The usefulness of static condensation in the finite element analysis of stiffened submersible cylindrical hulls

The usefulness of the static condensation technique in the finite element analysis of stiffened submersible. cylindrical hulls is examined in this paper. The finite element formulation used herein is essentially the same as outlined by the authors in an earlier paper wherein the stiffener is modeled rigorously using axisymmetric thin annular plate elements for the web and axisymmetric thin shell elements for the flange. The static condensation technique has been applied in this paper to reduce these stiffener finite elements so that their effect can be transferred to the shell node at the point of attachment of the stiffener with the shell. The advantage of such condensation of the stiffener elements is the smaller number of equations to be solved without the rigor of the stiffener modeling being lost in any way. The manner of incorporating the condensation in the computer program has been described. Examples of several stiffened submersible cylindrical hulls have been considered as an illustration of the use of the program.     

Analysis of laminated shells of revolution with laminated stiffeners using a doubly curved quadrilateral finite element

A finite element analysis of laminated shells of revolution reinforced with laminated stifieners is described here-in. A doubly curved quadrilateral laminated anisotropic shell of revolution finite element of 48 d.o.f. is used in conjunction with two stiffener elements of 16 d.o.f. namely: (i) A laminated anisotropic parallel circle stiffener element (PCSE); (ii) A laminated anisotropic meridional stiffener element (MSE).These stifiener elements are formulated under line member assumptions as degenerate cases of the quadrilateral shell element to achieve compatibility all along the shell-stifiener junction lines. The solutions to the problem of a stiffened cantilever cylindrical shell are used to check the correctness of the present program while it's capability is shown through the prediction of the behavior of an eccentrically stiffened laminated hyperboloidal shell.     

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.     

Minimum weight design of stiffened cylinders subjected to pure torsion

A minimum weight design procedure along with actual designs of two typical fuselage type of stiffened circular cylindrical shell geometries subjected to pure torsion is presented. By formulating the weight of the composite shell as the objective function, an optimization technique is adopted to minimize it against general instability. In the design, all other possible failure modes, i.e. panel instability, skin wrinkling, local instability of stringers, yielding of skin and stiffener materials as well as failure mode interactions have been avoided. Typical opened type, like rectangular, tee, I, etc. and closed type, hat stiffener with all possible combinations are studied. For each shell geometry the best suited combination of stiffener geometries-are shown. In the first trial without minimum gauge (WMG), a design is obtained utilizing rectangular stringers and rectangular rings. Then the procedure is extended for other stiffener geometries to obtain a minimum gauge (MG) design. The effect of relaxing the MG on the weight of the stiffened shell is shown graphically and in tabular form which will be of added advantage to the designer while deciding about the MG. Since the smeared technique is employed in the analysis of stiffened shells, an upper bound on the stiffener spacing is initially employed. Then this bound is relaxed and its effect on the minimum weight design is studied for those types of stiffeners which proved economically feasible with bounded spacings. The procedure adopted in this work can be used for any other shell and stiffener geometry.     

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.     

A practical method for the minimum weight design of stiffened plates under uniform lateral pressure

A method for the plastic analysis and minimum weight design of grillages and orthogonally stiffened plates subjected to lateral uniform pressure, and under a varying degree of rotational restraint at the supports, is presented. A computer aided design procedure for orthogonally stiffened rectangular plates based on this method is also described. This includes two optimization stages: overall optimization, in which the plastic moments for the beam in each set are found, and stiffener optimization, in which the optimum stiffener geometry is obtained. This design methodology has proved to be very efficient and easy to apply, and it is particularly valuable in the case of ship structures, where the maximum total number of panel stiffeners is in general not large.     

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.     

Geometric nonlinear analysis of stiffened plates by the spline finite strip method

Geometric nonlinear analysis of stiffened plates is investigated by the spline finite strip method. von Karman’s nonlinear plate theory is adopted and the formulation is made in total Lagrangian coordinate system. The resulting nonlinear equations are solved by the Newton–Raphson iteration technique. To analyse plates having any arbitrary shapes, the whole plate is mapped into a square domain. The mapped domain is discretised into a number of strips. In this method, the displacement interpolation functions used are: the spline functions in the longitudinal direction of the strip and the finite element shape functions in the other direction. The stiffener is elegantly modelled so that it can be placed anywhere within the plate strip. The arbitrary orientation of the stiffener and its eccentricity are incorporated in the formulation. All these aspects have ultimately made the proposed approach a most versatile tool of analysis. Plates and stiffened plates are analysed and the results are presented along with those of other investigators for necessary comparison and discussion.     

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.     

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.     

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 stress analysis technique for plate and shell built-up structures with junctions and its application to minimum-weight design of stiffened structures

A finite element formulation using the penalty function method to analyse exactly the junctions of plate and shell built-up structures is suggested for an isoparametric shell element. The connectivity condition at the junction is added to the potential energy functional by the penalty parameter and the interpolating function of displacements. This formulation yields an integral-type stiffness matrix of the special junction elements, which can directly evaluate the surface tractions at the junction. For applying the technique suggested here to the optimum design of structures with junction parts, a design sensitivity analysis formulation for the adhesive special element is also developed. The technique is applied to the minimum-weight design problems of isotropic and composite laminated plates with a stiffener subjected to stress constraints.     

Multilevel optimization applied to hull girder design using three panel forms

A multilevel design scheme for ship's hull girders (longitudinal members between two adjacent transverse frames) is presented in this paper. This design scheme handles, very conveniently, the complexity of using an optimization algorithm for such complex design problems having a large number of design variables, nonlinear constraints dealing with different failure modes and interactions among substructures, and nonlinear design objectives. The conventional multilevel design technique is modified by introducing an approach called constraint coordination to increase the probability of achieving the overall optimum very efficiently.The scheme is demonstrated by application to the structural design of hull girders with simple structural modelling to represent inland waterway ships on which there was a special emphasis in the original research project (Rahman 1991). Three possible panel (consisting of one stiffener and its attached plating) forms; tee stiffened, flat-bar stiffened and corrugated, are optimized to synthesize the hull girder in order to achieve the most efficient structure. The effect of price-structure (labour rate to material price ratio) on the design is also investigated.     

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.     

Eigenvalue sensitivity analysis of stiffened plates with respect to the location of stiffeners

This paper formulates the eigenvalue sensitivity of a stiffened plate with respect to stiffener location; the analysis is based on the generalized Rayleigh quotient of the combined platebeam system. The results show that the eigenvalue sensitivity is proportional to the force between the plate and the stiffener as well as to the slope of the eigenfunction at the interface between the plate and the stiffener.     

Instability analysis of Skylab structure

A computer program for the linear and nonlinear analysis of shell structures (STAGS) is applied to the instability analysis of a complex stiffened structure subjected to complicated sets of distributed and concentrated loads. Results obtained from two-dimensional bifurcation and nonlinear collapse analyses are presented. The nonlinear analysis includes the effects of an imperfection in a shell stiffened by heavy longerons. Several problems associated with the analysis of very large eigenvalue problems are discussed (one case involved a model with 20,910 degrees-of-freedom). These include the use of the appropriate finite-difference formulation to ensure convergence in a desired direction (approaching the asymptotic value either from above or from below) and the use of variable grid spacing to induce buckling in one part of the structure or the other. A technique of finding higher eigenvalues by removing the prestress at the point of bifurcation is described.     

Crack path bifurcation at a tear strap in a pressurized shell

Bifurcation of an initially longitudinal through crack in an internally pressurized cylindrical shell at a circumferential stiffener is investigated using a finite element analysis. The finite element model is developed from a fracture test of an aluminum shell having a 22.9 cm radius, a 1.02 mm wall thickness, and stiffened by two externally bonded circumferential straps spaced 40.6 cm apart. After initial stable crack growth in the longitudinal direction with increasing pressure, the crack propagated dynamically toward the strap, bifurcated near the strap into circumferential branches running parallel to the straps. Stable and unstable crack growth curves of pressure versus half-crack length are determined from the nonlinear analysis using a critical value of the crack tip opening angle as the criterion to predict crack growth. Although the crack growth curves are determined from a static analysis, they corroborate the test results for the location of crack path bifurcation. Also, the principal stress criterion for predicting crack turning is consistent with the test.     

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.     

Semi-analytical postbuckling analysis of stiffened imperfect plates with a free or stiffened edge

A large deflection, semi-analytical method is developed for pre- and postbuckling analyses of stiffened rectangular plates with one edge free or flexibly supported, and the other three edges laterally supported. The plates can have stiffeners in both directions parallel and perpendicular to the free edge, and the stiffener spacing can be arbitrary. Both global and local bending modes are captured by using a displacement field consisting of displacements representing a simply supported, stiffened plate and an unstiffened plate with a free edge. The out-of-plane and in-plane displacements are represented by trigonometric functions and linearly varying functions, defined over the entire plate. The formulations derived are implemented into a FORTRAN computer programme, and numerical results are compared with results by finite element analyses (FEA) for a variety of plate and stiffener geometries. Relatively high numerical accuracy is achieved with low computational efforts.