Experimental and Numerical Investigation of Post-Weld Heat Treatment on Residual Stress Relaxation in Orthotropic Steel Decks Welding

Abstract

Orthotropic steel decks (OSDs) serve as critical load-bearing components in long-span steel bridges, but high-amplitude welding residual stresses (WRSs) generated during the welding process pose significant threats to structural integrity. To mitigate these stresses, post-weld heat treatment (PWHT) has emerged as a promising technique. This investigation first establishes a semi-structural thermo-elasto-plastic finite element model of the Deck-U-rib-Diaphragm system with a six-pass welding sequence. The temperature field is modeled via a double-ellipsoidal heat source and birth–death element approach. Subsequently, thermo-mechanical coupling analysis is conducted to investigate the distribution characteristics of Von Mises residual stresses. The stress relief effect of PWHT is then explored by comparing different holding temperatures (T) and holding times (t), achieving a balance between stress reduction effectiveness and economic efficiency, when T = 550 °C and t = 40 min. Finally, full-scale experimental tests are designed, and the hole-drilling method is utilized to validate the numerical simulation results. This research provides valuable insights for the design of PWHT processes for OSDs.

1. Introduction

Orthotropic steel decks (OSDs), as one of the key load-bearing members for long-span steel bridges, combine the advantages of light weight, high strength, adaptability, and ease of construction [1,2,3,4]. They consist of a deck plate, U-rib, and diaphragm welded together to provide different stiffnesses in the longitudinal and transverse directions [5], thus optimizing the efficiency of bending and torsion resistance. The welding process is characterized by localized, non-uniform heat input and rapid cooling, driven by temperature gradients. The steel components undergo thermal expansion, plastic deformation, and phase transformation in the heat-affected zone (HAZ), thus generating self-equilibrating welding residual stresses (WRSs) [6]. Theoretically, WRS is comparable to the yield strength value of the metal material; furthermore, with the complex geometry of OSDs (as shown in Figure 1), WRS poses a significant threat to the service performance of long-span steel bridges. The main potential hazards associated with WRS include (1) reducing fatigue life [7,8,9]; (2) stress corrosion cracking [10,11]; (3) weakening the stability [12,13]. As traffic volumes increase, long-span steel bridges are subjected to more complex traffic loads, and mitigating the adverse effects of WRS will be a key consideration in steel bridge fabrication.

The precise control of WRS in OSDs necessitates a comprehensive understanding of their spatial distribution patterns, which has been extensively investigated in prior research: Wang et al. [14] conducted a comprehensive investigation on the WRS of U-rib stiffened plates through numerical simulation and experimental measurements, discussing the influence of dimensional parameters on WRS distribution patterns and providing a schematic diagram of WRS distribution; Wang et al. [15] compared the WRS distribution modes of OSDs under full penetration single- and double-sided welding, demonstrating that the WRS in longitudinal compressive stress regions was significantly higher in double-sided welding than in single-sided welding; Kong et al. [16] investigated the generation mechanisms and distribution characteristics of WRS in double-sided submerged arc-welded OSDs using a thermomechanical sequential coupling approach, with particular emphasis on the effects of plate thickness and welding sequence on WRS in the joint regions.

Since the primary origin of WRS is non-uniform plastic strain, which is notably influenced by dynamic softening and creep effects in steel [17,18,19], there remains significant potential for WRS relaxation through the optimized design of post-weld treatments. In order to address the hazards associated with WRS, a number of effective methods have been proposed: (1) hammer peening [20,21]; (2) vibrational stress relief [22,23]; (3) shot peening [24,25]; (4) explosive technique [26,27]. In addition to the aforementioned methods, post-weld heat treatment (PWHT) [28] is also a widely utilized technique for WRS relaxation. This method offers several advantages, including highly controllable automation integration, minimal dependence on manual labor, and low cost, making it a promising development prospect.

Specifically, Feng et al. [29] conducted tests on the welded joint specimens of OSDs subjected to PWHT, and found that the peak WRS reduction exceeded 75%, thereby demonstrating its potential in improving fatigue performance; Qiang et al. [30] introduced a creep constitutive model into ABAQUS, and discussed in detail the mechanism of the influence of heating cooling rate, holding temperature and time on WRS relaxation in OSDs, which provided a reference for process design and evaluation of PWHT; Li et al. [31] analyzed the effect of the parameters of PWHT on the stress relaxation of double-sided welded OSDs, utilizing both numerical simulations and blind hole tests, and proceeded to explore the feasibility of secondary continuous PWHT. Liu et al. [32] introduced the creep effect to analyze the WRS evolution of a 45 mm bridge steel butt-welded joint during PWHT, finding that the stress relaxation occurred mainly during the heating stage and was more sensitive to the temperature factor. In summary, the research on PWHT for OSDs has evolved from experimental testing to numerical simulation. Traditional experimental methods require the preparation of numerous specimens for parametric analysis, which significantly increases costs. In contrast, the development of numerical simulation techniques has effectively addressed this limitation. However, existing studies have rarely considered the scenario of PWHT in structures with the diaphragm. Notably, the welding of the diaphragm introduces high-amplitude WRS [33], and the complex stress states in OSDs at diaphragm locations increase the susceptibility to fatigue crack initiation. Therefore, it is essential to conduct further analysis on this critical aspect.

This investigation focused on a thermo-elasto-plastic finite element method (TEP-FEM) of the local semi-structural 6-pass welding sequence for OSDs made of S355J2G3 steel. The PWHT processes were simulated using the specialized welding software SYSWELD 2022. The temperature field and WRS distribution in the component were analyzed, along with the effect of holding temperature and holding time on stress relaxation. The numerical simulation results were validated using the hole-drilling method.

2. Theory of TEP-FEM and Creep

2.1. Welding Heat Transfer and Heat Resource Theory

The fundamental characteristics of metal material welding involve localized high heat input and rapid cooling, which can be categorized as a nonlinear transient heat conduction problem encompassing both temporal and spatial domains, and is described by Fourier’s law.

During the welding process, the welded component exchanges heat with the ambient medium, primarily categorized into convection heat transfer and radiation heat transfer. The former is calculated based on Newton’s law of cooling:

$$q_c=h_c(T_s-T_f)$$

(1)

where $q_c$ is the heat flow density in this heat exchange mode; $h_c$ is the convective heat transfer coefficient; $T_s$ is the surface temperature of the member; $T_f$ is the surrounding medium temperature, assumed to be 25 °C.

The latter is calculated according to the Stefan–Boltzmann law:

$$q_r=\epsilon \sigma (T_s^4-T_f^4)$$

(2)

where $q_r$ is the heat flow density in this heat exchange mode; $\epsilon$ is the surface emissivity of the material; $\sigma$ is the Stefan–Boltzmann constant taken as $5.669\times {10}^{-8} \mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$.

Currently, the dual ellipsoidal heat source model (DEHSM, Figure 2) proposed by Goldak [34] demonstrates favorable simulation performance of the submerged arc welding process, comprising the front $(q_f)$ and rear $(q_r)$ ellipsoid, respectively:

$$q_f={\displaystyle \frac{6\sqrt{3}{f}_f\eta UI}{a_{f}bc{\pi }^{3/2}}}\exp \left(-{\displaystyle \frac{3x^2}{a_f^2}}\right)\exp \left(-{\displaystyle \frac{3y^2}{b^2}}\right)\exp \left(-{\displaystyle \frac{3z^2}{c^2}}\right)$$

(3)

$$q_r={\displaystyle \frac{6\sqrt{3}{f}_r\eta UI}{a_{r}bc{\pi }^{3/2}}}\exp \left(-{\displaystyle \frac{3x^2}{a_r^2}}\right)\exp \left(-{\displaystyle \frac{3y^2}{b^2}}\right)\exp \left(-{\displaystyle \frac{3z^2}{c^2}}\right)$$

(4)

where $f_f, f_r$ are the energy distribution coefficients of the front and the rear ellipsoids of the DEHSM, respectively, satisfying $f_f+f_r=2$; η is the welding efficiency taken as 85%; U, I are the welding voltage and current; $a_f, a_r$ are the lengths of the front and rear half-axis of the DEHSM along welding direction; $b$ is the length of the half-axis in the width direction; $c$ is the length of the half-axis in the depth direction.

2.2. Elastic-Plastic Mechanics Theory

Due to the nonlinear TEP behavior inherent in welding mechanics, and the fact that WRS often reach the yield strength of the material, the Von Mises yield criterion is adopted to describe the yielding condition under complex stress states where plastic deformation begins:

$${\sigma }_{Von}={\displaystyle \frac{1}{\sqrt{2}}}\sqrt{{({\sigma }_x-{\sigma }_y)}^2+{({\sigma }_y-{\sigma }_z)}^2+{({\sigma }_z-{\sigma }_x)}^2}\ge f_y$$

(5)

where ${\sigma }_x$, ${\sigma }_y$, ${\sigma }_z$ are the principal stresses in the three orthogonal directions; $f_y$ is the yield strength of the material; ${\sigma }_{Von}\ge f_y$ represents that the material enters the yield state.

Correspondingly, the expression for the Von Mises equivalent strain is:

$${\epsilon }_{Von}={\displaystyle \frac{\sqrt{2}}{2(1+\nu )}}\sqrt{{({\epsilon }_x-{\epsilon }_y)}^2+{({\epsilon }_y-{\epsilon }_z)}^2+{({\epsilon }_z-{\epsilon }_x)}^2+{\displaystyle \frac{3}{2}}({\gamma }_{xy}^2+{\gamma }_{yz}^2+{\gamma }_{zx}^2)}$$

(6)

where $\nu$ is the Poisson’s ratio; ${\epsilon }_x$, ${\epsilon }_y$, ${\epsilon }_z$ are the principal strains in the three orthogonal directions; ${\gamma }_{xy}$, ${\gamma }_{yz}$, ${\gamma }_{zx}$ are the shear strains of the three adjacent surfaces.

Once the material enters the yield stage, the flow rule is employed to describe the relationship between plastic strain and yield strength, wherein the evolution of plastic deformation is captured in relation to the material’s yield behavior:

$$\mathrm{d}{\left\{\epsilon \right\}}_p=\mathrm{d}\lambda {\displaystyle \frac{\partial {\sigma }_{Von}}{\partial \left\{\sigma \right\}}}$$

(7)

where $\mathrm{d}{\left\{\epsilon \right\}}_p$; $\mathrm{d}\lambda$ is the plastic strain increment; is the plasticity factor; $\sigma$ is the stress function.

The hardening criterion is employed to describe how the yield behavior of a material evolves with deformation history, revealing the evolution of the size, center position, and shape of the subsequent yield surface. For strain hardening in metallic materials, the equivalent hardening criterion is adopted:

$$f(\sigma ,\xi )=0$$

(8)

where $\xi$ is the internal variable that remains constant within the yield surface, when changes in loading induce its variation, a new yield surface is formed.

2.3. Creep Theory

From the perspective of material behavior, the mechanisms of welding heat treatment include high-temperature creep: Under elevated temperatures, the strain of steel exhibits both nonlinear dependence on stress and time-dependent characteristics. The viscous plastic strain of steel is described using the creep law (kinematic approach) [35] inherent in the SYSWELD software:

$${\dot{\epsilon }}_p=K{(\sigma -1.5\alpha )}^n$$

(9)

$$\dot{\alpha }=H{\dot{\epsilon }}_p-{1.5}^{P-1}C{\alpha }^P$$

(10)

where $\alpha$ is the kinematic strain hardening variable; $\sigma$ is the stress function; $C,H,K,P,n$ are all temperature-dependent coefficients; $H{\dot{\epsilon }}_p$ is used to describe strain hardening and $-{1.5}^{P-1}C{\alpha }^P$ is used to describe viscous recovery; After specifying the initial plastic strain rate ${\dot{\epsilon }}_p^{(1)}$, the stable value of $\alpha$ can be obtained through the iterative calculation of Equation (10), with the iteration termination condition defined as ${\dot{\epsilon }}_p^{(k)}<{\dot{\epsilon }}_p^{(1)}$. Thereby implying that the secondary creep (steady-state creep) in this constitutive model arises from the equilibrium between strain hardening and viscous recovery. The material parameters were obtained from SYSWELD material database.

3. Numerical Simulation Detail and Experimental Procedure

3.1. Properties of Materials

In welding finite element analysis, significant temperature gradients are a distinctive feature, necessitating the incorporation of temperature-dependent material properties in the input parameters. Considering the requirements for completeness and accuracy of material parameters in the FEM, the temperature-dependent material properties of European Standard S355J2G3 [36] steel from the SYSWELD material library are utilized, as shown in

Table 1. The chemical composition of EN S355J2G3.

Composition	C	Mn	Si	P	S	Ni	Cr	Mo	Fe
%	0.05	1.47	0.23	0.02	0.004	0.03	0.03	0.004	residual amount

and Figure 3.

3.2. Finite Element Modeling

The welding-specific software SYSWELD (ESI Group, Bagneux, France), which integrates multi-physics coupling, precise material properties, and finite element computation principles, is extensively utilized in fields including aerospace, automotive, shipbuilding, and steel structures. Its primary modeling workflow is illustrated in Figure 4.

The detailed structure and dimensions of this local Deck-U-rib-Diaphragm component are as follows: the width of the deck is 600 mm, and the thickness is 20 mm; the upper opening width of the longitudinal U-rib is 300 mm, the lower opening width is 180 mm, the radius of the transition arc is 40 mm, the thickness is 8 mm, and it is welded on both sides; the bevel angle of the outer weld seam is 52°, and no bevel is set on the inner side, the angle between the U beam and the deck is 79°; the thickness of the diaphragm is 25 mm, and it smoothly transitions to 35 mm away from the lower opening of the U-rib at the cutting edge, the radius of the transition arc is 75 mm. The welding process and dual ellipsoidal heat source model parameters are shown in

Table 2. Welding Process and Dual Ellipsoidal Heat Source Model Parameters.

Weld Number	I/A	U/V	v/(mm·s−1)	a1/mm	a2/mm	b/mm	c/mm
WS_1	340	32	5.83	4.5	18.0	4.5	5.0
WS_2	620	32	6.67	7.0	28.0	5.0	6.0
WS_3~WS_6	630	30	6.67	7.0	28.0	5.0	6.0

.

Leveraging the symmetry of the structure, a semi-structural sequential 6-pass welding thermo-elasto-plastic finite element model of the OSDs (Figure 5) is established using the Visual Mesh module in SYSWELD. The mesh is primarily composed of hexahedral elements, with a small number of pentahedral elements, while the weld zone and heat-affected zone (HAZ) are entirely hexahedral to enhance the model’s adaptability to nonlinear phenomena and reduce the likelihood of computational non-convergence.

To balance computational accuracy and efficiency, a transition mesh is adopted: due to the high stress amplitude in the weld zone and its vicinity, the mesh size is finer, with the weld zone mesh size ranging 1~3 mm; conversely, in regions far from the HAZ, where stress levels are lower, the mesh size is coarser, reaching up to 20 mm for the diaphragm away from the HAZ. Figure 5 illustrates the semi-structural model after meshing, comprising 96,870 nodes and 122,772 elements.

Due to the symmetry of the structure, a semi-structural model is adopted. On the symmetry plane, a face constraint in the x-direction ($Ux=0$) is applied. Additionally, point constraints are imposed on two edge nodes on the opposite side of the top plate: $Uy=0$ and $Uy=Uz=0$. This constraint configuration restricts rigid body motion while allowing the structure to freely expand and contract under thermal loading. By establishing welding lines and reference lines at each weld seam cluster, the heat source position and the cross-sectional orientation can be determined. Subsequently, defining the welding speed enables the parameterization of the heat source’s uniform linear motion along the welding line. The model incorporates six sequentially arranged weld seams (WS_1 to WS_6).

3.3. PWHT Program

To relieve welding residual stresses, five PWHT schemes are formulated with holding temperature $T$ and holding time $t$ as variables, as summarized in

Table 3. PWHT scheme.

Scheme	(I)	(II)	(III)	(IV)	(V)
Holding temperature (°C)	500	550	600	600	600
Holding time (min)	40	40	40	60	80

. The specific operational procedure for applying the post-weld temperature field in SYSWELD is illustrated in Figure 6, along with the corresponding temperature–time curves for each scheme. After executing these procedures, the stress field after heat treatment can be obtained. Notably, sufficient time must be allowed for the component to cool down to ambient temperature after heat treatment to eliminate thermal stress interference; in this analysis, the cooling duration should not be less than 120 min.

3.4. Experiment

The WRS on the surface of the OSD specimen is measured using the hole-drilling method. This method was proposed by Mathar J. in 1934 [37], and is based on the principle of inferring residual stress by measuring the strain released during drilling. Before conducting the blind hole method test, a calibration test was carried out to obtain the release coefficients of the three corresponding strain gauges. The calibration specimens were the same as the material to be tested, and three specimens were cut along the rolling direction. They were prepared according to the GB/T31310-2014 specification, as shown in Figure 7a. Strain gauges were attached to both sides of the specimens to ensure uniform loading, and a resistance strain gauge was arranged at every 30 mm along the central axis. The specimens were loaded stepwise on a universal testing machine to the predetermined load, as shown in Figure 7b,c, and the strain readings were recorded. After unloading, holes were drilled, and the specimens were reloaded and the strains were recorded at each load level. The calculation formula for residual stress is as follows:

$$\left\{\begin{array}{l}{\sigma }_{1,2}={\displaystyle \frac{E}{4A}}({\epsilon }_1+{\epsilon }_3)\mp {\displaystyle \frac{E}{4B}}\sqrt{{({\epsilon }_1-{\epsilon }_3)}^2+{(2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3)}^2}\\ \tan 2\theta ={\displaystyle \frac{2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3}{{\epsilon }_3-{\epsilon }_1}}\end{array}\right.$$

(11)

where ${\sigma }_1, {\sigma }_2$ are the maximum and minimum principal stresses, respectively; ${\epsilon }_1, {\epsilon }_2, {\epsilon }_3$ are borehole release strains measured by sensitive grids $R_1, R_2, R_3$, respectively; $E$ is the Young’s modulus; $A$, $B$ are strain release coefficients, calculated as:

$$\left\{\begin{array}{l}A=-{\displaystyle \frac{(1+\nu )}{2E}}\cdot {\displaystyle \frac{d^2}{4r_1{r}_2}}\\ B=-{\displaystyle \frac{d^2}{2E{r}_1{r}_2}}\left[1-{\displaystyle \frac{1+\nu }{4}}\cdot {\displaystyle \frac{d^2(r_1^2+r_1{r}_2+r_2^2)}{4r_1^2{r}_2^2}}\right]\end{array}\right.$$

(12)

where $d$ is the diameter of the hole; $r_1, r_2$ are the distances from the proximal and distal ends of the strain gauges to the drilling center, respectively.

In addition, when opening holes with the drilling equipment, we calibrated the equipment, ensured the verticality of the drill rod, the rotational speed, and the hole depth control, and regularly calibrated the wear of the drill bit to avoid hole diameter deviations.

Finally, when collecting the data for the blind hole method test, the measure of taking the average of multiple measurements was adopted. Through the above processing methods, the uncertainty in the measurement of residual stress using the drilling method can be eliminated to the greatest extent and the repeatability can be guaranteed.

To validate the numerical simulation results, a full-scale specimen of the OSDs was fabricated in a steel structure manufacturing plant. The U-ribs were cold-formed, with internal and external welds processed using specialized internal welding machines and ship-position submerged arc welding equipment, respectively. The penetration rates of the welds met the design requirements, while the diaphragm welds were completed by welders. After visual inspection and non-destructive testing of the welds, it was confirmed that the specimen exhibited no significant defects, indicating excellent welding quality. Subsequently, the specimen was transferred to a mobile gas-fired heat treatment furnace for PWHT. After the holding period, the specimen was naturally cooled to ambient temperature, enabling subsequent welding residual stress testing (Figure 8).

4. Temperature, WRS, and PWHT Analysis

4.1. Temperature Field

The birth–death element technique is employed to simulate weld pool filling, where the weld seams are initially “killed” and progressively “activated” during the computational process. As shown in Figure 9, with the material’s melting point $T_m=1400 °\mathrm{C}$, the heat source region forms a molten pool boundary defined by $T_m$. The cross-sectional molten pool boundaries corresponding to the six weld seams exhibit an approximately arc-shaped profile.

As shown in Figure 10, six temperature observation points (Point_1~Point_6) are established at the intersection of the six weld seams and the base material. The complete thermal cycle curves for each point are extracted from the temperature field results. For each curve, when the center of the DEHSM approaches the observation point, the temperature rises sharply, exceeding the material’s melting point, and reaches a peak temperature of 1500~1800 °C, accompanied by material melting. After the heat source moves away, the temperature curve drops rapidly, falling below 300 °C within 100 s (a reduction of over 80%), and subsequently gradually decreases to the initial temperature over the next 300 s. Notably, the presence of adjacent measurement points, such as Point_1, 2 located on the inner and outer sides of the U-rib, respectively, leads to secondary temperature rises caused by heat transfer from adjacent welds. However, the temperature increases in the curves are all below 25% of the corresponding peak values. Overall, the thermal cycle curves of the six weld seams exhibit similar patterns, all accompanied by material melting and pronounced local high-temperature gradients, which align with the fundamental characteristics of welding heat input.

4.2. WRS Field

Under the localized heat source loading, the HAZ experiences a rapid temperature rise, causing thermal expansion of the material. However, constrained by surrounding regions, the internal stresses continuously increase and exceed the yield strength, resulting in irreversible plastic compression. Upon cooling, to maintain deformation compatibility, the surrounding constraints exert tensile forces on the plastic compression zone, forming a self-balanced stress state and ultimately generating WRS.

Figure 11 illustrates the Von Mises stress evolution history of the six weld seams after cooling. It can be observed that high-amplitude stresses are predominantly concentrated in the main regions of each weld seam. At the junction of multiple weld seams (near the middle of the top plate), the stress exhibits a localized minimum, which is attributed to thermal radiation from subsequent welds (WS_3~WS_6) approaching this area, resembling a localized heat treatment effect. By the final state, the U-rib exhibits the highest stress level, followed by the deck, while the majority of the diaphragm region remains in a low-stress state. This indicates that the U-rib demonstrates good overall load-bearing performance, whereas the diaphragm allows for the release of certain plastic strain.

4.3. PWHT Analysis

Figure 12 illustrates the temperature field evolution process of the finite element model during PWHT, indicating that the material undergoes softening and creep behavior, ultimately returning to ambient temperature. Taking Scheme (III) as an example, the time-dependent variations in stress and strain at the observation points (referenced as Point_1~6 in Figure 10) are plotted in Figure 13. It can be observed that Von Mises stress at each point exhibits a three-stage trend: rapid initial reduction, subsequent stabilization, and slow recovery followed by gradual stabilization. Meanwhile, Von Mises strain gradually increases and stabilizes under creep effects, which reflects the mechanism of PWHT.

The stress reduction in the rapid initial stage is primarily attributed to thermal softening, in which elevated temperatures cause partial recovery of the material’s microstructure, such as the rearrangement and annihilation of dislocations, reduction in residual strain energy, and localized restoration of lattice order. As PWHT proceeds, the holding temperature promotes creep deformation under constant thermal loading, allowing the redistribution of internal stresses through time-dependent plastic flow. During this stage, the sustained temperature accelerates the diffusion of atoms, enabling grain boundary sliding and subgrain growth, which further alleviate welding residual stresses. The subsequent slow recovery of stress upon cooling is mainly due to the gradual increase in elastic modulus as the material returns to ambient temperature, while plastic deformation accumulated during creep remains. Overall, the synergistic action of high-temperature softening and creep-induced stress relaxation constitutes the primary mechanism through which PWHT improves the structural integrity of OSDs.

The final Von Mises stress results for all schemes are presented in Figure 14, Figure 15, Figure 16 and Figure 17, revealing the following characteristics: (1) All schemes demonstrate significant improvements in Von Mises stress, with high-stress regions substantially reduced. The improvement becomes more pronounced as the holding temperature and holding time increase; (2) Before treatment, a considerable portion of the component surface exhibited stresses within the range of [442.4, 530.9] MPa. After PWHT, this stress range gradually narrows to undetectable levels. Specifically, when $T=600 °\mathrm{C}, t=80 \min$, the stress drops below the material’s yield strength (390 MPa). (3) However, at $T=600 °\mathrm{C}$, further increases in holding time do not significantly enhance stress relief, indicating that the relaxation effect is no longer increasing.

Further quantitative analysis of Von Mises stress under each scheme reveals the following: (1) Scheme (I) has a minimal impact on Paths 1~3, with only Path 4 showing a generalized stress reduction. The maximum stress reduction occurs near the end of Path 4, decreasing from 372.8 MPa to 242.5 MPa (a 35.0% reduction); (2) Scheme (II) significantly alleviates stress across all paths. The maximum stress values for Path 1~Path 4 are 321.2, 259.8, 332.5, and 337.9 MPa, respectively, representing maximum reductions of 151.9, 233.9, 146.9, and 178.5 MPa compared to the unprocessed state; (3) Scheme (III) demonstrates significant stress relief, but its improvement over Scheme (II) is minimal, with stress reductions of 0~20 MPa. This indicates that the optimization effect of holding temperature has reached a saturation point; (4) The stress curves of Schemes (III)~(V) exhibit high overlap, with differences maintained within 10 MPa, this suggests that extending the holding time has negligible additional benefits. (5) When comparing numerical simulation results with measured stress data, a high degree of correlation is observed, the majority of measured stresses agree with the simulations within ±30 MPa (error < 15%), as shown in

Table 4. Von Mises stress (FEM vs. experimental results).

Path	Measuring Point	${\mathit{\sigma }}_{\mathit{V}\mathit{o}\mathit{n}}$ (t = 40 min, EXP)	${\mathit{\sigma }}_{\mathit{V}\mathit{o}\mathit{n}}$ (t = 40 min)	Error	${\mathit{\sigma }}_{\mathit{V}\mathit{o}\mathit{n}}$ (t = 60 min)	Error	${\mathit{\sigma }}_{\mathit{V}\mathit{o}\mathit{n}}$(t = 80 min)	Error	Average Deviation
1	1	204	228	−24	223	−19	222	−18	−20.33
	2	237	227	10	221	16	221	16	14.00
	3	182	205	−23	201	−19	201	−19	−20.33
	4	189	174	15	167	22	169	20	19.00
2	5	222	205	17	202	20	202	20	19.00
	6	221	220	1	209	12	209	12	8.33
	7	208	240	−32	226	−18	226	−18	−22.67
	8	258	244	14	232	26	232	26	22.00
	9	212	243	−31	231	−19	231	−19	−23.00
	10	215	247	−32	233	−18	233	−18	−22.67
	11	241	225	16	216	25	216	25	22.00
	12	214	201	13	194	20	194	20	17.67
3	13	114	132	−18	132	−18	132	−18	−18.00
	14	103	129	−26	134	−31	134	−31	−29.33
4	15	233	218	15	211	22	211	22	19.67
	16	200	235	−35	227	−27	228	−28	−30.00
	17	215	244	−29	235	−20	237	−22	−23.67
	18	231	215	16	210	21	209	22	19.67

Unit: MPa.

, confirming that finite element analysis of PWHT can effectively reflect actual conditions.

In summary, Scheme (II) achieves the best balance between stress relief effectiveness and economic efficiency [38], followed by Scheme (III). These results can serve as a reference for designing PWHT schemes for OSDs.

5. Conclusions

In this investigation, a thermo-elasto-plastic finite element model of the OSDs was established using the welding simulation software SYSWELD, with a focus on six-pass welding sequences. Numerical simulations were conducted to analyze the temperature field, Von Mises WRS field, and the PWHT process. The numerical results were validated through experimental design, leading to the following conclusions:

A DEHSM was employed to simulate the temperature field evolution during OSDs welding. It was found that the thermal cycle curves at the weld seam locations exhibit high gradient change characteristics. The subsequent weld in adjacent seams exerts a localized heat treatment effect on the previously welded region, although the secondary temperature rise remains below 25% of the peak thermal cycle value.

High-amplitude Von Mises stress is concentrated in the main regions of each weld seam. At the junctions of multiple weld seams, local stress valleys are observed, attributed to the thermal radiation from subsequent welds near these areas, which mimics a localized heat treatment effect. After final cooling, the U-rib exhibits the highest stress level, followed by the deck, while the majority of the diaphragm region remains in a low-stress state.

The stress relief effect of PWHT on Von Mises stress was analyzed. By comparing stress results under different holding temperatures T (500, 550, 600 °C) and holding times t (40 min, 60 min, 80 min), it was found that when T = 550 °C and t = 40 min, an optimal balance between stress relief effectiveness and economic efficiency is achieved.