### Grant Program

Faculty Development Competitive Research Grant Program 2018-2020

### Project Description

The static and dynamic response analysis of beams and slabs on foundations under moving loads has been one of the research interests of railway engineers, especially in the last few decades. Much research has been carried out in this field because of application tin many practical fields of railway engineering like noise generation, wave propagation and track design optimization. The following is a proposal to investigate novel methods of simulation of beams resting on elastic and viscoelastic foundations. First, the simulations are novel in that new ways of treating non-linear terms, which arise in the governing equations will be explored looking for accuracy and efficiency of the analytical solutions. Secondly, current beam foundation models are unsatisfactory, mainly due to the fact that non-linear characteristics cannot be included easily or even at all. Therefore novel models, which will include non-linear terms will be explored with the view to obtaining accurate results for increasingly more realistic foundation types.

In all but a few references found in the literature, the foundation studied has been assumed to be linear, so as to simplify the mathematical model developed. In practice, however, the support structure of a railway track is highly non-linear because of the hardening characteristic of the ballast and also the rail-pad.

The proposed work here will be in five main phases:

1) Broadly the concentration of this phase’s work will be on developing, (a) non-linear beam models (e.g. Euler-Bernoulli beam or Timoshenko beam) (b) beam-foundation interaction models (i.e., bilateral and unilateral), and (c) appropriate non-linear viscoelastic models.

2) This phase will develop a range of analytical methods capable of solving fourth-order non-linear homogeneous and non-homogeneous equations representative of static and dynamic responses found by beams and slats under moving loads. Both free natural vibrations, and, forced and damped vibrations will be considered. The methods of solution to be developed are based on three recent novel methods of calculation. The first, and possibly the most promising, will be the Adomian Modified Decomposition Method (AMDM) in which Adomain polynomials can be employed for non-linear terms, and this method has proved in other applications to provide fast and accurate results. A second method, based on perturbation in conjunction with complex Fourier transformation will be investigated. It is intended that a closed-form solution presented in an integral form and based on Green’s function and the theorem of residues is used for the calculation of integrals. The last method to be investigated and developed is the Homotopy Perturbation Method (HPM), commonly known as He’s method. It is known that this method can handle non-linear equations, but it still has to be investigated how well the method will perform for the structures in question.

3) This phase will apply the solution methods and models of phases (1) and (2) above to high speed rail transport. Such research will include

a) Comparing models and methods developed for the analysis of a beam on a non-linear viscoelastic

foundation in the presence of a harmonic moving load which is directly applicable in the frequency

domain.

b) Time response histories of the beam will be graphically presented for various speeds of force.

c) Develop a formula for the critical velocity of the load, where critical velocity is defined as the phase

velocity of the slowest free wave.

d) Investigate the non-linear effects on the beam response and comparison of the results for a non-linear

and equivalent linear model.

e) Investigate the existence of higher harmonics in the high-order terms.

f) Study the influence of the load speed and frequency on the main and higher-order harmonics.

g) Compare the beam theories.

4) This phase will apply and expand on the methods and models developed in phases (1) and (2) and also build on the experience of phase (3). This phase will look at much slower velocities as the environment will be urban and will seek to simulate the linear and turning movements of light rail transit. From a mathematical point of view the original beam studies will now turn into those of a slab or plate, i.e. it is three dimensional as opposed to two. This may require a rethink on how actually to use the three fourth-order equation solution methods already mentioned, but generally similar linear and non-linear terms will re-appear in the development of the code. Similar research will be done for slabs as listed in phase (3) above, especially finding the critical formula for a moving load, but additional work will looking at optimized turning radii and rotary inertia of the slab

5) This phase will concentrate of packaging the developed codes to give a user-friendly environment for potential professional users. Java programming language will be used to design GUIs for easy code access.

In all but a few references found in the literature, the foundation studied has been assumed to be linear, so as to simplify the mathematical model developed. In practice, however, the support structure of a railway track is highly non-linear because of the hardening characteristic of the ballast and also the rail-pad.

The proposed work here will be in five main phases:

1) Broadly the concentration of this phase’s work will be on developing, (a) non-linear beam models (e.g. Euler-Bernoulli beam or Timoshenko beam) (b) beam-foundation interaction models (i.e., bilateral and unilateral), and (c) appropriate non-linear viscoelastic models.

2) This phase will develop a range of analytical methods capable of solving fourth-order non-linear homogeneous and non-homogeneous equations representative of static and dynamic responses found by beams and slats under moving loads. Both free natural vibrations, and, forced and damped vibrations will be considered. The methods of solution to be developed are based on three recent novel methods of calculation. The first, and possibly the most promising, will be the Adomian Modified Decomposition Method (AMDM) in which Adomain polynomials can be employed for non-linear terms, and this method has proved in other applications to provide fast and accurate results. A second method, based on perturbation in conjunction with complex Fourier transformation will be investigated. It is intended that a closed-form solution presented in an integral form and based on Green’s function and the theorem of residues is used for the calculation of integrals. The last method to be investigated and developed is the Homotopy Perturbation Method (HPM), commonly known as He’s method. It is known that this method can handle non-linear equations, but it still has to be investigated how well the method will perform for the structures in question.

3) This phase will apply the solution methods and models of phases (1) and (2) above to high speed rail transport. Such research will include

a) Comparing models and methods developed for the analysis of a beam on a non-linear viscoelastic

foundation in the presence of a harmonic moving load which is directly applicable in the frequency

domain.

b) Time response histories of the beam will be graphically presented for various speeds of force.

c) Develop a formula for the critical velocity of the load, where critical velocity is defined as the phase

velocity of the slowest free wave.

d) Investigate the non-linear effects on the beam response and comparison of the results for a non-linear

and equivalent linear model.

e) Investigate the existence of higher harmonics in the high-order terms.

f) Study the influence of the load speed and frequency on the main and higher-order harmonics.

g) Compare the beam theories.

4) This phase will apply and expand on the methods and models developed in phases (1) and (2) and also build on the experience of phase (3). This phase will look at much slower velocities as the environment will be urban and will seek to simulate the linear and turning movements of light rail transit. From a mathematical point of view the original beam studies will now turn into those of a slab or plate, i.e. it is three dimensional as opposed to two. This may require a rethink on how actually to use the three fourth-order equation solution methods already mentioned, but generally similar linear and non-linear terms will re-appear in the development of the code. Similar research will be done for slabs as listed in phase (3) above, especially finding the critical formula for a moving load, but additional work will looking at optimized turning radii and rotary inertia of the slab

5) This phase will concentrate of packaging the developed codes to give a user-friendly environment for potential professional users. Java programming language will be used to design GUIs for easy code access.

Status | Active |
---|---|

Effective start/end date | 3/20/18 → 12/31/20 |

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Dynamic response

Rails

Light rail transit

Java programming language

Phase velocity

Graphical user interfaces

Green's function

Nonlinear equations

Wave propagation

Tin

Hardening

Packaging

Polynomials

Mathematical models

Decomposition

Engineers

Design optimization