n_{3}L)} \\ 0 &{} 0 &{} {m_{1}}^2 &{} {m_{2}}^2 &{} {m_{3}}^2 &{} {n_{1}}^2&{} {n_{2}}^2 &{} {n_{3}}^2\\ 0 &{} 0 &{} t_{3}&{}t_{4}&{}t_{5}&{}t_{6}&{}t_{7}&{}t_{8}\\ 0 &{} 0 &{} t_{3}{e}^{(m_{1}L)}&{}t_{4}{e}^{(m_{2}L)}&{}t_{5}{e}^{(m_{3}L)}&{}t_{6}{e}^{(n_{1}L)}&{}t_{7}{e}^{(n_{2}L)}&{}t_{8}{e}^{(n_{3}L)}\\ 0 &{} 0 &{} m_{1}^{3} &{} m_{2}^{3} &{} m_{3}^{3} &{} n_{1}^{3}&{} n_{2}^{3} &{} n_{3}^{3} \\ 0 &{} 0 &{} m_{1}^{3}{e}^{(m_{1}L)} &{} m_{2}^{3}{e}^{(m_{2}L)} &{} n_{1}^{3}{e}^{(n_{1}L)} &{} n_{2}^{3}{e}^{(n_{2}L)} &{} n_{3}^{3}{e}^{(n_{3}L)} &{} n_{4}^{3}{e}^{(n_{4}L)}\\ \end{bmatrix} \end{aligned}$$
(C.1)

(b) Cantilever beam:

$$\begin{aligned}{}[G(P)]= \begin{bmatrix} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1&{} 1 &{} 1\\ 0 &{} 1 &{} {m_{1}} &{} {m_{2}} &{} {m_{3}} &{} {n_{1}}&{} {n_{2}} &{} {n_{3}}\\ 0 &{} 0 &{} {m_{1}}^2 &{} {m_{2}}^2 &{} {m_{3}}^2 &{} {n_{1}}^2&{} {n_{2}}^2 &{} {n_{3}}^2\\ 0 &{} 0 &{} {m_{1}}^3 &{} {m_{2}}^3 &{} {m_{3}}^3 &{} {n_{1}}^3&{} {n_{2}}^3 &{} {n_{3}}^3\\ 0 &{} 0 &{} p_{3}{e}^{(m_{1}L)}&{}p_{4}{e}^{(m_{2}L)}&{}p_{5}{e}^{(m_{3}L)}&{}p_{6}{e}^{(n_{1}L)}&{}p_{7}{e}^{(n_{2}L)}&{}p_{8}{e}^{(n_{3}L)}\\ 0 &{} 0 &{} r_{3}{e}^{(m_{1}L)}&{}r_{4}{e}^{(m_{2}L)}&{}r_{5}{e}^{(m_{3}L)}&{}r_{6}{e}^{(n_{1}L)}&{}r_{7}{e}^{(n_{2}L)}&{}r_{8}{e}^{(n_{3}L)}\\ 0 &{} 0 &{} q_{3}{e}^{(m_{1}L)}&{}q_{4}{e}^{(m_{2}L)}&{}q_{5}{e}^{(m_{3}L)}&{}q_{6}{e}^{(n_{1}L)}&{}q_{7}{e}^{(n_{2}L)}&{}q_{8}{e}^{(n_{3}L)}\\ 0 &{} 0 &{} m_{1}^{4}{e}^{(m_{1}L)} &{} m_{2}^{4}{e}^{(_{2}L)} &{} m_{3}^{4}{e}^{(m_{3}L)} &{} n_{1}^{4}{e}^{(n_{1}L)} &{} n_{2}^{4}{e}^{(n_{2}L)} &{} n_{3}^{4}{e}^{(n_{3}L)}\\ \end{bmatrix} \end{aligned}$$
(C.2)

(c) Clamped beam:

$$\begin{aligned}{}[G(P)]= \begin{bmatrix} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1&{} 1 &{} 1\\ 1 &{} L &{} {e}^{(m_{1}L)} &{} {e}^{(m_{2}L)} &{} {e}^{(m_{3}L)} &{} {e}^{(n_{1}L)}&{} {e}^{(n_{2}L)} &{} {e}^{(n_{3}L)} \\ 0 &{} 1 &{} {m_{1}} &{} {m_{2}} &{} {m_{3}} &{} {n_{1}}&{} {n_{2}} &{} {n_{3}}\\ 0 &{} 1 &{} {m_{1}}{e}^{(m_{1}L)} &{} {m_{2}}{e}^{(m_{2}L)} &{} {m_{3}}{e}^{(m_{3}L)} &{} {n_{1}}{e}^{(n_{1}L)}&{} {n_{2}}{e}^{(n_{2}L)} &{} {n_{3}}{e}^{(n_{3}L)} \\ 0 &{} 0 &{} {m_{1}}^2 &{} {m_{2}}^2 &{} {m_{3}}^2 &{} {n_{1}}^2&{} {n_{2}}^2 &{} {n_{3}}^2\\ 0 &{} 0 &{} m_{1}^{2}{e}^{(m_{1}L)} &{} m_{2}^{2}{e}^{(m_{2}L)} &{} m_{3}^{2}{e}^{(m_{3}L)} &{} n_{1}^{2}{e}^{(n_{1}L)} &{} n_{2}^{2}{e}^{(n_{2}L)} &{} n_{3}^{2}{e}^{(n_{3}L)}\\ 0 &{} 0 &{} {m_{1}}^3 &{} {m_{2}}^3 &{} {m_{3}}^3 &{} {n_{1}}^3&{} {n_{2}}^3 &{} {n_{3}}^3\\ 0 &{} 0 &{} m_{1}^{3}{e}^{(m_{1}L)} &{} m_{2}^{3}{e}^{(m_{2}L)} &{} m_{3}^{3}{e}^{(m_{3}L)} &{} n_{1}^{3}{e}^{(n_{1}L)} &{} n_{2}^{3}{e}^{(n_{2}L)} &{} n_{3}^{3}{e}^{(n_{3}L)}\\ \end{bmatrix} \end{aligned}$$
(C.3)

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Ishaquddin, M., Gopalakrishnan, S. Static, stability and dynamic analyses of second strain gradient elastic Euler–Bernoulli beams. Acta Mech 232, 1425–1444 (2021). https://doi.org/10.1007/s00707-020-02902-5

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