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Minimisation of Railhead Edge Stresses through Shape Optimisation

Nannan Zong, Manicka Dhanasekar*

School of Civil Engineering & Built Environment, Queensland University of

Technology, Brisbane, Australia.

*Corresponding Author: Professor M. Dhanasekar, School of Civil Engineering and

Built Environment, Queensland University of Technology, Brisbane, QLD4000. Ph.

+61 7 3138 6666; Fax +61 7 3138 1170; Email: m.dhanasekar@qut.edu.au

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Minimisation of Railhead Edge Stresses through Shape Optimisation

Abstract: Railhead is severely stressed under the localised wheel contact patch

closer to the gaps of the insulated rail joints (IRJs). Modified railhead profile in

the vicinity of the gapped joint through a shape optimisation model based on

coupled genetic algorithm and finite element method effectively alters the contact

zone and reduces the railhead edge stress concentration significantly. Two

optimisation methods, grid search method and genetic algorithm were employed

for this optimisation problem. The optimal results from these two methods are

discussed and their suitability for rail end stress minimisation problem is

particularly studied. The optimal profile is shown to be unaffected by either the

magnitude or the contact position of the loaded wheel through several numerical

examples. The numerical results are validated through a large scale experimental

study.

Keywords: gapped interfaces; Railhead-Wheel Contact; stress concentration;

shape optimisation; genetic algorithm; finite element modelling.

1. Introduction

Geometric discontinuities in supporting structures under moving loads generate a highly

localised, severe stress concentration capable of inflicting local damages. In insulated

rail joints, a purpose-made gap is formed to ensure electrical insulation for proper

identification of trains on the track with a view to appropriately controlling the

signalling system. The gap is defined in between two unsupported free edges and sharp

corners that form the source of stress concentration. A basic configuration of a gapped

edge subjected to contact loading under a rolling or stationary wheel as shown in

Figure 1. It can be seen that as the wheel approaches the free unsupported edge, the

elliptical contact pressure distribution modifies into a hyperbolic shaped distribution

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and the stress concentration that resides below the contact surface migrates upwards to

the sharp corner of the free unsupported edge.

Fundamental studies of the free-edge effects of contact problems have received

only a limited attention in the literature. Gerber (1968) evaluated a rigid frictionless

punch contacting a quarter plane using Hetenyi’s method (1970). Erdogan and Gupta

(1976) studied frictionless flat punch contacting with a two dimensional wedge at

arbitrary angles and discussed the singularity of the contact pressure when contact

region reaches the edge. The contact pressure close to the unsupported edge was studied

by Hanson and Keer (1989) for a frictionless contact of an elastic quarter plane. Results

show that the contact pressure distribution is significantly affected by the flexibility of

the free unsupported edge. Furthermore, for solving the contact problem of real-life

gap-jointed structures, such as the joints in gantry girders, expansion joints in bridge

girders or insulated bonded joints in rail signalling system circuitry, numerical tools are

necessary as the analytical formulation do not provide direct solution due to the

complex and variable contacting geometry involved.

The gap-jointed rail joints subjected to wheel loading were largely investigated

using finite element method. The unsupported rail edge under the static wheel load was

examined by Chen and Kuang (2002) for the frictionless contact using 3D finite element

method (FEM) and Chen (2003) using a 2D elastic-plastic plane strain FEM. The

contact pressure and stress variations in the proximity of the free edge was reported

with the maximum von Mises stress increased and shifted to the edge corner as the

wheel moved to the rail end.

Wen et al (2005) investigated dynamic contact-impact force, stresses and strains

in the railhead close to rail end region when a wheel passing over the gap-jointed rails;

Cai et al (2007) used FE analysis to study the distribution of dynamic stresses at the rail

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ends containing height difference in the vicinity of the gap. These studies reported that

the maximum equivalent Mises stress and equivalent plastic strain occurred on the rail

top surface of the unconstrained free edge, leading to severe early damage in the edge

vicinity.

A fatigue analysis of gap-jointed rail joints under static wheel loading was

conducted by Sandström and Ekberg (2008) using implicit FEM; their results reveal that

the free edge at the rail gap is severely stressed and the increased gap distance

accelerated the material deterioration.

As discussed above, the identified localised stress concentration and subsequent

early damage problem in the gap-jointed structures remains as a source of significant

problem to the rail industry. This problem is more acutely felt in heavy haul rail

systems, where static axle load of up to 300KN is routinely employed. Use of high

yield material can delay but cannot eliminate the occurrence of damages; further, high

yield materials are usually expensive and cannot be used for the whole railhead. The

lack of research on the solution of the stress minimisation through shaping of the

problematic zone in related gap-jointed structures is considered as an opportunity and

examined by the authors. In this paper, a simulation based optimisation model applied

to the gap-jointed rail edges subjected to wheel loads for minimisation of stress

concentration is reported. The optimisation model incorporates a single objective

function with multiple local minima due to nature of the problem compounded by the

wheel – rail contact nonlinearity. A Grid Search method is also used to illustrate the

existence of multiple local minima in the design space.

To deal with objective functions containing multiple local minima in engineering

applications, biologically inspired genetic algorithm (GA) is used widely (Lampinen

(2003), Cho and Rowlands (2007), Annicchiarico (2007) and Corriveau et al. (2010)).

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The robustness of the GA to identify the global optimal solution through the evaluation

of the objective function has encouraged its adaption in the optimisation model reported

in this paper. The paper provides the results from the GA and the grid search method,

which ensures the global optimality. The resulting optimal design parameters have

dramatically altered the magnitude of stress concentration with significant changes to

the load-transfer characteristics into the railhead. The effect of the optimal shape to the

dynamic rolling of wheels across the optimally shaped gap is also reported. The effect

of the gap distance to the optimal design variables is discussed. Lastly, the numerical

study and the performance of the optimal shape are validated through an elaborate

experimental study

2. Problem Description

2.1 Assumption

Sharp corners of the unsupported free edges in the insulated rail joints are proven to be

quite detrimental to loads that act in the vicinity of the sharp corner. In real-life

structural design, it is usually possible to minimise the stress concentration or

redistribute the stress field through appropriate profiling of the geometry without

intervention to the basic structural functions. Stress minimisation is often associated

with adoption of fillet/transitional curves or addition/removal of materials; the location

of stress concentration may or may not alter depending on the application considered.

Classic stress minimisation problems usually involve in the fillet curve design in

various structures, see, e.g. Refs. Pedersen (2008) and Le et al. (2011) or material

removal and redistribution, see, e.g. Refs. Li et al. (2006) and Xia et al. (2012).

Motivated by the design principles, stress minimisation coupled with genetic algorithms

has also been reported in a number of literatures, for example, see, e.g. Refs. Lampinen

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(2003), Annicchiarico (2007), Cho and Rowlands (2007) and Corriveau et al. (2010).

In this paper, stress concentration at the localised sharp corner of an insulated

rail joint at free edge is minimised by a smooth convex arc fillet using two local surface

parameters as shown in Figure 2. The configuration of the arc curve, which always

keeps tangent to the contact surface, can be uniquely defined using fillet length l and

fillet depth d. Subjected to contact loading, this fillet design is to be optimised to reduce

the stress minimisation. Unlike the other shape optimisation works cited above, where

the loading is far field, the complexity in the current problem is that the loading is near

field (i.e., applied on the modified shape itself) involving contact nonlinearity.

The objective function is to minimise the maximum magnitude of the von Mises

stress subjected to contact force close to the unsupported edge. The optimisation

problem can be expressed as in following equations:

Design variables: { , }TX l d (1)

Minimise: max( ) ( )ef X X (2)

Design constraints: l lL l U (3)

d dL d U (4)

in which:

max ( )e X is the maximum von Mises stress under the contact loading; X is the vector

consisting of two design variables, namely l and d; L={ lL , dL } and U={ lU , dU } are

lower and upper limit vectors for design variables l and d respectively.

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3. Optimisation Procedure

In this section, grid search method and genetic algorithms are described first. The

application of these two methods in the FE framework as the objective value solver is

then presented. The optimal results are presented and the efficiency of the two methods

compared.

3.1 Grid Search Algorithm

Grid search method (GSM) (Rao (1996), Rardin (1998) and Ataei and Osanloo (2004)),

belonging to one of the direct search methods, is based on simple optimal search

strategies and does not require the evaluation of derivative of the objective function.

Although generally it is less efficient than the class of the gradient-based methods,

which need the information of partial derivatives of the objective function, it still

remains popular due to its simplicity and the fact that many real optimisation problems

require the use of computationally expensive simulation packages to calculate the

values of objective functions, such as the wheel-rail contact problem examined in this

paper. However, direct search algorithms undergo the problem of being trapped into

local minima, and also somehow user-dependent and problem-dependent.

If the search space is known to be within a finite area defined by upper and lower

bounds of each of the independent design parameters, then the grid search method can

be applied. This search method starts with an initial grid over the space of the interest

and evaluates the objective values at each node of the grid. Figure 3 shows an example

of initial search grid of two design variables (x1, x2) with upper bound U=(U1 U2) and

lower bound L=(L1 L2). Initially, the objective value at each node marked as “o” is

evaluated, and the node with the minimum objective value is located. This node then

becomes the centroid of a smaller search space. Normally this smaller space is defined

by the adjacent nodes from the initial grid as shown in Figure 3. As a result, new grid

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nodes marked as “▲” is evaluated. In such a manner, successive narrowing of the

search space is continued until optimisation iterations meet the predefined converging

criterion such as the maximum number of iterations or the tolerance for the minimum

value is attained.

3.2 Genetic Algorithm

Genetic algorithms (GAs) are search algorithm based on the mechanics of natural

selection and Darwinian evolutionary theory. In the form of computer simulations, GA

is typically implemented through a population of n candidates (individuals) to an

optimisation problem and gradually evolves towards optimal solutions. The evolution

starts from a population of randomly generated individuals. In each generation, the

fitness value of each individual is evaluated in regard to the objective function and the

constraints. Based on the performance of these individuals, multiple individuals are

stochastically selected and subjected to crossover and mutation operations associated to

probabilities Pc and Pm respectively. Such evolution procedures are repeated in the

successive generations until the optimal solution converges to a prescribed level or the

maximum generation is reached. Genetic algorithms have a theoretical background

developed by Holland (1992). The efficiency of GAs is problem dependent and has

been proved positively in various structural design problems, see, e.g. Refs. Deb (2002)

and Renner and Ekart (2003).

Chromosome Representing Individual

In GAs, an individual is defined using a particular coding arrangement (chromosome)

for the design variables. A finite length of binary coded strings consisting of 0s and 1s is

widely used to describe the design variables for each solution. For the arc shape design

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defined by the two variables l and d as described in section 2, coding for each design

variable is concatenated into a complete string as shown in Figure 4. Each 8 digits can

be decoded as one of the design variables. The length of the chromosome can be

adjusted depending on the precision and the domain of design variables. The mapping

form a binary string back to a real number for the design variable is obtained by:

,max ,min 1

,min

1

22 1

ni i j

i i ijnj

x xx x k

(5)

where [0,1]ijk denotes the jth

digit of the ith

design variableix .

Genetic Operators

Selection

The selection procedure in GAs allows individuals to be retained in the basic pool for

the next generation prior to the crossover and mutation processes. Different selection

methods exist such as the proportional selection, tournament selection, ranking

selection, etc. In this paper, tournament selection is used. The tournament selection

randomly selects some individuals to compete with each other. The one with highest

fitness value survives. This process continues until the size of the required population is

reached. Furthermore, this paper also applied an elitist strategy, in which some

individuals of the current population with top-ranking fitness values are directly

retained for the subsequent generation to preserve desirable genetic information

throughout the entire evolution. During the implementation, all the individuals in the

previous generation are independently ranked and the top two (or, 10%) individuals of

the combined population are transferred to the new generation.

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Crossover and Mutation

Crossover and mutation are the next two processes following the selection. Different

types of crossover are used, including single-point, double-point, uniform and weighted

crossover. In this paper, a single point crossover was adopted. In the single crossover

operation, two parent chromosomes are initially selected from the population and a

random number between 0 and 1 is generated and compared with the crossover

probability Pc. If the number is less than Pc, a cross-point in the length of the parent

chromosomes is randomly determined and portions of the parent chromosomes are

swapped. The crossover probability Pc between 0.6 and 0.9 is generally recommended.

After the crossover operation, the uniform mutation operates and brings the diversity

into the population. The uniform mutation progresses over each bit through the length

of chromosome and change a bit from 0 to 1 or vice versa with a probability Pm

applied. The mutation probability Pm is normally very small (Pm<1%), since a high

mutation probability might destroy the efficiency and turn the GA into a simple random

walk.

3.3 Integrated FE Analysis for GSM/GA

The design optimisation models reported in this paper have integrated the finite

element (FE) simulation. Generally the optimisation model consists of two major

frames: One is the numerical simulation to obtain the objective value with regard to

different individuals in each generation; the other is to evaluate fitness value, prepare

for next iteration through GSM and GA operators respectively and interface with the

numerical analysis for a progressive optimisation evolution. To achieve this goal, an

interface scheme developed using object-oriented programming language (Python) and

MATLAB were employed.

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An application scripting interface in object-oriented programming language

(Python) available in ABAQUS was employed. The Python scripting in this present

work was interpreted through the kernel of the ABAQUS/CAE and can conduct the

following tasks without human intervention during the optimisation process:

Create the geometrical dimensions according to design parameters, define

material property, boundary conditions, meshing for preparing FE analysis file

(.inp file);

Submit to the FE solver for numerical analysis;

Access to the simulated results file (.odb file) and output the prescribed objected

value ( )f X .

These operations are instructed by GSM and GAs in MATLAB for the

calculation of the objective function with regard to each design parameters of the

iteration. In the GSM optimisation model, the objective values of each node of the grid,

namely the maximum von Mises stress is compared and the best candidate with the

minimum value is selected. A zoomed search grid is then generated around the minima

and objective evaluation is repeated for this newly updated search grid. The search grid

progressively contracts in the optimisation loop and terminates when the process is

converged. On the other hand, the integration with GA is implemented by using GA

operators as described in Section 3.2. By receiving the objective values of each design

individual in the current generation, the fitness evaluation and genetic operations are

implemented in the GAs for creating the next generation. Such optimisation loop is

formed and continued until optimal design converged. Figure 5 presents the

optimisation procedures of using GA coupled with FEA.

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4. Application to Shape Optimisation of Railhead Free Edge

The design objective in this example is to work out the optimal arc shape for the

railhead free edge to minimise the stress concentration maxe due to wheel contact. FEA

solution to this particular wheel rail contact problem was carefully developed. In Figure

6, a detail of the wheel rail contact configuration and local arc shape are presented.

The wheel is symmetrically positioned on the railhead and over the sharp corner

of the unsupported free rail edge (x=0). The rail bottom surface is assumed fully fixed

(for a local stress optimisation study it was considered non critical whether or not the

rail is suspended on sleepers) and the wheel carrying vertical load was allowed to

displace only in the vertical direction. Two design variables, namely l and d define the

arc shape close to the unsupported end. It is created by lofting the original top rail

contact surface uniformly along an arc towards to the rail edge. The ends of the arc have

the distance l in the longitudinal direction (x axis) and height difference d in the vertical

direction (z axis). The applied constraints were 0mm<l<100mm, 0mm<d<5mm.

Definition of contact interaction between the bodies is very sensitive to iteration

convergence, result accuracy and computational time. The master/slave contact surface

method in ABAQUS was employed. The lower surface of the wheel was defined as

master contact surface, while the rail head surface was defined as slave contact surface.

The contact surface pair was allowed to exhibit finite sliding. Friction coefficient

between them was set to 0.3. Hard contact was chosen for the contact pressure-over

closure relationship in ABAQUS/Standard. The penalty method was used to enforce the

contact constraints. To retain the finite element analysis efficient and accurate, the

regions close to the contact interaction were meshed using 8-node fully integrated

element (C3D8), while the region far away from the contact was meshed using 8-node

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reduced integration element (C3D8R). In a typical analysis, there were 47303 elements

in the rail, 53736 elements in the wheel and 303117 DOFs in the whole system.

In the optimisation procedure using GSM method, two initial search grids,

namely a fine search grid 20×20 and a coarse search grid 10×5 were employed and

searched. Based on the objective values, a new 5×5 local search grid in next iterative

search is generated. The termination criterion for GSM is that the difference of the

percentage in the optimal objective value between last iteration and the current iteration

is less than 0.5%. While in the method of GA, the key parameters used is shown in

Table 1.

4.1 Comparison of optimal result between GSM and GA

In this paper, a typical loaded wagon wheel F=130KN in the heavy haul

transport network is considered. Result by using GSM and GA is compared and

discussed. The wheel rail contact and stress variations under the optimal shape in the

following sections are nondimensionlised by the maximum contact pressure P0, major

semi-contact length a in x direction and minor semi-contact length b in y direction

calculated from Herzian contact theory (HCT), which is valid when the wheel is far

away from the unsupported rail edge.

In GSM optimisation procedure, the optimal result based on the initial fine

search grid 20×20 and coarse search grid10×5 are presented respectively first. Figure 7

shows the iso-stress plot based on the objective value from initial grid nodes. It can be

seen that the fine initial grid 20×20 has a better recognition of the stress response to the

shape of the rail end, three local minimum zones were found and the two nodes with the

best two objective values were located at two of the local minimum zones. However,

the coarse search grid 10×5 can only predict a single relative larger optimal zone, in

which two near-optimal nodes are selected. Subsequently, the optimal result based on

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the information provided by the initial grid could be entirely different. Result shows that

the optimal parameters based on the fine grid yield at l=40.00mm and d=1.50mm with

the optimal stress magnitude of 0.694P0, while the coarse gird has resulted in the final

solution of l=70mm and d=3.12 with the optimal stress, value at approximate 0.705 P0.

Such difference in terms of the final optimal design parameters indicates that the coarse

grid has provided a poor estimation of the objective function, where the global

minimum was missed out and optimisation tended to be diverted to the local minimum.

In the GA optimisation procedure, the optimisation evolution was limited to a

maximum of 20 generations and was found sufficient to provide stable and convergent

solutions. As shown in Figure 8 (a), for the load F, the initial stress concentration is

around 1.16P0. Gradually the optimal stress has decreased to 0.692 P0 in 15 generations

with less than 0.006% difference among the last 7 generations and finally reaches to

0.688P0. The optimal design parameter for l and d are 40.05mm and 1.47mm

respectively. It is found that the optimal result from GA is similar to the one obtained by

using fine search grid in GSM.

The optimum result is presented and discussed as above. It should be noted that

both of the GA method and fine grid search in GSM is CPU time intensive with the

numerical analysis consisting of large percentage of the optimisation iterations. In a

supercomputer of 4CPUs and 3GB memory, it took172 hours for the GSM to complete

and the GA took 167 hours. The coarse grid GSM took only 29.8 hours, but

unfortunately could not provide the best result as the fine grid GSM and GA. As the

objective is to obtain reliable optimum design parameters, and since the cost of CPU

time is progressively reducing with the advent of computers, GA coupled with FEA was

considered acceptable for the complex wheel – railhead problem considered in this

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paper.; more efficient genetic algorithms to minimise CPU time is a subject of further

research.

4.2 Optimal Shape at loaded and unloaded Wheel Loads

As discussed above, since GA performs better, compared with the classical grid search

method, the local minima are avoided and the final solution obtained is in the vicinity of

the global minimum. To further validate the reliability of the GA method as well as the

effect of the magnitude of the load to the optimal shape, two load levels were

considered, namely loaded wagon wheel F=130KN and unloaded wagon wheel 0.2F.

The evolution of the optimal maximum von Mises stress maxe under the loaded

wheel F and 0.2F at rail end is shown in Figure 8. From the first generation until the

final optimum, the algorithm has lead to the minimisation of the maximum von Mises

stress by modifying the two design variables l and d.

For the load F, the initial stress concentration is 1.16P0. Gradually the optimal

stress has decreased to 0.692 P0 in 15 generations with less than 0.006% difference

among the last 7 generations and finally reaches 0.688P0. While for the unloaded wheel

0.2F, the optimal stress is 1.12P0 in the initial generation. As the generation increases,

the stress concentration reaches a stable solution less than 0.3% deviation in the last five

generations. The final optimal solution yields at 0.682P0. The optimal arc shape defined

by the design variables l and d at the symmetrical plane of the rail is show in Figure 9

and Table 2.

It should be noted that the optimal stress and the optimal design variables at

these two loads values F and 0.2F are very similar to each other (as one would expect

for stress minimisation problems – however, these two analyses prove the stability and

dependability of the optimisation model developed in this study). The optimal stress is

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0.688P0 at F and 0.682P0 at 0.2F, with only 0.9% difference. The difference in the

design variables l and d are 5% and 2% respectively. The effect of these optimal shapes

on the contact and stress distribution at the unsupported rail end is discussed in the

following section.

4.3 Effect of Optimal Shape

In this section, the effect of the optimal shape to the wheel rail contact pressure

distribution and stress variation is compared with the ones at the original rail edge sharp

corner. As the wheel axis is located at the rail end (x/a= 0) subjected to load F, the

contact pressure distribution is shown in Figure 10.

With the optimal shape obtained from section 4.1, the maximum contact

pressure is 1.25P0, reduced by 25% compared to the one in the original rail end with

sharp corner, valued at 1.49P0. Moreover, the contact region at the optimal shape

exhibits approximately an elliptical shape and located away from the rail edge (although

the wheel axis is located at the rail edge). This is in contrast to the original rail edge

with sharp corner that has the maximum contact pressure bounded at the free

unsupported rail edge. The result indicates that under the same wheel loading condition,

the optimal shape has effectively changed the load transfer characteristics into the

railhead.

Under the same wheel loading F, the contact pressure distribution versus the

vertical deflection of the rail contact surface along the symmetric plane (z=0) in the end

vicinity is shown in Figure 11. It can be seen that the optimal shaped rail edge has

larger contact length than the one with the sharp corner. For the same load magnitude,

the larger contact length could lead to less average (also peak) contact pressure, which

has been confirmed as shown in Figure 10. More importantly, the spatial alteration in

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the position of the contact length at the optimal shape has also favoured reduction in the

deflection of the rail end. The maximum deflection point at the optimal shape is almost

50% reduced, compared to the original shape with the maximum deflection point

located at the sharp corner. This larger deflection is closely associated with larger

plastic deformation and early material deterioration. The loss of stiffness closer to the

free edge shown in Figure 11 is in agreement with the results discussed in Hanson and

Keer (1989) and Chen (2003).

Figure 12 shows the equivalent stress field close to the end at load F. The

optimal maximum von Mises stress is approximately 0.688P0, reduced from 0.692P0

compared to the one in the original shape with sharp corner. Apart from the significant

reduction of stress concentration, the position of the maximum equivalent stress is

translated away from the rail end sharp corner. The spatial alteration of the stress

concentration is consistent with the contact distribution as discussed in Figure 10 and

11; it can, therefore, be concluded that when the wheel approaches the rail end, the

optimal shape will effectively modify the contact load transfer characteristics into the

railhead, which will significantly improve its potential life through significantly reduced

levels of railhead stresses.

In some instances, the wheel axis might extend out to the geometric end,

resulting in more severe stress concentration. Therefore, the stress concentration at three

different wheel contact positions, namely x/a=0, -0.5 and -1.5 were also examined.

Results at loaded wheel F and unload wheel 0.2F are presented in Table 3.

With the wheel axis extended beyond the railhead edge, the maximum von

Mises stress in the optimal shape at load F is around 0.72P0 at x=-0.5a and 0.79P0 at x=-

1.5a, increased by only 3.8% and 15.3%, compared to the one at l/a=0. However, the

peak Mises stress in the original shape is around 2.23P0 at x=-0.5a and 3.73P0 at x=-

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1.5a, increasing by 61.6% and 169.6% compared to the one at l/a=0. Similar

observation can also be found for the wheel load 0.2F. The peak Mises stress increased

by 166%, from 1.37P0 to 3.65P0, whilst in the optimal shape, it just increased from

0.66P0 to 0.78P0. It is obvious that the stress increase is significantly less pronounced

compared to the original shape. Also invariantly the stress concentration and the peak

stress are located underneath of the rail head surface and shifted away from the edge.

4.4 Dynamic Wheel Rolling

In this section, the effect of the optimal shape obtained from the static wheel

load optimisation on contact pressure and railhead stress variations subjected to wheel

rolling contact is reported. The optimal shape used in the dynamic wheel rolling

analysis is formed by taking the average of the optimal design variables from the above

two load values (F and 0.2F), namely around l=39mm and d=1.5mm respectively. The

explicit FE modelling method for wheel dynamic rolling contact developed by Pang and

Dhanasekar (2006) was employed in this section. Detail of finite element model is

shown in Figure 13. In this dynamic analysis, the wheel velocity is set as 80km/h and

wheel vertical load is F. Gap distance between rail ends is 6mm. The FE model contains

139441 elements in the whole system. For the dynamic wheel rolling analysis, first the

implicit solution was obtained to establish contact between wheel and railhead. The

contact solution as well as the deformation and displacement of the whole system were

subsequently transferred to the explicit procedure for rolling of the wheel across the rail

gap.

The impact force history is shown in Figure 14. It shows that, at the beginning

of the explicit analysis, the imported static contact force increased further to 1.05F due

to the effect of the initial condition. After a short period of decline, the contact force

stabilised at F just after 2 millisecond of travel time. Afterwards, the wheel entered into

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the vicinity of the IRJ. The contact force decreased rapidly from F to 0.87F. The impact

happened at around 12 milliseconds and the vertical contact force peaked at 1.19F for

the sharp-cornered rails and 1.21F for the optimal shape during this period. Vibration

continued after the impact and stability was regained at 16.5 millisecond of travel.

The maximum von Mises stress at rail-1 and rail-2 versus the wheel rolling

position (x = 0 stands for the middle of the joint gap) are presented in Figure 15. Prior

to the wheel approaching rail-2, the maximum stress in rail-1stablised around 0.6P0. As

the wheel experiences two-point contact with both of the rail edges during the impact,

the maximum stress is peaked up to 0.72P0 in rail-1 and 0.86P0 in rail-2 for the optimal

shape (whilst for the original shape these stresses were 1.24P0 in rail-1 and 1.28P0 in

rail-2). After the impact, the stress regained stability at around 0.6P0. The stress history

indicates that the optimal shape obtained from the static wheel load can still

significantly contribute to the reduction of the stress concentration at gap-jointed rail

edges subjected to wheel rolling contact loading.

The von Mises stress variation as the wheel is at the middle of the joint gap

(x=0) is shown in Figure 16. Similar to the static analysis in the previous sections, it is

apparent that the stress concentration zone at the optimal shape is also shifted away

from the unsupported rail ends surface with lower stress value, compared with the

original shape, having the stress concentration crowded at the sharp corners. The

effectiveness of the optimal shape to stress minimisation and relocation in the wheel

rolling contact loading is also satisfied.

5. Experimental Study

To validate the numerical optimisation result, this paper reports laboratory testing as

part of an ongoing research test conducted at the Heavy Testing Laboratory,

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Rockhampton, Australia. The data collected from the test was processed and compared

with the numerical results to understand the optimal shape effect at the unsupported rail

end under the wheel loading.

5.1 Strain-gauged Rail Edge

The lab test involved three different loading positions as shown in Figure 17, namely

+3mm, 0mm and-3mm. Two static load values (50kN and 130kN) were applied

respectively. To acquire the rail edge strain response close to the railhead, which is of

interest for this study, a linear strain gauge was applied at the symmetrical axis of the

free rail edge, 3mm below the railhead. The railhead was carefully machined with an

arc shape with the averaged design variables l=39mm and d=1.5mm used in Section 4.3

as shown in Figure 18.

5.2 Laboratory Testing Setup

The test loading rig used for this lab test is shown in Figure 19. The vertical wheel

loads and the horizontal wheel movements are produced by two Servo-hydraulic

actuators respectively. The vertical actuator applies the vertical wheel load to generate

the wheel rail contact interaction, whilst the horizontal actuator moves the wheel to

prescribed distances in the vicinity of the rail edge. At each wheel position, the wheel

static load is gradually increased to required load value and retained for 10 seconds in

order to obtained stable strain data. The rail sample was manufactured into a reinforced

concrete block and mounted to the base frame of the loading rig. Such boundary

condition ensured the rail bottom is, to maximum extent, rigidly supported, similar to

the boundary definitions in FE model.

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5.3 Experimental Validation

The vertical strain magnitude Ezz at these three different loading positions (±3mm and

0mm) for each static wheel load value is recorded. At each static load value, strain data

was used to compare with the numerical model. It should be noted that the positive and

negative sign for the strain value stand for the tensile and compressive strain. The data

are presented in Table 4. The comparison indicates that FE result agrees well with the

tests for both of the optimal shape and original shape. It should be noted that the optimal

shape exhibits tensile vertical strain whilst the original shape exhibits high compressive

strain value.

More importantly, to further demonstrate the performance between the rail end

with optimal shape and the one with original shape (sharp corner), the vertical strain

values from both of the FE analysis and strain gauge indicates that the strain level with

the optimal shape is always lower than the original shape with sharp corner. As the

wheel approaches the unsupported rail end, the original shape has the maximum vertical

strain closer to the sharp corner area while the optimal shape still exhibit a tensile strain

and the magnitude is commonly less than 3% of the original one.

6. Conclusions

In this paper, an integrated optimisation model coupling GA and finite element analysis

was developed to conduct shape optimisation of the gap-jointed rail end subjected to

static wheel load. The following conclusions are drawn:

1) The GA method provides comparable results to the GSM and marginally

cheaper computationally to GSM for the problem considered in this research.

2) The optimisation model converges to the optimal solution within 20 generations.

22

3) The optimal curve shape can significantly reduce the magnitude of the stress

concentration. The rail edge effect resulted from the gap in the optimal shaped

joint is less pronounced, providing the contact pressure and stress concentration

shifted away from the railhead edge and similar to the situation in a continuous

rail under wheel loading.

4) The optimal shape is insensitive to wheel positions. The effectiveness of the

stress reduction as the wheel extends beyond rail end (x/a=-0.5 and -1.5) is

consistent with the wheel location over the rail edge (x/a=0).

5) The optimal shape provides marginally higher impact load compared to the

original shape; however, has maintained the significantly reduced stress levels

that is more predominant and desirable for delaying the material deterioration.

6) The beneficial effects of the optimal shape have been proved from laboratory

tests that exposed the free edge of the railhead in the joint. The experimental

data validated the FE model quite well.

It appears encouraging to apply the optimal shape into the free edge of the

insulated rail joint. The results can be adapted to other rail joints and similar

engineering structures. The sharp discontinuity lead loss of local structural stiffness is

the source of significant problem in engineering; by reducing the magnitude of the

stresses and relocating the maximum stresses within the body using shape optimisation,

the life of engineering joints where gaps are unavoidable can be potentially improved.

Acknowledgements

The CRC for Rail Innovation (established and supported under the Australian Government's

Cooperative Research Centre program) has funded this research through project R3.100 to the

second author. The first author acknowledges the fee waiver program in support of his

23

scholarship from the Queensland University of Technology (QUT). The support of Mark Barry

and other high performance computing (HPC) facility of QUT are sincerely thanked. The

support of Paul Boyd and the lab staff of the Centre for Railway Engineering, Rockhampton in

carrying out the lab testing are sincerely appreciated.

References

ABAQUS Scripting user's manual, Version 6.9, ABAQUS Inc., 2009.

Annicchiarico, W., 2007. Metamodel-assisted distributed genetic algorithms applied to

structural shape optimization problems. Engineering Optimization, 39(7), 757-

772.

Ataei, M. and Osanloo, M., 2004. Using a Combination of Genetic Algorithm and the

Grid Search Method to Determine Optimum Cutoff Grades of Multiple Metal

Deposits. International Journal of Surface Mining, Reclamation and

Environment, 18, 60-78.

Cai, W., Wen, Z., Jin, X. and Zhai, W., 2007. Dynamic stress analysis of rail joint with

height difference defect using finite element method. Engineering Failure

Analysis, 14, 1488-1499.

Chen, Y.C., 2003. The effect of proximity of a rail end in elastic-plastic contact between

a wheel and a rail. Proceedings of the Institution of Mechanical Engineers, Part

F: Journal of Rail and Rapid Transit, 217, 189–201.

Chen, Y.C., and Kuang, J.H., 2002. Contact stress variations near the insulated rail

joints. Proceedings of the Institution of Mechanical Engineers, Part F: Journal

of Rail and Rapid Transit, 216, 265-274.

Cho, H.K. and Rowlands, R.E., 2007. Reducing tensile stress concentration in

perforated hybrid laminate by genetic algorithm, Composites Science and

Technology, 67, 2877-2883.

Corriveau, G., Guilbault, R. and Tahan, A., 2010. Genetic algorithms and finite element

coupling for mechanical optimization. Advances in Engineering Software, 41,

422-426.

Deb, K., 2002. A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE

Trans Evol Comput., 6,181–197.

24

Erdogan, F. and Gupta, G.D., 1976. Contact and crack problems for an elastic wedge,

International Journal of Engineering Science, 14, 155-164.

Gerber, C. E., 1968. Contact problems for the elastic quarter plane and the quarter

space. Thesis (PhD). Stanford University, CA, U.S.A.

Hanson, M .T., and Keer, L. M., 1989. Stress analysis and contact problems for an

elastic quarter-plane. Q. J. Mech. Appl. Math., 42, 363-383.

Hetenyi, M., 1970. A general solution for the elastic quarter space. Journal of Applied

Mechanics, Transactions of the ASME, 39, 75–80.

Holland, J.H., 1992. Adaptation in natural and artificial systems, University of

Michigan Press, 1975 and First MIT Press edition, 121.

Lampinen, J., 2003. Cam shape optimisation by genetic algorithm. Computer-Aided

Design, 35, 727-737.

Le, C., Bruns, T. and Tortorelli, D., 2011. A gradient-based, parameter-free approach to

shape optimization. Computer Methods in Applied Mechanics and Engineering,

200, 985-996.

Li, Q., Steven, G.P. and Xie, Y.M., 2006. Evolutionary structural optimization for stress

minimization problems by discrete thickness design. Computers and Structures,

78, 769-780.

Pang, T. and Dhanasekar, M., 2006. Dynamic finite element analysis of the wheel-rail

interaction adjacent to the insulated rail joints. Proceedings of 7th international

conference on contact mechanics and wear of rail/wheel systems, 2, 509-516.

Pedersen, P., 2008. Suggested benchmarks for shape optimization for minimum stress

concentration. Structural Multidisciplinary Optimization, 35, 273-283.

Rao, S.S., 1996. Engineering Optimization (Theory and Practice). Wiley–Interscience

Publication, John Wiley & Sons Inc., New York, 3rd edition.

Rardin, R.L., 1998. Optimization in Operations Research. Prentice-Hall International,

Inc.

Renner, G., Ekart, A., 2003. Genetic algorithms in computer aided design. Computer

Aided Design, 35, 709–726.

Sandström, J., and A Ekberg, A., 2008. Numerical study of the mechanical deterioration

of insulated rail joints. Proceedings of the Institution of Mechanical Engineers,

Part F: Journal of Rail and Rapid Transit, 223, 265–273.

25

Wen, Z., Jin, X. and Zhang, W., 2005. Contact-impact stress analysis of rail joint region

using the dynamic finite element method, Wear, 258, 1301-1309.

Xia, Q., Shi, T., Liu, S. and Wang, Y., 2012. A level set solution to the stress-based

structural shape and topology optimization. Computers and Structures, 90-91,

55-64.

26

Table 1: GA parameter values used in optimisation

Table 2: Comparison of optimal design variables at wheel load F and 0.2F

Table 3: Comparison of optimal stress at different wheel positions

Table 4: Comparison of vertical strain magnitude

Figure 1: Contact Pressure and Stress Concentration at Unsupported Free Edge

Figure 2: Local arc shape defined by two design variables

Figure 3; Scheme of grid search method

Figure 4: Binary representation to design variables

Figure 5: Schematic diagram for the proposed optimisation model

Figure 6: Wheel rail contact configuration at rail edge and arc shape

Figure 7: Profile of optimal result using GSM

Figure 8: Evolution of optimal von Mises stress over generations

Figure 9: Optimal shape at the symmetrical plane of the rail edge

Figure 10: Distribution of contact pressure at rail edge

Figure 11: Contact pressure and deflection of the contact surface along the rail

symmetric plane (z=0)

Figure 12: Distribution of von Mises stress in optimal rail end shape and original shape

Figure 13: Detail of finite element model for dynamic analysis

Figure14: Impact force history at original and optimal rail ends shape.

Figure 15: Time history of maximum von Mises stress at rail bodies during wheel

impact

Figure 16: Snapshot distribution of von Mises stress under rolling wheel at x = 0.

Figure 17: Loading positions in lab testing

Figure 18: Strain gauges positions and optimal arc shape at rail end

Figure 19: Lab testing setup

27

Table 1: GA parameter values used in optimisation

GA parameters Parameter values

Chromosome 16-bit binary coding

Population size 20

Generation 20

Selection Tournament

Crossover One bit-point

Crossover rate 𝑃𝑐 0.7

Mutation rate 𝑃𝑚 0.005

Optimisation termination criteria Maximum evolution number

Table 2: Comparison of optimal design variabels at wheel load F and 0.2F

Wheel Load L (mm) D (mm)

Loaded wheel F 40.05 1.47

Unloaded wheel 0.2F 38.05 1.44

28

Table 3: Comparison of optimal stress at different wheel positions

Wheel

Positions

Load F Load 0.2F

Original

shape

Optimal

shape

Original

shape

Optimal

shape

0 1.38P0 0.688P0 1.37P0 0.66P0

-0.5 2.32P0 0.72P0 2.29P0 0.73P0

-1.5 3.73P0 0.79P0 3.65P0 0.78P0

Table 4: Comparison of vertical strain magnitude

Wheel Positions

Vertical Strain under Wheel Load

Static50kN Static130kN

Optimal shape Original shape Optimal shape Original shape

Expt. FE Expt. FE Expt. FE Expt. FE

3mm 24.8 30 -2869 -3055 49 67 -5586 -5722

0mm 33.1 39.5 -6933 -7011 81.3 96.9 -8265 -8185

-3mm 52 60 -9021 -8889 133.6 125.2 -10444 -10331

29

Figure 1: Contact Pressure and Stress Concentration at Unsupported Free Edge

Figure 2: Local arc shape defined by two design variables

UnsupportedFree edge

Contact surface

SharpCorner

Stress concentration

Contact pressure

30

Figure 3: Scheme of grid search method

Figure 4: Binary representation to design variables

x1

x2

U2

L2

U1L1

Min

Min

Initial Grid Zoomed Grid

31

Figure 5: Schematic diagram for the proposed optimisation model

Pre-processing:Geometry Definition;Material Description;

Contact definition;Boundary condition;Finite element mesh.

FEM analysis

FE Solver

Post-processing:Obtain objective value

GAs parameters:Generation;Population;

Selection method;Crossover/mutation.

StartGA Evolution

Initialise Population

Current Generation

Fitness evaluationElitism strategy

SelectionCrossover/mutation

Termination criterion

NO

YES

Optimal Design

Integrated FEA

Using Python Script

Optimisation Model Based on GA

32

Figure 6: Wheel rail contact configuration at rail edge and arc shape

R

F

H

x

z

O

z

y O

Rail Edge

Wheel

Rail

d

l

Arc shape at rail edge

l

d

Side viewIso view

33

(a) Iso-von Mises stress plot for 20×20 grid

(b) Iso-von Mises stress plot for 10×5 grid

Figure 7: Profile of optimal result using GSM

Curve Depth (d)

Cu

rve

Len

gth

(l)

0.73P0

0.83P00.73P0

0.73P0

0.83P0

0.97P0

1.08P0

1.23P0

1.42P0

1.42P0

0.97P0

Minima 1

Minima 2

Local search Grid

Curve Depth (d)

Cu

rve

Len

gth

(l)

Minima 1

(d=1.5 l=40)

Curve Depth (d)

Cu

rve

Len

gth

(l)

0.76P0

0.91P0

1.24P0

0.91P01.24P0Minima 1

Minima 2

Local search Grid

Curve Depth (d)

Cu

rve

Len

gth

(l)

Minima 1

(d=3.0 l=70)

34

(a) Static wheel load F

(b) Static wheel load 0.2F

Figure 8: Evolution of optimal von Mises stress over generations

0.6

0.7

0.8

0.9

1

1.1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Optimal stress

Generation

Op

tim

al s

tres

s σ

emax

/P0

1.16P0

Initial generation

5th generation

0.94P0

10th generation

0.75P0

15th generation

0.692P0

20th generation

0.688P0

0.6

0.7

0.8

0.9

1

1.1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Optimal stress

Generation

Initial generation

5th generation

0.99P0

10th generation

0.74P0

15th generation

0.684P0

20th generation

0.682P0

1.12P0

Opt

imal

str

ess σ

ema

x/P 0

35

Figure 9: Optimal shape at the symmetrical plane of the rail edge

(a) Optimal rail end with sharp corner

(b) Rail end with sharp corner

Figure 10: Distribution of contact pressure at rail edge

0

0.5

1

1.5

2

2.5

3

-10 0 10 20 30 40 50

Optimal shape at load F

Optimal shape at load 0.2FRail edge

Optimal arc shape

l

d

z co

ord

ina

tes

(mm

)x coordinates (mm)

1.5

1.0

0.5

0.0

-0.52b

b

0

-b-2b 0

a2a

3a4a

P/P0=1.25

y x

P/P

0

Rail Edgex=0

1.5

P/P

0

0.0

0b

0y x

P/P0=1.49

1.0

0.5

-0.5

2b

-b-2b

a2a

3a4a

Rail Edgex=0

36

Figure 11: Contact pressure and deflection of the contact surface along the rail

symmetric plane (z=0)

(a) Optimal shape (b) Original shape

Figure 12: Distribution of von Mises stress in optimal rail end shape and original shape

2.0

1.5

1.0

0.5

0

0.05

0.10

0.15

P/P

0D

efle

ctio

n U

z(m

m)

x coordinate (mm)

Contact pressure distribution

Deflected profile

923.32

838.02762.81687.54612.23536.96461.66

386.32311.02235.77160.45

85.130.00

Stress (MPa)von Mises

x y

z

Rail Symmetric Surface

Railhead Surface

Rail Edge x y

z

1852.86

1698.851544.831390.821236.811082.80928.78

774.77620.76466.74312.73

158.720.00

Stress (MPa)von Mises

Railhead Surface

Rail Edge Rail Symmetric Surface

37

Figure 13: Detail of finite element model for dynamic analysis

Figure14: Impact force history at original and optimal rail ends shape.

Travel direction Arc shape at

rail edges

Joint gapRail 1Rail 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.005 0.01 0.015 0.02

OriginalShape

Optimal shape

t (s)

Co

nta

ct im

pa

ct f

orc

e F

imp

act/F

38

Figure 15: Time history of maximum von Mises stress at rail bodies during wheel

impact

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-50 -40 -30 -20 -10 0 10 20 30 40 50

Rail-1 Original

Rail-2 Original

Rail-1 Optimal

Rail-2 Optimal

GapTravel direction

Max

imu

m v

on

Mis

esst

ress

/

P0

m

axe

Wheel position x coordinates (mm)

39

(a) With Original shape

(b) With optimal shape

Figure 16: Snapshot distribution of von Mises stress under rolling wheel at x = 0.

1305

11971088979.3870.6761.9653.2

544.6435.9327.2218.5

0.00109.8

Stress (MPa)von Mises

x y

z

Railhead Surface

Rail Symmetric Surface

Gap

40

Figure 17: Loading positions in lab testing

Figure 18: Strain gauges positions and optimal arc shape at rail end

Rail Edge

3mm

0mm

-3mm

Load positions

Arc shape at rail edge

41

Figure 19: Lab testing setup

Horizontal Actuator

Vertical Actuator

Wheel

Specimen Location

Concrete Supporting

Mounted on loading frame

Vertical load