Strength of damage ring and orthogonally stiffened shells—part I: Plain ring stiffened shells

This paper is the first of two parts describing the procedure of, and results from, a series of tests on ring and orthogonally stiffened thin-walled shells. The primary purpose of the work was to investigate the collapse behaviour of the shells subjected to simulated damage and then tested under a combination of external pressure and axial compressive loading.

The test specimens consisted of two five-bay cylinders stiffened with plain ring stiffeners; two three-bay cylinders stiffened with T-ring stiffeners and two three-bay orthogonally stiffened cylinders, one with 20 stringers, and the other with 40.

This Part I deals with the tests on the plain ring stiffened cylinders.

A major conclusion that can be drawn from the results of these few tests is that although the design of the plain rings was adequate to prevent general buckling of the undamaged shells, they were ineffective in limiting the area of initial damage when the shell was subjected to pressure loading.     

Multiobjective optimization of orthogonally stiffened cylindrical shells for minimum weight and maximum axial buckling load

In the present research, the weight and axial buckling optimization of orthogonally stiffened cylindrical shells is carried out by the Genetic Algorithm. Constraints include two nondimensional functions of weight and buckling load in such a way that the stiffened shell has no increase in the weight and no decrease in the buckling load with respect to the initial unstiffened shell. In analytical solution, the Rayleigh–Ritz energy procedure is applied and the stiffeners are treated as discrete members. The optimization is implemented for shells with simply supported end conditions stiffened by four shapes of stiffeners including rectangular-, cee-, I-, and hat-shaped ones. The results show that the I-section and rectangular-section stiffeners are, respectively, the most and the least efficient in designing stiffened cylindrical shells for minimum weight and maximum critical axial buckling load.     

Face layer wrinkling in sandwich shells— theoretical and experimental investigations

Wrinkling of the face layers is one of the typical failure modes in sandwich structures. Different aspects of this phenomenon are investigated in this paper: analytical considerations concern the determination of wrinkling loads for anisotropic elastic sandwiches under uniaxial as well as biaxial loading. Finite element calculations are carried out to investigate the postbuckling behaviour of isotropic sandwiches, with different combinations of nonlinear material combinations, under compressive and/or bending loading, and experimental studies are performed to verify the analytical and numerical results, respectively.     

Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods

In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e. rings with non-uniform stiffeners eccentricity and unequal stiffeners spacing is investigated using analytical, experimental and finite elements (FE) methods. Ritz method is applied in analytical solution while stiffeners treated as discrete elements. The polynomial functions are used for Ritz functions and natural frequency results for simply supported stiffened cylindrical shell with equal rings spacing and constant eccentricity is compared with other's analytical and experimental results, which showed good agreement. Also, a stiffened shell with unequal rings spacing and non-uniform eccentricity with free–free boundary condition is considered using analytical, experimental and FE methods. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes and damping in each mode. In FE method, two types of modeling, including shell and beam elements and solid element are used, applying ANSYS software. The analytical and the FE results are compared with the experimental one, showing good agreements. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained in this study could be as useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.     

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.     

Analysis and shape optimisation of variable thickness prismatic folded plates and curved shells — Part 2: Shape optimisation

This paper deals with the development of computational tools for structural shape optimisation of shells and folded plates in which the strain energy or the weight of the structure is minimised subject to certain constraints. Both thickness and shape variables defining the cross-section of the structure are considered. The analysis is carried out using curved, variable thickness finite strips formulated and tested in Part 1 of this paper. Optimal shapes are presented for several shells and folded plates of variable thickness including plates on elastic foundation. The changes in the relative contributions of the bending, membrane and shear strain energies are monitored during the whole process of optimisation. The tools developed in the present work can be used as an aid to structural engineers in designing novel forms for shells and folded plates and provide valuable insight into the structural behaviour. It is concluded that the finite strip method offers an accurate and inexpensive tool for the optimisation of a wide class of structures having regular prismatic-type geometry with diaphragm ends.     

Dynamic and static axial crushing of axially stiffened cylindrical shells

The axial impact of cylindrical tubes, which incorporate axial stiffeners, is examined in this paper. For comparison purposes, the effect of static loading is also studied. An examination is made into the influence of stiffener depth (T), number of stiffeners (N) and the effect of placing the stiffeners externally or internally.

The experimental results on mild steel specimens show that there are considerable differences between the static and dynamic modes of failure, and that an optimum T/D ratio may exist for a given value of N.     

Analysis and shape optimisation of variable thickness prismatic folded plates and curved shells — Part 1: Finite strip formulation

This paper deals with the linear elastic analysis of prismatic folded plate and shell structures supported on diaphragms at two opposite edges with the other two edges arbitrarily restrained. The analysis is carried out using curved, variable thickness, Mindlin-Reissner finite strips. The theoretical formulation is presented for a family of C (0) strips and the accuracy and relative performance of the strips are examined for curved situations. Some variable thickness and elastically supported plates are considered and the interesting phenomenon of the occurence of boundary layers in the twisting moments and shear forces is highlighted for a common boundary condition. Other examples analysed include box girders and cylindrical shells. In all cases transverse shear deformation effects are included and the contributions to the strain energy from membrane, bending and transverse shear behaviour noted. In a companion paper these accurate and inexpensive finite strips are used for structural shape optimisation.     

Precursors for rock fracturing and failure—Part II: IRR T-Curve abnormalities

There are many precursors including the abnormality of infrared radiation (IRR) for rock fracturing. The temporal evolution of surface IRR from loaded rock is the combined effect of rock thermoelasticity, pore gas desorbing–escaping, fracture propagation and extension, rock friction, heat transfer and environmental radiation. Being the integral indicator of the surface IRR energy, the mean IRR temperature (AIRT) is selected as the quantitative index to study the temporal evolution of IRR from loaded rock and to identify the precursors for rock fracturing and failure. This paper introduces some typical AIRT curves of uniaxially loaded, compressively sheared and biaxially loaded rock samples. The temporal evolution features of the AIRT curves and three precursor messages, which are short dropping, fast rising and dropping-to-rising, for rock fracturing are analyzed. For theoretical interpretation, the mechanism of AIRT abnormality is studied by taking the actuator, the rock sample and the environment as an independent system in an energy balance state. The heat transfer, as well as the thermoelastic effect based on the sum of principal stresses, has a positive or negative effect on AIRT; the desorbing and escaping of pore gas, as well as the production of new fractures and the expansion of initial fissures, joints and new produced fractures, have a negative effect on AIRT, while the friction action between fractures and between grains has a positive effect on AIRT. It is concluded that the occurrence of precursors is at 77–94% rock strength, i.e., stress peak, and the compressively loaded rock and biaxially loaded rock have clear AIRT precursors for rock fracturing and failure.     

Aluminum tubular sections subjected to web crippling—Part II: Proposed design equations

The web crippling design rules in the current American Aluminum Design Manual, Australian/New Zealand Standard, and European code for aluminum structures are assessed. Test strengths of aluminum square and rectangular hollow sections under end-two-flange (ETF) and interior-two-flange (ITF) loading conditions are compared with the design strengths (capacities) obtained using the aforementioned specifications. Furthermore, the test strengths are also compared with the design strengths obtained using the unified web crippling equation as specified in the North American Specification for cold-formed steel structural members. It is shown that the design strengths predicted by the aforementioned specifications are either quite conservative or unconservative, but in general the predictions are unreliable resulting from reliability analysis. Hence, two different unified web crippling equations for aluminum square and rectangular hollow sections under ETF and ITF loading conditions are proposed. The proposed unified design equation (A) uses the same technique as the North American Specification for the unified web crippling equation with new coefficients of C, C_N and C_h determined based on the test results obtained in this study. The proposed unified design equation (B) is similar to the unified web crippling equation in the NAS Specification, and the effect of the ratio N/h is also considered, where N is bearing length and h is the depth of the flat portion of web. Generally, it is shown that the proposed unified web crippling equation (B) compares well with the test results.     

Progressive collapse of multi-storey buildings due to sudden column loss—Part II: Application

The companion paper presents the principles of a new design-oriented methodology for progressive collapse assessment of multi-storey buildings. The proposed procedure, which can be implemented at various levels of structural idealisation, determines ductility demand and supply in assessing the potential for progressive collapse initiated by instantaneous loss of a vertical support member. This paper demonstrates the applicability of the proposed approach by means of a case study, which considers sudden removal of a ground floor column in a typical steel-framed composite building. In line with current progressive collapse guidelines for buildings with a relatively simple and repetitive layout, the two principal scenarios investigated include removal of a peripheral column and a corner column. The study shows that such structures can be prone to progressive collapse, especially due to failure of the internal secondary beam support joints to safely transfer the gravity loads to the surrounding undamaged members if a flexible fin-plate joint detail is employed. The provision of additional reinforcement in the slab over the hogging moment regions can generally have a beneficial effect on both the dynamic load carrying and deformation capacities. The response can be further improved if axial restraint provided by the adjacent structure can be relied upon. The study also highlights the inability of bare-steel beams to survive column removal despite satisfaction of the code prescribed structural integrity provisions. This demonstrates that tying force requirements alone cannot always guarantee structural robustness without explicit consideration of ductility demand/supply in the support joints of the affected members, as determined by their nonlinear dynamic response.     

Shear resistance of longitudinally stiffened panels — Part 2: Numerical parametric study

This paper presents the FE analysis of the influence of different parameters on the shear resistance of panels with different arrangements of longitudinal stiffeners. The studied parameters were the stiffener bending stiffness, the panel aspect ratio, the stiffener position, the web slenderness and the flange rigidity. Longitudinal stiffeners of trapezoidal shape were compared to open T-stiffeners. The former proved to be more efficient, since a larger panel resistance is achieved, for which in addition a smaller stiffness of trapezoidal stiffeners is needed. Different features of the new Eurocode rules were verified against the FEA results as well. Three different procedures for the determination of panel slenderness were tested and the reduction of stiffener bending stiffness, prescribed due to a better correlation with tests on open stiffeners, was verified for both closed and open stiffeners. The influence of bending moment was also considered and the verification of shear and bending interaction was discussed. Finally, the flange contribution to shear resistance was critically analysed.     

Shear resistance of longitudinally stiffened panels—Part 1: Tests and numerical analysis of imperfections

This paper deals with the results of four full-scale tests, numerical simulation of tests and initial geometric imperfection analysis for longitudinally stiffened panels in shear. The tests examine the influence of varying position and bending stiffness of one trapezoidal longitudinal stiffener on the panel shear resistance and its buckling behaviour. The stiffeners were designed such as to obtain both global and local buckling shapes. Numerical simulations (FEA), based on the test girder geometry, the measured initial geometric imperfections and elastic-plastic material characteristic from the tensile tests, demonstrate a very good agreement with the tests. The initial geometric imperfection study on different verified numerical models shows a limited sensitivity of the panel shear capacity to any kind of imperfection shape variation with amplitude at the allowable fabrication tolerances. Finally, the paper offers some ideas for modelling geometric imperfections with regard to the design or research demands.     

Aluminum alloy tubular columns—Part II: Parametric study and design using direct strength method

A parametric study of aluminum alloy columns of square and rectangular hollow sections was performed using finite element analysis (FEA). The columns were compressed between fixed ends. The parametric study included 120 columns with and without transverse welds at the ends of the columns. An accurate and reliable finite element model was used for the parametric study. Design approaches for aluminum alloy tubular columns with and without transverse welds were proposed. Column strengths predicted by the FEA were compared with the design strengths calculated using the current American, Australian/New Zealand and European specifications for aluminum structures. In addition, the direct strength method (DSM), which was developed for cold-formed carbon steel members, was used in this study for aluminum alloy columns. The design strengths calculated using the DSM were compared with the numerical results. Furthermore, design rules modified from the DSM were proposed. It is shown that the proposed design rules accurately predicted the ultimate strengths of aluminum welded and non-welded columns. The reliability of the current and proposed design rules was evaluated using reliability analysis.     

Non-probabilistic reliability-based design optimization of stiffened shells under buckling constraint

Stiffened shells are affected by numerous uncertainty factors, such as the variations of manufacturing tolerance, material properties and environment aspects, etc. Due to the expensive experimental cost of stiffened shell, only a limited quantity of statistics about its uncertainty factors are available. In this case, an unjustified assumption of probabilistic model may result in misleading outcomes of reliability-based design optimization (RBDO), and the non-probabilistic convex method is a promising alternative. In this study, a hybrid non-probabilistic convex method based on single-ellipsoid convex model is proposed to minimize the weight of stiffened shells with uncertain-but-bounded variations, where the adaptive chaos control (ACC) method is applied to ensure the robustness of search process of single-ellipsoid convex model, and the particle swarm optimization (PSO) algorithm together with smeared stiffener model are utilized to guarantee the global optimum design. A 3 m-diameter benchmark example illustrates the advantage of the proposed method over RBDO and deterministic optimum methods for stiffened shell with uncertain-but-bounded variations.     

Pentachlorophenol biodegradation—II : Anaerobic

The fate of pentachlorophenol (PCP) during anaerobic digestion of sewage sludge solids was tested in a three phase protocol. Phase I involved acclimation; Phase II investigated biodegradation in semicontinuous-flow, stirred-tank reactors at solids retention times of 10, 20 and 40 days; phase III assessed the importance of nonbiological removal mechanisms and collected additional data concerning the extent of biodegradation. PCP was found to inhibit methanogenesis in unacclimated cultures at concentrations in excess of 200 μg l⁻¹ and thus acclimation of the digesters to PCP required very gradual increases in the influent concentration thereby allowing enrichment of organisms capable of degrading PCP. Once acclimation was achieved, digesters receiving influent containing 5.0 mg l⁻¹ PCP achieved stable operation with effluent PCP concentrations below 5 μg l⁻¹ at all retention times studied. Sorption was shown to be unimportant in PCP removal. Volatilization was considered to be unimportant based upon results obtained in forced aeration studies. This suggested that PCP was subject to at least primary biodegradation. Other evidence collected during the study indicated that more complete biodegradation probably was occurring.     

Strength enhancements in cold-formed structural sections — Part II: Predictive models

Cold-formed structural sections are manufactured at ambient temperature and hence undergo plastic deformations, which result in an increase in yield stress and a reduction in ductility. This paper begins with a comparative study of existing models to predict this strength increase. Modifications to the existing models are then made, and an improved model is presented and statistically verified. Tensile coupon data from existing testing programmes have been gathered to supplement those generated in the companion paper [1] and used to assess the predictive models. A series of structural section types, both cold-rolled and press-braked, and a range of structural materials, including various grades of stainless steel and carbon steel, have been considered. The proposed model is shown to offer improved mean predictions of measured strength enhancements over existing approaches, is simple to use in structural calculations and is applicable to any metallic structural sections. It is envisaged that the proposed model will be incorporated in future revisions of Eurocode 3 [2,3].     

Stresses in thin spherical shells with imperfections. Part II: Influence of local imperfections

The changes in stress resultants in thin spherical shells, associated with a local imperfection introducing curvature errors in all directions, are investigated. An axisymmetric finite element model of the shell and imperfection is employed to carry out the linear elastic analysis. Parametric studies have been performed, to identify the main parameters controlling the response, for the case of internal pressure. The results are compared with those obtained in Part I for axisymmetric imperfections, and bounds for maximum elastic stress resultants are established to cover the possibility of both local and axisymmetric imperfections.     

Numerical investigation of particle breakage as applied to mechanical crushing—Part II: Interparticle breakage

A numerical approach to interparticle breakage is applied using the rock failure process analysis code, RFPA^2D. A 2D particle assembly in a container is first numerically simulated to obtain the fringe patterns of stress fields that resemble the photoelastic test. Then, in addition, the interparticle breakage of the particle assembly in a chamber is conducted. The chamber consists of a steel container and a steel platen for transferring the load, and contains 15 particles of arbitrary sizes and irregular shapes. A plane strain condition is assumed. The particle bed is loaded under form conditions, in which the size reduction and the applied force are a function of the displacement. The numerical results indicate that, during the crushing process, three principal regimes appear: (i) the elastic deformation regime, where each particle deforms elastically; (ii) the fragmentation regime, where the particle assembly is crushed in a particle-by-particle fashion; and (iii) the assembly hardening regime, where the densified assembly recovers a significant stiffness. The dominant mode of failure is at first splitting, which is more or less parallel to the loading direction, and then progressive crushing, which mainly depends on the confinement from the chamber walls. The analysis of the load–displacement curves of the assembly obtained from the simulations reveals a high undulating load plateau, which suggests a macro-ductile behaviour.     

Analysis of crack coalescence in rock-like materials containing three flaws—Part II: numerical approach

By using a Rock Failure Process Analysis code, RFPA^2D, numerical simulations on samples of rock-like material containing three flaws under uniaxial compressive loading are conducted to investigate the failure mechanism and crack coalescence modes. The pre-existing flaws are arranged in different systematic geometries. Friction in closed flaws is modelled by inserting ideal elasto-plastic materials into the flaws. As the uniaxial compression load is increased, new cracks propagate from the flaws and eventually coalesce. The simulations replicate most of the phenomena observed in actual experiments, such as initiation and growth of wing and secondary cracks, crack coalescence, and the macro-failure of the sample. For the samples containing three pre-existing flaws, four types of crack coalescence mode are obtained: the tensile mode (T); the compression mode (C); the shear mode (S); and the mixed tension/shear mode (TS). The C mode is mainly formed by coalescence of small tensile fractures in the form of a shear band, and is believed to correspond to the formation and growth of en-echelon cracks. The applied load required to initiate coalescence depends on which mode (tensile or shear) dominates the coalescence process. A higher load is required to cause coalescence in the shear mode (S) than that for coalescence in the tensile mode (T) and mixed mode (TS). A total of four types of samples containing three parallel inclined frictional flaws and nine samples containing two parallel inclined frictional flaws are numerically simulated. It is concluded that crack coalescence will occur following the weakest coalescence path among all possible paths between any two flaws. The results obtained in the simulations are qualitatively in good agreement with experiments presented in Part I of the accompanying paper and other reported experimental observations.