Free vibration analysis of joined conical shells: Analytical and experimental study

Natural frequencies and mode shapes of two joined isotropic conical shells are presented in this study. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder–plates or cone–plates, conical and cylindrical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell and Hamilton׳s principle. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions and all combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations and also modal testing. The effects of semi-vertex angles and meridional lengths on the natural frequency and circumferential wave number of joined shells are investigated.     

Buckling of tube-stiffened prolate domes under external water pressure

The paper describes experimental tests carried out on three tube-stiffened prolate hemi-ellipsoidal domes under external water pressure.The tubes were stuck to the internal surfaces of these three domes, in their flanks, and the experimental tests showed that the tubes increased the buckling pressures of these domes, especially in one case, when the tube was subjected to an initial internal pressure.All three domes failed by lobar buckling, in their flanks, and showed buckling resistances much larger than similar unstiffened domes.     

Vibration of a thin-walled carbon fibre corrugated circular cylinder under external water pressure

The paper reports on a theoretical and an experimental investigation carried out on a thin-walled corrugated carbon fibre circular cylinder in air and also under external water pressure. This corrugated circular cylinder was invented by Ross in 1987.The theoretical investigation was carried out using the finite element analysis to model both the structure and the fluid. The theoretical investigation used two different programs, one of which was the giant computer program ANSYS and the other was an in-house program developed by Ross and Little. For the shell structure, the ANSYS program used two different doubly curved thin-walled shell elements, while the in-house program used a simpler axisymmetric thin-walled shell element. This axisymmetric element allowed a sinusoidal variation of the displacements in the circumferential direction, thus, decreasing preparation and computational time. Agreement between the different finite elements was found to be quite good. The investigation also found that there was good correlation between experiment and theory for the in-house software, but was a little disappointing when using ANSYS. Errors may, however, have occurred with the experimental results, as the model was hand-made and neither its geometry nor its material properties were perfect. It was found particularly encouraging for the in-house software to give better results than ANSYS, as the in-house software only took a few hours to set up the computer model, and a few seconds to analyse the vessel, whereas the ANSYS software took several weeks to set up the computer model, and several minutes to analyse the shell. The ANSYS software, however, did have the advantage in producing excellent graphical displays in both the pre-processing and post-processing modes.     

Natural frequencies and mode shapes of an orthotropic thin shell of revolution

A method is developed to determine the free vibration characteristics of an orthotropic thin shell of revolution of arbitrary meridian. A solution is given within the context of the Sanders–Budiansky shell theory and using the differential quadrature method (DQM). Numerical examples for frequencies and mode shapes are given for a complete toroidal shell. Both completely free shells, and shells with circumferential line supports are considered. Close agreement is observed in comparisons with previously published results and with results obtained using the finite element method. The paper ends with a set of appropriate conclusions.     

Minimum weight of internally pressurised domes subject to plastic load failure

A series of eight mild steel domes subjected to internal pressure were tested up to burst in order to verify optimal solutions obtained by a zero order optimisation subroutine and a stress analysis code. Four classes of minimum weight domes were obtained. The first one was sought within torispherical shells of constant wall thickness. The second case corresponded to a variable meridional profile described by a generalised ellipse and a given wall thickness. In the third case, the uniform wall thickness became an additional design variable. In the fourth case both the meridional shape and variable wall thickness along the meridian were used as design variables. The plastic load was greater than 20 MPa in all four cases. The weight reduction obtained for the above mentioned four cases was 18, 19, 27 and 31%, respectively. Comparison with the fully stressed design is also provided.     

面内加载时弯管的数值计算方法

对平面内弯矩作用下的弯管的应力分析提出了一种新的计算方法,以替代现行的有限元网格法。新方法的提出基于有限弯曲单元,其位移由一系列三角函数或者五次多项式,与傅里叶级数共同计算而得。整体的壳位移由两类位移组合而成,一个是拱的弯曲位移,另一个则是环形的薄壁壳体的翘曲位移。梁模型的位移和平面内转角是不耦合的,分别应用三角函数,如Timoshenko或Mindlin梁理论计算得到。基于双曲线壳体的半膜单元概念,提出了一个简化的变形模型用于此计算方法。研究成果也与已有的试验和数值分析结果进行了对比。     

The shape of circumferential weld-induced imperfections in thin-walled steel silos and tanks

The strength of thin-walled cylindrical shell structures is highly dependent on the nature and magnitude of imperfections. Most importantly, circumferential imperfections have been reported to have an especially detrimental effect on the buckling resistance of these shells under axial load. Due to the manufacturing techniques commonly used during the erection of steel silos and tanks, specific types of imperfections are introduced into these structures, among them circumferential weld-induced imperfections between strakes of steel plates. The shape of such a localised circumferential imperfection has been shown to have a great influence on the degree of strength loss of thin-walled cylindrical shell structures. The results of a survey of imperfections in an existing silo at a location in Port Kembla, Australia in combination with linear elastic shell bending theory was used to develop and calibrate a shape function which accurately describes the geometric features of circumferential weld imperfections. The proposed shape function is the first function to combine shell theory with actual field imperfection measurements. It is a continuous function and incorporates all the necessary features to represent the geometry of a circumferential weld-induced imperfection. It was found that after filtering out the effects of overall imperfections three parameters governed the shape of the surveyed imperfections: the depth, the wavelength and the roundness.     

Investigation of pointed domes—II. Experiment

The aim of the experiment was to study the behavior of pointed domes subjected to point loading at the vertex. A concrete dome reinforceed with 1 mm bars running both in the meridional and parallel circular direction was constructed. Electrical resistance strain gages were mounted at several locations along the meridional and parallel direction of the shell both inside and outside. The structure was loaded within the elastic limit.The experimental values of membrane stress field are calculated and compared with the theoretical values as is described in part I of this report. (see pp. 81–85). These results are in close agreement in regions some distance away from the lower boundary and the pointed apex. But the difference between the two results becomes noticeable near the lower boundary and the vertex of the dome. Further analysis of the experiment findings indicates that the bending strain field is slight at regions that are some distance away from the lower boundary and to the apex of the dome.     

Strength and stiffness requirements for intermediate ring stiffeners on discretely supported cylindrical shells

Silos in the form of a cylindrical metal shell are often supported on a ring beam which rests on discrete column supports. This support condition produces a circumferential non-uniformity in the axial membrane stresses in the silo shell. One way of reducing the non-uniformity of these stresses is to use a very stiff ring beam which partially or fully redistributes the stresses from the local support into uniform stresses in the shell. A better alternative is to use a combination of a flexible ring beam and an intermediate ring stiffener. Recent research by the authors has identified the ideal location of the intermediate ring stiffener to provide circumferentially uniform axial membrane stresses above the stiffener. To be fully effective, this intermediate ring should locally prevent both radial and circumferential displacements in the shell. This paper explores the strength and stiffness requirements for this intermediate ring stiffener. Pursuant to this goal, the cylindrical shell below the intermediate ring stiffener is analysed using the membrane theory of shells and the reactions produced by the stiffener on the shell are identified. These reactions are then applied to the intermediate ring stiffener. Vlasov's curved beam theory is used to derive closed form expressions for the variation of the stress resultants around the circumference to obtain a strength design criterion for the stiffener. A stiffness criterion is then developed by considering the ratio of the circumferential stiffness of the cylindrical shell to that of the intermediate ring stiffener. The circumferential displacements of the ring and the shell are found for the loading condition previously obtained to determine the required strength. A simple algebraic expression is developed for this intermediate ring stiffness criterion. These analytical studies are then compared with complementary finite element analyses that are used to identify a suitable value for the intermediate ring stiffness ratio for practical design.     

Optimal design of shells against buckling under overall bending and external pressure

In this paper, the problem of optimal design of shells against instability is considered. A thin-walled shell is loaded, in general, by an external pressure and lateral forces causing overall bending moment (which varies along the axis of a shell) and the appropriate shearing force. We look for the shape of meridian as well as the thickness of a shell, which ensure the maximal critical value of the loading parameter. As the equality constraints, the volume of material and the capacity of a shell are considered. The concept of a shell of uniform stability is applied.     

On the shape of momentless tensionless masonry domes

In this paper the theory of shells is applied to find the shape of domes with bending and tension suppressed throughout the shell. As the result of this study the meridional shape and thickness variation of a tensionless masonry dome is obtained. The findings are plotted for some numerical values of parameters. A comparison is also made between the theoretically tensionless dome and a real historical masonry dome.     

Reliable and accurate prediction of the experimental buckling of thin-walled cylindrical shell under an axial load

Reliable and accurate method of the experimental buckling prediction of thin-walled cylindrical shell under an eccentric load is presented. The experimental arrangement and specimens are discussed in detail, including the measurement of the geometric imperfections of the specimen's surface using a coordinate measuring machine. Different FE models, in terms of complexity, are used to simulate the experiment arrangement in an attempt to get a good agreement with the experimental buckling loads and study the effect of measured initial geometric imperfections, load eccentricity, load eccentricity position along the shell's circumferential direction and different experimental arrangement that influence the boundary conditions. It has been demonstrated that FE models with simplified rigid support conditions overestimate the prediction of the experimental buckling load even though these models included the effects of the measured initial geometric imperfections and load eccentricity. By contrast, FE models with realistically modeled support conditions achieved the best result. The average deviation −1.59% from the experimental buckling loads was achieved using the FE model simulating the mounting devices as elastic bodies and with surface-to-surface contact interaction behavior on the support. The presented work also demonstrated the strong influence of the eccentric load position along the imperfect shell's circumferential direction on the buckling of the thin-walled shell.     

Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia

The governing strain-displacement and curvature-displacement equations for paraboloidal shells including shear deformation and rotary inertia are solved for free vibration of closed shells. The finite element method is used to obtain three-dimensional frequency of vibration solutions for a variety of boundary conditions, free, fixed and simply supported. Assumptions concerning the circumferential vibrational behavior are incorporated that reduce the analysis to a single coordinate and the element shape function is formulated using the meridional coordinate. The results for frequency of vibration compare favorably with the available literature. Selected results for frequency of vibration are presented in tabular form for several shell parameters, including free, pinned and fixed boundary conditions. Representative mode shapes are plotted for a fixed boundary condition.     

Dynamics of axisymmetric hyperbolic shell structures

Shell structures have many applications including cooling towers, liquid-retaining structures and roofs where large uninterrupted space is required. These structures are aesthetically pleasing, make efficient use of construction materials and can economically meet design criteria. A number of serious failures of shell structures during the last century has led to significant research into their behaviour, analysis, design and construction. Despite this, further research is still required to improve design, especially with respect to shell dynamics. While some research on free vibration has been carried out, very little or none has been documented to date on the response to earthquake loading. In earlier research carried out by the authors, it has been shown that the hyperbolic shape is very efficient for use as an axisymmetric shell. In view of this and other existing information, this paper treats the free vibration and seismic response of hyperbolic shells, and examines the influence of thickness, height and curvature on this response. It is found that the period of vibration decreases approximately linearly with increasing curvature, but at high curvatures this trend reverses. The early periods of the circumferential mode of vibration are also found to vary linearly with changes of height and thickness, with increasing thickness reducing the period and increasing height increasing the period. The response of the first lateral mode is also significantly affected by a change in the parameters. Transient earthquake analysis shows that height is the most important factor governing the dynamic response. However, significant changes in hoop and meridional stresses are also observed for structures with different shell thickness and curvature. Analysis has been undertaken using the finite element method, and results are presented in tabular and graphical format.     

Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.     

Fabrication of small models of large cylinders with extensive welding for buckling experiments

Cylindrical shells in large steel silos and tanks are commonly constructed from a large number of curved panels joined by many circumferential and meridional welds (referred to as the panel method hereafter). The extensive use of welding in these shells is a unique feature not previously studied in laboratory buckling experiments due to the great difficulty in fabricating realistic small-scale model shells. This paper presents an innovative technique for the fabrication of small models of such large steel cylindrical shells constructed from many welded panels. The experimental set-ups to implement this technique in the laboratory are also described. The new technique consists of two stages: (a) production of a high quality model by rolling two sheets (or a single sheet) and welding along the meridional seams; and (b) ‘welding’ in the form of controlled heat input in a required pattern of circumferential and meridional ‘welds’ on the central portion of the shell surface. The imperfections in an example specimen are also examined to show that they have a realistic pattern. The observed buckling behavior of this specimen is presented and discussed. The specimen buckled at a very low load, confirming that the welding-induced imperfections in such shells are severely detrimental to the buckling strength.     

Investigation of pointed domes—I. Theory

A theoretical method of shell analysis has been utilized and, with the help of certain innovations, the study of pointed domes subjected to pointed loading is presented in this paper.To derive an analytical expression for the pointed dome, the general theory of shell of revolution is used to take into account the effect of membrane and bending stress field. Additionally the general classical approach of analytical multi-shell analysis is utilized to obtain the solution.This solution is applied to a pointed dome with a given dimension and the membrane stress field was calculated. The results thus obtained give a good approximation of the internal force field in the central portion of the dome. However this is not true near the lower boundary and near the kinked vertex of the dome.The values of internal meridional bending moment and meridional shear have also been calculated. These indicate that the bending field diminishes in magnitude as we move away from the kinked region.     

Limit analysis of local failure in shallow spherical concrete caps subjected to uniform radial pressure

Many thin-walled shallow concrete shells (or caps or domes) have experienced structural collapse during or subsequent to their erection. Very few experimental investigations have been reported of shallow concrete spherical caps that allow for the effects of geometric and material non-linearities and imperfections to be identified, despite this information being essential for the validation of sophisticated numerical treatments. Classical thin-shell theories for axisymmetric domes predict a global buckling mode, but observations from experimental tests show that failure in concrete spherical caps is usually localised within a relatively small region and at a load significantly less than the classical buckling load. An investigation of the non-linear behaviour of thin-walled concrete spherical caps is currently being carried out both experimentally and theoretically at The University of New South Wales. As part of this study, an approach based on limit analysis has been developed on the basis of a local failure model and it is used for analysis of concrete/mortar spherical caps described in the published literature. The ultimate bending strength of a unit width of a spherical shell section is derived from a typical non-linear concrete stress-strain relationship and the in-plane thrust from shell membrane theory. The analytical results based on a local failure yield line model are compared with the available experimental results in the literature as well as with classical theoretical buckling results.     

Vibration of thin-walled ring-stiffened circular cylinders and cones

This paper presents a theoretical and an experimental investigation into the free vibration characteristics of thin-walled ring-stiffened circular cylinders and cones. Three ring-stiffened circular cylinders and three ring-stiffened circular cones were machined from a solid mild steel shaft. For each family of vessels, the only variable was the depth of the ring stiffeners. The theoretical investigation was based on the finite element method, and comparison between theory and experiment was found to be good.     

Elastic Buckling of Complete Toroidal Shells of Elliptical Cross-Section Subjected to Uniform Internal Pressure

It was predicted recently that some complete toroidal shells of elliptical cross-section would buckle when subjected to internal pressure. As yet there is no experimental evidence for this, so two independent shell buckling programs (BOSOR and INCA) were employed to calculate the internal buckling pressures for some test cases. The agreement between the results of the two programs is very good, with both programs predicting that buckling occurs.

Calculations were also carried out by using BOSOR on complete toroids having cross-sections in the form of prolate (k = a/b> 1.0) ellipses. The ranges of the parameters were: R/b = 4 and 10, b/t = 50, 100 and 200, and 1.3 < k < 2.5. The shells were assumed to be perfect, made from steel and to behave elastically. The buckling pressures and circumferential wavenumbers are given in tabular form and some are plotted graphically. The deformed shapes of some typical cross-sections prior to buckling are also illustrated, along with the buckling modes.