Study on the Mechanical Characteristics of Crack Propagation in 07MnMoVR Pressure-Bearing Steel Pipes Under Residual Stress

Abstract

Under long-term dynamic water pressure, weld zones in vertical shaft pressure-bearing steel pipes are prone to cracking induced by welding residual stresses (WRSs), which may further propagate and threaten structural safety. This study investigates the effects of initial crack angle and position on crack tip stress and propagation path under the influence of WRSs. Using the XFEM combined with a DFLUX-based thermomechanical simulation, a numerical model of crack growth in vertical shaft steel pipes is developed. Results indicate that increasing the initial crack angle raises the stress intensity factor, while crack-tip residual stress initially increases and then decreases, reaching a maximum value of 457.9 MPa when the initial crack angle is 30°. When WRSs are considered, localized stress concentration at the crack tip intensifies, leading to higher stress, stress amplitude, and stress intensity factor, with the amplitude peaking at 365.49 MPa. Moreover, cracks located outside the weld exhibit higher stress intensity factors than those inside. Overall, WRS, crack angle, and crack location all contribute to crack propagation, with crack angle being the dominant factor. Cracks within welds and oriented between 15° and 45° exhibit a significantly higher likelihood of propagation. These findings aid in identifying hazardous crack scenarios and provide guidance for the operation and monitoring of pressure pipelines.

1. Introduction

The safety and reliability of vertical shafts are particularly critical for the stable operation of hydropower stations. During operation, steel pipes are commonly employed as the internal lining material for vertical shafts to prevent seepage. Welded joints are recognized as the weak points where cracks initiate and propagate [1,2]. Weld quality is susceptible to various factors including the working environment, welding processes, and techniques, inevitably generating residual stresses. These stresses induce crack formation at different angles in various locations within the weld zone of steel pipes, altering the stress field distribution and influencing crack propagation paths and velocities [3,4,5]. This exacerbates stress concentration effects in the steel pipes, potentially leading to severe safety incidents. Therefore, based on the coupled analysis of welding residual stress redistribution and crack propagation, investigating the effects of different initial crack propagation in the weld zone of 07MnMoVR pressure steel pipes is essential for improving the reliability of 07MnMoVR welding and ensuring the safe operation of pressure vessels.

Currently, numerous scholars both domestically and internationally have conducted extensive research on the coupled mechanism between welding residual stresses and crack propagation using the finite element method. Gao et al. [6] proposed a finite element analysis method that couples residual stress redistribution from welding with crack propagation through fatigue testing of TC4 titanium alloy plates, thereby enhancing the accuracy of predicting crack propagation life in welded structures. Jie et al. [7] and Gu et al. [8] analyzed the influence of weld residual stress coupling on crack propagation in different welded joints by applying residual stresses and fatigue loads to plate welded joints. Their work provides theoretical support for reducing weld residual stresses, enhancing material fatigue performance, and optimizing welding processes. Guan et al. [9] introduced the initial crack angle effect while considering residual stresses and crack propagation in U-rib stiffener welds. Through theoretical derivation and experimental validation, they developed a mathematical model for the effective angle range within which an initial crack possesses propagation capability. While these studies reveal the coupled behavior between residual welding stresses and crack propagation, they primarily rely on flat plate welding models and have not fully addressed the influence of residual stresses on crack growth in curved structures such as pressurized water pipelines.

Research on initial crack propagation in curved structures has predominantly focused on isolated fatigue damage stress fields, neglecting the coupled effects of residual welding stresses in the weld zone and crack propagation. For instance, Liu et al. [10] comprehensively evaluated the failure modes of thick-walled welded pipes caused by pre-existing crack length and location by defining an initial crack. They established a fracture envelope for thick-walled welded pipes under high-temperature conditions, providing a reference for their safe design and operational maintenance. Hu et al. [11] focused on CFRP-reinforced obliquely welded steel plates, conducting an in-depth analysis of how initial crack angle and position affect the fatigue life of such plates. Zhang et al. [12] investigated the influence of crack size and location near spiral welds on crack initiation and propagation in X65 pipelines. Existing research on crack propagation in pressure-bearing steel pipes primarily concentrates on analyzing the effects of parameters such as temperature and material properties on transverse crack propagation in welded joints, failing to examine the development patterns of propagation paths and rates from multiple crack angles.

In the study of crack propagation in shaft structures, existing simulation models can generally be categorized into two major types: discrete crack methods and diffuse crack methods, in addition to numerical analysis methods based on the framework of fracture mechanics theory. The discrete crack method explicitly describes the crack initiation location and propagation path. Representative methods include the extended finite element method (XFEM) [13,14], and the displacement discontinuity method (DDM) [15,16]. These methods effectively characterize the geometric evolution of cracks and accurately reflect the complex interaction mechanisms between cracks and the surrounding stress field. Among these, XFEM demonstrates significant advantages in simulating crack initiation, propagation direction, and stress redistribution, as it does not require mesh refinement. In contrast, the diffuse crack approach, exemplified by the Continuous Damage Mechanics Model (CDMM) [17], characterizes material degradation by introducing distributed damage variables. It is primarily used to describe the overall stiffness degradation process of a structure rather than the explicit propagation path and geometric morphology of the crack. In summary, this study employs an extended finite element numerical simulation method based on fracture mechanics theory. By introducing initial cracks at varying angles and positions within the weld zone, it systematically investigates the coupling mechanism between residual welding stresses and crack propagation in curved pressure-bearing structures made of 07MnMoVR steel pipes. This research aims to provide theoretical guidance for curved structural design and welding techniques involving 07MnMoVR materials.

2. Materials and Methods

This chapter briefly introduces material parameters, welding residual stress simulation methods, and crack propagation analysis procedures to establish a research model and provide the necessary numerical foundation for subsequent results. First, fundamental material properties are presented as input parameters for numerical analysis. Subsequently, the welding residual stress field is obtained through thermal-mechanical coupled simulation. Finally, a crack propagation model is developed based on XFEM, incorporating various crack angles and positions.

2.1. Material Properties of 07MnMoVR

In the complex underground environment of vertical shafts, pressure-bearing steel pipes must fully account for operational conditions and performance requirements. 07MnMoVR is a low-alloy structural steel specifically designed for low-temperature pressure vessels. By reducing carbon content and adding nickel, it significantly enhances impact toughness in low-temperature environments while maintaining excellent weldability. With its outstanding comprehensive properties, 07MnMoVR steel has become a critical material for manufacturing key equipment such as pressure pipelines and reactors in hydropower, petroleum, and chemical industries. Therefore, this study focuses on 07MnMoVR pressure-bearing steel pipes used in the inner lining structure of vertical shafts. Based on the chemical composition in

Table 1. Chemical Element Composition of 07MnMoVR (10⁻¹ mas%).

Ingredient	C	Si	Mn	P	S	Cu	Ni	Cr	Mo	V	B	More
Max	-	1.5	12	-	-	-	-	-	1.0	0.2	0.02	Pcm ≤ 2
Min	0.9	4.0	16	0.2	0.1	2.5	4.0	3.0	3.0	0.6	-

Note: Pcm represents the weld crack sensitivity composition and is calculated using the following formula: Pcm = C + (Si/30) + [(Mn + Cu + Cr)/20] + (Ni/60) + (Mo/15) + (V/10) + 5B.

and calculations from Reference [18], the temperature-dependent patterns of mechanical and thermophysical properties for 07MnMoVR steel were determined, with results shown in Figure 1.

2.2. Numerical Simulation of Welding Residual Stresses

Accurate simulation of welding processes in the numerical domain largely depends on how appropriately the heat source model is defined and applied. A highly accurate heat source model can not only analyze the temperature field distribution characteristics during the welding process with precision but also effectively predict the quality characteristics and mechanical properties of the welded joints based on an accurate welding simulation process. Relevant studies have shown that welding residual stresses play a key role in welded joint performance and may influence the propagation of steel pipe cracks [19,20,21], which further emphasizes the importance of accurate heat source modeling in welding numerical simulation.

Among the numerous heat source models, the double ellipsoid distributed volume heat source model proposed by Goldak is widely recognized as more accurately reflecting real welding scenarios [22]. To represent the molten pool geometry in welding, a moving double-ellipsoid heat source model is employed, with its motion achieved via a DFLUX subroutine coded in FORTRAN. The welding sequence simulation is performed by progressively activating and deactivating elements, reflecting material deposition behavior. Since the volumetric heat source applied to the weld plate is divided into different sections along the axial direction, this method provides a more detailed representation of temperature gradient distributions throughout the welding process. The axial variation in heat flux for the double ellipsoid model is defined by Equation (1).

$$\begin{array}{c}\left\{\begin{array}{c}{q}_1\left(x,y,z\right)=\left(6\sqrt{3}\left(f_{1}Q\right)/a_{1}bc\pi \sqrt{\pi }\right)\exp \left[-3\left(x^2/a_1^2+y^2/b^2+z^2/c^2\right)\right], x\ge 0\\ q_2\left(x,y,z\right)=\left(6\sqrt{3}\left(f_{2}Q\right)/a_{2}bc\pi \sqrt{\pi }\right)\exp \left[-3\left(x^2/a_2^2+y^2/b^2+z^2/c^2\right)\right], x<0\\ f_1+f_2=2, f_1=2a_1/\left(a_1+a_2\right), f_2=2a_2/\left(a_1+a_2\right)\end{array}\right.\end{array}$$

(1)

In this model, f₁ (set to 2/3) and f₂ (set to 4/3) represent the energy distribution coefficients for the front and rear sections of the heat source, respectively. Q denotes the heat source power, which is determined by the welding current (I), arc voltage (U), and arc efficiency (E). The parameters a₁, a₂, b, and c define the shape of the double-ellipsoid heat source, which is influenced by the welding process. These parameters can be adjusted according to welding conditions to achieve the desired fusion zone characteristics. In this study, the values are set as a₁ = 0.009, a₂ = 0.018, b = 0.009, and c = 0.045.

The 07MnMoVR pressure pipe has an inner diameter (D) of 4.1 m and a wall thickness of 0.05 m, as shown in Figure 2a. Due to the large model size, single-pass welding was adopted to reduce computation time and improve efficiency. A non-uniform mesh was applied to the pipe model, employing a dense mesh in the weld region and a coarser mesh in areas distant from the weld to ensure computational efficiency in the numerical simulation. One-quarter of the pipe was selected as the computational model, as shown in Figure 2b. The welded model was constructed using hexahedral C3D8RT elements, with the mesh size in the weld region controlled at 1 mm. The weld width was 0.02 m, and the initial temperature was 20 °C. The welding parameters were set as follows: welding current of 210 A, voltage of 25 V, and welding speed of 2.3 mm/s.

Following the welding operation, in which the heat source traverses from left to right (as depicted in Figure 2b), the structure undergoes natural cooling until it reaches room temperature. Due to the highest temperature being concentrated at the weld center and the lowest at the edge regions, a temperature gradient forms, resulting in heat conduction from the weld centerline toward both ends. As shown in Figure 3, the typical thermal cycle curves at nodes perpendicular to the weld direction illustrate this behavior. The obtained results are consistent with those reported in Reference [23]. Therefore, the double-ellipsoidal heat source-based simulation method is validated for its accuracy in capturing welding behavior. Figure 4 displays the distribution contour of the stress field computed from the welding analysis. Circumferential and radial residual stresses were extracted along the weld centerline. The circumferential residual stress is referred to as longitudinal residual stress, while the radial residual stress is referred to as transverse residual stress, as shown in Figure 5. The figure reveals that longitudinal residual stress (S11) exhibits a compressive-tensile-compressive distribution. The tensile stress in the central region of the welded component reaches a maximum of 38.26 MPa, while compressive stress is present at both ends. Transverse residual stress (S22) also manifests as tensile stress in the central region of the welded component. During welding, longitudinal and transverse residual stresses are typically considered primary influencing factors, with residual tensile stresses believed to significantly impact crack propagation [24]. Therefore, the calculated residual stress field serves as the initial condition for crack propagation analysis, with the initial crack positioned within the tensile stress-dominated region of the weld.

2.3. Crack Propagation Analysis Method

2.3.1. Defining the Initial Microcrack Orientation and Location Within the Weldment

During steel pipe welding, localized heating and uneven cooling readily induce residual stresses and deformation, leading to the formation of initial cracks within the pipe. However, crack propagation characteristics are jointly governed by the initial crack geometry and the compressive state of the pipe wall. To ensure numerical simulation results accurately characterize engineering service conditions, the initial crack orientation and location were explicitly defined in the model, as shown in Figure 6. The model represents one-quarter of the circular pipe model in Figure 2a, consistent with Figure 2b. The mesh type is C3D8RT with a minimum mesh size of 0.3 mm, employing single-precision partitioning and totaling 88,800 mesh elements. Since the initial crack primarily originates from residual stresses formed during welding, its size is typically small, representing an early micro-defect within the weld zone. To ensure sufficient propagation space for the crack during numerical simulation and prevent the initial defect size from dominating the overall mechanical response, the initial crack size must be relatively small. Therefore, the initial crack is modeled as a two-dimensional rectangular structure. It is defined using standard fracture mechanics parameters: crack depth a = 2.0 mm, surface half-length c = 1.25 mm (corresponding to a total surface length 2c = 2.5 mm), and aspect ratio a/c = 1.60. Since the initial crack is predefined, no additional parameter settings or meshing are required for the crack structure. Based on the design flow rate specified in the engineering design, the hydraulic pressure exerted on the pipe wall was calculated to be 1.85 MPa. This value was applied to the initial pressure stress field of the pipe wall as a predefined condition for subsequent crack propagation simulations.

Based on the determination of crack geometric parameters and loading conditions, the crack propagation characteristics of vertical shaft steel pipe structures under welding residual stresses were investigated by designing different operating conditions. Specifically, the initial crack is embedded within the steel pipe, and the initial crack angle is defined as the angle between the crack plane and the XOZ plane (as shown in Figure 6). Seven angles were selected at 15° intervals to examine the influence of crack angle on propagation, specific operating conditions are shown in

Table 2. Calculation Conditions.

Case	WRS	Initial Crack Orientation	Initial Crack Location
1	Neglected	0°, 15°, 30°, 45°, 60°, 75°, 90°	Weld centerline
2	Considered		Weld centerline
3	Considered		Outside weld zone

. Additionally, by comparing crack locations (within and outside the weld), the study investigates the promoting and suppressing effects of welding residual stresses on crack evolution at different angles and positions.

2.3.2. Stress Intensity Factor Theory

During operation, vertical shaft water supply steel pipes must withstand not only the circumferential principal stresses generated by internal fluids but also the combined effects of residual tensile stresses introduced during welding. Although inclined cracks may induce mixed-mode stress states, the fracture behavior in the present study is dominated by Mode I loading due to internal pressure and tensile welding residual stresses; therefore, only Mode I stress intensity factors are considered. To rationally analyze crack propagation under initial conditions of varying angles and positions, this paper introduces a theoretical model for residual welding stress intensity factor (SIF). Based on linear elastic fracture mechanics theory, the mathematical expression for the crack stress intensity factor can be represented as:

$$\begin{array}{c}K=Y·\sigma ·\sqrt{\pi a}\end{array}$$

(2)

Here, Y represents the geometric correction factor (approximated as 1.0 in this study). It should be noted that the geometric correction factor Y is approximated as unity in this analytical formulation for qualitative interpretation purposes only. The absolute crack-driving forces and propagation behavior are evaluated using XFEM simulations. σ denotes the effective stress perpendicular to the crack plane, and a is the crack length. Within the weld region examined in this study, the total stress at the crack tip originates from both the circumferential stress σ_p induced by internal pipeline water pressure and the residual stress component σ_r(θ) along the crack direction. Consequently, the effective stress can be expressed as:

$$\begin{array}{c}\sigma ={\sigma }_{total}\left(\theta \right)={\sigma }_p+{\sigma }_r\left(\theta \right)\end{array}$$

(3)

The circumferential principal stress induced by internal pressure in a steel pipe can be expressed as:

$$\begin{array}{c}{\sigma }_p=pr/t\end{array}$$

(4)

Here, p represents the internal water pressure (constant in this study), r denotes the pipe radius (r = 2.05 m), and t indicates the wall thickness (t = 0.05 m). As shown in Section 2.2, residual stresses from welding reach their maximum in the weld normal direction and decrease as the crack deviates from this direction by an angle θ. Based on simplified assumptions, considering only the normal projection of welding residual stress acting on the crack plane and neglecting tangential components, the angular decay pattern of residual stresses can be approximated as:

$$\begin{array}{c}{\sigma }_r\left(\theta \right)={\sigma }_{r,max}·cos\left(\theta \right)\end{array}$$

(5)

It should be noted that the cosine-based angular decay function is introduced as a simplified representation to interpret numerical trends, rather than a rigorous description of the full tensorial welding residual stress field.

$$\begin{array}{c}{K}_{\theta }=Y\left[\left(pr/t\right)+{\sigma }_{r,max}cos\left(\theta \right)\right]·\sqrt{\pi a}\end{array}$$

(6)

Thus, the stress intensity factor expression incorporating the angle can be obtained:

This model comprehensively accounts for the coupled effects of internal pressure and residual stress, enabling it to reflect the variation trends of the local stress intensity factor at the crack tip under different crack angles. To facilitate comparison of crack propagation capabilities at various angles, a normalized coefficient η(θ) is introduced to evaluate both crack-prone directions and high-risk operating conditions. It is defined as the ratio of the stress intensity factor at a crack angle θ to that at θ = 0°:

$$\begin{array}{c}\eta \left(\theta \right)=K_{\theta }/K_0=\left[{\left(pr/t\right)+\sigma }_{r,max}cos\left(\theta \right)\right]/\left[\left(pr/t\right)+{\sigma }_{r,max}\right]\end{array}$$

(7)

3. Results

This chapter analyzes the variation patterns of residual stresses at crack tips under different initial crack angles and positions based on numerical simulation results. The analysis examines both crack orientation and crack location perspectives, revealing the influence characteristics of welding residual stresses on crack propagation driving forces and hazard levels, thereby providing data support for subsequent discussions.

3.1. Variation in Residual Stress Field at Crack Tip During Crack Propagation Under Different Initial Crack Orientations

Whether residual welding stresses are considered or neglected, crack propagation paths and morphologies generally exhibit similar trends, as shown in Figure 7. Since crack morphology changes minimally at 0°, it is not plotted in Figure 7. As the initial crack angle increases, crack propagation length follows an “increase-decrease-increase” pattern. Within the 30° to 45° range, stress concentration at the crack tip becomes most pronounced, causing the crack to deflect along complex paths. At 90°, the contact length between the crack edge and the hydraulic direction increases to approximately 2.5 mm. Expansion within the pressure zone intensifies the stress concentration effect, leading to a significant increase in crack propagation length. In summary, regardless of residual welding stresses, cracks originating within welds tend to propagate toward the weld. This occurs because weld zones typically exhibit lower mechanical properties within the structure, making them vulnerable points for crack growth. Therefore, during the service life of pressurized steel pipes, particular attention should be paid to the initiation and propagation of initial cracks in weld zones, treating them as critical control factors in fracture safety assessments.

Under both conditions, detailed evaluations of the transverse stresses in the crack tip region were conducted to better characterize its local stress environment. Stress data were obtained by extending 0.15 m to the left and right along the weld circumferential path, with the crack tip as the central axis. The corresponding distribution patterns are shown in Figure 8. Figure 8a depicts the transverse stress distribution for cracks oriented within the 0–90° range on the plate without welding residual stresses. Each curve reflects the circumferential variation in transverse stress at the crack tip corresponding to a specific initial crack angle. As the initial crack orientation angle increases, the residual compressive stress at the crack tip exhibits a trend of first increasing and then decreasing. When the initial crack orientation is 30°, the compressive stress reaches its peak. Figure 8b illustrates the variation in transverse residual stress distribution with crack orientation from 0 to 90°, considering the influence of welding residual stresses. Contrary to the distribution trend shown in Figure 8a, the stress profile under this condition exhibits a distinctly different pattern. Within the same crack region, the hydraulic stress component is tensile, whereas the transverse welding residual stresses component is predominantly compressive. Consequently, as the crack orientation increases, the transverse residual stress gradually transitions from compressive to tensile. Further analysis indicates that the transverse residual tensile stress progressively increases with the crack angle. When the crack orientation is 0°, the transverse residual tensile stress in the weld zone is minimal. At a crack orientation of 30°, the transverse tensile stress reaches its maximum. Concurrently, the fluctuation range of stress values significantly decreases, and the distribution pattern becomes more uniform and stable.

3.2. Variation in Residual Stress Field at Crack Tip During Crack Propagation Under Different Initial Crack Locations

Figure 9 illustrates the crack propagation behavior in 07MnMoVR steel pipes with initial cracks located outside the weld zone, accounting for residual welding stresses and varying initial crack angles. In comparison with Figure 7, the crack morphology demonstrates significant alterations: when the initial crack orientation is 0°, the crack propagation length attains its maximum value, with the crack propagating horizontally. As the initial crack angle increases, the crack length progressively decreases, exhibiting a distinct diminishing trend. When the initial crack orientation reaches 30° or greater, the crack tip extends to the weld boundary and interacts with the residual welding stress distribution zone. At this stage, it is observed that despite a substantial increase in tip stress, the crack does not continue propagating toward the weld center. This finding suggests that the stress field within the weld zone imposes critical constraints on the crack propagation path, preventing further extension beyond a specific location. The crack length progressively diminishes, while the 0° crack extends bilaterally without obstruction from residual welding stresses within the weld, resulting in the longest observed crack length. Analysis of the magnified crack section reveals that the propagation direction exhibits a strong correlation with the initial crack angle, typically extending along the original crack orientation. However, in Figure 7, the propagation of cracks originating within the weld zone remains unaffected by the weld boundary, enabling normal propagation toward both ends. This demonstrates a distinct crack propagation pattern and evolution mechanism compared to that observed in Figure 9.

In summary, residual welding stresses, the geometric characteristics of initial cracks, and their locations significantly influence crack propagation paths in curved steel pipe welds. Geometric features and position affect the direction of crack extension, while residual welding stresses determine the magnitude of stress at the crack tip. Furthermore, this phenomenon reflects the microstructural and mechanical property inhomogeneities between the weld and base metal regions in 07MnMoVR steel. The disparity in toughness and strength directly influences crack propagation stability, providing crucial insights for future pipeline material optimization and welding process improvements.

Figure 10 presents the distribution of transverse residual stresses near the crack tip outside the weld zone under multiple initial crack orientations, taking into account the residual stresses introduced by welding. Under this loading condition, a compressive stress of 25 MPa occurs when the initial crack orientation is 0°. This phenomenon arises because, at an initial crack angle of 0°, the crack propagates laterally toward both ends without intersecting regions of high residual stress at the crack tip. The overall trend of the stress curves in each direction corresponds with the stress distribution observed for cracks within the weld. Second, the transverse residual stress at the crack tip demonstrates a gradually increasing trend, reaching its maximum compressive value at the 75° orientation. This behavior differs from that observed when the crack is located within the weld zone (as shown in Figure 8). Additionally, during operation, the steel pipe continuously experiences outward water pressure, imposing significant compressive stress on the pipe wall. Consequently, compressive stress near the crack tip reaches its maximum at 0°. A comparative analysis of transverse residual stress curves at different orientations reveals that the overall trend of stress variation remains consistent across all angles.

To visually and accurately reflect stress changes following crack propagation and validate model accuracy, simulated stress results were substituted into Equation (7) for calculation and compared with analytical solutions (as shown in Figure 11). Under various operating conditions, both the initial crack initiation angle and stress intensity factor exhibited a monotonically increasing trend with increasing angle, consistent with the analytical solutions reported in [25]. When the crack angle is less than 30°, the increase is relatively slow; however, the growth accelerates within the 30° to 45° range. Beyond 45°, the rate of increase in the stress intensity factor slows down. When the crack is located within the weld and residual welding stresses are considered, this curve shows the highest agreement with the analytical solution and yields the maximum stress intensity factor value. When the initial crack is positioned within the weld, the crack tip simultaneously experiences hydrostatic pressure within the weld zone and tensile residual welding stresses. The superposition of these forces significantly increases the crack tip opening, thereby substantially enhancing the stress intensity factor. Conversely, when the initial crack originates outside the weld, crack propagation is primarily governed by hydrostatic pressure. Residual welding stresses only become influential once crack growth extends into the weld zone, resulting in a lower initial stress intensity factor. This discrepancy indicates that, when residual stresses are considered, the direct superposition effect of residual welding stresses at the crack tip is the primary cause of the significant increase in the stress intensity factor. Overall, across all angle ranges, the curve accounting for residual stresses consistently exceeds that without residual stress consideration. This indicates that neglecting residual stresses systematically underestimates the risk of surface cracks within the 30–60° range. In practical fracture assessments, the influence of residual stresses on initial cracks in steel pipe welds must be explicitly considered.

Figure 12 presents the distribution of maximum residual stress amplitude (hereafter referred to as “weld residual stress amplitude”) in the proximal region (the first integration point adjacent to the crack tip) for each initial crack angle after crack propagation under different working conditions, including the maximum and minimum residual stress values at the crack tip. Overall, the four operating conditions exhibit consistent trends: as the crack angle increases from 15° to 30°, the residual stress amplitude increases significantly; when the crack angle further increases to 45° and 90°, the residual stress amplitude gradually decreases. Notably, when welding residual stresses are considered and the initial crack is located within the weld (as shown in Figure 12A), the initial crack angle of 30° yields the maximum welding residual stress value of 457.9 MPa among the four operating conditions. This suggests that the 30° angle is the most sensitive to residual stresses and the most prone to crack propagation. However, the trend in minimum residual stress varies across conditions, with Figure 12A (inside the weld and accounting for welding residual stresses) exhibiting the most pronounced difference. This indicates that cracks within the weld are more susceptible to high-amplitude stress concentrations under residual stress fields, making this region the least favorable for crack propagation. In contrast, Figure 12B (inside the weld with residual stress ignored), Figure 12C (outside the weld with residual stress considered), and Figure 12D (outside the weld with residual stress ignored) show significantly reduced stress amplitude variations. This demonstrates that neither crack location nor residual stress consideration alone produces stress concentration effects comparable to those in Figure 12A. These findings further support the study’s conclusion: the combined effect of welding residual stresses and crack angle critically influences crack propagation, with the 30° crack orientation being the most hazardous within the weld.

To visually represent the magnitude changes under these four operating conditions,

Table 3. Amplitude of Residual Stress Variation at Initial Crack Orientation of 30°.

Figure Number	A	B	C	D
Stress Amplitude (MPa)	365.49	109.31	163.8	90.72
Share of total (%)	50.11	14.99	22.46	12.44

Note: A-Stress amplitude at the initial crack located within the weld when considering residual welding stresses; B-Stress amplitude at the initial crack located within the weld when welding residual stresses are not considered; C-Stress amplitude at the initial crack located outside the weld when considering residual welding stresses; D-Stress amplitude at the initial crack located outside the weld when welding residual stresses are not considered.

lists the stress amplitude values where welding residual stresses exert the greatest influence on the initial crack angle across different operating conditions. The maximum value occurs under condition A, indicating that residual stresses within the weld exhibit the most pronounced variation when a 30° crack is located internally. The next significant influence arises when the crack is externally positioned (condition C), demonstrating that weld residual stresses substantially affect crack propagation behavior even at identical initial crack angles. Therefore, stringent control of weld residual stresses is critical to mitigate their crack-promoting effects. Considering the material properties of steel, although 07MnMoVR steel possesses high strength, microstructural inhomogeneities and reduced toughness often exist in the weld and heat-affected zone. This makes residual tensile stresses more likely to drive crack propagation and reduce fracture toughness. Therefore, effectively controlling residual stress distribution through appropriate welding processes and subsequent heat treatment is vital for fully leveraging the steel’s strength advantages and enhancing the overall fracture safety of the structure.

4. Discussion

Research indicates that crack angle and welding residual stress exhibit significant coupled effects on crack propagation behavior. The crack angle determines the primary tensile stress direction at the crack tip, thereby influencing the energy release rate and the accumulation pattern of residual stress at the crack tip. At a 30° angle, both residual stress at the crack tip and the stress intensity factor reach their peak values, making this the most unfavorable crack orientation. This aligns with the theory that “the crack inclination angle influences stress flow direction and opening mode.” Simultaneously, introducing the actual welding residual stresses generally elevates local stress levels at the crack tip and significantly enhances stress concentration. This effect is most pronounced in the 30° condition within the weld, where stress amplitude increases markedly. This indicates that the welding residual stresses can be treated as an equivalent additional load during crack propagation, and neglecting its influence would lead to safety assessment deviations. Furthermore, when the crack angle ranges from 15° to 45°, crack propagation occurs through regions characterized by high residual tensile stresses and steep stress gradients. Both residual stress and stress amplitude at the crack tip remain in high-risk zones, indicating these conditions as high-risk propagation scenarios. Based on these findings, this study proposes using a crack length ≥ 2 mm and residual stress amplitude ≥ 365.49 MPa at the crack tip as potential failure criteria for crack monitoring and safety early warning in vertical shaft steel pipes.

Additionally, this study provides valuable insights for engineering design and operational maintenance. For welded areas, particularly regarding the direction of high-angle cracks within welds, priority should be given to reducing residual stress effects through methods such as stress relaxation, localized heat treatment, surface strengthening, or welding process improvements. In structural design, stress concentration at crack tips can be mitigated by adjusting weld placement, optimizing weld bevel angles, and enhancing local stiffness in curved components. For in-service structure inspections, prioritizing crack angle ranges between 15° and 45° during routine inspections can improve defect detection efficiency.

Although the wall thickness of the pressure-bearing steel pipe is 0.05 m and multi-pass welding is commonly adopted in practical engineering, a single-pass welding simulation was employed in this study to improve computational efficiency. When subjected to appropriate equivalent thermal input, single-pass welding simulations can reasonably capture the macroscopic characteristics of the residual stress field in welded structures, including tensile stress concentrations in the weld zone and stress gradients in the heat-affected zone. These macroscopic residual stress features play a dominant role in driving crack initiation and propagation. Therefore, the single-pass welding model adopted in this study is considered adequate for analyzing the coupling effects between residual stress and crack propagation. Nevertheless, it is acknowledged that multi-pass welding may lead to more complex local stress redistribution, which will be further investigated in future work.

Finally, it should be noted that this study obtained crack tip stresses and propagation trends based on numerical simulations without joint validation through fatigue crack propagation tests. Therefore, future research may consider integrating three-dimensional full-weld modeling, high-cycle fatigue testing, and full-scale model validation to further enhance the engineering applicability of the results. Overall, this study reveals the coupled mechanism between welding residual stresses and crack angle in influencing crack propagation behavior within curved pressure-bearing structures. It provides effective theoretical and engineering references for crack risk identification, safety assessment, and weld design optimization in vertical shaft steel pipes.

5. Conclusions

Based on a thermo-mechanical coupling model and an extended finite element method, crack propagation simulations were conducted on the curved pressure-bearing structure of the 07MnMoVR pressure-bearing steel pipe in shaft welding residual stress fields under typical engineering hydraulic pressure conditions. The study examined crack propagation under different initial crack angles and locations, yielding the following conclusions:

(1)

Crack propagation is significantly influenced by angle. The initial crack angle determines the distribution of residual stress at the crack tip, which follows a pattern of “initial increase followed by decrease” as the angle increases. At an angle of 30°, the maximum residual stress in the near-tip region (at the first integration point adjacent to the crack tip) reaches 457.9 MPa, which is significantly higher than at other angles, making this orientation the most unfavorable for crack propagation.

(2)

Residual stresses significantly enhance crack propagation. After introducing the actual welding residual stress field, localized stresses at the crack tip generally increased, and stress concentration intensified. This demonstrates the promoting effect of welding residual stresses on crack propagation. Under the 30° internal weld condition in particular, the stress amplitude at the crack tip reached a maximum of 365.49 MPa, representing an average increase of approximately 30% compared to other angles.

(3)

When cracks are located within the weld and their angles range between 15° and 45°, this condition presents a high risk of crack propagation. Defects with crack lengths equal to or exceeding the initial microcrack size (2 mm) and crack-tip residual stress amplitudes of 365.49 MPa or greater may be classified as high-risk propagation scenarios. This indicates that the crack has entered a stable growth phase under the combined influence of internal pressure and welding residual stress, warranting prioritization in safety assessments and monitoring.

In summary, during routine nondestructive testing (such as ultrasonic testing) in engineering, priority should be given to inspecting weld areas where such unfavorable crack propagation directions exist. From a design perspective, optimizing root geometry, controlling heat input, or adjusting welding sequences can all help mitigate crack initiation forces induced by residual stresses. Combined with stress-based inspection thresholds, these measures support more reliable maintenance planning—such as stress gauge placement—and enhance the long-term structural integrity of pressurized steel pipelines.