An improved dynamic relaxation method for the analysis of plate bending problems

An improved dynamic relaxation method is used to solve the set of simultaneous equations resulting from the application of finite element method for plate bending problems. Different weights are used as multiplying factors for each mass (m) and damping factor (C) in each equation representing one of the three degrees of freedom at each node. The optimum values of these weights are obtained for different cases of the bending of cantilever plates stiffened with edge beams with different sizes of stiffening edge beams.     

A finite element large displacement analysis of stiffened plates

A finite element analysis of the large deflection behaviour of stiffened plates using the isoparametric quadratic stiffened plate bending element is presented. The evaluation of fundamental equations of the stiffened plates is based on Mindlin's hypothesis. The large deflection equations are based on von Kármán's theory. The solution algmrithm for the assembled nonlinear equilibrium equations is based on the Newton-Raphson iteration technique. Numerical solutions are presented for rectangular plates and skew stiffened plates.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

Vibration and buckling analysis of plates of abruptly varying stiffness

The application of B-spline functions and the Rayleigh-Ritz procedure to analyze vibration and buckling of plates with abruptly varying stiffnesses is presented. Numerical examples of stepped thickness plates and perforated plates are presented and the results are discussed in comparison with those of other approximate methods.To demonstrate the versatility of the present method, vibration and buckling of skew plates of stepped thickness are also studied for various skew angles, ratios of thickness and ratios of width.     

Elastic limit loads of skew plates

An elastic limit load of skew plates is analyzed on the basis of the Rayleigh-Ritz method with B-spline functions and the Huber-Mises yield criterion. Dimensionless elastic limit loads, associated maximum deflections and the positions of the first yield points are presented for different skew angles, aspect ratios and boundary conditions.     

Combined stiffening and in-plane boundary conditions effects on the buckling of circular cylindrical stiffened-shells

The influence of in-plane boundary conditions on the critical loads of axially compressed simply supported stiffened cylindrical shells, stiffened by stringers and by combinations of rings and stringers is studied. It is observed that the axial restraint, u = 0, at the edges and the dimensions of either the stringers or the rings characterize the type of influence experienced. In shells stiffened by “medium” and “heavy” stringers the axial restraint is a predominant factor and the “weak in shear”, N_xφ = 0, B.Cs. have only a slight secondary effect. For such shells a “stiffening” effect is observed for SS2 and SS4 B.Cs. As the stringers become “weaker” the influence of the axial restraint diminishes and the isotropic or ring-stiffened like type of behavior, “sensitivity” to the vanishing of the circumferential restraint, overcomes the “stiffening” effect due to u = 0.In shells stiffened by combinations of rings and stringers the influence of the in-plane boundary conditions is governed by the relative magnitudes of the rings and stringers under consideration. Combinations of “heavy” stringers and “weak” rings behave like stringer-stiffened shells, exhibiting the “stiffening” effect due to u = 0 whereas shells stiffened by “heavy” rings and “light” stringers tend to behave like ring-stiffened shells, revealing their “sensitivity” to the “weak in shear” boundary conditions, N_xφ = 0.     

An integral-equation solution for the finite deflection of clamped skew sandwich plates

The large-deflection behaviour of skew sandwich plates is governed by a system of five coupled nonlinear partial differential equations which are highly complex in nature. In the reported study, this problem is analysed using an integral-equation approach. The integral equations of beams along the skew directions is used with appropriate boundary conditions to transform the governing nonlinear partial differential equations into a set of nonlinear algebraic equations. These equations are then solved using an iterative scheme suggested by Brown. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load-deflection characteristics have been presented for clamped skew sandwich plates.     

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.     

Large deflection of clamped skew plates

A theoretical analysis is presented for the large deflection elastic behaviour of clamped, uniformly loaded orthotropic skew plates. The governing nonlinear partial differential equations are transformed into a set of nonlinear algebraic equations. For this purpose a network of discrete points over the domain is considered, and the integral equation of beams along the skew directions is used with appropriate boundary conditions. The resulting nonlinear algebraic equations are then solved by a Newton-Raphson procedure.A convergence study of the solution has been made, and numerical results for a few cases of orthotropy and skew angles are presented in graphical form. The results obtained by this method are compared with available results of other investigators.     

Compressive buckling analysis and design of stiffened flat plates with multilayered composite reinforcement

This paper describes an analysis and its application in design for compressive buckling of flat stiffened plates considered as an assemblage of linked orthotropic flat plate and beam elements. Plates can be multilayered, with possible coupling between bending and stretching. Structural lips and beads are idealized as beams. The plate and the beam elements are matched along their common junctions for displacement continuity and force equilibrium in an exact manner. Buckling loads are found as the lowest of all possible general and local failure modes. The mode shape is used to determine whether buckling is a local or general instability and is particularly useful to the designer in identifying the weak elements for redesign purposes. Typical design curves are presented for the initial buckling of a hat stiffened plate locally reinforced with boron fiber composite.     

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.     

Macro-element analysis of skewed and triangular orthotropic thin plates with beam boundaries

A generalized macro-flexibility analysis of stiffened orthotropic skewed and triangular thin plates with beam stiffened boundaries subjected to bending and stretching is presented. The proposed method accounts for minimized number of elements in a domain, i.e. the element size is independent of the final results. This is accomplished by satisfying the equilibrium and compatibility conditions along the nodal lines interconnecting contiguous elements. The solution form and shape functions of element shapes are a combination of Fourier series and polynomials with undetermined coefficients. To obtain a solution of a general domain, the orthotropic triangular and parallelogram macro-plate elements with edge beams, satisfying moment and shear equilibrium conditions along nodal lines, were assembled and analyzed by utilizing compatibility of deflection and slope. The results from the proposed methodology were compared to the ones from the finite element method. Also, the convergence was checked by increasing the number of harmonics. The study indicates that good convergence is observed within the first four harmonics.     

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.     

A combined Rayleigh-Ritz/finite element method for the nonlinear analysis of stiffened plated structures

The paper describes a hybrid method for the non-linear analysis of steel plates and stiffened plating in which local (finite element) displacement functions are supplemented or replaced by global functions. The method is applied to the collapse analysis of box-girder bridges.     

Finite element free vibration of eccentrically stiffened plates

An isoparametric stiffened plate element is introduced for the free vibration analysis of eccentrically stiffened plates. The element has the ability to accommodate irregular boundaries. Moreover, the formulation considers shear deformation, hence, the formulation is applicable to both thick and thin plates. In the present formulation, the stiffeners can be placed anywhere within the plate element and they need not necessarily follow the nodal lines. In addition, the effects of lumped and consistent mass matrices on natural frequencies of stiffened plates are investigated. The effects of several parameters of the stiffener—eccentricity, shape, torsional stiffness etc.—on the natural frequencies of the stiffened plates are studied.     

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.     

Seminumerical solution for buckling of rectangular plates

A semianalytical, seminumerical method of solution is presented for the governing partial differential equation of rectangular plates subjected to in-plane loads. The basic functions in the y-direction are chosen as the eigenfunctions for straight prismatic beams. The classical method of separation of variables is employed to obtain an ordinary differential equation. The resulting equation is solved by a one-dimensional finite difference technique. The problem is then reduced to a typical eigenvalue problem which on solution yields the buckling coefficient of the plate. The method is applied on plates with different edge conditions and under various loading conditions. The results are compared with those of existing solutions. Results for the case when one loaded edge is fixed and the other simply supported were reported in the literature for the first time.     

An incremental galerkin method for plates and stiffened plates

In order to perform a detailed analysis of large deflection behavior of a rectangular plate or stiffened plate, an efficient semi-analytical method is developed. First, incremental forms of the governing differential equations of plates and stiffened plates with initial deflection are derived. These equations are linearized and may be easily solved. Secondly, these equations are solved for each load increment by the Galerkin method with a special consideration of simply supported boundaries.A procedure of equilibrium correction at intermediate load steps is presented such that good accuracy of the solution may be maintained with larger load steps.This method is successfully applied to plates with initial deflection subjected to in-plane as well as out-of-plane loads to obtain the whole histories of the behavior of these plates. Application of this method to stiffened plates with initial deflection is also presented. Comparisons of results obtained by this method with those obtained by other methods are made and the validity of the method is demonstrated.This incremental version of the Galerkin method is found to be extremely advantageous in certain types of plate and stiffened plate problems. These types are identified and the efficiency of the method is demonstrated.     

A new model for slab and beam structures––comparison with other models

In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models used by various researchers. The adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. According to this model the stiffening beams of the structure are isolated from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The adopted model describes better the actual response of the plate-beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. Four additional models neglecting the shear forces at the interfaces are presented and used for comparison reasons, while a three-dimensional elasticity model is also employed for the verification of the accuracy of the results of the examined models. The findings from this investigation, using the adopted model, which approximates better the actual response of the plate-beams system, necessitate the consideration of the inplane forces and deformations.     

Weight optimization of stiffened cylinders under axial compression

A procedure is developed for the design of a stiffened cylinder under a given uniform axial compression with minimum weight. The approach allows the consideration of various shapes of stiffening members. The effective stiffness of the skin in its post-buckled state is taken into account in the basic analysis. The buckling analyses are accomplished as a minimum problem in the buckling mode shape parameters space using the variable metric method. A mixed procedure which combines the exterior penalty function concept and random search is used to minimize the weight of the stiffened cylinders. The design examples demonstrate the validity of the present approach.