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Analysis of welding distortion using qualitative and semi-qualitative techniques Zhou, Ye 1998

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ANALYSIS OF WELDING DISTORTION USING QUALITATIVE AND SEMI-QUANTITATIVE TECHNIQUES  by  YE ZHOU B.A.Sc, Civil Engineering, University of British Columbia, 1995  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF  MASTER OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard  T H E UNIVERSITY  OFWITISH  September 1998 © Y e Zhou, 1998  COLUMBIA  In  presenting  this  degree at the  thesis  in partial fulfilment  University of  British Columbia,  freely available for reference copying  of  department  this or  and study.  by  his  or  her  &A/£»f/J &£g-)*7<gj  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  1 1  SepTfcHfreft I  that the  may be It  thesis for financial gain shall not  ClV'U  requirements  I further agree  representatives.  permission.  Date  the  I agree  thesis for scholarly purposes  publication of this  Department of  of  is  an advanced  Library shall make it  that permission for extensive granted by the understood  be  for  that  allowed without  head  of  my  copying  or  my written  ABSTRACT  In planning and design of engineering projects, engineers are often required to decide upon a course of action irrespective of the completeness and accuracy of available information. With the fast development of computer technology, many numerical analysis tools have arisen to assist engineering decision-making based on complete design information, which is, however, rarely available at most design stages. Little has been done to help engineers make sound decisions when complete design information is not available. Qualitative and SemiQuantitative Reasoning, a branch in the field of Artificial Intelligence, has the ability of analyzing "ill"-defined problems using sound and clear arguments which are based on facts. This thesis is an attempt to tackle "ill"defined engineering problems with the above mentioned reasoning techniques.  This thesis revolves around the topic of shrinkage and distortion in welded structures. Steel fabrication frequently involves the joining of components by welding. Each component must be fabricated to particular dimensional tolerances. Distortion caused by welding is a frequently occurring problem that makes it difficult to estimate the dimensions of the finished structures and thus increases the fabrication costs. Welding distortion is a poorly quantified phenomenon controlled by many factors that are difficult to describe numerically. The characteristics of the distortion problems make the traditional numerical analysis very arduous and expensive to apply. Due its high complexity, for many years, welding distortion problems have been approached primarily empirically or by trial and error, and uncertain design factors cannot be effectively considered. In this thesis, a computer software tool was developed using qualitative and semi-quantitative techniques, attempting to solve complex welding distortion problems with unclearly specified design factors. With the assistance of this software tool, it is hoped to make the efforts of predicting and controlling welding distortion become more of a science rather than an art.  11  CONTENTS  ABSTRACT  II  CONTENTS  Ill  TABLES  VI  FIGURES  VII  ACKNOWLEDGMENTS 1  INTRODUCTION  1  1.1  BACKGROUND  l  1.2  W E L D I N G DISTORTION  2  1.2.1  Development of Welding Distortion Analysis  3  1.2.2  Existing Approaches to Welding Distortion Analysis  4  1.3  Q U A L I T A T I V E A N D SEMI-QUANTITATIVE REASONING IN E N G I N E E R I N G  5  1.3.1  Applying Artificial Intelligence to Engineering  6  1.3.2  Qualitative and Semi-quantitative Analysis  8  1.4  2  IX  INTRODUCTION TO Q U A L I T A T I V E E N G I N E E R I N G S Y S T E M FOR W E L D I N G DISTORTION  WELDING DISTORTION 2.1  8  10 10  W E L D E D STRUCTURES  2.1.1  Advantages of Welded Structures over Riveted Structures  10  2.1.2  Problems with welded structures  11  2.2  INSIGHTS INTO W E L D I N G DISTORTION P R O B L E M S  12  2.2.1  Residual Stresses  12  2.2.2  Cause of Welding Distortion  14  2.2.3  Types of Welding Distortion  15  2.2.4  Analysis of Distortion in Weldment  17  2.3  N U M E R I C A L A N A L Y S I S OF W E L D I N G DISTORTION  20  2.3.1  One-dimensional Numerical Analysis  20  2.3.2  Finite Element Analysis of Welding Distortion  23  2.4  A N A L Y T I C A L A N D EMPIRICAL F O R M U L A S  25  2.4.1  Transverse Shrinkage - Butt Welds  26  2.4.2  Transverse Shrinkage in Fillet Welds  34  iii  2.4.3  Angular Changes of Butt Welds  35  2.4.4  Angular Distortion of Fillet Welds  36  2.4.5  Longitudinal Shrinkage of Butt Welds  42  2.4.6  Longitudinal Shrinkage of Fillet Welds  42  2.4.7  Longitudinal Bending Distortion  43  2.4.8  Buckling Distortion  46  2.5  M E T H O D S O F DISTORTION R E D U C T I O N IN W E L D M E N T S  2.5.1 2.6  Commonly Use Distortion Reduction Methods  46  M E T H O D S O F R E M O V I N G DISTORTION  49  2.6.1  Straightening by Flame Heating  50  2.6.2  Vibratory stress-relieving and electromagnetic-hammer technique  51  Q U A L I T A T I V E A N D SEMI-QUANTITATIVE ANALYSIS IN E N G I N E E R I N G  53  3 3.1  D E V E L O P M E N T O F Q U A L I T A T I V E A N A L Y S IS  53  3.1.1  Qualitative Reasoning  53  3.1.2  Semi-quantitative Analysis  55  3.2  S O L V I N G PREDICTION P R O B L E M S WITH Q U A L I T A T I V E A N A L Y S I S  57  3.2.1  Prediction  58  3.2.2  Refinement  59  3.2.3  Discussion  59  3.3  T H E O R I E S IN Q U A L I T A T I V E A N D S E M I - Q U A N T I T A T I V E A N A L Y S I S IN E N G I N E E R I N G  62  3.3.1  Components of Qualitative Calculus  62  3.3.2  Interval Analysis and Constraint-Satisfaction Methods  65  3.4  T H E Q U A L I T A T I V E E N G I N E E R I N G S Y S T E M QES  72  3.4.1  Structure of QES  73  3.4.2  Qualitative Analysis  74  3.4.3  Semi-Quantitative Reasoning  76  4  QES W D : A N A P P L I C A T I O N O F Q U A L I T A T I V E A N D S E M I - Q U A N T I T A T I V E A N A L Y S I S I N 78  WELDING DISTORTION 4.1  INTRODUCTION  78  4.2  S Y S T E M IMPLEMENTATION  79  4.2.1  System Structure  79  4.2.2  User Interface  83  4.2.3  Platform and Development Environment  85  4.3  5  46  SAMPLE ANALYSIS  86  89  CONCLUSION  iv  BIBLIOGRAPHY  TABLES  T A B L E 2.1  F O R M U L A E FOR PREDICTION O F T R A N S V E R S E S H R I N K A G E  27  T A B L E 2.2. E F F E C T O F V A R I O U S PROCEDURES O N T R A N S V E R S E S H R I N K A G E IN B U T T W E L D S  32  T A B L E 2.3  50  T A B L E 3.1  METHODS OF FLAME-STRAIGHTENING Q U A L I T A T I V E ADDITION, M U L T I P L I C A T I O N , A N D N E G A T I O N  63  T A B L E 3.2: R U L E S U S E D TO G E N E R A T E QUALITATIVE EQUATIONS  64  T A B L E 3.3: B A C K T R A C K I N G E X A M P L E  69  T A B L E 4.1  87  INPUT P A R A M E T E R S O F S A M P L E P R O B L E M  vi  FIGURES  F I G U R E 2.2  T Y P E S OF W E L D DISTORTION  16  F I G U R E 2.3  P R O C E D U R E O F W E L D I N G DISTORTION A N A L Y S I S  17  F I G U R E 2.4 STRIPPED E L E M E N T FOR O N E - D I M E N S I O N A L C A L C U L A T I O N  20  F I G U R E 2.5  LOADING A N D UNLOADING  21  F I G U R E 2.6  P R O C E D U R E OF C A L C U L A T I N G TRANSIENT D E F O R M A T I O N A T M I D - L E N G T H  22  F I G U R E 2.7  C O M P A R I S O N O F DISTORTION - C A L C U L A T I O N V S . E X P E R I M E N T  24  F I G U R E 2.8  C H A N G E S OF DEFLECTION DURING W E L D I N G  25  F I G U R E 2.9  S C H E M A T I C P R E S E N T A T I O N OF A T R A N S V E R S E S H R I N K A G E O F A B U T T W E L D IN A S I N G L E P A S S  28  F I G U R E 2.10  T E M P E R A T U R E DISTRIBUTION  31  F I G U R E 2.11  I N C R E A S E O F T R A N S V E R S E S H R I N K A G E D U R I N G M U L T I - P A S S W E L D I N G O F A B U T T JOINT  32  F I G U R E 2.12  DISTRIBUTION OF T R A N S V E R S E S H R I N K A G E O B T A I N E D IN S L I T - T Y P E S P E C I M E N S WITH D I F F E R E N T  WELDING SEQUENCE  34  F I G U R E 2.13  T R A N S V E R S E S H R I N K A G E IN F I L L E T W E L D S  34  F I G U R E 2.14  E F F E C T OF S H A P E OF G R O O V E O N A N G U L A R C H A N G E  35  F I G U R E 2.15  T H E M O S T S U I T A B L E G R O O V E S H A P E TO M I N I M I Z E A N G U L A R DISTORTION IN B U T T W E L D  36  F I G U R E 2.16  P A N E L S T R U C T U R E WITH STIFFENERS  37  F I G U R E 2.17  DISTORTION D U E TO F I L L E T W E L D S  38  F I G U R E 2.18  PLASTIC PREBENDING A N D ELASTIC PRESTRAINING  40  F I G U R E 2.19  E L A S T I C P R E S T R A I N I N G FOR R E D U C I N G A N G U L A R DISTORTION O F F I L L E T W E L D S  41  F I G U R E 2.20  A N A L Y S I S O F L O N G I T U D I N A L DISORTION IN A  43  F I G U R E 2.21  I N C R E A S E OF L O N G I T U D I N A L DISTORTION D U R I N G M U L T I - P A S S W E L D I N G  45  F I G U R E 2.22  C A L C U L A T E D FINAL CENTER DEFLECTION VS. PREHEATING T E M P E R A T U R E OF T H E WEB  49  F I G U R E 2.23  E F F E C T S O F T R A V E L S P E E D OF F L A M E O N A N G U L A R C H A N G E A N D T R A N S V E R S E S H R I N K A G E  51  F I G U R E 3.1  A P O L Y G O N SOLUTION O F I N T E R V A L A N A L Y S I S  vii  F I L L E T - W E L D E D JOINT  66  F I G U R E 3.2: A N I M P L E M E N T A T I O N OF T H E W A L T Z A L G O R I T H M [ D A V I S , 1987]  71  F I G U R E 3.4: A S A M P L E SOLUTION USING W A L T Z A L G O R I T H M [ D A V I S , 1987]  72  F I G U R E 3.3: G R A P H I C A L R E P R E S E N T A T I O N O F EXPRESSIONS  74  FIGURE 3.3A STRUCTURE OF Q E S  74  F I G U R E 3.4: F I N A L C O N S I S T E N C Y N E T W O R K  76  F I G U R E 3.5: A N E X A M P L E OF N U M E R I C A L C O N S T R A I N T SATISFACTION  77  F I G U R E 4.2  S C R E E N C A P T U R E S OF Q E S W D  81  F I G U R E 4.3  Q E S W D D A T A B A S E STRUCTURE  82  F I G U R E 4.4  S E C T I O N A L PROPERTIES O F T H E S A M P L E GIRDER  86  viii  ACKNOWLEDGMENTS  I would like to thank my thesis supervisor, Dr. Siegfried F. Stiemer, for his lucid advice and continual encouragement throughout my student career at U B C . He helped me keep my standard high and provided me with the opportunity to explore lots of interesting paths. Special appreciation goes to the engineers at Coast Steel Fabricators: David J. Halliday, Adjunct Professor, David S. K . Lo and Michael Gedig, in particular, are sincerely appreciated. I would like to thank my lovely wife, Jian, for putting up with the time I spent exploring all of those interesting paths, and also my parents, for directing me onto the correct path of life.  The financial assistance from Coast Steel Fabricators, Ltd., National Science and Engineering Research Council (NSERC), and the Science Council of British Columbia are gratefully acknowledged.  ix  1  INTRODUCTION  This thesis is concerned with an engineering software tool which, using sound logic techniques, can be used to draw as many conclusions as possible from an ill-defined engineering problem. It is intended for such a tool to be used by practicing engineers to make and collect guided design decisions, and by engineering students to comprehend complex engineering problems. This thesis applies the techniques in qualitative and semiquantitative reasoning to the problems in welding distortion control.  1.1  BACKGROUND  Because of the incomplete information in most engineering design processes, in particular in the early stage of design, numerical models are of limited value; instead, qualitative reasoning techniques, which have been studied in depth in the field of artificial intelligence, are in a systematic way to practical engineering problems. One of the objectives of this thesis is to incorporate these techniques within an accessible, expandable and simple framework. Qualitative reasoning techniques are augmented with interval analysis methods, which are sound procedures for reasoning with partially specified numerical information. In addition to qualitative reasoning and interval methods, a number of other reasoning techniques are also used. The emphasis of this thesis has been placed on the ability of the computer to use sound logic to gain insight into a welding distortion problem rather than just on its ability to manipulate numbers.  The focus of interest lies on the topic of shrinkage and distortion in welded structures. Steel fabrication frequently involves the joining of various components by welding. Each component must be fabricated to particular dimensional tolerances. Distortion caused by welding is a frequently occurring problem that makes it difficult to estimate the dimensions of the finished structure, and thus increases the fabrication costs. Welding distortion is also a poorly quantified or ill-defined phenomenon controlled by factors that are difficult to describe numerically. The characteristics of the distortion problems make the traditional numerical analysis difficult and expensive to apply. Due to the high complexity of welding distortion, for many years, the  1  distortion problems in structural engineering have been approached primarily empirically or by trial and error. Valuable knowledge from experienced technical staff is often lost because of changes of personnel, and furthermore of the lack of an effective mechanism for knowledge collection. Therefore, it will be advantageous to have a software system that is able to accept ill-defined numerical data as analysis input, having the ability of integrating welding knowledge into a flexible information database. For this reason, qualitative and semiquantitative methods for engineering, developed earlier at University of British Columbia ^ ' s  1 9 9 5  ^  a r e  used as  the main analysis tool; and a flexible knowledge database will be built to record engineering knowledge.  The focus of this research is to develop a computer software system which is able to provide information on welding distortion, estimate the quantity of welding distortion, and give advice on welding distortion control. The software system should also have the ability to gather engineering knowledge cognitively, and to construct a knowledge database for the improvement of future decision making and also personnel training. The core of the software system will consist of an analysis module, which will perform qualitative as well as semi-quantitative reasoning. The software will organize such diverse sources of information as text, numerical data, equations, and graphics into a coherent, expandable and easily accessible framework. With the development of this software tool, it is attempted that the prediction and control of welding distortion in structural engineering will become more of a science rather than an art.  1.2  WELDING DISTORTION  The welding process is used extensively in the fabrication of many structures, including buildings, bridges, pressure vessels, ships, airplanes, etc. It provides significant advantages over other joining techniques such as riveting, casting, and forging. Excellent mechanical properties, air and water tightness, and good joining efficiency are among its outstanding qualities.  There might be, however, possible problems such as residual stresses and shape distortion associated with the construction of welded structures. When a material is welded, it experiences local heat due to the welding heat source. The temperature field inside the weldment is not uniform and changes as the welding progresses. The  2  welding heat cycle gives rise to a complex strain field in the weld metal and in the base metal regions near the weld. These strains, along with the plastic upsetting, can create residual stresses that remain after the welding is completed. Consequently, shrinkage and distortion can be also produced.  Residual stresses and distortion are highly undesirable effects in welding technology. Thermal stresses during welding often cause cracking. High tensile residual stresses near the weld may lead to an immediate fracture or fatigue of the weldment. Compressive residual stresses in the welded plate, often combined with distortion, may reduce its buckling strength. If distortion occurs in the parts of complex structures that need to be joined, a mismatch problem may happen. The mismatch increases the possibility of weld defects. Furthermore, if the parts are forced into alignment and then joined, stresses are locked into the structures. This results in a strength reduction of the joint under either static loads or cyclical loading. In some extreme cases it may be necessary to remove any distortion completely before joining. However, correcting unacceptable distortion is often costly and in some cases impossible.  1.2.1  Development of Welding Distortion Analysis  Since welding distortion is accompanied by many undesirable as well as expensive consequences, it is important that design engineers can predict the amount of welding distortion before fabrication.  Studies of welding distortion have been conducted since the 1930s  [ M a s u b u c h l 1980]  , Spraragen and others have  summarized the results of these early studies in a series of reviews. Naka pioneered in the analytical study of shrinkage. In his work, most of which was carried out around 1940, he studied analytically and experimentally how a butt weld shrinks. To avoid mathematical complication, the analysis was kept one-dimensional, neglecting the change of shrinkage in the welding direction.  During the 1950s, several Japanese investigators, including Kihara, Watanabe, Masubuchi and Satoh, carried out extensive study programmes on residual stresses and distortion. They concentrated on stress and distortion in practical joints. In order to analyze their experimental data, they frequently used the concept of 3  incompatibility. A number of empirical formulae on various types of distortion were developed. Most of the efforts during this period, however, concentrated on the distortion remaining after the welding is completed. Most of the experiments were on weldments in low-carbon steel and covered electrodes. Since welding distortion is produced by complex mechanisms, studies performed before approximately 1960 were based on experiments and analyses of simple cases.  Since the 1970s, with the accessibility of modern computers, interest in analytical simulation has been revitalized. The analysis of thermal stress was the focus of the first computer-simulation projects on welding distortion  [Ta  "  i 9 6 4  \ Since then, the emphasis has been on transient metal movement and finite-element methods  for simulation. By using these computer-aided numerical analysis tools, it is now possible to analytically determine distributions of residual stresses and distortions of weldments in various shapes with reasonable accuracy.  1.2.2  Existing Approaches to Welding Distortion Analysis  Studies on transient welding distortion started in the 1930s. However, because the computation required for analyzing transient phenomenon is complex and involving, very limited studies were done. Since modern computers became accessible to researchers in the 1960s, more studies have been done to analyze welding distortion numerically.  The first significant attempt to use a computer in the analysis of thermal stresses during welding was done Tall " [Ta  19611  . Tall developed a simple programme on thermal stresses during bead-on-plate welding along the  centerline of a strip. In his analysis only the longitudinal stress was considered and was a function of lateral distance only. This type of analysis was later designated as one-dimensional. In 1968, based upon Tail's analysis, Masubuchi developed a F O R T R A N programme on the one-dimensional analysis of thermal stresses during welding  [ M a s u b u c h l l % 8  ^ T/ j programme was later modified and improved at Massachusetts Institute of n  s  Technology. However, due to the complexities of welding problems, particularly the complex effects of inelastic material response and material loading and unloading, this one-dimensional analysis could not be  4  further developed. The attention was the focused on numerical method, especially on the finite element method that could be applied to the highly nonlinear inelastic behaviour of weld structures.  Hibbitt and Marcal did the first application of the F E M to welding problems  [ H i b b i t t  '  M a r c a l 1 9 7 2 ]  . In their study a  thermo-mechanical model was developed to simulate the G M A W process. This model also accounted for temperature dependent material properties. Several investigators including Nickell and Hibbitt later used this model to investigate the welding phenomenon  [N,cke  "' "  , b b ml 9 7 5 ]  . Fridman  [ 1 9 7 7 )  developed finite element analysis  procedures for calculating stresses and distortion in longitudinal butt welds. I w a k i  [19711  developed a two-  dimensional finite element programme for the analysis of thermal stress during bead-on-plate welding.  Most of these analyses, however, share some common deficiencies: •  Being very basic, they are all based on relatively simple setup of weldments, which is rarely the case in actual engineering practice.  •  They require complete specifications of the welding material, setup, and ambient conditions, which are seldom available at the design phase of a structure.  Although some of these numerical methods give good predictions on welding distortion, their strict requirements on input make them less applicable in actual engineering design practice. Therefore, some other effective approaches should be sought to make existing knowledge in welding distortion available to design engineers. These new approaches, while adopting all current techniques in numerical analysis of welding distortion, should also be able to generate prediction results based on partially specified input.  1.3  QUALITATIVE AND SEMI-QUANTITATIVE REASONING IN ENGINEERING  Recent advances in the field of artificial intelligence gradually make the conversion from theories in the computer science to practical engineering application a reality. In the last two decades, the developments in qualitative and semi-quantitative reasoning, expert systems, fuzzy logic and artificial neural networks have created many exciting new areas for engineering applications, where traditional methods are difficult to apply. 5  1.3.1  Applying Artificial Intelligence to Engineering  In general, conventional engineering tools have a common weak area: they require a complete set of numerical input data before any analysis can carry on. During most engineering design processes, many design parameters are unknown at first and are gradually determined in the process of completing the design. Engineers are always contented with the fact that they have to provide parameters based on their experience or intuition, which may lead to some lengthen trial-and-error cycles and increase the design costs. The recent advances in artificial intelligence provide many possible solutions to the above problems, and some of them are yet widely adopted in the engineering applications. The following sections will give an introduction to some of these approaches: 1.  Automated Reasoning: make a computer prove theorems in some domain, say, geometry. Qualitative Reasoning and Semi-quantitative Reasoning are two branches in this area. In combination, these two approaches can solve incompletely specified engineering problems, and simulate loosely defined engineering processes.  2.  Expert Systems: Expert Systems are the first commercially viable applications of artificial intelligence. Expert systems have been implemented in many fields and make knowledge presentation more effective than traditional numerical-only computing tools. Now many Expert Systems perform in day-to-day operation throughout all areas of industry and government. They attempt to solve part, or perhaps all, of a practical, significant problem that previously required scarce human expertise.  3.  Learning: make a computer operating in areas such as the above learn to improve its performance over time. The advances in this area have been concentrating on Artificial Neural Networks in the recent years. By adopting Neural Networks, a computer programme can be trained to solve some engineering problems which are difficult to describe with equations.  4.  Natural Language Processing: make a computer communicate with humans in an everyday language, say, English or Chinese.  5.  Game Playing: make a computer play a game, say, chess.  6.  Vision: make a computer, by looking at the photographic image of a scene, interpret what the scene depicts, say, a kitchen with a running tap.  6  Among the above mentioned areas in Artificial Intelligence, automated thinking, expert systems and learning can be applied to solve engineering problems.  In the recent years, expert systems are the most frequently used A I applications in the field of engineering. However, expert systems typically rely on domain-specific heuristic knowledge, and tend to fail ungracefully when confronted with problems that fall even slightly outside the domain for which the system is intended. The key fault of most expert systems is their inability to reason using fundamental knowledge of the domain, such as conservation laws.  The techniques of artificial neural networks (NN) have been progressing quickly in the recently years. An artificial neural network simulates the mechanism of human brains in a very simplified way. It is a network consisting of input nodes, output nodes and one or more layers of processing nodes. Through learning processes, the processing nodes can be configured to direct input information to correct output, for which explicit relations between input and output are not needed to be specified. In dealing with engineering problems, N N techniques are most effective at analyzing problems for which equations or any other types of explicit relationships between input and output are difficult or impossible to obtain. However, a N N implementation requires a great amount of training before it becomes effectively functional, and its output is generally unstable before it is adequately trained.  Welding distortion analysis, the focus of this thesis, involves numerical calculations and use of engineering experience. It requires the adoption of fundamental knowledge and first principles, such as Hooke's Law, as most engineering analysis. It demands the accountability of any analysis output. For these reasons, qualitative and semi-quantitative reasoning is an ideal choice because of its ability to accept incompletely specified input, and its accountability from solid and well developed reasoning techniques.  7  1.3.2  Qualitative and Semi-quantitative Analysis  Qualitative analysis has been applied in diverse fields of the physical and social sciences, where precise mathematical models are difficult to solve analytically or just not available. Qualitative techniques are also often used in conjunction with precise mathematical models to determine bounds on the behaviour of the models.  Engineers often have to contend with complicated problems whereas only limit amount of information is available. This makes qualitative analysis an effective tool for engineering designs. In many cases, the engineers only need to know the bounds of certain behavioural properties instead of the complete, exact solution. In these cases it is unnecessary to carry out expensive, detailed numerical calculations for such modest requirements. When numerical analysis is necessary, it may not be warranted because of the amount of uncertainty within the input data. In many situations, it may be beneficial to use qualitative analysis in the initial stages of analysis, and to use quantitative analysis later when more detailed information is required. Qualitative analysis can also be used as a guide for selecting input parameters so as to reduce the number of repetitions of the detailed analysis.  Because engineers are rarely confronted with a situation where purely qualitative information is available, it is thus also necessary to be able to reason with partial numeric data. Semi-quantitative reasoning is the task of combining incomplete quantitative and qualitative knowledge. Semi-quantitative reasoning is important to model-based reasoning tasks such as design, monitoring and diagnosis. A l l of these tasks involve incomplete knowledge in both qualitative and quantitative forms. There are a number of different representations available for reasoning with incomplete knowledge of quantities, including bounding intervals, probability distribution functions, fuzzy sets, and order-of-magnitude relations. This thesis uses bounding intervals to represent partial knowledge of a real number.  1.4  INTRODUCTION TO QUALITATIVE ENGINEERING SYSTEM FOR WELDING DISTORTION  Qualitative Engineering System for Welding Distortion, or QESWD, is the software tool developed as part of this thesis. QESWD is a prototype software programme targeted to engineers, technicians and students who  8  need to deal with problems of welding distortion. Using the qualitative reasoning engine, Q E S W D is able to analyze welding distortion problems without complete or precise description of the input data. Due to the nature of welding distortion that the phenomenon cannot be easily accessed numerically, a flexible information database is set up and integrated into the software to assist users confine the problem and keep track of knowledge on welding distortion.  The main functionality of QESWD is its ability to compute welding distortion when input parameters are not yet available or cannot be accurately defined. At early stage of the design, engineers are often lack of such information as actual welding procedure, welding environment, etc. To worsen the problem, many welding parameters are impossible to precisely define at all. This scenario makes the conventional numerical analysis tools difficult to apply. QESWD, on the other hand, has the ability to accept incomplete information at the beginning of the design and derive all possible outcomes. When more information become available along with the design progress, users may feed the newly available information into QESWD, which, in turn, will generate more precise results from the better defined problem.  QESWD has the ability to cognitively store and present the knowledge on welding distortion. When Q E S W D is initiated for a new analysis, related knowledge retrieved from a database is presented to help users define the studied problem and select the methods for further analysis. After the analysis being carried out, users can get access to all the information related to the programme output. The flexible setup of the information database also enables the user to append the knowledge stored in QESWD by importing their own experience.  9  2  WELDING DISTORTION  This chapter provides the necessary background information on welding distortion and summarizes the current methods for control and reduction of welding distortion. The software product Q E S W D of this thesis was built upon the knowledge presented in this chapter.  2.1  WELDED STRUCTURES  Welded structures are superior in many aspects to riveted, castings, and forging structures. Therefore, welding is widely used in the fabrication of buildings, bridges, ships, oil-drilling rigs, pipeline, spaceships, nuclear reactors, and pressure vessels. Before World War II, most ships and other structures were riveted; today, almost all of them are fabricated by welding. In fact, many of the structures presently being built, e.g., space rockets, deep-diving submersibles, and very heavy containment vessels for nuclear reactors, could not have been constructed without the proper application of welding technology.  2.1.1  Advantages of Welded Structures over Riveted Structures  Welded structures are superior to bolted (riveted) structures in the following aspects: (1)  High joint efficiency. The joint efficiency is defined as: Fracture strength of a j oint  ^.  Q Q  Fracture strength of the base plate Values of joint efficiency of welded joints are higher than those of most bolted joints. For example, the joint efficiency of a normal, sound butt weld can be as high as 100%,  while the joint efficiency of  bolted joints, depending on the bolt diameter, the spacing, etc., can never reach (2)  100%.  Water and air tightness. It is very difficult to maintain perfect water and air tightness in a bolted structure during service. A welded structure is ideal of structures which require water and air tightness such as submarine halls and storage tanks.  10  (3)  Weight saving. The weight of a hull structure can be reduced as much as 10 and 20% if welding is used.  (4)  No limit on thickness. It is very difficult to efficiently rivet plates that are thicker than 2 inches. In welded structures there is virtually no limit to the thickness that can be employed.  (5)  Simple structural design. Joint designs in welded structures can be much simpler than those in riveted structures. In welded structures, members can be simply butted together or fillet welded. In riveted structures, complex joints are required.  (6)  Reduction in fabrication time and cost. By utilizing module construction techniques in which many assemblies are prefabricated in a plant and are assembled later on site, a welded structure can be fabricated in a short period of time. In a modern ship year, a 200,000-ton welded tanker can be launched in less than 3 months. If the same ship were fabricated with rivets and a similar effort in labour and tools would be made, more than a year would be needed l  2.1.2  M a s u b u c h l  m  o  \  Problems with welded structures  Welded structures are by no means free from all problems. Some of the major difficulties with welded structures are as follows: (1)  Difficult-to-arrest fracture. Once a crack starts to propagate in a welded structure, it is very difficult to arrest it. Therefore, the study of fracture in welded structures is very important. If a crack occurs in a bolted structure, the crack will propagate to the end of the plate and stop; and, though a new crack may be initiated in the second plate, the fracture has been at least temporarily arrested. For this reason that bolted joints are often used as crack arresters in welded structures.  (2)  Possibility of defects. Welds are often plagued with various types of defects including porosity, cracks, slag inclusion, etc.  (3)  Sensitive to materials. Some materials are difficult to weld. For example, steels with high strength are generally relatively difficult to weld without cracking and are very sensitive to even small defects. Aluminum alloys are prone to porosity in the weld metal.  11  (4)  Lack of reliable non-destructive-testing techniques. Although many non-destructive testing methods have been developed and are in use today, none are completely satisfactory in terms of cost and reliability.  (5)  Residual stress and distortion. Due to local heating during welding, complex thermal stresses occur during welding; and residual stress and distortion result after welding. Thermal stress, residual stress, and distortion cause cracking and mismatching; high tensile residual stresses in areas near the weld may cause fractures under certain conditions; distortion and compressive residual stress in the base plate may reduce buckling strength of structural members.  Consequently, in order to design and fabricate a soundly welded structure, it is essential to have: •  adequate design,  •  proper selection of materials,  •  adequate equipment and proper welding procedures,  •  good workmanship,  •  and strict quality control.  2.2  INSIGHTS INTO WELDING DISTORTION PROBLEMS  Welding is the process of joining two pieces of metal together by establishing a metallurgical bond between them. Heating the weld till the liquid state is reached and then allowing the liquid to solidify produces a continuous joint between the two metal pieces. Although the bond is seamless, the metallurgical properties of the weld are not the same as those of the original plates. These properties are different near the joint between the weld material and the host material. This area is called the heat-affected zone.  2.2.1  Residual Stresses  12  Residual stresses in metal structures occur during welding as well as in many manufacturing processes. Mathematically, they are caused by some singularities in a continuous body, which may be called "dislocations". Physically, they are those stresses that would exist in a body if all external loads were removed. Various technical term have been used to refer to residual stresses, such as internal stresses, initial stresses, inherent stresses, reaction stresses and lockup-in stresses. Residual stresses also occur when a body is subjected to a nonuniform temperature change; these stresses are usually called thermal stresses.  -:o'c  ;nsio  / £  —  z  1  h  I  ression  1  ,  \ \  \  \  \  \  a. E O  \  ,  E  Weld  /  =  Distortion Due to Heating by Solar Radioation  Residual Stresses Due to Grinding  Residual S t r e s s e s Due to Welding  Figure 2.1 Macroscopic residual stresses on various scales  A dislocation can be on a macroscopic or microscopic scale. Areas in which residual stresses exist vary greatly in scale from a large portion of a metal structure down to areas measurable only on the atomic scale. Figure 2.1 shows macroscopic residual stresses on several different scales. Residual stresses also occur on a microscopic scale. For example, residual stresses are produced in areas near martensitic structures in steel since the martensite transformation that takes place at relatively low temperatures results in the expansion of the metal. Residual stresses on the atomic scale exist in areas near dislocations. Welding distortion problems are concerned with macroscopic residual stresses.  The magnitude and distribution of residual stresses in a weld are determined by:  13  •  Expansion and contraction characteristics of the base metal and weld metal during the welding thermal cycle.  •  Temperature versus yield strength relationship of the base metal and weld metal.  Residual stresses in metal structures occur for many reasons during manufacturing. Residual stresses may be produced: •  In many materials including plates, bars, and sections during rolling, casting, forging, etc.  •  During forming and shaping of metal parts by such processes as shearing, bending, machining, and grinding.  •  During fabrication processes, such as welding.  Heat treatments during manufacturing can also influence residual stresses residual stresses. For example, quenching produces residual stresses while stress-relieving heat treatments reduce redisual stresses. Residual stresses may be classified according to the mechanisms which produce them: •  Those produced by structural mismatching.  •  Those produced by uneven distribution of non-elastic strains, including plastic and thermal strains.  The magnitude and distribution of residual stresses in a weld are determined by: •  Expansion and contraction characteristics of the base metal and the weld metal during the welding thermal cycle.  •  Temperature versus yield strength relationship of the base metal and the weld metal.  2.2.2  Cause of Welding Distortion  The temperatures required to melt the weld material cause a non-uniform heat distribution between the weld and the original plate. When the weld begins to cool, different phases of steel are produced in the heat-affected zone due to the different cooling rates across the section. The14new phases are harder and more brittle than the  original plate material. This property of welded joints is a concern for engineers, but it is not the only property that needs to be considered.  The large temperature differential between the weld material and the base material also produces residual stresses near the heat-affected zone. When the weld begins to cool, the hot metal tries to contract, while the surrounding, cooler parts of the base metal prevent it from shrinking. This causes the weld line to be in tension and the base metal to be in compression. Residual stresses have two major effects: they produce distortion or cause failure of the weld.  2.2.3  Types of Welding Distortion  Three fundamental dimensional changes that occur during the welding process cause distortion in fabricated structures: •  Transverse shrinkage perpendicular to the weld line.  •  Longitudinal shrinkage parallel to the weld line.  •  Angular distortion (rotation around the weld line).  15  (b) Angular C h a n g e  (a) Transverse Shrinkage  IiJjJiliiLLLJ:  ] 1 M )1  (c) Rotational Distortion  I  1 H 1 11 1 ] 11 ] 1 1 ] II 1  (d) Longitudinal Shrinkage  (e) Longitudinal Bending Distortion  (f) Buckling Distortion  Figure 2.2 Types of weld distortion  These dimensional changes are shown in Figure 2.2 and are classified by their appearance as follows: •  Transverse shrinkage. Shrinkage perpendicular to the weld line.  •  Angular change (transverse distortion). A non-uniform thermal distribution in the thickness direction causes distortion (angular change) close to the weld line.  •  Rotational distortion. Angular distortion in the plane of the plate due to thermal expansion.  •  Longitudinal shrinkage. Shrinkage in the direction of the weld line.  •  Longitudinal bending distortion. Distortion in a plane through the weld line and perpendicular to the plate.  •  Buckling distortion. Thermal compressive stresses cause instability when the plates are thin.  Shrinkage and distortion that occur during the fabrication of actual structures are far more complex than those shown in Figure 2.2. For example, when a long butt joint is welded by the step-back sequence, the transverse shrinkage is not uniform along the weld as shown in Figure 2.2 (a). When longitudinal shrinkage occurs in a  16  fillet-welded joint, the joint will bend longitudinally unless the weld line is located along the neutral axis of the joint. Whether or not a joint is restrained externally will also affect the magnitude and form of distortion.  2.2.4  Analysis of Distortion in Weldment  There are many factors that contribute to the total distortion in a weldment. These factors, their interaction, and their effect on the total distortion are shown in Figure 2.3. This figure depicts that distortion in a welded structure is a function of the structural parameters, the material parameters and the fabrication parameters.  Structural p a r a m e t e r s  Material p a r a m e t e r s  G e o m e t r y , of Structure  Base-Plate Material  Plate Thickness  FilierrMetal material  Fabrication parameters  Welding processes Procedure parameters  Joint Type  Assembty parameters  (1 ] D e t e r m i n e D i m e n s i o n a l C h a n g e s ih E a c h W e l d a . analysis of Heat Flow Extremely Difficult  Y b. Analysis of Thermal Stresses a n d I n c o m p a t i b l e Strains  c : Analysis of Residual Stress a n d Distortion  Angular .Change  Transverse Shrinkage  Longitudinal Shrinkage  Complex Weldment  Simple Weldment  (2) Determine.Distortion I n d u c e d in the Weldment  (3)' C o m b i n e All D i m e n s i o n a l C h a n g e s a n d I n d u c e d Distortion  Total Distortion  1  Figure 2.3 Procedure of welding distortion analysis  [Masubuchi 1980]  The structural parameters include the geometry of the structure (whether it is a panel stiffened with frames, a cylinder, a spherical structure, etc.), plate thickness and joint type (whether it is a butt joint, fillet joint, etc.).  17  The material parameters include types and conditions of base plate and filler-metal materials.  Among the fabrication parameters are the welding processes, including shielding metal-arc, submerged arc, G M A , G T A , and others; the procedure parameters: welding current, voltage, arc travel speed, preheat and interpass temperature, etc.; and the assembly parameters: welding sequence and degree of constraint, among others.  To determine residual stresses and distortion analytically, it is necessary to establish analytic relationships among these three sets of parameters and distortion. This can be done by: 1.  Determining dimensional changes produced in the structure by each weld.  2.  Determining distortion induced in the structure by these dimensional changes.  3.  Combining all dimensional changes and induced distortions.  For a simple weld, the second and third steps are not necessary.  The first step, the determination of dimensional change in each weld, can be further divided into the following: •  Analysis of heat flow.  •  Analysis of thermal stresses during welding to determine incompatible strains that do not satisfy the condition of compatibility of the theory of elasticity.  •  Determination of dimensional changes, including transverse shrinkage, longitudinal shrinkage, and angular change, induced by the incompatible strains.  In fusion welding a weldment is locally heated by the welding heat source. During the thermal cycle, the weldment is subjected to thermal stresses. When the weld is completed, incompatible strains remain in regions near the weld. Incompatible strains, which include dimensional changes associated with solidification of the weld metal, metallurgical transformations, and plastic deformation, are the sources of residual stresses and distortion. When welding processes and parameters are changed, the heat flow patterns are also changed. The  18  change in heat-flow pattern causes a change in the distribution of incompatible strains, and this causes changes in shrinkage and distortion. A number of articles have been published on the subject of heat flow, and, although not an easy problem, it can be handled analytically.  It is difficult to determine the distribution of incompatible strain. When a material undergoes plastic deformation, the stress-strain relationship is not linear and the plastic properties of the material change with the temperature. Even with the use of the computer, however, no complex geometric analysis has ever been made for practical weldments.  When the incompatible strains are determined, analytically or experimentally, the third stage in determining dimensional changes can be handled analytically. Moriguchi  119481  has developed a fundamental theory  concerning stress caused by incompatible strains, and Masubuchi has applied Moriguchi's theory to the study of residual stress and distortion due to welding.  Assuming that the dimensional changes in the welds are found either analytically or experimentally, the second step is to determine the distortion induced in the structure by these dimensional changes. The solution to this problem is rather straightforward. Although plastic deformation is produced in small areas near the weld, most of the remaining material in the structure is elastic. Consequently, the induced distortion can be analyzed by elastic theory. Solutions for a large number of boundary conditions are already available. The elastic theory equations used to determine the induced distortion are independent of fabrication parameters and involve only well-established material parameters. Thus, after the first experiments, the induced distortions can be readily calculated for all types of materials.  19  2.3  2.3.1  NUMERICAL ANALYSIS OF WELDING DISTORTION  One-dimensional Numerical Analysis  One-dimensional analysis employs the method of successive elastic solutions to calculate the transient strains, transient stresses, distortion, and residual stresses during welding. This method was first developed by Tall and later improved by Masubuchi.  X+AX  Figure 2.4 Stripped element for one-dimensional calculation  A simple model for a one-dimensional analysis is illustrated in Figure 2.4. To analyze the stress state of the plate cross section, a narrow strip element perpendicular to the weld line is cut out as shown. Both edges of the strip at x and x + Ax remain straight, the same as the assumption used in the simple beam theory.  A basic assumption inherent in the on-dimensional stress analysis is that a = x = 0. The stress equilibrium y  xy  equation in the absence of any external forces is thereby reduced to a single equation:  3a =0 3x r  (Formula 2.1)  20  This indicates that o~ cannot vary in the direction of the weld. It should be pointed out here, however, that the x  temperature distribution does vary in this direction and consequently so does o~. Hence, the one-dimensional x  model does not satisfy the equilibrium conditions. It is further assumed that at time t, the strip is a part of an infinitely long plate subject to the same temperature over its entire length.  If a single longitudinal position is considered, the entire welding process may be divided into a number of time steps. During each time step, the transverse temperature distribution is assumed to remain constant at the observed longitudinal position. At each new time step, the temperature is changed and a new stress distribution is obtained. Each time step is the fixed system corresponded to a given transverse strip in the moving system. The width of each strip is the product of the length of time step and the speed of the arc. The stress at each time step is calculated using the method of successive elastic solutions.  • e  • e  Figure 2.5 Loading and unloading  During the calculation of the total strains at each time step, the accumulated plastic strains from previous time steps are included to account for possible elastic loading and unloading (Figure 2.5). This is important in the case of welding, where the complex uneven temperature distribution present in the plate gives rise to complex stress histories.  21  2.4  @ x = mid length  Once the transient strains are calculated, it is possible to calculate the transient distortion of the weld plates. Figure 2.6 illustrates the procedure for determining transient distortion.  Computer programmes have been developed by Vitooraporn  [ l 9 9 0 1  and other researchers. These computer  programmes can take into account the temperature dependence of all material properties and any type of strain hardening, and can solve all bead-on-plate, bead-on-edge, and butt welds of flate plates with finite width. The output of each time step consists of the total strain, mechanical strain, plastic strain, and stress at each of the predetermined points located at various transverse distances from the weld line.  22  2.3.2  Finite Element Analysis of Welding Distortion  In order to improve the accuracy of the distortion prediction, the existence of the transverse stress, a near the y  weld line cannot be neglected. Thus, a two-dimensional model should be considered. In the plastic-elastic region, the only possible mean is the use of finite element method.  The governing incremental finite element equation for the problem can be written as:  K  [t + At] • AU  (M)  (0  = R[t + At] - F  ( M )  [t + At]  (Formula 2.2)  where  AT [f + Af] (M)  = tangent stiffness matrix at time t + At which includes the linear and nonlinear strain stiffness matrices.  R[t + At]  = vector of externally applied force at time t + At.  F  = vector of nodal point force due to element internal stress at time t.  (M)  [ f + At]  = increment in nodal point displacement in iteration i: U  (,)  The term F  ( M )  [t + At] - U ' [t + At]. (  l)  [ f + Af] can be evaluated for the materially nonlinear as follows: (Formula 2.3)  where = Constant strain-displacement transformation matrix.  B [t] T  L  <7  (l  !)  [t + At]  - Stress vector corresponding to the nodal point displacement /7' [r + Af]. (  23  !)  0.02  -0.02  1  0  1  100  1  1  1  200  300  400  1  500  Time, sec  Figure 2.7 Comparison of Distortion - Calculation vs. Experiment (  V i t o o r a  P°  r n  199  °1  Figure 2.7 shows an example of calculated distortion compared with the experimental results. The 1-D analytical analysis tends to shift the peak distortion further away from the one predicted by finite element method and experiment. Furthermore, the 1-D analytical analysis tends to overestimate the peak distortion. This result can be attributed to the higher temperature as well as slower cooling rate calculated from the analytical analysis. A good correlation can be obtained between the results from finite element calculation and experimental data. The degree of good correlation, however, varies with different materials. This can be attributed to the accuracy of the material property data obtained at elevated temperatures for each material.  In summation, the same physical behaviour of the specimen in experiments can be obtained from the numerical analysis. The discrepancy between the calculated results and the experimental data can be attributed to the accuracy of the input data such as material properties, heat source distribution, finite element mesh, assumptions made in the analysis, etc. It should be mentioned that the phase transformation does not include in both onedimension analysis and finite element analysis for this investigation. This will be the case, however, when poor agreement is observed between experimental data and calculated results. Nevertheless, this is another source of error.  24  2.4  A N A L Y T I C A L AND EMPIRICAL F O R M U L A S  There are several ways to analyze residual stresses and distortion. The orthodox method is analytical simulation. This approach makes it possible to study not only distortion after welding is completed, but also transient metal movement as well, which is desirable. It is important to follow the metal movement, because distortion during welding and distortion after welding is completed are quite different. For example, Figure 2.8 shows change of deflection during welding along the longitudinal edge of rectangular plate. Distortion during welding is opposite to the distortion after welding is completed.  However, analytical simulation is too complex a method to be useful in very many situations. Computer programmes with strict input requirements are needed to calculate the transient distortion, even in simple cases, such as a weld along the edge of a rectangular plate. The determination of the incompatible strains produced during welding in regions near the weld is the step that makes the analysis so complex.  25  If one is concerned only with the distortion that remains after the welding is completed, analytical simulation is unnecessary. In this case the distortion is treated as an elastic stress field containing incompatible strains. The mathematics involved is relatively simple, which makes this approach useful in analyzing actual practical joints.  In the following sections, analytical or empirical formulas for distortion calculation of all types of weld details are discussed.  2.4.1  Transverse Shrinkage  - Butt Welds  The mechanisms of transverse shrinkage have been studied by several investigators including Naka and Matsui jj  [1964]  i e  m o s t  j  m  p  0  r  t  a  n  t  f ] i g f their mathematical analyses is as follows: The major portion of transverse m c  n  0  shrinkage of a butt weld is due to contraction of the base plate. The base plate expands during welding. When the weld metal solidifies the expand base metal must shrink, and this shrinkage accounts for the major part of transverse shrinkage. Shrinkage of the weld metal itself is only about 10% of the actual shrinkage.  The major factors that cause this non-uniform transverse shrinkage in butt welds are: •  Rotational distortion. When welding is conducted progressively from one end of a joint to the other, the unwelded portion of the joint moves, causing a rotational distortion, as shown in Figure 2.2 (c). The rotational distortion is affected by the welding heat input and the location of tack welds.  •  Restraint. The amount of transverse shrinkage that occurs in welds is affected by the degree of restraint applied to the weld joint. The amount of shrinkage decreases as the degree of restraint increases.  The welding sequence has a complex effect on the rotational distortion and the distribution of restrain along the weld.  26  Empirical and Analytical Formulas Many investigators have proposed formulas for the estimation of transverse shrinkage of butt welds, which by and large are based on empirical information. These formulae are listed in Table 2.1.  Table 2.1 Formulae for prediction of transverse shrinkage Malisius's formula  119361  S: axial shrinkage perpendicular to the weld, mm. X\ \ linear thermal expansion of the bar from T to (T\-TQ)/2, about 0.004. 0  S=LK  — + X-yb S,  T : initial temperature of the bar 0  T\. temperature above which the material is no longer elastic (Ti>7b) Xq,: linear thermal expansion of the weld from To to Ti, about 0.0093 Q: cross-section of weld including reinforcement, mm , 2  Si: average thickness of bars, mm. B: average breadth of weld, mm. K: a constant depending on the thermal output of the welding process and the thermal conductivity. 43 for arc welding, bare electrodes (S=1.0 mm), 45 to 55 for coated electrodes (S=1.4 mm average), 64 for atomic-hydrogen welding (S=1.4 mm), 75 for oxyacetylene welding (S=1.7 mm). Capel's formula  119621  Al: transverse shrinkage, mm. s: thickness of layer of weld metal, mm.  . . KxWxlO Al = sxu  3  u: welding speed, cm/min. W: electric power of welding arc, 7xV I: welding current, amperes. V: arc voltage, volts. K: constant dependent on materials, 20.4 for aluminum, 22.7 for stainless steel, 17.4 for carbon steel.  Cline's formula  119651  Al: transverse shrinkage, in. t: plate thickness in.  Al = 0.1(77-0.230)  27  Analytical Analysis Figure 2.9 is a schematic presentation that shows the changes of transverse shrinkage in a single-pass butt weld in a free joint after welding. Shortly after welding, the heat of the weld metal is transmitted into the base metal. This causes the base metal to expand, with a consequent contraction of the weld metal. During this period the points of sections A and A ' do not move. When the weld metal begins to resist the additional thermal deformation of the base metal, parts of sections A and A ' , begin to move in response. The starting time of the movement of A and A ' is indicated by t . s  A'  x * -  A  mm  (a) t = 0  w* - L / 2  *-L/2—  w  i n n—s /2 in 1 ^ 1 :i" w  (b) t - t  A  c  1 *  (c) t > t  e  i  i  i  S  I  A 5  \  S/2-J  J-8,/2  Ti  S/2- -(L.S *Sr-lj (d) t =  (  •  2  !*  i  —L/2—  Figure 2.9 Schematic Presentation of a Transverse Shrinkage of a Butt Weld in a Single Pass '  Based on the above illustration, the transverse shrinkage can be calculated as followed:  28  M a s u b u c h i 1 9 8 0  '  • ,^ „ , a(T) • T(t , x) - a(T /2  s  =2j\  Sx  s  0  8 = 2J\a(T) • T(t, x)-a(T )0  l , )-T \-dx 0  &:  Thermal expansion of the base metal at t = t .  8:  Additional thermal deformation fo the base metal caused in A A ' at t > t . s  T(t , x)] • dx s  5 :  Thermal contraction of the weld metal at t > t .  -  Transverse shrinkage.  a{T):  Thermal expansion coefficient.  T(t,x):  Temperature.  TM'-  Melting temperature  T:  Initial and final (room) temperature.  W  S  = [a(T )-T -a(T )-T \L  w  M  M  0,  0  S w  for0<?<r  -8 + S ,  fort>i  w  8 +S S  0  W  s  , forf = c  0  s  (Formula 2.4)  From the above formulas, it can be found out that the shrinkage of weld metal, S , is less that 10% of the total w  weld. Therefore, the thermal expansion of the base metal is the most important factor in the final shrinkage of a single-pass butt weld in a free joint.  Effect of Plate Thickness Matsui  119641  studied analytically and experimentally how the plate thickness affects the transverse shrinkage in a  butt weld. On this basis, the approximate transverse shrinkage can be expressed using an error function erf(): Thin plate: c S =  [t :  LIAnXt ,  Q-  heat input,  c:  specific heat,  PA:  density, thermal diffusivity,  h:  plate thickness.  s  2_  , , erf (ji  , )  n  s  cph Thick plate: -(n/i)  5=  [l + 2^Te cp2nlt  s  s  A  h  2  ]-erf(p ) s  (Formula 2.5)  n=x  The following were assumed when the above formulas were derived: •  Thermal expansion coefficient was constant.  •  The thermal radiation would be neglected.  •  The thermal contraction of the weld metal would be neglected.  29  From the equation 2.5, it can be seen that the final shrinkage decreases with an increasing thickness, which was verified by Matsui's experimental data. But it should be emphasized that this is true only if the same amount of heat input is used, regardless of the joint thickness. Welding thicker plates may require more than one pass, which introduces more heat input.  Effect of Materials The amount of transverse shrinkage is different for the various materials. For example, compared to steel, aluminum alloys, because of their higher heat conductivity and thermal expansion coefficients, shrink more. It is well known that transverse shrinkage in aluminum welds is greater that that in steel welds.  Phase transformation of ferrous materials also plays an important role. M a t s u i  [1964!  has proposed that the  expansion due to phase transformation should be subtracted from the estimated shrinkage in order to predict the real shrinkage.  Effects of Restraint and Forced Chilling It is known that transverse shrinkage decreases when a joint is restrained. Iwamura  [ 1 9 7 4 1  investigated how  restraint and forced chilling affects the transverse shrinkage of butt welds in aluminum alloy. Both computer analysis and experiments showed that the restraint reduced the amount of shrinkage by about 30%. Chilling, however, were not proven to be a effective way to reduce shrinkage.  30  DISTANCE FROM C E N T E R L I N E (mm)  800  0  10 1  20  30  T  T"  400  r 6oou  H  < tr.  400^  Ul  0\t  • 10 sec.  t = 60 sec.  a. 5  300"  N  N  rr 200< tr ui a.  iool  200h  ^ CHILLED ZONE  _|o  0 0.5 1.0 DISTANCE FROM CENTERLINE (in.) Fee Joint  Figure 2.10 Temperature Distribution  Iwamura's tests were carried out on plated as illustrated in Figure 2.10. The chilling had little effect on the temperature distribution in the early stages of welding, e.g. the first 9 seconds, but lowered the temperatures at a later stage, e.g., after 60 seconds. The mathematical analysis indicates that the temperature distribution in the joint after the weld metal solidifies has a critical affect on transverse shrinkage. In order for the chilling to be effective, it is therefore important to alter the temperature distribution before the weld metal solidifies. But although this was possible, it was too late to effectively reduce transverse shrinkage.  Effects of Welding Procedures Table 2.2 shows the effects of various procedures on transverse shrinkage in butt welds.  31  0  5 10 15 20 25 Weight of Weld Metal perUnit Weld Length (w),gr/cm  0  0.5  1.0  1.5  Log w !0  a. Increase of Tronsverse Shrinkage in Multipass Welding  b. Relationship Between' log w and u  w: Weight of weld metal per unit weld length (H>), gr/cm. u: Transverse shrinkage (u), mm.  Figure 2.11 Increase of Transverse Shrinkage During Multi-pass Welding of a Butt J o i n t  t  [Masubuchi 1 9 7 0 1  Figure 2.11 shows schematically how the transverse shrinkage increases during multi-pass welding. Because the resistance against shrinkage increases as the weld gets larger, shrinkage was pronounced during the early weld passes but diminished during later passes.  Table 2.2. Effect of various procedures on transverse shrinkage in butt welds  Procedures  Effects  Root opening  Shrinkage increases as root opening increase. Effect is large.  Joint design  A single-vee joint produces more shrinkage than a double-vee joint. Effect is large.  Electrode diameter  Shrinkage decreases by using larger-seized electrodes.. Effect is medium.  Degree of constraint  Shrinkage decreases as the degree of constraint increases. Effect is medium.  Electrode type  Effect is minor.  Peening  Shrinkage decreases by peening. Effect is minor.  Rotational distortion of Butt Welds Rotational distortion is affected by both heat input and welding speed. When Vi-in. thick mild steel plates are welded using covered electrodes at a low welding speed, the unwelded portion of the joint tends to close. When  32  steel plates are welded using the submerged-arc process, the unwelded portion of the joint tends to open. This means that the tack welds used must be large enough to withstand the stresses caused by the rotational distortion.  Rotational distortion causes two problems: •  Rotational distortion is one component involved in the transverse shrinkage of a butt joint, especially in a long butt weld. When studying how the welding sequence affects the transverse shrinkage in a long butt weld, the effects of rotational distortion must be considered. The largest amount of rotational distortion occurs during the first pass, when the unwelded portions of the joint are relatively free.  •  The separating force produced by the rotational distortion can be large enough to fracture the tack welds and crack portions of the weld metal.  How Welding Sequence Affects Transverse Shrinkage Several steps are involved in the welding of a long butt joint. A variety of welding sequences may be used. These welding sequences are of two types: •  The block-welding sequence. The joint is divided into several blocks. Each block is welded separately, in turn.  •  The multi-layer welding sequence. Each layer is welded along the entire joint length before any of the next layer is begun.  Both types have many variations.  It is often found that rather uneven transverse-shrinkage distributions were obtained with block-welding sequence whereas the shrinkage distribution obtained with the multi-layer sequence was much more even. Welding using different arrangement of block sequencing often gives approximately the same distortion. Figure 2.12 shows an example of influence of weld sequencing.  33  ©  © © Welding Blocks  -©-©-©  1.0 \  0.8  /  -  -  y  -  (  i  K  i  H  2  )  V / ^  0.6  \  0.4  //  0.2  |  ^W»  T  Left .  . Center . Block  Right Block  Block  I  [-100 i  Figure 2.12 Distribution of Transverse Shrinkage Obtained in Slit-type Specimens with Different Welding Sequence  Shrinkage  Shrinkage  Figure 2.13 Transverse Shrinkage in Fillet Welds  2.4.2  Transverse Shrinkage in Fillet Welds  A fillet weld undergoes less transverse shrinkage than a butt weld (Figure 2.13). Only a limited amount of study has been done on transverse shrinkage in fillet welds. Spraragen and Ettinger  [19501  suggested the following  simple formula: •  For tee-joints with two continuous fillets: • , Shrinkage  leg of fillet thickness of plate  „„ x 1.016 mm  (Formula 2.6)  34  •  For intermittent welds, use correcting factor of proportional length of fillet to total length. For fillets in lap joint (two fillet welds): leg of fillet Shrinkage — xl.0\6mm thickness of plate  2.4.3  (Formula 2.7)  Angular Changes of Butt Welds  Angular change often occurs in a butt weld when the transverse shrinkage is not uniform in the thickness direction. A thorough investigation has been made of how various welding-procedure parameters, including the shape of the groove and the degree of restraint, affect the angular change in butt welds.  .60°.  Figure 2.14 Effect of Shape of Groove on Angular Change  During a groove welding, a mild increase of angular change was observed in the earliest stage of welding on the first side. The increase of angular change became greater in the intermediate stage, and then mild again in the final stage. The back chipping did not affect the angular change. Angular change in the reverse direction was produced during the welding of the second side. The angular change that remained after the welding was completed depended on the ratio of the weld metal deposited on the two sides of the plate. Since the angular change increased more rapidly during the welding of the second side, the minimum angular change was obtained in the specimen that had a little larger groove in the first side. Some researchers proposed that the angular change could be minimized for a butt joint having a (hi+ /2h ) to h ratio of approximately 0.6 (Figure 2.14). 1  3  35  An extensive programme was conducted by the Shipbuilding Research Association of Japan on angular change in butt welds. Figure 2.15 shows the groove shape that most successfully minimized angular change in butt welds of various thickness. Curves are shown for situations with and without strongbacks. For example, when the plate thickness is 20 mm, the ratio of hi and h that gives the minimum distortion when the joint is free is 7 2  to 3. In terms of the weight of the deposited metal, the W i / w ratio is approximately 49 to 9. 2  2.4.4  Angular Distortion of Fillet Welds  The panel structure, a flat plate with longitudinal and transverse stiffeners fillet welded to the bottom, is a typical structural component in ships, aerospace vehicles, and other structures. A n example of such panel structures is shown in Figure 2.16.  36  Figure 2.16 Panel Structure with Stiffeners  The major distortion problem in the fabrication of panel structures is related to out-of-plane distortion caused by angular changes along the fillet welds. Corrugation failures of bottom shell plating in some welded cargo vessels are believed to be caused when excessive initial distortion reduces the buckling strength of the plating.  When longitudinal and transverse stiffeners are fillet welded as shown in Figure 2.16, the deflection of the panel, 8, changes in both the x-direction and y-direction. Because of the mathematical difficulties involved in twodimensional analysis, most studies conducted so far have been one-dimensional.  Distortion Calculation Figure 2.17 shows the typical out-of-plane distortion found in two types of simple fillet-welded structures. In both cases, the distortion is one-dimensional. When a fillet joint is free from external constraint, the structure bends at each joint and forms a polygon. But if the joint is constrained by some means, a different type of distortion is produced. For example, if the stiffeners are welded to a rigid beam, the angular changes at the fillet welds will cause a wavy, or arc-form, distortion of the bottom plate.  37  A. Free joint RIGID BEAM ///////////////////////////////////a  BOTTOM PLATE B. Constrained joint (framed structure )  Figure 2.17 Distortion Due to Fillet Welds  Masubuchi et al  i m t  ' found that the wavy distortion and resulting stresses could be analyzed as a rigid-frame 1  stress problem. In the simplest case in which the sizes of all welds are the same, the distortion of all spans are equal and distortion, 5, can be expressed as follows:  5 = a  . I_(£_O.5) ].0 4 a  0=-  2  [  0o 2D 1+ aC  0:  angular change at a fillet weld, radians,  fo-  angular change of a free fillet weld (ref.0.280)  ci:  length of span  D:  rigidity of bottom plate  C:  coefficient of rigidity for angular changes (Formula 2.8)  Out-of-plane Distortion Out-of-plane distortion reduces the buckling strength of a panel. It is believed that the initial distortion and the residual stresses are the major reasons for the corrugation damage in the bottom plates of a number of transversely framed welded cargo ships.  38  Structural designs and welding procedures have rarely been considered at the same time, although both need to deal with out-of-plane distortion. This can be understood because normally they are conducted by different specialists, structural and welding engineers. In addition, each subject is rather complicated and an integrated study has to involve complicated computations that might not be possible to be handled manually. In practice, however, it is desirable to combine the two analyses. For example, a simple way to reduce the amount of distortion is to reduce the size of the fillet welds. But if the fillet size is too small, the welds may fail and floors may be ripped from the plating during service. On the other hand, if the fillet size is increased too much, distortion of the plate will become excessive and the plate may buckle during service. In order to achieve the optimum design, it is important to analyze both weld distortion and its effects on the service behaviour of the structure.  A design procedure to satisfy the above requirements may consist two parts. The first part calculates values of allowable initial distortion, for a given set of structural parameters, including plate thickness, frame spacing, aspect ratio of panel, and compressive in-plane stresses, while the second part calculates the amount of weld allowed to produce the maximum distortion at panel centre.  Parameters Affecting Angular Distortion The parameters affecting the angular distortion of fillet welds is discussed as follows: (1)  Welding current, speed and plate thickness.  Watanabe and Satoh  119611  proposed the following formula: /:  \m+l  0o  1  = i| c  h^fvh  Mvh  welding current, amperes,  V:  welding speed, cm/sec,  Iv.  plate thickness, cm.  C\, C and m: coefficients determined by the type of electrodes. For an ilmenite electrode: C,=0.0885xl0" , C =6.0xl0" , m=1.5. 2  (Formula 2.9)  6  3  2  (2) Preheating. Preheating can reduce the angular distortion. Preheating the back of the plate proves more effective in reducing angular distortion than preheating the front. It is an additional expense during fabrication.  39  (3) Prestraining. The angular distortion of a fillet weld can be reduced if an initial angular distortion is provided in the negative direction. There are basically two methods for this: (1) plastic prebending and, (2) elastic prestraining (See Figure 2.18).  Q. PLASTIC PREBENDING  b. ELASTIC PRESTRAINING  Figure 2.18 Plastic Prebending and Elastic Prestraining  If an exact amount of plastic prebending could be used, a fillet weld with no angular distortion whatsoever would be the result. In elastic pre-straining a restraining jig is used. Often this is simply a bar of a certain size placed under the weld and the plate clamped in a jig. If the proper amount of prestraining is used, the fillet weld will have no angular distortion after release.  There are advantages and disadvantages with both methods. It is generally believed that in practice elastic prestraining is more reliable than plastic prebending. Since the weldment is clamped, the angular distortion is always much less than it would be if it were free. Even if an error is made in the amount of prestraining used,  40  the angular distortion is always reduced. If plastic prebending is used, the amount of prebending used must be exact if a joint without distortion is to be produced. The amount of adequate prebending changes with the plate thickness, the welding conditions, and other parameters, and the bending-line must exactly match the weld line.  A. Experimental set-up used  ,<3  z  *"/  6 o'  2  4  {ft / c LEG LENGTH 10 "SFAN 380™ ~ AX LEG LENGTH 75 SFttN 380 ' to  » LEG LENGTH 6 SPAN 380 • LEG LENGTH 75 SPAN 760  /  6  8  10  12  14  16  18  20  THICKNESS OF PLATE, mm B. Relationship between plate thickness and skin stress required for obtaining zero angular change  Figure 2.19 Elastic Prestraining for Reducing Angular Distortion of Fillet Welds  Figure 2.19 depicts a sample setup of elastic prestraining. Kumose et al.  119541  proposed a relationship between  the bottom plate thickness and the skin stress required to produce a zero angular distortion:  41  D: diameter of the bar placed under the bottom of the plate, t: plate thickness, E' =  L: length of free span,  E  E: Young's modulus, v: Poisson's ratio. (Formula 2.10)  2.4.5  Longitudinal Shrinkage of Butt Welds  The longitudinal shrinkage in a butt weld is approximately 1/1000* the weld length, much less than the transverse shrinkage. Only limited studies have been made of longitudinal shrinkage in a butt weld. King proposed the following formula:  AL =  0.12/L  /: welding current, amps,  100,000?  L: length of weld, in., t: plate thickness, in. (Formula 2.11)  For example, when t = A in . (6.4mm) and I = 250 amperes, A L / L = 1.2 x 10" . X  2.4.6  3  Longitudinal Shrinkage of Fillet Welds  Guyot [b716] conducted an extensive study on the longitudinal shrinkage of fillet welds in carbon steel. We found that longitudinal shrinkage is primarily a function of the total cross-section of the joints involved. Restraint is more effective when the plates are thicker and wider. The total cross-section of the welded plates in the transverse section is called the resisting cross-section. The following formula may be used to predict the longitudinal shrinkage of fillet welds:  /  42  8: longitudinal shrinkage (mm) per 1 m of weld.  x25  A : area of the weld metal. w  A : resisting cross-sectional area. p  (Formula 2.12)  2.4.7  Longitudinal Bending Distortion  When the weld line does not coincide with the neutral axis of a weld structure, the longitudinal shrinkage of the weld metal induces bending moments, resulting in longitudinal distortion of the structure. This type of distortion is of special importance when fabricating T-bars and I-beams.  x Figure 2.20 Longitudinal Distortion in a Fillet-welded joint  Take the welding of an I beam as an example (Figure 2.20). When the welding proceeds, the deformation increases with the welding of the underside fillet, and decreases with the welding of the upper side. The deformation due to the welding of this second fillet is generally smaller than that of the first, causing some residual deformation to remain, even when the weight of the deposit metal of both fillet welds is equal and the geometry of the joint is symmetric. This occurs because the effective resisting area of the joint differs between the two; the upper flange does not effectively constrain the deformation during the welding of the underside of the fillet, since the upper flange is only tack welded to the web plate, but both flanges effectively constrain the welding of the upper side fillet, since the lower flange has already been welded to the web.  43  Sasayama et al  analyzed some experimental results and developed a theory similar to the bending-beam  y m i i  theory. In the case of the bending distortion of a long, slender beam, longitudinal residual stress (a ) and the x  curvature of longitudinal distortion (\/R) are given by the following equation:  "  o=-Ee  r  M * p * +—— z + -^— I A y  A: sectional area of the joint, y  y  P *: apparent shrinkage force, P  ~ El  y  x  / : moment of inertia of the joint around the neutral axis,  l _M *_ El  e ": incompatible strain,  x  y  x  -^Ee  x  dydz  My*: apparent shrinkage moment, M " = ^Ee "zdydz = P *l* x  (Formula 2.13)  x  L*: distance between the neutral axis and the acting axis of apparent shrinkage force  Formula 2.13 shows that it is necessary to know the distribution of incompatible strain (e ") in order to know x  the distribution of residual stress (o ) but the information about moment (M *) is sufficient only for determining x  y  the amount of distortion (l/R). The moment (M *) can be determined when the magnitude of the apparent y  shrinkage force (P *) and the location of its acting axis are known. Through experiments, it was found that the x  acting axis of "x' is located somewhere in the weld metal. It is believed that the apparent shrinkage force (P *) x  causes residual stress and distortion. More information can be obtained when the P * value rather than the %  distortion value itself is used in the analysis of experimental results. With this it is possible to separate the various factors that affect the magnitude of distortion into those caused by changes in geometry (A, I , or L*) y  and those caused by changes in the value of P** itself.  44  150 T-Bar  I - Beam  0- 100 r -  .c IS)  o a  Q. 50 r<  0 0.5 1.0 1.5 2.0 Weight of Electrode Consumed Per Unit Weld Length , g r / m m I l i i i i i 0 6 8 10 12 14 Length of Leg , mm  2. I  Figure 2.21 Increase of Longitudinal distortion During Multi-pass Welding ^ y " ™  1 9 5 5  ]  The increase of longitudinal distortion, i.e. the apparent shrinkage force P * , during multi-pass welding is as x  shown in Figure 2.21. A l l of the plate specimens were made from mild steel 1200 mm long and 12 to 13 mm thick. The P * values increased proportionally with the weight of the electrode consumed per weld length, x  except for the first layer. The large amount of distortion obtained in the first layer was due to the lack of resisting-area during that stage of welding; the flange plate was not yet attached firmly to the web plate.  Practically no distortion was produced during the intermittent welding (Specimen 1-4). This is due to the fact that longitudinal residual stress does not reach a high value in a short intermittent weld.  45  2.4.8  Buckling Distortion  When thin plates are connected by welding, residual tensile stresses occur in the weld and compressive stresses occur in areas away from the weld which cause buckling. Buckling distortion occurs when the specimen length exceeds the critical length for a given thickness in a given size specimen. In studying weld distortion in thinplated structures, it is important to first determine whether the distortion is being produced by buckling or by bending. Buckling distortion differs from bending distortion in that: •  There is possibly more than one stable deformed shape.  •  The amount of deformation in buckling distortion is much greater than bending distortion.  Since the amount of buckling distortion is large, the best way to avoid it is to select appropriate plate thickness, stiffener spacing, and welding parameters. Any plate has a critical buckling load. In order to avoid failure, the welding stresses must remain below this level. This can be achieved by less welding, using less heat, or removing the heat. One way to reduce the amount of weld is to use intermittent welding; by halving the amount of welding, the critical load is approximately doubled. Another way is to decrease the weld-bead size, which results in smaller heat requirements during welding and hence in lower stress levels. The alternate way to reduce the stress levels is to remove the welding heat from the plate using chill bars, water-cooled backing plates, etc.  2.5  M E T H O D S OF DISTORTION REDUCTION IN W E L D M E N T S  This section presents several methods of reducing distortion in weldments.  2.5.1  Commonly Use Distortion Reduction Methods  The common methods of reducing weld distortion are reviewed as follows:  Weldment Dimension The length, the width, and the thickness of a weldment all influence the amount of distortion. The plate thickness greatly influences the angular distortion in a fillet weld. Since the angular change of a fillet weld is  46  caused by temperature differences between the top and bottom surfaces of the plate, at a certain plate thickness (about 3/8 in.for steel and A in. for aluminum respectively) the angular change is maximum. When the X  thickness is greater than this, the angular change is less because of the rigidity of the plate and also because the temperature differential between the top and bottom surfaces is less. However, this does not mean that when engineers fabricate thin-plated structures they will have fewer distortion problems. Buckling governs and since buckling distortion, if present, is always serious. It is best to avoid it by a careful selection of structural parameters, e.g. plate thickness, stiffener spacing, and welding parameters.  Joint Design Distortion is affected by joint design. As a general rule, distortion can be reduced by keeping the amount of weld metal used at a minimum. Sections 2.4.3 shows the groove shape that gives a zero angular distortion in butt welds.  Welding Processes and Welding Conditions Since residual stresses and distortion are the result of uneven heating during welding, it is generally true that the less total heat a process uses in joining, the less distortion will be produced. Weldments produced using narrowgap welding, electron beam welding, and laser welding all exhibit less distortion than those produced using arc welding. Generally, a weld made using a low heat input generally exhibits less distortion than a weld made using a high heat input.  It must also be recognized that the influence of the temperature distribution and the heat input on various types of distortion can be rather complex. For example, the transverse shrinkage of a butt weld is greatly affected by the temperature distribution in the base plates when the weld metal solidifies. The best way to reduce transverse shrinkage in butt welds, therefore, is to reduce the heat-spread before the weld metal solidifies.  47  The angular change of a fillet weld is greatly influenced by the temperature difference between the top and bottom surfaces, which are closest to and opposite to the heat source, respectively. The longitudinal shrinkage of the weld zone induces bending distortion, which can be reduced by placing the weld near the neutral axis of the weldment and by reducing the amount of weld metal.  Multi-pass Welding A large percentage of the transverse shrinkage that takes place in butt welds occurs during the first and the second pass. And it has been found that the use of larger diameter electrodes will result in less shrinkage.  Constraints The use of external constraints to reduce distortion is a common practice in the fabrication of welded structures. Through the selection of appropriate strong-backs, jigs, clamps and rollers, induced distortions can be reduced.  Welding Sequence It has been found that welding sequence affects transverse distortion. Block welding sequences were generally found to cause less shrinkage than multi-layer sequences.  Intermittent Welding Longitudinal bending distortion can be significantly reduced when intermittent fillet welds are used. However, it is found that it is not as effective to angular distortion.  Peening When metal that has shrunk during welding is stretched by peening, some of the distortion can be removed.  48  50 100 TEMPERATURE *C  200  Figure 2.22 Calculated Final Center Deflection vs. Preheating Temperature of the web  Differential Heating to reduce Longitudinal Distortion of built-up Beams This technique refers to a distortion reduction method that intentionally create temperature differences between the parts to be welded. The preheated part cools and contracts more than the part that is not preheated. The thermal stress generated can partially cancel out the residual bending stresses that would have resulted if the parts had been at the same temperature when joined. The result is distortion reduction. Figure 2.22 illustrates an example of the welding of a tee beam, where by adjusting preheat temperature to about 50°C will create the least distortion.  2.6  M E T H O D S OF R E M O V I N G DISTORTION  Even though there are many distortion reduction methods in regular use, distortion often exceeds tolerable levels. Distortion is also produced during service, from collision for example. It becomes necessary to utilize methods of removing distortion.  49  2.6.1  Straightening by Flame Heating  The most common distortion removal technique is to flame heat a plate at selected spots or along certain lines and then to water cool it. Sometims plates are heated and hammered, a technique that requires great intuitive skill on the part of the workmen, since little scientific information, either analytical or experimental, is available on the distortion removal mechanisms operative when this treatment is used. Although several papers have been published on flame-heating, they are primarily of an empirical nature. Scientific data on flame-straightening is still scarce. Table 2.3 illustrates some methods of flame straightening.  Table 2.3 Methods of Flame-straightening Line Heating In line heating, heat from the torch is applied along a line or a set of parallel lines. This methods is frequently used for removing angular distortion produced by the fillet welds attaching a plate to its stiffeners. Line heating can also be used as a method for bending plates. Figure 2.23 shows some results from some researchers. [bf754,755]  • • •  Pine-needle heating In pine-needle heating, heat is applied along two short lines crossing each other as shown. This method is half-way between line heating and spot heating. Since the shrinkage and angular distortion occur in two directions, this method produces a uniform distortion-removal effect. Checkerboard heating W W  w w  Heat is applied along a system of two lines crossing each other as shown. This method is often used to remove sever distortion. One must be careful not to overheat the metal. Spot Heating  ®  9  ®  ®  ®  Heat is applied on a number of spots. This method is widely used for distortion removal, especially in thin-plated structures.  Triangular Heating Heat is applied on a triangular-shaped area. This method is useful for the removal of bending-distortion in frames. A  A  o  Red-hot Heating When severe distortion occurs in a localized area, it may be necessary to heat the area to a high temperature and beat it with a hammer. This method can cause metallurgical changes, however.  50  2.5  5.0  7.5  10.0  12.5  15.0  12.5  15.0  TRAVEL SPEED  '/sec.  a. ANGULAR CHANGE  0.7 0.6 0.5  \  0.4 0.3 0.2  \\  \  0.1 0,  2.5  5.0  7.5  10.0  TRAVEL SPEED  mm  /s|  b. TRANSVERSE SHRINKAGE  Figure 2.23 Effects of Travel Speed of Flame on Angular Change and Transverse Shrinkage  2.6.2  [ S a t o h 1 9 6 9 1  Vibratory stress-relieving and electromagnetic-hammer technique  Recently attempts have been made to develop new distortion-removal techniques. Vibratory stress-relieving and electromagnetic-hammer are two of these techniques  [ C h e e v e r  1977]  .  Vibratory stress relieving This technique reduces residual stress and distortion by means of vibrating the weldments. The main equipment consists of a variable-speed vibrator clamped to the work piece. By varying the speed of the vibrator motor, the frequency can be varied until a resonant frequency has been reached for the work piece. The piece is then allowed to vibrate for a period which varies in length roughly in relation to the weight of the work piece and usually ranges from 10 to 30 minutes.  51  The vibratory technique is much more economical than thermal stress-relieving techniques. It involves little in the way of expensive equipment and achieves its objective in a relatively short time. However, there is almost a complete lack of scientific information on how it works. It is believed that the vibratory energy introduced in the work piece realigns the lattice structure to relieve stress and stabilize the part without distortion.  Electromagnetic Hammer The direct application of electromagnetic forces to metal working is a recent innovation. The electromagnetic hammer is named after hammer because it can be used in place of a hammer. The material to be straightened must be considered a part of a total magnet-forming system; its characteristics can significantly change the amount of deformation resulting from a given amount of stored energy. The conductivity of the material determines the effectiveness of energy conversion to magnetic forces. Electromagnetic hammers have successfully been used to remove the distortion in welded tanks and bulk heads in rockets.  52  3  QUALITATIVE  AND SEMI-QUANTITATIVE ANALYSIS  IN  ENGINEERING  This chapter first introduces the basic concepts in qualitative and semi-quantitative analysis, which is followed by brief description on QES  [ G e d  '  s 19951  , providing qualitative reasoning techniques to welding distortion analysis  in this research.  3.1  D E V E L O P M E N T OF QUALITATIVE ANALYSIS  Qualitative analysis can be considered as consisting of qualitative reasoning and semi-quantitative analysis. The techniques of qualitative reasoning have been applied in diverse fields in the last few decades. The latest advances in qualitative reasoning are embodied in an area of artificial intelligence called qualitative physics. Semi-quantitative analysis enables the acceptance of numerical information in qualitative analysis, by adopting such well-developed techniques as interval analysis.  3.1.1  Qualitative Reasoning  Qualitative analysis has been applied in diverse fields of science and engineering, including economics, applied mathematics, ecology and control theory. Qualitative analysis can be superior to quantitative analysis in many cases, such as when precise mathematical models are not available. Even if mathematical equations are available, it may not be possible to solve them analytically so that direct relationships between various parameters can be inferred. When the mathematical model can be solved analytically, it might be still difficult to specify the parameters precisely to reflect the real-world situations. When dealing with these cases, qualitative techniques can be used in conjunction with more precise mathematical models, in order to efficiently guide the solution of such models, or to determine bounds on the behaviour of models.  Qualitative analysis has been applied in the fields where the exact form of the mathematical relations linking various parameters is not known, or the mathematical models are too difficult to solve. For example, in population ecology, it may be observed that the populations of foxes and rabbits obey a predator-prey  53  relationship. If the population of foxes increases, the population of rabbits in the same area tends to decrease. Using qualitative analysis techniques, it is still possible to reason with such information even though the exact mathematical form of the relationships is not known. Some qualitative methods, such as the loop analysis developed by Puccia and Levins  [ 1 9 8 5 1  have been used for solving problems in diverse areas such as applied  ecology, population ecology, and social epidemiology. Qualitative analysis has been also effectively applied in the field of economics, which often concerns with complex, multi-variable systems. For example, Samuelson 119831  describes a calculus of qualitative relations, which uses a matrix of signs to represent the qualitative  changes in the parameters of an economic equilibrium system.  In the field of engineering, qualitative methods are useful in ascertaining the general behaviour of the solutions to mathematical equations, which concern issues such as the existence and bounds of solutions. In the study of elasticity, for example, qualitative methods are used to bound the energy and deformation of a body from above and below ' " ° ' ' [ v  8  0  l 9 7 7 )  . In contact mechanics and elastic buckling, qualitative methods are often useful when the  solutions of boundary value problems cannot be represented in closed form  [ 0 d e n  1986]  .  Recently, qualitative analysis has been applied to a field of artificial intelligence called qualitative reasoning about physical systems, or qualitative physics. Qualitative is a formal representation schemes developed for reasoning qualitatively about the behaviour of physical systems. Physical systems include any systems, both natural and fabricated, which operate under the laws of physics. Qualitative physics evolved from the study of "commonsense" reasoning. Early in the study of artificial intelligence, researchers realize that humans are able to function with seemingly little effort when faced with situations characterized with uncertainty. The field of commonsense physical reasoning, or naive physics, sought to formalize the types of knowledge required to reason about such everyday situations. Qualitative physics, while embodying the goals of commonsense reasoning, also seeks to predict the behaviour of physical artifacts, and to generate plausible explanations for their behaviour. Qualitative physics deals not only with everyday devices, but also with complex physical systems.  54  Qualitative physics provides a formal, systematic approach to qualitative analysis, and constitutes a key part of the qualitative reasoning application QESWD, the software product of this thesis. This discussion in this chapter will focus on the topic of qualitative dynamics, which deals with the behaviour of physical systems over certain parameters, such as time. There are other areas of qualitative reasoning, such as qualitative spatial reasoning, which is of interest in robotics, but this topic involves a more complex range of issues than need be described here.  3.1.2  Semi-quantitative Analysis  Semi-quantitative reasoning is the task of combining incomplete quantitative and qualitative knowledge. The engineer is rarely confronted with a situation where purely qualitative information is available. Almost invariably there exists some knowledge of certain quantities, even though these may be incomplete or partially specified. In order to make full use of incomplete information, it is necessary to be able to reason in some way with partial numeric data. Semi-quantitative reasoning is important to model-based reasoning tasks such as design, monitoring and diagnosis. A l l of these tasks involve incomplete knowledge in both qualitative and quantitative forms.  There are a number of different representations available for reasoning with incomplete knowledge of quantities, including bounding intervals, probability distribution functions, fuzzy sets, and order-of-magnitude relations. This chapter focuses on the use of bounding intervals to represent partial knowledge of a real number. The discipline of interval analysis is a well-developed areas in applied mathematics and provides a rich array of techniques for working with intervals.  Interval analysis Interval analysis was originally developed as a means of providing rigorous error bounds on the results of machine computations. It provides important methods with which one can reason on the range of values of variables. Interval analysis is now a well-developed branch of applied mathematics. Interval computations are  "55  used in a wide range of applications where numeric uncertainty is concerned. For example, in performing a costbenefit analysis of a project, interval methods have been used by the World Bank to find upper and lower bounds on the rates-of-return.  Since interval methods provide a method for dealing with uncertainty, interval analysis includes techniques that are useful in engineering. Engineers use mathematical equations to model the behaviour of physical systems. Even if it is possible to solve the mathematical equation exactly, which is not usually the case, the result is still only an approximate description of the behaviour of the real system. Mathematical models used in engineering often contain constants that are determined experimentally by measurements. The numerical values assigned to such constants will therefore have limited precision. Using interval methods enable the computation of bounds on the set of solutions corresponding to intervals of possible values for the measured quantities.  Numeric Constraint Satisfaction Systems of interval equations can be formulated as numeric constraint-satisfaction problems. A constraintsatisfaction problem is defined by a set of variables, each with an associated domain of possible values and a set of constraints on the variables. In numeric constraint-satisfaction problems, the domains are continuous while the constraints are numeric relations. Since the domains are continuous, it is impossible to enumerate all of the solutions, which is feasible with finite domains. Even if the domains are finite integer domains, it is generally not possible to find all solutions because of the large number of combinations that may be formed. This difficulty may be overcome by associating a dynamic domain with each variable and by propagating this domain through the constraints.  Constraint propagation is a technique used in enforcing consistency in networks of constraints. Consistency techniques have been used to solve numeric constraint-satisfaction problems  [ L h o m m e I 9 9 3 )  . The goal of these  numeric consistency techniques is neither to enumerate all solutions nor to algebraically solve a system of constraints, but to determine the exact ranges of values of the different variables. A key concept of numeric consistency techniques is that the domains of variables are represented by intervals, and that as constraints are  56  propagated from one variable to another, the bounds of the intervals are updated dynamically. By using established techniques for dealing with intervals, numeric consistency techniques are able to guarantee that the domains of each variable will bound the correct result.  3.2  SOLVING PREDICTION P R O B L E M S WITH Q U A L I T A T I V E ANALYSIS  Much work in engineering and science involves the construction of models. For continuous physical systems, equation systems, such as Ordinary Differential Equations (ODE) models, are a common representation. Such models consist of a system of equations that describe the trajectory of a set of state variables. Making predictions from such models is quite straightforward and efficient given the availability of many numerical equation solvers. However, this approach to modeling and simulation requires that the modeler determine the precise equations for the physical system of interest. For many problems, a precise set of equations may be difficult to find.  While considering the task of designing a physical device, e.g., a chemical plant or electronic circuit, the designer will typically construct a model of the device with precise values for its components determined by detailed analysis, but real components have manufacturing tolerances, so it is unlikely that the real device will behave exactly the same as the model. Consider the task of modeling an existing physical device, perhaps for the purpose of monitoring or diagnosis. While possibly possessing a good understanding of the physical principles involved in the device, the modeler will not have precise knowledge of the values of every parameter and function that describes the system. Thus even if the modeler creates a precise model, its predictions may not agree with the behaviour of the real system.  In each of these cases, there is an inherent imprecision in the modeling task. This imprecision defines a model space that covers many different precise models. Models are typically used to describe and make predictions about the behaviour of the modeled system. If we view a model space M as a set of equation models, we can define a trajectory space T(M)(t) as the set of trajectories starting from some initial state, where t is a controlling parameter such as time. We can then describe two different tasks associated with imprecise models:  57  •  The prediction task determines T(M)(t) from M. Ideally, we would like to use models that explicitly represent model imprecision and can produce a useful behavioural prediction, i.e., a prediction that is precise enough to distinguish desired from undesired behaviour.  •  The refinement task determines M  R  c M from T(M)(t) and a stream of possibly noisy measurements.  Ideally, we would like M to be as small as possible for a given set of measurements. R  3.2.1  Prediction  Considering the importance of reasoning about imprecise models, it is not surprising that there has been much work in this area. Some approaches to imprecise modeling are: •  Simply ignore the problem. For some classes of systems (for example, systems with tight feedback), small amounts of parametric uncertainty do not affect the results. For systems with larger amounts of uncertainty or nonlinearities, however, the model-device mismatch may be large.  •  Use MonteCarlo analysis  [ K a h a n e r 19891  . By running repeated simulations of the Ordinary Differential  Equation system using different combinations of parameter values, it is possible to get some idea of the behaviour of the models in the model space. Unfortunately, in addition to being slow, this approach may miss certain key combinations of parameters that produce behaviour not captured by the simulations. Furthermore, it is difficult to represent functional uncertainty with this method. •  Represent uncertainty using probability distributions for imprecisely known parameters and use variance propagation methods to make predictions  [Reckhow 1 9 8 ? l  . This approach fails when the uncertainty cannot be  modeled as probabilities; for example, if we have functional uncertainty. Furthermore, linearization is required to solve the variance equations which leads to a mismatch between the behaviour of the true model and its linearization. •  Use a specialized model structure that separates the uncertainty from the equation system  [ L u n z e 19891  . The  equation model and the uncertainty model can be solved separately and then combined to bound the model space. This method yields very precise predictions, but requires very specific model structures and is thus not generally applicable.  58  •  Use intervals to bound parameters and use an interval equation solver to compute predictions  [ C o r l l s s  19951  .  This method is fast and general, but produces weak bounds.  3.2.2  Refinement  In contrast to prediction, refinement is a learning task given prior information in the form of a model space M. Traditional system identification methods define M using a parametric model and then search the parameter space for a point that minimizes some criterion, e.g., sum of squared error between model and measurements. Traditional system identification methods make the following problem assumptions: •  There exists a precise parameterization of the model space. If the desired model space cannot be spanned by a single parameterization then multiple models must be considered.  •  The model space is easily searchable. This may not be the case if M has high dimensionality.  •  The data is informative. Because search methods must converge to a precise point in the parameter space, measurements which do not exhibit the dynamic behaviour to be modeled can cause the identifier to converge on the wrong model.  These assumptions require that adequate information about the model space and control over the measurement set is possible. For some cases, such as monitoring an uncertain system, such detailed prior knowledge and an informative measurement stream may not be available. This thesis presents a model refinement method that is robust in the face of uncertain model structure and uninformative data.  3.2.3  Discussion  By investigating methods for modeling, simulation, and refinement of imprecisely-defined systems, several key concepts are summarized as follows: •  Knowledge about both models and predictions can be represented at a variety of levels from qualitative to quantitative. By selecting a level where a particular type of knowledge is precise, inference methods tailored to that level can provide powerful predictions.  59  •  Both simulation and refinement can be viewed as operations that transform model spaces into trajectory spaces and vice versa. This view provides a way of examining the generation of spurious behaviours in a simulation method. It also clarifies the role of inverse simulation as a method for generating models, or more specifically, refinements of a model, from a trajectory space.  •  Raw data can be represented as a semi-quantitative trend. This allows significant savings in matching data to models, since the determination of qualitative properties of the trend is inexpensive and can rule out large portions of the trajectory space.  •  Monitoring can be viewed as a form of identification of an imprecise model. By focusing on refinement, monitoring becomes more efficient since the predictions from a refined model are more specific than from the original model.  •  Identification can be viewed as a refutation process rather than a search process. Since refutation is insensitive to search problems such as local maxima, it is easier to construct refutation-based identifiers that are robust in the face of highdimensional parameter spaces and uninformative data.  The Limits of Semi-quantitative Simulation A key component of this thesis is the simulation of imprecisely or " i l l " defined equation systems. A central tenet of QES is that the simulator should guarantee that all real behaviours of the equation system are predicted. Unfortunately, this idea does not come without a cost. While qualitative simulation and interval propagation can provide such a guarantee, they also produce spurious behaviours.  By examining each component of QES we see a different source of spurious behaviour and inefficiency. •  QES produces spurious behaviours due to ambiguity in qualitative arithmetic. Such ambiguity can result in intractably large behaviour trees because unimportant qualitative distinctions cannot be eliminated by the constraint satisfaction algorithm.  •  QES produces conservative predictions because interval propagation ignores correlation between variables (e.g., X-X&0  when X is a non-zero width interval). It is inefficient because the propagation simply  60  follows changes until afixed-pointis reached. Thus, constraints are not applied in an optimal manner which may result in loops.  The key insight of QES is that these sources of conservatism are in some sense orthogonal so the results of each type of simulation can reduce ambiguity in other parts. Unfortunately, in some cases this is still not sufficient to generate a useful prediction.  Since QES can only predict as well as its underlying methods (qualitative constraint satisfaction and interval arithmetic) allow, it makes sense to look for ways that might improve these methods: Improve the Simulation Methods. Over the years, the algorithms used in QES have undergone continual improvement. In addition to the semi-quantitative improvements described in this chapter, other research related to taming intractably branching behaviours has been conducted. In particular, work by Clancy [Clancy and Kuipers, 1993] on elimination of uninteresting distinctions, behaviour abstraction, and focusing techniques show promise in reducing the number of spurious behaviours in a qualitative simulation. There have been a variety of methods developed for reducing spurious behaviour in interval simulation of qualitative equation systems. Much of this work, including methods for reducing the effect of ignored correlation, would improve the predictions of QES. It is also possible to tackle overly wide dynamic envelopes by using search to rule out sections of the trajectory space that are inconsistent with the trajectory and event information.  Apply More Knowledge. Another way to tighten predictions is to provide more information to the simulation engine. Of particular interest here is to work on constraining the qualitative behaviour by way of trajectory descriptions  m [  B  R  A  J  M  K  6  \ This technique can also be used to provide a method for specifying the behaviour  of exogenous variables in a qualitative simulation. The software system Q E S W D constructs such an interface that encourages and assists users to provide as much knowledge as possible to the qualitative reasoning engine.  61 /  Simplify the Model Space Representation. The main representational ingredient that distinguishes QES from parametric models is the inclusion of static envelopes around monotonic functions. While extremely useful for providing a method for abstracting families of functions, they are also very difficult to compute with. By eliminating such descriptions and sticking to purely parameterized models, existing methods for predicting behaviour become available..  3.3  THEORIES IN QUALITATIVE AND SEMI-QUANTITATIVE ANALYSIS IN ENGINEERING  Qualitative analysis is carried out using qualitative calculus, which is constructed upon such key components as intervals and equations. QESWD focuses on the use of bounding intervals to represent partial knowledge of a real number. Interval analysis is proceeded by solving interval equations. A n effective way to solve systems of interval equations is to use numeric constraint-satisfaction methods.  3.3.1  Components of Qualitative Calculus  Qualitative calculus describes the concepts and rules in qualitative arithmetic. The signs, relative magnitude and directions of change in quantities are all types of information that can be used to reason qualitatively, thus have to be described in qualitative calculus. At University of British Columbia, a language has been developed in which it is possible to express imprecise knowledge of quantities and the relations between quantities [Gedig 1995]. The terms qualitative variables, qualitative equations, and qualitative inequalities are used to describe concepts which are analogous to concepts in algebra. The qualitative calculus is targeted at solving two type of qualitative reasoning: qualitative arithmetic and interval algebra.  Qualitative arithmetic is constructed upon that functional relationships between quantities may also be described qualitatively. For example, Hooke's law of ideal springs can be expressed qualitatively as: "The greater the force on the spring, the larger the displacement from the equilibrium position."  62  Interval algebra is built upon intervals. Qualitative variables may be constructed from continuous variables. The entire domain of a continuous variable is subdivided into a finite number of non-overlapping subintervals, and all the values in the same interval are treated as equivalent. A variable can have any number of landmark values at which behaviour changes significantly, but qualitative reasoning is most efficient when their number is finite. The most common subdivision of continuous domains is into three subintervals, labeled negative, zero, and positive. This subdivision will be used to explain some of the general concepts of the qualitative calculus. Table 3.1 Qualitative Addition, Multiplication, and Negation [ +  m  +  o  +  0  X  1 9 9 5  l  1 0  -  1  -  -  1  +  -  0  0  1  -  +  1  7  ?  1  1  [X]  1  [X]  + [Y]  1  -  +  1  7  s  -  [X]  +  +  Gedi  0 7  +  0  -  1  +  0  -  1  0 ?  0  0  0  0  +  ?  0  ?  ?  Qualitative calculus is composed of several components and concepts described as follows:  The Domain of Signs The domain of signs is the set of qualitative values which results when the landmark zero is used, dividing the positive numbers from the negative numbers on the real number line. Arithmetic operations, such as multiplication and unary negation, introduce ambiguous results, as illustrated in Table 3.1.  Viewing addition and multiplication as a relation gives us no more information than when it is viewed as a function, it, however, helps to prevent the spread of uncertainty. When qualitative addition and multiplication are viewed as functions, uncertainty can propagate through a chain of calculations. For example, if we wish to evaluate the expression [Z\ x ([X] + [Y]) when the value of [X] is + and the value of [Y] is -, the expression in  63  brackets evaluates to "?", so the result of the multiplication operation. When addition is viewed as a relation, three signs are evaluated and a value of true or false is returned. Evaluating the expression equals to asking the question: "is it true or false that adding [X] to [Y] will result in the value '+','-' or zero?"  Qualitative Equations Qualitative equations act as qualitative constraints. Qualitative equations are derived from models of physical systems in much the same way as the usual quantitative equations. In general, a quantitative equation can be derived in the following ways: •  transform a quantitative equation into a qualitative equation using rules of the form shown in Table 3.2;  •  derived from qualitative knowledge of how variable values depend on others, and,  •  linearize a quantitative equation using a Taylor series expansion  [ d e K l e e r  '  1 9 8 4 1  .  Table 3.2: Rules used to generate qualitative equations [e,+e ] => [e,] + [e ]  [e e ]  [0] + [e] => [e]  [0] [e]  [+][>] [e]  [-][e] => -[e]  2  2  x  2  [ei][e ] 2  [0]  Solving Systems of Qualitative Equations Solving systems of qualitative equations targets to locate qualitative values of variables that satisfy the equations. Unlike systems of equations in real variables, a system of independent qualitative equations in general does not have a unique solution, even when the number of variables is the same as the number of equations.  The problem of solving a system of qualitative equations is a constraint-satisfaction problem where the equations are the constraints to be satisfied. Local constraint propagation is a commonly used technique to solve the equation systems. However, this technique often does not work. Even though they are able to find a solution, they cannot guarantee that they will find all results. Another approach to constraint satisfaction is to use a generate-and-test approach, where all possible combinations of assignments to variables are made. Each possible  64  set of assignments is checked against the equations in order to determine if the assignment implies a contradiction. This method is guaranteed to find all possible solutions to the system of equations, however it is inefficient. More efficient means of obtaining all solutions deal with constraint-satisfaction methods, which are discussed later in section 3.3.4.  Systems of qualitative equations have multiple solutions because the field axioms of qualitative algebra are weaker than their quantitative counterparts. The ambiguity of qualitative calculus causes the effort of searching for solutions to grow very rapidly with the number of variables in the system and the number of possible assumptions. There have been several suggestions for dealing with the problems of ambiguity and search complexity in qualitative calculus. Most involve using additional knowledge, including quantitative knowledge: •  Search efficiency can be improved by using domain-specific and problem-specific knowledge.  •  Allowing qualitative variables a greater number of possible values is another approach to resolving ambiguity in qualitative analysis. Some of the ambiguities may be removed using knowledge about ordinal relations between variables.  •  Dormoy and Raiman  [ 1 9 8 8 ]  developed a technique for solving qualitative equations in a manner analogous to  the Gaussian elimination procedure that is used to solve sets of conventional linear equations.  It should be noted that multiple solutions are an important part of the theory. Each solution describes a behaviour that can potentially occur in the operation of the system being modeled. Since the qualitative model is an abstraction of the physical device being modeled, it is possible that the model may represent some other realizable device. Thus qualitative analysis can be useful in design applications, because it is capable of finding solutions to systems of constraints which are not obvious but nonetheless valid.  3.3.2  Interval Analysis and Constraint-Satisfaction Methods  Techniques in interval analysis provide important methods with which one can reason on the range of values of variables. Interval analysis is a well-developed branch of applied mathematics and many efficient techniques  65  have been developed. The following paragraphs give an overview of interval analysis and introduce to the key techniques in interval reasoning, such as constraint satisfaction methods.  Simply, an interval of real numbers can be described as a pair of real numbers, representing the ending points of the intervals. For example, an interval describing a closed bounded set of real numbers can be expressed as: [a, b] - { x : a<x<b }  Interval arithmetic has many similarities with the traditional arithmetic involving real numbers. However, there are a number of complexities which are introduced in dealing with intervals rather than with real numbers. For example, the solution set to a two-dimensional equation with interval coefficients can be represented as a polygon, as shown in Figure 3.1. Although it is possible to solve this equation set by hand calculation, the solution set is not particularly easy to represent, a eight-vertex polygon in this case. In higher-dimensional cases, the difficulty is obviously compounded. Fortunately, systems of interval equations can be formulated as numeric constraint-satisfaction problems.  (-120,240)  A\\X\ + A x - B] i2  2  A xi + A x = B 2l  22  2  2  where the coefficients A , A , A , A , as well as B\ and B , are intervals: n  1 2  2 1  (60,90);  22  2  An = [2, 3]  ^.2=[0, 1]  A  A = [ 2 , 3]  2 1  = [l,2]  Bi = [0, 120]  2 2  B = [60, 240]. 2  (30, -60) •100  Figure 3.1 A Polygon Solution of Interval Analysis  66  [Gedig  1995]  A constraint-satisfaction problem is defined by a set of variables, each with an associated domain of possible values and a set of constraints on the variables. It is impossible to solve a constraint-satisfaction problem because the domains are continuous while the constraints are numeric relations. Since the domains are continuous, it is impossible to enumerate all of the solutions, which is feasible with finite domains. Even if the domains are finite integer domains, it is generally not possible to find all solutions, because of the large number of combinations which may be formed. However, this difficulty may be overcome by associating a dynamic domain with each variable and by propagating this domain through the constraints, which brings in constraintbased reasoning.  Constraint-satisfaction problems (CSPs) follow a general formulation, which consists of a finite set V of n variables {V  h  V ,..., V„},each associated with a domain of possible values, D D ,... D„and a set of 2  u  2  constraints { C C , . . . , C,}. Each of the constraints is expressed as a relation defined on some subset of b  2  variables, given that there are subsets of the Cartesian product of the domains of the variables involved. The set of solutions is the largest subset of the Cartesian product of all the given variable domains such that each in that set satisfies all the given constraint relations. A n assignment of a unique domain value to a member of some subset of variables is called an instantiation. A n instantiation is said to satisfy a given constraint C, if the partial assignment specified by the instantiation does not violate C,. A n instantiation is legal or locally consistent if it satisfies all the constraints in which it involves.  Constraint-based reasoning is a model for formulating knowledge as a set of constraints without specifying the method by which these constraints are to be satisfied. Many problems are more naturally expressed in terms of what is allowed or, conversely, what is not allowed. A variety of techniques have been developed for finding partial or complete solutions for different kinds of constraint expressions. The techniques used in solving constraint-satisfaction problems can be classified into three categories  [ D e c h t e r 1992]  . The first category consists of  search techniques which are used to systematically explore the space of all solutions. The most common algorithm in this class is backtracking, which traverses the search space in a depth-first fashion. The second category is consistency algorithms, which are generally used to perform preprocessing in order to improve the  67  efficiency of the backtracking search, as well as be incorporated in the search. The third technique, structuredriven algorithms, can be used to support both the consistency algorithms and the backtrack search.  Search By Backtracking Consistency Algorithms The fundamental technique for solving any constraint-satisfaction problems is to assume finite discrete domains, with which the assignment space D - D\X D x ... x D is finite, so one may evaluate each constraint on each 2  3  element of D. Once a legal instantiation is found the problem is solved. This generate-and-test algorithm guarantees correct results but is slow. The backtracking technique may be used to "prune" away significant portions of the assignment space.  Backtracking algorithms systematically explore the assignment space D by sequentially instantiating the variables in some order. As soon as any constraint has all its variables instantiated, its true value is determined. If the constraint is not satisfied, that partial assignment is excluded from the total valid assignment. Backtracking then falls back to the last variable with unassigned values remaining in its domain, if any.  As an example of backtracking search, consider three variables V , V , and V . The only valid assignment to x  2  3  each of these variables is one of the integers 1, 2, or 3, so that the domains of these variables may be specified as D\ = D = D = {1, 2, 3}. The constraints and solutions for this problem are shown in Table 3.3. 2  3  68  Table 3.3: Backtracking example. Constraints  Solution by backtracking search  V, X> 1  1  C\%(X\, X ) :  x >x  2  C2i(X\, X ) :  Xi>X  Ci(X): 2  2  x  2  v,v  2  V,V V 2  3  21  2  2 1 1 2 1 2 2 1 3 22 2 3 3 31 3 1 1 3 1 2 3 1 3 32 321 3 2 2 3 2 3 33  At times, backtracking can be very inefficient. A variety of consistency-enforcing algorithms have been developed which may be performed prior to the search in order to improve the efficiency  [ F r e u d e r  '  1 9 7 S 1  , These  algorithms transform a given constraint network into an equivalent, yet more explicit network by deducing new constraints to be added on to the network.. In practice, it has been found more efficient to run the algorithm for a few steps as a preprocessor to simplify subsequent backtracking search.  Numerical Constraint-satisfaction In numeric constraint satisfaction problems, domains of variables in a numeric constraint-satisfaction problem are represented as intervals on the real numbers, in contrast to constraint-satisfaction problems involving qualitative variables. Traditional constraint-satisfaction problems involve variables with finite domains which support the enumeration of all solutions to the problems. When constraint propagation provides no further pruning, one can always resort to exhaustive search using generate-and-test. In contrast, numeric constraintsatisfaction problems are generally highly under-constrained, resulting in a very large number of solutions.  69  Therefore, exhaustive search is generally infeasible with quantitative domains. The key to avoiding explosive growth in complexity in reasoning with continuous domains is to represent domains as intervals, and to handle domains only by their bounds. The approach is to attribute a dynamic domain to each variable, and to propagate this domain through the constraints. The constraint propagation process is similar to that used to attain consistency in constraint networks involving variables with finite, qualitative domains.  The formulation of numeric constraint propagation problems is very similar to conventional constraintsatisfaction problems. An NCSP consists of a set of numeric variables, V-  {V], V ,... V„}, a set of domains D = 2  {Di, D ,... £>„} where D „ a set of numerical values, is the domain associated with variable V„ and a set of 2  constraints C-  [C  u  C ,..., C ), 2  m  where a constraint C, is defined by any numeric relation linking a set of  variables.  Several techniques have been developed for solving numeric constraint-satisfaction problems.  [ D a v , s 19871  A  consistency technique called label propagation can be used for satisfying constraints where the quantities involved are either the extended signs (+, 0, -, ?) or intervals. Another consistency technique for numeric constraint-satisfaction problems, called arc B-consistency (bounds consistency), can be applied to both continuous and finite interval domains [  Lhomme 1993  ]  pj  o t n  t n e  s e techniques use a modified version of Waltz  algorithm (Figure 3.2). While conventional consistency techniques are only used to simplify a CSP before going on to enumerate the solutions, consistency methods for numeric constraint-satisfaction problems may be used to generate the solution.  Waltz filtering algorithm by David Waltz  [1972]  , can be used to filter out inconsistencies and find a complete,  exact solution to a NCSP with finite domains. In the Waltz algorithm, the domains of variables are repeatedly updated, and the changes are propagated to all constraints associated with the given variable. The procedure continues until the inconsistencies in the network are eliminated. Figure 3.2 shows an efficient implementation of the Waltz algorithm. The Waltz algorithm may also be illustrated with the following example in Figure 3.2a. Some problems exist in using Waltz algorithm, for example, some sets of constraints may cause the algorithm to  70  go into an infinite loop. However, the problems can be solved using some additional techniques such as Lhomme  I1993]  ' s B-consistency methods.  procedure R E V I S E ( C ( X j , X * ) begin set CHANGED to 0 for each argument X, do begin D <r- {xj e Dj\3 (xi e D i = 1,... , k, i * j), C(x h  h  x x )} j t  k  if D= 0 then stop else if D^Z), then begin Di <- D add Xi to CHANGED end end return CHANGED end REVISE procedure WALTZ begin Q <— a queue of all constraints while Q#0 do begin remove constraints C from Q CHANGED <- REVISE(C) for each X in CHANGED do f  for each constraint C'*C which has X,- in its domain do add C to Q end end WALTZ Figure 3.2: A n implementation of the Waltz algorithm  71  Variables:  Constraints  Vj = [1, 10]  C,:V, + V = V  V = [ 3 , 8]  C : V < V.  2  2  2  2  3  x  V^ = [2, 7] Solution procedure: 1.  Cj is removed from the queue.  - Since V > 1 and V ^ 3, C, gives V > 4, so reset V to [4, 7]. {  2  3  3  - Since V < 7 and V > 3, C gives V, < 4, so reset V, to [1, 4]. 3  2  x  2.  Since V and V have changed, add C to the queue.  3.  C is removed from the queue.  3  x  2  2  - Since V < 4, C gives V < 4, so reset V to [3, 4]. 2  x  2  2  - Since V > 3, C gives V] > 3, so reset V to [3, 4]. 2  2  x  4.  Since V\ and V have changed, add C\ to the queue.  5.  Q is removed from the queue.  2  - Since V > 3, V > 3, C, gives V > 6, so reset V to [6, 7]. x  6. 7.  2  3  3  Since only V has been changed and V has no other constraints besides C\, nothing is added to the 3  3  queue.  Since the queue is empty, stop. Figure 3.4: A sample solution using Waltz algorithm  3.4  THE QUALITATIVE ENGINEERING SYSTEM Q E S  In developing a practical engineering tool for reasoning with partially-specified information, it would be desirable to incorporate both the power of abstraction inherent in qualitative methods, and the elegance of interval analysis. The Qualitative Engineering System (QES), developed by Michael Gedig ' [  9951  , features a  coherent framework which integrates both qualitative and semi-quantitative reasoning techniques. QES incorporate the techniques in qualitative and semi-quantitative analysis, and adjust these techniques towards the applications in the field of engineering. For this reason, this thesis adopts QES as the main reasoning engine for the analysis of welding distortion problems. With some enhancements on QES platform, the software system Q E S W D has been developed for analyzing welding distortion problems using qualitative and semi-quantitative methods.  The following sections discuss the structure of QES and its basic operations, which utilize the qualitative and semi-quantitative techniques described in the previous sections. Gedig provided detailed descriptions on the 72  internal structure of QES in his master thesis. The following paragraphs summarizes the main components in QES.  3.4.1  Structure of QES  QES is constructed upon several key components: intervals, expressions, equations and inequalities. In QES, constraints are represented as equations and inequalities, which are composed of expressions, which consist of variables and constants. Expressions evaluate to quantities and are used to represent variables and constants. The interval representation is used to express quantities to varying degrees of precision. These components are described by the following data types: interval, bound, expression, operator, relation and relationLink, as shown in Figure 3.3a. Upon these components, constraint satisfaction techniques are used to find qualitative and semi-quantitative solutions.  As depicted in Figure 3.3a, expressions compose the fundamental building blocks of a QES problem. Equations or inequalities are formed by connecting expressions with relationLinks. One or several such equations or inequalities construct an equation system, which defines the entire QES problem. An expression can be a single variable, or be composed of sub-expressions linked by such operators as add, multiply, etc. An important property of an expression is its interval, which is described by bounds.  For illustration, consider the equation A + BxC = 0, which is depicted in Figure 3.3. The relation link is shown as the dotted arc between the number zero and the expression node labeled "+". Both sides of the equation are composed of expressions. The right hand side is an expression described by a number zero. On the left hand side, the expression is A + B x C, which is actually composed of two child expressions: A and B x C. The child expressions are connected by hierarchical links to compose their parent expression. During a reasoning process, the status of an expression is indicated by its properties represented by intervals.  73  Relational Link  Hierarchical Link  Figure 3.3: Graphical representation of expressions  Hierarchical Link  Relational Link  Expression  Expression  Expression  Interval Bound CO  c O O  H—  ZD  CT Expression  Expression  Expression  Figure 3.3a Structure of QES  3.4.2  Qualitative Analysis  In QES, the constraint equations, composed by expressions, are solved using constraint satisfaction methods. The constraint equations are expressed as trees, as shown in Figure 3.3; these trees subsequently compose a  74  constraint network for constraint satisfaction reasoning. QES uses a constraint synthesis process to build up successively higher level of consistency in the constraint network.  The following example illustrates the constraint satisfaction process:  F +F ]  =0  2  Fx - F = 0. 3  F >0. 2  The problem is to determine which assignments to the qualitative variables F\, F and F satisfy the three 3  2  constraints. The constraint network built up with expression trees is shown in Figure 3.4. The steps of solving the problem are summarized as: 1.  Add initial constraints to the network. In this example, add F > 0;  2.  Enforce the equality constraint, e.g., equalize two expressions F\ + F and F\ - F ;  3.  Update the domains of all variables.  2  2  3  Although this problem has one solution, this is not generally the case when systems of qualitative constraints are concerned. If the constraint on F (F > 0) had not been presented, there would be three legal instantiations 2  2  instead of one.  75  -  + -  Figure 3.4: Final consistency network  3.4.3  [ G e d l g 1  Semi-Quantitative Reasoning  In QES, systems of numeric constraints equations are formulated as numeric constraint-satisfaction problems. The bounds consistency techniques described in section 3.3.2 were implemented to solve numeric constraintsatisfaction problems. Consistency techniques are applied to numeric constraint-satisfaction problems by associating a dynamic domain with each variable and by propagating this domain through the constraints. The domains of variables are represented by intervals, so that as constraints are propagated from one variable to another, the bounds of the intervals are updated dynamically. This procedure is explained in the example shown in Figure 3.5.  76  Constraints C,:Y  = X+  1  C :Y = 2 x X 2  The constraint network  Constraint propagation procedure: 1.  The change in D propagates from X to X + 1 through a parent link of X.  2.  The constraint interval of X + 1 is updated to [ 1, 11], using the current values of its children.  3.  The change in X + 1 is propagated through a relation link to Y.  4.  Variable Y is updated to [1, 11].  5.  The change in Y causes 2 x X to be updated to [1, 11], because 2 x X shares a relation link with Y.  6.  X is recalculated using the new value of 2 x X and the right child, the numeric constant 2. The constraint  x  interval of X is updated to [0.5, 5.5] 7.  The change in X propagates to X + 1 by way of the hierarchical link.  The domain changes propagate cyclically through the constraint network, and asymptotic convergence toward the solution (X = 1, Y = 2) occurs. Figure 3.5: An Example of numerical constraint satisfaction  Several constraint inference techniques are implemented in QES to complement constraint propagation. Constraint inference is a way of deriving new constraints from existing ones. These techniques are relational arithmetic, constant elimination arithmetic, and graph search.  77  4  QESWD:  A N A P P L I C A T I O N OF Q U A L I T A T I V E A N D SEMI-  Q U A N T I T A T I V E A N A L Y S I S IN W E L D I N G  DISTORTION  The Qualitative Engineering System (QES) described in the Chapter 3 provides an effective tool to approved complexity and uncertainty, such as in problems of welding distortion. The outcome of this research is a software tool, that can be used to analyze welding distortion problems when the problems cannot be precisely described.  4.1  INTRODUCTION  Qualitative Engineering System for Welding Distortion, or QESWD, is the software tools developed as part of this thesis. QESWD is a prototype software programme targeted at engineers, technicians and students who need to deal with the problems of welding distortion. Using the qualitative reasoning engine, QESWD is able to analysis welding distortion problems without complete or precise description of input data. Due to the nature of welding distortion that the phenomenon cannot be easily assessed numerically, an information database is set up and integrated into the software to help users confine the problem and keep track of knowledge on welding distortion. When developing a practical engineering tool for predicting and controlling weld distortion, it would be ideal to incorporate both the power of abstraction inherent in qualitative methods, and the flexibility of information storage of database. QESWD features such a coherent framework which integrates both qualitative analysis tools and knowledge databases.  The level of uncertainty presented in a welding distortion analysis is not static. In the early design stage, there is more uncertainty than in the final stages. A reasoning framework is constructed which is capable of accepting new information, precise or vague, as it becomes available, and updating stored information with new reasoning conclusions. QESWD uses the powerful notion of constraints as a means of expression engineering problems. The constraints, along with literal notations of engineers' knowledge and experience, are packed in a knowledge database. The literal contents of the database provide information for engineers to understand and initiate an  78  analysis of welding distortion. Upon a successful completion of the analysis, the database can be expanded by recording engineers' newly-gained knowledge or experience, and the reasoning result can be reused in the future analyses. Because of the high complexity of welding distortion, the methods and theories on its analysis and control are being expanded every year, thus creating many new constraints available for welding distortion analysis. The use of a database to store qualitative constraints provides a convenient way of maintenance and upgrade the ability of the software in order to give more effective assessment on welding distortion problems.  4.2  4.2.1  SYSTEM IMPLEMENTATION  System Structure  Q E S W D is built upon several information databases, which are used to store variables and constraints for the reasoning engine, and to provide knowledge to assist users with the analyses. The system adopts two types of databases, the first type of which is used to store reasoning information such as constants, variables and constraints (equations), which are the basic input component of Q E S reasoning engine. The other type of database is used to maintain text and graphical information on welding distortion, which is to be used for training users as well as increasing knowledge input level for the reasoning engine. Both types of the databases can be easily appended to accommodate additional research results as time passing by. The database framework also enables users to expand the built-in standard designs with their customized designs, which suits the multiformity nature of engineering problems.  79  Output  Input Variables G eometries . Arrangement . Neutral Axis Plate Thickness Joint Type  Constraints: rules & equations Distortion = K x Neutral Axis Offset  Amount of Distortion Orientation ot Distortion  Qualitative Reasoning Engine  Actual Distortion Actual Behaviour  Base-plate material Filler-metal Material Welding Procedure . Preheat Procedure Para. Assembly Para.  Knowledge Database  Suggestions for controlling welding distortion  Figure 4.1 Structure of Q E S W D  The structure of Q E S W D is depicted in Figure 4.1. Some of the input and output screen are shown in Figure 4.2. As illustrated, the core of the system is a qualitative and semi-quantitative analysis engine, which provides reasoning ability of the system. The input side of the system consists of controlling variables and equations on welding distortion, which are to be fed into the QES engine by users. Upon the start of an analysis, users are provided with relevant information on a specific welding distortion problem. Aided by the information, users can define the studied problems in further depth and identify its controlling parameters and constraints. Provided with the parameters and constraints, Q E S W D carries out the reasoning and derives all the possible conclusions of the studied problem. Upon the completion of the reasoning, Q E S W D searches through the knowledge database and retrieves all information related for the reasoning conclusions. At the output side, users will be given all possible outcomes on the specified welding distortion problem and the available methods to control the distortion. Seeing the outcome, users may update the input parameters and constraints if further insight of the problem is available. Users can keep working on this define-reason-redefine cycle till all available information has been utilized and satisfactory outputs have been obtained.  80  / • Un.ilil.ilivr t miiniMMing  File  |dit  Option  System tat Welding Otstniium  ii  ilelp  c  Oi&oitiDnTjve JAnalucirl Log | frnriadg* Managmmt |  ,  'i^-l  r  JJP',«  J!dJI£3  "  Available Distortion Typet  (a) f  [Longitudinal Bending Distortion  J  Lo-~nDefrutiongtu i an i ai Bendn i g D<sto i -• Cortrol of Longitudinal Wtfdtitg Stqueaci -~ Distortion Prediction of iB has been found that welding sequence affects transverse distortion:- Blocl Cooholing Facte** of L r ! generally .tound to cause'less "shrinkage than mu«l ijsequences:. Re&ed Topics^ ?  welClin9 s e q u e n c e s  rnrr  w e r o  i i  ~3  .n v_E . m n . v ^ X  - .n V j . I i  > -100;' < U O O -  >• - 1 0 0 0  a . n / . v _ I « <  1000  Longitudinal hending distortion can be significantly reduced when Intermittent i / / e q u a t i o n s ; -Beam Theory -:.Methods ot Removing i welds are usediiBut Hirai and Nakamura [b724).found that ft is not.as effectiv | ( c o n s t r a i n ^ - ; - v-__t angular, distortion: \ . f t r » ± n : : d U v - c£dU""+ v _ d t t P*titing  m  | c o n * t r a i n - dl' - c ^ d L • +v j d L : o n s t r a i n ^ L : * e^l-•'+" V _ L '  tWen metal that has shrunk during welding Is stretched by^peening some of :onstiain-..B>c^B <+ v__B distortion can be removed. ' •"' ' m. | c o n s t r a i n - I x . = c _ ^ I x + v^Ix constrain  Du: = £ * p...» dU * L D I - « • & •** P ' * d L •* L D •<* Du + D I  Differential HeatfttQ to reduce Longitudinal Distortion of built-uo he o n s jt r a m This technique refers to a distortion reduction methodttiat intentionally create^ c o n s t r a i n .Itemperature differences between the parts to bo welded; 'The preheated pariM? /*  ' 'rtiu'.'i p r i n t D , D u , D l  (JiMhl.i tiVf (.niiini'iMinr) '>utli m Itn W* Idinq l l i s l i n l n i n £dit Option - -Help  fjle  * -t /, - > L "/.  s /;ix  I  D i U o t f i O T l y p e i f | Log" j-Knowtedge Management ] Scrplt  P L Du DI  Illlllil  'j |Normal termination  1  li  Mil •  Numerc constrant satisfaction. Modify D to C-3S9 027916.-25 51771) Jormal termination  784 8e3 •Tooio 651  11 Iii" SlniiZZZZZZ.jJiL'_,l 200100 . 7 3956939c9  ••• i * •» XPRESSIOND (25 51771,389.027316)  17^3957prie9"  Printing Variable Du EXPRESSION Uu (482 886695,695.674092)  bl  printrg Variable DI EXPRESSION 01 (-457 370985 -306 646176)  Curvature Due lo longitudinal  Analysis time 1 43 sec Total time 1 59 sec  Description  Krov^edge and Informinn"  t Previous*.  Ready ,  Figure 4.2 Screen Captures of Q E S W D  81  (c)  keyword  De;crip:ion  Topic  distortion, type  3 fundamental /veld ng distortion types  1 Transvershrirkagc perpend cula'to the weld line. 2 Longitucinal ;hrin<afle parallel ti the weld line. 3 Angular distortion (rotation jrourd the weld line).  jppejrart<-«, d i m e n ' o  Dimensional change*  1  change  dassilied ty appearance  weld line. 2 Anyuldt ult*jny« (1 dii^uui^u Ji 'u Lion;. A iiuii-unifvim tl" e rm a I distribution in thethicfcnessdirerfion causes d stortlon fjngular orangey close to the meld line. 3 Rotational distortion. Angu ar distortion inthe ilane o' the plate due to trermil expansion. 4 Longitucinal *hrin<age Sh'inkaje inthe direction of the weld line. 5 Longitucinal bencing distortion. Distortion in a plane through the weld line and perpendicula' to the plate. 6 Buocling distention, thermal compressive stresses cause instability when the plates ara thin.  0 approaches i* 30 welding distortion  1 Th« d e v e l o p m e n t of w e l d i n g p r o t e s t s a n d f i b r i c i t i o n procedures th*t m i n i m i z e diftorticn.  Transverse shrinkage.  O h r i r k a g e p e r p e n d i c u l a r to the  2 The eiLdbliiliinen. uf uliuiia! sldiiddid> t\t au  field roster  ovlnc.  1111111  Field Name  distorion leduction  ,r s" ;  PAi l :-  2' Keyword  !i  *M  Type o f function Inherent shrinkage  Be a:  \ Plate  Conventional formulas  250 50  Physical meaning o f constants  fitfl  Inherent angular change  i l f l i C cc  exp(-wQh- ") J  Inherent angular clungs  S oc C,(—L)""* x  Ditto  One pass:  S oc fQh^^y*  x  exp(-n0r ") J  Tee-fillet joint  multi pass:  SozN(Qh-" )- x 1  i  60c — Ci(—=•) > W Wv>! 0  Figure 4.3 Q E S W D Database Structure  Knowledge on welding distortion has been categorized and imported into a table that is used as a knowledge database in the system. Figure 4.3 illustrates the database structure and some of its sample contents. The main fields in the database table consist of ID, Keywords, Topic and Description. The ID is used as the links to the other parts of the system and also as an index to speed up knowledge searching. The Keywords act as the keys for knowledge allocation and linking, while Topic and Description fields store the knowledge contents.  82  The second database contains the variable and constraint table. The required input of the qualitative reasoning system consists of variables (or constants) and constraints. In Q E S W D , the variables are the controlling parameters of welding distortion, such as plate geometry and energy input, and the constraints are the equations. Since the qualitative reasoning is able to handle imprecise and incomplete information, users can describe the studied problems with only the currently available information, instead of supplying data to everything input parameter. Figure 4.3 also illustrates the structure of the database tables storing the variables and constraints. Because of the complexity of welding distortion problems, different sets of formulae and variables are needed to handle different types of welding distortion. Via the Keyword field, the software can suggest all the relevant sets of formulae needed to solve the specific problems based on the description provided by users. Several formulae sets can be applied simultaneously to narrow the range of reasoning outcome.  The database setup of Q E S W D makes it easy to expand and update the system when more research results are available, or after more insight information is obtained through welding practice. Users' newly gained knowledge or experience after each use of Q E S W D can be packed into the database and will assist the analysis in the future uses. With the scheme of expandability, the software system becomes cognitive: the more it is used, the more precise analysis results it provides.  4.2.2  User Interface  Figure 4.2 illustrates screen captures of the main components of the Q E S W D user interface. The main user interface is composed of "Distortion Type" and "Analysis" screen.  The screen "Distortion Type", as shown in Figure 4.2 (a), is for initiating a new analysis. At this step, Q E S W D retrieves all available distortion types from its information database. For each distortion type, the system searches through the database and provides the user with the related distortions to a specified distortion type. After user chooses a distortion of interest, the system explores the database further and displays all.the methods, data and formulas available for the studied problem. Figure 4.2 (a) illustrates a sample screen as just described: when longitudinal bending distortion is studied, the system shows that "Angular Change" and "Longitudinal 83  Shrinkage" are the related distortions; when the user chooses to analyze the "Angular Change" in detail, the system prompts five methods or formulas available for the studied distortion "angular change". The information database of Q E S W D is tightly integrated with every step of the system operation. When the user read through the distortion information on the screen using a computer mouse, the definition of the distortion pointed by the mouse cursor is displayed immediately on the screen. More comprehensive information is also available in the "Knowledge and Information" window any time by clicking "Knowledge and Information" button.  Figure 4.2 (b) shows the "Knowledge and Information" window displayed for longitudinal Bending Distortion. As illustrated, the knowledge topics are listed in tree-shaped hierarchies, which provides concise knowledge presentation and ease of browsing. All knowledge displayed is retrieved from the database, as explained in the previous section, on the fly by searching through the keywords and indices.  The Analysis screen is shown in Figure 4.2 (d). After the user chooses the methods to the applied, the formulas and analysis procedures are loaded from the Constraint database. On the Analysis screen, the user is provided with all information that is needed to proceed with the analysis: types of parameters, explanations of all parameters with their suggested value ranges, and formulas. It should be noted that the user does not have to specify the exact values of the parameters, as opposite in other engineering software. The user can start the analysis at any completion level of input parameters. However, the more precise specification of the input parameter, the more accurate reasoning results the system will generate. In the sample screen shown in Figure 4.2 (d), the distortion value based the input parameters is between 25.5 and 389.0 mm. Special techniques are employed to display formulas in true math symbols instead of those as in the hard-to-read computer scripts.  While the user is provided with a simple input screen , Q E S W D carries out a complex process of script conversion so that the reasoning variables and constraints can be fed into the QES analysis engine. Figure 4.2(c) shows a sample of some converted QES scripts. All the QES scripts in the system can be customized and saved for later uses.  84  4.2.3  Platform and Development Environment  The software system Q E S W D is developed to be user friendly for both end users and system developers. Q E S W D is implemented for Microsoft Windows® environment, which runs on personal computers and is readily available to most engineering offices. Following the Windows® application development guideline, the user interface of Q E S W D features a logically arranged tab-like layout (Figure 4.2). Online help and explanations are available on all screen components, such as buttons and text fields, by right clicking of the mouse button. A l l information displayed in the text fields can be exchanged with other W i n d o w s ® applications via "Copy & Past" or D D E . The interface design streamlines the programme operations and makes it possible for users to start using the programme at the first sight, without first going through any operating manuals.  Q E S W D is written using a W i n d o w ® based programming environment called Delphi®, from Borland International. Delphi was chosen because of its ability of fast prototyping, strong database functioning, as well as compiling of true machine code which enables fast programme execution. Although qualitative analysis is effective in approaching engineering problems, very often it is still a tedious process to prepare scripts for the qualitative reasoning engine. In order to change the process from being tedious to as easy as possible, many interface elements such as buttons, table grids, have to be provided to simplify the interaction. Delphi makes the implementation of these interface elements easier than other development platforms such as Visual C++. As explained in previous sections, all the system information, such as knowledge of welding distortion and constraints for qualitative reasoning, is stored in databases to provide flexibility and ease of manipulation. The strong functionality in database operation thus makes Delphi the ideal choice.  The database format used in Q E S W D is Paradox®, a relational desktop database originally developed by Borland International. Paradox is chosen because of its fast data manipulation, good integration with Delphi and its ability of storing binary data. Most engineering software displays formulas as computer scripts, which are hard to understand and often hinder the engineering intuition of users. Using the paradox B L O B field, a binary storage field, Q E S W D is able to display formulas and equation in true mathematical symbols  85  4.3  SAMPLE ANALYSIS  To illustrate how Q E S W D analyzes welding distortion problems, an example is given as follows to predict the welding distortion of a built-up steel girder.  The sectional properties of the girder are shown in Figure 4.4. The length of the girder is 10 m and it is assumed that it has one free end and one fixed. The girder is built with four plates welded together, and is symmetrical to y axis but asymmetrical to x axis. Since the welds are not coincident with the x axis, the shrinkage force from welding will introduce bending into the girder, i.e. longitudinal bending distortion.  To analyze such distortion, it is possible to construct a finite element model to compute the distortion. Although a finite element programme may result at an accurate prediction based on a well specified set of input, it is difficult to ensure that all inputs actually precisely describe the real scenario, such as ambient conditions and skill level of welders. Thus the finite element analysis may not be considered giving accurate output when an actual welding case cannot be precisely described. Q E S W D , however, makes it possible to accept loosely defined inputs and reason with incomplete information. Although Q E S W D will not provide a single exact  86  solution in most cases, it can point out the range that the result falls in and the qualitative nature of the problem, which give designers an effective evaluation tool at the conceptual design stage.  The Q E S W D information database suggests that Sasayama's  119551  method, which is described in detail in section  2.4.7., can be effectively deployed for the studies case. Sasayama et al. proposed that the welding distortion of a built-up girder can be predicted using the formula:  Curvature $ ~~ R =  ^  =  ^  EI  X  1=1  x  and it is known that from Beam Theory:  after applying the boundary conditions, we have:  8{x) =  fr =x ^ » 2  1  P  di  Since the exact input values are not known, we feed Q E S W D with a set of value ranges as shown in Table 4.1. These input data is also shown in Figure 4.3. Table 4.1 Input Parameters of Sample problem Parameter  Value  P, Shrinkage force per weld  588 ~ 784.8 kN  Corresponding to weld leg size 10-15 mm (Refer to section 2.4.7. for details) E, Young's Modulus  20000 ± 100 MPa  I , Moment of inertia  7.3957 x 10 + 1000 mm  du, Distance between neutral axis to top weld centroid  631 + 20 mm  d , Distance between neutral axis to bottom weld centroid  406 + 20 mm  L, Length of girder  10000+ 10 mm  9  x  L  4  After receiving the input data, Q E S W D converts the data into a set of variable definitions and constraints, as required by the qualitative reasoning engine QES. This converted script is shown in Figure 4.3.  87  After the reasoning using numeric constraint satisfaction, Q E S W D shows that the cambering created by welding distortion locates in the range of (25.5,389.0) mm. The user can also modify the variation of the input parameters and see their reflections on the finally distortion. For example, if we reduce the lower bound of the shrinkage force from 588 kN to 500 kN, the final distortion range will be changed to (-46.7,434.9) mm.  88  5  CONCLUSIONS  A computer software system, Qualitative Engineering System for Welding Distortion (QESWD), has been developed to analyze problems of welding distortion, targeting to students, technicians and engineers. By employing techniques of qualitative and semi-quantitative reasoning, Q E S W D is able to analyze problems of welding distortion with incomplete or vaguely- described information. A knowledge base framework is built into Q E S W D to enable efficient reasoning of qualitative information, as well as to organize the knowledge on welding distortion in a hierarchical and relational fashion. In summary, Q E S W D provides the following two key functions: •  Analyze problems of welding distortion with incomplete or vaguely defined data;  •  Manage and present knowledge on welding distortion in a hierarchical as well as relational network.  Welded structures are in many aspects superior to bolted or riveted structures, whereas there exist many problems, among which welding distortion is very difficult to tackle with. Over decades, research has been done in hope to solve the problems of welding distortion. Most of the available techniques to predict welding distortion fall into two categories: empirical or analytical formulae, and numerical analysis. However, both types of the techniques have shortcomings that make them difficult to apply to real engineering practice. Numerical methods must deal with the complexity from residual stress and heat transfer before any distortion analysis can start, thus constrain their scopes to only simple weldment setups. Although empirical and analytical methods greatly simplify the prediction problem by concentrating on only key factors, the complexity of the problem still persists due to the difficulty of obtaining the precise specification of every key factor. Using techniques in interval analysis, Q E S W D solves this problem by its ability of using incomplete or vaguely specified input factors. Another difficulty in analyzing welding distortion problem is the lack of organized knowledge on the subject. Q E S W D sets up a framework that can be used to collect and manage such knowledge. A large amount of information on welding distortion has already been integrated into the framework.  89  Techniques in qualitative and semi-quantitative reasoning have been employed in Q E S W D to handle incomplete numerical information. Concepts in constraint satisfaction are used to handle qualitative and semi-quantitative reasoning. It is found that qualitative analysis provides an effective tools for the domain of engineering problem solving, where uncertainty exists over most of the design duration. However, the conclusions from the qualitative methods become rapidly weaker as the number of unknowns in the problem rises. This program is partially dealt with in Q E S W D by providing a knowledge assisting mechanism which encourages as much information as possible to be entered into the system to reduce the uncertainty.  Further research is recommended in more effective techniques in qualitative reasoning, so that the ability of Q E S W D in handling incomplete information can be more efficient. It is also recommended to continue collecting knowledge on welding distortion so as to further complete the Q E S W D knowledge base.  90  BIBLIOGRAPHY  Alefeld, G., and Herzberger, J. (1983) Introduction to Interval Computation. Academic Press, New York. Benjamin, J.R. (1959) Statically Indeterminate Structures. 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