PSI - Issue 81

Borys Shelestovskyi et al. / Procedia Structural Integrity 81 (2026) 162–169

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The greatest technological challenges arise when welding flanges, nozzles, or patches are inserted into existing products. A specific feature of such insertions is the presence of circular welds, where transverse shrinkage cannot be compensated by simple displacements, unlike the stiffener welding. Taking it into account the transverse shrinkage deformations in welds significantly exceed the longitudinal ones (Kasatkin, et al., 1981), the stresses and deformations induced by circular welds often dominate all other stress and strain components. In fact, the design and fabrication of welded structures testifies that, due to the proper design and technological measures, these stresses and deformations can be reduced to acceptable levels, ensuring structural integrity. At the same time, the availability of analytical relations for determining residual stresses and deformations under different welding parameters is of great importance. The relevance of these research results is in the need to enhance the reliability of welded structures while minimizing distortion and crack formation, as well as to develop new analytical and numerical approaches to model thermomechanical processes. Residual stresses originate due to non-uniform heating and cooling of the material, which results in the localized plastic deformations. These processes significantly affect the stress distribution not only in the weld area but also at a considerable distance from it. In the modeling of welding processes, one of the main methodological features is the difference between analytical and numerical approaches. Analytical methods primarily use the classical theory of heat conduction and thermoelasticity, where the problem is stated for simplified geometries — for instance, an infinite or semi-infinite plate, or a thin circular plate with localized heating. Such models are of the closed nature or semi-analytical expressions that describe the temperature field, the evolution of plastic strains, and the resulting residual stress distributions. The analytical approach provides a clear physical interpretation of how the main parameters, plate thickness, heat input, and heat source radius in particular, influence the final stress state. It also makes it possible to identify specific trends, such as the transition from plane stress to three-dimensional stress states with increasing plate thickness. However, the analytical solution needs a series of idealizations. Homogeneous and isotropic material properties, simplified boundary conditions, stationary or axisymmetric heat sources are assumed, and the time-dependent nature of heat transfer during welding is not taken into account. On the contrary, numerical methods, the finite element method (FEM) in particular, provide a much more flexible and comprehensive framework. FEM makes it possible to solve the fully coupled thermo-mechanical problem of welding, taking into account transient heat flow, material plasticity, creep, and even metallurgical phase transformations. At the same time, this is a result of expensive computational complexity. High-fidelity FEM simulations require fine spatial and temporal discretization, careful mesh design near the weld zone, and accurate definition of material properties over a wide temperature range. Moreover, the results of numerical simulations must be proved experimentally, since small errors in heat source parameters or boundary conditions can significantly affect the predicted stress field. Therefore, both approaches are worth being treated as equipollent. Analytical solutions are indispensable for understanding the underlying physical mechanisms and for verifying the plausibility of numerical results. Numerical models, on the other hand, extend these insights to practical engineering applications, enabling the optimization of welding parameters, the design of residual stress mitigation techniques, and the prediction of component performance under operation. Numerous aspects of the residual stress problem in welding have been investigated by both domestic and international researchers. Classical analytical models were developed by Rosenthal (1946), who first described the temperature field generated by a moving point heat source. Later papers, such as those by Goldstein and Gorshkov (1972), improved the thermoelastic analysis of welding problems, taking into account the rheological properties of metals. Modern approaches are based on numerical modeling. For instance, Lingen et al. (2020) testified the capabilities of the finite element method (FEM) for predicting residual stress fields in thick-walled components. Zhang et al. (2019) testified the effectiveness of combining thermal analysis with experimental X-ray diffraction techniques. Kim and Hong (2018) conducted experimental validation of numerical models for welded plates under various cooling conditions. Ukrainian researchers such as Makhnenko and Kudriavtsev (2020) investigated the influence of arc welding parameters on residual stress levels in steel structures. A significant contribution to the development of residual stress evaluation methods was made by Mikhaylov (2021), who researched nonlinear thermoplasticity models in welding and Yasniy et al. (2017) of residual stress in aircraft alloy plates with cold expanded holes. Previous papers by these authors were also devoted to the analytical modeling of pre-stressed solids. In fact, Habrusiev et al. (2018) developed a model analyzing the influence of elastic pre-deformations on contact interaction. Shelestovskyi and Habrusiev (2003) investigated the contact behavior of a layer containing residual deformations caused by a circular weld. Despite substantial progress, several issues have not been sufficiently solved. There is a great need for refined analytical models that enable rapid evaluation of residual stress distributions without the computational expense of full-scale numerical analysis, as well as of parametric studies examining the influence of plate thickness and heating geometry on the stress field characteristics. The aim of this study is to develop an analytical model for assessing residual stresses in a plate under the localized heating, taking into account spatial effects and plastic deformations. The obtained results can be applied to optimize welding regimes and improve the durability of welded structures.