Chapter 3

HEAT FLOW IN WELDINGTheoretical considerations

Thick plates

Traditionally analytic solutions to the problem of heat flow in welding have been obtained by considering heat flow from an ideal moving point heat source. Although solutions to this problem have been available for many years they are usually associated with the work of Rosenthal. Consider the case of an ideal point source moving on the surface of a semi-infinite block as shown in Fig. 3.1. The heat flow equation to be solved is

Figure 3.1Ideal point heat source moving on the surface of a semi-infi

where it is assumed the material constants are independent of temperature. For a moving source this can be solved by transforming to a set of coordinates moving with the source. Let the moving coordinate in the direction of travel be such that

= xvt

The heat flow equation then transforms to

31

If the source is well away from the edge of the plate a steady state is reached where the temperature relative to the moving coordinates does not change with time. Thus,

0Tv L c T = - 2 X v - g ^

The solution to this equation isT-T= - - - exp [-Xv(r+ 1)] where f + y* + z*

2irk r(3.1)

which gives the temperature distribution around the moving source.

This equation may the weld centreline

be differentiated to find the cooling rate. For example ony = 0 and equation 3.1 becomes

Q 1T-To =Z'jTk I

(3.2)IDEAL LINE HEAT SOURCE MOVING IN X DIRECTION WITH VELOCITY V

Figure 3.3Heat flow in plate of intermediate thickness.

differentiating

dtZirkv

(T-T)'

(3.3)

The term C l/v is often written H being the conventional heat input to the weld. Equation 3.3 is recognized as the familiar equation for cooling rate on the weld centreline for 3D flow in thick plates.

Thin platesFor thin plates consider a line heat source through the thickness of the plate

moving with velocity v as shown in Fig. 3.2.

32

H I

Using the same approach as before the temperature distribution relative to a set of moving coordinates is

e x p ( - X v i ) Ko (\vr)

(3.4)where Ko is a modified Bessel function of the second kind and zero order. If the argument of the modified Bessel function is large, i.e. if \vr>10 (valid for most conditions of welding except close to the arc or for very slow welding speeds) the function may be approximate by

Ko ( \v r ) ~ e x p (Xvr)Thus 3.4 becomes

(3.5)The cooling rate on the weld centreline (i.e. y=0) can be found by differentiating this equation giving

8T _2ffkpCgV . 3at 0^

(3.6)which is recognized as the equation for cooling rate for 2D heat flow.

For many practical purposes it is possible to consider plates as either thick or thin and use the appropriate equation (3.3 or 3.6) for cooling rate. More accurate equations for heat flow in intermediate plates (sometimes termed 2-5D heat flow) can be found by using the method of images.

X I Images spaced a t 2g

reol so&rce

plate of interm edite thiekneee

Images spaced a t 2g

Figure 3.4 Position of images for determining heat flow in a plate of intermediate thickness.

33

Intermediate platesConsider the point heat source on the surface of a plate of intermediate

thickness as shown in Fig. 3.3. The bottom surface of the plate can be considered to reflect heat back into the plate. This will have the same effect as if the surface were absent and an imaginary heat source were placed an equal distance away from the surface. Thus boundary surfaces may be taken into account by placing imaginary heat sources at appropriate positions. The temperature at any point may now be found by adding the temperature contributed by all the sources where the sources now are considered to exist in an infinite medium. I

For the bead-on-plate case the images will be placed as shown in

When the plate becomes thin, large numbers of terms have to be taken and the summing of terms is facilitated by a computer. The equation may also be differentiated to obtain the cooling rate and a typical result is shown in Fig. 3.5 where cooling rate is plotted against thickness. For heat sources moving with high velocity the results approach those obtained by Adams who assumed that the weld could be approximated by an instantaneous line heat source deposited along the weld line. It can be seen that the maximum error incurred by neglecting the 2 5D mode and using either 2D or 3D equations is about 20%.

Fig. 3.4.

The temperature at any point is therefore given by

O

0^ 30

Surface transfer

The foregoing approach has assumed that heat is lost by conduction only. This is a good approximation for thick plates and low heat input. For high heat inputs and thin plates, however, considerable losses of heat may occur from the surface by radiation, convection, and conduction to the air. Solutions may be obtained for the heat flow equation by assuming that surface losses are proportional to temperature. For thin plate (2D conduction) the temperature distribution is given by

(3.7)

Figure 3.6 Influence of surface transfer on the cooling rate (after Adams).

H H eat input (K J/ln )

Figure 3.7 Effect of surface transfer on cooling rate at 540C [h taken as 0.0004 cals/sec/cm^/C).

where h is the coefficient of surface transfer. The cooling rate is then obtained by differentiation. Adams has presented these results in the form shown in Fig. 3.6. From this diagram it is seen that surface losses can be neglected when

35

Values for h my be easily obtained experimentally by heating plates uniformly, allowing them to cool and measuring the temperature as a function of time. It is found that h is not independent of temperature as assumed above but decreases slightly with decreasing temperature. Some typical results for steel and for aluminum are given in Table 3.1. The effect of surface transfer on the cooling rate at 540C is shown in Fig. 3.7 for a range of plate thicknesses.

The value of h has been taken as 4X l(r cal/sec/cmVC for steel cooling in still air.

Tabl 3.1 Values of coefficient of surface transfer

Temperature C 100 200 300

Steel 2.7 3.7 4.7Aluminium 1.4 1.8 2.3

Values in cals/sec/cmV Cxio-"

Note: The values are for overall surface transfer. Since the value increases with temperature it is necessary to take a value appropriate for an average temperature of the cooling weld. For calculating cooling rates at 540 C a value of h at about 200 to 250C is appropriate. This is about 4 X 1 0 - ' cals/sec/cmVC for steel in still air.

PLATE OF FINITE WIDTH

Figure 3.8 Heat flow in a plate of finite width can be determined using the method of images.

Finite size platesThe foregoing theory of heat flow in welding has assumed that the plates

are infinitely large in the x and y directions. For a finite width plate with a moving source along the centre of the plate the method of images can again be used

to obtain a solution. To take account of the two edges of the plate image heat sources are placed on either side of the real source with distance from the edge as shown in Fig. 3.8. The temperature at any point is now given by the sum of the contributions from all the sources. Thus at a point on the weld line the temperature is

where r is the distance of the n" source from the point.

This basic method may be used to determine heat flow in more complex structures.

Effect of joint geometry

Simple cases of differing joint geometry may be considered by simply dividing the heat input according to the available heat sinks. For example the cooling rate in a flllet weld between plates of equal thickness made with energy H may be expected to be the same as that of a bead-on-plate weld made with energy %H.

Similarly a weld at the bottom of a deep groove in thick plate would be expected to cool twice as fast as the same weld on the surface of a thick plate.

Figure 3.9 Division of heat input between plates in fillet welds.

37

Fillet weldsThe case of fillet welds can be generalized to cover the condition of welds

between plates of differing thickness by using the method of images. Firstly, the two plates are considered as essentially separate and the heat input from the weld is divided between the plates. The division of heat H is assumed to be such that equal temperatures would be created at the position of the weld in the two plates. Clearly in the case of plates of equal thickness this is done by allowing %H to enter the bottom plate and enter the top. Using the equations presented for temperature distributions would lead to the same temperature in the bottom plate as in the top at the weld.

If the plates are of unequal thickness the division of heat must be found by iteration, i.e. assume a given division of heat, calculate the temperatures in the top and bottom plates and adjust the division of heat until these are equal. To calculate the temperature on the top and bottom plates for any given thickness the method of images can be used. Calculations have been made in this manner and are discussed more fully in a later section.

SPACED w

X XFigure 3.11 Distribution of images for calculating heat flow

in a twin fillet weld.

38

Twin fillet weldsThe method for calculating heat flow in fillet welds may be extended by

the method of images to the case of two fillet welds deposited simultaneously on either side of a plate. This is a common procedure in practice and is often used, for example, when welding a stiffener into a girder.

The temperature history in one of the welds may be found by adding the contributions from both the real sources plus the contribution from appropriate images placed to take account of the surfaces. The distribution of images is shown in Fig. 3.11.

Cooling rates using this method have been calculated for a range of plate thicknesses. The effective thermal properties derived from measured cooling rates for bead-on-plate tests were used in the equations and will be discussed

later. It is of interest to express the results in terms of the ratio of the cooling rate (at 540C) to the cooling rate of a single fillet weld made with twice the individual energy, i.e.

______cooling rate in twin fillet weld each weld energy H____cooling rate in single fillet of energy 2H (same plate thickness)

Results are shown in Fig. 3.12 and give the value of K plotted against the top plate thickness for two energy inputs. The ratio changes from 1 to 0.5 as the thickness increases. This means that when the plate is thin the two heat inputs add to each other and the joint cools like a weld with twice the individual heat. When the plate is very thick the two welds are independent and each

. cools as if it were deposited separately.

These results suggest that for thin top plates it is possible to simply add the two heat inputs together to calculate the cooling rate and this offers important practical opportunities for controlling the cooling rate without the need to resdrt to the use of preheat.

WEB THICKNESS (In.)

f l a n g e 1/4 INCH

Figure 3.12 Ratio of cooling rate in twin fillet weld to that in single fillet weld with twice the energy input plotted against thickness of 'web' member.

39

Corner welds

The treatment of twin fillet welds also provides the solution for corner welds since the twin fillet case is symmetrical about a plane through the middle of the upper plate and one side of this exactly represents the corner weld (Fig.

Figure 3.13 Treatment of corner welds.

Effect of plate thickness

A popular way of taking in account the effect of thickness and joint geometry is by using the concept of combined plate thickness. This is the sum of the thicknesses of the heat paths frm the joint - there being three heat paths from

COMBINED THICKNESS ( tn i j

Figure 3.14 Cooling rate plotted against combined plate thickness. For thin sections the cooling rate is uniquely determined by the combined plate thickness.

a fillet weld and two from a butt weld. The previous results of cooling rates in fillet welds between plates of differing thickness have been plotted against combined plate thickness in Fig. 3.14. Up to a combined plate thickness of about

in. the cooling rate is uniquely given but beyond that the cooling rate depends on the individual plate thicknesses. Combined plate thickness is therefore a useful concept for thin material (in fact where heat flow is 2D) but may give misleading results for thick material where the plates are of greatly differing thickness.

TEMPERATURE C

Figure 3.15 Variation of thermal conductivity with temperature in steels.

Limitations to the theory

There are several, limitations to the point heat source theory. First it is assumed that the thermal properties of the material are independent of temperature. This is not the case and the wide variation in properties over the temperature range involved in welding is seen from the figures. A further complication is that heat is evolved or absorbed during phase changes. In steel a number of phase changes occur (liquid to solid, ferrite to austenite, etc.) which will affect the heat flow in a weld. A major assumption in the theory is that the weld can be represented by a point source. This is a useful approximation when considering points well away from the weld or at long times after welding. It leads to considerable errors, however, at points close to the weld and does not lead to

41

an accurate description of the behaviour at high temperatures. Many of these effects can be included by using numerical methods to obtain solutions. Modern computer techniques have been used to include the effects of changing material properties, phase changes and distributed heat sources.

Figure 3.1 6 Variation of specific heat with temperature in steel.

To apply heat flow solutions it is necessary to know the heat input. Normally this is done by assuming it to be a fixed percentage of the energy input, i.e. the electrical energy consumed per unit length of weld. This arc efficiency is assumed to be constant for a given process. Many attempts have been made to measure the efficiency and some typical results are shown in Fig. 3.17. They show a wide variation between processes but some processes such as submerged arc show a high efficiency presumably because the heat used to melt the flux is conducted into the weld because of the close contact between the flux and the weld and the insulating effect of the loose flux above.

Measured values of cooling rate |Many experiments have been carried out to determine the cooling rate in |

welds. The results have generally been presented either as empirical equations or have been compared with the theoretical solutions previously discussed. Some of the more recent results are given below.

Bead-on-plateDorschu studied cooling rates in beads made with the argon shielded metal

arc process on a 50 mm (2 in.) plate. His results are essentially applicable to thick plate. He found that the cooling rate could be expressed as

dT _ 3.77X10-niOOO-To)^ 1000Fdt E

42

kVA

I 2 4 6 10 20 40

ARC ENERGY (EI/4*I8), Cal/sec.

Figure 3.17 Typical values for arc efficiency of various processes (after Christensen).

The results showed good agreement with the theoretical form of the equations.

In a comprehensive study Signes measured cooling rates in plates of various thicknesses using various initial plate temperatures (preheats) and several different processes. The results were plotted in parameters suggested by the dimensionless parameters used by Adams. The cooling rate relative to that in a thick plate was plotted against a function of the thickness. For plates where surface transfer became important the cooling rate relative to that where no

43

surface losses occurred is also plotted as a function of the thickness and energy input. These data enabled the cooling rate for argon gas metal arc welding tobe expressed in the form

dTd t l 3 0 0

= 3.02X10-

ACT

UA

L CO

OLI

NG

RAT

E 3-

D C

OO

LIN

G R

ATE

Figure 3.19 Correction factor , for 2-D heat transfer (after Signes).

= p2 ( | 3 0 0 - T o) / E v^

Figure 3.20 Correction factor ^ when surface transfer is significant (after Signes).

45

Gravine measured cooling rates at various temperatures for a range of plate thicknesses and energy inputs using the submerged arc process. He found good agreement with the theoretical form of the equation when written

d T _ (T-ToV

and

D zd

for thick plate

dt

(T -T of

dT

dtfor thin plate

at 700C is shown in Fig. 3.21 and at 200C is shown in

l / E ( k J / i n ) " ' X 10-2

Figure 3.21 Value of A 3 0 at 700C against a function of energy input (log plot) (after Graville).

Fig. 3.22. The values of Bao and Bzd were found to be temperature dependent and changed abruptly during the transformation of the weld metal from austenite to a ferritic structure. This is shown in Figs. 3.23 and 3.24, and can be interpreted as an increase in the effective thermal conductivity during transformation. In this work the change between 3D cooling and 2D cooling was fairly abrupt and all data could be treated as either 3D or 2D.

46

2D

ot

20

0'C

C-2

M

C.

g ^ / I n ^ . kJ X 1 0 ^

Figure 3.22 Value of .42o at 200C against a function of E and g (after Graville).

8 0 0 6 0 0 4 0 0 2 0 0TEMPERATURE C

Figure 3.23 Value of Bzo as a function of temperature (after Graville).

Figure 3.24 Value of 8 2 0 as a function of temperature (after Graville).

47

Fillet weldsCottrell, Bradstreet and co-workers made measurements of the cooling rate

in controlled thermal severity (CTS) tests. These tests are described in detail in Chapter 6 but in simple terms consist of a fillet weld about 80 mm long between two plates. The thickness of the plates controls the thermal severity of the joint. An empirical equation for cooling rate at 300C was determined from the results and applied to manual covered electrodes.

Bailey has made measurements of cooling rates in CTS tests and his results agree well with Cottrells formula. The results were also compared with cooling rates measured in longer welds in joint simulation tests (see Chapter 6). Although the weld in the CTS test was only 80 mm in length good agreement was found between the two sets of measurements. It thus appeared valid to apply the CTS cooling rate formula to real structures containing longer welds.

From the previous theoretical discussions, it would be expected that a fillet weld would cool 1.5 times as fast as a bead-on-plate weld for thick material. For thin material a factor of 2.25 would be expected since the cooling rate is proportional to 1/E^ Inagaki et al presented data showing a factor of 1.4 which is close to the theoretical but their data indicates a value of 1.4 even for thin material which is not in agreement with the theory. Results of tests by Ried fitted the theoretical values quite well supporting the idea that a fillet weld between plates of equal thickness can be treated by taking % of the heat input and regarding it as a bead-on-plate weld.

Cooling rates for fillet welds between plates of differing thickness have been calculated using the method of images. In the theoretical equations for temperature distributions the thermal properties of the material were replaced by effective values taken from the data shown in Figs. 3.23 and 3.24. It was assumed that the effective values remained constant down to a temperature of 500C. Cooling rates determined in this manner for fillet welds between plates of equal thickness are presented in Figs. 3.25 and 3.26, and compared with the experimental results of McParlan. Good agreement exists and suggests that the method of calculation is valid for fillet welds. Cooling rates for various combinations of plate thicknesses determined by this method are given in Figs. 3.28 and 3.29.

Cooling rate chartsThe equations described above can be conveniently presented in graphical

form involving the arc energy input, and thickness and the initial plate temperature. The graphs presented are those based on the experimental results with submerged arc welding. These are likely to be more reliable than other results because of the more consistent arc efficiency. Cooling rates for other welding processes can be determined by assuming a certain arc efficiency. The cooling rates quoted are at 540C and the reason for this will become apparent in Chapter 5. The corresponding cooling time between 800-500C is also given approximately by

s800-500 217R 5 4 0

for 3D heat flow

andSaooj for 2D heat flow

I 5 4 0

These relations assume the initial plate temperature is 20C and should be regarded only as approximations.

Charts are given for the cases of bead-on-plate, fillet welds, and twin fillet welds. The foregoing discussions enable approximate extension to other cases such as groove welds.

For completeness the initial plate temperature is included in the graphs for bead-on-plate welds. For moderate levels of initial plate temperature this may be interpreted as equivalent to the preheat level. Since the cooling rates are at 540C they will also be accurate enough for practical purposes even if the preheat is a local one limited to the immediate weld zone. The lines for preheat are only valid for thick plate.

Figure 3.25 Cooling rates in fillet welds for thick plates of equal thickness (after McParlan).

49

Figure 3.26 Cooling rates in fillet welds in thin plates of equal thickness (after McParlan).

Cooling rates in groove welds

Despite the practical importance of groove welds very few results of cooling rates in such welds have been published. Details of available data in the literature will not be given but in general the results show that the cooling rate in the bottom of a single V is approximately the same as for a bead-on-plate. For the later passes of groove welds it also appears that the cooling rate is not much different from that in a bead-on-plate.

For double Vee or Y grooves the root pass cooling rate may be higher than a bead-on-plate. One report shows the double Vee to b.e similar to the bead-on- plate. Comparison of HAZ hardness in a Y groove with that in a bead-on-plate in the same steel indicated that the cooling rate in the root run of a Y groove was about 1.6 times that of a bead-on-plate. The theoretical considerations previously discussed suggest that in a very deep sharp groove the cooling rate could be as high as 2 times the bead-on-plate rate.

The effect of thickness is likely to be more complex in a groove weld than a bead-on-plate. If the root run is in the centre of the plate, the thickness seen by the weld is half the plate thickness. If cooling rates are to be calculated for groove welds by dividing the energy by a fixed factor (say 1.6) and then using bead-on-plate equations, the thickness term should therefore be half the real plate thickness. For very thin plates or cooling rates at very low temperatures the weld can be treated as a simple bead-on-plate.

50

I

Coo

ling

Rot

e ot

54

0C

C

*C/S

ec)

200

150

100

70

50

40

30

20

10 9

8

7

Figure 3.27 Graph to determine cooling rate in bead-on-plate for submerged arc process.

I

51

30

40

50

60

70 8

0 90

100

150

200

98

76

54

3

2E

nerg

y In

put

For

Sub

ore

(KJ/

inJ

Coo

ling

Rat

e at

54

0C

(

C/S

ec)

Co)

COOLING RATE AT 5 4 0 C { C / * e 4

Singit FliUt W tldt Wtb and Plonge o f equol

thlcknese

Figure 3.28 Graphs to determine cooling rates in fillet welds, (a-f)

(t>)

COOLING RATE AT S 4 0 C (C/tAC.)

Single Fillet Welds Flange 1/4 in.

For vorious web thicknesses

52

COOLING RATE AT 5 4 0 * C ( C / m c J

(0

SIngiG F illet Welds Flange 1/2 In.For various web thicknesses

t 2 S 4 S S 7 S S I 0 2 0 3 0 4 0 9 0 iOO 200

COOLING RATE AT 9 4 0 * C ( * C / t c J

(d)

Single Fillet Welds Flange I In.For various web thicknesses

COOLINO A TE AT 5 4 0 * C ( C /tM O

SIngli FIIU1 WI

^ Cooling rate at the end of a weld

The previous theoretical considerations and the majority of reported values for cooling rates have been applicable to the middle of very long welds. Under these conditions steady state conitions exist with respect to co-ordinates moving with the source. At the edges of plates or at the beginning or ends of welds non-steady state conditions exist and the cooling rates will be different. The cooling rate at the end of a weld away from the edge of a plate will be twice that at the middle of the weld.

Cooling in the low temperature range

The cooling rate in the lower temperature range (typically 300-100C) is of great importance because of its relation to cracking and the effect of preheat. Heat flow theory for determining cooling rates or times in this range will differ from that presented above in one main respect in that the initial plate temperature (i.e. the preheat) is generally not uniform (i.e. the preheat is local). There is, therefore, continuous loss of heat from the plate by conduction in the plates and surface transfer until the plate eventually reaches the same temperature as the environment, i.e. room temperature. For the majority of cases the problem can be considered 2D, particularly for thin plate. Thick plate, preheated from one side bcomes a 3D problem.

55

ENER6Y

INPUT

Kl/ln.

ENERGY

INPUT

R J /ln .

COOLING RATE AT 9 4 0 * C (* C /t e .)

(C )

Twin F illet Welds

Plonge 1/2 In.

For vorloue web thicknesses

COOLING RATE AT 9 4 0 * C ( C / l c )

( d )

Twin Fillet Welds Flange I in.

For vorious web thicknesses

57

Uniform preheat

If the plate has a uniform preheat a weld deposited on the plate will cool until it reaches the preheat level. It is assumed that the preheat itself does not decrease. This case can be treated using the equations developed earlier for 3D and 2D where surface heat losses are included. The effective values of the physical properties of steel at the lower temperatures can be obtained from Figs. 3.23 and 3.24. For example, using the equation for cooling time between two temperatures in the 2D case and taking a value of h = 0.0004 the cooling time between 300C and 100C is shown in Fig. 3.30 a) and b) for an initial plate temperature of 20C and 50C. The equations would be invalid for higher preheat temperatures.

For preheats higher than 100G the cooling time obviously becomes infinite.

59

Figure 3.30 Cooling time in uniformly preheated plates, a)

60

Cooling of preheated plates

If the plates are heated uniformly and the heat from welding is neglected (i.e. long after the weld is made so that the heat from welding has spread well away from the joint) the preheat will decrease because of loss of heat by surface transfer, and conduction in the plates will not play a part. The time to cool between 300C and 100C is therefore simply given by

. 200 Cpg 180 h

For example the time to cool from 300-100C for a 300C preheat is shown as a function of thickness in Fig. 3.31.

61

Figure 3.31 Cooling of preheated plates in still air. Preheat of 300C.

Locally preheated plates

For the case of locally preheated plates the thermal cycle can be found by determining the temperature due to the preheat and that due to the weld separately as a function of time. These can then be added to give the total temperature as a function of time. Consider the case where an infinitely large plate is preheated along a line and a weld deposited on this line. The preheat and the weld both cool by conduction into the plate and by surface transfer to the surroundings. An approximate expression for the temperature due to the weld (thin plate equation) is

The temperature due to the preheat cooling isht (yu)^

----------------- ^ o c i ------

T=e ^ = 1 f(u)e 4't

where f(y) is the initial temperature distribution due to the preheat. Attention is restricted to the weld line where y=0 and the initial distribution is assumed to be gaussian; _yz

f(y) = Tpe

where Tp is the preheat temperature and a is the characteristic width of preheating. The temperature is then simply given by

htT = T Akg (2a't-Ho )^

The total temperature is then the sum of that due to the weld and that due to the preheat. These equations have been used to determine the effect of preheat on cooling time and are shown in Fig. 3.32. A width of preheated region of 2a = 30 cm has been used.

LO

CA

L

PR

EH

EA

T

"Cr 2000I

COOLING TIME 3 0 0 - lO O 'C ( s e c s . )100 1000

1 ~~i-----1 1 1 1 Ml \ 1---- 1Ir i~i I I

COOLING TIME TO l O O ' C

Figure 3.32 Relation between preheat temp, and cooling time for locally preheated plates of various thickness for various heat inputs, (a-e)

63

LO

CA

L

PRE

HE

AT

*C

LO

CA

L PR

EH

EA

T

*C

(000 2000 I

COOLING TIME 3 0 0 - IOO*C ( s tc s .)

(00I' I I-------- r I I i I I t ......f................ , nn I , , ---- 1 r 1 1

COOLING TIME 3 0 0 - l 0 0 * C ( s e c s .)100 1000 2000

I ' I 1-------------1---------1------- ! I I I I ----------------~ T 1

64

(c) COOLING TIM E TO lO O 'C

LO

CA

L

PRE

HE

AT

*C

COOLING TIM E 3 0 0 - IOO C ( s e c s .)100 1000 2000

r I-----------1-------1------ 1 I rTn -------------------------------1 1 I I I I I I

( d ) COOLING TIM E TO I 0 0 C

COOLING TIM E 3 0 0 - IO O C ( s e c s .)

100 1000 2000T -I r -T r -T - i-------------------------1 1 1------- 1 I I T - r i 1-1

65

References for Chapter 3

Carsiaw, H.S., and Jaegar, J.C., Conduction of Heat in Solids, 2nd ed.. Oxford Press, London, U.K. (1959).

Rosenthal, D., ASME Trans., pp 849-866 (November 1946).

Myers, P.S., Uyehara, O.A., and Borman, G.L., "Fundamentals of Heat Flow in Welding WRC Bulletin No. 123 Quly 1967).

Jhaveri, P., Moffat, W.G., and Adams, Jr. CM., Welding Journal 41 (1) Res. Suppl, pp 12-s to 16-s (January 1962).

Wells, A.A., Welding Journal 31 (5) Res. Suppl, pp 263-s to 267-s (May 1952).Christensen, N., Davies, V., and Gjermundsen, K., Brit. Weld. Jnl. 12 (2) pp 54-75

(February 1965).

Dorschu, K.E., Welding Journal 47 (2) Res. Suppl, pp 49-s to 62-s (February 1968).Signes, E., Welding Journal 51 (10) Res. Suppl, pp 473-s to 484-s (October 1972).Graville, B.A., Welding Journal 52 (9) Res. Suppl, pp 377-s to 385-s (September

1973).Cottrell, C.L.M., and Bradstreet, B.J., Brit. Weld. Jnl. 1 (7) pp 322-326 (July 1954). McParlan, M., Dominion Bridge Report 280C/6/72.