E. FilEDiAN Westinghouse Electric Corp.,

Bettis Atomic Power Laboratory, West Miff l in, Pa.

Thermomechanical Analysis i f t ie Welding Process Using t ie Finite Element Method Analytical models are developed for calculating temperatures, stresses and distortions resulting from the welding process. The models are implemented in finite element formulations and applied to a longitudinal butt weld. Nonuniform temperature transients are shown to result in the characteristic transverse bending distortions. Residual stresses are greatest in the weld metal and heat-affected zones, while the ac-cumulated plastic strain is maximum at the interface of tliese two zones on tlie under-side of the weldment.

Introduction The welding of metal structures is aimed at providing a means

of joining together a number of components in such a way as to minimize the impairment of the properties of those components. Considerable effort has been expended on developing effective welding techniques for a large number of metals and alloys. Con-current with these developments have been efforts aimed at identifying the various problems that result from welding proces-ses, determining the strength of welded joints and structures subject to specified loading conditions, and establishing guide-lines and criteria for most effective joint design. The information accumulated in these areas over the years has been overwhelming-ly experimental. Though attempts have been made to establish empirical approaches toward understanding the complex be-havior of materials due to welding, comparatively little effort has been expended in developing and applying analytical models to explain and predict the transient thermal response and the transient and residual mechanical response. Survey papers that deal with many of the analysis methods that do exist have been published by the Welding Research Council on heat flow due to welding [l]1 and welding-induced stresses and distortions [2].

Analytical treatments of temperature transients during welding have most often been considered by assuming the effective thermal energy supplied by the heat source to be deposited in such a narrow band of material that it may be idealized as a point, or a line source, depending on the geometry of the weld. The early work done in this area was concerned with the quasi-stationary transient temperature solution (steady-state with respect to a moving coordinate system) resulting from a line source of heat traveling on a straight path at a constant speed

Numbers in brackets designate References at end of paper.

Contributed by the Pressure Vessels and Piping Division and presented at the Second National Congress on Pressure Vessels and Piping, San Francisco, Calif., June 23-27, 1975, of THE AMERICAN SOCIETY OF MECHANICAL EN-GINEERS. Manuscript Teeeived at ASME Headquarters, March 28, 1975. Paper No. 75-PVP 27.

through an infinite plate. The classical line source solution, which was determined independently by Boulton and Lance-Martjn [3] and Rosenthal and Schmerber [4], has been used by Tallp] in the development of a numerical model to calculate the longitudinal stresses (stresses in the direction of the weld line) at a section of the plate normal to the weld line. Plastic flow near the weld line was included in the calculation, and mechanical propertiesthough not thermal propertieswere taken to be temperature-dependent. Masubuchi, Simmons, and Monroe [6] used Tail's method of analysis to write a digital computer pro-gram to calculate temperatures and stresses in welded plates.

Because of computational difficulties encountered when at-tempting to include inelastic strains in an analysis in which rapid temperature changes take place, little analytical work has been carried out which goes significantly beyond Tail's one-dimensional analysis of longitudinal stress. The complexity of performing analytical studies of welding stresses and distortions for various weldment configurations, when considering factors such as tran-sient multidimensional temperature distributions, effects of weldment fixturing and surrounding structure on heat sink and constraint characteristics, and realistic models of inelastic ma-terial behavior in the vicinity of the weld line, among others, suggests the use of numerical methods of analysisin particular, the widely adaptable finite element methodfor predicting tran-sient and residual stresses and distortions.

Though a determination of the complete thermomechanical response due to welding would require a full three-dimensional, inelastic analysis of mechanical behavior, accompanied by com-putation of the three-dimensional transient temperature dis-tribution, computer run times would be very large and the resulting costs perhaps prohibitive. I t is therefore reasonable at this time to first develop, for weldments under quasi-stationary thermal conditions, numerical techniques for planar analysis in sections normal to the weld direction. Applications of these techniques yield valuable data that have heretofore been un-obtainable, both in the fusion and heat-affected zones, and in the weldment itself.

The recent paper by Hibbitt and Marcal [7] represents an initial step in the development of numerical analysis techniques

206 / A U G U S T 1975 Transactions of the ASME Copyright 1975 by ASME

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that will simultaneously account for many welding parameters iind factors that previously had been considered piecemeal or not a t all. I " this work, the thermal input for the finite element stress analysis is developed by use of a finite element transient thermal analysis. The thermal model formulated by Hibbitt and Marcal is designed to simulate the gas-metal arc welding process, in which consumable electrodes are employed. The numerical thermal analysis is required to account for such phenomena as temperature-dependent material properties, phase change, through-the-thickness temperature variations, irregular weldment geometries, nonuniform distribution of energy from the heat source, and deposition of filler metal, among others. The line source solution [3-6] clearly cannot be used to model any of these phenomena.

The weld thermomechanical model described in the present work has been developed generally along the lines of Hibbitt and Marcal [7]; i.e., the finite element method is utilized to model both the thermal and the mechanical behavior. Emphasis, however, is placed on applying the analysis technique to gas-timgsten arc welding, although suitable modification of the thermal model will enable the method to be applied to other welding processes, as well. Though the purpose here is to develop a capability for predicting the weld-induced transient and residual stresses and distortions, the nature and characteristics of the thermal model by itself are obviously of great importance, since the model can be used, for example, to predict cooling rates, peak temperature distributions, changes in metallurgical struc-ture, and effects of chill block placement and other fixturing, as well as to provide input for the mechanical model. Since strain rates induced by welding are expected to be of the same order of magnitude as the free thermal expansion rates and since the (hermoelastic coupling coefficient [8] is small, thermomechanical coupling is neglected. On this basis, descriptions of first the thermal model and then the procedure for calculating stresses and distortions are presented. An application of the thermo-mechanical welding analysis method is then given for a simple butt weld.

Thermal Model Calculation of the transient temperature distribution is based

on the attainment of quasi-stationary conditions, which are de-veloped when the welding heat source is moving at constant speed on a regular path (i.e., a straight line in a planar weld, or a circle in an axisymmetric weld), and end effects resulting from either initiation or termination of the heat source are neglected. The temperature distribution is then stationary with respect to a moving coordinate system whose origin coincides with the point of application of the heat source. Consider the planar weld illustrated in Fig. 1. The temperature at any point in the weldment is expressed functionally as:

T(xs, x%, Xz, t) = T(xi, xi, x3 vt), (1)

where v is the welding speed. Thus, given the transient tempera-ture distribution at any one section of the weldment defined, say, by x3 = 0, the temperature at any other section is determined by an appropriate shift of the time scale as follows:

T(xu xt, x3, t) = T(xu xi, 0, t - x3/i>). (2)

The problem is, therefore, reduced to finding the two-dimen-sional, unsteady temperature field at a section normal to the weld line. A planar analysis may be used for this purpose when the weld speed, relative to a characteristic diffusion rate for the material, is sufficiently high so that the amount of heat con-ducted ahead of the weld torch is very small relative to the total heat input. In this case, the net heat flow across any infmitesi-mally thin slice of the weldment normal to the weld line is as-sumed to be negligible relative to the heat being diffused within

the slice itself; tha t is, the term dx3

k(T) dxz

/ _ - - WELD LINE

/ f /

/ / /

DIRECTION OF / / ELECTRODE TRAVEL / / //

I I

T

-AAJSA-YAAAA^

-SECTION ANALYZED

150 mm mrf

Flf. 1 Weldment configuration

the heat conduction equation. Two-dimensional thermal analysis at the section x3 = 0, normal to the direction of welding, is thus treated in the present work.

Perhaps the most critical input data required for welding thermal analysis are the parameters necessary to describe the heat input to the weldment from the arc. Not only the magni-tude, but the distribution of heat input will influence the dimen-sions of the weld metal and heat-affected zones, the cooling rates, and the peak temperature distribution, as well as the temperature gradients necessary to calculate stresses and distortions. The welding parameters required to formulate the heat flux boundary conditions are the magnitude of heat input from the arc Q, the distribution of the heat input, characterized by a length param-eter f, and the weld speed v. Introducing the arc efficiency !j, Q is found simply from the formula

Q = r]EI , (3)

where E and / are the arc voltage and current, respectively. The heat deposited per linear inch of weld is merely Q/v. I t must be recognized that an a priori determination of the magnitude and distribution of heat input cannot, in general, be made due to the lack of knowledge regarding energy transfer from the arc to the workpiece. Investigations of the physics of the welding arc are required to shed light on this area.

Following the approach of Pavelic, et al. [9], the heat from the welding arc is, at any given time, assumed to be deposited on the surface of the weldment as a radially symmetric normal distribu-tion function. Letting r be the distance from the center of the heat source, which is coincident with the axis of the electrode, the heat flux q is given by:

q(r) = qoC -Cr2 (4)

where q0 and C are constants determined by the magnitude and distribution of the heat input. The heat input parameters Q and f are defined by:

is neglected in

Q '2w I q(r) r dr */ o

q(f) = 0.05 g0.

(5)

(6)

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Equation (4) then becomes

9 ( r ) = ^

jV-1 increments. Consider yielding to be governed by the Mises criterion and plastic flow by the Prandtl-Reuss law. Then, as for example derived by Hill [14], the increment of plastic strain may be expressed in the form

3 < 3 . .

2er (15)

where $,/ = a,/ 1/3 ov* 8,7 is a component of deviatoric stress,

is the generalized stress, and &ePL is an in-* = (fSSj crement of the generalized plastic strain. A conventional power law relation is postulated to characterize the generalized plastic strain increment, such that

AePL(l) = maximum {(a(t)/KY -^PL(t - At); 0)] , (16)

where K and n are material parameters obtained from the uni-axial stress-strain curve and, in general, are temperature-depend-ent. Defining the yield strength cr, as the stress level at which the plastic strain in a uniaxial, isothermal test is 0.2 percent, K is evaluated by

K(T) =

Solid lines are representative of known data, while dashed lines depict assumed high temperature properties. The density of Alloy 600 is 8430 kg/m 3 and its melting range is 1630 K - 1690 K [16]. The latent heat of nickel, which is 309 kj /kg, is used for phase change calculations.

The thermal properties are extrapolated linearly to the melting temperature. The adequacy of this assumption is of course open to question, and only the experimental determination of the required high temperature data can serve to test its validity. The characterization of heat transfer in the weld puddle itself is an even more formidable problem. Until more is known about the mechanisms of heat flow in the puddle, only estimates of the molten metal thermal characteristics can be made by specifi-cation of appropriate values of conductivity and specific heat to be used in the finite element analysis. The heat capacity of the molten metal is assumed to be equal to its value at the melting point. The thermal conductivity, on the other hand, has been taken to be half that of the solid material at the solidus tempera-ture. This is based on the expected behavior of molten metal that is not in motion. The motion of the weld puddle, however, results in temperatures that do not greatly exceed the melting temperature. The relatively low puddle conductivity used ap-pears to have caused excessive temperatures to be computed, as will be seen in the next section. The effective conductivity above the melting point should realistically be somewhat greater than that of the solid material, in order to simulate the heat transfer mechanisms in the puddle, and thus to lower the molten weld metal temperatures to more realistic levels.

Since the strength and elastic modulus are negligible at the melting point, both these properties are assumed to decrease linearly from their respective values at the highest temperatures at which data are available, to extremely low values at 1630 K, The strain hardening characteristics of the material, character-ized by the parameter n, are not known at very high tempera-tures. A value of n 11, however, yields an analytical stress-strain relationship in good agreement with test data at 590 K, and is used to characterize hardening at all temperatures. Though the uniaxial stiffness becomes negligible as the melting range is approached, the material is relatively incompressible in the molten state. In order to model this behavior and, at the same time, to avoid possible ill-behavior of the solution process and thus facilitate convergence, the elastic modulus approaches zero and the Poisson's ratio one-half, in such a way that the bulk modulus is maintained at its room temperature value through the entire high temperature range, including the phase transition and the liquid state. Since compressibility has no effect on the generalized plastic strain, no plastic strain is, as expected, ac-cumulated in the weld puddle.

Analysis of a Butt Weld Consider a 2.54-mm thick flat weldment whose configuration

is shown in Fig. 1. Since the temperature distribution is sym-metric about the weld line xi = 0, only half the weldment need be modelled. The plane section normal to the direction of travel of the electrode is assumed to be in a state of plane strain. Thus motion is completely restrained in the weld, or longitudinal, direction. To illustrate transverse bending distortions that are characteristic of thin weldments of this type, the weldment is completely free to expand and bend in the transverse direction.

Though the plane strain approach prevents characterization of the longitudinal bow mode of distortion also common to these types of welds, the method nevertheless enables one to determine complete residual stress distributions, transverse distortions, and estimates of damage accumulated as a result of the welding process. I t should be noted, however, tha t longitudinal extension and bowing could be considered without resorting to a three-dimensional analysis by introducing up to three additional degrees of freedom to the system. This approach, called generalized plane strain, and used by Marcal in his analysis of curved pipe

bends [17], consists of relaxing the plane strain requirement by letting the longitudinal strain be a linear function of these added degrees of freedom, which represent longitudinal extension, and rotation about the xt and x2 axes of Fig. 1. For the present, however, the constraints associated with plane strain are main, tained.

The welding parameters chosen for this analysis are as follows: heat input Q = 703 W, characteristic radius of heat flux distribn. tion f = 5.08 mm, weld speed v = 2.12 mm/s. An accurate determination of both the magnitude and distribution of heat input from the welding arc is predicated upon knowledge of the heat transfer mechanisms from the arc to the workpiece. These values, however, were selected to illustrate the thermomechanical response of a full-penetration weld with a weld metal cross-section that varies in width through the weldment thickness.

The transient thermal analysis was performed using the finite element mesh shown in Fig. 3(a). A total of 47 time steps was used in the analysis from the time the section being analyzed (x3 = 0) is initially subject to heating from the welding arc, to complete cool-down. Temperature transients are plotted in Fig. 4, while the resulting weld metal zone, which is by definition that portion of the weldment in which the peak temperatures exceed the liquidus temperature, is depicted in Fig. 5. Tliese two figures demonstrate the variation through the thickness of the thermal response, resulting from the thermal energy from the welding arc being distributed over a finite portion of the

^ ^ H E A T FLUX ^ ^ - ^ X ^ ^ ,

(a)

y i i i i r i ~i ~i

(b)

Fig. 3 Finite element mesh: (a) Temperature analysis; (fa) Stress and distortion analysis

\ I ^ TOP SURFACE OF WELD J 1 __ BOTTOM SURFACE OF WELD

-0.1 0 0.1 0.2 0.3 0.4 0.5 TIME-MIN

f i g . 4 Temperature histories at various distances from the we IP certterlins

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WELD

WELD

Fig, 5 Weld metal zone

-ORIGINAL GEOMETRY

^-DISTORTED GEOMETRY

F ig . 6 T r a n s v e r s e d i s t o r t i o n of w e i d m e n t

heated surface of the weldment (as opposed to a line source of heat, which yields no thickness variations whatsoever, employed in previous investigations [3-6]). This localized temperature non-uniformity yields a variation of shrinkage through the thickness during cool-down, thus causing the characteristic bending dis-tortions, which are accompanied by a nonuniform accumulation of plastic strain in the weld metal and heat-affected zones.

In the weld puddle, calculated peak temperatures are con-siderably higher than the melting temperature (Fig. 4). As pointed out in the previous section, these apparently excessive temperatures arise from the relatively low value of molten metal conductivity used in the analysis. The cooling curves for weld metal material passing through the liquid-solid phase transition range are characteristic of solidifying alloys, in that the latent heat liberated as a result of the phase transformation results in a momentary cooling rate decrease during the freezing period.

The incremental elastoplastic stress and distortion analysis was carried out using the finite element mesh shown in Fig. 3(6). A total of 24 loading increments was used for the entire weld cycle, from initial heatup to final cooldown. Proper modelling of the bending distortion mode in the region in close proximity to the weld centerline requires a relatively fine grid both in the thickness and transverse directions. The final (residual) dis-torted shape of the weldment cross-section, evaluated when cool-down is complete and the temperature is uniform at its room temperature value, is shown in Fig. 6. The maximum deflection (at the weld centerline) is 0.066 mm.

The unconstrained motion of the weldment in the plane normal to the weld line produces transient and residual stress patterns that are dominated by the longitudinal stress. Longitudinal stress distributions at a number of points in time, including the residual stress distribution at final cool-down, are plotted in Fig. 7. Stress histories at various locations are plotted in Fig. 8. I t is of interest to note some of the characteristics of the transient stress distribution.

Upon initial heatup (prior to the time at which the center of the electrode passes over the section x3 0, being analyzed), the localization of severe temperature gradients in the immediate vicinity of the weld line produces compressive yielding in this

O 10 20 30 40

DISTANCE FROM WELD LINE-mm

F ig . 7 T r a n s i e n t a n d r e s i d u a l l o n g i t u d i n a l s t ress d i s t r i b u t i o n s

region. As more energy is being supplied by the arc and tem-peratures increase, the yield strength quickly decreases until, at the melting point, it is negligible. During the time period prior to solidification of the weld metal, all material outside the puddle is in compression with that region immediately adjacent to t h e molten zone in a plastic state of stress. Since temperatures in this region are extremely nonuniform, the yield stresses vary from zero at the melting point to about 260 MN/m 1 , which is only slightly lower than the room temperature yield strength (291 MN/m 2 ) . Plastic deformation in molten material has been completely relieved.

Upon solidification, the fusion zone material yields initially in compression at rather low stress levels. Upon further cooling, yield strength is increased and unloading from the yield surface proceeds elastically until the material yields in tension. In the weld metal and heat-affected zones, unloading and reversed yielding occur over a very narrow stress range. Hence, as cool-down proceeds, tha t portion of the weldment that is in tension longitudinally grows steadily until, at final cool-down, the residual longitudinal stresses, which are appreciable only in the region within about 50 mm of the weld centerline, are completely tensile. This is due to the plane strain restrictions placed on the analysis. Though relaxation of this constraint would yield residual stresses in compression outside this region, it is important to note that the tensile stresses in the highly stressed region of interest (within about 25 mm of the centerline) are governed primarily by plastic behavior and should change little.

The residual stress distribution is such that there is little variation of stress through the thickness. The residual stresses are essentially uniform at 390 M N / m 2 in the weld metal and heat-affected zones and then decrease uniformly to the room tempera-ture yield strength at a location 23 mm from the weld centerline. In the elastic region beyond this location, the stresses drop off quite rapidly. The residual stresses in the plastic region exceed the room temperature yield strength because of material strain hardening, which is such that the initial short-time compressive yielding produces an expansion of the yield surface followed, during cool-down, by yielding in tension at a stress level higher than the mono tonic yield strength. This type of behavior is consistent with the residual stresses measured for bead-on-plate welds by Nagaraja Rao and Tall [18]. I t was noted in this study that residual stresses in (or in the close vicinity of) the weld metal zone are usually about 50 percent above the yield strength of the base metal. The results presented here, therefore, appear to be in good qualitative agreement with these test data.

Plastic deformations are confined to a localized region as

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300

ZOO

100

0

100

200

-

-

-

STRESSES AT MfOSURFACE

SOLIDIFICATION, / i f MELTfNG^ \ /Jf

.,-0 25-, \ X/jr

\\ \ 1 f\ / "i"3 72 \

\ \ \ / / V / *,-U.43v /

v ~ r ~~ i

> ' ' ' ^ H ^ ^ ^ i i U U

/ / / - "

. y.1-31.75 /

\ _ _ ^ / , , , , , , , i , , , |

Fig. 8 Longitudinal stress histories at various distances from the weld center!?!?

0 5 IO 15 20 25 DISTANCE FROM WELD L I N E - m m

Fig. 8 Transient and residua! generalized plastic strain distributions

illustrated by Fig. 9, in which transient and residual generalized plastic strain distributions are plotted. Consistent with the plastic zone associated with the residual stress distribution, plastic strains greater than 0.2 percent are generated within 23 mm of the weld centerline. The highest plastic strain levels exist in the weld metal and heat-affected zones throughout the entire welding cycle, with the exception of that time period in which the weld metal is molten and the plastic strain nonexistent. Beyond the heat-affected zone the plastic strain decreases quite rapidly. The plots of plastic strain against time at various loca-tions in Fig. 10 illustrate the manner in which plastic deforma-tion is accumulated in the weld. In the weld metal zone, plastic strain is accumulated rapidly as the material yields in compres-sion. At the melting point, the plastic strain, which has reached a level of 2.3 percent, is completely relieved. Upon solidification, the plastic strain very rapidly reaches a level in close proximity to that existing just prior to melting. As cooling continues and yielding in tension progresses, the plastic strain steadily in-creases to its final value of 4.3 percent. In the heat-affected zone, plastic strain is accumulated during heat-up in much the same way as in the weld metal zone, but not quite as rapidly. At the very high temperature levels just below the solidus, the accumu-lated plastic strain is partially relieved. As the material cools, it yields in tension and further plastic strain is accumulated. In regions removed from the weld metal and heat-affected zones, there is no relief of plastic strain and elastic withdrawal from a plastic state of compressive stress is characterized by no accumu-lation of plastic strain until yielding in tension commences.

The generalized plastic strain, which characterizes permanent deformation, may be used as an indicator of cumulative "dam-age" in the weld metal and heat-affected zones during the welding process. Contours of the accumulated generalized plastic strain at final cool-down are plotted in Fig. 11 and, superposed with

curves defining the farthest extent of the liquidus and solidus isotherms (recall that the former defines the weld metal zone), indicate that there is considerable variation of plastic strain through the thickness of the weld in the fusion and heat-affected zones. The maximum residual plastic strain exists on the under-side of the weldment at the weld metal/heat-affected zone inter-face and reaches a level of 4.5 percent. Maximum damage can thus be expected to occur in this vicinity. This agrees qualita-tively with conclusions based on plastic strain distributions cal-culated by Hibbitt and Marcal [7],

Conclusions

A thermomechanical model of the welding process has been developed using the finite element method of analysis. The finite element approach has been shown to be a powerful tool both for determining the welding thermal cycle and for evaluating the stresses and distortions generated as a result of the tempera-ture transients. The analysis procedures are applicable to planar or axisymmetric welds under quasistationary conditions.

The method used for determining temperatures is featured by a direct iteration procedure to accurately account for the latent heat liberated during solidification of the weld. The finite ele-ment calculations enable, in particular, the effects of the heat input distribution on the heat flow patterns through the thick-ness of the weldment to be determined. The short-time thermal response, which yields the dimensions of the fusion and heat-affected zones, thus greatly affects the resulting nonuniform shrinkage in these zones.

The mechanical response is calculated by implementing the equations of time-independent incremental plasticity with tem-perature-dependent material properties. The transient longi-tudinal stress history of compressive loading followed by load reversal and a tensile residual stress state is produced. Peak residual stresses 30 percent in excess of the room temperature yield strength are predicted both in the fusion zone and in the heat-affected zone. A significant portion of the weldment outside these zones is in a plastic state of stress as well. Nonuniform shrinkage in the vicinity of the weld centerline produces the transverse bending distortions typical of these types of welds.

The plastic strains, or permanent deformations, accumulated during welding are maximum at the interface of the weld metal and heat-affected zones on the underside of the weldment. With plastic strain as an indicator of "damage" generated during welding, the analytical prediction that maximum damage occurs in this region is consistent with expected damage in full-penetra-tion welds. Thus, in addition to predicting residual stresses and distortions due to welding, the analytical technique can poten-tially be used for damage assessments as well.

Though experimental studies of weld temperatures, residual stresses and distortions have been made, none appears to de-scribe the weld process and the material properties in sutficient detail to adequately define the finite element thermal and me-

212 / A U G U S T 1975 Transactions of the ASME

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~

1 1 ' 1 PLASTIC STRAINS AT MIDSURFACE

MELTING.

,=0 25-v / /

J/y

SOLIDIFICATION-v /

^ - - . , - 2 79

^ , = 2 ? 9 mm

. , 3 ! 75-,

_ _ ^ ^ H

Fig. U Generalized plastic strain histories at various distances from th wId e tnMf -Htte

FARTHEST EXTENT OF UOUIDUS (I), SOUDUS 12) CONTOURS

043 .042 .Q4* 04g \ \ 043 .041 .038 .036 .034 .032 .030

Fig. U Contours of residual generaiized plastic strain

chanical models. I t is clear that further assessments of both models can be made only when numerical results are correlated with experimental data generated from test welds designed specifically for this purpose. Nevertheless, the potential of utilizing finite element, methods of analysis for determining the effects of the welding parameters, weldment geometry, material properties, heat sink and constraint conditions, etc. on the thermomechanical response, is evident.

Development of the weld thermomechanical model has also served to identify other areas of work that would yield a more thorough understanding of the welding process, in addition to providing more reliable input data to the finite element analyses. Among these are determinations of material properties at very high temperatures, studies of heat transfer mechanisms in the weld puddle, and investigations of the physics of the welding arc to determine the means by which thermal energy is trans-ferred from the arc to the workpiece.

Acknowledgment The author acknowledges the contributions of Dr. C. M.

Friedrich in the development of the incremental plasticity formulations employed for stress and distortion analysis.

References 1 Myers, P. S., Uyehara, 0 . A., and Borman, G. L., "Fun-

damentals of Heat Flow in Welding," Welding Research Council Bulletin No. 123, July 1967.

2 Masubuchi, K., "Control of Distortion and Shrinkage in Welding," Welding Research Council Bulletin No. 149, Apr. 1970.

3 Boulton, N. S., and Lance Martin, H. E., "Residual Stresses in Arc Welded Plates," Proc. Inst. Mech. Engrs., Vol. 133, 1936, pp. 295-339.

4 Rosenthal, D., and Schmerber, R., "Thermal Study -of Arc Welding," Welding Journal, Research Supplement, Vol. 17, No. 4, Apr. 1938, pp. 2s-8s.

5 Tall, L., "Residual Stresses in Welded PlatesA Theo-retical Study," Welding Journal, Research Supplement, Vol. 43, No. 1, Jan. 1964, pp. 10s-23s.

6 Masubuchi, K., Simmons, F . B., and Monroe, R. E., "Analysis of Thermal Stresses and Metal Movement, During Welding," Battelle Memorial Institute, RSIC-820, Redstone Scientific Information Center, NASA-TM-X-61300, N68-37857, July 1968.

7 Hibbitt, H. D., and Marcal, P. V., "A Numerical Thermo-Mechanical Model for the Welding and Subsequent Loading of a Fabricated Structure," Computers and Structures, Vol. 3, No. 5, Sept. 1973, pp. 1145-1174.

8 Boley, B. A., and Weiner, J. H., Theory of Thermal Stresses, Wiley, New York, 1960, pp. 42-44.

9 Pavelic, V., et al., "Experimental and Computed Tem-perature Histories in Gas Tungsten-Arc Welding of Thin Plates," Welding Journal, Research Supplement, Vol. 48, No. 7, July 1969, pp. 295s-305s.

10 Friedman, E., "A Direct Iteration Method for the In-corporation of Phase Change in Finite Element Heat Conduction Programs," WAPD-TM-1133, Bettis Atomic Power Laboratory, Mar. 1974.

11 Zienkiewicz, O. C , and Parekh, C. J., "Transient Field Problems: Two-Dimensional and Three-Dimensional Analysis by Isoparametric Finite Elements," Intl. J. Num. Methods Engr., Vol. 2, 1970, pp. 61-71.

12 Wilson, E. L., and Nickell, R. E "Application of the Finite Element Method to Heat Conduction Analysis," Nuclear Engr. and Design, Vol. 4, 1966, pp. 276-286.

13 Fung, Y. C., Foundations of Solid Mechanics, Prentice Hall, Inc., Englewood Cliffs, N. J., 1965, pp. 131-152.

14 Hill, It., The Mathematical Theory of Plasticity, Oxford University Press, London, 1967, pp. 38-39.

15 Zienkiewicz, O. C , The Finite Element Method in En-gineering Science, McGraw-Hill, London, 1971, pp. 374-377.

16 "Engineering Properties of Inconel Alloy 600," Technical Bulletin T-7, International Nickel Co., Huntington, W. Va,, 1964.

17 Marcal, P. V., "Elastic-Plastic Behavior of Pipe Bends with In-Plane Bending," J. Strain Analysis, Vol. 2, No. 1, 1967, pp. 84-90.

18 Nagaraja Rao, N. It., and Tall, L., "Residual Stresses m Welded Plates," Welding Journal, Research Supplement, Vol. 40, No. 10, October 1961, pp. 468s-480s.

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