1.　Introduction

Steel angles are widely used as truss members for bridges, housings, etc. Steel angles are classified as equal-leg angles and unequal-leg angles, and which is selected for use in any given application depends on what is mechanically reasonable in the operating environment. Steel angles are rarely used singly and are usually joined into truss structures. The joining methods are classified as mechanical joining and welding joining. Although welding joining is advantageous in terms of reducing the number of parts, it is subject to welding deformation due to thermal strain. The risk of large welding deformation such as buckling is considered to be particularly high in a long truss structure of thin angles. Since dimensional accuracy is essential when assembling truss members on site, welding deformation becomes a serious problem. In order to suppress welding deformation and reduce the post-welding correction process, it is necessary to understand the three-dimensional welding deformation behavior of the truss structure.

In order to understand the welding deformation behavior, it is essential to conduct a theoretical investigation and apply the finite element (FE) method. Research into the prediction of welding deformation has been pursued since the 1940s, with the theory of linear elasticity being developed in early work1-6). In the 1970s, thermal elastic-plastic analysis using the FE method was applied to the nonlinear problem of welding phenomena7),8). Subsequently, the heat source model9), material modeling10) and calculation techniques11) have been developed, making it possible to clarify complicated phenomena. Recently, taking advantage of the

progress of computers and algorithms, thermal elastic-plastic analysis has predicted the residual stress and the welding deformation of large structures that could not be clarified previously12-22). Since large structures often consist of multi-pass welding and three-dimensionally curved welding lines, the welding deformation behavior cannot be determined solely from simple angular distortion or transverse shrinkage. Although research on the development of mechanical models for large structures has been pursued23-25), the literature includes few studies on welding deformation in the truss structure of steel angles, employing either real-scale models or joint models.

Since steel angles are generally joined by fillet welding on flat plate, the deformation behavior appears to resemble the behavior in fillet welding of T joints and bead-on-plate welding of flat plate. However, unlike in the case of T joints and flat plate, the neutral axis of the steel angle is not located inside the heat-input plate (plane). Although mechanical bending behavior due to external load in such a cross-sectional shape of steel angles has been studied, it has yet to be clarified how internal stresses and thermal strain due to the welding cause the deformation.

The objective of this study is to clarify the welding deformation behavior and the generation mechanism in the truss structure of steel angles.

The investigation was conducted numerically and experimentally using a joint model and a real-scale truss structure model where steel angles are welded on the same plane. We attempted to develop a 3D thermal elastic-plastic FE model with a Goldak heat source. Comparison with experimental results confirmed the validity of the FE analysis (FEA). In the experiment, the temperature history during the welding was measured by a thermocouple, and the three-dimensional deformation distribution was measured by a laser displacement meter. Furthermore, in order to understand the deformation mechanism of the whole truss structure, a theoretical model based on the inherent strain was proposed in the bead-on-plate model of a steel angle. Comparison with the FEA results confirmed the validity of this theoretical

Experimental and Numerical Study of Welding Deformation in Truss Structure of Steel Angles＊

by TADANO Satoshi**, MIZUUCHI Rieko***, SUGIURA Kouji**** and NAKATANI Yujiro**

It is necessary to understand the formation characteristics of welding deformation that occurs during the manufacturing process of the truss structure of steel angles. We evaluated the welding deformation behavior experimentally and numerically using a joint model and a real-scale truss structure model. As a result, we revealed that the main beams of the welded truss structure were arched against the direction of angular distortion observed in the welded joint model. The deformation tendency was explicable on the basis of theory in terms of uniformly distributed inherent strain within the mechanical melting zone. This analysis suggested a new theoretical insight that can facilitate suppression of welding deformation in the truss structure.

Key Words: Welding deformation, Truss structure, Steel angle, Fillet weld, Finite element analysis, Goldak heat source, Inherent strain

   *受付日　平成30年9月26日　受理日　平成31年2月6日  **正  員　 東芝エネルギーシステムズ株式会社　Member, Toshiba

Energy Systems & Solutions Corporation ***      　 東芝エネルギーシステムズ株式会社　Toshiba Energy

Systems & Solutions Corporation****      　 東芝エレベータ株式会社　Toshiba Elevator and Building

Systems Corporation

[溶接学会論文集　第 37 巻　第 1号　p. 52-58（2019）]

model, and the welding deformation mechanism of steel angles was clarified.

2.　Methodology

2.1　Experimental method

Figures 1 and 2 show the configuration of the fillet welded joint and the truss structure to be welded, respectively. The joint consists of two steel angles of L-125 x75 x10 mm and L-50 x50 x6 mm, each of which was joined by fillet welding (flux-cored arc welding) with a leg length of 6 mm. The truss structure consists of two L-125x75x10mm (L-125 angle: main beam) and seven L-50x50 x6mm (L-50 angle: sub-beam) with a total length of 3540mm in the longitudinal direction. Each angle was joined by fillet welding with a leg length of 6mm, the same as that of the joint model. The temperature history during the welding process was measured using the joint model. The K-type thermocouple was attached to the surface at a distance of 5, 10 and 15 mm from the weld bead. The welding deformation was measured using a laser displacement meter. The deformation of the joint was evaluated by a reflection-type displacement measurement method using “KEYENCE LK-080”, and the deformation of the truss structure was evaluated by a light sectioning method using “Nikon MM Dx100”. The displacement and deformation of the model were calculated from the difference of surface shape before and after welding.

2.2　FEA method

A 3D thermal and mechanical FE model was developed using the commercial software “SYSWELD”. Figure 3 shows the FE model and the welding direction of the joint and the truss structure. The model size was the same as the experimental size. The cross-sectional element size around the welding trajectory was approximately 1 x1 mm, which was sufficiently smaller than the heat source size. The model was almost entirely meshed with eight-node linear hexahedron solid elements. The model of the joint and the truss structure were composed of 38,210 and 392,270 nodes, respectively. The material properties are shown in Fig.4, which had temperature dependency26). The model assumed elastic-perfectly plasticity. Transient thermal analysis was conducted by means of a Goldak heat flux distribution profile27-28), described by Eq. (1).

(1)

where q is the absorbed heat flux, η is the beam efficiency coefficient, qf,r is the absorbed heat flux in front and rear of the heat source, af,r , b and c are the heat source size, and x, y and z are the distance from the welding trajectory. The heat input parameter is shown in Table 1. The radiation plane as the heat flux boundary condition was defined on the surface of the model. The value of the coefficient of convective heat transfer was 25W/m2K. The heat flow density for radiation qrad was governed by the Stefan-Boltzmann law. σ is the Stefan-Boltzman constant (5.67*10-8 Wm-2K-4), e is the surface emissivity (0.8), t is the surface temperature of parts,

Fig.1 Configuration of the fillet welded joint.

Fig.2 Configuration of the truss structure.Fig.3 FE models and schematic illustration of welding lines.

((a) The fillet welded joint, (b) The truss structure)

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and t0 is the ambient temperature (293K). Only rigid body motion was restricted in the finite element model. In order to consider the transformation phenomenon through melting to solidification, an element activation/deactivation subroutine was used. In the step before melting (e.g., before heat source passing), the welding bead elements had the weak material properties. When the heat source passed through and temperature became above melting temperature, the material properties changed to the values shown in Fig.4.

Figure 5 shows the FE model of the bead-on-plate welded L-125 angle. The shape of the angle was L-125 x75 x10 mm, and the length was 500 mm, which was longer than the joint shown in Fig.1. The length of the welding line and the material properties were the same as those of the joint model. The heat input parameter was defined based on the parameter in Table 1, and the only heat input was changed from 313 to 1053J/mm.

3.　Results and discussion

3.1　Experiment and FEA in fillet welded joint

In order to validate the FEA, the analysis results in the fillet welded joint were compared with the experimental results. In addition, the welding deformation behavior of the joint model was evaluated.

Figure 6 shows the results of the temperature history during the welding process in the fillet welded joint. The evaluation was performed at the positions 5, 10 and 15mm from the weld bead in a cross section 60mm from the welding start point. The temperature of the right welding line was evaluated. As a result, the peak temperature and temperature history at each point of the FEA almost agreed with the experimental values, indicating the validity of the Goldak heat source model in the FEA.

Regarding the welding deformation, Fig.7 shows the results of the Z displacement contour in the fillet welded joint at ambient temperature after the welding. The welded plane of the L-125 angle was noticeably displaced to the minus direction around the weld bead. That is, it was found that the welded plane of the L-125 angle showed the angular distortion typically observed in bead-on-

Fig.5 FE bead-on-plate model.

Fig.6 Measured and analyzed transient temperature profiles of the fillet welded joint.

Fig.4 Material properties used in the FEA. ((a) Physical properties, (b) Mechanical properties)

Table 1 Parameters concerning heat input of the FEA.

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plate welded joints. Figure 8 shows the result of comparing the angular distortion of the FEA and that of the experiment in the fillet welded joint along the welding line from the start point. The amount of the distortion was calculated from the inner product of coordinate vectors. The coordinates were read from the position of the scribe lines that were drawn along the welding lines at a distance of 5mm and 30mm from the L-125 angle edge. As shown in Fig.8, the angular distortion reached the maximum value at the welding start point and tended to monotonically decrease almost linearly toward the welding end point. This tendency is considered to be due to structural internal restraint, and it can be inferred that the backside (non-welded surface) of L-125 angle contributes as a restraint. It was found that the results of experiment and the FEA at each point quantitatively agreed well (-1.2 – +0.14 x10-3 rad. error), which validated the FEA result.

3.2　Experiment and FEA in truss structure

In order to verify the FEA in the truss structure, the FEA results was compared with the experimental results. In addition, the behavior of the welding deformation of the truss structure was evaluated.

Figure 9 shows the FEA result of the displacement contour in

the Z direction (out-of-plane direction) of the truss structure after welding. Both the upper and lower L-125 angles had large displacements in the plus direction at the center. In order to understand the displacement of L-125 angles quantitatively, the line profiles of the Z displacement are shown in Fig.10 (a) and (b). The displacement was evaluated along the longitudinal direction at a position approximately 15 mm from the welding start point. Although there were fine irregularities locally, both the experimental and the analytical results showed a convex tendency overall. In the upper angle, the displacement of the experiment and the FEA was

-0.4 – +2.6mm and -0.3 – +2.7mm, respectively (Fig.10 (a)). On the other hand, in the lower angle, the displacement of the experiment and the FEA was -1.3 – +2.0mm and -1.4 – +2.5mm, respectively (Fig.10 (b)). The deformation behaviors of the experiment and the FEA quantitatively agreed well, and the validation of the FEA was indicated also in the truss structure.

Fig.7 Analyzed deformation (displacement Z) contour after the welding of the fillet welded joint.

Fig.8 Measured and analyzed angular distortion of the fillet welded joint.

Fig.9 Analyzed deformation (displacement Z) contour after the welding of the truss structure.

Fig.10 Measured and Analyzed deformation (displacement Z) of the truss structure in the L-125 angles. ((a) upper L-125 angle, (b) lower L-125 angle)

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In regard to the behavior of the welding deformation, the convex tendency in L-125 angle was opposite to the angular distortion of the joint model shown in Section 3.1 (Fig.7). In order to understand the welding deformation behavior of the truss structure theoretically, a fundamental study is discussed in the next section.

3.3　FEA and inherent strain theory in bead-on-plate model

To further discuss the generation mechanism of the truss structure, the welding deformation in the bead-on-plate welded L-125 angle was investigated using the FEA and inherent strain theory.

Figure 11 shows the FEA result of displacement Z line profiles after welding in the basic heat condition (the heat input: 794J/mm). The line crossed at the center of the welding line was defined as A line, and the line on the top of the non-bead plane (back plane) was defined as B line. The displacement of A and B line was symmetrical to the welding line. Regarding A line, the displacement reached the minimum value around the welding line, and monotonously increased with distance from the welding line. The displacement at a position +150 mm or more (-150 mm or less) from the welding line was almost zero. Regarding B line, the displacement reached the maximum value at the welding line, and monotonically decreased toward the edges.

These results suggest an insight that can facilitate understanding of the welding deformation in the fillet welded joint and the truss structure. The deformation of the A line around the welding line (-150 mm < y < +150 mm) was similar to the angular distortion observed in the joint model, whereas the deformation of the B line showed the convex pattern observed in the L-125 angle (main beam) of the truss structure. From these results, the deformation of the joint (Fig.7) and the truss structure (Fig.9), which seemed different, actually represented the characteristics of the welding deformation of the steel angle.

It is the convex deformation in the main beam that plays a critical role with respect to the dimensional accuracy of the

products. To understand the deformation mechanism and obtain an insight as to how to suppress the welding deformation, we attempted to describe this deformation behavior of the angle by the inherent strain theory. Watanabe and Satoh4) calculated bending deformation (angular distortion) by the inherent strain theory in the case that strain was distributed uniformly in the elliptical region of the cross-section. Here, we tried to develop a model assuming that strain was distributed uniformly within the circular region as shown in Fig.12. Since the cause of angular distortion is the inherent strain in the direction orthogonal to the welding line (i.e., the Y direction), the deformation behavior in the Y direction will be discussed below.

The inclination (angular change) of AB (CD) with respect to the O’z’ axis is described by Eq. (2).

(2)

where ρ( y) is the radius of curvature.Here, in accordance with the Euler-Bernoulli beam theory, the bending moment acting on the cross section of the rod is zero (Eq. (3)). In addition, the relational expression of stress and strain is expressed by Eq. (4), and the relational expression between the strain and the radius of curvature is described by Eq. (5).

(3)

(4)

(5)

From Eq. (3)-(5), Eq. (6) is derived.

(6)

where ε0 is the strain of the neutral axis (z = 0), and gy is the inherent strain value in the Y direction and the region of the

Fig.12 Schematic illustration about uniformly distributed inherent strain. ((a) perspective view of the model, (b) aa cross section, (c) bb cross section)

Fig.11 Analyzed deformation (displacement Z) of the bead-on-plate model.

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inherent strain is defined as a function of the x, y, z coordinates. Here, the first moment of area in the section with the X axis passing through the centroid as the axis of rotation is represented by Eq. (7), and the second moment of area is expressed by Eq. (8). Further, the right side of Eq. (6) is expanded as shown in Eq. (9a) and (9b).

(7)

(8)

(9a)

(9b)

where t is the plate thickness, h1 and h2 are the dimensions of the angle steel, Cz is the position of the centroid in the Z axis direction, L is the length of the welding line and g0 is the uniformly distributed inherent strain value in the Y direction. To summarize the above, the angular distortion is calculated by Eq. (10) and can be described as a function of g0 and b.

(10)

Okano et al.1),24) evaluated the angular distortion of bead-on-plate welded joint in JIS 490A by the inherent strain theory. They proposed a simplified method for predicting the amount of angular distortion by setting the zone of uniformly distributed inherent strain as the mechanical melting zone (800 °C) and setting the value of the strain as approximately αTmelt. In this research as well, we tried to calculate the angular distortion based on this concept where g0 is defined as 0.0234 which is equal to thermal strain from 20 °C to Tmelt. (i.e., 1500 °C) and b is defined as the size of the mechanical melting zone (800 °C) obtained by the thermal FEA. A list of parameters used for the calculation is shown in Table 2.

Figure 13 shows the amount of angular distortion of the bead-

on-plate model with the heat input as a variable. The results of the FEA and the inherent strain theory are shown. Both results showed a similar tendency where the amount of the angular distortion decreased with decreasing heat input and were zero at approximately 200J/mm. Although the value of the inherent strain theory tended to be slightly larger than that of the FEA, the degree of the difference was similar to the previous study by Okano et al., and the effectiveness of this theoretical model was validated.

Based on these results, the mechanism of the convex deformation observed in the main beam of the truss structure and the method of reducing the deformation are discussed. As shown in Fig.12, it is the positional relationship between the welding line and the neutral axis that dominates macroscopic deformation. In the steel angle, since the neutral axis is not located in the plate but in the space, the center of curvature is located in the direction of the welding line (inherent strain region). As shown in Eq. (10), the amount of the convex deformation is parametrically dependent on the second moment of area, the radius of the induced inherent strain, the inherent strain value and the welding line length. Therefore, it is considered that reducing the radius and the value of the induced inherent strain, increasing the second moment of area, or reducing the welding line length would be remarkably effective in suppressing the welding deformation of the angle.

4.　Conclusion

In order to keep dimensional accuracy in the truss structure of steel angles, the welding deformation behavior and the generation mechanism were investigated. The 3D thermal elastic-plastic FE models of the joint and the real-scale truss structure were developed, and the deformation behavior was evaluated experimentally and numerically. The generation mechanism was discussed, using the bead-on-plate FE model and the inherent strain theory. The major conclusions are summarized below.

Fig.13 Angular distortion of the bead-on-plate model calculated by the FEA and the inherent strain theory.

Table 2 Parameters used in the inherent strain theory.

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1. The temperature history of the FEA with a Goldak heat source showed good agreement with the experimental results in the fillet joint. In addition, the welding deformation calculated by the FEA was also in good agreement quantitatively with the experimental results for the joint model and the truss structure model, indicating the validity of the 3D thermal elastic-plastic FE models.

2. The main beams of the welded truss structure were arched against the direction of angular distortion observed in the welded joint model.

3. This welding deformation tendency was explicable on the basis of theory in terms of uniformly distributed inherent strain within the mechanical melting zone. The theory indicated that the amount of the deformation is parametrically dependent on the second moment of area, the radius of the induced inherent strain, its value and the welding line length, which suggested theoretical knowledge that can facilitate suppression of the welding deformation in steel angles.

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