MASTER'S THESIS. Robustness Analysis of Welding Simulations by Using Design of Experiments. Pirjo Koivuniemi. Luleå University of Technology

Transcription

1 MASTER'S THESIS 28:1 CIV Robustness Analysis of Welding Simulations by Using Design of Experiments Pirjo Koivuniemi Luleå University of Technology MSc Programmes in Engineering Engineering Physics Department of Applied Physics and Mechanical Engineering Division of Material mechanics 28:1 CIV - ISSN: ISRN: LTU-EX--8/1--SE

2 Preface This work has been conducted as a Master s Thesis at Luleå University of Technology at the division of Material Mechanics. I would like to thank Professor Lars-Erik Lindgren and my supervisor Andreas Lundbäck. Finally, I would like to thank my sister Jessica; the words are not enough to describe what you mean to me, I love you, and Jens, for your outstanding encouragement, support and help throughout this work. Pirjo Koivuniemi Luleå, May 28 i

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4 Abstract In this master s thesis robustness in welding has been investigated by means of simulations. A simple welding case has been simulated with the FE program MSC.Marc. If stability can be discovered during a simulation, then the welding process can be redesigned. However, instability in the simulation must not be due to instability in the real welding process. It can be due to pure numerical and modelling causes also and therefore care must be exercised when evaluating robustness. The study also includes a review part about the meaning of robustness and different approaches for evaluating this. In this work, a parametric study was used to investigate the robustness for an existing model. The material used in the simulations is called Greek Ascoloy, a martensitic stainless steel and the geometry consists of two plates which are welded together without additional filler material. The aim was to investigate which of the chosen parameters has the largest influence on the model, i.e. how sensitive or robust the model is when it is subject to disturbances. The chosen parameters to investigate were Poisson s ratio, Young s modulus, thermal expansion, initial yield stress, minimum yield stress, welding power, and the heat source geometrical parameters. To evaluate the results and determine whether or not the model is robust for a given set of parameters, the responses of the residual stresses in the sheet plane as well as gap and out of plane deformations were studied. The study was performed by using a factor screening method and each investigated parameter has been investigated at three levels. The methodology has been to change one parameter at a time while keeping the other parameters constant. The results have shown that the simulation model is quite sensitive to changes in welding power which resulted in increased deformations when the power was set to a high level. This will of course also occur in a real welding operation since the welding power has a big influence on the deformations of the welded component. The mechanical activation and thermal expansion, both resulting in large differences of stresses and deformations between the different levels, have also shown to have a negative influence on the robustness of the model. The heat source geometrical parameters have not influenced the model to large extent, although reducing the c r parameter showed an increased gap deformation. The model has also shown to be robust regarding changes in Young s modulus, Poisson s ratio and yield stress since very small changes in stresses and deformations occurred when changing these parameters. One possible continuation of this work could be to study the influence of interaction between the different parameters on the robustness of the model. Also, an in-depth investigation to explain why the model is more sensitive to changes in certain parameters would also be of interest. ii

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6 Table of contents 1. Introduction Background Literature review Method Finite element model Welding simulation Design of experiment Experimental plan Results Influence of different parameters on stresses and deformations Discussion Conclusions Future work References Appendix iii

7 Nomenclature ν Poisson s ratio Ε Young s modulus [MPa] Th ε Thermal expansion σ y Yield stress [MPa] σ y min Minimum yield stress [MPa] Q Welding power [W] a Heat source parameter [mm] b Heat source parameter [mm] c f Heat source parameter [mm] c r Heat source parameter [mm] iv

8 1. Introduction This section describes the background to this work and the methodology to how the work was conducted. A literature review has also been performed and is presented here. 1.1 Background This work is a part of an EU-project called Virtual Engineering for Robust Manufacturing with Design Integration (VERDI) and is coordinated by Volvo Aero. The aim of the project is to develop the virtual manufacturing of structural aero engines components in cooperation with other European manufacturers, universities and research institutes. The goal is to contribute to global leadership for the European aeronautics industries. Virtual simulation means that by using computer aided design and simulation at an early stage it will be possible to see how the components are affected in the manufacturing process. This enables the possibility to study all stages in the development of engines in a virtual environment which saves development time and money. A lot of work has been done within the field of weld simulation and research in this area has been conducted for over 3 years. Despite this fact there are still many questions that remain unanswered. As the performance of computers has increased and the software has improved more industrial problems have become possible to solve. One problem in weld simulations is that during the process, instability problem can occur, such as buckling. After the simulation is finished the structure can be in stable equilibrium. This makes it necessary to investigate if there are any instability problems during the entire simulation. The aim of the current work is to evaluate different methods that can be used for this purpose. The work consists of a literature review about robustness, an investigation whether a welding procedure is robust by using finite element (FE) analysis and to find which material parameters affected the deformations and stresses. The simulations are performed in a commercial FE software called MSC.Marc. 1.2 Literature review Evaluation of robustness When evaluating the robustness and stability in welding simulations it mainly concerns the mechanical behaviour. The concept of robustness and stability has no clear borderline [1]. There are different methods which can be used to evaluate robustness in a process; one method is called Design of Experiments, which is a statistical method to plan and conduct an experiment. The second method is to perform an eigenvalue analysis which can predict when instability occurs in the process. A process is robust if the performance of the system is within the 1

9 specifications made for the system and small changes in input gives small changes in output for the process. A process which is not robust because of unexpected variations can result in unsatisfactory product performance, low production yields, and increased product cost Design of Experiments An experiment can be defined as a test or series of tests in which controlled changes are made to the input variables of a process or system with the objective to analyze the variation induced by these changes in the output response. Experiments are used in order to simulate a part of, or an entire real process with the aim of increasing the knowledge about how it works. The benefits of using experiments are e.g. higher output from the process, lowered costs and reduced variability. A process or system can be visualized as a combination of operations, machines, methods, people and other resources that transform some input into an output as shown in Figure 1. Controllable factors x 1 x 2 x 3 Input Process Output z 1 z 2 z 3 Uncontrollable factors Figure 1. Schematic of a process with an input, output and different factors The statistical design of experiments (DOE) is an efficient method for planning and conducting experiments so that the data obtained can be analyzed by statistical methods to yield valid and objective conclusions. DOE is based on three experimental design principles such as randomization, replication and blocking, and they are part of every experiment. Randomization is the cornerstone in experimental design. Randomization means that the trials in an experiment are running in random order. This allows the effects of extraneous factors to averaging out each other. Replication means that each factor combination is independently repeated. This allows the experimenter to estimate the experimental error, and to improve the precision in the estimated effect of a factor. Blocking is a method which is dealing with nuisance factors and is extensively used in industrial experimentation [2]. 2

10 Identifying the process DOE begins with recognition and statement of the problem, where all ideas about the objectives of the experiment are developed, the input should be collected from all concerned parties. For this reason a team approach is recommended. It is important to keep the overall objective in mind; for example, if the process or system is new, then the initial objective is likely to be a screening experiment. Characterization or screening experiment are used in the early stages of experimental work, when many factors are investigated, the objective is to identify the influential factors which have large effects on the response. If a system already has been characterized the objective may be optimization. When optimizing a process the aim is to determine the region in the important factors that leads to the best possible response. Another possible objective of an experiment is to test robustness for products and processes, that is, under what conditions do the response variables of interest degrade or how can the variability be reduced in the response variable that arises from external sources. When formulating the problem it is important to be conscious that it is unwise to design a single, large, comprehensive experiment at the start of a study. A single comprehensive experiment requires knowledge of the influential factors, the ranges over which these factors are varied, the appropriate number of levels for each factor, and the proper methods and units of measurement for each factor and response. Generally, we do not perfectly know the answers to these questions at the beginning of the experiment, but we learn the answers as an experimental program progresses. Throughout the entire process it is important to keep in mind that experimentation is a part of a learning process where the hypotheses are tentatively formulated about a system. Experiments are performed to investigate these hypotheses and on the basis of the results formulate new hypotheses and so on. This suggests that experimentation have iterative or sequential approach. Consequently, and as a general rule, no more than about 25 percent of the resources of experimentation (runs, budget, time, etc.) should be invested in the initial experiment. Often these first efforts are just learning experiments, and this will ensure that sufficient resources are available to perform confirmation runs and ultimately accomplish the final objective of the experiment [2]. Selection of factors, levels, range and response variables When the objective of an experiment is determined, the next step is to make the choice of factors, levels, and ranges, and to make the selection of the response variable, or the output of the experiment. A factor is the variable which is studied in the experiment, that is, which has influence on the output response of the process. Factors may be quantitative or qualitative. A quantitative factor is one whose levels can be associated with points on a numerical scale, such as temperature, pressure or time. The levels for the qualitative factors cannot be arranged in order of magnitude. Typical qualitative factors are operators, batches of raw material, and shifts, and the factors are not ranked in any particular numerical order. These factors can be classified as either design factors or nuisance factors. The design factors are the variables which are selected for study in the experiment. A nuisance factor is a design factor that probably has an effect (variability) on the response, but these 3

11 effects are not of interest. For example, batches of raw material, days of the week etc. These factors are often classified as known and controllable, known and uncontrollable, or noise factors. Blocking is used when the nuisance factor is controllable, that is, the levels may be set by the experimenter. When the factor is uncontrollable, but it can be measured, that is, the value for the factor can be observed on each run of the experiment, its effect can be compensated in the statistical analysis by using the analysis of covariance. A noise factor is unknown and uncontrollable, that is, the experimenter does not know that the factor exists and it may be changing levels while the experiment is progressing. In such situations, the objective is to find the settings of the controllable design factors that minimize the variability transmitted from the factor. This is called a process robustness study or a robust design problem. When design factors have been selected, the number of levels for each factor, and the specific levels at which runs will be made must be chosen. If the objective of the experiment is factor screening or process characterization, the number of factor levels should be held low. Generally, two levels work very well in factor screening studies. Thought must also be given to how these factors are to be controlled at the desired values and how they are to be measured. The ranges or region over which each factor will be varied must also be chosen. In factor screening the region should be large, that is, the range over which each factor is varied should be broad. When the experimenter learn more about which factors are important and which levels gives the best results, these region will become narrower. When selecting the response variable, or the output response, the experimenter has to be certain that this variable really provides useful information about the process under study. Multiple responses can occur. A response variable may be discrete or continuous. If the set of all possible values of the variable is either finite or countable infinite, then the variable is discrete, whereas if the set of all possible values of the response variable is an interval, then the variable is continuous. To organize the generated information the cause-and-effect diagram or the fishbone diagram can be a useful method to use. In this diagram the response variable of interest is drawn along the spine of the diagram and the design factors are organized in a series of ribs [2]. Selection of experimental design If the pre-experimental planning above are done correctly, then the next step is the choice of experimental design. Choice of design involves determining of sample size, that is, number of replicates to run. Generally, increasing the number of replicates increases the sensitivity of the design. The experimenter must also select the suitable run order for the experimental trials and determine whether or not blocking or other randomization restrictions are involved. There are different types of experimental designs, one of the most important and widely used in industrial experimentation are the factorial designs. These designs investigates the effects of two or more k factors, each at two levels, the design is called a 2 k factorial design. In these designs the factors are varied together instead of one at a time which means that possible interaction between the factors can be observed. The interaction cannot be observed in a one-factor-at-a-time approach. If the number of factors is increased the total number of experiments will become very high since it increases 4

12 exponentially. To reduce the number of runs it is possible to use something called a fractional factorial design where only a part of all the possible runs are performed. This method is widely used in industrial R&D for e.g. screening experiments. The opposite, i.e. performing all possible runs, is called full factorial design. As an example consider two factors, A and B, each at two levels. The levels are denoted low (-) and high (+) and all possible combinations of the two factors across their levels are used in the design. This type of factorial experiment is called a 2 2 factorial design. The design is geometrically represented as a square where the four runs are forming the corners of the square as shown in Figure 2. The effects of interest in this design are the main effects A and B and the interaction AB. The main effects are defined as the change in mean response when the factor is changed from low to high level [2]. Factor B High (+) Low (-) Low (-) High (+) Factor A Figure 2. Example of a 2 2 factorial experiment and the response shown in the corners Conducting the experiment When the actual experiments are to be conducted it is extremely important that everything is done according to the previously determined experimental plan. This is crucial to ensure that the results obtained are valid. To reduce the risk of producing useless results it is a good idea to perform some trial runs. This will, besides gaining experience on the experimental equipment, also provide information about e.g. the experimental error that can be expected and the consistency of the materials being used [2]. Statistical analysis of the data In order to provide objective conclusions based on the obtained data it is recommended that a statistical method is utilized to analyze the data. This can very often be accomplished by using some software package. Some useful ways of analyzing the data could be hypothesis testing, confidence intervals and residual analyses. There are of course several ways of presenting the obtained results and some examples are graphical representation and deriving empirical models [2]. 5

13 Conclusions and recommendations Finally some conclusions or practical recommendations are to be drawn based on the statistical analysis results. If there is still some uncertainty connected to the obtained results some additional tests to confirm that results can be performed. These suggestions will hopefully lead to a greater knowledge and/or optimization of the process in question [2] Design of Experiments in welding simulation The DOE methodology can also be used in welding simulations. For example, the objective of the experiment is factor screening, i.e. to identify the influential factors which have large effects on the response. First influencing parameters (factors) have to be selected. Thereafter the number of levels for each factor and the specific levels at which the runs will be made must be chosen. The number of factor levels should be held low and two levels for each factor work well. The ranges over which each factor will be varied must also be chosen and in this case the region should be large. Selection of the response variable, or the output, is also made. Following the DOE methodology, a set of experiments are determined [1] Eigenvalue Analysis Eigenvalue analysis can be used to find critical load factors at which a structure becomes elastically unstable. In most buckling analysis the classical Euler buckling analysis is used to predict the theoretical buckling strength of an ideal elastic structure. The buckling problem is in the form of a linear eigenvalue problem in which the buckling load is the eigenvalue and the corresponding eigenvector is the buckling mode, i.e. where buckling is expected to occur. Definitions of stability and instability The terms stability and instability are used in various contexts by many people in a variety of occasions. In mechanical systems the terms refer to structures carrying loads where stability and instability are associated with the behaviour of the system or the particular equilibrium state of the system. The system is in stable equilibrium state if relatively small changes in the loading or perturbations in the system parameters, including changes in the system geometry or in its boundary conditions, causes only small changes on the state of the system and it tend to return to its original equilibrium state. In an unstable system slight changes in the system parameters or the conditions related to that state would cause major changes in the existing state and force the system away from that equilibrium state. To illustrate the concepts of stability and instability equilibrium in a mechanical system consider the system of balls as shown in Figure 3 [3]. 6

14 B A C Figure 3. Stability and instability of equilibrium The ball at A is in neutral equilibrium or is neutrally stable. The ball at B is unstable and at C it is stable. These definitions define the static systems, i.e. systems which can maintain a state of equilibrium for a period of time. Buckling When a slender structure is subjected to an increasing compressive axial load and reaches a critical load value where the structure undergoes visibly large deformations in geometry and loose its ability to carry the load then it is said to buckle. For small loads the process is elastic since the deformations are hardly noticeable when the load is removed [4]. Buckling is a special mode of elastic instability of equilibrium which can occur in columns, beams, arches, rings, plates and shells. The behaviour of a buckling system is reflected in the shape of its load-deflection curve referred to as the equilibrium path [4]. Generally, there are two types of structural instability: (1) bifurcation buckling and (2) limit load buckling. In bifurcation buckling, the deflection under compressive load changes from one direction to a different direction, e.g. from axial shortening to lateral deflection. Under certain conditions the structure has two possible equilibrium configurations. The load at which the bifurcation occurs is called the critical buckling load or critical load and marks the critical state of behaviour of an elastic system. The deflection path that exists prior to bifurcation is called the primary path or prebuckling path which is the systems initial equilibrium regime, the deflection path after bifurcation is called the secondary or postbuckling path. Depending on the structure and loading, the secondary path may be symmetric (stable or unstable) or asymmetric, and it may rise or fall below the critical buckling load as shown in Figure 4. In limit load buckling, the structure attains a maximum load without any previous bifurcation, i.e. with only a single mode of deflection as shown in Figure 5. The snapthrough buckling which occurs in shallow arches and spherical caps are an example of limit load buckling [4]. 7

15 Stable Load, P a) Secondary path P cr Unstable Transverse Deflection, Δ Primary path + Load, P b) Secondary path P cr Rotation, θ Primary path + Figure 4. Bifurcation buckling a) symmetric bifurcation and stable and unstable postbuckling curve; b) asymmetric bifurcation Figure 5. Limit load buckling with single mode of deflection [4] 8

16 Buckling load Stability criteria are one of the main design considerations in the field of structural engineering. Design of structures is often based on strength and stiffness considerations. Strength is defined to be the ability of the structure to withstand the applied load, while stiffness is the resistance to deformation, i.e. the structure is sufficiently stiff not to deform beyond permissible limits. However, a structure may become unstable long before the strength and stiffness criteria are violated. Therefore, it is important in the design procedure to perform a buckling analysis to assure the stability of an equilibrium configuration and determine the critical value of the loading under which the instability occurs, especially when the structure is slender and lightweight [4]. Linear elastic bifurcation buckling of structural members is the most elementary form of buckling to understand the buckling behaviour of complex structures, including structures incorporating inelastic behaviour, initial imperfections, residual stresses, etc. The load at which linear elastic buckling occurs is important, because it provides the basis for commonly used buckling formulas used in design codes [4]. To perform a classical buckling analysis numerical methods are necessary to use when the structural geometries, loads and boundary conditions are complicated. In such cases it is impossible to obtain exact analytical solutions. There are several techniques to determine the elastic buckling load. These techniques may be grouped under two general approaches: (a) the vector approach and (b) the energy approach. In the vector approach, Newton s second law is used to obtain the governing equations, whereas in the energy approach the total energy, which is the sum of internal energy and potential energy due to the loads, is minimized to obtain the governing equations. The governing equations are in the form of a linear eigenvalue problem [4]. ([ K] + cr[ Kσ] ref ){ dd} = { } λ (1) [ ] where K is the stiffness matrix, and [ Kσ ]ref is the stress stiffness matrix. The solution of the eigenvalue problem gives the eigenvalues, λ cr, which is the buckling loads and the corresponding eigenvectors, { dd }, which is the buckling modes. In the classical buckling analysis, deformations prior to buckling are neglected [5]. The smallest buckling load is termed the critical buckling load. This load is associated with the state of neutral equilibrium, i.e. characterized by the stationary condition of the load with respect to the deflection. In order to ascertain whether the equilibrium position is stable or unstable, the perturbation technique for the vector approach are used or by examining the second derivative of the potential energy [4]. Leonhard Euler [ ] was the first to study elastic stability; he used the theory of calculus variations to obtain the equilibrium equation and buckling load of a compressed elastic column [4]. 9

17 In the elastic linear stability analysis the linear theory of column buckling is sometimes referred to as the classical Euler buckling theory of columns. The Euler buckling theory assumes that the column is perfectly straight and uniform, perfectly free of end moments and lateral loads, and the forces are perfectly centred and perfectly axial. The perfect column models are in common use in structural mechanics, and are the simplest type of buckling [5]. For example, the simplest cases of loading and boundary conditions are that of a column pinned at both ends as shown in Figure 6. The column has length L, Young s modulus E, area A and moment of inertia I, and is loaded by an axial force P. L, EI Figure 6. Column under compressive load P The linear differential equation 4 2 d w d w EI + P = (2) 4 2 dx dx Bending moment equation 2 d w M ( x) = Pw( x) = EI (3) 2 dx Equation (2) and (3) yields 2 d w 2 + n w( x) = 2 dx (4) where 2 n = P EI (5) The general solution to equation (3) is w( x) = Asin nx + B cosnx (6) The boundary conditions are w ( x = ) = B = (7) w ( x = L) = Asin nl = (8) 1

18 Boundary condition (7) has two solutions A = w( x) = (trivial solution, no buckling occurs) (9) sin nl = (1) This is an eigenvalue problem, to have nontrivial solutions, i.e. to have buckling, we must have sin nl = nl = mπ m = 1,2,3... (11) The lowest, nontrivial, root is obtained for m = 1 (12) That is nl = π (13) This gives the critical bifurcation buckling load or the critical Euler load P EI 2 π EI L = π Pcr = (14) 2 L The corresponding buckling mode is π w( x) = Asin x (15) L The same procedure may be used for cases with other boundary conditions [6]. The behaviour for a perfect column can be described as shown in Figure 7. At first the column is elastically stable at the primary path, i.e. the column is straight, which last until the increasing load reaches the critical Euler load. At this point two equilibrium paths are possible, the column could remain straight (primary path) or the column could buckle (secondary path) [5]. Caution has to be taken when computing buckling load because in reality there are always imperfections so a real structure may collapse at a load different that predicted by a linear bifurcation analysis. For example, if the column is not initially straight, deflection starts from the beginning of the loading and there is no sudden buckling by bifurcation, but a continuous increase of the deformations. The magnitude for the imperfections is denoted by the eccentricity parameter, e, as shown in Figure 7 [5]. 11

19 Figure 7. Deformation behaviour for a perfect column and column with imperfections [5] 12

20 2. Method This section describes how the work was performed according to the design of experiments approach. The finite element model used in the simulations is also described. 2.1 Finite element model An elastic-plastic 3D solid model based on von Mises theory has been used for the welding simulation using the FE program MSC.Marc. The model used in the simulations consists of two plates which are welded together without additional filler material. The dimension of the welded plate is 19x1x1.7 mm (length x width x thickness). To prevent rigid body motion the plates are clamped in both x- and z- direction. In front of the clamping area the plate is free to move in all directions. Figure 8 shows the two plates with the clamped area shaded with ends at location 1. Location 2 and 3 is the start and end of the weld respectively. Locations 4 and 5 indicate the position of nodes 7 and 86 respectively where the residual stresses (x- and z-directions) was measured. Location 6 indicates the position where the bending behaviour in the y-direction (out of plane) and the gap was measured y z x Figure 8. Schematic of the plates showing clamping area, start (2) and end (3) of the weld, positions of nodes 7 (4) and 86 (5) and location for measurement of the out of plane and gap deformations (6) 13

21 Table 1. Positions for the indicated locations in Figure 8 Point number X [mm] Z [mm] The material and thermal properties for the model are implemented by user subroutines and linked to the FE code. The material in the plates is Greek Ascoloy, a martensitic stainless steel which is a high-strength, corrosion and creep resistant material and the hardening behaviour of the material is isotropic and linear. The model consist of linear solid elements and nodes. The mesh geometry for the finite element model is shown in Figure 9. Figure 9. Mesh geometry for the finite element model 2.2 Welding simulation In the welding simulation the heat input is a moving heat source where the energy input is distributed as a double ellipsoid as shown in Figure 1. The parameters a, b, c f, and c r gives the desired melted zone. 14

22 Figure 1. The moving heat source defined as a double ellipsoid [7] When performing welding simulations, different kind of deformations can be studied. In this work the deformations of interest is the bending behaviour in the y-direction (out of plane) and the gap between the welded plates as shown in Figure 11. y z (a) z x (b) Figure 11. Deformation behaviour of the plates in different directions, (a) out of plane and (b) gap opening 15

23 2.3 Design of experiment The aim with the parametric study is to investigate which of the chosen parameters that has the largest influence on the existing model, i.e. how sensitive or robust the model is when it is subject to disturbances. The chosen parameters are Poisson s ratio, Young s modulus, thermal expansion, initial yield stress, yield limit stress, welding power, and the heat source geometrical parameters. To evaluate the results and determine weather or not the model is robust for a given set of parameters, the responses of residual stresses in the sheet plane as well as gap and out of plane deformations are studied. The residual stresses (x- and z-directions) for two specific nodes (indicated in Figure 8) are compared to the stress value obtained from the first simulation which is considered to be the nominal value. The gap deformation is also compared to the nominal value. The out of plane deformation is compared to both the nominal value and to the results from an experiment performed on sheet of the same material as in the simulations. 2.4 Experimental plan Since the aim of this study is to investigate which parameters that have a large influence on the stresses and deformations (response) during a welding simulation, the chosen method is a factor screening where one parameter has been changed at a time while keeping the other parameters constant. Each parameter has been investigated at three levels (nominal, high and low). The experimental plan that has been used is shown in Table 2. The cells that contains e.g. the expression 1 table indicates that the parameter values is obtained by multiplying a table value by a scalar which is changed (± 2%) in order to get a high or low value. The values obtained from the tables are used to introduce a temperature dependence of the parameters ν, E and σ y. The first simulation is the reference to which the high and low runs are compared to. With a view to keep the number of simulations at reasonable level the possibility of interactions between different parameters has not been investigated. 16

24 Table 2. Values of investigated parameters for each simulation Sim. nr ν Ε Th ε σ y σ y min Mechanical activation Q a b c f c r 1 1 table 1 table 1. 1 table table 1. 1 table table 1. 1 table table.8 table 1. 1 table table 1.2 table 1. 1 table table 1 table.8 1 table table 1 table table table 1 table table table 1 table 1..8 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table 5. 1.E table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table table 1 table 1. 1 table

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26 3. Results The obtained results from the simulations will be presented in this section. All figures are found in appendix Influence of different parameters on stresses and deformations The influence of the investigated parameters on the stresses and deformations in the sheets is presented in this section. The parameters are varied in high and low levels and compared to the results from the original model termed nominal level. The out of plane deformations will also be compared to the results obtained from welding experiments performed at Volvo Aero Poisson s ratio The effect of Poisson s ration on the stresses perpendicular to the welding direction at nodes 7 and 86 are shown in Figure 12. As seen the effect at node 86 is negligible when Poisson s ratio is varied between the three different levels. However, when looking at node 7, it can be seen that a higher Poisson s ration results in larger compressive stresses in the sheet. When looking at the stresses in the welding direction, Figure 13, a similar behaviour is seen although node 86 is experiencing tensile stresses. The effect of the Poisson s ratio on the gap deformation, Figure 14, is negligible. The out of plane deformation is not affected by the changing the level but it shows a large difference compared the results from the experiments, Figure Young s modulus The effect of Young s modulus on the stresses perpendicular to the welding direction at nodes 7 and 86 is shown in Figure 16. A similar behaviour as for the Poisson s ratio is seen for node 7 but a high Young s modulus level results in lower compressive stresses as compared a high Poisson s ratio level. The difference between the levels is also smaller in the case of Young s modulus. At node 86 there is very little difference. In the welding direction, Figure 17, also shows a similar behaviour with increasing compressive stresses when the Young s modulus is increased. When comparing node 86 for the Poisson s ration and Young s modulus it can be seen that the latter case shows larger difference between the different levels. When changing the Young s modulus the effect on the gap deformation and out of plane deformation is very small, Figure 18 and Figure 19. However, out of plane deformation results from experiments shows a higher value compared to the simulations. 19

27 3.1.3 Thermal expansion When looking at the thermal expansion, Figure 2 and Figure 21, a different behaviour compared the previous parameters is observed. In both directions at node 7 a low thermal expansion results in lower compressive stresses and the nominal and high levels results increased compressive stresses since the high and nominal levels has shifted place. The behaviour at node 86 is very similar to that seen for the Young s modulus. The thermal expansion yields larger differences on the deformations compared to the previous parameters, Figure 22 and Figure 23. A higher thermal expansion will result in larger gap and out of plane deformations. In the latter case, compared to the experiments, all levels result in larger deformations Yield stress In Figure 24 and Figure 25, the results from varying the yield stress are shown. In the direction perpendicular to the welding direction the behaviour is very similar to that observed for the Young s modulus for both nodes. In the welding direction at node 7 and 86, a change in yield stress results in very small differences. It should be pointed out that there is a difference between the two directions since a lower value gives higher stresses in the welding direction and vice versa in the perpendicular direction. The influence of yield stress on the two deformations is very small, as seen in Figure 26 and Figure 27, and the results are very similar those obtained for the Young s modulus. Compared the experiments, the simulations results show larger out of plane deformations Minimum yield stress The minimum yield stress is the yield stress value that is used during the simulations when the temperature dependant yield stress passes a certain point (a minimum value). As shown in Figure 28 and Figure 29, the effect of changing this value has very minor influence on the stresses in both nodes. These results are very similar to those for the yield stress as seen in Figure 3 and Figure 31. The only noticeable difference is that varying the cut off yield stress results in smaller differences between the levels Mechanical activation The effect of changing the mechanical activation, which is the distance that the mechanical activation is lagging relative to the arc, is shown in Figure 32 and Figure 33. At node 86 the effect is negligible but node 7 experiences some interesting behaviour. In the perpendicular direction, Figure 32, the nominal and high levels results in high compressive stresses. A low level results in small tensile stresses and an intermediate level, in between the high and low, shows quite high tensile stresses. 2

28 In the welding direction, Figure 33, a similar behaviour is seen but the low level shows small compressive stresses and the intermediate level results in smaller tensile stresses. The influence of the mechanical activation shows quite large differences between the different levels. A decrease in the mechanical activation results in larger deformations, gap and out of plane, as shown in Figure 34 and Figure 35. Compared to the experiments, the simulations results show larger deformations Welding power A change in welding power is shown in Figure 36 and Figure 37. In the perpendicular direction at node 7 the nominal level yields high compressive stresses and a further increase of the power actually decreases the compressive stresses. The low level only results in small compressive stresses. The effects at node 86 are very small. In the welding direction, a similar behaviour is seen at node 7. At node 86, an increase in power results in higher tensile stresses. The welding power has also shown to have a large influence on the deformations, Figure 38 and Figure 39. An increase in welding power has resulted in larger gap and out of plane deformations. As seen previously, the experiments show smaller deformations compared to the simulations Heat source geometrical constant a As seen in Figure 4, the change in a yields large differences between the levels at node 7. The nominal level shows high compressive stresses and the low level decreases this stress level by about 5%. The highest a-value results in tensile stresses. Node 86 is not affected. In the welding direction, Figure 41, the same behaviour is observed but with higher compressive stresses for the low and nominal levels and the high level shows almost no resulting stresses. No difference is observed for the deformations when changing the a-value, Figure 42 and Figure 43. The out of plane deformations are larger for the simulations compared to the experiments Heat source geometrical constant b For the b-value, Figure 44 and Figure 45, the effect on node 7 in both directions is similar as that for the a-value. However, the difference between the low and nominal levels is almost zero. Node 86 is not affected in either direction. No difference is observed for the deformations when changing the b-value, Figure 46 and Figure 47. The out of plane deformations are larger for the simulations compared to the experiments. 21

29 3.1.1 Heat source geometrical constant c f Changing the c f -value gives almost identical results, Figure 48 and Figure 49, as those for the a-value. The effect of changing the c f -value on the gap deformation, Figure 5, is small but a low level yields slightly larger deformation. The out plane deformation, Figure 51, shows the same behaviour regarding the levels and larger deformations compared to the experiments Heat source geometrical constant c r The c r -value results in a different behaviour as compared to the other geometrical constants. In the perpendicular direction, Figure 52, the nominal level shows high compressive stresses whereas the high and low levels results very small or almost no stresses. In the welding direction, Figure 53, the nominal level shows high compressive stresses again and the high and low levels results in similar compressive stresses. The c r -value appears to have a bigger influence on the gap deformations compared to the other geometrical constants. As seen in Figure 54, a low c r -value results in decreased deformations. The same behaviour is observed for the out of plane deformation, Figure 55. The experiments show smaller deformations compared to the simulations in this case too. Summarised results A summary of the above described results can be found in Table 3. The influence of each parameter on stresses and deformations is graded on a scale where: means no influence and +++ means high influence. The stresses are denoted σ T for transversal stresses (x-direction) and σ L for longitudinal stresses (z-direction). The indexes C and F indicate if the node is close (node 7) or far away (node 86) from the weld respectively. The deformations (gap and out of plane) are denoted δ. 22

30 Table 3. Summary of how the different parameters influence stresses and deformations σ T, C σ L, C σ T, F σ L, F δ Poisson s ratio Young s modulus Thermal expansion Yield stress Minimum yield stress Mechanical activation Welding power Heat source geometrical constant a Heat source geometrical constant b Heat source geometrical constant c f Heat source geometrical constant c r

31 24

32 4. Discussion When looking at the stresses from the simulations with varying Poisson s ratio it can be seen that the nominal level, which is temperature dependant, gives similar results during welding as the high level since Poisson s ratio increases with temperature. The low value will therefore correspond to the nominal level after cooling, which it does. If the effects on the deformations are considered, the results show that the simulation model is robust with respect to changes in Poisson s ratio. The results have shown that the stresses are reduced with decreased Young s modulus. This is however expected since they are proportional to each other, in the elastic region, according to Hooke s law. Changing the Young s modulus has resulted in small differences of the deformations but still larger than those for Poisson s ratio. The increased deformations with decreased Young s modulus can be attributed to the lower stiffness of the material. The simulation model is quite robust with respect to changes in Young s modulus. The thermal expansion is a parameter that has shown large influence on both the stresses and deformations. The stress level obtained with low thermal expansion can be explained by less expansion/deformation of the material during heating and cooling. This is also the reason for the reduced deformations that occurred. It seems that the simulation model is more sensitive to changes in thermal expansion compared to the other material parameters. The gap and out of plane deformations observed when increasing the yield stress can be attributed to higher compressive stresses, perpendicular to the welding direction, which will deform the sheets more. The robustness with respect to the yield stress is quite good since only small differences are noticed between the levels. The minimum yield stress has shown very similar results for the stresses in the welding direction. These values are slightly lower than obtained from the low value yield stress simulation which is expected. Regarding the gap and out of plane deformations, similar results are obtained as in case of the yield stress. The difference is that less scatter is observed indicating that the model is more robust to changes in cut off yield stress. The mechanical activation is one of the parameters that have resulted in large differences between the levels. The nominal and high levels, which activates elements at distance of 6.5 and 5 mm from the arc respectively, shows the highest compressive stresses in both directions. The intermediate level, corresponding to a distance of 3 mm, results in only tensile stresses in both directions which yields larger gap and out of plane deformations. The largest deformations are obtained in the case when the activation distance is very small (1-6 mm). This results in a combination of tensile and compressive stresses (initial compressive in the x- direction and tensile in the z-direction which is reversed after about 5 s) which in turn will give larger deformations. The simulation model appears to be quite sensitive to changes in the mechanical activation. 25

33 The welding power is another parameter that has resulted in large differences when changed. A low power has resulted in small compressive stresses and small deformations possibly due the lower heat input and consequently faster cooling. The large deformations obtained in case of high power can possibly be attributed to the combination of tensile and compressive stresses that occurred. Regarding the robustness, the model seems to be most sensitive to changes in welding power of all the investigated parameters. The parameters that describe the geometry of the heat source have shown to have minor influence on the deformations. Parameters a, b and c f have resulted in differences of the stress level in both directions but in despite of this hardly any difference was observed on the deformations. In case of c r, the low value has resulted in decreased gap and out of plane deformations which can be explained by the very low stresses that occurred perpendicular to the welding direction. Apparently the geometry of the heat source gives similar stresses for high and low values of c r (increased and decreased volume of the melted zone) and there seems to exist some geometry, in-between the large and small, that results in high compressive stresses. In general two different behaviours of the parameters have been observed. In one case the change in levels results in expected response of the stresses, i.e. high level results in high stress and vice versa. In the other case there is a shift between the how the stress responds to a change in level of a parameter, i.e. a high level can lead to a reduced stress compared to the nominal level. Since the FE software MSC.Marc Mentat 25r3 does not have an integrated feature for combining the welding simulation and the eigenvalue analysis, it will require too much manual work to perform a buckling analysis by using this technique. The initial intention with this work was, besides the literature review, to investigate different methods to evaluate robustness during a welding simulation. However, due to the above mentioned complication with eigenvalue analysis and the simulation time of approximately four hours the scope of work was limited to only involve a factor screening of different parameters and to identify which ones that has the largest influence on stresses and deformations during a welding simulation. 26