Dynamic analysis of rectangular plates crossed by distributed moving loads

Abstract

This study investigates the dynamic behaviour of plates crossed by distributed moving gravitational and inertial loads, in the case in which the relative magnitude of the moving mass introduces a coupling effect with the structure, with possible applications to the vibration analysis of railway bridges. A rectangular Kirchhoff plate is considered, simply supported on two opposite edges and free on the other two edges, loaded by a partially distributed mass travelling in the parallel direction with respect to the free edges. The formulation includes damping, and it is accomplished by the Rayleigh–Ritz method, expressing the solution in semi-analytical form. The shape functions for describing the transverse displacement field of the plate are selected as tensor products of linearly independent eigenfunctions of homogeneous uniform beams in flexural vibration, yielding a low-order model with time-dependent coefficients. Numerical examples are then presented and discussed, aimed at investigating the effects of each of the model governing parameters.

Keywords

Rectangular plate distributed moving inertial load Rayleigh–Ritz method

1. Nomenclature

A	Amplification factor	U	Potential of strain energy
a	Acceleration of the load	V	Potential of applied loads
C	Viscous damping matrix	v	Velocity of the load
D	Stiffness of the plate	w	Displacement
E	Young’s modulus	x	Spatial coordinate
f	Shape of the moving strip	y	Spatial coordinate
g	Gravity acceleration	α	Velocity ratio
H	Unit step distribution	β	Frequency parameter
h	Thickness of the plate	δ	Distance of the moving strip from one edge of the plate
K	Stiffness matrix	ε	Relative error
L	Length	Φ	Modal matrix
l	Width	ϕ	Beam eigenfunction
M	Mass matrix	µ	Length ratio
N	Modal order	ν	Poisson’s ratio
n	Modal index	Π	Total potential
p	Load per unit area	ρ	Mass per unit area
q	Modal coordinate	ω_n	Modal natural frequency
r	Mass ratio	ξ	Moving coordinate
t	Time	ζ	Damping factor

2. Introduction

In structural dynamics, the analysis of elements such as beams or plates loaded by moving masses is an important issue, especially when the magnitude of the moving mass introduces a coupling effect with the structure [1]. The inertia of the load changes the dynamic properties of the base structure and the response differs significantly from that of the system only subjected to the gravitational component of the load, as for instance in the case of railway bridges [2,3].

The most commonly studied models consist of continuous Euler–Bernoulli beams, or Timoshenko beams, traversed by either concentrated or distributed moving loads. The problem of an Euler–Bernoulli beam carrying a moving concentrated mass was solved by Lin [4] using the finite element method, while in Stancioiu et al. [5] a four-span beam was studied by imposing continuity conditions at the supports for the single-span eigenfunctions. A partially distributed moving load (acting instantaneously on part of the spatial domain of the beam) was considered by Adetunde [6], with the solution obtained by adopting the assumed modes method. The effects of vibration absorbers were analysed by Lin and Cho [7], but neglecting the inertial action of the moving concentrated masses. The problem of a concentrated mass moving along a Timoshenko beam was studied by Dyniewicz and Bajer [8], with the solution obtained numerically using the space-time finite element method, an approach in which the interpolation of nodal displacements is performed by using shape functions of both space and time variables [9].

The dynamic behaviour of plates excited by concentrated moving loads has also been extensively investigated. In Nikkhoo et al. [10], the authors solved in a semi-analytical form the problem of a Kirchhoff plate vibrating under two series of moving concentrated inertial loads traversing the plate surface along parallel rectilinear trajectories with opposite directions. A Kirchhoff plate on multiple supports was studied by Marchesiello et al. [11], loaded by travelling vehicles (modelled as concentrated loads due to sprung masses), adopting the Rayleigh–Ritz method coupled with an iterative dynamic substructuring method. Concentrated moving masses on a Rayleigh beam and a non-Mindlin plate (taking into account rotary inertia, but not shear deformation) were considered by Gbadeyan and Oni [12], providing a solution in series form via generalized finite integral transform and Struble’s method. In De Faria and Oguamanam [13], a numerical solution was found for a Mindlin plate crossed by concentrated masses, using a finite element method with adaptive meshes at low speed. In Dyniewicz et al. [14], a Mindlin plate subjected to a concentrated inertial load travelling at a variable speed along an arbitrary trajectory was considered; the problem of two concentrated inertial loads travelling in opposite directions along the same trajectory was also investigated, obtaining a numerical solution using the above-mentioned space-time finite element method.

Few contributions are focused on the particular case of plates crossed by distributed moving masses. In Gbadeyan and Dada [15], the dynamic response of a Mindlin rectangular plate under a partially distributed moving load was investigated using the finite difference method, transforming the differential equations into a set of linear algebraic equations. In other contributions, the problem was tackled by means of applications of the finite element method, as for instance in Wu [16], introducing a moving distributed mass element along with the definition of appropriate shape functions. Finally, in Amiri et al. [17], a Mindlin plate subjected to a distributed load was studied with the method of separation of variables, expressing the solution for different mass trajectories in a semi-analytical form on the basis of the unloaded Mindlin plate eigenfunctions.

In the present study a rectangular Kirchhoff plate is considered, simply supported on two opposite edges and free on the other two edges, loaded by a partially distributed mass (acting instantaneously on part of the spatial domain of the plate) travelling in parallel direction with respect to the free edges. With this configuration, the model is suitable for application to simplified analyses of bridges crossed by vehicles, in particular double-track railway bridges. The moving loads are assumed to travel with constant speed, and to be always in contact with the loaded structure.

The formulation is accomplished by the Rayleigh–Ritz method [18], expressing the solution in semi-analytical form in terms of a linear combination of functions. Instead of considering unloaded plate eigenfunctions, as in Amiri et al. [17], in the present study the functions for describing the transverse displacement field of the plate are selected as tensor products of linearly independent eigenfunctions of homogeneous uniform beams in flexural vibration [19,20]. The proposed approach, novel with respect to the application to moving distributed inertial loads, has the advantage of a simpler formulation (with respect to the above-mentioned references) in conjunction with the possibility of easily including in the analysis important features, such as non-homogeneous plate properties (variable thickness, density and stiffness) and non-homogeneous boundary conditions, as well as different kinds of damping distributions (for modelling the structural damping of the plate, which in this case is not a trivial issue). It is also suitable for extension to consider non-rectangular plates, since it can be directly integrated in a coordinate mapping procedure, as described by Catania and Sorrentino [20].

The proposed method yields a reduced-order model with time-dependent coefficients, allowing a parametric analysis of the problem. Different example cases are presented and discussed in detail, analysing the effects of velocity, mass and length of the distributed load on the plate’s dynamic response with respect to the mass, stiffness and damping of the plate itself.

3. Description of the model

A homogeneous isotropic Kirchhoff rectangular plate is considered [21], simply supported on two opposite edges, free on the other two edges and loaded by a distributed mass travelling in a direction parallel to the free edges. The moving load and the loaded structure are assumed to be always in contact. The load per unit area p over the plate can be expressed as

p (x, y, t) = - ρ_{t} f (ξ, y) [g + \overset{\cdot\cdot}{w} (x, y, t)], ξ = x - v t

(1)

where dots denote total derivatives with respect to time, w is the out-of-plane displacement of a point of the plate or of the load, ρ_t is the equivalent mass per unit area of the load, g is the gravity acceleration, v = v_(x) is the travelling speed in the x direction, ξ is a moving coordinate in the same direction and f models the translating strip representing the instantaneous position of the load

f (ξ, y) = [H (ξ + L_{t}) - H (ξ)] [H (y - δ) - H (y - δ - l_{t})]

(2)

Note that in the present studyρ_t is assumed to be constant; however, piecewise-constant or other distributions ρ_t (ξ ) may be considered and adopted within the proposed method.

Equation (2) contains the Heaviside unit step distribution H(·), L_t and l_t are the length and the width of the strip modelling the train, respectively, and δ is the distance between the side of the strip and the edge y = 0 of the plate, as shown in Figure 1. The second term on the right-hand side of Equation (1) describes the inertial action of the load. The total acceleration can be expressed in the following general form

\overset{\cdot\cdot}{w} = w_{tt} + v_{(x)}^{2} w_{xx} + 2 v_{(x)} v_{(y)} w_{xy} + v_{(y)}^{2} w_{yy} + 2 v_{(x)} w_{xt} + 2 v_{(y)} w_{yt} + a_{(x)} w_{x} + a_{(y)} w_{y}

(3)

where subscripts to w denote its partial derivatives with respect to t, x, y, while v_(x), v_(y), a_(x), a_(y) express the velocities and accelerations of the travelling load in the x and y directions respectively. Considering a moving load travelling at constant speed v in the x direction, Equation (3) reduces to

\overset{\cdot\cdot}{w} = w_{tt} + 2 v w_{xt} + v^{2} w_{xx}

(4)

The first term of the right-hand side of Equation (4) expresses the influence of vertical acceleration of the moving load, the second term the influence of Coriolis acceleration and the third term the influence of centripetal acceleration, due to the influence of the curvature of the plate [1].

The functional of the total potential energy of the coupled system can be written as the sum of a term U due to the strain energy plus a term V representing the potential of all applied loads (including the inertial forces)

Π = U + V

(5)

The potential of the strain energy can be written in terms of second-order derivatives of the out-of-plane displacement w

U = \frac{1}{2} \int_{0}^{l_{b}} \int_{0}^{L_{b}} D [w_{xx}^{2} + w_{yy}^{2} + 2 ν w_{xx} w_{yy} + 2 (1 - ν) w_{xy}^{2}] dxdy, D = \frac{E h^{3}}{12 (1 - ν^{2})}

(6)

where D is the flexural stiffness of the plate, expressed as a function of Young’s modulus E, Poisson’s ratio ν and thickness h [21]. In the adopted formulation, the inertial forces are included in the potential of applied loads V as follows

V = \int_{0}^{l_{b}} \int_{0}^{L_{b}} (ρ_{b} w w_{tt} - wp) dxdy

(7)

where ρ_b is the mass per unit area of the plate and p is the load given by Equation (1).

The equations of motion of the undamped system can be obtained after imposing the stationarity of the functional Π in Equation (5). After that, they can be modified to include the effects of a damping distribution, as described in the next section.

4. Solution method

The problem is solved with weak formulation adopting the Rayleigh–Ritz method, expressing the out-of-plane displacement w by means of a linear combination of shape functions. Each of these functions must respect only the essential conditions at the boundary of the plate, also known as principal or kinematic conditions [18]. Therefore, in this case the boundary conditions are implicitly set with an appropriate choice of shape functions. However, it would also be possible considering further external elastic constraints on w by introducing additional terms ΔV in the expression of the functional Equation (5), as for instance explained by Catania and Sorrentino [20].

In the present study, the shape functions are selected as tensor products of homogeneous uniform Euler–Bernoulli beam eigenfunctions ϕ, each of them given in general by a linear combination of four complex exponential functions of a space variable (x or y) [18]. Consequently, the displacement w can be expressed in the form

w (x, y, t) = \sum_{i = 1}^{N_{x}} \sum_{j = 1}^{N_{y}} q_{ij} (t) ϕ_{i} (x) ϕ_{j} (y)

(8)

where N_x and N_y are the number of beam eigenfunctions ϕ_i(x) and ϕ_j(y) in the x and y directions, respectively. If an integer n is assigned to any combination i, j, then

w (x, y, t) = \sum_{n = 1}^{N} q_{n} (t) ϕ_{n} (x, y) \Rightarrow w = ϕ^{T} q

(9)

where ϕ_n(x,y) is the nth eigenfunction product, N = N_x×N_y, ϕ is a vector comprising the N eigenfunction products and q is a vector comprising the N generalized coordinates q_n(t).

Introducing the displacement expansion in the quadratic functional Π Equation (5) and imposing its stationarity yields the following system of linear ordinary differential equations

[M + r Δ M] \overset{\cdot\cdot}{q} + 2 r α β [Δ C] \overset{\cdot}{q} + β^{2} [K + r α^{2} Δ K] q = - rg f

(10)

In Equation (10) dots denote differentiation with respect to time, with

r = \frac{ρ_{t}}{ρ_{b}}, α = \frac{v}{L_{b} β}, β = \frac{1}{L_{b}^{2}} \sqrt{\frac{D}{ρ_{b}}}

(11)

where β is a frequency parameter and α is a dimensionless parameter depending on the speed v. The matrices in square brackets in Equation (10) can be regarded as dimensionless quantities, and they can be computed according to the following integrals

\begin{matrix} M = \int_{0}^{l_{b}} \int_{0}^{L_{b}} (ϕ ϕ^{T}) dxdy, Δ M = \int_{δ}^{δ + l_{t}} \int_{x_{0}}^{x_{1}} (ϕ ϕ^{T}) dxdy, Δ C = L_{b} \int_{δ}^{δ + l_{t}} \int_{x_{0}}^{x_{1}} (ϕ ϕ_{x}^{T}) dxdy \\ K = L_{b}^{4} \int_{0}^{l_{b}} \int_{0}^{L_{b}} [ϕ_{xx} ϕ_{xx}^{T} + ϕ_{yy} ϕ_{yy}^{T} + ν (ϕ_{xx} ϕ_{yy}^{T} + ϕ_{yy} ϕ_{xx}^{T}) + 2 (1 - ν) (ϕ_{xy} ϕ_{xy}^{T})] dxdy \\ Δ K = 2 L_{b}^{2} \int_{δ}^{δ + l_{t}} \int_{x_{0}}^{x_{1}} (ϕ ϕ_{xx}^{T}) dxdy, f = \int_{δ}^{δ + l_{t}} \int_{x_{0}}^{x_{1}} ϕ dxdy \end{matrix}

(12)

In Equations (12) the integration interval [x₀, x₁] is time-dependent, where x₀ and x₁ are respectively the smallest and the largest values of x in which the moving load (translating strip) is instantaneously acting. Introducing the ratio between L_t and L_b

μ = \frac{L_{t}}{L_{b}}

(13)

then x₀ and x₁ vary according to the laws given in Table 1. When µ < 1, there are three different loading cases: firstly, the action area of the moving load on the plate is progressively growing (entering phase), in which case x₀ = 0 and x₁ = vt; secondly, the action area lays entirely on the plate, then x₀ = vt – L_t and x₁ = vt; thirdly, the action area is progressively reducing (leaving phase), hence x₀ = vt – L_t and x₁ = L_b. When, on the contrary, µ > 1, the first (entering) and third (leaving) loading cases present the same integration intervals as before (see Table 1); the second loading case is different, since now the action area of the moving load spans the whole length L_b of the plate. Finally, when µ = 1, the second loading case does not occur, with direct transition from the first loading case to the third one.

Table 1.

Time-dependent interval of integration.

Case $μ < 1$	Case $μ = 1$	Case $μ > 1$
$0 \leq v t < L_{t} \Rightarrow {\begin{matrix} x_{1} = v t \\ x_{0} = 0 \end{matrix}$ $L_{t} \leq v t < L_{b} \Rightarrow {\begin{matrix} x_{1} = v t \\ x_{0} = v t - L_{t} \end{matrix}$ $L_{b} \leq v t < L_{b} + L_{t} \Rightarrow {\begin{matrix} x_{1} = L_{b} \\ x_{0} = v t - L_{t} \end{matrix}$	$0 \leq v t < L_{b} \Rightarrow {\begin{matrix} x_{1} = v t \\ x_{0} = 0 \end{matrix}$ $L_{b} \leq v t < 2 L_{b} \Rightarrow {\begin{matrix} x_{1} = L_{b} \\ x_{0} = v t - L_{b} \end{matrix}$	$0 \leq v t < L_{b} \Rightarrow {\begin{matrix} x_{1} = v t \\ x_{0} = 0 \end{matrix}$ $L_{b} \leq v t < L_{t} \Rightarrow {\begin{matrix} x_{1} = L_{b} \\ x_{0} = 0 \end{matrix}$ $L_{t} \leq v t < L_{t} + L_{b} \Rightarrow {\begin{matrix} x_{1} = L_{b} \\ x_{0} = v t - L_{t} \end{matrix}$

Since the excitation is not harmonic, in this case the structural or hysteretic damping model cannot be adopted for modelling energy dissipation within the structure [18]. On the other hand, the so-called internal viscous damping model, in fact a viscous damping model proportional to the stiffness [18], applied to a homogeneous plate would lead to modal damping factors ζ_n growing linearly with natural frequencies ω_n, which is not realistic.

A simple method for attempting to overcome this difficulty consists of an application to a distributed parameter model (plate) of the equivalent viscous damping (or real damping) method [18]. For this purpose, a dimensionless damping matrix C is defined by means of the modal matrix Φ (mass normalized), the eigenvalues ω_n of the plate (computed from M and K) and forcing a modal damping ratio ζ equal for all modes

2 ζ C = 2 ζ Φ^{- T} diag (\frac{ω_{n}}{β}) Φ^{- 1} = Φ^{- T} diag (2 ζ \frac{ω_{n}}{β}) Φ^{- 1}

(14)

which, introduced in Equation (10), yields the equations of motion of the damped system in the form

[M + r Δ M] \overset{\cdot\cdot}{q} + 2 β [ζ C + r α Δ C] \overset{\cdot}{q} + β^{2} [K + r α^{2} Δ K] q = - rg f

(15)

where the time-dependent matrices ΔM, ΔC and ΔK are due to the vertical inertial force, the Coriolis force and the centrifugal force, respectively. The presence of an inertial Coriolis term ΔC and of an inertial circulatory term ΔK make the system non-natural [18].

Equation (15) is a reduced-order discretized model with time-dependent coefficients, which can be solved numerically.

5. Numerical results

Numerical examples are presented for studying the dynamic behaviour of the model described in Section 3. The plate is assumed to be simply supported on two opposite edges (parallel to y) and free on the other two (parallel to x), modelling a single span bridge. Hence, the boundary conditions for the beam eigenfunctions defining the shape functions of the plate can be set in the form

\begin{matrix} {ϕ |}_{x = 0} = 0, {ϕ_{xx} |}_{x = 0} = 0, {ϕ |}_{x = L_{b}} = 0, {ϕ_{xx} |}_{x = L_{b}} = 0 (simply supported at both ends) \\ {ϕ_{yy} |}_{y = 0} = 0, {ϕ_{yyy} |}_{y = 0} = 0, {ϕ_{yy} |}_{y = l_{b}} = 0, {ϕ_{yyy} |}_{y = l_{b}} = 0 (free at both ends) \end{matrix}

(16)

where the subscripts x and y of ϕ denote partial derivatives with respect to x and y, yielding

\begin{matrix} ϕ_{i} (x) = \sin (i π L_{b}^{- 1} x), i \geq 1 \\ ϕ_{1} (y) = a_{1}, ϕ_{2} (y) = a_{2} y, \\ ϕ_{j} (y) = a_{j} [\sin (ψ_{j} y) + \sinh (ψ_{j} y)] - b_{j} [\cos (ψ_{j} y) + \cosh (ψ_{j} y)], j \geq 3 \\ a_{j} = \cos (ψ_{j} l_{b}) - \cosh (ψ_{j} l_{b}), b_{j} = \sin (ψ_{j} l_{b}) - \sinh (ψ_{j} l_{b}), j \geq 3 \end{matrix}

(17)

where ψ_j are solutions of the characteristic equation [18]

\cos (ψ l_{b}) + \cosh (ψ l_{b}) = 1

(18)

In all the following numerical simulations, homogeneous initial conditions are assumed with the plate at rest when the moving load starts acting on it.

The influence of parameters v, r, µ, β, ζ governing Equation (15) is highlighted by studying time responses w(x, y, t) and velocity response functions A of the dimensionless frequency α (amplification factor as a function of the velocity, as adopted by Dyniewicz and Bajer [8]), defined according to

A (x, y; α) = {\frac{{max}_{t} [w (x, y, t)]}{w_{s}} |}_{x, y; α}

(19)

where w_s is the static deflection due to the load centred in L_b / 2 with v = 0.

The solution w, in the form of Equation (8), is expanded using 4 × 3 beam eigenfunctions: four simply supported eigenfunctions along the x direction and three free–free eigenfunctions along the y direction, according to the expressions given in Equations (17). The equations of motion Equation (15) are then rewritten in the state-space getting a linear time-variant system of first-order differential equations, solved numerically using the standard fourth-order Runge–Kutta algorithm (RK4) [22].

As a reference case study, focusing on possible applications to the dynamics of railway bridges, the following values for the parameters are assumed.

Plate: L_b = 50 m, l_b = 10 m, β = $\sqrt{5}$ rad/s, ζ = 0.05.

Moving load: µ = 1.4, L_t = 70 m, l_t = 2.5 m, δ = 1.5 m, r = 0.5.

Realistic values of parameter β for different kinds of bridges, based on large collections of experimental data [2], are reported as functions of L_b in Figure 2: the reference case was selected accordingly.

The effects of the speed v of the moving load are first studied considering the time response function w(x, y, t). The two contour plots reported in Figure 3, computed in the reference case with v = 40 m/s, represent the deflection w in the domains (x, t; y = l_b / 2) and (y, t; x = L_b / 2), respectively, showing that the maximum deflection area is around half the length of the plate. The plots w(t) displayed in Figure 4, highlighting the effects of v, are therefore computed at coordinates x = L_b / 2 and y = l_b / 2. The speed v varies from 20 to 100 m/s (72 to 360 km/h), and the other parameters are as in the reference case: a moderate increase with v of both maximum deflection and amplitude of free oscillation is observed.

The relative mass parameter r can produce important shifts in the magnitude of A(α), but not in its shape, as shown in Figure 5 (parameters other than r as in the reference case, with v = 40 m/s). On the contrary, the relative length parameter µ strongly affects both the magnitude and shape of A(α), producing large shifts in the position of the peaks, as shown in Figure 6 (parameters other than µ as in the reference case, with v = 40 m/s). However, this can be observed only in the case 0 < µ≤ 1, since A(α) is insensitive to any value of µ > 1. The damping parameter ζ has the obvious effect of progressively smoothing the oscillations of A(α), the reduction in amplitude becoming particularly significant at high speed, as shown in Figure 7 (parameters other than ζ as in the reference case, with v = 40 m/s). It can be noticed that, adopting the equivalent viscous damping model as described in Equation (14), A(α) becomes, as expected, monotonic with ζ = 1. The frequency parameter β, at least within the considered range of values, scarcely affects the behaviour of A(α), so it may be considered independent from β.

The effects of piecewise distributed loads are highlighted by comparison of a continuous strip (as represented in Figure 1) with piecewise distributions consisting of two shorter sections. The assumed piecewise distributions are given by

0 \leq ξ \leq \frac{L_{t}}{χ} and (L_{t} - \frac{L_{t}}{χ}) \leq ξ \leq L_{t}

(20)

with χ < 0.5 (χ = 0.5 yields the continuous strip). Since for the continuously distributed load it is assumed r₀ = 0.5, for the piecewise distributed load described by Equation (20) r₀ increases to r = 1/(2χ) ×r₀. Response functions w(t) are computed in x = L_b / 2, y = l_b / 2 for different values of χ (1/6, 1/48, 1/480), as reported in Figure 8 (L_t = 24 m; other parameters as in the reference case, with v = 40 m/s). The dynamic response of the structure is strongly influenced by different distributions of the moving load.

The effects of neglecting the time-dependent matrices ΔM, ΔC, ΔK on the solution w(t) are studied by introducing a relative error, according to

ε_{M} = \frac{w (t) - {[w (t)]}_{Δ M = 0}}{{max}_{t} | w (t) |}

(21)

where [w(t)]_{ΔM = 0} refers to the solution computed assuming ΔM = 0 in Equation (15). Similarly, ε_C and ε_K can be defined, considering ΔC = 0 and ΔK = 0. The error functions ε_M, ε_C and ε_K are plotted versus time in Figure 9, where ε_Tot represents the total error, assuming that ΔM, ΔC and ΔK are equal to 0 at the same time (parameters as in the reference case, with v = 40 m/s). The smallest and negligible contribution to the error ε_Tot appears to be ε_C (<1 %), while the main contribution (as expected) is due to ε_M. The effect on A(x, y;α) of neglecting the time-dependent matrices ΔM, ΔC, ΔK is similar: the effect of neglecting ΔC is very small, while the total error can be significant. It can be concluded that the total contribution of ΔM, ΔC and ΔK to the dynamic response of the structure is not negligible, ΔM being the most important component, while ΔC may be disregarded.

6. Conclusions

The dynamical behaviour of a rectangular Kirchhoff plate crossed by partially distributed travelling loads was investigated by means of the application of the Rayleigh–Ritz method, expressing the solution in semi-analytical form. The shape functions for the transverse displacement field of the plate were selected as tensor products of linearly independent eigenfunctions of homogeneous uniform beams in flexural vibration.

The proposed approach, novel with respect to the application to moving distributed inertial loads, has the advantage of a simple formulation in conjunction with the possibility of including important features, such as non-homogeneous plate properties and boundary conditions, as well as different kinds of damping distributions. A reduced-order discretized model with time-dependent coefficients was obtained, in a compact form suitable for fast numerical computations.

The effects of each of the model governing parameters were studied comparing time histories as well as amplification factors, functions of the travelling speed of the load. Such velocity response functions can be an effective tool for studying the dynamic behaviour of structures crossed by moving loads, since the travelling speed is the most important parameter influencing the structural stresses, which in general increase with increasing speed. In particular, it was shown how different (continuous versus piecewise) distributions of the moving distributed mass can strongly influence the dynamic response of the structure. The specific contributions of the inertial force, the Coriolis force and the centrifugal force due to coupling between the structure and travelling mass were also put in evidence, showing that their total contribution to the dynamic response of the structure is not negligible. The inertial force is the most important component, while the Coriolis force component may be disregarded.

Applications of the proposed method may regard simplified analyses of bridges crossed by vehicles, in particular double-track railway bridges.

Footnotes

Funding

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

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