Experimental investigations on the collapse of spiral-stiffened cylindrical shells under external hydrostatic pressure

This paper presents a comprehensive experimental investigation into the collapse behavior of spiral stiffened cylindrical shells under external hydrostatic pressure. Spiral stiffened cylindrical shells with two different pitches and cylindrical shells were designed and fabricated. The fabricated specimens were divided into three groups. Each group has two identical specimens. Test specimens were welded from stainless steel. The shell was formed by cold bending, and the spiral ribs were welded to the surface of the shell to form a spiral stiffened cylindrical shell. Measurements of initial geometric imperfections and shell thickness were performed on the fabricated specimens to verify the accuracy of fabricating. Subsequently, the specimens were placed in a pressure chamber and pressurized until collapsed to obtain the ultimate collapse pressure and collapse modes. In addition, a uniaxial tensile test was conducted on the shell material to obtain its material properties. Experimental results indicated that the fabricated specimens had excellent repeatability and reasonable precision. The presence of a spiral rib effectively boosted the ultimate collapse pressure of the cylindrical shell with a more pronounced strengthening effect observed as the rib density increased.


Introduction
Cylindrical shells are widely used in the design of pressure pipelines, hydraulic pressure-resistant structures and oil silos due to their good load-bearing capacity and mature processing technology.These cylindrical shells are typically used to withstand complex external or internal loads.How to improve the stability of cylindrical shells under external loads or internal has been a hot topic of concern in the engineering community.
To improve external load capacity and structural efficiency, cylindrical shells are usually strengthened by welding circumferential or longitudinal ribs.For example, Cho et al. experimentally studied various collapse modes of cylindrical shells strengthened by ring ribs under external hydrostatic pressure.After the experiments, a reliable numerical model was developed and a failure indicator was proposed to distinguish the collapse modes [1].Yang et al. experimentally studied the ultimate buckling strength of composite cylindrical shells reinforced with ring ribs and unreinforced cylindrical shells.The results indicated that cylindrical shells could achieve enormous buckling strength with a small increase in structural mass.Then, they performed numerical analysis to show the evolution process of the buckling failure of the reinforced cylindrical shell [2].Do et al. performed a collision damage experiment on a cylindrical shell reinforced with longitudinal ribs, and then placed it in a pressure chamber for hydrostatic testing.The result indicated that collision damage had very little effect on the ultimate strength of the cylindrical shell reinforced with longitudinal ribs [3].
As for cylindrical shells reinforced by spiral ribs, researchers mainly focused on evaluating the axial ultimate strength of cylindrical shells.For example, Alreben and Ekmekyapar performed axial compression tests on concrete-filled steel tubes strengthened by continuous spiral ribs.The results indicated that the spiral rib could effectively improve the axial bearing capacity of the cylindrical shell [4].Few studies have investigated the collapse behavior of spiral stiffened cylindrical shells under external hydrostatic pressure.Shokrzadeh and Sohrabi conducted a numerical study investigating the reinforcing effect of spiral stairways on metal liquid storage tanks under wind and vacuum pressures.The spiral stairway has improved the lateral wind pressure resistance of the tank.However, since it is only strengthening local areas, the buckling resistance of the tank has not been significantly improved under vacuum pressure [5].The experimental quantitative verification of the strengthening effect of spiral ribs on cylindrical shells is still relatively lacking.
This paper presents a comprehensive experimental investigation into the collapse behavior of spiral stiffened cylindrical shells under external hydrostatic pressure.Spiral stiffened cylindrical shells with two different pitches and cylindrical shells were fabricated, geometrically measured and hydrostatically tested.The initial geometrical imperfections of the test specimens were quantitatively analyzed.The ultimate collapse pressure and collapse mode of the test specimens were obtained.The results indicated that the fabricated specimens had excellent repeatability and reasonable precision.The presence of a spiral rib effectively boosted the ultimate collapse pressure of the cylindrical shell with a more pronounced strengthening effect observed as the rib density increased.

Specimen fabrication
Spiral stiffened cylindrical shells with two different pitches and cylindrical shells were designed and fabricated.The fabricated specimens were divided into three groups, in which each group had two identical specimens.The first group was cylindrical shells without stiffeners, and the latter two groups were spiral stiffened cylindrical shells with different pitches.The three groups were named CYS, SCS-A, and SCS-B, respectively, followed by a number suffix to represent two identical species in one group.For the designed cylindrical shell, L, r and t denote the total length, outer radius, and wall thickness respectively.The spiral stiffened cylindrical shell had a single spiral rib with a pitch of P, width of L s , and thickness of t s .A circular plate was welded at the two ends of the cylindrical shell specimens with a diameter of D and thickness of T. Schematics and dimensions of the cylindrical shells and spiral stiffened cylindrical shells are presented in Figure 1 and Table 1.
For the fabrication process of cylindrical shells, first, a CNC laser was used to cut the required design dimensions from a stainless steel flat plate.Second, the cut flat plate was rolled into a cylindrical shell by using a special rolling machine.Third, the cylindrical shell and circular plates were assembled by using TIG spot welding.Finally, the ends and longitudinal boundaries of the cylindrical shell were closed by using seam welding, and the weld seams were polished.For the fabrication process of spiral stiffened cylindrical shells, the multi-segment spiral ribs were joined together by spot welding to form a continuous spiral rib whose axial length is equal to the length of the cylindrical shell.The spiral rib was cold-bent from a ring-shaped rib using a dedicated cold bender.The material of the spiral rib was the same as that of the cylindrical shell.Then, the continuous spiral rib was affixed to the outer surface of the cylindrical shell through spot welding.The spiral rib end start was collinear with the longitudinal weld of the cylindrical shell to ensure fabrication consistency.Finally, the spiral stiffened cylindrical shell was obtained by completely welding both sides of the spiral rib.Table 1.Specimens geometric parameters. Specimen

Material properties
To acquire the material properties of stainless steel, three flat tension coupons were subjected to uniaxial tensile tests by the ISO 6892-1 standard.The three tensile coupons and the cylindrical shell material were taken from the same stainless steel plate, and the mechanical properties were tested by using an electronic universal testing machine (MZ-5001D1, Yangzhou, Jiangsu, China).The geometrical dimensions and mechanical property tensile tests of the three coupons are shown in Figure 2.
The true stress-strain curves of the three tensile coupons are displayed in Figure 3.These coupons exhibited similar mechanical properties, featuring bilinear stress-strain relationships.The Young's modulus was determined from the slope of the linear elastic phase, and the yield strength corresponded to a plastic strain of 0.2%.Material properties for the three coupons are summarized in Table 2 with the average values representing the stainless steel's properties.Young's modulus E = 199.5 GPa and yield strength σ y = 304 MPa.The Poisson's ratio for stainless steel is 0.3, as provided by the manufacturer.imperfections played a crucial role in influencing the ultimate pressure of cylindrical shells.Therefore, before the hydrostatic test, the geometric imperfections of the specimens were measured by using a handheld three-dimensional (3D) optical scanner (EinScan HX, Hangzhou, Zhejiang, China).Figure 4a shows a schematic scan of specimen SCS-A-2.First, mark points were evenly pasted on the surface of the specimen, and then the hand-held scanner scanned uniformly along the circumferential direction and axial direction of the specimen.Figure 4b displays the acquired point cloud data for geometric imperfections.The scanner has a precision of 0.05 mm, and the point cloud consisted of 441, 168 to 819, 392 data points for the specimens.
The circular plates and spiral ribs were assumed to be geometrically perfect.Hence, the point cloud data of the circular plates at both ends and spiral ribs were deleted by using Geomagic studio software.Only point cloud data of the cylindrical shells was retained.Then, the processed point cloud data and the CAD model of the nominal cylindrical shell were imported into the GOM Inspect software for surface fitting alignment and geometric imperfections analyses.Finally, the 3D coordinates of the geometric imperfection point cloud data were exported which can be used as imperfection data in finite element analysis.
To visually represent the amplitude and distribution range of geometric imperfections in cylindrical shells, the geometric imperfections' point cloud data was flattened into a planar format along the circumferential direction, as depicted in Figure 5.The amplitude of geometric imperfections in all specimens predominantly fell within the -1 to 1 mm range.Positive values denoted a real model diameter which is larger than the nominal diameter, while negative values indicated the opposite.It should be noted that in the geometric imperfection cloud diagram of each specimen, a relatively obvious outward protruding area can be seen, which is the position of the axial weld seam of the cylindrical shell.There are also obvious outward protruding areas on the surface of the shell with welded spiral ribs, indicating that welding increases the initial geometric imperfections on the surface of the shell.

Shell wall thickness measurement.
Considering the fabrication accuracy, there are always slight deviations between the wall thickness of the actual specimen and the ideal design wall thickness.To establish a reliable finite element model, we measured the actual wall thicknesses of all specimens.Specifically, we measured the wall thicknesses of the spiral stiffened cylindrical shells before the spiral rib was welded.An ultrasonic thickness gauge (DAKOTA/PX-7, USA) was used to perform ultrasonic measurements on the wall thickness of the fabricated specimens with an accuracy of ± 0.001 mm.The acoustic velocity was measured at 5664 m/s. Figure 6 illustrates the measurement process with 180 points measured on each specimen.A reference line was divided into 30° intervals in the circumferential direction, and 15 points were evenly distributed along the axial direction of this reference line.Table 3 presents the measured wall thickness results for the specimens.The average wall thickness value for each specimen will be utilized in the subsequent finite element modeling.

Hydrostatic test
Hydrostatic pressure tests were performed on six specimens to assess the impact of the spiral rib on the collapse strength of the cylindrical shell.All fabricated specimens were tested in a vertical pressure chamber until their ultimate failure.The pressure chamber, situated at Jiangsu University of Science and Technology, features an internal diameter and height of 500 cm each, with a maximum pressure capacity of 8 MPa.The cover of the pressure chamber is provided with a water inlet, an exhaust port and a pressure monitoring port respectively.A manual water pump (SRB-30X, Zhenhuan Hydraulic Equipment, China) was utilized for hydrostatic pressure application.The pressure in the pressure chamber was monitored by using a digital pressure gauge (SUP-P300, Hangzhou, Zhejiang, China) with accuracy levels and ranges of 0.5 and 0 to 10 MPa respectively.The data of the digital pressure gauge was collected by the dynamic data acquisition instrument (DH5902N, Jiangsu, China) in real time and transmitted to the computer system.The data acquisition frequency was set to 50 HZ.Figure 7 displays the configuration for the hydrostatic pressure test.

Cylindrical shells
A sharp pressure decrease was documented, accompanied by a loud noise when the cylindrical shell reached its ultimate collapse pressure.Figure 8a illustrates the pressure-time curves of the CYS specimens.Since the application of the pressure is with a manual water pump, the pressure curve resembled a staggered slope.For specimen CYS-1, when the pressure increased to 0.757 MPa, the pressure dropped sharply, and the cylindrical shell collapsed with a loud noise.The cylindrical shell lost its bearing capacity after the first collapse, so the pressure was stopped after a loud noise was heard.The highest point of pressure was the ultimate collapse pressure, which was 0.757 MPa. Figure 10a shows the post-collapse configurations of specimen CYS-1.A local dent appeared on the right side of the longitudinal weld seam with an axial length almost equal to the length of the cylindrical shell.
For specimen CYS-2, when the pressure increased to 0.593 MPa, the pressure dropped sharply, indicating that the cylindrical shell collapsed for the first time, but no obvious noise was heard.At this point, we continued to apply pressure to the pressure chamber.The pressure rose slowly, and when it increased to 0.585 MPa, the pressure dropped again with a loud noise, indicating that a secondary collapse of the cylindrical shell had occurred.Immediately after hearing the noise, we stopped applying pressure.Comparing the two collapse pressures, the ultimate collapse pressure of specimen CYS-2 was determined to be 0.593 MPa.A local dent appeared on each side of the longitudinal weld, as displayed in Figure 10a.The high stiffness of the longitudinal weld on the cylindrical shell significantly impeded the expansion of the two dimples in the circumferential direction.

Spiral stiffened cylindrical shells of SCS-A series
The recorded pressure history curve of spiral stiffened cylindrical shells of the SCS-A series is shown in Figure 8b.For specimen SCS-A-1, the spiral stiffened cylindrical shell experienced the first collapse when the pressure reached 0.913 MPa, with a slight decrease in pressure.However, stiffened cylindrical shells can withstand higher pressure, which is different from unstiffened cylindrical shells.Continuing to pressurize the pressure chamber to 1.021 MPa and 1.119 MPa respectively, a second and third collapse occurred.After the pressure reached 1.119 MPa, the pressure curve showed a linear decline with a negative slope, and the obvious sound of water leakage in the pressure chamber was heard, indicating that the final failure of the specimen occurred.The ultimate collapse pressure of the specimen SCS-A-1 was the third collapse pressure, which was 1.119 MPa.After the hydrostatic test, the specimen SCS-A-1 was removed from the pressure chamber.It could be found that the deformation of the collapse tore the longitudinal weld seam, resulting in the eventual failure of the specimen.Figure 9 shows the tear failure of the longitudinal weld of specimen SCS-A-1 after the experiment.The post-collapse mode of specimen SCS-A-1 is shown in Figure 10b.The local collapse of the shell between the spiral rib occurred.The stiffness of the spiral rib of specimen SCS-A-1 was sufficient to avoid the global collapse of the stiffened cylindrical shell.There were a total of three dents distributed on both sides of the longitudinal weld.The number of dents was consistent with the findings of the pressure curve.
For specimen SCS-A-2, the first collapse occurred when the pressure reached 0.941 MPa.Subsequently, the longitudinal weld of the specimen suffered a tearing failure and loss of loadcarrying capacity.The ultimate collapse pressure of specimen SCS-A-2 was 0.941 MPa.The variation in ultimate strength between the two SCS-A specimens was primarily due to differences in the quality of their longitudinal welds.Specimen SCS-A-2 also failed due to local collapse.A dent appeared next to the longitudinal weld.

Spiral stiffened cylindrical shells of SCS-B series
The recorded pressure history curve of spiral stiffened cylindrical shells of the SCS-B series is shown in Figure 8c.Before the two specimens reached the ultimate collapse pressure, the pressure curves fluctuated and noises were heard, indicating that the specimens had collapsed many times.However, the time interval between collapses was significantly shorter than that of specimen SCS-A-1.The ultimate collapse pressures of sample SCS-B-1 and sample SCS-B-2 were 2.287 MPa and 2.168 MPa, respectively.Similar to specimen SCS-A, specimen SCS-B also had multiple dents between the spiral ribs, which were randomly distributed on both sides of the longitudinal weld, as shown in Figure 10c.The average ultimate collapse pressures of specimens CYS, SCS-A, and SCS-B were 0.675 MPa, 1.030 MPa, and 2.228 MPa, respectively.The average ultimate collapse pressures of specimens SCS-A and SCS-B were 1.526 and 3.301 times greater than those of specimens CYS respectively.It was concluded that the presence of a spiral rib effectively boosted the ultimate collapse pressure of the cylindrical shell with a more pronounced strengthening effect observed as the rib density increased.

Conclusions
This paper presents an experimental investigation into the collapse behavior of spiral stiffened cylindrical shells under external hydrostatic pressure.The primary findings from this study are as follows.
(1) For the specimen manufactured by TIG welding technology, the geometric imperfections were mainly distributed around the weld seam.Welding-induced shell deformation was the primary cause of the initial geometric imperfections in the specimens.(2) The fabricated specimens had excellent repeatability and reasonable precision.The amplitude of geometric imperfections in the specimens primarily fell within the -1 to 1 mm range, and the actual wall thickness closely matched the nominal value.
(3) The average ultimate collapse pressure of the SCS-A and SCS-B series spiral stiffened cylindrical shells were 1.526 and 3.301 times greater than that of the non-reinforced cylindrical shells.The result confirmed the effectiveness of spiral reinforcement.All specimens locally collapsed into one or more dents next to the longitudinal weld.
Future research should carry out numerical model modeling based on experimental data and verify its accuracy for more extensive parametric studies.

Figure 2 .
Figure 2. Material properties tests for the three tensile coupons: (a) schematic geometry of a tensile coupon (mm), and (b) experimental site photograph.

Figure 3 .
Figure 3. True stress-strain curves for the three tensile coupons.

Figure 4 .
Figure 4. Measurement of initial geometric imperfections for specimen SCS-A-2: (a) the schematic diagram of the scanning experiment and (b) point cloud data of measured geometric imperfection.

Figure 5 .
Figure 5.The cloud diagrams of the geometric imperfections of six specimens.

Figure 6 .
Figure 6.The schematic diagram of shell thickness measurement.

Figure 7 .
Figure 7. Configuration for the hydrostatic pressure test.

Figure 8 .
Figure 8. Record of the applied history pressure for the specimens (a) cylindrical shells, (b) spiral stiffened cylindrical shells of SCS-A series, and (c) spiral stiffened cylindrical shells of SCS-B series.

Figure 9 .
Figure 9.The tear failure of the longitudinal weld of specimen SCS-A-1.

Figure 10 .
Figure 10.The collapse deformation photos of the specimens (a) cylindrical shells, (b) spiral stiffened cylindrical shells of SCS-A series, and (c) spiral stiffened cylindrical shells of SCS-B series after the hydrostatic test.

Table 2 .
Material properties for the three tensile coupons.

Table 3 .
Measured wall thickness of six specimens.
Specimen t nom [mm] t min [mm] t max [mm] t ave [mm] t std [mm] nom : nominal thickness, t min : minimum thickness, t max : maximum thickness, t ave : average thickness, and t std : standard deviation.