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The arc-length Riks strategy has rapidly become a standard tool for path-following analysis of nonlinear structures due to its theoretical ability to surpass limit points. The aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle ‘locking’ effect intrinsic to the nonlinear nature of the problem. As a consequence, its sanitization has to be pursued within a reformulation of the structural model. The use of a mixed (stress-displacement) variant of the algorithm, in particular, appears very promising in this respect.The topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in [39]. It is shown that, with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation, it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm.

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... M and µ being some suitable metric factors [49,50], ∆ξ an assigned increment of ξ and ...

... The load-controlled scheme is obtained assuming g[u, λ] = λ (see [49] for further details) while keepingK = K[u 1 ] we have the modified Newton scheme. The solution of Eq.(3.5) is conveniently performed as follows ...

... The convergence of the iterative process (3.5) has been widely discussed by Garcea et al. [49] and can be expressed by the condition ...

A brand-new design philosophy tends to harness the load-carrying capacity
hidden beyond the onset of buckling phenomena in shell structures. However,
when designing in the postbuckling range, among other effects, attention should
be given at imperfection sensitivity which may generate catastrophic and un�expected consequences on the optimised structures. Therefore, what would be
necessary is an optimisation strategy able to deal with the complex geometries
of full-scale structures and, meanwhile, efficiently gather the complexity of
their postbuckling response. The aim of this work is to meet this demand by
proposing numerical methods that face the problem from different sides, namely
the geometrically nonlinear description of the shell, the solution algorithm and
the optimisation strategy.
As a starting point, a convenient format to describe geometrically nonlinear
shell structures is identified in the solid-shell model. On the basis of this
model, a discretised environment is constructed using isogeometric analysis
(IGA) that, by taking advantage from the high continuity of the interpolation
functions, leads to a reduced number of variables with respect to standard
finite elements. Afterwards, an IGA-based multimodal Koiter’s method is
proposed to solve the geometrically nonlinear problem. This method meets the
aforementioned requirements of efficiency, accuracy and is capable of providing
information on the worst-case imperfection with no extra computational cost
with respect to the analysis of a perfect structure. Additionally, a new strategy
for improving the accuracy of the standard version of Koiter’s algorithm in
the presence of geometrical imperfections is devised. The last part of the
thesis concerns the optimal design of full-scale structures undergoing buckling
phenomena. In particular, the design focuses on variable angle tow laminates,
namely multi-layered composites in which fibre tows can describe curvilinear
paths, thereby providing great stiffness-tailoring capacity. Two optimisation
strategies are proposed, both based on the use of Koiter’s method to evaluate the
postbuckling response. The first one makes use of a fibre path parameterisation
and stochastic Monte Carlo random search as a global optimiser. The second
one is based on direct stiffness modelling using lamination parameters as
intermediate optimisation variables that lead to a reduction of the nonlinearity
of the optimisation problem and remove the direct dependence from the number
of layers.

... In path-following methods for geometrically nonlinear analyses of beams and shells, many authors observed a more robust and efficient iterative solution for mixed formulations [8]. The performance of Newton's method drastically deteriorates in displacement formulations when the membrane/flexural stiffness ratios get higher [9,10]. Mixed formulations are not affected by this drawback as the stress unknowns are used as independent variables in the iterative process. ...

... Eq. (7) and (10) in Cartesian components become respectively ...

... In [10,9], it is shown that the Newton's method convergence for displacement-based formulations gets slower and requires a smaller step size when the slenderness of the structure increases. This fact is unrelated to the accuracy of the interpolation and occurs because the stresses σ σ σ g (d e ), used to evaluate the tangent stiffness matrix K e (σ σ σ g (d e ), d e ), are forced to satisfy the constitutive equations at each iteration. ...

... In path-following methods for geometrically nonlinear analyses of beams and shells, many authors observed a more robust and efficient iterative solution for mixed formulations [8]. The performance of Newton's method drastically deteriorates in displacement formulations when the membrane/flexural stiffness ratios get higher [9,10]. Mixed formulations are not affected by this drawback as the stress unknowns are used as independent variables in the iterative process. ...

... Eq. (7) and (10) in Cartesian components become respectively ...

... In [10,9], it is shown that the Newton's method convergence for displacement-based formulations gets slower and requires a smaller step size when the slenderness of the structure increases. This fact is unrelated to the accuracy of the interpolation and occurs because the stresses σ σ σ g (d e ), used to evaluate the tangent stiffness matrix K e (σ σ σ g (d e ), d e ), are forced to satisfy the constitutive equations at each iteration. ...

... When comparing mixed and displacement formulations in path-following methods for geometrically nonlinear analyses of beams and shells, many authors observed a more robust and efficient iterative solution for mixed formulations. The reason for this is explained in [20,21] where it is shown that the performance of Newton's method drastically deteriorates in displacement formulations when the membrane/flexural stiffness ratios get higher. On the contrary mixed formulations are not affected by this drawback because the stress unknowns are used as independent variables in the iterative process. ...

... However, efficiency and robustness of a nonlinear analysis do not only depend on the number of unknowns and integration points, but also on the iterative effort, that is on the capability of the Newton's method to converge using a low number of iterations and to withstand large step sizes (increments). In [20,21], it is shown that the Newton's method exhibits a slow convergence and requires a small step size for slender elastic structures undergoing large displacements when any purely displacement-based formulation is adopted. This could be considered as a sort of "locking" of the Newton's method, since its performance gets worse when the slenderness of the structure increases. ...

... Conversely, mixed (stress-displacement) formulations are not affected by this phenomenon, because the stresses are directly extrapolated and corrected in the iterative process, allowing a faster convergence of the Newton's method also with very large steps, independently of the slenderness of the structure. We refer readers to [20,21] for further details on this. ...

Isogeometric Kirchhoff–Love elements have been receiving increasing attention in geometrically nonlinear analysis of thin shells because they make it possible to meet the C1 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.

... When comparing mixed and displacement formulations in path-following methods for geometrically nonlinear analyses, many authors observed a more robust and efficient iterative solution for mixed formulations. The reason for this is explained in [34,35] where it is shown that the performance of Newton's method drastically deteriorates in displacement formulations when the slenderness of a structure gets higher. To eliminate this inconvenience in displacement-based finite elements, the Mixed Integration Point (MIP) strategy has been recently proposed in [36]. ...

... However, the efficiency and the robustness of a nonlinear analysis do not only depend on the number of unknowns and integration points, but also on the iterative effort, that is on the capability of the Newton's method to converge using a low number of iterations and to withstand large step sizes (increments). In [34,35], it is shown that the Newton's method exhibits a slow convergence and requires a small step size for slender elastic structures undergoing large displacements when any purely displacementbased formulation is adopted. This could be considered as a sort of "locking" of the Newton's method, since its performance gets worse when the slenderness of the structure increases. ...

... Conversely, mixed (stress-displacement) formulations are not affected by this phenomenon, because the stresses are directly extrapolated and corrected in the iterative process, allowing a faster convergence of the Newton's method and very large steps, independently of the slenderness of the structure. We refer readers to [34,35] for further details on this. ...

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 C2 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

... However the reasons for these better performances are, in our opinion, not clear, as they are often wrongly attributed to the properties of the finite element interpolation. One of the goals of this chapter is therefore to clarify the true reason and origin of this phenomenon, extending the results presented some years ago [32,33] in the context of path-following and Koiter analyses of 2D framed structures. ...

... Mixed and displacement descriptions, while completely equivalent at the continuum level, behave very differently when implemented in path-following and Koiter solution strategies even when they are based on the same finite element interpolations, that is when they are equivalent also at the discrete level. This is an important, even if frequently misunderstood, point in developing numerical algorithms and it has been discussed in [32,33,6] to which readers are referred for more details. ...

... M and µ being some suitable metric factors [32], ∆ξ an assigned increment of ξ andJ ...

This thesis aims at developing a reliable and efficient numerical framework for
the analysis and the design of slender elastic shells, in particular when composite
materials are adopted, taking account of the geometrically nonlinear behaviour.
Different aspects of this challenging topic are tackled: discretisation techniques,
numerical solution strategies and optimal design. The first chapter, after a short
summary of the Riks and Koiter methods, discusses the important advantages
of using a mixed (stress-displacement) solid model for analysing shell structures
over traditional shell models and the implications of this on the performances of
the solution strategies. The second chapter introduces a mixed solid-shell model
and reformulates the Koiter method to obtain an effective tool for analysing imperfection
sensitive structures. This approach is the starting point of the third
chapter, which proposes a stochastic optimisation strategy for the layup of composite
shells, able to take account of the worst geometrical imperfection. The
fourth chapter extends the benefits of the mixed formulation in the Newton iterative
scheme to any displacement-based finite element model by means of a novel
strategy, called Mixed Integration Point. The fifth chapter illustrates an efficient
implementation of the novel Koiter-Newton method, able to recover the equilibrium
path of a structure accurately with a few Newton iterations, combining an
accurate Koiter predictor with the reduced iterative effort due to a mixed formulation.
The solid-shell discrete model is reformulated in the sixth chapter, following
the isogeometric concept, by using NURBS functions to interpolate geometry and
displacement field on the middle surface of the shell in order to take advantage
of their high continuity and of the exact geometry description. The approach
is made accurate and efficient in large deformation problems by combining the
Mixed Integration Point strategy with a suitable patch-wise reduced integration.
The resulting discrete model proves to be much more convenient than low order
finite elements, especially in the analysis of curved shells undergoing buckling.
This is shown in the seventh chapter, which proposes an efficient isogeometric
Koiter analysis.

... However, when comparing mixed and displacement formulations in path-following geometrically nonlinear analyses, many authors observed that the mixed ones can withstand much larger step sizes (increments) with a reduced number of iterations to obtain an equilibrium point and then the equilibrium path. The reason for this is explained in [25,26] where it is shown that the performances of the Newton method drastically deteriorate in displacement formulations when the slenderness of the structure increases. Conversely, the Newton method in mixed formulations is unaffected by this phenomenon, which depends only on the format of the iterative scheme adopted (mixed or purely displacement based) and also holds when a mixed and a displacement formulation provide the same discrete accuracy. ...

... M and µ being some suitable metric factors [26,35], ∆ξ an assigned increment of ξ and ...

... The load-controlled scheme is obtained assuming g[u, λ] = λ (see [26] for further details) while keepingK = K[u 1 ] we have the modified Newton scheme. The solution of Eq. (30) is conveniently performed as follows ...

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.

... However the reasons for these better performances are, in our opinion, not clear, as they are often wrongly attributed to the properties of the finite element interpolation. One of the goals of this paper is therefore to clarify the true reason and the origin of this phenomenon, extending the results presented some years ago [24,25] in the context of path-following and Koiter [26,27,28] asymptotic analyses of 2D framed structures. ...

... Mixed and compatible description, while completely equivalent at the continumm level, behave very differently when implemented in path-following and asymptotic solution strategies even when they are based on the same finite element interpolations, that is when they are equivalent also at the discrete level. This is an important, even if frequently misunderstood, point in developing numerical algorithms and it has been discussed in [24,25,29] to which we refer readers for more details. To represent the strain energy in a smooth enough way in order to make the truncation error of its Taylor expansion up to a given order as small as possible is crucial: in path-following analysis, this ensures a fast convergence of the Newton (Riks) iterative process; in asymptotic analysis, which is based on Taylor series the expansion of the strain energy [26,28] implies an accurate recovery of the equilibrium path. ...

... In this paper we intentionally use the compatible description derived from the mixed finite element in order to have the same discrete approximation for both the description, i.e. the finite element is the same but the format of the problem changes. This allows us to focus of the way as the problem description affects its solution in large deformation problems (see also [24,25]) and then to show the reasons of the better performance of the use of a mixed description. The conclusions are general and hold for any other compatible or mixed finite element. ...

... When comparing mixed and displacement finite elements many authors observe that the mixed ones are more robust and allow larger steps in path-following geometrically non-linear analyses [13][14][15]. This fact was investigated for the first time some years ago in [16], when it was shown that the robustness and the efficiency (in terms of number of iterations) of the Newton iterative scheme are penalized in displacement formulations because of a phenomenon that was called "extrapolation locking". This is not a locking of the FE discretization, but of the iterative scheme usually found in beam/shell problems, when the axial/membranal stiffness is much higher than the flexural one. ...

... M and µ being some suitable metric factors [16,19], ∆ξ an assigned increment of ξ and ...

... The load-controlled scheme is obtained assuming g[u, λ] = λ (see [16] for further details) while keepingK = K[u 1 ] we have the modified Newton-Raphson scheme. The solution of Eq. (5) is conveniently performed as follows ...

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.

... However, the reasons for these better performances are, in our opinion, not clear, as they are often wrongly attributed to the properties of the FE interpolation. One of the goals of this paper is therefore to clarify the true reason and origin of this phenomenon, extending the results presented some years ago [24,25] to the context of path-following and Koiter [26-28] asymptotic analyses of two-dimensional (2D) framed structures. ...

... Mixed and displacement descriptions, while completely equivalent at the continumm level, behave very differently when implemented in path-following and asymptotic solution strategies even when they are based on the same FE interpolations, that is when they are equivalent also at the discrete level. This is an important, even if frequently misunderstood, point in developing numerical algorithms, and it has been discussed in [24,25,29] to which we refer readers for more details. ...

... A convergence condition similar to Equation (5e) but limited to the subspace of nonsingular values of Q K holds also for the arc-length scheme [24] that, like for the load controlled case, is as faster as ...

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

... The other difficulty with this method is, because of To overcome these difficulties of conventional N-R method, various arc length based methods have been developed where the load increment and displacement increment are both unknown parameters and a constraint equation is added to the global set of equations to help pass the limit points in the load-displacement plot. The details can be found in References [7] [14][15] [16]. ...

... We also briefly describe the convergence issues with Riks method (References [14] [15]). ...

... Residual difference between two successive iterations, for the Arc Length method specializes as [15] (the reader is referenced to [14] [15] for additional detail): ...

Bends are an integral part of a piping system. Because of the ability to ovalize and warp they offer more flexibility when compared to straight pipes. Piping Code ASME B31.3 [1] provides flexibility factors and stress intensification factors for the pipe bends. Like any other piping component, one of the failure mechanisms of a pipe bend is gross plastic deformation. In this paper, plastic collapse load of pipe bends have been analyzed for various bend parameters (bend parameter = Display Formula t R b r m 2 ) under internal pressure and in-plane bending moment for various bend angles using both small and large deformation theories. FE code ABAQUS version 6.9EF-1 has been used for the analyses.
Copyright © 2015 by ASME Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature
Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

... Vale la pena ricordare che questa forma di lockingè tipica di strutture snelle analizzate attraverso l'aggiornamento della configurazione edè presente anche nell'analisi path-following, legato alla grossa differenza esistente tra la matrice di it-erazioneK ed il valore "medio" effettivo che essa assume nel corso dell'analisi, e dove, comunque, implica solo un alto deterioramento della convergenza nel processo di soluzione iterativa [30]. Le piastre sottili, essendo generalmente caratterizzate da un elevato rapporto tra rigidezza assiale e flessionale, sono particolarmente sensibili a tale fenomeno. ...

... Le piastre sottili, essendo generalmente caratterizzate da un elevato rapporto tra rigidezza assiale e flessionale, sono particolarmente sensibili a tale fenomeno. Pertanto un processo di soluzione affidabile richiede un'opportuna La rilevanza del locking da estrapolazione, di cui si discute in dettaglio in [30], e strettamente legata alla scelta delle variabili primarie per l'estrapolazione, cioè, dalla scelta delle quantità ottenute dall'estrapolazione diretta rispetto a quelle ottenute dall'estrapolazione stessa. Si osservi che la curva soggetta a estrapolazione nonè altro che una particolare rappresentazione del percorso di equilibrio, sono possibili diverse rappresentazioni dello stesso percorso. ...

... In tale contesto una formulazione compatibile (in cui il percorso di equilibrioè descritto attraverso l'estrapolazione degli spostamenti mentre le tensioni sono ottenute come quantità derivate) può presentare, a causa di piccole rotazioni precritiche, un elevato grado di nonlinearità tale da rendere completamente inattendibile a distanza finita la valutazione asintotica di quantità rilevanti (variazione 2 a , 3 a e 4 a dell'energia di deformazione) richieste nell'analisi producendo un incremento patologico sia del carico di buckling che della curvatura post-critica del percorso, tutto ciò porta ad un abbattimento dell'accuratezza della soluzione. Tuttavia, come riportato in [67,81,30], una formulazione mista, in cui le tensioni sono assunte anche come variabili primarie, fornisce una rappresentazione del percorso molto meno nonlineare e quindi esente dal problema di locking. ...

... Moreover, a careful combination of the displacement and stress approximations generally allows to reduce locking and increase accuracy. Finally, the use of a mixed stress approach connects directly with the mixed solution format 8,9 to increase the e±ciency and robustness of geometrically nonlinear solution procedures. ...

... As already pointed out, if the stress approximation satis¯es the equilibrium equation (the details about the assumed stress approximation are given in Sec. 3.3), the internal work can be obtained by integrating on the element contour through Eq. (8) and, hence, its evaluation only needs that the contour displacements are de¯ned. On this regard, the displacement interpolation along the generic side k, u k , is de¯ned as a sum of three contributions: ...

... As regards the compatibility matrix, since assumed stresses (35) are self-equilibrated, then, according to Eq. (8), matrix Q f can be conveniently evaluated performing the integration on the element contour as a sum of the four side contributions: ...

... Moreover, a careful combination of the displacement and stress approximations generally allows to reduce locking and increase accuracy. Finally, the use of a mixed stress approach connects directly with the mixed solution format 8,9 to increase the e±ciency and robustness of geometrically nonlinear solution procedures. ...

... As already pointed out, if the stress approximation satis¯es the equilibrium equation (the details about the assumed stress approximation are given in Sec. 3.3), the internal work can be obtained by integrating on the element contour through Eq. (8) and, hence, its evaluation only needs that the contour displacements are de¯ned. On this regard, the displacement interpolation along the generic side k, u k , is de¯ned as a sum of three contributions: ...

... As regards the compatibility matrix, since assumed stresses (35) are self-equilibrated, then, according to Eq. (8), matrix Q f can be conveniently evaluated performing the integration on the element contour as a sum of the four side contributions: ...

... The element directly uses the equilibrium equations to improve the accuracy and reduce the number of interpolation parameters. A block elimination of the variables that do not require inter-element continuity allows, at the global level, only 6 þ n kinematical parameters per node to be exposed and the use of a pseudo-compatible format for the analysis [33,34]. ...

... where the generalized stresses α c ½s and α ω ½s are evaluated by employing the definition in Eq. (33). In particular introducing the vector of the resultants t ¼ ft c ; t ω g with t ω ¼ fB; T g and using the properties in Eqs. ...

... At the global level a pseudo-compatible format, in terms of the standard nodal values fu½0; φ½0; μ½0; u½ℓ; φ½ℓ; μ½ℓg, is exploited. As the computational cost is closely related to the factorization of the pseudocompatible stiffness matrix, the finite element proposed has almost the same computational cost as the compatible one with the same global variables (see [33,41,34]). ...

A geometrically nonlinear model for isotropic and homogeneous beams including the non-uniform out of plane warping deformation of the cross-section due to torsion and shear is derived using the implicit corotational method (ICM). The basic idea of ICM is the application of the corotational method, initially proposed for a whole finite element, at level of the continuum geometrical description. This enable the definition of a general tool to obtain objective models for structural finite elements. The generalized variational principle of Hellinger-Reissner, requiring a separate description of the stress and displacement fields, is exploited. For each section, a corotational frame is introduced where statics and kinematics are derived from a 3D linear elastic formulation. The latter exactly reproduces the Saint-Venant solution while variable warping effects are added in a simplified but accurate way, making the model suitable for the analysis of beams with either thin-walled or compact crosssections. A mixed finite element is implemented and used inside a Koiter-like analysis algorithm which is highly sensitive to the geometrical coherence of the formulation. The numerical simulations performed demonstrate the accuracy and effectiveness of the proposed nonlinear beam model.

... It is also worth mentioning that the proposed approach does not require any adhoc assumption about the structural model at hand, nor depends on any particular parametrization of the rotation tensor, but actually behaves as a blackbox tool able to translate known linear modelings into the corresponding nonlinear ones. Moreover, the direct use of a mixed (stress/strain) description provides an automatic and implicitly coherent methodology for generating models free of nonlinear locking eects [51,52] in a format directly suitable for use in FEM implementations. ...

... The equation format (i.e. mixed or purely compatible format) also plays an important role in the convergence of the iterative solution processes and in the accuracy of the extrapolation results (see [51,52]). These topics will be discussed and the use of mixed FEM discretizations and a mixed format is suggested. ...

... We also know (see [51,52] , which also implicitly denes the stress σ as workassociated with ε. The invariance from superposed rigid body motions, that is objectivity, is an essential prerequisite for function ε [u]. ...

Such we call Implicit Corotational Method, is proposed as a tool to obtain geo-metrically exact nonlinear models for structural elements, such as beams or shells, undergoing nite rotations and small smooth strains starting from the basic solutions for the 3D Cauchy continuum used in the corresponding linear modelings. The idea is to use a local corotational description to decompose the deforma-tion gradient in a stretch part followed by a nite rigid rotation. Referring to this corotational frame and using standard change in the observer algebra we can derive, from the linear stress tensor and the deformation gradient provided by the linear theory, an accurate approximation for the nonlinear Biot stress and strain tensors which implicitly assure the frame invariance of the description. The stress and strain elds recovered in this way are then entered in a mixed variational formulation to ob-tain, nally, a nonlinear modeling in terms of standard generalized stress and strain parameters, in a form directly suitable for FEM implementations. The great potential of the method lies in its ability to recover objective nonlinear structural models by fully reusing information gained from its linear counterpart and so exploit in a quite automatic way the great experience and many results already available from linear theories. The applications regard the construction of 3D beam and plate nonlinear models starting from the Saint Venánt rod and Kirchho and MindlinReissner plate linear theories, respectively. A nite element implementation of these models, suitable for both pathfollowing and asymptotic postbuckling analysis, is reported, showing the eectiveness of the proposed approach for obtaining numerical solutions in nonlinear analysis. Dierent aspects of the FEM modeling are discussed in detail, including the nu-merical handling of nite rotations, interpolation strategies and the equation formats. Two mixed nite elements are presented, suitable for nonlinear analysis: a 3D beam element, based on interpolation of both the kinematic and static elds, and a rotation free thin plate element, based on a biquadratic spline interpolation of the displacement and piecewise constant interpolation of stress. Both are frame invariant and free from nonlinear locking. A numerical investigation has been performed, also comparing beam and plate solutions in the case of thinwalled beams. The good agreement between the re-covered results with available theoretical solutions and/or numerical benchmarks, clearly shows the correctness and robustness of the proposed approach as a general strategy for numerical implementations.

... The finite element formulation is obtained from the Hellinger-Reissner functional introducing a suitable interpolation of both the stress and displacement fields. By means of a block elimination of the variables that do not require inter-element continuity, the element, at the global level, exposes only a reduced number of kinematical parameters and uses a pseudo-compatible format to perform the analysis [35,36,29]. It will be shown that very accurate results can be obtained considering few generalized warpings of the cross section also for complex buckling modes containing a localization of displacements. ...

... The mixed beam finite element proposed exploits separate interpolations for stresses and displacements with a Gauss point integration to compute the discrete quantities. Due to the high number of variables for each element, a certain cure has been posed in the assemblage procedures, while a block elimination for the unknowns that do not require inter-element continuity [35,36,29] has been exploited. ...

... The element behavior is defined in terms of the 17 þ 4n kinematical and 11 þ 6n statical parameters. The last ones, not At the global level a pseudo-compatible format, in terms of the nodal values fυ i ; φ i ; μ i ; υ j ; φ j ; μ j g, is exploited (see [35,46,36,29]). ...

... The ICM does not require any ad-hoc assumptions about the structural model at hand, nor depends on any particular parametrization of the rotation tensor, but actually behaves as a black-box tool able to translate known linear models into the corresponding nonlinear ones. Moreover, the direct use of a mixed (stress/displacements) description provides an automatic and implicitly coherent methodology for generating models free of nonlinear locking effects, as described in [27][28][29][30], and in a format directly suitable for use in FEM implementations. ...

... By means of a block elimination of the variables that do not require inter-element continuity the element, at the global level, exposes only 9 kinematical parameters per node and uses a pseudo-compatible format to perform the analysis [27,42]. The equilibrium path is recovered by means of a FEM formulation of the Koiter asymptotic approach [43][44][45][46][47][48], that has shown to be highly sensitive to the geometrical coherence of the structural model and of its finite element interpolation [18,19]. ...

... At the global level a pseudo-compatible format, in terms of the remaining standard 18 nodal values fu i ; u i ; l i ; u j ; u j ; l j g, is exploited. As the computational cost is closely related to the factorization of the pseudocompatible stiffness matrix, the finite element proposed has almost the same computational cost as the compatible one with the same global variables (see [27,28,42]). ...

A new geometrically nonlinear model for homogeneous and isotropic beams with generic section including non-uniform warping due to torsion and shear is derived. Each section is endowed with a corotational frame where statics and kinematics are described using a 3D linear elastic model which extends the Saint-Venant solution to non-uniform warping cases. The algebra of change of observer and a mixed variational principle give the model in terms of generalized parameters. Using a mixed interpolation the model is implemented within a FEM Koiter analysis highly sensitive to the geometrical coherence of the formulation.

... The iterative advantages of MFEs can be observed also in implicit large deformation dynamics as shown in [21]. Nevertheless, it is necessary to mention a precursory work [22], where for the rst time it was shown that the path-following analysis of slender elastic beams discretized via DFEs is aected by an iterative burden that increases with the axial/exural stiness ratio in large rotation problems. According to that paper, this feature is due to the fact that the stress used to evaluate the iteration matrix can be aected by a large error when computed from wrong displacements predicted during the iterations. ...

... The linear system in Eq. (22) is solved at global level after a standard FE assemblage so obtainingḋ. The strain correctionsε g are then retrieved by local back-substitutions. ...

In large rotation analyses of beams and shells with displacement-based discretization, the iterative burden grows considerably with the membrane/flexural stiffness ratio. For linear elastic materials, it was shown that a stress-displacement iteration solves this issue and reduces the computational cost even of several times. Convergence problems occur also when large rotations are coupled to material nonlinearity, even if this case is not well addressed in the literature. The extension of the mixed iteration to large deformation problems with nonlinear constitutive laws is faced in this work, focusing on elastoplasticity. New iterative schemes are derived for both displacement-based finite elements (DFEs) and mixed finite elements (MFEs), each featuring further variables in the linearization along with displacements. For DFEs, it is shown that the most convenient approach is to impose the constitutive law for an independent integration point strain, with the strain-displacement compatibility solved together with the global equilibrium. For MFEs, as an extension of this strategy, the most performing scheme solves element compatibility and global equilibrium simultaneously, with the element state evaluated for independent strains work-conjugate of the stress DOFs. In both cases, global linear systems in displacements only are required and strain-driven material laws are easily considered. Numerous tests validate the proposed iterative schemes showing a significant reduction of the computational cost compared to standard approaches.

... The equilibrium paths are reconstructed using the Riks arc-length method [71], with the implementation details on the constraint surface given in [72]. When necessary, to achieve post-critical branches, small imperfection loads work-conjugated to the critical modes are added. ...

... In any case, in view of a greater efficiency, it would be worth asking how much we could gain by using a less nonlinear representation of the equilibrium paths, abandoning the choice of a compatible formulation in favour of the general Hu-Washizu framework or of the Hellinger-Reissner mixed framework, the latter in the event that the use of simple harmonic potentials is shown to be sufficiently accurate in this context. In both cases, this would require the introduction of further variables for describing the interactions, to be managed by local condensation procedure [67,72]. ...

This paper deals with a numerical model for the buckling and post-buckling analysis of single-wall carbon nanotubes. Reasons of efficiency lead to the choice of a simple molecular statics model, wherein binary, ternary and quaternary atomic interactions are accounted for and described using Morse and cosine potential functions. The equations of the model are discussed in depth and the parameters of the potential functions are justified in the light of a comparison with ab-initio results. Several case studies regarding zigzag and armchair tubes of different aspect ratios, under compression, bending and torsion, are addressed with the aim of investigating the efficacy of the model and the role of the quaternary interactions, in contexts of both global and local behaviours.

... In the past, it has been observed in several studies that mixed (stress and displacement) finite elements show superior properties compared to pure displacement-based elements in the context of a standard Newton approach for nonlinear analyses. 17,18 In general, the mixed formulation allows for larger step sizes and requires less iteration steps to regain equilibrium as highlighted in Garcea et al 19 for beam elements. A more recent work 20 observed the beneficial convergence properties of mixed-model formulations in a general context, which can be extended to displacement-based finite element formulations following the mixed integration points strategy proposed in Magisano et al. ...

... Details of the Newton iteration with the Riks constraint can be found in previous studies. 19,21 The construction of the reduced-order model is dominated by the factorization of the governing system of equations of the augmented problem (15) and (16), respectively. It is important to note that both systems of equations of dimension (N + m + 1) have an identical system matrix, hence factorization is needed only once. ...

The Koiter-Newton method had recently demonstrated a superior performance for nonlinear 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 postbuckling theory that 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.

... The assumed stress approach also seems promising for the applications in nonlinear analysis [29,30], lower bound strategy for shakedown, limit analysis and elastoplastic analysis [31]. It is worth noting that the simplicity of the interpolation fields leads to an essential and well-working element with few requirements in the case of general nonlinear applications. ...

... The piece-wise constant stress [17] or strain [12] over the triangles of the grid, allowing discontinuities within it, are assumed in the initial formulations and give coincident models in linear elasticity [28]. Here the finite element model is constructed by assuming independent interpolations to approximate the displacement and the stress field, making the use of the present model in geometrically nonlinear models simpler and effective [29,30]. The preliminary numerical experiments show that the enriched kinematic (6) makes the model with constant stress interpolation too soft. ...

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.

... It is important to note as the linear extrapolation of the fundamental path gives completely different results when a mixed (stress and displacement) or a displacement description of the problem are adopted. The mixed description allows, in fact, the elimination of the extrapolation locking phenomenon discussed in detail in [49,26] in the context of path-following analysis and in [31,13,50] for asymptotic analyses. The extrapolation locking affects any displacement model and consists in an overestimated stiffness evaluated in an extrapolated point. ...

... Finally note that the mixed formulation is required to sanitize an underhand locking effect, called in [49,26] extrapolation locking affecting both path-following and asymptotic analyses. ...

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 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.
Keywords: multilayer composite structures, mixed solid-shell, Koiter method,
post-buckling behavior, finite elements

... The mixed interpolation strategy, based on the same pattern as the ES-FEM, seems the most natural and effective approach in lower bound strategies for shakedown and limit analysis [29] and has proved to be the most effective approach to increase the accuracy and robustness of geometrically nonlinear analysis [27,28]. It is also worth mentioning that this simple assumption makes it possible to use self-equilibrated stress interpolations in Cartesian coordinates directly as stated in [34]. ...

... For application in plastic analysis [38], the method is expected to behave well as the discontinuous interpolation for the stress field addresses discontinuities in the plastic deformation field and the plastic admissibility is also imposed in a simple manner. Furthermore in geometrical nonlinear analysis, it has been shown [28,27] that the mixed format significantly reduces the nonlinearities of the problem, allowing fast and robust solution algorithms. ...

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.

... The element directly uses the equilibrium equations to improve the accuracy and reduce the number of interpolation parameters. A block elimination of the variables that do not require inter-element continuity allows, at the global level, only 6 þ n kinematical parameters per node to be exposed and the use of a pseudo-compatible format for the analysis [33,34]. As a final comment observe how the mixed model adopted here is particularly suitable for the extension to geometrically nonlinear analyses using corotational formulations such as those proposed in353637. ...

... At the global level a pseudo-compatible format, in terms of the standard nodal values fu½0; φ½0; μ½0; u½ℓ; φ½ℓ; μ½ℓg, is exploited. As the computational cost is closely related to the factorization of the pseudocompatible stiffness matrix, the finite element proposed has almost the same computational cost as the compatible one with the same global variables (see [33,41,34]).Fig. 21. ...

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 this section an asymptotic algorithm capable of treating single or multiple, also not coincident, bifurcations and of considering the effects of a nonlinear precritical behaviour is presented. Further details can be found in [16][17][18][19][20][21][22][23][24][25][26][27][28]. ...

... Mixed or compatible formats, while completely equivalent in principle, behave very differently when implemented in asymptotic but also in path-following solution strategies. This is an important, even if frequently misunderstood, point in practical computations which has been widely discussed in [8,13,14,18,19]. By referring readers to these papers for more details, we only recall here that both numerical strategies need function Φ and its Hessian K[u] to be appropriately smooth in its controlling variables. ...

The analysis of slender structures, characterized by complex buckling and postbuckling phenomena and by a strong imperfection sensitivity, is heavily penalized by the lack of adequate computational tools. Standard incremental iterative approaches are computationally expensive and unaffordable, while FEM implementation of the Koiter method is a convenient alternative. The analysis is very fast, its computational burden is of the same order as a linearized buckling load evaluation and the simulation of different imperfections costs only a fraction of that needed to characterize the perfect structure. In this respect it can be considered as a direct method for the evaluation of the critical and post-critical behaviour of geometrically nonlinear elastic structures. The main objective of the present work is to show that finite element implementations of the Koiter method can be both accurate and reliable and to highlight the aspects that require further investigation. © 2014 Springer Science+Business Media Dordrecht. All rights are reserved.

... A brief overview of the FEM implementation of Koiter's asymptotic approach is presented here, for the convenience of the reader and to summarize the main notation and equations involved. Further details can be found in [18][19][20][21][22][23][24][25][26][27][28][29][30][31]. ...

... Mixed or compatible formats, while completely equivalent in principle, behave very differently when implemented in asymptotic but also in path-following solution strategies. This is an important, even if frequently misunderstood, point in practical computations which has been widely discussed in [21,20,15,16,6]. By referring readers to these papers for more details, we only recall here that both numerical strategies need function and its Hessian d u to be appropriately smooth in its controlling variables. ...

The analysis of slender structures, characterized by complex buckling and
postbuckling phenomena and by a strong imperfection sensitivity, is heavily penalized
by the lack of adequate computational tools. Standard incremental iterative approaches are
computationally expensive and unaffordable, while FEM implementation of the
Koiter method is a convenient alternative. The analysis is very fast, its computational
burden is of the same order as a linearized buckling load evaluation and the
simulation of different imperfections costs only a fraction of that needed to
characterize the perfect structure. The main objective of the present work is to show
that finite element implementations of the Koiter method can be both accurate and
reliable and to highlight the aspects that require further investigation

... For these reasons, an efficient tool for computation of these derivatives is required. The isogeometric shell model described in this section with the patch-wise reduced integration rule proposed in [31] is accurate and efficiently solved exploiting the so called called Mixed Integration Point strategy [16,35], proposed in order to overcome the inefficiency of standard displacementbased finite element problems and then extended and tested in displacement-based isogeometric formulations [28,38,39]. ...

Numerical simulation based on FEM/IGA methods is the standard approach for the approximated solution of applied physical problems. In this context, the differentiation of the numerical counterpart of mechanical fields is required. Moreover, the differentiated function can have a complicated shape, depend on many variables and change within the process. Many state-of-the-art numerical differentiation methods are not suitable for this kind of applications and the common way is to exploit analytical differentiation. Thus, an on-the-fly differentiation method is desirable particularly when the process is complicated and when new mechanical models are under development. In this paper, a new method is proposed for a precise computation of the gradient and Hessian. This method has been applied to nonlinear analysis of Kirchhoff–Love shells, which can be considered as an appropriate test bench to prove the reliability in relevant physical context. Numerical experiments show the advantages of the proposed techniques with respect to standard approaches.

... Among these, are worth mentioning: constraint equations on the rate of variation of selected sets of DOFs (De Borst, 1987, May andDuan, 1997); on strain measures (Chen and Schreyer, 1990, Geers, 1999, Pohl et al., 2014; or quantities associated with the energy dissipation occurring in the system during the development of non-linearities , Verhoosel et al., 2009b, Lorentz and Badel, 2004, Singh et al., 2016, Stanić and Brank, 2017, Barbieri et al., 2017. Mixed path-following approaches in terms of stresses, strains, displacements, damage, and other variables were also proposed (Garcea et al., 1998, Formica et al., 2002, Bilotta et al., 2012, Magisano et al., 2017 to improve the convergence of standard path-following methods. ...

Damage, cracking, and strain localization mechanisms often lead to unstable structural responses characterized by snap-backs (i.e., force and displacement decrease simultaneously). Standard nonlinear Newton-based solution algorithms with displacement/force control cannot capture the equilibrium curve in its entirety. This can be overcome using path-following formulations. A general (i.e., valid for any finite element code) implementation framework can be designed and applied to the Cast3M software by collecting the essential concepts and the formalism of partitioned path-following arc-length algorithms. Thanks to these developments, Cast3M is now capable of processing path-following equations without any major modifications. Three path-following constraints were selected to demonstrate the applicability of this framework: a first one on the combination of the displacement increment at a given set of nodes, a second one on the maximum strain increment over the computational domain, and a third one on the maximum elastic predictor of the damage/plastic criterion function over the computational domain. Two-and three-dimensional strain localization simulations show that the proposed framework behaves in a stable and convergent manner, even when multiple severe snap-back instabilities are present. Users of Cast3M shall find the proposed study helpful in that it allows them to focus on developing new path-following equations for the software. Cast3M is developed by the French Alternative Energies and Atomic Energy Commission (CEA) and freely available for research purposes. The developments discussed in this paper have been made available to the user/developer community along with Cast3M 2021 (release date: June 2021).

... The equilibrium equations are solved through the arc-length method. More details about it are given in [42,43], while the second variation of the potential, which implicitly defines the tangent operator, is reported in Appendix B. ...

The paper deals with a numerical model for hexagonal boron nitride nanostructures. The model is developed in finite kinematics and takes into account binary, ternary and quaternary atomic interactions, described through the UFF potential with an ad-hoc set of parameters. Once validated with respect to ab-initio results of the tensile failure of boron nitride sheets under periodic conditions, the model is used to study the response of a sheet of finite size under traction, compression and shear.

... In order to obtain the equilibrium paths describing both stable and unstable equilibrium states, the adopted numerical strategy implements a method based on the Riks scheme [9] thus allowing to follow the pre-and postcritical equilibrium paths. Such a method has been validated in several previous works and efficiently emplyed in nonlinear shell problems (see, e. g., [10,11]).In addition, for the finite element implementation a non-standard formulation is here adopted, namely, the mixed solid-shell finite element, first proposed by Sze et al. in 1993 [12], and later developed and utilized in composite shells in [13,14]. ...

... Although the finite element method can be identified as a different version of the assumed modes method, it can be generalized to be used in a wider context, in particular, when nonlinear effects arise as for a multilink manipulator [Sharf 1996;Eugster et al. 2014;Luongo and D'Annibale 2013]. To address some issues related to failures in convergence that are occasionally experienced, some authors have proposed a mixed formulation, based on both stress and displacement degrees of freedom, which appears very promising in this respect [Hodges 1990;Garcea et al. 1998]. However, this reformulation of the problem involves a greater complexity of modeling. ...

In this paper, a discrete model is adopted, as proposed by Hencky for Elastica based on rigid bars and lumped rotational springs, to design the control of a lightweight planar manipulator with multiple highly flexible links. This model is particularly suited to deal with nonlinear equations of motion as those associated with multilink robot arms, because it does not include any simplification due to linearization, as in the assumed modes method. The aim of the control is to track a trajectory of the end effector of the robot arm, without the onset of vibrations. To this end, an energy-based method is proposed. Numerical simulations show the effectiveness of the presented approach.

... The equilibrium paths and the strain and stress tensors are numerically evaluated through custom-written computer programs, implemented in MATLAB language, while the output results are displayed through the Gmsh open source post-processor [55]. The non-linear equilibrium paths are obtained through the Riks arc-length incremental iterative strategy [53,54], with the implementation details given in [56]. The tangent operator for the numerical solution of Eq.(12) is given in Appendix B. Fig. 6 shows the reference configuration of a specimen of SLGS subjected to two different periodic boundary loading conditions which we will refer to as zigzag and armchair tensile test, respectively. ...

The tensile behaviour and the pure shear behaviour of pristine and perforated single-layer graphene sheets are numerically investigated through a stick-and-spring model including both material and geometric non-linearities. The model is formulated in finite kinematics and the atomic interactions are modelled through the modified Morse potential, tuned with an improved set of parameters. The progression of the failure process of the sheets is numerically reconstructed using the arc-length strategy. The failure profiles are displayed and discussed. A continualization of the obtained results is made. The engineering strains and stresses and the second Piola and Green–Lagrange tensors are computed and compared with results given in the literature.

... Lo schema path-following alla Riks [6,7]è attualmente la strategia di analisi più efficace e robusta per l'analisi evolutiva di strutture a comportamento nonlineare. Lo schema considera un percorso di carico ...

Slides del Corso di Formazione "Introduzione all'analisi nonlineare" tenuto presso l'ordine degli Ingegneri di Genova, gennaio 2019

... Moreover, being intrinsically nonlinear, it does not introduce any approximation due to some kind of linearization [21,22]. Although the finite element method converges faster than that of lumped parameters and requires a lower number of degrees of freedom to obtain the same accuracy, to deal with nonlinear cases, it is necessary to make use of a specific, rather complex formulation, which could be familiar to experts of computational mechanics, but less accessible to a larger audience (see, e.g., [23]). It should be noted, indeed, that the commercial FEM codes are rather lacking in dealing with the problem of large deflections of beams, to the best of the authors' knowledge. ...

The problem of the trajectory-tracking and vibration control of highly flexible planar multi-links robot arms is investigated. We discretize the links according to the Hencky bar-chain model, which is an application of the lumped parameters techniques. In this approach, each link is considered as a kinematic chain of rigid bodies, and suitable springs are added in order to model bending resistance. The control strategy employed is based on an optimal input pre-shaping and a feedback of the joint angles to treat the effects of undesired disturbances. Some numerical examples are given to show the potentialities of the proposed control, and a comparison with a standard collocated Proportional-Derivative (PD) control strategy is performed. In particular, we study the cases of a linear and a parabolic trajectory with a polynomial time law chosen to minimize the onset of possible vibrations.

... representing a circumference of radius Dn and center in the initial point of the step {u (k) ,a (k) }, K and l being suitable metric factors (Riks 1992;Garcea et al. 1998). At each step of the analysis, starting from a trial solution {u j , a j } corresponding to the value of Dn, the unknown parameters {u (k?1) ,a (k?1) } can be evaluated through the Newtonian scheme ...

The adoption of a reliable shear model for predicting brittle failure modes of a reinforced concrete (r.c.) beam-column joint, beyond ductile flexural mechanisms at member level, is essential to retrofit r.c. framed buildings properly. The retrofitting of existing structures by means of the insertion of hysteretic damped braces (HYDBs) turns out to be a highly effective means of improving seismic response. In the present work, a Displacement-Based Design (DBD) procedure to proportion the HYDBs to attain, for a specific level of seismic intensity, a designated performance level has been revisited in order to take into account the effects of the nonlinear shear response of beam-column joints. To this end, two-, four- and eight-storey r.c. framed structures, representative of low-, mid- and high-rise r.c. framed buildings, are designed in line with a former Italian seismic code for a medium-risk seismic zone. These are then to be retrofitted by inserting HYDBs to attain performance levels imposed by the current Italian code in a high-risk seismic zone. A computer code for the nonlinear static analysis of r.c. framed structures has been developed, involving local shear response of beam-column joints. A path-following analysis based on the arc-length method has been adopted to obtain the pushover curves of primary and retrofitted test structures, with and without nonlinear shear modelling of the beam-column joints, and the HYDB response is idealized by a bilinear law on the assumption that buckling of the steel braces is prevented.

... Successivamente (1998) si realizzò che l'inconveniente era dovuto ad un fenomeno di locking (locking da estrapolazione) che inficiava in realtà la condizione Kt < 2K . ...

Testo della presentazione tenuta in Roma presso l’Ordine degli Ingegneri della Provincia

... Eq. (21) can be solved using standard path-following techniques [46,47,27] for an assigned imperfectionũ. Note that in the hybrid solid-shell FE model, the internal force vector of the imperfect structure is obtained by simply subtracting a constant vectorp, evaluated once and for all at the beginning of the analysis, to the internal forces vector s[u] of the perfect structure. ...

The Koiter method recovers the equilibrium path of an elastic structure using a reduced model, obtained by means of a quadratic asymptotic expansion of the finite element model. Its main feature is the possibility of efficiently performing sensitivity analysis by including a-posteriori the effects of the imperfections in the reduced non-linear equations. The state-of-art treatment of geometrical imperfections is accurate only for small imperfection amplitudes and linear pre-critical behavior. This work enlarges the validity of the method to a wider range of practical problems through a new approach, which accurately takes into account the imperfection without losing the benefits of the a-posteriori treatment. A mixed solid-shell finite element is used to build the discrete model. A large number of numerical tests, regarding non-linear buckling problems, modal interaction, unstable post-critical and imperfection sensitive structures, validates the proposal. This article is protected by copyright. All rights reserved.

... Il vantaggio dello schema di Riks risiede, in definitiva, nella presenza del "filtro" [I − ω j B j ] (per maggiori dettagli si veda [67]). Nelle vicinanze dei punti limite,û tende al modo critico (quel modo, cioè, che rende singolare la matrice) e l'effetto filtro consiste proprio nel troncare le componenti di r j in questa direzione. ...

Progetto MECOM, Programma Operativo Plurifondo 94/99 Misura 4.4 " Ricerca scientifica e tecnologica Sviluppi ed applicazioni della meccanica computazionale nella progettazione strutturale in campo civile ed industriale ". Analisi nonlineare di pannelli murari soggetti a fenomeni di tipo fessurativo G. Formica, R. Casciaro

... [89] ...

The paper describes different computational approaches and solution methodologies that can be used in nonlinear structural analysis, in particular, the so called path–following anal-ysis, the linearized stability analysis, the asymptotic analysis, the imperfection sensitivity analysis and the transient dynamic analysis, for each showing the main problems, peculiar aspects, possible failures and computational convenience. Nonlinear solutions are, by their nature, sensitive to small variations in data, so a performance–based analysis must include an extensive investigation, which takes into account all possible loading imperfections and geometrical defects. Great care has to be taken to assure the reliability of the results and, if possible, any analysis should be repeated using an alternative approach.

We present a novel geometrically nonlinear EAS element with several desirable features. First, the Petrov‐Galerkin ansatz significantly improves the element's performance in distorted meshes without loosing the simple strain‐driven format. Second, the recently proposed mixed integration point strategy is employed to improve the element's robustness in the Newton‐Raphson scheme. Finally and most importantly, we enhance the spatial displacement gradient instead of the usually modified deformation gradient. This allows to construct an element without the well‐known spurious instabilities in compression and tension as shown numerically and supported by a corresponding hypothesis. All in all, this leads to a robust, stable, locking‐free and mesh distortion insensitive finite element successfully applied in a wide range of examples.

The computational efficiency of an enhanced version of a pseudo-arclength pathfollowing scheme tailored for general multi-degree-of-freedom (multi-dof) nonlinear dynamical systems is discussed. The pathfollowing approach is based on the numerical computation of the Poincaré map and its Jacobian in order to tackle nonautonomous systems with discontinuous vector fields. The scheme is applied to obtain frequency response curves of multi-dof hysteretic systems with a state vector size up to 120, as well as various reduced-order models of single and multiple cantilever beams on a shuttle mass. The proposed approach is shown to drastically increase the speed of convergence in the modified Newton–Raphson scheme thanks to a Krylov sub-space iteration which makes use of the LU decomposition of a frozen Jacobian matrix, which, upon convergence, becomes the monodromy matrix. Several numerical tests performed on mechanical systems with material or geometric nonlinearities corroborate the efficiency of the numerical strategy.
The C++ code implementing the proposed methodology is freely available at https://doi.org/10.5281/zenodo.6616482.

Collection of the Proceedings of the Third Edition of the International Workshops MULTISCALE INNOVATIVE MATERIALS AND STRUCTURES. Cetara, Amalfi Coast - February 28-March 2, 2019

The enhanced assumed strain (EAS) method is one of the most frequently used methods to avoid locking in solid and structural finite elements. One issue of EAS elements in the context of geometrically non‐linear analyses is their lack of robustness in the Newton‐Raphson scheme, which is characterized by the necessity of small load increments and large numbers of iterations. In the present work we extend the recently proposed mixed integration point (MIP) method to EAS elements in order to overcome this drawback in numerous applications. Furthermore, the MIP method is generalized to generic material models, which makes this simple method easily applicable for a broad class of problems. In the numerical simulations in this work, we compare standard strain based EAS elements and their MIP improved versions to elements based on the assumed stress method in order to explain when and why the MIP method allows to improve robustness. A further novelty in the present work is an inverse stress‐strain relation for a Neo‐Hookean material model.

Seconda parte delle slides del corso di approfondimento tenuto presso l'Ordine degli Ingegneri di Cosenza nei giorni 27 e 28 febbraio 2020

A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in a FEM context, is presented. From the optimization point of view, standard arc–length strain driven elastoplastic analysis, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other nonlinear programming methods, simpler convergence proofs and duality to be exploited. Due to the unified approach in terms of total stresses, the strain driven algorithms become more effective and less nonlinear with respect to a self equilibrated stress formulation and easier to implement in existing codes performing elastoplastic analysis.

Bends are an integral part of a piping system. Because of the ability to ovalize and warp they offer more flexibility when compared to straight pipes. Piping Code ASME B31.3 [1] provides flexibility factors and stress intensification factors for the pipe bends. Like any other piping component, one of the failure mechanisms of a pipe bend is gross plastic deformation. In this paper, plastic collapse load of pipe bends have been analyzed for various D/t ratios (Where D is pipe outside diameter and t is pipe wall thickness) for internal pressure and in-plane bending moment, internal pressure and out-of-plane bending moment and internal pressure and a combination of in and out-of-plane bending moments under varying ratios. Any real life component will have imperfections and the sensitivity of the models have been investigated by incorporating imperfections as scaled eigenvectors of linear bifurcation buckling analyses. The sensitivity of the models to varying parameters of Riks analysis (an arc length based method) and use of dynamic stabilization using viscous damping forces have also been investigated. Importance of defining plastic collapse load has also been discussed. FE code ABAQUS version 6.9EF-1 has been used for the analyses.

A new iterative algorithm to evaluate the elastic shakedown multiplier is proposed. On the basis of a three field mixed finite element, a series of mathematical programming problems or steps, obtained from the application of the proximal point algorithm to the static shakedown theorem, are obtained. Each step is solved by an Equality Constrained Sequential Quadratic Programming (EC-SQP) technique that retain all the equations and variables of the problem at the same level so allowing a consistent linearization that improves the computational efficiency. The numerical tests performed for 2Dproblems showthe good performance and the great robustness of the proposed algorithm. © Springer International Publishing Switzerland 2015.

Over the last decades powerful numerical methods have been developed to carry out one of the oldest and most important tasks of design engineers, which is to determine the load carrying capacity of structures and structural elements. Particularly attractive among these methods are the so-called “Direct Methods”, embracing Limit—and Shakedown Analysis because they allow rapid and direct access to the requested information in mathematically constructive manners without cumbersome step-by-step computation.
This collection of papers is devoted to this subject. It is the outcome of a workshop hosted by the University of Reggio Calabria in October 2013, in line with previous workshops at RWTH-Aachen University, University of Technology and Sciences of Lille, and National Technical University of Athens and give an excellent insight into the state of the art in this broad and growing field of research.
The individual contributions stem namely from the areas of new numerical developments rendering the methods more attractive for industrial design, extensions of the general methodology to new horizons of application, probabilistic approaches and specific technological applications. The papers are arranged in the
order as presented in the workshop.
It might be worth noting that the success of the workshops and the growing interest in Direct Methods in the scientific community were motivations to create the association IADiMe (http://www.iadime.unirc.it/) as a platform for exchange of ideas, advocating scientific achievements and not least, promotion of young scientists working in this field. It is open for all interested researchers and engineers.
The editors warmly thank all the scientists who have contributed by their outstanding papers to the quality of this edition.
—We hope you enjoy reading it!
Reggio Calabria, August 2014 Paolo Fuschi
Reggio Calabria Aurora Angela Pisano
Aachen Dieter Weichert

The lectures provide an introduction to the computational treatment of Koiter’s asymptotic strategy for post-buckling analysis of thin elastic structures.
The analysis of slender structures characterized by complex buckling and post-buckling phenomena and by a strong imperfection sensitivity, suffers from a lack of adequate computational tools. Standard algorithms, based on incremental-iterative approaches, are computationally expensive: it is practically impossible to perform the large number of successive runs necessary for the sensitivity analysis, that is, to evaluate the reduction in load due to all possible imperfections. Finite element implementations of Koiter’s perturbation method give a convenient alternative for that purpose. The analysis is very fast, of the same order as a linearized stability analysis and new analyses for different imperfections only require a fraction of the first analysis time.
The main objective of the present course is to show that finite element implementations of Koiter’s method can be both accurate and reliable.

The buckling and post-buckling analysis of elastic planar frames is considered and the use of geometrically exact beam models is thereby advocated. It is shown that usual technical beam models fail to predict correctly the curvature of the post-buckling curve at bifurcation even for standard problems of elastic stability theory. It is also argued that versatile and efficient computational procedures for bifurcation analysis of general planar frames are to be based on unconstrained beam models. Some remarks on finite element representation of nonlinear beam models are passed in conclusion.

This paper refers to the analysis of the postbuckling behaviour of thin-walled structures by means of an asymptotic approach based on a finite element implementation of Koiter's non-linear theory of instability.The analysis has been accomplished by using the following assumptions: (i) the structure is described as an assemblage of flat slender rectangular panels; (ii) a non-linear Kirchhoff-type plate theory is used to model each panel; (iii) HC finite elements discretization is used; (iv) linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths, respectively; (v) multimodal buckling is considered; and (vi) imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest– descent path criterion.Several numerical results are presented and discussed. Comparisons with numerical solution obtained by standard incremental codes are given, which show the accuracy and reliability of the proposed approach.

It is known that the evolutional analysis of nonlinear structures can be conveniently accomplished by incremental step by step procedures.Two fundamental problems take place in this approach:a) the step by step incremental scheme that can be used;b) the way in which the nonlinear implicit equations recurring in each step of the incremental process can be solved.The latter problem is discussed in this paper. In particular, through some very broad hypotheses on the structural behavior, a class of convergent iterative methods is shown (a direct extension of the well-known initial stress method is a member of this class).Furthermore, an improved iterative method is proposed, which utilizes all information obtained in the process and proves to be very convenient for the user. ben noto come l'analisi evolutiva di structure non lineari possa essere convenientemente eseguita mediante processi incrementalistep by step.In tale approccio sono presenti due problemi fondamentali:a) quale procedimentostep by step pu essere usato?;b) come possono essere risolte le equazioni implicite non lineari ricorrenti a ogni passo del processo incrementale?Nel presente lavoro viene affrontato questo secondo problema. In particolare, sulla base di ipotesi estremamente larghe sul comportamento della struttura, viene mostrata una classe di metodi iterativi convergenti di soluzione (una diretta estensione del ben notoinitial stress method membro di tale classe).Viene inoltre discusso un miglioramento dello schema iterativo, che consente di tener conto di tutte le informazioni che man mano vengono raccolte nel corso delle iterazioni e risulta particolarmente conveniente per l'utente.

This paper is focussed on path following methods which are derived from consistent linearizations. The linearization procedure leads to some well-known constraint equations—like the constant arc length in the load-displacement space—and to different formulations than those given in the literature. A full Newton scheme for the unknown quantities (displacements and load parameter) can be formulated. A comparison of the derived algorithms with other path following methods is included to show advantages and limits of the methods.Using the linearization technique together with scaling a family of path following methods is introduced. Here, the scaling bypasses physical inconsistencies associated with mixed quantities like displacements and rotations in the global vector of the unknowns. Several possible scaling procedures are derived from a unified formulation. A discussion of these methods by means of numerical examples shows that up to now the choice of the scaling procedure is problem-dependent.If the arc-length methods are combined with a modified Newton method, an enhancement of the algorithms is achieved by line search techniques. Here, a simple but efficient line search was implemented and compared with a numerical relaxation technique. Both methods improve the convergence rate considerably.

The modification proposed concerns the search and computation of critical points, leading to the method studied. A number of experimental computations were carried out to evaluate the merits of the modification suggested. The results of the tests show the feasibility and potential of the calculations presented.

In this paper we discuss new strategies for path-following algorithms. For this purpose we derive, based on a parametrization of the solution path, a refined, which means a quadratical, predictor step. Under certain assumptions the necessary higher order derivations can be calculated for the associated finite element formulation in an efficient and simple way. Alternatively a numerical differentiation procedure is introduced. The examples show that based on the refined predictor a reduction of 20–30% of corrector iterations is possible. Furthermore the derived algorithm is much more robust in critical situations.

The numerical solution of problems of elastic stability through the use of the iteration method of Newton is examined. It is found that if the equations of equilibrium are completed by a simple auxiliary equation, problems governed by a snapping condition can, in principle, always be calculated as long as the problem at hand is properly formulated. The effectiveness of the proposed procedure is demonstrated by means of an elementary example.

This Note presents the results of an investigation of stability of nonshallow circular arches whose one end is clamped and the other hinged. The analysis is based on Euler's nonlinear theory of the inextensible elastica. This theory is exact in the sense that no restrictions are placed on the magnitudes of deflections. The results presented herein were obtained by means of the nonlinear equations.

The perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena. This paper presents a summary of the principles involved and describes some trends in the development of these techniques.

For the prebuckling range an extensive literature of effective solution techniques exists for the numerical solution of structural problems but only a few algorithms have been proposed to trace nonlinear response from the pre-limit into the post-limit range. Among these are the simple method of suppressing equilibrium iterations, the introduction of artificial springs, the displacement control method and the “constant-arc-length method” of Riks/Wempner. It is the purpose of this paper to review these methods and to discuss the modifications to a program that are necessary for their implementation. Selected numerical examples show that a modified Riks/Wempner method can be especially recommended.

Interpolation and extrapolation are employed to approximate the fields in a nonlinear theory of solid bodies. Nodal points are employed in the space of position and load, and the continuous fields are essentially replaced by nodal values. Interpolation between nodes (extrapolation in load) defines a continuous approximation. The differential equations of the continuum are replaced by algebraic equations of the discrete system. Nonlinear equations are replaced by a succession of linear equations, as a nonlinear path is approximated by linear segments. Variational theorems are used as the bases of the algebraic formulations which govern the discrete approximation. The algebraic equations are related to their differential counterparts. A generalized arc-length is introduced in the configuration-load space in order to facilitate the incremental computations near limit points. The arc-length is used as the loading parameter in some illustrative problems. An appendix describes the viewpoint of finite elements and the continuity conditions which insure the equivalence of the methods.

A new solution procedure is presented, based on the arc-length method, for passing limit points (load or displacement peaks) in nonlinear finite element analysis of structures. In addition to the usual equilibrium equations, a quadratic arc-length constraint equation is specified so that the nonlinear solution is sought on a small ellipsoidal surface in load-deflection space. The main new feature of the proposed procedure is the resolution of the out-of-balance loads into a parallel and an orthogonal component with respect to the vector of applied external loads. Methods for avoiding complex roots to the quadratic constraint equation, and for selecting the appropriate root, are presented. Applications of the proposed procedure to two reinforced concrete problems give satisfactory results. The method is effective and versatile in handling both snap-through and snap-back problems.

The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + 1} \to R^n $, for given $b \in R^n $. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third o...

This paper summarizes a part of the first author's Ph.D. Thesis completely devoted to multimode elastic buckling within an FEM strategy. The theoretical arguments unfold among critical points on radial paths (the unique post-critical paths variationally defined), algebraic characterizations, proposition demonstrations and so on, by aiming to prove that the complexity of the phenomenon of multimode buckling (secondary bifurcations, post-critical attractive paths) can be theoretically explained. © 1997 by John Wiley & Sons, Ltd.

Shells of revolution subject to axisymmetric loads often fail by non-symmetric bifurcation buckling after non-linear axisymmetric deformations. A number of computer programmes have been developed in the past decades for these problems, but none of them is capable of bifurcation analysis on the descending branch of the primary load–deflection path following axisymmetric collapse/snap-through. This paper presents the first finite element formulation of post-collapse bifurcation analysis of axisymmetric shells in which a modified arc-length method, the accumulated arc-length method, is developed to effect a new automatic bifurcation solution procedure. Numerical examples are presented to demonstrate the validity and capability of the formulation as well as the practical importance of post-collapse bifurcation analysis. The accumulated arc-length method proposed here can also be applied to the post-collapse bifurcation analysis of other structural forms. © 1997 by John Wiley & Sons, Ltd.

The paper describes how several procedures for higher-order predictions have been introduced in order to improve the convergence speed in a general finite element program for non-linear structural analysis. In addition to higher-order Taylor expansions earlier discussed, Lagrangian extrapolations and some methods commonly used for the integration of initial value problems have been introduced. The methods are used for improved predictions in the stepwise solution of equilibrium states and for accurate descriptions of the initial post-bifurcation behaviour. They are used in a general solution algorithm, based on a parameter formulation. The methods are discussed in the light of the strategies for re-creation of the tangential stiffness matrix, used for equilibrium iterations.
Numerical examples, exhibiting different limit and bifurcation behaviours for trusses, frames and shells, are used to evaluate the numerical properties and efficiencies of the methods.
The paper concludes that the overall efficiency in the algorithm can be improved by introduction of more accurate predictions than the standard Euler prediction. In terms of reliability combined with efficiency, an implicit generalized Simpson method is the preferred method.

This paper describes a method for introducing line searches into the arc-length solution procedure. Such line searches may be used at each iteration to calculate an optimum scalar step-length which scales the normal iterative vector. In practice, a loose tolerance is provided so that on many iterations the line searches are avoided. However on ‘difficult iterations’, the line searches are shown to lead to a substantial improvement in the convergence characteristics. A simple single-parameter acceleration is also developed using line search concepts. The new arc-length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs. Significant improvements are demonstrated in relation to the standard arc-length method.

The present paper extends the finite element perturbation approach already presented for pin-jointed and framed structures15 to rectangular thin plates. Koiter's asymptotic strategy is coupled with a High-Continuity finite element discretization of the plate. The consistency of the discrete model is discussed from the kinematical and numerical points of view and several numerical tests are reported. It appears that use of the HC elements makes the perturbation algorithm insensitive to the locking phenomenon occurring in the evaluation of the postbuckling behaviour. It also allows the use of very fine discretization meshes at low computational cost.

A unified presentation of some popular continuation procedures used in the non-linear finite element analysis of structural mechanics is introduced. An extension of the elliptical constraint equation proposed by Crisfield is given. It is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter. Updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur. For bifurcation points a modification of Rheinboldt's branching procedure is presented. Post critical response after limit and bifurcation points is determined in some numerical examples.

The paper discusses the introduction of constraining equations in the tangential stiffness relation used to calculate the responses to different load cases in solution algorithms for non-linear mechanical Finite Element (F.E.) problems. An alternative to the normal two-phase solution method is discussed. This method is used to represent different iteration constraints, and in conjunction with the search for critical solution points. Numerical tests are presented, evaluating the efficiency of different iteration constraints for a model problem. Practically useful criteria for critical points are discussed. The basic methods for search of such points and some numerical aspects are discussed and evaluated for three different problems.

An adaptive procedure for selecting the step size when incremental or continuation methods are used to solve sets of non-linear equations is presented. The increment size is limited by requiring the corrective iteration procedure employed to reduce the drifting error to be within a contractive boundary at each level. The usefulness of the procedure is extended by the development of a set of conditions for detecting impending divergence of the corrective iteration process. These conditions, used in conjunction with the step size selection procedure permit the continuation of the solution through highly non-linear regions and also provide a simple means of isolating a limit point, if one exists. Two additional benefits of the procedure are an effective convergence criterion for terminating the iteration process and a simple means for switching between the Newton-Rephson and modified Newton-Raphson iteration procedures. The paper concludes with a number of example problems, three are hyperelastic bodies at finite strain and the final example is the large displacement analysis of an elastic beam. The results illustrate that the procedure is computationally inexpensive, using only information normally obtained in a non-linear analysis, flexible in the sense that a dense or sparse distribution of points along the total solution curve may be obtained, and effective, normally requiring less total computational effort than a constant step size procedure.

An asymptotic method directly derived from Koiter's theory and suitable for the solution of elastic buckling problems and its natural adaptation to a numerical solution by means of a finite element technique are presented here. The order of the extrapolation of the equilibrium equations has been intentionally kept very low because attention has been entirely devoted to all those features (theoretical definitions, eigenproblem numerical techniques, suitable FEM implementation) which make such an approach competitive with respect to the classic step-by-step methods. For plane frames and 3D pin-jointed trusses, the performances of the algorithm (numerical accuracy and computational cost) are compared with those of Riks' are-length method.

The paper describes a study of incremental-iterative solution techniques for geometrically non-linear analyses. The solution methods documented are based on a modified Newton-Raphson approach, meaning that the tangent stiffness matrix is computed at the commencement of each load step but is then held constant throughout the equilibrium iterations. A consistent mathematical notation is employed in the description of the iterative and load incrementation strategies, enabling the simple inclusion of several solution options in a computer program. The iterative strategies investigated are iteration at constant load, iteration at constant displacement, iteration at constant ‘arc-length’, iteration at constant external work, iteration at minimum unbalanced displacement norm, iteration at minimum unbalanced force norm and iteration at constant ‘weighted response’. The load incrementation schemes investigated include strategies based on the number of iterations required to achieve convergence in the previous load step, strategies based on the ‘current stiffness parameter’ and a strategy based on a parabolic approximation to the load-deflection response. Criteria for detecting when the applied external load increment should reverse sign are described.A challenging example of a circular arch exhibiting snap-through (load limit point) behaviour and snap-back (displacement limit point) behaviour is solved using several different iterative and load incrementation strategies. The performance of the solution schemes is evaluated and conclusions are drawn.

A solution strategy for the analysis of nonlinear structures is described. The strategy is a simple extension of existing Newton-type procedures, and can easily be incorporated into existing computer programs.
Earlier work which contributed to the development of the strategy is reviewed and the theory of the procedure is presented. Six examples, covering several different types of structural behaviour are described. These examples suggest that the strategy is remarkably stable and efficient.

The stability of symmetric structural systems with first order and second order imperfections is investigated. The analysis concerns a discrete system with n degrees of freedom. Resulting expressions relating reduction in critical load to the magnitudes of the imperfections indicate that a complex interaction exists between these quantities. Second order imperfections have a relatively minor effect on the critical load. Experiments on a rigidly jointed roof truss are discussed. The experimental results show good qualitative agreement with theory.РефератИccлeдyeтcя ycтoйчивocть cиммeтpичecкич cтpoитeльныч cиcтeм c нeтoчнocтями пepвoгo и втopoгo poдa. Aнaлиз кacaeтcя диcкpeтнoй cиcтeмы c n cтeпeнями cвoбoды. Peзyльтиpyющиe выpaжeния, кacaющиecя yмeньшeния кpитичecкoй нaгpyзки oтиocитeльиo вeличин нeтoчиocтeй, yкaзывaют нa cyщecтвoвaииe кoмплeкcнoгo взaимoдeйeтвия мeждy зтими вeличинaми, Heтoчнocти втopoгo poдa имeют oтнocитeльнo мeньшиe влияниe нa кpитичecкyю нaгpyзкy, Oбcyждaютcя зкcпepимeнты нa жecткo coeдинeaныч cтpoпильныч пepeкpытяч. зкcпepимeнтaльныe peзyльтaты пoкaзывaют чopoмyю кaчecтвeннyю cчoдимocть c тeopиeй.

The matrix displacement analysis of geometrically nonlinear structures becomes an intricate task as soon as finite elements in space with rotational degrees of freedom are considered. The fundamental reason for these difficulties lies in the non-commutativity of successive finite rotations about fixed axes with different directions. In order to circumvent this difficulty, a new definition of rotations — the so-called semitangential rotations — is introduced in this paper. Our new definition leads to a reformulation of the theory of [1,2]which in itself is clearly consistent and correct.In contrast to rotations about fixed axes these semitangential rotations which correspond to the semitangential torques of Ziegler [3]possess the most important property of being commutative. In this manner, all complexities involved in the standard definition of rotations are avoided ab initio.A specific aspect of this paper is a careful exposition of semitangential torques and rotations, as well as the consequences of the semitangential definitions for the geometrical stiffness of finite elements. In fact, these new definitions permit a very simple and consistent derivation of the geometrical stiffness matrices. Moreover, the semitangential definition automatically leads to a symmetric geometrical stiffness which clearly expresses that the nonlinear strain-displacement relations must satisfy the condition of conservativity of the structure itself — independently of any loading.The general theory of geometrical stiffness matrices as evolved in this paper is applied to beams in space. The consistency of the theory is demonstrated by a large number of numerical examples not only of straight beams but also of the lateral and torsional buckling and post-buckling behaviour of stiff-joined frames. Most of the former developments appear to be inadequate.

Concerning the incremental strategies to solve the nonlinear set of algebraic equations deriving from the application of the finite element method, this paper deals with the arc-length-type methods. Two modified Risks-Wempner methods and a new algorithm to choose the appropriate root of a nonlinear constraint equation are presented. A geometrical interpretation of the consistent linearization method is also provided and on the basis of a quadratic constraint equation, a new arc-length-type method is proposed, which consistent linearization leads to both the Riks-Wempner and to the Riks-Wempner-Ramm methods as particular cases. To compare the proposed methods to other existing ones, numerical results are quoted by the use of the simplest nonlinear model for which a limit point exists, i.e. the equation for a parabola. Furthermore a discussion is presented concerning the interpretation of the typical operation failures which happen in the application of the several methods: no convergent solution, oscillations, snap-back, complex roots of Crisfield's constraint equation, etc. These failures are related to the following causes: 1.(I) failure of the constraint-type-equation2.(II) failure of the linearization of the equilibrium equations3.(III) failure of the linearization of the constraint equation4.(IV) failure related to the computer fixed precision and/or to the reliability of the numerical algorithms.

The path following technique known as ‘The Arc Length Method’ has evolved over the past decade into a commonly used tool in nonlinear finite element analysis. Variations of this method have orginated from a number of other workers. This paper presents a new general formulation for all arc length procedures. A derivation is given based on orthogonality principles which provides a new perspective illustrating the relationship between the existing and proposed path following techniques. A simplified procedure, obtained directly from the new general formulation, provides the same results as Crisfield's explicit iteration procedure (Comput. Struct.13, 55–62 (1981)) with a reduction in computational effort. Practical application of this theory is demonstrated using a numerical example of a reticulated shell structure.

The arc-length method has been extensively applied in the last decade to trace solution paths in nonlinear static structural problems. The sign of the loading parameter increment in the predictor phase of the method, however, must be determined according to some criteria which are not well established. By further investigation of the approach within the framework of continuation methods in this paper, a new criterion has been proposed, which can be incorporated into both direct and iterative solvers, and has proved to be very successful for the test problems considered.

The paper deals with the basic requirements in the construction of a reliable continuation procedure. Adaptive step length determination and calculation of critical equilibrium states are discussed. For simple critical points an algorithm, which does not need classification between different types of bifurcations or even distinction between limit vs bifurcation point, is described. Situations where the extension of the parameter space could reveal vital information concerning the behaviour of the structure being analysed, are addressed.

An algorithm for the automatic incremental solution of nonlinear finite element equations in static analysis is presented. The procedure is designed to calculate the pre- and post-buckling/collapse response of general structures. Also, eigensolutions for calculating the linearized buckling response are discussed. The algorithms have been implemented and various experiences with the techniques are given.

An alternative to the Riks-Wempner-Crisfield iterative correction scheme is presented that does not require an explicit displacement-load accession path to the nonlinear equilibrium curve, nor a known equilibrium point. Its symmetry with respect to the displacement and load assures success in rounding limit points as well as turning points.

This paper is concerned with the numerical solution of systems of equations of discrete variables, which represent the nonlinear behaviour of elastic systems under conservative loading conditions. In particular, an incremental approach to the solution of buckling and snapping problems is explored.The topics that are covered can be summarized as follows:—The computation of nonlinear equilibrium paths with continuation through limit points and bifurcation points.—The determination of critical equilibrium states.Characteristic to the procedures employed is the use of the length of the equilibrium path as control parameter. This feature, together with the second order iteration method of Newton, offers a reliable basis for the procedures described. Actual computations, carried out on a finite element model of a shallow circular arch, illustrate the effectiveness of the methods proposed.

A nonlinear finite element of plane beam oriented to Koiter's perturbation analysis of frames is presented. The locking problem arising in the calculation of the post-critical curvature is thoroughly examined and solved at the continuum level, by means of a suitable definition of the kinematic settings. The proposed strain measures are Lagrangian and rationally founded (geometrically exact). The exact solutions of the critical problem have been chosen as shape functions of the finite element, so obtaining an immediately convergent finite element in the calculation of the critical load and of the post-critical slope.