binary behavior of fuzzy programming with piecewise linear membership functions the nature of vagueness imprecision and uncertainly is fuzzy rather than crisp and or random especially for a multiple objectives decision making problem a key component of fuzzy programming is the membership function that represents a mathematical expression of level function for the decision maker s preference in fact a decision making problem involves the achievement of fuzzy goals some of which are met while others are not because these fuzzy goals are subject to real world constraints to represent this situation the binary piecewise linear membership function is then employed in order to solve the problem we propose a new idea of how to formulate the binary piecewise linear membership function the formulated problem can be easily solved using common integer programming packages in addition an illustrative example is included for demonstrating the usefulness of the proposed model finally the analytical superiority of the proposed method in terms of the execution time can be seen through a computation experiment conducted on a set of generated test examples

p version plate and curved shell element for geometrically non linear analysis this paper presents a p version geometrically non linear formulation based on the total lagrangian approach for a nine node three dimensional curved shell element the element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element the element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction the element approximation functions and the corresponding nodal variables are derived from the lagrange family of interpolation functions the resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures c continuity the element properties are established using the principle of virtual work and the hierarchical element approximation in formulating the properties of the element complete three dimensional stresses and strains are considered hence the element is equally effective for very thin as well as extremely thick shells and plates incremental equations of equilibrium are derived and solved using the standard newton raphson method the total load is divided into increments and for each increment of load equilibrium iterations are performed until each component of the residuals is within a preset tolerance numerical examples are presented to show the accuracy efficiency and advantages of the present formulation the results obtained from the present formulation are compared with those available in the literature

comments on fuzzy programming with nonlinear membership functions abstract yang et al in their paper fuzzy programming with nonlinear membership functions published in fuzzy sets and systems 41 1991 declared that their model can solve a fuzzy program with an s shaped membership function by adding only one 0 1 variable this paper indicates that their declaration is correct only for a specific type of s shape membership functions we propose another model to treat the fuzzy programs which cannot be solved effectively by yang et al

co rotational flat facet triangular elements for shell instability analyses abstract this paper investigates the formulation of co rotational flat facet triangular elements for the numerical analysis of instability phenomena in shell structures the elements have three nodes with six degrees of freedom at each node the term co rotational relates here to the provision of a local system that continuously rotates and translates with the element following mainly nour omid and rankin b nour omid and c c rankin finite rotation analysis and consistent linearization using projectors comput methods appl mech engrg 93 1991 353 384 the definition of an element resorts to a change of variables from the local frame to the global one this is done through the use of a projector matrix which relates the variations of the local displacements to the variations of the global ones by extracting the rigid body modes from the latter the main difference from the original formulation lies in the parameterization of 3d finite rotations in contrast to the paper by nour omid and rankin a parameterization based on the rotational vector is here adopted and thus an additional change of variables has to be performed as a result the rotational variables become additive and the necessity of a special updating procedure is avoided the main feature of the adopted formulation is its independence of the local assumptions used to derive the internal forces and tangent stiffness in local coordinates for a certain class of elements i e elements with the same number of nodes and degrees of freedom the main co rotational framework is the same using this property three types of local formulations are considered a set of carefully chosen test problems is used in order to assess the performances of the three element types

updated lagrangian formulation of a flat triangular element for thin laminated shells an updated lagrangian formulation of a three node flat triangular shell element is presented for geometrically nonlinear analysis of laminated plates and shells the flat shell element is obtained by combining the discrete kirchhoff theory plate bending element and a membrane element that is similar to the allman element but a derivative of the linear strain triangular element results are presented for large rotation static response analysis of a cantilever beam under end moment cylindrical shell under pinching and stretching loads a hemispherical shell under pinching and stretching loads and a ring plate under a line load for dynamic response analysis of a cylindrical panel and for thermal postbuckling analysis of an imperfect square plate and a cylindrical panel to estimate the accuracy of the present formulation in predicting the nonlinear response of large flexible structures static analysis of an apex loaded circular arch is performed the arch is a building block of a large inflatable structure the results are in good agreement with those available in the existing literature and those obtained using the commercial finite element software abaqus demonstrating the accuracy of the present formulation he two most widely adopted approaches in the finite element analysis of shells are use of curved shell elements based on a suitable shell theory and approximation of the curved structure by an assemblage of flat shell elements in which the membranebending coupling is brought about as a result of material anisotropy and transformation of the element stiffness matrices computed in a local coordinate system to the global coordinate system prior to assembly the curved shell elements can be computationally very expensive especially in the case of nonlinear analysis because of the complexity of the formulation and the need to compute the curvature information flat shell elements are more attractive because of their simplicity and the ease with which they can be built from alreadyexisting familiar membrane and plate bending elements though a large number of elements are required to accurately model curved structures the analysis is computational ly less expensive because of extremely simple formulation updated lagrangian formulations have been predominantly used in the flat shell formulations available in the existing literature in an updated lagrangian formulation all of the variables are referred to a known configuration the reference configuration which is updated continuously during the deformation process if the rigid body modes are removed from the total or incremental displacements the resulting deformational translations and rotations are very small and hence a linearized incremental formulation1 can be used in a linearized incremental formulation the stresses are computed using linear strain displacement relations the tangent stiffness matrix contains only the linear stiffness matrix and the initial stress matrix the stiffness matrices resulting from the nonlinear terms in the strain displacement relations are neglected resulting in very economical analysis if the rigid body modes are not removed all nonlinear terms in the strain displacement relations will have to be considered for computing the stresses and the tangent stiffness matrix a three node flat triangular shell element was introduced by argyris et al 2 for nonlinear elastic stability problems this formula

corotational nonlinear dynamic analysis of thin shell structures with finite rotations a new formulation for corotational nonlinear dynamic analysis of thin shell structures involving large displacements and finite rotations has been presented in this paper by introducing the kinematic description of the geometrically exact quasi static shell model to describe the motion of an arbitrary material point in a triangular shell element the new expressions for the inertial force vector and tangent inertia matrix were derived systematically the elastic deformation was handled by using the consistent symmetrizable equilibrated corotational formulation which is independent with respect to the local finite element modeling an energy momentum conserving algorithm and an energy decaying momentum conserving algorithm were developed for nonlinear dynamic analysis by applying the generalized energy momentum method in the corotational formulation three classic numerical examples including the dynamic response free motion and dynamic buckling of thin shell structures were computed to verify the ac

on a non linear formulation for curved timoshenko beam elements considering large displacement rotation increments summary an incremental total lagrangian formulation for curved beam elements that includes the effect of large rotation increments is developed a complete and symmetric tangent stiffness matrix is obtained and the numerical results show in general an improvement over the standard formulation where the assumption of infinitesimal rotation increments is made in the derivation of the tangent stiffness matrix

generalization of yang et al s method for fuzzy programming with piecewise linear membership functions li and yu fuzzy sets and systems 101 1999 109 argued that the model of yang et al for fuzzy programming fuzzy sets and systems 41 1991 39 is correct only for a specific type of piecewise linear membership function and proposed their model for other type of membership functions this study generalizes the model of yang et al a numerical example indicates that the model of li and yu is inapplicable to a fuzzy programming problem that involves more than one membership function whereas the proposed model gives the optimal solution

vibration analysis of 3 d composite beam elements including warping and shear deformation effects abstract in this paper the dynamic analysis of 3 d composite beam elements restrained at their edges by the most general boundary conditions and subjected in arbitrarily distributed dynamic loading is presented for the solution of the problem at hand a boundary element method is developed for the construction of the 14 14 stiffness matrix and the nodal load vector of a member of an arbitrarily shaped composite cross section taking into account both torsional warping and shear deformation effects which together with the corresponding mass and damping matrices lead to the formulation of the equation of motion to account for shear deformations the concept of shear deformation coefficients is used defining these factors using a strain energy approach eight boundary value problems with respect to the variable along the bar angle of twist to the primary warping function to a fictitious function to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure bem approach both free and forced vibrations are considered taking also into account effects of transverse longitudinal rotatory torsional and warping inertia and damping resistance numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy

solving fuzzy inequalities with piecewise linear membership functions this paper deals with systems of fuzzy inequalities it shows that a system of fuzzy inequalities with piecewise linear membership functions can be converted to a one constraint nonlinear programming problem by employing the concepts of surrogate constraints and maximum entropy an augmented lagrangean algorithm is then applied to solve the resulting problem some computational results are included

a total lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements part ii arches frames and axisymmetric shells la formulation tient compte des grands deplacements grandes rotations de la structure ainsi que des effets de deformation de cisaillement

a hybrid mixed model for non linear shell analysis and its applications to large rotation problems adopting an updated lagrange approach the general framework for the fully non linear analysis of curved shells is developed using a simple quadrilateral c0 model hmsh5 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified hellinger reissner type with independent assumptions for displacement and strain fields emphasis is placed on devising effective solution procedures to deal with large rotations in space finite stretches and generalized rate type material models in particular a geometrically exact scheme for configuration update is developed by making use of the so called exponential mapping algorithm and the resulting element was shown to exhibit a quadratic rate of asymptotic convergence in solving practical shell problems with newton raphson type iterative schemes for the purpose of updating the spatial stress field of the element an objective generalized midpoint integration rule is utilized which relies crucially on the concept of polar decomposition for the deformation gradient and is in keeping with the underlying mixed method finally the effectiveness and practical usefulness of the hmsh5 element are demonstrated through a number of test cases involving beams plates and shells undergoing very large displacements and rotations

a shear flexible facet shell element for large deflection and instability analysis abstract a facet shell element based on an anisoparametric plate bending element and a quadratic plane stress element with vertex rotations is formulated for geometrically nonlinear analysis of shells the updated lagrangian formulation which proves to be effective for three node elements is employed the restriction of small rotation between the increments is removed by using hsiao s finite rotation method in which the rigid body motion is eliminated from the total displacement the displacement control is used to alleviate the singularity of the tangential stiffness matrix in the limit point type problems numerical solutions are presented for beams plates and shells to evaluate the performance of the element

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

genuinely resultant shell finite elements accounting for geometric and material non linearity the underlying theory is statically and geometrically exact and it naturally includes small strain and finite strain problems of thin as well as thick shells this paper examines the development of computational procedures employing the newton kantorovich linearization process and the galerkin type discretization method the treatment of finite rotations through an arbitrary parametrization of the rotation group the interpolation procedure of so 3 valued functions underlying the construction of finite element basis

a rotation free shell formulation using nodal integration for static and dynamic analyses of structures summary this paper proposed a rotation free thin shell formulation with nodal integration for elastic static free vibration and explicit dynamic analyses of structures using three node triangular cells and linear interpolation functions the formulation is based on the classic kirchhoff plate theory in which only three translational displacements are treated as the filed variables based on each node the integration domains are further formed where the generalized gradient smoothing technique and green divergence theorem that can relax the continuity requirement for trial function are used to construct the curvature filed with the aid of strain smoothing operation and tensor transformation rule the smoothed strains in the integration domain can be finally expressed by constants the principle of virtual work is then used to establish the discretized system equations the translational boundary conditions are imposed same as the practice of standard finite element method while the rotational boundary conditions are constrained in the process of constructing the smoothed curvature filed to test the performance of the present formulation several numerical examples including both benchmark problems and practical engineering cases are studied the results demonstrate that the present method possesses better accuracy and higher efficiency for both static and dynamic problems copyright 2015 john wiley sons ltd

geometrically non linear formulation for three dimensional curved beam elements with large rotations this paper presents a geometrically non linear formulation gnl for the three dimensional curved beam elements using the total lagrangian approach the element geometry is constructed using co ordinates of the nodes on the centroidal or reference axis and the orthogonal nodal vectors representing the principal bending directions the element displacement field is described using three translations at the element nodes and three rotations about the local axes the gnl three dimensional beam element formulations based on these element approximations are restricted to small nodal rotations between two successive load increments the element formulation presented here removes such restrictions this is accomplished by retaining non linear nodal terms in the definition of the element displacement field and the consistent derivation of the element properties the formulation presented here is very general and yet can be made specific by selecting proper non linear functions representing the effects of nodal rotations the details of the element properties are presented and discussed numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements a comparison of the results obtained from the present formulation with those available in the literature using a linearized element approximation clearly demonstrate the superiority of the formulation in terms of large load steps large rotations between two load steps and extremely good convergence characteristics during equilibrium iterations the displacement approximation of these elements is fully compatible with the isoparametric curved shell elements with large rotations and since the elements possess offset capability these elements can also serve as stiffeners for the curved shells

on the posynomial fractional programming problems abstract the posynomial fractional programming pfp problem arises from the summation minimization of several quotient terms which are composed of posynomial terms appearing in the objective function subject to given posynomial constraints this paper proposes an approximate approach to solving a pfp problem a linear programming relaxation is derived for the problem based on piecewise linearization techniques which first convert a posynomial term into the sum of absolute terms these absolute terms are then linearized by some linearization techniques the proposed approach could reach a solution as close as possible to a global optimum

geometrically nonlinear formulation of a 48 d o f quadrilateral shell element with rational b spline geometry a geometrically nonlinear formulation for a 48 degrees of freedom d o f quadrilateral shell element is presented each of the three displacement functions of u v and w is based on the bicubic hermitian polynomial the surface of the element is modelled using linearized ratronal b spline functions which may be linked to the geometric data bases generated by the computer aided design system the displacement functions and the 48 d o f are expressed in both curvilinear and cartesian forms whereas the strain displacement equations are expressed in curvilinear co ordinates the use of cartesian displacement functions and degrees of freedom allow for the proper representation of the six rigid body motions even for the element with nonzero gaussian curvature such as a bellows two examples are demonstrated snap through buckling of a spherical cap and large deflection of a semi toroidal bellows shell results compare well with alternative solutions

necessary and sufficient conditions for solvability of the riccati inequalities in the general case abstract this paper presents necessary and sufficient conditions for solvability of riccati inequalities in the general case in terms of purely imaginary eigenvalues of associated hamiltonian matrices

dynamic analysis of 3 d beam elements including warping and shear deformation effects abstract in this paper the dynamic analysis of 3 d beam elements restrained at their edges by the most general linear torsional transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting bending transverse or longitudinal loading is presented for the solution of the problem at hand a boundary element method is developed for the construction of the 14 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section taking into account both warping and shear deformation effects which together with the respective mass and damping matrices lead to the formulation of the equation of motion to account for shear deformations the concept of shear deformation coefficients is used defining these factors using a strain energy approach eight boundary value problems with respect to the variable along the bar angle of twist to the primary warping function to a fictitious function to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure bem approach that is only boundary discretization is used both free and forced transverse longitudinal or torsional vibrations are considered taking also into account effects of transverse longitudinal rotatory torsional and warping inertia and damping resistance numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy the influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame moreover the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect especially in thick walled cross section members

a simple element for static and dynamic response of beams with material and geometric nonlinearities a unified simple 6 degrees of freedom beam finite element and the associated computational procedures have been developed for the fast and efficient solution of a wide class of static and dynamic response problems of the beam type with material and or geometrical nonlinearities the material nonlinearity is treated by including its effect in the governing equations by forming the stiffness matrix of each element using a two dimensional grid of gauss points and using the material properties at each point corresponding to the uniaxial strain at that point examples are provided for metal and reinforced concrete beams a powerful yet straightforward method for the solution of elastica problems of beams and frames using the beam element developed by the senior author has been extended for determining the dynamic response of beams undergoing large displacements including large rotations the solution procedure involves piecewise linearization of response equations and iterations at each incremental step to achieve equilibrium the solution procedure is simple and easy to apply a variety of problems is solved to determine the applicability of the proposed simple formulations excellent agreement with existing analytical solutions which employ higher order elements demonstrates the efficiency and versatility of the present simple beam element in nonlinear analysis

a solution algorithm for fuzzy linear programming with piecewise linear membership functions abstract this paper deals with the fuzzy linear program with continuous piecewise linear membership functions a fuzzy linear program with piecewise linear membership functions has been considered by hannan and nakamura hannan s method can solve the problems only when all membership functions are concave in the range 0 1 and nakamura s method needs to use the linear programming technique repetitionally we propose a technique to solve the problem using a standard linear programming when membership functions are strictly quasiconcave and the minimum operator is adopted for aggregating fuzzy goals the repetitional use of linear programming technique is not required in our method

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

spatial stability analysis of thin walled space frames a clearly consistent finite element formulation for spatial stability analysis of thin walled space frames is presented by applying linearized virtual work principle and introducing vlasov s assumption the improved displacement field for unsymmetric thin walled cross sections is introduced based on inclusion of second order terms of finite rotations and the potential energy corresponding to the semitangential moments is consistently derived in the present formulation displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis and all bending torsional coupling effects due to unsymmetric cross sections are taken into account for finite element analysis cubic hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of hermitian space frame element also load correction stiffness matrices for off axis point loadings are derived based on the second order rotation terms finite element solutions for the spatial buckling analysis of thin walled space frames are compared with available solutions and other researcher s results

homogeneous riccati equations versus quadratic inequalities it is proved that under certain hypotheses the existence of hermitian solutions of a homogenized algebraic riccati equation of continuous type is guaranteed if the corresponding quadratic inequality admits a hermitian solution both finite dimensional matrices and infinite dimensional hilbert space operators riccati equations are considered

dissipativity and the lur e problem for parabolic boundary control systems dissipativity is studied for a class of boundary control systems whose free evolution is described by a holomorphic semigroup it is proved that the system is dissipative if and only if there exists a bounded selfadjoint solution of the corresponding linear operator inequality in addition the lur e problem is solved under particular assumptions

inverse scattering experiments structured matrix inequalities and tensor algebra this paper deals with a general matrix extension problem with structural constraints and provides a recursive solution in terms of an inverse scattering experiment both the stationary and nonstationary cases are considered in addition to connections to tensor algebra

on finding compromise solutions in multicriteria problems using the fuzzy min operator abstract this paper presents an application of fuzzy approaches to the linear vectormaximum problem it shows that using the fuzzy min operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem

degenerated isoparametric finite elements using explicit integration the kinematics of degenerated finite elements using explicit integration are investigated in particular for geometric nonlinear applications two representations of the nodal rotational degrees of freedom are investigated the first adopts the infinitesimal representation of rotations while the second uses a valid large rotation representation a method of explicit integration is presented which satisfies rigid body rotations under arbitrary large rotations the accuracy and convergence of these formulations are investigated

torsion constant for matrix analysis of structures including warping effect abstract in this research an effective torsion constant j eff for a wide flanged member with different warping restraint conditions at the ends is developed in order to derive j eff the boundary conditions for different warping restraint conditions at the ends as imposed by the use of different types of connections in steel structures are employed in the general equation of torsional rotation o this effective torsion constant can be used directly instead of the st venant torsion constant j in the conventional member stiffness matrix the use of j eff will account for the effect of warping when using the commercial computer programs which employ either a 6 x 6 member stiffness matrix for grid or a 12 x 12 member stiffness matrix for space frame a table for factor f which is the ratio of j eff to j is presented for different member properties and warping restraint conditions for the easy and rapid torsional analysis of structures composed of wide flanged members by the matrix method the solutions of a sample grid problem using j eff are compared with well known solutions to demonstrate its application

finite rotation elements for the non linear analysis of thin shell structures abstract for the numerical analysis of shells undergoing finite rotations doubly curved finite shell elements are developed via the displacement formulation the derivation starts from a consistent finite rotation shell theory which is transformed by a variational procedure into an incremental formulation thus the non linearity can be treated by an incremental iterative technique the non linear element matrices are obtained by a tensor oriented procedure permitting a direct transformation of the initial equations into efficient numerical models unlike in the usual procedure the kirchhoff love assumption is treated as a subsidiary condition at the element level this computer oriented approach permits the elimination of the dependent rotational degrees of freedom without loss of accuracy finally some examples are presented to demonstrate the ability of the resulting finite elements to deal with finite rotation problems

a total lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements part i two dimensional problems shell and plate structures a total lagrangian finite element formulation for the geometrically nonlinear analysis large displacement large rotations of shells is presented explicit expressions of all relevant finite element matrices are obtained by means of the definition of a local co ordinate system based on the shell principal curvature directions for the evaluation of strains and stresses a series of examples of nonlinear analysis of shell and plate structures is given

solution to the positive real control problem for linear time invariant systems in this paper we study the problem of synthesizing an internally stabilizing linear time invariant controller for a linear time invariant plant such that a given closed loop transfer function is extended strictly positive real necessary and sufficient conditions for the existence of a controller are obtained state space formulas for the controller design are given in terms of solutions to algebraic riccati equations or inequalities the order of the constructed controller does not exceed that of the plant

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

multi choice goal programming with utility functions abstract goal programming gp has been and still is the most widely used technique for solving multiple criteria decision problems and multiple objective decision problems by finding a set of satisfying solutions however the major limitation of goal programming is that can only use aspiration levels with scalar value for solving multiple objective problems in order to solve this problem multi choice goal programming mcgp was proposed by chang 2007a following the idea of mcgp this study proposes a new concept of level achieving in the utility functions to replace the aspiration level with scalar value in classical gp and mcgp for multiple objective problems according to this idea it is possible to use the skill of mcgp with utility functions to solve multi objective problems the major contribution of using the utility functions of mcgp is that they can be used as measuring instruments to help decision makers make the best appropriate policy corresponding to their goals with the highest level of utility achieved in addition the above properties can improve the practical utility of mcgp in solving more real world decision management problems

a procedure for analysing space frames with partial warping restraint stiffness matrices for three dimensional beam elements that include the effect of warping restraint on elastic torsional response have been derived by various investigators using one of the available stiffness matrices and assuming that the warping boundary conditions can be specified on a member by member basis an elastic warping support is introduced to represent conditions of partial warping restraint at the member ends the concept of a warping indicator is then introduced to facilitate use of warping springs following this static condensation is used to eliminate the restrained warping degrees of freedom the condensed stiffness matrices for the elements can then be assembled to yield a global stiffness matrix in the global matrix continuous warping degrees of freedom that is those internal to a member represented by several elements are expressed in local co ordinates the remainder are expressed in global co ordinates in the force recovery phase it is shown that an indirect method yields most accurate results for the bimoment and warping torsion when the twist function is represented by a cubic polynomial solutions to examples of linear elastic analysis are compared with well known analytical solutions to demonstrate the application of the method

nonlinear dynamic analysis with a 48 d o f curved thin shell element the developments of an existing 48 degrees of freedom d o f curved quadrilateral thin shell element for materially and geometrically nonlinear static analysis of shell structures are extended for the study of dynamic responses of nonlinear shells the variable order polynomial representations of the shell surface and the non axisymmetric definition of the shell boundaries allow the study of the dynamic behaviour of a class of shell structures more general than those treated by using flat plate elements and elements with assumptions of axisymmetry the equations of motion are based on a lagrangian frame of reference a combination of step by step and iterative procedures is used for the solution of nonlinear equations the incremental equations of motion are linearized for computation purposes and an algorithm for numerical integration based on newmark s generalized operator for dynamic analysis using optional iteration is adopted the flow theory of plasticity is used in the inelastic range and perfectly plastic or isotropic strain hardening materials are considered the spread of plastic zones in the thickness direction is treated by using a layered model numerical examples presented include the dynamic analyses of a square plate a circular annulus a cylindrical panel and a spherical cap comparisons with existing solutions demonstrate the validity and accuracy of the present developments

geometrically nonlinear formulation for the axisymmetric shell elements a geometrically nonlinear formulation using total lagrangian approach is presented for the axisymmetric shell elements the basic element is formulated using the co ordinates of the mid surface nodes and the mid surface nodal point normals an important aspect of the formulation presented here is that the restriction on the magnitude of the nodal rotations is eliminated this is accomplished by retaining true nonlinear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties based on this displacement field the element properties are derived and presented in detail numerical examples are also presented to demonstrate the element behaviour and the accuracy

non linear analysis of thin walled structures using plate elements this paper presents a simple and efficient rectangular thin plate element for elastic geometric non linear analysis of thin walled structures based on the updated lagrangian formulation and small strain hypothesis this element can accurately predict the non linear pre and post buckling load deflection paths of a plate structure using symbolic manipulation the linear and geometric stiffness matrices for the rectangular thin plate element are derived explicitly thereby eliminating the need for numerical integration the element contains 30 degrees of freedom d o f 14 d o f for the in plane membrane action and 16 d o f for the out of plane bending action several examples involving the large deflection behaviour of thin walled structures are presented to demonstrate the accuracy efficiency and versatility of the method

fuzzy goal programming with multiple priorities via generalized varying domain optimization method this paper proposes a generalized varying domain optimization method for fuzzy goal programming incorporating multiple priorities according to the three possible styles of the objective function the varying domain optimization method and its generalization are corresponding proposed in contrast to the previous method the proposed method can make that the higher priority achieving the higher satisfaction degree in this way the decision maker can get the optimal solution as well as guarantee the priorities of the multiple objective optimization problem we demonstrate the power of this proposed method by three illustrative examples and a practice application

geometrically non linear analysis of plates and shallow shells by dynamic relaxation abstract this paper investigates the application of dynamic relaxation dr method to the geometrically non linear analysis of plates and shells involving large deflections small rotations and strains the merits and demerits of two types of approaches suggested for the evaluation of parameters of the dr method are reviewed an accurate shallow shell element is developed in the present work using marguerre s shallow shell theory total lagrangian tl approach is used for an explicit derivation of element internal force vector by energy approach considering all higher order terms both in the membrane strain displacement relations and in curvature expressions the efficiency of the proposed shallow shell element in combination with the dr method for preand post buckling analysis of structures is illustrated through various numerical examples also the effect of these higher order terms in membrane and curvature expressions on overall accuracy of the solution is studied for the class of geometrically non linear problems considered in the present study

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

some extensions of fuzzy linear programming abstract in the formulation of fuzzy linear programming proposed by zimmermann it is assumed that fuzzy goals and fuzzy constraints are explicitly given as fuzzy sets on appropriate real lines respectively and are actually characterized by membership functions in linear form for relaxing those assumptions two kinds of extensions are derived in this paper in the first extension it is shown that fuzzy goals and fuzzy constraints expressed as fuzzy relations with fuzzy parameters can be treated as fuzzy sets on different real lines under some assumptions in the second extension it is shown that optimization in the case where the membership functions of the fuzzy goals and the fuzzy constraints are given in piecewise linear form can be achieved by using a standard linear programming technique repetitionally

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

geometrically nonlinear formulation for the curved shell elements this paper presents a geometrically nonlinear formulation using total lagrangian approach for the three dimensional curved shell elements the basic element geometry is constructed using the coordinates of the middle surface nodes and the mid surface nodal point normals the element displacement field is described using three translations of the mid surface nodes and the two rotations about the local axes the existing shell element formulations are restricted to small nodal rotations between two successive load increments the element formulation presented here removes such restrictions this is accomplished by retaining nonlinear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties the formulation presented here is very general and yet can be made specific by selecting proper nonlinear functions representing the effects of nodal rotations the element properties are derived and presented in detail numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements

a fuzzy multiobjective program with quasiconcave membership functions and fuzzy coefficients abstract a method is proposed for solving a fuzzy multi objective linear programming problem fmp with quasiconcave membership functions and fuzzy coefficients the proposed method first expresses a piecewise function as the summation of absolute terms then we search for the interval where the optimal solution is allocated by finding the corresponding points with same value of membership functions after that the problem is solved by goal programming techniques comparing with other fmp methods the proposed method does not need to add extra zero one variables to divide the original problem into several sub problems or transforming all original quasiconcave functions into concave functions in addition the proposed method could solve a fmp problem with fuzzy coefficients to obtain a solution closing to a global optimum

large deformation tests and total lagrangian finite element analyses of flexible beams presented here is a total lagrangian displacement based finite element formulation for general anisotropic beams undergoing large displacements and rotations the theory fully accounts for geometric nonlinearities large rotations general initial curvatures and extensionality by using jaumann stress and strain measures an exact coordinate transformation and a new concept of orthogonal virtual rotations moreover transverse shear deformations are accounted for by using a first order shear deformation theory to verify the accuracy of the finite element model two test fixtures have been built for bending and twisting experiments large static deformation tests of beams with different loading conditions have been performed the finite element results agree closely with the experimental results and numerically exact solutions obtained by using a multiple shooting method to solve for post buckling deformations of highly flexible beams undergoing large static rotations and displacements in three dimensional space

an approximation approach for representing s shaped membership functions in general to formulate a fuzzy linear programming problem with n s shaped utility membership functions traditional methods require n or more extra binary variables because s shaped curves are neither convex nor concave in all places adding binary variables does not improve the bound of the linear programming relaxation on the contrary added binary variables increase the computational burden in the solution process if problems get large therefore a formulation without binary variables should be more efficient accordingly this study proposes a piecewise linear approach to formulate an s shaped membership function mf without adding any extra binary variables which improves the efficiency of fuzzy linear programming in solving decision management problems with s shaped mfs finally a computational experiment is provided to demonstrate the superiority of the proposed models an illustrative example is also provided to show the usefulness of the proposed method

a curved quadrilateral element for static analysis of shells with geometric and material nonlinearities a 48 degrees of freedom quadrilateral element including the effect of both material and geometric nonlinearities is formulated and appropriate numerical procedures are adopted for the development of a systematic and efficient approach for the static nonlinear analysis of general shell structures the element surface is described by a variable order polynomial in curvilinear co ordinates the displacement functions are described by bicubic hermitian polynomials in curvilinear co ordinates without being confined to the assumption of axisymmetry this formulation allows for the treatment of shells with a more general shape and with a complex spread of plastic zones in the formulation for geometric nonlinearity the total lagrangian approach is adopted only small strains and small rotations are allowed in the formulation for plastic deformation the concept of a layered element model is used in the inelastic range the material is assumed to obey the von mises yield criterion and the prandtl reuss flow rule a tangential stiffness formulation is combined with the modified newton raphson iteration method for the solution of nonlinear problems a systematic choice of examples ranging from fiat plates to cylindrical panels and to spherical caps is solved and compared with available solutions to evaluate the recommended formulations and procedures in terms of their accuracy and efficiency

geometric non linear substructuring for dynamics of flexible mechanical systems a substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non linear effects large relative displacement is permitted between points within bodies that undergo small strain elastic deformation components are divided into substructures on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure compatibility conditions are derived and imposed as constraint equations at boundary points between substructures system equations of motion that include geometric non linear effects of large rotation in terms of generalized co ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure are assembled the method is implemented in an industry standard flexible multibody dynamics code with minimal modification use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation

zeros of spectral factors the geometry of splitting subspaces and the algebraic riccati inequality in this paper it is shown how the zero dynamics of not necessarily square spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic riccati inequality the notion of output induced subspace of a minimal markovian splitting subspace which is the stochastic analogue of the supremal output nulling subspace in geometric control theory is introduced through this concept the analysis can be made coordinate free and straightforward geometric methods can be applied it is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces finally the well known characterization of the solutions of the algebraic riccati equation is generalized in terms of lagrangian subspaces invariant under the corresponding hamiltonian to the larger solution set of the algebraic riccati inequality

non linear behaviour of the rfns element large displacements and rotations the reformulated four node shell element has been shown to exhibit an excellent behaviour with respect to a well known set of benchmark tests for shell elements recently the performance of the rfns element was re confirmed in terms of its asymptotic behaviour by means of classical and new limit tests in the present paper the rfns formulation is extended to incorporate geometric non linear behaviour the total lagrangian formulation is employed the rfns element non linear formulation allows for large displacements and large rotations an efficient performance of the non linear rfns element formulation is confirmed for the applications investigated

a new one point quadrature quadrilateral shell element with drilling degrees of freedom abstract in this paper we discuss the development of a new four node general shell element with single point quadrature used for the analysis of non linear geometrical and material problems one of the main features of the present development is the implementation of a rotation component around the shell normal i e drilling rotation to belytschko s family of shell elements thus at each node six degrees of freedom i e three translations and three rotations make the element easy to connect to space beams stiffeners or intersecting shells a projection scheme for warping correction is proposed so that the element is accurate for both flat and warped configurations all locking phenomena such as transverse shear in plane shear and membrane locking are controlled by an assumed strain method also a physical stabilization approach for control of spurious zero energy modes is proposed without requiring any artificial stabilization parameters this approach is inexpensive and accurate because it updates and stores hourglass stresses only at the mid surface rather than at all the integration points through the shell thickness to demonstrate the features of the new shell element it was applied to the analysis of large scale non linear static dynamic and impact problems using elastic isotropic anisotropic elastoplastic and composite damage models the element performed well for the problems that sometimes cause difficulties for other shell element techniques

direct evaluation of the worst imperfection shape in shell buckling abstract for the evaluation of the imperfection sensitivity of elastic and elastic plastic shells a fully nonlinear finite element method has been developed that directly gives the worst imperfection shape connected to the ultimate limit load the approach uses no approximations such as asymptotical theories or computations in the deep postcritical range the key point of the method is the description of imperfections as nodal degrees of freedom at the element level these unknown quantities are implemented by isoparametric shape functions in a finite shell element including finite rotations and thickness stretch the ultimate buckling load and the corresponding worst imperfection shape is defined by two different criteria and numerically determined by an extended system of nonlinear equations in the computation of the structurally stable collapse the deflections the imperfection shape the eigenvector and the lowest possible load level are obtained the method is illustrated by several numerical examples

nonlinear dynamic finite element analysis of composite cylindrical shells considering large rotations a new nonlinear finite element formulation is developed for the nonlinear dynamics of shell structures using a vectorial approach the kinematics that can be used to describe very large displacements and large rotations are derived green strain measures and second piola kirchhoff stress measures are used in the determination of the total potential a displacement based eight noded cylindrical shell element with a total of 36 degrees of freedom is developed several numerical examples are dynamically analyzed to observe the characteristics of large displacements and large rotations a convergence study was carried out to establish the numerical accuracy of the model the most efficient and accurate model was used he nonlinear vibration analysis of shells has been the focus of research for the past few years thin shells subjected to dynamic loads could encounter deflections of the order of the thick ness of the shell dynamic response of thin shells also could lead to the phenomena of dynamic snapping or dynamic buckling be cause these kinds of responses cannot be determined accurately using small displacement and small rotation theories large defor mation and large rotation theories are required the complex nature of these theories requires solving numerous simultaneous nonlin ear differential equations even with simplifying assumptions these equations remain complex and extensive as background a few relevant past efforts are reviewed earlier work regarding the nonlinear vibrations for isotropic shell structures can be attributed to carr 1 yeh 2 clough and wilson 3 belytschko and marchertas 4 and belytschko and tsay 5 these researchers used flat plate elements saigal and yang6 developed a curved shell element for isotropic shells for composite shell structures simitses7 provided an elegant analytical solution of thin laminated shells subjected to sudden loads he ignored through the thickness shear effects and used donnell type kinematic relations raouf and palazotto8 9 developed a perturbation procedure to derive a set of asymptotically consistent nonlinear equations of motion for an arbi trary laminated composite cylindrical shell in cylindrical bending they also neglected transverse shear effects simo and tarnow10 de veloped an energy and momentum conservating algorithm for shell analysis to analyze shells undergoing large rigid body motions palazotto and dennis11 presented the simplified large rotation slr theory to analyze shells experiencing moderately large ro tations the slr theory determines the equilibrium path of or thotropic shells using a total lagrangian approach and includes a parabolic transverse shear stress distribution this approach cap tures the appropriate kinematics through displacement polynomi als green strain displacement relations are used and all of the final displacement functions are carried into the expression for the total potential without making any attempt at separating the rigid body movement using this approach the features classically pre sented for including through the thickness shear for example can be exploited 12 smith and palazotto13 discussed eight higher order shear theories to the nonlinear finite element analysis of compos ite shells they also used the green lagrange strains and second piola kirchhoff stresses these theories were successful for large displacements and moderately large rotations a reason for the fail ure of these theories in the modeling of large rotation can be at tributed to the approximations used in the kinematics gummadi and palazotto14 modified the kinematics incorporated in the slr theory and successfully carried out the solutions to problems ex hibiting the characteristics of very large displacements and rota tions in their work a vectorial approach was followed to derive the appropriate kinematics that can capture the characteristics of large displacements and large rotations using appropriate assumptions the kinematics for the slr theory can be obtained as a special case the large rotation kinematics was introduced through the use of sine and cosine functions a 36 degree of freedom composite shell element was developed based on this theory based on the slr theory tsai and palazotto15 formulated a nonlinear dynamic finite element analysis procedure for composite cylindrical shells in their work a linear mass matrix was incorpo rated along with the nonlinear stiffness matrix that was originally developed in slr theory this theory again has the same drawbacks as the slr theory in terms of realizing the large displacements and large rotations in the present paper a theory that can capture the large displacement and large rotations is discussed the present work is an extension of the authors earlier work on large rotation theory for static analysis a new mass matrix is generated using hamilton equations the mass matrix thus generated is a nonlin ear mass matrix a fi m method15 is used to solve the nonlinear algebraic equations resulting from the finite element technique se lected numerical examples are solved to determine the validity of the developed theory

dissipativity of uncontrollable systems storage functions and lyapunov functions dissipative systems have played an important role in the analysis and synthesis of dynamical systems the commonly used definition of dissipativity often requires an assumption on the controllability of the system in this paper we use a definition of dissipativity that is slightly different and less often used in the literature to study a linear time invariant possibly uncontrollable dynamical system we provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not mixed i e no pair of uncontrollable poles is symmetric about the imaginary axis this condition is known to be related to the solvability of a lyapunov equation we link lyapunov functions for autonomous systems to storage functions of an uncontrollable system the set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature we show that for an uncontrollable system the set of storage functions is unbounded and that the unboundedness arises precisely due to the set of lyapunov functions for an autonomous linear system being unbounded further we show that stabilizability of a system results in this unbounded set becoming bounded from below positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive in this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions in most of the results in this paper we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity

the algebraic riccati equation a polynomial approach abstract polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic riccati equation

fuzzy programming with nonlinear membership functions piecewise linear approximation one promising approach that has been proposed for dealing with multicriteria i e multiple objective programming models has been fuzzy linear programming flp in essence the flp approach involves the replacement of the multiple objectives with goals by means of assigning an aspiration level to each objective fuzzy membership functions are then introduced to represent the measure of the achievement of the actual solution for each goal relative to its aspired level it has been shown that if the flp problem is linear and if linear membership functions are employed to reflect goal achievement then the flp problem may be transformed into a conventional i e single objective linear programming model as such one may use any conventional linear programming algorithm to solve the transformed model however in the more general case of nonlinear membership functions the transformation process becomes considerably more involved and has in the past typically led to less desirable formulations in this paper we present what we believe to be a straightforward and computationally efficient procedure for dealing with the flp problem with any general class of nonlinear membership functions conversion of such a flp will result in either a regular linear programming model or a linear integer programming model depending upon the specific characteristics of the membership function in either case existing commercial software is readily available to solve such models

on a quadratic matrix inequality and the corresponding algebraic riccati equation abstract the purpose of this paper is to report some new results concerning the geometry of the set x1d4ab 0 of all solutions of a quadratic matrix inequality qmi arising in stochastic realization theory and the subset x1d4ab 0 of all solutions of the corresponding algebraic riccati equation are the boundary interior and extreme points of x1d4ab are studied some new results linking x1d4ab 0 to x1d4ab are presented a complete characterization of some subsets of x1d4ab 0 and x1d4ab are given additional properties of the feedback matrix corresponding to any element in x1d4ab 0 are obtained

algebraic riccati equation and symplectic algebra questions of existence uniqueness and the parametric dependence of solutions of the algebraic riccati equation are considered the different criteria for the solubility of this equation are obtained with the help of symplectic algebra

a simplified co rotational method for quadrilateral shell elements in geometrically nonlinear analysis summary this paper presents a simplified co rotational formulation for quadrilateral shell elements inheriting the merit of element independence from the traditional co rotational approach in literature with the objective of application to nonlinear analysis of civil engineering structures the authors further simply the formulation of the geometrical stiffness using the small strain assumption which is valid in the co rotational approach with the warping effects considered as eccentricities compared with the traditional element independent co rotational method the projector is neglected both in the tangent stiffness matrix and the internal force vector for simplicity in formulation meanwhile a quadrilateral flat shell element allowing for drilling rotations is adopted and incorporated into this simplified co rotational algorithm for geometrically nonlinear analysis involved with large displacements and large rotations several benchmark problems are presented to confirm the efficiency and accuracy of the proposed method for practical applications

the general nonstrict algebraic riccati inequality abstract if a b is stabilizable it is well known how to verify the solvability or the existence of positive definite solutions of the nonstrict algebraic riccati inequality ari a x xa xbb x q 0 this paper serves to demonstrate how far it is presently possible to generalize these results if the uncontrollable modes of a b are not restricted we prove a new reduction principle which may be formulated as follows the ari has a solution iff a certain reduced order riccati equation for a stabilizable system a linear equation and a reduced order nonstrict lyapunov inequality corresponding to the uncontrollable modes on the imaginary axis have solutions basically the quadratic riccati inequality is hence reduced to a linear lyapunov inequality we discuss how this principle may be applied to actually check the solvability of the ari as the main motivation for our work we briefly explain the consequences of these results for the general state feedback h optimization problem

linear programming with multiple fuzzy goals abstract this paper describes the use of fuzzy set theory in goal programming gp problems in particular it is demonstrated how fuzzy or imprecise aspirations of the decision maker dm can be quantified through the use of piecewise linear and continuous functions models are then presented for the use of fuzzy goal programming with preemptive priorities with archimedean weights and with the maximization of the membership function corresponding to the minimum goal examples are also provided

a geometrically non linear tensorial formulation of a skewed quadrilateral thin shell finite element a 48 degree of freedom d o f skewed quadrilateral thin shell finite element including the effect of geometrical non linearity is formulated and appropriate numerical procedures are adopted for the development of an efficient approach for the static and dynamic analysis of general thin shell structures the element surface is described by a variable order polynomial in curvilinear co ordinates the displacement functions are described by bicubic hermitian polynomials in curvilinear co ordinates the directions of the curvilinear co ordinates at each nodal point are uniquely defined to coincide with the directions of the boundaries of the element in the present case of a skewed quadrilateral with non orthogonal curvilinear coordinates the coupling terms of the metric tensor and curvature tensor of the surface no longer vanish such as in the case of orthogonal co ordinates the tensor form is used in the setup of the shape functions geometric derivatives stiffness matrix and computer code this allows for the treatment of shells with irregular shapes and variable curvatures to evaluate the efficiency and accuracy of this formulation a systematic list of examples is chosen i linear and non linear static analysis of square and rhombic plates cylindrical and spherical shells ii linear vibrations of trapezoidal flat and curved plates iii large amplitude vibrations of a rhombic plate for the square plate and cylindrical and spherical shell shewed element meshes with various distortion angles are used to study the effect of the distortion angles on the accuracy of the results and to demonstrate the versatility of the present element all results are compared with alternative available solutions including those obtained using regular rectangular meshes pinched thin cylindrical and spherical shells are studied using different skewed meshes and various gauss integration meshes and no membrane locking phenomenon is observed

geometrically nonlinear transient analysis of laminated composite shells using the finite element method abstract the nonlinear transient response of composite shells with without cutouts and initial geometric imperfection is investigated using the finite element method the present formulation considers doubly curved shells incorporating von karman type nonlinear strains into the first order shear deformation theory the analysis is carried out using quadratic c0 eight noded isoparametric element the governing nonlinear equations are solved by using the newmark average acceleration method in the time integration in conjunction with modified newton raphson iteration scheme the validity of the model is demonstrated by comparing the present results with those available in the literature parametric studies are carried out varying the radius of curvature to width ratio and amplitude of initial geometric imperfection of laminated composite cylindrical spherical and hyperbolic paraboloid shells with without cutouts