New Nonlinear First-Order Shear Deformation Beam Model Based on Geometrically Exact Theory

Abstract

In this paper, a new geometrically exact model of a first-order shear deformation beam is developed. By use of the derived equations, free vibration and primary resonance of a simply supported beam are studied. In this formulation, the transverse displacements and bending rotations are considered completely independent and this causes the nonlinear behavior of the system to be predicted more accurately than other models. Nonlinear equations of motion are discretized by the Galerkin method and then in order to obtain an analytical solution the method of multiple scales was applied to the resulting equations. The results obtained from the present first-order shear deformation model are verified with other first-order shear deformation models. It will be shown that the present formulation is superior to other nonlinear first-order shear deformation beam theories. The effects of linear and nonlinear shear terms and slenderness ratio are studied on amplitude, linear and nonlinear frequencies of the system, frequency response, and locus of bifurcation points, which are more noticeable in higher vibration modes.

Appendices

Appendix A

The proof of $\delta \gamma ,\delta \kappa$:

If $\overline{\delta q}^{B}$ is virtual displacement in the deformed frame and $\overline{\delta u}$ is in the undeformed frame, then:

$$\overline{\delta u} = C^{T} \overline{\delta q}^{B}$$

(A1)

Virtual rotation is defined as follows:

$$\widetilde{{\overline{\delta \psi } }}^{BA} = - \delta CC^{T} \overline{\delta \psi } =$$

(A2)

From the first part of (10), one can write:

$$\gamma = C\left( {e_{1} + u^{\prime} + \tilde{k}u} \right) - e_{1}$$

(A3)

It can be written as:

$$\begin{gathered} \delta \gamma = \delta C\left( {e_{1} + u^{\prime} + \tilde{k}u} \right) + \widetilde{{\overline{\delta \psi } }}^{BA} C\left( {e_{1} + u^{\prime} + \tilde{k}u} \right) + \hfill \\ \,\,\,\,\,C\left( {\delta u^{\prime} + \tilde{k}\delta u} \right) + C\widetilde{{\overline{\delta \psi } }}^{BA} \left( {e_{1} + u^{\prime} + \tilde{k}u} \right) \hfill \\ \,\,\,\,\,\,\, = \widetilde{{\overline{\delta \psi } }}^{BA} CC^{T} \left( {\gamma + e_{1} } \right) + \widetilde{{\overline{\delta \psi } }}^{BA} \left( {\gamma + e_{1} } \right) + \overline{\delta q}^{{B^{\prime}}} - \hfill \\ \,\,\,\,\left( {C\tilde{k} - \tilde{K}C} \right)C^{T} \overline{\delta q}^{B} + C\tilde{k}C\overline{\delta q}^{B} + C\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} \left( {\gamma + e_{1} } \right) \hfill \\ \,\,\,\,\,\,\, = \overline{\delta q}^{{B^{\prime}}} + \tilde{K}\overline{\delta q}^{B} + C\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} \left( {\gamma + e_{1} } \right) = \overline{\delta q}^{{B^{\prime}}} + \hfill \\ \,\,\,\,\,\,\tilde{K}\overline{\delta q}^{B} - \widetilde{{\overline{\delta \psi } }}^{BA} \left( {\gamma + e_{1} } \right) \hfill \\ \end{gathered}$$

(A4)

Then:

$$\delta \gamma = \overline{\delta q}^{{B^{\prime}}} + \tilde{K}\overline{\delta q}^{B} + \left( {\tilde{e}_{1} + \tilde{\gamma }} \right)\overline{\delta \psi }^{BA}$$

(A5)

From the second part of (10), one can write:

$$\tilde{K} = - C^{\prime}C^{T} + C\tilde{k}C^{T}$$

(A6)

Given that the initial curvature is always constant, it can be written as:

$$\begin{aligned} \delta \tilde{\kappa } &= - \delta C^{\prime}C^{T} - \widetilde{{\overline{\delta \psi } }}^{BA} C^{\prime}C^{T} - C^{\prime}\delta C^{T} - C^{\prime}\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} \hfill \\ &\quad + \delta C\tilde{k}C^{T} + \widetilde{{\overline{\delta \psi } }}^{BA} C\tilde{k}C^{T} + C\tilde{k}\delta C^{T} + C\tilde{k}\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} \hfill \\ & \quad= \widetilde{{\overline{\delta \psi } }}^{{BA^{\prime}}} - \widetilde{{\overline{\delta \psi } }}^{BA} \tilde{K} + \widetilde{{\overline{\delta \psi } }}^{BA} C\tilde{k}C^{T} + \widetilde{{\overline{\delta \psi } }}^{BA} (\tilde{K}C - \hfill \\ & \quad C\tilde{k})C^{T} + (\tilde{K}C - C\tilde{k})C^{T} \widetilde{{\overline{\delta \psi } }}^{BA} \hfill \\ & \quad + (\tilde{K}C - C\tilde{k})\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} - \widetilde{{\overline{\delta \psi } }}^{BA} C\tilde{k}C^{T} + \widetilde{{\overline{\delta \psi } }}^{BA} C\tilde{k}C^{T} \hfill \\ & \quad + C\tilde{k}C^{T} \widetilde{{\overline{\delta \psi } }}^{BA} + C\tilde{k}\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} \hfill \\ &\quad = \widetilde{{\overline{\delta \psi } }}^{{BA^{\prime}}} + \tilde{K}\widetilde{{\overline{\delta \psi } }}^{BA} + \tilde{K}C\widetilde{{\overline{\delta \psi } }}^{BA} C^{T} \hfill \\ &\quad = \widetilde{{\overline{\delta \psi } }}^{{BA^{\prime}}} + \tilde{K}\widetilde{{\overline{\delta \psi } }}^{BA} - \widetilde{{\overline{\delta \psi } }}^{BA} \tilde{K} \hfill \\ \end{aligned}$$

(A7)

then:

$$\delta \kappa = \overline{\delta \psi }^{{BA^{\prime}}} - \tilde{K}\widetilde{{\overline{\delta \psi } }}^{BA}$$

(A8)

Appendix B

$$\alpha = - \left( {\lambda_{2}^{2} k_{s} n^{2} \pi^{2} - \beta_{w}^{2} } \right)/\left( {\pi {\mkern 1mu} nk_{s} \lambda_{2}^{2} } \right)$$

(B1)

$$\delta = - \left( {\lambda_{2}^{2} k_{s} n^{2} \pi^{2} - \beta_{\theta }^{2} } \right)/\left( {\pi {\mkern 1mu} nk_{s} \lambda_{2}^{2} } \right)$$

(B2)

$$Y_{1} = - I_{2}^{2} \pi^{2} n^{2} k_{s} \lambda_{2}^{2} - I_{2} {\mkern 1mu} n^{2} \pi^{2} - I_{2} {\mkern 1mu} k_{s} \lambda_{2}^{2}$$

(B3)

$$Y_{2} = I_{2} \sqrt {\mathop k\nolimits_{s}^{2} \left( {I_{2} {\mkern 1mu} n^{2} \pi^{2} + 1} \right)^{2} \mathop \lambda \nolimits_{2}^{4} - 2{\mkern 1mu} n^{2} \pi^{2} k_{s} \left( {I_{2} {\mkern 1mu} n^{2} \pi^{2} - 1} \right)\mathop \lambda \nolimits_{2}^{2} + \pi^{4} n^{4} }$$

(B4)

$$\Lambda_{1} = 3/2{\mkern 1mu} n^{2} \pi^{2} \alpha^{2} \left( {\lambda_{1}^{2} - \lambda_{2}^{2} k_{s} } \right) - \alpha^{2} \beta_{u}^{2}$$

(B5)

$$\Lambda_{2} = 3/2{\mkern 1mu} n^{2} \pi^{2} \delta^{2} \left( {\lambda_{1}^{2} - \lambda_{2}^{2} k_{s} } \right) - \delta^{2} \beta_{u}^{2}$$

(B6)

$$\Lambda_{3} = - \beta_{u}$$

(B7)

$$\Pi_{1} = - \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w} c_{2} k_{s} \lambda_{2}^{2} + \pi {\mkern 1mu} n\beta_{w} c_{1} k_{s} \lambda_{2}^{2} + \alpha {\mkern 1mu} \beta_{w}^{3} c_{2}$$

(B8)

$$\begin{gathered} \Pi_{2} = - 9/2{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - 9/4{\mkern 1mu} \pi^{3} \alpha^{2} n^{3} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - \hfill \\ 73/8{\mkern 1mu} \pi {\mkern 1mu} \alpha^{2} n\beta_{w}^{2} k_{s} \lambda_{2}^{2} - 9/2{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w}^{2} k_{s} \lambda_{2}^{2} - 3/2{\mkern 1mu} \pi^{2} \hfill \\ \alpha^{3} n^{2} \beta_{w}^{2} - 9/2{\mkern 1mu} \alpha^{3} \beta_{w}^{2} k_{s} \lambda_{2}^{2} + 9{\mkern 1mu} /8\pi^{2} \alpha^{3} n^{2} k_{s}^{2} \lambda_{2}^{4} + \hfill \\ 9/2{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w}^{2} \lambda_{1}^{2} + 27/4{\mkern 1mu} \pi {\mkern 1mu} \alpha^{2} n\beta_{w}^{2} \lambda_{1}^{2} + 45{\mkern 1mu} /8\pi^{3} \alpha^{2} \hfill \\ n^{3} k_{s}^{2} \lambda_{2}^{4} + 9/2{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} k_{s}^{2} \lambda_{2}^{4} + 3/2{\mkern 1mu} \pi^{4} \alpha^{3} n^{4} k_{s} \lambda_{2}^{2} + \hfill \\ 9/4{\mkern 1mu} \alpha^{3} \beta_{w}^{2} \lambda_{1}^{2} + 3/2{\mkern 1mu} I2{\mkern 1mu} \alpha^{3} \beta_{w}^{4} - 3/2{\mkern 1mu} I2{\mkern 1mu} \pi^{2} \alpha^{3} n^{2} \beta_{w}^{2} \hfill \\ k_{s} \lambda_{2}^{2} \hfill \\ \end{gathered}$$

(B9)

$$\begin{gathered} \Pi_{3} = 9/2{\mkern 1mu} \alpha {\mkern 1mu} \delta^{2} \beta_{w}^{2} \lambda_{1}^{2} + 3/2{\mkern 1mu} I2{\mkern 1mu} \alpha {\mkern 1mu} \delta^{2} \beta_{w}^{4} - 3{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} k_{s} \hfill \\ \lambda_{1}^{2} \lambda_{2}^{2} - 3{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w}^{2} k_{s} \lambda_{2}^{2} - 3{\mkern 1mu} \pi^{3} \alpha {\mkern 1mu} \delta {\mkern 1mu} n^{3} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - \hfill \\ 27/2{\mkern 1mu} \pi \alpha {\mkern 1mu} \delta {\mkern 1mu} n\beta_{w}^{2} k_{s} \lambda_{2}^{2} + 15/2{\mkern 1mu} \pi^{3} \alpha \delta {\mkern 1mu} n^{3} k_{s}^{2} \lambda_{2}^{4} + 9{\mkern 1mu} \pi {\mkern 1mu} \alpha {\mkern 1mu} \delta \hfill \\ {\mkern 1mu} n\beta_{w}^{2} \lambda_{1}^{2} - 19{\mkern 1mu} /4\pi {\mkern 1mu} \delta^{2} n\beta_{w}^{2} k_{s} \lambda_{2}^{2} - 6{\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{w}^{2} k_{s} \lambda_{2}^{2} + \hfill \\ 3{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} \delta^{2} n^{4} k_{s} \lambda_{2}^{2} + 9/4{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} \delta^{2} n^{2} k_{s}^{2} \lambda_{2}^{4} - 3/2{\mkern 1mu} \pi^{3} \delta^{2} n^{3} \hfill \\ k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - 6{\mkern 1mu} \pi^{4} \delta {\mkern 1mu} n^{4} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - 3/2{\mkern 1mu} I2{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} \delta^{2} n^{2} \beta_{w}^{2} k_{s} \hfill \\ \lambda_{2}^{2} - 3/2{\mkern 1mu} I_{2} {\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} \delta^{2} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} + 3/2{\mkern 1mu} I2{\mkern 1mu} \alpha {\mkern 1mu} \delta^{2} \beta_{\theta }^{2} \beta_{w}^{2} \hfill \\ - 9{\mkern 1mu} \alpha {\mkern 1mu} \delta^{2} \beta_{w}^{2} k_{s} \lambda_{2}^{2} + 15{\mkern 1mu} /4\pi^{3} \delta^{2} n^{3} k_{s}^{2} \lambda_{2}^{4} + 6{\mkern 1mu} \pi^{4} \delta {\mkern 1mu} n^{4} k_{s}^{2} \hfill \\ \lambda_{2}^{4} - 3{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} \delta^{2} n^{2} \beta_{w}^{2} + 9/2{\mkern 1mu} \pi {\mkern 1mu} \delta^{2} n\beta_{w}^{2} \lambda_{1}^{2} + 6{\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \hfill \\ \beta_{w}^{2} \lambda_{1}^{2} + 3{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w}^{2} \lambda_{1}^{2} + 3{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} k_{s}^{2} \lambda_{2}^{4} \hfill \\ \end{gathered}$$

(B10)

$$\begin{gathered} \Pi_{4} = 3/4{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w}^{2} k_{s} \lambda_{2}^{2} + 3/4{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - \hfill \\ 3/4{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} k_{s}^{2} \lambda_{2}^{4} - 3/4{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w}^{2} \lambda_{1}^{2} \hfill \\ \end{gathered}$$

(B11)

$$\Pi_{5} = - 2{\mkern 1mu} I_{2} {\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{w} k_{s} \lambda_{2}^{2} + 2{\mkern 1mu} \pi {\mkern 1mu} n\beta_{w} k_{s} \lambda_{2}^{2} + 2{\mkern 1mu} I_{2} {\mkern 1mu} \alpha {\mkern 1mu} \beta_{w}^{3}$$

(B12)

$$\Pi_{6} = 1/2{\mkern 1mu} \pi {\mkern 1mu} nk_{s} \lambda_{2}^{2}$$

(B13)

$$\Delta_{1} = - \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta } c_{2} k_{s} \lambda_{2}^{2} + \pi {\mkern 1mu} n\beta_{\theta } c_{1} k_{s} \lambda_{2}^{2} + \delta {\mkern 1mu} \beta_{\theta }^{3} c_{2}$$

(B14)

$$\begin{gathered} \Delta_{2} = - 3/2{\mkern 1mu} I2{\mkern 1mu} \pi^{2} \delta^{3} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} - 3/2{\mkern 1mu} \pi^{2} \delta^{3} n^{2} \beta_{\theta }^{2} - 9/2{\mkern 1mu} \hfill \\ \delta^{3} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} + 9/2{\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} \lambda_{1}^{2} + 27{\mkern 1mu} /4\pi {\mkern 1mu} \delta^{2} n\beta_{\theta }^{2} \lambda_{1}^{2} + \hfill \\ 45/8{\mkern 1mu} \pi^{3} \delta^{2} n^{3} k_{s}^{2} \lambda_{2}^{4} + 9/2{\mkern 1mu} \pi^{4} \delta {\mkern 1mu} n^{4} k_{s}^{2} \lambda_{2}^{4} + 3/2{\mkern 1mu} \pi^{4} \delta^{3} n^{4} \hfill \\ k_{s} \lambda_{2}^{2} + 9/8\pi^{2} \delta^{3} n^{2} k_{s}^{2} \lambda_{2}^{4} - 73/8{\mkern 1mu} \pi {\mkern 1mu} \delta^{2} n\beta_{\theta }^{2} k_{s} \lambda_{2}^{2} - \hfill \\ 9/4\pi^{3} \delta^{2} n^{3} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - 9/2{\mkern 1mu} \pi^{4} \delta {\mkern 1mu} n^{4} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - 9/2{\mkern 1mu} \hfill \\ \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} + 9/4{\mkern 1mu} \delta^{3} \beta_{\theta }^{2} \lambda_{1}^{2} + 3/2{\mkern 1mu} I_{2} {\mkern 1mu} \delta^{3} \beta_{\theta }^{4} \hfill \\ \end{gathered}$$

(B15)

$$\begin{gathered} \Delta_{3} = - 3/2{\mkern 1mu} I_{2} {\mkern 1mu} \pi^{2} \alpha^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} - 3/2{\mkern 1mu} I_{2} {\mkern 1mu} \pi^{2} \alpha^{2} \delta {\mkern 1mu} n^{2} \beta_{w}^{2} \hfill \\ k_{s} \lambda_{2}^{2} - 6{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} + 9{\mkern 1mu} \pi {\mkern 1mu} \alpha {\mkern 1mu} \delta {\mkern 1mu} n\beta_{\theta }^{2} \lambda_{1}^{2} + 3{\mkern 1mu} \pi^{4} \alpha^{2} \delta \hfill \\ {\mkern 1mu} n^{4} k_{s} \lambda_{2}^{2} + 9/4{\mkern 1mu} \pi^{2} \alpha^{2} \delta {\mkern 1mu} n^{2} k_{s}^{2} \lambda_{2}^{4} - 19{\mkern 1mu} /4\pi {\mkern 1mu} \alpha^{2} n\beta_{\theta }^{2} k_{s} \lambda_{2}^{2} \hfill \\ - 3{\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} - 27/2{\mkern 1mu} \pi {\mkern 1mu} \alpha {\mkern 1mu} \delta {\mkern 1mu} n\beta_{\theta }^{2} k_{s} \lambda_{2}^{2} - 9{\mkern 1mu} \alpha^{2} \delta {\mkern 1mu} \beta_{\theta }^{2} \hfill \\ k_{s} \lambda_{2}^{2} + 3/2{\mkern 1mu} I_{{2{\mkern 1mu} }} \alpha^{2} \delta {\mkern 1mu} \beta_{\theta }^{2} \beta_{w}^{2} - 3{\mkern 1mu} \pi^{2} \alpha^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} + 9/2{\mkern 1mu} \pi {\mkern 1mu} \alpha^{2} \hfill \\ n\beta_{\theta }^{2} \lambda_{1}^{2} + 6{\mkern 1mu} \pi^{2} \alpha {\mkern 1mu} n^{2} \beta_{\theta }^{2} \lambda_{1}^{2} + 3{\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} \lambda_{1}^{2} - 6{\mkern 1mu} \pi^{4} \alpha {\mkern 1mu} n^{4} \hfill \\ k_{s} \lambda_{1}^{2} \lambda_{2}^{2} - 3/2{\mkern 1mu} \pi^{3} \alpha^{2} n^{3} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} \hfill \\ \end{gathered}$$

(B16)

$$\begin{gathered} \Delta_{4} = 3/4{\mkern 1mu} \pi^{4} \delta {\mkern 1mu} n^{4} k_{s} \lambda_{1}^{2} \lambda_{2}^{2} + 3/4{\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} k_{s} \lambda_{2}^{2} - 3/4{\mkern 1mu} \hfill \\ \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta }^{2} \lambda_{1}^{2} - 3/4{\mkern 1mu} \pi^{4} \delta {\mkern 1mu} n^{4} k_{s}^{2} \lambda_{2}^{4} \hfill \\ \end{gathered}$$

(B17)

$$\begin{gathered} \Delta_{5} = - 2{\mkern 1mu} I_{2} {\mkern 1mu} \pi^{2} \delta {\mkern 1mu} n^{2} \beta_{\theta } k_{s} \lambda_{2}^{2} + 2{\mkern 1mu} \pi {\mkern 1mu} n\beta_{\theta } k_{s} \lambda_{2}^{2} + \hfill \\ 2{\mkern 1mu} I_{{2{\mkern 1mu} }} \delta {\mkern 1mu} \beta_{\theta }^{3} \hfill \\ \end{gathered}$$

(B18)