Imperfection sensitivity of externally pressurised shells

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.

Online damage detection and structural health monitoring (SHM) of aerospace structures are having great scientific and economical importance, since maintenance costs can be considerably reduced by introducing condition-based structural maintenance instead of time-linked preventive maintenance. Active structural health monitoring using guided wave modes is gaining importance after the introduction of lightweight piezoelectric wafer transducers (PWTs) since the mass penalty associated with these systems is considerably less. Multiple research papers are generated worldwide on the use of guided Lamb wave modes for active defect detection in thin platelike structures. Since so many civil and aerospace structures are cylindrical in shape, studies on the use of guided wave modes for SHM of cylindrical shells are gaining importance. Studies on the use of guided wave modes for SHM of cylindrical shells and pipes are done by a few authors by pulse echo method. In this chapter, the methods to generate different guided wave modes in metallic isotropic cylindrical shells, dispersion characteristics of wave modes in cylindrical shells, their interaction with defects, etc., are studied. Moreover, the structural health monitoring of cylindrical shells using the time reversibility concept adopted from guided Lamb wave mode is also studied. Numerical modelling of propagation of guided wave modes in plates and cylindrical shells is done using finite element methods, and comparisons are made. Experimental evaluation on the propagation of guided waves in cylindrical shells is carried out. Dispersion characteristics of guided wave modes in cylindrical shells and the signature of guided wave modes for different defects are studied by numerical modelling and experiments. A tomography scheme is worked out which can predict the presence of defects and its propagation, which can be used for condition-based structural maintenance of aerospace structures. A comparison between the numerical modelling and experimental studies is done, and damage indices based on time-reversed reconstruction are generated.

ABSTRACTSteel products which have various characteristic (e.g. high ductility, high yield stress, low yield stress and high weldability) are used as main structures by diversification of demanded application, and technology for steel manufacturing has developed. Because steel square tubular member is processed by cold‐forming and welding in the process of manufacturing, residual stress will exist in the products. Moreover, due to the cold‐forming process, geometrical initial imperfection, i.e., global and local, will exist. These initial imperfection are still remained even so the technology for steel manufacturing is developed. However, effect of the initial imperfection on elasto‐plastic behaviour of square hollow section columns under compressive axial force with bending moment is not clarified. In this paper, firstly, the effect of individual initial imperfection (e.g. initial global imperfection, initial local imperfection or residual stress) on elasto‐plastic performance is investigated independently; comparison between the experimental results are shown. It is shown that initial local imperfection has a significant impact on elasto‐plastic performance. Secondly, the effect of composite initial imperfections is investigated. The amplitude of the local imperfection is the most influential initial imperfection on the deformation capacity of the column, and the deformation capacity decreased with an increase of the amplitude of local imperfection. Validation of the numerical simulation were also performed and recommend values of imperfections are shown.

Purpose: the existing problem of detected damages to tank car boilers during operation due to loss of stability requires new approaches to both theoretical and practical assessment of possible causes of such damages. The evaluation of the combination of various factors, including the impact of shape deviation of the boiler during its manufacture, on the possibility of loss of stability during operation is of particular relevance. Method: to determine the causes of boiler deformation, stability calculations were performed on the boiler. All calculations were conducted for actual thicknesses determined by measurement and thickness gauging.A tank car model with the minimum measured thicknesses was selected for modeling. Calculations were carried out under the action of the standard external pressure value and under excessive external pressure values. Results: stability calculations of the boiler with the actually measured thicknesses under the influence of external excess pressure showed that loss of stability in the boiler: without shape deviation under the action of the standard external excess pressure of 40 kPa was not detected; with local defects from welding under the action of the standard external excess pressure of 40 kPa was not detected; without local welding defects but with a global shape deviation of the boiler under the action of external excess pressure of 40 kPa is possible with a relative ovality of about 2.7 %; without global shape deviation of the boiler, it is possible under the influence of excessive external pressure of 79 kPa and above. Practical significance: possible causes of boiler instability during operation were identified. The cause of boiler instability may be a global shape deviation of the boiler with a relative ovality of about 2.7 %; the effect of excessive external pressure of 79 kPa and above, which is possible due to a violation of the tank car unloading process; a combination of global shape deviation of the boiler and the effect of excessive external pressure due to a violation of the tank car unloading process.

Experiments on cylindrical shells under pure bending and external pressure

Based on a systematic literature review, no design guidelines addressing practical issues related to steel circular hollow section (CHS) members under hydrostatic pressure have been documented thus far, except for the design recommendations proposed by the European standard EN 1993-1-6 concerning the strength and stability of shell structures. Considering that the existing codes are often outdated and subject to improvement, there is a growing demand for modern solutions that can accurately replicate the stability loss response of equipment subject to external pressure. The development of a more direct and simple method, augmented by reliable computational techniques, is advisable to avoid the numerous challenges associated with manual calculations. This approach will streamline the design process and mitigate the complexity of current practical or technical approaches. Thus, this research has formulated a method to predict design strength through the direct application of the critical elastic buckling stress, determined from Generalised Beam Theory (GBT) analyses, induced by hydrostatic pressure in custom-crafted equations, finely tuned for this specific objective. The authors present a formulation of GBT to analyze the elastic buckling behavior of CHS subject to both combined axial compression and external pressure. This work provides an extensive study of the behavior of circular cylindrical shells under external loading conditions. It first introduces the analyses of GBT and the theoretical and empirical formulas applicable to this objective. It then describes the steps involved in the development and validation of a numerical model that simulates the response of circular cylindrical shells under hydrostatic pressure, conducting an extensive parametric study using finite element analyses (FEAs). The results from available tests validate the finite element (FE) model. Upon achieving adequate agreement between the generated numerical and experimental results, new resistance reduction curves for cylindrical shell members are developed based on the results of the parametric study. Additionally, the study compares the existing provisions of EN 1993-1-6 for the design of cylindrical shells under hydrostatic pressure. Finally, a statistical approach is adopted to compare the responses of cylindrical shells calculated according to the EN 1993-1-6 code recommendations with those obtained through the newly proposed method.

The effects of localized imperfections on the collapse strength of plates are studied by means of finite element analysis performed by different authors. First a benchmark study is performed to assess the accuracy of the results obtained using different codes on the same case study. Afterwards, a parametric study on the effect of the localized imperfections on the collapse strength is performed. Finally, the effect of combining local and global imperfections was studied. It is found that the addition of a localized imperfection changes the collapse strength of the plate, which depends on the amplitude and size of the localized imperfection change.

This paper proposes a new type of shell, similar to a cylindrical shell, which has significantly higher buckling strength when subjected to an arbitrary combination of uniform external pressure and axial compression. The underlying principle consists in a slight modification of the perfect cylinder in order to counteract the natural deformations which get larger and larger and lead to the collapse of the structure. Such shells are called ASTER shells. The concept has been validated through experiments, then analyzed numerically in order to explain what was observed and to propose avenues for improvements. The shells were made of electrodeposited nickel. The material was characterized. The chosen specimens were carefully measured to characterize their thickness and initial imperfections, then tested under the various types of loading. Then they were analyzed using finite elements. Thus, we were able to compare the finite element predictions with the experimental results. This comparison shows that plasticity has a decisive influence on the critical load and that linear elastic dimensioning leads to a serious overestimation of the experimental critical load. Contrary to perfect cylindrical shells, this type of shell is not significantly affected by geometric imperfections: this is another advantage of this type of design. Finally, we propose a numerical analysis in order to optimize the choice of the shape and propose shapes which resist buckling much better than a smooth cylinder when subjected to uniform external pressure, axial compression or a combination of both.

This paper considers the stability of a toroidal shell subjected to uniform external pressure. Stability equations for a toroidal shell are solved by use of series expansions in the circumferential and meridional directions for the displacement components that develop during buckling. Axially symmetric as well as asymmetric buckling modes are considered. Design curves that give nondimensional buckling pressures for a wide range of the toroidal shell's geometric parameters are presented. In addition, the variation of the mode shapes with the geometric parameters is illustrated. In a comparison between the results of the present theory and the few available tests on toroidal shells, it is shown that test and theory agree to within 10%. For the limiting case of a sphere under external pressure, the well-known classical result (p — 1.21 Eh/a) is obtained numerically for both the asymmetric and axially symmetric buckling modes.

Vertical cylindrical containers are widely used for storage of granular bulk materials. The enclosing side surface of such containers is made in the form of cylindrical metal circular shells with a wall thickness that is constant or piecewise constant along the height of the shell. Known designs for storage of bulk materials of the reservoir type with the installation of a cylindrical shell on the annular foundation with hinged fixed attachment of the shell to the foundation. Thin-walled shells are made suspended from the supporting structures of the storage facilities. To stabilize the cylindrical shape of the shell and its position in space, the suspended shells are pre-tensioned in the vertical direction. During operation, storage facilities are empty and filled with bulk materials, exposed to the environment in the form of wind and temperature climatic influences. The object of study of this work is the enclosing structures of storages of granular bulk materials in the form of vertical circular cylindrical thin-walled metal shells, in the general case of piecewise constant thickness, subject to temperature and climatic influences of the environment ‒ changes in the temperature of the outside air, direct and diffuse solar radiation. The subject of the study is the components of the stress-strain state of the shell due to changes in temperature and climatic influences. The performed studies of the temperature fields of storage shells on models and full-scale objects made it possible to substantiate the assignment of the temperature field of cylindrical storage shells by Fourier series. One-sided solar heating of cylindrical storage shells completely illuminated by the sun induces in the shell wall a flat temperature field symmetrical with respect to the normal of incidence of sunlight, which can be represented by a Fourier series with five terms of the cosine expansion series. In the presence of a structure located next to the shell, which covers half of the shell along the entire height in the circumferential direction, the temperature field is described by a Fourier series containing 10 harmonics of expansion in sines and cosines. The small thickness of the shell, the significant radius of curvature of the shells, the large thermal diffusivity of metals provides a small variability of the temperature of the shell over the thickness, the ability to describe the stress-strain state of the shell, the momentless theory and a simple edge effect. Formulas are obtained in Fourier series for the forces of a momentless state, the residuals of which at the joints of shell chords of different thicknesses and in the support zones are eliminated by a simple edge effect.

In this paper, FGM toroidal shell segments surrounded by elastic foundation and subjected to uniform external pressure are investigated by analytical method. A novelty of the study is that the Reddy’s third-order shear deformation shell theory (TSDT) with von Karman geometrical nonlinearity combined with deflection function selected with three terms is used to investigate the nonlinear stability of thicker FGM toroidal shell segments. In addition, the thermal element in the shell is also taken into account. The FGM shell is a convex and concave toroidal shell segment. It is a general form for a circular cylindrical shell. Closed-form expressions for determining the static critical external pressure load and postbuckling load–deflection curves are obtained. Effects of temperature field, foundations, material and dimensional parameters on the stability of shells are considered. This paper also shows that the use of TSDT to analyze the nonlinear stability of thicker toroidal shell segments is necessary and more suitable.

9 - Design of submarine pressure hulls to withstand buckling under external hydrostatic pressure

A systematic computational study to understand the effect of metals (Mg, Ca, Sr) doping and external isotropic static pressure on phase stability, electronic band structure and optical properties of KNbO3

Longitudinally stiffened corrugated cylindrical shells under uniform external pressure

Experiments on dented cylindrical shells under peripheral pressure