Flexural motions under moving concentrated masses of elastically supported rectangular plates resting on variable winkler elastic foundation

Flexural motions under moving concentrated masses of elastically supported rectangular plates resting on variable winkler elastic foundation

T. O. Awodola^*

Department of Mathematical Sciences Federal University of Technology, Akure, Nigeria

ABSTRACT

The flexural motions of elastically supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work In order to solve the fourth order partial differential equation governing the problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. These equations are then solved using a modification of the Struble's technique and method of integral transformations. Numerical results are then presented in plotted curves. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor R_o increases and for fixed value of R_o, the displacements of the elastically supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus F_o increases. Also, for fixed R_o and F_o, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Therefore, the moving force solution is not a safe approximation to the moving mass problem. Hence, safety is not guaranteed for a design based on the moving force solution. Furthermore, the results show that the critical speed for the moving mass problem is reached prior to that of the moving force for the elastically supported rectangular plates on Winkler elastic foundation with stiffness variation.

Keywords: Winkler Foundation, Foundation Modulus, Rotatory Inertia, Resonance, Critical Speed, Moving Force, Moving Mass.

1 INTRODUCTION

The analyses of elastic structures, such as beams and plates, acted upon by moving loads and resting on a foundation constitute an important part of Engineering and applied Mathematics literatures. In general, such analyses are mathematically complex due to the difficulty in modeling the mechanical response of the subgrade which is governed by many factors.

When the vehicle-track interaction is completely neglected, we have the so called 'moving for-ce' problem which has been shown by several researchers that it is a crude approximation to the 'moving mass' problem where the vehicle-track interaction is considered, Muscolino and Palmeri (2007). Several researchers have considered the vehicle-track interaction in their analyses. These researchers include Stanisic et al (1974), Milornir et al (1969), Clastornic et al (1986), Sadiku and Leipholz (1981) and Gbadeyan and Oni (1995). Douglas et al (2002) solved the problem of plate strip of varying thickness and the center of shear. In their work, they considered a free-vibrating strip with classical boundary conditions, precisely, they assumed the plate strip clamped at one end and free at the other end. Pesterev et al (2001) came up with a series expansion method for calculating bending moment and shear force in the problem of vibration of a damped beam sub-ject to an arbitrary number of moving loads. This kind of solution, though could be accurate, cannot account for vital information such as the phenomenon of resonance in the dynamical sys-tem.

Recently, several other researchers have made tremendous efforts in the study of dynamics of structures under moving loads, these include Oni (2004), Oni and Omolofe (2005), Oni and Awodola (2003), Omer and Aitung (2006), Adams (1995), Savin (2001), Jia-Jang (2006). In all of these, considerations have been limited to cases of one-dimensional (beam) problems. Where two-dimensional (plate) problems have been considered, the foundation moduli are taken to be con-stants. No considerations have been given to the class of dynamical problems in which the foun-dation is the type with stiffness variation. In an attempt to solve such two-dimensional problem, all the methods used in the above works break down due to the variation of the foundation mod-el.

Generally, the dynamical problems of structures under moving load and resting on a founda-tion is complex, the complexity increases if the foundation stiffness varies along the structure. Aside the problem of singularity brought in by the inclusion of the inertia effects of the moving load, the coefficients of the governing fourth order partial differential equation are no longer con-stant but variable. Earlier researchers into beam member on variable elastic foundation include Franklin and Scott (1979) who presented a closed-form solution to a linear variation of the foun-dation modulus using contour-integrals. In a recent development, Oni and Awodola (2005) inves-tigated the dynamic response to moving concentrated masses of uniform Rayleigh beams resting on variable Winkler elastic foundation.

However, in all these, the problem of determining the dynamic response of structures under the action of moving concentrated masses has been almost exclusively reserved for elastic struc-tures having the normal ideal boundary conditions. Such ideal boundary conditions include among others, Clamped edge, Free edge, Simply supported edge and Sliding edge boundary condi-tions. For practical applications in many cases, it is more realistic to consider non-classical boundary conditions because the ideal boundary conditions can seldom be realized. A common

example is the elastically supported end conditions. As a problem of this kind, Wilson (1974) studied the response of a cantilever plate strip restrained elastically against rotation and subject-ed to a moving normal line load.

More recently, Oni and Awodola (2010) considered the dynamic response under a moving load of an elastically supported non-prismatic Bernoulli-Euler beam on variable elastic foundation. The technique was based on the generalized Galerkin's method and integral transformations.

In all these previous investigations, extension of the theory to cover two-dimensional (plate) problem has not been effected, when the plate is on variable foundation. Therefore, this study concerns the response to moving concentrated masses of elastically supported rectangular plate resting on Winkler elastic foundation with stiffness variation.

2 GOVERNING EQUATION

Consider a rectangular plate carrying an arbitrary number (say N) of concentrated masses M_i moving with constant velocities c_i, i = 1, 2, 3, ... , N along a straight line parallel to the x - axis ( no difficulty arises by assuming that masses travel in an arbitrary path ) issuing from point y = s on the y - axis. The equation governing the dynamic transverse displacement W(x,y,t) of an elastically supported rectangular plate when it is resting on a variable Winkler foundation and traversed by several moving concentrated masses is the fourth order partial differential equation given by; Oni and Awodola (2011),

where

is the bending rigidity of the plate, ∇^{2 is the two-dimensional Laplacian operator, h is the plate's thickness, E is the Young's Modulus, v is the Poisson's ratio (v < 1) , μ is the mass per unit} area of the plate, R₀ is the Rotatory inertia correction factor, F₀ is the foundation's stiffness, g is the acceleration due to gravity, x and y are respectively the spatial coordinates in x and y direc-tions and t is the time coordinate. δ(.) is the Dirac - Delta function.

The initial conditions, without any loss of generality, is taken as

In this paper, in the first instance, we consider rectangular plate resting on a variable Winkler elastic foundation elastically supported at edges y = 0, y = L_Y with simple support at edges x = 0, x = L_X, the boundary conditions can be written as; Oni and Awodola (2010)

and for normal modes

where k₁ is the stiffness against rotation and k₂ is the stiffness against translation.

Secondly, we consider an elastic rectangular plate resting on a variable Winkler elastic founda-tion and having elastic supports at all its edges, the boundary conditions are given by; Oni and Awodola (2010)

and for normal modes

where k₁ and k₂ are the stiffness against rotation and the stiffness against translation respectively.

3 ANALYTICAL APPROXIM ATE SOLUTION

The method of analysis involves expressing the Dirac - Delta function as a Fourier cosine series. Because of the variable foundation term, the elegant method of the generalized integral transform breaks down while the generalized Galerkin's method used in one-dimensional structural problems (Beam problems) could not handle the two-dimensional structural problem (Plate problems). Thus, In order to solve equation (1), in the first instance, the deflection is written in the form; Shadnam et al (2001)

where φ_n are the known eigenfunctions of the plate with the same boundary conditions. The φ_n have the form of

where

Ω_n, n = 1, 2, 3, ... , are the natural frequencies of the dynamical system and T_n(t) are amplitude functions which have to be calculated.

At this juncture, the right hand side of equation (1) is written in the form of a series and we have

Substituting equation (20) into equation (23) we have

where

Multiplying both sides of equation (24) by φ_p(x,y) and integrating on area A of the plate, we have

Considering the orthogonality of φ_n(x,y)

where

Using (27), equation (1), taken into account (21), can be written as

Equation (28) must be satisfied for arbitrary x, y (that is, each point of the plate) and this is possible only when

The system in equation (29) is a set of coupled ordinary differential equations.

Considering the property of the Dirac-Delta function and expressing it in the Fourier cosine series as

and

equation (29) becomes

Equation (32) is the transformed equation governing the problem of an elastically supported rectangular plate on a variable Winkler elastic foundation. This is a coupled second order diffe-rential equation.

In what follows, φ_n(x,y) are assumed to be the products of the functions ψ_ni(x) and ψ_nj(y) which are the beam functions in the directions of x and y axes respectively, Lee and Ng (1996). That is

Since each of these beam functions satisfies the boundary conditions in its direction, the kernel (the product of these beam functions) in the above integrals satisfies all boundary conditions for any plate problem of practical interest. In particular, these beam functions can be defined respectively, as

and

where A_ni, A_nj, B_ni, B_nj, C_ni and C_nj are constants determined by the boundary conditions. Ω_ni and Ω_nj are called the mode frequencies.

In order to solve equation (32) we shall consider a mass M traveling with constant velocity c along the line y = s. The solution for any arbitrary number of moving masses can be obtained by superposition of the individual solution since the governing differential equation is linear. Thus for the single mass M₁ equation (32) reduces to

where

Equation (36) is the fundamental equation of our problem. In what follows, we shall discuss two special cases of the equation (36) namely; the moving force and the moving mass problems.

CASE I: RECTANGULAR PLATE TRAVERSED BY A M OVING FORCE

Setting Γ = 0 in equation (36) gives an approximate model of the differential equation describing the response of a rectangular plate resting on a variable Winkler elastic foundation and traversed by a moving force. Thus, if Γ = 0 in equation (36), we have

Evidently, an exact analytical solution to this equation is not possible. Consequently, the ap-proximate analytical solution technique, which is a modification of the asymptotic method of Struble discussed in Gbadeyan and Oni (1995) shall be used.

To solve equation (38), first, we neglect the rotatory inertial term and rearrange the equation to take the form

where

Consider a parameter λ < 1 for any arbitrary ratio Γ ^* defined as

so that

Substituting equation (42) into the homogenous part of equation (39) yields

When λ is set to zero in equation (43), a situation corresponding to the case in which the ef-fect of the foundation is regarded as negligible is obtained.

Struble's technique requires that the asymptotic solution of the homogenous part of equation (39) be of the form

where A_n(t) and Φ_n(t) are slowly varying functions of time or equivalently

where → implies " is of "

Thus, equation (43) can be replaced with

where

represents the modified frequency due to the effect of the foundation. It is observed that when λ = 0, we recover the frequency of the moving force problem when the effect of the foundation is neglected.

Thus; using (47), equation (38) can be written as

The homogenous part of equation (48) is rearranged to take the form

where

Now consider the parameter ε₀ < 1 for any arbitrary mass ratio λ₀ defined as

It can be shown that

Following the same argument, equation (49) can be replaced with

where

is the modified frequency corresponding to the frequency of the free system due to the presence of the rotatory inertia. It is observed that when ε₀ = 0, we recover the frequency of the moving force problem when the rotatory inertia effect is neglected.

In order to solve the non-homogenous equation (48), the differential operator which acts on T_n(t) is replaced by the equivalent free system operator defined by the modified frequency γ_sf. Thus

where

Therefore, the moving force problem is reduced to the non-homogeneous ordinary differential equation given as

where

When equation (56) is solved in conjunction with the initial conditions, one obtains expression for T_n(t). Thus in view of equation (20), one obtains

Equation (57) represents the transverse displacement response to a moving force of a rectangu-lar plate resting on variable Winkler elastic foundation.

CASE II: RECTANGULAR PLATE TRAVERSED BY A M OVING M ASS

If the mass of the moving load is commensurable with that of the structure, the inertia effect of the moving mass is not negligible. Thus Γ ≠ 0 and one is required to solve the entire equation (36) when no term of the coupled differential equation is neglected. This is termed the moving mass problem.

Thus, equation (36) can be rewritten in the form

where

we rearrange equation (58) to take the form

where ε has been written as a function of the mass ratio μ_o,

Considering the homogeneous part of the equation (59) and going through the same arguments and analysis as in the previous case, the modified frequency corresponding to the frequency of the free system due to the presence of the moving mass is

retaining terms to o(μ_o) only.

Thus, to solve the non-homogeneous equation (59), the differential operator which acts on T_n(t) and T _q(t) is replaced by the equivalent free system operator defined by the modified fre-quency β_sf. Therefore, taking into account equations (34) and (35), we have

where

It is noticed that equation (61) is analogous to equation (56) with β_sf and G₀ replacing γ_sf and K₀ respectively. Therefore, when equation (61) is solved in conjunction with the initial conditions, one obtains expression for T_n(t) and in view of equation (20), one obtains

Equation (63) is the transverse displacement response to a moving mass of a rectangular plate resting on variable Winkler elastic foundation. The constants A_ni, A_pi, A_nj, A_pj, B_ni, B_pi, B_nj, B_pj, C_ni, C_pi, C _nj and C_pj are to be determined from the choice of the end support condition.

4 ANALYSIS OF THE SOLUTION

Next, the phenomenon of resonance is examined. Equation (57) clearly shows that the rectangular plate on a variable Winkler elastic foundation and traversed by a moving force reaches a state of resonance whenever

while equation (63) shows that the same plate under the action of a moving mass experiences resonance effect whenever

where

Equations (65) and (66) imply that

Consequently from equations (64) and (67), for the same natural frequency, the critical speed (and the natural frequency) for the system traversed by a moving mass is smaller than that of the same system traversed by a moving force. Thus, for the same natural frequency of the plate, the resonance is reached earlier when we consider the moving mass system than when we consider the moving force system.

5 ILLUSTRATIVE EXAM PLES

a. Rectangular plate elastically supported at edges y = 0, y = L _Y with simple support at ed-

ges x = 0, x = L _{X .}

At x = 0 and x = L_X, the plate is taken to be simply supported and at the edges y = 0 and y = L_Y, it is taken to be elastically supported.

Using the conditions (4-11) in equations (34) and (35), the following values of the constants and the frequency equation are obtained for the elastic edges.

where

Equation (68) when simplified yields

which is termed the frequency equation for the elastic edge, such that

For the simple edges, it can be shown that

Similarly,

Using (68), (69), (71), (72) and (73) in equations (57) and (63) one obtains the displacement response respectively to a moving force and a moving mass of a simple-elastic rectangular plate resting on a variable Winkler elastic foundation.

b . Elastic support at all edges .

Using the conditions (12-19) in equations (34) and (35), one obtains

where

and

where

Equations (74) and (76) when simplified yield

and

Using (74), (75), (76), (77), (78) and (79) in equations (57) and (63) one obtains the transver-se-displacement response respectively to a moving force and a moving mass of an elastically sup-ported rectangular plate resting on a variable Winkler elastic foundation.

6 NUM ERICAL CALCULATIONS AND DISCUSSION OF RESULTS

In order to carry out the calculations of practical interests in dynamics of structures and engi-neering design for the elastically supported plate resting on variable Winkler elastic foundation, a rectangular plate of length L_Y = 0.914m and breadth L_X = 0.457m is considered. It is assumed that the mass travels at the constant velocity 0.8123m/s. Furthermore, values for E, S and Γ are chosen to be 2.109x10⁹kg/m², 0.4m and 0.2 respectively. For various values of the foundation modulus F₀ and the rotatory inertia correction factor R₀, the deflections W(x,y,t) of the elastically supported plate are calculated in meters at and plotted against time t

in seconds.

a . Simple - elastic rectangular plate on variable W inkler foundation .

Figures 6.1 - 6.3 present the responses of the plate simply supported at the edges x = 0 and x = L_X and elastically supported at the edges y = 0 and y = L_Y. Figure 6.1 displays the effect of foundation modulus F₀ on the transverse deflection of moving force for simple-elastic rectangular plate, while Figure 6.2 displays the effect of rotatory inertia correction factor R₀ on the transverse displacement of moving mass for the simple-elastic plate at a fixed value of F₀ = 1000N/m³. It is shown that as both F₀ and R₀ increase the amplitude of the deflection decreases respectively for the simple-elastic rectangular plate resting on variable Winkler elastic foundation.

For the purpose of comparison, Figure 6.3 compares the displacement curves of moving force and moving mass for the simple - elastic plate for fixed F₀ and R₀. It is evident from the graph that the response amplitude of a moving mass is greater than that of a moving force problem.

b . Elastically supported rectangular plate on variable W inkler foundation .

The responses of the plate elastically supported at all its edges are presented in figures 6.4 - 6.6. observed in figures 6.4 and 6.5 that as the values of R₀ and F₀ increase the deflection amplitude of the plate decreases for both cases of moving force and moving mass respectively for fixed F₀ = 1000N/m³ in Figure 6.4. Figure 6.6 compares the displacement response of the moving force and moving mass for an elastically supported rectangular plate for fixed values of F₀ and R₀. It is evident that the displacement response of the moving mass problem is greater than that of the moving force problem.

6 CONCLUSION

The problem of the dynamic behaviour under moving concentrated masses of rectangular plates resting on variable Winkler elastic foundation is considered in this work. The governing fourth order partial differential equation is a non-homogenous equation with variable and singular coeffi-cients. The objective of the work has been to study the problem of the dynamic response to mo-ving concentrated masses of rectangular plates on variable Winkler elastic foundations. In parti-cular, the closed form solutions of the fourth order partial differential equations with variable and singular coefficients of the rectangular plate is obtained for both cases of moving force and mo-ving mass. The method is based on (i) Separation of variables (ii) The modified Struble's techni-que and (iii) The method of integral transformations.

These solutions are analyzed and resonance conditions are obtained for the problem. The nu-merical analysis for both moving force and moving mass problems carried out show that the mo-ving force solution is not an upper bound for the accurate solution of the moving mass problem and that as the rotatory inertia correction factor increases, the response amplitudes of the plates decrease for both cases of moving force and moving mass problem. When the rotatory inertia correction factor is fixed, the displacements of the elastically supported rectangular plates resting on variable Winkler elastic foundations decrease as the foundation modulus increases.

Furthermore, for fixed values of rotatory inertia correction factor and foundation modulus, the response amplitude for the moving mass problem is greater than that of the moving force problem implying that resonance is reached earlier in moving mass problem than in moving force problem of the elastically supported rectangular plate resting on variable Winkler foundation. Hence, it is dangerous to rely on the moving force solutions.

Finally, for the elastically supported rectangular plate resting on Winkler elastic foundation with stiffness variation, for the same natural frequency, the critical speed for moving mass pro-blem is smaller than that of the moving force problem, and as rotatory inertia correction factor and the foundation modulus increase, the critical speeds increase showing that risk is reduced.

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Publication Dates

• Publication in this collection
  16 June 2014
• Date of issue
  2014