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The optimal design of the postbuckling response of variable angle tow composite structures is an important consideration for future lightweight, high-performing structures. Based on this premise, a new optimisation tool is presented for shell-type structures. The starting point is an isogeometric framework which uses NURBS interpolation functions to provide a smooth description of the deformed shapes, thereby reducing the number of degrees of freedom with respect to standard finite elements. The stiffness variation is obtained by exploiting the same NURBS interpolation to describe lamination parameters, employed as intermediate optimisation variables. This choice allows the design space to be thoroughly explored with relatively few design variables in a smooth optimisation space. Therefore, the optimisation strategy is divided into two stages. Firstly, the optimal distribution of lamination parameters is determined using a gradient based algorithm. Afterwards, an actual distribution of fibre orientation is retrieved. The viability of the tool is tested firstly onto a cylindrical panel under compressive loading. Then, the postbuckling optimisation of a composite wingbox is given. For both structures, the optimised postbuckling response is compared with those of the corresponding quasi-isotropic baselines showing significant improvements.

It is with profound sadness that we share the news that Prof. Raffaele Casciaro passed away on 27 July 2020. He is buried in his beloved Rossano, a small city in southeast Italy, where he was born in 1943. It is not easy to describe, in just a few sentences, the extraordinary human and scientific personality of Raffaele. He was an unattainable model to imitate, a mentor and a friend not only for us, but also for many other colleagues from outside the computational mechanics group of the University of Calabria. He left an indelible mark on the professional and personal lives of all of us and his absence arouses a deep sense of emptiness. We attempt to describe the scientific character of Raffaele and what he symbolizes to those who knew him and held him in high esteem.

Isogeometric Kirchhoff–Love elements have received an increasing attention in geometrically nonlinear analysis of elastic shells. Nevertheless, some difficulties still remain. Among the others, the highly nonlinear expression of the strain measure, which leads to a complicated and costly computation of the discrete operators, and the existence of locking, which prevents the use of coarse meshes for slender shells and low order NURBS, are key issues that need to be addressed. In this work, exploiting the hypothesis of small membrane strains, we propose a simplified strain measure with a third order polynomial dependence on the displacement variables which allows an efficient evaluation of the discrete quantities. Numerical results show practically no difference to the original model, even for very large displacements and composite structures. Patch-wise reduced integrations are then investigated to deal with membrane locking in large deformation problems. An optimal integration scheme for third order C 2 NURBS, in terms of accuracy and efficiency, is identified. Finally, the recently proposed Newton method with mixed integration points is used for the solution of the discrete nonlinear equations with a great reduction of the iterative burden with respect to the standard Newton scheme.

peer-reviewed The stiffness-tailoring capability of Variable Angle Tow (VAT) laminates gives enhanced freedom to design thin-walled structures. One key advantage of tow steering is the ability to redistribute stresses improving buckling performance, leading to reduction in material weight and costs. The aim of this work is to optimise the initial postbuckling behaviour of a recently proposed VAT composite wingbox. The optimisation process is based on a fibre path parameterisation. It involves seeking the stacking sequence that minimises the displacements occurring in the postbuckling regime. This problem is solved by coupling the multi-modal Koiter asymptotic approach implemented with a solid-shell Finite Element environment through stochastic optimisation strategies. Results obtained regarding different optimisation scenarios show a much improved performance for the buckling and postbuckling response of the wingbox with respect to the initial VAT design. Additionally, manufacturing constraints are readily included in the optimisation program. The possibility of performing an efficient and robust optimisation process of a complex structure with a multi-modal Koiter asymptotic approach is demonstrated, showing its viability as a design tool for buckling dominated structures. A parametric study regarding the influence of steering radii shows that overcoming the current manufacturing constraint on minimum radius is worthy of investigation.

A new computational strategy for the elasto-plastic analysis of framed structures is proposed. The approach is composed of two levels of analysis: the frame level, based on a 3D mixed beam element, and the cross-section level, described through the Generalized Eigenvectors approach whose formulation is here extended for the analysis of materials with elasto-plastic behaviour. The proposed method, suitable for analyzing single beams as well as assemblages, can tackle composite cross-sections by taking into account all the stress components of the 3D solution.

Lightweight thin-walled structures are crucial for many engineering applications. Advanced manufacturing methods are enabling the realization of composite materials with spatially varying material properties. Variable angle tow fibre composites are a representative example, but also nanocomposites are opening new interesting possibilities. Taking advantage of these tunable materials requires the development of computational design methods. The failure of such structures is often dominated by buckling and can be very sensitive to material configuration and geometrical imperfections. This work is a review of the recent computational developments concerning the optimisation of the response of composite thin-walled structures prone to buckling, showing how baseline products with unstable behaviour can be transformed in stable ones operating safely in the post-buckling range. Four main aspects are discussed: mechanical and discrete models for composite shells, material parametrization and objective function definition, solution methods for tracing the load-displacement path and assessing the imperfection sensitivity, structural optimisation algorithms. A numerical example of optimal material design for a curved panel is also illustrated.

Isogeometric Kirchhoff-Love elements have received an increasing attention in geometrically nonlinear analysis of thin walled structures. They make it possible to meet the C1requirement in the interior of surface patches, to avoid the use of finite rotations and to reduce the number of unknowns compared to shear flexible models. Locking elimination, patch coupling and iterative solution are crucial points for a robust and efficient nonlinear analysis and represent the main focus of this work. Patch-wise reduced integrations are investigated to deal with locking in large deformation problems discretized via a standard displacement-based formulation. An optimal integration scheme for third order C2NURBS, in terms of accuracy and efficiency, is identified, allowing to avoid locking without resorting to a mixed formulation. The Newton method with mixed integration points (MIP) is used for the solution of the discrete nonlinear equations with a great reduction of the iterative burden and a superior robustness with respect to the standard Newton scheme. A simple penalty approach for coupling adjacent patches, applicable to either smooth or non-smooth interfaces, is proposed. An accurate coupling, also for a nonmatching discretization, is obtained using an interface-wise reduced integration while the MIP iterative scheme allows for a robust and efficient solution also with very high values of the penalty parameter.

The optimal design of shell structures undergoing buckling phenomena is nowadays an open problem, whose solution would provide interesting answers to both academy and industry. The main difficulty is represented by the high computational cost required for optimising a full-scale structure characterised by a relatively complex postbuckling behaviour. In this work, this topic is addressed by proposing an isogeometric solid-shell model. The equilibrium path is evaluated through a multimodal Koiter's method. The resulting postbuckling analysis turns out to be efficient and accurate, as shown by numerical examples.

The starting point of this work is the definition of an automatic procedure for evaluating the axial force-biaxial bending yield surface of steel and reinforced concrete sections in fire. It provides an accurate time-dependent expression of the yield condition by a section analysis carried out once and for all, accounting for the strength reduction of the materials, which is a function of the fire duration. The equilibrium state of 3D frames with such yield conditions, once discretized using beam finite elements, is then formulated as a nonlinear vectorial equation defining a curve in the hyperspace of the discrete variables and the fire duration. An incremental-iterative strategy is proposed for tracing this curve evaluating a sequence of safe states at increasing fire durations up to the limit fire duration, that is the time of exposure which leads to structural collapse. The procedure represents a global fire analysis able to take account of the stress redistribution over the frame. Numerical examples are given to illustrate the proposal.

Numerical formulations of the Koiter theory allow the efficient prediction, through a reduced model, of the behavior of shell structures when failure is dominated by buckling. In this work, we propose an isogeometric version of the method based on a solid-shell model. A NURBS interpolation is employed on the middle surface of the shell to accurately describe the geometry and the high continuity typical of the displacement field in buckling problems and to directly link the CAD model to the structural one. A linear interpolation is then adopted through the thickness together with a modified generalized constitutive matrix, which allows us to easily eliminate thickness locking and model multi-layered composites. Reduced integration schemes, which take into account the continuity of the shape functions, are used to avoid interpolation locking and make the integration faster. A Mixed Integration Point strategy makes it possible to transform the displacement model into a mixed (stress–displacement) one, required by the Koiter method to obtain accurate predictions, without introducing stress interpolation functions. The result is an efficient numerical tool for buckling and initial post-buckling analysis of composite shells, characterized by a low number of DOFs and integration points and by a simple and quick construction of the reduced model.

A numerical stochastic strategy for optimising composite elastic shells undergoing buckling is presented. Its scope is to search for the best stacking sequence that maximises the collapse load considering the post-buckling behaviour. Its feasibility is due to a reduced order model built for each material setup starting from a hybrid solid-shell finite element model exploiting a multimodal Koiter method. The approach has no limitations concerning geometry, boundary conditions and material properties distribution. The collapse load is evaluated using a Monte Carlo simulation able to detect the worst imperfection shape, including a posteriori the imperfections in the reduced order model. For a limited number of parameters the proposal allows to analyse all the possible layups. In the general case, it uses a Monte Carlo scanning of the design parameters with different levels of adaptability. The optimisation of curved panels, also with stiffeners, confirms the feasibility and reliability of the proposed strategy.

In this work an isogeometric solid-shell model for geometrically nonlinear analyses is proposed. It is based on a linear interpolation through the thickness and a NURBS interpolation on the middle surface of the shell for both the geometry and the displacement field. The Green–Lagrange strains are linearized along the thickness direction and a modified generalized constitutive matrix is adopted to easily eliminate thickness locking without introducing any additional unknowns and to model multi-layered composite shells. Reduced integration schemes, which take into account the high continuity of the shape functions, are investigated to avoid interpolation locking and to increase the computational efficiency. The relaxation of the constitutive equations at each integration point is adopted in the iterative scheme in order to reconstruct the equilibrium path using large steps and a low number of iterations, even for very slender structures. This strategy makes it possible to minimize the number of stiffness matrix evaluations and decompositions and it turns out to be particularly convenient in isogeometric analyses.

An efficient continuation method is proposed for a direct sensitivity analysis of the critical point to geometrical imperfections, e.g. expressed as combination of given shapes, for thin-walled structures prone to buckling. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc-length equation is added to follow a path of critical points in the imperfection space. Remarkable novelties are achieved. Firstly, the Jacobian of the extended system is entirely computed analytically by means of a solid-shell model and a strain-based modeling of the geometrical deviation. Moreover, a mixed scheme with independent element-wise stress variables is devised for a more efficient and robust iterative solution compared to the standard null vector method. The mixed algorithm speeds up the sensitivity analysis allowing larger imperfection steps and much fewer factorizations of the condensed stiffness matrix, only one per imperfection in its modified version with constant matrix over the step. Finally, the critical point derivatives with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient-based critical path for a quick search of the worst-case imperfection.

Summary The Koiter-Newton method had recently demonstrated a superior performance for non-linear analyses of structures, compared to traditional path-following strategies. The method follows a predictor-corrector scheme to trace the entire equilibrium path. During a predictor step a reduced order model is constructed based on Koiter's asymptotic post-buckling theory which is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces. In this manuscript, we introduce a robust mixed solid-shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger-Reissner variational formulation facilitates the reduced order model construction omitting an expensive evaluation of the inherent fourth order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out-of-balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems. This article is protected by copyright. All rights reserved.

In this paper a semi-analytic solution for the post-critical behavior of compressed thin walled members with generic cross sections is presented. It is based on the Koiter approach and the method of separation of variables. The buckling solution is exactly evaluated using a single sinusoidal function and the initial post-critical behavior is obtained as a decoupled sinusoidal series solution along the beam axis. A specialized integration scheme allows obtaining the solution with only a few terms and a very low cost, with respect to standard finite element analyses. This tool is then used to highlight the reason of the poor behavior of beam models enriched with cross section deformation modes in reconstructing the post-critical solution. Successful strategies are proposed in order to overcome these limitations. A series of numerical tests are reported.

In this paper, an alternative formulation of the NS-FEM based on an assumed stress field is presented to include drilling rotations. Within each triangular element the displacement field is described by a revised Allman triangle interpolation, while the stress field is assumed as linear or linear reduced on the conflict domain of the background grid. The elastic solution is constructed through the stationarity condition of a constrained mixed Hellinger–Reissner principle. The numerical experiments show that the proposed model performs well in elastic problems, in particular in the case of incompressibility, and takes advantage of the enrichment of the interpolation functions from quadratic contributions to the displacement field. The paper also shows a way to improve the description of the stress field.

This work presents an efficient continuation strategy based on the Riks method to describe the stable and of unstable branches of the response of carbon nanotubes (CNT)/polymer nanocomposite shells. Exploiting the superior properties of this class of nanostructured materials in the context of elastic instabilities and multistability has the potential to pave the way towards a variety of novel smart engineering applications. The equilibrium paths and the static bifurcations of CNT nanocomposites are numerically investigated highlighting the effects of material parameters such as the orientation and weight fraction of high aspect ratio CNTs integrated in a thermoplastic polymer.

The optimisation of the structural behaviour of the wing is one of the key aspects in the design of future aircraft. Enhanced freedom to designers has been offered by the stiffness-tailoring capability of Variable Angle Tow (VAT) laminates. Efficient and robust optimisation strategies are, consequently, of great importance to fully explore such an increased design domain. Simultaneously, taking account of the initial postbuckling behaviour can provide hidden load-bearing capability, leading to reduction in weight and costs. In this work we propose an optimisation strategy of the postbuckling behaviour of a recently-proposed VAT wingbox. Being based on an efficient reduced order model for the evaluation of the equilibrium path and on robust stochastic algorithms for the solution of the optimisation problem, the approach shows its viability as a general design tool for buckling dominated structures. Manufacturing constraints are included and the influence of geometrical imperfections is efficiently handled during the optimisation. Different optimisation scenarios are investigated and results show a much improved solution with respect to the initial VAT design.

Different strategies based on rotation vector and exact strain measure have been proposed over the years for analyzing beams undergoing large rotations. The interpolation of the total rotation vector is path independent but also singular at the full angle and non objective. The interpolation of the incremental rotation vector avoids the singularity but is also non objective and leads to path dependence. In this work, we develop a novel strategy which uses the incremental nodal rotation vectors to define corotational nodal rotations, which are then interpolated for the evaluation of the nonlinear strains. This choice makes the approach singularity-free, allows for additive updates within each increment and preserves all the features of the theoretical problem for any mesh and interpolation: rotational variables, objectivity, exact strain measure, path independence and symmetric stiffness matrix for conservative loads.

This work presents a cost-effective and reliable numerical framework for geometrically nonlinear thermoelastic analyses of thin-walled structures. Firstly, the structure is discretised using an isogeometric solid-shell formulation, that allows an accurate approximation of geometry and kinematics avoiding the parameterisation of finite rotations. Focus is given to an efficient modelling of thermal strains, temperature-dependent materials and general temperature profiles. Then, a generalised arc-length method is developed for solving the discrete equations with the temperature amplifier as additional unknown. A consistent definition of the tangent operators and a mixed integration point strategy lead to a robust analysis with a reduced iterative burden, even in case of complex nonlinear behaviours where standard simulation tools are outperformed or even fail. Finally, an accurate eigenvalue analysis is derived for a quick estimate of critical temperatures. The overall framework is validated by a number of applications with isotropic and multi-layered composite materials, including variable angle tow laminates.

This work introduces a mathematical problem named limit fire analysis for estimating the structural safety of 3D frames in case of fire, taking into account the stress redistribution. It is a generalization of the classic limit analysis to a fire event, where the load factor is replaced by the time of fire exposure that reduces the strength of the materials. A lower bound theorem is derived, making it possible to define the limit fire duration, i.e. the maximum time of exposure for which the structure is safe, unaffected by initial plastic deformations, loading history, thermal strains and variations of elastic properties. A numerical framework is also given for evaluating the limit time. The structure is discretized using an equilibrated mixed finite element for each beam and column. The time-dependent admissibility of the stress is imposed through a fiber approach at multiple sections along the elements. An arc-length continuation method with the fire duration as additional unknown provides a sequence of safe time of exposure converging to the limit one. Newton’s iterations are used at each step to determinate the element state and to impose global equilibrium, with all tangent operators obtained analytically. Reinforced concrete frames are considered as example of application. Numerical tests show an efficient analysis also for large buildings.

In this paper we show how to significantly improve the robustness and the efficiency of the Newton method in geometrically non-linear structural problems discretized via displacement-based finite elements. The strategy is based on the relaxation of the constitutive equations at each integration point. This leads to an improved iterative scheme which requires a very low number of iterations to converge and can withstand very large steps in step-by-step analyses. The computational cost of each iteration is the same as the original Newton method. The impressive performances of the proposal are shown by many numerical tests. In geometrically non-linear analysis, the proposed strategy, called MIP Newton, seems worthy to replace the standard Newton method in any finite element code based on displacement formulations. Its implementation in existing codes is very easy.

In this paper we propose a powerful tool for the evaluation of the initial post-buckling behavior of multi-layered composite shells and beams in both bifurcation and limit load cases, including mode interaction and imperfection sensitivity. This tool, based on the joint use of a specialized Koiter asymptotic method and a mixed solid-shell finite element model, is accurate, simple and characterized by a computational cost far lower than standard path-following approaches and many advantages with respect to asymptotic analysis performed with shell elements. The method is very simple and easy to include in existing FE codes because it is based on the same ingredients of a linearized buckling analysis, with very light formula due to the presence of displacement degrees of freedom only. Due to its efficiency it is suitable for layup design when geometrical nonlinearities have to be considered.

The element is simple accurate and robust in the analysis of elastic plane problems.The displacement interpolation is enriched by a revised Allman interpolation.The choice of the displacement interpolation keeps out zero-energy modes.The assumed stress is important in reducing the over-stiffness of linear triangles. We propose a mixed smoothed finite element model for plane elasticity. Within each triangular element the displacement field is described by a revised Allman interpolation, while the stresses are assumed to be piece-wise constant on a background grid associated with the edges of the triangle. A straightforward implementation of the element, in order to make it easy to readily incorporate it in existing FE packages, is described. The numerical experiments show that the proposed model performs well and takes advantage of the enrichment of the displacement field. Moreover the drilling rotation parameters makes simple the extension to form shell elements.

Summary The paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed (stress and displacement) format with respect to the more commonly used displacement one: (i) the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and (ii) the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used. The smaller extrapolation error produces a lower number of iterations and larger step length in path-following analysis and a greater accuracy in Koiter asymptotic method. To focus on the origin of the phenomenon, the two formats are derived for the same finite element interpolation. The reduced polynomial dependence improves the Koiter asymptotic strategy in terms of both computational efficiency, accuracy and simplicity. Copyright © 2016 John Wiley & Sons, Ltd.

The paper proposes a 3D mixed finite element and tests its performance in elasto-plastic and limit analysis problems. A composite tetrahedron mesh is assumed over the domain. Within each element the displacement field is described by a quadratic interpolation, while the stress field is represented by a piece-wise constant description by introducing a subdivision of the element into four tetrahedral regions. The assumptions for the unknown fields make the element computationally efficient and simple to implement also in existing codes. The limit and elasto-plastic analyses are formulated as a unified mathematical programming problem allowing the use of Interior Point like algorithms. A series of numerical experiments shows that the proposed finite element is locking free and has a good plastic behavior. HighlightsThe element is simple and easy nevertheless is accurate and robust in plasticity.The displacement and stress interpolations produces an isostatic element.The element is locking free.The piece-wise constant stress addresses the discontinuities in the plastic strain.T.The numerical experiments show that the element is stable also with irregular meshes.

This paper compares two distinct approaches for obtaining the cross-section deformation modes of thin-walled members with deformable cross-section, namely the method of Generalized Eigenvectors (GE) and the Generalized Beam Theory (GBT). First, both approaches are reviewed, emphasizing their differences and similarities, as well as their resulting semi-analytical solutions. Then, the GE/GBT deformation modes for four selected cross-sections are calculated and examined in detail. Subsequently, attention is turned to the efficiency and accuracy of the GE/GBT mode sets in typical benchmark problems, namely the calculation of the global–local–distortional first-order and buckling (bifurcation) behaviors of bars with the previously analyzed cross-sections. It is concluded that GE and GBT, both based on the method of separation of variables, yield accurate results although they use different structural models and mode selection strategies. Therefore they offer complementary advantages, which are put forward in the paper.

In large rotation analysis of shear deformable 3D beams, some features characterize the ideal discretization method: locking-free solution, small number of unknowns, accurate description of curved geometries, no singularity for large rotations, objectivity, additivity and robustness in the iterative solution, path independence and symmetric matrix for conservative loads. This work shows how to achieve all these features using a novel isogeometric weak formulation. Incremental rotation vectors are associated to the control points of quadratic NURBS describing the geometry. The control point vectors are used to define a novel path-independent, objective and singularity-free approximation of the rotation matrix used for the evaluation of the exact strain measure, based on an element-wise corotational system. The inter-element NURBS regularity is preserved also for the rotation. Additivity and symmetry are assured by the vector-like parameterization. Imposing constant spatial forces over each patch results in a locking-free discretization without displacement DOFs within the patch. Boundary conditions can be imposed directly without any ad hoc procedure. Accurate results are obtained using coarse meshes with reduced number of unknowns compared to displacement-based isogeometric analyses and locking-free finite elements.

Isogeometric Kirchhoff–Love elements have been receiving increasing attention in geometrically nonlinear analysis of thin shells because they make it possible to meet the C 1 requirement in the interior of surface patches and to avoid the use of finite rotations. However, engineering structures of appreciable complexity are typically modeled using multiple patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. Simple penalty approaches for coupling adjacent patches, applicable to either smooth or non-smooth interfaces and either matching or non-matching discretizations, have been proposed. Although the problem dependence of the penalty coefficient can be reduced by scaling factors which take into account geometrical and material parameters, only high values of the penalty coefficient can guarantee a negligible coupling error in all possible cases. However, this can lead to an ill conditioned problem and to an increasing iterative effort for solving the nonlinear discrete equations. In this work, we show how to avoid this drawback by rewriting the penalty terms in an Hellinger–Reissner form, introducing independent fields work-conjugated to the coupling equations. This technique avoids convergence problems, making the analysis robust also for very high values of the penalty coefficient, which can be then employed to avert coupling errors. Moreover, a proper choice of the basis functions for the new fields provides an accurate coupling also for general non-matching cases, preventing overconstrained solutions. The additional variables are condensed out and then not involved in the global system of equations to be solved. A highly efficient approach based on a mixed integration point strategy and an interface-wise reduced integration rule makes the condensation inexpensive preserving the sparsity of the condensed stiffness matrix and the coupling accuracy.

In this work an automatic procedure for evaluating the axial force-biaxial bending yield surface of reinforced concrete sections in fire is proposed. It provides an accurate time-dependent expression of the yield condition by a section analysis carried out once and for all, accounting for the strength reduction of the materials, which is a function of the fire duration. The equilibrium state of 3D frames with such yield conditions, once discretized using beam finite elements, is formulated as a nonlinear vectorial equation defining a curve in the hyperspace of the discrete variables and the fire duration. A generalized path-following strategy is proposed for tracing this curve and evaluating, if it exists, the limit fire duration, that is the time of exposure which leads to structural collapse. Compared to the previous proposals on the topic, which are limited to local sectional checks, this work is the first to present a global analysis for assessing the fire resistance of 3D frames, providing a time history of the fire event and taking account of the stress redistribution. Numerical examples are given to illustrate and validate the proposal.

Shakedown analysis of 3D frames subjected to complex load combinations is presented.The strategy is suitable to treat n-dimensional load space (n of order of hundred).The algorithm avoids redundant constraints improving computational performances.The method deals with seismic actions coming from response spectrum analysis. A strategy for an efficient treatment of load combinations is proposed with the aim of making the shakedown analysis an affordable design tool to be used in practical applications. The paper refers to 3D frames subjected to complex combinations of statical and dynamical loads defined according to Eurocode rules. The yield surface of the sections is defined by its support function values associated to presso-flexural mechanisms and it is approximated as Minkowski sum of ellipsoids. An algorithm capable of detecting the significant vertexes of the elastic envelope and suitable for use in the case of response spectrum analyses, is proposed.