The study of fatigue is linked by many academics to the work of August Wohler, a German railway engineer, who conducted the first systematic investigation on fatigue in the middle 1850s. Peterson reviewed the discussion on fatigue problems during meetings of the Institution of Mechanical Engineers (IMechE) in Birmingham, held just before 1850. The study on fatigue has grown with developments in materials engineering. The development of new engineering alloys over the last century required engineers to look at fatigue performance of these alloys and this led to a much better understanding of their fatigue behavior in engineering applications. Peterson also refers to the concept of the 'endurance limit' as already defined by Wohler. Wohler is responsible for most of the concepts that form part of fatigue analysis today in engineering. The improvements in fatigue analysis became possible due to the development of essentially new experimental facilities, computers and numerical stress analysis. However, some of the basic concepts remained, such as that fatigue in metallic materials is due to cyclic slip and stress concentrations contribute to reduced fatigue endurance. The prediction of fatigue properties of structures and avoiding structural fatigue were recognized as engineering problems in the early decades
The study of fatigue is linked by many academics to the work of August Wohler, a German railway engineer, who conducted the first systematic investigation on fatigue in the middle 1850s. Peterson reviewed the discussion on fatigue problems during meetings of the Institution of Mechanical Engineers (IMechE) in Birmingham, held just before 1850. The study on fatigue has grown with developments in materials engineering. The development of new engineering alloys over the last century required engineers to look at fatigue performance of these alloys and this led to a much better understanding of their fatigue behavior in engineering applications. Peterson also refers to the concept of the 'endurance limit' as already defined by Wohler. Wohler is responsible for most of the concepts that form part of fatigue analysis today in engineering. The improvements in fatigue analysis became possible due to the development of essentially new experimental facilities, computers and numerical stress analysis. However, some of the basic concepts remained, such as that fatigue in metallic materials is due to cyclic slip and stress concentrations contribute to reduced fatigue endurance. The prediction of fatigue properties of structures and avoiding structural fatigue were recognized as engineering problems in the early decades
The study of fatigue is linked by many academics to the work of August Wohler, a German railway engineer, who conducted the first systematic investigation on fatigue in the middle 1850s. Peterson reviewed the discussion on fatigue problems during meetings of the Institution of Mechanical Engineers (IMechE) in Birmingham, held just before 1850. The study on fatigue has grown with developments in materials engineering. The development of new engineering alloys over the last century required engineers to look at fatigue performance of these alloys and this led to a much better understanding of their fatigue behavior in engineering applications. Peterson also refers to the concept of the 'endurance limit' as already defined by Wohler. Wohler is responsible for most of the concepts that form part of fatigue analysis today in engineering. The improvements in fatigue analysis became possible due to the development of essentially new experimental facilities, computers and numerical stress analysis. However, some of the basic concepts remained, such as that fatigue in metallic materials is due to cyclic slip and stress concentrations contribute to reduced fatigue endurance. The prediction of fatigue properties of structures and avoiding structural fatigue were recognized as engineering problems in the early decades
The study of fatigue is linked by many academics to the work of August Wohler, a German railway engineer, who conducted the first systematic investigation on fatigue in the middle 1850s. Peterson reviewed the discussion on fatigue problems during meetings of the Institution of Mechanical Engineers (IMechE) in Birmingham, held just before 1850. The study on fatigue has grown with developments in materials engineering. The development of new engineering alloys over the last century required engineers to look at fatigue performance of these alloys and this led to a much better understanding of their fatigue behavior in engineering applications. Peterson also refers to the concept of the 'endurance limit' as already defined by Wohler. Wohler is responsible for most of the concepts that form part of fatigue analysis today in engineering. The improvements in fatigue analysis became possible due to the development of essentially new experimental facilities, computers and numerical stress analysis. However, some of the basic concepts remained, such as that fatigue in metallic materials is due to cyclic slip and stress concentrations contribute to reduced fatigue endurance. The prediction of fatigue properties of structures and avoiding structural fatigue were recognized as engineering problems in the early decades 48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to of the 20 th century. It was understood that high stress concentrations could be harmful and should be avoided. The significance of stress concentration factors was known before 1950 and designers realized that the fatigue performance of a structure was dependent on improved detail design. Various models were developed for the prediction of notch and size effects. The proposed models included a great deal of empirism. One specific goal was to predict the fatigue limit, an important fatigue property for many products of the industry. Prediction problems can be defined in two categories. The similarity concept (sometimes called the similitude concept) is characteristic for the first category. Fatigue damage accumulation is based on the second category [1].
The current study has looked at various concepts and theories currently used in fatigue life prediction of engineering materials. Fatigue life prediction is a specialist field that should be taken seriously, and should not be solely based on prediction models but on material behavior as well. Different materials behave differently due to metallurgical differences and the effects of residual stresses within these materials. The effect of residual stresses on fatigue life of commercially pure grade 2 titanium alloy is discussed in great detail in [2] .
This paper focusses on the similarity concept and according to Schijve et al; the physical argument of the similarity approach is that "similar conditions, applied to similar systems should produce the same consequences". The physical principle is the basis of many predictions of properties of materials and structures and it can also be applied to fatigue prediction problems. At the same time, it should be realized that this physical principle does not necessarily imply that the physical mechanism of the fatigue phenomenon should be understood [1] .
The determination of fatigue life prediction models is a challenging undertaking and conventional methods were used to properly understand and explain the behaviour of the titanium alloy used in this study. The S-N curve formed the basis of fatigue life prediction conducted in this study. In terms of application, the findings from this study will prove useful to structural and mechanical design engineers making use of formed plates in their designs more especially when using the titanium alloy evaluated here. Table 1 shows the chemical composition of commercially pure grade 2 titanium alloy which consists of elements like carbon, oxygen, nitrogen, iron, hydrogen, and titanium. The percentage amount of these elements varies from manufacturer to manufacturer but the maximum percentage amounts should not be exceeded. The grade is weld-able, formable and has excellent corrosion resistance properties. The thickness of the material used was 3mm. 
The machine shown in figure 1 was used for the mechanical forming of commercially pure grade 2 titanium alloy plates used in this study. The prepared samples were affixed on jaws and then formed to the shape shown. The plates were formed using the stretch draw forming method to form an equivalent radius of curvature of 120 mm in order to make comparisons with the laser formed plate samples shown in figure 2 . 
Laser forming involves heating a sheet metal workpiece along a certain path with a defocused laser beam directed normal to the surface, and with or without a jet stream of cold gas or water emulsion cooling the path after the beam passes. During laser forming, a transient temperature field is caused by the irradiation and travelling of a laser beam. Consequently, thermal expansion and contraction take place, which give rise to deformation of the work piece. Laser forming has its roots in a traditional sheet metal forming method -flame bending, where the sheet metal is heated and deformed by an oxyacetylene torch [3] .
The laser-forming process is realised by introducing thermal stresses (without melting) into the surface of a workpiece with a high-power laser beam. These internal stresses induce plastic strains that bend the material or result in local elastic/plastic buckling. The laser-forming process is principally used at the macro level to form metallic sheet material. Research at the University of Liverpool (UK) and other research groups has shown that it can produce both parts with 2D bends or 3D, spatially formed parts [4-9].
The Rofin DY 044 Nd: YAG shown in figure 2 was used in this study. During this study only 10 samples could be irradiated per batch and the beam diameter was kept constant at 12mm to assess the influence of line energy discussed in [10] on the formation of the 120 mm radius of curvature. Table 2 shows the most important variables considered in this study. 
Residual stress measurement was carried out on the parent material, mechanically and laser formed plate samples in accordance with ASTM E837 which makes use of the hole drilling method. The hole drilling method relies on stress relaxation when a hole is drilled into the centre of the rosette strain gauge. The Average Stress Method was used in the analysis because this exercise was comparative and required no quantitative analysis.
The residual stresses found in these plates result from the manufacturing process and those introduced by the two forming processes. The surface on some of the samples in particular those irradiated at a line energy higher than 576 J/m were difficult to polish due to the effects of laser irradiation on the surface. The laser forming process caused surface hardening on the plates and an endmill designed for titanium was used to drill the holes, during the measurement of these stresses. Residual stress data is shown in table 3 and is for all the plate samples evaluated in this study.
The Avery 7303 fatigue testing machine used to determine fatigue life in this study is shown in figure 4 and is made up of an electrical motor with an adjustable double eccentric connecting rod with a bending arm transducer. A rigid bending arm is pivoted at one end to the machine frame at the main axis of rotation. The other end of the bending arm is given an oscillatory motion by means of a double eccentric flywheel mounted on the motor shaft through the medium of a connecting rod. This applies an alternating load to the specimen, the magnitude of which can be varied by adjustment of the eccentric. A static superimposed load can be applied to the specimen by raising or lowering the motor backing plate on its slide rails.
This alters the mean position of the bending arm. A rigid bending measuring arm is supported by flexible steel strips so that it is in effect pivoted at the main axis of rotation during operation. At the same time, the shortening of the bending specimens under load is allowed for, thus avoiding the superimposition of the axial load. The measuring arm lies normally in the same plane as the bending arm. The CMS Tecnocut Idro line water jet cutting machine was used in the preparation of fatigue samples. On average it took about three minutes to finish one batch of fatigue samples and minimal heat is generated during the cutting operation making this an ideal process for preparing samples. The hour-glass fatigue specimens were cut using a water-jet machine to the same dimensions for comparison purposes according to ASTM B593-96. After the water jet cutting process, the fatigue samples were polished using a flapper wheel and sand paper resulting in a mirror-like surface finish. The main aim of polishing the samples was to get rid of sharp corners and indentation marks formed due to the water jet cutting process. The plates obtained from the parent material were used to benchmark the performance of all the formed plates.
It should be noted that most fatigue failures occur at stresses below the yield strength of the material. Other variables that affect the propensity of fatigue include stress concentration, corrosion, temperature, overload, metallurgical structure and residual stresses. The aim of fatigue testing therefore was to: (A) determine the fatigue life of specimens subjected to alternating bending loads; (B) determine the number of cycles to failure of the plates on the three stress levels (95%; 85% and 65% of the yield stress), and (C) be able to predict the influence of the forming processes on the fatigue life of the material. The fatigue plate samples were subjected to varying constant amplitude stress, which alternated between equal positive and negative values. Fatigue data was then collected, analysed and used in plotting the S-N curve diagram. Figure 6 shows some of the fatigue samples used in this study, while The tensile testing of the parent material was done in accordance to the American Society for Testing and Materials (ASTM) E8/E8M test method. A Hounsfield tensile machine was used to test the samples and the resulting data made available using computer software of the machine.
According to the Soderberg failure criteria, if a point on the combined stress is below the Soderberg line then the component will not fail. With the Goodman line if the point on the combined stress is below the relevant Goodman line, then the component will not fail. If the point on the combined stress is below the Gerber line then the component will not fail. If the alternating stress is low and constant but the mean stress continues to increase, failure is bound to happen in the fatigue plates.
If the alternating stress is varied and the mean stress remains constant, failure will occur. Also if the load ratio is maintained, and there is an increase in alternating and mean stresses failure will occur in the plates. The variables used in fatigue testing need to be balanced in all the failure methods used and it must be ensured that these do not go over the respective lines to avoid possible failure. It should also be noted that these failure methods do not consider the effect of residual stresses which are now considered in this study. Figure 7 shows the Soderberg, Gerber and Goodman design criteria. 
Regression constant distance (a) Appendix A (Goodman's criterion) and B (Gerber's criterion) both have the regression constants, the actual applied stress, the mean stress and the actual average number of cycles to failure in common. Appendix A shows the Goodman's criterion which includes the predicted number of cycles to failure and the actual average number of cycles to failure for parent and mechanically formed plates. There is an underestimation in the predicted number of cycles to failure, which would force the design engineer to maximise the factor of safety in engineering components to be made.
The theoretical total fatigue stress results from adding the bending, fatigue and residual stresses. The concept of total fatigue stress is discussed in great detail in [2] . When all the above-mentioned stresses are considered, the predicted number of cycles to failure changes resulting in a reduction in the predicted number of cycles to failure. Figure 8 shows the Goodman's criterion for the mechanically formed plates and the figure enables us to see the differences between actual and predicted fatigue life values. The lowest stress level had a prediction that was 34% of the actual number of cycles to failure. The stress of 270MPa had a predicted fatigue life of 11% of the actual number of cycles to failure. The highest stress had a predicted fatigue life of 9% of the actual number of cycles to failure. All these values are lower than those obtained from actual testing. The parent and mechanically formed graphs look similar when one considers the predicted number of cycles to failure. The laser formed plates followed a similar pattern where the predicted number of cycles to failure were all lower than the actual number of cycles to failure. This also applied to both Goodman's and Gerber's criterion evaluated here. It should be noted that for the Goodman's criterion, the theoretical total fatigue stress was used in all the predicted values and this gave a true reflection of what the samples were exposed to during fatigue testing as all the stresses are considered in the analysis. actual number of cycles to failure respectively as indicated in appendix A. The parent material predicted values were higher than all the plates evaluated. This is due to low residual stress values in the parent material when compared to the rest of the plates. (Laser formed at 2.5kW) The predicted number of cycles to failure at 2.5kW is also less than 15% of the actual number of cycles to failure between the highest stress (320MPa) and a stress of 270MPa. The lowest stress level of 220MPa has a predicted number of cycles of less than 8%, while the highest stress level has a predicted number of cycles of less than 5% when compared to the actual number of cycles to failure.
The high laser powers all fall under 10 000 cycles at 320 and 270MPa respectively. This also shows that the stress applied approaches the ultimate tensile strength of the material and also higher than the yield strength of the material. This is due to the inclusion of the theoretical total fatigue stress in the analysis. The inclusion of this stress was a way of acknowledging the effect of physical changes in the material used in this study.
For the Goodman's criterion, the inclusion of the theoretical total fatigue stress results in a low number of predicted number of cycles to failure as a result. The behavioural pattern of samples analysed by Gerber's and Goodman's criterion is similar with minor differences in the predicted number of cycles to failure at various stress levels. High cycle fatigue requires more than 10 000 cycles to failure and values obtained point to low cycle fatigue as most values are lower than this number for the actual stress amplitudes of 321 and 270MPa especially in laser formed plates. The low-cycle fatigue regime now prevalent in most laser formed plates comes into effect as a result of using the Goodman's and Gerber's life prediction models which assume high load amplitudes. This means the stress levels used in testing are higher than what they should have been, therefore negatively influencing the outcome of the prediction in both prediction methods used.
As a result of considering the theoretical total fatigue stress, the stress values used in the analysis are higher than the yield stress (338MPa) of the material which makes the analysis a bit complicated, as it changes the initial intentions of the study. The Gerber's criterion uses a different formula to calculate fatigue strength and the number of cycles to failure but this formula is an adaptation of the Goodman's formula as can be observed from the results. The higher number of cycles to failure at low stress levels is to be expected in most engineering metals undergoing fatigue testing. The life prediction models, do not take into consideration physical material changes due to the forming processes on the fatigue samples. Perhaps this information could be used in future studies of titanium and other engineering materials if there is not enough time to do extensive testing of samples. The use of the prediction methods together with the theoretical total fatigue stress help in the understanding of the underlying concepts.
Predicting the life of parts stressed above the endurance limit is at best a difficult procedure. The Juvinall and Marshek method can be used in the prediction of fatigue life for a variety of engineering materials [10] . In this method a point at 1000 cycles [S1000] is established on the vertical axis [Sn] and then a straight line from this point is drawn to an estimated fatigue limit [Se] at a specified number of cycles (usually 1x10 6 cycles). Using the applicable mean stress relationship, the completely reversed stress amplitude [Sn] is determined. Although the 10 3 cycle fatigue strength is computed as being 90% of the ultimate strength, the actual strength is not high. Loads large enough to cause failure in 10 3 cycles usually cause significant yielding, resulting in actual stresses that are lower than calculated values. For the large percentage of mechanical and structural parts subjected to randomly varying stress cycle intensity (for example, automotive suspension and aircraft structural components), the prediction of fatigue life is further complicated [11] . Failure of engineering structures under cyclic loading occurs mostly due to fatigue. The prediction of fatigue life is very important for such structures and since real structures nearly always experience an irregular stress history it seems more reasonable to predict their fatigue lives by a probabilistic analytic approach [12] . Titanium and its alloys behave like steel in that they tend to exhibit a true endurance limit in the range of 10 6 to 10 7 cycles with the endurance limit between 0.45 and 0.65 of the tensile strength [11] .
In predicting the fatigue life, information gathered from various life prediction methods was considered and the Juvinall and Marshek method was chosen as it gives a more realistic predicted number of cycles to failure when compared to the actual results of the parent plate and the forming processes investigated in this study. The fatigue test is made to determine the behaviour of materials under fluctuating loads, simulating what could happen in service and the data obtained from this test is tabulated and presented in an S-N diagram. Fatigue plates were obtained from the parent material and also processed plates (laser and mechanically formed plates) to determine the fatigue life of the plates. The plates obtained from the parent material were used to benchmark the performance of all formed plates. The parent material plate samples were more ductile, had lower hardness values making it possible to achieve a higher number of cycles to failure compared with laser formed plates. Each forming process introduces residual stresses into the parent plate. The amount of these stresses within the parent plate was less than those obtained in all the formed plates evaluated and this influenced the fatigue performance. The presence of higher values of residual stresses was detrimental to the fatigue life of laser formed plates. As far as the laser formed plates are concerned, the amount of residual stresses within plates differs with changes in line energy (as a result of changes in processing parameters). Figure 11 shows a plot resulting from the evaluation of parent plate specimens during fatigue testing. Fatigue life prediction is therefore based on the straight line relationship of the evaluated parent plate samples. The following formula shows the resulting line obtained from using the Juvinall and Marshek method for the parent material:
Where: a S = Alternating stress in MPa and, f N = Number of cycles to failure By using this formula, fatigue life of the parent material can be predicted to give an approximation when no fatigue data is available. The alternating stress ( a S ) is interchangeably used with the theoretical total fatigue stress (TFS) or test stress for the purpose of fatigue life prediction in this study. For an alternating stress of 332MPa, a fatigue life of approximately 38600 cycles should be expected. A lower stress value will give a higher fatigue life while a higher stress will result in lower fatigue life. In this study residual stress was measured in samples obtained from batches of laser and mechanically formed plates as well as the parent material. The concept of the theoretical total fatigue stress is discussed in detail in [2] .
When residual stresses are considered the actual stress the plate is exposed to during fatigue testing increases considerably. The increase in the percentage of stress applied in plates exposed to fatigue testing resulted in a reduced number of cycles to failure for the commercially pure grade 2 titanium alloy plates. The same concept could also be applied to other engineering metals leading to better fatigue life prediction on plates subjected to fatigue testing. In determining the fully reversed alternating stress values, the residual stress values obtained from parent plate and formed plates were used in conjunction with the alternating stress. This incorporation of residual stresses increased the overall fatigue stress in the samples evaluated. The tensile and compressive residual stresses were incorporated in the calculation of the mean residual stress [2] .
Engineering components and structures are often subjected to cyclically varying loads with mean stress (or mean strain). It is well known that the fatigue process is sensitive to superposed tensile mean stress in both the high cycle fatigue [13, 14, 15, 16] and low cycle fatigue regimes [17, 18] . This can be explained by noting that a positive mean stress increases the crack opening and accelerates the fatigue damage accumulation, Xia et al [19] . The preceding statement justifies the inclusion of mean residual stress in this fatigue life prediction exercise. The theoretical total fatigue stress is a combination of the bending stress, mean stress and mean residual stress. The fatigue life results obtained from the testing of laser formed plate samples show a major influence of the residual stresses on the fatigue life of titanium. This is due to the process parameters used, the cooling method and physical characteristics of this alloy.
Appendix C shows life prediction on the parent, mechanical and laser formed plates where the actual values are compared with prediction 1 and 2. Prediction 1 includes the theoretical total fatigue stress while prediction 2 is made up of just the test stress. Prediction 1 for the parent material has an over estimation of 2% when compared to actual test values at the highest stress. There is also an under estimation of 40% at the intermediate stress whilst the lowest stress has an over estimation of 22%. The mechanically formed plate samples on the other end have an under estimation for the highest, intermediate and lowest stresses of 7%, 30% and 3% respectively. Considering only the test stress results in an over prediction of fatigue life when compared to actual test data. The changes and unpredictability in material behaviour is due to the fact that some plates were tested at a stress much higher than the yield stress of the material. The theoretical total fatigue stress results in an under prediction in fatigue life in the plates evaluated. On initiating the study none of this information was considered as all the plates were tested at values lower than the yield stress of the material. Had this been considered in the test, the results could have been different. The actual results could have reflected the fatigue life shown by plates where the theoretical total fatigue stress is considered. The samples in this study were evaluated in the same manner and the inclusion of the theoretical total fatigue stress reduced fatigue life as all the active stresses in fatigue testing are included. In instances where there is no theoretical total fatigue stress there is a marked improvement in the predicted fatigue life (Prediction 2). The importance of the inclusion of both the alternating and residual stresses in fatigue analysis ensures that all the participating stresses are considered thereby reflecting a true picture of fatigue life in the plates analysed.
The objective of this study was to test fatigue plate samples at 95%, 80% and 65% of the yield strength and therefore the inclusion of the total fatigue stress reduced fatigue life as it increased the theoretical load on the samples. This meant that the tests were performed at values much higher than the yield strength of the material. The theoretical loading on other samples was higher than the yield stress and on the verge of approaching the ultimate tensile strength of the material. The inclusion of all the stresses gives a realistic predicted number of cycles to failure and is therefore necessary in the field of fatigue life prediction.
The fatigue life results obtained from the laser formed plates show a lower fatigue life when compared to those tested below the yield stress of the material in most of the plates evaluated. Beyond the yield stress, the behaviour is no longer linear elastic and the material behaves plastically, which means that anything that happens subsequently to this depends on whether the material is ductile or brittle. Plates that received little or no laser irradiation behave in a ductile manner while those that were fully irradiated behave in a brittle manner and in the process affecting the fatigue life. This observation is based on the results obtained from this study and quite obvious when looking at the influence of residual stress values which are directly proportional to the amount of irradiation each plate received.
Residual stress values were the main drivers of the theoretical total fatigue stress and based on this analysis the consideration of the theoretical total fatigue stress in [2] is of paramount importance in the fatigue life prediction of commercially pure grade 2 titanium alloy plates. From the perspective of applied cyclic stresses, fatigue damage of a component correlates strongly with the applied stress amplitude or applied stress range and is also influenced by the mean stress (a secondary factor). In the high cycle fatigue region, normal mean stresses have a significant effect on fatigue behaviour of components and these are responsible for the opening and closing of micro cracks. Since the opening of micro cracks accelerates the rate of crack propagation and the closing of micro cracks retards the growth of cracks, mean stresses are beneficial in terms of 
Fatigue Life (220 MPa) Fatigue Life (270 MPa) Fatigue Life (321MPa) fatigue strength. There is little or no effect of mean stress on fatigue strength in the low-cycle fatigue region in which the large amounts of plastic deformation erase any beneficial or detrimental effect of a mean stress [14] .
The high theoretical total fatigue stress values result in a lower average number of cycles to failure in all the formed plates. The average number of cycles to failure obtained change the research approach from being that of a stress-based approach to a strain-based one. The strain-based approach differs significantly from the stress-based which emphasizes nominal (average) stresses, rather than local stresses and strains. The strain-based approach employs elastic stress concentration factors and empirical modifications thereof. Employment of the cyclic stress-strain curve is a unique feature of the strainbased approach as it is the use of a strain versus life curve, instead of a nominal stress versus life (S-N) curve.
The strain-based approach was developed in response to the need to analyze fatigue problems involving fairly short fatigue lives. The main focus of the study was the stress-life approach which was unfortunately changed by the results obtained resulting in a combination of the two life approaches.
The following observations were noted in the study of fatigue life prediction of laser and mechanically formed plates.
The theoretical inclusion of all the stresses involved in fatigue testing ensures realistic results which resemble actual fatigue data.
The predicted average number of cycles to failure in all the forming processes investigated is considered acceptable due to the fact that these give realistic values when compared to actual test data.
The theoretical inclusion of both mean and residual stress in fatigue life prediction models makes provision for the most influential factors in fatigue testing. The Gerber's and Goodman's fatigue life prediction methods do not provide realistic fatigue life values. This is evident when results from these methods are compared to actual test results.
The incorporation of the theoretical total fatigue stress presents a new dimension in the fatigue life prediction methods presented here and needs to be analysed in future studies of engineering materials. Unlike the Goodman's and Gerber's failure methods which consider only the mean and alternating stresses, the theoretical total fatigue stress includes all known stresses available during fatigue testing including those within the material.
• The Nelson Mandela University (NMU) for technical assistance and the National Research Foundation (NRF) for financial assistance.
• Professor Patrick J McGrath (NMU) for all the motivation to do my best in my studies.
Appendix B