Dynamic analysis of multiple cracked Timoshenko beam under moving load

Abstract

When cracks start to surface in the surrounding areas of the structure, they create a local softness zone and influences on the dynamic response of the structure. The beams are more susceptible to shear and flexural cracks because of being subjected to shear and bending stress. In this study, the dynamic response of the single-span and multi-span damped beam under moving load with multi-crack and elastic boundary condition is studied based on Timoshenko’s theory. The Green’s function method is used to calculate the dynamic response of the cracked beam. In addition, the Green’s function method provides a solution for the differential equations. Moreover, the effects of the crack on the essential characteristics of the multi-span beams, especially the natural frequencies, are investigated. In this study, crack by itself is modeled in different situations and its effect on the behavior of the beam is analyzed. Also, the elastically restrained beam is modeled and its effect on the behavior of the beam is assessed. Because of the fact that the Euler–Bernoulli theory is also used in most beams, in this study, the results of the numerical examples are compared with the Euler–Bernoulli theory. Several examples are analyzed for a better understanding of the Timoshenko cracked beam.

Keywords

Crack cracked beam Green’s function dynamic analysis Timoshenko beam

1. Introduction

Basically, “crack” by itself is a kind of damage that often occurs in structures, which are subjected to shear and bending stress. Detection of the cracked beam behavior is both safe and economically important. Several studies have been conducted to identify cracked beam behavior and the changes that they bring to the properties of the cracked beam because of the presence of crack on structural elements. In reference to Fernandez-Saez et al. (1999) the Rayleigh method to evaluate the fundamental frequency of the cracked beam, by adding the polynomial functions, the longitudinal deformation and transverse deformation were considered. Also, by comparing their results with the results of finite element modeling, a small percentage error was observed. Obtaining the natural frequency of the cracked beam by using the finite element method was carried out by Zheng and Kessissoglou (2004). By substituting the general flexible matrix in a local flexible matrix in the crack region, it was concluded that the results are more accurate than the local matrix. The vibrations of the beam with several cracks that were subjected to axial load have been studied by Binici (2005). By applying the continuity conditions at the crack section, Binici could calculate the mode shapes for a beam with different supporting conditions and compare the results with a finite element method. Investigations on cracked beam based on Timoshenko theory were conducted by Loya et al. (2006). To achieve the natural frequency of the Timoshenko cracked beam, the beam was divided into two parts in the cracked region and connected to each other with a spring. This method was used to extend the discontinuity in the vertical and rotational directions. The differential equation for the free vibration of the beam for each separate part was solved, and with the special continuity conditions at the cracked section, the two parts were connected. Kisa and Gurel (2007) provided a numerical method for analyzing the free vibration of a cracked beam with a uniform and nonuniform circular cross section. The finite element method and component mode synthesis were used together and the cracked section was modeled by the matrix. By using this technique, the natural frequency and mode shape were obtained. Sekhar (2008) assessed different studies in the field of multiple cracks beam. Then, by comparing them, a method that is more effective in identifying the behavior of the cracked beam led to the selection and examination of the effects of cracks in the beam. The calculation of the stiffness of the reinforced concrete cracked beam under static loading for evaluating and determining the dynamic response was carried out by Xu and Castel (2016). In this research, the effect of bending crack in the concrete, as well as failure in the connection between steel and concrete were considered. Moreover, this carried out a static and dynamic analysis of the beam and concluded that the failure of the bond between steel and concrete had an insignificant impact on the natural frequency and dynamic stiffness of the beam. The study of the forced vibration of the Euler–Bernoulli cracked beam was carried out by Zhao et al. (2016). To reach this goal, the Green’s function was used to obtain the dynamic response of cracked beam and also in regard to a torsion spring applied for modeling the cracked cross section. Panigrahi and Pohit (2018) used Timoshenko theory and the Ritz method to obtain the governing differential equation of cracked beam. The effects of rotational speed, depth and location of cracks, and properties of the used materials in the beam were analyzed. Also, the cracked Timoshenko beam was analyzed based on wave vibration approach by Mei et al. (2006). Forced and free vibration of cracked beam, which was subjected to axial load, were completed by using this method. Shafiei and Khaji (2011) investigated multi-cracked Timoshenko beam under moving load. The modal expansion approach was applied to obtain the dynamic response of the beam. At the same time, the nonuniform multi-cracked Timoshenko beam was analyzed based on the differential quadrature element method by Torabi et al. (2014). The effects of depth and location of the crack on the natural frequency were studied. Al-Said et al. (2006) proposed a method based on Lagrange’s method to derive motion equations of the Timoshenko beam. The influences of shear deformation and rotation speed on the dynamic response of the cracked Timoshenko beam were studied and the results were compared with Euler–Bernoulli theory. A closed form of multi-crack Timoshenko beam response with various boundary conditions was introduced by Aydin (2007). Vibrational frequency and effect of buckling load were investigated. The new shape function of displacement was introduced by Stojanović et al. (2013). Using this new function leads to p-version finite element method to vibration analysis damaged Timoshenko beam. Chouiyakh et al. (2017) researched Timoshenko’s thin and thick with several numbers of cracks under moving mass. To study the forced and free vibration of multi-cracked Timoshenko beam, the “methodological” method was applied. Sarvestan et al. (2017) investigated the cracked Timoshenko beam by using spectral finite element procedure. The dynamic response of the cracked beam under moving load with various load velocities was obtained. Ghannadiasl and khodapanah Ajirlou (2019) investigated on cracked multi-span Euler–Bernoulli beam. The damping effect on natural frequency and displacement is studied. The influence of partial elastic foundation and elastic boundary condition on cracked beam features by using the Green function was studied by Ghannadiasl and khodapanah Ajirlou (2018a, 2018b).

In previous studies, various approaches were used to determine the dynamic response of cracked Timoshenko and Euler–Bernoulli beam. Li et al. (2014) investigated on Timoshenko noncracked beam by using the Green’s function. Zhao et al. (2016) also applied Green’s function in the Euler–Bernoulli cracked beam based on Abu-Hillal (2003) studies. In this study, dynamic analysis of single-span and multi-span cracked Timoshenko beam with general boundary condition is presented by using a Green’s function approach based on Li et al. (2014).

2. Mathematical modeling of Timoshenko beam

The governing differential equations of the uniform Timoshenko beam are defined as (Lueschen et al., 1996)

E I θ_{xx} + κ G A (w_{x} - θ) - c_{r} θ_{t} - ρ I θ_{t t} = 0

(1)

κ G A (w_{x x} - θ_{x}) - c_{t} w_{t} - ρ A w_{t t} = F (x, t)

(2)

where

F (x, t)

can be defined as follows

F (x, t) = F_{0} δ (x - x_{F}) e^{i Ω t}

(3)

where

x_{F}

is the equation of the position of the moving load and may be calculated from the following relation (Ghannadiasl and Mofid, 2014)

x_{F} = x_{0} + v_{x} t + \frac{1}{2} a_{x} t^{2}

(4)

where

x_{0}

v_{x}

, and

a_{x}

are the initial position, speed, and acceleration of the moving load in the x-direction. The position function can be described as the decelerating or accelerating motion. In addition,

Ω

, the circular frequency of the moving load, is calculated as

Ω = π v / L

, where

v = \sqrt{v_{x}^{2} + 2 a_{x} (x_{F} - x_{0})}

is the speed of the moving load. The equation of forced vibration of the Timoshenko beam can be written in the form

\begin{array}{l} [\begin{array}{l} E I \frac{\partial^{4}}{\partial x^{4}} - ρ I (1 + \frac{E}{κ G}) \frac{\partial^{4}}{\partial t^{2} \partial x^{2}} - (c_{r} + \frac{E I c_{t}}{κ A G}) \frac{\partial^{3}}{\partial t \partial x^{2}} + c_{t} \frac{\partial}{\partial t} + (ρ A + \frac{c_{t} c_{r}}{κ A G}) \frac{\partial^{2}}{\partial t^{2}} \\ + \frac{ρ}{κ A G} (A c_{r} + I c_{t}) \frac{\partial^{3}}{\partial t^{3}} + \frac{ρ^{2} I}{κ G} \frac{\partial^{4}}{\partial t^{4}} \end{array}] w (x, t) \\ = [\frac{E I}{κ A G} \frac{\partial^{2}}{\partial x^{2}} - \frac{c_{r}}{κ A G} \frac{\partial}{\partial t} - \frac{ρ I}{κ A G} \frac{\partial^{2}}{\partial t^{2}} - 1] F (x, t) \end{array}

(5)

where

c_{t}

and

c_{r}

denote the damping coefficients, which are assumed as

\begin{array}{l} c_{t} = α_{t} ρ A \\ c_{r} = α_{r} ρ I \end{array}

where

α_{i}

and

α_{e}

are the proportionality constants. Now, consider that the

w (x, t)

can be assumed as the harmonic function, that is,

w (x, t) = W (x) e^{i Ω t}

On substituting the harmonic function in equation (5), one may get

\begin{array}{l} [\begin{array}{l} \frac{\partial^{4}}{\partial x^{4}} + ϕ^{2} (ϕ^{2} (r^{2} + α^{2}) - i ϕ β (c_{r} + α^{2} c_{t})) \frac{\partial^{2}}{\partial x^{2}} - ϕ^{2} (1 - ϕ^{2} r^{2} α^{2}) \\ + i c_{t} ϕ β (1 - ϕ^{2} α^{2} (\frac{c_{r}}{c_{t}} + r^{2})) - \frac{ϕ^{2} α^{2} β^{2}}{E I} (c_{t} c_{r}) \end{array}] W (x) \\ = \frac{F_{0}}{E I} ([ϕ^{2} r^{2} α^{2} - i c_{r} \frac{α^{2} ϕ β}{E I} - 1] δ (x - x_{F} (t)) + α^{2} F_{0} δ_{x x} (x - x_{F} (t))) \end{array}

(6)

where

ϕ

β, α

, and

r

are the parameter proportional to the natural frequency of the beam

(ϕ^{2} = Ω^{2} ρ A / EI)

, the parameter proportional to the beam rigidity

(β^{2} = E I / ρ A, α^{2} = E I / κ A G)

, and the radius of gyration (

r^{2} = I / A

), respectively. The superposition principle by considering Green’s function is defined as follows

W (x) = \int_{0}^{L} f (x_{0}) G (x, x_{0}) d x_{0}

(7)

where

G (x, x_{0})

represents the Green’s function. By using the physical meaning of Green’s function in equation (6), the following differential equation may be written

\begin{array}{l} [\begin{array}{l} \frac{\partial^{4}}{\partial x^{4}} + ϕ^{2} (ϕ^{2} (r^{2} + α^{2}) - i ϕ β (c_{r} + α^{2} c_{t})) \frac{\partial^{2}}{\partial x^{2}} - ϕ^{2} (1 - ϕ^{2} r^{2} α^{2}) \\ + i c_{t} ϕ β (1 - ϕ^{2} α^{2} (\frac{c_{r}}{c_{t}} + r^{2})) - \frac{ϕ^{2} α^{2} β^{2}}{E I} (c_{t} c_{r}) \end{array}] G \\ = \frac{1}{E I} ([ϕ^{2} r^{2} α^{2} - i c_{r} \frac{α^{2} ϕ β}{E I} - 1] δ (x - x_{F} (t)) + α^{2} δ_{x x} (x - x_{F} (t))) \end{array}

(8)

The above expressions may be simplified and rewrite as

G_{x x x x} + a_{1} G_{x x} + a_{2} G = b_{1} δ_{x x} (x - x_{F} (t)) + b_{2} δ (x - x_{F} (t))

(9)

where

\begin{array}{l} a_{1} = ϕ^{2} (ϕ^{2} (r^{2} + α^{2}) - i ϕ β (c_{r} + α^{2} c_{t})) \\ a_{2} = - ϕ^{2} (1 - ϕ^{2} r^{2} α^{2}) + i c_{t} ϕ β (1 - ϕ^{2} α^{2} (\frac{c_{r}}{c_{t}} + r^{2})) - \frac{ϕ^{2} α^{2} β^{2}}{E I} (c_{t} c_{r}) \\ b_{1} = \frac{α^{2}}{E I} \\ b_{2} = \frac{1}{E I} [ϕ^{2} r^{2} α^{2} - i c_{r} \frac{α^{2} ϕ β}{E I} - 1] \end{array}

(10)

The Laplace transform method is used to obtain the relation, which satisfies boundary conditions (Ghannadiasl and Khodapanah Ajirlou 2019). Finally, the relation based on the Green’s function method is presented in the following equation (Li et al., 2014)

G (x, x_{0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (\begin{array}{l} H (x - x_{F} (t)) Γ_{1} (x - x_{F} (t)) + Γ_{2}^{'} (x) (w {(x) |}_{x = 0}) + Γ_{2} (x) (w^{'} {(x) |}_{x = 0}) \\ + {Γ^{'}}_{3} (x) (w^{″} {(x) |}_{x = 0}) + Γ_{3} (x) (w^{‴} {(x) |}_{x = 0}) \end{array})

(11)

where

w (0), w^{'} (0), w^{''} (0)

, and

w^{'''} (0)

depend on the boundary conditions of the beam which are chosen for each beam with its boundary conditions. In the above equation,

Γ_{i} (x) (i = 1,2,3)

may be calculated from the following relations

\begin{array}{l} Γ_{1} (x) = (\frac{b_{1} s_{1}^{2} - b_{2}}{s_{1}}) Sinh (s_{1} x) - (\frac{b_{1} s_{2}^{2} - b_{2}}{s_{2}}) Sinh (s_{2} x) \\ Γ_{2} (x) = (\frac{a_{1} + s_{1}^{2}}{s_{1}}) Sinh (s_{1} x) - (\frac{a_{1} + s_{2}^{2}}{s_{2}}) Sinh (s_{2} x) \\ Γ_{3} (x) = \frac{1}{s_{1}} Sinh (s_{1} x) - \frac{1}{s_{2}} Sinh (s_{2} x) \end{array}

(12)

In equation (12), $s_{1}$ and $s_{2}$ , according to equation (8), can be presented as follow

s_{1} = \sqrt{\frac{- a_{1} - \sqrt{a_{1}^{2} - 4 a_{2}}}{2}} s_{2} = \sqrt{\frac{- a_{1} + \sqrt{a_{1}^{2} - 4 a_{2}}}{2}}

(13)

3. Green’s function for Timoshenko cracked beam

Consider one-cracked beam, which applied a concentrated load at $x_{10}$ as shown in Figure 1. Existence of crack along the length of the beam produces a discontinuity in the slope of the cracked beam deflections; thus, the beam is divided into two parts and for each part, equation (11) based on boundary condition is expanded (Zhao et al., 2016). In an instance, $w (0)$ and $w^{''} (0)$ must be vanished for the simply supported beam. The Green’s function for simply supported one-cracked beam is expressed in equation (14)

\begin{array}{l} G (x_{1}, x_{10}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (H (x_{1} - x_{10}) Γ_{1} (x_{1} - x_{10}) + Γ_{2} (x_{1}) (w^{'} {(x) |}_{x = 0}) + Γ_{3} (x_{1}) (w^{'''} {(x) |}_{x = 0})), x_{1} \in [0 - L_{1}) \\ G (x_{2}, x_{20}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (H (x_{2} - x_{20}) Γ_{1} (x_{2} - x_{20}) + Γ_{2} (x_{2}) (w^{'} {(x) |}_{x = 0}) + Γ_{3} (x_{2}) (w^{‴} {(x) |}_{x = 0})), x_{2} \in [0 - L_{2}) \end{array}

(14)

As mentioned, crack creates a discontinuity in the length of the beam; therefore, displacement, moment, and shear force of the beam on the left- and right-hand sides of the crack, respectively, may be written as

\begin{array}{l} {G_{1} |}_{x_{1} = L_{1}} = {G_{2} |}_{x_{2} = L_{2}} \\ {(\frac{\partial G_{1}^{2}}{\partial x_{1}^{2}} + η_{1} G_{1}) |}_{x_{1} = L_{1}} = {(\frac{\partial G_{2}^{2}}{\partial x_{2}^{2}} + η_{1} G_{2}) |}_{x_{2} = L_{2}} \\ {(\frac{\partial G_{1}^{3}}{\partial x_{1}^{3}} + η_{2} \frac{\partial G_{1}}{\partial x_{1}}) |}_{x_{1} = L_{1}} = {(- \frac{\partial G_{2}^{3}}{\partial x_{2}^{3}} - η_{2} \frac{\partial G_{2}}{\partial x_{2}}) |}_{x_{2} = L_{2}} \end{array}

(15a)

To consider crack effects, the cracked section may be modeled as local flexibility. According to jump in the bending slope caused by local flexibility, the discontinuity in the slope of the beam can be defined as follows (10)

{(\frac{\partial G_{1}^{3}}{\partial x_{1}^{3}} + η_{3} \frac{\partial G_{1}}{\partial x_{1}}) |}_{x_{1} = L_{1}} = {- \frac{E I}{k_{c}} \frac{\partial}{\partial x} (\frac{\partial G_{2}^{3}}{\partial x_{2}^{3}} + η_{3} \frac{\partial G_{2}}{\partial x_{2}}) |}_{x_{2} = L_{2}} - {(\frac{\partial G_{2}^{3}}{\partial x_{2}^{3}} + η_{3} \frac{\partial G_{2}}{\partial x_{2}} |) |}_{x_{2} = L_{2}}

(15b)

where

\begin{array}{l} η_{1} = ϕ (ϕ - i \frac{α^{2} β}{E I} c_{t}) \\ η_{2} = ϕ (ϕ (1 + r^{2}) - i \frac{β}{E I} (α^{2} c_{t} + c_{r})) \\ η_{3} = \frac{1}{α^{2}} + η_{1} \end{array}

(16)

By substituting equation (14) into (15), the following equation is obtained. By calculating the determinate of the following matrix, the natural frequencies of Timoshenko for one-cracked beam are determined

| \begin{matrix} Γ_{2, L_{1}} & Γ_{3, L_{1}} & - Γ_{2, L_{2}} & - Γ_{3, L_{2}} \\ η_{1} Γ_{2, L_{1}} + Γ_{2, L_{1}}^{″} & η_{1} Γ_{3, L_{1}} + Γ_{3, L_{1}}^{″} & - η_{1} Γ_{2, L_{2}} - Γ_{2, L_{2}}^{″} & - η_{1} Γ_{3, L_{2}} - Γ_{3, L_{2}}^{″} \\ η_{2} Γ_{2, L_{1}}^{'} + Γ_{2, L_{1}}^{″'} & η_{2} Γ_{3, L_{1}}^{'} + Γ_{3, L_{1}}^{″'} & η_{2} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} & η_{2} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} \\ η_{3} Γ_{2, L_{1}}^{'} + Γ_{2, L_{1}}^{″'} & η_{3} Γ_{3, L_{1}}^{'} + Γ_{3, L_{1}}^{″'} & η_{3} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} + \frac{E I}{k_{c}} (η_{3} Γ_{2, L_{2}}^{″} + Γ_{2, L_{2}}^{″''}) & η_{3} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} + \frac{E I}{k_{c}} (η_{3} Γ_{3, L_{2}}^{″} + Γ_{3, L_{2}}^{″''}) \end{matrix} | = 0

(17)

where

Γ_{i, L_{j}} = Γ_{i} (L_{j})

(18)

In this section, Figure 2 is presented for modeling the multi-cracked Timoshenko beam. The presence of n cracks divides the beam into n + 1 parts. For each part, the relation (11) is used separately, and by applying the special continuity relationships for each part, the continuity along the beam is created. For the first and last segments of the simply supported multi-cracked beam, the Gxreen’s function is defined as

\begin{array}{l} G (x_{i}, x_{i 0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (H (x_{i} - x_{10}) Γ_{1} (x_{i} - x_{i 0}) + Γ_{2} (x_{i}) ({w^{'} (x) |}_{x = 0}) + Γ_{3} (x_{i}) ({w^{'''} (x) |}_{x = 0})), x_{i} \in [0 - L_{i}) \\ G (x_{n}, x_{n 0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (H (x_{n} - x_{n 0}) Γ_{1} (x_{n} - x_{n 0}) + Γ_{2} (x_{n}) ({w^{'} (x) |}_{x = 0}) + Γ_{3} (x_{n}) ({w^{'''} (x) |}_{x = 0})), x_{n} \in [0 - L_{n}) \end{array}

(19)

And for the middle segments (i.e. $1 < i < n - 1$ ), the following relation is expressed

\begin{array}{l} G (x_{i + 1}, x_{(i + 1) 0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (\begin{array}{l} H (x_{i + 1}, x_{(i + 1) 0}) Γ_{1} (x_{i + 1}, x_{(i + 1) 0}) + {Γ^{'}}_{2} (x_{i + 1}) ({w (x) |}_{x = 0}) + Γ_{2} (x_{i + 1}) ({w^{'} (x) |}_{x = 0}) \\ + Γ_{3}^{'} (x_{i + 1}) ({w^{″} (x) |}_{x = 0}) + Γ_{3} (x_{i + 1}) ({w^{‴} (x) |}_{x = 0}) \end{array}) \\ x_{i + 1} \in [L_{i} - L_{i + 1}) \end{array}

(20)

For the first and ith segments, the following continuity equations in the cracked section are considered

\begin{array}{l} {G_{i} |}_{x_{i} = L_{i}} = {G_{i + 1} |}_{x_{i + 1} = 0} \\ {(\frac{\partial G_{i}^{2}}{\partial x_{i}^{2}} + η_{1} G_{i}) |}_{x_{i} = L_{i}} = {(\frac{\partial G_{i + 1}^{2}}{\partial x_{i + 1}^{2}} + η_{1} G_{i + 1}) |}_{x_{i} = 0} \\ {(\frac{\partial G_{i}^{3}}{\partial x_{i}^{3}} + η_{2} \frac{\partial G_{i}}{\partial x_{i}}) |}_{x_{i} = L_{i}} = {(\frac{\partial G_{i + 1}^{3}}{\partial x_{i + 1}^{3}} + η_{2} \frac{\partial G_{i + 1}}{\partial x_{i + 1}}) |}_{x_{i + 1} = 0} \\ {(\frac{\partial G_{i}^{3}}{\partial x_{i}^{3}} + η_{3} \frac{\partial G_{i}}{\partial x_{i}}) |}_{x_{i} = L_{i}} = {- \frac{E I}{k_{c}} (\frac{\partial G_{i + 1}^{4}}{\partial x_{i + 1}^{4}} + η_{3} \frac{\partial G_{i + 1}^{2}}{\partial x_{i + 1}^{2}}) |}_{x_{i + 1} = 0} + {(\frac{\partial G_{i + 1}^{3}}{\partial x_{i + 1}^{3}} + η_{3} \frac{\partial G_{i + 1}}{\partial x_{i + 1}}) |}_{x_{i + 1} = 0} \end{array}

(21)

Also in the cracked section, which is located between n and n − 1 segments, equation (22) is presented to make continuity

\begin{array}{l} {G_{n - 1} |}_{x_{n - 1} = L_{n - 1}} = {G_{n} |}_{x_{n} = L_{n}} \\ {(\frac{\partial G_{n - 1}^{2}}{\partial x_{n - 1}^{2}} + η_{1} G_{n - 1}) |}_{x_{n - 1} = L_{n - 1}} = {(\frac{\partial G_{n}^{2}}{\partial x_{n}^{2}} + η_{1} G_{n}) |}_{x_{n} = L_{n}} \\ {(\frac{\partial G_{n - 1}^{3}}{\partial x_{n - 1}^{3}} + η_{2} \frac{\partial G_{n - 1}}{\partial x_{n - 1}}) |}_{x_{n - 1} = L_{n - 1}} = {(- \frac{\partial G_{n}^{3}}{\partial x_{n}^{3}} - η_{2} \frac{\partial G_{n}}{\partial x_{n}}) |}_{x_{n} = L_{n}} \\ {(\frac{\partial G_{n - 1}^{3}}{\partial x_{n - 1}^{3}} + η_{3} \frac{\partial G_{n - 1}}{\partial x_{n - 1}}) |}_{x_{n - 1} = L_{n - 1}} = {- \frac{E I}{k_{c}} (\frac{\partial G_{n}^{4}}{\partial x_{n}^{4}} + η_{3} \frac{\partial G_{n}^{2}}{\partial x_{n}^{2}}) |}_{x_{n} = L_{n}} - {(\frac{\partial G_{n}^{3}}{\partial x_{n}^{3}} + η_{3} \frac{\partial G_{n}}{\partial x_{n}}) |}_{x_{n} = L_{n}} \end{array}

(22)

By substituting equations (19) and (20) in (21) and (22), the following matrix is obtained

| \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} | = 0

(23)

More details of equation (23) are given in the Appendix 1. By solving equation (23), the natural frequencies of the simply supported two-cracked beam are determined.

4. Green’s function for multi-span Timoshenko beam

Multi-span beams are most commonly used in bridges and other large structures. The multi-span cracked beam with elastic boundary conditions is exhibited in Figure 3.

For the first and last segments of the beam, the Green’s function with respect to elastic supports is defined as

\begin{array}{l} G (x_{i}, x_{i 0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (\begin{array}{l} H (x_{i} - x_{i 0}) Γ_{1} (x_{i} - x_{i 0}) + A_{i} Γ_{2}^{'} (x_{i}) + B_{i} Γ_{2} (x_{i}) \\ + Γ_{3}^{'} (x_{i}) ({w^{''} (x) |}_{x = 0}) + Γ_{3} (x_{i}) ({w^{'''} (x) |}_{x = 0}) \end{array}), x_{i} \in [0 - L_{i}) \\ G (x_{n}, x_{n 0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (\begin{array}{l} H (x_{n}, x_{n 0}) Γ_{1} (x_{n}, x_{n 0}) + A_{n} Γ_{2}^{'} (x_{n}) + B_{n} Γ_{2} (x_{n}) \\ + Γ_{3}^{'} (x_{n}) ({w^{''} (x) |}_{x = 0}) + Γ_{3} (x_{n}) ({w^{'''} (x) |}_{x = 0}) \end{array}), x_{n} \in [0 - L_{n}) \end{array}

(24)

The constants $A_{i}, B_{i}, A_{n}$ , and $B_{n}$ provide the elastic conditions which are defined as follows

\begin{array}{l} A_{i} = \frac{- E I}{k_{L T}} w_{i}^{″'} (0), B_{i} = \frac{E I}{k_{L R}} w_{i}^{″} (0) \\ A_{n} = \frac{- E I}{k_{R T}} w_{n}^{″'} (0), B_{n} = \frac{E I}{k_{R R}} w_{n}^{″} (0) \end{array}

(25)

The Green’s function for the middle segments is presented in the following form

\begin{array}{l} G (x_{i + 1}, x_{(i + 1) 0}) = \frac{1}{s_{1}^{2} - s_{2}^{2}} (\begin{array}{l} H (x_{i + 1}, x_{(i + 1) 0}) Γ_{1} (x_{i + 1}, x_{(i + 1) 0}) + Γ_{2}^{'} (x_{i + 1}) ({w (x) |}_{x = 0}) + Γ_{2} (x_{i + 1}) ({w^{'} (x) |}_{x = 0}) \\ + Γ_{3}^{'} (x_{i + 1}) ({w^{''} (x) |}_{x = 0}) + Γ_{3} (x_{i + 1}) ({w^{'''} (x) |}_{x = 0}) \end{array}) \\ x_{i + 1} \in [L_{i} - L_{i + 1}) \end{array}

(26)

The following continuity relations are used to make continuity in the middle-span location

\begin{array}{l} {G_{i} |}_{x_{i} = L_{i}} = {G_{i + 1} |}_{x_{i + 1} = 0} \\ {\frac{\partial G_{i}^{2}}{\partial x_{i}^{2}} |}_{x_{i} = L_{i}} + η_{1} {G_{i} |}_{x_{i} = L_{i}} = {\frac{\partial G_{i + 1}^{2}}{\partial x_{i + 1}^{2}} |}_{x_{i} = 0} + η_{1} {G_{i + 1} |}_{x_{i + 1} = 0} \\ {(\frac{\partial G_{i}^{3}}{\partial x_{i}^{3}} + η_{2} \frac{\partial G_{i}}{\partial x_{i}}) |}_{x_{i} = L_{i}} - {(\frac{\partial G_{i + 1}^{3}}{\partial x_{i + 1}^{3}} + η_{2} \frac{\partial G_{i + 1}}{\partial x_{i + 1}} |) |}_{x_{i + 1} = 0} = {\frac{k_{M}}{E I} G_{i + 1} |}_{x_{i + 1} = 0} \\ {(\frac{\partial G_{i}^{3}}{\partial x_{i}^{3}} + η_{3} \frac{\partial G_{i}}{\partial x_{i}}) |}_{x_{i} = L_{i}} = {(\frac{\partial G_{i + 1}^{3}}{\partial x_{i + 1}^{3}} + η_{3} \frac{\partial G_{i + 1}}{\partial x_{i + 1}}) |}_{x_{i + 1} = 0} \end{array}

(27)

According to Figure 3, the continuity relations for the cracked section are given as follow

\begin{array}{l} {G_{n - 1} |}_{x_{n - 1} = L_{n - 1}} = {G_{n} |}_{x_{n} = L_{n}} \\ {(\frac{\partial G_{n - 1}^{2}}{\partial x_{n - 1}^{2}} + η_{1} G_{n - 1}) |}_{x_{n - 1} = L_{n - 1}} = {(\frac{\partial G_{n}^{2}}{\partial x_{n}^{2}} + η_{1} G_{n}) |}_{x_{n} = L_{n}} \\ {(\frac{\partial G_{n - 1}^{3}}{\partial x_{n - 1}^{3}} + η_{2} \frac{\partial G_{n - 1}}{\partial x_{n - 1}}) |}_{x_{n - 1} = L_{n - 1}} = {(- \frac{\partial G_{n}^{3}}{\partial x_{n}^{3}} - η_{2} \frac{\partial G_{n}}{\partial x_{n}}) |}_{x_{n} = L_{n}} \\ {(\frac{\partial G_{n - 1}^{3}}{\partial x_{n - 1}^{3}} + η_{3} \frac{\partial G_{n - 1}}{\partial x_{n - 1}}) |}_{x_{n - 1} = L_{n - 1}} = {- \frac{E I}{k_{c}} (\frac{\partial G_{n}^{4}}{\partial x_{n}^{4}} + η_{3} \frac{\partial G_{n}^{2}}{\partial x_{n}^{2}}) |}_{x_{n} = L_{n}} - {(\frac{\partial G_{n}^{3}}{\partial x_{n}^{3}} + η_{3} \frac{\partial G_{n}}{\partial x_{n}}) |}_{x_{n} = L_{n}} \end{array}

(28)

By considering the two-span cracked beam and substituting equations (24) and (26) in (27) and (28), the following equation can be obtained

| \begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix} | = 0

(29)

More details of equation (29) are given in the Appendix 1. By solving equation (29), the natural frequencies of the two-span cracked beam with elastic boundary conditions are determined. To perform the modeling of the cracked section, a massless torsional spring with stiffness $k_{c}$ is used. The term $k_{c}$ in continuity conditions has the following expression

k_{c} = \frac{E I}{h \times D_{h^{'}}}

(30)

where

D_{h^{'}}

is the nondimension constant and calculated from fracture mechanics. For the rectangular cross-section, once

h^{'} \in [0 - 0.6]

, from there Fernandez-Saez et al. (1999) proposed the following relation

D_{h^{'}} = 2 {(\frac{h^{'}}{1 - h^{'}})}^{2} (5.93 - 19.69 h^{'} + 37.14 h^{' 2} - 35.84 h^{' 3} + 13.12 h^{' 4})

(31)

Bilello (2001) presented an equation, which has more precision in $h^{'} \in [0.6 - 1]$

D_{h^{'}} = \frac{h^{'} (2 - h^{'})}{0.9 {(2 - h^{'})}^{2}}

(32)

5. Numerical examples

5.1. Timoshenko beam with a crack in a different location

A cracked cantilever beam with $E = 210 GPa, ρ = 7860 kg / m^{3}$ , $ν = 0.20$ , and $κ = 0.83$ is illustrated in Figure 4. The natural frequencies of the cantilever cracked beam are provided in Table 1. In addition, natural frequencies of the cantilever cracked beam are also obtained from Krawczuk et al. (2003) study and comparisons are made between the two methods. Evidently, the comparisons indicate that there are insignificant errors, which show the level capability of the present method.

Table 1.

Natural frequencies of cantilever cracked beam.

Crack ratio	Mode	Crack location
		Present study			Krawczuk et al. (2003)			Error (%)
		0.25L	0.5L	0.75L	0.25L	0.5L	0.75L	0.25L	0.5L	0.75L
$h^{'} = 0$	1	26.232	26.232	26.232	26.138	26.138	26.138	0.3583	0.3583	0.3583
	2	164.357	164.357	164.357	164.24	164.24	164.24	0.0712	0.0712	0.0712
	3	459.961	459.961	459.961	459.62	459.62	459.62	0.0741	0.0741	0.0741
	4	900.571	900.571	900.571	899.75	899.75	899.75	0.0912	0.0912	0.0912
$h^{'} = 0.05$	1	26.226	26.231	26.232	26.138	26.138	26.138	0.3355	0.3545	0.3583
	2	164.355	164.31	164.341	164.24	164.24	164.24	0.0700	0.0426	0.0615
	3	459.875	459.961	459.827	459.62	459.62	459.55	0.0554	0.0741	0.0602
	4	900.378	900.321	900.336	899.69	899.63	899.63	0.0764	0.0768	0.0784
$h^{'} = 0.25$	1	26.091	26.194	26.229	26.075	26.138	26.138	0.0613	0.2138	0.3469
	2	164.320	163.320	163.999	164.18	163.36	163.93	0.0852	0.0245	0.0421
	3	458.042	459.959	456.969	458.04	456.62	457.16	0.0004	0.7259	0.0418
	4	896.337	895.063	895.426	896.30	895.23	895.54	0.0041	0.0187	0.0127
$h^{'} = 0.5$	1	25.489	26.027	26.215	25.196	25.887	26.138	1.1495	0.5379	0.2937
	2	164.163	158.985	162.412	163.99	157.21	161.67	0.1054	1.1165	0.4569
	3	450.088	459.950	444.365	446.8	459.62	439.26	0.7305	0.0717	1.1488
	4	879.896	873.158	875.579	873.43	864.50	867.90	0.7349	0.9916	0.8770

The effect of crack position and crack depth on the natural frequency of the Timoshenko beam is displayed in Figures 5–7.

5.2. Multi-span Timoshenko cracked beam

The two-span beam with $L = 2 m$ , $E = 210 GPa, ρ = 7860 kg / m^{3}$ , $ν = 0.20$ , and $κ = 0.83$ is illustrated with a crack in each span in Figure 8. The crack ratios $(h^{'})$ of A and B are 0.3 and 0.6, respectively. Moreover, the natural frequencies of the beam when there is only crack A, and when both exist, based on Euler–Bernoulli and Timoshenko theories, are shown in Table 2. The natural frequencies of the beam, based on the Euler–Bernoulli theory, are more than the natural frequencies of Timoshenko’s beam. The first three modes of multi-span cracked beam are presented in Figure 9. Because of the stiffness of the supports, the cracks have no significant effect on the first mode. However, the cracks influence is clear in the second mode.

Table 2.

Natural frequencies of two-span cracked beam $(r / L = 0.002887)$ .

Theory	Condition	Natural frequencies
Theory	Condition	I	II	III	IV	V
Timoshenko	No crack	459.778	666.555	1486.97	1832.29	3092.33
	Crack A $(h^{'} = 0.3)$	456.927	662.248	1485.25	1832.29	3068.02
	Crack B $(h^{'} = 0.6)$	437.829	657.494	1443.24	1767.25	3082.40
	Crack A and B	435.450	652.688	1441.760	1776.15	3057.44
Euler–Bernoulli	No crack	460.120	667.679	1491.09	1840.48	3111.03
	Crack A $(h^{'} = 0.3)$	457.261	663.349	1489.35	1840.48	3086.33
	Crack B $(h^{'} = 0.6)$	438.129	658.587	1447.04	1774.58	3101.04
	Crack A and B	435.744	653.760	1445.54	1774.48	3075.66

The influence of moving load on the cracked Timoshenko beam is displayed in Figure 10.

5.3. Force vibration of clamped-elastic cracked beam

The Timoshenko cracked beam with different crack locations is illustrated in Figure 11. At the same time, the considered constants in this example are based on Lueschen et al. (1996) data are presented in Table 3. The effect of damping ratio on cracked Timoshenko beam is presented in Table 4. By increasing the stiffness of the spring, the natural frequencies of a cracked beam in every position enhanced; in addition, by raising the damping ratio, the natural frequency decreased.

Table 3.

Constants used in the clamped-elastic cracked beam.

Parameter	Value
$ρ (lb - s^{2} / {in}^{4})$	728 × 10^-6
$E (lb / {in}^{2})$	3 × 10⁷
$ν$	0.3
$A ({in}^{2})$	4
$I ({in}^{4})$	$1.33$
$κ$	0.83
$L (in)$	20

Table 4.

Natural frequencies of damped and undamped cracked clamped elastic beam.

		No damped				Damped (5%)				Damped (10%)
$k_{T}$	Mode	No cracks	Crack location			No crack	Crack location			No crack	Crack location
$k_{T}$	Mode	No cracks	0.15L	0.25L	0.35L	No crack	0.15L	0.25L	0.35L	No crack	0.15L	0.25L	0.35L
$0$	1	52.422	42.855	45.058	47.063	52.356	42.801	45.0019	47.004	52.1585	42.6395	44.8322	46.8267
	2	320.90	271.21	278.86	270.02	320.506	270.878	278.519	269.685	319.323	269.882	277.494	268.693
	3	856.23	801.97	736.35	714.87	855.218	801.026	735.492	714.037	852.173	798.195	732.907	711.541
	4	1568.85	1503.64	1454.98	1538.85	1567.07	1501.94	1453.34	1537.1	1561.71	1496.82	1448.44	1531.86
$0.00001 E I$	1	53.094	43.667	45.863	47.857	53.0278	43.6121	45.805	47.797	52.8278	43.447	45.633	47.617
	2	321.01	271.35	278.99	270.13	320.614	271.017	278.646	269.804	319.43	270.019	277.621	268.811
	3	856.27	801.99	736.38	714.90	855.257	801.054	735.518	714.07	852.211	798.223	732.933	711.574
	4	1568.87	1503.66	1455.00	1538.87	1567.09	1501.96	1453.37	1537.13	1561.73	1496.85	1448.47	1531.89
$0.0001 E I$	1	58.757	50.317	52.489	54.422	58.683	50.254	52.422	54.353	58.462	50.064	52.225	54.148
	2	321.99	272.60	280.14	271.21	321.592	272.266	279.798	270.875	320.404	271.264	278.768	269.878
	3	856.62	802.25	736.62	715.20	855.609	801.305	735.756	714.369	852.562	798.473	733.17	711.872
	4	1569.06	1503.88	1455.26	1539.13	1567.28	1502.18	1453.63	1537.38	1561.91	1497.07	1448.72	1532.14
$0.001 E I$	1	96.292	89.859	92.687	94.757	96.172	89.747	92.571	94.638	95.809	89.409	92.222	94.282
	2	332.099	285.73	292.26	282.58	331.692	285.384	291.902	282.236	330.466	284.333	290.826	281.196
	3	860.19	804.79	739.04	718.25	859.168	803.849	738.176	717.417	856.107	801.007	735.582	714.909
	4	1570.92	1506.13	1457.80	1541.68	1569.13	1504.43	1456.16	1539.94	1563.76	1499.31	1451.25	1534.68
$0.01 E I$	1	188.07	165.37	172.94	174.59	187.84	165.168	172.723	174.375	187.14	164.554	172.081	173.726
	2	435.77	416.28	415.53	401.01	435.238	415.767	415.014	400.513	433.627	414.227	413.477	399.033
	3	899.47	833.77	767.77	754.44	898.398	832.787	766.872	753.565	895.182	829.839	764.173	750.921
	4	1590.46	1529.38	1484.17	1567.99	1588.64	1527.64	1482.5	1566.2	1583.19	1522.42	1477.47	1560.83
$0.1 E I$	1	221.23	185.70	195.07	195.23	220.961	185.472	194.831	194.991	220.143	184.787	194.11	194.27
	2	654.92	652.74	619.60	586.69	654.136	651.959	618.859	585.993	651.788	649.619	616.644	583.906
	3	1201.61	1093.84	1035.87	1065.23	1200.2	1092.58	1034.68	1064.06	1195.98	1088.79	1031.12	1060.39
	4	1818.26	1745.73	1734.75	1804.23	1816.16	1743.76	1732.79	1802.17	1809.83	1737.83	1726.91	1795.96
$E I$	1	224.78	187.80	197.134	197.29	224.506	187.571	197.092	197.044	223.675	186.878	196.365	196.316
	2	692.00	684.76	642.67	608.58	691.18	683.95	641.909	607.865	688.717	681.513	639.626	605.715
	3	1344.63	1232.44	1169.75	1222.03	1343.1	1231.06	1168.44	1220.68	1338.5	1226.91	1164.53	1216.6
	4	2118.95	1920.34	1953.69	1970.52	2116.62	1918.3	1951.63	1968.39	2109.62	1912.15	1945.41	1961.98
$10 E I$	1	225.14	188.01	197.56	197.49	224.86	187.78	197.317	197.248	224.028	187.087	196.588	196.519
	2	695.47	687.58	644.69	610.56	694.644	686.765	643.924	609.839	692.169	684.319	641.635	607.683
	3	1357.17	1245.83	1182.77	1237.49	1355.63	1244.44	1181.45	1236.12	1351.004	1240.25	1177.51	1232.001
	4	2148.71	1935.18	1982.74	1982.74	2146.36	1933.13	1970.802	1980.61	2139.31	1926.97	1964.56	1974.19

Figure 12 displays a two-dimensional contour graph of forced vibration of a fixed-elastic cracked beam when the moving load with $v_{x} = 1 L$ and $a_{x} = 1 L$ pass over the cracked beam from left-hand side to the end of the beam. The deflection of the cracked beam is changed by increasing the stiffness of the spring. The maximum deflection occurs at the end of the beam when there is no stiffness. By enhancing the stiffness of the spring, the maximum deflection transfers to inside of the beam.

6. Conclusion

In this study, the dynamic response of the multi-span Timoshenko cracked beam studied. The Green’s function method used to calculate the dynamic response of the cracked beam. The effects of the existence of a crack or multiple cracks in a multi-span beam along with the important characteristics of the beam itself relevant to such as natural frequencies were also investigated. The natural frequencies of the beam were calculated and compared to Euler–Bernoulli and Timoshenko theory. It is noticed that by increasing the depth of the crack, its effects on the natural frequencies are more pronounced. Thus, displacement of the Timoshenko beam in various levels of the stiffness of elastic supports was derived. Also, the mode shapes of a multi-span cracked beam were achieved.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Amin Ghannadiasl

Saeid Khodapanah Ajirlou

Appendix 1

The submatrix of equation (23) is obtained from following equations: (33)

\begin{array}{l} A_{11} = [\begin{matrix} Γ_{2, L_{1}} & Γ_{3, L_{1}} & - Γ_{2,0} & - Γ_{2,0} \\ η_{1} Γ_{2, L_{1}} + Γ_{2, L_{1}}^{″} & η_{1} Γ_{3, L_{1}} + Γ_{3, L_{1}}^{″} & - η_{1} Γ_{2,0} - Γ_{2,0}^{″} & - η_{1} Γ_{2,0} - Γ_{2,0}^{″} \\ η_{2} Γ_{2, L_{1}}^{'} + Γ_{2, L_{1}}^{″'} & η_{2} Γ_{3, L_{1}}^{'} + Γ_{3, L_{1}}^{″'} & - η_{2} Γ_{2,0}^{'} - Γ_{2,0}^{″'} & - η_{2} Γ_{2,0}^{'} - Γ_{2,0}^{″'} \\ η_{3} Γ_{2, L_{1}}^{'} + Γ_{2, L_{1}}^{″'} & η_{3} Γ_{3, L_{1}}^{'} + Γ_{3, L_{1}}^{″'} & - η_{3} Γ_{2,0}^{'} + \frac{E I}{k_{c 1}} η_{3} Γ_{2,0}^{″} - Γ_{2,0}^{″'} + \frac{E I}{k_{c 1}} Γ_{2,0}^{″''} & - η_{3} Γ_{2,0}^{'} + \frac{E I}{k_{c 1}} η_{3} Γ_{2,0}^{″} - Γ_{2,0}^{″'} + \frac{E I}{k_{c 1}} Γ_{2,0}^{″''} \end{matrix}] \\ A_{12} = [\begin{matrix} - Γ_{3,0} & - Γ_{3,0} (0) & 0 & 0 \\ - η_{1} Γ_{3,0} - Γ_{3,0}^{″ ɛ} & - η_{1} Γ_{3,0} (0) - Γ_{3,0}^{″} (0) & 0 & 0 \\ - η_{2} Γ_{3,0}^{'} - Γ_{3,0}^{″'} & - η_{1} Γ_{3,0} (0) - Γ_{3,0}^{″} (0) & 0 & 0 \\ - η_{3} Γ_{3,0}^{'} + \frac{E I}{k_{c 1}} η_{3} Γ_{3,0}^{″} - Γ_{3,0}^{″'} + \frac{E I}{k_{c 1}} Γ_{3,0}^{″''} & - η_{3} Γ_{3,0}^{'} + \frac{E I}{k_{c 1}} η_{3} Γ_{3,0}^{″} - Γ_{3,0}^{″'} + \frac{E I}{k_{c 1}} Γ_{3,0}^{″''} & 0 & 0 \end{matrix}] \\ A_{21} = [\begin{matrix} 0 & 0 & Γ_{2, L_{2}} & Γ_{2, L_{2}} \\ 0 & 0 & η_{1} Γ_{2, L_{2}} + Γ_{2, L_{2}}^{″} & η_{1} Γ_{2, L_{2}} + Γ_{2, L_{2}}^{″} \\ 0 & 0 & η_{2} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} & η_{2} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} \\ 0 & 0 & η_{3} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} & η_{3} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} \end{matrix}] \\ A_{22} = [\begin{matrix} Γ_{3, L_{2}} & Γ_{3, L_{2}} & - Γ_{2, L_{3}} & - Γ_{3, L_{3}} \\ η_{1} Γ_{3, L_{2}} + Γ_{3, L_{2}}^{″} & η_{1} Γ_{3, L_{2}} + Γ_{3, L_{2}}^{″} & - η_{1} Γ_{2, L_{3}} - Γ_{2, L_{3}}^{″} & - η_{1} Γ_{3, L_{3}} - Γ_{3, L_{3}}^{″} \\ η_{2} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & η_{2} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & η_{2} Γ_{2, L_{3}}^{'} + Γ_{2, L_{3}}^{″'} & η_{2} Γ_{3, L_{3}}^{'} + Γ_{3, L_{3}}^{″'} \\ η_{3} Γ_{3, L_{2}}^{''} + Γ_{3, L_{2}}^{″'} & η_{3} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & η_{3} Γ_{2, L_{3}}^{'} + \frac{E I}{k_{c 2}} η_{3} Γ_{2, L_{3}}^{″} + Γ_{2, L_{3}}^{″'} + \frac{E I}{k_{c 2}} Γ_{2, L_{3}}^{″''} & η_{3} Γ_{3, L_{3}}^{'} + \frac{E I}{k_{c 2}} η_{3} Γ_{3, L_{3}}^{″} + Γ_{3, L_{3}}^{″'} + \frac{E I}{k_{c 2}} Γ_{3, L_{3}}^{″''} \end{matrix}] \end{array}

where

k_{c 1}

and

k_{c 2}

provide the specifications of cracks.

The submatrix of equation (29) is determined from following equations: (34)

\begin{array}{l} B_{11} = [\begin{matrix} \frac{E I}{k_{L R}} Γ_{2, L_{1}} + Γ_{3, L_{1}} & \frac{- E I}{k_{L T}} Γ_{2, L_{1}} + Γ_{3, L_{1}} & - Γ_{2,0} & - Γ_{2,0} \\ \frac{E I}{k_{L R}} η_{1} Γ_{2, L_{1}} + η_{1} Γ_{3, L_{1}} + \frac{E I}{k_{L R}} Γ_{2, L_{1}}^{″} + Γ_{3, L_{1}}^{″} & \frac{- E I}{k_{L T}} η_{1} Γ_{2, L_{1}} + η_{1} Γ_{3, L_{1}} - \frac{E I}{k_{L T}} Γ_{2, L_{1}}^{″} + Γ_{3, L_{1}}^{″} & - η_{1} Γ_{2,0} - Γ_{2,0}^{″} & - η_{1} Γ_{2,0} - Γ_{2,0}^{″} \\ \frac{E I}{k_{L R}} η_{2} Γ_{2, L_{1}}^{'} + η_{2} Γ_{3, L_{1}}^{'} + \frac{E I}{k_{L R}} Γ_{2, L_{1}}^{″'} + Γ_{3, L_{1}}^{″'} & \frac{- E I}{k_{L T}} η_{2} Γ_{2, L_{1}}^{'} + η_{2} Γ_{3, L_{1}}^{'} - \frac{E I}{k_{L T}} Γ_{2, L_{1}}^{″'} + Γ_{3, L_{1}}^{″'} & \frac{- k_{m}}{E I} Γ_{2,0} - η_{2} Γ_{2,0}^{'} - Γ_{2,0}^{″'} & \frac{- k_{m}}{E I} Γ_{2,0} - η_{2} Γ_{2,0}^{'} - Γ_{2,0}^{″'} \\ \frac{E I}{k_{L R}} η_{3} Γ_{2, L_{1}}^{'} + η_{3} Γ_{3, L_{1}}^{'} + \frac{E I}{k_{L R}} Γ_{2, L_{1}}^{″'} + Γ_{3, L_{1}}^{″'} & \frac{- E I}{k_{L T}} η_{3} Γ_{2, L_{1}}^{'} + η_{3} Γ_{3, L_{1}}^{'} - \frac{E I}{k_{L T}} Γ_{2, L_{1}}^{″'} + Γ_{3, L_{1}}^{″'} & - η_{3} Γ_{2,0}^{'} - Γ_{2,0}^{'''} & - η_{3} Γ_{2,0}^{'} - Γ_{2,0}^{'''} \end{matrix}] \\ B_{12} = [\begin{matrix} - Γ_{3,0} & - Γ_{3,0} & 0 & 0 \\ - η_{1} Γ_{3,0} - Γ_{3,0}^{″} & - η_{1} Γ_{3,0} - Γ_{3,0}^{″} & 0 & 0 \\ \frac{- k_{m}}{E I} Γ_{3,0} - η_{2} Γ_{3,0}^{'} - Γ_{3,0}^{″'} & - η_{1} Γ_{3,0} - Γ_{3,0}^{″} & 0 & 0 \\ - η_{3} Γ_{3,0}^{'} - Γ_{3,0}^{″'} & - η_{3} Γ_{3,0}^{'} - Γ_{3,0}^{″'} & 0 & 0 \end{matrix}] \\ B_{21} = [\begin{matrix} 0 & 0 & Γ_{2, L_{2}} & Γ_{2, L_{2}} \\ 0 & 0 & η_{1} Γ_{2, L_{2}} + Γ_{2, L_{2}}^{″} & η_{1} Γ_{2, L_{2}} + Γ_{2, L_{2}}^{″} \\ 0 & 0 & η_{2} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} & η_{2} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} \\ 0 & 0 & η_{3} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} & η_{3} Γ_{2, L_{2}}^{'} + Γ_{2, L_{2}}^{″'} \end{matrix}] \\ B_{22} = [\begin{matrix} Γ_{3, L_{2}} & Γ_{3, L_{2}} & \frac{- E I}{k_{R R}} Γ_{2, L_{3}} (L_{3}) - Γ_{3, L_{3}} (L_{3}) & \frac{E I}{k_{R T}} Γ_{2, L_{3}} - Γ_{3, L_{3}} \\ η_{1} Γ_{3, L_{2}} + Γ_{3, L_{2}}^{″} & η_{1} Γ_{3, L_{2}} + Γ_{3, L_{2}}^{″} & \frac{- E I}{k_{R R}} η_{1} Γ_{2, L_{3}} - η_{1} Γ_{3, L_{3}} - \frac{E I}{k_{R R}} Γ_{2, L_{3}}^{″} - Γ_{3, L_{3}}^{″} & \frac{E I}{k_{R T}} η_{1} Γ_{2, L_{3}} - η_{1} Γ_{3, L_{3}} + \frac{E I}{k_{R T}} Γ_{2, L_{3}}^{″} - Γ_{3, L_{3}}^{″} \\ η_{2} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & η_{2} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & \frac{- E I}{k_{R R}} η_{2} Γ_{2, L_{3}}^{'} + η_{2} Γ_{3, L_{3}}^{'} + \frac{E I}{k_{R R}} Γ_{2, L_{3}}^{″'} + Γ_{3, L_{3}}^{″'} & \frac{E I}{k_{R T}} η_{2} Γ_{2, L_{3}}^{'} + η_{2} Γ_{3, L_{3}}^{'} - \frac{E I}{k_{R T}} Γ_{2, L_{3}}^{″'} + Γ_{3, L_{3}}^{″'} \\ η_{3} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & η_{3} Γ_{3, L_{2}}^{'} + Γ_{3, L_{2}}^{″'} & ψ_{1} & ψ_{2} \end{matrix}] \end{array}

where (35)

\begin{array}{l} ψ_{1} = \frac{E I}{k_{R R}} η_{3} Γ_{2, L_{3}}^{'} + η_{3} Γ_{3, L_{3}}^{'} + \frac{{(E I)}^{2}}{k_{c} k_{R R}} η_{3} Γ_{2, L_{3}}^{″} + \frac{{(E I)}^{2}}{k_{c} k_{R R}} η_{3} Γ_{2, L_{3}}^{″} + \frac{E I}{k_{c}} η_{3} Γ_{3, L_{3}}^{″} + Γ_{3, L_{3}}^{″'} + \frac{{(E I)}^{2}}{k_{c} k_{R R}} Γ_{2, L_{3}}^{″''} + \frac{E I}{k_{c}} Γ_{3, L_{3}}^{″''} \\ ψ_{2} = \frac{- E I}{k_{R T}} η_{3} Γ_{2, L_{3}}^{'} + η_{3} Γ_{3, L_{3}}^{'} - \frac{{(E I)}^{2}}{k_{c} k_{R T}} η_{3} Γ_{2, L_{3}}^{″} - \frac{{(E I)}^{2}}{k_{c} k_{R T}} η_{3} Γ_{2, L_{3}}^{″} + \frac{E I}{k_{c}} η_{3} Γ_{3, L_{3}}^{″} + Γ_{3, L_{3}}^{″'} - \frac{{(E I)}^{2}}{k_{c} k_{R T}} Γ_{2, L_{3}}^{″''} + \frac{E I}{k_{c}} Γ_{3, L_{3}}^{″''} \end{array}

References

Abu-Hilal

(2003) Forced vibration of Euler–Bernoulli beams by means of dynamic Green functions. Journal of Sound and Vibration 267(2): 191–207.

Al-Said

Naji

Al-Shukry

(2006) Flexural vibration of rotating cracked Timoshenko beam. Journal of Vibration and Control 12(11): 1271–1287.

Aydin

(2007) Vibratory characteristics of axially-loaded Timoshenko beams with arbitrary number of cracks. Journal of Vibration and Acoustics 129(3): 341–354.

Bilello

(2001) Theoretical and experimental investigation on damaged beams under moving systems. PhD Thesis, University degli Studi di Palermo, Palermo, Italy.

Binici

(2005) Vibration of beams with multiple open cracks subjected to axial force. Journal of Sound and Vibration 287(1–2): 277–295.

Chouiyakh

Azrar

Alnefaie

, et al. (2017) Vibration and multi-crack identification of Timoshenko beams under moving mass using the differential quadrature method. International Journal of Mechanical sciences 120: 1–11.

Fernández-sáez

Rubio

Navarro

(1999) Approximate calculation of the fundamental frequency for bending vibrations of cracked beams. Journal of Sound and Vibration 225(2): 345–352.

Ghannadiasl

Khodapanah Ajirlou

(2018a) Dynamic response of multi-cracked beams resting on elastic foundation. International Journal of Engineering 31(11): 1830–1837.

Ghannadiasl

Khodapanah Ajirlou

(2018b) Analytical solution of dynamic analysis of cracked Euler–Bernoulli beam with elastic boundary condition by GFM. Romanian Journal of Acoustics and Vibration 15(2): 100–107.

10.

Ghannadiasl

Khodapanah Ajirlou

(2019) Forced vibration of multi-span cracked Euler-Bernoulli beams using dynamic Green function formulation. Applied Acoustics 148: 484–494.

11.

Ghannadiasl

Mofid

(2014) Dynamic Green function for response of Timoshenko beam with arbitrary boundary conditions. Mechanics Based Design of Structures and Machines 42(1): 97–110.

12.

Kisa

Gurel

(2007) Free vibration analysis of uniform and stepped cracked beams with circular cross sections. International Journal of Engineering Science 45(2–8): 364–380.

13.

Krawczuk

Palacz

Ostachowicz

(2003) The dynamic analysis of a cracked Timoshenko beam by the spectral element method. Journal of Sound and Vibration 264(5): 1139–1153.

14.

Zhao

(2014) Green’s functions of the forced vibration of Timoshenko beams with damping effect. Journal of Sound and Vibration 333(6): 1781–1795.

15.

Loya

Rubio

Fernández-Sáez

(2006) Natural frequencies for bending vibrations of Timoshenko cracked beams. Journal of Sound and Vibration 290(3–5): 640–653.

16.

Lueschen

GGG

Bergman

McFarland

(1996) Green’s functions for uniform Timoshenko beams. Journal of Sound and Vibration 194(1): 93–102.

17.

Mei

Karpenko

Moody

, et al. (2006) Analytical approach to free and forced vibrations of axially loaded cracked Timoshenko beams. Journal of Sound and Vibration 291(3–5): 1041–1060.

18.

Panigrahi

Pohit

(2018) Effect of cracks on nonlinear flexural vibration of rotating Timoshenko functionally graded material beam having large amplitude motion. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232(6): 930–940.

19.

Sarvestan

Mirdamadi

Ghayour

(2017) Vibration analysis of cracked Timoshenko beam under moving load with constant velocity and acceleration by spectral finite element method. International Journal of Mechanical Sciences 122: 318–330.

20.

Sekhar

(2008) Multiple cracks effects and identification. Mechanical Systems and Signal Processing 22(4): 845–878.

21.

Shafiei

Khaji

(2011) Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load. Acta Mechanica 221(1–2): 79.

22.

Stojanović

Ribeiro

Stoykov

(2013) Non-linear vibration of Timoshenko damaged beams by a new p-version finite element method. Computers and Structures 120: 107–119.

23.

Torabi

Afshari

Haji Aboutalebi

(2014) A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions. Computers & Mathematics with Applications 67(3): 527–541.

24.

Castel

(2016) Modeling the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration loads. Journal of Sound and Vibration 368: 135–147.

25.

Zhao

Gao

, et al. (2016) Green’s functions for the forced vibrations of cracked Euler–Bernoulli beams. Mechanical Systems and Signal Processing 68-69: 155–175.

26.

Zheng

Kessissoglou

(2004) Free vibration analysis of a cracked beam by finite element method. Journal of Sound and vibration 273(3): 457–475.

Dynamic analysis of multiple cracked Timoshenko beam under moving load–analytical method

Abstract

Keywords

1. Introduction

2. Mathematical modeling of Timoshenko beam

3. Green’s function for Timoshenko cracked beam

4. Green’s function for multi-span Timoshenko beam

5. Numerical examples

5.1. Timoshenko beam with a crack in a different location

5.2. Multi-span Timoshenko cracked beam

5.3. Force vibration of clamped-elastic cracked beam

6. Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Appendix 1

References