Deflection profile analysis of beams on two-parameter elastic subgrade

A procedure involving spectral Galerkin and integral transformation methods has been developed and applied to treat the problem of the dynamic deflections of beam structure resting on bi-parametric elastic subgrade and subjected to travelling loads. The case of the response to moving constant loads of this slender member is first investigated and a closed form solution in series form describing the motion of the beam while under the actions of the travelling load is obtained. The response under a variable magnitude moving load with constant velocity is finally treated and the effects of prestressed, foundation stiffness, shear modulus and damping coefficients are investigated. Results in plotted curves indicate that these structural parameters produce significant effects on the dynamic stability of the load-beam system. Conditions under which the beam-load system may experience resonance phenomenon are also established some of these findings are quite useful in practical applications.


INTRODUCTION
Study concerning the subject of moving load systems has become increasingly important owing to its range of applications in transportation industries, aerospace engineering and related fields.Elastic structures are very useful in many engineering fields, thus their dynamic behaviours when under the action of travelling loads of different forms have received extensive attention in the open literature [1][2][3][4][5][6][7][8].When these important engineering structures are resting on an elastic foundation, the structure-foundation interaction effects play significant roles in their response behaviour and alter the dynamic states of the structures from those vibrating in the absence of foundation [9].Hence, the dynamic behaviour of structures on elastic foundation is of great importance in structural, aerospace, civil, mechanical and marine engineering applications.
Consequently, it is important to clarify the influence of the foundation on the behaviour of elastic structures in engineering designs.Furthermore, to accurately assess the dynamic response of any structural member on elastic foundations, a mechanical model is required to predict the interaction effects between such structures and foundations.Beams on elastic foundation and under the actions of the moving loads have received a considerable attention in literature; see for example references [10][11][12][13][14][15][16][17][18].However, most of these works employed the simplest mechanical model which was developed by Winkler and generally referred to as a one-parameter model.The deficiency of this model is that it assumes no interaction between the springs, so it does not accurately represent the characteristics of many practical foundations [19].Thus to overcome the deficiencies inherent in Winkler formulation, a two-parameter foundation models which takes into account the effect of shear interactions between springs has been suggested.Nonetheless, dynamic analysis of elastic structures on a two-parameter foundation model has received little attention in the open literature.It is observed that, when the parameters of the foundation mechanical models are constant along the span of the structure, the differential equation has constant coefficients, and the solution can be given as a linear combination of elementary functions, but if the foundation parameters vary along the structures, it is difficult to obtain the exact solutions of these differential equations in most cases, and a numerical technique is resorted to.Nevertheless, an exact analytical procedure is desirable as solution so obtained sheds more light on some vital information about the vibrating system.Thus, in this paper, an analytical approach is developed to assess the dynamic response behaviour of elastic beams resting on two-parameter elastic foundation and subjected to travelling loads.To classify the influence of the foundation model and other important structural parameters on the dynamic response of the beams to moving loads, several numerical examples will also be presented.Effects of different types of moving loads on the beams will be investigated.

THE MATHEMATICAL FORMULATION
Consider a structurally damped elastic beam resting on a two-parameter elastic subgrade and under the actions of concentrated travelling load.The differential equation governing the motion of such beam is given by where the prime and the over-dot are the partial derivatives with respect to the spatial coordinate x and the time t respectively, E is the modulus of elasticity, I is the constant moment of inertia, ( ) , Z x t is the transverse deflection of the beam, N is the axial force, m is the mass of the beam per unit length, 0 e is the damping coefficient, F(x) is the variable elastic foundation, G(x) is the non-uniform shear rigidity of the foundation and P(x,t) is the time dependent concentrated travelling load.
The boundary conditions at the end x = 0 and end x = L are given as and the initial conditions are also given as Furthermore, the variable elastic foundation F(x) and the non-uniform shear rigidity of the foundation are given as and where F 0 and G 0 are the foundation and shear rigidity constants respectively.Substituting ( 4) and ( 5) into (1) one obtains In what follows, we seek to calculate the dynamic deflection Z x,t ( )of the vibrating system for different type of dynamic load P x,t ( ) .

FORCED VIBRATIONS OF BEAMS SUBJECTED TO CONSTANT MAGNITUDE TRAVELLING LOAD
If the travelling concentrated load is assumed to be of the constant magnitude then, the dynamic load ( ) , P x t can be written as Thus, in view of ( 7) equation ( 6) can be rewritten as Equation ( 8) describes the motions of a prestressed homogeneous beam resting on a twoparameter elastic subgrade and subjected to fast travelling forces.Closed-form solution to the fourth order partial differential equation (8) governing the motion of the elastic thin member under the action of concentrated moving forces is now sought.

Solution procedures
The approximate analytical method due to Galerkin extensively discussed in [4] is employed to approximate the solution of the boundary-initial-value problem (1), (2).According to this technique, the M th term approximate solution of (2), ( 8) is sought in the form where ( ) i Q t are coordinates in modal space and ( ) i P x are the normal modes of vibration written as No difficulty arises at all to show that for a beam with simply supported end conditions, taking into account equation (10), equation ( 9) can be written as Substituting equation (11) into the governing equation ( 8), one obtains which after some simplifications and rearrangements yields Latin American Journal of Solids and Structures 10(2013) 263 -282 To determine the expression for ( ) i Q t , the expression on the LHS of equation ( 13) is required to be orthogonal to the function sin jπx L .Thus, multiplying equation ( 13) by sin jπx L and integrating with respect to x from x=0 to x=L, leads to where Noting the property of the dirac delta function and considering only the i th particle of the system, equation ( 14) can then be written as where and To obtain the solution of the equation ( 17), it is subjected to a Laplace transform defined as where s is the Laplace parameter.Applying the initial conditions (3), one obtains the simple algebraic equation given as where θ = jπv i L which when simplified further yields where In what follows, we seek to find the Laplace inversion of equation (21).To this effect, the following representations are adopted So that the Laplace of ( 21) is the convolution of f i s ( ) and g s ( ) defined as Thus, the Laplace inversion of ( 21) is given by Latin American Journal of Solids and Structures 10(2013) 263 -282 where Thus, in view of (25), taking into account (26) one obtains Substituting equation ( 27) into equation ( 9) leads to Equation ( 28) represents the transverse displacement response of the damped beam resting on a two-parameter elastic subgrade and under the actions of constant magnitude moving loads.

FORCED VIBRATIONS OF BEAMS SUBJECTED TO EXPONENTIALLY VARYING MAGNITUDE TRAVELLING LOAD
In this section, the dynamic response of the elastic thin beam resting on a two-parameter elastic subgrades to exponentially varying load is scrutinized.Thus, for the purpose of example in this section the dynamic load ( ) , P x t is taken in this section to be of the form Substituting equation ( 29) into equation ( 6), following the same arguments as in the previous section, after some simplification and rearrangements one obtains where all parameters are as previously defined.
Using the property of Dirac delta already alluded to, equation (30) can be rewritten as Solving equation (31) in conjunction with the initial conditions, yields ) Substituting equation (31) into equation ( 9) one obtains which represents the transverse displacement response of the damped beam resting on a twoparameter elastic subgrade and under the actions of exponentially varying magnitude moving loads.

FORCED VIBRATIONS OF BEAMS SUBJECTED TO HARMONICALLY VARYING MAGNITUDE TRAVELLING LOAD
In this section, the dynamic response of the elastic thin beam resting on two-parameter elastic subgrades to exponentially varying load is scrutinized.Thus, for the purpose of example in this work the traversing load ( ) , P x t tis taken to be of the form Substituting equation (34) into equation ( 6), following the same arguments as in the previous section, after some simplification and rearrangements one obtains where all parameters are as previously defined.Again, using the property of dirac delta , quation (35) can be rewritten as Using trigonometric identity, equation (36) can further be written as Following the same arguments and procedures listed in section 3.0, one obtains the solution of equation (37) as where ) Equation (38) on inversion leads to which represents the transverse displacement response of the damped beam resting on a twoparameter elastic subgrade and under the actions of harmonic variable magnitude travelling loads.

DISCUSSION OF ANALYTICAL SOLUTIONS
This section seeks to examine and establish the conditions under which the vibrating system may grow without bound.This phenomenon constitutes great concern in dynamical system problems.
From equation (28), it is evidently clear that, an axially prestressed damped beam resting on a two-parameter elastic subgrade and under the actions of constant magnitude moving load will experience a state of resonance whenever and the velocity, known as critical velocity at which this occur is given as Similarly, equation (33) depicts that an axially prestressed damped beam resting on a twoparameter elastic subgrade and under the actions of exponentially varying magnitude travelling load will experience a state of resonance whenever and the velocity at which this occur is given as While, equation (40) depicts that an axially prestressed damped beam resting on a twoparameter elastic subgrade and under the actions of harmonic variable magnitude travelling load will experience resonance phenomenon whenever and the critical velocity for this system is given by Latin American Journal of Solids and Structures 10(2013) 263 -282

RESULTS AND DISCUSSION
In this section, the foregoing analysis is illustrated by considering an isotropic beam structure of modulus of elasticity E = 2.10924 ×10 9 N/m 2 , the moment of inertia I = 2.87698 ×10 −3 m 4 , the beam span L = 12.192m and the mass per unit length of the beam m=2758.291Kg/m.The moving loads in both the cases of travelling constant loads and varying magnitude moving load is assumed to be V = 3.128m / s .The values of foundation moduli is varied between 0 3 / N m and 400000 3 / N m , the values of axial force N is varied between 0 N and 2 ⋅ 0 ×10 8 N and the values of shear modulus G is varied between 0 3 / N m and 6000000 N / m 3 .
Figure 1 depicts the dynamic deflections of structurally damped beam resting on elastic foundation and subjected to constant magnitude loads travelling at constant velocity.It is clearly seen that for fixed values of foundation rigidity K, shear stiffness G and damping coefficient e, the transverse displacement response of the beam decreases as the values of the prestress function N increases.Similarly, figures 5 and 9 display for fixed values of foundation rigidity K, shear stiffness G, damping coefficient e and for various values of prestress function N the transverse displacement response of structurally damped beam to variable harmonic magnitude and exponentially varying moving loads respectively.It is found also that the dynamic deflections of the beam increases as the values of axial force N reduces for fixed values  Figure 4 illustrates the dynamic behaviour of structurally damped beam resting on elastic foundation and subjected to constant magnitude loads travelling at constant velocity.It is shown that for fixed values of foundation rigidity K, shear stiffness G and the prestress function N, the transverse displacement response of the beam decreases as the values of the damping coefficient e increases.Similarly, figures 8 and 12 display for fixed values of foundation rigidity K, shear stiffness G, prestress function N and for various damping coefficient e the transverse displacement response of structurally damped beam to variable harmonic magnitude and exponentially varying moving loads respectively.It is observed that the dynamic deflections of the beam increases as the values of damping coefficient e dereases for fixed values of foundation rigidity K, shear stiffness G and prestress function N.
The comparison of the dynamic behaviour of the axially prestressed beam resting on twoparameter elastic foundation and under the actions of constants and variable magnitude loads is shown in figure 13.From this figure, it is observed that higher values of the structural parameters namely axial force N, foundation rigidity K, shear stiffness G and damping coefficient e are required for a more noticeable effects in the case of the dynamical systems involving variable magnitude loads than those involving constant magnitude moving loads.

CONCLUSION
The dynamic response of a prestressed homogeneous beam system continuously supported by elastic foundation to fast travelling loads of different forms was investigated.The dynamic deflections of this slender member when under the actions of moving loads are obtained in closed forms.Conditions under which the beam-load system may experience resonance phenomenon are established.The calculated deflections are clearly presented in plotted curves and discussed.The effects of the damping, pretrsssed, and stiffness of the viscoelastic layer on the beam deflections

Figure 1 Fig 1 :Figure 2 Figure 3
Figure 1 Transverse displacement of structurally damped beam for various values of axial force N and for fixed value of Foundation stiffness K, shear modulus G and damping coefficient e

Fig 2 :Fig 3 :Figure 4 Figure 5 Fig 4 :Fig 4 :Figure 6 Figure 7 Fig 5 :Fig 6 :Figure 8 Figure 9 Fig 7 :Fig 8 :
Fig 2: Deflection profile of a structurally damped beam for various values of foundation modulus K and for fixed values of Axial force N, shear force G and damping coefficient e

Figure 10
Figure 10 Transverse displacement of structurally damped beam subjected to harmonic variable magnitude load for various values of axial force N and for fixed value of Foundation stiffness K, shear modulus G and damping coefficient e

Fig 9 :Figure 11 Figure 12
Fig 9: Transverse displacement of structurally damped beam subjected to harmonic variable magnitude load for various values of axial force N and for fixed value of Foundation stiffness K , shear modulus G and damping coefficient e -0.00002

Fig 10 :Fig 11 :Figure 13 Figure 14
Fig 10: Deflection profile of a structurally damped beam under the actions of exponentially varying loads for various values of foundation modulus K and for fixed values of Axial force N, shear modulus G and damping coefficient e -0.000025

Fig 12 :Fig 13 :
Fig 12: Transverse response of a structurally damped beam to harmonic variable magnitude moving load for various values of damping coeficient e and for fixed values of Axial force N, Foundation stiffeness K and Shear modulus G