Minimization of welding residual stress and distortion in large structures
P. Michaleris ­ The Penn State University, University Park, PA J. Dantzig and D. Tortorelli ­ University of Illinois at Champaign Urbana, Urbana, IL Abstract
Welding distortion in large structures is usually caused by buckling due to the residual stress. In cases where the design is fixed and minimum weld size requirements are in place, the thermal tensioning process has proven effective to reduce the welding residual stress below the critical level and eliminate buckling distortion. In this work, a systematic design approach using conventional finite element analysis, analytic sensitivity analysis, and nonlinear programming is implemented to investigate and optimize the thermal tensioning process. 1 Introduction
The trend in current design and manufacturing practice is to reduce product weight through the use of weldable high strength materials in thin sections. However, use of thin section materials increases the susceptibility of a structure to buckling during manufacturing due to the welding residual stress. Buckling distortion can also degrade the product performance, increase manufacturing cost due to the poor fit-up and the need for straightening, reduce structural integrity and cause excessive product rejection. Buckling distortion can be eliminated by either increasing the rigidity of the structure through improved designs or by reducing the welding residual stress through process modifications. Over the past fifteen years, the finite element method has been used to predict distortion and residual stress due to welding. Simulations of welding processes involve thermo-mechanical finite element analyses of the weld zone. Many investigators (Refs. 1- 5) have performed transient nonlinear thermal analyses and small deformation quasi-static elasto-plastic analyses. Following such analyses, Michaleris and DeBiccari (Refs. 6-7) have demonstrated that the welding residual stress can be accurately predicted and consequently applied as a pre-stress in a buckling analysis of a structure. Reducing the welding heat input and modifying the structural configuration reduces the occurrence of buckling (Refs. 6-9). Design considerations, however, may impose limits on such modifications. In this case, new manufacturing techniques such as thermal tensioning can be used to eliminate buckling due to welding. The thermal tensioning technique for controlling welding residual stress and distortion as discussed by Burak et al. (Refs. 10-11) involves generating a tensile strain at the weld zone, prior to and during welding, by imposing a steady state temperature differential. Recently, Michaleris and Sun (Ref. 12) used a thermo-elastoplastic model to demonstrate that the thermal tensioning process minimizes the welding residual stress by reducing the plastic strain accumulation during welding. However, generating a steady state temperature differential prior to welding requires the use of heat sinks (cooling devices) that are impractical, costly, and environmentally unfriendly. Therefore, the development of a thermal tensioning process (referred to here transient thermal tensioning) that uses transient temperature 1 differentials is desirable. Such transient differentials can be generated by a moving heat source. Effective implementation of this tensioning process requires determining the appropriate intensity, size and location of the heat source such to minimize the welding residual stress. Unfortunately, even for simplified geometries, a conventional parametric study for determining the appropriate process parameters is impractical as the number of process configurations is prohibitively large. To resolve this problem we use, sensitivity analysis and optimization that has proven successful in determining optimum designs for linear (Refs. 13-14) and nonlinear (Refs. 15-16) problems with large numbers of design variables. In this work, a systematic design approach using conventional finite element analysis, analytic sensitivity analysis, and nonlinear programming is implemented to investigate and optimize the transient thermal tensioning process. 2 Transient Thermal Tensioning Process
The transient thermal tensioning device investigated here consists of two heating bands traveling along with the welding torches (Figure 1). Flame heating is investigated due to the low acquisition and operational cost. Then, the design of a transient thermal tensioning device becomes the determination of the width of the heating bands (d1), length of the heating bands (d2), offset from the first torch (d3), and offset from the weld centerline (d4) that minimize the welding residual stress without degrading material performance with undesirable metallurgical transformations. A separate set of design variables (d1- d4) must be determined for each weld heat input, panel and stiffener thickness combinations. 3 Optimization of Thermal Tensioning
A parametric investigation to identify the desired combination of the four design variables d1 to d4 would require numerous analyses. For example, if ten values of each design variable are considered, then ten thousand combinations will need to be investigated. Each combination includes the preparation, computation, and interpretation of ten thousand welding simulations. Furthermore, a discrete (fixed values for the design variables) parametric space will most likely miss an optimum combination. Finally, this exercise will need to be repeated for each weld heat input, panel and stiffener thickness combination. Such a parametric investigation is practically and economically unfeasible. Rather than performing a parametric design analysis, the transient thermal tensioning process is investigated here by solving an optimization problem stated as follows: Identify di, i=1, n, such that: min f (d 1 , d 2 ,..., d n ) for:
g j (d1 , d 2 ,..., d n ) 0 (1) (2) (d i )min d i (d i )max 2 where, f is the objective function which quantifies the welding residual stress, n is the number of design variables, di are the design variables, (di)min and (di)max are the minimum and maximum values allowed for each of the design variables, and gj are a set of inequality constraints. The problem defined by equations (1) and (2) is solved by the steepest descent method for simplicity (Ref. 17). In the steepest descent method, an initial selection of the design variables is iteratively modified until convergence is reached (Figure 2). At each iteration i, the objective function gradient si is initially computed to check for convergence. If convergence is not achieved, a line search is performed in the - si direction to determine the ai that minimizes where: f (d i + ai s i ) f f f , ,K , ) s i = -( d1 d 2 d n (3) The line search direction of equation (3) is determined by computing the derivative (sensitivity) of the objective function with respect to each design variable. 4 Numerical Approach
4.1 Analysis of the Forward problem The residual stress is evaluated by performing a 2D transient heat conduction analysis followed by a quasi-static elasto-plastic analysis assuming generalized plane strain conditions. In the generalized plain condition, a model cross section is assumed to deform between two rigid planes. This condition is acceptable for small sections (narrow panels). However for large sections (wide plates), a loading on one part of the cross section will generate finite strains over the entire section leading to inaccuracies. For example, Michaleris and DeBiccari, (Refs. 6-7) show that under the generalized condition the computed welding residual stresses extend to locations away from the weld. However, experimental measurements indicate that, in large sections (wide panels) the residual stress gradually diminishes away from the weld (Ref. 8). The analysis plane is set to a cross-section of the model perpendicular to the welding direction. The governing equations consist of the energy balance equation (which neglects the stress power so that the thermal analysis is independent from the mechanical analysis):
r- h - (-kT) = 0 t (4) the mechanical equilibrium equation:
+ b = 0 (5) and the mechanical constitutive law: 3 · æ· · · ö = Cç t - p - th è p = e p F (e p , , T )
· (6) 0 Y (e p , , T )
where, r is the internal heat generation, h is the enthalpy, k is the thermal conductivity, t is time, is stress, C the elastic constitutive tensor, t, p, and th are the total, plastic and thermal expansion strains, b is the body force, ep is the equivalent plastic strain, F is the plastic flow rule, and Y is the yield function. The system of equations (4-6) is solved using the finite element method. The welding heat input is modeled by a double ellipsoid body heat flux distribution (Ref. 3): r= 6 3Qb f - æ 3X2b 2 + 3Yb2 2 + 3(Zb +vt )2 ö ç e ç a b c2 è abc (7) where, Qb is the welding heat input,= is the welding efficiency (set here to 80%) Xb, Yb, and Zb are the local coordinates of the double ellipsoid model, a is the weld width, b is weld penetration, c = a and f = 0.6 before the torch passes the analysis plane and c = 4a and f = 1.4 after the torch passes the analysis plane, v is the torch travel speed, and t is the clock time. The thermal tensioning heat source is applied on the top surface of the plate (Figure 1), and is defined by an ellipsoid surface heat flux q:
q(x) = 4 3Qs -3æ X s22 + Z s22 ö ç e ç d1 d2 è (8) where Xb and Zb, are the local coordinates of the surface ellipsoid defines as follows: X s = x - d 4 - 0.5d1 Z s = vt + d 3 - 0.5d 2
Qs is the flame heat flux per unit area, is the heating efficiency (set here to 30%), x is the horizontal distance from the weld centerline, v is the welding torch travel speed, t is the clock time, and d1 to d4 are the design variables (i.e. the width of the heating bands, length of the heating bands, offset from the first torch, and offset from the weld centerline), respectively.
Generalized plane strain conditions are assumed in the mechanical analysis to account for the out-of-plane expansion in the model. The out-of-plane strain z is assumed to have a linear distribution over the analysis plane:
z = e - x y + y x (10) (11) 4 where e is the out-of-plane strain at the coordinate origin and x and y are the strain variations in the y and x axes respectively. 4.2 Sensitivity Analysis The use of optimization drastically improves the ability to effectively design the thermal tensioning process. However, optimization requires the computation of the derivative (sensitivity) of the residual stress with respect to the design variables. The computational approach of the sensitivity analysis is analogous to the approach presented Michaleris et al. (Ref. 15). The gradient of the objective function is computed using the direct differentiation method (Ref. 15). First, the objective function f is expressed as a generalized response function of the temperature, displacement, and plastic strain field: f = f (T , u , p ) (12) Then, the design derivative of the generalized function is computed by chain rule differentiation of equation (12): f f T f u f p = + + (13) di T di u di p di The sensitivities of the temperature T, displacement u, and plastic strain p are computed by differentiating the governing equations (4-6) which are now written in discretized residual form as: W (T ) = 0
R ( p , u, T ) = 0 (14) H( p , u, T ) = 0
Differentiating and rearranging equations (14) yields the following expressions for the sensitivities: T æ W ö é W ù = -ç ÷ di è T ø ê di ë
p æ H = -ç ç di è p ö é H u H H T ù ÷ + + ÷ ê u di di T di ë ø
-1 -1
-1 -1 (15) é R R u = -ê - di ê u p ë æ H ö H ù é R R T R ç ÷ ú ê + - ç ÷ u ú ê di T di p è pø ë æ H ç ç è p ö H ù ÷ ÷ di ø
-1 The inverse operators in equations (15) are equivalent to the consistent tangent operators that are computed and inverted in the forward problem (Ref. 15), therefore the numerical overhead of 5 computing the design sensitivities is a very small fraction of that required for the forward problem. 5 Computational Results
The proposed design algorithm is implemented to determine the optimum transient thermal tensioning device for welding a 4" x 4" (101.6 x 101.6mm) stiffener with 3/16" (4.76 mm) thick flange and web on a 3/16" (4.76 mm) thick, 2' (0.6 m) wide plate with 3/16" (4.76 mm) fillet welds. Both plate and stiffener are made of AH36 steel. This configuration is identical to the one selected by Michaleris and Sun (Ref. 12) in their investigation of steady state thermal tensioning. The 2D finite element mesh used in this study is illustrated in Figure 3. Radiation and convection boundary conditions are assigned for all free surfaces. A temperature dependent free convection coefficient is used here and is plotted on Figure 4. The emissivity is set to 0.2. The material properties and free surface boundary conditions are identical to those discussed in Michaleris and DeBiccari (Ref. 7). The temperature dependent thermal conductivity K, and specific heat Cp are plotted on Figure 4. The latent heat of fusion is set to 247 kJ/kg/°C and the density to 7.86 x 103 kg/m3. Elastic-plastic material response is assumed with isotropic work hardening. Figures 5 and 6 illustrate the temperature dependent mechanical properties. The welding heat input Qb is set to 6682 J (25.7V, 260 A) for the first torch and 6180.5 J (26.3 V, 235 A) for the second. The thermal tensioning heat input is Qs set to 8.3e-4 J/in2 (0.5 J/mm2). The welding travel speed is 24.72 ipm (10.5 mm/s). The thermal forward problem and sensitivity analyses are performed in an enhanced version of the commercial code FIDAP (Ref. 18), while the mechanical forward problem and sensitivity analyses are performed in a FORTRAN finite element code developed by the authors. The bounds of the design variables are selected based on practical limitations (Table 1). The initial design selected for the optimization consists of two square heaters located 2" away from the weld centerline, and 5" ahead of the first welding torch. Table 1. Bounds of design variables Design variable d1 d2 d3 d4 Minimum (in.) 1 1 0 2 Maximum (in.) 12 12 12 4 The computed longitudinal welding residual stress for the initial selection of design variables is illustrated in Figure 7. The weld region is under yield level tension which agrees with the conventional welding results of Michaleris and DeBiccari (Refs. 6-7). The compressive residual stress at the plate's edge is 8.8 ksi (60.7 MPa) which is slightly higher than the conventional welding case where the compressive stress is 8.062 ksi (55.9 MPa). Thus, the initial set of design variables has an adverse effect on the residual stress. 6 The following two sections describe the optimization results for two choices of the objective function. 5.1.1 Minimum Stress at the Plate's Edge
The objective function is set to the absolute value of the longitudinal residual stress of the element located at the edge of the plate (location A in Figure 3). After six iterations, the optimization converges towards the heating elements that are illustrated in Figure 8. The heater width (d1) is 8.72" (221.6 mm), the heater length (d2) is 8.19" (208 mm), the offset form first torch (d3) 1.3" (33.2 mm), and the offset from the weld centerline (d4) is 3.6" (90.5 mm). The corresponding longitudinal residual stress is illustrated in Figure 9. As seen in the figure, the residual stress at the plate's edge is negligible, however, a band of tensile residual stress is generated next to plate's edge. This is caused by the plastification of the plate due to excessive pre-heat. To avoid optimization solutions that may plastify the plate, the optimization problem can be modified by: 1) Imposing constraints to limit the peak temperature over the application region of the heating bands 2) Imposing constraints to limit the plastic strain over the application region of the heating bands 3) Modifying the objective function to include the effects of residual stress in application region of the heating bands The following section explores the option 3 above. 5.2 Minimum Sum of Squares Stress on the Plate The objective function is selected as the sum of squares of the residual stress of all the elements on the plate located two inches away from the weld centerline (location B in Figure 3). An optimum design is reached after ten design iterations resulting to an approximately zero value objective function, which corresponds to negligible residual stress over the plate two inches away from the weld centerline. The optimal heating elements are illustrated in Figure 10 where it is seen that the heat width (d1) expands over the entire plate width and the heater length (d2) is 7.35" (186.7 mm). A zero offset form the first torch (d3) is required and the minimum allowable offset from the weld centerline (d4) is computed. This result is attributed to the thermal diffusion from both heating pads and welding torches. The computed residual stress using the optimum heater configuration of Figure 10 is illustrated in Figure 11. The residual stress at the weld region is tensile with magnitude of about half the room temperature yield stress. A small compressive stress region exists around the weld to balance the tension of the weld. The residual stress over the plate two inches away from the weld is negligible. Figure 12 illustrates the computed temperature history at seven points (P1 though P7) located on the bottom of the plate (Figure 3). As seen in the figure, the peak temperature in the plate is 530 oC. This temperature does not cause adverse metallurgical transformations on the 7 material. However, if high temperatures were found to cause undesirable transformations on the material, constraints on peak temperature could be added in the optimization set up (equation (2)) to exclude such heating configurations from the solution. 6 Discussion
This study demonstrates that numerical optimization and finite element analysis can be combined to minimize residual stress and distortion using the thermal tensioning process. The generalized plane condition has been used in this study to minimize model size and computational time. In comparisons with experimental measurements in narrow panels (2' wide), the generalized plane strain condition has given accurate results for modeling conventional welding (Ref. 7) and welding with thermal tensioning (Ref. 12). However, additional numerical and experimental research is needed to verify the general applicability of the generalized plane strain condition for modeling welding under thermal tensioning, especially for wide panels. In wide panels, inaccuracies may be introduced due the fact that in the generalized plane strain condition a loading on one part of the cross section generates finite strains over the entire section. Therefore, the restraint caused by a wide panel will be overestimated and therefore it will artificially reduce the tension generated by the heaters. Using the generalized plane condition, a wide panel is expected to require heaters extending over the entire panel width. The numerical approach presented here can be expanded to 3D finite element formulations to investigate wide panels. 7 Conclusions
The numerical investigation presented here demonstrates the effectiveness of using numerical analysis and optimization to design the thermal tensioning process. The transient thermal tensioning using no cooling and localized heating can produce panels with zero residual stress on the plate. Such panels will have no buckling distortion (Refs.6-7). They will also have improved structural integrity. Furthermore, they will have negligent longitudinal shrinking. Thus they will facilitate the implementation of a neat cut manufacturing approach. The computational approach can easily accommodate design or material limitations by introducing constraints on the optimization set up. For example, constraints on peak temperatures can ensure that heating does not cause adverse metallurgical transformations. The approach can also be extended to 3D finite element formulations to investigate wide size panels. 8 Acknowledgment
This work was funded by the Navy Joining Center, Columbus, Ohio. The United States Government, Navy Joining Center, and Edison Welding Institute make no warranties and assume no legal liability or responsibility for the accuracy, completeness, or usefulness of the information disclosed in this report. Reference to commercial products, processes, or services does not constitute or imply endorsement or recommendation. 8 9 References
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