1 Introduction

The dynamic analysis of axially loaded thin-walled beams is of paramount importance in the field of structural mechanics, particularly for applications involving complex load-bearing structures, such as those in aerospace, civil, and mechanical engineering systems. Traditional beam theories, including the Bernoulli–Euler and classical Timoshenko–Ehrenfest models, often fall short in capturing the intricate coupled behaviors observed in thin-walled beams subjected to bidirectional bending and torsional vibrations. This limitation necessitates the development of more comprehensive analytical frameworks that can accurately predict the dynamic response of such structures under various loading conditions.

Over the years, the theory of coupled bending-torsion vibrations in beams has been extensively studied by many researchers. Among the pioneers, Bishop and Price [1] introduced theories for Timoshenko–Ehrenfest beams under axial loads. Li et al. [2, 3] further refined these theories by incorporating the stiffness coupling between bending and torsion.

Earlier models, however, did not include warping effects. This gap was addressed by Bercin and Tanaka [4], who integrated warping effects into their research. Yaman and Özdemïr [5] investigated the theory of coupled bidirectional bending and torsion vibrations in beams. Li et al. significantly advanced the analysis of bending-torsion coupled vibrations in monosymmetric thin-walled beams. They introduced a dynamic transfer matrix method incorporating axial load, warping stiffness, shear deformation, and rotary inertia, enhancing accuracy in determining natural frequencies and mode shapes [6, 7]. Additionally, they expanded analytical methods for stochastic vibration analysis, comprehensively including these effects to accurately predict mean-square displacement responses under various random excitations [8,9,10]. These contributions significantly improved the precision of structural vibration analyses and practical engineering designs. Furthermore, Jun et al. [11] examined the coupled vibration theory of bidirectional bending and torsion in Bernoulli–Euler beams subjected to axial loads. Prokic [12] explored the coupled vibration theory for bidirectional bending and torsion in Timoshenko–Ehrenfest beams, continuing this work in 2012 [13] with a focus on axial load effects. Yadav et al. [14] studied the dynamic response of beams under axial loads and their relationship with natural frequencies. Jrad et al. [15] delved into the coupled bending-torsion vibration theory for Vlasov beams. Burlon and Failla [16] provided a comprehensive framework for the dynamic response of beams experiencing coupled bidirectional bending and torsion. Most recently, Cai et al. [17] presented analytical solutions for the coupled bending-torsion vibrations of simply supported beams.

Toward potential engineering applications, Bozyigit et al. [18,19,20,21] established an exact theoretical framework for the vibration characteristics of axially pre-stressed Timoshenko–Ehrenfest beams and addressed related engineering challenges. In addition, frame-type structures with coupled axial and bending effects were investigated [22,23,24], thereby laying a solid foundation for precise dynamic modeling in engineering applications.

J.R. Banerjee’s pioneering work on the Timoshenko–Ehrenfest beam theory [25,26,27,28,29,30,31,32,33] has progressively established an exact dynamic stiffness framework for beam elements, encompassing coupled bending-torsion formulations, axial-load extensions, extensional-torsional coupling, warping effects, deterministic and random load responses, explicit modal analysis, transverse-lateral and axial-bending interactions, and eccentricity corrections, and collectively provides a comprehensive analytical foundation for precise dynamic modeling of beam and frame structures under a diverse range of coupled loading and geometric conditions.

Timoshenko–Ehrenfest beam theory improves the Euler Bernoulli model by accounting for shear deformation as well as considers the rotational inertia of the cross-section which makes the Ehrenfest formulation more accurate for high frequency vibrations and wave propagation in deep beams. These two combatively predict the free vibration behavior of the beam [34].

This paper presents a novel analytical framework for the coupled bidirectional bending and torsional vibrations of non-symmetric, axially loaded thin-walled Timoshenko–Ehrenfest beams. Our approach incorporates axial loads, shear deformation, rotational inertia, and warping stiffness, extending traditional Timoshenko–Ehrenfest theory to encompass complex bending-torsion interactions, resulting in a robust and precise analysis tool.

Using Hamilton’s principle, we derive five coupled differential equations and twelve boundary conditions describing the beam’s forced vibration. By formulating total strain and kinetic energies and deriving the Lagrangian, we account for various deformation modes. Orthogonality conditions for the coupled modes transform these partial differential equations into uncoupled equations via modal analysis, essential for efficiently solving dynamic responses under arbitrary harmonic loads.

Furthermore, this paper introduces an original framework for the coupled bidirectional bending and torsional vibrations of non-symmetric axially loaded thin-walled Timoshenko–Ehrenfest beams. This framework includes the derivation of analytical solutions for the exact frequency response under arbitrary harmonic loads using the normal mode method, the establishment of appropriate orthogonality conditions for the coupled bending-torsion modes, and the construction of precise modal impulse and frequency response functions for any load. The computations for frequency response and eigenvalue problems involve 12\(\times \)12 matrices, providing precise and computationally efficient solutions for all response variables, and are broadly applicable to a wide range of engineering problems. This study also explores the impact of various parameters, such as axial loads, on the natural frequencies of beams, offering practical guidance for engineering design.

In conclusion, this paper presents a rigorous framework for the dynamic analysis of axially loaded thin-walled Timoshenko–Ehrenfest beams, providing a valuable tool for engineers and researchers in advanced structural design. Our findings enhance the understanding of complex loading conditions, promising improved performance and reliability for various engineering applications. Future work will focus on experimental validation and extending the framework to include nonlinear effects and different boundary conditions.

2 Governing equations of motion and the boundary conditions at the ends

The uniform and straight thin-walled beam of length L is shown in Fig. 1. The cross-section of the beam has no symmetrical axis. The shear center and the centroid of the cross-section are denoted by SC and MC, respectively, separated by a distance \(\sqrt{{z_c}^2+{y_c}^2}\). In the right-handed Cartesian coordinate system shown in Fig. 1, the x-axis is assumed to coincide with the elastic axis (i.e. locus of the SC of the cross-section of the thin-walled beam). The bending translations in the z-direction and y-direction, and the torsional rotation about the x-axis of SC, are denoted by \(u_z(x,t),u_y(x,t)\) and \(\psi (x,t)\), respectively, where x and t denote the distance from the origin and time, respectively. The axial deformation along the x-axis is represented by \(u_x(x,t)\). The cross-sectional rotations due to bending in the y and z directions are denoted by \(\theta _y(x,t)\) and \(\theta _z(x,t)\), respectively. A constant compression axial force P is assumed to act through the centroid of the cross-section of the thin-walled beam. P can be positive or negative so that tension is included.

The dynamic response analysis considers the effects of shear deformation, rotational inertia, and warping stiffness. The external excitations acting on the thin-walled beam are represented by a force \(f_x(x,t),f_y(x,t),f_z(x,t)\), per unit length that parallel to sx-axis, sy-axis, sz-axis and applied to the SC together with a torque \(m(x,t),m_y(x,t),m_z(x,t)\), per unit length about sx-axis, sy-axis, sz-axis, respectively (Fig. 1).

In the context of small torsion, the classical relationships used in Vlasov’s model are straightforward:

$$\begin{aligned} \begin{aligned} u_X&= u_x - z \theta _z - y \theta _y - \omega \psi ', \\ u_Y&= u_y - z \psi , \\ u_Z&= u_z + y \psi . \end{aligned} \end{aligned}$$
(1)

Component deformation for the defined displacement field are:

$$\begin{aligned} \begin{aligned} \varepsilon _{xx}&= \frac{\partial u_X}{\partial x} = u_x' - z \theta _z' - y \theta _y' - \omega \psi '', \\ \varepsilon _{xy}&= \frac{\partial u_Y}{\partial x} + \frac{\partial u_X}{\partial z} = u_y' - z \psi ' - \theta _y - \psi ' \frac{\partial \omega }{\partial z}, \\ \varepsilon _{xz}&= \frac{\partial u_Z}{\partial x} + \frac{\partial u_X}{\partial z} = u_z' + y \psi ' - \theta _z - \psi ' \frac{\partial \omega }{\partial z}. \end{aligned} \end{aligned}$$
(2)

For a linear elastic material the stress–strain relationship is defined by Hooke’s law:

$$\begin{aligned} \begin{aligned} \sigma _{xx}&= E \varepsilon _{xx} = E \left( u_x' - z \theta _z' - y \theta _y' - \omega \psi '' \right) , \\ \tau _{xy}&= G \varepsilon _{xy} = G \left( u_y' - z \psi ' - \theta _y - \psi ' \frac{\partial \omega }{\partial y} \right) , \\ \tau _{xz}&= G \varepsilon _{xz} = G \left( u_z' + y \psi ' - \theta _z - \psi ' \frac{\partial \omega }{\partial z} \right) . \end{aligned} \end{aligned}$$
(3)
Fig. 1
Fig. 1
Full size image

Beam with asymmetric cross-section subjected to unsteady distributed load: a mechanical model, b loading conditions, and c cross-section

The governing equations and boundary conditions for the forced vibration of an axially loaded Timoshenko–Ehrenfest thin-walled beam, which includes bidirectional bending and torsional vibration coupling with damping, can be expressed through five coupled differential equations and twelve boundary conditions. These equations and conditions are systematically derived using Hamilton’s principle as outlined below.

The total strain energy U of an axially loaded Timoshenko–Ehrenfest thin-walled beam shown in Fig. 1 is given by:

$$\begin{aligned} \begin{aligned} U&= \frac{1}{2} \int _0^L \left[ \int _A \left( \sigma _{xx} \varepsilon _{xx} + \tau _{xy} \varepsilon _{xy} + \tau _{xz} \varepsilon _{xz} \right) \hbox {d}A \right] - P \left[ \left( \frac{\partial u_Y}{\partial x} \right) ^2 + \left( \frac{\partial u_Z}{\partial x} \right) ^2 \right] \\&\quad - (2 u_x f_x + 2 u_z f_z + 2 u_y f_y + 2 m \psi + 2 m_y \theta _z + 2 m_z \theta _y ) \hbox {d}x, \end{aligned} \end{aligned}$$
(4)

which can be approximated as

$$\begin{aligned} \begin{aligned} U \approx \quad&\frac{1}{2} \int _0^L \Bigg \{ \int _A \Bigg [ E \left( u_x' - z \theta _z' - y \theta _y' - \omega \psi '' \right) ^2 \\&\quad \quad \quad + k_z G \left( u_z' - \theta _z \right) ^2 + k_y G \left( u_y' - \theta _y \right) ^2 + G \left( z^2 + y^2 \right) (\psi ')^2 \Bigg ] \hbox {d}A \\&\quad \quad \quad - P \left[ \left( u_z' + y \psi ' \right) ^2 + \left( u_y' - z \psi ' \right) ^2 \right] \\&\quad \quad \quad - \Big ( 2 u_x f_x + 2 u_z f_z + 2 u_y f_y + 2 m \psi + 2 m_y \theta _z + 2 m_z \theta _y \Big ) \Bigg \} \hbox {d}x \\ \approx \quad&\frac{1}{2} \int _0^L \Bigg \{ EA (u_x')^2 + EI_y (\theta _z')^2 + EI_z (\theta _y')^2 + 2EI_{zy} \theta _y' \theta _z'\\&\quad \quad \quad \quad - P \Big [ (u_z')^2 + 2y_c u_z' \psi ' + (u_y')^2 - 2z_c u_y' \psi ' + \frac{I_s}{\mu } (\psi ')^2 \Big ]\\&\quad \quad \quad \quad + k_z AG(u_z' - \theta _z)^2 + k_y AG(u_y' - \theta _y)^2 + GJ (\psi ')^2\\&\quad \quad \quad \quad + E\Gamma (\psi '')^2 + 2E\psi '' \Big ( -{\tilde{S}}_\omega u_x' + J_{y \omega } \theta _y' + J_{z \omega } \theta _z' \Big ) \\&\quad \quad \quad \quad - \Big ( 2 u_x f_x + 2 u_z f_z + 2 u_y f_y + 2 m \psi + 2 m_y \theta _z + 2 m_z \theta _y \Big ) \Bigg \} \hbox {d}x. \end{aligned} \end{aligned}$$
(5)

The total kinetic energy T of an axially loaded Timoshenko–Ehrenfest thin-walled beam is given by

$$\begin{aligned} \begin{aligned} T&= \frac{1}{2} \int _0^L \int _A \rho \left( {\dot{u}}_x^2 + {\dot{u}}_y^2 + {\dot{u}}_z^2 \right) \hbox {d}A \hbox {d}x \\&= \frac{1}{2} \int _0^L \int _A \rho \Bigg [ ( {\dot{u}}_z + y {\dot{\psi }} )^2 + ( {\dot{u}}_y - z {\dot{\psi }} )^2 + ( {\dot{u}}_x - y {\dot{\theta }}_y - z {\dot{\theta }}_z - \omega {\dot{\psi }}' )^2 \Bigg ] \hbox {d}A \hbox {d}x \\&\approx \frac{1}{2} \int _0^L \Bigg \{ \mu \Big [ ({\dot{u}}_z)^2 + 2 y_c {\dot{u}}_z {\dot{\psi }} + ({\dot{u}}_y)^2 - 2 z_c {\dot{u}}_y {\dot{\psi }} \Big ] + I_s ({\dot{\psi }})^2 + \rho A ({\dot{u}}_x)^2 + 2 \rho I_{zy} {\dot{\theta }}_z {\dot{\psi }} \\&\quad + \rho I_y ({\dot{\theta }}_z)^2 + \rho I_z ({\dot{\theta }}_y)^2 + \rho \Gamma ({\dot{\psi }}')^2 + 2 \rho {\dot{\psi }}' \left( -{\tilde{S}}_\omega {\dot{u}}_x + J_{y\omega } {\dot{\theta }}_y + J_{z\omega } {\dot{\theta }}_z \right) \Bigg \} \hbox {d}x. \end{aligned} \end{aligned}$$
(6)

The Lagrangian is given by \(L=T-U\).

The governing equations of motion and the boundary conditions can be derived conveniently by means of the Hamilton’s principle, which is formulated as

$$\begin{aligned} & \delta \int _{t_1}^{t_2} L \, \hbox {d}t = 0, \end{aligned}$$
(7)
$$\begin{aligned} & \delta u_x = \delta u_y = \delta u_z = \delta \theta _y = \delta \theta _z = 0 \quad \text {at} \quad t = t_1, t_2, \end{aligned}$$
(8)
$$\begin{aligned} & \begin{aligned} \delta \int _{t_1}^{t_2} L \, \hbox {d}t&= \delta \frac{1}{2} \int _{t_1}^{t_2} \int _0^L \Bigg \{ \mu \left[ ({\dot{u}}_z)^2 + 2 y_c {\dot{u}}_z {\dot{\psi }} + ({\dot{u}}_y)^2 - 2 z_c {\dot{u}}_y {\dot{\psi }} \right] + I_s ({\dot{\psi }})^2 + \rho A ({\dot{u}}_x)^2 \\&\quad + 2 \rho I_{zy} {\dot{\theta }}_z {\dot{\theta }}_y + \rho I_y ({\dot{\theta }}_z)^2 + \rho I_z ({\dot{\theta }}_y)^2 + \rho \Gamma ({\dot{\psi }}')^2 + 2 \rho {\dot{\psi }}' \left( -{\tilde{S}}_\omega {\dot{u}}_x + J_{y \omega } {\dot{\theta }}_y + J_{z \omega } {\dot{\theta }}_z \right) \\&\quad -\Bigg [ EA (u_x')^2 + EI_y (\theta _z')^2 + EI_z (\theta _y')^2 + 2 EI_{zy} \theta _y' \theta _z' + k_z AG (u_z' - \theta _z)^2\\&\quad + k_y AG (u_y' - \theta _y)^2 - P \left[ (u_z')^2 + 2 y_c u_z' \psi ' + (u_y')^2 - 2 z_c u_y' \psi ' + \frac{I_s}{\mu } (\psi ')^2 \right] \\&\quad + GJ (\psi ')^2 + E \Gamma (\psi '')^2 + 2 E \psi '' \left( -{\tilde{S}}_\omega u_x' + J_{y \omega }\theta _y' + J_{z \omega } \theta _z' \right) \\&\quad - (2 u_x f_x + 2 u_z f_z + 2 u_y f_y + 2 m \psi + 2 m_y \theta _z + 2 m_z \theta _y) \Bigg ] \Bigg \} \hbox {d}x \, \hbox {d}t. \end{aligned} \end{aligned}$$
(9)

A coordinate system is established with the SC as the origin. The x-axis aligns with the beam’s longitudinal axis, while the yz-plane represents the beam’s cross-section. It is designated as the principal sectoral pole, and the principal sectoral zero is chosen so that holds. At the same time, we can choose an orthogonal coordinate system yOz such that the product of inertia \(I_{zy}=0\). Because if \(I_{zy}>0\), we can rotate the orthogonal coordinate system yOz counterclockwise by \(90 ^{\circ }\). According to the definition of the product of inertia, this rotation will result in \(I_{zy}<0\). Given that the product of inertia varies continuously with the rotation angle, the intermediate value theorem guarantees the existence of an angle at which \(I_{zy}=0\).

Substituting Eqs. (5), (6) and (7) into Eq. (9) and following the standard procedures, we obtain the governing equations of motion and the boundary conditions. The boundary conditions at the ends \(x=0\) and \(x=L\):

$$\begin{aligned} & [-EA(u_x^\prime )]\delta ux=0, \end{aligned}$$
(10)
$$\begin{aligned} & [P(u_y^\prime -z_c\psi ^\prime )-k_yAG(u_y^\prime -\theta _y)] \delta uy=0,\end{aligned}$$
(11)
$$\begin{aligned} & [P(u_z^\prime +y_c\psi ^\prime )-k_zAG(u_z^\prime -\theta _z)] \delta uz=0, \end{aligned}$$
(12)
$$\begin{aligned} & \left[ P\left( \frac{I_s}{m}\psi ^\prime -z_cu_y^\prime +y_cu_z^\prime \right) +E\Gamma \psi ^{\prime \prime \prime }-GJ\psi ^\prime -\rho \Gamma {\ddot{\psi }}^\prime \right] \delta \psi =0, \end{aligned}$$
(13)
$$\begin{aligned} & (-EI_y\theta _z^\prime )\delta \theta _z=0, \end{aligned}$$
(14)
$$\begin{aligned} & (-EI_z\theta _y^\prime )\delta \theta _y=0, \end{aligned}$$
(15)
$$\begin{aligned} & (-E\Gamma \psi ^{\prime \prime \prime })\delta \psi ^\prime =0. \end{aligned}$$
(16)

Therefore, it can be concluded that:

$$\begin{aligned} \left\{ \begin{aligned}&\text {Axial force: } F = -EA(u_x'), \\&\text {Shear force in y-direction: } S_y = P(u_y' - z_c \psi ') - k_y AG(u_y' - \theta _y), \\&\text {Shear force in z-direction: } S_z = P(u_z' + y_c \psi ') - k_z AG(u_z' - \theta _z), \\&\text {Torque: } T = P\left( \frac{I_s}{m} \psi ' - z_c u_y' + y_c u_z' \right) + E\Gamma \psi ''' - GJ \psi ' - \rho \Gamma \ddot{\psi }', \\&\text {Bending moment in y-direction: } M_y = -EI_z \theta _y', \\&\text {Bending moment in z-direction: } M_z = -EI_y \theta _z', \\&\text {Bimoment: } B = -E\Gamma \psi ''. \end{aligned} \right. \end{aligned}$$
(17)

The governing equations of motion are given by:

$$\begin{aligned} & \rho A\frac{\partial ^2u_x(x,t)}{\partial t^2}-EA\frac{\partial ^2u_x(x,t)}{\partial x^2}=f_x(x,t), \end{aligned}$$
(18)
$$\begin{aligned} & \begin{aligned}&\mu \frac{\partial ^2u_y(x,t)}{\partial t^2}-\mu z_c\frac{\partial ^2\psi (x,t)}{\partial t^2}-k_yAG\left( \frac{\partial ^2u_y(x,t)}{\partial x^2}-\frac{\partial \theta _y(x,t)}{\partial x}\right) \\ &\quad +P\left( \frac{\partial ^2u_y(x,t)}{\partial x^2}-z_c\frac{\partial ^2\psi (x,t)}{\partial x^2}\right) =f_y(x,t);\\&\mu \frac{\partial ^2u_z(x,t)}{\partial t^2}+\mu y_c\frac{\partial ^2\psi (x,t)}{\partial t^2}-k_zAG\left( \frac{\partial ^2u_z(x,t)}{\partial x^2}-\frac{\partial \theta _z(x,t)}{\partial x}\right) \\ &\quad +P\left( \frac{\partial ^2u_z(x,t)}{\partial x^2}+y_c\frac{\partial ^2\psi (x,t)}{\partial x^2}\right) =f_z(x,t); \end{aligned} \end{aligned}$$
(19)
$$\begin{aligned} & \begin{aligned}&\rho I_z\frac{\partial ^2\theta _y(x,t)}{\partial t^2}-EI_z\frac{\partial ^2\theta _y(x,t)}{\partial x^2}-k_yAG\left( \frac{\partial u_y(x,t)}{\partial x}-\theta _y(x,t)\right) =m_z(x,t),\\&\rho I_y\frac{\partial ^2\theta _z(x,t)}{\partial t^2}-EI_y\frac{\partial ^2\theta _z(x,t)}{\partial x^2}-k_zAG\left( \frac{\partial u_z(x,t)}{\partial x}-\theta _z(x,t)\right) =m_y(x,t), \end{aligned} \end{aligned}$$
(20)
$$\begin{aligned} & \begin{aligned}&GJ\frac{\partial ^2\psi (x,t)}{\partial x^2}+\rho \Gamma \frac{\partial ^4\psi (x,t)}{\partial x^2\partial t^2}-P\left( \frac{I_s}{\mu }\frac{\partial ^2\psi (x,t)}{\partial x^2}-z_c\frac{\partial ^2u_y(x,t)}{\partial x^2}+y_c\frac{\partial ^2u_z(x,t)}{\partial x^2}\right) \\&\quad -\mu y_c\frac{\partial ^2u_z(x,t)}{\partial t^2}+\mu z_c\frac{\partial ^2u_y(x,t)}{\partial t^2}-I_s\frac{\partial ^2\psi (x,t)}{\partial t^2}-E\Gamma \frac{\partial ^4\psi (x,t)}{\partial x^4}=m(x,t). \end{aligned} \end{aligned}$$
(21)

Including the damping term, the equations of motion can be modified as follows:

$$\begin{aligned} & \rho A\frac{\partial ^2u_x(x,t)}{\partial t^2}-EA\frac{\partial ^2u_x(x,t)}{\partial x^2}+c_4\frac{\partial u_x(x,t)}{\partial t}=f_x(x,t), \end{aligned}$$
(22)
$$\begin{aligned} & \begin{aligned}&\mu \frac{\partial ^2u_y(x,t)}{\partial t^2}-\mu z_c\frac{\partial ^2\psi (x,t)}{\partial t^2}-k_yAG\left( \frac{\partial ^2u_y(x,t)}{\partial x^2}-\frac{\partial \theta _y(x,t)}{\partial x}\right) \\&\quad +P\left( \frac{\partial ^2u_y(x,t)}{\partial x^2}-z_c\frac{\partial ^2\psi (x,t)}{\partial x^2}\right) +c_{1z}\left( \frac{\partial u_y(x,t)}{\partial t}-z_c\frac{\partial \psi (x,t)}{\partial t}\right) =f_y(x,t),\\&\mu \frac{\partial ^2u_z(x,t)}{\partial t^2}+\mu y_c\frac{\partial ^2\psi (x,t)}{\partial t^2}-k_zAG\left( \frac{\partial ^2u_z(x,t)}{\partial x^2}-\frac{\partial \theta _z(x,t)}{\partial x}\right) \\&\quad +P\left( \frac{\partial ^2u_z(x,t)}{\partial x^2}+y_c\frac{\partial ^2\psi (x,t)}{\partial x^2}\right) +c_{1y}\left( \frac{\partial u_z(x,t)}{\partial t}+y_c\frac{\partial \psi (x,t)}{\partial t}\right) =f_z(x,t); \end{aligned} \end{aligned}$$
(23)
$$\begin{aligned} & \begin{aligned} \rho I_z\frac{\partial ^2\theta _y(x,t)}{\partial t^2}-EI_z\frac{\partial ^2\theta _y(x,t)}{\partial x^2}-k_yAG\left( \frac{\partial u_y(x,t)}{\partial x}-\theta _y(x,t)\right) +c_{3z}\frac{\partial \theta _y(x,t)}{\partial t}&=m_z(x,t),\\ \rho I_y\frac{\partial ^2\theta _z(x,t)}{\partial t^2}-EI_y\frac{\partial ^2\theta _z(x,t)}{\partial x^2}-k_zAG\left( \frac{\partial u_z(x,t)}{\partial x}-\theta _z(x,t)\right) +c_{3y}\frac{\partial \theta _z(x,t)}{\partial t}&=m_y(x,t), \end{aligned} \end{aligned}$$
(24)
$$\begin{aligned} & \begin{aligned}&GJ\frac{\partial ^2\psi (x,t)}{\partial x^2}+\rho \Gamma \frac{\partial ^4\psi (x,t)}{\partial x^2\partial t^2}-P\left( \frac{I_s}{\mu }\frac{\partial ^2\psi (x,t)}{\partial x^2}-z_c\frac{\partial ^2u_y(x,t)}{\partial x^2}+y_c\frac{\partial ^2u_z(x,t)}{\partial x^2}\right) \\&\quad -\mu y_c\frac{\partial ^2u_z(x,t)}{\partial t^2}+\mu z_c\frac{\partial ^2u_y(x,t)}{\partial t^2}-I_s\frac{\partial ^2\psi (x,t)}{\partial t^2}-E\Gamma \frac{\partial ^4\psi (x,t)}{\partial x^4}+c_5\frac{\partial ^3\psi (x,t)}{\partial x^2\partial t}\\&\quad -c_2\frac{\partial \psi (x,t)}{\partial t}-c_{1y}y_c\frac{\partial u_z(x,t)}{\partial t}+c_{1z}z_c\frac{\partial u_y(x,t)}{\partial t}=m(x,t), \end{aligned} \end{aligned}$$
(25)

where all the variables and symbols are defined in Appendix A.

Equation (18), which describes axial vibration, is decoupled from the rest of the system and can be analyzed independently. It should be noted that Eqs. (19)–(21) form a set of coupled system dynamics equations as a whole. Although P does not appear explicitly in Eq. (20), its influence is transmitted through the coupling terms. In reference [7], the control equations in the paper similarly do not include the effect of P on the corresponding equations, which further verifies this point. It is apparent that the various coupled vibration equations for beams suggested by previous researchers [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] are specific instances of this broader set of equations. Under different conditions, these equations simplify to the coupled vibration equations for various beams as developed by earlier researchers.

3 Free vibration of thin-walled Timoshenko–Ehrenfest beams

For undamped free vibration of axially loaded Timoshenko–Ehrenfest thin-walled beam, the external excitations \(f_y(x,t)\), \(f_z(x,t)\), \(m_y(x,t)\), \(m_z(x,t)\) and m(x, t) are set to zero, as are the damping coefficients \(c_{1y}\), \(c_{1z}\), \(c_2\), \(c_{3y}\), \(c_{3z}\) and \(c_4\), in order to determine the natural frequencies and mode shapes of the thin-walled beam. A sinusoidal variation of \(u_y(x,t)\), \(u_z(x,t)\), \(\theta _y(x,t)\), \(\theta _z(x,t)\) and \(\psi (x,t)\) with circular frequency on is assumed to be of the forms:

$$\begin{aligned} \begin{aligned} u_y\left( x,t\right)&=V_y(x)\sin {\omega }t,\\ u_z\left( x,t\right)&=V_z(x)\sin {\omega }t, \\ \theta _y\left( x,t\right)&=\Theta _y(x)\sin {\omega }t,\\ \theta _z\left( x,t\right)&=\Theta _z(x)\sin {\omega }t,\\ \psi \left( x,t\right)&=\Psi (x)\sin {\omega }t, \end{aligned} \end{aligned}$$
(26)

where \(n=1,2,3\ldots \), \(V_y\), \(V_z\), \(\Theta _y\), \(\Theta _z\) and \(\Psi \) are the amplitudes of the sinusoidally varying flexural translation, flexural rotation and torsional rotation, respectively.

Substituting Eqs. (26) into Eqs. (23)–(25) gives the five simultaneous differential equation for \(V_y\), \(V_z\), \(\Theta _y\), \(\Theta _z\) and \(\Psi \):

$$\begin{aligned} & \begin{aligned} -\mu \omega ^2V_y(x)+\mu \omega ^2z_c\Psi (x)-k_yAG\left( \frac{d^2V_y(x)}{\hbox {d}x^2}-\frac{\hbox {d}\Theta _y(x)}{\hbox {d}x}\right) +P\left( \frac{d^2V_y(x)}{\hbox {d}x^2}-z_c\frac{d^2\Psi (x)}{\hbox {d}x^2}\right)&=0 \end{aligned}\end{aligned}$$
(27)
$$\begin{aligned} & \begin{aligned} -\mu \omega ^2V_z(x)-\mu \omega ^2y_c\Psi (x)-k_zAG\left( \frac{d^2V_z(x)}{\hbox {d}x^2}-\frac{\hbox {d}\Theta _z(x)}{\hbox {d}x}\right) +P\left( \frac{d^2V_z(x)}{\hbox {d}x^2}+y_c\frac{d^2\Psi (x)}{\hbox {d}x^2}\right)&=0 \end{aligned} \end{aligned}$$
(28)
$$\begin{aligned} & \begin{aligned} -\rho I_z\omega ^2\Theta _y(x)-EI_z\frac{d^2\Theta _y(x)}{\hbox {d}x^2}-k_yAG\left( \frac{\hbox {d}V_y(x)}{\hbox {d}x}-\Theta _y(x)\right)&=0 \end{aligned}\end{aligned}$$
(29)
$$\begin{aligned} & \begin{aligned} -\rho I_y\omega ^2\Theta _z(x)-EI_y\frac{d^2\Theta _z(x)}{\hbox {d}x^2}-k_zAG\left( \frac{\hbox {d}V_z(x)}{\hbox {d}x}-\Theta _z(x)\right)&=0 \end{aligned} \end{aligned}$$
(30)
$$\begin{aligned} & \begin{aligned}&GJ\frac{d^2\Psi (x)}{\hbox {d}x^2}-\rho \Gamma \omega ^2\frac{d^2\Psi (x)}{\hbox {d}x^2}-P\left( \frac{I_s}{\mu }\frac{d^2\Psi (x)}{\hbox {d}x^2}-z_c\frac{d^2V_y(x)}{\hbox {d}x^2}+y_c\frac{d^2V_z(x)}{\hbox {d}x^2}\right) \\&\quad +\mu \omega ^2y_cV_z(x)-\mu \omega ^2z_cV_y(x)+I_s\omega ^2\Psi (x)-E\Gamma \frac{d^4\Psi (x)}{\hbox {d}x^4} {=0} \end{aligned} \end{aligned}$$
(31)

Let’s assume \(\frac{\hbox {d}}{\hbox {d}x}=D\) is a linear differential operator. By combining Eqs. (27) and (31) , we can form a system of linear equations as follows:

$$\begin{aligned} \underset{(5 \times 5)}{{\textbf{A}}}\underset{(5 \times 1)}{\vec {x}}=\underset{(5 \times 1)}{\vec {0}}. \end{aligned}$$
(32)

For details in regard to the matrix \(\underset{(5 \times 5)}{{\textbf{A}}}\), see Appendix C.

In this system of linear equations \(\underset{(5 \times 1)}{\vec {x}}=(V_y,V_z,\theta _y,\theta _z,\Psi )^T\) cannot be identically zero, so we obtain the following \(\text {det}\underset{(5 \times 5)}{{\textbf{A}}}=0\):

$$\begin{aligned} (a_1D^{12}+a_2D^{10}+a_3D^8+a_4D^6+a_5D^4+a_6D^2+a_7)R=0, \end{aligned}$$
(33)

where \(a_i\) are the coefficients derived from the system of Eq. (32), and \(R=V_y,V_z,{\Theta }_y,{\Theta }_z,{\Psi }\). The solution of the differential Eq. (33) can be obtained by substituting the trial solution \(R=e^{kx}\) to give the characteristic equation:

$$\begin{aligned} a_1k^{12}+a_2k^{10}+a_3k^8+a_4k^6+a_5k^4+a_6k^2+a_7=0. \end{aligned}$$
(34)

Let

$$\begin{aligned} \chi =k^2, \end{aligned}$$
(35)

substituting Eq. (35) into Eq. (34) gives

$$\begin{aligned} a_1\chi ^6+a_2\chi ^5+a_3\chi ^4+a_4\chi ^3+a_5\chi ^2+a_6\chi +a_7=0. \end{aligned}$$
(36)

Using the technique employed by Bishop et al. [35] and combining the solutions for the coupled bending-torsion vibration equations of beams provided by previous researchers. It can be shown that all six roots of Eq. (36) are real, three of them negative and the other three positive. Suppose that the six roots of Eq. (36) are \(\chi _1\), \(\chi _2\), \(\chi _3\), \(-\chi _4\), \(-\chi _5\), \(-\chi _6\), where are \(\chi _j~(j=1-6)\) real and positive. Then the twelve roots of the characteristic Eq. (34), including \(\pm \beta _1\), \(\pm \beta _2\), \(\pm \beta _3\), \(\pm i\beta _4\), \(\pm i\beta _5\), \(\pm i\beta _6\), where \(\beta _j=\sqrt{\chi _j}~(j=1-6)\). It follows that the solution of Eq. (34) is of the following forms:

$$\begin{aligned} \begin{aligned} R(x)=&A_1\cosh {\beta _1}x+A_2\sinh {\beta _1}x+A_3\cosh {\beta _2}x+A_4\sinh {\beta _2}x\\&+A_5\cosh {\beta _3}x+A_6\sinh {\beta _3}x+A_7\cos {\beta _4}x+A_8\sin {\beta _4}x\\&+A_9\cos {\beta _5}x+A_{10}\sin {\beta _5}x+A_{11}\cos {\beta _6}x+A_{12}\sin {\beta _6}x, \end{aligned} \end{aligned}$$
(37)

which further describes the solutions for \(V_y\), \(V_z\), \({\Theta }_y\), \({\Theta }_z\), \({\Psi }\):

$$\begin{aligned} & \begin{aligned} V_y(x)=&A_1\cosh {\beta _1}x+A_2\sinh {\beta _1}x+A_3\cosh {\beta _2}x+A_4\sinh {\beta _2}x\\&+A_5\cosh {\beta _3}x+A_6\sinh {\beta _3}x+A_7\cos {\beta _4}x+A_8\sin {\beta _4}x\\&+A_9\cos {\beta _5}x+A_{10}\sin {\beta _5}x+A_{11}\cos {\beta _6}x+A_{12}\sin {\beta _6}x, \end{aligned} \end{aligned}$$
(38)
$$\begin{aligned} & \begin{aligned} V_z(x)=&B_1\cosh {\beta _1}x+B_2\sinh {\beta _1}x+B_3\cosh {\beta _2}x+B_4\sinh {\beta _2}x\\&+B_5\cosh {\beta _3}x+B_6\sinh {\beta _3}x+B_7\cos {\beta _4}x+B_8\sin {\beta _4}x\\&+B_9\cos {\beta _5}x+B_{10}\sin {\beta _5}x+B_{11}\cos {\beta _6}x+B_{12}\sin {\beta _6}x, \end{aligned} \end{aligned}$$
(39)
$$\begin{aligned} & \begin{aligned} {\Theta }_y(x)=&C_1\cosh {\beta _1}x+C_2\sinh {\beta _1}x+C_3\cosh {\beta _2}x+C_4\sinh {\beta _2}x\\&+C_5\cosh {\beta _3}x+C_6\sinh {\beta _3}x+C_7\cos {\beta _4}x+C_8\sin {\beta _4}x\\&+C_9\cos {\beta _5}x+C_{10}\sin {\beta _5}x+C_{11}\cos {\beta _6}x+C_{12}\sin {\beta _6}x, \end{aligned} \end{aligned}$$
(40)
$$\begin{aligned} & \begin{aligned} {\Theta }_z(x)=&D_1\cosh {\beta _1}x+D_2\sinh {\beta _1}x+D_3\cosh {\beta _2}x+D_4\sinh {\beta _2}x\\&+D_5\cosh {\beta _3}x+D_6\sinh {\beta _3}x+D_7\cos {\beta _4}x+D_8\sin {\beta _4}x\\&+D_9\cos {\beta _5}x+D_{10}\sin {\beta _5}x+D_{11}\cos {\beta _6}x+D_{12}\sin {\beta _6}x, \end{aligned} \end{aligned}$$
(41)
$$\begin{aligned} & \begin{aligned} {\Psi }(x)=&E_1\cosh {\beta _1}x+E_2\sinh {\beta _1}x+E_3\cosh {\beta _2}x+E_4\sinh {\beta _2}x\\&+E_5\cosh {\beta _3}x+E_6\sinh {\beta _3}x+E_7\cos {\beta _4}x+E_8\sin {\beta _4}x\\&+E_9\cos {\beta _5}x+E_{10}\sin {\beta _5}x+E_{11}\cos {\beta _6}x+E_{12}\sin {\beta _6}x, \end{aligned} \end{aligned}$$
(42)

where \(A_1-A_{12}\), \(B_1-B_{12}\), \( C_1-C_{12}\), \(D_1-D_{12}\), and \(E_1-E_{12}\) are six different sets of constants.

By substituting Eq. (38) and Eq. (40) into Eq. (29), the relationships between the constants A and C are derived. Similarly, substituting Eqs. (38), (40) and (42) into Eq. (27) reveals the relationships among the constants A, C and E. Subsequently, substituting Eqs. (38–40) and (42) into Eq. (31) establishes the relationships among the constants A, B, C and E. Finally, substituting Eqs. (38–42) into Eq. (30) elucidates the relationships among the constants A, B, C, D and E. Consequently, it can be seen that the constants B, C, D and E can all be expressed in terms of the constants A.

By applying the particular boundary conditions to the Eqs. (10)–(16), a set of 12 homogeneous algebraic equations

$$\begin{aligned} \underset{(12\times 12)}{\Pi (\omega )}\underset{(12\times 1)}{\vec {a}}=\underset{(12\times 1)}{\vec {0}} \end{aligned}$$
(43)

are obtained, where \(\underset{(12\times 12)}{\Pi (\omega )}\) is a \(12\times 12\) matrix, specified by the boundary conditions, and \(\vec {a}\) is a \(12\times 1\) vector of unknown constants (\(A_1-A_{12}\)). For nontrivial solutions to exist the matrix of coefficients must vanish, yielding a transcendental equation from which the frequencies of the beam can be found. The frequency equation, which is a \(12\times 12\) determinantal equation, is

$$\begin{aligned} \det \bigl [\Pi (\omega )\bigr ]_{12 \times 12}=0, \end{aligned}$$
(44)

in other words, the rank is lower than 12. Together, Eqs. (44) and (34) must be solved numerically for the eigenvalues of the given modes. Once they are known, the mode shapes are specified by Eq. (43).

According to the method introduced in reference [1] and combined with the orthogonality conditions provided by previous research [2, 3, 8,9,10, 16], it can be inferred that the mode functions of the Timoshenko–Ehrenfest beam under axial load, involving bidirectional bending and torsion coupled vibration, must satisfy the following orthogonality conditions.

The first orthogonality condition is:

$$\begin{aligned} \begin{aligned} \int _{0}^{L}&\left( \rho \Gamma \frac{d^2{\Psi }_m}{\hbox {d}x^2}\frac{d^2{\Psi }_n}{\hbox {d}x^2}+\mu {V_y}_m{V_y}_n+\mu {V_z}_m{V_z}_n+\rho I_z{\Theta }_{ym}{\Theta }_{yn}+\rho I_y{\Theta }_{zm}{\Theta }_{zn}+I_s{\Psi }_m{\Psi }_n\right) \\&\qquad -(\mu z_c{V_y}_m{\Psi }_n+\mu z_c{V_y}_n{\Psi }_m+\mu y_c{V_z}_m{\Psi }_n+\mu y_c{V_z}_n{\Psi }_m)\hbox {d}x\\&=m_n\delta _{mn}, \end{aligned} \end{aligned}$$
(45)

where \(m_n\) is the generalized mass in the \(n\text {th}\) mode and \(\delta _{mn}\) is Kronecker delta function.

The second orthogonality condition is

$$\begin{aligned} \begin{aligned}&\left( \omega _m-\omega _n\right) \int _{0}^{L}\Bigg [{-EI_z\frac{\hbox {d}{\Theta }_{yn}}{\hbox {d}x}\frac{\hbox {d}{\Theta }_{ym}}{\hbox {d}x}-k_yAG\left( \frac{\hbox {d}V_{ym}}{\hbox {d}x}-{\Theta }_{ym}\right) \left( \frac{\hbox {d}V_{yn}}{\hbox {d}x}-{\Theta }_{yn}\right) }-EI_y\frac{\hbox {d}{\Theta }_{yn}}{\hbox {d}x}\frac{\hbox {d}{\Theta }_{ym}}{\hbox {d}x}\\&\qquad -k_zAG\left( \frac{\hbox {d}V_{zm}}{\hbox {d}x}-{\Theta }_{zm}\right) \left( \frac{\hbox {d}V_{zn}}{\hbox {d}x}-{\Theta }_{zn}\right) +GJ\frac{d{\Psi }_n}{\hbox {d}x}\frac{d{\Psi }_m}{\hbox {d}x}-E{\Gamma }\frac{d^2{\Psi }_n}{\hbox {d}x^2}\frac{d^2{\Psi }_m}{\hbox {d}x^2}\Bigg ]\hbox {d}x\\&\qquad -\omega _m\omega _n\left( \omega _m-\omega _n\right) \int _{0}^{L}\left( \rho \Gamma \frac{d^2{\Psi }_m}{\hbox {d}x^2}\frac{d^2{\Psi }_n}{\hbox {d}x^2}+\mu {V_y}_m{V_y}_n\right. +\mu {V_z}_m{V_z}_n+\rho I_z{\Theta }_{ym}{\Theta }_{yn}+\rho I_y{\Theta }_{zm}{\Theta }_{zn}\\&\qquad \left. +I_s{\Psi }_m{\Psi }_n\right) -(\mu z_c{V_y}_m{\Psi }_n+\mu z_c{V_y}_n{\Psi }_m+\mu y_c{V_z}_m{\Psi }_n+\mu y_c{V_z}_n{\Psi }_m)\hbox {d}x\\&\quad =0. \end{aligned} \end{aligned}$$
(46)

The orthogonality conditions are introduced to convert the five partial coupled differential equations governing the beam motion under an impulsive load into a set of uncoupled differential equations leading to the modal impulse and modal frequency response functions of the beam. The latter can be used to calculate the response to arbitrary loads as described in Sect. 4.

4 Forced vibration analysis of axially loaded thin-walled Timoshenko–Ehrenfest beams

The modal expansion method employed herein assumes small deformations and linear elastic behavior, and is therefore valid only under that assumption. The solution to the forced vibration problem can be expressed as

$$\begin{aligned} \begin{aligned} u_y\left( x,t\right) =\sum _{n=1}^{\infty }{q_n(t)}V_{yn}(x),\\ u_z\left( x,t\right) =\sum _{n=1}^{\infty }{q_n(t)}V_{zn}(x),\\ \theta _y\left( x,t\right) =\sum _{n=1}^{\infty }{q_n(t)}{\Theta }_{yn}(x),\\ \theta _z\left( x,t\right) =\sum _{n=1}^{\infty }{q_n(t)}{\Theta }_{zn}(x),\\ \psi \left( x,t\right) =\sum _{n=1}^{\infty }{q_n(t)}{\Psi }_n(x), \end{aligned} \end{aligned}$$
(47)

where \(q_n(t)\) is a time-dependent generalized coordinate for the \(n\text {th}\) mode. Substitution of Eq. (47) into Eqs. (23)–(25), and using Eqs. (27)–(31) yields

$$\begin{aligned} & \begin{aligned}&\sum _{n=1}^{\infty }{[\mu (V_{yn}-z_c{\Psi }_n){\ddot{q}}_n+c_{1z}(V_{yn}-z_c{\Psi }_n){{\dot{q}}}_n}+\mu {\omega _n}^2(V_{yn}-z_c{\Psi }_n)q_n]=f_y(x,t),\\&\sum _{n=1}^{\infty }{[\mu (V_{zn}+y_c{\Psi }_n){\ddot{q}}_n+c_{1y}(V_{zn}+y_c{\Psi }_n){{\dot{q}}}_n}+\mu {\omega _n}^2(V_{zn}+y_c{\Psi }_n)q_n]=f_z(x,t),\\&\sum _{n=1}^{\infty }{[(\rho I_z{\Theta }_{yn}){\ddot{q}}_n+(c_{3z}{\Theta }_{yn}){{\dot{q}}}_n}+(\rho I_z{\omega _n}^2{\Theta }_{yn})q_n]=m_z(x,t),\\&\sum _{n=1}^{\infty }{[(\rho I_y{\Theta }_{zn}){\ddot{q}}_n+(c_{3y}{\Theta }_{zn}){{\dot{q}}}_n}+(\rho I_y{\omega _n}^2{\Theta }_{zn})q_n]=m_y(x,t),\\ \end{aligned} \end{aligned}$$
(48)
$$\begin{aligned} & \begin{aligned}&\sum _{n=1}^{\infty }\left[ \left( \rho \Gamma \frac{d^2{\Psi }_n}{\hbox {d}x^2}-\mu y_cV_{zn}+\mu z_cV_{yn}-I_s{\Psi }_n\right) {\ddot{q}}_n\right. \\&\quad +\left( c_5\frac{d^2{\Psi }_n}{\hbox {d}x^2}-c_{1y}y_cV_{zn}+c_{1z}z_cV_{yn}-c_2{\Psi }_n\right) {{\dot{q}}}_n\\&\quad \left. +{\omega _n}^2\left( \rho \Gamma \frac{d^2{\Psi }_n}{\hbox {d}x^2}-\mu y_cV_{zn}+\mu z_cV_{yn}-I_s{\Psi }_n\right) q_n\right] =m(x,t), \end{aligned} \end{aligned}$$
(49)

where superscript dot denotes differentiation with respect to time.

Multiplying Eqs. (48)–(49) by \(V_{ym}\), \(V_{zm}\), \({\Theta }_{ym}\), \({\Theta }_{zm}\) and \({\Psi }_m\), respectively, then summing up these five equations and integrating from 0 to L, and using orthogonality condition (45)–(46) gives

$$\begin{aligned} {\ddot{q}}_n\left( t\right) +2\zeta _n\omega _n{{\dot{q}}}_n(t)+{\omega _n}^2q_n(t)=[P_{yn}(t)+P_{zn}(t)+M_n(t)+M_{yn}(t)+M_{zn}(t)], \end{aligned}$$
(50)

where

$$\begin{aligned} \begin{aligned} P_{yn}(t)&=\frac{1}{m_n}\int _{0}^{L}{V_{yn}(x)f_y(x,t)}\hbox {d}x,\\ P_{zn}(t)&=\frac{1}{m_n}\int _{0}^{L}{V_{zn}(x)f_z(x,t)}\hbox {d}x,\\ M_n(t)&=\frac{1}{m_n}\int _{0}^{L}{{\Psi }_n(x)m(x,t)}\hbox {d}x,\\ M_{yn}(t)&=\frac{1}{m_n}\int _{0}^{L}{{\Theta }_{yn}(x)m_y(x,t)}\hbox {d}x,\\ M_{zn}(t)&=\frac{1}{m_n}\int _{0}^{L}{{\Theta }_{zn}(x)m_z(x,t)}\hbox {d}x,\\ \zeta _n&=\frac{c_{1y}}{2\mu \omega _n}=\frac{c_{1z}}{2\mu \omega _n}=\frac{c_2}{2I_s\omega _n}=\frac{c_{3y}}{2\rho I_y\omega _n}=\frac{c_{1z}}{2\rho I_z\omega _n}=\frac{c_5}{2\rho \Gamma \omega _n}. \end{aligned} \end{aligned}$$
(51)

Here, \(\zeta _n\) is a nondimensional quantity known as the viscous damping factor. Here the assumption, \(c_{1z}=c_{1y}\), \(c_2=\frac{I_s}{\mu }c_{1y}\), \(c_{3y}=\frac{\rho I_y}{\mu }c_{1y}\), \(c_{3z}=\frac{\rho I_z}{\mu }c_{1y}\), \(c_5=\frac{\rho \Gamma }{\mu }c_{1y}\) have been made to take advantage of the orthogonality conditions Eqs. (45)–(46) in order to avoid having coupling terms \({\dot{q}}_n\) in Eq. (50).

By using Duhamel’s integral, the general solution of Eq. (50) can be obtained as

$$\begin{aligned} \begin{aligned}&q_n(t)=\ e^{-\zeta _n\omega _nt}\left[ q_n(t_0)\cos {(}\omega _{nd}t)+\frac{{{\dot{q}}}_n(t_0)+\zeta _n\omega _nq_n(t_0)}{\omega _{nd}}\sin {(}\omega _{nd}t)\right] \\&\quad + \frac{1}{\omega _{nd}} \int _{t_0}^{t_e} e^{-\zeta _n \omega _n (t-\tau )} \,\sin \!\bigl [\omega _{nd}(t-\tau )\bigr ]\, \Bigl [ P_{y n}(\tau ) + P_{z n}(\tau ) + M_{n}(\tau ) + M_{y n}(\tau ) + M_{z n}(\tau ) \Bigr ] \,\hbox {d}\tau , \end{aligned} \end{aligned}$$
(52)

where \(\omega _{nd}=\omega _n\sqrt{1-\zeta _n^2}\), \(A_n\) and \(B_n\) are coefficients related to the initial conditions:

$$\begin{aligned} A_n=q_n(t_0),~B_n=\frac{{\dot{q}}_n(t_0)+\zeta _n\omega _n q_n(t_0)}{\omega _{nd}}. \end{aligned}$$
(53)

Substitution of Eq. (52) into Eq. (47) gives the general solutions for bending deflections in the y-direction \(u_y\left( x,t\right) \), bending deflections in the z-direction \(u_z\left( x,t\right) \), cross-sectional rotation due to bending in the y-direction \(\theta _y(x,t)\), cross-sectional rotation due to bending in the z-direction \(\theta _z(x,t)\), torsional rotation about the x-axis of the SCs \(\psi \left( x,t\right) \) in the following form, letting \(t_0=0\):

$$\begin{aligned} & \begin{aligned}&u_y\left( x,t\right) =\sum _{n=1}^{\infty }{V_{yn}(x)}\left\{ e^{-\zeta _n\omega _nt}\left[ A_n\cos {(}\omega _{nd}t)+B_n\sin {(}\omega _{nd}t)\right] \right. \\&\quad + \frac{1}{\omega _{nd}} \int _{t_0}^{t_e} e^{-\zeta _n \omega _n (t-\tau )} \,\sin \!\bigl [\omega _{nd}(t-\tau )\bigr ]\, \Bigl [ P_{y n}(\tau ) + P_{z n}(\tau ) + M_{n}(\tau ) + M_{y n}(\tau ) + M_{z n}(\tau ) \Bigr ] \,\hbox {d}\tau , \end{aligned} \end{aligned}$$
(54)
$$\begin{aligned} & \begin{aligned}&u_z\left( x,t\right) =\sum _{n=1}^{\infty }{V_{zn}(x)}\left\{ e^{-\zeta _n\omega _nt}\left[ A_n\cos {(}\omega _{nd}t)+B_n\sin {(}\omega _{nd}t)\right] \right. \\&\quad + \frac{1}{\omega _{nd}} \int _{t_0}^{t_e} e^{-\zeta _n \omega _n (t-\tau )} \,\sin \!\bigl [\omega _{nd}(t-\tau )\bigr ]\, \Bigl [ P_{y n}(\tau ) + P_{z n}(\tau ) + M_{n}(\tau ) + M_{y n}(\tau ) + M_{z n}(\tau ) \Bigr ] \,\hbox {d}\tau , \end{aligned} \end{aligned}$$
(55)
$$\begin{aligned} & \begin{aligned}&\theta _y\left( x,t\right) =\sum _{n=1}^{\infty }{{\Theta }_{yn}(x)}\left\{ e^{-\zeta _n\omega _nt}\left[ A_n\cos {(}\omega _{nd}t)+B_n\sin {(}\omega _{nd}t)\right] \right. \\&\quad + \frac{1}{\omega _{nd}} \int _{t_0}^{t_e} e^{-\zeta _n \omega _n (t-\tau )} \,\sin \!\bigl [\omega _{nd}(t-\tau )\bigr ]\, \Bigl [ P_{y n}(\tau ) + P_{z n}(\tau ) + M_{n}(\tau ) + M_{y n}(\tau ) + M_{z n}(\tau ) \Bigr ] \,\hbox {d}\tau , \end{aligned} \end{aligned}$$
(56)
$$\begin{aligned} & \begin{aligned}&\theta _z\left( x,t\right) =\sum _{n=1}^{\infty }{{\Theta }_{zn}(x)}\left\{ e^{-\zeta _n\omega _nt}\left[ A_n\cos {(}\omega _{nd}t)+B_n\sin {(}\omega _{nd}t)\right] \right. \\&\quad + \frac{1}{\omega _{nd}} \int _{t_0}^{t_e} e^{-\zeta _n \omega _n (t-\tau )} \,\sin \!\bigl [\omega _{nd}(t-\tau )\bigr ]\, \Bigl [ P_{y n}(\tau ) + P_{z n}(\tau ) + M_{n}(\tau ) + M_{y n}(\tau ) + M_{z n}(\tau ) \Bigr ] \,\hbox {d}\tau , \end{aligned} \end{aligned}$$
(57)
$$\begin{aligned} & \begin{aligned}&\psi \left( x,t\right) =\sum _{n=1}^{\infty }{{\Psi }_n(x)}\left\{ e^{-\zeta _n\omega _nt}\left[ A_n\cos {(}\omega _{nd}t)+B_n\sin {(}\omega _{nd}t)\right] \right. \\&\quad + \frac{1}{\omega _{nd}} \int _{t_0}^{t_e} e^{-\zeta _n \omega _n (t-\tau )} \,\sin \!\bigl [\omega _{nd}(t-\tau )\bigr ]\, \Bigl [ P_{y n}(\tau ) + P_{z n}(\tau ) + M_{n}(\tau ) + M_{y n}(\tau ) + M_{z n}(\tau ) \Bigr ] \,\hbox {d}\tau . \end{aligned} \end{aligned}$$
(58)

The solutions for shear force in y-direction \(S_y(x,t)\), shear force in z-direction \(S_z(x,t)\) and torque T(x, t), bending moment in y-direction \(M_y(x,t)\), bending moment in z-direction \(M_z(x,t)\), bimoment B(x, t) are given by

$$\begin{aligned} \begin{aligned} S_y&= P\!\bigl (u_y' - z_c\psi '\bigr ) - k_y A G\!\bigl (u_y' - \theta _y\bigr ),\\ S_z&= P\!\bigl (u_z' + y_c\psi '\bigr ) - k_z A G\!\bigl (u_z' - \theta _z\bigr ),\\ T&= P\!\Bigl (\tfrac{I_s}{m}\psi ' - z_c u_y' + y_c u_z'\Bigr ) + E\Gamma \psi ''' - GJ\psi ' - K_y\theta _y' - K_z\theta _z' - \rho \Gamma \ddot{\psi }',\\ M_y&= -EI_z \theta _y' - EI_{zy}\theta _z' - K_y\psi ' + EAy_c u',\\ M_z&= -EI_y \theta _z' - EI_{zy}\theta _y' - K_z\psi ' + EAz_c u_x',\\ B&= -E\Gamma \psi ''. \end{aligned} \end{aligned}$$
(59)

To obtain the solution for a series of concentrated loads, the externally applied loading is assumed in the forms

$$\begin{aligned} \begin{aligned} f_y\left( x,t\right)&={P_y}_i\delta \left( x-a_i\right) \cos {{\Omega }_i}(t-t_0)\\ f_z\left( x,t\right)&={P_z}_i\delta \left( x-b_i\right) \cos {{\Omega }_i}(t-t_0)\\ m\left( x,t\right)&=M_i\delta \left( x-c_i\right) \cos {{\Omega }_i}(t-t_0)\\ m_y\left( x,t\right)&=M_{yi}\delta \left( x-d_i\right) \cos {{\Omega }_i}(t-t_0)\\ m_z\left( x,t\right)&=M_{zi}\delta \left( x-e_i\right) \cos {{\Omega }_i}(t-t_0) \end{aligned} \end{aligned}$$
(60)

which represent a system of concentrated simple harmonic forces and torques with circular frequency \({\Omega }_i\) acting at points \(a_i\), \(b_i\), \(c_i\), \(d_i\), \(e_i\), respectively, where \(\delta \left( x\right) \) is the Dirac delta function.

Then the dynamic response at \(t_0=0\) becomes:

$$\begin{aligned} & \begin{aligned} u_{y}(x,t)=&\sum _{n=1}^{\infty } V_{yn}(x)\Bigl \{e^{-\zeta _{n}\omega _{n}t}\bigl [A_{n}\cos (\omega _{nd}t)+B_{n}\sin (\omega _{nd}t)\bigr ] +\sum _{i=1}^{N} X_{i}\cos (\Omega _{i}t-\alpha )\\&-\sum _{i=1}^{N} X_{i} e^{-\zeta _{n}\omega _{n}t}\bigl [\xi _{n}\omega _{n}\cos \alpha +\Omega _{i}\sin \alpha \,\omega _{nd}\sin (\omega _{nd}t) +\cos \alpha \cos (\omega _{nd}t)\bigr ] \Bigr \}, \end{aligned} \end{aligned}$$
(61)
$$\begin{aligned} & \begin{aligned} u_{z}(x,t)=&\sum _{n=1}^{\infty } V_{zn}(x)\Bigl \{e^{-\zeta _{n}\omega _{n}t}\bigl [A_{n}\cos (\omega _{nd}t)+B_{n}\sin (\omega _{nd}t)\bigr ] +\sum _{i=1}^{N} X_{i}\cos (\Omega _{i}t-\alpha ) \\&-\sum _{i=1}^{N} X_{i} e^{-\zeta _{n}\omega _{n}t}\bigl [\xi _{n}\omega _{n}\cos \alpha +\Omega _{i}\sin \alpha \,\omega _{nd}\sin (\omega _{nd}t) +\cos \alpha \cos (\omega _{nd}t)\bigr ] \Bigr \}; \end{aligned} \end{aligned}$$
(62)
$$\begin{aligned} & \begin{aligned} \theta _{y}(x,t)=&\sum _{n=1}^{\infty } \Theta _{yn}(x)\Bigl \{e^{-\zeta _{n}\omega _{n}t}\bigl [A_{n}\cos (\omega _{nd}t)+B_{n}\sin (\omega _{nd}t)\bigr ] +\sum _{i=1}^{N} X_{i}\cos (\Omega _{i}t-\alpha )\\&-\sum _{i=1}^{N} X_{i} e^{-\zeta _{n}\omega _{n}t}\bigl [\xi _{n}\omega _{n}\cos \alpha +\Omega _{i}\sin \alpha \,\omega _{nd}\sin (\omega _{nd}t) +\cos \alpha \cos (\omega _{nd}t)\bigr ] \Bigr \}, \end{aligned} \end{aligned}$$
(63)
$$\begin{aligned} & \begin{aligned} \theta _{z}(x,t)=&\sum _{n=1}^{\infty } \Theta _{zn}(x)\Bigl \{e^{-\zeta _{n}\omega _{n}t}\bigl [A_{n}\cos (\omega _{nd}t)+B_{n}\sin (\omega _{nd}t)\bigr ]+\sum _{i=1}^{N} X_{i}\cos (\Omega _{i}t-\alpha ) \\&-\sum _{i=1}^{N} X_{i} e^{-\zeta _{n}\omega _{n}t}\bigl [\xi _{n}\omega _{n}\cos \alpha +\Omega _{i}\sin \alpha \,\omega _{nd}\sin (\omega _{nd}t) +\cos \alpha \cos (\omega _{nd}t)\bigr ] \Bigr \}, \end{aligned} \end{aligned}$$
(64)
$$\begin{aligned} & \begin{aligned} \psi (x,t)=&\sum _{n=1}^{\infty } \Psi _{n}(x)\Bigl \{e^{-\zeta _{n}\omega _{n}t}\bigl [A_{n}\cos (\omega _{nd}t)+B_{n}\sin (\omega _{nd}t)\bigr ]+\sum _{i=1}^{N} X_{i}\cos (\Omega _{i}t-\alpha ) \\&-\sum _{i=1}^{N} X_{i} e^{-\zeta _{n}\omega _{n}t}\bigl [\xi _{n}\omega _{n}\cos \alpha +\Omega _{i}\sin \alpha \,\omega _{nd}\sin (\omega _{nd}t) +\cos \alpha \cos (\omega _{nd}t)\bigr ] \Bigr \}, \end{aligned} \end{aligned}$$
(65)

where

$$\begin{aligned} & X_i=\frac{F_i}{\omega _n}\frac{1}{\sqrt{(1-\gamma ^2)^2+(2\xi _n\gamma )^2}},~\alpha =\hbox {arctan}{\frac{2\xi _n\gamma }{1-\gamma ^2}},~ \gamma =\frac{{\Omega }_i}{\omega _n}, \end{aligned}$$
(66)
$$\begin{aligned} & F_i=\frac{P_{yi}V_{yn}(a_i)+P_{zi}V_{zn}(b_i)+M_i{\Psi }_n(c_i)+M_{yi}{\Theta }_{yn}(d_i)+M_{zi}{\Theta }_{zn}(e_i)}{m_n}. \end{aligned}$$
(67)

Finally, Eqs. (61)–(65) provide the general solutions for bending deflections in the y-direction \(u_y\left( x,t\right) \), bending deflections in the z-direction \(u_z\left( x,t\right) \), cross-sectional rotation due to bending in the y-direction \(\theta _y(x,t)\), cross-sectional rotation due to bending in the z-direction \(\theta _z(x,t)\), torsional rotation about the x-axis of the SC \(\psi \left( x,t\right) \) for simple harmonically varying multi-point force and torque loading at given locations.

5 Numerical examples

5.1 Natural frequency

The previous section provides an in-depth explanation of the dynamic theory for the Timoshenko–Ehrenfest beam. Due to the intricate nature of the calculations involved, a simplified approach is adopted to find the natural frequencies of the Timoshenko–Ehrenfest beam. This part of the study concentrates on the scenario of a beam with simple supports, as outlined in Prokic’s study [12]. For a beam supported at both ends by fork supports (which prevent rotation and allow free warping), the boundary conditions are specified as:

$$\begin{aligned} \left( V_y,V_z,\frac{\hbox {d}{\Theta }_y}{\hbox {d}x},\frac{\hbox {d}{\Theta }_z}{\hbox {d}x},{\Psi },\frac{d^2{\Psi }}{\hbox {d}x^2}\right) ^T=\vec {0}. \end{aligned}$$
(68)

These requirements are satisfied by taking

$$\begin{aligned} \begin{bmatrix} V_y(x) \\ V_z(x) \\ \Psi (x) \end{bmatrix} = \begin{bmatrix} C_{V_y} \sin (\lambda _n x) \\ C_{V_z} \sin (\lambda _n x) \\ C_{\Psi } \sin (\lambda _n x) \end{bmatrix} \quad \text {and} \quad \begin{bmatrix} \Theta _y(x) \\ \Theta _z(x) \end{bmatrix} = \begin{bmatrix} C_{\Theta _y} \lambda _n \cos (\lambda _n x) \\ C_{\Theta _z} \lambda _n \cos (\lambda _n x) \end{bmatrix}, \end{aligned}$$
(69)

where \(C_{V_y}\), \(C_{V_z}\), \(C_{{\Theta }_y}\), \(C_{{\Theta }_z}\), \(C_{\Psi }\) are unknown constants and \(\lambda _n=\frac{n\pi }{L},\ n=1,2,3\cdots \).

Substituting Eq. (69) into Eqs. (27)–(31) gives the five homogeneous differential equations for \(C_{V_y}\), \(C_{V_z}\), \(C_{{\Theta }_y}\), \(C_{{\Theta }_z}\), \(C_{\Psi }\), the resulting matrix equation is expressed as

$$\begin{aligned} \underset{(5 \times 5)}{{\textbf{B}}(\omega )}\underset{(5 \times 1)}{\vec {x}}=\underset{(5 \times 1)}{\vec {0}}. \end{aligned}$$
(70)

For details in regard to \(\underset{(5 \times 5)}{{\textbf{B}}(\omega )}\), see Appendix D. For

$$\begin{aligned} \underset{(5 \times 1)}{\vec {x}}=(C_{V_y},C_{V_z},C_{{\Theta }_y},C_{{\Theta }_z},C_{\Psi })^T, \end{aligned}$$
(71)

this system of linear equations cannot have \(\underset{(5 \times 1)}{\vec {x}}\) identically equal to zero, thus \(\det [ ~\underset{(5 \times 5)}{{\textbf{B}}(\omega )}~]=0\) holds, i.e.:

$$\begin{aligned} b_1\left( \omega ^2\right) ^5+b_2\left( \omega ^2\right) ^4+b_3\left( \omega ^2\right) ^3+b_4\left( \omega ^2\right) ^2+b_5\left( \omega ^2\right) ^1+b_6=0. \end{aligned}$$
(72)

The coefficients \(b_1\), \(b_2\), \(b_3\), \(b_4\), \(b_5\), \(b_6\) are derived from the system of Eq. (70). This implies that for each n, which specifies an external vibration mode along the beam’s axis, there exist five corresponding internal vibration modes within the cross-section. Here, the external vibration mode refers to the vibration pattern where the integer n represents the number of half-waves along the beam’s centroid axis under simply-supported conditions, indicating the peaks and valleys along the beam’s centerline. This mode describes the global curvature distribution of the beam. The internal vibration mode, for a given external half-wave n, allows each cross-section to deform in five independent ways: bending about two principal axes, shear-related warping rotations about two principal axes, and torsional warping.

For n = 1, 2 and 3, the natural frequencies of a simply supported thin-walled beam with various cross-sections were identified. These frequencies correspond to different internal vibration modes, including bending vibrations in the y-direction and z-direction, and torsional vibrations around the x-, y-, and z-axis. These results were then compared to those predicted by traditional theories for thin-walled beams with varying slenderness ratios.

Fig. 2
Fig. 2
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Cross-section for the numerical example

The calculation example directly applies the simply supported beam model provided by Prokic in [12]. The material properties and geometry can be retrieved from original technical paper, including \(E,G,\rho ,L,A\), \(I_z, I_y, k_y,k_z,{\Gamma }, J,y_c,z_c\) (see Appendix B for more details). It is worth noting that, when the axial load is not considered, i.e., \(P=0~\hbox {kN}\), our model becomes Prokic’s, and comparability is applicable. A thin-walled beam with the cross-section illustrated in Fig. 2 is considered, and the corresponding results are presented in Table 1.

Table 1 Natural frequencies (rad/s) of beam studied as numerical example
Full size table

It can be seen that for different values of n, Eq. (72) can derive the five natural vibration frequencies of a simply supported beam, corresponding to the natural vibration frequencies of \(V_y,V_z\), \({\Theta }_y\), \({\Theta }_z\), \({\Psi }\). The natural frequencies will be calculated using classical theory and compared with the results presented in this paper. The classical theory starts from the torsional vibration of the rod. Applying \({\Psi }(x)=C_{\Psi }\sin {\lambda _n}x\), it is evident that this function satisfies the boundary conditions

$$\begin{aligned} \left. {\Psi }\right| _{x=0}=\left. {\Psi }\right| _{x=L}=0, \end{aligned}$$
(73)

and as for the frequency, \(\omega _n=\frac{n\pi }{L}\sqrt{\frac{G}{\rho }}\) holds. For the vibration of a Timoshenko–Ehrenfest beam with simply supported ends, the frequency equation is

$$\begin{aligned} \omega _n=\left( \frac{n\pi }{L}\right) ^2\sqrt{\frac{EI}{\rho A}}\left[ 1+\frac{I}{A}\left( \frac{n\pi }{L}\right) ^2\right] ^{-1/2}, \end{aligned}$$
(74)

in which, only the effect of the rotational inertia of the cross-section is considered by conventional method.

The comparing results for various length L are listed in Table 2 (\(L=10\) m), Table 3 (\(L=20\) m), and Table 4 (\(L=40\) m). The impact of axial load variations on the lowest natural vibration frequency of the Timoshenko–Ehrenfest beam is examined for beams of three different lengths, and depicted in Fig. 3. The natural frequency \(\omega _n\) becomes zero at the following critical axial loads for a simply supported beam of lengths \(L=10\ \textrm{m}\), \(20\ \textrm{m}\) and \(40\ \textrm{m}\) (mode numbers \(n=1,2,3\)):

$$\begin{aligned} \begin{aligned} L=10\ \textrm{m}:\quad P&=13392.6\ \textrm{kN},\ 47069.0\ \textrm{kN},\ 97000.2\ \textrm{kN};\\ L=20\ \textrm{m}:\quad P&=4553.3\ \textrm{kN},\ 13392.6\ \textrm{kN},\ 27749.5\ \textrm{kN};\\ L=40\ \textrm{m}:\quad P&=1641.4\ \textrm{kN},\ 4553.3\ \textrm{kN},\ 8264.1\ \textrm{kN}. \end{aligned} \end{aligned}$$

Consequently, the smallest critical loads (\(13392.6\ \textrm{kN}\), \(4553.3\ \textrm{kN}\) and \(1641.4\ \textrm{kN}\) for \(L=10\ \textrm{m}\), \(20\ \textrm{m}\) and \(40\ \textrm{m}\) respectively) constitute the elastic–buckling loads of the beam. In addition, substantial discrepancies exist between the present Timoshenko-based solution, which accounts for biaxial bending and the coupling between bending and torsion, and classical uncoupled analytical formulations that treat these modes independently. These discrepancies could stem from the interaction of axial force with the coupled deformation field. An asymmetric cross-sectional geometry, characterized by the offset between the shear center and the mass center (SC-MC offset), enables transfer of internal forces between bending about the y- and z-axes and torsion, an effect omitted in decoupled theories. Moreover, the classical Saint-Venant torsion model neglects warping restraint under axial stress, so it cannot reproduce the dynamically coupled response governed by the interplay of axial loading and geometry.

Table 2 Natural frequencies (rad/s) of the beam with length \(L=10\) m
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Table 3 Natural frequencies (rad/s) of the beam with length \(L=20\) m
Full size table
Table 4 Natural frequencies (rad/s) of the beam with length \(L=40\) m
Full size table
Fig. 3
Fig. 3
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The impact of axial load variations on the lowest natural vibration frequency: a \(L=10\) m, b \(L=20\) m, and c \(L=40\) m

5.2 Mode-shape components

Introducing

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\omega } \det \bigl ({\textbf{B}}(\omega )\bigr ) = \operatorname {tr}\!\left( \operatorname {adj}\bigl ({\textbf{B}}(\omega )\bigr )\cdot {\textbf{B}}'(\omega )\right) , \end{aligned}$$
(75)

the derivation of which is provided in Appendix E.

If the computed \(\det \bigl (\underset{(5 \times 5)}{{\textbf{B}}(\omega )}\bigr )=0\) (Eq. (72)), and \(\omega \) is not a repeated root, then the rank of matrix \(\underset{(5 \times 5)}{{\textbf{B}}(\omega )}\) is 4. This is because that if \(\omega \) is not a repeated root, \(\frac{\hbox {d}}{\hbox {d}\omega } \det \bigl ({\textbf{B}}(\omega )\bigr ) \ne 0\) holds, which implies \(\hbox {RHS} \ne 0\). Generally, \({\textbf{B}}'(\omega )\) is nonzero, as many elements of the matrix \({\textbf{B}}(\omega )\) explicitly depend on \(\omega \). Therefore, in most cases, \(\operatorname {adj}\bigl ({\textbf{B}}(\omega )\bigr ) \ne 0\) must hold to satisfy \(\operatorname {tr}\!\left( \operatorname {adj}\bigl ({\textbf{B}}(\omega )\bigr )\cdot {\textbf{B}}'(\omega )\right) \ne 0\).

Since \(\operatorname {adj}({\textbf{B}}(\omega )) \ne 0\), using matrix properties, each element of \(\operatorname {adj}({\textbf{B}}(\omega ))\) is an \((n-1)\text {th}\) order algebraic cofactor of \({\textbf{B}}(\omega )\). Therefore, \({\textbf{B}}(\omega )\) has an \((n-1)\text {th}\) order minor not equal to zero, so the rank of the matrix satisfies \(4 \le \operatorname {rank}\!\big ({\textbf{B}}(\omega )\big )\le 5\). According to the requirement of Eq. (72), \(\det \!\big ({\textbf{B}}(\omega )\big )=0\). Hence \(\operatorname {rank}\!\big ({\textbf{B}}(\omega )\big )<5\). It can be concluded that \(\operatorname {rank}\!\big ({\textbf{B}}(\omega )\big )=4\), using the Rank–Nullity Theorem in linear algebra

$$\begin{aligned} \operatorname {nullity}\!\big ({\textbf{B}}(\omega )\big )=\dim {\mathcal {N}}\!\big ({\textbf{B}}(\omega )\big )=n-\operatorname {rank}\!\big ({\textbf{B}}(\omega )\big ). \end{aligned}$$
(76)

Therefore, we can know that the solution space of \(\big (\underset{(5 \times 5)}{{\textbf{B}}(\omega )}\big )\,\underset{(5 \times 5)}{\vec {x}}=\vec {0}\) is 1. This explains why, by substituting \(\omega _{n}\) into Eq. (70), the five vectors of constants \(\vec {x}\) may be determined up to a multiplicative constant factor. This is also consistent with the earlier mention in Eq. (43) that the constants B, C, D, and E can all be expressed in terms of the constant A. This point is similarly verified in Prokic’s study [12], wherein \(\vec {c}_{5\times 1}=\big (C_{v_y},\,C_{v_z},\,C_{\theta _y},\,C_{\theta _z},\,C_{\psi }\big )^{T}\), \(C_{v_y}\) is set to 1, and the other constants \(C_{v_z},\,C_{\theta _y},\,C_{\theta _z},\,C_{\psi }\) can be computed as constants.

As can be seen in our earlier calculations in Table 1, for every n, the natural frequencies \(\omega \) obtained from Eq. (72) are all distinct, with no repeated roots. Therefore, based on the earlier theoretical argument, we can naturally conclude that the solution space of \({\textbf{B}}(\omega )\,\vec {x}=\vec {0}\) is 1. Hence, the mode shape \(\vec {x}_{5\times 1}=\big (C_{v_y},\,C_{v_z},\,C_{\theta _y},\,C_{\theta _z},\,C_{\psi }\big )^{T}\) can be represented using the same method.

This paper adopts the same computational approach as Prokic’s study [12], first setting \(C_{v_y}\) to 1, and then computing the other constants \(C_{v_z},\,C_{\theta _y},\,C_{\theta _z},\,C_{\psi }\) as constants. Specific expressions are provided in Appendix F. For n = 1, 2, and 3, and L = 10 m, 20 m, and 40 m, the corresponding modes of vibration amplitudes are provided in Tables 5, 6, 7, 8, 9, 10, 11, 12, and 13 of Appendix G. It should be noted that the amplitudes of the displacement of the beam in the y-direction and the rotation due to displacement in the y-direction are relatively small, while those of the displacement in the z-direction and the rotation due to displacement in the z-direction are larger. The torsional angle about the x-direction exhibits the largest amplitude.

Figure 4 shows the variation of \(\vec {x}\) as the axial load P varies continuously. As previously calculated, buckling occurs for beams with lengths L = 10 m, 20 m, and 40 m when reaches 13392.6 kN, 4553.3 kN, and 1641.4 kN, respectively. Therefore, this study focuses specifically on the cases where P = 13392.6 kN, 4553.3 kN, and 1641.4 kN for beam lengths L = 10 m, 20 m, and 40 m. The modes of vibration amplitudes increase consistently with increasing axial load. Among these, the amplitude of the torsional angle about the x-direction exhibits the most rapid and significant increase. This trend reflects the progressive loss of stability in the beam as the axial load increases, ultimately leading to buckling.

Fig. 4
Fig. 4
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The impact of axial load variations on the mode-shape components at lowest natural vibration frequency: a \(L=10\) m, b \(L=20\) m, and c \(L=40\) m

6 Conclusions

By comparing the natural frequencies of the Timoshenko–Ehrenfest beam with those derived from traditional vibration theories, we identify notable differences across various vibration modes. The model developed in this study addresses the coupled bidirectional bending and torsional vibrations of a Timoshenko–Ehrenfest beam, distinguishing itself from traditional models that typically focus on isolated vibration modes. Our findings demonstrate that vibration behavior is significantly influenced by these coupling effects. The impact of coupling factors on bending vibrations is relatively minor, as the natural frequencies computed using the Timoshenko–Ehrenfest beam model show only slight deviations from those predicted by models that consider bending vibrations in isolation. This indicates that bending vibrations are less affected by the complexities introduced by coupling effects. In contrast, torsional vibrations are substantially influenced by coupling factors, with the frequencies calculated by the Timoshenko–Ehrenfest beam model showing considerable differences compared to those predicted by traditional, decoupled vibration theories. This significant discrepancy highlights the crucial importance of accounting for coupled effects in the accurate analysis of torsional vibrations.

We also examined the impact of axial load variations on the lowest natural frequency of the Timoshenko–Ehrenfest beam. Our analysis reveals that as the axial load increases, the natural frequency of the beam gradually decreases, consistent with findings in the existing literature [3, 6, 11, 13, 14, 36]. As the axial load continues to increase, the lowest natural frequency eventually drops to zero, indicating the onset of elastic buckling, which also aligns with previous research [14, 36].

What’s more, in our numerical example, the axial load values that cause the natural frequency to reach zero increase with the mode number n and decrease with the beam length L. This behavior is consistent with the characteristics predicted by the classic Euler formula for the stability of a compression member \(F_{cr}={n^2\pi ^2EI}{\mu ^{-2} l^{-2}}\). This relationship provides a robust theoretical and practical foundation for tackling modern structural engineering challenges, particularly in the design and analysis of thin-walled beam structures subjected to complex dynamic loads.

Finally, we computed various vibration mode components, and the results demonstrate that the modes of vibration amplitudes are significantly larger in the z-direction and in torsion compared to the y-direction. We also examined the variations of different vibration modal components under continuously increasing axial load. As the axial load approaches the critical buckling value, all mode amplitudes increase, with torsion exhibiting the most pronounced growth.

This paper introduces an original framework for the coupled bidirectional bending and torsional vibrations of non-symmetric axially loaded thin-walled Timoshenko–Ehrenfest beams. This framework encompasses: (a) the derivation of analytical solutions for the exact frequency response under arbitrary harmonic loads; (b) the establishment of appropriate orthogonality conditions for the coupled bending-torsion modes, which are employed to construct precise modal impulse and frequency response functions for any load. The computations for frequency response and eigenvalue problems involve 12\(\times \)12 matrices. Modal responses are derived in analytical form upon the determination of eigenvalues. The proposed framework provides precise and computationally efficient solutions for all response variables, encompassing both bending and torsional responses. This framework is broadly applicable to a wide range of engineering problems. Future research effort will be devoted to carry out experimental validations of the proposed method.