A THERMO - METALLURGICAL MODEL PREDICTING THE STRENGTH OF WELDED JOINTS USING THE FINITE ELEMENT METHOD Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY GARY W. KRUTZ 1976 w‘fi‘ This is to certify that the thesis entitled ‘ fl 7H£IMO~METfluuzémAL WMSZ, fflED'CT/A/é THE STKEIVchI OF I (MEI-060 To/m (JIM/c TH E Fla/r!" ELWr presented by 2'” 5771/00 é’q’é/V fit a 7’1 has been accepted towards fulfillment of the requirements for f4 '4: degree in 464.6%: flaws; Date L’ m° '- 7“ 0-7639 s .. .‘U‘TAT TONI . , Isl N'EMT lllzl’i‘lvl‘llll "! LA. ‘”3 FINITE ELEMENT METHOD ABSTRACT A THERMO-METALLURGICAL MODEL PREDICTING THE STRENGTH OF WELDED JOINTS USING THE BY Gary W. Krutz The object of this work was to use finite element analysis in predicting the thermo-history of a welded joint and provide design engineers with a reliable method of optimizing the metallurgical properties of a welded joint. A finite element computer model was developed and used to determine the time temperature history of a butt weld as related to the heat input. Variables affecting this cooling time include metallurgical constituents, radius of the heat flux, velocity of the arc, and thermal conductivity of molten steel. Thermal conductivity and specific heat were considered as a function of tempera- ture. The latent heat of fusion, and radiation and convec- tion heat transfer were included in the model. A Gaussian distributed heat flux was assumed. Time-temperature computer results were verified using experimental thermalcouple data available in the liter- \ ature. Computer determined weld pool size correlated with experimental values. Joint metallurgy was Optimized using a critical cooling time. Past cooling in the heat affected zone creates hard but brittle metallurgical properties lacking in bend angle strength. Longer cooling times reduce the hardness and increase the bend angle of a weld joint. For this low carbon steel model a definite cooling time, called critical, exists between fast and slow cooling. epartment airman A THERMO-METALLURGICAL MODEL PREDICTING THE STRENGTH OF WELDED JOINTS USING THE FINITE ELEMENT METHOD By .L"\ . \\ ‘0‘ Gary W. Krutz A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1976 ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. Larry J. Segerlind, whose guidance and helpfulness were indispensable during this study. In addition, thanks are in order for Sandy Clark for typing this manuscript. A special dedication of this work is intended for my wife, Barbara, for her continual encouragement throughout this program. ii TABLE OF CONTENTS List of Figures I. II. III. IV. Introduction Literature Review 2. 2 2. 2 2 U1-hMN 1 Past History Time-Temperature Relationships Metallurgical Aspects Residual Stresses and Deformation Finite Element Applications Theoretical Background GUI-hm .1 .2' \l Objective . . . . .- Finite Element Formulation of the Welding Process Triangular Elements . Temperature Dependent Properties Radiation Boundary Condition Specific heat as a function of temperature . . . Latent heats Heat Flux Model . . Optimizing Weld Joint Strength . Finite Element Program 4. 4. 4. 4. 1 2 3 Iterative Procedure Input Requirements and Output Thermal Conductivity and Specific Heat as Program Variables Additions to the Force Vector iii Page \J-b use» hi < 11 14 16 16 20 25 31 36 38 42 46 53 61 61 62 64 69 V. Verification and Sensitivity Analysis 5.1 Verification with Christensen's and Hess' Work 5.2 Sensitivity of Some Variables 5.2.1 Radius of the Welding Arc . 5.2.2 Thermal Conductivity in the Weld Pool . . . 5.2.3 Convection Coefficient and Emissivity Value 5.2.4 Grid Coarseness and Time Step VI. Application to Welding Design . 6.1 Joint Strength . . . 6.2 Effect of Arc Velocity on Cooling Time . . . . . . . . VII. Summary and Conclusions VIII. Recommendations for Further Study . References iv 88 93 95 97 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10. LIST OF FIGURES Weld Heat Affected Zones Temperature Dependent Properties of Low-Carbon Steel The Two-Dimensional Simplex Element (Triangular Element) Triangular Element Weld Joint Model Triangular Element and associated latent heat surfaces . . Heat Flux Distribution . Model of constant q over an element side . Heat Flux Model Example Continuous Cooling Trans- formation Diagram for Steel Ref (12) Page 1096 . . . . . CCT Diagram showing critical pts, critical cooling curves, and critical cooling times . Cooling time versus hardness, bend andle and absorbed energy Computer Program Flow Diagram Storage in "A" Column Vector Verification with Hess's work (time- temperature curve) Element Model of Hess' Work 27 32 43 48 50 51 56 57 59 65 66 72 73 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Cross Section Temperature Verifi- cation Verifying Model Weld Pool and HAZ Sizes . . . . . . . Sensitivity to Welding Arc Radius Sensitivity to Weld Pool Thermal Conductivity . . Sensitivity of Convection Coefficient Sensitivity of (e) Emissivity in Molten Area . . . . Fine Grid Model . Coarse Versus Fine Grid . . . . Comparison of Two Time Steps Minimum Cooling Time Determination Arc Velocity Effects on a Node Cooling . . . . . . Effect of Velocity on C% vi 75 78 80 81 82 83 85 86 89 90 92 I. Introduction Welding is probably the most popular manufacturing process used for joining metals in structural applica- tions. It is a common method used in fabricating farm equipment and many times the size of the weld area will govern the size of the components. Because of its wide and flexible use, weld strength has been the concern of designers and researchers who have studied it both analytically and empirically in an effort to maximize the joint strength. This maximization when achieved would reduce the weight of machines, structures, components and save material cost and processing energy. The goal of this work was to develop a general numerical model of the welding process which is capable of predicting the strength of a weld joint via the metal- lurgical viewpoint. Since many variables enter the pro- cess and this is the first attempt at a particular metallurgical model the study will be limited to low carbon steel. The properties of low carbon steel are not constant but vary from piece to piece as a result of process variations (15). The problems of distortion, residual stresses, and reduced strength of the material in the joint area are 1 2 results of the large amount of heat energy applied at the weld site over a short period of time. This heat gradient at elevated temperatures causes dynamic changes in the weld metallurgy and affects the joint toughness. The finite element method was used to predict the transient thermo-history of a welded composite. The joint strength was predicted from the temperature history. It is hoped that this method will provide prac- ticing engineers with a technique that will be widely used in weld-joint design. II. Literature Review 2.1 Past History The problem of fast cooling rates and high tempera- tures that change the metallurgy and distort the joint have been concerns of the engineers for years. D. Rosen- thal (1,2,3) deveIOped the mathematical theory of heat distribution in welding during the late 1930's. By using the conventional heat transfer differential equa- tion for a Quasi-Stationary State Rosenthal developed equations for point, linear and plane sources of heat. These equations closely approximated actual tests and presented a method of predicting time and rate of cool- ing with some accuracy for a wide variety of steel thicknesses, ranges of temperature and welding conditions. azr 32T 32T _ dT Quasi-Stationary 3X2 + 3Y2 + 322 - DC .5? State eqn. Further work on time-temperature relationships was needed because Rosenthal's work was primarily applicable to butt joints and assumed many constant inputs and boundaries. 4 2.2 Time-Temperature Relationships Adams (16) took the basic equations of Rosenthal and derived relationships to provide cooling rates and peak-temperatures as a function of geometry, thermal and welding variables. This thermal history is needed to make the metallurgical interpretations in the weld heat affected zone. Using a point heat source Adams developed a three-dimensional conduction model. T1 - T = 8V EXPLm -\/m2 + n2] 0 4nKa -\/;7—:j;; T1 = steady state temperature at point (X, R) To = initial temperature of the plate X = distance behind source parallel to movement on axis centerline a = thermal diffusivity (5§—) P = rate of heat flow from source thermal conductivity of plate <7<.D u = velocity of source 9 = density of plate Cp = specific heat of plate m = VX/ZG n = VR/Za R a radial distance (cylindrical coordinates) For thin plates he predicts heat flow with a two- dimensional steady state temperature distribution given as: t = thickness of plate K0(P) = modified Bessel function of the second kind and zero order From these equations he expressed peak-temperature as a function of distance from the fusion zone. Paley (17) g£_al.expanded on Adams' work and devel- oped a numerical relationship to provide peak-temperature distributions for submerged-arc welding. They deter- mined that peak temperature, not heating or cooling rates, was the important variable in determining the metallurgy of the heat affected zone (HAZ). I Paley (18) recently used a numerical method (finite differences) to determine the thermal history of welds. This computer program solved the heat flow equation and showed good correlation with experiments in determining the macrostructure of welds. Graphic displays of both the maximum temperature and the moving temperature field in horizontal and vertical planes were some of the out- put. Computerized plotting of thermal cycles allowed comparison of metallurgy with this numerical technique. He also did an experiment to show that the surface condition (emissivity) of the steel plate had no effect on the shape of isotherms. Hess (19) did an extensive experimental procedure in determining cooling rates and provides an enormous 6 amount of data. From his studies he concluded that arc travel speed, are voltage, and arc current can be con- veniently grouped into a single factor called energy in- put per unit of weld (joules per inch). Then for low carbon steels the only factors affecting weld cooling rates are energy input, plate thickness, plate temper- ature, and joint geometry. A study of cooling curves was done by Paschkis (20) by means of electrical analogy. He concluded that the physical properties of steel varied with temperature and more information in this area is needed. Since his study Goldsmith (11) has provided this information needed by Paschkis. Christensen (49) developed charts to determine weld and HAZ width using basic theory. Roberts (50) studied the relationship between plate size and temper- ature distribution. Most of these analytical theories were summa- rized by Meyers et a1.(53) where he states the assumptions.which make their usefulness limited. The assumptions are: l. ~The material is solid at all times and at all .temperatures, no phase changes occur, and is isotropic and homogenous. 2. The thermal conductivity, density, and specific heat are constant with temperature. 7 3. There are no heat losses at the boundaries, i.e., the piece is insulated. 4. The piece is infinite except in the directions specifically noted. 5. Conditions are steady with time, i.e., in the middle of a long weld, heat input, travel speeds, etc. are steady. 6. The source of heat is concentrated in a zero- volume point, line or area. 7. There is no Joule (12R electric) heating. Because of these limitations researchers of the 1970's have gone to numerical methods which can handle the above seven items. 2,3 Metallurgical aspects Mahla (4) e£_31: noted cracking in the base metal was caused by metallurgical changes introduced during the welding process. This cracking results when austen- ite steel is cooled too rapidly forming brittle martens- ite. Rates of transformation of austenite can be pre- dicted from Continuous Cooling Transformation Curves (CCT) and the resultant metallurgical properties deter~ mined. Therefore the range of properties for a given weld can be predicted if four influencing factors are known: (1) Rate of heating 8 (2) Maximum temperature and time at that temperature (3) Rate of cooling (4) Composition of the base metal Mahla's experiments with SAE 1020 steel plates of 1.9 cm thickness or less were entirely free of martensite. He also determined that as the plate size increased from .6 cm to 1.9 cm the cooling rate increased from 10°K per second to 29°K per second respectively. These rates were measured in a range of temperature of 767-870°K. Dolby and Sanders (5) investigated the metallurgi- cal factors controlling the toughness of the heat affected zone for a range of steels. Dolby's results were similar to Mahla et al. A specimen with the ab- sence of martensite showed the embrittlement to be in the HAZ region where peak temperatures were close to the Ac; temperature. With martensite present the regions adjacent to the fusion boundary were the weakest. Strain concentrators existing in the parent metal and HA2 were the only significant variable on toughness for sub- critical temperature regions of the HAZ. In their tests refining the HAZ after welding did increase the joint toughness. Lancaster (6) generalized that the larger the weld pool the coarser the grain where larger grains are associated with poor impact properties. With special additives suitable grain refinement can be achieved. 9 Stout and Doty (7) define the HA2 in steels as occurring in four distinct areas as shown in Figure 1. Area A is heated slightly above the transformation range causing some of the pearlite to go to austenite. Upon cooling austenite transforms to fine pearlite. The ferrite is not disturbed in this area. In area B during the welding process pearlite goes to austenite and some ferrite is dissolved. Upon cooling pearlite forms in a scattered grain sense. Area C which is closer to the weld pool has complete transformation to austenite. This area is moderately fast cooled limiting ferrite and forming fine pearlite. The area next to the fusion line (area D) has coarser grains than area C being at a higher temperature. Stout and Doty include micrographs that depict the HA2 area and its metallurgical structure in their book. Henry (8) gives a very good overview of what happens metallurgically during the heating and cooling of the areas around the weld metal. The ll75°K temper- ature is the beginning point of grain growth. Henry points out that these grown grain structures will not break up into smaller grains unless work is applied or unless the grains undergo a transition. This transition is the cooling below ll75°K where gamma grains form alpha iron resulting in larger alpha grains transform- ing from larger gamma iron. The structure of the weld metal is discussed by 10 Figure l. Weld Heat Affected Zones. 11 Linnert (9). The four common features observed are: l. Macrostructure columnar growth which follows direction of thermal flow from the weld area. 2. Fine grain size. 3. Substructure contains a large number of dislocations. 4. A Widmanstfitten pattern develops. The properties of the weld metal are dependent on their metallurgical composition which varies greatly with different processes, base metals, and etc. The orienta- tion of grains tends to lower the: ductility and tough- ness. Inagaki (10) g£_§l. developed monographs to deter- mine cooling time given heat input and plate thickness. Then from CCT curves they concluded existence of a minimum critical cooling time (Cf) which the weld HA2 should cool over to provide the best ferrite-pearlite composition in alloy steels. Using the CCT diagrams they approximated the microstructure hardness at any point in the HA2. In researching metallurgy of welding steels ma- terial properties are needed to be known and are avail- able in various sources (11, 12, 23, 14, 15). 2.4 Residual Stresses and Deformation Faukel's book (26) exemplified the nature and 12 significance of residual stresses and thermal deforma- tion. Residual stresses are induced whenever a material is non-uniformly plastically deformed. In such an operation the permanent strain produced prevents the elastic component from completely recovering. These deformations may be induced by plastic bending, shot peening, grinding, welding, and other operations. Distortion is a problem in welding and Iwamura (21) studied the thermal stresses of flame forming. A two- dimensional model was used to determine the stresses via a finite element program. The authors concluded that the model could be expanded to the three—dimensional case (moving point source). Rybicki §£_al, (22) developed a method of non- linear stress analysis when the physical properties of a material vary with such items as temperature. He (23) later used the finite element method to evaluate de- formation due to welding. The Welding Institute has conducted some research relative to residual stresses and Dawes (24) explains how and what kind of residual stresses result from welding. In arc welding, a pool of molten metal is surrounded by a relatively cold solid metal. The peak temperature at the fusion zone is approximately 177S°K. Three mm away this temperature will have dropped to ll75°K 13 and approximately 100 mm away it will be down to around 575°K. Because the surrounding colder material will restrain local expansion and contraction and the heated portion will go into compression elastically then plastically. Cooling will accompany elastic recovery leaving yield magnitude residual tension stresses in the weld metal. The surrounding material will be balanced by compressive stresses. Dawes states that these resid- ual stresses and strains will not have a significant effect on the ultimate tensile strength of a low carbon steel welded joint because they are highly localized. Schofield (25) reviewed the many techniques to measure residual stresses. Masubuchi et al. (28, 29, 48) have a two-dimensional finite element program that calculates thermal stress and distortion of bead-on-plate and butt welds. In their program they include material property temperature dependence and yield criterion and have the ultimate goal of minimizing joint distortion. The control of residual stresses can be accomplished by many means. Metallurgical annealing or deformation of the metal are a couple of examples. Recently many researchers (21, 22, 23, 24, 25, 28, S7, and 61) have become interested in this problem and are making worth- while contributions. Comparison of numerical results with actual tests in their research usually fails be- cause of the extreme difficulty in measuring subsurface l4 residual stresses. Also, restraining weldments, ambient temperature variations, and subsequent loading of the structure can increase or completely decrease these stresses. In the future such problems will be resolved and a more representative residual stress weld model developed. The optimal model will come from the joining of the elastic-plastic (residual stress) models and the thermo-metallurgical model. 2.5 Finite Element Applications The finite-element method is a mathematical modeling technique from which the solution is obtained numerically (computer applications). The superiority of this method is that it can increase the quality and quantity of results, is fast, accurate and economical as compared to other techniques. It resulted from the need to analyze complicated structures and has spread to other areas of engineering and the sciences. Other advantages that may be incorporated in its application are: (l) the handling of complicated geometries, loading and support characteristics before and during numerical solving. (2) the combining of various structures and shapes. (3) the changing of material characteristics, cross section characteristics and wall thickness from element to element. 15 Whiteman (30) has compiled a book of Finite Element publications. References (31-47) show that an extensive amount of research has been done in the area of thermal stresses. My goal is to apply this method and deter- mine weld strength. Major hurdles include the varying of material properties with temperature and the deter- mination of temperature as a function of time. The basis of time-dependent finite-element analysis has been pro- posed by Kohler (58), Zienkiewicz (44, 51) and Singh (52). The thermo-mechanical model has been worked on by researchers such as Friedman (55), Hibbit (56) and Westby (57). Westby gives an excellent detailed matrix analysis of a residual stress model varying yield stress, Young's Modulus and other parameters with temperature. Lubkin (27, 54) developed a finite-element shell model of a welded joint in his study of car frame vibrations. III. Theoretical Background 3.1 Introduction In this section, the finite element model used for the joint strength analysis is developed from the fundamental quasi-stationary state heat transfer equa- tion. The approach adopted is of a more detailed type, so the assumptions being made may be critically examined during the development. The model of the welding process was done in a two part analysis: first, the temperature was determined as a function of location and time; subsequently, all points in the heat-affected zone are checked against the critical cooling time from which revisions to the process are recommended or final data acquired. There are several boundary conditions and physical phenomena which complicate this non-linear problem. These are discussed here pointing out which ones are essential to the model. 1. The materials are subjected to a wide range of temperatures. All properties were considered as functions of temperature with the exception of density. Figure 2 depicts these 16 17 02' SPECIFIC. HEAT .¥ O.I— / CD 3 oo- 5 4i. .l75- z THERMAL “5.150- cowouc'rlv ITY U M25- 5") .IOO- “'34 3 & 5.075- 0 9 16'00 ' zobo TEMPERATUREC°K) Figure 2. Temperature Dependent Properties of Low- Carbon Steel 18 properties for Low-Carbon Steel over the temper- ature range in question. The important properties affected are thermal conductivity and specific heat. It was assumed that the thermal history (i.e. thermal expansion) has little or no effect on the shape of the elements. In actuality the molten and plastic zones deform due to shrinkage and restraints. 2. The phase changes are another important phenom~ enon affecting this work. The heat of transforma- tion and latent heat of fusion represent a storage of heat during the welding process and subsequently greatly affect the cooling time and microstructure metallurgy. This heat is large enough that it may create favorable recrystallized fine-grained ferrite/pearlite or unfavorable brittle structures. An implicit heat absorption feature to simulate the phase change over a temperature range was included. Most models assumed that the phase change takes place at a specific temperature which is not the case for alloys. 3. Shrinkage distortion and residual stresses are existent and noticeable in the work-piece, but con- siderable analytical progress must be achieved to pro- vide accurate and meaningful results. All current methods are unable to predict cracking, a common occurrence in many types of welds, and therefore 19 this model has taken another approach. 4. Input heat flux has been passed over lightly by some researchers who have used voltage times current multiplied by an efficiency factor. A more complete description of the heat flux as a function of position and time will be presented. The addition of filler metals does affect heat input, but will not be added to the finite-element grid because of the difficulty in element modeling. 5. Boundary conditions must account for both radiation (quartic Stefan-Boltzman) and linear Newton convective cooling. No forced convection was assumed and the effects of gas diffusion in the weld pool were not considered. Also, slag formation and its effects were assumed small and negligible because the width of slag is dependent on the process or rod coating and its formation at lower temperatures should have a reduced effect on convection and element conductiVity. 6. Weld pool size was determined by the location of the solidus line. The model was assumed two- dimensional and because of symmetry only one half was constructed and analyzed. The second part is the determination of the critical cooling time factor. Inagaki (10) did a 20 large amount of experimental work determining a relation- ship between energy absorbed during a weld joint bend test and the cooling time of the weld joint. It was concluded that the energy absorbed decreased as the cool- ing time decreased beyond a certain value. Therefore, it was recommended that a minimum critical cooling time (Cf) should be preserved to give maximum joint strength. This Ck was related empirically to the equivalent carbon content of alloy steels thus providing a guideline for a wide range of materials. Since hardness and tensile strength are related, the Critical cooling time becomes an optimum for joint strength seeing multiple loading conditions. The finite element model will be geared to maximize joint strength using the critical cooling time as the optimization criteria. 3.2 Finite Element Formulation of the WeldingAProcess The predominant mechanism controlling temperature distribution in the welded joint is thermal conduction. Heat conduction in a solid body is expressed three- dimensionally in the following general form: 3T 3 3T cp'gfg'afLK‘a—X')+ Yg—EUK) '7%@+ Q where T = temperature (K°) K = thermal conductivity (cal/cm sec °K) 21 co = volumetric specific heat (cal/cm3°K) Q = exchange of heat [convection, radiation, latent heats, and heat flux] (cal/cm’sec) When analyzing a two-dimensional body this three- dimensional equation can be modified by deleting the unwanted coordinate direction. In the finite element method for field problems (heat conduction) a Ritz approximation in variational form replaces the differential equation. The exact solution is then obtained by minimizing this Ritz variational form which also satisfies the boundary con- ditions. This minimization is obtained by differentiat- ing the variational function and setting it equal to zero. To check that the derivative of the functional equal to zero is a minimum the second derivative is taken and its value must be positive for all ranges of the functions to rule out maximums and points of inflections. This variational process is covered in various publications (62, 63) which provide greater detailed background. The heat conduction functional for an individual finite element has been determined by mathematicians and is expressed as: re = I 1 {T V 2 e + Ie q[Ne]{T hTmlNe] where a I Q + Q N I“ IF. U H1 0 H *‘3 A \l \1 U1 constant for T Z 77S°K Again by taking the derivative and setting it equal to zero a minimum for this governing portion of the variational form is arrived at. d{T} N N dV (1 6[ IT I I ’3?" d1 T 0 = - [N] dV + A Q “A HTTT 3.. da dLI} A [N]{T}[N]T dV 40 The derivative of GA for temperatures less than 775°K is: daA a2 TPF'T‘EE' Going back to the minimized variation form the first term (including only Q) remains the same as derived earlier. The other two terms then become 1 d{T} “2 dfT} \flaAm] [dev 1? dV + T ‘f,[N]{T}[N] 1t— dV Using area coordinates for triangular elements and noting that: [N] - [L1, L2, L3] the evaluation of the a2 integral portion of the final two terms is T. 1 “2 T£[L19 L2, L3] Tj [L1, L2, L3] dA 431—}. = Tk Ti 2 1 1 £12 A Tj 1 2 1 Tk 1 1 2 where [L1, L2, L3]{T}[L1, L2, L3] equalled 41 [leTi + L1L2Tj + LlLaTk, LILZTi + L221]. + LzLaTk, L1L2Ti + LzLaTJ. + Lasz] The GA term was presented earlier and when combined with the a2 term, specific heat adds only to the conductance matrix [C] VGAA : 2 I + WazAtriT i 2 I Eli—U— dt 1 1 2 1 1 Z and for T < 775°K 1 [C]=aA+asz [152:] ”N or for T Z 775°K I-Z l 1 0‘13 l l 2 I_. Note that the additional term containing {T}T in the temperature range<775°K becomes unsymmetric. 42 3.7 Latent heats Heat conduction in welding must include the change of phase. The dominant latent heat is that of fusion. The latent heat transformation has a value for steel of approximately 1/10 that of fusion and is considered minor in importance. Fusion usually occurs in temper- ature range for alloy steels from about 1700°K to 1755°K and therefore must be considered as a factor of heat in- put or withdrawal over that range. During the welding process,as heat is input from the arc,there exists a liquid front surface, a solid front surface, and the in-between transition zone where latent heat is being absorbed (see Figure 5). The opposite phenomenon takes place during the solidification process. Heat is given off in the trans- ition zone as the metal is cooled which will increase the cooling time. The solid surface is represented by a 1700°K isotherm for steels while the liquid surface consists of the 1755°K isotherm. A method to incorporate latent heat was developed for the triangular ele- ment. The possibility of both the liquid and solid front intersecting a triangular element exists. This would create a complicated internal boundary problem to be solved for latent heat effects. By decreasing ele- ment sizes in this latent affected zone the number of elements affected by this internal boundary problem can be minimized. Then only a solid front or liquid front 43 !.._A LIQUID FRONT TRANSITION ZONE Figure 5. Triangular Element and associated latent heat surfaces. 44 would be intersecting a single element. This also creates a complex computational problem along with all the increased computer calculations used with temper- ature dependent properties. Therefore, a compromising assumption was made to approximate the true latent heat effects. The latent heat generated or given off within an element will be allotted equally to all three nodes of the triangle. To determine if an element is affected by latent heat an average temperature for the element is calculated: If TZV falls within the latent heat range of l700°K to e 1755°K this particular element will then have a term added or subtracted to the force matrix. The sign of this term is either negative or positive corresponding to heat absorption and heat generation, respectively and the computer program was coded to a sign convention which is regulated by increasing or decreasing average element temperatures. The portion of the functional affected is: I={,-(Q-%—})Tdv Taking the derivative with respect to {T} yields: 4S 3%, = - \[INTTQ dV Since Q was assumed constant within the element this term becomes: T Qf [N1 dv V now Q(watts) can be replaced by %% cm3 where: p = density (for steel = 7.87 g/cm ) L = latent heat (for steel = 65.5 cal/gram) At = length of a time step (seconds) . . pL . watts g1v1ng —— units of ————— At cm3 The portion added to the ferce vector {f} is: L T g—t {,[N] dV Which is: pLV 3At After the integral is evaluated. 46 Assuming a constant thickness this term is: 1 pLA 1 3At l Friedman (60) has a direct iteration method for phase change in finite element programs. He includes the pL term as an addition to the pc term and controls the implementation of the latent zone using a large number of coefficients. 3.8 Heat Flux Model Friedman (55) states more reliable input data is needed in the finite element analysis to model the heat flux portion. This would yield a more thorough under- standing of the welding process. He asks for more in- vestigation of the physics of the welding arc. Andrews (48) and Muraki (28) express the intensity of the heat source, Q (Joule/m-sec) as: 1 Q =H'TIVI where V = arc voltage (volts) I = welding current (amperes) n = arc efficiency 47 h = plate thickness (cm) Again, the assumption of efficiency was made to the in- put data which because of the unsurety of its value can greatly vary results. Friedman (55) assumed that heat of the welding arc to be a radially symmetric normal distribution function: g 2 q (r) = qoe Cr where qo and C are constants determined by the magnitude and distribution of the heat input and r = distance from center of arc (m) Westby' (69) concurs with this distribution of specific heat flux, q (cal/sec cmz), as being an approximation expressed by Gauss's distribution law. But, he simpli- fies it to a line or point source. He also gives efficiency factors for various welding processes which include heat losses due to radiation. Verification of the heat flux distribution was done by Wilkinson (59). He used photographic techniques and pressure sensors to show this radial distribution (see Fig. 6). Assuming this radial distribution and setting the heat input equal to: ” q = = ___01r = qtotal £q(r)2nrdr C Q 48 ELECTRODE 0.----Z/\\ WORK g” PIECE Figure 6. Heat Flux Distribution 49 where Q = Voltage x Amperage x Efficiency The constant C can be solved for after assuming any- thing outside 5% of qa is negligible or for q(?)= 5% of maximum value (qo) q = qoec = .05 QOCE't—E‘S') (T) m 3.0 C = f:—- r But at present F can only be determined by photo- graphic techniques. Only after F’is determined or estimated can the distribution of heat flux be modeled. Now r is the region in which 95% of the heat flux is transferred to the body. watts m2 element boundary the portion added to the force matrix {F} is: Assuming a constant q ( ) across a triangular q L. l 11 1 watts m and heat flux was modeled as shown in Figure 7. An improvement approaching exact values of q at element nodes can be achieved by using many small elements (see Figure 8). This approximation approaches a Guass distributed heat flux qur,t). SO ELEMENT 2‘s HEA'T aux”; (ME‘ /’ :}1/--.-- 29 vs I at F'v‘lz I ‘.\ \ I E--\r-H‘N3 I I I \I <:- : : 1 k : J=>I-——I* I \| j ‘1 as." : \ A 0‘ Figure 7. Model of constant q over an element side. 51 Figure 8. Heat Flux Model 52 q is given by: qicrnz) = 1.2- epr-SCr/F)2] EXPI-3(vt/?)2] wr where r = distance from center for a particular qi (m) v 8 velocity of the arc (m/second) Q a total heat input (watts) t = time (seconds) q1 = watts/m at point i ? maximum radius At time equal to zero this equation has a heat flux distribution at its maximum for the two-dimensional plane being studied. Therefore, the above expression was backed up in time to model a passing arc by intro- ducing a lag factor I. The modified equation becomes: qi(r,t)= -9- EXP[- 3r/r)2 ] EXP[ avg—Lg” ] 11”.?2 In this planar analysis it has been assumed the Speed of the arc is high compared to the diffusion rate of the material so the amount of heat conducted ahead of the arc is relatively small to the total heat input. In order to make this a two-dimensional problem it has been assumed that the heat flow across the plane in the third dimension (direction of electrode travel) is again very small and negligible. Therefore, the 37[K% T] 53 term goes to zero in the heat conduction equation. 3.9 Optimizingweld Joint Strength The design of a weldment has many variables affect- ing the outcome of how the joint can withstand various loading and environmental conditions. The design should consider that welding in some cases will result in significantly poorer properties than found in the base metal. The welded joint may be handicapped with lower fatigue strength caused by weld bead geometry, decreased toughness produced by microstructural changes in the heat affected zone or lowered corrosion resistance in certain environments caused by residual stresses. The control of stress concentration can be accomplished by varying the process, type of electrode, or doing post welding operations such as grinding. These are more Of a state of the art fix to this problem. Residual stresses have no effect on lower carbon steel structures, so the problem of engineering a low carbon steel joint for maximum strength resolves to manipulation of the microstructure metallurgy in the heat affected zone. By controlling the high heat gradients over time the metallurgy of this area is controlled. Catastrophic failure of weldments by brittle fracture have been found in bridges, ships and other structures are usually caused by a reduced fracture toughness of the steel section. Again, the toughness of a weld joint is 54 affected by the high weld heat input over a short period of time. These heat gradients for any process are now determined by the thermo-finite element model previously developed. Additional alloying elements added to steels help to increase strength such as grain size inhibitors of columbium or nitrogen. Deoxidation elements like aluminum or titanium secure fine grain sizes which aid in increasing steel toughness. Success in welding carbon steels is chiefly the ability to avoid the development of an unsuitable structure in or adjacent to the weld joint. Microstructural changes involving transformation of austenite, ferrite and martensite are the most important part and the cooling rate on the final structure being the most important parameter. Since heat input and metal mass of the joint have a wide range, the actual cooling rate may vary from fast rates to low rates. Other factors like joint design, electrode size, core wire composition, type of flux covering, kind of current and polarity of direct current have been found to have little influence on cooling rates. Because of these fast cooling times in a welded joint the iron-carbon diagram can no longer be used to determine the metallurgical structure. Continuous cooling transformation diagrams have been developed which more closely predict the microstructure of the 55 heat-affected zone. See Figure 9. Two researchers, Inagaki and Sekiguchi (10), studied the hardness and microstructure of the heat affected zone. They con- firmed that the microstructure and hardness adjacent to the fusion line may be predicted from CCT diagrams. An example of how their prediction is approximated is shown in Figure 10. If the heat input during welding is small and cooling takes place at a rapid rate, the Z-cooling curve will depict the heat-affected-zone structure which will be entirely martensite and the hardness will be about 415 DPH. If the cooling rate should fall between the Z and F cooling curve some intermediate structure will result. Slower cooling of the structure resulting in values passing to the right of the F-cooling curves sees formation of some proeutectoid ferrite yielding a final structure of ferrite, an intermediate structure, and martensite. If cooling is carried out at a slower rate than the P- curve some pearlite structure develops. A further in- crease in cooling time beyond the E cooling curve en- ables the heat affected zone to be entirely ferrite and pearlite. Inagaki, et_al, (10) found that the heat- affected zones which cooled at a sufficiently slow rate to produce some proeutectoid-ferrite did not crack Spontaneously and exhibited a good degree of ductility and toughness. They concluded that this cooling rate which marked the appearance of proeutectoid ferrite be TEMPERATURE,°F Figure 9. 56 \ ‘\ AUSVENITE” GAIN! \ AUSrENlrf——'-HAPTENSITE FIFPIT TRANSFORMATION TIME - SECONDS Example Continuous Cooling Transformation Diagram for Steel Ref (12) Page 1096. S7 g 750-- L \ \‘ 01 2500/7 < / , I5 / /// ~/ ,6? I/// 02-250... MA RT\E N SITE LIJ I \ I— \ \ \ O i ' i i i ‘r I 4 IO IOO IOOO TI ME(s ECONDs) Figure 10. CCT Diagram showing critical pts, critical cooling curves, and critical cooling times. 58 marked by the symbol "CL" representing a limiting welding parameter. In their study total bend angle and Charpy Impact tests were done on samples to verify that cooling time longer than Ck created larger bend angles and more absorbed energy (see Figure 11). These curves show that hardness decreases with cooling times greater than C%. Therefore, to keep the toughness at a maximum use a minimum Ck. To keep the strength of the steel up, attempt to keep CE at a maxi- mum. It then can be concluded that C% is a cooling time to optimize weld joint strength for various loadings and steels. From experimental results Inagaki (10) deter- mined the minimum cooling time from the equivalent carbon content Ceq' log C% = 8.59 Ceq - 1.69 where c:eq = a c + 112 M“ + 71;” It is known that the A3 transformation point is affected by the metallurgy of the steel and the CCT diagram affected by how long the structure is heated above the A3 temperature. To determine C%, the time to cool between 1075°K to 775°K was considered to be the only general practical boundaries. The A:5 trans- formation temperature point is approximately lO75°K for mild and high tensile steels. For various other 59 O CRAM) 25005.,” ABSORBED ‘0 ENERGY ~«m3 7 z A 0 2 I\ ”3001 :2 6'" I HARDNE E r I me o 5.. 1.1 h I z z ’ Y- m 4-1 T4 > I ‘ IO Io'O IOOO COOLING TIME (I075 TO 775 °K) (SECONDS) Figure 11. Cooling time versus hardness, bend angle and absorbed energy. I.J ” 1 (“OH-"w . rt .. f": a . -— .w 1' 'V "v“‘ I i v- . , . m “,iw .O'Vhflwd-v‘J-fi‘gw v’wwfm' v; a r - “.' .7 _— ‘ I'V :' "Z 60 alloys the cooling time will have to be determined over a different range with 775°K still being the lower bound by definition of CE. Different welding processes didn't affect a typical hardness—cooling curve (see Figure 11). The hardness at C% has been found to be equivalent to Hv = 350 which is a maximum guideline set up by Welding Standards. At this hardness,no underbead cracks result in the heat affected zone and weld joint exhibits enough ductility. A weld joint can be optimized by varying the input parameters to achieve the critical cooling time Ck which will give maximum hardness at the fusion line without sacrificing bend tests angle. This can be calcu- lated in a finite element program. For metallurgy determination away from the fusion line CCT diagrams will have to be referred to. IV. Finite Element Program 4.1 Iterative Procedure A Crank-Nicolson iterative method is used in solv- ing the system of equations generated in this model. The implicit equation in matrix form is: '{Fl} + {F0} 2 _ 2 (IKI + AT [CI){T1} - (AT [CI ' IKI){T0} ‘ 2C 2 ) During each iteration the Old temperature (prior iter- ation) {To} is a known value along with values for [K], [C], and {F}. The solution gives the new temperatures {T1}. Because some properties vary with temperature, [K] and [C] are reconstructed for each iteration. This process adds to computer execution time. A 539 element (303 nodes) model with a time step of .3 seconds required 13 minutes of computer execution time to simulate 75 seconds of real time. The time step variable (AT) is important because its magnitude can affect the stability of the FEM solution. Fujii (72), Yalamanchili (71) and Chu (71) discuss stability in finite element schemes. Fujii re- 61 62 lates stability for the consistent mass case as re- stricted by the inequality: < Z — Am(m+1) ("I”) max (0, do ({1-26}-$:— ) Kmin 2 for simplex acute triangle I’m m 2 for Z-dimensional case Kmin = minimum altitude of the triangle element AT = time step 00 = K/DC But when 6 Z l/Z,the finite element scheme is un- conditionally stable as is the case for this weld model because 9 = 1/2 (Crank-Nicolson). 4.2 Input Requirements and Output The largest amount of input data consists Of the element node numbers and their respective coordinates. To minimize punching over 500 cards (one for each element) and creating possible data errors an automatic grid generation program was used (62). This program included a node renumbering and optimizing subroutine which minimizes the bandwidth of the final sys- tem of equations (Computer.storage area needed for a matrix). Besides element related information this weld program 63 requires the node numbers affected by radiation and heat flux. Their related element and coordinate values are subsequently supplied. The number of nodes, number of elements, and bandwidth are input and used to supply subroutines with variables regulating calculations and the building of [K], [C] and {F}. Convection and radiation heat transfer are surfaces keyed Off element grid data. Initial values for physical constants of thermal conductivity and specific heat are provided for these variables in subroutines. Other physical constants needed in calculations are ambient temperature, initial temperature, and the heat transfer convection coefficient. Iteration requirements include time step, number of iterations, and iterations between printed output. The arc velocity, maximum heat flux at the centerline, and 95% radius are the only variables needed to model the welding arc. The output consists of the nodal temperatures at specific times after the beginning of welding. Also, during the arcing process heat flux is given at the element mid-pOint. Element number and appropriate surface of radiation or convection heat losses are printed for each iteration. All important physical quantities and element data are printed for convenience, as an analysis aid, and checking. 64 Figure 12 is a general flow diagram of this finite element weld model program. Figure 13 shows how the complete set of matrices are stored in the column {A} vector constructed in subroutine SETFL. ApprOpriate subroutines have been written to be compatible with manipulations of this A vector. A brief description of the subroutines is as follows. SETFL - Sets the dimension of the column vector {A} SETMAT - Constructs [C], [K] and {F} in {A}. DCMPBD - Decomposes {A} into an upper triangle matrix. TRANSNT - Heat flux calculated and output written MULTBD - Combines right side terms of Crank-Nicol- son equation SLVBD - Solves for {Tnew} by backward substitution. 4.3 Thermal Conductivity and Specific Heat as Program variables It is difficult to choose the values of thermal conductivity, K, and specific heat, C, because they vary with temperature and material. As carbon content increases or alloying elements are changed the slope of the thermal conductivity versus temperature curve can change and even become positive (see Figure 2). The material was therefore restricted to low carbon steel (11). 65 IINPUTiDATA I LCALL SETFLI WRITEFQATAI ECALL S ETMAT It ICAIfI. DCMPBD I—>[CALL TRAN§NT |<——H ->LCALI. MULTBD I— CALL SLVBD I— I: WRITE IOUT FUTI L’ICALI. S E'TMAT I-—*' Figure 12. Computer Program Flow Diagram Figure 13. Storage in "A" Column Vector 67 Another variable, convection in the weld pool, changes the thermal conductivity from 1/5 of the value at room temperature to 3 or 4 times those values. Mizikar (73) has provided an estimate value of K in molten steel. Because Of the need for further infor- mation U.S. Steel Corporation was contacted. Mr. Moore, U.S. Steel Corporation, (74) suggested using a value of approximately 2 times that of room temperature; a value commonly used in the industry. The finite element model incorporated thermal con- ductivity as a function of temperature by uSing three relationships: Kxx = .795 - .0004 * TAVE for T < 1300 °K = watts o 0 xxx .3 (m) for 1775 K > T Z 1300 K = o Kxx .6 for T 2 1775 K where TAVE = average of element nodal temperatures Ti + Ti+ T 3 k These equations would have to be changed when a differ- ent material is modeled or when welding under other convective patterns in the molten pool. A single constant value for thermal conductivity was calculated for each element. Chapter 3 states thermal conductivity creates a 68 second term from the variational form when varied with temperature. [K] = [KC] + girBITIKBI second term An estimate of the second term's influence on temperature results was performed. Analysis showed: 1. A minor (5%) importance when nodal tempera- tures vary by more than 300°K per element. 2. Decreased importance for small differences in nodal temperatures. 3. Further decreased importance for small ele- ments . 4. Increased computer costs and execution time because of [K] becoming unsymmetric (4 times or more). These results, especially the non-symmetry property, increase the cost of obtaining a solution and the additional second term was assumed negligible and not included in constructing [K]. The previous three Kxx equations then become part of [KC]. A similar second term results when specific heat is a function of temperature. 69 2 1 1 2 1 1 “AA 121 +°czA 121 1 1 2 \ii‘v—~:;j>2 second term Again, because of non-symmetry and a similar importance analysis yielding negligibility for the second term, this term was not included in constructing [C]. Specific heat was modeled as: pc 1.3 + .0053 * TAVE for T < 775°K pc 8 5.0 for T.z 775°K density assumed constant '0 II The average of nodal temperatures determines specific heat which was assumed constant over an ele- ment. 4.4 Additions to the Force Vector Heat flux is added to the force vector by a distri- bution presented in Chapter 3 for qi(r,t). Heat flux was calculated for a radius equaling the midpoint of an element and divided equally between the pair of nodes and added to the force matrix. Using the heat flux model in Chapter 3, r was determined by T = F/V F = radius including 95% of q 70 V = arc velocity This heat flux model provided a gauss distribution approaching Figure 8 as the size of the elements gets very small in the arc area. When heat flux isn't an input to the force vector either convection or radiation from the surface is dominant. For steels 67S°K is an approximate break- even point between these two surface related prop- erties. Latent heat was added to the force matrix during the solidification process if the average temperature of the element was between 1700-l7SS°K (common for alloy steels). The derivation of this addition is given in Chapter Three. If melting occurred, latent heat was sub— tracted from the force vector. The program limited the occurrence of an addition or subtraction of latent heat to a one time phenomenon” The reinforcement of a weld pool (filler added) was not modeled by additional elements, but temperatures of melted nodes were allowed to go to 4000°K providing accountability for this phenomenon. Other modeling where reinforcement wasn't present the maximum allowable nodal temperature was 3500°K. This is an important limitation because the elements are finite in size and radiation influence does cause instability with higher than 4000°K surface temperatures. V. Verification and Sensitivity Analysis 5.1 Verification with Christensen's and Hess' Work In order to be confident that the finite element model is approximating an actual weld, a comparison with actual thermocouple experiments was made. Two groups of researchers, Hess (l9) and Christensen (49), have done extensive recording of time-temperature relation- ships for a range of welding applications. Their data was used to verify the accuracy of this numerical model. Three comparisons which show close agreement between experimental and finite element approximation are given in Figures l4, l6, 17. Figure 14 is a comparison of thermocouple readings and the finite element model (Figure 15) in the HA2. After a considerable time-lapse (50 seconds from arc passing) the two begin to differ by as much as 10% with the experimental readings being lower. This could very well be the effect of the thermocouple's drilled hole creating a heat loss boundary lowering local temperatures. The key area of the curve, critical cooling time, Correlates well and the values of Cf are almost equal. This alone shows that this finite element 71 72 ___HESS »-«FEM 1W7OWMTHSH' ”WZOOT é V= .3 CM/SEC LIJ \ :: '._ < m 800- Id 0. 2 _ u 600 f—- f 0 2'0 4'0 . 60 80 TIME (SEC) Figure 14. Verification with Hess's work (time- temperature curve) FINITE ELEMENT GRID 3.00 C3O'.OO 2.00 4.00 X-RXIS CI’I Figure 15. Element Model of Hess' Work 74 Q=3 6568 WATTS. V=.5CM/5Ec_ TlME=II.75£C 2 . A x—xCHR ISTENSEN x “"FEM Q. I800: LIJ g . I400 52 £3 0. I000" 2 . LIJ I— 600 " ‘ I10 210 sfo 4b WIDTH (CM) 19- -o-r-+ Figure 16. Cross Section Temperature Verification I i WIDTH(CM) O //I./ V I O A / :-: ,1 w\\ I: w“ z\ a ‘\ w 07.4.”. .// 2.0 o—oF‘EM HCHRISTENSEN '9 LIMITS Q= 6568 WATTS '1 (‘b F" < < =.5CM/5 EC. 76 model gives a close approximation to actual welding conditions. Figure 15 relates Christensen's work with the modeling of a submerged-arc process. The FEM model is in close agreement with experimental data in the heat affected zone (1.5-2.0 cm) which is the most important correlation area. Only in the weld p001 does the finite element result vary by more than 5% from experimental. In this area of variation thermal con- ductivity of molten steel being erratic might provide reasoning for the difference. But, Christensen stated that his thermocouple measurements were very inaccurate in the pool. Values ranged from 2500°K to 3500°K and depended how measurements were taken. Adding another dimension to this finite element model's versatility approximate solidus line and HA2 can be determined. This is depicted in Figure 17. Again, Christensen's data had scatter in it relating the difficulty in measuring this process. The FEM approximation of the heat affected zone and weld p001 correlate well with experimental results falling. within Christensen's established bounds. 5.2 Sensitivity of Some Variables 5.2.1 Radius of the Welding Arc The area affected by the heat flux is regulated by the 95% containment radius T. This radius is currently 77 determined by photographic estimation. A sensitivity analysis was done on the influence of E on the finite element solution because of the un— surety of its value. This analysis was done by vary- ing E from .5 cm to 1.5 cm using Christensen's welding parameters. A node in the heat affected zone was chosen to determine the effects of various welding arc radii. Figure 18 shows a plot of time-temperature values and how they are affected by varying EL From this graph it can be depicted that E affects the size of the weld pool, size of the heat affected zone and critical cool- ing time. Therefore, the welding arc must be accurately modeled for meaningful results to be Obtained from numerical methods. Since little research has been done in arc model- ing and its shape highly influences finite element results, more work is needed. Possible use of inverse conduction methods might provide accurate arc dimensions and heat transfer efficiencies of welding processes. 5.2.2 Thermal Conductivity in the Weld Pool A heat affected node was used as the basic location for estimating the sensitivity of the FEM solution to variations in weld pool thermal conductivity. Three different values Of thermal conductivity were used and the effect on temperature relationships are shown in 78 mswemm uy< mcwwaez ow >uw>wuwmcom Guam: :. oe oe pm .wH chewed UU 2U 2.557% MOOZ uun\_zun.u> >> w omomuo 209.131. zoo._&.::. :um._&1 o I. m. :08 .e 3 w :08. I. n no :8: n mm :00». 79 Figure 19. As K varied from .2 watts/cm°K to a value of 1.2 watts/cm°K, the temperature history experienced by the surface location varied from one extreme of not even getting hot enough to enter the HA2 to reaching melting temperatures. The relationship of higher thermal conductivity values creating higher maximum temperatures makes FEM solutions very sensitive to this variable. 5.2.3 Convection Coefficient and Emissivity Value Boundary conditions can vary and their effect on the finite element model was checked. Figures 20 and 21 are plots of various common values for the heat transfer coefficient (free convection) and the emissivity of steel in the molten area (T > l700°K). Variations in each of these variables changed the solution by minor amounts. The heat transfer coefficient produced only a 6°K difference while the emissivity is a little greater with a difference in nodal temperatures up to 19°K in the HA2. The solution results are considered insensitive to these variables. 5.2.4 Grid Coarseness and Time Step The element grid used to compare Christensen's work is shown in Figure 22. This grid model has a fine mesh in the area of high thermal gradients. A comparison 80 xuw>fluusvcou Hmaposb Hood Cam: op zufi>fiuwmcom .mH ONSMHm 0.0 0.? 0.N O I. .HHIHWIII n1llllll add TIIIIw/l \1-ooew 0/ Id IIAFI.I\\\ s add MU I l x 1000 “a 20o~nfirux \ _ W. MOOZ fl Oun\s_um..u> uefiwwswfl av. .00: w" . \<’ “H 01.01. .\II .. o e 3 mwmmmIO KoEU\>>N._uzI :00». W 81 unoauwmmoou :OMuoo>:oo,wo Aufi>wuwmcom .oN ogswfim Guava}; 0+. 3 oh 0 . . # II— . m. 1.000 Id mufimaw 19....» m“ zuwnwofa heoz .009 w um n\2u ewm....> m 2’ N_NMNO 4 .IOO.V- ‘30 M 31...: o\3 800.111 v.32 o} mooo.I_._..- .... mop< :OHHoz :H >HH>HMMHEm new we sz>HuHmcom .HN oyswflm GNOME: 0.0 0.1. om o - - l 3 I W 60m A. 2 3 8 ”Sign .19..» / W. 20383“ . .00» m. “002 a . . .. .600. mm. uun\2u omm e> A ”weed I 3856 maul 83 FINITE ELEMENT GRID O.OO 8'.OO I'E.OO X—RXIS CM Figure 22. Fine Grid Model 84 of results with a coarser grid (Figure 4) is graphed in Figure 23. The coarser grid does not adequately transmit heat in the weld area and results in a lower temperature curve. The choice of grid size is an engineering judgment when using finite element approximations. It plays an important part in getting accurate results with small elements a must criteria for high gradient areas. Figure 24 depicts the sensitivity of two different time steps on the finite element solution. AT of .3 and .1 seconds were compared. Little variation resulted in the HAZ temperatures, but some instability at high temperatures (1800°K on up) was seen when the smaller time step was used. Increasing the time step to determine its effects was ruled out by the need to plot results using small steps. 85 Q=36568W v=.SC M/S EC. NODE 'X=2.0 CM Y= TOP SURFACE ‘ x—-—xFINE GRID r-WCOURSE GRID 5 o ‘? 800- 600- TE MPERATURE(°K) 6 2'0 40 TIME(SEC) Figure 23. Coarse Versus Fine Grid 86 Q=36568W V=.5CM SEC TIM E= ".7 SEC 22 00' HDT=.I SEC A ~ I: o----oDT=.3 SEC 3E Iaoo-T m g ,_ I400I- 4: 0:- 31! I000" 2 :1." 600- ‘E : I : l 0 Lo 2.0 3.0 4.0 .9. Figure 24. WIDTH(CM) Comparison of Two Time Steps VI. Application to Welding Design 6.1 Joint Strength The total joined area regulates the strength in the static loading of welded joints. This area can be determined from the finite element model. The only variable needed in static load calculations not deter- mined by the FEM model is the weld reinforcement stress concentration which is a state of the art function primarily related to the type of welding rod, the angle of the arc, and the arc velocity. For dynamic loading cases, such as fatigue, the toughness of a joint regulates its strength over time. Inagaki's (10) report lays the ground work for maximizing the energy absorbed by a joint (toughness) with his critical cooling time factor. This C% is the optimum cooling time to obtain maximum strength at maximum tough- ness. % is estimated by Inagaki's experimental equation: log 0% = 8.59 ceq - 1.69 The finite element model results were used to calculate the cooling time for a proposed set of welding parameters. 87 88 If the cooling time varies from Inagaki's optimum, the welding parameters can be varied to optimize the joint strength by meeting the C% criteria. Some parameters that affect CE are arc velocity, size of the electrode, type of welding process, backup plates, physical size of the base metal, and material constit- uents. The advantage of using the finite element model in optimizing CE is the ease of changing variables and the ability to obtain results of the temperature history in a matter of minutes. 6.2 Effect of Arc Velocity on Cooling Time To depict how one of the aforementioned vari- ables can be changed to optimize C', the arc velocity effect was looked at. Critical cooling time is de- fined as the time to cool the HA2 from 1075° K to 775°K. The examples of Ck location in Figure 25 are in agree- ment with Inagaki's critical location (the fusion line). As a HAZ node (or location) approaches the fusion line the cooling rate becomes faster and more critical. Little difference in C% is seen between internal and surface nodes on the fusion line. An example of arc velocities' effect of C% is shown in Figure 26. The velocity was varied from .2 to 1.5 cm/sec causing the temperature history of one HA2 sur- face node to change considerably. To show the effect of different velocities on C’ :oHumcHEHouom OEHH wcHHoou ESEHcHz .mm ousmHm 0.000 02:. 0.0 0_m 0 \ I. 4 N M .I I o L w I. I I HI .I \1.\ C 000 T0 4:225... I .7 I I H M. o w . gmenemmo / (L m uuéman L07» so EMMA .II. 001$. Sign 19.1, 20000.me I2... mm 02... 20.9.1...» 200::le .00». 0 9 mcHHoou 0002 m :o muoomwm >0H00H0> 0H< 0.40.03 02.... .2... 0.33m Om O. N O - IhIV — a, II. . .9: I: II. ILWIJHH. 1111111. \\ ..AvAnvmw .I. I I, / I. I / \ / 01‘ uuIII OMQEO mail... .000. (We)30lan83dW3.L 91 temperature histories for fusion line nodes were plotted in Figure 27. It can be concluded from these plots that increasing C% can be accomplished by slowing the speed of travel. The speed of travel also inter- acts with melt depth, where slower speeds result in larger depths. Before the use of these conclusions become a recommendation, economic factors must be considered. Items such as welding time costs may lead to a compromise in final joint conditions. 92 m0 :0 H0H00H0> mo uuemmm .HH eesmHm 1.003 0.2:. 0.. e. e... e I._ 83... 0 1 m. ”J / 3 I000 d . I .I I E W F . X, / :000. I. 2U+Nm0u> QTHX r, 0 20.0001,” 00.; X / \ _ a 12..., 02.; / 1 .004. n 000 «.mumu 000;. 03.071 x. 1 mm .005 m0." .0 009200.07--- , oummemLU 00m\20m...>T1. C a .000. VII. Summary and Conclusions The finite element method was used to obtain the time-temperature relationship in a butt welded joint. This transient thermal history influences the metallur- gical properties in the heat affected zone. By varying certain welding parameters, the weld joint strength can be optimized when a certain metallurgical structure is achieved. The finite element computer program used simplex triangular elements, included specific heat and thermal conductivity as functions of temperature, incorporated latent heat of fusion, and convection and radiation heat losses from the surface. The following conclusions were drawn from this study: 1. The non-linear finite element method model closely approximates actual welding conditions but must be used cautiously because the results are sensitive to the arc radius and the thermal conductivity of molten steel. 2. The fusion line-top surface intersection is an acceptable location to use in estimating critical cooling time. 93 94 Slowing the arc velocity increases the cooling time of the fusion line. Heat affected zone and penetration can be determined with this method. If joint strength is insufficient in weld depth, the correct value can be arrived at by varying heat input or arc velocity. VIII. Recommendations for Further Study This study is the beginning of the application of the finite element numerical technique in the area of welding and casting. Only the surface of a large volume of possible research paths has been scratched. Results presented should help those continuing onward in this area. By pursuing some of the recommendations, future researchers assist in solving a vast number of practical and theoretical engineering problems. 1. Basic physical property research is needed on thermal conductivity in weld pool temperature ranges. 2. Weld arc distributions should be determined for all processes. This might be done numerically using an inverse heat conduction approach (65). 3. Material could be saved in the casting in- dustry if a three-dimensional finite element model could be developed. 4. Expand this model to include slag and weld re- inforcement. The use of higher order elements may improve results and reduce computer costs. Their use should be investigated. 95 10. 96 Fatigue test various joints to assure the critical cooling time is an optimizing parameter. Incorporate the non-symmetric temperature dependent terms of thermal conductivity and specific heat into [K] and [C] and determine if the results have been improved in the high heat gradient areas. Develop mathematical bounds for time steps of various finite elements to assure stability of the solution. Model other weld configurations such as spot, fillet, and pre-notched joints. Develop a FEM model for the welding of other materials including aluminum, plastics and high strength steel. Apply these thermal history results to the elastic-plastic FEM models. REFERENCES (1) (Z) (3) (4) (5) (6) (7) (8) (9) (10) REFERENCES Rosenthal, D. "Mathematical Theory of Heat Dis- tribution During Welding and Cutting." The Welding Journal, pp. 2205-2348. May 1941. Rosenthal, D. "The Theory of Moving Sources of Heat and Its Application to Metal Treatments." Transactions of ASME, pp. 849-866. Nov. 1946. Rosenthal, D. and R. Schmerber. "Thermal Study of Arc Welding. Experimental Verification of Theoretical Formulas." The Weldipngournal (17) pp. 2-8. April 1938. Mahla, E. M., M. C. Rowland, C. A. Shook, G. E. Doan. "Heat flow in Arc Welding." The Welding Journal, pp. 4595-4688. Oct. 1941. Dolby, R. E. and G. G. Sanders. "Metallurgical Factors Controlling the Heat Affect Zone Frac- ture Toughness of Carbon: Manganese and Low Alloy Steels." Welding Institute Report II WDoc. IX—891-74. Lancaster, J. F. Metallur y of Welding, Brazing; and Soldering. G. Allené Unwih, London, p. 4 -69 1965. Stout, R. D. and W. D. Doty. Weldability of Steels. Welding Research Council, New York, NY. p. 831. 1953. Henry, 0. H. Weldin Metallur y. American Weld- ing Society, New YorK, NY. 5rd Edition. 1965. Linnert, G. E. Weldin Metallur . Vol. 2, Ameri- can Welding Soc1ety, New YorE, NY. 3rd Edition. pp. 309-395. 1957. Inagaki, M. and H. Sekiguchi. "Continuous Cooling Transformation Diagrams of Steels for Welding and their Applications." TransaCtions of the National Research Institute for Metals (Japan) V01} 2, N 2. pp. 102-125, 1960. 97 (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) 98 Goldsmith, A., I. E. Watermour and A. J. Hisschhorn. Handbook of Thermophysical Properties of Solid Material§,Vol.II, Pergamon Press, 1963T" McGannon, H. The Makin , Shaping and Treating of Steel, 9th Edition 19 II Printed'by Herbick and HeId Pittsburgh, Pennsylvania for US Steel Corporation. Re ublic Allo Steels. Tana, OH IZIDI. 1968. Materials Properties Handbook. LondOn. Vol. II, I966. Republic Steel Corp., Cleve- Hartford House, 1975 SAE Handbook. Warrendale, PA. Society of Automotive Engineers, Vol. 1. Adams, Clyde M. "Cooling Rate and Peak Tempera- tures in Fusion Welding." The Welding Journal, pp. 2108-2158, May 1958. Paley, Zui, J. N. Lynch and C. M. Adams. "Heat Flow in Welding Heavy Steel Plate." The Welding Journal, pp. 715-795, Feb. 1964. Paley, Z. and P. D. Hibbert. "Computations of Temperatures in Actual Weld Designs," Report Pm-m-74-2. Dept. of Energy, Mines and Resources, Ottawa, Canada, March 12, 1974. Hess, W. F., L. L. Merrill, E. F. Wippes, and A. P. Bunk. "The Measurement of Cooling Rates Associ- ated with Arc Welding, and Their Application to the Selection of Optimum Welding Conditions." The Welding Journal, pp. 3773-4228, Sept. 1943. Paschkis, V. "Establishment of Cooling Curves of Welds by Means of Electrical Analogy," IBID, pp. 4628-4388, Sept. 1943. Iwamura, Y. and E. F. Rybicki. "A Transient Elastic-Plastic Thermal Stress Analysis of Flame Forming." Journal of Engineering for Industry, pp. 163-171, Feb. 1973. Rybicki, E. F. and L. A. Schmit, Jr. "An Incremental Complementary Energy Method of Non-Linear Stress Analysis," AIAA Journal Vol. 8 No. 10, pp. 1805- 1812, Oct. 1970. (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) 99 Ghadiali, N. D. and E. F. Rybicki. "An Analytical Technique for Evaluating Deformation Due to welding." Battelle Lab Report, Columbus, Ohio. Aug. 1974. Dawes, M. G. "Residual Stresses and Strains in Weldments." The Weldin Institute Research Bul- letin, pp. 264—268, Sept. 1974. Schofield, K. G. "Residual Stress Measurement Techniques." IBID, pp. 74-79, March 1975. Faupel, J. En ineerin Design. John Wiley and Sons, New YorE, NY, pp. 658-909, 1964. Lubkin, J. L. "Structural Modeling of a Welded Joint Typical for Vehicle Frames." Michigan State University, Div. of Engr. Research, Dec. 31, 1973. Muraki, T., J. J. Bryan, and K. Masubuchi. "Analysis of Thermal Stresses and Metal Movement During Welding. Part I - Analytical Study." Trans- actions of ASME. Journal of Engineerin Materials and Technolpgy, 74-WA/MATe4, pp. 1-4, 1 74. Muraki, T., J. J. Bryan and K. Masubuchi. "Analysis of Thermal Stresses and Metal Movement during Welding. Part II Comparison of Experimental Data and Analytical Results." IBID, pp. 1-7, 1974. Whiteman, J. R. A Bibliography for Finite Elements. Academic Press, LondOn, 1975? Wilson, E. L. and R. E. Nickel. "Application of the Finite Element to the Heat Conduction Equa- tion." Nuclear Engineering and Designs 3. pp. 1-11, 1966. Aguirre-Ramirfz G. and J. T. Oden. "Finite Ele- ment Technique Applied to Heat Conduction in Solids with Temperature Dependent Thermal Con- ductivity." ASME Ppper 69-WA/HT-34 N.P. 13, 1969. Donea, J. and S. Giuliani. "Code Tafest Numerical Solution to Transient Heat Conduction Problems using Finite Elements in Space and Time." Report ER-5049, Joint Nuclear Research Centre, European Atomic Energy Commission, ISPRA, Italy, 1974. (34) (35) (36) (37) (38) (39) (40) (41) (42) 100 Visser, W. "A Finite Element Method for the Determination of Non-Stationary Temperature Distribution and Thermal Deformations." Proc. lst Conf. Matrix Methods in Structural Mechanics, wright Patterson AFB, OHlO, A FFDL Tr 66'80, 1965. Visser, W. "The Finite Element Method in De- formation and Heat Conduction Problems." DELFT. 1968. Wilson, E. L., K. J. Bathe and F. E. Peterson. "Finite Element Analysis of Linear and Non Linear Heat Transfer." Paper Ll/4, Proceedings and Structural Mechanics in Reactor Tech. Conf. Berlin, 1973. Hutula, D. N., B. E. Wiancko and S. M. Zeiler. "Apache a Three-Dimensional Finite Element Pro- gram for Steady State or Transient Heat Conduc- tion Analysis. Report WAPD-IM-IOBO, Bettis Atomic Power Laboratory, Pittsburgh, 1973. szernik, A. and A. Leech. "The Comparison of Iterative and Direct Solution Techniques in the Analysis of Time-Dependent Stress Problems, In- cluding Creep, by the Finite Element Method." pp. 449- 462 of J. R. Whiteman (F. D. ) The Mathematics of Finite Elements and Applications. Academic Press, London, 1973. Segerlind, L. J., R. P. Singh, J. G. P. De Baerde- maeker and R. J. Gustafson. "Theoretical Aspects of the Finite Element Method." ASAE Paper 74-5501 pp. 1-28, Dec. 1974. Segerlind, L. J., R. P. Singh, J. G. P. De Baerde- maeker and R. J. Gustafson. "Computer Implementa- tion of the Finite Element Method." IBID. Paper 74-5502, pp. 1-10, Dec. 1974. Segerlind, L. J., R. P. Singh, J. G. P. De Baerde- maeker and R. J. Gustafson. "Element Data Gen- erator for Some Two-Dimensional Finite Element Programs." IBID 74-5503 pp. 1-19, Dec. 1974. Segerlind, L. J., R. P. Singh, J. G. P. De Baerde- maeker, and R. J. Gustafson. "Some Finite Ele- ment Programs for Agricultural Engineering In- struction." IBID. No. 74-5504 pp. 1-39, Dec. 1974. (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) 101 Bruch, J. C. and G. Zyvoloski. "Transient Two- Dimensional Heat Conduction Problems Solved by the Finite Element Method."' Int.‘J;‘Numer. Meth. Egg. 8, pp. 481-494, 1974. Comini, G., S. D. Guidice, R. W. Lewis and O. C. Zienkiewicz. "Finite Element Solution of Non- Linear Heat Conduction Problems with Special Reference to Phase Change." Int.‘J. Numer. Meth. Egg. 8, pp. 613-624, 1974. Fujino, T. and K. Chsaka. "Heat Conduction and Thermal Stress Analysis by the Finite Element Method." Proc. 2nd Conf. Matrix Methods in in Structural Mechanics, Wright-Patterson AFB., Ohio AFFDL-TR-68-150, 1968. Fullard, K. "The Computation of Temperature Dis- tributions and Thermal Stresses using Finite Element Techniques." Paper M5/3 Proc. lst Struc- tural Mech. in Reactor Tech. Conf. Berlin 1971. Naehrig, T. H. and J. P. Gaschen. "The Calculation of Three-Dimensional Temp. Dist. and Thermal Stresses using Finite Element Method." Paper M5/2, IBID. Andrews, J. B., M. Arita, and K. Masubuchi. "Anal- ysis of Thermal Stress and Metal Movement During Welding." NTIS Report N7l-26l43., Dec. 1970. Christensen, N., V. de L. Davies, and K. Gjermuns- den. "Distribution of Temperatures in Arc Weld- ing." British Welding Journal, Vol 12, pp. 54-75, Feb. 1965} Roberts, Doris K., A. A. Wells. "Fusion Welding of Aluminum Alloys." Part V. IBID. Vol. 1 pp. 553-560. 1954. Zienkiewicz, 0. C. The Finite Element Method in Engineering_Science. *MCGraw-HiIl, London, 1971. Singh, R. P. and L. J. Segerlind. "The Finite Element Method in Food Engineering." ASAE Paper No. 74-6015. pp. 1-18, June 1974. Meyers, P. S., Uyehara, O. A., and Borman, G. L. "Fundamentals of Heat Flow in Welding," Welding Research Council Bulletin, No. 123, July 1967, pp. 1-46. (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) 102 Finite Element Application to Vehicle Design, Inter- national Conference on Vehicle Structural Mech- anics SAE COnference Proceedings Detroit, Michi- gan. MérEh 26¥28, 1974. pp. 1-285. Friedman, E. "Thermomechanical Analysis of the Welding Process Using the Finite Element Method." Transactions of ASME. Paper 75-PVP-27, 1975, pp. 148C Hibbitt, H. D., and Marcal, V. P. "A Numerical Thermo-Mechanical Model for the Welding and Sub- sequent Loading of a Fabricated Structure," Com- W. Vol- 3. pp- 1145-1174. Perganon Press 1973. Westby, Ola, Element Methods for Welding Deformations and Residual Stresses, Special Lecture, 1975, Technical’University of Norway. Kohler, W. and Pittr, J. "Calculations of Transient Temperature Fields with Finite Elements in Space and Time Dimensions," International Journal for Numerical Methods in Engineering, V01. , 1974, pp. 625-631. Milner, D. R. and Wilkinson, J. B. "Heat Transfer From Arcs," British Welding Journal, February 1960, pp. 115-128. Friedman, E. "A Direct Iteration Method for the Incorporation of Phase Change in Finite Element Heat Conduction in Programs," Report WAPD-TM-ll33, U.S. Atomic Energy Commission, March 1974. Toyooka, T., and Watanabe, M. Embrittlement in Weld Stain-Affected Zone in Carbon Steel, Special Report - Kawasaki Heavy Industry, LTD. Kobe, Japan. Segerlind, L. J.,.Applied Finite Element Analysis, John Wiley and Sons, Néw York, NYil976. Strang, G. and Fix, G. An Analysis of the Finite Element Method. Prentice-Hall Inc.,‘l973. Beck, J. V. "Nonlinear Estimation Applied to the Nonlinear Inverse Heat Conduction Problem," International Journal of Heat and Mass Transfer, V01. 13, pp. 7O3-716T7 Pergamon Press, 1970. (65) (66) (67) (68) (69) (70) (71) (72) (73) 103 Beck, J. V. "Criteria for Comparison of Methods of Solution of the Inverse Heat Conduction Problem," ASME Paper 75-WA/HT-82. pp. 1-11, 1975. Beck, J. V. "Determination of Optimum Transient Experiments for Thermal Contact Conductance," International Journal of Heat and Mass Transfer, V01. 12. pp. 621-633. Perganon Press 1969. Report on Physical Properties of Metals and Alloys from Cargogenic to Elevated Temperatures, ASTM, Publication Number 296, 1960. pp. 51-58. Report on Elevated-Temperature Properties of Wrought Medium-Carbon Alloy Steels, ASTM, Publi- cation No. 199, 1957. Westby, 01a "Temperature Distribution in the Work- Piece by Welding," Institutt for Mekanisk Teknologi, Technical University of Norway, March 7, 1968. pp. 1-64. Winterton, K. "Weldability Prediction from Steel Composition to Avoid Heat Affected Zone Cracking," Welding Journal, Vol. 40. 1961. pp. 253-5 to 258-5. Yalamanchili, R. V. S. and Chu, S. U. "Stability and Oscillation Characteristics of Finite Ele- ment, Finite-Difference, and Weighted-Residuals Methods for Transient Two-Dimensional Heat Con- duction in Solid." Journal of Heat Transfer, May 1973, pp. 235-239. Fujii, H. "Some Remarks on Finite Element Analysis of Time-Dependent Field Problems," Department of Computer Sciences, Kyoto Sangyo Univ., Japan. Mizikar, Eugene A. "Mathematical Heat Transfer Model for Solidification of Continuously Cast Steel Slabs," Transactions of the Metallur ical Societyof AIME, VOlume 239, Nov. 1967, pp. 1747- 1753. (74) Moore, Michael, Phone conversation, March 1976, U.S. Steel Corporation, Pittsburgh, PA.