Introduction

Over the past few decades, there has been an interest in infrastructure monitoring and damage detection. It emphasizes incorporating technological developments to make it more advanced [1]. The dynamic behavior of structures generally follows the laws of non-linearity, which are often associated with geometry and boundary conditions [2]. The degree of damage can be determined by the decrease in dynamic stiffness and the increase in damping. It does not matter whether this damage is localized like a crack or distributed in most samples by the number of micro-cracks. Local or distributed stiffness changes can lead to changes in the natural frequency of the structure [3, 4]. The traditional modal analysis method is often appropriate for structures, which can be modeled by discrete elements of grouped parameters, where the presence of damage leads to a low-frequency change in the system's overall behavior. On the other side, small defects, such as cracks, are hidden by modal approaches, since such phenomena are high-frequency effects that cannot easily be discovered by examining changes in modal mass, stiffness, or damping parameters [5, 6].

Many analytical methods have been developed to solve wave propagation problems. This includes Fourier synthesis, popularly known as the spectral finite-element method (SFEM) [7,8,9,10,11,12].

F. Pind et al. [13] have developed a method for simulating room acoustics in the time domain using a combination of a spectral element method and an implicit–explicit Runge–Kutta time stepping method. This method has low dispersion and dissipation, can handle adaptive, unstructured meshes with curvilinear elements, and is suitable for parallel implementation. Wu Z et al. [14] present a new type of lattice structure with a two-order hierarchical periodicity, inspired by the structure of butterfly wings, has been designed and the band-gap properties have been investigated using the spectral element method. The dynamic behavior of the structure has been analyzed, and it was found to have distinct macro- and micro-stop-bands in the given frequency domain. The mechanisms for these band-gaps and the relationship between the hierarchical periodicities and the stop-bands were also explored. The efficiency of the spectral element method was validated by comparing the results with those obtained using the finite-element method. ZJ Wu et al. [15] study the dynamic properties of three-dimensional piezoelectric Kagome grids were analyzed to understand their ability to transform mechanical energy into electric energy and their performance in vibration isolation. The spectral element method (SEM) was used to solve the equation of motion for the structure and accurately determine the frequency-domain responses. The results were compared with those obtained using the finite-element method (FEM) and with those from previous literature, and it was found that the SEM was effective in studying 3D piezoelectric Kagome grids. ZJ Wu [16] also studied about the vibration characteristics of sandwich panels with corrugated cores were analyzed using the spectral element method (SEM). The SEM was used to establish spectral equations for plate elements with both in-plane and out-of-plane components, and the spectral stiffness matrix for the entire structure was assembled. The frequency responses were then obtained by calculating the spectral equations. The results showed that the SEM was effective in studying the vibration band-gap properties of sandwich panels with corrugated cores, and that it was able to produce more accurate results than the finite-element method (FEM). WJ Wu [17] presents a study to investigate the vibration band-gap properties of three-dimensional (3D) Kagome lattices. The SEM was able to accurately and effectively analyze the vibration characteristics of the 3D Kagome lattices by deriving the dynamic stiffness matrix of the 3D element and establishing the spectral equations of motion for the entire lattice structure. The results were compared with those obtained using the finite-element method (FEM) and it was found that the SEM was suitable for analyzing the vibration band-gap properties of 3D Kagome lattices. A Mukherjee et al. [18] propose numerical method to solve the nonlinear eigenvalue problem (NLEP) in SEM. NLEP cannot be solved using linear numerical eigen-solvers, so two distinct numerical methods were proposed to compute the eigenvalues of the problem more efficiently using SEM. These methods were a root finding method of rational polynomial functions and a linearization of Lagrange matrix interpolating polynomials. The proposed methods were able to solve NLEP in a stable, efficient, and accurate way, even in the presence of singularities. The accuracy of the methods was numerically evaluated by comparing the solutions obtained with those from modal analysis using the finite-element method (FEM). Loya et al. [19] present a view on methods for detecting cracks and knowing about their nature by finding natural frequencies for bending vibrations of the cracked beam. The beam is modeled as two segments connected by an extensional and rotational spring. The results from the perturbative method agree with those obtained with the direct method. A study is proposed to examine the vibrational properties of axially loaded Euler–Bernoulli and Timoshenko beams with arbitrary cracks. The cracks in beams are modeled as transversely open and non-propagating. A massless rotational spring models the cracking on a beam under axial loading [20, 21]. A new crack localization algorithm based on a mathematical model is proposed, which describes the lateral vibration of a rotating cracked Timoshenko beam. This model is derived using Lagrange equations and the assumed mode method [22]. Shi et al. [23] present an accurate solution method for the free vibrations of Timoshenko beams with general elastic restraints at the endpoints. Generally, the method is applied to a wide range of boundary conditions without modifying solution algorithms and procedures.

Efficient vibration analysis of one-dimensional (1D) lattice (or periodic) structures, the so-called spectral transfer matrix method (STMM), is introduced by combining the good features of SFEM with a well-known transfer matrix method. The efficiency of the proposed method is proven in the solution of Bernoulli–Euler beams and large planar lattice structures [24]. Unlike traditional finite elements, the length of spectral elements is not a limiting factor. Each element is formulated exactly, regardless of its length. Therefore, structural connection and discontinuities control the length of the element [25]. A new matrix formulation is presented to analyze the dynamic response and propagation of elastic waves in the structure consisting of beams and rods connected by rigid or pinned joints. Their methods showed higher accuracy than the existing FEM in evaluating the natural frequencies of framed structures [26].

A new plate element is developed using SFEM with cracks along its length. The elastic behavior of the plate at the crack position is considered a line spring with various stiffnesses along the crack length [27, 28]. An approach is developed for the flow characteristics of the vibrational force of open cracks in the Timoshenko beam structure. The results indicate that the flow characteristics are unusually related to the crack position and depth, especially at high frequencies [29]. Kudela et al. [30] present a new spectral finite element of the composite plate. The wave propagation analysis using transverse elastic waves is carried out on the element for various orientations and relative volume fractions of reinforcing fibers. The modeling of wave propagation in plate structure using 3D-SFEM is studied for damage identification [31]. The Love rod theory proposes the formulation for coupled axial–flexural wave interaction in a sagged rod using SFEM [32]. Structural damage detection through axial–flexural wave interaction using SFEM is established for rod and beam structure [33]. The experimental investigation using piezoelectric transducers is proposed under narrowband amplitude-modulated excitation force for coupled axial–flexural wave propagation in a sagged rod with structural discontinuity [34]. The dynamic stiffness matrix of the Timoshenko beam under moving load with constant velocity and acceleration subjected to an open crack is derived using SFEM. The modeling is done by two rotational and extensional massless springs [35].

Numerous civil/mechanical structures are demonstrated as beams for the dynamic analysis, employing classical Euler–Bernoulli or Timoshenko beam theories. However, several structural members have coupling interactions from various fundamental deformation modes, which conventional beam theories do not deal with reasonably [36, 37]. Recently, the stiffness theory was established by developing the governing differential equations of motion of the coupled axial-bending beam using Hamilton’s principle [38, 39]. Still, substantial literature is unavailable for the dynamic analysis of the coupled axial-bending wave propagation problem in SFEM. This paper has developed a spectral element model for the coupled axial-bending Timoshenko beam element. Equally, the main focus is to understand the cracked Timoshenko beam element. From the previous literature, it can be understood that the Timoshenko beam theory makes an excellent candidate to combine with crack mechanics, making it a general theory for any spectral analysis.

Firstly, this article attempts to unify the theory of the SFEM for the Timoshenko beam and fracture mechanics in the frequency domain, making it simpler for wave propagation and modal analysis with coupled axial-bending effects. An open non-propagating transverse-single edge crack opening is investigated in the present work. Furthermore, the variation in wave propagation analysis is studied in the presence of crack by comparing responses from damaged and undamaged coupled axial-bending Timoshenko beams without coupling coefficients. Then, the effect of crack depth and location is studied for the various wave propagation phenomena. Correspondingly, the modal analyses are carried out to investigate the crack formation effect with the coupling coefficients using STMM. Finally, the parametric study is carried out to understand the variation of coupling coefficients for crack formation. The proposed spectral element for the coupled axial-bending cracked Timoshenko beam model has been shown to analyze the damaged structures and precisely identify and evaluate the crack location.

Equation of Motion for a Coupled Axial-Bending Timoshenko Beam

The following equation expresses the planar motion of a coupled axial-bending Timoshenko beam as:

$$\left[\begin{array}{c}{N}_{t}^{^{\prime}}\\ {Q}_{t}^{^{\prime}}\\ {M}_{t}^{^{\prime}}\end{array}\right]= \left[\begin{array}{ccc}\rho A& 0& \rho R\\ 0& \rho A& 0\\ \rho R& 0& \rho I\end{array}\right] \left[\begin{array}{c}\ddot{u}\\ \ddot{w}\\ \ddot{\theta }\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ {Q}_{t}\end{array}\right].$$
(1)

Here, \({N}_{t}\left(x,y\right)\), \({Q}_{t}\left(x,y\right)\), and \({M}_{t}\left(x,y\right)\) are axial tensile forces, lateral shear forces, and bending moments, respectively, where the prime (‘) represents the derivative with respect to the spatial coordinate along the longitudinal axis of the beam, say \(x\); \(u\left(x,y\right)\), \(w\left(x,y\right)\), and \(\theta \left(x,y\right)\) are axial and lateral displacement and rotation, respectively; \(\rho\) A, \(\rho R\), and \(\rho I\) are the effective mass per length, the first-order moment of inertia, and the second-order moment of inertia. It is expressed as

$$\rho A= \int \rho dA, \rho R= \int \rho zdA, \rho I= \int \rho {z}^{2}dA,$$
(2)

where \(\rho\) is the mass density per unit volume and \(A\) is the cross-section area. The force–displacement relationship for a coupled axial-bending Timoshenko beam is represented by

$$\left[\begin{array}{c}{N}_{t}\\ {Q}_{t}\\ {M}_{t}\end{array}\right]= \left[\begin{array}{ccc}EA& {C}_{1}& {C}_{2}\\ {C}_{1}& GA& {C}_{3}\\ {C}_{2}& {C}_{3}& EI\end{array}\right] \left[\begin{array}{c}{u}^{^{\prime}}\\ {w}^{^{\prime}}-\theta \\ {\theta }^{^{\prime}}\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ {Q}_{t}\end{array}\right],$$
(3)

where \(EA\), \(GA,\) and \(EI\) are axial, shear, and flexural rigidity, respectively; \({C}_{1}\), \({C}_{2}\), and \({C}_{3}\) are the coupling stiffnesses representing the interaction between axial, lateral shear, and bending deformations. The decoupled equations of motion of the classical Timoshenko beam and longitudinal rod are obtained when all \(\rho R\), \({C}_{1}\), \({C}_{2}\), and \({C}_{3}\) parameters are ignored [40]. The coupled axial-bending Timoshenko beam theory represented by Eq. (1) is applied to a homogenized beam model of a 1D periodic lattice structure whose cross-section is symmetric with respect to the vertical axis (\(z\) -axis). The beam model takes the extension-transverse shear-bending coupled in-plane vibration in the \(x - z\) plane without torsion about the elastic axis (\(x\) -axis), as shown in Fig. 1.

Fig. 1
figure 1

The coordinate system for a coupled axial-bending Timoshenko beam

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Mathematical Framework: Spectral Finite-Element Method

According to SFEM, analyzing frequencies with one spectral element in any range is possible. It is important to analyze high natural frequencies, as slight imperfections could cause noticeable changes. In such cases, the classical FEM involves a very dense grid and is time-consuming to calculate. For the classical FEM models, elements of the dynamic stiffness matrix in the frequency domain are nearly constant for higher frequencies, and there is an additional mathematical computation needed. The spectral approach provides a dynamically changing stiffness matrix in a whole frequency range. There are three specific cases for wave propagation in the connected waveguide. The first is that members are connected at both ends, called a double-nodded element. It conducts energy in both directions. The second is when members connect at one point and expand to infinity by conducting energy in one direction without reflection, called a single-nodded or a throw-off element. These elements act as an infinite medium to transfer energy out of the system. Both behaviors are fundamentally different and need to be handled separately. The third is the presence of local discontinuity along the length of the structure. It is complex to model waveguide elements with local discontinuity, such as holes, cracks, or joints.

Finite Spectral Element Formulation

The spectral displacement field is expressed to construct the spectral elements of a coupled axial-bending Timoshenko beam, based on the discrete Fourier transform theory, as follows:

$$\begin{gathered} u\left( {x,t} \right) = \mathop \sum \limits_{n = 0}^{N - 1} \overline{u}_{n} \left( x \right)e^{{i\omega_{n} t}} \hfill \\ w\left( {x,t} \right) = \mathop \sum \limits_{n = 0}^{N - 1} \overline{w}_{n} \left( x \right)e^{{i\omega_{n} t}} \hfill \\ \theta \left( {x,t} \right) = \mathop \sum \limits_{n = 0}^{N - 1} \overline{\theta }_{n} \left( x \right)e^{{i\omega_{n} t}} , \hfill \\ \end{gathered}$$
(4)

where \({\overline{u} }_{n}\), \({\overline{w} }_{n}\), and \({\overline{\theta }}_{n}\) (n = 0, 1, 2, …., N-1) are the Fourier component of \(u\left(x,y\right)\), \(w\left(x,y\right)\), and \(\theta \left(x,y\right)\), respectively. Each corresponds to the discrete frequency defined by \({\omega }_{n}=2\uppi n/{T}_{t}\), where \({T}_{t}\) is the time window (period). It is related to the number of samples as \(N=2{f}_{NQY}{T}_{t}\); where \({f}_{NQY}\) is the Nyquist frequency. Similarly, the axial tensile force \(T\left(x,y\right)\), lateral shear force \(Q\left(x,y\right)\), and bending moment \(M\left(x,y\right)\) are expressed in a spectral format as follows:

$$\begin{gathered} T(x,t) = \sum\limits_{(n = 0)}^{N - 1} {\overline{T}_{n} (x)e^{{(i\omega_{n} t)}} } \hfill \\ Q\left( {x,t} \right) = \mathop \sum \limits_{n = 0}^{N - 1} \overline{Q}_{n} \left( x \right)e^{{i\omega_{n} t}} \hfill \\ M\left( {x,t} \right) = \sum\limits_{n = 0}^{N - 1} {\overline{M}_{n} \left( x \right)e^{{i\omega_{n} t}} } . \hfill \\ \end{gathered}$$
(5)

The subscript n for the nth Fourier component is omitted in the following derivations to simplify the process. Substitution of Eqs. (4) and (5) into Eqs. (1) and (3) gives

$$\left[\begin{array}{c}{\overline{T} }^{^{\prime}}\\ {\overline{Q} }^{^{\prime}}\\ {\overline{M} }^{^{\prime}}\end{array}\right]= -{\omega }^{2}\left[\begin{array}{ccc}\rho A& 0& \rho R\\ 0& \rho A& 0\\ \rho R& 0& \rho I\end{array}\right] \left[\begin{array}{c}\ddot{u}\\ \ddot{w}\\ \ddot{\theta }\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ {Q}_{t}\end{array}\right]$$
(6)

and

$$\left[ {\begin{array}{*{20}c} {\overline{T}} \\ {\overline{Q}} \\ {\overline{M}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {EA} & {C_{1} } & {C_{2} } \\ {C_{1} } & {GA} & {C_{3} } \\ {C_{2} } & {C_{3} } & {EI} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\overline{u}^{^{\prime}} } \\ {\overline{w}^{^{\prime}} - \overline{\theta }} \\ {\overline{\theta }^{^{\prime}} } \\ \end{array} } \right].$$
(7)

By substituting Eq. (6) into Eq. (7), the governing equations are derived as

$$\left[ {\begin{array}{*{20}c} {EA} & {C_{1} } & {C_{2} } \\ {C_{1} } & {GA} & {C_{3} } \\ {C_{2} } & {C_{3} } & {EI} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\overline{u}^{{^{\prime\prime}}} } \\ {\overline{w}^{{^{\prime\prime}}} } \\ {\overline{\theta }^{{^{\prime\prime}}} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 & 0 & { - C_{1} } \\ 0 & 0 & { - GA} \\ {C_{1} } & {GA} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\overline{u}^{^{\prime}} } \\ {\overline{w}^{^{\prime}} } \\ {\overline{\theta }^{^{\prime}} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & {GA} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\overline{u}} \\ {\overline{w}} \\ {\overline{\theta }} \\ \end{array} } \right] + \omega^{2} \left[ {\begin{array}{*{20}c} {\rho A} & 0 & {\rho R} \\ 0 & {\rho A} & 0 \\ {\rho R} & 0 & {\rho I} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\overline{u}} \\ {\overline{w}} \\ {\overline{\theta }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \right].$$
(8)

Assuming the general solution of Eq. (8) as

$$\overline{u }\left(x\right)= \alpha W{e}^{-ikx}, \overline{w }\left(x\right)= W{e}^{-ikx}, \overline{\theta }\left(x\right)= \beta W{e}^{-ikx},$$
(9)

where k is the wavenumber. Substitution of Eq. (9) into Eq. (8) gives

$$X\left(k\right)W\left[\begin{array}{c}\alpha \\ 1\\ \beta \end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],$$
(10)

where

$$X\left(k\right)=\left[\begin{array}{ccc}{-k}^{2}EA+{\omega }^{2}\rho A& -{k}^{2}{C}_{1}& ik{C}_{1}{-k}^{2}{C}_{2}+{\omega }^{2}\rho R \\ -{k}^{2}{C}_{1}& {-k}^{2}GA+{\omega }^{2}\rho A& ikGA{-k}^{2}{C}_{3}\\ -ik{C}_{1}{-k}^{2}{C}_{2}+{\omega }^{2}\rho R& -ikGA{-k}^{2}{C}_{3}& {-k}^{2}EI+{\omega }^{2}\rho I-GA\end{array}\right].$$
(11)

From Eq. (11), the dispersion equation is written as

$$X(k)={a}_{1}{k}^{6}+{a}_{2}{k}^{4}+{a}_{3}{k}^{2}+{a}_{4}=0,$$
(12)

where

\({a}_{1}=EI{C}_{1}^{2}-2{C}_{1}{C}_{2}{C}_{3}+GA{C}_{2}^{2}+EA{C}_{3}^{2}-EAEIGA\)

$${a}_{2}=2{C}_{1}{C}_{3}\rho R{\upomega }^{2}-{C}_{3}^{2} \rho A{\upomega }^{2}-{C}_{1}^{2}\rho I{\upomega }^{2}-2{C}_{2}GA\rho R{\upomega }^{2}+EAEI\rho A{\upomega }^{2}+EAGA\rho I{\upomega }^{2} + EIGA\rho A{\upomega }^{2}$$
$${a}_{3}=GA\rho {\mathrm{R}}^{2}{\upomega }^{2}-EI\rho {A}^{2}{\upomega }^{4}-{C}_{1}^{2}\rho A{\upomega }^{2}+EAGA\rho A\rho R{\upomega }^{4}-EA\rho A\rho I{\upomega }^{4}+EIGA\rho A{\upomega }^{2}$$
$${a}_{4}=\rho I\rho {A}^{2}{\upomega }^{6}-GA\rho {A}^{2}{\upomega }^{4}-\rho A\rho {R}^{2}{\upomega }^{6}.$$

The six roots of Eq. (12) are obtained as follows:

$$\begin{gathered} k_{1} = - k_{4} \quad k_{2} = - k_{5} \quad k_{3} = - k_{6} \hfill \\ k_{1} = \left( { - \frac{1}{3}b_{2} - \frac{1}{2} \left( {B + C} \right) - i\frac{\sqrt 3 }{2}\left( {B - C} \right)} \right)^{\frac{1}{2}} \hfill \\ k_{2} = \left( { - \frac{1}{3}b_{2} - \frac{1}{2} \left( {B + C} \right) - i\frac{\sqrt 3 }{2}\left( {B - C} \right)} \right)^{\frac{1}{2}} \quad k_{3} = \left( { - \frac{1}{3}b_{2} + \left( {B + C} \right) } \right)^{\frac{1}{2}} , \hfill \\ \end{gathered}$$
(13)

where

$$\begin{gathered} B = \left( { P + \sqrt D } \right)^{\frac{1}{3}} ,\quad C = \left( { P - \sqrt D } \right)^{\frac{1}{3}} ,\quad D = O^{3} + P^{2} \hfill \\ O = \frac{{3b_{1} - b_{2}^{2} }}{9},\quad P = \frac{{9b_{2} b_{1} - 27b_{0} - 2b_{2}^{3} }}{54} \hfill \\ b_{0} = \frac{{a_{4} }}{{a_{1} }},\quad b_{1} = \frac{{a_{3} }}{{a_{1} }},\quad b_{2} = \frac{{a_{2} }}{{a_{1} }}. \hfill \\ \end{gathered}$$
(14)

From Eq. (10), \({\alpha }_{n}\) and \({\beta }_{n}\) are determined as

$$\begin{array}{c}{\alpha }_{n}=\left(\rho A\rho R{\omega }^{4}-{C}_{1}{C}_{3}{k}^{4}+{C}_{2}{GAk}^{4}-{C}_{2}{k}^{2}\rho A{\omega }^{2}-{GAk}^{2}\rho R{\omega }^{2}+{iC}_{1}k\rho A{\omega }^{2}\right)/\Delta \\ {\beta }_{n}=\left({C}_{1}^{2}{k}^{4}-\rho {A}^{2}{\omega }^{4}-{EAGAk}^{4}+{EAk}^{2}\rho A{\omega }^{2}+{GAk}^{2}\rho A{\omega }^{2}\right)/\Delta \\ \\ \Delta =\left(k\left({iC}_{1}^{2}{k}^{2}-{C}_{2}{C}_{1}{k}^{3}+\rho {RC}_{1}k{\omega }^{2}+{C}_{3}E{Ak}^{3}-{iEAGAk}^{2}-{C}_{3}\rho Ak{\omega }^{2}+iGA\rho A{\omega }^{2}\right)\right).\end{array}$$
(15)

Using the six wavenumbers computed in Eq. (13), the general solution for an element is expressed in the form of

$$\begin{array}{c}\overline{u }(x)={\alpha }_{1}{W}_{1}{e}^{-{ik}_{1}x}+{\alpha }_{2}{W}_{2}{e}^{-{ik}_{2}x}+{\alpha }_{3}{W}_{3}{e}^{-{ik}_{3}x}+{\alpha }_{4}{W}_{4}{e}^{-{ik}_{1}\left(L-x\right)}+{\alpha }_{5}{W}_{5}{e}^{-{ik}_{2}\left(L-x\right)}+{\alpha }_{6}{W}_{6}{e}^{-{ik}_{3}\left(L-x\right)}\\ \overline{w }(x)={W}_{1}{e}^{-{ik}_{1}x}+{W}_{2}{e}^{-{ik}_{2}x}+{W}_{3}{e}^{-{ik}_{3}x}+{W}_{4}{e}^{-{ik}_{1}\left(L-x\right)}+{W}_{5}{e}^{-{ik}_{2}\left(L-x\right)}+{W}_{6}{e}^{-{ik}_{3}\left(L-x\right)}\\ \overline{\theta }(x)={\beta }_{1}{W}_{1}{e}^{-{ik}_{1}x}+{\beta }_{2}{W}_{2}{e}^{-{ik}_{2}x}+{\beta }_{3}{W}_{3}{e}^{-{ik}_{3}x}+{\beta }_{4}{W}_{4}{e}^{-{ik}_{1}\left(L-x\right)}+{\beta }_{5}{W}_{5}{e}^{-{ik}_{2}\left(L-x\right)}+{\beta }_{6}{W}_{6}{e}^{-{ik}_{3}\left(L-x\right)}.\end{array}$$
(16)

Figure 2 illustrates the sign conventions assumed for nodal displacements and nodal forces for finite spectral element formulation. Substituting the boundary conditions of nodal displacements gives

Fig. 2
figure 2

Sign conventions assumed for a nodal displacements and b nodal forces for finite spectral element formulation

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$$\left[\begin{array}{c}{\overline{u} }_{1}\\ {\overline{w} }_{1}\\ {\overline{\theta }}_{1}\\ {\overline{u} }_{2}\\ {\overline{w} }_{2}\\ {\overline{\theta }}_{2}\end{array}\right]=\left[\begin{array}{c}\overline{u }(0)\\ \overline{w }(0)\\ \overline{\theta }(0)\\ \overline{u }(L)\\ \overline{w }(L)\\ \overline{\theta }(L)\end{array}\right]=\left[{{\varvec{T}}}_{1}\right]\left[\begin{array}{c}{W}_{1}\\ {W}_{2}\\ {W}_{3}\\ {W}_{4}\\ {W}_{5}\\ {W}_{6}\end{array}\right].$$
(17)

By substituting Eq. (16) into Eq. (7), the balancing nodal forces for an element gives

$$\left[\begin{array}{c}{\overline{T} }_{1}\\ {\overline{Q} }_{1}\\ {\overline{M} }_{1}\\ {\overline{T} }_{2}\\ {\overline{Q} }_{2}\\ {\overline{M} }_{2}\end{array}\right]=\left[\begin{array}{c}-\overline{T }\left(0\right)\\ -\overline{Q }\left(0\right)\\ -\overline{M }\left(0\right)\\ \overline{T }\left(L\right)\\ \overline{Q }\left(L\right)\\ \overline{M }\left(L\right)\end{array}\right]=\left[{{\varvec{T}}}_{2}\right]\left[\begin{array}{c}{W}_{1}\\ {W}_{2}\\ {W}_{3}\\ {W}_{4}\\ {W}_{5}\\ {W}_{6}\end{array}\right].$$
(18)

Combining Eq. (18) and Eq. (17) gives

$$\left[\begin{array}{c}{\overline{T} }_{1}\\ {\overline{Q} }_{1}\\ {\overline{M} }_{1}\\ {\overline{T} }_{2}\\ {\overline{Q} }_{2}\\ {\overline{M} }_{2}\end{array}\right]=\left[{{\varvec{T}}}_{2}\right]{\left[{{\varvec{T}}}_{1}\right]}^{-1}\left[\begin{array}{c}{\overline{u} }_{1}\\ {\overline{w} }_{1}\\ {\overline{\theta }}_{1}\\ {\overline{u} }_{2}\\ {\overline{w} }_{2}\\ {\overline{\theta }}_{2}\end{array}\right].$$
(19)

From Eq. (19), the frequency-dependent dynamic stiffness matrix for a finite spectral element is expressed as

$$S\left(\upomega \right)=\left[{{\varvec{T}}}_{2}\right]{\left[{{\varvec{T}}}_{1}\right]}^{-1}.$$
(20)

\(\left[{{\varvec{T}}}_{1}\right] and \left[{{\varvec{T}}}_{2}\right]\) are presented in the Appendix section; refer to Equation (A2) and Equation (A3), respectively.

Throw-off Spectral Element Formulation

A throw-off element is developed to capture only either forward or backward propagating waves. Figure 3 represents the nodal displacements and nodal forces for the throw-off spectral element. Similarly, the frequency-dependent dynamic stiffness matrix can be derived for the throw-off spectral element.

Fig. 3
figure 3

Sign conventions defined for a nodal displacements and b nodal forces for throw-off spectral element formulation

Full size image

From Eq. (16), the general solution reduces to: (as \(L=\infty\))

$$\begin{array}{c}{\overline{u} }_{t}(x)={\alpha }_{1}{W}_{1}{e}^{-{ik}_{1}x}+{\alpha }_{2}{W}_{2}{e}^{-{ik}_{2}x}+{\alpha }_{3}{W}_{3}{e}^{-i{k}_{3}x}\\ {\overline{w} }_{t}(x)={W}_{1}{e}^{-{ik}_{1}x}+{W}_{2}{e}^{-i{k}_{2}x}+{W}_{3}{e}^{-{ik}_{3}x}\\ {\overline{\theta }}_{t}(x)={\beta }_{1}{W}_{1}{e}^{-{ik}_{1}x}+{\beta }_{2}{W}_{2}{e}^{-i{k}_{2}x}+{\beta }_{3}{W}_{3}{e}^{-{ik}_{3}x}.\end{array}$$
(21)

Substituting nodal displacement shown in Fig. 3 gives

$$\left[\begin{array}{c}{\overline{u} }_{t1}\\ {\overline{w} }_{t1}\\ {\overline{\theta }}_{t1}\end{array}\right]=\left[\begin{array}{c}{\overline{u} }_{t}(0)\\ {\overline{w} }_{t}(0)\\ {\overline{\theta }}_{t}(0)\end{array}\right]=\left[{{\varvec{T}}}_{{\varvec{t}}1}\right]\left[\begin{array}{c}{W}_{1}\\ {W}_{2}\\ {W}_{3}\end{array}\right].$$
(22)

By substituting Eq. (22) in Eq. (8), the balancing nodal forces are expressed as

$$\left[\begin{array}{c}{\overline{T} }_{t1}\\ {\overline{Q} }_{t1}\\ {\overline{M} }_{t1}\end{array}\right]=\left[\begin{array}{c}-{\overline{T} }_{t}\left(0\right)\\ -{\overline{Q} }_{t}\left(0\right)\\ -{\overline{M} }_{t}\left(0\right)\end{array}\right]=\left[{{\varvec{T}}}_{{\varvec{t}}2}\right]\left[\begin{array}{c}{W}_{1}\\ {W}_{2}\\ {W}_{3}\end{array}\right].$$
(23)

Combining Eq. (23) and Eq. (22) gives

$$\left[\begin{array}{c}{\overline{T} }_{t}\\ {\overline{Q} }_{t}\\ {\overline{M} }_{t}\end{array}\right]=\left[{{\varvec{T}}}_{{\varvec{t}}2}\right]{\left[{{\varvec{T}}}_{{\varvec{t}}1}\right]}^{-1}\left[\begin{array}{c}{\overline{u} }_{t}\\ {\overline{w} }_{t}\\ {\overline{\theta }}_{t}\end{array}\right].$$
(24)

Om Eq. (24), the frequency-dependent dynamic stiffness matrix for a throw-off spectral element is expressed as

$${S}_{t\left(\omega \right)}=\left[{{\varvec{T}}}_{{\varvec{t}}2}\right]{\left[{{\varvec{T}}}_{{\varvec{t}}1}\right]}^{-1}.$$
(25)

\(\left[{{\varvec{T}}}_{{\varvec{t}}1}\right] and \left[{{\varvec{T}}}_{{\varvec{t}}2}\right]\) are explained in the Appendix section; refer to Equation (A5) and Equation (A6), respectively. The spectral elements in Eqs. (20) and (25) can be analogously assembled in the conventional FEM. After applying the relevant boundary conditions, a global dynamic stiffness matrix equation can be obtained.

Coupled Axial-Bending Cracked Timoshenko Beam Formulation

Figure 4 shows a finite spectral element of a coupled axial-bending Timoshenko beam with an open transverse and non-propagating crack. The length of the element is L, and its cross-sectional area is A. The crack is replaced by a dimensionless-massless spring, the flexibility of which is calculated using the Castigliano Theorem and laws of fracture mechanics [41, 42]. The spectral nodal displacements \(\overline{u }\) and \(\overline{w }\), and rotations \(\overline{\theta }\) are presumed for the left and right parts of the beam as follows:

$$\begin{gathered} \overline{{u_{1} }} =\; A_{1} \alpha_{1} e^{{\left( { - ik_{1} x} \right)}} + B_{1} \alpha_{2} e^{{\left( { - ik_{2} x} \right)}} + \in_{1} \alpha_{3} e^{{\left( { - ik_{3} x} \right)}} \;\;+ D_{1} \alpha_{4} e^{{\left( { - ik_{1} \left( {L_{1} - x} \right)} \right)}} + E_{1} \alpha_{5} e^{{\left( { - ik_{2} \left( {L_{1} - x} \right)} \right)}} \;\;+ F_{1} \alpha_{6} e^{{\left( { - ik_{3} \left( {L1 - x} \right)} \right)}} for x \in \left( {0,L_{1} } \right) \hfill \\ \overline{{w_{1} }} \;\;= A_{1} e^{{\left( { - ik_{1} x} \right)}} + B_{1} e^{{\left( { - ik_{2} x} \right)}} + \in_{1} e^{{\left( { - ik_{3} x} \right)}} + D_{1} e^{{\left( { - ik_{1} \left( {L_{1} - x} \right)} \right)}} + E_{1} e^{{\left( { - ik_{2} \left( {L_{1} - x} \right)} \right)}} + F_{1} e^{{\left( { - ik_{3} \left( {L_{1} - x} \right)} \right)}} for x \in \left( {0,L_{1} } \right) \hfill \\ \overline{{\theta_{1} }} \;\;= A_{1} \beta_{1} e^{{\left( { - ik_{1} x} \right)}} + B_{1} \beta_{2} e^{{\left( { - ik_{2} x} \right)}} + \in_{1} \beta_{3} e^{{\left( { - ik_{3} x} \right)}} + D_{1} \beta_{4} e^{{\left( { - ik_{1} \left( {L_{1} - x} \right)} \right)}} + E_{1} \beta_{5} e^{{\left( { - ik_{2} \left( {L_{1} - x} \right)} \right)}} + F_{1} \beta_{6} e^{{\left( { - ik_{3} \left( {L_{1} - x} \right)} \right)}} for x \in \left( {0,L_{1} } \right) \hfill \\ \overline{{u_{2} }} = A_{2} \alpha_{1} e^{{\left( { - ik_{1} \left( {L_{1} + x} \right)} \right)}} + B_{2} \alpha_{2} e^{{\left( { - ik_{2} \left( {L_{1} + x} \right)} \right)}} + \in_{2} \alpha_{3} e^{{\left( { - ik_{3} \left( {L_{1} + x} \right)} \right)}} + D_{2} \alpha_{4} e^{{\left( {ik_{1} \left( {L_{1} - L + x} \right)} \right)}} + E_{2} \alpha_{5} e^{{\left( {ik_{2} \left( {L_{1} - L + x} \right)} \right)}} + F_{2} \alpha_{6} e^{{\left( {ik_{3} \left( {L_{1} - L + x} \right)} \right)}} for x \in \left( {0,L - L_{1} } \right) \hfill \\ \overline{{w_{2} }} \;\;= A_{2} e^{{\left( { - ik_{1} \left( {L_{1} + x} \right)} \right)}} + B_{2} e^{{\left( { - ik_{2} \left( {L_{1} + x} \right)} \right)}} + \in_{2} e^{{\left( { - ik_{3} \left( {L_{1} + x} \right)} \right)}} + D_{2} e^{{\left( {ik_{1} \left( {L_{1} - L + x} \right)} \right)}} + E_{2} e^{{\left( {ik_{2} \left( {L_{1} - L + x} \right)} \right)}} + F_{2} e^{{\left( {ik_{3} \left( {L_{1} - L + x} \right)} \right)}} for x \in \left( {0,L - L_{1} } \right) \hfill \\ \overline{{\theta_{2} }} = A_{2} \beta_{1} e^{{\left( { - ik_{1} \left( {L_{1} + x} \right)} \right)}} + B_{2} \beta_{2} e^{{\left( { - ik_{2} \left( {L_{1} + x} \right)} \right)}} + \in_{2} \beta_{3} e^{{\left( { - ik_{3} \left( {L_{1} + x} \right)} \right)}} + D_{2} \beta_{4} e^{{\left( {ik_{1} \left( {L_{1} - L + x} \right)} \right)}} + E_{2} \beta_{5} e^{{\left( {ik_{2} \left( {L_{1} - L + x} \right)} \right)}} + F_{2} \beta_{6} e^{{\left( {ik_{3} \left( {L_{1} - L + x} \right)} \right)}} for x \in \left( {0,L - L_{1} } \right), \hfill \\ \end{gathered}$$
(26)

where \({L}_{1}\) is the location of the crack from the left, and the amplitude ratios \({\alpha }_{n}\) and \({\beta }_{n}\) are determined from Eq. (15), respectively.

Fig. 4
figure 4

A coupled axial-bending Timoshenko beam model with a transverse open and non-propagating crack simulated by an elastic hinge

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The coefficients in Eq. (26),\(A_{1} ,B_{1} , \in_{1} ,D_{1} ,E_{1} ,F_{1} ,A_{2} ,B_{2} , \in_{2} ,D_{2} ,E_{2} ,F_{2}\) are calculated from the following boundary conditions:

  • At the left end of the element (\(x=0)\)

    $$\begin{gathered} u_{1} \left( x \right) = q_{1} \hfill \\ u_{1} \left( x \right) = q_{1} \hfill \\ w_{1} \left( x \right) = q_{2} \hfill \\ \theta_{1} \left( x \right) = q_{3} . \hfill \\ \end{gathered}$$
    (27)

  • At the right end of the element (\(x=L\))

    $$\begin{gathered} u_{2} \left( x \right) = q_{4} \hfill \\ w_{2} \left( x \right) = q_{5} \hfill \\ \theta_{2} \left( x \right) = q_{6} . \hfill \\ \end{gathered}$$
    (28)

  • At crack location (a total change of displacements and rotation angle, compatibility of bending

Moments and shear forces)

$$\begin{gathered} x = L_{1}\,\, for \,\,u_{1} \left( x \right),w_{1} \left( x \right),\theta_{1} \left( x \right)\,\, and\,\, x = 0 for u_{2} \left( x \right),w_{2} \left( x \right),\theta_{2} \left( x \right) \hfill \\ u_{2} \left( x \right) - u_{1} \left( x \right) - \phi \frac{{\partial u_{1} \left( x \right)}}{\partial x} = 0 \hfill \\ EA\frac{{\partial u_{1} \left( x \right)}}{\partial x} + C_{1} \left( {\frac{{\partial w_{1} \left( x \right)}}{\partial x} - \theta_{1} \left( x \right)} \right) + C_{2} \frac{{\partial \theta_{1} \left( x \right)}}{\partial x} - \left( {EA\frac{{\partial u_{2} \left( x \right)}}{\partial x} + C_{1} \left( {\frac{{\partial w_{2} \left( x \right)}}{\partial x} - \theta_{2} \left( x \right)} \right) + C_{2} \frac{{\partial \theta_{2} \left( x \right)}}{\partial x}} \right) = 0 \hfill \\ w_{2} \left( x \right) - w_{1} \left( x \right) - \xi \left( {\frac{{\partial w_{1} \left( x \right)}}{\partial x} - \theta_{1} \left( x \right)} \right) = 0 \hfill \\ C_{1} \frac{{\partial u_{1} \left( x \right)}}{\partial x} + GA\left( {\frac{{\partial w_{1} \left( x \right)}}{\partial x} - \theta_{1} \left( x \right)} \right) + C_{3} \frac{{\partial \theta_{1} \left( x \right)}}{\partial x} - \left( {C_{1} \frac{{\partial u_{2} \left( x \right)}}{\partial x} + GA\left( {\frac{{\partial w_{2} \left( x \right)}}{\partial x} - \theta_{2} \left( x \right)} \right) + C_{3} \frac{{\partial \theta_{2} \left( x \right)}}{\partial x}} \right) = 0 \hfill \\ \theta_{2} \left( x \right) - \theta_{1} \left( x \right) - \Omega \frac{{\partial \theta_{1} \left( x \right)}}{\partial x} = 0 \hfill \\ C_{2} \frac{{\partial u_{1} \left( x \right)}}{\partial x} + C_{3} \left( {\frac{{\partial w_{1} \left( x \right)}}{\partial x} - \theta_{1} \left( x \right)} \right) + EI\frac{{\partial \theta_{1} \left( x \right)}}{\partial x} - \left( {C_{2} \frac{{\partial u_{2} \left( x \right)}}{\partial x} + C_{3} \left( {\frac{{\partial w_{2} \left( x \right)}}{\partial x} - \theta_{2} \left( x \right)} \right) + EI\frac{{\partial \theta_{2} \left( x \right)}}{\partial x}} \right) = 0, \hfill \\ \end{gathered}$$
(29)

where \(\phi\), \(\xi\), and \(\Omega\) are the non-dimensional form for the axial, bending, and shear flexibilities, respectively. By considering the formulae describing the nodal spectral displacements in Eqs. (27) - (29) for the left and right parts of the Timoshenko beam, the boundary conditions in the form of a matrix are expressed as

$$\left[ {cT_{1} } \right]_{12 \times 12} \left[ {\begin{array}{*{20}c} {A_{1} } \\ {B_{1} } \\ { \in_{1} } \\ {D_{1} } \\ \cdot \\ \cdot \\ { \in_{2} } \\ {D_{2} } \\ {E_{2} } \\ {F_{2} } \\ \end{array} } \right]_{12 \times 1} = \left[ {\begin{array}{*{20}c} {q_{1} } \\ {q_{2} } \\ {q_{3} } \\ {q_{4} } \\ {q_{5} } \\ {q_{6} } \\ 0 \\ . \\ . \\ 0 \\ \end{array} } \right]_{12 \times 1} .$$
(30)

The nodal spectral forces (\(P\)- axial force, \(T\)- shear force, and \(M\)- bending moment) are obtained by differentiating the spectral displacements; and formerly written in the matrix system as

$$\left[ {cT_{2} } \right]_{6 \times 12} \left[ {\begin{array}{*{20}c} {A_{1} } \\ {B_{1} } \\ { \in_{1} } \\ {D_{1} } \\ \cdot \\ \cdot \\ { \in_{2} } \\ {D_{2} } \\ {E_{2} } \\ {F_{2} } \\ \end{array} } \right]_{12 \times 1} = \left[ {\begin{array}{*{20}c} {P_{1} } \\ {V_{1} } \\ {M_{1} } \\ {P_{2} } \\ {V_{2} } \\ {M_{2} } \\ \end{array} } \right]_{6 \times 1} .$$
(31)

From \(\left[{{\varvec{c}}{\varvec{T}}}_{1}\right]\) and \(\left[{{\varvec{c}}{\varvec{T}}}_{2}\right]\), the square matrix (6 × 6), which signifies the frequency-dependent dynamic stiffness for a coupled axial-bending Timoshenko beam spectral element with transverse open and non-propagating crack, is obtained. It is represented in the Appendix section; refer to Equation (A7) and Equation (A8).

Flexibilities in a Crack Region

The beam flexibilities at the crack region are obtained from Castigliano’s theorem by the following equation:

$${c}_{ij}=\frac{{\partial }^{2}U}{\partial {P}_{i}\partial {P}_{j}} i=1, 2,..6 , j=1, 2,..6,$$
(32)

where \(U\) represents the elastic strain energy of the element caused by the presence of the crack, and P is an independent nodal force acting on the element. The elastic strain energy due to the crack is expressed as [43]

$$U=\frac{1}{E}{\int }_{{A}_{a}} \left({K}_{I}^{2}+{K}_{II}^{2}\right)d{A}_{a},$$
(33)

where \({K}_{I}\) and \({K}_{\mathrm{II}}\) denote stress intensity factors corresponding to the first and second crack growth mode, and \({A}_{a}\) denotes crack area. Figure 5 shows the first (I) and second (II) crack propagation modes and cross-section of beam element at the crack location.

Fig. 5
figure 5

Flexibility at the crack location: a first (I) and second (II) crack propagation modes; b cross-section of beam element at the crack location

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Axial Flexibility

The stress intensity factor corresponding with the first mode of the crack formation due to the load \({P}_{1}\) is obtained as

$${K}_{a}=\frac{{P}_{1}}{{A}_{a}}\sqrt{\pi \varkappa }{F}_{I}\left(\frac{\varkappa }{H}\right),$$
(34)

where \(\varkappa\) and \(H\) are illustrated in Fig. 5 (b). \({F}_{I}\) is the correction factor expressed as [37]

$${F}_{I}\left(\frac{\varkappa }{H}\right)=\sqrt{\frac{tan\left(\pi \varkappa /2H\right)}{\pi \varkappa /2H}}\frac{0.752+2.02(\varkappa /H)+0.37[1-sin(\pi \varkappa /2H){]}^{3}}{cos\left(\pi \varkappa /2H\right)}.$$
(35)

The flexibility of the elastic element modeling of the cracked cross-section of the rod spectral finite element is rewritten as

$$\begin{gathered} c_{a} = \;\frac{{2\pi (1 - v^{2} )}}{EB}\;\int\limits_{0}^{{\overline{a}}} {\overline{\varkappa }{\text{F}}_{I}^{2} \left( {\overline{\varkappa }} \right)d\overline{a}} \hfill \\ \phi = EA_{a} c_{a} , \hfill \\ \end{gathered}$$
(36)

where \({c}_{a}\) is the axial flexibility due to the presence of a crack; \(\overline{a }=\frac{a}{H}\) and \(\overline{\varkappa }=\frac{\varkappa }{H}\); \(\phi\) is the non-dimensional form of axial flexibility\({c}_{a}\); \({A}_{a}=BH\) (refer to Fig. 5).

Bending Flexibility

Similar to the axial flexibility, the stress intensity factors due to the bending moment M and shear force V are obtained as

$${K}_{I}=\frac{6M}{B{H}^{2}}\sqrt{\pi \varkappa }{F}_{I}\left(\frac{\varkappa }{H}\right),{ K}_{II}=\frac{\vartheta V}{BH}\sqrt{\pi \varkappa }{F}_{II}\left(\frac{\varkappa }{H}\right),$$
(37)

where \(\vartheta\) is the shear factor [44]; \({F}_{\mathrm{I}}\) (refer to Eq. (35)) and \({F}_{\mathrm{II}}\) are correction functions expressed as

$${F}_{II}\left(\frac{\varkappa }{H}\right)=\frac{1.30-0.65(\varkappa /H)+0.37(\varkappa /H{)}^{2}+0.28(\varkappa /H{)}^{3}}{\sqrt{1-\left(\varkappa /H\right)}}.$$
(38)

The modeling of the flexibilities of the elastic elements cracked cross-section of a coupled axial-bending Timoshenko beam finite spectral element is written as

$${c}_{b}=\frac{72\pi \left(1-{\upnu }^{2}\right)}{B{H}^{2}E}{\int }_{0}^{\overline{a} } \overline{\varkappa }{F }_{I}^{2}\left(\overline{\varkappa }\right)d\overline{a }, {c}_{s}=\frac{4\vartheta \pi \left(1-{\upnu }^{2}\right)}{BE}{\int }_{0}^{\overline{a} } \overline{\varkappa }{F }_{II}^{2}\left(\overline{\varkappa }\right)d\overline{a }$$
$$\xi = \frac{{EIc_{b} }}{L},{ }\Omega = \frac{{GA_{a} c_{s} }}{L},$$
(39)

where \({c}_{b}\) and \({c}_{s}\) denote bending and shear flexibilities, respectively; \(\overline{a }=\frac{a}{H}\) and \(\overline{\varkappa }=\frac{\varkappa }{H}\) (refer to Fig. 5); \(\xi\) and \(\Omega\) are the non-dimensional form for the bending and shear flexibilities, respectively.

Numerical Investigations

Several numerical investigations have been performed to demonstrate the effectiveness of the proposed model on beams. Figure 6 shows the input excitation signal in the time domain and its FFT in the frequency domain. The beam has dimensions of 2.0 m in length, 0.02 m in height, and 0.02 m in width; Young's modulus is 210 GPa, and a mass density of 7860 kg/m3 is considered for the analysis. The beam model is excited by a signal of 0.7 MHz. It is the product of a Hanning window lasting for 125 μs, performed through convolution in the frequency domain, which distributes the effects of the pulse. It is understood that a signal that lasts for a short time can excite a wider range of frequencies in the frequency domain for the tested signal. The dynamic stiffness matrix derived for coupled axial-bending cracked Timoshenko beam in Eqs. (30) and (31) is used for numerical investigations.

Fig. 6
figure 6

Excitation signal in a time domain and b frequency domain

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The first and second numerical tests validate the proposed Timoshenko beam model for capturing forward and backward moving waves. All the coupling coefficients are considered zero (\({C}_{1},{C}_{2},{C}_{3}=0\)), and the first moment of inertia is also zero (\(\rho R=0\)). Figure 7 (a) shows that the first numerical test model (Test-I) consists of classical Timoshenko beam and throw-off elements with the position of measurement points. The second numerical test model (Test-II) consists of only one classical Timoshenko beam element, as shown in Fig. 7 (b). It is exciting from one end, and the other has a fixed boundary condition.

Fig. 7
figure 7

Proposed Timoshenko beam model: a for Test-I and b for Test-II, respectively

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Figure 8 shows the wave propagation for the model in the axial and bending part of the Timoshenko beam, respectively. There is no reflection from the beam's other end due to the throw-off element's presence, as it acts like a conduit that throws the wave energy out of the system. Figure 9 shows the wave propagation for the model in the axial and bending part of the Timoshenko beam, respectively. There is a reflection from the fixed end, as expected. Thus, both numerical tests show that the model can capture forward and backward propagating waves. From the reasons mentioned above, it can be seen that with the proposed spectral Timoshenko beam element with a crack, the dynamically changing stiffness matrix can be obtained without any additional mathematical computation. In this figure, these are the reflections of the second mode of bending part of coupled axial-bending Timoshenko beam. Coupled axial-bending Timoshenko beam is a higher order beam theory. The bending part of coupled axial-bending Timoshenko beam has two modes. In paper \({k}_{2}\left(\omega \right)\) and \({k}_{3}\left(\omega \right)\). \({k}_{2}\left(\omega \right)\) is always real, and \({k}_{3}\left(\omega \right)\) is initially imaginary, but it also becomes real after crossing the cut-off frequency. The excitation frequency in this paper is beyond the cut-off frequency; hence, the second mode propagates. Compared to the Euler–Bernoulli beam, it also has two modes, but the second mode is equal to the first mode and always imaginary, hence no propagation. The wave propagation in structures depends upon the material and cross-sectional properties. The axial and bending waves have different propagating speeds. The axial wave propagation speed is equal to \(\surd \frac{E}{\rho }\).

Fig. 8
figure 8

Wave propagation in Test-I for a axial and b bending parts

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Fig. 9
figure 9

Wave propagation in Test-II for a axial and b bending parts, respectively

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In contrast, in the Timoshenko beam, the expression for group speed is more complex and frequency-dependent, and the speed is slower than the axial part. The axial part has only one propagating mode, which is not observed in the axial part. Only the bending part has a reflection of the first and second modes from the crack.

The third numerical test model (Test-III) illustrates the influence of the presence of a crack in the beam, as shown in Fig. 10a. The crack introduced is a transverse open and non-propagating crack. It is located at 25% of the beam length, and the crack depth is 15% of the beam depth. The proposed beam model is excited with the input signal, as shown in Fig. 7. All the coupling coefficients are considered zero (\({C}_{1},{C}_{2},{C}_{3}=0\)), and the first moment of inertia is also taken as zero (\(\rho R=0\)). The model consists of one cracked Timoshenko beam element in the middle and two throw-off elements at both ends. The results are compared in Fig. 10 (b) and 10 (c), with a cracked rod for the axial part [35] and a cracked Timoshenko beam for the bending part [36].

Fig. 10
figure 10

a Wave propagation in Test-II for a axial and b bending parts, respectively, compared with FEM

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Figure 10b shows that the first wave packet stands for the excitation signal, and the second is reflected from the crack. There is no third reflection as it is a throw-off spectral element. The Timoshenko beam axial response agrees with the cracked rod results. Likewise, the Timoshenko beam bending response closely matches the cracked Timoshenko beam model.

To validate the wave propagation results obtained above, tests were also carried out in the time domain (FEM). The lack of throw-off elements makes it challenging to analyze when both forward and moving waves are present. Therefore, infinite boundary conditions are difficult to implement. Therefore, semi-infinite beam is not modeled in FEM.

Wave reflection from a fixed boundary condition is performed in FEM and compared with the SEM in Fig. 10. To perform this analysis, 1000 elements in 2 m were discretized. The time step was very small equal to 1.5e-8 s. Compared to SEM, to obtain the same results, only one element is required, and the analysis is not required for the whole range; since the beam is excited at 700 kHz, a band around that specific frequency is enough. This also saves lots of computation cost for integrations making it faster than FEM. From this analysis, it is evident that SEM is advantageous over FEM for high-frequency analysis. As shown in Fig. 10b, the results were distorted and exploded at the end. This result was obtained without adding any damping in an explicit time integration scheme. This is a disadvantage in the time domain as the same result is obtained accurately using SEM when no damping is applied. When a small damping is applied to the explicit scheme to make it stable, the reflections of the second propagating mode (higher frequency mode) disappear, as shown in Fig. 10b. This shows another advantage of SEM that it can do theoretical simulations when there is no damping. The explicit time domain is sensitive to two things: a) CFL condition and b) absence of damping in the model leads to inherent instability.

Modeling the same crack in the conventional FEA software has the following challenges: (a) stiffness of the crack is frequency-dependent, (b) non-orthogonal due to the equation of coupled axial-bending wave propagation, and (c) coupled to other degrees of freedom. Figure 11 shows the variation of dynamic stiffness of axial degree of freedom of axial-bending coupled Timoshenko beam with respect to the frequency at low and high frequencies. Figure shows stiffness reduction when the crack (a/H = 0.5) is present.

Fig. 11
figure 11

Dynamic stiffness for axial degree of freedom at a low frequency and b high frequency

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To validate the results for wave propagation in FEM and SEM, the following workaround is proposed to validate wave propagation in axial direction. In FEM, the crack is modeled as tiny element, whose stiffness is obtained from the dynamic stiffness matrix of SEM element at low frequency corresponding to the axial degree of freedom. The results obtained are shown in Fig. 12. The time taken for the wave to return is half the time to reflect from the boundary for the axial part of coupled axial-bending Timoshenko beam.

Fig. 12
figure 12

Reflection from element representing a crack in simple bar element

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The influences of varying crack locations on the wave propagation in the beam for axial and bending parts are presented in Fig. 14a, b, respectively. The location of the crack is changed to find out how the model will behave when the location of the crack moves away from the excitation node. Correspondingly, Figs. 13c and 14b show the presence of a second propagating mode [10], which is faster than the first mode.

Fig. 13
figure 13

Proposed beam model for numerical Test-III and its wave propagation phenomenon

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Fig. 14
figure 14

Influence of varying crack position on the wave propagation in beam for a axial part and b bending part, respectively

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The fourth and fifth numerical tests are examined to show that the proposed coupled axial-bending cracked Timoshenko beam model is useful for modal analysis. For the free-vibration analysis, the model is considered without coupling coefficients (Test-IV) and with coupling coefficients (Test-V). The investigation is carried out on the beam models, namely, clamped-free and simply–simply supported boundary conditions, with and without crack, to estimate modal properties. Table 1 illustrates the structural properties of the beam model with coupling coefficients. The proposed spectral elements are used in a planar beam with five repeating elements (i.e., 2 m overall length) to assess the natural frequency when exposed to different boundary conditions. The length of the repeating element unit is 40 cm. Figure 15 illustrates the beam model in clamped-free and simply–simply supported boundary conditions.

Table 1 Properties of a coupled axial-bending Timoshenko beam model with coupling coefficients
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Fig. 15
figure 15

Proposed beam models with different boundary conditions utilized for Tests-IV and V

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For the numerical Tests-IV and V, the beam models are considered without and with coupling effects, respectively. The crack is located in the middle of the beam model. The parametric study is conducted for modal analysis to understand the crack depth effect by varying it from 5 to 25% of the beam height (\(\frac{a}{H}\) ratio). The natural frequencies for both support conditions are calculated using the STMM for all the crack depths. The beam model is excited with the respective natural frequencies. Figure 16a, b shows the first four normal bending mode shapes of the beam model without coupling effects for clamped-free and simply–simply supported boundary conditions, respectively.

Fig. 16
figure 16

Lowest four bending normal modes without coupling effects

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The investigation for Test-V is executed with a shear-bending coupling coefficient (\({C}_{3}\)) taken into account. Figure 17a, b shows the first four normal bending mode shapes of the beam model with coupling effects for clamped-free and simply–simply supported boundary conditions, respectively. The computed natural frequency results considering coupling effects with varying crack depths are tabulated for beam models with clamped-free and simply–simply supported boundary conditions in Tables 2 and 3. The results are validated and found to agree with the values presented in Ref. [42] for the non-coupling and clamped-free part. Table 4 summarizes and compare with ref. [42] for SEM.

Fig. 17
figure 17

Lowest four bending normal modes for the simply supported beam with coupling effects

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Table 2 Estimated natural frequencies for the clamped-free beam with various crack formations
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Table 3 Estimated natural frequencies for the simply–simply supported beam with various crack formations
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Table 4 Comparison of natural frequencies for clamped-free and without coupling effects
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The numerical tests are investigated to show that the proposed element works fine and is verified with various parameters. The flexibility at the crack locations and beam with a coupling coefficient is also explained well. It is observed that these coupling coefficients are essential for the coupled axial-bending Timoshenko beam element, and the influence of coupling on an overall response needs to be understood in detail. For this, a parametric study is carried out by varying the coefficient value to observe its effects on the overall structural response of the element in its fundamental mode. As the varying coupling coefficients change the element's mode shapes, it directly influences the overall response of the beam. The same material properties explained in Table 1 are used for the parametric study. Tests-I and II are investigated with varying \({C}_{3}\) value for clamped-free and simply supported beams. The crack is located at the mid-span of the beam, and the crack depth is considered half of the beam's depth. Figure 18 (a) and 18 (b) shows the variation of first natural frequencies with the change in \({C}_{3}\) for clamped-free and simply supported beams, respectively. Here, \({\omega }_{n}\) and \({\omega }_{n0}\) are the natural frequency of the beam without and with crack and coupling effect, respectively.

Fig. 18
figure 18

Variation of \({\omega }_{n}/{\omega }_{n0}\) with the \({C}_{3}\) value

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The ratio (\({\omega }_{n}/{\omega }_{n0}\)) first decreases and increases for higher \({C}_{3}\) values. At first, there is a slight discontinuity at the mid-span of the beam due to the presence of a crack. As \({C}_{3}\) is increased, the shear deformation is also increased. Since the transverse crack deformation will be very high, all the discontinuities focus on the crack location. This beam behaves in dual nature: one part being a cantilever and the other being supported and connected by the crack. Hence, the beam will not bend; it becomes straight. Therefore, more shear deformation is observed than bending deformation. A few mode shapes are shown in Fig. 19 to display the difference in mode shape in the absence and presence of coupling coefficients.

Fig. 19
figure 19

Mode shape comparison for a simply supported beam with crack

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Conclusions

This paper presents a novel spectral element model for the coupled axial-bending cracked Timoshenko beam element. The work attempts to unify the theory of the SFEM for Timoshenko beam and fracture mechanics in the frequency domain, making it simpler for wave propagation and modal analysis with the presence of coupled axial-bending effects. It is easy to see that the spectral approach provides more information and is more appropriate for detecting damage. The results show that the current approach can calculate higher natural frequencies without additional computational time. An open non-propagating transverse-single edge crack opening is examined in the present research. The inferences derived from the five numerical test investigations are given below:

  • Tests-I and II are conducted to investigate whether the coupled axial-bending Timoshenko beam element and the throw-off element are capable of wave propagation as intended. For these numerical models, all the coupling coefficients and the first moment of inertia were set to zero for minimal complications. It is concluded from the investigations that the element allow both forward and backward wave propagation. Test-II was also analyzed in FEM and it showed the few of the many advantages of SEM over FEM in high-frequency analysis.

  • In Test-III, the variation in wave propagation analysis is studied in the presence of crack by comparing responses from damaged and undamaged coupled axial-bending Timoshenko beams without coupling coefficients. Then, the effect of crack depth and location is studied for the various wave propagation phenomena. Due to challenges of modeling crack in FEM, to validate wave propagation, analysis for axial degree of freedom was performed. The results were compared for SEM and FEM.

  • The modal analyses are carried out in Tests-IV and V on the beam models to obtain natural frequencies and mode shapes to understand the variation of the coupling coefficient (\({C}_{3}\)) for crack formation. The natural frequencies for both support conditions are calculated using the STMM for all the crack depths.

  • A parametric study is carried out by varying the coefficient value to observe its effects on the overall structural response of the element in its fundamental mode. The ratio (\({\omega }_{n}/{\omega }_{n0}\)) is well understood for varying \({C}_{3}\) values and the mode shapes are compared for a simply supported beam with a crack present.

The proposed spectral element for the coupled axial-bending cracked Timoshenko beam model has been shown to analyze the damaged structures and precisely identify and evaluate the crack location. Future research will extend the proposed spectral element system to damaged structures of more complex shapes.