Yueyang Wang·Jian Zhang·Wenxian Tang,2

Abstract This study aims to experimentally and numerically examine the buckling performances of stainless steel spherical caps under uniform external pressure.Three laboratory-scale caps were fabricated,measured,and tested.The buckling behaviors of these caps were investigated through experiments and three numerical methods,namely,nonlinear Riks algorithm,nonlinear bifurcation,and linear elastic analysis.The buckling of equal-radius caps was numerically analyzed with different methods to identify their applicability under different wall thicknesses.The results obtained from the nonlinear Riks algorithm are in good agreement with the experimental results,which means the nonlinear Riks algorithm can accurately predict the buckling performances of spherical caps,including the magnitude of critical buckling loads and the deformation of post-buckling modes.The nonlinear bifurcation algorithm is only suitable for predicting the buckling loads of ultra-thin or large-span caps,and the linear buckling method is inappropriate for predicting the buckling of metal spherical caps.

Keywords Spherical cap .Stainless steel .Buckling .External pressure .Bifurcation buckling .Critical buckling

1 Introduction

Domed caps have long received considerable research attention owing to their high pressure-supporting capacity.Accordingly,they have been widely used as end closures on cylindrical pressure vessels or as hatches to cover the access holes of subsea pressure hulls(Krivoshapko 2007;Jasion and Magnucki 2015a; Ifayefunmi 2016; Tripathi et al. 2016).However,such caps are prone to lose stability,characteristics et al. 2017b). Moreover, most studies are entirely numerical investigations.

In the present study, the buckling behaviors of stainless steel spherical caps were experimentally and numerically examined under uniform external pressure. Three laboratoryscale spherical caps were fabricated,measured,and tested to collapse.Comparisons between the collapse loads and corresponding buckling loads computed with different numerical methods were carried out.Furthermore,the buckling of spherical caps with different radius-to-thickness ratios was numerically studied to obtain the best method for the buckling prediction of caps with different wall thicknesses.The results of this study may have some potential applications in the deepsea field.

2 Material and Methods

On the basis of a previous study, the optimal configuration of the spherical cap with a height-to-diameter ratio of h/d ≈0.274 was conducted (Zhang et al. 2018). A total of 304 stainless steel spherical caps were fabricated with the same nominal base diameter (d) of 145 mm, uniform wall thickness (t) of 1 mm, height (h) of 39.73 mm, and radius(r) of 86 mm. To ensure a nearly fixed constraint, each cap was welded on a heavy metal plate with a nominal diameter of D=170 mm and nominal wall thickness of T=20 mm (Figure 1).

The wall thickness of each cap was ultrasonically measured at 10 equidistant points on each of eight equally spaced meridians,yielding a total of 73 measurement sites on each cap. Table 1 lists the obtained minimum tmin,maximum tmax, and average tavemagnitudes of wall thickness and the corresponding standard deviations tstdand nominal values tnom.The wall thickness of the caps varied from 1.013 to 1.086 mm, and the average value was 1.040 mm, which was close to the nominal value of 1 mm. Thus, the nominal wall thickness of the spherical caps can be used to study their pressure-supporting capacity.

Figure1 Sketch of a spherical cap

Before the hydrostatic test,the outer surface of each fabricated cap was scanned in the form of a point cloud using Open Technologies Corporation (accuracy ≤0.02 mm) to acquire geometric shapes. The radius deviations between the fabricated and perfect caps are very small (Figure 2);only the base of the cap shows a relatively large deviation owing to the welding deformation.These caps exhibit similar radius deviations,indicating repeatability.

After the pretests,three spherical caps were sequentially immersed into a hydrostatic test rig, which was fabricated for the experiments.To avoid the concentrated force created by the dive,the heavy plate affixed to the experimental cap was required to make contact with the bottom of the chamber to prevent any external force from affecting the experimental results. The pressure inside the vessel was recorded using a pressure sensor, and the pressure was slowly applied through the medium of water in increments of ～0.1 MPa using a programmable logic controller.

To characterize the parent material properties,four flat tension coupons were designed in accordance with a modified version of ASTM D638 (ASTM 2003). The measurement approaches are detailed in Zhang et al.(2018).The properties of the experimentally measured cap material (304 stainless steel)are as follows:Young’s modulus E=159.208GPa,yield stress σy=335.408 MPa,and Poisson’s ratio μ=0.291.

3 Results and Discussion

3.1 Experimental Analysis

As indicated in Figure 3, all the curves have a similar trend; that is, they first increase up to the peak value (the collapse load), beyond which the pressures drop drastically. The collapse loads can be easily identified in Figure 3,and all caps take the form of a single local dimple near the base, indicating a reasonable repeatability. The collapse loads pcollfrom the tests are listed in the seventh column of Table 1. The collapse loads of the spherical caps range from 5.036 to 5.709 MPa.The variations in wall thickness,shape, and material hardening during the stamping and welding processes may have caused the range in collapse loads. Nevertheless, this slight change indicates the adequate repeatability of the hydrostatic tests.

3.2 Analysis of the Numerical Methods

To evaluate the buckling behaviors of the tested spherical caps, their critical, bifurcation, and linear bucklingcomputations were performed. The finite elements of the caps were freely and uniformly generated on the measured geometries (Figure 4). The S4 shell elements and few S3 elements were chosen to prevent hourglassing.Mesh convergence studies were performed to establish approximately 15 000 elements for each model (Zhang et al. 2018). The material properties were the same as those presented in Section 2. For each spherical cap, the wall thickness was assumed to be the average wall thickness presented in Table 1. Fixed boundary conditions were applied, which were consistent with the test conditions presented in Section 2.

Table1 Nominal tnom, maximum tmax, minimum tmin, and average tave wall thicknesses of all the fabricated spherical caps and the corresponding standard deviations tstd and experimental and numerical buckling loads

Figure2 Geometric deviations of the fabricated spherical caps from the corresponding perfectly geometrical counterparts

3.3 Buckling of Caps with Various Wall Thicknesses

To investigate the effects of the wall thickness on the buckling of spherical caps, equal-radius spherical caps of various wall thicknesses were established.These caps have the same geometrical configurations and material properties as the fabricated caps presented in Section 2, except for the wall thickness. The cap radius-to-thickness ratio ranges from 50 to 1000, which are detailed in Table 2. For each cap, the nonlinear Riks algorithm, nonlinear bifurcation,and linear elastic analysis were conducted. The buckling loads of the three methods are listed in the last three columns in Table 2,and the ratios of the bifurcation(pbif)and linear (plin) buckling loads to the critical buckling loads(pcri) are listed in the parentheses in the corresponding column.In all the analyses,the element type and boundary conditions are the same as those in Section 3.2.

Figure3 Curves of the relationships between experimental pressure ptest and time t for three tested caps, along with the vertical views of their experimental results

Figure4 FE mesh and boundary condition of a spherical cap

As shown in Table 2,as the radius-to-thickness ratio r/t increases,the ratio of pbifto pcridecreases.When this ratio becomes large enough, such that r/t > 300, the two buckling loads are basically the same.This result indicates that the bifurcation buckling algorithm is more suitable for the buckling prediction of ultra-thin or large-span caps compared with the Riks method because the calculation is simpler and faster. Linear buckling loads share similar trends with bifurcation buckling loads. However, even when the ratio r/t is as large as 1000, the magnitude of plinis still much larger than pcri, indicating that the linear buckling method may not be suitable for the buckling prediction of metal spherical caps. In addition, as indicated in Figure 6,except for the buckling mode of the Riks method that is well consistent with the experimental results,the other two methods cannot predict the damage form of the spherical cap.

Figure5 Equilibrium paths of the tested spherical caps

Table2 Critical, bifurcation, and linear buckling loads for spherical caps with different radius-to-thickness ratios

4 Conclusions

In this work, the buckling behaviors of three spherical caps with medium wall thickness were experimentally and numerically investigated, and the accuracy of different numerical methods was measured. Furthermore, the numerical results for the buckling of equal-radius spherical caps with various wall thicknesses were obtained.

Figure6 Post-,bifurcation,and linear buckling modes of the perfect and fabricated caps

The results presented in this work show that the critical buckling load, obtained from the Riks method available in the ABAQUS code, can accurately predict the pressuresupporting capacity of spherical caps.Moreover,the bifurcation buckling algorithm is more suitable for predicting the magnitude of buckling loads of ultra-thin or large-span caps.However,the linear buckling method cannot be used to predict the buckling of metal spherical caps.Experiments on the buckling of thin-walled spherical caps can lay a solid foundation for further studies.

Funding InformationThis study is supported by the National Natural Science Foundation of China (No. 51709132), Natural Science Foundation of Jiangsu Province (No. BK20150469), and Jiangsu Provincial Government Scholarship Programme.