3d finite element modeling of the welding process ...

3D FINITE ELEMENT MODELING OF THE WELDING PROCESS USING ELEMENT BIRTH AND ELEMENT MOVEMENT TECHNIQUES

Ihab F. Z. Fanous

Dr. Maher Y. A. Younan

Dr. Abdalla S. Wifi

Mechanical Engineering Department The American University in Cairo, Egypt [email protected]

Professor of Mechanics and Design Mechanical Engineering Department The American University in Cairo, Egypt [email protected] Tel: (202) 797-5336, Fax: (202) 795-7565 (contact author)

Professor Mechanical Design and Production Department Cairo University, Egypt [email protected]

ABSTRACT The modeling and simulation of the welding process has been of main concern for different fields of applications. Most of the modeling of such a problem has been mainly in 2D forms that may also include many sorts of approximation and assumptions. This is due to limitations in the computational facilities as the analysis of 3D problems consumes a lot of time. With the evolution of new finite element tools and fast computer systems, the analysis of such problems is becoming in hand. In this research, a simulation of the welding process with and without metal deposition is developed. A new technique for metal deposition using element movement is introduced. It helps in performing full 3D analysis in a shorter time than other previously developed techniques such as the element birth.

INTRODUCTION Many models for the welding simulation have been developed in the past few years. Most of the models had to include some approximations in order to avoid long computing time and geometrical non-linearity. Also, most of them were intended for special applications in which reducing the model from 3D to 2D, for example, could be a valid assumption. On the other hand, some 3D models were developed in which approximation were applied to the material behavior at elevated temperatures, and without including the metal deposition. Nguyen, Ohta, Matsuoka, Suzuki and Maeda [1] have developed an analytical procedure for evaluating the transient temperature profile during the welding process. They used the Goldak’s [2] formulation of the double-ellipsoidal heat source and compared the results with an experiment that he conducted. They assumed that there is no heat loss neither through convection nor radiation from the surfaces of the plate, which lead to some discrepancy in temperature predictions. To calculate the residual stresses, Ueda and Yuan [3] and Mochizuki and Hattori [4] used the inherent strain method. It is based on the assumption that the inherent strain of a complicated welded structure can be approximated by another of a similar simpler structure. The inherent strain is affected by other parameters such as the material of the base metal, the difference between the material of the base metal and that of the filler, the welding speed, the amount of heat input, etc. Hence, to use the inherent strain technique, a database of the profile for different parameters should be developed. However, for a complicated structure, the use of inherent strain method becomes inaccurate. Dong [5] has developed a model utilizing the element birth technique to simulate the metal deposition that is valid in the cases with no thermal analysis of the sudden large temperature variation. This model is a combined two 2D models with a small allowable variation in the results along the thickness. This concept eliminates the effect of the heat transfer through the thickness of the plate, which is of great

NOMENCALTURE Heat flux at a distance r from the centre of the q (r ) 2 heat source (W/m ) Total heat input rate (W) Q The radius at which the total heat input is 95% of rb the actual value (m) Convection coefficient (W/m²·ºC) hconvection

ε em

Emissivity

σ bol

Stefan-Boltzman constant (5.669×10 W/m²·ºK)

Ti

Expected temperature for current increment (ºK)

-8

Ta

Ambient temperature (ºK)

ti v

Time of step i (s) Welding speed (m/s)

1

model shall be used as a reference for verification of “Model 3” utilizing the element movement technique that is developed in this research.

effect in case of the thick plate welding. The technique used for developing a cross-sectional model applies only for a straightline welding. However, Dong’s technique could give a general overview of the residual stresses due to welding. Hibbit and Marcal [6] have performed an advanced modeling procedure for a complete 3D simualtion of the welding process. The loading of the welded structure due to operation conditions, temperature dependant material properties and phase transformation were accounted for in their model. Friedman [7] developed a comprehensive two-dimensional analysis. Noting that the temperature profile does not vary with time but moves at constant speed along the welding line, the problem size was reduced so as to evaluate the temperature profile at a section perpendicular to the welding line. Detailed analyses for 2D and 3D welding using the ADINA software were shown by Wilkening and Snow [8]. They considered most of the non-linear aspects of the analysis including temperature dependant material properties and metal deposition. The element birth technique was used to simulate the metal deposition. The moving heat source was modeled using a time curve for nodal heat flux. In the present work, a new technique of element movement is developed for full 3D simulation of the welding process. The standard commercial code ABAQUS is used to model this problem as it has been used extensively in such a highly non-linear problem in previous researches. In the first model considered in this research, the dimensions and thermal load values are acquired from the research done by Friedman [7] for comparison and verification. In this case, simple butt welding of two coplanar plates without considering metal deposition is simulated. In the other two models that are considered in this research, larger thicknesses are assumed to include the effect of adding filler material in the weld pool between the plates. Three different boundary conditions shall be modeled comparing their effect on the welding.

Verification Model (Model 1) The model simulates basic arc welding of two coplanar plates along the parting line as illustrated in Fig. 1. The model is developed similar to that of Friedman [7] so as to be able to verify the subroutine of the heat source and heat loss in comparing the thermal history, and the structural boundary conditions in comparing the residual stresses.

z

y

x

Figure 1: The diagram of the welding process of case 1. Each plate has a length of 100 mm (x-direction), width of 50 mm (y-direction) and height of 2.5 mm. The welding speed is 2 mm/s. The electric input is 24 V and 30 A, and the arc efficiency is assumed to be 90%. The process is modeled using one plate upon which symmetry loading and boundary conditions are applied. Parametric meshing is used in order to easily track the results along a certain predefined path in any direction. The element used is an 8-noded brick element that can perform a coupled displacement-temperature analysis. For thermal symmetry, the heat flux passing across the surface of symmetry shown in Fig. 2 is assumed to be zero, and, for structural symmetry, the translation in the y direction of the same surface is also zero. Besides, for structural stability of the model, fixture point 1 is constrained in the x and z directions, and fixture point 2 is constrained in the z direction.

FINITE ELEMENT MODELS In developing a general purpose model for the welding process, it is important to consider the moving heat source, heat loss, temperature dependant material properties and metal deposition. A moving heat source is modeled by setting a heat flux distribution that varies with time applied to the top surface of the weld pool zone. Using ABAQUS, it is simulated by developing a user-defined subroutine to which time and position of interest are passed as parameters and returns the heat flux accordingly. Similarly, another subroutine is developed to simulate the heat lost from the top surface of the body. These subroutines are verified in “Model 1” which simulates the welding process of two plates, without considering metal deposition, and compares to the results of Friedman [7]. Also, the material properties are entered to the model for different temperature values as described later in this paper. The verified subroutines of the heat load and heat loss and material data are then used to simulate another welding process that includes the addition of filler material to the base plate. Initially, “Model 2” is developed to simulate the process utilizing the element birth technique to simulate the metal deposition as has been used in previous researches. This

Fixture point 2

Monitoring points

Welding line Surface of symmetry

Far surface Fixture point 1

z Mid section

x y

Figure 2: General boundary conditions The moving heat load is applied as distributed heat flux to the top surface of the model, Fig. 3. The region within which the heat is applied has a circular shape assuming the heat source is applied perpendicularly to the plate without any inclination. A user subroutine named DFLUX is developed in [9] using the FORTRAN language and included in the model to calculate the heat flux at a certain time and location within the surface of elements upon which the load is applied according to equation (1).

2

⎛

⎞

3Q −3⎜ r r ⎟ q (r ) = 2 e ⎝ b ⎠ πrb

Therefore, when the value of r is less than or equal to r b , the heat flux is calculated according to (1). Otherwise, the heat load is set to zero. r b is set to 5 mm according to Friedman [7]. The thermal boundary conditions include the radiation and convection to the environment from all sides of the welded plate except the symmetry surface and the area upon which the heat is applied. For all sides of the plate that loose heat, the heat lost is calculated by

2

(1)

The total heat input Q is evaluated according to the type of heat source. For example, in electric arc welding, Q = η VI

(2)

(

q = hconvection (Ti − Ta ) + ε emσ bol Ti 4 − Ta4 0s

1s

2s

3s

4s

5s

(

htotal = hconvection + ε emσ bol Ti 3 + Ti 2Ta + TiTa2 + Ta3

q = htotal (Ti − Ta )

(3)

(4)

The value of t0 is the time taken for the centre point of the heat to reach the first node along the welding line that has a value of zero in Model 1. This way, as the time increases, xh increases simulating the motion of the circle of the heat load zone as shown in Fig. 3.

Heat source area

Heat loss area

y

(6)

(7)

Element Birth Technique (Model 2) To broaden the application of the modeling procedure, the problem of Model 2 simulates arc welding of two coplanar plates with the addition of a filler material between them. By developing several trial models, it was found that a length of 100 mm (x-direction), a width of 50 mm (y-direction) and a thickness of 5 mm (z-direction) are adequate for a single pass welding process. The welding speed is assumed to be 1 mm/s. The metal deposition of the filler material is considered using the element birth technique. This technique is based on deactivating and reactivating the elements of the weld pool as the welding progresses. The meshing of the base plate and the weld pool has a clear parting surface between them. That way, when the elements of the weld pool are deactivated, the remaining elements would have the initial shape of the base plate with its modified edge that forms the cavity of the weld pool. The meshing in the weld pool is fine enough to account for the high temperature gradient calculations and that in the base plate has similar meshing characteristics described in Model 1 with the elements near the fusion surface having a size close to the ones in the weld pool. The elements defining the weld pool are grouped to form slices. Having the elements of the weld pool initially deactivated, every slice is then reactivated as the heat sources moves along the welding line as shown in Fig. 5. The nodes that appear due to the reactivation of the elements have an initial temperature above the liquidous temperature.

where

x h = (t − t 0 )v

)

such that the total heat loss in equation (5) becomes

This distribution, according to Friedman [7], represents 95% of the total heat Q when applied within a circle with radius r b . The distance r in equation (1) shown in Fig. 4 is the distance from the center point of the heat source to the point for which the heat flux is being calculated and is given by

(x − x h )2 + y 2

(5)

The convection coefficient is 8 W/m2·ºC, which assumed to be constant as varies primarily with ambient temperature, and the emissivity is 0.5. Another subroutine called FILM is developed in [9] in conjunction with DFLUX to account for the variation of the heat loss coefficients with time for the top surface. In this code, the location under consideration is checked if it lies within the circle of application of the load using equations (3) and (4), and, if it is true, there is no heat loss. Otherwise, heat is lost by the same coefficients as the other sides from the area of the top surface other than that of the heat load. The coefficient calculated in the FILM subroutine is a combination of both the convection and radiation coefficients which is given by

Figure 3: The moving heat source of Model 1.

r =

)

r xh x

Figure 4: Zones of heat load and heat loss.

3

Three different sets of boundary conditions are considered in order to check the response to the change in the structural and thermal boundary conditions. In the first boundary conditions set, the plates to be welded are intended to be help in place at the far surface end indicated in Fig. 6, after which the plates are released and checked for residual stresses. In the second boundary conditions set, the plates to be welded are simulated to be part of a large structure so that the plate may expand freely but with no rotation allowed even after the welding process is finished. The same thermal load in set 1 and set 2 is applied. In the third set, the structural boundary conditions are the same as that of set 1 but with an decrease in the thermal load. When considering the structural boundary conditions, in boundary condition set 1, the far surface shown in Fig 6 is totally constrained in the x, y and z-directions throughout the whole welding process and then released after complete cool down to check for the residual stress at no load. For structural stability, fixture point 1 is constrained in the x and z directions, and fixture point 2 is constrained in the z direction after the complete constrained of the far surface being released. Although this might affect the shape of the weld pool cavity before metal deposition since the base plate is free to expand towards the weld pool (y-direction), the newly activated elements take the new shape since they share common nodes with the base plate. In boundary conditions set 2, slight movement of the far surface in Fig. 6 is allowed but with no rotation about any axis which is simulated by forcing the displacement of all the nodes on the far surface to be the same. This is modeled by using a constraint equation that couples the y-component of the displacement of each two consecutive nodes on the surface. Therefore, with reference to Fig. 6, the y-displacement of node 1 is set to be equal to that of node 2 and that of node 2 is set to be equal node 3, etc… This way y-displacement of the all the nodes would be equal starting from node 1 to the last node on the surface. Finally, in boundary condition set 3, the structural boundary conditions are similar to those of set 1. However, the thermal load is changed by altering the total heat input from 1300 to 900 watts.

0s

2s

4s

6s

8s

The heat source in this model is similar to that used in Model 1. However, it initially stays a while at the beginning of the welding process for the filler material to start melting. Then the heat flux starts to gradually impose its effect as shown in Fig. 5. Note that there is no heat load at the initial state. This effect is modeled by using the DFLUX and FILM subroutines as developed in Model 1 having the initial time t 0 greater than zero, t0 =

rb v

(8)

At time zero, x h equals to −rb . This makes the heat source just out of the model at the beginning of the analysis with the first group of elements being active. As the heat source moves and starts to go out of the current active element slice, the next slice of elements is activated, as illustrated in Fig. 5. The total heat input rate is 1300 W. Last node

Fusion line

Welding line

Node 3

Surface of symmetry

Node 2

z

Node 1

x Far surface

Mid section

Monitoring point

y

Figure 6: General boundary conditions

The analysis procedure is divided into several steps. In each step, a new group of elements is activated. Which means that every step performs analysis over a time period t i = x i v where x i is the length of the group of elements (along the xdirection) to be activated in that step. A step is added when all elements are activated to allow for heat loss only over a long period of time (40 minutes) simulating cool down. Finally, an extra step is added in which the fixation of the far surface is released, as mentioned before, to be able to check for the residual stresses at no load. It is important to note that the automatically estimated time increment in the analysis drops to a very small value due to the vast difference in the temperatures of consecutive nodes in the reactivated elements. This effect can be dramatic for small element sizes of the weld pool at the fusion surface (the parting surface). In addition, the part of the heat passed to the base plate is neglected in this model. Therefore, it is important to gradually transfer heat from the filler material to the base plate to smoothen the contact between the two parts.

Element Movement Technique (Model 3) The meshing in this model is similar to that in Model 2. However, the elements of the weld pool are separated from those of the base plate so as to be free to move as shall be discussed in the next section. Parametric meshing is used in both parts in order to be able to handle nodes along a certain path or on a certain plane within the two parts. In order to impose the gradual heat transfer effect, the part of the weld

10 s

Figure 5: The moving heat source of Model 2.

4

temperature change limit may cause loss of accuracy and smaller value would lead to longer analysis time). Therefore, the thermal conductivity of the gap elements gradually increases from the initial value of zero at a certain gap clearance to the maximum value at zero clearance according to the values shown in Fig. 8 (b). This way, when the elements of the weld pool reach those of the base plate, the temperature of the two coincident nodes shall become approximately equal. This early interaction accounts for the heat lost from the molten metal to the base plate while falling into the weld pool. Also, by increasing the value of thermal conductivity in the range before contact, the transferred amount of heat can be increased and, thus, account for heat being transmitted from the heat source to the base plate directly. This is important in the model simulating arc welding where the heat is generated through the electric arc between the electrode (the filler material) and the base plate. Also, in gas welding, there is some heat not subjected to the filler material and is applied directly to the base plate. However, it is necessary to keep the conductivity at zero clearance to be at a value similar to that of the base plate material so that the flow of heat across the gap element would be equal to that through the material itself. Due to the possible expansion of the base plate, elements of the weld pool are allowed to penetrate the elements of the base plate (a criteria know as over-closure in ABAQUS explained in [10]) when they are moved towards it. When modeling the element movement, the nodes lying on the same y-z plane move together towards the base plate by applying translational boundary condition equivalent to the initial gap clearance. With reference to Fig. 9, the nodes that are yet to move in the following steps are held in position in order not to allow any deformation in the remaining portion of the weld pool. In this case, the elements formed by the nodes that moved towards the base plate and the next group of nodes posses some strain. Therefore, the group of nodes that has just reached the base plate is held in position until those of the next step follow them. This way, the strain in the translated elements shall tend back to zero. When the second group of nodes is lowered, the first one is released to deform freely without being affected with the strain previously generated in neighboring element. In order to reduce the strain generated in the element, the nodes move only half of the gap distance every step, which implies that the nodes reach the base plate after two subsequent steps. The welding speed is the length of each element divided by the time of each step. When the nodes of the weld pool reach the base plate, a coupling equation is activated between the coincident nodes of both parts to simulate the fusion process. These coupling equations force the deformation of the coincident nodes to be equal. Hence, the weld pool and the base plate act as one body at this point. A user subroutine MPC is developed in [9] to activate the coupling between every group of nodes that come into contact at a specific time according to the welding speed. In this subroutine, the x-coordinate of every group of nodes is checked if it is less than the x-coordinate of the centre point of the heat source, and, if it is true, the coupling equation is activated. Usually, the fusion at different points between the weld pool and the base plate depends on the peak temperature and the time during which the coincident points stay in the liquid state. However, this criterion is not included in

pool is shifted in the z-direction a certain distance from the base plate as shown in Fig. 7. This way, the thermal and structural interaction between the two parts is made dependant on the distance d between them. Base plate elements

Weld pool elements

d

Gap elements GAPUNIT z

y

Figure 7: Gap clearance.

To avoid large skewing of the elements due to the moving ones simulating metal deposition, the initial gap between the weld pool and the base plate is set to be small compared to the element size in the z-direction. The type of element used is the same as that in the previous models with structural and thermal degrees of freedom in order to perform coupled displacement-temperature analysis. Having the weld pool and the base plate parts separated required the introduction of gap thermal and structural interaction between the two bodies. The gap links, named GAPUNIT in ABAQUS, are used between the two parts to take care of the interaction by joining adjacent nodes together. These links are two-noded elements and are modeled such that they allow heat to be transmitted between nodes of the weld pool and the adjoining nodes of the base plate, which implies that the meshing of the adjoining surfaces must be the same in the horizontal and vertical directions of the surfaces as illustrated in Fig. 7. The gap elements are assigned thermal conductivity, which is at the room temperature. Its effective area which is the contact area that each gap element represents on the contact surfaces as shown in Fig. 8 (a). 60 Conductivity

50 40 30 20 10 0 0

Effective area

0.2

0.4

0.6

0.8

1

Gap distance (mm)

(a) (b) Figure 8: (a) The effective area and (b) thermal conductivity vs. the gap clearance of the gap elements. To avoid heat to be transmitted from the weld pool to the base plate before any deposition, the thermal conductivity of the gap elements are initially set to zero. However, since the base plate is at room temperature and the weld pool is at the melting point, the conductivity cannot be set to its maximum value just at the time of contact since there will be coincident points having temperatures greatly different. This may cause problems in the automatic incrementing process since a major parameter for calculating the time increment is the maximum change in temperature per increment. This should not exceed a certain limit selected to be 200ºC (larger value of the

5

10 shows the motion of the heat load whose center (the darkest point) reaches the first node of the welding at time = 4 seconds. Finally, the thermal heat loss is modeled such that heat is lost from all sides of the plate and the weld pool except those surfaces that are in contact. Just as it is mentioned in the previous case, the heat from the top surface is modeled using the previously developed FILM subroutine with the value of t 0

the research and all coincident points on the contact surfaces are assumed to have full fusion because the model is designed to check for the residual stresses.

identical to that just calculated for DFLUX. However, since the topmost elements of the weld pool that are yet to be deposited are considered part of the top surface, they must not be considered in the heat loss area. Therefore, the FILM subroutine is slightly modified so that elements with xcoordinate larger than that of the center point of the heat source must have a heat loss coefficient of zero. Figure 11 shows the heat load and heat loss regions in Model 3 according to the modifications in the FILM subroutine. The analysis steps in this case is similar to that of Model 2 with two extra steps added to the procedure in order to account for the depositing groups of nodes, and that the deposition of each group is done in two steps

(0, 0, 0) Initial shape

Step 1

(0, 0, - d/2)

(0, 0, 0) (0, 0, 0) (0, 0, -d/2) Step 2

Step 3

(0, 0, - d)

released

(0, 0, -d/2) (0, 0, - d)

released

Heat loss area (light shaded) (0, 0, 0) Step 4

(0, 0, - d)

(0, 0, 0)

(0, 0, -d/2)

Step 5

(0, 0, - d)

(0, 0, -d/2)

Fusion line

Figure 9: The steps of the element movement technique. y x

0s

Figure 11: Zones of heat load and heat loss.

2s

4s

6s

8s

10 s

Heat source area (dark shaded)

Material Properties The material used in Model 1 is Inconel Alloy 600 that is used by Friedman [7]. The material properties used with metal deposition in Model 2 and Model 3 were acquired from Brown [11] whose properties are shown in Table 1. The latent heat indicated is included by ABAQUS in the specific heat variation with temperature between the liquidous and solidous levels. Table 1: Material properties versus temperature Temperature (ºC) Young’s modulus (GPa) Poisson ratio Yield strength (MPa) Yield strength at strain 1.0 (MPa) -6 Thermal expansion (1/ºC × 10 ) Thermal conductivity (W/m.ºC) Specific heat (J/kg.ºC) Latent heat (J/kg)

Figure 10: The moving heat source of Model 3.

The heat source has the same concept as that of Model 2 in which the heat source must stay a while at the start of the welding process before it moves along the welding. However, when applying the DFLUX subroutine developed earlier, the value of t0 will be different to match the deposition process. It is assumed that the center point of the heat source should be at the first node of the welding line just as the first group of nodes comes into contact with the base plate. In other words, since the first group of nodes reaches the base plate in two consecutive steps, the value of t0 shall be twice the time taken for each step, which is 2 seconds for the model in hand. Figure

20 1550 200 0.2 0.25 0.25 290 1 314 1 10 15 50 30 450 400 260000

1650 -5 2×10 0.25 0.01 0.01 15 30 400

2000 -5 2×10 0.25 0.01 0.01 15 30 400

RESULTS AND DISCUSSION Verification Study Good agreement appeared between the thermal and structural results of Model 1 and those of Friedman [7]. Figure

6

12 shows the temperature history at 3 monitoring points indicated in Fig. 2 along the mid transverse line on the top and bottom surface. Figure 13 shows the history of the variation of the longitudinal stress along the transverse direction (ydirection) along the top of the mid-section. y = 2.5 mm (top) y = 2.5 mm (bottom)

y = 5 mm (top) y = 5 mm (bottom)

Temperature (ºC)

y = 0 mm (top) y = 0 mm (bottom)

Element birth (top) Element movement (top)

Liquidous level Solidous level

Temperature (ºC)

2500 2000

Element birth (bottom) Element movement (bottom)

2000 1800 1600 1400 1200 1000 800 600 400 200 0 0

1500

50

100

150

200

Time (s) 1000

(a)

500

Element birth (top) Element movement (top)

0 0

10

20

30

40

50

Temperature (ºC)

Time (s)

Figure 12: Temperature history. 10 s

20 s

30 s

40 s

50 s

2500 s 400

200 100 0 -100

Stress (MPa)

300

2000 1800 1600 1400 1200 1000 800 600 400 200 0 0

30

20

10

100

150

200

(b) Element birth (top) Element movement (top)

-300 40

50

Time (s)

-200 50

Element birth (bottom) Element movement (bottom)

Element birth (bottom) Element movement (bottom)

1800

0

1600

Temperature (ºC)

Distance from the welding line (mm)

Figure 13: Stress distribution along the mid-section Boundary Conditions Set 1 versus Set 2 The thermal response of boundary conditions set 2 is similar to that of boundary conditions set 1 since the heat source and, thus, the heat input rate did not change. This is illustrated in comparing the peak temperature and the temperature at a time after complete welding process (showing the cooling rate) in thermal history of the monitoring point in Fig. 14 (a) with Fig. 14 (b). In addition, only the transient behaviors of the longitudinal and transverse stresses during the welding process were similar in both boundary conditions sets. However, the response of the stresses during and after cooling had some differences. This is illustrated when comparing the residual stress distribution shown in Fig. 16 (a) with that of Fig. 16 (c) and Fig. 16 (b) with that of Fig. 16 (d). Note that, in the distribution in Fig. 16 (a) and (b) of boundary conditions set 1, the high stresses occur inside the welded plate and reduces tremendously at the surfaces, which is not the case in Fig. 16 (c) and (d) of boundary conditions set 2. Also, the difference in the response to the change in the structural boundary conditions can be noted when comparing the stress history at the monitoring points in Fig. 17 (a) with Fig. 18 (a) and Fig. 17 (b) with Fig. 18 (b).

1400 1200 1000 800 600 400 200 0 0

50

100

150

200

Time (s)

(c)

Figure 14: Temperature history at the top and bottom surfaces of Model 1 and 2 in boundary conditions (a) set 1, (b) set 2, and (c) set 3.

The generated longitudinal residual stresses (σx) in boundary conditions set 2 are larger than those in set 1 since they are affected by the translation of the far surface as a result of welding the whole part. Also, the transverse residual stresses (σy) in this set 2 are much higher than those in set 1 since the transverse constraint in the latter is released at the end of the analysis relieving a large part of the elastic strain. Table 2 shows a comparison of the maximum stress values between the two boundary conditions sets. It can be noted that the change in the structural boundary conditions has changed the points with maximum longitudinal stresses from near the mid point of the weld pool in boundary conditions set 1 to near the edges of the base plate boundary conditions set 2. In

7

addition, the locations of the maximum tensile and compressive transverse stresses are significant along the welding line in boundary conditions set 1 and at the fusion surface in boundary conditions set 2. This difference is illustrated when comparing the variation of the stresses along the welding line and the fusion line shown Fig. 17 (e) to (h) of boundary conditions set 1 with Fig. 18 (e) to (h) of boundary conditions set 2. It must be noted that the longitudinal stresses near the welding line increase in tension and those far away from the welding line increase in compression. This is shown when comparing Fig. 17 (c) of boundary conditions set 1 with Fig. 18 (c) of boundary conditions set 2 showing the variation of the stresses along the top and bottom path of the mid section perpendicular to the welding line. Note that the fluctuations in the stresses near the weld pool in Fig. 18 (c) and Fig. 18 (d) have the same range shown in Fig. 17 (c) and Fig. 17 (d) of boundary conditions set 1. This verifies the fact that these fluctuations may be considered to be due to the assumption of the material properties and their variations with temperature near the melting point.

17 (a) of boundary conditions set 1 with Fig. 19 (a) of boundary conditions set 3 and the transverse stress history in Fig. 17 (b) with Fig. 19 (b). Similarly, the residual stress variation along the different paths had very slight changes from those of boundary conditions set 1 as illustrated in Fig. 19 (e) and Fig. 19 (f) for the welding line, Fig. 19 (g) and Fig. 19 (h) for the fusion line and Fig. 19 (c) and Fig. 19 (d) for the mid section paths. This is also illustrated in Table 2 showing that the location of the maximum stresses did not change significantly with the change of the thermal load

Element Birth versus Element Movement Techniques The results of the element movement technique (Model 3) had a very close match with those of the element birth (Model 2). The slight reduction in the peak temperature predicted using the element movement is due to the gradual flow of heat from the weld pool to the base. Besides, due to the difference in the node arrangement between the two techniques in having two separate bodies in Model 3 instead of one in Model 2, there is a difference in the arrangement of the integration points representing similar nodes. This caused a difference in the temperature of the monitoring points due to the high temperature gradient. In addition, it can be observed that the temperature of the monitoring point at the top is lower than that at the bottom since the distance of the former from the welding line is larger than that of latter, since the weld pool is wider at the top than at the bottom. Also, the gradual heat flow from one body to the other in the element movement technique, as shown in Fig. 15, causes a slight reduction in the temperature when compared to the element birth techniques. On the other hand, it is noted that the both techniques gave the same response to the change in the structural and thermal boundary conditions although the models are different. A very close match in the history of the stress components at the monitoring point (Fig. 16) is shown in parts (a) and (b) of Figs. 17, 18 and 19. As for the residual stresses, some discrepancies had occurred at the contacting node due to the existence of the gap elements as shown in parts (c) and (d) of Figs. 17, 18 and 19. Table 4 shows the analysis steps and timing of Model 1 and Model 2. The table shows the total number of iterations which is the sum of iterations performed for all increments including those that were attempted with several time increments to achieve convergence. This illustrates the huge reduction in the time spent in performing the simulation using the element movement technique. It is observed that, in the element birth technique, the main disadvantage is the time increment of the analysis. According to thermal conduction, the temperature change in any node is directly proportional to the difference between its temperature and those of the adjacent nodes, and its distance from these nodes. Therefore, in the element birth techniques, the newly activated elements are formed with nodes that are in the base plate and others from the weld pool. Those in the base plates will have a very small temperature and those in the weld pool would have temperature above the melting point. The rate of change in the temperature of these nodes depends on the initial temperature difference between them and the length of the element. Therefore, the incremental change in

Table 2: Comparison of the maximum stress values

Longitudinal tensile stress (MPa) Longitudinal comp. stress (MPa) Transverse tensile stress (MPa) Transverse comp. stress (MPa)

Boundary conditions set 1 129 20,3.78,4.75 101 82,0.48,1.26 90 96,0.22,3.5 155 4,0.22,3.5

Boundary conditions set 2 340 60,5.55,2.5 187 48,30.6,4.74 256 9,3.62,2.5 323 99,1.35,1.26

Boundary conditions set 3 133 20,2.9,3.5 111 82,1.35,1.26 62 96,0.34,5.74 122 4,1.6,5.74

Boundary Conditions Set 1 versus Set 3 In boundary conditions set 3, the peak temperature at the monitoring points was reduced due to the reduction in the heat input rate which can be illustrated when comparing Fig. 14 (a) with Fig. 14 (c). In addition, it can be noted that the response of both the element birth and element movement technique to the change in the heat source were very close. Table 3 shows the percentage drop in the temperature at the monitoring point at the top and the bottom when reducing the heat load from that in boundary conditions set 1 to set 3 for both the element birth and element movement techniques. Table 3: Comparison of the thermal results

Heat input rate (W)

Boundary Boundary Difference conditions conditions (%) set 1 (˚C) set 3 (˚C) 1300 900

Element birth (top) Element movement (top)

1698.14 1677.45

1413.17 1428.50

20.2 % 17.4 %

Element birth (bottom) Element movement (bottom)

1916.02 1927.43

1605.21 1644.99

19.4 % 17.2 %

As for the stress history, there were slight differences observed when comparing the longitudinal stress history in Fig.

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temperature, and thus the time increment, would depend on the mesh density. Higher mesh densities may reduce the time increment below an acceptable limit.

the analysis time. Also, the stress history and the residual stress distribution resulting from both techniques compared well, with an acceptable difference when evaluated versus the computing cost.

Table 4: Comparison of the analysis steps Boundary Boundary Boundary conditions conditions conditions set 1 set 2 set 3 Element birth -3 Minimum time increment (s x 10 ) Number of increments Number of iterations Total analysis time (hours)

0.01149 1827 4928 ~36

0.00287 1753 4758 ~48

0.01879 2107 5614 ~41

Element movement -3 Minimum time increment (s x 10 ) Number of increments Number of iterations Total analysis time (hours)

2.344 957 3401 ~20

0.11 966 3471 ~24

9.499 848 2927 ~18

REFERENCES [1] Nguyen, N. T., A. Ohta, K. Matsuoka, N. Suzuki, and Y. Maeda, “Analytical Solutions for Transient Temperature of Semi-Infinite Body subjected to 3-D Moving Heat Sources,” Welding Journal – Welding Research Suplement, August 1999, pp. 265-274. [2] Goldak, John, “Keynote Address: Modeling Thermal Stresses and Distortions in Welds,” Recent trends in welding science and technology, ASM International, 1990 [3] Ueda, Y., and M. G. Yuan, “Prediction of Residual Stresses in Butt Welded Plates Using Inherent Strains,” Journal of Engineering Materials and Technology, Vol. 115, October 1993, pp. 417-423, Vol. 116, July 1994, pp. 285. [4] Mochizuki, Hayashi and Hattori, “Residual Stress Analysis by Simplified Inherent at Welded Pipe Junctures in a Pressure Vessel,” Journal of Pressure Vessel Technology, Vol. 121, November 1999, pp. 353-357 [5] Dong, P., “Residual Stress Analyses Multi-Pass Birth Weld: 3-D Special Shell versus Axisymmetric Models,” Journal of Pressure Vessel Technology, Vol. 123, May 2001, pp. 207-213 [6] Hibbit, Hugh D., and Pedro V. Marcal, “A Numerical, Thermo-Mechanical Model for the Welding and Subsequent Loading of a Fabricated Structure,” Computers and Structures, Vol. 3, 1973, pp. 1145-1174. [7] Friedman, E., “Thermomechanical Analysis of the Welding Process Using the Finite Element Method,” Journal of Pressure Vessel Technology, Vol. 97, August 1975, pp. 206-213. [8] Wilkening, W. W. and J. L. Snow “Analysis of WeldingInduced Residual Stresses With The Adina System,” Computers and Structures, Vol. 47, No. 4/5, pp. 767-786, 1993 [9] Fanous, Ihab F. Z., “3D Modeling of the Welding Process Using Finite Elements,” M.Sc. thesis, The American University in Cairo, February 2002 [10] Hibbitt, Karlsson and Sorensen, “ABAQUS/Standard User’s Manual”, v. 6.2, 2001 [11] Brown, S., and H. Song, “Finite Element Simulation of Welding Large Structures,” Journal of Engineering and Industry, Vol. 114, November 1992, pp. 441-451.

However, in the element movement technique, the highly thermally affected nodes due to deposition of the weld pool element are those of the fusion surface. The incremental change in temperature of these nodes, in this case, depends on the variation of the conductance of the gap element versus their length which usually starts from zero and increases gradually as explained earlier. This reduces the incremental temperature change, and, thus, the time increment, in the analysis of the model. It can be noted that the highly affected nodes are those on the boundary surfaces of the weld pool and the base plate. Therefore, the mesh density will hardly affect the time increment in the element movement technique. This way, there will be more flexibility in increasing the mesh density if required using the element movement technique without reducing the time increment.

CONCLUSION The stresses generated during the welding process are not highly affected by the change in the boundary conditions. However, the residual stresses had some differences. Welding a free plate caused a much lower residual stresses than welding a plate that is part of a larger structure. On the other, the change in the thermal load had a very slight effect on the residual stresses. In the comparing the element movement technique versus the element birth technique, it can be observed that the former showed to be very effective. It allowed for early thermal interaction between the weld pool and the base plate reducing

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(a)

(b)

Figure 15: Comparison between the heat flow in the (a) element birth technique and the (b) element movement technique. Max.: 197.9 MPa Min.: -134.3 MPa

Max.: 124.7 MPa Min.: -165.1 MPa

(a)

(b)

Max.: 353.0 MPa Min.: -213.9 MPa

Max.: 310.2 MPa Min.: -323.3 MPa

(c)

(d)

Figure 16: Comparison between residual (a) σx (longitudinal stress) and (b) σy (transverse stress) of boundary conditions set 1 and (c) σx (longitudinal stress) and (d) σy (transverse stress) of boundary conditions set 2

10

Element birth (bottom) Element movement (bottom)

Element birth (top) Element movement (top) 300 250 200 150 100 50 0 -50

1

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(a) Element birth (top) Element movement (top)

(b) Element birth (bottom) Element movement (bottom)

Element birth (top) Element movement (top)

100

Transverse Stress (MPa)

Longitudinal Stress (MPa)

120 80 60 40 20 0 -20 -40 -60 0

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Distance from the welding line (mm)

(c)

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Element birth

Element movement

150 Transverse Stress (MPa)

150 Longitudinal Stress (MPa)

Element birth (bottom) Element movement (bottom)

35 30 25 20 15 10 5 0 -5 -10 -15 -20

Distance from the welding line (mm)

Element birth

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Distance along the welding line (mm)

(e) Element birth

(f)

Element movement

Element birth

Element movement

150 Transverse Stress (MPa)

150 Longitudinal Stress (MPa)

Element birth (bottom) Element movement (bottom)

350

300 250 200 150 100 50 0 -50 -100 -150 -200 -250

Transverse Stress (MPa)

Longitudinal Stress (MPa)

Element birth (top) Element movement (top)

100 50 0 -50 -100 -150

100 50 0 -50 -100 -150

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0

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Distance along the welding line (mm)

Distance along the welding line (mm)

(g)

(h)

100

Figure 17: Comparison between the element birth and element movement techniques according to boundary conditions set 1 for (a) σx (longitudinal stress) and (b) σy (transverse stress) history at the monitoring points, (c) σx and (d) σy distribution along the mid section, (e) σx and (f) σy distribution along the welding line, and (g) and (h) along the fusion line

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Element birth (top) Element movement (top)

Element birth (bottom) Element movement (bottom)

400

140

300

120

Transverse Stress (MPa)

Longitudinal Stress (MPa)

Element birth (top) Element movement (top)

200 100 0 -100 -200

100 80 60 40 20 0 -20 -40

-300 1

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1000

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10000

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(a) Element birth (top) Element movement (top)

(b) Element birth (top) Element movement (top)

Element birth (bottom) Element movement (bottom) Transverse Stress (MPa)

Longitudinal Stress (MPa)

Element birth (bottom) Element movement (bottom)

200

400 300 200 100 0 -100 -200 -300

150 100 50 0 -50 -100

0

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Distance from the welding line (mm)

(c) Element birth

(d) Element birth

Element movement

350

300

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Transverse Stress (MPa)

Longitudinal Stress (MPa)

10000

Time (s)

Time (s)

250 200 150 100 50 0 -50 -100

Element movement

100 0 -100 -200 -300 -400

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Distance along the welding line (mm)

(e) Element birth

(f) Element birth

Element movement

350

300

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Transverse Stress (MPa)

Longitudinal Stress (MPa)

Element birth (bottom) Element movement (bottom)

250 200 150 100 50 0 -50 -100

Element movement

100 0 -100 -200 -300 -400

0

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40

60

80

Distance along the fusion line (mm)

Distance along the fusion line (mm)

(g)

(h)

100

Figure 18: Comparison between the element birth and element movement techniques according to boundary conditions set 2 for (a) σx (longitudinal stress) and (b) σy (transverse stress) history at the monitoring points, (c) σx and (d) σy distribution along the mid section, (e) σx and (f) σy distribution along the welding line, and (g) and (h) along the fusion line

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Welding line

Element birth (bottom) Element movement (bottom)

Element birth (top) Element movement (top)

300 250 200 150 100 50 0 -50 -100 -150 -200 -250

250 200 150 100 50 0 -50 -100

1

10

100

1000

10000

10

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Time (s)

(b) Element birth (bottom) Element movement (bottom)

Element birth (top) Element movement (top) Transverse Stress (MPa)

Longitudinal Stress (MPa)

Element birth (bottom) Element movement (bottom)

50

120 100 80 60 40 20 0 -20 -40

40 30 20 10 0 -10

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Distance from the welding line (mm)

(c) Element birth

(d)

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Element birth

Element movement

150 Transverse Stress (MPa)

150 Longitudinal Stress (MPa)

10000

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(a) Element birth (top) Element movement (top)

100 50 0 -50 -100 -150

100 50 0 -50 -100 -150

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Distance along the welding line (mm)

(e) Element birth

(f)

Element movement

Element birth

Element movement

150 Transverse Stress (MPa)

150 Longitudinal Stress (MPa)

Element birth (bottom) Element movement (bottom)

300 Transverse Stress (MPa)

Longitudinal Stress (MPa)

Element birth (top) Element movement (top)

100 50 0 -50 -100 -150

100 50 0 -50 -100 -150

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Distance along the welding line (mm)

Distance along the welding line (mm)

(g)

(h)

100

Figure 19: Comparison between the element birth and element movement techniques according to boundary conditions set 3 for (a) σx (longitudinal stress) and (b) σy (transverse stress) history at the monitoring points, (c) σx and (d) σy distribution along the mid section, (e) σx and (f) σy distribution along the welding line, and (g) and (h) along the fusion line

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