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Stresses in heavy section electroslag joining Ivo, Paulo Silveira 1982

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STRESSES IN HEAVY SECTION ELECTROSLAG JOINING by PAULO SILVEIRA IVO B.A.Sc.,Universidade F e d e r a l de Minas G e r a i s , B r a s i l , l 9 7 8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of M e t a l l u r g i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © Paulo S i l v e i r a Ivo, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of M e t a l l u r g i c a l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 05, 1982 i i A b s t r a c t A study of the thermal s t r e s s e s . r e s u l t i n g i n the E l e c t r o s l a g J o i n i n g Process as a p p l i e d to heavy gauge f o r g i n g s has been undertaken, s i n c e a survey of p u b l i s h e d r e p o r t s on the process i n d i c a t e s that although s o l i d i f i c a t i o n c r a c k i n g ought to be a problem, a p p a r e n t l y i t i s not r o u t i n e l y observed. Welding c o n d i t i o n s which are known to produce s o l i d i f i c a t i o n c r a c k s i n E l e c t r o s l a g Welding with wire e l e c t r o d e s were repo r t e d to make c r a c k - f r e e welds using ESJ p l a t e e l e c t r o d e s . T h i s study r e p o r t s work on the thermal and s t r e s s f i e l d s developed d u r i n g ESJ of 150 mm t h i c k A36 s t e e l p l a t e s . C o n d i t i o n s p r e d i c t e d by pre v i o u s workers to form cr a c k s with wire e l e c t r o d e s were e s t a b l i s h e d and found not to form cr a c k s with p l a t e e l e c t r o d e s . Measurements of s t r e s s and temperature durin g welding were made and found to agree w e l l with a simple numerical model of the process. I t i s concluded that the thermal f i e l d of ESJ i s s u f f i c i e n t l y d i f f e r e n t to ESW that i t can r e l a x the s t r e s s f i e l d developed, even i n a f u l l y - c o n s t r a i n e d j o i n t , to the po i n t at which s o l i d i f i c a t i o n c r a c k i n g i s no longer observed. < i i i Table of Contents A b s t r a c t , i i L i s t of Tables v L i s t of F i g u r e s v i Acknowledgement v i i i I. INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 Process D e s c r i p t i o n and A p p l i c a t i o n 3 1.3 Previous Work 6 1.4 S o l i d i f i c a t i o n Cracking 9 1.5 Present O b j e c t i v e s 12 I I . MATHEMATICAL MODELING 13 2.1 Temperature D i s t r i b u t i o n C a l c u l a t i o n .14 2.1.1 Assumptions 14 2.1.2 D e r i v a t i o n Of Equations 16 2.1.3 Numerical S o l u t i o n 17 2.2 Thermal S t r e s s And S t r a i n C a l c u l a t i o n 18 2.3 Computer Production Runs 21 I I I . EXPERIMENTAL WORK 23 3.1 Furnace Design 23 3.2 C o o l i n g Shoe Design 23 3.3 E l e c t r o d e And Slag P r e p a r a t i o n 24 3.4 Weld Set-up And C o n s t r a i n i n g ....26 3.5 Sequence of Operation - Welding Procedure 28 3.6 R e s i d u a l S t r e s s 29 i v 3.7 Temperature Measurements 29 IV. DISCUSSION AND RESULTS 31 V. CONCLUSIONS 35 VI. SUGGESTIONS FOR FUTURE WORK 36 BIBLIOGRAPHY 37 APPENDIX A - BOUNDARY CONDITIONS .....68 APPENDIX B - COMPUTER PROGRAM SAMPLE 76 APPENDIX C - EFFICIENCY FACTOR AND HEAT SINK CALCULATIONS . 88 APPENDIX D - RESIDUAL STRESS EVALUATION 90 V L i s t of Tables I. Computer Model Parameters 40 I I . ESJ T y p i c a l Log Sheet 41 v i L i s t of F i g u r e s 1. Schematic Layout of ESJ Equipment 42 2. E l e c t r o s l a g T y p i c a l Weld S t r u c t u r e ( R e f . 35) 43 3. ESJ Thermal P r o f i l e - C a l c u l a t e d and Measured ........44 4. Boundary C o n d i t i o n s 45 5. Nodal Arrangement 46 6. Model Flowchart 47 7. S t r e s s A n a l y s i s Schematic Diagram 48 8. UBC E l e c t r o s l a g Unit 49 9. C o o l i n g Shoe - Water Channels 50 10. C o o l i n g Shoe Top View 51 11. Copper recess 52 12. C o o l i n g Shoes i n P o s i t i o n 53 13. C o o l i n g Shoe Close-up - Water connections 54 14. P l a t e E l e c t r o d e i n P o s i t i o n 55 15. Aluminum Feeder 56 16. E l e c t r o d e and Copper Stub 57 17. Run-in Copper Tabs 58 18. C o n s t r a i n i n g Rod with S t r a i n Gauges 59 19. S t r a i n versus Time P l o t 60 20. Boxed I-Beam 61 21. Hardened 4340 D i s c Spacer 62 22. I n f e r i o r I-beam Placement 63 23. Strain-gauge Set-up 64 v i i 24. Thermal S t r e s s Curve 65 25. Thermal S t r e s s Curve ...66 26. ESJ Thermal Gradient 67 v i i i Acknowledgement S i n c e r e thanks to Dr. A l e c M i t c h e l l f o r h i s guidance throughout the d u r a t i o n of t h i s work. Thanks are a l s o due to Dr. E. B. Hawbolt and f e l l o w graduate students f o r innumerable h e l p f u l d i s c u s s i o n s . The a s s i s t a n c e of the t e c h n i c a l s t a f f , i n p a r t i c u l a r Mr. E. Barry and Mr. G. S i d l a i s g r e a t l y a p p r e c i a t e d . The f i n a n c i a l support provided by the Canadian I n t e r n a t i o n a l Development Agency and by E l e t r o m e t a l Acos Fi n o s S . A . / M i n i s t e r i o da I n d u s t r i a e Comercio - S e c r e t a r i a de T e c n o l o g i a I n d u s t r i a l , B r a s i l i s g r a t e f u l l y acknowledged. A s p e c i a l thanks to Consuelo for her care and encouragement. 1 I. INTRODUCTION 1 .1 I n t r o d u c t i o n The manufacture of l a r g e s t e e l f o r g i n g s v i a the c o n v e n t i o n a l route r e q u i r e s s t a r t i n g i n g o t s having a low in g o t -t o - f o r g i n g y i e l d ( 3 0 to 60%) and a low equipment p o t e n t i a l u t i l i z a t i o n time due to the heavy weights i n v o l v e d . A l t e r n a t i v e p r o d u c t i o n routes have, t h e r e f o r e , to be sought. S e v e r a l techniques have been proposed in an e f f o r t to present v i a b l e s o l u t i o n s . The MHKW(Midvale-Heppenstall-Klockner-Werke) p r o c e s s 1 i s one whereby a c o n v e n t i o n a l l y c a s t ingot i s trepanned and subsequently core remelted, thus improving the q u a l i t y of the ingot c e n t r a l p a r t through enhanced i s o t r o p i c ESR p r o p e r t i e s . The B.E.S.T.(Boehler E l e c t r o s l a g Hot Topping) t e c h n i q u e 2 i s a l s o a p o t e n t i a l process f o r improving the i n g o t - t o - f o r g i n g y i e l d . A f u r t h e r promising route i s the use of E l e c t r o s l a g J o i n i n g f o r welding two or more p i e c e s of s t e e l before f o r g i n g to make a l a r g e preform or j o i n i n g a l r e a d y forged products to t h e i r f i n a l 2 shape. Russian workers have developed a method f o r j o i n i n g l a r g e s e c t i o n s used i n the manufacture of r o t o r f o r g i n g s f o r atomic power s t a t i o n t u r b o g e n e r a t o r s . " 7 Four l a r g e s e c t i o n consumable e l e c t r o d e s were employed and the welding equipment used a b i f i l a r c o n f i g u r a t i o n which i s claimed to be very e f f i c i e n t . In t h i s i n v e s t i g a t i o n i t was concluded that the process would be a p p l i c a b l e to the production of heavy r o t o r f o r g i n g s using h i g h * a l l o y Cr-Ni-Mo-V s t e e l s and a comparative assessment was made of the f r a c t u r e r e s i s t a n c e of the weld and parent metal with good mechanical p r o p e r t i e s obtained. I t was a l s o found that p r e l i m i n a r y and concurrent h e a t i n g of the p a r t s being j o i n e d were e l i m i n a t e d . L i t t l e has been done as to the a p p l i c a t i o n of the process in the case of carbon s t e e l f o r g i n g s in the range 20-100 t , which r e p r e s e n t s the bulk of the open d i e f o r g i n g market. 9 Here the main concern i s with r e p e a t a b i l i t y , r e l i a b i l i t y , u l t r a s o n i c t e s t i n g and product q u a l i f i c a t i o n . In other words, the process behaves very s i m i l a r l y to the more c o n v e n t i o n a l welding techniques and, t h e r e f o r e , must be c a r e f u l l y c o n t r o l l e d to a v o i d d e f e c t s . The process i s c a r r i e d out with ease and r e l a t i v e l y f a s t i n one s i n g l e pass and when compared with, f o r example, submerged-arc welding, time savings are s i g n i f i c a n t . The r e s u l t a n t coarse 3 s t r u c t u r e due to long thermal c y c l e s i s l i a b l e to produce lower mechanical p r o p e r t i e s and b e t t e r equipment c o n t r o l and welding procedures are needed. 1 .2 Process D e s c r i p t i o n and A p p l i c a t i o n E l e c t r o s l a g J o i n i n g i s one of the s e v e r a l a p p l i c a t i o n s of the E l e c t r o s l a g Remelting p r i n c i p l e . I t i s a f u s i o n welding process whereby heat i s generated by Jo u l e e f f e c t when an e l e c t r i c c u r r e n t passes through an e l e c t r o r e s i s t i v e f l u x ( s l a g ) . Both the t i p of the e l e c t r o d e and the su r f a c e of the work are melted by heat coming from the s l a g -source. The e l e c t r o d e molten d r o p l e t s t r a n s f e r by g r a v i t y to the weld pool by f a l l i n g through the molten f l u x . As can be seen in F i g . 1, the a x i s of the j o i n t i s v e r t i c a l but the welding i s performed i n a h o r i z o n t a l p o s i t i o n . The remelted product i s surrounded by the parent m a t e r i a l and by c o o l i n g shoes e i t h e r s t a t i o n a r y or movable. The most popular v e r s i o n of the process i s the one employing f l u x - c o a t e d or bare wire e l e c t r o d e f e e d i n g as w e l l as consumable or non-consumable e l e c t r o d e guides. In t h i s case, the f i l l e r m a terial(wire+guides) and the parent metal o f t e n have d i s s i m i l a r compositions and when s e v e r a l wires are used, 4 depending on the t h i c k n e s s of the p a r t s being welded, proper monitoring of the feeding system becomes rather c r i t i c a l . The weld metal a c q u i r e s a c a s t i n g s t r u c t u r e wherein the s i z e s and o r i e n t a t i o n s of the g r a i n s are c o n t r o l l e d by heat removal. The presence of long columnar g r a i n s i s c h a r a c t e r i s t i c and four types of g r a i n s t r u c t u r e s have been d e f i n e d by P a t o n . 3 5 Because the process i n t r o d u c e s a l a r g e amount of heat i n t o the p a r t s being j o i n e d , a l a r g e heat a f f e c t e d zone can be expected. F i g . 2 i l l u s t r a t e s these m i c r o s t r u c t u r e s . Segregation along the g r a i n boundaries of these columnar g r a i n s can cause crack appearance as d i s c u s s e d below. The t y p i c a l slow c o o l i n g r a t e s i n the process when c o o l i n g from the a u s t e n i t e range form p e a r l i t e : the p r o e u t e c t o i d f e r r i t e normally forms a network along the p r i o r - a u s t e n i t e g r a i n boundaries and Widmanstatten s i d e p l a t e s that extend from the g r a i n boundaries i n t o the m a t r i x . 5 Due to f a s t c o o l i n g , r e s i d u a l s t r e s s e s are present i n the weld and heat a f f e c t e d zone but are somewhat r e l i e v e d in the l o n g i t u d i n a l d i r e c t i o n . The use of p l a t e e l e c t r o d e as f i l l e r metal has r e c e n t l y been c o n s i d e r e d and i t i s b e l i e v e d to enable more e f f i c i e n t heat t r a n s f e r . Because the el e c t r o m a g n e t i c f o r c e s do not produce as s t r a y f l u i d motions, s l a g v e l o c i t i e s are slower(2 to 4 cm/s) and the t o t a l weld heat may be reduced by as much as 30% when p l a t e e l e c t r o d e s are used i n s t e a d of wire welding. A l s o the heat f l u x 5 to the base metal from the s l a g i s more un i f o r m . 6 The hydrogen content of the s t e e l has been found t o decrease with i n c r e a s i n g c a l c i u m f l u o r i d e content i n the f l u x . A system c o n t a i n i n g CaF + A l 0 i s recommended f o r e f f e c t i v e 2 2 3 hydrogen c o n t r o l . 7 A l s o the hydrogen content has been found to be l a r g e r at the weld s t a r t than elsewhere. To le s s e n t h i s e f f e c t , as d i s c u s s e d below, proper p r e - h e a t i n g of the f l u x components was normal procedure throughout these experiments. A l i q u i d s l a g s t a r t procedure should improve the hydrogen c o n t r o l even f u r t h e r , should t h i s prove to be a problem. E l e c t r o s l a g Welding and E l e c t r o s l a g J o i n i n g have u s u a l l y been used to j o i n heavy t h i c k n e s s , l a r g e welds and a p p l i e d to the manufacture of generator r o t o r s f o r the nuclear i n d u s t r y , to the o n - s i t e welding of t h i c k v e r t i c a l j o i n t s where Arc Welding co u l d not be employed, 3 5 and to the c o n s t r u c t i o n of b r i d g e s , b u i l d i n g s and storage t a n k s . 4 High pressure v e s s e l s and heavy b o i l e r drums used i n power p l a n t s have a l s o been f a b r i c a t e d . 3 5 8 Welding of f o r g i n g presses and a n v i l s up to 2000 X 2000 mm and of r o l l i n g m i l l frames as w e l l as s h i p rudder p a r t s has been performed." 3 Recently, main p r o p u l s i o n s h a f t i n g and other machinery s h a f t s f o r use i n Cla s s e d Ships made by E l e c t r o s l a g J o i n i n g have been q u a l i f i e d under L l o y d ' s R e g i s t e r r equirements. 3 6 1 .3 Previous Work The theory of moving sources i n welding(Arc Welding) heat conduction presented by R o s e n t h a l 1 0 and the a n a l y t i c a l m o d e l l i n g proposed by R y k a l i n 1 1 set b a s i s f o r a l l subsequent work done in the f i e l d of weld m o d e l l i n g . L i t t l e a t t e n t i o n has been given to the thermal s t r e s s c a l c u l a t i o n s i n E l e c t r o s l a g J o i n i n g except fo r some p u b l i s h e d Czech work 2 2 2 5 done with wire e l e c t r o d e welding. The r e s u l t i n g complex equations from a n a l y t i c a l s t u d i e s on welds have always hindered a b e t t e r understanding of the d i f f e r e n t processes, e s p e c i a l l y i n studying thermal s t r e s s e s 1 7 . Gray et a l 2 0 have improved Okerblom's 1 2 theory of one-dimensional s t r e s s a n a l y s i s based on a two-dimensional heat flow treatment and a p p l i e d to t h i n gauge m a t e r i a l s . L i t t l e e xperimental i n f o r m a t i o n was given which c o u l d be used to t e s t the theory and i t made use of in-pla n e c u r v a t u r e s rather than c o n t r a c t i o n s . These authors have, however, succeeded in c a r r y i n g out experiments intended to t e s t the theory f o r l o n g i t u d i n a l c o n t r a c t i o n s . 7 With the ever i n c r e a s i n g speed of modern computers, s o l u t i o n techniques have been de v i s e d which enable more accurate and f a s t e r c a l c u l a t i o n s . The f i r s t attempt to make use of computers i n a n a l y s i n g welding s t r e s s e s dates as f a r back as 1961. 1 8 M a s u b u c h i 1 7 r e p o r t e d some e f f o r t s being made at a n a l y s i n g t r a n s i e n t l o n g i t u d i n a l s t r e s s e s d u r i n g multipass welding of a heavy p l a t e , but no r e s u l t s were g i v e n . The m a j o r i t y of experimental s t u d i e s of thermal s t r e s s e s d u r i n g welding, however, has used m a t e r i a l s i n the t h i c k n e s s range of 0.30 mm to 25.4 mm(0.0l2" to 1"), which were a l l done using Arc Welding w h i l s t experiments i n the present work were performed using much t h i c k e r m a t e r i a l . N i s h i d a 2 1 reviewed s e v e r a l methods for c a l c u l a t i n g thermal s t r e s s e s and compared t h e o r e t i c a l r e s u l t s with experimental data, again working with t h i n m a t e r i a l s . E r i k s s o n et a l 3 0 have developed a h o t - c r a c k i n g t e s t to assess the weld metal composition i n f l u e n c e on the h o t - c r a c k i n g tendency i n heavy E l e c t r o s l a g welds and have concluded that to avoid s o l i d i f i c a t i o n c r a c k i n g , the carbon content should be kept as low as p o s s i b l e and the Mn/S r a t i o should exceed 45. In t h i s t e s t , however, the s t r e s s f i e l d s are not known e i t h e r i n a b s o l u t e terms or i n r e l a t i o n to those present in heavy s e c t i o n welds. There e x i s t s e v e r a l other t e s t s to evaluate w e l d a b i l i t y 8 c r a c k i n g problems and two of them seem to be e s p e c i a l l y s u i t a b l e f o r s o l i d i f i c a t i o n c r a c k i n g : V a r e s t r a i n t and T r a n s v a r e s t r a i n t t e s t s . They are the same i n p r i n c i p l e and o p e r a t i o n , except f o r the d i r e c t i o n of the a p p l i e d s t r a i n with respect to the welding d i r e c t i o n . They have not been a p p l i e d to e l e c t r o s l a g welding c o n f i g u r a t i o n s . Ueda et a l 1 9 have developed a method f o r t h e o r e t i c a l a n a l y s i s of thermal s t r e s s e s , t a k i n g i n t o c o n s i d e r a t i o n e f f e c t s of changes i n modulus of e l a s t i c i t y , y i e l d s t r e s s and the c o e f f i c i e n t of l i n e a r expansion on the metal with temperature. At the i n s t a n t of welding, a l i m i t e d p o r t i o n of the p a r t s being j o i n e d such as the weld bead and the parent m a t e r i a l c l o s e to the hot face i s heated up to a very high temperature and t h e r e a f t e r c o o l e d down to room temperature. As t h i s thermal c y c l e proceeds, the temperature d i s t r i b u t i o n changes with time and the mechanical performance of the welded assembly i s a l s o a f u n c t i o n of temperature. It i s , t h e r e f o r e , imperative to assess the temperature d i s t r i b u t i o n d u r i n g welding. P e r t s o v s k i i et a l 2 6 have c a l c u l a t e d the thermal c y c l e i n the heat a f f e c t e d zone d u r i n g E l e c t r o s l a g Welding of t h i c k s t e e l p l a t e s and have r e a l i z e d that the nonuniform g e n e r a t i o n of heat in the l i q u i d pool and complex pool o u t l i n e , make i t extremely hard to study the heat flow d i s t r i b u t i o n at the boundary between s o l i d and l i q u i d phases. However, they have r e p l a c e d the above 9 mentioned complex volumes by a c o l l e c t i o n of three l i n e a r heat sources at d i f f e r e n t l e v e l s i n the pool f o r a s e m i - i n f i n i t e system. The r e s u l t s presented seem to agree w e l l with measured va l u e s and are a p p l i c a b l e to CGESW(Consumable Guide E l e c t r o s l a g Welding). More r e c e n t l y , B a c o n 2 3 s t u d i e d the heat flow i n both systems(wire and p l a t e e l e c t r o d e ) . High d e p o s i t i o n r a t e s , shallow s l a g depth requirements, smooth he a t i n g and c o o l i n g r a t e s and g r e a t e r degree of p e n e t r a t i o n are r e a l i z e d i n E l e c t r o s l a g J o i n i n g as compared to wire e l e c t r o d e welding. Temperature d i s t r i b u t i o n measurements and c a l c u l a t i o n s done f o r the present p r o j e c t followed the same trend found by B a c o n 2 3 and had s i m i l a r h e a t i n g and c o o l i n g r a t e s , r e s p e c t i v e l y 0.5 -2.0°C/s and 0.2 - 1.0°C/s.(See F i g . 3) 1.4 S o l i d i f i c a t i o n C racking Work developed both by Brown et a l 2 7 and by P h i l l i p s et a l 2 8 have i n d i c a t e d that p rovided the same s t r e s s f i e l d i s a p p l i c a b l e , E l e c t r o s l a g J o i n i n g should not be more or l e s s s u s c e p t i b l e to s o l i d i f i c a t i o n c r a c k i n g than c o n v e n t i o n a l welding. The c r i t i c a l welding speed which w i l l ensure a sound weld has been d e f i n e d by Semenov 2 9, t a k i n g i n t o account thermal s t r e s s e s and mainly the volume change duri n g m e t a l l u r g i c a l 10 t r a n s f o r m a t i o n s . The technique c o n s i s t s i n imposing a t e n s i l e s t r e s s on the s o l i d i f y i n g r e g ion of a f u l l y - c o n s t r a i n e d weld (which r e p r e s e n t s the worst c a s e ) . According to these authors, s t a b l e welding v o l t a g e assures the c o r r e c t p e n e t r a t i o n during the weld and together with the c r i t i c a l welding speed determine the proper shape f a c t o r ( r a t i o of the weld gap to the metal pool depth) f o r a sound weld. With i n c r e a s i n g heat input the depth of the metal pool i n c r e a s e s s h a r p l y , the shape f a c t o r v a r i e s and the tendency to form c r a c k s a l s o i n c r e a s e s . A c c o r d i n g to Makara et a l " " , an i n c r e a s e i n the weld v o l t a g e leads to improvement of the shape of the pool p r o f i l e and r e s u l t s i n crack tendency r e d u c t i o n . B e n d i s * 3 has concluded that the most c r i t i c a l p e r i o d d u r i n g E l e c t r o s l a g Welding i s when one-quarter to one-half of the j o i n t i s welded. The parameters that c o n t r o l the r e s i s t a n c e of the weld metal to hot c r a c k i n g as r e p o r t e d by P a t o n 3 5 are: chemical composition, r i g i d i t y of the welded j o i n t and the shape f a c t o r . Brown et a l 2 7 , however, found no cracks when a p p l y i n g the suggested welding c o n d i t i o n s given by Semenov et a l 2 9 and even d e l i b e r a t e l y a l t e r e d the Mn/S r a t i o to a value lower than the recommended minimum of 4 5 which a c c o r d i n g to E r i k s s o n et a l 3 0 should have r e s u l t e d i n s o l i d i f i c a t i o n c r a c k i n g . It i s , t h e r e f o r e , c l e a r that E l e c t r o s l a g Welding(wire e l e c t r o d e ) behaves q u i t e d i f f e r e n t l y with respect to s t r e s s b u i l d - u p as compared to E l e c t r o s l a g J o i n i n g ( p l a t e e l e c t r o d e ) The high heat input observed i n E l e c t r o s l a g Welding which keeps the s l a g pool i n the molten c o n d i t i o n can be very 11 e f f e c t i v e i n a v o i d i n g the uptake of hydrogen from moisture but i t i s known to enhance c o n d i t i o n s l e a d i n g to c r a c k s i n the welds. The p o s s i b i l i t y of s o l i d i f i c a t i o n c r a c k i n g ( h o t c r a c k s ) e x i s t s both i n the weld m e t a l ( s o l i d i f i c a t i o n c r a c k i n g ) and i n the heat a f f e c t e d z o n e d i q u a t i o n c r a c k i n g ) . 3 3 S t e e l s are known to f a i l i n a b r i t t l e manner at temperatures c l o s e to the m e l t i n g p o i n t and t h i s behaviour i s a s c r i b e d t o i n c i p i e n t m e l t i n g of s o l u t e r i c h r egions i n the s t e e l which r e s u l t s from g r a i n boundary segregation or m i c r o s e g r e g a t i o n d u r i n g s o l i d i f i c a t i o n . " 6 For continuous c a s t i n g s t e e l s with 0.25% to 1.0% C the b r i t t l e range s t a r t s at 40°C below the s o l i d u s temperature as reported by Weinberg." 2 When the deformations i n t h i s temperature range exceed the deformation c a p a c i t y of the metal a hot tear d e v e l o p s . 3 3 Lower me l t i n g p o i n t secondary phases such as s u l p h i d e s and oxides at the heat a f f e c t e d zone at the g r a i n boundaries fuse l o c a l l y and produce a weaker bonding that f a i l s under the e f f e c t of shrinkage s t r e s s e s c a using what i s known as l i q u a t i o n c r a c k i n g . In both cases, however, rupture of the metal occurs i n an i n t e r g r a n u l a r form, c o n t r a s t i n g with t y p i c a l lower temperature, i n t r a c r y s t a l l i n e path c o l d c r a c k s , r e s u l t i n g mainly from hydrogen e m b r i t t l e m e n t . " 8 Rymkevich et a l 3 2 have pursued the d e t e r m i n a t i o n of the b r i t t l e temperature range i n E l e c t r o s l a g Welding of carbon s t e e l s and found i t to be 1380-1450°C. 1 2 1.5 Present O b j e c t i v e s As d e s c r i b e d above, c o n d i t i o n s i n the welds l e a d i n g to cra c k s are r e a l i z e d i n the process d e s p i t e the improvements experienced by a higher heat input and s u i t a b l e welding f l u x i n the p l a t e e l e c t r o d e technique. An understanding of the nature and extent of s o l i d i f i c a t i o n c r a c k i n g and, t h e r e f o r e , of the proper welding c o n d i t i o n s has prompted the need f o r f u r t h e r s t u d i e s of the thermal s t r e s s e s produced i n the p r o c e s s . 13 I I . MATHEMATICAL MODELING The inherent d i f f i c u l t i e s a s s o c i a t e d with performing experimental work, e s p e c i a l l y i n the case of s t r e s s d e t e r m i n a t i o n i n welds, make i t a l l the more i n t e r e s t i n g to use a model i n order to study how the process behaves. The f o l l o w i n g pages c o n t a i n the development of a computer program which was used to c a l c u l a t e the thermal s t r e s s e s r e a l i z e d d u r i n g the E l e c t r o s l a g J o i n i n g of t h i c k b l o c k s , based upon a temperature p r o f i l e i n p u t . Weld c r a c k i n g i n E l e c t r o s l a g J o i n i n g appears to be caused by thermal s t r e s s - f i e l d induced by nonuniform temperature changes. I t i s e s s e n t i a l l y a q u e s t i o n of how much heat flows through the p l a t e s being j o i n e d . Although r a d i a t i o n and conve c t i o n take p l a c e while E l e c t r o s l a g J o i n i n g , conduction i s the dominant heat t r a n s f e r mode i n the bl o c k s , except d u r i n g hot topping when r a d i a t i o n and c o n v e c t i o n s t a r t p l a y i n g a more s i g n i f i c a n t r o l e . According to L i b y et a l . 3 " two aspects of heat flow need to be c o n s i d e r e d : heat generation i n the s l a g and heat conduction through the parent b l o c k s . The former i s assign e d a me l t i n g p o i n t temperature as the s t a r t i n g value f o r c a l c u l a t i o n s i n the f i r s t time step of the model and from the second time step on, 1 4 i t i s c a l c u l a t e d f o r every time step, t a k i n g i n t o c o n s i d e r a t i o n the e l e c t r o d e l a t e n t heat, heat c o n d u c t i v i t y and d e n s i t y . The l a t t e r w i l l c o n s t i t u t e the u n d e r l y i n g p r i n c i p l e based upon which the temperature d i s t r i b u t i o n w i l l be determined. 2.1 Temperature D i s t r i b u t i o n C a l c u l a t i o n 2.1.1 Assumpt ions The general heat conduction F o u r i e r equation i n two dimensions i s thought to d e s c r i b e the phenomena i n v o l v e d and i s used to c a l c u l a t e the thermal p r o f i l e s . In order to so l v e i t , s e v e r a l assumptions had to be made so that the boundary c o n d i t i o n s c o u l d be p r o p e r l y a p p l i e d : ( S e e F i g . 4 and F i g . 5) i ) No-flux boundary c o n d i t i o n at the top of the s l a g . According to P a t o n 3 5 only 1.3% of the t o t a l heat i s r a d i a t e d to the s u r f a c e s and 1.2% i s a c t u a l l y l o s t throught r a d i a t i o n to the atmosphere. i i ) Symmetry a x i s — because the welding process i s symmetrical, a no-f l u x boundary c o n d i t i o n i s thought to be a v a l i d assumption. i i i ) Block w a l l s -- the i n t e r n a l w a l l s were a l s o assumed a 1 5 n o - f l u x c o n d i t i o n as the heat r a d i a t i o n mentioned i n i ) i s n e g l i g i b l e . i v ) A l l the heat reaching the slag-metal i n t e r f a c e leaves i t and flows to the blocks being welded. v) P o s i t i o n s i n the block away from the hot f a c e ( f u s i o n l i n e ) are c o n s i d e r e d to be at room temperature and the temperature of the heat source (slag) to be the m e l t i n g p o i n t temperature f o r carbon steel(1520°C.) v i ) P h y s i c a l p r o p e r t i e s were c o n s i d e r e d to be constant and the e l e c t r o d e l a t e n t heat was taken i n t o c o n s i d e r a t i o n . Such an assumption i s not too f a r from r e a l i t y as the energy coming from the i d e a l i z e d heat source(slag+metal pool) a c t u a l l y reaches e l e c t r i c a l e q u i l i b r i u m i n l i g h t of the f a c t t h a t , i f the r i g h t volume of s l a g i s chosen, the welding equipment performing the weld operates i n a s t a b l e manner. Values f o r Cp, k and are l i s t e d i n Table I. v i i ) The heat s o u r c e ( s l a g + metal pool) t r a v e l s " i n d e f i n i t e l y " along the height of the block, although an unsteady s t a t e model was used to p r e d i c t the thermal p r o f i l e s . v i i i ) As i n the case of welding l a r g e assemblies, no heat flow in the z d i r e c t i o n i s assumed. ix) Symmetry i s invoked and the modelling i s b u i l t on one sid e of the j o i n i n g assembly as i l l u s t r a t e d i n F i g . 4. 16 2.1.2 D e r i v a t i o n Of E q u a t i o n s The two d i m e n s i o n a l heat c o n d u c t i o n e q u a t i o n 3 2T ^ 3 2T q P Cp 3_T ( 1 ) " H ? + ~ ~ 3 ^ k k 3 t Where: T.= temperature t = time q = heat f l u x k = heat c o n d u c t i v i t y p = d e n s i t y Cp = s p e c i f i c heat L e t t i n g a =p Cp/k and knowing that t h e r e i s n e i t h e r h e a t g e n e r a t i o n nor consumption i n t h e b l o c k s ( q / k = 0 ) , e q u a t i o n (1) becomes: 3 2T f 3x' + 3 2T a 3T (2) 3y 3t 17 E q u a t i o n (2) i s s o l v e d n u m e r i c a l l y s u b j e c t e d t o d i f f e r e n t boundary c o n d i t i o n s ( S e e Appendix A ) . 2.1.3 N u m e r i c a l S o l u t i o n U s i n g a heat b a l a n c e approach f o r an i n t e r i o r node, the g o v e r n i n g e q u a t i o n was e x p r e s s e d in f i n i t e d i f f e r e n c e form as:(See F i g . 2) For 1<i<IM-1 1<j<IN For the f i r s t h a l f time s t e p : -a T. . . , ( 2 . 2 a , * _a T . 2 T n m A x ^ A c A x z 1,2 A x ^ J A t J 2 a I , ., - T. . 2 a(T. .- T. . .) + i , i + l 1,1 _ i , i i , i - l /  A y . ( A y . + A y . ) A y . ( A y . + A y ) 3 3 J + l 3 3 J-1 S i m i l a r l y f o r the second h a l f time s t e p : m " + l o o 1 Tn+1 - 2 a T. ._. , _2_ 2 a . 2 a T. A y j ( A y j + A y j _ 1 ) 1 , 3 X + ^ A t A V j ( A y . + A y j + 1 ) A y j ( A y j + A y j _ 1 ) ' J * 2 T. . ) (4) 18 and s o l v e d n u m e r i c a l l y by using an I m p l i c i t A l t e r n a t e D i r e c t i o n F i n i t e D i f f e r e n c e method i n 2-D. The other nodes were determined s i m i l a r l y ( S e e Appendix A). The unsteady s t a t e c o n d i t i o n s , t y p i c a l of E l e c t r o s l a g J o i n i n g ( m a i n l y the heat d i s t r i b u t i o n i n the blocks) c a l l s f o r an i m p l i c i t method such as the I.A.D.F.D., making the s o l u t i o n independent of any s t a b i l i t y c r i t e r i a . T h i s technique s o l v e s a 2-D problem by using a 1-D approach in one d i r e c t i o n ( i m p l i c i t by columns) f o r the f i r s t h a l f time step and again a 1-D s o l u t i o n f o r the other d i r e c t i o n ( i m p l i c i t by rows) and second h a l f time s t e p . 3 6 A t r i d i a g o n a l system of equations i s simultaneously s o l v e d without any r e s t r i c t i o n s as to the s p a t i a l and time increments. F i g . 6 presen t s the main flowchart f o r the model. 2.2 Thermal S t r e s s And S t r a i n C a l c u l a t i o n S t r e s s e s appear as a r e s u l t of nonuniform h e a t i n g of the elements of a body which cannot expand f r e e l y . In E l e c t r o s l a g p l a t e j o i n i n g the heating and c o o l i n g r a t e s are t y p i c a l l y low but, as the process i s a long one, the t o t a l heat input i s ra t h e r l a r g e and, t h e r e f o r e , bound to c r e a t e thermal s t r e s s e s and s t r a i n s . 19 C o n s i d e r i n g a r e c t a n g u l a r beam of depth 2h, t h i c k n e s s TH and l e n g t h L(See F i g . 7), assuming L t o be much l a r g e r than the o t h e r dimensions(See Appendix C) and knowing t h a t T = T ( x ) , a s i m p l e model was used to c a l c u l a t e thermal s t r e s s e s and s t r a i n s . Because the beam i s c o n s i d e r e d t h i n ( m i d - s e c t i o n p l a n e i n F i g . 4) a p l a n e s t r e s s assumption i s made: z z y z x y a = a = 0 (6) X X z x and a = a ( x ) (7) y y y y For the a n a l y t i c a l s o l u t i o n of the r e s u l t i n g two-d i m e n s i o n a l problem which by a p p l y i n g e q u a t i o n (6) becomes a o n e - d i m e n s i o n a l p r o b l e m ( e q u a t i o n (7)) a l l bounding s u r f a c e s a r e f r e e of t r a c t i o n except the end f a c e s y=+L/2 and y=-L/2 and the f o l l o w i n g e q u a t i o n s s a t i s f y e q u i l i b r i u m and c o m p a t i b i l i t y c o n d i t i o n s 3 8 : S t r e s s Components: 2 0 ° y y = " ^ 1 + - ^ K T - H - f ^ M T ( 8 ) c = a = 0 . <9> XX ZX S t r a i n C o m p o n e n t s : W h e r e NT i s t h e i n t e g r a l o f a E T d z f r o m - h t o +h a n d MT i s t h e i n t e g r a l o f a E T z d z f r o m - h t o +h T h e v a l u e s o f E ( Y o u n g ' s M o d u l u s ) w e r e c a l c u l a t e d a s a f u n c t i o n o f t e m p e r a t u r e a c c o r d i n g t o t h e e x p e r i m e n t a l r e s u l t s p u b l i s h e d by M i n a k a m i e t a l . 3 7 f o r c a r b o n s t e e l . T h e i n t e g r a l s w h i c h a p p e a r i n t h e a n a l y t i c a l e x p r e s s i o n a b o v e w e r e c a l c u l a t e d n u m e r i c a l l y a n d w e r e u s e d i n e q u a t i o n s ( 8 ) a n d ( 1 0 ) t o c a l c u l a t e t h e t h e r m a l s t r e s s e s i n t h e y - d i r e c t i o n f o r e a c h c o r r e s p o n d i n g n o d a l t e m p e r a t u r e t h u s , c o v e r i n g t h e e n t i r e h e i g h t b e i n g w e l d e d . 21 A sample of the computer program used to c a l c u l a t e the s t r e s s e s and s t r a i n s i s shown in Appendix B. 2.3 Computer Production Runs The thermal s t r e s s c a l c u l a t i o n s were performed f o r s e v e r a l welding c o n d i t i o n s ( d i f f e r e n t temperature d i s t r i b u t i o n s ) , a l l of which simulated r e a l j o i n i n g s . P r o v i s i o n s were made in the program to allow f o r changes i n the furnace parameters which would a f f e c t the f i n a l welded s t r e s s s t a t e . Thus, v a r y i n g the heat input namely amperage and v o l t a g e , produced a d i f f e r e n t welding speed which, i n t u r n , caused a d i f f e r e n t s t r e s s d i s t r i b u t i o n . Since the process i s very s t a b l e e l e c t r i c a l l y , the welding v e l o c i t y was c o n s i d e r e d to be constant throughout the model. The i n i t i a l temperature c o n d i t i o n s f o r the weld region i s r e p l a c e d every subsequent time step by a new temperature c a l c u l a t e d upon a heat balance performed at the f u s i o n boundary. Only a small amount of energy r e l e a s e d by the e l e c t r o d e m e l t i n g e f f e c t i v e l y goes i n t o the j o i n i n g b l o c k s . The e f f i c i e n c y f a c t o r F used to determine how much of the a v a i l a b l e incoming energy flows through the blocks gi v e s an i n d i c a t i o n of the thermal e f f i c i e n c y of the p r o c e s s . A massive heat e x t r a c t i o n i s performed by the copper and aluminum c o o l i n g shoes. T h i s very e f f i c i e n t heat sink absorbs 22 approximately 50% of the energy a v a i l a b l e and l i t t l e i s l o s t as r a d i a t i o n energy, except u n t i l the very end of the p r o c e s s . 23 I I I . EXPERIMENTAL WORK 3.1 Furnace Design The equipment used f o r a l l runs was the U B C E l e c t r o s l a g f a c i l i t y as shown in F i g . 8. T h i s u n i t can c a s t s t e e l up to 1 t i n weight and operates on AC. The furnace power supply i s a 250 KVA step-down transformer which i s connected to a 600 V primary s i n g l e phase l i n e . T h i s dry type transformer operates with a high v o l t a g e of 600 V and a low i n the range of 25-60 V, AC c u r r e n t of up to 8000 A. The e l e c t r o d e feeding system c o n s i s t s of a) an e l e c t r o d e holder c a r r i a g e which s l i d e s on aluminum r a i l s i n s i d e the furnace framework(Fig.8 ) and i s suspended from b) an e l e c t r o d e d r i v e c a r r i a g e coupled to a v a r i a b l e speed reductor that enables speeds from 0 to 163 mm/min. The o p e r a t i o n a l parameters are r e a d i l y monitored from s e v e r a l instruments i n a c o n t r o l p a n e l . More d e t a i l e d i n f o r m a t i o n on the furnace design i s given elsewhere. 3 9 3.2 C o o l i n g Shoe Design E l e c t r o s l a g J o i n i n g i s fundamentally s i m i l a r to E l e c t r o s l a g 24 Remelting.The r e a d i l y n o t i c e a b l e d i f f e r e n c e l i e s i n the remelted product: i n the j o i n i n g process the parent m a t e r i a l i s pa r t of the heat e x t r a c t i o n system w h i l s t i n p l a i n r e m e l t i n g the product i s completely surrounded by water-cooled copper c r u c i b l e s . Heat flows by conduction through the blocks and a s u b s t a n t i a l p o r t i o n i s e x t r a c t e d v i a the c o o l i n g shoes. T h e r e f o r e , t h i s p a r t of the set-up p l a y s a s i g n i f i c a n t r o l e i n c o n t r o l l i n g the d i r e c t i o n a l s o l i d i f i c a t i o n . The design used f e a t u r e s r e c t a n g u l a r aluminum s l a b s 25.4 mm X 254 mm X 1066.8 mm(1"X10"X42"), with three 11 mm deep c o o l i n g groves(channels) per shoe on the outer aluminum s e c t i o n ( S e e F i g . 9 ) copper and which was coupled to the inner copper s l a b ( F i g . 10).This recess in the copper p a r t i n c r e a s e s the su r f a c e area c o n t a c t and enables proper p o s i t i o n i n g of the e l e c t r o d e . C o n s i d e r i n g that the copper has a higher heat c o n d u c t i v i t y , i t r e s u l t e d i n a rather e f f i c i e n t heat removal, thus ensuring proper c o o l i n g and good wear r e s i s t a n c e . F i g 12 and 13 show the shoes i n p l a c e and a cl o s e - u p . The water enter s through the bottom, c i r c u l a t e s through the channels and leaves through the top. 3.3 E l e c t r o d e And Slag P r e p a r a t i o n The e l e c t r o d e ( i . e . , the f i l l e r metal) i n E l e c t r o s l a g J o i n i n g has the same composition(whenever p o s s i b l e even from the 25 same s t e e l shop run) as the parent m a t e r i a l b e i n g welded. T h i s a s s u r e s c h e m i c a l homogeneity and weld r e p e a t a b i l i t y . 8 The commercial carbon s t e e l used f o r both t h e f i l l e r and p a r e n t m a t e r i a l was ASTM A36 whose compo s i t i o n i s : C=0.29% Mn=0.90% Si=0.15% P=0.04% S=0.05% A t y p i c a l e l e c t r o d e s e c t i o n would be two 152.0 mm(6") X 19 mm(3/ 4") X 2930 mm(115" ) , 3 8 mm t h i c k , as i l l u s t r a t e d i n F i g s . 14 and 15. It i s c o n n e c t e d t o a water-c o o l e d s t u b t h a t i s p e r f e c t l y a l i g n e d w i t h t h e f u r n a c e frame. ( F i g . 16) A s m a l l rod 25 mm(l") i n d i a m e t e r and 152.4 mm(6") long i s welded t o the bottom o f the e l e c t r o d e so as to have a f a s t e r weld s t a r t . Two 152 mm X 457 mm X 914 mm p l a t e s w e i g h i n g a p p r o x i m a t e l y 500 kg each were used i n the experiments. The w e l d i n g f l u x used was of c o m p o s i t i o n : 55F/15/15/15, i . e . , CaF 2 =55% Ca0=l5% 0.-15% S i 0 2 = l 5 % (% i n weight) and p r e - h e a t e d at a temperature of about 600°C i n o r d e r t o 26 a v o i d any moisture r e t e n t i o n . A t o t a l of 7.0 kg per weld was enough to maintain s t a b l e welding c o n d i t i o n s . 3.4 Weld Set-up And C o n s t r a i n i n g A l l the weld p r e p a r a t i o n was done on a movable c o l o r l i t h p l a t f o r m that s l i d e s on a m o n o r a i l ( F i g . 8). A p r o t e c t i v e l a y e r of asbestos i s put on top of the c o l o r l i t h where the s t a r t e r p l a t e s are p o s i t i o n e d . In order to ensure proper weld p e n e t r a t i o n s i n c e the s t a r t of the weld and a l s o to make set-up and s t r i p p i n g o p e r a t i o n s more e f f i c i e n t , r u n - i n water-cooled copper sumps, as shown i n F i g . 17 were used. At the top of the blocks c l o s e to the weld, 76.2 mm(3") run-out sumps were tack-welded to extend the block height and thus, accommodate the s l a g volume past the j o i n i n g s e c t i o n s . The c o o l i n g shoes are h e l d i n p o s i t i o n by braces which are ti g h t e n e d on to b o l t s welded onto the b l o c k s ( F i g . 1 2 ) . The blocks were set 100 mm apart(which i s the recess length) ( F i g . 11). An a i r s e t t i n g high temperature mortar(SAIRSET) i s spread along the shoe s i d e s touching the blocks to provide a safe s e a l i n g a g a i n s t any p o s s i b l e s l a g and/or metal leakage. When t h i s was completed the assembly was ready to be pl a c e d in the welding p o s i t i o n . At t h i s stage the e l e c t r o d e was i n s e r t e d in the weld gap using the overhead crane, a l i g n e d and t i g h t e n e d 27 to the stub. When a weld i s f u l l y - c o n s t r a i n e d and no c r a c k i n g i s observed, i t may be a good i n d i c a t i o n that the weld can accommodate c o n t r a c t i o n movements and w i l l l i k e l y be a sound one. T h e r e f o r e , c o n s t r a i n e d set-ups were prepared so as to produce a weld crack. I n i t i a l l y an I-beam with I ( i n e r t i a moment) of approximately 23 i n " was thought to be strong enough to stand up to the pressures i n the system and was p o s i t i o n e d at the top of the b l o c k s . A s t u r d i e r I-beam was needed and i t was prepared so as to have an i n e r t i a moment of 126 i n " , as shown i n F i g . 20. The weld on the I-beam d i d f a i l as the set up became r i g i d and the s t r a i n gauges were unable to r e c o r d the a c t u a l s t r a i n undergone by the weld metal.(See F i g . 19) At t h i s stage i t was r e a l i z e d that as the weld went on the system(blocks being j o i n e d + weld region) became more r i g i d below the welding p o i n t ( m e t a l p o o l ) , to an extent that the welded I-beams c o u l d not take the pressure and, t h e r e f o r e , d i d not p r o p e r l y c o n s t r a i n the weld. I t was then thought t h a t i f the c o n s t r a i n i n g rods were p l a c e d at the bottom a lower system s t i f f n e s s would be e f f e c t e d thus, e n a b l i n g a stronger c o n s t r a i n i n g c a p a b i l i t y which would r e s u l t i n a weld c r a c k . The I-beams were arc welded on to the ends of the blocks away from the hot face, i n i t i a l l y at the top and l a t e r at the bottom. A rod 76.2 mm(3") i n diameter, 965.2 mm(38") i n len g t h and s l i g h t l y tapered o f f at the ends was p o s i t i o n e d i n f r e e compression r e s t r a i n i n g the p a r t s from c l o s i n g in(See F i g . 18). F i g . 20 and F i g . 21 show the boxed I-28 beam with i n c r e a s e d i n e r t i a moment and the hardened 4340 d i s c spacer to a v o i d any l o c a l i z e d deformation. The strain-gauges were set on both c o n s t r a i n i n g rods and monitored by a T r a n s d u c e r / S t r a i n I n d i c a t o r 8-channel STRAINSERT - Model TN8C. For c o r r e c t measurements a gauge f a c t o r was used and the apparatus was operated i n f u l l b ridge mode. F i g . 22 presents the same set-up d e s c r i b e d above except f o r the f a c t t h at the c o n s t r a i n i n g rods were p o s i t i o n e d at the bottom. T h i s way, the same c o n s t r a i n i n g r i g i d i t y would be experienced w h i l s t the assembly s t i f n e s s would not be as l a r g e as when the rods were pl a c e d at the top and, t h e r e f o r e , the rod s t r e n g t h would be comparable to the weld s t r e n g t h . Once the s t r a i n s are known, the s t r e s s e s can be c a l c u l a t e d as the rods remain in the e l a s t i c r e g i o n . 3.5 Sequence of Operation - Welding Procedure A f t e r the e l e c t r o d e i s p r o p e r l y a l i g n e d , a mixture of c a l c i u m f l u o r i d e and s t e e l shavings(the 'compact') i s p o s i t i o n e d on top of the s t a r t e r p l a t e and under the e l e c t r o d e t i p ; the 29 c o o l i n g system i s checked f o r leaks and the pre-heated f l u x i s poured i n t o the gap. I n i t i a l l y an arc i s s t r u c k and the small 25 mm diameter rod i s r e a d i l y melted. At t h i s stage,some of the f l u x s t a r t s to melt and soon the volume of s l a g i s such that no more a r c i n g occurs and heat i s generated by s l a g r e s i s t a n c e o n l y . Table II shows the furnace parameters f o r a t y p i c a l E l e c t r o s l a g J o i n i n g experimental run. Aluminum d e o x i d a t i o n was e f f e c t e d throughout a l l the runs at a r a t e of 1 g/min.(See F i g . 15) 3.6 R e s i d u a l S t r e s s A f t e r the weld was. c o o l e d down to room temperature, r e s i d u a l s t r e s s measurements were performed at the top of the assembly i n the parent metal, i n the heat a f f e c t e d zone and i n the weld. The equipment used was s u p p l i e d by P h o t o l a s t i c Inc., and the r e s u l t s were a r r i v e d at by using the B l i n d Hole D r i l l i n g M ethod." 5Fig. 23 shows the schematic s t r a i n gauge set-up. 3.7 Temperature Measurements Chromel-alumel thermocouples were p l a c e d i n the block to measure temperatures i n order to compare them with the 30 model.(Fig. 3) The r e s u l t s obtained as w e l l as the p e n e t r a t i o n depth p r e d i c t e d agreed w e l l with the observed v a l u e s . 31 IV. DISCUSSION AND RESULTS The Russian and Czech work developed i n E l e c t r o s l a g Welding(wire e l e c t r o d e ) had i n d i c a t e d that hot cra c k s were, a s s o c i a t e d mostly with the weld chemical composition l e a d i n g to t r a n s f o r m a t i o n a l volume changes which would cause d e f e c t s and tha t there e x i s t e d a c r i t i c a l height range wherein c r a c k s were more prone to appear. In t h e i r work i t was never made c l e a r whether the thermal s t r e s s f i e l d had been f u l l y understood or even i n v e s t i g a t e d . The magnitude of the s t r e s s f i e l d s i n ESJ was not d e f i n e d and even i n a q u a l i t a t i v e sense was not known. The f i r s t attempts to study thermal s t r e s s e s d u r i n g p l a t e e l e c t r o d e welding in t h i s work, r e v e a l e d that a great amount of the s t r e s s was being r e l a x e d as the heat source t r a v e l l e d along the weld h e i g h t . The l a r g e heat input(1.7 kW/cm), t y p i c a l of t h i s process, c r e a t e s a ra t h e r broad temperature f i e l d ( h e a t a f f e c t e d zone) capable of absorbing the high s t r e s s e s that develop. F u r t h e r experimentation confirmed the i n a b i l i t y of the I-beams and rods to e f f e c t i v e l y c o n s t r a i n the bl o c k s at the top and the f a c t that the thermal f i e l d was accommodating the deformations. A much s h o r t e r weld was made with the c o n s t r a i n i n g rods p o s i t i o n e d a t the bottom. The p r o p o s a l was t h a t , as mentioned above, the weld s t r e n g t h at a smal l e r height i n terms of weld 3 2 c r o s s - s e c t i o n a l area would be comparable to the area of the rods and, t h e r e f o r e , the l a t t e r would be able to take up the weld s t r e s s e s . I t was found that even a f t e r that procedure the weld d i d not present any c r a c k s , the thermal f i e l d having presumably r e l a x e d them even i n a true f u l l y - c o n s t r a i n e d weld. Both w e l d s ( c o n s t r a i n e d at the top and at the bottom) were thoroughly t e s t e d u l t r a s o n i c a l l y and no cr a c k s were found. A f t e r that they were sent out to be i n s p e c t e d by radiography. The p i e c e s were exposed to Co-60 r a d i a t i o n f o r approximately 15 hours and again no i n d i c a t i o n of c r a c k s was found. The thermal s t r e s s e s p r e d i c t e d by the model and c a l c u l a t e d fo r s e v e r a l welding c o n d i t i o n s were c o n s i s t e n t with experimental o b s e r v a t i o n . In F i g . 24 the c o n d i t i o n s set f o r the computer run were the same as the ones undergone by the a c t u a l j o i n i n g experiment. I t i s noted that d u r i n g most part of the weld the r e s u l t i n g thermal s t r e s s e s f o r d i f f e r e n t d i s t a n c e s away from the block hot face are t e n s i l e i n nature. When the l i q u i d pool approached the end, new boundary c o n d i t i o n s were imposed to account for r a d i a t i o n and c o n v e c t i o n whereupon compressive s t r e s s e s s t a r t to appear, subsequently, i n d u c i n g t e n s i l e s t r e s s e s underneath the l i q u i d pool where metal i s s o l i d i f y i n g . I f those compressive s t r e s s e s are higher than the m a t e r i a l y i e l d s t r e n g t h , then they should induce the high t e n s i l e s t r e s s e s that would l e a d to c r a c k s . 33 In order to i n v e s t i g a t e f u r t h e r the r e s u l t s above, a run with i d e n t i c a l welding c o n d i t i o n s ( e x c e p t f o r the weld height which was doubled up) was t r i e d and the same t r e n d was again observed.(See F i g . 25). In other words, should the computed values f o r the s t r e s s e s at the end of of the weld be exc e e d i n g l y high f o r a p a r t i c u l a r set of welding c o n d i t i o n s then, by a d j u s t i n g the l a t t e r a l e s s severe s t r e s s f i e l d would r e s u l t which, i n t u r n , provided enough p e n e t r a t i o n i s e f f e c t e d , c o u l d in p r i n c i p l e i n d i c a t e the optimum welding c o n d i t i o n s to produce a c r a c k - f r e e weld. The r e s i d u a l s t r e s s e s remaining i n the system a f t e r c o o l i n g to room temperature were measured and found to show t e n s i l e s t r e s s e s develop during or a f t e r welding. The r e s i d u a l s t r e s s e s in a l l three p o s i t i o n s ( p a r e n t metal, heat a f f e c t e d zone and weld metal) were t e n s i l e and below the m a t e r i a l y i e l d p o i n t . See Appendix D. Based on the modelling r e s u l t s and on the a c t u a l experiments performed, i t i s p o s s i b l e to say that the c h a r a c t e r i s t i c broad thermal f i e l d of E l e c t r o s l a g J o i n i n g i n the p a r t s being j o i n e d enhances thermal s t r e s s e s i n the weld(as w e l l as i n the heat a f f e c t e d zone) which are r e l a x e d by the thermal f i e l d to a c o n s i d e r a b l e extent as the weld p r o g r e s s e s . 34 These f i n d i n g s seem to c o n t r a d i c t the r e s u l t s presented by the Czech l i t e r a t u r e on E l e c t r o s l a g Welding c r a c k s . I t has been observed and p r e d i c t e d that the c r a c k s i n E l e c t r o s l a g J o i n i n g , i f at a l l present, w i l l always occur at the top of the j o i n t when compressive s t r e s s e s s t a r t mounting due to thermal c o n t r a c t i o n , wherever the top may be. If the Czech/Russian approach to p r e d i c t i n g hot c r a c k i n g were to be followed, then a s t e e l which transforms at high temperatures would have c r a c k s when a s t e e l of low tr a n s f o r m a t i o n temperature would not. These workers have p r e d i c t e d that h i g h a l l o y grades such as Ni-Cr-Mo-V f o r g i n g s do not crack when carbon s t e e l s do. I f so, the s t e e l used f o r the present study should have r e s u l t e d i n s o l i d i f i c a t i o n c r a c k i n g once i t i s a l r e a d y a 'good' crack-former i n terms of tr a n s f o r m a t i o n volume changes. 35 V. CONCLUSIONS 1) C o n s i d e r i n g the t h e o r e t i c a l c a l c u l a t i o n s and experimental evidence, i t can be s a i d that E l e c t r o s l a g Welding and E l e c t r o s l a g J o i n i n g d i f f e r q u i t e s i g n i f i c a n t l y as to the thermal s t r e s s b u i l d - u p and hot cracky tendency. I t i s c l e a r that the d i s s i m i l a r thermal p r o f i l e s experienced i n each of these processes r e s u l t in d i f f e r e n t s t r e s s f i e l d s . The r e s u l t i n g crack formation tendency i s t h e r e f o r e q u i t e d i f f e r e n t . 2 ) The r e s i d u a l s t r e s s measurement confirms the r e s u l t s p r e d i c t e d by the model and observed e x p e r i m e n t a l l y . The r e s i d u a l s t r e s s values are below the weld y i e l d s t r e n g t h l e v e l and, t h e r e f o r e , no c r a c k s were d e t e c t e d i n ESJ under c o n d i t i o n s which would have produced c r a c k s i n ESW. 3) E l e c t r o s l a g J o i n i n g i s more a p p l i c a b l e to the welding of heavy gauge f o r g i n g s than E l e c t r o s l a g Welding due to i t s i n t r i n s i c f e a t u r e s . I t should probably be regarded more as a shop f a b r i c a t i o n technique than as a f i e l d welding process. 4 ) A simple mathematical model was developed which enables the s e m i - q u a n t i t a t i v e v i s u a l i s a t i o n of the thermal and s t r e s s trends r e a l i z e d d u r i n g welding 5) The t h e o r e t i c a l p r e d i c t i o n s can be used when a s s e s s i n g the a p p l i c a t i o n of t h i s heavy t h i c k n e s s j o i n i n g method. 36 VI. SUGGESTIONS FOR FUTURE WORK 1) Produce a weld having a narrow thermal f i e l d , i . e . , a narrow weld gap, using wire e l e c t r o d e with narrow consumable guide i n order to v e r i f y the thermal s t r e s s f i e l d approach to e x p l a i n i n g hot c r a c k s i n t h i s kind of weld. 2) Use higher a l l o y s t e e l s having d i f f e r e n t lower t r a n s f o r m a t i o n temperatures under the same c o n d i t i o n s c i t e d i n 1) t o check on the i n f l u e n c e of volume change i n crack forming tendency. 3) Develop a more s o p h i s t i c a t e d mathematical model i n order to be able to a r r i v e at more accurate values when e v a l u a t i n g s t r e s s e s . Such.an approach would i n v o l v e a lengthy and complex f i n i t e element thermal s t r e s s a n a l y s i s coupled with experimental support i n r e l a t i o n to the boundary c o n d i t i o n s . 37 BIBLIOGRAPHY 1. Austel,W.;Heyman, H. and Maidorn, Ch., 6th I n t e r n a t i o n a l Vacuum M e t a l l u r g y Conference,San Diego, Ca., 1979, p. 747-756 2. Machner, P.,. 6th I n t e r n a t i o n a l Vacuum M e t a l l u r g y Conference, San Diego, Ca., 1979, p. 757-773 3. V i e i r a , E. M. and Guimaraes, A. A., I n t e r n a l Report, E l e t r o m e t a l Acos Fin o s S.A., Sumare, SP B r a s i l , 1982 4. Raman, A., Weld. J . , v o l . 60, (12), Dec. 1981, p. 17-21 5. S c h i l l i n g , C. G. and Benter, W. P., N a t i o n a l Cooperative Highway Research Program - Report 201 Transp. Res. Board, NRC, Washington, DC, May 1979 6. D i l a w a r i , A.; Eagar, T. W. and Szekely, J . , Weld. J . , v o l . 57, (1), 1978, p. 24s-30s 7. Masumoto, N. et a l . , Yosetsu G a k k a i s h i , v o l . 46, (12), 1977, p. 869-875 8. Naganathan, S.; S c r e e n i v a s a l u , A. and Rao, A. S., Weld. J . , v o l . 52, (11), 1973, p. 125s-234s 9. V i e i r a , E. M. and M i t c h e l l , A., Metals Technology, Oct. 1981, p. 405-410 10. Rosenthal, D., Trans. ASME, v o l . 68, 1946, p. 849-866 11. R y k a l i n , N. N., " C a l c u l a t i o n of Heat Flow in Welding", t r a n s l a t e d by Z. Paley and C. M. Adams, US c o n t r a c t number UC-19-060-3817, 1951 12. Okerblom, N. 0., "The C a l c u l a t i o n s of Deformations of Welded Metal S t r u c t u r e s " , t r a n s l a t e d by DSIR, HMSO, London, 1958 13. E r e g i n , L. P. and M a l a i , A. E., Svar. P r o i z . , v o l . 10, 1978, p. 26-27 14. W i l l i a m s , N. T.; Smith, C. J . and T o f t , L. H., Proceedings of the I n t e r n a t i o n a l Conference on R e s i d u a l S t r e s s e s i n Welded C o n s t r u c t i o n and t h e i r E f f e c t , The Welding I n s t i t u t e , London, Nov. 1977 15. A s a i , Y. and Nakamura, U., " E l e c t r o s l a g Welding of S t e e l Slabs with P l a t e E l e c t r o d e s " , Nippon S t e e l Co., Nagoya Works, 2nd I n t e r n a t i o n a l Symposium of the JWS, Osaka, 1975 38 16. Prokhorov, N.N. et a l . , S v a r . P r o i z . , vol.1,1972, p. 2-4 17. Masubuchi, K., Proceedings of the I n t e r n a t i o n a l Conference on R e s i d u a l S t r e s s e s i n Welded C o n s t r u c t i o n and t h e i r E f f e c t , The Welding I n s t i t u t e , London, Nov. 1977 18. T a l l , L., Weld. J . , v o l . 43, (1), 1964, p. 10S-23S 19. Ueda, Y. and Yamakawa, T., Trans. of the JWS, v o l . 2, (2), Sept. 1971, p. 90-99 20. Gray, T. G. F. and Wickramasinghe, D. M. G., Welding Research I n t e r n a t i o n a l , v o l . 8, (5), 1978, p. 409-421 21. N i s h i d a , M., Master T h e s i s , MIT, March 1976 22. Becka, J . and Kupka, I., Zvaranie, v o l . 25, (3), 1976, p. 72-77 23. Bacon, W. G., Ph.D. T h e s i s , UBC, 1979 24. S i l v a , A. C , Metl 560 P r o j e c t , UBC, 1978 25. Becka, J . , Zvaranie, v o l . 29, 1970 26. P e r t s o v s k i i , G. A. and Pugin, A. I., Avt. Svarka, v o l . 6, 1963, p. 14-23 27. Brown, R. and M i t c h e l l , A., S t e e l Seminar 1980, UBC, Sept. 1980 28. P h i l l i p s , R. H. and Jordan, M. F., Metals Technology, Aug. 1977, p. 396-405 29. Semenov, V. M.; Gel'man, A. S. and Rymkevich, A. I., Svar. P r o i z . , v o l . 11, 1973, p. 49-50 30. E r i k s s o n , L. and Ostensson, B., J . Scan. Met., v o l . 2, 1973, p. 282-284 31. Pense, A. W.; Wood, J . D. and F i s h e r , J . W., Weld. J . , v o l . 60, (12), Dec. 1981, p. 33-42 32. Rymkevich, A. I.; Gel'man, A. S. and Semenov, V. M., Svar. P r o i z . , v o l , 10, 1973, p. 10-11 33. Homberg, G. and W e l l n i t z , G., Schweissen un Schneiden, v o l . 27, (3), 1975, p. 90-93 34. L i b y , A. L.; M a r t i n s , G. P. and Olson, D. L., "Modeling of C a s t i n g and Welding Processes", Conference Proceedings, The Met. Soc. of AIME, Rindge, NH, Aug. 1980, p. 161-196 39 35. Paton, B. E., " E l e c t r o s l a g Welding", A W S, N. Y., 1962 36. Carnahan, B.; Luther, H. A. and Wilkes, J . 0., " A p p l i e d Numerical Methods", John Wiley & Sons, 1969 37. Minakami, H. et al.,Tetsu-to-Hagane, v o l . 63, 1973, s562 38. Boley, A. and Weiner, J . H., "Theory of Thermal S t r e s s e s " , John Wiley & sons, Inc., 1960 39. S i d l a , G. and M i t c h e l l , A., "The Design, C o n s t r u c t i o n and Operation of an ESC I n s t a l l a t i o n " , S p e c i a l Report to DREP/DSS, Vancouver, BC, June 1980 40. F r o s t , R. H. et a l . , Weld. J . , Jan. 1981, p. 1s-6s 41. Mosny, J . and Slabon, I., Proceedings of an I n t e r n a t i o n a l Conference on Welding Research r e l a t e d to Power P l a n t , U. of Southampton, England, Sept. 1972, p. 456-463 42. Weinberg, F., Met.- Trans. B, v o l . 10B, June 1979, p. 219-227 43. Bendis, A., Zvaranie, v o l . 16, (10), 1967, p. 365-370 44. Makara, A. M.; G o t a l ' s k i i , Yu. and Nuvikov, I. V., Avt. Svarka, v o l . 8, (4), 1955, p. 3-12 45. Redner, S., "Measurement of R e s i d u a l S t r e s s e s by B l i n d Hole D r i l l i n g Method", B u l l e t i n TDG-5, P h o t o l a s t i c Inc., May 1974 46. Weinberg, F., Met. Trans. B, v o l . 10B, Dec. 1979, p. 513-522 47. Paton, B. E. et al.., Proceedings of a Conference on ESR, The ISI , U. of S h e f f i e l d , Jan. 1973, p. 105-112 48. G r a v i l l e , B. A., "The P r i n c i p l e s of Cold C r a c k i n g C o n t r o l i n Welds", Dominion Bridge Company, L t d . , Montreal, 1975 40 T a b l e I - C o m p u t e r M o d e l P a r a m e t e r s T M P * 1 5 2 0 . 0 ( D e g C ) I N I T I A L T E M P - 2 5 . 0 ( D e g C) T I M E S T E P = 3 0 . 0 ( s ) S P E C I F I C H E A T = 0 . 1 0 7 0 ( c a l / g . C ) D E N S I T Y - 7 . 8 6 0 ( g / c m * * 3 ) C O N D U C T I V I T Y * 0 . 0 7 4 0 ( c a l / c m . s . C ) D X = 2 . 0 ( c m ) H E A T F A C T O R * 0 . 1 4 2 H E A T S O U R C E D E P T H = 1 2 . 0 0 0 ( c m ) P R I N T C Y C L E - 6 0 0 . 0 ( s ) E N D O F C A L C U L A T I O N * 4 2 0 0 . 0 ( s ) D Y = 0 . 5 1 . 0 1 . 0 1 . 0 1 . 0 1 . 0 3 . 0 3 . 0 3 . 0 3 . 0 D Y = 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 D Y = 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 3 . 0 N U M B E R O F D I V I S I O N S I N X - D I R E C T I O N = 5 0 N U M B E R O F D I V I S I O N S I N Y - D I R E C T I O N = 3 0 W E L D I N G P A R A M E T E R S : W E L D G A P * 9 . 5 c m T H I C K N E S S = 1 5 . 0 c m C U R R E N T * 5 0 0 0 . 0 A V O L T A G E * 3 3 . 0 V E L E C T R O D E S U R F A C E A R E A * 5 7 . 9 ( c m * * 2 ) W E L D G A P A R E A * 1 5 4 . 8 ( c m * * 2 ) L A T E N T H E A T * 6 5 . 0 ( c a l / g ) H O T T O P P I N G T I M E = 3 9 9 0 . 0 ( S ) A I R T E M P E R A T U R E = 3 0 . 0 ( D E G C ) H O T T O P C U R R E N T * 5 0 0 0 . O A H O T T O P V O L T A G E * 3 3 . 0 V W E L D I N G V E L O C I T Y * 0 . 0 2 3 ( c m / s ) 41 Table I I - ESJ T y p i c a l Log Sheet Time NMT Prim Sec V o l t MS WT (s) " (A) (A) (V) (rps) (Deg. ' 300 1468 320 4800 34 19 1 1 .0 500 2118 330 5000 34 19 11.5 700 2881 330 5000 34 20 11 .5 1 000 4042 330 5000 34 21 11.5 1 400 5631 330 5000 34 21 11.5 1700 6839 340 5100 33 21 12.0 2200 • 8847 340 5000 34 21 13.0 2400 9672 330 4900 33 21 13.0 2700 1 0987 340 5000 33 21 13.0 3000 1 2275 340 5000 33 21 13.0 3400 1 4058 330 4900 33 21 13.0 3700 1 5451 330 4900 33 21 13.0 4000 16805 310 4700 34 21 13.0 NMT = Number of Motor Turns MS = Motor Speed WT = Water Temperature \ 42 movement water-cooled shoes B 250 KVA F i g u r e 1 - Schematic Layout of ESJ Equipment 43 fusion line zone 2 fusion line Type I Type II zone 1 zone 2 zone 3 fusion line zone 1 fusion line Type Type IV F i g u r e 2 - E l e c t r o s l a g T y p i c a l Weld S t r u c t u r e ( R e f . 35) \ 44 F i g u r e 3 - ESJ Thermal P r o f i l e - C a l c u l a t e d and Measured 45 run-in t a b ^ ^ F i g u r e 4 - B o u n d a r y C o n d i t i o n s 46 Boundary III Boundary V Boundary II-Heat Source, (slag) x (i) y(j) Boundary VIII IM. IN • \ \ \ \ \ W \ \ \ \ • '//////; V/////-//////> • '/////* • </////> • V////S • •Ti +1 •j • '/////> T.,j-1 .Ti.j •Ti.j+1 '/////A • •THj '////// • •TAX VMM Arf AYj« V////A //////, \ ^ \ \ \ ^ \ \ \ \ \ \ \ \ \ \ V Boundary I U N Boundary VI Boundary IV F i g u r e 5 - Nodal Arrangement 47 f S T A R T ) SUB ROUTINE 0 INPUT SUBROUTINE INITL SUBROUTINE V E L SUBROUTINE CONST t = t + A t SUBROUTINE S N D S SUBROUTINE HEAT SUBROUTINE PRINT TB(I.J) =T(I.J) SUBROUTME] S T R E S S SUBROUTINE F S T S ^ R E S U L T S ^ / SUBROUTINE S N D S Q E N D ) F i g u r e 6 - Model Flowchart 48 L 2 x L. 2 I 1 TH F i g u r e 7 - S t r e s s A n a l y s i s Schematic Diagram 4 9 F i g u r e 8 - UBC E l e c t r o s l a g U n i t 50 86.5 mm. F i g u r e 9 - C o o l i n g Shoe - Water Channels 51 Copper A l u m i n u m F i g u r e 10 - C o o l i n g S h o e T o p V i e w 77 mm. 100mm. 25.£mm. BnomV - N 0 5 2 3 * / 77 mm. 86.5 mm. 81 mm. 86.5 mm. 254 mm. F i g u r e 11 - Copper r e c e s s F i g u r e 12 - C o o l i n g Shoes in P o s i t i o n F i g u r e 13 - C o o l i n g Shoe Close-up - Water connections 5 5 F i g u r e 14 - P l a t e E l e c t r o d e i n P o s i t i o n F i g u r e 15 - Aluminum Feeder 5 7 F i g u r e 16 - E l e c t r o d e and Copper Stub Figure 17 - Run-in Copper Tabs 59 60 0 60.8 121.6 1324 243.2 304 364.8 TIME , min. F i g u r e 19 - S t r a i n versus Time P l o t 61 Fi g u r e 20 - Boxed I-Beam 62 F i g u r e 21 - Hardened 4340 Disc Spacer 6 3 F i g u r e 2 2 - I n f e r i o r I-beam Placement 64 strain gauge 914 mm F i g u r e 2 3 - S t r a i n - g a u g e S e t - u p 6 5 300 240 nG 180 r : 120 Q. 60 CO LU rr 0 175 -60 rr x -120 -180 -240 h -300 1 1 1 1 1 1 1 1 — ELECTROSLAG JOINING Time = 4200 s Distance from hot face = 0.5 / 1.5 / 3.5 cm. Voltage = 33 V Current = 5000 A 0 . 5 1 . 5 3 . 5 J 1 1 i_ J 1 • 0 33.6 672 100.8 134.4 168 201.6 2352 268.8 3024 336 369.6 403.2 TIME . s ( 1 x io1) F i g u r e 24 - Thermal S t r e s s Curve 66 300 240 180 00 •_ 120 o X 60 CO 0 CO LU rr . t— -60 CO _ l < -120 cc LU X -180 1— -240 -300 ~l 1 1 I ELECTROSLAG JOINING Time = 8400 s Distance from hot face =0.5 cm. Voltage = 33 V Current = 5000 A Weld height = 193 cm. 0.5 0 672 134.4 201.6 268B 336 403.2 470.4 5376 604.8 672 739.2 806-4 TIME , s (1 x 10 1) F i g u r e 25 - Thermal S t r e s s Curve 6 7 ELECTROSLAG JOINING weld height = 96 cm. Welding conditions : voltage = 33 v current = 5000 A welding speed = 0.023 c m / s 64 128 192 256 320 384 448 512 576 640 704 768 DISTANCE FROM HOT FACE (mm.) F i g u r e 2 6 - E S J T h e r m a l G r a d i e n t 4 6 8 APPENDIX.A - BOUNDARY CONDITIONS  BOUNDARY I ( i = l , j = l ) i ) I f e l e m e n t I c o n t a c t s t h e h e a t s o u r c e : 1 S t ^ T 1 , 1 = TMI 2 n d - A | i i ) I f e l e m e n t I d o e s n o t c o n t a c t t h e h e a t s o u r c e , t h e f o l l o w i n g e q u a t i o n c a n b e o b t a i n e d b y a p p l y i n g a h e a t b a l a n c e : * n 1 s t A t A y A x C p p ^ l , ! " T l , l ) _ k A y _ l ( T * - T* ) + 2 2 A t ~ x Z , X i ' i 2 k A x + 2 2 " T* ) o r / 2 2 . * 2ou T* 2 Tlf: a _ n n ( A T + A x ^ ) T l , r ^ ' A t 1 , 1 + A y i(A y ; L+Ap ) t T l , 2 ~ T l , l } n+1 * CT - T ^ kAv * * 2nd A t Ay A x C p p v 1 , 1 1,1 yl (T - T ) + 2 1 2 A t " A x Z y ± 1 , 1 k A x . i j_i + ~ 2 ( T n + 1 - T D + 1 ) 1 2 1 l ' Ay +Ay2 X " L 2 o r a T n + 1 2 ) T n + 1 _ 1 » 2 - 2 * 2 a * * ^At + A y i ( A y i + A y 2 ) ; A l , l ^ ( A y ^ A y p A t ^ . l & 2 U 2 , r i , l J 69 BOTNDARY_II ( i=2-IM-l, j=1) i ) I f element I I c o n t a c t s the heat source: 1st At T * ) 1 = 2 n d ^ T t l = T M F i i ) I f element I I does not contact the heat source, the f o l l o w i n g equation can be obtained by a p p l y i n g a heat balanc e : ( T l r T i , i } k A y i , * * * 1st A^ Ay^xpCp • ^  = - - ( T . ^ ^ ^ ^ ^ ^ T . ^ ) 2 • kAx n n . y l y 2 1 , 2 1 ' 1 * 2 2 * a T* 2 T n + ^—2" 1-1,1 At A x z / i , l Ax^ ' At ( T ? , - T ° ) + A y ^ A y ^ A y ^ 1 ' i n+1 * 9 ( T i l ~ T i 1> k A y l * •2nd At Ay A x p C p — ± * ± — — = A (T..,+,T. , r 2 T . . ) —  Jl r At Ax x+1,1 1-1,1 i , j 2 A y l + A y 2 or , 2 a \ T n + 1 2 r n + 1 = 2 T * + g „ f T * 4.T* 9T*S 1 At +' A y i ( A y i + A y 2 ) ; T i , l " Ay 1 ( A y 1 + A y 2 ) " , 1 l , 2 " S P i , ! 1 " A ^ T i 4 a , l ^ i ^ i ; i 2 T i J l 7 0 BOUNDARY I I I ( i = I M , j = l ) i ) I f e l e m e n t I I I c o n t a c t s t h e h e a t s o u r c e : 1 S t ^ V l = T M P 2 2 n d A t T m ; i = T M i i i ) I f e l e m e n t I I I d o e s n o t c o n t a c t t h e h e a t s o u r c e , t h e f o l l o w i n g e q u a t i o n c a n b e o b t a i n e d b y a p p l y i n g a h e a t b a l a n c e : 1st A | ^ P ^ ^ ' V - ^ ( T ; ; 1 - + k A x n * 2 + 2 I M , 2 I M . r o r ( A y l + % ) 2 2 " T * . + r _ 2 _ . + 2 a = _2_rn a n n , Ax2 I M - 1 , 1 v A t A x ^ ' I M , 1 A t I M , 1 A y ^ ( A y ^ + A y ^ ) I M , 2 " I M , 1 n+1 * I M 1~ IM 1 k A y - i * * 2 n d ^ | x p C P ( ^ M' 1)- - - ^ T m > 1 - T ^ ^ ) + 2 k A x f T n + l n+1 . + 2 v I M . 2 I M , 1 ; o r A y l + A y 2 + A t A y 1 ( A y 1 + A y 2 ) •)T n+1 I M , 1 A y 1 ( A y 1 + A y 2 ) T n + 1 = 2 T * -- I M , 2 - V l ^ f T * T * ^ A x 2 v I M , 1 I M - 1 U 71 BOUNDARY I V ( i = l , j = 2 - I N - l ) * n * * T . - T , . k A y . ( T . - T . ) 1st A t A x A y .pCpC-^^ i t l ) = 3 7^ ± - a J -—x 3 A t ' A x ^ 2 k A x ( T ? . - T n . ) k A x ( T ? . - T ? . , ) -r- l , j + l i , j — l , j 1 , J -1 ' + z " * • A y . + Ay A y . + Ay 1 -1+1 .1 .1-1 o r 2 a ( T n - T n ) 2 a ( T l i ~ T l 1-1> ( -2- + - = ~ r i T * - - ^ T * - - 2 - T n + l>.r ~ Av (Ay +Av ) ( A t + ^ T l , j ^ 2 , j A t T l , j + Ay ( A y +Ay ) ( A y j + t * i - l > n+1 * * * . . _ T " T - T . . k A y . ( T „ . - T ^ . ) . 2 n d A t -AxAy pCp l t , i l t i 1 y i a 2 ? , 1 1?,T + — 2 J A t A x k A x n+1 _ n+1 k A x . n+1 n+1 . + 2 U l , j + 1 1 l , . j J _ 2 U l , j ; A y . + A y . . . A y . + Ay *1 1+1 _ J .1-1 2 2 o r -2 a -n+1 A y . ( A y . +Avi_j_^) l , j - l + (" 2 2 a  A t A y ( A y +Ay ^ 2 a A y . ( A y . + A y . J 3 3" - ) T n + 1 " 2 a n+1 2 * 2a * * T , 4 + 7 3 Z ( T o 4 - T , 4 ) Ay ( A y +Ay ^ X j + l A t to? V A 2 , j 72 BOUNDARY Y(i=IM,j =IN-1) -1st At AxAy.pCpf-TIM,i " ^ M,j,= T (.. 3M,j ~ TIM-l,j) + 2 2 3 • At - Ax 2 A y j + A yj+1 ' 4 y 1 +Ayj-1. 2 " 2 " or &x^  n t AxZ flC Ay4 (Ay. + Ay' ) 2n Cm,A - ^ . j - i) Ay.. (Ay.. +Ay.._1) n+1 * * * 2nd At AxAy.p Cp(TIM,.i-TIM,j) = - KAyj (TIM,3-TIM-1,j) 2 2 3 At Ax kAxd^.^-T^ 1.) kAxCT^.-T?: 1. n) -J IM,j+l IM,j' — IM.,2 IMo-l' + Ay. +Ay... Ay. + Ay. 1 -1+1 -1 -I-* or -2 a ^n+1 . , 2 . 2 a 2a et 1. + A Y j(A y j +Ay j_ 1) iIM,j-l v At A 7 j (A y j+Ay j + 1) y.. (Ay^ .+Ay.._^ ) IM,j 2_a n+1 2 * 2a * * Ay j(Ay j+Ay j + 1) TIM,j+l = At TTM,j" Ax? CTTM,j " T I M - l , j ) •BOUNDARY V I ( i = l , j = I N ) 1st j& j g A v T L I P C » F T 1. I N- T1, I N : - J^S T2,IH-T1,IN 2 A t ' Ax ' 2 _ R ^ ( T l > IN-^1 > ' A y I N + A 5 ? I N -1 o r ;_2 + 2 k A t ) T 1 , IN - ^ T 2 , I N = f ^ T l . I N A x Ax a r i . I N - 1,IN-1) A y l N ( A y I N ^ I N - l ) n+1 * * 2nd A t A x A Y I N p C p , T l , I N - l . I N . = ^ I M , T 2 , I N - T 1 , I N 2 2 ^ A t ; A k A x n+1 „ n + l K 2C1 .IN - l . I K -1) A y I N + Ay I N - l o r r n + l l . I N - l + _2 +.. r n + l 1 , IN A y l N ( A y I N + A y I N - l ) A t A y I N A y I N + A y I N - l , * * *, _2 T 1 , I N + 2o_ ( T2 , IN - T 1 , I N ) A t Ax 74 •BOUNDARY V I I ( i = 2 - I M - l , j = I N ) * * 1 s t A t AxAy pCp * T l t I N - T l , H p = 2 i N j& 2 & A tk jx xi K A y I N ( T i + l , I N + T i - 1 , I N - 2 T 1 , I N ) - K A x ( T i , I N - T i , I N - 1 ^ A y I N + A y I N - l o r * * * n -a T i - 1 , I N + , _ 2 + 2a v T i , I N - _ a _ T i + 1 , I N = 2 _ T 1 , I N + A 2 V A t . 2} . 2 A t Ax Ax Ax ( T n T n ) - a 1 , I N - i , I N - r A y i N ( A y I N + A y l N - l } ( T n + 1 T * ) ( T n + 1 T n + 1 ) 2 n d A t AxAy p C p v i , I N - 1 , I N ; = - K A x v i , I N - i . I N - r * 2 f- A ? I N + A y ! N - l 2 * + K A y l N ( T i + l , I N + T i - 1 , I N - 2 T i , I N ) Ax o r T n + 1 T n + 1 i , I N - l + ; ( _ 2 + a ) A i , I N A y i N ( A y i N + ^ I N - l 5 A t A y i N ( A y l N + A y l N - l } * * * * _ 2 T i , I N + _ a _ ( T I , I N + T i - I , I N - 2 T i , I N ) A t ... 2 Ax 'BOUNDARY VIII(i=IM,1=IN) * n (T - T ) •1st At AxAy pCp ^ IM,IN IM,nr = 2 2 N t * * K A x C T n - T n 1 ^ I N (TTM,IN " TIM-1,IN> - "2 m ^ ** A y I N + A y I N - l Or T * + - 2a IM-1,IN Ax 2' • i.2. At + Ax 2a) 2 K n C = 2 T I M , I N 7 - I M , I N + _.At (T n T n ) v I M , I N - I M , I N - r A y I N ( A y I N A y I N - l > n+1 * (T - T ) 2nd At AxAy pCp v T M , I N I M . I N ^ = 2 2 ' At K A Y I N (TIM,IN ~ TIM-1,IN ) Ax K Ax . n + 1 _ n+1 . IM.IN IM,IN-r W + A v  y I N y I N - l o r A y I N ( A y l N + A y l N - l ) Tn+1 I N , I n - l ( 2 + Tn+1 a . I M , I N A t A y I N ( A W y I N - l > * * A _2 T I M , I N " _2a ( T I M , I N ~ ^ I M - l . I N * At Ax' 7 6 APPENDIX B ~ COMPUTER PROGRAM SAMPLE C CALCULATION OF THERMAL STRESSES IN HEAVY SECTION C ELECTROSLAG JOINING C C C C PAULO SILVEIRA IVO C C C C MAIN PROGRAM C DIMENSION T(100,100),TS(100,100),TB(100,100),DY(100),ID(100) CALL DlNPUT(DY,DX,TMP,TBO,DT,CP,RO,AK,DPH,PRNT,TLAST,IM, *IN,XI,VO,TH,F,G,SE,SWG,AL,FR,CPW,DELT,ROW,HEB) C C C C C C C C c c CALL INITL(T,TS,TB,TBO,DPH,DX,ID,IM,IN,PRNT,PRNTO) CALL VEL(V,F,XI,VO,Q1,IM,DT,DX,TH,T,DY,DPH,AK,IN,TBO,TMP,ID) TIME=0.0 1 TIME=TIME+DT CALL HSPOS(DPH,V,TIME,DX,ID,IM) CALL CONST(DT,DX,CP,RO,AK,ALPHA,A 1 ,A2,A3) CALL FSTS(TB,TS,DY,DX,DT,ALPHA,IN,IM,TMP,ID,A1,A2,A3) CALL SNDS(TS,T,DY,DX,DT,ALPHA,IN,IM,TMP,ID,A1,A2,A3) CALL HEAT(XI,VO,AK,CP,RO,DX,DPH,TH,T,TMP,ID,G,F,IM,IN,TBO,DY, *DT,V,TIME,SE,SWG,AL,FR,CPW,DELT,ROW,HEB) CALL PRINT(T,TIME,IN,IM,IP,PRNT,PRNTO,TLAST,ID) IFdP.EQ. 1 ) GO TO 2 DO 10 I = 1 ,1M DO 20 J=1,IN TB ( I , J ) = T ( I , J ) 20 CONTINUE 10 CONTINUE GO TO 1 2 CALL STRESS(T,TT,TTT,IT,IM,IN,ALFA,H) STOP END C C READ DATA AND PRINT HEADINGS C SUBROUTINE DlNPUT(DY,DX,TMP,TBO,DT,CP,RO,AK,DPH,PRNT,TLAST,IM, *IN,XI,VO,TH,F,G,SE,SWG,AL,FR,CPW,DELT,ROW,HEB) C DIMENSION DY(100) READ(5,100) TMP,TBO,DT,CP,RO,AK,DX,F,DPH,PRNT,TLAST 100 FORMAT(F6.1,9F6.3,F10.0) READ(5,110) IM.IN 110 FORMAT(2I3) 7 7 READ(5,120) DY(1),DY(2),DY(3),DY(4),DY(5),DY(6),DY(7) 120 FORMAT(7F5.0) DO 10 I=8,IN DY(I)=DY(7) 10 CONTINUE READ(5,130) G,TH,XI,VO,SE,SWG,AL 130 FORMAT(7F6.1) READ(5,140) FR,CPW,DELT,ROW,HEB 140 FORMAT(4F6.3,F7.4) WRITE(6,200) 200 FORMAT(1H ,///,5X,' ESW THERMAL PROFILE *,3X, *'PAULO S. IVO',/,6X,l9('*'),///) WRITE(6,210) TMP,TBO,DT,CP,RO,AK,DX,F,DPH,PRNT,TLAST 210 FORMAT(1H ,IX,'TMP=',F6.1,1X,'(Deg C)',/,2X,'INITIAL TEMP=', *F6.1,1X,*(Deg C)',/,2X, *'TIME STEP=',F6.1,IX,'(s)',/,2X,'SPECIFIC HEAT*',F6.4,IX, * ' ( c a l / g . C ) ' *,/,2X,'DENSITY=',F6.3,IX,'(g/cm**3)',/,2X, *'CONDUCTIVITY=",F6.4,1X,'(cal/cm.s.C)',/,2X,'DX=', *F6.1,IX,'(cm)',/,2X,'HEAT FACTOR=',F6.3,IX, */,2X,'HEAT SOURCE DEPTH*',F6.3,IX,'(cm)',/, *2X,'PRINT CYCLE=',F7.1,1X,'(S)',/,2X,'END OF CALCULATION' * , F l 0 . 1 , 1 X , ' ( s ) ' ) WRITE(6,220) (DY(I),1=1,IN) 220 FORMATOH , 1X, 'DY= ' , 1 0F6. 1 ) WRITE(6,230) IM,IN 230 FORMATOH ,IX,'NUMBER OF DIVISIONS IN X-DIRECTION =',13,//, *2X,'NUMBER OF DIVISIONS IN Y-DIRECTION =',I3) WRITE(6,240) G,TH,XI,VO,SE,SWG,AL 240 FORMAT(1H ,///,2X,'WELDING PARAMETERS',///,2X, *'WELD GAP=',F5.1,IX,'cm',2X,'THICKNESS=',F5.1,1X,'cm',// *2X,'CURRENT=',F7.1,1X,'A',2X,'VOLTAGE=',F5.1,IX,'V,//, *2X,'ELECTRODE SURFACE AREA=',F5.1,IX,'(cm**2)',2X, *'WELD GAP AREA =',F6.1,IX,'(cm**2)',2X,//,2X,'LATENT HEAT=*, *F 5 . 1 , I X , * ( c a l / g ) ' , / / ) WRITE(6,250) FR,CPW,DELT,ROW,HEB 250 FORMAT ("IH , IX,'WATER FLOW RATE= ' , F6 . 1 , 1X, ' (cm** 3/s ) * , */,2X,'WATER SPECIFIC HEAT=',F6.1,1X,'(cal/g.C)',/, *2X,'TEMP. DIFF. IN MOULD=' ,F6.1,1X,' (C) ' ,/,2X, *'WATER DENSITY=',F6.1,1X,'(g/cm**3)',/,2X, *'HEAT EFF. TO THE BLOCKS'',F7.4,1X,/) RETURN END C C C C C c SUBROUTINE INITL(T,TS,TB,TBO,DPH,DX,ID,IM,IN,PRNT,PRNTO) C DIMENSION T(100,100),TS(100,100),TB(100,100),ID(100) DO 10 1=1,IM DO 20 J=1,IN T(I , J)=TBO TS(I,J)=TBO TB(I,J)=TBO 20 CONTINUE 10 CONTINUE NO=IFIX(DPH/DX+0.49)+1 78 DO 30 1=1,NO 30 ID(I)=1 NP=NO+1 DO 40 I=NP,IM 40 ID(I)=0 PRNTO=PRNT RETURN END C C C CALCULATION OF THE WELDING SPEED C C C SUBROUTINE VEL(V,F,XI,VO,Q1,IM,DT, DX, TH, T, DY,DPH,AK,IN,TBO, *TMP,ID) DIMENSION ID(100),T(100,100),DY(100) CALL HEAT 1 (AK,DX,TH,T,DY,ID,Q1,1M,DPH,DT,IN,TBO,TMP) FACTOR=0.24*0.95*0.35*XI*VO/(.023*Q1) V=((.35*XI*VO*0.24*.95)/(FACTOR*Q1)) WRITE(6,88B) V,FACTOR 888 FORMAT(1H ,/,'WELDING VELOCITY*',F6.3,2X,'(cm/s)',F7.2,/) RETURN END DECISION ON WHETHER THE HEAT SOURCE IS CONTACTING THE BLOCK SUBROUTINE HSPOS(DPH,V,TIME,DX,ID,IM) DIMENSION ID(100) NO=IFIX(V*TIME/DX+0.51) + 1 NS=IFIX((DPH+V*TIME)/DX+0.49)+l NN=NO-1 IF(N0-1) 1,1,2 2 DO 10 I=1,NN ID(I)=0 10 CONTINUE 1 DO 20 I=NO,NS ID(I)=1 20 CONTINUE NNS=NS+1 DO 30 I=NNS,IM ID(I)=0 30 CONTINUE RETURN END C C C C C C SUBROUTINE CONST(DT,DX,CP,RO,AK,ALPHA,A1,A2,A3) ALPHA=AK/(CP*RO) A1=2.0*(1.0/DT+ALPHA/(DX**2)) A2=2.0*ALPHA/(DX**2) A3=2.0/DT RETURN END C C C 79 c C CALCULATION OF THE FIRST HALF-TIME STEP C C C c SUBROUTINE FSTS(TB,TS,DY,DX,DT,ALPHA,IN,IM,TMP,ID,Al,A2,A3) C DIMENSION TB(100,100),TS(100,100),DY(100),ID(100) DIMENSION A(100),B(100),C(100),D(100),TPRIME(100) C I F ( I D ( 1 ) - 1 ) 1,2,2 2 CALL S0URCE(A(1),B(1),C(1),D(1),TMP) GO TO 3 1 A(1)=0.0 B(1)=A1 C(1)=-A2 D(1)=A3*TB(1,1)+ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) **(TB(1,2)-TB(1,1)) 3 CONTINUE IJ=IM-1 IK=IN-1 DO 10 I=2,IJ I F ( I D ( I ) - 1 ) 4,5,5 5 CALL•SOURCE(A(I),B(l),C(I),D(l),TMP) GO TO 10 C 4 A(I)=-0.5*A2 B(I)=A1 C(I)=-0.5*A2 C D(I)=A3*TB(I,1)+(ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) * * ( T B ( I , 2 ) - T B ( I , 1 ) ) ) C 10 CONTINUE C IF(ID(IM)-1) 6,7,7 7 CALL SOURCE(A(IM),B(IM),C(IM),D(IM),TMP) GO TO 8 C 6 A(IM)=-A2 B(IM)=A1 C(IM)=0.0 C D(IM)=A3*TB(IM,1)+(ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) **(TB(IM,2)-TB(IM,1))) C 8 CALL TRIDAGd , IM, A, B ,C ,D,TPRIME) C DO 15 1 = 1 ,IM 15 TS(I,1)=TPRIME(I) DO 20 J=2,IK C A(1)=0.0 B(1)=A1 C(1)=-A2 D(1)=A3*TB(1,J)+(2.0*ALPHA/(DY(J)*(DY(J)+DY(J+1)))) **(TB(1,J+1)-TB(1,J))-(2.0*ALPHA/(DY(j)*(DY(j) *+DY(J-1))))*(TB(1,J)-TB(1,J-1)) 80 c DO 25 1=2,IJ C A(I)=-0.5*A2 B(I)=A1 C(I)=A(I) C D(I)=A3*TB(I,J)+(2.0*ALPHA*(TB(I,J+1)-TB(I,3))/ *(DY(J)*(DY(J)+DY(J+1 ).) )) *-(2.0* A L P H A * ( T B ( I , J ) - T B ( I , J - 1 ) ) / ( D Y ( J ) * ( D Y ( J ) + D Y ( J - 1 ) ) ) ) C 25 CONTINUE C C A(IM)=-A2 B(IM)=A1 C(IM)=0.0 C D(IM)=A3*TB(IM,J)+(2.0*ALPHA*(TB(IM,J+1)-TB(IM,J)) */(DY(J)*(DY(J)+DY(J+1))))-(2.0*ALPHA*(TB(IM,J) * - T B ( l M , J - 1 ) ) / ( D Y ( J ) * ( D Y ( J ) + D Y ( J - 1 ) ) ) ) C CALL TRIDAG(1,IM,A,B,C,D,TPRIME) C DO 30 1=1,IM 30 TS(l', J)=TPRIME(I) 20 CONTINUE C A(1)=0.0 B(1)=A1 C(1)=-A2 C D(1)=A3*TB(1,IN)-(ALPHA*(TB(1,IN)-TB(1,IN-1)) */(DY(IN)*(DY(IN)+0.5*DY(IN-1)))) C DO 35 1=2,IJ A(I)=-0.5*A2 B(I)=A1 C(I)=-0.5*A2 C D(I)=A3*TB(I,IN)-(ALPHA*(TB(I,IN)-TB(I,IN-1)) */(DY(IN)*(DY(IN)+0.5*DY(IN-1)))) C 35 CONTINUE C A(IM)=-A2 B(IM)=A1 C(IM)=0.0 C • D(IM)=A3*TB(IM,IN)-(ALPHA*(TB(IM,IN)-TB(IM,IN-1)) */(DY(IN)*(DY(IN)+0.5*DY(IN-1)))) C CALL TRIDAG(1,IM,A,B,C,D,TPRIME) DO 40 1 = 1 ,IM 40 TS(I,IN)=TPRIME(I) RETURN END C C c 81 SUBROUTINE SNDS(TS,T,DY,DX,DT,ALPHA,IN,IM,TMP,ID,A1,A2,A3) C DIMENSION TS(100,100),T(100,100),DY(100),ID(100) DIMENSION A(100),B(100),C(100),D(100),TPRIME(100) I F ( I D O ) - I ) 1,2,2 2 CALL SOURCE(A(1),B(1),C(1),D(1),TMP) GOTO 3 1 A(1)=0.0 B(1)=A3+ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) CO)=-ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) C D(1)=A3*TS(1,1)+A2*(TS(2,1)-TS(1,1)) 3 CONTINUE C IJ=IM-1 IK=IN-1 DO 10 J=2,IK C A(J)=-2.0*7\LPHA/(DY(J)*(DY(J)+DY(J-1))) C B(J)=A3+(2.0*ALPHA/(DY(J)*(DY(J)+DY(J+1)))) *+(2.0*ALPHA/(DY(J)*(DY(J)+DY{J-1)))) C C(J)=-2.0*ALPHA/(DY(J)*(DY(J)+DY(J+1))) C D(J)=A3*TS(1,J)+A2*(TS(2,J)-TS(1, J ) ) 10 CONTINUE C A(IN)=-ALPHA/(DY(IN)*(DY(IN)+0.5*DY(IN-1))) C B(IN)=A3+ALPHA/(DY(IN)*(DY(IN)+0.5*DY(IN-1))) C C(IN)=0.0 C D(IN)=A3*TS(1,IN)+A2*(TS(2,IN)-TS(1,IN)) C C C CALL TRIDAG(1,IN,A,B,C,D,TPRIME) C DO 15 J=1,IN 15 T(1,J)=TPRIME(J) C DO 20 1=2,IJ I F ( I D ( I ) - 1 ) 4,5,5 5 CALL S0URCE(A(1),B(1),C(1),D(1),TMP) GO TO 6 C 4 A(1)=0.0 B(1)=A3+ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) C C(1)=-ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) C D(1)=A3*TS(I,1)+0.5*A2*(TS(I+1,1)+TS(1-1,1)-*2.0*TS(I,1)) 6 CONTINUE C DO 30 J=2,IK C A(J)=-2.0*ALPHA/(DY(J)*(DY(J)+DY(J-1))) 8 2 B(J)=A3+2.0*ALPHA/(DY(J)*(DY(,3)+DY(J+1 )))+ *2.0*ALPHA/(DY(J)*(DY(J)+DY(J-1))) C(J)=-2.0*ALPHA/(DY(J)*(DY(J)+DY(J+1))) D(J)=A3*TS(I,J)+(0.5*A2*((TS(I+1,J)+TS(I-1,J))-*2.0*TS(I,J))) 30 CONTINUE A(IN)=-ALPHA/(DY(IN)*(DY(IN)+0.5*DY(IN-1))) B(IN)=A3-A(IN) C(IN)=0.0 D(IN)=A3*TS(I,IN)+0.5*A2*(TS(I+1,IN)+TS(I-1, IN)-*2.0*TS(I,IN)) CALL TRIDAG(1,IN,A,B,C,D,TPRIME) DO 40 J=1 ,IN T(I,J)=TPRIME(J) 40 CONTINUE 20 CONTINUE IF(ID(IM)-1) 7,8,8 8 CALL SOURCE(A(1),B(1),C(1),D(1),TMP) GO TO 9 7 A(1)=0.0 B(1)=A3+ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) C(1)=-ALPHA/(DY(1)*(DY(1)+0.5*DY(2))) D(1)=A3*TS(IM,1)-A2*(TS(IM,1)-TS(IM-1,1)) 9 CONTINUE DO 50 J=2,IK A(J)=-2.0*ALPHA/(DY(J)*(DY(J)+DY(J-1))) C(J)=-2.0*ALPHA/(DY(J)*(DY(J)+DY(J+1))) B(J)=A3-A(J)-C(J) D(J)=A3*TS(IM,J)-A2*(TS(IM,J)-TS(IM-1,J)) 50 CONTINUE A(IN)=-ALPHA/(DY(IN)*(DY(IN)+0.5*DY(IN-1))) B(IN)=A3-A(IN) C(IN)=0.0 D(rN)=A3*TS(IM,IN)-A2*(TS(IM,IN)-TS(lM-1,IN)) CALL TRIDAG(1,IN,A,B,C,D,TPRIME) DO 60 J=1,IN 83 60 T(IM,J)=TPRIME(J) RETURN END C C C SUBROUTINE SOURCE(A,B,C,D,T) C A=0.0 B=1 .0 C=0.0 D=T RETURN END C C C SUBROUTINE PRI NT (T, T l ME, IN , IM, IP,PRNT,PRNTO,TLAST, ID) C C c DIMENSION T(100,100),ID(100) IP=0 IF(TIME.GT.TLAST) GO TO 1 C IF(TlME.LT.PRNT) GO TO 2 C C WRITE(8,100) TIME C 100 FORMAT(1H ,21X,'TIME=' ,F10 . 1 ) C DO 10 1 = 1 ,IM K=I IF(ID(K)-1) 10,20,20 20 WRITE(8,200) (T(I,J),J=1,8),K 200 FORMAT(8(F6.1,IX),13) 10 CONTINUE PRNT= PRNT+ PRNTO GO TO 2 1 IP=1 2 RETURN END C C c c C SUBROUTINE "TRIDAG" FROM 'APPLIED NUMERICAL METHODS' BY C CARNAHAN, LUTHER AND WILKES C C C SUBROUTINE FOR SOLVING A SYSTEM OF LINEAR SIMULTANEOUS C EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX. C THE EQUATIONS ARE NUMBERED IF THROUGH L AND THEIR C SUB-DIAGONAL, DIAGONAL AND SUPER-DIAGONAL COEFFICIENTS C ARE STORED IN THE ARRAYS A, B AND C. THE COMPUTED C SOLUTION VECTOR IS STORED IN THE ARRAY V. C C C C SUBROUTINE TRIDAG(IF,L,A,B,C,D,V) 84 DIMENSION A(100),B(100),C(100),D(100),V(100) DIMENSION BETA(101),GAMMA(101) C C c C#### COMPUTE INTERMEDIATE ARRAYS BETA AND GAMMA... C BETA(IF)=B(IF) GAMMA(IF)=D(IF)/BETA(IF) IFP1=IF+1 DO 10 I=IFP1,L BETA(I)=B(I)-A(I)*C(I-1)/BETA(I-1) GAMMA(I)=(D(I)-A(I)*GAMMA(1-1))/BETA(I) 10 CONTINUE C C*##« COMPUTE FINAL SOLUTION VECTOR V... C V(L)=GAMMA(L) LAST=L-IF DO 20 K=1,LAST I=L-K V(I)=GAMMA(I)-C(I)*V(I+1)/BETA(I) 20 CONTINUE RETURN END C C C C C C HEAT. BALANCE USED TO CALCULATE THE NEW BOUNDARY TEMPERATURE C AT EVERY TIME STEP C SUBROUTINE HEAT(XI,VO,AK,CP,RO,DX,DPH,TH,T,TMP,ID,G,F,IM,IN, *TBO,DY,DT,V,TlME,SE,SWG,AL,FR,CPW,DELT,ROW,HEB,BF) DIMENSION ID(100),T(100,100),DY(100) Q1=0.0 NO=IFIX(V*TIME/DX+0.51)+1 NS=IFIX((DPH+V*TIME)/DX+0.49)+1 NN=NO-1 IF(N0-1) 1,1,2 2 DO 15 K= 1,NN ID(K)=0 15 CONTINUE 1 DO 25 M=NO,NS ID(M)=1 25 CONTINUE NNS=NS+1 DO 35 J=NNS,IM ID(J)=0 35 CONTINUE BETA=((SE/SWG)/(1.-(SE/SWG))) DO 10 1=1,IM IF(ID(I)-1) 10,20,20 20 Q1=Q1-((AK*DX*TH)*((4*T(I,2)-T( I ,3)-3*TMP)/ *(DY(1)+DY(2)))) 10 CONTINUE WRITE(6,990) Q1,BETA 990 FORMAT(1H ,'HEAT INPUT=',1X,F15.2,/,F8.5,IX,/) CE=CP*RO*DPH*TH*G 85 TS=TMP-(((Q1*DT*HEB)/(CE))-((XI*VO*F*DT*.228)/(CE)) *+((AL*(1,/BETA)*V*DT)/(CP*DPH))) TMP=TS WRITE(6,999) TMP,V 999 FORMAT(1H ,'TMP=',1X,F7.2,/,F7.2) RETURN END C C C SUBROUTINE HEAT1(AK,DX,TH,T,DY,ID,Q1,IM,DPH,DT,IN,TBO,TMP) DIMENSION ID(100),T(100,100),DY(100) DO 4 I=1,IM DO 6 J=1,IN . T(I,J)=TBO 6 CONTINUE 4 CONTINUE Q1=0.0 TIME=DT NO=IFIX(0.05*TIME/DX+0.51)+1 NS=IFIX((DPH+0.05*TIME)/DX+0.49)+1 NN=NO-1 IF(NO-I) 1,1,2 2 DO 10 K=1,NN ID(K)=0 10 CONTINUE 1 DO 20 M=NO,NS ID(M)=1 20 CONTINUE NNS=NS+1 DO 30 J=NNS,IM iD(J)=0 30 CONTINUE DO 5 1=1,IM I F ( I D ( I ) - 1 ) 5,50,50 50 Q1=Q1-((AK*DX*TH)*((4*T(I,2)-T(I,3)-3*TMP)/ *(DY(1)+DY(2)))) 5 CONTINUE WRITE(6,897) Q1 897 FORMAT(1H ,IX,'HEAT INPUT=',F15.1) RETURN END C C THERMAL STRESS CALCULATION PERFORMED FOR EACH NODAL C TEMPERATURE C SUBROUTINE STRESS(T,TT,TTT,IT,IM,IN,ALFA,H) DIMENSION T(500,20),TT(500),TTT(500) DO 10 1=1,IT READ(5,100) (T(IT,IN),IN=1,8) 100 FORMAT(8(F6.1,1X)) • 10 CONTINUE C CALCULATION OF E(YOUNG*S MODULUS) AS A FUNCTION OF TEMPERATURE C DO 11 L=1,IM DO 12 J=1,8 DO 14 1=1,IT TT(I ) = T ( I , J ) TTT(I)=TT(I)*L I F ( T T ( I ) .LT. 1000.) GO TO 1 86 E=0.000012 1 I F ( T T ( I ) .GE. 1000. .AND. TT(I) .LE. 1400.) GOTO 2 E=(2000.-(1.875*(TT(I)-l000.)*100000.)) 2 IF(TT(I) .GE. 1400. .AND. TT(I) .LE. 1475.) GO TO 3 E=((1250.*(l475.-TT(l))/75.)*10000.) 3 IF(TT(I) .GT. 1475.) GO TO 14 E=0.0 AREA1=QINT4P(I tTT(I),465,1,465) AREA2=QINT4P(I,TTT(I),465,1,465) SIGMA=(-ALFA*E*TT(I)+((0.5*H)*(ALFA*E*AREA1)) *+(((1.5*L)/(H**3.))*(ALFA*E*AREA2))) WRITE(7,200) SIGMA 200 FORMAT(1H ,2X,'THERMAL STRESS=',FB.3) 14 CONTINUE 12 CONTINUE 11 CONTINUE RETURN END C C C C C C LIST OF SYMBOLS USED IN THE MODEL C C C C C C TMP = Melti n g p o i n t temperature(Deg. C) C C TBO = Parent metal i n i t i a l temperature(Deg. C) C C DT = Time step(s) C C CP = S p e c i f i c heat of steeKcal/g.deg.C) C C RO = Density of steel(g/cm**3) C C AK = Thermal c o n d u c t i v i t y of steel(cal/cm.s.deg C C DX = Space increment i n X-direction(cm) C C DY = Space increment i n Y-direction(cm) C C V = Welding v e l o c i t y ( c m / s ) C C DPH = Slag + l i q u i d metal depth(cm) C • C PRNT = P r i n t c y c l e ( s ) C C TLAST = End of c a l c u l a t i o n ( s ) C C IM = # of d i v i s i o n s in the X - d i r e c t i o n C C IN = # of d i v i s i o n s i n the Y - d i r e c t i o n C C T = Temperature(deg. C) C C XI = Current(A) 8 7 c C VO = Vo l t a g e ( V ) C C TH = Thickness(cm) C C G = Weld gap(cm) C C F = E f f i c i e n c y f a c t o r C C SE = E l e c t r o d e area(cm**2) C C SWG = Weld gap area(cm**2) C C AL = Latent h e a t ( c a l / g ) C C FR = F i l l r a t i o C C ALFA = C o e f f i c i e n t of expansion(/deg. C) C C H = Weld height(cm) C C ID = Heat source c o n t a c t index 88 A P P E N D I X C - E F F I C I E N C Y F A C T O R A N D H E A T S I N K C A L C U L A T I O N S P a t o n 3 5 r e p o r t s t h a t a b o u t 5 8 . 6 % o f t h e a v a i l a b l e h e a t g o e s i n t o t h e b l o c k s w h e n E l e c t r o s l a g W e l d i n g ( w i r e e l e c t r o d e ) . D u e t o t h e d i f f e r e n t t h e r m a l c h a r a c t e r i s t i c s a l r e a d y d i s c u s s e d , f o r E l e c t r o s l a g J o i n i n g t h a t n u m b e r w o u l d n o t a p p l y . T h e r e f o r e , new c a l c u l a t i o n s h a d t o b e p e r f o r m e d b a s e d o n some m e a s u r e m e n t s : T h e c o o l i n g s h o e w a t e r f l o w r a t e w a s m e a s u r e d a n d f o u n d t o b e 3 6 9 8 c m 3 / s . The e l e c t r o d e m e l t r a t e c a n b e c a l c u l a t e d a s f o l l o w s : M e l t R a t e = E l e c t r o d e F e e d R a t e x A r e a x D e n s i t y T h e e l e c t r o d e f e e d r a t e c a n b e a s c e r t a i n e d u s i n g t h e f o l l o w -i n g e x p r e s s i o n g i v e n b y F r o s t e t a l . 1 * 1 : w h e r e : E F R = e l e c t r o d e f e e d r a t e F R = f i l l r a t i o ( e l e c t r o d e a r e a / w e l d a r e a ) V = w e l d i n g v e l o c i t y F o r t h e m a t e r i a l d i m e n s i o n s u s e d i n m o s t e x p e r i m e n t s - 1 . 6 7 3 5 F o r a n e x p e r i m e n t a l w e l d i n g v e l o c i t y o f 0 . 0 2 3 c m / s , t h e EFR was f o u n d t o b e : E F R = 1 . 6 7 3 5 x 0 . 0 2 3 = 0 . 0 3 8 5 c m / s T h e r e f o r e , M e l t R a t e = 0 . 0 3 8 5 c m / s x 5 7 . 9 1 2 c m 2 x 7 . 8 6 g / c m 3 = 1 7 . 5 2 g / s I f a p p r o x i m a t e l y 4 0 0 KWH a r e n e e d e d t o m e l t 1 0 0 0 k g o r 1 x 1 0 ^ g o f s t e e l , t h e n t h e p o w e r f o r m e l t i n g w o u l d b e : T>_ 1 7 . 5 2 g / s x 4 0 0 KWH x 3 6 0 0 s _ . . . . 1 x 10o g 89 The h e a t f l u x p e r m o u l d w o u l d t h e n b e : 1 ^ = F l o w r a t e x C p y x AT x p y x ^ Q Q Q 8 = 8 5 . 1 6 K W T h e r e f o r e , 2 5 . 2 3 = 1 6 . 1 % = 5 4 . 3 % \ " 3 3 x 5 0 0 0 x . 9 5 8 5 . 1 6 1 5 6 . 7 5 H R = - 1 % • ^ l o c k " 1 0 0 " Z ( H M + H W + V " b l o c k * 1 5 % A n d , t h e r e f o r e , o n l y a b o u t 15 % o f t h e a v a i l a b l e e n e r g y f l o w s t h r o u g h t h e b l o c k s a n d i s a c c u m u l a t e d t h e r e . T h i s i s t h e f a c t o r u s e d i n t h e m o d e l w h e n c a l c u l a t i n g t h e h e a t f l o w . ***************** THE BLOCK AS A HEAT S I N K The a m o u n t o f h e a t l o s t t h r o u g h t h e b l o c k c o l d f a c e i s v e r y s m a l l w h e n c o m p a r e d w i t h t h e h e a t a v a i l a b l e f r o m t h e e l e c t r o d e t h a t i s e n t e r i n g t h e b l o c k v i a t h e o p p o s i t e f a c e . I n o r d e r t o s e e how t h a t i s e f f e c t e d a p l o t o f t e m p e r a t u r e v e r s u s d i s t a n c e f r o m t h e h o t f a c e h a s b e e n g e n e r a t e d a n d i l l u s t r a t e d i n F i g . 26 T h e p o i n t a t w h i c h t h e t e m p e r a t u r e d r o p s t o r o o m t e m p e r a t u r e h a s b e e n f o u n d t o b e 61 cm away f r o m t h e h o t f a c e . ( A p p r o x i m a t e l y 2 f e e t ) 9 0 A P P E N D I X D - R E S I D U A L S T R E S S E V A L U A T I O N . (A + B c o s 2 g ) e a - (A - B c o s 2 g ) e c  0 1 4 A B c o s 23 (A + B c o s 20) e c - (A - B c o s 20) e a ° 2 4 A B c o s 20 ca - 2 e b + e c t a n 20 = -e a - e c P o s i t i o n 1 - P a r e n t m e t a l D e p t h = 120 t h o u e a = - 3 5 u e eb = - 6 2 u e e c = - 1 4 3 y e 2 - 5 5 4 A - - 1 . 3 5 * 1 0 " * 4 B = - 3 . 5 5 x 10 ( A f t e r R e d n e r ) T h e r e f o r e , A = - 3 . 3 7 5 x 1 0 ~ 9 a n d B = - 8 . 8 6 8 x 1 0 ~ 9 a i = + 9 7 8 1 p s i a 2 = +16589 p s i P o s i t i o n 2 - H e a t a f f e c t e d z o n e D e p t h = 120 t h o u e a = +67ue eb = +51ue e c = - 2 2 5 u e 0 . 2 0 4 5 0 , . i n/o i n - 8 r 2 = Q Q 7 1 = 2 . 8 7 4 A = - 1 . 1 1 4 2 x 10 4 B = - 2 . 9 9 2 0 x 1 0 ~ 8 T h e r e f o r e , A = - 2 , 7 8 5 5 x 1 0 ~ 9 a n d B = - 7 . 4 8 x 1 0 ~ 9 01 = + 1113 p s i 0 2 = + 2 7 2 4 8 p s i 91 P o s i t i o n 3 - W e l d m e t a l D e p t h = 1 2 0 t h o u £ a = - 1 5 2 y e eb = - 1 5 7 u e e c = - 1 0 7 y e " n'lfS5 = 2 ' 8 4 A = - 1 . 1 ^ 2 x 1 0 " 8 3 U . u o / 4 B = - 3 . 0 x 1 0 ~ 8 T h e r e f o r e , A - - 2 . 7 8 5 4 x 1 0 ~ 9 a n d B = - 7 . 5 x 1 0 ~ 9 oi = + 2 5 6 1 5 p s i a 2 = + 2 0 8 7 7 p s i T h e r e s i d u a l s t r e s s m e a s u r e m e n t w a s c a r r i e d o u t u s i n g t h e B l i n d 45 H o l e D r i l l i n g t e c h n i q u e w h i c h i s a s e m i - d e s t r u c t i v e m e t h o d w h e r e b y a s m a l l h o l e , 3 . 1 7 5 m m ( l / 8 " ) i n d i a m e t e r i s d r i l l e d t o a d e p t h a p p r o x i m a t e l y e q u a l t o i t s d i a m e t e r . T h e r e l a x e d s t r a i n s a r e t h e n m e a s u r e d a r o u n d t h e h o l e . 02' i s t h e l o n g i t u d i n a l s t r e s s r e m a i n i n g i n t h e w e l d e d a s s e m b l y a n d a j i s a t a 90 d e g r e e a n g l e t o 02- T h e y w e r e m e a s u r e d i n t h e p a r e n t m a t e r i a l , h e a t a f f e c t e d z o n e a n d w e l d m e t a l . 

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