Comparison of residual stress on different three-dimensional multi-layer welding methods of welded hollow sphere joint in large-span structure by considering latent heat

Abstract

This article quantitatively evaluated the influence of three-dimensional multi-layer welding on the welding residual stress of super-big diameter welded hollow sphere joint in the large-span spatial steel structure considering the influence of material state change using finite element method. First, this article proposed a conception of equivalent state change time—the virtual time of state change in the analysis of welding and cooling—which represents the heat absorbed or released during the change of solid and fluid states in welding material. The equivalent state change time increases the accuracy of the thermal analysis considering the latent heat of fusion in welding and cooling process. Second, this article simulated the numerical experiments of three welding methods including multi-layer symmetric welding, multi-layer sequential welding, and single-layer symmetric welding, which show that multi-layer symmetric welding not only increases the balance of supporting ability of structure but also decreases the value of maximum tensile residual stress, and it is better than the other two welding methods.

Keywords

ANSYS APDL equivalent state change time large-span spatial steel structure latent heat three-dimensional multi-layer welding welded hollow sphere joint welding residual stress

Introduction

With wide application of the large-span spatial steel structure to the architectural construction, there has been increasing demand for safety of joints which are the important supporting components in the structure.

The welded hollow sphere joint (WHSJ) devised by Liu and Chen¹ is one of the various kinds of joints in the large-span structure and has been widely used as it has simple shape and strong ability of supporting loads. The WHSJ is the welding joint of welded hollow sphere and steel tube span. The welded hollow sphere is manufactured by heating two thick steel circle plates up to high temperature, pressing and deforming them into half-spheres, and last, joining them by welding. Needless to say, big residual stresses are produced not only in the heat plastic treatment such as heating and pressing the steel plates and in the welding process of two half spheres but also in the process of welding the welded hollow sphere and tube span. Those residual stresses have influence on the supporting ability, deformation, and fatigue life of the large span structure. Particularly, reducing the residual stress of the joint between the sphere and the span plays a decisive role in enhancing the supporting ability. The essence of research on the super-big diameter WHSJ is to predict correctly the quantitative influence of welding residual stress and residual distortion on the supporting ability of the long-span structure and to propose the decisive assessment criteria for estimating the advantage or disadvantage of various welding methods.

The welding residual stress is present in the material after welding, which may result in loss of dimensional control, costly rework, and production delays. Messler,² Feng,³ and Deng and Murakawa⁴ introduced that the welding residual stress distribution depends on the temperature distribution, which affects the material state change of solid-fluid-solid, microstructure mechanism (austenite, ferrite, pearlite, bainite, and martensite), hardness, distortion, and so on.

John and Mehdi,⁵ Singh⁶ and Michaleris⁷ investigated that the temperature field depends on heat source of weld, thermo-mechanical properties of welding material and matrix, the geometric shapes of the welding objects, the procedure of welding sector, pass or layer, and so on.

Ueda et al.⁸ conducted that the heat source model depends on the kind of welding process such as manual metal arc welding (MMAW), gas tungsten arc welding (GTAW), gas metal arc welding (GMAW), submerged arc welding (SAW), laser welding (LW), electron beam welding (EBW), and so on. The MMAW is also known as shielded metal arc welding (SMAW) and still referred to as “Stick” welding in some countries of the world, which has been one of the most common techniques used in the fabrication of steels for many years. It has advantages that the equipment requirements are simple, a large range of consumables are available, and the process is extremely portable, and for these reasons, the process has been traditionally used in structural steel fabrication.

Phillips,⁹ Abid and Siddique,¹⁰ and Weman¹¹ used the heat source model as the product of current, voltage, welding time, and thermal efficiency according to Joule’s Law. Needless to say, the efficient can never be 100%. Not all the generated heat is used in welding and there are significant loss of heat caused by spatter and heat dissipation through the three routes of thermal transfer—convection, conduction, and radiation. Mohammad and Norbert^12,13 reviewed different solutions to the heat flow problem in welding and proved that Rosenthal’s and Nguyen’s analytical solutions along with finite element method suffer lack of accuracy, and finite element method is often too costly; therefore, a new method was proposed based on finding an adaptive function which can require no knowledge about the thermo-physical properties of the material and the heat source boundary conditions which are difficult to determine.

Michaleris⁷ investigated the efficiency in arc welding, which is the effective use of the generated heat, and is expressed as percentage of the generated heat against the actually used. This can range from 20% to 85% in various arc-welding processes. The welding processes such as GTAW (40%), MMAW (60%–85%), and SAW (95%) are sorted in the ascending order of efficiency. The most important problem here is how to suppose the efficiency, while Murugan et al.¹⁴ and Lindgren et al.¹⁵ explained that the value of efficiency generally depends on experience, has not yet made sufficient theoretical basis, that is, the current, voltage, and welding time can be measured directly with instruments; however, the prediction of efficiency usually depends on experience and does not still have enough theoretical and practical accuracy. In general, the heat efficiency for welding process is determined by measurement. Mohammad et al.¹⁶ and Hurtig et al.¹⁷ proposed methods to determine the arc efficiency by calorimetric method with water-cooled anode calorimeter. Mohammad et al.¹⁶ determined the arc efficiency of GTAW as a function of current, arc length, polarity, and gas flow rate and proved that the arc efficiency decreases as the arc length increases, and it is not significantly affected by the shielding gas flow rate.

The heat efficiency is also concerned with the maximum temperature in the welding pool. The maximum temperature in the welding pool of arc welding is 1500°C–2500°C in experience, which can be approximately measured with various instruments in practice. Some studies conducted by Murugan et al.¹⁴ and Lindgren et al.¹⁵ supposed the maximum temperature in the welding pool is in the range of 1500°C–1750°C in analysis of temperature field during arc welding, but Singh⁶ and Wang et al.¹⁸ investigated that the maximum temperature in the welding pool is over 2000°C in MMAW.

And some obstacles exist in considering the latent heat in applying the heat source model to welding objects using finite element analysis (FEA), because common FEA programs, such as ANSYS, ABAQUS, SYSWELD, and so on, cannot consider the phenomena of the fusion and solidification process (effect of latent heat of fusion), so called solid–fluid–solid state change process, but can simulate the change of thermal or structural properties only in the solid state or liquid state. In order to overcome these difficulties, many researchers such as Murugan et al.¹⁴ and Singh⁶ proposed various methods considering the effect of latent heat of fusion, for example, the release or absorption of latent heat of fusion could be simulated by an artificial increase in the value of specific heat over the melting temperature range, or in another method, the liquid–solid state change and associated latent heat were modeled using an artificial heat flow method. However, it is still difficult to decide which method is the most accurate.

Not only the material properties but also the result concerning residual stress and distortion are also strongly dependent on solid-state phase transformation, from the austenite to martensite, respectively. This strain incompatibility results in stresses between the grains of different phases, which may have a significant influence on the state of residual stresses in the weld metal and heat-affected zone (HAZ) in joining certain types of material such as martensitic weldments.³ Multi-layer welding produces more complicated problems than single-layer welding. Recently, with the expansion of spans of the large-span structures, the welded hollow spheres with super-big diameters have appeared, and the width and depth of weld joint have been continually increased, so single-layer weld cannot be applied to welding of super-big WHSJ, and multi-layer welding of WHSJ is imperative in construction of large-span structures. Up to now, most of the studies on WHSJ such as Zhao et al.¹⁹ and Yuan²⁰ are based on the single-layer welding of WHSJ with relatively small diameters, but very few of them have been described on three-dimensional (3D) multi-layer welding or symmetric welding. Weingrill et al.,²¹ Rong et al.,²² and Zhang and Yang²³ proved that the distribution of welding deformation and residual stress in multi-layer welding is much different with that in single-layer welding, while interlamination stress is surely inhomogeneous. Particularly, the effect of welding procedure and method on the residual stress and distortion cannot be ignored, and various welding methods such as 3D multi-layer symmetric welding method have been suggested to reduce the residual stress.

Based on the above research results, this article proposed a conception of equivalent state change time considering the effect of the latent heat of fusion during the state change in the welding process and estimated the quantitative effect of 3D multi-layer welding process on welding residual stress of WHSJ with super-big diameter in large-span spatial steel structure with numerical experiment.

Basic principle and welding model for FEA

Basic principle for transient thermo-mechanics analysis

A sequentially uncoupled thermo-mechanical analysis is used to obtain the residual stress. The solution procedure consists of two steps. First, the temperature distribution and its history in welding model are computed by the heat conduction analysis. Then, the temperature history is employed as a thermal load in the subsequent mechanical calculation of the residual stress field.

Thermal transfer process and temperature field evaluation

The welding process is simulated as a nonlinear transient heat transfer process. Equations (1) and (2) show welding heat transfer equation and boundary condition

\begin{array}{l} ρ (T) C (T) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (k_{x} (T) \frac{\partial T}{\partial t}) \\ + \frac{\partial}{\partial y} (k_{y} (T) \frac{\partial T}{\partial t}) + \frac{\partial}{\partial z} (k_{z} (T) \frac{\partial T}{\partial t}) + Q \end{array}

(1)

k_{n} \frac{\partial T}{\partial n} + \bar{q} + h (T - T_{0}) + σ ς (T^{4} - T_{0}^{4}) = 0

(2)

where $T_{0}$ is welding temperature, °C; $k_{x} (T)$ , $k_{y} (T)$ , and $k_{z} (T)$ are coefficients of heat conduction in every direction, W/(m °C); $ρ (T)$ is density, kg/m³; $C (T)$ is specific heat, J/(kg °C); $Q$ is heat source, J/(m³ s); $k_{n}$ is coefficient of heat conduction on boundary, W/(m °C); $n$ is distance in normal direction of isothermal surface; $\bar{q}$ is heat flow density on boundary, J/(m² s); $ς$ is emissivity of the surface of the body (ranging from 0 to 1); $h$ is convective heat transfer coefficient of the surface of the body, W/(m² °C); and $σ$ is Stefan–Boltzmann constant which is taken as 5.67 × 10⁻⁸ W/(m² °C).⁴

The matrix shape of equations (1) and (2) is written as follows

\begin{array}{l} [C_{e}^{T}] {{\dot{T}}_{e}} + ([K_{e}^{t m}] + [K_{e}^{t b}] + [K_{e}^{t c}]) {T_{e}} \\ = {Q_{e}^{f}} + {Q_{e}^{c}} + {Q_{e}^{g}} \end{array}

(3)

where $[C_{e}^{T}]$ is element specific heat (thermal damping) matrix, $[K_{e}^{t m}]$ is element mass transport conductivity matrix, $[K_{e}^{t b}]$ is element diffusion conductivity matrix, $[K_{e}^{t c}]$ is element convection surface conductivity matrix, ${Q_{e}^{f}}$ is element mass flux vector, ${Q_{e}^{c}}$ is element convection surface heat flow vector, and ${Q_{e}^{g}}$ is element heat generation load.

From equation (3), temperature according to every time and position can be calculated, and from the decided temperature, thermal deformation according to every time and position can be calculated.

Stress evaluation from thermal strain

The thermo-elastic strains and stresses are calculated depending on the time and the position from the temperature result.

The governing equation of thermo-elastic deformation is as follows

{σ} = [D] {ε} - {β} Δ T

(4)

Q = T_{0} {β}^{T} {ε} + ρ C_{v} Δ T

(5)

where ${σ}$ is stress vector, $[D]$ is elastic stiffness matrix, ${ε}$ is strain vector, ${β} = [D] {α}$ is vector of thermo-elastic coefficients, ${α}$ is vector of coefficients of thermal expansion, $Δ T = T - T_{R E F}$ is subtract of temperatures, $T$ is current temperature, $T_{R E F}$ is reference temperature, $Q$ is heat density, $T_{0} = T_{R E F} + 273 K$ is absolute reference temperature, and $C_{v}$ is specific heat at constant strain or volume.

Equations (1) and (5) are derived as follows

\frac{\partial Q}{\partial t} = T_{0} {β}^{T} \frac{\partial {ε}}{\partial t} + ρ C_{v} \frac{\partial (Δ T)}{\partial t} - [K^{T}] \nabla^{2} T

(6)

where $[K^{T}]$ is thermal conductivity matrix.

Applying the variation principle to the stress equation (equation (4)) and the heat transfer equation (equation (6)) coupled by the thermo-elastic constitutive equations, the finite element matrix equation is produced as follows

\begin{array}{l} [\begin{matrix} [M] & 0 \\ 0 & 0 \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{u}} \\ {\overset{\cdot\cdot}{T}} \end{matrix}} + [\begin{matrix} [C] & [0] \\ [C^{t u}] & [C^{t}] \end{matrix}] {\begin{matrix} {\dot{u}} \\ {\dot{T}} \end{matrix}} \\ + [\begin{matrix} [K] & [K^{u t}] \\ [0] & [K^{t}] \end{matrix}] {\begin{matrix} {u} \\ {T} \end{matrix}} = {\begin{matrix} {F} \\ {Q} \end{matrix}} \end{array}

(7)

where $[M]$ is element mass matrix, $[C]$ is element damping matrix, $[K]$ is element stiffness matrix, $[C^{t}]$ is element specific heat matrix, $[K^{t}]$ is element diffusion conductivity matrix, $[C^{t u}]$ is element elastic damping matrix, and $[K^{u t}]$ is element elastic stiffness matrix.

Thermoplastic strain equation can be produced by considering plastic deformation heat on the thermo-elastic strain equation. That is

\begin{array}{l} [\begin{matrix} [M] & 0 \\ 0 & 0 \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{u}} \\ {\overset{\cdot\cdot}{T}} \end{matrix}} + [\begin{matrix} [C] & [0] \\ [0] & [C^{t}] \end{matrix}] {\begin{matrix} {\dot{u}} \\ {\dot{T}} \end{matrix}} \\ + [\begin{matrix} [K] & [0] \\ [0] & [K^{t}] \end{matrix}] {\begin{matrix} {u} \\ {T} \end{matrix}} = {\begin{matrix} {F} \\ {Q} + {Q^{p}} \end{matrix}} \end{array}

(8)

where ${Q^{p}} = \int_{v o l} {\dot{Q}}_{n}^{p} {N} d v$ , ${\dot{Q}}_{n}^{p}$ is heat generation velocity of plastic strain per unit volume on nth load step, and ${N}$ is shape function. From above formula, stress according to every time and position can be calculated.

Finite element model and welding sectors

Finite element model

This article estimates the residual stress of the joint between the welded hollow steel sphere, the outer diameter and thickness of which is 30 and 3 cm, and the steel cylinder tube, the outer diameter, thickness and length of which is 9 cm, 1.5 cm, and 1.5 cm. The study ignores the influence of residual stress produced in heat plastic treatment and weld of two half spheres in previous manufacturing process. And the joint is not preheated before and after welding.

The temperature fields and the evolution of residual stresses are investigated by means of the finite element method. In order to accurately capture temperature fields and residual stresses in sphere-tube weld, a three-dimensional (3D) finite element model is developed as shown in Figure 1. The mesh in welding zone is refined, where the element size is 0.4–0.7 mm, and the transitional pyramid mesh is used to make mesh coarsen in the area that is far from the welding zone.

Figure 1.

Finite element model of welded hollow sphere joint.

Welding sectors

There are three welding layers including fillet layer and every layer has 12 equal interval welding sectors, and the whole number of welding sectors is 36. Supposing that the voltage and current is constant in the welding process, welding volume and time of every sector is divided equally (Figures 1 and 3).

In this study, all analyses are performed using ANSYS APDL code of ANSYS Inc.,²⁴ and all graphs are drawn by MATLAB of MathWorks Inc.²⁵

Physical properties of material dependent on temperature

Thermal properties of material dependent on temperature

The material type of considered welded hollow sphere and tube span are Q235, and the type of weld material is D802 (C 0.7–1.44, Si ⩽ 2, Mn ⩽ 2, Cr 25–32, Fe ⩽ 4, W 3–6, and other ⩽4 remainder Co).

Matrix and weld materials are heated up to much higher temperature above the fusing point in the welding process, which produces not only heat deformation but also state change of solid-fluid-solid in the weld material, and finally large residual stress is produced.

The thermal properties of matrix and weld material including density, specific heat, heat conductivity, convection coefficient, and so on depend on temperature and jump at the boundary of fusing point, where the material state changes from solid to liquid (Figure 2).

Figure 2.

Thermal properties change according to temperature (matrix, weld material).

Structural properties of material dependent on temperature

The yield strengths of welded hollow sphere and tube span, Q235, are 235 MPa at ordinary temperature, and the yield strength of weld material, D802, is 450 MPa at ordinary temperature.

The structural properties of matrices and weld material including Young’s ratio, Poisson’s ratio, expansion coefficient, yield stress, and so on depend on temperature and jump at the boundary of fusing point, where the material state changes from solid to liquid (Figure 3).

Figure 3.

Structural properties change according to temperature (matrices, weld material).

On the contrary, during welding phase transformation, a given volume of material contains several kinds of volume fraction, namely, liquid, austenite, pearlite, and martensite, and other structures induced by precipitation by recovery effect, say, during the cooling process. However, most of martensite in the welding material generally appears in the process of critical quenching austenite in cooling period after welding.³ In this work, the cooling process does not use critical quenching treatment but uses natural convection treatment, so the effect of solid-state phase transformation on the residual stress is ignored. If it is necessary to analyze the effect of phase transformation on the residual stress, the new material properties during solidification stage (from austenite to martensite) in cooling process should be added.

Welding heat efficiency and heat source model predicting the heat efficiency

Based on measuring the electrical power of MMAW, welding speed of butt weld (melting volume per second), and the maximum temperature of welding liquid, the heat efficiency and heat generation rate per unit volume can be predicted.

Model of heat source is one of the most important input parameters in thermal numerical analysis. During welding process, the heat absorbed into matrices and weld material is

Q = η \int_{t_{w r}} u i d t

(9)

where $η$ is efficiency; $u, i$ is voltage and current depending on time; and $t_{w r}$ is real welding time.

Now the value of $η$ is considered. $Q$ is the heat absorbed into weld material and matrix, and the heat absorbed into weld material occupies the most of $Q$ .

The efficient heat $Q$ includes the heat amount for heating the solid of welding material on the extrude from ambient temperature up to the fusion temperature, for melting the welding material from solid to liquid on the fusion temperature and for reheating the liquid of welding material in the bead from the fusion temperature up to maximum temperature, and excludes the heat amount of releasing into ambient air through heat conduct, convection, radiation, and so on.

In fact, the welding material is not ideal crystal and has its melting range of about 1450°C and 1550°C; however, according to the 2nd law of thermodynamics, regardless of whether the welding material is considered to be crystal or amorphous, the whole heat amount absorbed or released during state change process does not change. Therefore, supposing the considered weld material to be ideally perfect crystal so as to save the calculation amount, during the state change, the heat transformation must not produce any temperature change, so the temperature of weld material is constant at the fusing point and only the state of material changes—melting or solidification. Therefore, $η$ must be satisfied

\frac{ρ_{w} V_{w} [\int_{T_{a i r}}^{T_{λ}} C_{s} d T + λ + \int_{T_{λ}}^{T_{\max}} C_{l} d T]}{\int_{t_{w r}} u i d t} < η < 1

(10)

where $C_{s}, C_{l}$ are the specific heats of solid and liquid of weld material depending on temperature, $T_{a i r}$ is the temperature of ambient air (20°C), $T_{λ}$ is fusing point (1535°C in the case of D802), $T_{\max}$ is the value of the maximum temperature of weld liquid in every welding sector (2010°C in this work), $λ$ is specific melting heat (3.772 × 10⁵ J/kg), $ρ_{w}$ is density of welding material, and $V_{w}$ is sector volume of weld material melted in the welding time $t_{w r}$ .

Supposing that the voltage and current are constant in the welding process and using the average specific heat in solid and liquid of weld material, equation (10) is simplified as

\frac{ρ_{w} V_{w} [\bar{C_{s}} (T_{λ} - T_{a i r}) + λ + \bar{C_{l}} (T_{\max} - T_{λ})]}{U I t_{w r}} < η < 1

(11)

where

\bar{C_{s}} = \frac{1}{(T_{λ} - T_{a i r})} \int_{T_{a i r}}^{T_{λ}} C_{s} d T, \bar{C_{l}} = \frac{1}{(T_{\max} - T_{λ})} \int_{T_{λ}}^{T_{\max}} C_{l} d T

They are estimated to be 678 and 1512 J/kg °C in this work.

Substitutin, into equation (11), voltage, current ( $U$ = 20 V, $I$ = 200 A), and sector volume of weld material melted in the welding time, the numerical range of $η$ is approximately decided. In this project, considering that a sector volume $V_{w}$ is about 1.02 × 10⁻⁶ m³ and the welding time $t_{w r}$ is 5.3 s, then the minimum of $η$ is estimated to be about 78.23%, which is consistent with the result of Michaleris.⁷

Welding heat source model

In order to evaluate the effect of multi-layer symmetric welding in this work, and MMAW is used, and the heat load from the moving welding arc is not used, respectively, the ellipsoid heat source model for straight weld, but the heat load from static welding arc is used as a body heat source. The Gaussian distribution was commonly used to simulate heat source model and the satisfactory accuracy can be achieved for the usual welding method such as MMAW. From the value of heat efficiency $η$ in equation (11), the heat flux is calculated according to the Gaussian distribution function not related to time as equation (12)¹⁹

q (r) = \frac{η U I}{2 π r_{a}^{2}} \exp (- \frac{r^{2}}{2 r_{a}^{2}})

(12)

where $q (r)$ is heat flux at distance $r$ from center of heat source, W/m²; $η$ is overall heat input efficiency, %; $U$ is arc voltage, V; $I$ is welding current, A; $r_{a}$ is Gaussian distribution parameter which is the radius of area; and $r$ is polar coordinate. The parameters of the heat source model are shown in Table 1.

Table. 1.

Parameters of heat source model.

Property name	Value
Voltage, V	20
Current, A	200
Efficiency, %	82
Radius of area, mm	1.5
Number of fill sectors	36

A new method of considering latent heat based on equivalent state change time

The heat transformation in the change of material state should be considered.

Actually, the finite element simulation cannot reflect the influence of absorbing or emitting heat in change process of state, because there is no temperature change in the melting or solidification process. Therefore, the heat quantity simulated in the melting process is smaller and the heat quantity simulated in the solidification process is larger than the actual fusion heat. That is the reason why this article proposes a conception of equivalent state change time, by reversing the heat absorbed or emitted in the change of state into the equivalent state change time. This equivalent state change time represents the time to be subtracted from actual welding time in melting process or to be added to actual cooling time in solidification process.

Then the equivalent state change time is calculated.

Supposing that heat transfer velocity is constant in welding process, the equivalent state change time $τ_{e q}$ can be defined as

τ_{e q} = \frac{λ}{[\int_{T_{a i r}}^{T_{λ}} C_{s} d T + λ + \int_{T_{λ}}^{T_{\max}} C_{l} d T]} t_{w r}

(13)

Considering equations (11) and (13) can be rewritten as

τ_{e q} = \frac{λ}{[\begin{array}{l} \bar{C_{s}} (T_{λ} - T_{a i r}) + λ \\ + \bar{C_{l}} (T_{\max} - T_{λ}) \end{array}]} t_{w r} = η_{m} t_{w r}

(14)

where $η_{m}$ is the ratio of melting time on the real weld time. In this project, considering the initial condition and the material properties, the $η_{m}$ = 14.6%. Therefore, the equivalent weld time and equivalent cooling time are

{\begin{matrix} t_{w e} = t_{w r} - τ_{e q} = (1 - η_{m}) t_{w r} \\ t_{c e} = t_{c r} + τ_{e q} = t_{c r} + η_{m} t_{w r} \end{matrix}

(15)

where $t_{w e}$ is equivalent weld time, $t_{c e}$ is equivalent cooling time, and $t_{c r}$ is real cooling time.

After all, the equivalent weld time is about 85.4% of the actual weld time and the equivalent cooling time is 1.146 times of real weld time longer than the real cooling time. In this work, considering that the real weld time is 5.3 s and the real cooling time is 4.2 s, the equivalent state change time $τ_{e q}$ is calculated to be about 0.8 s from equation (15). Then the equivalent weld time is 4.5 s and the equivalent cooling time is 5 s from equation (15).

Welding methods

This article uses three different welding methods (Figure 4).

Figure 4.

Welding methods.

Method 1 is a multi-layer symmetric welding method to perform symmetric weld in the first layer and go on to the other layers in this way. Method 2 is a multi-layer sequential welding method to perform sequential weld in the first layer and go on to the other layers in this way. And method 3 is a single-layer symmetric welding method to perform sequential multi-layer weld at one sector and go on to symmetric weld at the other sectors.

In this project, simulation welding time at one sector is 4.5 s, simulation cooling time under the natural convection after each welding is 5 s, and it takes 337 s to complete welding 36 sectors. Then, cooling time under the natural convection after completion of all the welds is 4 h; after all, the entire simulation time is 14742 s (4 h 5 min 42 s).

For convergence of solution, the time interval during welding process must be extremely short (0.001–0.01 s), and for saving the analysis time, the time interval after completion of welding is adjusted to about 10 s.

As temperature boundary condition, the temperature is constant on the opposite end side of tube span against the sphere; the outer surfaces of sphere, span and weld layers are under the natural convection condition; and ambient air temperature is 20°C. As structure boundary condition, the opposite end side of tube span against the sphere is completely fixed.

For comparing temperature field and stress field according to three welding methods under the same conditions, it is assumed that the welding parameters do not change accordingly.

Comparison of residual stress in different welding methods considering latent heat

Comparison of temperature field

In using welding method 1, the maximum temperature of weld joint is 1685.9°C (Figure 5) at the moment of completion of weld (337 s) and is 25.7246°C after 4 h of cooling by natural convection (14742 s) (Figure 6, Supplemental Material Online Resource 1).

Figure 5.

Temperature distribution at the moment of completion of weld (337 s).

Figure 6.

Temperature distribution after cooling 4 h by natural convection (14742 s).

In using welding method 2 (Supplemental Material Online Resource 2)and method 3 (Supplemental Material Online Resource 3), the maximum temperature of weld joint is similar to in using method 1 at the moment of completion of weld (337 s) and is 25.7726°C, 25.7585°C after the same hours of cooling (14742 s).

In welding process, the time, the position, and the temperature, when the temperature increases up to the maximum value, are different in using each of the three welding methods. In using method 1, the temperature is 1968.96°C in the 11th sector on the first layer at 99.5 s. In using method 2, the temperature is 2032.13°C in the 12th sector on the first floor at 109 s. And in using method 3, the temperature is 2073.96°C in the 32nd sector on the third floor at 299 s (Figure 7). Therefore, the maximum temperature at the weld pool is approximately 2000°C, which is the same as the results in Wang et al.¹⁸

Figure 7.

Maximum temperature and position in each welding method.

This shows that the maximum temperature is the lowest in using method 1 out of the three welding methods. In the figures, unlinked partitions are sectors that have not yet been welded.

The temperature changes are considered at a surveying point (the center of the first sector on the first layer) according to welding time (0–400 s) (Figure 8).

Figure 8.

Temperature change according to the time.

The three graphs show that the temperature change at the surveying point is different in using each welding method. In using method 1, during the whole welding, the temperature increases for three times up to about 1820°C, 1820°C, and 1120°C at each interval time of one layer welding time (114 s), and in one layer welding time, the temperature increases twice, when welding the neighbor sectors, up to about 850°C and 980°C influenced by symmetric welding. In using method 2, during the whole welding, the temperature increases for three times up to about 1820°C, 1820°C, and 1120°C at each interval time of one layer welding time (114 s) as in using method 1, but in one layer welding time, the temperature slowly decreases in the former half part of the layer and increases again in the later half part of the layer influenced by sequential welding. In using method 3, throughout the welding, the temperature increases for three times, when welding the present sector and the next sectors, up to about 1820°C, 870°C, and 1030°C influenced by symmetric welding (Figure 8).

Comparison of stress field

In using welding method 1, von-Mises stress distribution at the last moment of welding (337 s, max 333 MPa) and after 4 h of cooling by natural convection (14742 s, max 400 MPa) are shown in Figures 9 and 10.

Figure 9.

Von-Mises stress distribution at the last moment of welding (337 s).

Figure 10.

Von-Mises stress distribution after 4 h of cooling by natural convection (14742 s).

The von-Mises stress change at the surveying point according to welding time (0–400 s) is shown in Figure 11.

Figure 11.

Von-Mises stress change according to time.

These graphs show that the stress at surveying point in using each method increases up to 350–380 MPa and thoroughly depends on the temperature change graphs. The stress decreases every time when the temperature increases in welding and increases every time when the temperature decreases in cooling. The time point of local maximum stress on stress change graph (Figure 11) equals the time point of local minimum temperature on temperature change graph. (Figure 8) In other words, the shape of stress graph is upside-down, compared with the temperature graph.

Then, the residual stress in welding material after 4 h of cooling (14742 s) is described more detailedly.

First, von-Mises residual stress is described. In using every method, the maximum von-Mises residual stresses all approaches to the yield strength of weld material, but the distributions of stress are different from one other (Figure 12).

Figure 12.

Von-Mises residual stress distribution according to welding method.

In using methods 1 and 3, residual stress distributions are symmetric on the link axis of sphere and span, but in using method 2, the residual stress distribution is asymmetric and high stresses are concentrated at the first location of weld, and the residual stress distribution decreases along the welding route. Therefore, in using method 2, needless to say, supporting ability of structure will be unbalanced.

Second, principal residual stress vector and maximum principal residual stress are described. In using every method, the location of maximum tensile stress is the outer surface of upper layer in the direction of beginning the welding. The residual stress distribution, in using methods 1 and 3, consists of four tension–compression partitions and the value of maximum tensile stress in using method 3 is 1.36 times, 140 MPa higher than in using method 1. In using method 2, the stress slowly decreases from tensile down to compress along the welding route (Figures 13 and 14).

Figure 13.

Principle residual stress vector in using each of the three welding methods.

Figure 14.

Principle tensile residual stress distribution in using each of the three welding methods.

Finally, the component residual stresses are described. For considering the change of stress according to the positions, three paths are set up (Figure 15).

Figure 15.

Considering paths.

The first path AB is the radial direction linear path passing through the surveying point (center of the first welding sector), where A is the point at the bottom of the first welding sector in the first floor, and B is the point on the outer surface of the same sector in the third floor. The second path is the hoop direction circular path passing C, where C is the center of second sector in the second floor and rotating direction is welding direction in using method 2. The last third path DE is the axial direction linear path passing through C, where D is the point in the welded hollow sphere and E is the point in the circular tube (Figure 15).

Radial stress distribution and graphs of component stresses on radial direction path AB are shown in Figures 16 and 17. Radial stress distribution is similar to the other stress distributions, but absolute value is the smallest of the three component stresses. Graphs of component stresses on the path explain that stress values increase from A up to B on the path in using methods 1 and 2, but decrease in using method 3.

Figure 16.

Radial residual stress distribution according to welding method.

Figure 17.

Graphs of component stresses on radial direction path AB (o: radial stress, *: hoop stress, +: axial stress).

Hoop stress distribution and graphs of component stresses on hoop direction path C are shown in Figures 18 and 19. Hoop stress distribution is similar to von-Mises stress distribution, but the absolute value is smaller. Graphs of component stresses on the path explain that in using methods 1 and 3, there are four repeated partitions of tension-compression on the circular path, and the difference of each peak value in using method 1 is smaller than in using method 3, and in using method 2, the result of which is similar to the experiment result of Wang et al.,¹⁸ the stress slowly decreases along the weld route.

Figure 18.

Hoop residual stress distribution according to welding method.

Figure 19.

Graphs of component stresses on hoop direction path C (o: radial stress, *: hoop stress, +: axial stress).

Axial stress distribution and graphs of component stresses on axial direction path from D to E are shown in Figures 20 and 21. Axial stress distribution is similar to hoop distributions, but the stress size in using method 3 is the largest. Graphs of component stresses on the path explain that stress sizes in weld material are the largest and the smallest in the linked part of the circular tube nearby the welding part.

Figure 20.

Axial residual stress distribution according to welding method.

Figure 21.

Graphs of component stresses on axial direction path DE (o: radial stress, *: hoop stress, +: axial stress).

After all, in using every method, the axial stress and hoop stress out of the three component stresses play an important role, and the influence of radial stress is the smallest. Stress distribution is also symmetric in using methods 1 and 3 and asymmetric in using method 2. The stress value in using method 3 is much higher than in using method 1. In using method 3, radial stress is 1.26 times (46 MPa), hoop stress is 1.17 times (57 MPa), and axial stress is 1.34 times (130 MPa) higher than in using method 1, and it means that, in using the multi-layer symmetric welding method, the radial stress, the hoop stress, and the axial stress are smaller than in using the single-layer symmetric welding method—79.3%, 85.4%, and 74.6%, respectively.

Conclusion and discussion

Based on the results of the study, the following conclusions can be made:

In analyzing the welding process by the finite element method, the heat during the material state change in the simulation process must be subtracted from the real welding heat and added to the real cooling heat in order to evaluate the effects of melting and solidification processes correctly. The amount of heat during the change in melting or solidification can be simulated by reversing the heat absorbed or released during the state change into an equivalent state change time.

The equivalent state change time is 14.6% of the real welding time. From this, the simulation time in the melting process becomes smaller than the real welding time—the equivalent state change time, and the simulation time in the solidification process becomes larger than the real cooling time—the equivalent state change time.

The residual stress distribution and its value, produced during 3D multi-layer welding process of super-big WHSJ in large-span structure, are much different in using each of the three welding methods. In using every method, the location of maximum tensile stress is the outer surface of upper layer in the direction of beginning the welding, regardless of choosing any of the three welding methods. This shows that the distortion of the joint shape after welding has the tendency to be swollen at the first point of welding.

The axial stress and hoop stress play an important role, and the influence of radial stress is the smallest of the three component stresses in analysis of WHSJ residual stress in the large-span structure.

The components of residual stresses in the multi-layer symmetric welding method are more balanced than in the multi-layer sequential welding method, and are smaller than in the single-layer symmetric welding method—74% to 85%, respectively.

Footnotes

Acknowledgements

I would like to express my wholehearted gratitude to my conductor Prof. Chen Zhihua and Prof. Ma Zhiyao of the Tianjin University for their selfless advices to the research. I also thank Dr Sim Song-jun, the Chief of English Department of Pyongyang University of Foreign Studies, for his great help to my paper in English.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is sponsored by the National Natural Science Foundation of China (Grant No. NSFC61272264).

ORCID iD

Nam-Hyok Ri

Supplemental material

Supplemental material for this article is available online.

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