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.     

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.     

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.     

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.     

Orthotropic bridge decks with closed stiffeners—Analysis and behaviour

The behaviour of stiffened bridge decks is investigated. The deck is treated not as an equivalent continuum but as a monolithic unit in which all the interactions are considered. A rectangular shell finite element is developed which is suitable for the idealization and efficient analysis of closed type stiffeners. Solutions are compared with three experimental tests of closed type stiffeners of trapezoidal shape. These and other examples show excellent agreement with the analysis. An investigation of the effect of one of the principal parameters entering the problem is presented. This concerns the rib spacing for torsionally stiff ribs, and results obtained indicate that present design methods are conservative in some areas, but are slightly unconservative in others.     

Topographical design of stiffener layout for plates against blast loading using a modified ant colony optimization algorithm

The stiffened plates are of demonstrable advantages and potential in offering high resistance to such extreme loading scenarios as blast. Since the distribution of the stiffeners has considerable effect on their performance, its design signifies an important topic of research. However, existing research has mainly focused on empirical design, and the configurations were largely experience based, which limits structural explosion-proof capacity. In order to improve the performance of stiffened plates against blast loading, we introduced here two new structural configurations of stiffened plates. In this study, the modified ant colony optimization (MACO) algorithm which introduces the mass constraint factor to the pheromone update function and integrates the idea of crossover and mutation was used to design the subjected to given working conditions. Specifically, material distribution of stiffeners is taken to be the design variables, and minimization of the maximum deflection of the center point of the plate to be the design objective under predetermined mass constraints. Compared with the baseline structure, the optimal designs largely improved the explosion-proof performance through distributing stiffener topology on the plates. The results showed that the optimum designs all present the reinforcement stiffeners to link with the fixed boundaries against the deformation. Moreover, the optimum designs placed more reinforcement materials in the central regions instead of four angles, and with the increase of the mass fraction, the reinforcement placement gradually extends from the center to the edges. The proposed method and new topological configurations are expected to provide some insights into design for novel protective structures.

     

Reliability-based optimum design of a welded stringer-stiffened steel cylindrical shell subject to axial compression and bending

The economy of stiffened shells vs the unstiffened version depends on loading, type of stiffening and stiffener profile. The stiffening is economic when the shell thickness can be decreased in such a measure that the cost savings caused by this decreasing is higher than the additional cost of stiffening material and welding. The present work deals with cylindrical shell columns fixed at the bottom and free at the top subject to axial compression and horizontal force acting on the top of the column. The shell is stiffened outside with stringers welded by longitudinal fillet welds. Half rolled I-section (UB) stiffeners are used to reduce welding cost. The cost function to be minimized includes the costs of the materials, forming of shell elements into the cylindrical shape, assembly, welding and painting. The design variables are the shell thickness, number and profile of stiffeners for the stiffened shell, but only the first type of variable in the unstiffened case. Randomness is considered both in loading and material properties. A level II reliability method (first-order reliability method) is employed. Individual reliability constraints related with shell buckling, stringer panel buckling and the limitation of the horizontal displacement of the column top are considered. The overall structural reliability is obtained by using Ditlevsen's method of conditional bounding. The costs of both the stiffened and unstiffened shells designed to ensure a stipulated probability of failure will be compared with the solutions obtained for a code-based method, which employs partial safety factors. Results are given illustrating the influence of the constraint on the horizontal displacement.     

Finite element free vibration analysis of eccentrically stiffened plates

A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach.     

一种改进的叶脉建模方法

提出了一种改进的叶脉建模方法。该方法将L-system加入到Runions的叶脉建模方法当中,较好地解决了主侧脉生长比较慢的问题。作为整体算法中的一步,采用一种新的dart-throwing算法得到均匀分布的营养点集,有效地提高了整个算法的绘制速度。实验结果表明,改进后的方法能够以更快的速度生成更加逼真的叶脉模型。     

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.     

Numerical methods for optimization of nonlinear shell structures

A numerical method for the optimal design of nonlinear shell structures is presented. The nonlinearity is only geometrical and the external load is assumed to be conservative. The nonlinear shell is analysed using standard nonlinear shell finite elements with the displacements and the rotation of the shell normals as independent analysis variables. Shell thicknesses and cross-sectional dimensions of beam stiffeners are used as design variables. The nonlinear optimization problem is solved using a Newton barrier method. The usefulness of the proposed method is demonstrated on shallow stiffened shell structures exhibiting significant nonlinear response.Presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Sept. 23 – Oct. 4, 1991     

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.

     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

Shape optimization of stiffeners in stiffened composite plates with thermal residual stresses

The finite element analysis method is used to examine the influence of manufacturing-induced thermal residual stresses on the optimal shape of stiffeners in stiffened, symmetrically laminated plates. Three stiffener arrangements are studied via an optimization process in which the objective is to maximize the first natural frequency of the stiffened plate. The optimization problem is solved using the method of moving asymptotes (MMA). The numerical simulations indicate that thermal residual stresses can either cause a dispersion of stiffeners along the perimeter or a concentration around the centre. Further, the optimum fundamental frequency tends to increase with increasing temperature difference.     

Stiffened plate plane stress elements for the analysis of ships' structures

A new approach has been suggested to account for the stiffeners within the plate element under plane stress. It caters accurately the spacing and orientation of the stiffeners. The derivation of the stiffeness matrix of the stiffeners placed within isoparametric and triangular elements has been presented. The advantages of the proposed approach over the conventional models have been indicated. The paper incorporates the results of a stiffened deep beam, a rectangular ship plating and a typical web frame of a tanker obtained by using these elements.     

Finite element analysis of eccentrically stiffened plates in free vibration

A compound finite element model is developed to investigate eccentrically stiffened plates in free vibration. The plate elements and beam elements are treated as integral parts of a compound section, and not as independent bending components. The derivation is based on the assumptions of small deflection theory. In the orthogonally stiffened directions of the compound section, the neutral surfaces may not coincide. They lie between the middle surface of the plate and the centroidal axes of the stiffeners. The results of this study are compared with existing ones and with those of the orthotropic plate approximation. Modifications to the existing equivalent orthotropic rigidities are proposed.     

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.     

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     

Combined shape and reinforcement layout optimization of shell structures

This paper presents a combined shape and reinforcement layout optimization method of shell structures. The approach described in this work is applied to optimize simultaneously the geometry of the shell mid-plane as well as the layout of surface stiffeners on the shell. This formulation involves a variable ground structure, since the shape of the shell surface is modified in the course of the process. Here we shall consider a global structural design criterion, namely the compliance of the structure, following basically the classical problem of distributing a limited amount of material in the most favourable way.The solution to the problem is based on a finite element discretization of the design domain. The material within each of the elements is modelled by a second-rank layered Mindlin plate microstructure. By a simple modification, this type of microstructure can be used to find the optimum distribution of stiffeners on shell structures. The effective stiffness properties are computed analytically through a smear-out procedure. The proposed method has been implemented into a general optimization software called Odessy and satisfactorily applied to the solution of some numerical examples, which are illustrated at the end of the paper.