**JM Conradie, 2014**. "Finite Element Modelling of Off-Road Tyres"

Most tyre models developed to date require a fair amount of data before an accurate representation

of the tyre can be obtained. This study entails the development of a simplified, yet accurate, nonlinear

Finite Element (FE) model of an “off-road” tyre to study the behaviour of the tyre due to radial

loading conditions. The study aims to develop a FE tyre model that can solve fast and be accurate

enough to be used in multibody dynamic vehicle simulations. A model that is less complex than

conventional detailed FE models is developed.

The work explores the use of superimposed finite elements to model the varying stiffness in the

respective orthogonal directions of the sidewall and tread of the tyre. Non-linear elements defined

by Neo-Hookean or Ogden models and elements with different linear orthogonal stiffnesses are

superimposed onto each other to simulate the global material properties of the tread and the

sidewall of the tyre investigated.

The geometry of the tyre studied was measured experimentally using laser displacement transducers

and digital image correlation techniques. Material properties of segments of the tyre were obtained

by performing tensile tests on samples. Since the rubber slipped against the clamps during the

experiment, deformation of the segments was also measured using digital image correlation. These

geometrical and material properties were used as input to develop a finite element model of an “offroad”

tyre.

Measurements were conducted using laser displacement transducers, load cells mounted to

actuators, etc. to obtain accurate sidewall deformation profiles and global radial load vs.

displacement curves for different radial loading conditions. The data obtained from the results was

used to validate the tyre model developed.

Numerous analyses are performed with different combinations of moduli of elasticity in the

respective orthogonal directions of the sidewall stiffness and the tread to investigate its influence on

the global behaviour of the tyre model.

The main focus of the project was to develop a tyre model from data obtained from laser and

photogrammetry measurements in a laboratory that accurately represents tyre behaviour due to

radial forces. A finite element model that can simulate the effect of radial forced and obstacles on a

tyre was developed. The use of two subsets of elements, superimposed onto each other to simulate

global material properties of the rubbers, steel wires, polyester and nylon threads, was investigated.

The combination of material properties that gave the best fit for all the load cases investigated were

determined. The finite element model correlated well with the load vs. displacement graphs and

sidewall displacement profiles determined experimentally.

The solving time is still fairly high and is still not quite suitable for real-time dynamic simulation.

However, it solves faster than more complex tyre models where details of steel wires, etc. are

included in the model.

For future studies it is recommended that different element types be investigated in the tyre model.

The study proves that equivalent material properties can be used to simulate the composite

properties of the materials in tyres. Most tyres can be divided into a few regions that each has its

own material structure right through the region. These regions can be characterized by simple tests

and the input can be used as a first estimation of the tyre’s material properties for the model.

Accurate validation criteria should be used to validate the tyre model if time does not allow for

excessive testing of the material properties of all the rubber, steel wires, polyester threads, etc.

Geometric displacement data at various loading conditions can be used for validation of the tyre

model.

The model developed can be used to investigate the effect of different stiffnesses and other material

changes in the sidewall or tread of a tyre. Useful insight can be obtained from the finite element

model developed for dynamic simulation where the force vs. global displacement data is important.

Keywords: finite element analysis, finite element modelling, model updating, experimental testing,

non-linear modelling, non-linear materials, tyres, multibody, ride simulation

**Supervisors: Professors P.S. Heyns & P.S. Els**

**C Booysen, 2014**. "Fatigue Life Prediction of Steam Turbine Blades during Start-up Operation Using Probabilistic Concepts"

Fatigue in low pressure turbine blades has been recognised to be one of the primary causes of steam turbine blade failures worldwide. As a result various methods for predicting the fatigue life of the blades have been proposed. Application of these methods has traditionally been performed using deterministic models which often require overly conservative assumptions. However, given the range of uncertainty in key variables such as material properties, loading and damping; the question is then raised about the subjectivity in the selection of these parameters. An alternative approach is to incorporate probabilistic modelling which can eliminate overly conservative assumptions and allow for uncertainty in key variables to be accounted for. This necessitates the need for development of a probabilistic model which can be used in fatigue life calculations.

The objective of this work is to develop a methodology which can be used to assess the high cycle fatigue life of low pressure steam turbine blades during resonance conditions encountered during a turbine start-up by making use of probabilistic principles. Material fatigue properties are determined through experimental testing of used blade material X22CrMoV12-1 along with statistical modelling using regression analysis to interpret the stress-life diagram. A finite element model of a free-standing LP blade is developed using the principle of sub-structuring which enables the vibration characteristics and transient stress response of the blade to be determined for variations in blade damping. Rainflow cycle counting and a linear damage accumulation rule enabled fatigue damage accumulation during start-up condition to be quantified. Random curve fitting routines are performed on the fatigue and FEM stress data to ensure that the selection of the random variables used in fatigue life calculations is stochastic in nature. The random vectors are selected from a multivariate normal distribution.

Forced response analysis revealed that increased levels of blade damping significantly decreased the peak dynamic stresses which had a significant effect on the overall fatigue life. The use of confidence intervals in the probabilistic fatigue life model worked effectively in being able to account for uncertainty in the material fatigue strength parameters and varying stress in the blade root. Probabilistic modelling was shown to be successful in being able to predict the fatigue life of the blade as results are shown to reinforce those of the discrete life model.

**Supervisor:** Prof PS Heyns

**E.Jansen, 2014**. "Thermodynamic optimisation of an open-air solar thermal Brayton cycle"

This project mainly focused on the implementation of the second law of thermodynamics

relating to the design of heat-exchanging components in an open-air solar thermal Brayton cycle.

These components include one or more regenerators (in the form of cross-flow heat exchangers)

and the receiver of the parabolic dish where the system heat was absorbed. The generation of

entropy is under close investigation since the generation of entropy goes hand in hand with the

destruction of exergy, or available work. This phenomenon is caused by two factors, namely the

transfer of heat across a finite temperature difference and also the friction that is caused by the flow

of a working fluid in a system consisting of components and ducts. The dimensions of some

components were used to optimise the cycles under investigation. Entropy Generation Minimisation

(EGM) was employed to optimise the system parameters by considering their influence on the total

generation of entropy. Various assumptions and constraints were considered and discussed to aid in

the solution process, making it simpler in some cases and more feasible in others. The total entropy

generation rate and irreversibilities were determined by considering all of the individual components

and ducts of the system, and their respective inlet and outlet conditions such as temperature and

pressure. The major system parameters were evaluated as functions of the mass flow rate to allow

for proper discussion of the system performance. Ultimately, conclusions and recommendations

were made, which state the optimum system to be used in this type of solar application, where the

amount of net power output is the main driving factor.

Supervisors: Dr T Bello-Ochende and Prof JP Meyer

**MC Beckley, 2014**. "Comparison of Sampling Methods for Kriging Models"

This study aims to generate from a three-dimensional data set of carbon dioxide flux in the

Southern Ocean, a sample set for use with Kriging in order to generate estimated carbon

dioxide flux values across the complete three-dimensional data set. In order to determine

which sampling strategies are to be used with the three-dimensional data set, a number of

a-priori and a-posteriori sampling methods are tested on a two-dimensional subset. These

various sampling methods are used to determine whether or not the estimated error variance

generated by Kriging is a good substitute for the true error as a measure of error.

Carbon dioxide is a well known ”greenhouse gas” and is partially responsible for climate

change. However, some anthropogenic carbon dioxide is absorbed by the oceans and as such,

the oceans currently play a mitigating role in climate change by acting as a sink for carbon

dioxide. It has been suggested that if the production of carbon dioxide continues unabated

that the oceans may become a source rather than a sink for carbon dioxide. This would mean

that the oceanic carbon dioxide flux (exchange of carbon dioxide between the atmosphere

and the surface of the ocean) would invert. As such, modelling of the carbon dioxide flux is

of clear importance. Additionally as the Southern Ocean is highly undersampled, a sampling

strategy for this ocean which would allow for high levels of accuracy with small sample sizes

would be ideal.

Kriging is a geostatistical weighted interpolation technique. The weights are based on the covariance

structure of the data and the distances between points. In addition to an estimate at

a point, Kriging also produces an estimated error variance which can be used as an indication

of uncertainty. This study made use of model data for carbon dioxide flux in the Southern

Ocean. This data covers a year by making use of averaged data for 5 day intervals. This

results in a three-dimensional data set covering latitude, longitude and time. This study used

this data to generate a covariance structure for the data after the removal of trend and using

this covariance structure, tested various sampling strategies in two dimensions, sampling

approximately 10% of the two-dimensional data subset. These sampling strategies made use

of either the estimated error variance or the true error and included two simple heuristics,

genetic algorithms, the Updated Kriging Variance Algorithm and Random Sampling. Two

of the genetic algorithms tested were selected to maximise the error measure of interest, in

order to determine the full range of errors that could be generated. The percentage absolute

errors obtained across these methods ranged from 2.1% to 64.4%.

Based on these strategies, the estimated error variance was determined to not be an accurate

surrogate for true error and that in cases where absolute error is available, such as data

minimisation, absolute error should be used as the measure of error. However, if no data is

available then it does provide an easy to calculate measure of error. This study also concluded

that Addition of a Point at Point of Maximum Absolute Error does provide a good validation

sampling method to which other methods may be compared.

Additionally, based on true errors and computational requirements, three methods were selected

to be implemented on a three-dimensional subset of the data. These methods were

Random Sampling, Addition of a Point at Point of Maximum Absolute Error and Addition of

a Point at Point of Maximum Estimated Error Variance. Each of these methods for sampling

were performed twice on the data, sampling up to approximately 5% of the data. Random

Sampling produced percentage absolute errors of 21.02% and 20.98%, Addition of a Point at

Point of Maximum Estimated Error Variance produced errors of 18.54% and 18.55% while

Addition of a Point at Point of Maximum Absolute Error was able to produce percentage

absolute errors of 14.33% and 14.32%.

Supervisor: Prof. S Kok

**J. Crous, 2014.** " The influence of a coupled formulation on the fluid dynamics in a large scale journal bearing"

In the pursuit of more accurate diagnostics of turbo machinery sophisticated rotor and bearing models are to be developed in order to better understand the dynamics of the rotor-bearing system. This study is concerned with such bearing models.

Four distinct fluid models are developed: The first two have a viscous fluid formulation, where fluid dependencies enter the momentum equations primarily through the viscosity of the fluid. The last two have a viscoelastic fluid formulation where dependencies enter the equations through an additional differential constitutive relation. This constitutive relation is strongly coupled with the momentum equation.

The dependencies included in the formulation of the fluid are: pressure, shear rate and temperature. The coupling of the fluid models is subject to the dependencies present in the formulation. Uncoupled, weakly coupled and strongly coupled formulations are compared in this work.

The formulated models are solved numerically using the Finite Volume Method in the open source program OpenFOAM. These models were newly implemented in OpenFOAM as part of this study. The models are validated by comparing results with various known analytical solutions.

A region of the bearing is subsequently analysed, where the dependencies of the lubricant are most prominent. In this region the influence of a weak and strong coupled formulation of the fluid dynamics in the oil film was considered.

In this study it is shown that both weak and strong couplings influenced the fluid behaviour significantly. It is shown that when these dependencies are no longer isolated in the mere adjustment of fluid properties is inadequate to account for the influence of dependencies.

The weak coupled formulation shows the difference between the coupled and uncoupled formulations. The weak coupling influence the fluid dynamics to the same extent as the pressure dependency in the region considered. The departure from the classical formulation is however observed to be uniform in the case of a weak coupling.

The difference between the uncoupled and strongly coupled formulation was not as great as in the weakly coupled case. Although the difference was less, it was seen that the presence of the strong coupling was about 40% of that of the temperature dependency in the region considered. The change in flow, for the strong coupled formulation, was non-homogenous compared to the classical formulation.

The influence of the coupling is therefore different in nature. The weak coupling changes the flow more than the strong coupling compared to the classical formulation. The strong coupling introduce a new characteristic to the fluid behaviour not seen with the weak coupled formulation.

Lastly it is shown that in order to model the bearing adequately, the fluid model and the coupling of the governing equations are not trivial. Great care must be taken in both the fluid model used as well as the formulation of the coupled equations.

**Supervisors:** Prof PS Heyns, Dr J. Dirker

Natasha Botha, 2014. "**Effect of numerical modelling assumptions on the simulated corneal response during Goldmann applanation tonometry"**** **

It is widely known that Central Corneal Thickness (CCT) and Radius of Curvature (RoC) influence the estimated IntraOcular Pressure (IOP) obtained from Goldmann Applanation Tonometry (GAT). However, not much is known about the influence of corneal material properties, especially in a clinical setting.

Several numerical studies have been conducted in an attempt to quantify the influence of corneal material properties on the IOP. These studies agree that corneal material properties do influence the estimated IOP, which contradict the initial premise on which GAT was designed, namely that material properties do not influence the obtained GAT readings. Also, there is no consensus among these studies with respect to corneal material properties, thus a wide range of proposed properties exist.

A possible explanation for this range of available corneal properties is the numerical modelling assumptions used, which seem to be quite different. Different sets of experimental inflation test data were used to calibrate the constitutive models and different limbal boundary conditions were applied to simulate the experimental setup as well as *in vivo* conditions during GAT simulations. Therefore the purpose of this study is to determine whether these modelling assumptions influence the obtained IOP and ultimately the overall conclusions.

A Finite Element (FE) model of the human cornea is developed, implementing a constitutive model to represent the complex corneal structure and two limbal boundary conditions. This model is then calibrated using two different sets of experimental inflation test data. During calibration of the fibre reinforced elastic constitutive model it is found that independent of the assumptions made regarding the material coefficients, that the numerical inflation data compare well with the experimental data for all cases.

Using this model a GAT simulation is conducted to estimate the IOP and the influence of the modelling assumptions, cornea geometry and material properties are then investigated. The results indicate that the modelling assumptions, cornea geometry and material properties do influence the estimated IOP. However, when assuming the cornea ground substance stiffness to be constant, it is found that the influence on IOP due to material properties is not as significant. A correction equation is also proposed to account for the corneal geometric properties by calibrating the numerical model for a numerically normal cornea. This is done by utilising the various data sets which are obtained during the calibration of the constitutive model with the experimental inflation test data.

It is concluded that using only inflation data to calibrate the constitutive model is not sufficient to uniquely describe the corneal material. This is evident as different material data sets are obtained, even though the experimental inflation data is matched well for a variety of considered cases. Each of these material data sets, in conjunction with geometric properties, yield different estimates for IOP during GAT simulations.

This study therefore recommends the use of additional experimental data, such as strip extensometry, along with inflation test data to adequately calibrate a numerical model. It should also be noted that when modelling GAT care should be taken when considering the choice of limbal boundary condition, experimental data for calibration and assumptions made with regards to material coefficients, as these choices could potentially influence the outcomes and conclusions of a study.

**Supervised by Prof. Schalk Kok and Ms. Helen M. Inglis**

**FH Burger, 2014**. "Three-dimensional conductive heat-spreading layouts obtained using topology optimisation for passive internal electronic cooling"

In this study, topology optimisation for heat-conducting paths in a three-dimensional domain was investigated. The governing equations for the temperature distribution were solved using the finite volume method, the sensitivities of the objective function (average temperature) were solved using the adjoint method, and finally, the optimal architecture was found with the method of moving asymptotes (MMA) using a self‑programmed code. A two-dimensional domain was evaluated first as a validation for the code and to compare with other papers before considering a three-dimensional cubic domain.

For a partial Dirichlet boundary, it was found that the converged architecture in three dimensions closely resembled the converged architectures from two dimensions, with the main branches extending to the outer corners of the domain. However, the partial Dirichlet boundary condition was not realistic, and to represent a more realistic case, a full Dirichlet boundary was also considered.

**Supervisors:** Dr J. Dirker and Prof J.P. Meyer

**R.Scheepers, 2014**."A COMPARATIVE STUDY OF FINITE ELEMENT METHODOLOGIES FOR TORSIONAL VIBRATION RESPONSE CALCULATIONS OF BLADED ROTORS"

Turbo-generator trains are susceptible to torsional vibration which can lead to fatigue cracking

and failure. Methods are available for the measurement and calculation of the torsional natural

frequencies of these systems for the purpose of design, monitoring and life prediction.

Calculation methods are conventionally based on one dimensional (1D) finite element (FE)

methodologies which require the simplification of a number of aspects including the participation

of flexible blades in torsional vibration modes.

The accuracy of 1D, three dimensional (3D) and three dimensional cyclic symmetric (3DCS) FE

methods was investigated by the application thereof on a small test rotor. Experimental

measurements of static and dynamic vibration responses were conducted with rotation and

torsional forcing accomplished through the use of a DC motor and a digital control system

optimised for fast transient and stable steady state response. Blade stagger angle was

demonstrated to have a significant effect on torsional frequencies although no stress stiffening

effects were noted in the speed range considered. Similarly, damping was measured to decrease

with blade stagger angle but not with rotational speed. Step changes in torsional frequencies due

to the activation of the motor field and armature currents required optimisation of the motor

models for static and dynamic conditions. Shaft torsional vibration responses were found not to

include all blade modes and vice versa.

**Supervisor:** **Professor P.S. Heyns**

**EF Williams, 2014**. "Design and Analysis of a Practical Large-Force Piezoelectric Inchworm Motor with a Novel Force Duplicator"

The work presented in this dissertation on piezoelectric inchworm motors (IWM) is part of a process to gain an understanding of the design, analysis and testing of this smart actuator technology. This work will form the foundation of what will hopefully lead to the realisation of a production-ready IWM design to be used in energy-scarce, battery-operated Unmanned Aerial Vehicles (UAVs), and forms part of a larger national drive to expand the UAV industry in South Africa. Although the principles used in the design of IWMs are well known, a new innovation is employed. A novel way to increase the force capacity of IWMs without compromising on the speed or displacement when compared to conventional methods is shown to be effective, and was used for the first time on IWMs. The use of a simple design equation is demonstrated to be useful in predicting the load limits and step displacements. Challenges of finding a correlation between predicted and measured performance values are discussed and solutions are presented. The history of IWMs and some background on piezoelectricity are given for the reader not familiar with these. The use of micro ridges on the clamp mechanisms is explored. The effects of the control signals on the mechanism of the motor are discussed in detail and some important comments on electrical controllers are made. The emphasis is on designing a strong motor that capitalises on the high-force density of piezoelectric material.

**Supervisor: Prof NJ Theron
Co-supervisor: Dr Philip Loveday **

**Darshik Garach,2014.** "**Heat Transfer and Pressure Drop in Microchannels with Different Inlet Geometries for Laminar and Transitional Flow of Water"**

An experimental investigation of heat transfer and pressure drop in rectangular microchannels was conducted for water in the laminar and transitional regimes for three different inlet configurations. The inlet types under consideration were the sudden contraction, bellmouth, and swirl inlet types, and hydraulic diameters of 0.57 mm, 0.85 mm, and 1.05 mm were covered. The sudden contraction inlet type was investigated for both adiabatic, as well as diabatic cases. For this inlet type, adiabatic friction factors were predicted well by the laminar Shah and London correlation, while for diabatic cases this correlation resulted in over predictions of up to 15%. The critical Reynolds numbers were found to be between 1 800 and 2 000 for adiabatic cases, while for diabatic cases the transition regime commenced at a Reynolds number of about 2 000. The bellmouth inlet type was investigated for diabatic cases only. Here, laminar friction factors were also over-predicted by up to 15% by the Shah and London correlation and an early onset of transition was observed at a Reynolds number of 1 600. Correcting for the wall to bulk fluid temperature viscosity ratio improved on friction factor prediction accuracy. The swirl inlet type, which was only investigated for a hydraulic diameter of 1.05 mm and with a diabatic wall, exhibited the highest friction factors. Unlike the trends of conventional theory, Nusselt numbers in the laminar flow regime exhibited an increase with the Reynolds number. The axial conduction effect was found to affect results as the value for M was greater than 0.01as described by the literature. During transition, the bellmouth inlet type exhibited an increase in the friction factor and Nusselt number of up to 30% and 70% respectively, compared to the sudden contraction inlet type. For the swirl inlet type, increases of up to 72% and 120% for friction factor and Nusselt numbers respectively were achieved compared to the sudden contraction inlet type. After the onset of transition, Nusselt numbers approached the Gnielinski correlation predictions for turbulent flow. Based on the experimental data obtained in this study, a correlation was developed which describes the relation between the friction factor and Colburn j-factor. The correlation has a mean absolute error of 6% and predicts 92% of the results for all inlet types to within 15%.

**Supervisors: Dr. J. Dirker, Prof. J.P. Meyer**

**Christian M. Kangaj**,2014. "Pull-out of hooked end steel fibres from epoxy matrix: Experimental and Numerical study"

The reinforcement of concrete with steel fibres changes the failure of the composite material from catastrophic brittle failure to pseudo-ductile behaviour as a result of crack-bridging by the fibres, and the additional work which is absorbed by fibre pull-out. A good understanding of the properties of fibre-reinforced concrete depends on an understanding of fibre pull-out process. The main aim of the current study is to investigate, both experimentally and numerically, the pull-out behaviour of a single hooked end steel fibre from epoxy matrix, where epoxy was chosen to replace concrete in order to enable visualisation of the pull-out process. The experimental and numerical results both contribute to the development of a physical understanding of the mechanism of pull-out.

Experimental studies included the evaluation of the mechanical properties of hooked end steel fibre and epoxy matrix by means of tensile tests, the manufacturing of pull-out specimens consisting of a single hooked end steel fibre embedded in epoxy matrix, and the experimental characterisation of the fibre pull-out. The significant features (peaks and minima) of the load vs. displacement graph were correlated to stills taken from a video of the pull-out process, in which the plastic deformation of the fibre is evident. Small deformations (spalling) were also observed in the matrix. A model is proposed for the mechanisms which interact during the pull-out process.

Numerical analysis has been performed using a 3D finite element analysis implemented in ANSYS, incorporating nonlinear contact between the fibre and the matrix and plastic deformation of the fibre. Sensitivity studies investigated the effects of material and interface properties on the resultant load vs. displacement curve of the pull-out. Results of these studies demonstrated the importance of accurately modelling the plastic behaviour of the steel fibre in predicting the pull-out response of the system.

The numerical and experimental pull-out behaviours capture the same fundamental behaviour. The fibre displacement at critical intervals is well-correlated, and the numerically-predicted load is within the range of the experimental results at these intervals. The finite element analysis allows investigation of the stress and strain distributions. The plastic strain accumulated in different regions of the fibre, and the maximum stress experienced in the matrix are studied, and correlated with the global response, and further understanding is gained into the mechanisms of fibre pull-out.

**Supervisor: Mrs Helen M. Inglis**

**Co-supervisor: Prof Schalk Kok
**

**Mj Stallmann, 2014. " Tyre model verification over off-road terrain"**

Vehicle dynamic simulations form a significant part of the design and development process of vehicles. These simulations are used to study and improve the vehicle’s durability, ride comfort and handling capabilities. All forces acting on the vehicle are either generated in the tyre-road interface or are due to aerodynamic effects, where at low speeds the latter one can be ignored. The accuracy of the tyre model describing the forces on the tyre-road interface is thus of exceptional importance. It ensures that the simulation model is an accurate representation of the actual vehicle.

Various approaches are adopted when developing mathematical tyre models. Many of these models are developed to study the handling capabilities of passenger cars over a smooth road. Passenger car tyres are the focal point as larger tyres introduce some difficulties due to their size and load rating. Off-road truck tyres also differ in their construction which will influence force and moment generation of the tyre. Research efforts are increasing to meet the need of tyre models that can describe the behaviour of the tyre over uneven terrain with sufficient accuracy. This thesis addresses the question of whether existing mathematical tyre models can accurately describe the forces and moments generated by a large off-road tyre while driving over rough terrain.

The complexity of different mathematical tyre models varies greatly, as does the parameterisation efforts required to obtain the model parameters. The parameterization of most tyre models relies on some experimental test data that is used to extract the necessary information to fit model parameters. The selection of a suitable tyre model for a simulation is often dependent on the availability of such experimental data and the effort to identify the required parameters. In this study the parameterisation process for four different tyre models, are discussed in detail to highlight the difficulties in acquiring the test data and the effort to parameterize the model. The models considered are the One Point Contact, 3D Equivalent Volume contact, 3D Enveloping Contact and FTire model.

Experimental measurements are conducted on a 16.00R20 Michelin XZL tyre. Laboratory tests, as well as field tests, over discrete obstacles and uneven hard surfaces are used for parameterisation and validation purposes. Simulation results are compared to experimental test data to determine whether the models could be used to describe the tyre road interactions with sufficient accuracy. Recommendations are made for tyre model selection and model accuracy for simulations over rough off-road surfaces.

**Supervisor: ** **Prof. P.S. Els**

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