Distribution of Cutting and Welding Thermal Residual Stress in Steel Bone with and without Holes in the Partially Encased Composite Wall

Abstract

In order to ensure the mechanical reliability of a partially encased composite wall (the PEC wall), the distribution law of the thermal residual stress of the steel bone of the PEC wall needs to be deeply studied. In this study, we first conducted a cutting thermal conductivity analysis and stress analysis of the flange and webs. Second, based on the cutting residual stress of different sheet steel, the welding thermal conductivity analysis and stress analysis of I-shaped calculation units with and without a hole in the web were carried out. Finally, the difference in the distribution of the stress on the section during welding (or cutting) and after the cooling of the welding (or cutting) was clarified, the shape and size of the residual stress distribution in the two types of steel bone formed by the cutting and welding thermal processing were evaluated, and the approximate distributions of the residual stress of the flange and webs were obtained, respectively.

1. Introduction

Partially encased composite sections (PEC sections) have been extensively studied and used due to their good mechanical properties caused by the concrete–steel interaction and as they provide better fire protection for steel members [1,2]. The partially encased composite wall (the PEC wall, Figure 1a) proposed, based on the study of PEC columns [3,4,5] and walls with PEC end columns [6,7,8], consists of steel bone welded from steel plates, horizontal links and concrete, which are mainly used to bear horizontal action and vertical loads. To control the welding deformation, the welding seams and welding splicing sequence of the steel bones are shown in Figure 1b,c. The good seismic performance of PEC walls has been verified by experiments and numerical simulations [9,10].

However, it was found that when comparing the numerical simulations with the experimental data of the cyclic loading experiments, the stiffness of the finite element model of the PEC wall was much higher than that of the test specimens, even if the model was built according to the actual material properties. The results of the finite element analysis of the PEC wall with holes in the steel web (Figure 2) do not agree with its experimental results, mainly in the sense that the left side of the wall (where the holes are located lower than on the other side) prematurely buckled within the height of the lowest hole at the edge of the web when it was compressed in the experiment. This resulted in a large difference between the peak bearing capacities corresponding to the positive and negative loading obtained from the pseudostatic loading experiments, and the specimen finally failed following the bending and twisting damage mode. In the numerical simulation, this damage pattern cannot be simulated regardless of whether the initial defect was added or not, and the peak bearing capacity corresponding to the positive and negative loading was similar [11].

Based on the above phenomena, considering the characteristics of the narrow plates and many splice welds in the steel bone of PEC walls, the proposition that the cutting and welding thermal residual stresses in the plates are the cause of these phenomena is put forward.

There are various methods for cutting and welding. Different methods may introduce a different pattern and density of the heat source to the workpiece, which directly leads to differences in heat distribution and thus affects the distribution patterns of the thermal residual stresses in the steel members.

The plates of steel bone in PEC walls are mainly cut by means of gas cutting, while the holes in the web are cut using a plasma arc cutting method. There have been many studies on the thermal conduction and residual stress of the gas cutting process. Jokiaho et al. [12] analyzed the heat transfer process and the distribution pattern of residual stresses in plates with different gas cutting temperature histories, different cutting speeds, different plate thicknesses and different preheating temperatures. The heat input generated during the gas cutting was quantified, and the characteristics of the heat input distribution along the thickness were clarified by Terasaki et al. [13]. The cutting thermal conductivity analysis of steel plates was carried out by Bae et al. [14] in the form of a Gaussian heat source to simulate the gas cutting process. Arenas et al. [15] proposed a body heat source model for simulating gas cutting and validated its reliability. However, most studies on the plasma cutting process have focused on the plasma torch characteristics [16,17], current parameters [18], etc., while the thermal–stress coupling analysis of the plasma cutting process is rarely studied. The combination of the double ellipsoidal model and cone [19] or cylinder [20,21] model is usually used in plasma welding thermal conductivity analysis as the heat source model. However, the high energy density and jet velocity required for cutting makes the cutting plasma have a larger radial temperature gradient than the welding plasma [22], and there is no plasma welding of the carbon steel heat source model available.

The welding of steel bone in PEC walls is conducted by automatic arc welding. As a common welding method, the heat–stress distribution of arc welding has been widely studied. Lee et al. [23] conducted welding simulations of butt welds on stainless steel and carbon steel plates of an SM400 grade, respectively. Gao et al. [24] analyzed the distribution of the residual stress fields in double-sided double arc welding, conventional welding and single arc welding and measured the actual residual stresses in the experiments. An experimental investigation into the effects of the welding process on the residual stress distributions of HSS built-up box columns was carried out by Jiang et al. [25]. Peric et al. [26] conducted the numerical simulation and experimental investigation of a buried-arc welding process on a butt-welded plate sample and got the heat input efficiency for the buried-arc welding process using parametric analysis.

In summary, although the thermal–stress analyses of gas cutting and arc welding have been studied widely, few studies have been conducted to comprehensively evaluate the combined residual stresses in the members after successive gas cutting, plasma cutting and arc welding processes. In addition, the differences in the material properties [27,28], cutting and welding sequences [29] and plate shapes (boundary conditions) [30,31] directly affect the distribution of the residual stresses in the components. To understand the thermal residual stresses in the steel bone—especially in the steel bone with holes in the web—of PEC walls, a special analysis is required. In this paper, the thermal–stress coupling analysis of the cutting and welding of steel bones with and without holes in the web of PEC walls was performed according to the actual situation of steel processing, respectively. First, the cutting thermal–stress coupling analysis of the plates was carried out separately for the steel bone with and without holes in the PEC walls, where the simulation included gas cutting at the linear boundary and plasma cutting at the web holes. Subsequently, the welding thermal–stress coupling analysis of the steel bone was conducted on the basis of importing and not importing cutting residual stresses, respectively, and the influence of the cutting thermal effects and welding thermal effects on the residual stresses in the PEC steel bone was comprehensively evaluated. Finally, the distribution law of thermal residual stresses in the PEC steel bone with and without holes was given, which can be used as initial stresses for the performance analysis of the PEC wall.

2. Fundamentals

The temperature field present in an object is a function of position and time, i.e.,

(1)$t=f\left(x,y,z,\tau \right),$

If the temperature field in the object is an unsteady temperature field, a spontaneous transfer of heat energy from the high temperature region to the low temperature region will proceed according to the second law of thermodynamics, which follows Fourier’s law and can be expressed in terms of the heat flow density vector:

(2)$\mathit{q}=-\mathsf{\lambda }\mathrm{grad}t,$

where q is the heat flux density vector (W/m³), which is the heat flux through a unit area per unit time and can be decomposed into heat flux density components along three coordinate directions. λ is the thermal conductivity (W/(m·K)), and grad t is the temperature gradient at a point of the object.

According to Fourier’s law and the first law of thermodynamics, the differential equation for the unsteady thermal conductivity of any infinitesimal parallel hexahedron with an internal heat source in Cartesian coordinates (as shown in Figure 3) can be obtained as follows:

(3)$\rho c\frac{\partial t}{\partial \tau }=\frac{\partial }{\partial x}\left(\mathsf{\lambda }\frac{\partial t}{\partial x}\right)+\frac{\partial }{\partial y}\left(\mathsf{\lambda }\frac{\partial t}{\partial y}\right)+\frac{\partial }{\partial z}\left(\mathsf{\lambda }\frac{\partial t}{\partial z}\right)+\dot{\Phi },$

where ρ, c and $\dot{\Phi }$ are the density (kg/m³), specific heat (J/(m·K)) and the heat (W/(m³·s)) generated in a unit volume per unit time by the internal heat source, respectively, all of which can be variables. τ is the time (s). The differential equation for thermal conductivity is solved to obtain the temperature data at each moment on each node of the finite element model.

3. Model Establishment

3.1. Model Description

3.1.1. The Objects of Analysis

When cutting (or welding), the area near the cutting edge (or the welding seam) will experience a sharp rise and fall in temperature for a short period of time. In order to accurately obtain the temperature change in this area, the finite element mesh of this area needs to be particularly fine. If the thermal conductivity analysis is performed towards the whole member, the number of meshes will be huge, and the analysis will take a long time. As cutting and welding is a continuous operation at the boundary, the temperature and thermal stress are continuously distributed inside the plate. In order to shorten the calculation time, an appropriate analysis object of the member can be chosen for analysis, as shown in Figure 4. One thing to keep in mind when selecting the analysis object: the analysis object needs to be large enough that the boundary conditions have a negligible effect on the interior of it.

The analysis object, from the steel bone with and without holes in the web of the PEC walls, is spliced by the flange plate and the web plate, which are of different dimensions and heat treatment regions, as shown in Figure 5. Due to these differences between the two kinds of plate, the cutting simulation of the flange plate and the web plate need to be separately carried out, and the overall welding simulation should be carried out after the respective cutting residual stresses have been obtained to acquire the total thermal residual stress distribution of the member.

3.1.2. Material Properties

A Q235B grade of mild steel was used in the simulation. The yield stress of the thermal–stress analysis model is 235 MPa at ambient temperature. It has been shown that the material properties of steel at high temperatures are significantly different from its properties at ambient temperatures. As temperature-dependent material properties have a significant impact on the distribution of cutting and welding residual stresses, the mechanical properties of materials at an ambient temperature cannot be used as a generalized substitute for the mechanical property parameters that vary with temperature required in a thermal–force analysis.

In this paper, the ideal elastoplastic model was used to simulate the stress–strain relationship of mild steel corresponding to different temperatures. The relationship between yield stress and temperature was calculated according to the Chinese Code (GB 51249-2017) [32]. Following the method of the literature [33] and summarizing the data between 20 °C and 2500 °C from the literature [34,35,36], the data of the elasticity modulus E, Poisson’s ratio μ, thermal expansion coefficient α, thermal conductivity λ and specific heat C which were applied in finite element simulation are shown in Figure 6.

Phase transition was not considered in the simulation due to its small effect on the thermal–stress relationship of the mild steel.

3.1.3. Element and Meshing

The elements DC3D8 and C3D8I were used in the thermal conductivity analysis and stress analysis, respectively.

Given the small size of the heat source, the finite element mesh of the thermal conductivity model near the heat source needs to be very fine to capture a more stable moving heat flux. Dimension differences exist between the cutting heat source and the welding heat source, and the mesh of the different models needs to be adjusted accordingly. The mesh size nearby the cutting line and welding seam were about 0.5 mm and 2 mm, respectively.

Away from the heat source, the mesh size can be increased appropriately to reduce the number of elements and nodes. The mesh size between the fine mesh and the larger mesh should be smoothly transitioned to reduce the negative impact on the simulation results due to mesh distortion. The mesh of the thermal conductivity analysis models can be seen in Figure 7 and Figure 8. In addition, in the stress analysis model, the grid of the model can be increased appropriately.

3.1.4. The Boundary Conditions

• 1.. The boundary conditions of the thermal conductivity analysis models

In order to accurately simulate the thermal conduction of the steel cutting and welding process, the boundary conditions of the finite element model should be consistent with the heat transfer conditions of the steel member processing. The initial temperature of the simulation (T₀) was defined as room temperature (20 °C). The cutting edge and welding seams were applied with a moving heat source. The surface heat transfer coefficient β between the steel plate and air was defined as 10 W/(m²·K) for all surfaces of the plate. The air temperature T_0a = 20 °C. Furthermore, the emissivity of radiation heat dissipation (ε) was defined as 0.85 for all surfaces of the plate [37].

• 2.. The boundary conditions of the stress analysis models

It is necessary to ensure that the boundary conditions in stress analysis are imposed only to constrain the rigid body displacement of the model while minimizing the restriction on its own heat deformation to ensure the uniformity of the simulation results with the practical conditions. As the change in temperature will lead to inhomogeneous deformation of the steel members, the boundary constraints need to be imposed in the region away from the cutting and welding heat sources. The boundary conditions to meet the above requirement are set as shown in Figure 9.

3.2. Heat Source Model

3.2.1. Cutting Heat Source Model

The straight edge of the plate is cut using the gas cutting method, and the hole in the web is cut using the plasma cutting method. As there is no universally accepted heat source model for gas and plasma cutting, the heat source model is selected as described below according to the principle and characteristics of cutting.

• 1.. The heat source of gas cutting

When gas cutting is proceeding, the metal on the place where the cutting torch reaches achieves the ignition point (about 960 °C [38]) by the heating of the flame and burns at the contact surface with the cutting oxygen beam. During this time, the flame of the cutting torch acts on the surface of the plate, and combustion occurs at the cutting slit of the plate.

A combined model of the Gaussian surface heat source model [13] with heat flow density distribution as shown in Equation (4) and a cylindrical homogeneous body heat source model are used to simulate the heat input mode according to the implementation of gas cutting [15,39], as shown in Figure 10.

(4)$q\left(r\right)=q_{max}\exp \left(-k{r}^2\right)$

where $q_{max}$ is the maximum value of the heat flux density, calculated according to the method mentioned in the literature [40]. k is the concentration factor of the heat source, and $k=3/r_0^2$, $r_0$ is the effective radius of the Gaussian heat source.

• 2.. The heat source of plasma cutting

In plasma cutting, the metal heated by the plasma arc reaches the melting point of mild steel (about 1537 °C [15]) at the cutting slit, and a high-temperature air stream blows through the cut to remove the molten metal to form the cutting slit.

Based on the cutting principle of a plasma arc, the cylindrical uniform body heat source model is used to simulate the plasma cutting method (as shown in Figure 11).

3.2.2. Welding Heat Source Model

The double ellipsoidal heat source model [41,42], which is widely approved in the thermal conductivity analysis of arc welding, consists of two ellipsoidal bodies stitched together at the front and back to approximate the shape of the welding pool. The schematic diagram of the heat source model is shown in Figure 12. The expressions of the heat source distribution within the two ellipsoids are:

(5)$\{\begin{array}{c}The front part:q\left(x,y,z\right)=\frac{6\sqrt{3}{f}_{1}Q}{\pi a_{1}bc\sqrt{\pi }}exp(-3\frac{x^2}{a_1^2}-3\frac{y^2}{b^2}-3\frac{z^2}{c^2})\\ The back part:q\left(x,y,z\right)=\frac{6\sqrt{3}{f}_{2}Q}{\pi a_{2}bc\sqrt{\pi }}exp(-3\frac{x^2}{a_2^2}-3\frac{y^2}{b^2}-3\frac{z^2}{c^2})\end{array}$

where Q = ηUI, η is heat source efficiency, which is generally measured by the test. The heat source efficiency of the submerged arc welding used in the model is about 0.77~0.90 [43]. U is the welding voltage (V), and I is the welding current (A). a₁, a₂, b and c are the shape parameters of the ellipsoid. f₁ and f₂ are the component distribution coefficients of the front and back half of the ellipsoid, respectively; f₁ + f₂ = 2. The parameters used in the simulation are shown in Table 1.

When the axis of the welding torch is perpendicular to the welded workpiece, the double ellipsoidal heat source can simulate a high temperature region with a shape similar to the actual shape of the molten pool and can simulate the rapid heating in the local area in front of the torch and slow cooling in the local area behind the torch due to the advance of the torch. However, the welded workpiece simulated in this paper is the T-shaped joint, and the angle between the welding torch and the welding seam is 45° when welding is applied, as shown in Figure 13. Therefore, the double ellipsoidal heat source model needs to be rotated by 45° around the x axis (the direction of advancing of welding torch) as shown in Figure 14. According to the transformation equations, the final expression of heat source distribution is as follows:

(6)$\{\begin{array}{c}The front part:q\left(x,y,z\right)=\frac{6\sqrt{3}{f}_{1}Q}{\pi a_{1}bc\sqrt{\pi }}exp(-3\frac{{(x+y)}^2}{2a_1^2}-3\frac{{(x-y)}^2}{2b^2}-3\frac{z^2}{c^2})\\ The back part:q\left(x,y,z\right)=\frac{6\sqrt{3}{f}_{2}Q}{\pi a_{2}bc\sqrt{\pi }}exp(-3\frac{{(x+y)}^2}{2a_2^2}-3\frac{{\left(x-y\right)}^2}{b^2}-3\frac{z^2}{c^2})\end{array}$

4. Thermal–Stress Simulation

The sequentially coupled thermal–stress analysis method was used in this paper. The coupled thermal–stress analysis of the analysis object, which was made by the software of ABAQUS, was carried out in two parts in accordance with the practical heat treatment process, as follows:

• The thermal–stress analysis of the cutting process: the cutting thermal conductivity analysis and the corresponding stress analysis were performed in succession for the flange plate, the web and the web with a hole (as shown in Figure 5), respectively, to gain the residual stress distribution of different plates caused by cutting heat.

• The thermal–stress analysis of the welding process: based on the cutting residual stress, the thermal conductivity analysis and stress analysis of the processing of the double-sided fillet welds were carried out on the two I-shaped welding analysis objects (as in Figure 4), respectively, to obtain the distribution of the total residual stresses of the cutting and welding process.

4.1. Thermal–Stress Simulation of Cutting Process

4.1.1. Analysis Steps

The analysis steps in the thermal conductivity analysis model of the flange, the web and the web with a hole were set as follows.

• Cutting simulation of the flange

Step 1: To simulate the change in the temperature field of the flange when two straight edges were undergoing consecutive gas cutting in succession.

Step 2: To simulate the change in the temperature field of the flange during the cooling process after cutting.

When conducting the flange cutting simulation, it was found that the temperature at the mid-plane of the flange increased by about 20 degrees Celsius at most during the whole period of the cutting and cooling process. Therefore, it can be deduced that for a steel web with a larger width than the flange, the application of a single-sided cutting heat source has little effect on the temperature change in the material on the other side of its vertical neutral plane. As the material properties of the steel change little in the temperature range from 20 to 100 degrees Celsius, it can be inferred that the unilateral cutting operation has less effect on the residual stress distribution in the region on the other side of the neutral axis of the web. In the thermal conductivity analysis of the web and the web with a hole, the whole web (with a width of 274 mm) would be taken as the analysis object for the thermal–stress analysis considering the boundary effect on heat transfer, but only a single side cutting (welding) heat source will be applied to it.

• 2.. Cutting simulation of the web

Step 1: To simulate the change in the temperature field of the web when one straight edge was undergoing consecutive gas cutting.

Step 2: To simulate the change in the temperature field of the web during the cooling process after cutting.

• 3.. Cutting simulation of the web with a hole

Step 1: To simulate the change in the temperature field of the web with a hole when one straight edge was undergoing consecutive gas cutting.

Step 2: To simulate the change in the temperature field of the web with a hole during the cooling process before circular plasma cutting.

Step 3: To simulate the change in the temperature field of the web with a hole during the circular plasma cutting.

Step 4: To simulate the change in the temperature field of the web with a hole during the cooling process after circular plasma cutting.

The analysis step settings in the stress analysis were the same as that in the corresponding thermal conductivity analysis.

4.1.2. Analysis Results

The time-varying temperature field and stress field of the flange and webs were gained, respectively, as shown in the figure below (Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19). When the plate thickness is less than 20 mm, the welding stress in the thickness direction was so small that the welding thermal residual stress in the plate can basically be regarded as biaxial [44]. The stresses perpendicular to the cutting direction were much smaller and had less effect on the bending bearing performance than that parallel to the cutting direction, so only the distribution of the stress parallel to the cutting direction was given. The direction of the heat source movement changes during circular hole cutting, so the Mises stress distribution maps are given in addition.

Conclusions can be drawn from the above figures as follows:

• As shown in Figure 15, the width of the temperature-affected zone was about 2 cm in the gas cutting processing. The cutting surface was inclined, and the area affected by the cutting oxygen flow on the cutting surface was approximately in the shape of an elongated nail, while the cutting surface in the plasma cutting process was vertical, and the width of the heat-affected area was only about 8 mm.

• From the picture (a) of Figure 16, Figure 17, Figure 18 and Figure 19 we could know that the steady-state temperature of the flange with cutting operations on two edges was lower at the four corners and higher in the middle of the plate, with a temperature difference of about 0.25 °C. The distribution of the steady-state temperature field of the web with one edge cutting was higher on the side near the heat source and lower on the side away from the heat source. The opening had a serious effect on the heat transfer inside the web, resulting in higher temperatures in the narrow area between the cutting edge and the opening boundary after both cuttings compared with other areas.

• As exhibited in picture (b) and (c) of Figure 16, Figure 17, Figure 18 and Figure 19, the stress map on the flange plate was approximately symmetrical, while the stress distribution on the web was characterized by a gradient variation. The stresses in the narrow area between the opening and the cutting edge were much greater than in other areas at the same distance from the cutting edge.

4.2. Thermal–Stress Simulation of the Welding Process

4.2.1. Analysis Steps

The analysis steps of the thermal conductivity analysis model of the I-shaped welding analysis objects were the same as that of I-shaped welding analysis objects with a hole in the web, as shown below.

Step 1: To simulate the change in the temperature field of the model when one fillet weld was welding.

Step 2: To simulate the change in the temperature field of the model during the cooling process after the first fillet weld was completed.

Step 3: To simulate the change in the temperature field of the model when the other fillet weld was welding.

Step 4: To simulate the change in the temperature field of the model during the cooling process after the second fillet weld was completed.

4.2.2. Analysis Results

The time-varying temperature field and stress field of the models were obtained, respectively, as shown in the figures below (Figure 20 and Figure 21).

The distribution character of the temperature and residual stress can be summarized according to the above figures as follows:

• As shown in Figure 20, the shape of the temperature distribution near the welding heat source was a double ellipse with a flat front and steep back, and the steel melting point was reached within the whole width of the welding seam and within part of the thickness of the web.

• As shown in Figure 21, after cooling, the temperature distribution of the I-shaped models exhibited a slightly higher temperature of the plate on the near side of the welding seam than that of the plate on the far side of the welding seam. The hole in the web affected the path of heat transfer, resulting in higher temperatures in the narrow area near the welding seam than in other areas at the same distance from the welding seam and in lower temperatures in the narrow area far from the welding seam than in other areas at the same distance from the welding seam.

• As shown in figures in the Section 5, the stress distribution near the upper and lower boundaries of the plate significantly differed from the stress distribution close to the middle height of the plate. As only the heat source of the two left welding seams were applied to the I-shaped welding unit, the flanges of the unit can be divided into two categories—the flange near the welding seam and the flange away from the welding seam, and the residual stress of the former was much larger than that of the latter. The stress map on the flange exhibited an approximate symmetrical distribution pattern, while the stress distribution on the web was characterized by a gradient variation. The stress in the narrow area of the web between the hole and the welding seam was much greater than in other areas at the same distance from the welding seam.

5. Thermal Residual Stress Distribution in the Member

The temperature gradient lines in the middle height of both the flange and the web are shown as smooth curves with small curvature, so it could be inferred that the influence of the boundary effect on the heat transfer in the middle height of both plates can be neglected. Therefore, the stress distribution in the cross-section at the middle height of the plate in the models can be used to represent the distribution of the residual stress distribution in the steel bone of the PEC wall. As can be seen from Figure 22, Figure 23 and Figure 24, the distribution and magnitude of the stress along the welding seam and of the Mises stress were similar in the plates except for the web with a hole. Therefore, the residual stress distribution along the welding seam of the flange and web are shown, while both the residual stress along the welding seam and the Mises residual stress of the web with a hole are exhibited, as shown below.

5.1. Cutting Thermal Residual Stress Distribution

The distribution of the cutting residual stress in the flange and webs along the cross-section are shown in Figure 25.

It was indicated from the above figure that:

• The material where the torch reached bore compressive stress because its expansion due to high temperatures was limited by the surrounding material, while the stress here turns to be tensile after cooling. The cutting residual tensile stress in the flange was rapidly reduced from yield stress to zero within 6 mm of the cutting edge and was expressed as a very small compressive stress in the middle of the sheet. The simplified distribution of the cutting residual stress on the flange is shown in Figure 26a.

• As only a single side cutting heat source was applied in the simulation of the web, the stress near the cutting edge rapidly changed from tensile stress to compressive stress and then gradually transitioned to very small tensile stress (about 2 MPa) over the width of the plate. It was evident that the heat source of the gas cutting on the left side of the web had a negligible effect on the right region of the web. Both sides of the web in the steel bone of the PEC wall experienced gas cutting, and the distribution of the longitudinal residual stress in the web can be obtained by mirroring and simplifying the residual stress on the cutting half, as shown in Figure 26a.

• On the web with a hole, the heat source of gas cutting on the left border had little effect on the narrow area on the right side of the hole due to the influence of the hole on the plate, and the residual stress of this area was only about 3 MPa. However, the residual stress in the narrow area on the left side was all yielded in tension within 6 mm of the cutting line, and the compressive stress in the rest of the width was about 35 MPa. After the cooling of the plasma cutting, the total cutting residual stress in the narrow area of the left of the hole can be considered as a superposition of the gas cutting residual stress and the plasma cutting residual stress. As the residual stress perpendicular to the gas cutting edge was much smaller than the residual stress parallel to the gas cutting edge, the distribution and magnitude of the Mises stress was mainly influenced by the latter. The web with a hole in the steel bone of the PEC wall undergoes gas cutting on the right edge as well as on the left edge, and its longitudinal residual stress on the cross-section through the center of the hole can be regarded as the superposition of the gas cutting residual stress and the plasma cutting residual stress, and the simplified cutting residual stress distribution is shown in Figure 26b.

• Comparing the distribution of the cutting residual stress in the web with and without a hole, it could be seen that the hole reduced the area of the web and increased the heat input to the web, resulting in the cutting residual stress in the narrow area of the web on both sides of the hole being much larger than the cutting residual stress at the same region in the web without a hole.

5.2. Welding Thermal Residual Stress Distribution

The distribution of the welding residual stress in the flange and webs along the cross-section are shown in Figure 27, Figure 28, Figure 29 and Figure 30.

Through the above pictures, some conclusions can be drawn:

• The effect of the web with or without a hole on the distribution of the welding residual stress in the flange was negligible. The simplified distribution of the welding residual stress in the flange can be obtained, as shown in Figure 31a.

• The cutting residual stress in the plates was much smaller than the residual stress of the welding. The total residual stress of a plate can be approximated as the superposition of the residual stress obtained from each heat treatment.

• In the web with a hole, the welding seam on the left side of the hole had little effect on the narrow area on the right side of the hole, which was reflected by the fact that the total residual stress of the cutting and welding in the narrow area on the right side were coincident with the cutting residual stress curve. In the steel bone of the PEC wall, there are double fillet welds on both sides of the web with a hole, and the simplified distribution of the cutting and welding residual stresses in the web with a hole is shown in Figure 31b.

• In the web, the welding process of the left border had little effect on the right side of it. According to the stress balance within the cross-section, the simplified distribution of the cutting and welding residual stress in the web of the PEC steel bone can be obtained by mirroring and simplifying, as shown in Figure 31c.

• Comparing the distribution of the total residual stress in the web with and without a hole, it can be drawn that the presence of the hole resulted in the total residual stress induced by the cutting process and the welding process in the narrow area of the web on both sides of the hole being much larger than the total residual stress at the same region in the web without a hole.

6. Conclusions

In this paper, the following conclusions were obtained from the coupled thermal–stress analysis of the cutting of plates and the coupled thermal–stress analysis of the welding of I-shaped analysis objects with and without a hole in the web:

• The welding residual compressive stresses in both the flange and webs were much greater than the cut residual compressive stresses, while the residual tensile stresses caused by both the cutting process and welding process yielded.

• The total residual stress of a plate can be approximated as the superposition of the residual stress obtained from each heat treatment.

• The presence of the hole in the web greatly affects the distribution pattern of the residual stress in the web. Both the cutting residual compressive stress and the welding residual stress in the narrow area are about 2~3 times larger than the corresponding residual compressive stress at the same location on the web without a hole. This will result in the premature yielding of the narrow region of the web in the cross-section passes through the center of the hole when subjected to compressive stress and buckling under the influence of slight out-of-plane deformation.

• The distribution of the residual stress in the plate is approximately symmetrical around the plate centerline. The approximate distribution of residual stresses is obtained from the analysis results, which can be introduced into the finite element analysis of the PEC shear wall as the initial stress state of the steel bones.

Author Contributions

Methodology, validation, formal analysis, investigation, writing—original draft, visualization and resources, Y.H.; conceptualization, supervision, project administration, and writing—review and editing, Q.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. The PEC wall: (a) the configurations of PEC walls; (b) welding sequence diagram of steel bone in PEC walls (I → II); (c) welded seams in the steel bone of PEC walls.

Figure 2. The PEC wall with holes in the steel web: (a) the configurations; (b) distribution of holes in the web.

Figure 3. Thermal conductivity of infinitesimal parallel hexahedron.

Figure 4. The analysis object of welding thermal–stress analysis: (a) the analysis object without hole on web; (b) the analysis object with hole on web.

Figure 5. The analysis object of cutting thermal–stress analysis: (a) the flange plate; (b) the web plate without hole; (c) the web plate with hole.

Figure 6. Material properties related to temperature of steel with strength grade of Q235B: (a) yield stress; (b) thermal coefficient of expansion, elasticity modulus and Poisson’s ration; (c) specific heat and thermal conductivity.

Figure 7. The meshing of cutting thermal conductivity analysis model: (a) the flange; (b) the web; (c) the web with hole.

Figure 8. The meshing of welding thermal conductivity analysis model: (a) narrow flange I-shape member; (b) narrow flange cellular I-shape member.

Figure 9. Boundary conditions in stress analysis models: (a) the cutting simulation model of flange; (b) the cutting simulation model of web; (c) the cutting simulation model of web with hole; (d) the welding simulation model of steel bone without hole; (e) the welding simulation model of steel bone with hole.

Figure 9. Boundary conditions in stress analysis models: (a) the cutting simulation model of flange; (b) the cutting simulation model of web; (c) the cutting simulation model of web with hole; (d) the welding simulation model of steel bone without hole; (e) the welding simulation model of steel bone with hole.

Figure 10. Heat source of gas cutting.

Figure 11. Heat source of plasma cutting.

Figure 12. Double ellipsoidal heat source of welding.

Figure 13. Relative angle of the welding torch to the workpiece.

Figure 14. Double ellipsoidal heat source at fillet welds of T-joint.

Figure 15. The distribution of field around the heat source during cutting: (a) temperature field of gas cutting; (b) temperature field of plasma cutting; (c) stress distribution; (d) strain distribution.

Figure 16. The distribution of field on flange after cooling of cutting process: (a) temperature field; (b) the stress parallel to cutting line (S33); (c) the Mises stress.

Figure 17. The distribution of field on web after cooling of cutting process: (a) temperature field; (b) the stress parallel to cutting line (S33); (c) the Mises stress.

Figure 18. The distribution of field on web with hole after cooling of gas cutting: (a) temperature field; (b) the stress parallel to cutting line (S33); (c) the Mises stress.

Figure 19. The distribution of field on web with hole after cooling of plasma cutting: (a) temperature field; (b) the stress parallel to cutting line (S33); (c) the Mises stress.

Figure 20. Temperature distribution and weld pool near the welding heat source during the welding process: (a) the I-shaped welding analysis objects with hole on web; (b) the I-shaped welding analysis objects.

Figure 21. The temperature field of welding models after cooling of welding process: (a) the I-shaped welding analysis objects with hole on web; (b) the I-shaped welding analysis objects.

Figure 22. The distribution of stress field on flanges of the I-shaped welding analysis model after cooling of welding process: (a) the stress parallel to weld seam (S33) on the flange near weld seam; (b) the Mises stress on the flange near weld seam; (c) the stress parallel to weld seam (S33) on the flange away from weld seam; (d) the Mises stress on the flange away from weld seam.

Figure 23. The distribution of stress field on the web of the I-shaped welding analysis model with hole in web after cooling of welding process: (a) the stress parallel to weld seam (S33); (b) the Mises stress.

Figure 24. The distribution of stress field on the web of the I-shaped welding analysis model with hole in web after cooling of welding process: (a) the stress parallel to weld seam (S33); (b) the Mises stress.

Figure 25. The distribution of cutting residual stress along the path in the middle of the steel plate: (a) the stress path on the flange; (b) the stress path on the web; (c) the stress path on the web with hole; (d) the stress of the flange parallel to cutting line (S33); (e) the stress of webs parallel to cutting line (S33); (f) the Mises stress of webs.

Figure 26. Approximate distribution of cutting thermal residual stress across the section of the flange and webs: (a) the cutting thermal residual stress on the flange and the web; (b) the cutting thermal residual stress on the web with a hole.

Figure 27. The stress path on flange of different initial stress conditions: (a) the stress path on the flange without cutting residual stresses; (b) the stress path on the flange with cutting residual stresses.

Figure 28. The comparison of the stress parallel to welding seam (S33) on the flange near to welding seams under different conditions: (a) effects of hole in the web on the distribution of welding residual stress on flanges near welding seams; (b) the comparison of welding residual stress and cutting residual stress on the flange near welding seams.

Figure 29. The stress path on webs with and without a hole: (a) the stress path on the web; (b) the stress path on the web with hole.

Figure 30. Distribution of residual stress and cutting residual stress parallel to cutting line (S33) on webs: (a) the comparison of cutting residual stress and welding residual stress of web with hole; (b) the comparison of cutting and welding residual stress of web with and without hole.

Figure 31. Approximate distribution of welding thermal residual stress across the section of the flange and webs: (a) the welding thermal residual stress on the flange; (b) the welding thermal residual stress on the web; (c) the welding thermal residual stress in the cross-section of the web with a hole which passes through the center of the hole.

Figure 31. Approximate distribution of welding thermal residual stress across the section of the flange and webs: (a) the welding thermal residual stress on the flange; (b) the welding thermal residual stress on the web; (c) the welding thermal residual stress in the cross-section of the web with a hole which passes through the center of the hole.

Table 1

The parameters of the double ellipsoidal heat source for the simulation.

I	U	η	f 1	f 2	a 1	a 2	b	C
400 A	33 V	0.85	0.4	1.6	0.004 m	0.016 m	0.01 m	0.01 m

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