Dent-to-Stiffener Evaluation Concept for Thin-Walled Steel Cylinders

Buckling is one of the worst failure modes of unstiffened cylindrical shells under external pressure. Thus, it is crucial to design such cylinders according to the buckling conditions. Defects/imperfections can occur during manufacturing, assembly, welding, and other processes, which can reduce the critical buckling load. Dented cylinders can no longer support the load acting on the structure before buckling. Theoretical and experimental studies on shell structures have been conducted since the 18^th century [1,2,3]. Koiter pioneered the investigation of the stability of cylindrical shells and the effect of imperfections on buckling [4]. During this period, different codes and standards were developed. Different codes and standards regarding critical buckling pressures are included in Jawad, Ross, Ventsel and Krauthaer's theory, the British Standards Institution Code, the European Recommendations Code, and NASA SP-8007. Several studies have been conducted on the buckling reaction of cylindrical shell structures with dent-related imperfections under external pressure [5,6,7,8,9,10,11,12]. Ghazijahani et al. and Rathinam and Prabu that certain imperfections increase the critical buckling pressure of the cylindrical shell slightly more than that of a perfect cylindrical shell [13, 14]. The main factors affecting the collapse pressure of circular cylindrical shells are the H/R aspect ratio (height of the shell to the radius) and slenderness ratio (the ratio of the radius to the thickness of the shell) [8, 15].

In this paper, a parametric study based on the dent shape (dent angle, dent length, dent depth, and dent width) is carried out to determine which type of dents can be artificially improved to increase the critical buckling load of the thin-walled steel cylindric shell. Numerical models are generated with centrally located double dents for the symmetry concept. The models are grouped as “0, 45, 90, <-< and <->” according to their dent shapes. The models were designed according to four different dents heights, two different dents widths, and four different dents depths. All the models had an H/R value of 2 and an R/t value of 250. FE results were compared with Jawad, Ross, Ventsel and Krauthaer's theory and NASA SP-8007. The equations used in the calculations are as follows:

Jawad Theory [16]: (1) Pj=0.92E(tR)2,5hc/R {{\rm{P}}_{\rm{j}}} = {{0.92{\rm{E}}{{({{\rm{t}} \over {\rm{R}}})}^{2,5}}} \over {{{\rm{h}}_{\rm{c}}}/{\rm{R}}}}

Ross Theory [17]: (2) Pr=2,6E(t2R)2,5hc2R−0,45 (t2R)0,5 {{\rm{P}}_{\rm{r}}} = {{2,6{\rm{E}}{{\left( {{{\rm{t}} \over {2{\rm{R}}}}} \right)}^{2,5}}} \over {{{{{\rm{h}}_{\rm{c}}}} \over {2{\rm{R}}}} - 0,45\,{{\left( {{{\rm{t}} \over {2{\rm{R}}}}} \right)}^{0,5}}}}

Ventsel and Krauthaer [18]: (3) Pr=2,6E(t2R)2,5hc2R−0,45(t2R)0,5 {{\rm{P}}_{\rm{r}}} = {{2,6{\rm{E}}{{\left( {{{\rm{t}} \over {2{\rm{R}}}}} \right)}^{2,5}}} \over {{{{{\rm{h}}_{\rm{c}}}} \over {2{\rm{R}}}} - 0,45{{\left( {{{\rm{t}} \over {2{\rm{R}}}}} \right)}^{0,5}}}}

NASA SP-8007 [19]: (4) PSP−8007=0,9PLBA {{\rm{P}}_{{\rm{SP}} - 8007}} = 0,9{{\rm{P}}_{{\rm{LBA}}}}

The circumferential buckling wave number is calculated through approximations by Teng et al. [20]. (5) n=2,74 RhCRt {\rm{n}} = 2,74\,\sqrt {{{\rm{R}} \over {{{\rm{h}}_{\rm{C}}}}}\sqrt {{{\rm{R}} \over {\rm{t}}}} }

This study is a continuation of a previous study that investigated both the experimental and numerical defects [21]. The cylindrical shells are modelled using shell elements (with an element size of 7.5 mm and a structured quadrilateral mesh), along with triangular elements, in ANSYS [22]. Also, more precise mesh sizes were used for dents. By utilizing this element, it is possible to handle the membrane, bending, and transverse shear effects, as well as satisfactorily form curvilinear surfaces. Furthermore, this element exhibited plasticity, stress stiffening, and large deflection and strain capabilities. The analytical figures for the models are 10 times greater.

2.1.

Geometry and Material

In this section, a series of dented shells are introduced which represent cylindrical shells. The cylindrical shells are burdened by external pressure. The height and diameter of the cylindrical models are fixed in all models, the height is determined as 1000 mm and the radius is 250 mm. In the models, 5 different types of dents were examined. Different parameters such as dent depth, width, height, angle, and shape were selected as dent types. The dent shapes are shown in Fig. 1. In all models, two dents are centrally and symmetrically placed. The dent depth in models in all groups was taken as t (shell thickness) and 2t. The dent lengths in the models were determined as h/4, h/3, h/2, 3h/4. The first group has dents at 0 degrees and includes models M1 to M6. Dents are intertwined in models with 0 degree and 3h/4 length. Therefore, it is not included in the results. The second group has dents at a 45-degree angle and includes models M7 to M12. In models with 45 degrees group, V, and inverted V dents, the dent lengths are considered for the vertical distance. The third group has 90-degree dents and includes models M13 to M20. V-shaped dents were examined in the fourth and fifth groups. While the V shape is in the same direction in the fourth group, they are in the opposite direction in the fifth group. Information regarding the geometries of the models is given in Table 4, Table 5, Table 6, Table 7, and Table 8. Two factors were carefully considered when modelling localized geometric imperfections, such as dents. The first is the accurate modelling of the dent shape, and the second is the need for careful modelling of the surrounding region in shell structures, where there is a change in curvature, as this can cause local bending stresses [23,24,25]. The core concepts that underlie contemporary design codes for metallic structures were primarily established through the examination of bilinear material behaviour, which entails an elastic response followed by a perfectly plastic one [26]. The properties of AISI 316 stainless steel, characterized as a bilinear isotropic material with a tangent modulus (ET), are provided in Table 1. The values presented in the table were derived from the investigation conducted by Korucuk (2019).

Figure 1.

Table 1.

Material Properties [12]

Material	Yield stresses (MPa)	Tangent Modulus (MPa)	Young's modulus (MPa)	Poisson's ratio
Cylinder sheet	198.8	1450	210 000	0.29

2.2.

Model Validation

The critical buckling pressures were compared with the Combescure and Gusic [27] and Windernburg and Trilling [28] studies given in the Rathinam and Prabu [13] study to validate the FE model with the current study results. The model geometries, material properties and Critical buckling pressure are listed in Table 2 and Table 3. Fig. 2 shows the results of the analysis.

Table 2.

Comparison with buckling pressure of the cylindrical shell given in Combescure and Gusic [27] and Rathinam and Prabu [13]

Cylindrical shell	R (mm)	L (mm)	t (mm)	Young's modulus (GPa)	Poisson's ratio (γ)	FE eigen buckling pressure (MPa)
						Combescure and Gusic	Rathinam and Prabu	From present analysis
A	50	20	0.15	200	0.3	0.2936	0.2927	0.2985

Table 3.

Comparison with buckling pressure of the cylindrical shell given in Windernburg and Trilling [28] and Rathinam and Prabu [13]

Cylindrical shell	R (mm)	L (mm)	t (mm)	Young's modulus (GPa)	Poisson's ratio (γ)	FE eigen buckling pressure (MPa)
						Windernburg and Trilling	Rathinamn and Prabu	From present analysis
53	203.2	406.4	0.8	193	0.3	0.096 (8)	0.108 (8)	0.111 (8)

Figure 2.

2.2.1.

Boundary condition

All the nodes in the bottom edge (i.e., at y=0, with reference to the coordinate system shown in Fig. 2(a) were restrained from moving in the x, y, and z-directions, and all the nodes in the top support edges were restrained from moving in the x and z-directions only.

2.3.

Boundary Conditions and Solution Algorithm

Radial restraint is known as the main effective edge restraint in thin-walled cylindrical shells. The top and bottom edges in all models were fixed supported with only radial restraint. Clamped boundary conditions (see Fig. 3) are applied. The bottom and top edges of the cylindrical models were restrained against radial displacements. Uniform external pressure was applied to the entire surface of the cylindrical shells. Mesh and convergence correction studies were performed to ensure the high accuracy of the analysis.

Figure 3.

Geometric and material nonlinear analysis with imperfections (GMNIA) was utilized to determine the buckling resistance of the cylindrical shells under investigation. Nonlinear analysis was carried out using the full Newton-Raphson approach. Energy-based stabilization techniques available in ANSYS were used to ensure convergence. The energy dissipation ratio was assumed to be in the stabilization range, with the stabilization energy never exceeding 1% of the total potential energy. Within nonlinear numerical computations, geometric imperfections and nonlinear material models were employed. The outcome of the nonlinear analysis represents the behaviour of the structure by means of a load-displacement path related to the selected boundary conditions and the analyzed load while considering the effect of elastoplastic instability. Nonlinear materials were used in all the models. Except for the M37 model, all models were geometrically imperfect, and the M37 model was designed perfectly.

First, the critical buckling results of the perfect model were obtained. The results of the perfect model (M37) are given in Fig. 4 and Fig. 5. Theoretical formulas are evaluated for perfect shells. The critical buckling pressures calculated according to the theoretical formulations were between 48.9–65.2. The critical buckling pressure of the perfect model was approximately 60.1 kPa. The wave number was obtained from Teng's formula as approximately 6. The wave number of the perfect model (M37) was 7. Fig. 4 demonstrates the FE results of the perfect model. In Fig. 5, the results of the perfect model before-initial-post buckling are shown.

Figure 4.

Figure 5.

Consequently, three dent forms that enhance the critical buckling pressure were determined according to the perfect model. These were the M3 and M6 models with a dent along the entire circumference and the M8 model with a 45-degree angle. The FE results showed that the dent shapes with the highest critical buckling pressures were the 0°-angle dent shapes. If we evaluate the dent widths, in the first group (0°-angle dent), the critical buckling pressure value decreased as the dent width increased between M1 and M4, M2 and M5. From M3 to M6, the increase in the dent width increased the critical buckling pressure by 6%. In this group, the increase in dent length from h/4 to h/3 (from M1 to M2 and M4 to M5) caused a decrease in the critical buckling pressure in the models. But as it increased from h/3 to h/2 (from M2 to M3 and M5 to M6), the critical buckling pressure increased. The buckling wave numbers ranged from 6–7. Load-deformation and load-strain graphs of the first group are given in Fig. 6. The M3 and M6 models are quite distinctive in the group. The FE results of this first group are shown in Table 4.

Figure 6.

In the second group (45°-angle dent), only the critical buckling pressure of the M8 model is higher than the perfect model. Among the models with the same dent length and different dent widths, the dent width had a reducing effect on the critical buckling pressure. The dent length increased the critical buckling pressure from h/4 to h/3 and decreased it from h/3 to h/2. The buckling wave numbers ranged from 6–8. Load-deformation and load-strain graphs of the second group are given in Fig. 7. The FE results of this second group are shown in Table 5.

Figure 7.

The critical buckling pressure decreased from 0 to 90°-dent angles. In the third group (90°-angle dent), the critical buckling pressure decreased as the dent widths increased, similar to the other two groups. The dent length reduced the critical buckling pressure from h/4 to 3h/4. The buckling wave numbers were 6 and their shapes were nearly identical. Load-deformation and load-strain graphs of the third group are given in Fig. 8. The FE results of this third group are shown in Table 6. In models with depth t, the angle increased from 0° to 90°, while the critical buckling pressures decreased at h/4 and h/2 dent lengths, while the critical buckling pressure increased from 0° to 45° in dents of length h/3, and the critical buckling pressure decreased from 45° to 90°. In models with 2t depth, the critical buckling pressures decreased from 0° to 45 °, whereas the critical buckling pressure increased from 45° to 90° for dent lengths h/4, h/3, and h/2.

Figure 8.

The last two groups had <-shaped dents. The dents belonging to the fourth group were both in the same direction. In models with the same dent length, increasing the dent width in models with h/3 (M21 and M25) and 3h/4 (M24 and M28) lengths decreased the critical buckling pressure. In other models (M22 and M26, M23 and M27), the dent width increased the critical buckling pressure. Increasing the dent length decreased the critical buckling pressure in h/4-h/3-3h/4 lengths in t-width models, while it increased the critical buckling pressure in the h/2 model. In models with a 2t width, the critical buckling pressure increased from h/4 to h/2 and decreased again at 3h/4. The buckling wave numbers ranged from 6–8. In the fifth group, the <-dents were designed in different directions. The critical buckling pressure of the H/4 dent and t-width model was nearly identical to the critical buckling pressure of the 2t-width model. In models with h/3 and 3h/4 dent sizes, increasing dent width increased the critical buckling pressure. In models with h/2 dent length, the critical buckling pressure decreased with increasing width. Increasing the dent length decreased the critical buckling pressure in h/4-h/3-3h/4 models, while it increased the critical buckling pressure in h/2 models. The buckling wave numbers were 6. Load-deformation and load-strain graphs of the fourth and fifth groups are given in Fig. 9. The FE results of both groups are shown in Table 7 and Table 8.

Figure 9.

In general, dent geometries with the highest critical buckling pressure within each group were determined. These were the M6 (0°-2t-h/2), M8 (45°-th/3), M13 (90°-t-h/4), M27 (<<-2t-h/2) and M31 (<>-t-h/2) models (<-< : symmetrical and concentric <-shaped dent, <-> : symmetrical and reverse <-shaped dent).

In this section, critical buckling pressure values in FE are compared with theoretical formulas and a perfect model. As can be seen in Fig. 10, the P_cr value of the perfect model in FE was among the P_cr values obtained from the formulas produced for perfect shells.

Figure 10.

In Fig. 11, the variation in P_cr values of all models compared to the perfect model is shown. When the results obtained from the theoretical formulas given for the perfect models are examined, it is seen that the M37 (perfect model) model is the closest NASA-SP8007 formula to the critical buckling pressure value. The critical buckling pressure values of the M3 and M6 models were found to be higher than the values obtained from all the theoretical formulas given for the perfect models. Models with critical buckling pressure values exceeding the values obtained from Jawad, Ross, Ventsel and Krauthaer formulas are M1, M2, M3, M4, M6, M7 and M8. The critical buckling pressure values of all other models are lower than both M37 (perfect model) and theoretical pressure values. While an increase was observed in the overall buckling pressures in the M3, M6, and M8 models, a decrease of 1.61–47.3% was detected in all other models compared to the overall buckling pressure of the perfect model.

Figure 11.

This present paper is a numerical study of the ultimate strength of dented cylindrical shells under external pressure of 37 thin-walled cylinders, 36 dented and a perfect model, with 5 different notch groups as reference. The conclusions are made based on FE analysis carried out on thin cylindrical shells with a dent of different shapes, sizes and angles of dent at 1/4-1/3/1/2-3/4 of the height of cylindrical shell taken for study.

Overall, the study provides important insights into the behaviour of thin-walled steel cylinders with dents and offers a promising approach to enhancing the longevity and safety of structures through the transformation of dents into artificial stiffeners. The findings of this research can have significant implications for various industries, including aerospace, automotive, and construction.