Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

Optimal design of underwater stiffened shells

The optimal design parameters of stiffened shells are determined using a rational multicriteria optimization approach. The adopted approach aims at simultaneously minimizing the shell vibration, associated sound radiation, weight of the stiffening rings as well as the cost of the stiffened shell. A finite element model is developed to determine the vibration and noise radiation from cylindrical shells into the surrounding fluid domain. The production cost as well as the life cycle and maintenance costs of the stiffened shells are computed using the Parametric Review of Information for Costing and Evaluation (PRICE) model. A Pareto/min-max multicriteria optimization approach is then utilized to select the optimal dimensions and spacing of the stiffeners. Numerical examples are presented to compare the vibration and noise radiation characteristics of optimally designed stiffened shells with the corresponding characteristics of plain un-stiffened shells. The obtained results emphasis the importance of the adopted multicriteria optimization approach in the design of quiet, low weight and low cost underwater shells which are suitable for various critical applications. Received September 14, 2000 Communicated by J. Sobieski     

Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion

This paper investigated the general instability of cylindrical shells in which the stiffeners formed spirals along the length and at an arbitrary angle with the axis. Two loading conditions were considered: uniform axial and lateral compressions and torsion. The stress-strain relations of the stiffeners were developed by rotation of the strain tensor. The buckling determinate was obtained by introducing into the equilibrium equations the admissible displacement functions consistent with the end constraints, thereby enforcing equilibrium by satisfying the characteristic equations.

The buclking equations were programmed for a computer which rearched through a finite set of stress resultants for assigned values of spiral angle and modes and printed out the buckling load. The optimum structure weight of the stiffened shell was determined by iterating the design parameters at the required spiral angle so that the buckling load approached the applied load as a limit until the difference between these loads was within the design allowance.     

Analysis of laminated shells with laminated stiffeners using rectangular shell finite elements

A finite element analysis of laminated shells reinforced with laminated stiffeners is described in this paper. A rectangular laminated anisotropic shallow thin shell finite element of 48 d.o.f. is used in conjunction with a laminated anisotropic curved beam and shell stiffening finite element having 16 d.o.f. Compatibility between the shell and the stiffener is maintained all along their junction line. Some problems of symmetrically stiffened isotropic plates and shells have been solved to evaluate the performance of the present method. Behaviour of an eccentrically stiffened laminated cantilever cylindrical shell has been predicted to show the ability of the present program. General shells amenable to rectangular meshes can also be solved in a similar manner.     

Bifurcation and collapse analysis of stringer and ring-stringer stiffened cylindrical shells with cutouts

This paper presents results for cylindrical shell configurations using the STAGS computer program. Discontinuities have been imposed upon the shell's skin by incorporating symmetrical cutout openings. In addition, the surface is stiffened with both stringer and ring-stringer arrangements.The cutout problem has been shown to be highly nonlinear for smooth surface shells, but the author has found that bifurcation and collapse loads are close when one is considering stiffened skin configurations. In order to arrive at this conclusion, it was necessary to evaluate the following:—comparison between smeared and discrete stiffener theory for linear solutions—numerical finite difference convergence as directed toward buckling determination—collapse load results with the various skin stiffeners.This paper also includes a linear bifurcation study relating to stiffening effects around cutout areas present within stringer and ring-stringer shell surfaces. Comparisons have been made between a variety of geometric positions considering cutout frame and thickened skin additions. The investigation points toward an optimum positioning.     

Effect of boundary conditions on natural frequencies for rotating composite cylindrical shells with orthogonal stiffeners

The effect of the boundary conditions on the natural frequencies for rotating composite cylindrical shells with the orthogonal stiffeners is investigated using Love’s shell theory and the discrete stiffener theory. The frequency equation is derived using the Rayleigh–Ritz procedure based on the energy method. The considered boundary conditions are four sets, namely: (1) clamped–clamped; (2) clamped–simply supported; (3) clamped–sliding; and (4) clamped–free. The beam modal function is used for the axial vibration mode and the trigonometric functions are used for the circumferential vibration mode. The composite shells are stiffened with uniform intervals and the stiffeners have the same material. By comparison with the previously published analytical results for the rotating composite shell without stiffeners and the orthogonally stiffened isotropic cylindrical shells, it is shown that natural frequencies can be determined with adequate precision.     

Finite element analysis of bimodulus composite stiffened thin shells of revolution

This paper is a sequel to the work published by the first and third authors[l] on stiffened laminated shells of revolution made of unimodular materials (materials having identical properties in tension and compression). A finite element analysis of laminated bimodulus composite thin shells of revolution, reinforced by laminated bimodulus composite stiffeners is reported herein. A 48 dot doubly curved quadrilateral laminated anisotropic shell of revolution finite element and it's two compatible 16 dof stiffener finite elements namely: (i) a laminated anisotropic parallel circle stiffener element (PCSE) and (ii) a laminated anisotropic meridional stiffener element (MSE) have been used iteratively.The constitutive relationship of each layer is assumed to depend on whether the fiberdirection strain is tensile or compressive. The true state of strain or stress is realized when the locations of the neutral surfaces in the shell and the stiffeners remain unaltered (to a specified accuracy) between two successive iterations. The solutions for static loading of a stiffened plate, a stiffened cylindrical shell. and a stiffened spherical shell, all made of bimodulus composite materials, have been presented.     

Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation

In this paper, an analytical method is used to study the nonlinear primary resonance of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal stiffeners. The SSFG cylindrical shell is surrounded by linear and nonlinear elastic foundation and the effect of structural damping on the system response is also considered. The material properties of the shell and stiffeners are assumed to be continuously graded in the thickness direction. Three-parameter nonlinear elastic foundation model is consists of two-parameter linear elastic foundation (Winkler and Pasternak) and one hardening/softening cubic nonlinearity parameter. Based on the von Kármán nonlinear equations and the classical plate theory of shells, the strain–displacement relations are derived. The smeared stiffener technique is used to the model of the internal stiffeners. Using the Galerkin method, the partial differential equations of motion are discretized. The nonlinear primary resonance is analyzed by means of the multiple scales method. The effects of various geometrical characteristics, material parameters and elastic foundation coefficients are investigated on the nonlinear primary resonance.

     

Ultimate strength of stiffened symmetrical welded steel beam-to-column flange connections

Several studies have confirmed the importance, in beam-column connections, of reinforcing the column web with horizontal stiffeners opposite the beam flanges welded to the column web and flanges. Present design codes, besides being conservative, do not adequately reflect the role of the stiffeners as lateral support to the column web aginst buckling. Non-linear finite element formulations, employing an incremental version of Stowell's deformation theory and a scaled version of the inverse power method, allowed accurate predictions of the bifurcation load of general concentrically stiffened steel plates in the elasto-plastic range. In this paper, the analytical procedure developed has been used to provide qualitative and quantitative design information on the strength of this type of connection by conducting a parameteric study, the parameters considered being the effect of stiffener size on column web buckling capacity and the effect of column axial load on the stiffening requirements of a connection. It has been demonstrated that proper reinforcing of the column web can, in practice, diminish the harmful effects of high column axial loads on connections.     

Optimum design of stiffened cylindrical shells with natural frequency constraints

The design optimization of axially loaded, simply supported stiffened cylindrical shells for minimum mass is considered. The design variables are thickness of shell wall, thicknesses and depths of rings and stringers, number/spacing of rings and stringers. Natural frequency, local and overall buckling strengths and direct stress constraints are considered in the design problems. Three different combinations of stiffeners are considered. In each case, the independent effects of behaviour constraints are also studied. The optimum designs are achieved with one of the standard nonlinear constrained optimization techniques (Davidon-Fletcher-Powell method with interior penalty function formulation) and few optimal solutions are checked for the satisfaction of Kuhn-Tucker conditions.     

Elastic buckling of double walled cylindrical storage tanks—numerical and analytical estimates

This paper is concerned with estimating the elastic buckling pressures of large liquid natural gas (LNG) storage tanks which are used by the British Gas Corporation for seasonal demand peak shaving. They consist of two concentric ring stiffened cylindrical shells separated by substantial thermal insulation which maintains the LNG within the inner shell at − 165°C with minimal boil off. There is natural gas vapour above the LNG and throughout the tank interior which is normally at just above atmospheric pressure. The shell walls increase in thickness from the top to the bottom and are fabricated from very thin steel or aluminium alloy plates (diameter to thickness ratio ~4000 at the top) since they are usually in hoop tension, but under certain conditions this can become compressive making elastic buckling a possible mode of failure. The individual buckling pressures for the two shells can be estimated using standard procedures but in these LNG tanks the annular insulation transfers loads between the shells enhancing their individual strengths. A numerical method using the finite difference code BOSOR4 and a simple analytical method have been used to estimate these pressures.     

Instability of short stiffened composite cylindrical shells under bending with prebuckling displacements

This paper presents results for buckling of a stiffened cylindrical shell with cutouts and both isotropic and composite shells without cutouts acting under end bending moments. The STAGS-C program has been used in the analysis.     

Effect of cold bending and welding on buckling of ring-stiffened cylinders

The BOSOR5 computer program for elastic plastic buckling of shells of revolution is used for calculation of bifurcation buckling of cold bent and welded ring-stiffened cylinders under external pressure. Residual stresses and deformations from cold bending and welding are included in the model for buckling under service loads by introduction of these manufacturing processes as functions of a time-like parameter which ensures that the material in the analytical model experiences the proper sequence of loading prior to and during application of the service loads. The cold bending process is first simulated by a thermal loading cycle in which the temperature varies linearly through the shell wall thickness, initially increasing in time to simulate cold bending around a die and then decreasing in time to simulate springback to a final somewhat larger design radius. The welding process is subsequently simulated by the assumption that the material in the immediate neighborhoods of the welds is cooled below the ambient temperature by an amount that leads to weld shrinkage amplitudes typical of those observed in tests. Buckling loads are calculated for a configuration including and neglecting the cold bending and welding processes. These predictions are compared to values obtained from tests on two nominally identical specimens, one carefully machined and the other fabricated by cold bending the shell and then welding machined ring stiffeners to it.     

Minimum cost design of a cellular plate loaded by uniaxial compression

Cellular plates are constructed from two base plates and an orthogonal grid of stiffeners welded between them. Halved rolled I-section stiffeners are used for fabrication aspects. The torsional stiffness of cells makes the plate very stiff. In the case of uniaxial compression the buckling constraint is formulated on the basis of the classic critical stress derived from the Huber’s equation for orthotropic plates. The cost function contains the cost of material, assembly and welding and is formulated according to the fabrication sequence. The unknown variables are the base plate thicknesses, height of stiffeners and numbers of stiffeners in both directions. The cellular plate is lighter and cheaper than the plate stiffened on one side. The Particle Swarm Optimization and the IOSO techniques are used to find the optimum. PSO contains crazy bird and dynamic inertia reduction criteria, IOSO is based on a response surface technology.     

Minimum cost design of a welded stiffened square plate loaded by biaxial compression

The optimized plate structure consists of a simply supported square base plate stiffened with an orthogonal grid of flat stiffeners welded to the base plate by fillet welds. The uniformly distributed compressive load acts biaxially in the plane determined by the centre of gravity of T-sections, which consist of a part of the base plate and of a stiffener. In the optimization process the number of stiffeners as well as the thicknesses of the base plate and flat stiffeners, which minimize the cost function and fulfil the design constraints, as sought. The cost function includes the cost of material, assembly, welding and painting. Constraints relate to the global buckling, local buckling of base plate parts and stiffeners as well as to the deflection due to shrinkage of welds. To illustrate the effectiveness of the mathematical methods, the problem is solved by the Rosenbrocks hill-climb algorithm as well as by entropy-based unconstrained minimization.     

Free vibration analysis of multi-symmetric stiffened shells

The application of structural symmetry techniques to the free vibration analysis of cylindrical and conical shells for the prediction of natural frequencies and mode shapes is described. Appropriate boundary conditions have been developed for the analysis of only a part of the shell and have been shown to yield results comparable to the full shell analysis. Half and quarter models of the shell have been developed and analysed using semi-loof and facet shell finite elements. Unstiffened and stiffened circular cylindrical shells and stiffened conical shells have been considered.     

A novel snap-through buckling behaviour of axisymmetric shallow shells with possible application in transducer design

In this study, the snap-through buckling behaviour of axisymmetric shells, subjected to axisymmetric horizontal peripheral load or displacement for various shell parameters and various boundary conditions, is investigated. Results obtained seem not to have been reported previously. An application of peripheral displacement type of loading is seen in metal-ceramic composite transducers developed by sandwiching a piezoelectric (PZT) ceramic between two metal end caps which serve as mechanical transformers for converting and amplifying the lateral displacement of the ceramic into an axial motion normal to the metal cap. In our numerical search, we have observed that snap-through and snap-back buckling is possible for shallow spherical caps for a very narrow range of the shell parameter used. When a hole is opened around the apex of the cap, buckling is possible for a larger range of the shell parameter. Obtaining the displacement amplification and the blocking or generative force for various material and geometric properties is necessary for the possible application of the findings in transducer design. The numerical results are presented in graphical forms.     

Global minimum cost design of a welded square stiffened plate supported at four corners

This paper presents the application of a refined version of the original Snyman–Fatti (SF) global continuous optimization algorithm (Snyman and Fatti, J Optimiz Theory Appl 54:121–141, 1987) to the optimal design of welded square stiffened plates. In particular we investigate square plates of square symmetry subjected to uniformly distributed normal static loads, supported at four corners, and stiffened by a square symmetrical orthogonal grid of ribs. Halved rolled I-section stiffeners are used welded to the base plate by double fillet welds. Profiles of different size are used for internal and edge stiffeners. A cost calculation method, developed by the first two authors and mainly used for welded structures (Farkas and Jármai 2003), allows for the computation of cost for different proposed designs of the welded stiffened plates. The cost function includes material, welding as well as painting costs, and is formulated according to the fabrication sequence. Design variables include base plate thickness as well as the dimensions of the edge and internal stiffeners. Constraints on stress in the base plate and in stiffeners, as well as on deflection of edge stiffeners and of internal stiffeners are considered. For this purpose the Snyman–Fatti (SF) global unconstrained trajectory method is adapted to handle constraints of this type. For control purposes a particle swarm optimization (PSO) algorithm is also applied to confirm the results given by the SF algorithm. Since the torsional stiffness of open section stiffeners is very small, the stiffened plates are modelled as a torsionless gridwork. We present an algorithm for calculating the moments and deflections for torsionless gridworks with different number of internal stiffeners, using the force method.     

B0S0R5—program for buckling of elastic-plastic complex shells of revolution including large deflections and creep

B0S0R5 can handle segmented and branched shells with discrete ring stiffeners, meridional discontinuities, and multi-material construction. The shell wall can be made up of as many as six layers, each of which is a different nonlinear material. In the prebuckling analysis large-deflection axisymmetric behavior is presumed. Bifurcation buckling loads are computed corresponding to axisymmetric or nonaxisymmetric buckling modes. The strategy for solving the nonlinear prebuckling problem is such that the user obtains reasonably accurate answers even if he uses very large load or time steps. B0S0R5 has been checked by means of numerous runs in which the results have been compared to other analyses and to tests.The prebuckling and plastic bifurcation (eigenvalue) analyses are described, with the most important equations given. These equations are derived from a finite difference energy method. The strategy for solving problems simultaneously involving large deflections, elastic-plastic material behavior, and primary and secondary creep permits the use of rather large time and load steps without undue sacrifice in accuracy. This strategy is based on a subincremental iteration method in which the size of the subincrement is automatically determined such that the change in stress is less than a certain prescribed percentage of the effective stress. The theoretical treatment of discrete ring stiffeners, the material of which is elastic-plastic and can creep according to a primary or secondary creep law, is also given. Discrete rings of arbitrary cross-section are considered to be assemblages of thin rectangular elements. The structure of the B0S0R5 computer program, which runs on the CDC 6600 and on the UNIVAC 1108 and 1110, is described.The paper gives comparisons between test and theory for many configurations, including axially compressed cylinders and internally and externally pressurized shells of various shapes with and without ring stiffeners. The results of sensitivity studies are given in which the effect on predicted critical load of various analytical models of the ring-shell wall intersection area are explored. A method of predicting the effect of welding on buckling load is described, and an example involving a ring-stiffened doubly-curved shell is given. Welding the ring stiffeners to a shell introduces residual stresses and geometrical imperfections, both of which reduce the load-carrying capability.     

Buckling of cylindrical shells under the action of nonuniform external pressure

Over the last few decades, storage tanks have become bigger and thinner. Because of this, the buckling capacity of these cylindrical shells may well be the determining factor of shell thickness. In this paper, the critical buckling load of isotropic and orthotropic cylinders subjected to different types of wind load distributions is investigated. The prebuckling displacements are obtained by using the membrane theory of shell analysis. The principle of minimum potential energy in conjunction with Ritz's approach is used to obtain the stability matrix. The size of the stability matrix in this analysis is (81 × 81). By solving the stability matrix as an eigenvalue problem, the critical pressures are obtained as eigenvalues and the deflection shapes as eigenvectors. In the present study cylindrical shells of various dimensions, which are fixed at the base and free at the top, are investigated. The buckling load curves for isotropic and orthotropic cylinders of various dimensions are given for practical use.