Dynamic response of cracked Timoshenko beams on elastic foundations under moving harmonic loads

Abstract

A thorough understanding of the dynamic behavior of one-dimensional structural members such as beams plays a crucial role in specialized disciplines including ocean, bridge and railway engineering. The vibratory response of an in-service beam-like component may deviate from that expected from the intact structure when defects are present. In this work, we present a semi-analytical approach to predict the forced response of a multi-cracked Timoshenko beam traversed by a moving harmonic load with constant speed. The beam is fully or partially supported by the viscoelastic foundation, where the normal stiffness and shear modulus of the subgrade are considered. The effects of transverse open cracks are modeled by massless rotational springs with a linear moment-rotation constitutive law to account for the local flexibility induced by the damage. Based on the transfer matrix method, the defective structure is treated as an assembly of sub-beams to derive the eigenvalue solution of the system. The time response is then obtained by utilizing identical generalized coordinates for lateral and rotational displacement components when applying the modal expansion technique. The use of general elastic end constraints allows us to recover all possible boundary conditions. Numerical examples are also provided to demonstrate the robustness and accuracy of the proposed method, and also to investigate the influence of important parameters on the dynamic behavior of the damaged structure.

Keywords

Moving harmonic load cracked beam Timoshenko theory elastic foundation transfer matrix method

1. Introduction

The dynamic characteristics of in-service mechanical members can be influenced significantly by defects like cracks. Pre-existing flaws within the material may exist in the structural elements as a source of stress concentration or cracks can occur due to the fatigue phenomenon in the component. Nonetheless, cracks generated by either of the mentioned mechanisms deteriorate the structural integrity and change the dynamic properties of the structure such as natural frequencies and vibration mode shapes. As a result, for a reliable assessment of the component safety, it is crucial to utilize predictive simulations that provide the tools to deal with the dynamic analysis of damaged members.

In practice, the phenomenon of moving-load-induced vibration occurs in a number of physical problems, especially in ocean, bridge and railway engineering. The response of the rail and subgrade media to the traveling train at different speeds is a good example of a practical application which can be modeled by the vibrational response of an elastically supported beam due to a moving load. Another example is modeling the complex mechanical behavior of a pipeline–soil interaction problem in offshore industry, which can also be treated, under certain assumptions, as a beam-like structure resting on an elastic foundation. Therefore, to design systems that incorporate one-dimensional flexible members excited by traveling loads, it is clearly desirable to understand and predict their vibration behavioral properties in details.

In the context of the classic problem of intact beams subjected to moving loads, the earliest work was pioneered by Timoshenko (1922) to study the vibratory response of continuous single-span beams due to moving loads. Various analytical solutions based on the integral transformation were also well documented in an excellent monograph by Fryba (1972). Since then, owing to the basic theoretical interest and practical importance, a variety of methods have been used in the literature to solve this problem. These techniques include the finite element method (Andersen et al., 2001; Nguyen and Duhamel, 2008), using the Fourier transform method to solve the differential equations of motion (Mallik et al., 2006), employing a modal expansion method and the direct integration method (Dugush and Eisenberger, 2002), utilizing the regular perturbation method in conjunction with a Fourier integral transformation (Kargarnovin et al., 2005), developing a semi-analytic methodology involving discretization of the beam by conventional two nodes elements (Martínez-Castro et al., 2006) and using the Newmark time integration scheme and lattice spring model to obtain the nonlinear dynamic response of beams on tensionless foundations (Attar et al., 2014b).

The literature also contains numerous papers dedicated to the investigation of the dynamic behavior of cracked beam-like structures using different one-dimensional beam theories. Open nonpropagating cracks in a beam have been described as the source of local flexibility by many researchers to evaluate the effects of damage on the eigenvalues and eigenvectors of structural elements. Investigations of the vibratory response of damaged members have highlighted the role of crack sizes, crack locations, boundary conditions and material properties in the changes of natural frequencies. Characterizing the edge crack effects emerged from the pioneering works by Papadopoulos and Dimarogonas (1987) and Ostachowicz and Krawczuk (1991). Among many works on this topic, Takahashi (1999) investigated the effect of a transverse surface crack upon the vibration and stability of a nonuniform Timoshenko beam subjected to a follower force. Matbuly et al. (2009) applied the differential quadrature method (DQM) to formulate the eigenvalue problem of a multi-cracked beam made of functionally graded material (FGM) on the Pasternak-type foundation based on the Euler–Bernoulli theory. For an inverse problem of determining crack locations and sizes from measured data of the damaged structure, Khaji et al. (2009) developed a closed-form method within the framework of the Timoshenko beam theory. In this context, Attar (2012) exposed the details of an analytical scheme for the inverse problem in stepped beams with elastic end constraints employing the transfer matrix method (TMM). Feklistova and Hein (2013) used the Haar wavelet transform and neural networks to identify depth and location of a defect in the beam. Moreover, in order to investigate the effects of inclined edge cracks on the dynamic behavior of a cantilever beam, Behera et al. (2014) performed finite element analysis and experimental studies. Recently, a lattice spring model was developed by Attar et al. (2014a) to investigate free vibrations of damaged beam-like structures resting on two-parameter foundations. They utilized the Timoshenko beam model to treat the defective component and substrate as an assembly of discrete units interacting through springs where the local stiffness reduction induced by the edge crack is simulated by an additional rotational spring at the damage position.

However, in the case of moving loads crossing over damaged beam-like structures, the number of contributions in the literature is limited. Mahmoud and Abouzaid (2002) treated a simply supported Euler–Bernoulli beam with a transverse crack and utilized an iterative modal analysis approach to show how the deflection pattern of the damaged structure alters compared to the undamaged one. A theoretical and experimental study was conducted by Bilello and Bergman (2004) to investigate the response of the damaged Euler–Bernoulli beam traversed by a moving mass and weakened by an arbitrary number of cracks. They simulated the damaged sections through rotational massless springs and used the modal expansion theory (Galerkin approximation) to obtain the forced vibratory response. They also validated their solution through a series of experimental tests simulating a prototype bridge with a small-scale model. The same method has been applied by Lin and Chang (2006) to explore the effect of an edge crack on the dynamic response of a cantilever Euler–Bernoulli beam. In order to detect defects on a bridge deck, Zhu and Law (2006) utilized a modal expansion approach and performed spatial wavelet analysis for the operational dynamic deflection of a single measuring point on the Euler–Bernoulli beam model. Yang et al. (2008) treated slender FGM beams subjected to an axial compressive force and a lateral point load traveling at constant speed. The classical Euler–Bernoulli theory was used by them for damaged inhomogeneous beams with three different types of boundary conditions. Recently, the modal expansion theory was utilized by Yan et al. (2011) to study forced vibration of FGM Timoshenko beams traversed by nonharmonic moving loads. Different generalized coordinates for displacement components in the solution are assumed to characterize the dynamic response of a single-cracked beam. Khorram et al. (2012) also used this theory in a wavelet-based damage detection technique to find the location and size of a crack in a simply supported beam.

Therefore, the results reported so far in the literature focus mostly on free vibration analysis (Bakhtiari-Nejad et al., 2014), nonharmonic moving loads (Museros et al., 2013), Euler–Bernoulli beam theory (which loses its accuracy at higher frequencies; Wang, 1997), components with a limited number of cracks and basic typical boundary conditions (Reis and Pala, 2012). The role of an elastic foundation fully or partially supporting the cracked beam-like structures excited by moving loads has not been addressed explicitly in the previous models. The primary objective of the present study is to fill this gap and develop a general semi-analytical scheme based on the combination of the TMM and modal expansion scheme to obtain the vibratory response of multi-cracked Timoshenko beams on shear elastic foundations (fully or partially supported) induced by harmonic moving loads. The use of general elastic end constraints at two ends of the beam permits the model to cover all classical boundary conditions which can be recovered if appropriate values for stiffness of the springs are chosen. These elastic end constraints may also be used to model nonclassical boundary conditions which are defined as the damaged or imperfect end supports and they can be represented by a combination of translational and rotational springs at two ends of the beam (Sari and Butcher, 2010). Following the procedure of the TMM, the beam-like structure is treated as an assembly of sub-beams to obtain the free vibration data of the system. The computed natural frequencies and corresponding mode shapes are subsequently used to evaluate the dynamic response of the structure by implementing the modal expansion technique with the same time functions for displacement components. One advantage of the proposed approach is that it allows us to easily tackle partially supported beams under harmonic loads with any number of edge cracks and generic boundary conditions. Hence this semi-analytical basis can provide a very accurate and promising benchmark for verification of other analytical and numerical methods. Some illustrative numerical examples are also provided not only to demonstrate the versatility and robustness of the analytical tool developed, but also to investigate the effects of physical and geometrical parameters on the dynamic response of the damaged structure, such as maximum dynamic deflection, v_max (the velocity corresponding to the peak value of the maximum deflection when the structural member is traversed by a single moving load) and v_cr (the critical velocity which leads to the resonance phenomenon when the component is subjected to a stream of moving loads).

The present paper is divided into the following sections. In Section 2, we outline the theoretical formulation for the dynamics of the damaged beam-like structure under moving loads. Section 3 is dedicated to the numerical results and discussion, and Section 4 includes the concluding remarks.

2. Theoretical formulation

2.1. Equations of motion

Let us consider a straight homogeneous beam-like component of length L and uniform cross-sectional area A as illustrated in Figure 1. Generic elastic restraints are considered at both ends of the beam to cover all possible boundary conditions and it is also perfectly bonded to a viscoelastic foundation (Pasternak-type) to model the effects of the subgrade medium. The supporting reaction of the foundation is delineated by Pasternak-type foundation equation as $P (x, t) = k w w - k p (\partial 2 w / \partial x 2) + c w (\partial w / \partial t)$ where P is the foundation reaction per unit length (N/m), w is the transverse deflection of the beam (m), k_w is the Winkler modulus (N/m²), k_p is the Pasternak shear modulus (N) and c_w is the foundation normal damping coefficient (Nċs/m²) of the elastic foundation (corresponding to the normal and shear interactions of the beam and foundation). The beam is only subjected to a harmonic concentrated load with constant amplitude F₀ and excitation frequency $ω \land$ , which is traveling at a uniform speed v from the left end to the right end. The transverse moving load is always in contact with the structure during the forced vibration period 0 ≤ t ≤ L/v and it can be described with the aid of the Dirac delta function $δ (\cdot)$ as

F (x, t) = F 0 P \land (t) δ (x - vt), 0 \leq x \leq L, 0 \leq t \leq L / v

(1)

where

P \land (t)

is a harmonic function of time with a period of

2 π / ω \land

such as

P \land (t) = \cos (ω \land t)

The Timoshenko theory describes the displacement field components (u₁,u₂,u₃) of an arbitrary point (x,y,z) on the beam cross-section as

{u 1 (x, y, z, t) = u (x, t) + z ϕ (x, t) u 2 (x, y, z, t) = 0 u 3 (x, y, z, t) = w (x, t)

(2)

where u and w are respectively the x- and z-components of the total displacement vector of the point (x,0,0) on the beam neutral axis at time t, and ϕ is the cross-section rotation (about the y-axis) with respect to the shown rectangular Cartesian coordinate system. The x-coordinate is defined along the beam length, the y-coordinate is along the width and the z-coordinate is along the height. In order to derive the governing differential equations of motion and corresponding boundary conditions Hamilton’s principle may be applied for this mechanical system as

δ \int_{0}^{t 1} (T b - Π b + R_{b}^{e} + R_{b}^{d}) d t = 0

(3)

where δ denotes the variation symbol, T_b is the kinetic energy of the beam, Π _b is the total potential energy including the strain energy of the beam and potential energy of the elastic foundation and elastic end supports, R_b^e is the virtual work done by external forces applied to the beam in a time interval

[0, t 1]

and

R_{b}^{d}

is the virtual work due to the nonconservative damping force of the foundation. The first variation of the beam kinetic energy on the time interval

[0, t 1]

has the form

δ \int_{0}^{t 1} T b 1 mu d t = δ \int_{0}^{t 1} {\frac{1}{2} \int V ρ [(\frac{\partial u 1}{\partial t}) 2 + (\frac{\partial u 2}{\partial t}) 2 + (\frac{\partial u 3}{\partial t}) 2] d V} d t

(4)

where V and ρ denote the volume and density of the beam, respectively. The strain energy U in an isotropic linear elastic material occupying volume V (in this case the volume of the beam) is given by

U = \frac{1}{2} \int V (σ ij ɛ ij) d V

(5)

where σ_ij and ɛ_ij are the stress and strain tensors, respectively. The total potential energy is the sum of the beam strain energy U and the potential energy of the elastic foundation and end supports. Thus, the first variation of Π _b can be expressed as

δ \int_{0}^{t 1} Π b 1 mu d t = δ \int_{0}^{t 1} \frac{1}{2} {\int V (σ xx ɛ xx + σ xz γ xz) d V + \int_{0}^{L} (k w w 2 + k p (\frac{\partial w}{\partial x}) 2) d x + K L 1 [w | x = 0] 2 + K R 1 [ϕ | x = 0] 2 + K L 2 [w | x = L] 2 + K R 2 [ϕ | x = L] 2} d t

(6)

where K_L₁, K_L₂, K_R₁ and K_R₂ are the stiffnesses of the elastic constraints at the ends of the beam, as depicted in Figure 1. The variation of virtual work done by the moving load and the normal damping force applied to the component on the time interval

[0, t 1]

can also be written as

δ \int_{0}^{t 1} R_{b}^{e} 1 mu d t = \int_{0}^{t 1} \int_{0}^{L} {F (x, t) δ w} d x d t

(7)

δ \int_{0}^{t 1} R_{b}^{d} 1 mu d t = \int_{0}^{t 1} \int_{0}^{L} (- c w \frac{\partial w}{\partial t} δ w) d x d t

(8)

Using the first variation of the kinetic energy, total potential energy and virtual work done by the external force in equation (3), the governing equations of the beam can be obtained as (as shown in Appendix A)

\frac{\partial Q}{\partial x} - k w w + k p \frac{\partial 2 w}{\partial x 2} - c w \frac{\partial w}{\partial t} + F (x, t) - m 0 \frac{\partial 2 w}{\partial t 2} = 0

(9)

\frac{\partial M}{\partial x} - Q - m 2 \frac{\partial 2 ϕ}{\partial t 2} = 0

(10)

where m₀ and m₂ are the inertial terms (m₀ is the mass per unit length and m₂ is the rotary inertia per unit length; for more details, see equations (82) and (83)), M is the internal bending moment (see equation (86)) and Q is the internal shear force (see equation (87)). The associated boundary conditions are also derived as

Q + Q \land - K L 1 w = 0 or w = w \land 0 at x = 0

(11)

Q + Q \land + K L 2 w = 0 or w = w \land L at x = L

(12)

M - K R 1 ϕ = 0 or ϕ = ϕ \land 0 at x = 0

(13)

M + K R 2 ϕ = 0 or ϕ = ϕ \land L at x = L

(14)

Note that

Q \land = k p \frac{\partial w}{\partial x}

in equations (11) and (12) is the internal shear force associated with the Pasternak shear layer (Kerr, 1976; Zhaohua and Cook, 1983), while Q is the contribution of the bending effects to the total shear force. Therefore, the total transverse shear force (S) at each cross-section involves a combination of Q and

Q \land = k p \frac{\partial w}{\partial x}

. It is worth noting that the role of internal shear force associated with the shear layer

Q \land

has been overlooked in the previous literature (Matbuly et al., 2009; Yan et al., 2011) for cracked beams securely attached to the Pasternak elastic foundation. In order to solve the equations of motion, a set of initial conditions are also required to be specified as

w (x, 0) = w 0 (x) w \cdot (x, 0) = w \cdot 0 (x)

(15)

ϕ (x, 0) = ϕ 0 (x) ϕ \cdot (x, 0) = ϕ \cdot 0 (x)

(16)

where

w 0 (x)

w \cdot 0 (x)

ϕ 0 (x)

and

ϕ \cdot 0 (x)

are the prescribed functions. By substituting for the internal bending moment and shear force into equations (9) and (10), the dimensionless governing equations of motion of the Timoshenko beam supported on the viscoelastic foundation can be achieved in terms of displacements as (for 0 ≤ ζ ≤ 1 and 0 ≤ t ≤ 1/v)

\frac{1}{s} (\frac{\partial 2 w ¯}{\partial ζ 2} + \frac{\partial ϕ}{\partial ζ}) - k ¯ w w ¯ + k ¯ p \frac{\partial 2 w ¯}{\partial ζ 2} - c ¯ w \frac{\partial w ¯}{\partial t ¯} = \frac{\partial 2 w ¯}{\partial t ¯ 2} - f 0 P \land (t ¯) δ (ζ - v ¯ t ¯)

(17)

\frac{\partial 2 ϕ}{\partial ζ 2} - \frac{1}{✓ r ✓ s} (ϕ + \frac{\partial w ¯}{\partial ζ}) = \frac{\partial 2 ϕ}{\partial t ¯ 2}

(18)

where w = w/L is the dimensionless lateral displacement of the beam which is a function of dimensionless axial coordinate ζ = x/L and time

t ¯ = t \sqrt{E / (ρ L 2)}

. The harmonic function

P \land (t ¯)

is defined with a dimensionless period of

2 π / λ \land

such as

P \land (t ¯) = \cos (λ \land t ¯)

, where

λ \land = ω \land \sqrt{ρ L 2 / E}

. The other dimensionless quantities in equations (17) and (18) are

k ¯ w = \frac{k w L 2}{EA}, k ¯ p = \frac{k p}{EA}, c ¯ w = \frac{c w L}{A \sqrt{ρ E}}, ✓ r = \frac{I}{AL 2}, ✓ s = \frac{E}{κ s G}, v ¯ = v \sqrt{\frac{ρ}{E}}, f 0 = \frac{F 0}{EA}

(19)

where E is the Young’s modulus, G is the shear modulus, κ_s is the Timoshenko shear correction factor and I is the second moment of the cross-sectional area about the y-axis (see equation (84)). It should be pointed out that by letting P(

t ¯

) = 1 in equation (17), the governing equations of motion reduce to the model under a constant moving load, and by changing

δ (ζ - v ¯ t ¯)

δ (ζ - ζ 1)

, we can obtain the governing equations for the beam subjected to the static harmonic point load (v = 0) applied at ζ = ζ₁. After the moving force leaves the beam span (t > 1/v or in dimensional form t > L/v), the beam-like component experiences residual free vibrations where the motion is governed by equations (17) and (18) with setting f₀ = 0. Therefore, the remaining free vibrations are described as the solution of the governing equations (with f₀ = 0 or in dimensional form F₀ = 0) subject to the initial conditions prescribed by the forced vibratory response at t = L/v.

2.2. Edge-cracked beam

If an elastic structure is weakened by an edge crack, the strain energy concentration occurs in the surrounding area around the crack tip. This surface damage can thus be considered as a source of local flexibility at its location. The idea of modeling the crack by a lumped massless spring is presented to establish the relation between the strain energy concentration and applied loads. Based on Castigliano’s theorem, the flexibility coefficients can be expressed in terms of the stress intensity factors (SIF). Generalized loading conditions for a rectangular sectional beam element with a transverse surface crack are shown in Figure 2. The crack has a tip line parallel to the y-axis and the bar is loaded with axial load, P₁, shear loads, P₂ and P₃, torsional torque, P₄, and bending moments, P₅ and P₆. The additional displacement $u_{i}^{c}$ caused by the crack in the direction of the loading P_i can be determined with the aid of Castigliano’s theorem as

u_{i}^{c} = \frac{\partial U c}{\partial P i} = \frac{\partial}{\partial P i} [\int A c J c 1 mu d A]

(20)

where U_c is the strain energy due to the crack, A_c is the crack section and J _c is the strain energy density function given by Papadopoulos and Dimarogonas (1987) as

J c = \frac{1}{E'} [(\sum_{i = 1}^{6} K I i) 2 + (\sum_{i = 1}^{6} K II i) 2 + (1 + ν) (\sum_{i = 1}^{6} K III i) 2]

(21)

where ν denotes Poisson’s ratio,

E' = E

for plane stress and

E' = E / (1 - ν 2)

for plane strain. In equation (21), K_I _i , K_II _i and K_III _i denote the crack SIF for modes I, II and III, respectively, corresponding to the generalized loading P_i. The SIF K _εi (ε = I,II,III) can be expressed as

K ε i = σ i \sqrt{π a} ψ ε i

(22)

in which σ_i is the stress at the crack cross-section corresponding to the ith independent load, a is the crack size and ψ_εi denotes a geometry-dependent nondimensional crack configuration factor. The components of the compliance matrix, by definition, can be derived as

C ij = \frac{\partial u_{i}^{c}}{\partial P j} = \frac{\partial 2}{\partial P i \partial P j} [\int A c J c 1 mu d A]

(23)

According to equations (21) to (23), it can be deduced that the components of the local flexibility matrix depend only on the degrees of freedom being considered for the generalized loading applied to the crack section and thus the full compliance matrix in the general case is 6 × 6. However, in this study the flexural vibration of the beam in the x−z-plane is investigated in particular, so P₅ (the bending moment about the y-direction) is the only significant load in equation (23) and the contribution of the other components to the strain energy of the system is small and negligible. Consequently, C₅₅ is the only nonzero element of the flexibility matrix and the SIF corresponding to P₅ in equation (22) takes the form

K I 5 = σ 5 \sqrt{π ξ} ψ I 5, σ 5 = \frac{P 5 b}{2 I}

(24)

where ξ (and η) is a spatial variable to measure the crack length (and width) along the crack plane, as shown in Figure 3. The origin of the ξ- and η-axes in Figure 3 is different from the origin of the z- and y-axes in Figure 2. Now, using equations (21) and (23), the following equation can be obtained for C₅₅:

C 55 = \frac{\partial 2}{\partial P_{5}^{2}} [\int_{0}^{a} {\int_{- \frac{c}{2}}^{\frac{c}{2}} (\frac{K_{I 5}^{2}}{E'}) d η} d ξ]

(25)

Finally, using equation (24) in equation (25) and integrating along the crack edge, as shown in Figure 3, the local flexibility (C = C₅₅) for a rectangular cross-section can be achieved as (Zheng and Fan, 2003)

C = \frac{6 π b}{EI} \int_{0}^{\frac{a}{b}} \times χ {\frac{\sqrt{\frac{2}{π χ} tan (\frac{π χ}{2})} (0.923 + 0.1999 [1 - sin (\frac{π χ}{2})] 4)}{cos (\frac{π χ}{2})}} 2 d χ

(26)

Ostachowicz and Krawczuk (1991) proposed an equation to approximate the local flexibility due to the crack as

C = \frac{6 π b}{EI} (\frac{a}{b}) 2 {0.6384 - 1.035 (\frac{a}{b}) + 3.7201 (\frac{a}{b}) 2 - 5.1773 (\frac{a}{b}) 3 + 7.553 (\frac{a}{b}) 4 - 7.332 (\frac{a}{b}) 5 + 2.4909 (\frac{a}{b}) 6}

(27)

Similar equations are derived by Zheng and Fan (2003) in order to compute the local flexibility for cracked beams with other forms of cross-section.

2.3. Natural frequencies and mode shapes of the damaged beam-like structure

The modal expansion method is based on the eigenvalue solution of the undamped system under moving load. Thus, to predict the forced vibratory response of the structure, the first step is to obtain the undamped natural frequencies and corresponding normal mode shapes. In the present study, the TMM is proposed to perform free vibration analysis of a beam weakened by multiple edge cracks and partially supported by an elastic foundation. This method is a prevalent and accurate technique for eigenvalue solution of beam-like structures with nonuniform mechanical properties (Attar, 2012; Boiangiu et al., 2014).

The procedure of the TMM is based on treating the beam as an assembly of N sub-beams separated by N − 1 discontinuity points. These points may be either edge cracks or partial foundation boundaries, as is demonstrated in Figure 4 for a partially supported single-cracked beam where sub-beam boundaries are $X ¯ = {X ¯ 0, X ¯ 1, X ¯ 2, X ¯ 3, X ¯ 4}$ . Each sub-beam is regarded as an independent intact beam and its governing differential equations of motion are given in equations (17) and (18). For free vibration analysis, by setting c_w = 0 and f₀ = 0, and manipulating equations (17) and (18), the uncoupled equations of motion can be extracted as

✓ r ✓ p \frac{\partial 4 w ¯}{\partial ζ 4} - (✓ r ✓ s k ¯ w + k ¯ p) \frac{\partial 2 w ¯}{\partial ζ 2} + k ¯ w w ¯ - ✓ r (✓ s + ✓ p) \frac{\partial 4 w ¯}{\partial ζ 2 \partial t ¯ 2} + (1 + ✓ r ✓ s k ¯ w) \frac{\partial 2 w ¯}{\partial t ¯ 2} + ✓ r ✓ s \frac{\partial 4 w ¯}{\partial t ¯ 4} = 0

(28)

✓ r ✓ p \frac{\partial 4 ϕ}{\partial ζ 4} - (✓ r ✓ s k ¯ w + k ¯ p) \frac{\partial 2 ϕ}{\partial ζ 2} + k ¯ w ϕ - ✓ r (✓ s + ✓ p) \frac{\partial 4 ϕ}{\partial ζ 2 \partial t ¯ 2} + (1 + ✓ r ✓ s k ¯ w) \frac{\partial 2 ϕ}{\partial t ¯ 2} + ✓ r ✓ s \frac{\partial 4 ϕ}{\partial t ¯ 4} = 0

(29)

where

✓ p = 1 + \frac{k p}{κ s GA}

(30)

One may prescribe the following displacement and rotation fields for harmonic motion by using the separation of variables method to solve the uncoupled equations:

w ¯ (ζ, t ¯) = W ¯ (ζ) exp (i λ t ¯)

(31)

ϕ (ζ, t ¯) = Φ (ζ) exp (i λ t ¯)

(32)

where

i = \sqrt{- 1}

and

λ = ω \sqrt{ρ L 2 / E}

with ω denoting the natural frequency of the beam. Insert equations (31) and (32) into equations (28) and (29) and after some manipulations, one obtains

\frac{\partial 4 W ¯}{\partial ζ 4} + 2 Θ \frac{\partial 2 W ¯}{\partial ζ 2} + Λ W ¯ = 0

(33)

\frac{\partial 4 Φ}{\partial ζ 4} + 2 Θ \frac{\partial 2 Φ}{\partial ζ 2} + Λ Φ = 0

(34)

where

Θ = \frac{✓ r λ 2 (1 + ✓ s + ✓ s k ¯ p) - ✓ r ✓ s k ¯ w - k ¯ p}{2 ✓ r ✓ p} Λ = \frac{k ¯ w - λ 2 (1 + ✓ r ✓ s k ¯ w) + ✓ r ✓ s λ 4}{✓ r ✓ p}

(35)

The solutions of equations (33) and (34) and the corresponding dimensionless bending moment and shear force for the jth sub-beam can be expressed in the following form:

{Ψ j (ζ)} = [t \land (ζ; k ¯ w, k ¯ p)] \cdot {Θ j}

(36)

where

{Θ j} = {✓ A j, ✓ B j, ✓ C j, ✓ D j} T

denote unknown constants and

{Ψ j (ζ)} = {W ¯ j (ζ), Φ j (ζ), M ¯ j (ζ), S ¯ j (ζ)} T

represent the dimensionless displacement, rotation, bending moment and total shear force corresponding to the jth sub-beam, respectively. In equation (36),

[t \land (ζ; k ¯ w, k ¯ p)]

is expressed by

[t \land (ζ; k ¯ w, k ¯ p)] = [\sin (α ζ) \cos (α ζ) \sinh (β ζ) \cosh (β ζ) - δ 1 \cos (α ζ) δ 1 \sin (α ζ) δ 2 \cosh (β ζ) δ 2 \sinh (β ζ) α δ 1 \sin (α ζ) α δ 1 \cos (α ζ) β δ 2 \sinh (β ζ) β δ 2 \cosh (β ζ) (α p \lor - δ 1) \cos (α ζ) (- α p \lor + δ 1) \sin (α ζ) (β p \lor + δ 2) \cosh (β ζ) (β p \lor + δ 2) \sinh (β ζ)]

(37)

The dimensionless quantities are defined as the following:

α = \sqrt{\sqrt{Θ 2 - Λ} + Θ}, β = \sqrt{\sqrt{Θ 2 - Λ} - Θ}

(38)

δ 1 = \frac{α}{1 + (α 2 - λ 2) ✓ r ✓ s}, δ 2 = \frac{β}{- 1 + (β 2 + λ 2) ✓ r ✓ s}

(39)

The dimensionless bending moment and total shear force are shown by

M ¯ (ζ)

and

S ¯ (ζ)

, respectively, and they are defined by

M ¯ (ζ) = \frac{L}{EI} M (ζ) = \frac{\partial Φ}{\partial ζ} S ¯ (ζ) = \frac{1}{κ s GA} S (ζ) = ✓ p \frac{\partial W ¯}{\partial ζ} + Φ

(40)

It must be emphasized that the role of internal shear force in the shear layer is incorporated in the TMM, making the present scheme different from previous treatments of cracked beams bonded with the Pasternak foundation.

With the definition of equation (36),

[t \land (ζ; k ¯ w, k ¯ p)]

for each sub-beam is defined with parameter ζ representing the axial coordinate in the global coordinate system and arguments

k ¯ w, k ¯ p

denoting the normalized stiffnesses of the foundation supporting the sub-beam. Consequently, utilizing equation (36) for the jth sub-beam, we may relate the vector of deflection, rotation, bending moment and shear force at the left end of this sub-beam (

X ¯ j

) to the right end (

X ¯ j + 1

) as

{Ψ j (X ¯ j + 1)} = [t \land (X ¯ j + 1; k ¯ w, k ¯ p)] \cdot [t \land (X ¯ j; k ¯ w, k ¯ p)] - 1 \cdot {Ψ j (X ¯ j)}

(41)

Then, using the compatibility conditions at connection points of the sub-beams, one may extract the overall transfer matrix for the whole length of the beam which relates the left end of the beam (

X ¯ 0

= 0) to the right end (

X ¯ N

= 1). If sub-beams are separated by a partial foundation boundary, the compatibility conditions are continuity of displacement, rotation, bending moment, and total shear force at the common point of two sub-beams. The compatibility conditions if sub-beams are separated by an edge crack are continuity of displacement, bending moment, and total shear force and discontinuity of rotation at crack location. The discontinuity of rotation is proportional to the bending moment at the cracked section and rotation of the massless spring representing the open edge crack. Thus, the compatibility conditions for a crack located at ζ =

X ¯ j

take the form

{Ψ j + 1 (X ¯ j)} = [C \land j] \cdot {Ψ j (X ¯ j)}

(42)

where

[C \land j]

is given by

[C \land j] = [100001 C ¯ j 000100001]

(43)

with

C ¯ j

= EIC_j/L the dimensionless local flexibility due to the crack located at ζ =

X ¯ j

. Finally, equation (41) and compatibility conditions can help us to extract the overall transfer matrix which furnishes the relationships between displacements and rotations at two ends of the beam as

{W ¯ N (1) Φ N (1) - k r 2 Φ N (1) - k l 2 W ¯ N (1)} = [T] \cdot {W ¯ 1 (0) Φ 1 (0) k r 1 Φ 1 (0) k l 1 W ¯ 1 (0)}

(44)

where dimensionless elastic end parameters are

k r 1 = \frac{L}{EI} K R 1, k l 1 = \frac{L}{κ s GA} K L 1, k r 2 = \frac{L}{EI} K R 2, k l 2 = \frac{L}{κ s GA} K L 2

(45)

and

[T]

represents the overall transfer matrix which can be written as

[T] = [T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44]

(46)

with T_ij denoting the components of the overall transfer matrix. For instance, the overall transfer matrix

[T]

for the partially supported single-cracked beam in Figure 4 may be determined by considering four sub-beams. Using equation (41) and compatibility conditions at common points of the sub-beams (

{X ¯ 1, X ¯ 2, X ¯ 3}

), the following relation for the overall transfer matrix can be deduced:

[T] = [t \land (1; k ¯ w 2, k ¯ p 2)] \cdot [t \land (X ¯ 3; k ¯ w 2, k ¯ p 2)] - 1 \cdot [t \land (X ¯ 3; k ¯ w 1, k ¯ p 1)] \cdot [t \land (X ¯ 2; k ¯ w 1, k ¯ p 1)] - 1 \cdot [C \land 2] \cdot [t \land (X ¯ 2; k ¯ w 1, k ¯ p 1)] \cdot [t \land (X ¯ 1; k ¯ w 1, k ¯ p 1)] - 1 \cdot [t \land (X ¯ 1; 0, 0)] \cdot [t \land (0; 0, 0)] - 1

(47)

By rearrangement of equation (44), the general form of the characteristic (eigenvalue) equation for the cracked beam takes the form

[T 31 + k l 1 T 34 + k r 2 (T 21 + k l 1 T 24)] \cdot [T 42 + k r 1 T 43 + k l 2 (T 12 + k r 1 T 13)] - [T 41 + k l 1 T 44 + k l 2 (T 11 + k l 1 T 14)] \cdot [T 32 + k r 1 T 33 + k r 2 (T 22 + k r 1 T 23)] = 0

(48)

It should be noted that the general characteristic equation is a function of natural frequencies, material properties, elastic parameters of the foundation and end supports, and the locations and depths of the edge cracks. The present TMM is exact and solving the characteristic equation can deliver closed-form analytical solutions for natural frequencies of the damaged structure. By choosing appropriate values of elastic end parameters we may model all possible boundary conditions (the elastic end parameters for some common boundary conditions are provided in Table 1). For example, equation (48) for a simply supported beam reduces to

T 12 T 34 - T 14 T 32 = 0

and if the beam is fully supported by a two-parameter substrate while it is weakened by one edge crack (i.e.

X ¯ = {0, X ¯ 1, 1}

), the explicit form of the eigenvalue equation can be derived as follows:

2 (α δ 2 + β δ 1) ✓ p \sin (α) \sinh (β) + α δ 1 (δ 2 + β ✓ p) \times C ¯ 1 \sinh (β) {\cos (α) - \cos (α (1 - 2 X ¯ 1))} - β δ 2 (δ 1 - α ✓ p) C ¯ 1 \sin (α) \times {\cosh (β) - \cos (β (1 - 2 X ¯ 1))} = 0

(49)

It is obvious that equation (49) for the intact beam (

C ¯ 1

= 0) reduces to

\sin (α) = 0

which is the well-known eigenvalue equation of the simply supported Timoshenko beam. The transverse and rotational mode shapes associated with natural frequencies can also be calculated through eigenvector solution of each sub-beam. For this purpose, one can use equations (36), (44) and compatibility conditions to compute the unknown constants corresponding to each sub-beam and form the flexural eigenmodes of the entire beam. The calculated free vibration data including natural frequencies and vibration mode shapes provide the necessary information to evaluate the dynamic behavior of the one-dimensional system under the moving load.

Table 1.

Common boundary conditions modeled with appropriate values for end supports.

Boundary conditions	Stiffness parameters of the end constraints
	K_L ₁	K_R ₁	K_L ₂	K_R ₂
Clamped–free (C–F)	∞	∞	0	0
Simply supported (H–H)	∞	0	∞	0
Clamped–hinged (C–H)	∞	∞	∞	0
Clamped–clamped (C–C)	∞	∞	∞	∞
Clamped–free shear (C–S)	∞	∞	0	∞
Free–free (F–F)	0	0	0	0

2.4. Forced response

Utilizing the modal expansion technique, the dynamic response of the beam-like structure for the forced vibration period 0 ≤ t ≤ L/v can be obtained in terms of the vibrational mode shapes. Following the method elaborated by Dadfarnia et al. (2005), identical time functions are assumed to express the transverse and rotational response of each sub-beam as (total number of sub-beams = N)

w ¯ j (ζ, t ¯) = \sum_{k = 1}^{✓ n} W ¯ jk (ζ) η k (t ¯) j = 1, 2, \dots, N ϕ j (ζ, t ¯) = \sum_{k = 1}^{✓ n} Φ jk (ζ) η k (t ¯) j = 1, 2, \dots, N

(50)

where ň is the total number of truncated terms (eigenmodes) used to approximate the solution, η_k(

t ¯

) are the unknown time functions (generalized coordinates) which have to be computed, and

W ¯

_jk and Φ _jk are the transverse and rotational vibration mode shapes corresponding to the kth mode of the jth sub-beam obtained in the previous section. Substituting the dynamic responses from equation (50) into equations (17) and (18), multiplying by

W ¯ jh

and

Φ jh

(

j = 1, 2, \dots, N

and

h = 1, 2, \dots, ✓ n

), respectively, and integrating over the beam length, leads to

\sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} \sum_{k = 1}^{✓ n} {\frac{1}{✓ s} (W ¯ jk'' + Φ jk') η k - k ¯ w W ¯ jk η k + k ¯ p W ¯ jk'' η k - c ¯ w W ¯ jk η \cdot k - W ¯ jk η ·· k} W ¯ jh d ζ} = \sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} {f 0 P \land (t ¯) δ (ζ - v ¯ t ¯)} W ¯ jh d ζ}

(51)

\sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} \sum_{k = 1}^{n \lor} {Φ jk'' η k - \frac{1}{r \lor s \lor} (W ¯ jk' + Φ jk) η k - Φ jk η \cdot\cdot k} Φ jh d ζ} = 0

(52)

where

X ¯ j

is the

(j + 1)

th member of the vector of sub-beam boundaries

X ¯ = {X ¯ 0, X ¯ 1, \dots, X ¯ N}

, a superscript ′ represents the derivative with respect to the normalized axial coordinate ζ and a superimposed dot denotes the derivative with respect to the normalized time t. On the other hand, by recalling equations (31) and (32) for undamped free vibration of the jth sub-beam in the kth mode and inserting them into equations (17) and (18) (when

c ¯ w

and f₀ = 0), one may obtain

\frac{1}{✓ s} (W ¯ ″ jk + Φ' jk) - k ¯ w W ¯ jk + k ¯ p W ¯ ″ jk = - λ_{k}^{2} W ¯ jk

(53)

Φ ″ jk - \frac{1}{✓ r ✓ s} (W ¯' jk + Φ jk) = - λ_{k}^{2} Φ jk

(54)

Adding each side of equations (51) and (52) and using equations (53) and (54) in them results in

\sum_{k = 1}^{✓ n} {(η ·· k + λ_{k}^{2} η k) \sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} (W ¯ jk W ¯ jh + ✓ r Φ jk Φ jh) d ζ} + c ¯ w η \cdot k \sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} W ¯ jk W ¯ jh d ζ}} = f 0 P \land (t ¯) \sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} δ (ζ - v ¯ t ¯) W ¯ jh d ζ}

(55)

Applying the principle of the orthogonality of the mode shapes through the total length of the beam it can be shown that (Dadfarnia et al., 2005)

\sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} (W ¯ jk W ¯ jh + ✓ r Φ jk Φ jh) d ζ} = ✓ O h δ kh k, h = 1, 2, \dots, ✓ n

(56)

in which δ_kh is the Kronecker delta, and

✓ O h

is a parameter for normalizing the hth eigenfunctions which can be defined for each mode from equation (56) by setting k = h. The sifting property of the Dirac delta function can also be represented in the form

\int_{X ¯ j - 1}^{X ¯ j} W ¯ jh (ζ) δ (ζ - v ¯ t ¯) d ζ = W ¯ jh (v ¯ t ¯) [H ~ (t ¯ - \frac{X ¯ j - 1}{v ¯}) - H ~ (t ¯ - \frac{X ¯ j}{v ¯})]

(57)

introducing Heaviside step function

H ~ (x)

, which is defined to be zero if x < 0, and one if x ≥ 0. Thus, making use of equations (56) and (57), equation (55) is simplified as follows:

η ·· h + \frac{c ¯ w}{✓ O h} \sum_{k = 1}^{✓ n} {\sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} W ¯ jh W ¯ jk d ζ} η \cdot k} + λ_{h}^{2} η h = R h (t ¯) h = 1, 2, \dots, ✓ n

(58)

where

R h (t ¯) = \frac{f 0}{✓ O h} P \land (t ¯) \sum_{j = 1}^{N} {W ¯ jh (v ¯ t ¯) [H ~ (t ¯ - \frac{X ¯ j - 1}{v ¯}) - H ~ (t ¯ - \frac{X ¯ j}{v ¯})]} h = 1, 2, \dots, ✓ n

(59)

Expanding equation (58) and rewriting in matrix form yields

{η ·· 1 η ·· 2 : η ·· ✓ n} + [c 11 c 12 \dots c 1 ✓ n c 21 c 22 \dots c 2 ✓ n : : : c ✓ n 1 c ✓ n 2 \dots c ✓ n ✓ n] \cdot {η \cdot 1 η \cdot 2 : η \cdot ✓ n} + [λ_{1}^{2} 0 \dots 00 λ_{2}^{2} \dots 0 : : : 00 \dots λ_{✓ n}^{2}] \cdot {η 1 η 2 : η ✓ n} = {R 1 R 2 : R ✓ n}

(60)

where

c hk = \frac{c ¯ w}{✓ O h} \sum_{j = 1}^{N} {\int_{X ¯ j - 1}^{X ¯ j} W ¯ jh W ¯ jk d ζ} h, k = 1, 2, \dots, ✓ n

(61)

The transient dynamic response problem now consists in finding the generalized coordinates $η h (t ¯)$ by utilizing a standard time-integration scheme to solve equation (60) with appropriate initial conditions. Making use of equation (50), the initial transverse and rotational displacements and speeds ( $t ¯$ = 0) can be expressed in the form

w ¯ j (ζ, 0) = \sum_{k = 1}^{✓ n} W ¯ jk (ζ) η k (0) ¯ \cdot w j (ζ, 0) = \sum_{k = 1}^{✓ n} W ¯ jk (ζ) η \cdot k (0)

(62)

ϕ j (ζ, 0) = \sum_{k = 1}^{✓ n} Φ jk (ζ) η k (0) ϕ \cdot j (ζ, 0) = \sum_{k = 1}^{✓ n} Φ jk (ζ) η \cdot k (0)

(63)

Multiplying both sides of the equations for initial transverse displacement and initial transverse velocity in equation (62) by

W ¯ jh

and also both sides of the equations for initial rotation and initial rotational speed in equation (63) by

Φ jh

(

j = 1, 2, \dots, N

and

h = 1, 2, \dots, ✓ n

), integrating over the length of each sub-beam, adding the corresponding sides of the resulting equations, and making use of the orthogonality principle (equation (56)), the necessary initial conditions to solve equation (60) can be derived as

η h (0) = \frac{1}{✓ O h} \sum_{j = 1}^{N} \times {\int_{X ¯ j - 1}^{X ¯ j} {w ¯ j (ζ, 0) W ¯ jh (ζ) + ✓ r ϕ j (ζ, 0) Φ jh (ζ)} d ζ} h = 1, 2, \dots, ✓ n

(64)

η \cdot h (0) = \frac{1}{✓ O h} \sum_{j = 1}^{N} \times {\int_{X ¯ j - 1}^{X ¯ j} {¯ \cdot w j (ζ, 0) W ¯ jh (ζ) + ✓ r ϕ \cdot j (ζ, 0) Φ jh (ζ)} d ζ} h = 1, 2, \dots, ✓ n

(65)

While we can use any standard numerical time-integration technique to solve the system of differential equations (60), the convolution integral theorem can be employed to derive the explicit analytical solution for the transient dynamic response of systems without damping. For the undamped model (

c ¯ w = 0

), equation (58) reduces to

η ·· h (t ¯) + λ_{h}^{2} η h (t ¯) = R h (t ¯) h = 1, 2, \dots, ✓ n

(66)

Assuming s as the Laplace transform variable and taking the Laplace transform, equation (66) can be cast in the s-domain as

✓ η h (s) = \frac{s}{λ_{h}^{2} + s 2} η h (0) + \frac{1}{λ_{h}^{2} + s 2} η \cdot h (0) + \frac{1}{λ_{h}^{2} + s 2} ✓ R h (s) h = 1, 2, \dots, ✓ n

(67)

where

✓ η h (s)

and Ř_h(s) are the transform functions of η_h(

t ¯

) and R_h(

t ¯

), respectively, in the s-domain. Finally, with the aid of the convolution integral theorem and taking the inverse Laplace transform, an analytical expression for the unknown generalized coordinates in the time domain and their time derivative can be derived from equation (67) as the following:

η h (t ¯) = \cos (λ h t ¯) η h (0) + \frac{1}{λ h} \sin (λ h t ¯) η \cdot h (0) + \frac{f 0}{✓ O h λ h} \sum_{j = 1}^{✓ m} {\int_{τ j - 1}^{τ j} {\sin [λ h (t ¯ - τ)] P \land (τ) W ¯ jk (v ¯ τ)} d τ}

(68)

η \cdot h (t ¯) = - λ h \sin (λ h t ¯) η h (0) + \cos (λ h t ¯) η \cdot h (0) + \frac{f 0}{✓ O h} \sum_{j = 1}^{✓ m} {\int_{τ j - 1}^{τ j} {\cos [λ h (t ¯ - τ)] P \land (τ) W ¯ jk (v ¯ τ)} d τ}

(69)

where τ_j is the member of τ referring to the nondimensional times at which the moving load has passed the sub-beam boundaries (

X ¯ j

) as

τ = {τ 0, τ 1, τ 2, \dots, τ ✓ m - 1, τ ✓ m} τ 0 < τ 1 < τ 2 < \dots < τ ✓ m - 1 < τ ✓ m τ 0 = \frac{X ¯ 0}{v ¯} = 0, τ 1 = \frac{X ¯ 1}{v ¯}, τ 2 = \frac{X ¯ 2}{v ¯}, \dots, τ ✓ m - 1 = \frac{X ¯ ✓ m - 1}{v ¯}, τ ✓ m = t ¯

(70)

Note that τ at each time instant has

✓ m + 1

components where

✓ m

denotes the number of sub-beam boundaries which are less than

v ¯ t ¯

. This means that the minimum possible value for

✓ m

is one (it occurs when the moving load is traversing the first sub-beam) and the maximum is N (it occurs when the moving load is in the last sub-beam). The forced vibration response due to the moving load on the beam-like structure can therefore be calculated from equation (50) by utilizing the analytical expression for generalized coordinates in equations (68) and (69) as well as the vibration mode shapes from the previous section. In general, we can rewrite the beam response due the single moving load in dimensional form as

z (x, t) = [w (x, t), w \cdot (x, t), ϕ (x, t), ϕ \cdot (x, t)]

(71)

where for a specific system z involves two parts: the forced vibratory response (0 ≤ t ≤ L/v) when the beam is subjected to the moving load, and the free vibration response (t > L/v) when the component experiences residual free vibrations generated by the load after it exits the beam span from the right end. The forced period solution depends on the initial conditions at t = 0 and the load applied on the structure. In order to depict the dependence on the initial conditions and the traversing load, the forced response can be specified as

z 1 = z 1 (w 0, w \cdot 0, ϕ 0, ϕ \cdot 0, F, x, t) 0 \leq x \leq L, 0 \leq t \leq L / v

(72)

The free vibration response of the system (t > L/v) can also be obtained by utilizing the forced solution at t = L/v (i.e.

z 1 (x, L / v)

) as the initial conditions and setting F = 0 as

z 2 = z 2 (w (x, L / v), w \cdot (x, L / v), ϕ (x, L / v), ϕ \cdot (x, L / v), 0, x, t) 0 \leq x \leq L, t > L / v

(73)

Finally, we can portray the system response due to the single moving load including both the forced and free vibratory response as

z = {z 1 0 \leq x \leq L, 0 \leq t \leq L / v z 2 0 \leq x \leq L, t > L / v

(74)

z = z 1 {H ~ (t) - H ~ (t - L / v)} + z 2 H ~ (t - L / v)

(75)

The Timoshenko beam formulation and its solution process can be reduced to a system with an Euler–Bernoulli beam resting on the Pasternak foundation. This can be done by neglecting the role of shear deformation (G →∞) and rotary inertia (m₂ = 0) in the beam component (note that the formulation corresponding to the Rayleigh beam theory can also be derived in a similar way if we only neglect the shear deformation and keep the rotary inertia term). Thus, the nondimensional governing equation for the Euler–Bernoulli beam in the forced period takes the following form:

✓ r \frac{\partial 4 w ¯}{\partial ζ 4} + k ¯ w w ¯ - k ¯ p \frac{\partial 2 w ¯}{\partial ζ 2} + c ¯ w \frac{\partial w ¯}{\partial t ¯} + \frac{\partial 2 w ¯}{\partial t ¯ 2} = f 0 P \land (t ¯) δ (ζ - v ¯ t ¯) 0 \leq ζ \leq 1, 0 \leq t ¯ \leq 1 / v ¯

(76)

The semi-analytical scheme presented above (combination of TMM and modal expansion technique) can be utilized in a similar way for dynamic analysis of a multi-cracked slender beam on the viscoelastic substrate. For instance, employing TMM for an intact simply supported beam on the full foundation allows writing the ith nondimensional natural frequency as

λ i = \sqrt{✓ r i 4 π 4 + k ¯ w + i 2 π 2 k ¯ p}

and the corresponding linear normal mode as

W ¯ i (ζ) = \sin (i π ζ)

. Therefore, after using the free vibration data in the modal expansion method, we can express the forced transient response of the Euler–Bernoulli beam due to the pulsating moving force

f = f 0 \cos (λ \land t ¯) δ (ζ - v ¯ t ¯)

in explicit form as given in Appendix B.

Due to the linear character of the system, the general solution for a single moving load (equation (74)) enables us to predict the dynamic behavior of a beam under a series of moving loads as well. To this end, we can use the superposition principle to express the overall response to successive traveling forces as the sum of accumulated vibrations generated in the system by every single load. For example, the transient solution for a beam-like component subjected to N_F equidistant loads with d as the characteristic length (i.e. the distance between two consecutive forces) can be obtained as

z g (x, t) = \sum_{i = 0}^{N F - 1} z (x, t - \frac{id}{v}) H ~ (t - \frac{id}{v})

(77)

3. Numerical results and discussion

The semi-analytical methodology presented in this study has been implemented in a computer code to solve the problem for interaction of the moving load, damaged structure and elastic foundation. This code is capable of accommodating several edge cracks and partial elastic foundations. The convergence of the numerical results is examined in each case to find the required sufficient number of modes for the modal expansion technique, and thus we are able to obtain the solution of forced response with a high degree of accuracy. Also, the elastic boundaries considered for the beam allow us to use the proposed approach for a variety of problems, including different types of boundary conditions.

In order to assess the performance of the proposed scheme and highlight its potential, several numerical examples will be studied in this section. At the first stage, we reduce our model to particular cases which have been addressed in the literature to demonstrate the overall validity of the present work. Unless stated otherwise, zero initial conditions and damping coefficient are assumed for the beams in the following examples.

We start with an intact beam by setting crack sizes equal to zero (see Figure 4 if a = 0) and compare the computed dimensionless natural frequencies ( $λ k = ω k \sqrt{ρ L 2 / E}$ ) from the present TMM with available results from the literature. The length-to-thickness ratio (L/b), shear correction factor (κ_s), Poisson’s ratio (ν) and dimensionless elastic foundation parameters ( $k ¯ w, k ¯ p$ ) are considered in accordance with Yokoyama (1991) and Matsunaga (1999). Table 2 depicts a good agreement with the previous studies for the first three modes of vibration. Figure 5 also provides the variation of the first three dimensionless frequencies against $k ¯ w / ✓ r$ when the beam is partially ( $α ¯ 1$ = 0 and $α ¯ 2$ = 0.5) and fully ( $α ¯ 1$ = 0 and $α ¯ 2$ = 1) supported by the foundation. The present results closely agree with those obtained by Yokoyama (1991) from finite element analysis. Note that increasing the length of the elastic foundation leads to an increase in the natural frequencies. However, for small values of k_w, this increase becomes more apparent if the role of subgrade shear modulus is taken into account, but once $k ¯ p = 0$ and $k ¯ w$ is small (when $\frac{k ¯ w}{✓ r} < 35$ for the first mode and $\frac{k ¯ w}{✓ r} < 100$ for the second and third modes, approximately), an increase in the foundation length from $α ¯ 2$ = 0.5 to $α ¯ 2$ = 1 leads to only a relatively small change in the natural frequencies.

Table 2.

Dimensionless natural frequencies (λ_k) of the simply supported intact beam ( $α ¯ 1$ = 0).

							Dimensionless natural frequencies
$\frac{L}{b}$	ν	κ_s	$α ¯ 2$	$k ¯ w$	$k ¯ p$	Method	λ ₁	λ ₂	λ ₃
5	0.3	$\frac{5}{6}$	0	0	0	Matsunaga (1999) ^a	0.5364 (0.5354)	1.867	3.580
						TMM	0.5354	1.857	3.548
5	0.3	$\frac{5}{6}$	1	$\frac{1}{30}$	0	Matsunaga (1999) ^a	0.5653 (0.5649)	1.875	3.584
						TMM	0.5649	1.866	3.553
$\frac{25}{\sqrt{12}}$	$\frac{1}{4}$	$\frac{2}{3}$	1	$\frac{4}{25}$	0	Yokoyama (1991)	0.5506	1.435	2.782
						TMM	0.5499	1.437	2.783
5	0.3	$\frac{5}{6}$	1	$\frac{1}{30}$	$\frac{π 2}{300}$	Matsunaga (1999) ^a	0.7927 (0.7970)	2.155	3.898
						TMM	0.7970	2.166	3.911
$\frac{25}{\sqrt{12}}$	$\frac{1}{3}$	$\frac{2}{3}$	1	$\frac{4}{25}$	$\frac{π 2}{625}$	Yokoyama (1991)	0.6757	1.633	3.014
						TMM	0.6753	1.633	3.011

The value in parentheses shows the exact solution using Timoshenko beam theory.

Table 3 lists the dimensionless natural frequencies of damaged beams with two different slenderness ratios and with no foundation support (see Figure 4 for

k ¯ w = 0

and

k ¯ p = 0

). The Euler–Bernoulli beam model has been utilized by Khorram et al. (2012) for the length-to-thickness ratio of L/b = 50, while the natural frequencies of the damaged beam-like component with the slenderness ratio L/b = 5 are obtained by Khaji et al. (2009) within the framework of Timoshenko beam theory. The values computed by the present TMM are compared to the results from the previous studies indicating a reasonable agreement.

Table 3.

Dimensionless natural frequencies (λ_k) of the beam with no elastic foundation $(k ¯ w = 0, k ¯ p = 0, X ¯ 1 = 0.5, ν = 0.3, κ s = 5 / 6)$ .

				Dimensionless natural frequencies
Boundary conditions	$\frac{L}{b}$	$\frac{a 1}{b}$	Method	λ ₁	λ ₂	λ ₃	λ ₄	λ ₅	λ ₆
H–H	50	0	TMM	0.0569	0.2273	0.5097	0.9020	1.401	2.003
H–H	50	0.5	Khorram et al. (2012)	0.0535	0.2279	0.4838	0.9117	1.351	2.051
			TMM	0.0536	0.2273	0.4826	0.9020	1.334	2.003
C–F	5	0.5	Khaji et al. (2009) (exact)^a	0.1858	0.8947	2.508	3.848
			Khaji et al. (2009) (FEA)^b	0.1861	0.8875	2.522	3.820
			TMM	0.1858	0.8947	2.508	3.848	5.939	7.297
H–C	5	0.5	Khaji et al. (2009) (exact)^a	0.6905	2.115	3.409	5.520
			Khaji et al. (2009) (FEA)^b	0.6964	2.065	3.324	5.207
			TMM	0.6905	2.115	3.409	5.520	9.201	11.24
C–C	5	0.5	Khaji et al. (2009) (exact)^a	0.9429	2.378	3.586	5.663
			Khaji et al. (2009) (FEA)^b	0.9610	2.400	3.618	5.708
			TMM	0.9429	2.378	3.586	5.663	9.225	11.95

Exact solution using Timoshenko beam theory. ^bFinite element analysis.

The dynamic response of the single-cracked H–H beam is also presented by Khorram et al. (2012) when it is traversed by a constant load traveling at two different dimensionless speeds $(v ¯ = 5 \sqrt{\frac{262}{7 \times 109}}$ and $40 \sqrt{\frac{262}{7 \times 109}})$ . Based on the present approach, the necessary free vibration data for dynamic analysis of this problem are given in Table 3. We also need the flexural eigenmodes which are evaluated and indicated in Figure 6 for the first four natural modes of the cracked and intact beams. Notably, the surface crack cannot influence the second and fourth transverse mode shapes since it is located at the mid-point of the structure and the cross-sectional rotation is equal to zero at ζ = 0.5 for these modes. This also agrees with the predicted results in Table 3 for H–H boundary conditions where the second, fourth and sixth natural frequencies of the damaged model are identical to the intact one. Subsequently, in order to illustrate the accuracy of the proposed technique, the dynamic mid-span deflections under the uniformly traveling load are demonstrated in Figure 7. The magnitude of the dynamic lateral displacement $w ¯ (ζ = 0.5, t ¯)$ is normalized by (note that $F 0 L 2 / 48 EI$ is the mid-span dimensionless static deflection of the intact Euler–Bernoulli beam without elastic foundation due to the concentrated load applied at ζ = 0.5). There is an excellent match between the present Timoshenko beam solution and the results reported by Khorram et al. (2012) based on the Euler–Bernoulli theory.

In order to study the influence of rotary inertia and shear deformation, we now investigate the system depicted in Figure 8(a). The intact simply supported beam is traversed by two consecutive harmonic forces with a distance of d = 0.5L between them. This means that the first load enters the beam span from the left end at t = 0 and exits from the right end at t = L/v, while the second load enters at t = L/(2v) and exits at t = 3L/(2v). The dynamic lateral displacement of the mid-span obtained from Timoshenko theory is compared with the Euler–Bernoulli model for three different slenderness ratios L/b = 3,5,10 in Figure 8(b) to (d) for 0 ≤ t ≤ 2L/v. As expected, it is evident that the Euler–Bernoulli model loses its accuracy in predicting the system forced and free vibratory responses for L/b < 10. Moreover, the effects of rotary inertia and shear deformation become more pronounced for both regions in higher applied velocities and frequencies.

			Crack sizes	Dimensionless natural frequencies
H–H	0	0	0	0	0	0.2802	1.071	2.256	3.714
	$\frac{1}{30}$	0	0	0	0	0.3341	1.086	2.263	3.718
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0	0	0	0.4383	1.223	2.412	3.878
	0	0	0	0.5	0	0.2192	1.071	1.942	3.714
	$\frac{1}{30}$	0	0	0.5	0	0.2848	1.086	1.950	3.718
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0	0.5	0	0.4049	1.223	2.129	3.878
	0	0	0.4	0.5	0	0.2047	0.9465	1.876	3.577
	$\frac{1}{30}$	0	0.4	0.5	0	0.2738	0.9635	1.884	3.581
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0.4	0.5	0	0.3967	1.121	2.073	3.751
	0	0	0.4	0.5	0.4	0.1931	0.8403	1.789	3.504
	$\frac{1}{30}$	0	0.4	0.5	0.4	0.2652	0.8593	1.797	3.508
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0.4	0.5	0.4	0.3898	1.032	2.002	3.679

The lowest four natural frequencies of the damaged Timoshenko beam with simply supported boundary conditions are computed and listed in Table 4 by employing the TMM. The slenderness ratio is L/b = 10 and the beam-like components are weakened by one, two and three open edge cracks which are located at $X ¯ 1$ = 0.3, $X ¯ 2$ = 0.5 and $X ¯ 3$ = 0.7. In what follows, a Poisson’s ratio of ν = 0.3 and a shear correction factor of κ_s = 5/6 are assumed for the beams. The presence of defects yields less structural integrity and lower natural frequencies, while this reduction depends on the size, location and number of cracks. Moreover, the supporting reaction of the elastic foundation can change the sensitivity of dynamic characteristics to the presence and size of cracks, compared to the structure with no subgrade support. The influence of imperfections on the vibratory mode shapes of an H–H beam in the presence of cracks are demonstrated in Figure 9 (flexural modes of the intact beam are identical to Figure 6(a)).
Figure 9.
Flexural modes of the simply supported beam in the presence of (a) one crack a₁/b = 0, a₂/b = 0.5, a₃/b = 0, (b) two cracks a₁/b = 0.4, a₂/b = 0.5, a₃/b = 0, (c) three cracks a₁/b = 0.4, a₂/b = 0.5, a₃/b = 0 (L/b = 10, $X ¯ 1$ = 0.3, $X ¯ 2$ = 0.5, $X ¯ 3$ = 0.7, ν = 0.3, κ_s = 5/6).

Table 4.
Dimensionless natural frequencies $(λ i)$ of the intact and damaged beams $(X ¯ 1 = 0.3, X ¯ 2 = 0.5, X ¯ 3 = 0.7, α ¯ 1 = 0, α ¯ 2 = 1, L / b = 10, ν = 0.3, κ s = 5 / 6)$ .

Crack sizes
Dimensionless natural frequencies

Boundary condition $k ¯ w$ $k ¯ p$ $\frac{a 1}{b}$ $\frac{a 2}{b}$ $\frac{a 3}{b}$ λ ₁ λ ₂ λ ₃ λ ₄

H–H 0 0 0 0 0 0.2802 1.071 2.256 3.714

$\frac{1}{30}$ 0 0 0 0 0.3341 1.086 2.263 3.718

$\frac{1}{30}$ $\frac{π 2}{1200}$ 0 0 0 0.4383 1.223 2.412 3.878

0 0 0 0.5 0 0.2192 1.071 1.942 3.714

$\frac{1}{30}$ 0 0 0.5 0 0.2848 1.086 1.950 3.718

$\frac{1}{30}$ $\frac{π 2}{1200}$ 0 0.5 0 0.4049 1.223 2.129 3.878

0 0 0.4 0.5 0 0.2047 0.9465 1.876 3.577

$\frac{1}{30}$ 0 0.4 0.5 0 0.2738 0.9635 1.884 3.581

$\frac{1}{30}$ $\frac{π 2}{1200}$ 0.4 0.5 0 0.3967 1.121 2.073 3.751

0 0 0.4 0.5 0.4 0.1931 0.8403 1.789 3.504

$\frac{1}{30}$ 0 0.4 0.5 0.4 0.2652 0.8593 1.797 3.508

$\frac{1}{30}$ $\frac{π 2}{1200}$ 0.4 0.5 0.4 0.3898 1.032 2.002 3.679

The beam-like structure is then subjected to a single point load with dimensionless velocity of $v ¯$ = 0.004 which enters the beam from the left end at $t ¯$ = 0 and exits the beam domain at $t ¯ = \frac{1}{v ¯}$ . The traveling load is assumed to induce excitation frequencies of $λ \land = {0, 0.5 λ_{1}^{III}, λ_{1}^{III}}$ where the harmonic function is defined by $P \land (t ¯) = \cos (λ \land t ¯)$ . These excitation frequencies can be taken from Table 4. Here, $λ_{1}^{III}$ denotes the fundamental frequency of the damaged beam with three cracks. Figure 10 shows the time history of the normalized mid-span deflection, $\frac{w ¯ (ζ = 0.5, t ¯)}{F 0 L 2 / (48 EI)}$ , corresponding to the intact and damaged beams. In order to assess the effect of subgrade foundation on the dynamic response, we selected three different cases including a beam with no foundation ( $k ¯ w = 0$ and $k ¯ p = 0$ ), with only Winkler elastic support ( $k ¯ w = \frac{1}{30}$ and $k ¯ p = 0$ ) and with a Pasternak elastic foundation ( $k ¯ w = \frac{1}{30}$ and $k ¯ p = \frac{π 2}{1200}$ ). We clearly observe that the damaged structures feature an increase in the dynamic deflection, compared to the intact beam, and the magnitude of this increase depends on the excitation frequency, foundation parameters, crack sizes and locations. The results show that, regardless of the type of the foundation, the effects of surface damage become more pronounced by increasing the excitation frequency from $λ \land = 0$ to $λ \land = λ_{1}^{III}$ . For instance, the maximum dynamic deflection of the beam sitting on the Pasternak foundation increases 41.7% in the presence of three cracks at $λ \land = 0$ (Figure 10(c.1)), whereas it increases 49.0% at $λ \land = 0.1949$ (Figure 10(c.2)) and a very high value of 592% at $λ \land = 0.3898$ (Figure 10(c.3)). Hence, an insight into the critical influence of cracks on the vibratory behavior of the excited beam can be obtained when the applied frequency $λ \land$ becomes close to the first natural frequency of the damaged beam. The presence of damage has the potential to cause a significant alteration of the structural dynamic properties and thus the vibration amplitude starts to increase monotonically at lower excitation frequencies. By comparing Figure 10(a), (b) and (c), we also notice that the subgrade foundation generally yields more oscillation in the dynamic response of the beam, with or without the presence of surface damage. In particular, more oscillation can be observed once the beam-like component is supported by the Winkler foundation and it becomes even more oscillatory if the role of interaction between Winkler springs is taken into account.
Figure 10.
Normalized mid-span dynamic deflection of the intact and damaged simply supported beam with one (a₁/b = 0, a₂/b = 0.5, a₃/b = 0), two (a₁/b = 0.4, a₂/b = 0.5, a₃/b = 0) and three (a₁/b = 0.4, a₂/b = 0.5, a₃/b = 0.4) edge cracks ( $L / b = 10, P \land (t ¯) = \cos (λ \land t ¯), v ¯ = 0.004, X ¯ 1 = 0.3, X ¯ 2 = 0.5, X ¯ 3 = 0.7, ν = 0.3, κ s = 5 / 6$ ). (a1) $λ \land = 0$ , (a2) $λ \land = λ_{1}^{III} / 2 = 0.1931 / 2$ , (a3) $λ \land = λ_{1}^{III} = 0.1931$ , (b1) $λ \land = 0$ , (b2) $λ \land = λ_{1}^{III} / 2 = 0.2652 / 2$ , (b3) $λ \land = λ_{1}^{III} = 0.2652$ and (c1) $λ \land = 0$ , (c2) $λ \land = λ_{1}^{III} / 2 = 0.3898 / 2$ , (c3) $λ \land = λ_{1}^{III} = 0.3898$ .

To further explore the dynamic behavior of the damaged structure, we have calculated the maximum transverse displacement of the beam’s mid-point as a function of the load velocity. The results are displayed in Figure 11. When comparing the dynamic response of the damaged and healthy beams under moving loads, generally it can be seen that due to the adverse effects of surface damage the highest dynamic displacement becomes larger and this peak value occurs at a lower velocity. However, we observe that the cracked beam-like member supported by the elastic subgrade demonstrates a smaller drop in the structural dynamic strength, independent of the moving load frequency. For example, Figure 11(a.1) shows that the maximum lateral deflection of the beam increases up to 122% when it is weakened by three cracks and there is no elastic foundation (from 1.77 at ${v ¯}_{max}^{0} = 0.053$ to 3.93 at ${v ¯}_{max}^{III} = 0.038$ , where ${v ¯}_{max}^{0}$ and ${v ¯}_{max}^{III}$ denote the velocity corresponding to the peak value of the maximum deflections in intact and triple-cracked beams, respectively). In contrast, if the effects of the Winkler springs are incorporated (Figure 11(b.1)), this increase is 67.1% (from 1.24 at ${v ¯}_{max}^{0} = 0.065$ to 2.08 at ${v ¯}_{max}^{III} = 0.049$ ) and when the Pasternak foundation is supporting the cracked component (Figure 11(c.1)), it is 31.9% (from 0.721 at ${v ¯}_{max}^{0} = 0.086$ to 0.951 at ${v ¯}_{max}^{III} = 0.080$ ). Thus, the elastically supported beam exhibits a higher defect tolerance for selected loading conditions.
Figure 11.
Maximum normalized mid-span deflection of the intact and damaged simply supported beam with one (a₁/b = 0, a₂/b = 0.5, a₃/b = 0), two (a₁/b = 0.4, a₂/b = 0.5, a₃/b = 0) and three (a₁/b = 0.4, a₂/b = 0.5, a₃/b = 0.4) edge cracks versus dimensionless moving load velocity ( $L / b = 10, P \land (t ¯) = \cos (λ \land t ¯), X ¯ 1 = 0.3, X ¯ 2 = 0.5, X ¯ 3 = 0.7, ν = 0.3, κ s = 5 / 6$ ). (a1) $λ \land = 0$ , (a2) $λ \land = λ_{1}^{III} / 2 = 0.1931 / 2$ , (a3) $λ \land = λ_{1}^{III} = 0.1931$ , (b1) $λ \land = 0$ , (b2) $λ \land = λ_{1}^{III} / 2 = 0.2652 / 2$ , (b3) $λ \land = λ_{1}^{III} = 0.2652$ and (c1) $λ \land = 0$ , (c2) $λ \land = λ_{1}^{III} / 2 = 0.3898 / 2$ , (c3) $λ \land = λ_{1}^{III} = 0.3898$ .

Taking a closer look at Figure 11, two other important observations are apparent. First, when the roles of Winkler and Pasternak foundations are considered, the peak values of the dynamic deflections shift to the right and their corresponding speeds ( $v ¯$ _max) increase (for example, compare Figure 11(a.1), (b.1) and (c.1)). Second, as the forced frequency $λ \land$ is increased (up to $λ_{1}^{III}$ ) we find a shift toward the lower values of the load velocity for the peaks (for example, compare Figure 11(c.1) with (c.2)). Importantly, it means that once the moving load frequency approaches $λ_{1}^{III}$ , the response amplitudes of the damaged beams become substantial at low velocities (visible in Figure 11(a.3), (b.3) and (c.3)). This implies that the load velocity plays a crucial role in the response of the defective beam-like structure subjected to a moving load pulsating with a frequency close to $λ_{1}^{III}$ . In other words, if $λ \land$ is close to the fundamental natural frequency of the damaged component, the lower the moving load speed, the larger the deflection amplitude, and hence the higher the possibility of catastrophic failure due to the surface cracks. In Figure 12(a) to (d) the dimensionless deformed shape, rotation, bending moment and shear force of the single cracked beam (a/b = 0.5 and $X ¯$ = 0.5) with no foundation are displayed along the beam length respectively. The rotation is normalized by the static slope of the corresponding intact Euler–Bernoulli beam at ζ = 0 with the point force applied at the mid-span, while the bending moment and shear force are normalized with respect to the maximum static bending moment and shear force at ζ = 0.5. The results are displayed for two moving load speeds $v ¯$ = λ₁/(20π), λ₁/(2π) and at four different time instants t = L/v, 2L/(5v), 3L/(5v), 4L/(5v). The effect of the edge crack can be clearly observed as the jump in rotation, while the jump in the shear force diagram occurs at the location of the applied moving load. It is important to note that increasing the moving load speed not only leads to a change in the maximum value of the bending moment and shear force, but also alters the distribution of them throughout the beam length.
Figure 12.
(a) Dynamic deflection, (b) rotation, (c) bending moment distribution, and (d) shear force distribution of the damaged simply supported beam with one edge crack (a/b = 0.5 and $X ¯$ = 0.5). The results are displayed at four time instants t = L/v, 2L/(5v), 3L/(5v), 4L/(5v) and two different values for the moving load speed ( $L / b = 10, P \land (t ¯) = 1, k ¯ w = 0, k ¯ p = 0, ν = 0.3, κ s = 5 / 6, λ 1 = 0.2192$ ). (a1) $ν ¯ = λ 1 / (20 π) = 0.2192 / (20 π)$ , (a2) $ν ¯ = λ 1 / (2 π) = 0.2192 / (2 π)$ , (b1) $ν ¯ = λ 1 / (20 π) = 0.2192 / (20 π)$ , (b2) $ν ¯ = λ 1 / (2 π) = 0.2192 / (2 π)$ and (c1) $ν ¯ = λ 1 / (20 π) = 0.2192 / (20 π)$ , (c2) $ν ¯ = λ 1 / (2 π) = 0.2192 / (2 π)$ .

It is important to mention that the shift to lower velocities occurs not only for $v ¯ max$ but also for $v ¯ cr$ , owing to the presence of surface damage. As noted earlier, $v ¯ max$ is the velocity corresponding to the peak value of the maximum deflection when the structural member is traversed by a single moving load. But $v ¯ cr$ represents the critical velocity (resonant speed) which leads to the resonance phenomenon when several successive loads are regularly passing the beam span. This mechanism makes the beam deflection increase continuously with time and consequently leads to unbounded oscillation. We can evaluate $v ¯ max$ readily from Figure 11, but determining the critical velocity of the system is more complicated. Adams (1995) has derived a relationship for v_cr associated with the simply supported beam as
$v ¯ cr = \frac{d}{L} \frac{λ k}{2 π m}, m = 1, 2, 3, \dots$
(78)
where λ_k is the kth natural frequency of the system and d is the distance between two consecutive loads. In practical cases, the first resonant speed (m = 1) is more crucial and should be avoided in order to optimize the design and implementation of the system. Thus, assuming d = L, the critical speed associated with the first natural frequency (k = 1) can be computed as $v ¯ cr = \frac{λ 1}{2 π}$ . We employed the present semi-analytical scheme in order to explore the effects of damage on the time response of the system under successive loads.

Figure 13 represents the time response of the simply supported beam sitting on the Pasternak foundation to a series of equidistant loads (d = L). We have considered two different sets of speeds and pulsation frequencies ( $v ¯ = {v ¯}_{cr}^{III}, λ \land = 0$ and $v ¯ = 0.02, λ \land = λ_{1}^{III}$ ). The critical velocity corresponding to the cracked beam can be directly calculated by using the first natural frequency from Table 4 ( ${v ¯}_{cr}^{III} = \frac{λ_{1}^{III}}{2 π} = 0.062$ ). As expected, the resonance phenomenon occurs at a lower speed compared to the intact structure owing to the presence of surface damage. From Figure 13(a) it can be seen that when the dimensionless velocity is equal to ${v ¯}_{cr}^{III}$ , the deflection amplitude of the cracked beam grows unboundedly while the intact structural element exhibits bounded behavior at this speed. Figure 13(b) can also clearly show how repeating the action of moving loads can enhance the edge crack effects and amplify the oscillation amplitude of the defective structure provided that the excitation frequency is close to the system’s natural frequency. In particular, the maximum normalized deflection of the present intact beam is 2.50 which occurs in the first time step (when the first moving load is traversing the beam span from $t ¯ = 0$ to $t ¯ = \frac{1}{v ¯}$ ), while the successive loading process results in increasing the maximum lateral displacement of the cracked beam from 2.79 in the first time step to 10.59 after passage of about five loads. Finally, the harmonic repetitive loads traveling at a velocity different from $v ¯ cr$ lead to bounded response, even though their frequency is equal to the natural frequency of the faulty component.
Figure 13.
Normalized mid-span dynamic deflection of the intact and damaged simply supported beam with three edge cracks under successive loads ( $L / b = 10, d / L = 1, P \land (t ¯) = \cos (λ \land t ¯), k ¯ w = \frac{1}{30}, k ¯ p = \frac{π 2}{1200}, a 1 / b = 0.4, X ¯ 1 = 0.3, a 2 / b = 0.5, X ¯ 2 = 0.5, a 3 / b = 0.4, X ¯ 3 = 0.7, ν = 0.3, κ s = 5 / 6$ ). (a) $ν ¯ = {ν ¯}_{cr}^{III}, λ \land = 0$ and (b) $ν ¯ = 0.02, λ \land = λ_{1}^{III}$ .

4. Conclusions

We have presented a semi-analytical solution for the dynamic response of a multi-cracked beam posed on the Pasternak foundation due to the passing of moving harmonic loads with constant velocity. The effects of shear and rotational inertia have been captured by employing the Timoshenko beam theory. Identical time functions for the transverse and rotational components of the displacement field have been utilized when applying the modal expansion technique to obtain the forced and free transient solutions. Elastic end constraints have also been considered in the model, which allows us to recover all possible boundary conditions. The effects of edge cracks are expressed in terms of additional local flexibilities at the crack locations which are represented by massless rotational springs. It is assumed that these cracks are always open and the additional localized flexibilities due to the damage do not change with time.

The analytical investigation reveals that the loss of structural integrity due to the defects leads to an increase in the dynamic deflection of beam-type components, the level of which is dependent on the load excitation frequency and foundation parameters, as well as crack sizes and positions. The damaging effects of the edge cracks are exacerbated when the moving load excitation frequency is increased (up to the fundamental frequency of the cracked structure). However, we observe that the supporting reaction of the elastic subgrade can influence the drop in the dynamic deflection due to the cracks. Exploring the maximum dynamic response of the healthy and damaged beams at different load speeds also shows that as the load pulsation frequency increases, the vibration amplitude of the cracked component becomes more substantial at lower velocities. This implies that when a damaged beam-like structure is traversed by a load which is pulsating with a frequency close to the beam fundamental natural frequency, very large dynamic deflections along with detrimental effects are likely to occur in the low velocity zone of the traveling load.

For damaged beam-type components under successive traveling loads, the dynamic response of the system to a single moving load and the superposition principle are utilized to derive the general semi-analytical solution. The numerical examples reveal that the resonance phenomenon occurs at lower critical velocities, which is a direct result of open edge cracks. Also, the repeated action of the harmonic moving loads with a frequency close to the beam fundamental natural frequency can intensify the adverse effects of defects even though the moving load velocity is less than the resonant speed.

Footnotes

Acknowledgements

The first author would like to thank Professor A Dyskin and Professor E Pasternak (both UWA) for insightful discussions during the initial stages of this work.

Funding

The first author gratefully acknowledges scholarship support from The University of Western Australia (SIRF and UPIAS) and the Australian Government towards successful completion of the present work.

References

1.
Adams G (1995) Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads. International Journal of Mechanical Sciences 37(7): 773–781.

2.
Andersen L Nielsen S Kirkegaard P (2001) Finite element modeling of infinite Euler beams on Kelvin foundations exposed to moving loads in convected co-ordinates. Journal of Sound and Vibration 241(4): 587–604.

3.
Attar M (2012) A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions. International Journal of Mechanical Sciences 57(1): 19–33.

4.
Attar M Karrech A Regenauer-Lieb K (2014a) Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model. Journal of Sound and Vibration 333(11): 2359–2377.

5.
Attar M Karrech A Regenauer-Lieb K (2014b) A novel technique for dynamic analysis of beam-like structures on tensionless elastic foundations subjected to moving loads. Advanced Materials Research 1016: 192–197.

6.
Bakhtiari-Nejad F Khorram A Rezaeian M (2014) Analytical estimation of natural frequencies and mode shapes of a beam having two cracks. International Journal of Mechanical Sciences 78: 193–202.

7.
Behera RK Pandey A Parhi DR (2014) Numerical and experimental verification of a method for prognosis of inclined edge crack in cantilever beam based on synthesis of mode shapes. Procedia Technology 14: 67–74.

8.
Bilello C Bergman L (2004) Vibration of damaged beams under a moving mass: Theory and experimental validation. Journal of Sound and Vibration 274(3–5): 567–582.

9.
Boiangiu M Ceausu V Untaroiu CD (2014) A transfer matrix method for free vibration analysis of Euler-Bernoulli beams with variable cross section. Journal of Vibration and Control. Epub online ahead of print 22 September 2014. DOI: 10.1177/1077546314550699.

10.
Dadfarnia M Jalili N Esmailzadeh E (2005) A comparative study of the Galerkin approximation utilized in the Timoshenko beam theory. Journal of Sound and Vibration 280(3–5): 1132–1142.

11.
Dugush Y Eisenberger M (2002) Vibrations of non-uniform continuous beams under moving loads. Journal of Sound and Vibration 254(5): 911–926.

12.
Feklistova L Hein H (2013) Crack identification in vibrating beams using Haar wavelets and neural networks. Applied Mechanics and Materials 420: 62–67.

13.
Fryba L (1972) Dynamics of Solids and Structures Under Moving Loads, Groningen, The Netherlands: Noordhoff International Publishing.

14.
Kargarnovin M Younesian D Thompson D (2005) Response of beams on nonlinear viscoelastic foundations to harmonic moving loads. Computers & Structures 83(23–24): 1865–1877.

15.
Kerr AD (1976) On the derivation of well posed boundary value problems in structural mechanics. The International Journal of Solids and Structures 12(1): 1–11.

16.
Khaji N Shafiei M Jalalpour M (2009) Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions. International Journal of Mechanical Sciences 51(9–10): 667–681.

17.
Khorram A Bakhtiari-Nejad F Rezaeian M (2012) Comparison studies between two wavelet based crack detection methods of a beam subjected to a moving load. International Journal of Engineering Science 51: 204–215.

18.
Lin HP Chang SC (2006) Forced responses of cracked cantilever beams subjected to a concentrated moving load. International Journal of Mechanical Sciences 48(12): 1456–1463.

19.
Mahmoud M Abouzaid M (2002) Dynamic response of a beam with a crack subject to a moving mass. Journal of Sound and Vibration 256(4): 591–603.

20.
Mallik A Chandra S Singh AB (2006) Steady-state response of an elastically supported infinite beam to a moving load. Journal of Sound and Vibration 291(3–5): 1148–1169.

21.
Martínez-Castro A Museros P Castillo-Linares A (2006) Semi-analytic solution in the time domain for non-uniform multi-span Bernoulli-Euler beams traversed by moving loads. Journal of Sound and Vibration 294(1–2): 278–297.

22.
Matbuly M Ragb O Nassar M (2009) Natural frequencies of a functionally graded cracked beam using the differential quadrature method. Applied Mathematics and Computation 215(6): 2307–2316.

23.
Matsunaga H (1999) Vibration and buckling of deep beam-columns on two-parameter elastic foundations. Journal of Sound and Vibration 228(2): 359–376.

24.
Museros P Moliner E Martínez-Rodrigo M (2013) Free vibrations of simply-supported beam bridges under moving loads: Maximum resonance, cancellation and resonant vertical acceleration. Journal of Sound and Vibration 332(2): 326–345.

25.
Nguyen VH Duhamel D (2008) Finite element procedures for nonlinear structures in moving coordinates. Part II. Infinite beam under moving harmonic loads. Computers & Structures 86(21–22): 2056–2063.

26.
Ostachowicz W Krawczuk M (1991) Analysis of the effect of cracks on the natural frequencies of a cantilever beam. Journal of Sound and Vibration 150(2): 191–201.

27.
Papadopoulos C Dimarogonas A (1987) Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. Journal of Sound and Vibration 117(1): 81–93.

28.
Reis M Pala Y (2012) Vibration of a cracked cantilever beam under moving mass load. Journal of Civil Engineering and Management 18(1): 106–113.

29.
Sari M Butcher EA (2010) Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials. International Journal of Engineering Science 48(10): 862–873.

30.
Takahashi I (1999) Vibration and stability of non-uniform cracked Timoshenko beam subjected to follower force. Computers & Structures 71(5): 585–591.

31.
Timoshenko S (1922) On the forced vibrations of bridges. Philosophical Magazine Series 6 43(257): 1018–1019.

32.
Wang RT (1997) Vibration of multi-span Timoshenko beams to a moving force. Journal of Sound and Vibration 207(5): 731–742.

33.
Yang J Chen Y Xiang Y (2008) Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. Journal of Sound and Vibration 312(1–2): 166–181.

34.
Yan T Kitipornchai S Yang J (2011) Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load. Composite Structures 93(11): 2992–3001.

35.
Yokoyama T (1991) Vibrations of Timoshenko beam-columns on two-parameter elastic foundations. Earthquake Engineering & Structural Dynamics 20(4): 355–370.

36.
Zhaohua F Cook RD (1983) Beam elements on two-parameter elastic foundations. Journal of Engineering Mechanics 109(6): 1390–1402.

37.
Zheng D Fan S (2003) Vibration and stability of cracked hollow-sectional beams. Journal of Sound and Vibration 267(4): 933–954.

38.
Zhu X Law S (2006) Wavelet-based crack identification of bridge beam from operational deflection time history. The International Journal of Solids and Structures 43(7–8): 2299–2317.

			Crack sizes			Dimensionless natural frequencies
Boundary condition	$k ¯ w$	$k ¯ p$	$\frac{a 1}{b}$	$\frac{a 2}{b}$	$\frac{a 3}{b}$	λ ₁	λ ₂	λ ₃	λ ₄
H–H	0	0	0	0	0	0.2802	1.071	2.256	3.714
	$\frac{1}{30}$	0	0	0	0	0.3341	1.086	2.263	3.718
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0	0	0	0.4383	1.223	2.412	3.878
	0	0	0	0.5	0	0.2192	1.071	1.942	3.714
	$\frac{1}{30}$	0	0	0.5	0	0.2848	1.086	1.950	3.718
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0	0.5	0	0.4049	1.223	2.129	3.878
	0	0	0.4	0.5	0	0.2047	0.9465	1.876	3.577
	$\frac{1}{30}$	0	0.4	0.5	0	0.2738	0.9635	1.884	3.581
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0.4	0.5	0	0.3967	1.121	2.073	3.751
	0	0	0.4	0.5	0.4	0.1931	0.8403	1.789	3.504
	$\frac{1}{30}$	0	0.4	0.5	0.4	0.2652	0.8593	1.797	3.508
	$\frac{1}{30}$	$\frac{π 2}{1200}$	0.4	0.5	0.4	0.3898	1.032	2.002	3.679