# Strength Analysis of Cylindrical Shells with Tangential Nozzles under Internal Pressure
## 1 Introduction  
The pressure vessel is a closed container under pressure. In certain processes, it is necessary to install tangential nozzles on the pressure vessels. Compared to the conventional orthogonal nozzle configuration, the stress distribution in pressure vessels with tangential nozzles becomes more complex and exhibits a more severe stress concentration [1]. However, there are currently no suitable design specifications or codes available for the opening structure of tangential nozzles on the cylinder shells. With advancements in finite element technology, numerous scholars have conducted extensive research on pressure vessel openings using the limit-load method for analysis purposes. [PERSON] et al. [2] employed elastic-plastic finite element analysis to conduct parametric studies encompassing a wide range of vessel geometries and materials; they proposed a novel model capable of accurately predicting burst strength in both thin-walled and thick-walled pressure vessels. [PERSON] et al. [3] employed the elastic-plastic large deflection analysis method and nonlinear finite element analysis to determine the bursting pressure and fracture location at the intersection of the oblique nozzle and cylindrical shell. [PERSON] et al. [4] utilized the finite element inelastic analysis method to investigate the connection structure between the nozzle and shell. The findings reveal that the ultimate load of radial oblique nozzles is significantly lower than that of radial nozzles. Additionally, parametric analysis results demonstrate the influence of nozzle inclination angle and diameter ratio on the ultimate load value. [PERSON] et al. [5] analyzed variations in ultimate pressure for cylindrical pressure vessels with a nozzle when the inclination angle of the nozzle in the longitudinal and radial planes is changed by employing an elastic-plastic limit-load method, which holds significance for optimizing pressure vessel design. Focusing on parametric analysis of nozzle structures, [PERSON] et al. [6] designed three dimensionless parameters (\(d_{o}/D_{i}\), \(t/T\) and \(D_{i}/T\)) and conducted a limit-load-based parametric study on cylinder radial nozzles under compound loads, obtaining an empirical equation describing their ultimate load relationship under such conditions through regression analysis to solve for their bearing capacity, effectively within the shells' radial nozzles context. [PERSON] [7] examined four dimensionless parameters (circumferential nozzle inclination angle \(\beta\), opening ratio \(d/D\), wall thickness ratio \(\delta_{et}/\delta_{e}\) and diameter-to-thickness ratio \(D/\delta_{e}\)) influencing ultimate pressure in circumferential nozzle structures with cylindrical shells while designing an orthogonal test model for further regression analysis and fitting a formula for the limit load. [PERSON] [8] not only investigated the relationship between various geometric parameters (\(d/D\), \(D/T\) and \(t/T\)) and blasting pressure, but also derived the corresponding equations. Additionally, four different materials were employed to validate the proposed equations. Focusing on the open nozzle structure of cylindrical shells under external pressure, [PERSON] [9] systematically examined the variation pattern of critical pressure \(P_{cr}\) under different opening ratios \(d/D\), nozzle-to-cylinder thickness ratio \(\delta_{et}/\delta_{e}\), cylinder diameter-to-thickness ratio \(D_{o}/\delta_{e}\) and cylinder length-to-diameter ratio \(L/D_{o}\) through finite element nonlinear buckling analysis. [PERSON] [10] conducted finite element analysis on orthotropic titanium cylindrical shells with perforated nozzle structures. The results revealed that as \(\delta_{et}/\delta_{e}\), \(D/\delta_{e}\) and \(L/D\) decrease, the weakening effect of the opening ratio \(\rho\) on the \(P_{cr}\) of the cylindrical shell increases. A regression equation for critical instability pressure \(P_{cr}\) was obtained through regression analysis.