O crescente uso de lajes sem vigas (lisa com ou sem capitel) como solução estrutural de edificações, justificado por questões econômicas e construtivas, têm destacado a importância do estudo desses elementos estruturais. Já as desvantagens desse sistema estrutural estão associadas às altas tensões geradas na região próxima dos pilares, originando o fenômeno de punção. As normas atuais especificam uma seção íntegra resistente para dimensionamento à punção, entretanto, devido principalmente à passagem de instalações hidráulicas e elétricas através dos pavimentos, é necessária a execução de furações, em alguns casos muito próximas ao contorno dos pilares, o que implica diretamente na redução dessa seção e da resistência das lajes. Portanto, são necessários modelos de cálculo específicos para isso. A NBR 6118 tem critérios para esse problema, mas são antigos e precisam ser modernizados. Assim, comparando os dados experimentais de algumas pesquisas com as cargas últimas estimadas para normas vigentes (fib MC2010:2013, EC2:2004, ACI-318:2014 e NBR 6118:2014), este trabalho teve como objetivo identificar limitações das normas estudadas e verificar a eventual possibilidade de melhorias. Com base nos resultados observados foi proposta uma alteração para a NBR 6118: 2014 no cálculo da punção para lajes lisas com furo(s) na região do pilar, de modo que seja considerado no dimensionamento o momento gerado pela excentricidade do perímetro crítico devido à presença de furo(s) na laje. / The increasing use of flat slabs (slabs without beams with or without drop panel) as a structural solution of buildings, justified by economic and constructive issues, has highlighted the importance of the study of these structural elements. The disadvantages of this structural system are associated with high stresses generated in the region near of the columns, causing the punching shear phenomenon. The current standards specify a resistant section for punching shear design, however, due mainly to the passage of hydraulic and electric installations through the floors, it is necessary to execute holes, in some cases very close to the boundaries of the columns, which reduces this section and the slab strength. Therefore, it is necessary to specify calculation models for that. The NBR 6118 has criteria for this problem, nonetheless it is old and needs to be modernized. Comparing experimental data from some researches, with the estimated ultimate loads for current standards (fib MC2010: 2013, EC2: 2004, ACI-318: 2014 and NBR 6118: 2014), this work had as objective to identify limitations of the standards studied and verify possible improvements. Based on the observed results, it was proposed a modification for NBR 6118: 2014 in the design of the punching shear of flat slabs with hole(s) near of the column, considering the moment caused by the eccentricity of the critical perimeter due to the presence of hole(s) in the slab.

Na área de análise de pavimentos, muitos trabalhos já foram realizados, levando-se em conta diferentes considerações sempre com o objetivo de aprimorar os modelos que representam o comportamento estrutural. O método dos elementos finitos tem se mostrado um dos principais métodos na resolução deste problema, onde as lajes podem ser discretizadas por elementos finitos de placa e as vigas por elementos finitos de barra. Dentre os diversos tipos de elementos finitos de placa, o elemento finito DKT (discrete Kirchhoff triangle), tem sido largamente utilizado na consideração da rigidez à flexão das lajes. O objetivo deste trabalho é apresentar o recém-formulado elemento finito de placa P15N na análise de pavimentos, bem como apresentar as diferenças e semelhanças na utilização deste elemento com o elemento finito DKT, na discretização das lajes, na determinação dos deslocamentos e esforços em pavimentos. Para a realização das análises descritas, desenvolveu-se um programa computacional em linguagem de programação FORTRAN 90. Vários exemplos numéricos são apresentados / In the area of floor analysis, many studies have already been carried out taking into account different considerations and always aiming to improve the models which represent the structural behavior. The finite element method has shown to be one of the principal methods for the resolution of this problem, where the slabs and the beams may be meshed by plate finite elements and bar finite elements, respectively. Among the various kinds of plate finite elements, the DKT (discrete Kirchhoff triangle) finite element has been largely used in the consideration of slabs bending stiffness. The objective of this work is to present the P15N as a plate finite element in the analysis of floors, as well as to point out the differences in the utilization of this element with the DKT finite element in slab discretization and in the determination of displacements and stresses in floors. For such analysis, a computational program was developed in FORTRAN 90 programming language. Several numerical examples are presented

Para calcularem-se lajes de concreto armado considerando todos os esforços resistentes, incluindo o momento volvente, é necessário o conhecimento de procedimentos para determinar momentos normais a duas direções escolhidas, segundo as quais serão dispostas as armaduras, que cubram o terno de esforços Mx, My e Mxy. No presente trabalho são estudados e comparados dois métodos para armaduras ortogonais e um método para o caso de armaduras oblíquas entre si. Com a finalidade de se considerar a resistência do concreto ao momento volvente no detalhamento das armaduras, são estudadas as resistências do concreto às tensões cisalhantes devidas à força cortante e ao momento torçor e as maneiras de se combinarem essas tensões. Outra finalidade do trabalho foi a de desenvolver pré-processadores para facilitar a modelagem de pavimentos por elementos finitos e um pós-processador que permitisse a análise dos resultados utilizando os conceitos estudados. Utilizando estes \"softwares\" fez-se um pequeno estudo de lajes retangulares apoiadas nos quatro lados, procurando avaliar as recomendações existentes na bibliografia especializada, sobre as armaduras de canto necessárias a esse tipo de laje. Fez-se, também uma comparação de detalhamentos das lajes de um pavimento tipo calculadas pelo método das charneiras plásticas, pelo método dos elementos finitos sem a consideração da rigidez à torção e pelo método dos elementos finitos considerando a rigidez à torção. / In order to design reinforced concrete slabs considering all the resisting moments, including the torsional moment, it is necessary to know procedures to determine normal moments, in two chosen reinforcement directions, that are equivalent to the field of moments Mx, My and Mxy. In this work, two methods for orthogonal reinforcement are studied and compared and one for skew reinforcement is studied. In an attempt to consider the resistance of concrete to the torsional moment, the shear stress due to shear carried by concrete and the shear stress due to torsion carried by concrete and the ways of combining them are studied. Another aim of this work was to develop pre-processors to facilitate the process of modeling building floors in finite elements and one post-processor that permits the analysis of the results, using the concepts of this work. Using these softwares, a short study of simply supported rectangular slabs was made, in an attempt to evaluate the recommendations, found in the specialized bibliography, about the corner reinforcement necessary in these kinds of slabs. A comparison of the slabs design of one building floor, calculated using the yield line theory, using the finite element method without the torsional strength and using the finite element method with the torsional strength was also made.

Este trabalho apresenta uma associação dos cálculos elástico e plástico, para lajes retangulares com carga uniforme e uma borda livre. O cálculo elástico é usado para pré-dimensionamento, na determinação dos momentos negativos e da razão dos momentos positivos nas duas direções. Esses momentos fletores são obtidos através de funções aproximadoras, determinadas a partir de tabelas. Após compatibilização dos momentos elásticos, são adotados os momentos de plastificação negativos, sendo os positivos obtidos através da teoria das chameiras plásticas, que é utilizada também no cálculo do comprimento das barras sobre os apoios. Conhecidas as armaduras, os momentos elásticos são novamente utilizados na verificação das flechas. Nos exemplos são comparadas as armaduras do cálculo elástico com as obtidas através do cálculo plástico, no qual se pôde constatar a economia deste último em relação ao primeiro. / This work presents an association of elastic and plastic analysis, for rectangular slabs with uniform load and a free edge. The elatic theory is used to estimate the negative moments and the relation between positive moments in the two directions. These bending moments are obtained from approached functions, based on tables. After compatibilization of elastic moments, the negative plastic moments are adopted and the positive moments are obtained from yield line theory, which will be used again in the calculation of the lenght of the bars over the supports. Once the reinforcements are known, the elastic moments are used again for checking the deflexions. In the examples, the results of elastic plate theory are compared with those obtained by plastic analysis, where it is possible to evidence the economy of the later over the former.

A análise estrutural dos pavimentos de edifícios vem se constituindo de recursos cada vez mais refinados, com a difusão de inúmeros \"softwares\", dotados de pré e pós-processadores que possibilitam análises precisas, com a consideração da interação entre as rigidezes de todos os elementos que compõem o sistema estrutural do pavimento. Entretanto, há que se ressaltar que, diante desta ampla disponibilidade de recursos, os cálculos simplificados não podem ser deixados de lado nas análises dos projetistas, uma vez que propiciam confirmações dos resultados fornecidos por um \"software\". Assim sendo, o presente trabalho apresenta alguns aspectos da análise estrutural das lajes de edifícios, abordando conceitos relativos a ações e disposições construtivas a considerar nos projetos, ao pré-dimensionamento das lajes e aos tipos de análise estrutural possíveis de serem realizadas. São apresentados alguns modelos estruturais para análise linear das lajes de edifícios, a saber, o cálculo aproximado mediante tabelas do método elástico, a técnica da analogia de grelha e o método dos elementos finitos, mostrando as particularidades de cada um e definindo até que ponto cada um deles pode ser aplicado com segurança, apresentando resultados coerentes para o comportamento estrutural. No regime elasto-plástico, é apresentado um modelo estrutural de análise através da teoria das charneiras plásticas. São abordadas, para cada um dos modelos estruturais propostos, questões relativas à distribuição das armaduras e aos valores de esforços solicitantes a considerar para o correto dimensionamento das lajes do pavimento. Apresentam-se, ainda, os detalhamentos das armaduras positivas e negativas resultantes das análises realizadas. / The structural analysis of building pavements has been made through the use of very sophisticated instruments, with the dissemination of severa! softwares, characterized by pre- and post-processors which enable accurate analysis, considering the interaction between the stiffness of ali the elements that constitute the pavement structural system. lt is important to notice that even with the large number of available resources, simplified calculations cannot be disregarded in a designer\'s analysis, since they provide the confirmation of the results yielded by the software. So, this study presents some aspects of building slabs structural analysis, regarding concepts related to actions and constructive dispositions to be considered in such projects, to slabs pre-design and kinds of possible structural analysis. Some structural models for linear analysis of building slabs are presented: (1) the simplified calculation through the use of tables of elastic method; (2) crossing beams analogy technique and (3) the finite element method, showing the characteristics of each one and defining up to what point each of them can be used safely, presenting coherent results to structural behavior. In the elastic-plastic regime, a structural model of analysis through the yield line theory is presented. For each proposed structural model, questions about arm distribution and values of requested efforts are considered, regarding the correct pavement slabs design. The details regarding positive and negative arms resulting from the analysis performed is also presented.

O principal objetivo deste trabalho é calcular esforços e deslocamentos de estruturas tridimensionais de edifícios de andares múltiplos, sujeitos às ações verticais e laterais, considerando a rigidez transversal à flexão das lajes, em teoria de 2ª ordem. O elemento finito de placa adotado na discretização do pavimento, responsável pela consideração da rigidez transversal das lajes na análise do edifício é o DKT (Discrete Kirchhoff Theory). Para os pilares o equilíbrio de forças é verificado na sua posição deformada, ou como é conhecido da literatura técnica, análise em teoria de 2ª ordem, considerando a não linearidade geométrica. Para o cálculo dos esforços e deslocamentos na estrutura são aplicadas as técnicas de subestruturação em série e paralelo na matriz de rigidez global da estrutura. Elaborou-se um programa de computador para o processo de cálculo, utilizando a linguagem computacional Fortran Power Station 90 e pré e pós processadores em Visual Basic 4.0 para ambiente Windows. Finalmente são apresentados alguns exemplos para comprovar a validade do processo de cálculo utilizado / The main aim of this work is to calculate stresses and displacements of threedimensional structures of multistory buildings, subjected to vertical and lateral loads, considering the transverse bending stiffness of slabs, in second order theory. The plate finite element adopted in floor discretization, responsible for considering the bending stiffness contribution of slabs in the analysis of buildings, is the DKT (Discrete Kirchhoff Theory). For columns the forces equilibrium is verified for the columns in their deformed position, which is known in the technical literature as 2nd order analysis, considering the geometric non-linearity. The techniques of serial and parallel analysis of substructures are applied to the global stiffness matrix for the calculus of forces and displacements in the strucuture. A computer program was developed for the calculation process, using the computer language Fortran Power Station 90 and pre and post-processors in Visual Basic 4.0 for a Windows environment. Finally, some examples are presented to check the validity of the employed calculus process

[pt] Este trabalho apresenta um estudo de um novo sistema de laje mista com o uso de perfis metálicos de chapa dobrada e isopor seqüencialmente dispostos lado a lado e preenchidos com concreto de armado. A principal motivação para este estudo foi o desenvolvimento de um eficiente sistema de laje fácil e rápida de ser construído, e que fosse econômica. O sistema de laje desenvolvido maximiza o espaço livre útil da construção, eliminando a necessidade de formas e escoras tendo assim como conseqüência direta a minimização das perdas construtivas. Os resultados obtidos indicaram que os custos do novo sistema de laje são equivalentes aos outros métodos de construção de lajes, mas que outras características como o tempo e as perdas economizadas na fase construtiva viabilizam a utilização do novo sistema.O trabalho foi consta de três fases principais: primeiramente uma otimização do estudo dos perfis de chapa dobrada foi conduzida analisando as suas características estáticas e dinâmicas para o sistema de laje. Variáveis como peso, altura, espessura,tensão de escoamento, vão máximo, carga suportada por vão e por espessura da chapa metálica, momento máximo, capacidade de carga, minimização dos custos de fabricação e possibilidades de uso de vãos múltiplos foram validados. Quando a seção do perfil metálico estava definida e otimizada, foi estabelecido um estudo com uma série de 5 (cinco) testes em escala real. O propósito principal dos testes foi obter os estados limites último da seção do perfil adotado associado ao sistema de laje para validação do sistema estrutural. Os testes avaliaram a real aderência entre concreto e perfil metálico e sua influência no estado limite último de flexão, cisalhamento e deslizamento. Outra solução além da aderência química natural do concreto com o perfil metálico envolveram o uso de parafusos auto atarrachantes como conectores de cisalhamento. A fase final compreendeu a comparação dos resultados experimentais com as normas de aço para averiguar a validade dos resultados do sistema de laje mista. / [en] This work presents an investigation of an innovative composite slab system made of a steel deck and a styrofoam plate, side by side, filled with reinforced concrete. The objective of the present study was the development of an efficient composite slab system that is easy to manufacture, erect and also economic.The investigation started with a parametric design study of the steel deck profile.When an optimum steel deck profile was established the investigation proceeded with a series of full-scale- experiments. The tests enable the establishment of the ultimate and serviceability limit states associated with the investigated system.Another aim of the experiments was to investigate the concrete versus steel deck connection. Solutions using only the natural concrete versus steel adherence and others utilising shear connectors made of self-drilling bolts in the steel deck were considered. Finally current steel design codes were used to compare the experimental results and access the accuracy of the composite slab system In this paper a composite slab system developed for low-cost building construction is presented. Results and conclusions from a theoretical and experimental study were summarized below the work. Finally these composite slab systems can be implemented in large scale construction, just by the development of portable rolling equipment that can produce very long and continuous profiles, as required by the construction teamwork.

Long span flat slab systems with internal spherical void formers have been used in Europe for a decade now. Cobiax® is the brand name of a successful system, recently introduced in South Africa. It is a bi-axial reinforced concrete flat slab system, with a grid of internal spherical void formers. The main advantage is the possibility of long spans due to the significant reduction in own weight, as well as the fast construction sequence with the use of flat slab formwork systems. Design requirements of SANS 10100:2000 are affected. Vertical shear capacity is a concern due to loss of aggregate interlock. Research in Germany proved a factor of 0.55 to be a conservative shear resistance reduction factor for Cobiax slabs. Theoretical and preliminary laboratory South African research suggests that a greater factor of 0.85 might be used when considering the shear capacity of the steel cages. These cages’ vertical legs also cross the cold joint caused by the two concrete pours required for Cobiax slabs, and proved to provide sufficient horisontal shear resistance if the correct cage diameters are used. Laboratory tests in Germany supported by theoretical calculations further showed reduced deflections for Cobiax slabs. Although stiffness and own weight are reduced due to the voids, Cobiax slabs had smaller absolute deflections than solid slabs with the same thickness. Cobiax research factors are safe to apply to SANS 10100-01:2000. The economy of Cobiax slabs was tested against that of coffer and post-tensioned slabs. Different span lengths and loads were considered. Based on 2007 material costs in South Africa, Cobiax slabs subject to the same loads and span lengths will be slightly more expensive than that of coffer slabs and post-tensioned slabs when considering only direct slab construction costs. Cobiax will be most appropriate where a flat soffit is required for high multi-storey buildings, requiring large spans with a light load application. Copyright / Dissertation (MEng)--University of Pretoria, 2010. / Civil Engineering / unrestricted

Following some incidents in high-rise buildings, such as Ronan Point London 1968, in which collapse of a limited number of structural elements progressed to a failure disproportionate to the initial cause, consideration of robustness was introduced in British Standard. The main method of preventing progressive collapse for providing robustness to steel framed buildings with precast concrete floor slabs focuses on the allowable tying forces that the reinforcement in between the slabs and in hollowcores should carry. However there are uncertainties about the basis of the practical rules associated with this method. This thesis presents the results of numerical and analytical studies of tie connection behaviour between precast concrete floor slabs (PCFS). It is shown that under current design regulations the tie connection is not able to resist the accidental load limit applied on the damaged floor slabs. By establishing the capability of a finite element model to depict and predict the behaviour of concrete members in situations such as arching and catenary action against several experimental tests, an extensive set of parametric studies was conducted in order to identify the effective parameters in enhancing the resistance of the tie connection between PCFSs. These parameters include: tie bar diameter, position, length, yield stress and ultimate strain; the slab’s height, length; and the compressive strength of the grouting concrete in between the slabs that encases the tie bar. Recommendations are made based on the findings of this parametric study in order to increase the resistance of the tie connection. Based on the identified effective parameters in the parametric study a predictive analytical relationship is derived which is capable of determining the maximum vertical displacement and load that the tie connection is able to undergo. This relationship can be used to enable the connection to capture the accidental limit load on a damaged slab. The identified parameters are examined in a three dimensional finite element model to assess their effect when columns of the structure are lost in different locations such as an edge, corner or internal column. Based on the findings of this study methods for improving the connections performance are presented. Also the effect of alternative transverse tying method is evaluated and it is concluded that although this kind of tie increases the load carrying capacity of the connection, its effect on the catenary action is not significant.

Cette thèse est consacrée au développement de modèles semi-analytiques précis pour le calcul numérique de dispositifs nanophotoniques résonants. Il s'agit en particulier de membranes à cristaux photoniques, qui supportent des résonances avec des très grands facteurs de qualité, et d’ensembles composés de plusieurs nano-antennes plasmoniques, qui présentent des résonances avec des faibles facteurs de qualité. La thèse est divisée en deux parties.La première partie présente un modèle semi-analytique pour le calcul des modes supportés par des membranes à cristaux photoniques. Les modes à fuite (leaky modes) supportés par ces membranes structurées sont modélisés comme une résonance Fabry-Perot transverse composée de quelques ondes de Bloch propagatives qui vont et viennent verticalement à l'intérieur de la structure. Ce modèle est appliqué à l'étude des états liés dans le continuum (bound states in the continuum, ou BIC). Nous montrons que le modèle Fabry-Perot multimode est parfaitement adapté pour prédire l'existence des BICs ainsi que leur position dans l'espace des paramètres. Grâce à la semi-analyticité du modèle, nous étudions la dynamique des BICs avec l'épaisseur de la membrane pour des structures symétriques et asymétriques. Dans ce dernier cas, nous étudions des objets présentant soit une symétrie horizontale brisée, soit une symétrie verticale brisée (ajout d'un substrat). Le modèle Fabry-Perot nous permet d’obtenir des informations importantes sur la nature et le comportement des BICs. Nous démontrons que lorsque la symétrie miroir horizontale est brisée, les BICs dus à la symétrie du système, qui existent dans les structures symétriques au point Gamma du diagramme de dispersion, restent des BICs malgré l’absence de symétrie mais changent de nature. Ils deviennent des BICs dus à des interférences destructives entre les ondes de Bloch. La deuxième partie est consacrée au développement d'une théorie modale originale pour modéliser la diffusion de la lumière par des structures complexes composées d'un ensemble de plusieurs nano-antennes. L'objectif est de pouvoir modéliser la diffusion de la lumière par des métasurfaces à partir de la seule connaissance des modes de leurs constituants individuels. Pour ce faire, nous combinons un formalisme modal basé sur l’utilisation des modes quasi-normaux (QNM) avec la théorie multipolaire de la diffusion multiple basée sur le calcul de la matrice de transition (matrice T) d'un diffuseur unique. La matrice T fournit la relation entre le champ incident et le champ diffusé dans la base des harmoniques sphériques vectorielles. Elle contient toutes les propriétés de diffusion intrinsèques à l'objet. Le calcul de cette matrice représente une charge numérique lourde car elle nécessite de nombreux calculs rigoureux du champ diffusé. L'utilisation d'une décomposition modale avec des QNMs nous permet d’une part de rendre une partie du calcul analytique et d’autre part d'apporter une meilleure compréhension physique. Nous dérivons une décomposition modale de la matrice T et testons sa précision sur le cas de référence d'une nanosphère métallique.Enfin, la décomposition modale de la matrice T est appliquée à des cas pratiques d'intérêt en nanophotonique. A partir de la seule connaissance de quelques modes d'un nanocylindre plasmonique unique, nous calculons analytiquement la diffusion multiple de la lumière par un dimère et par une antenne Yagi-Uda composés de ces nanocylindres. Nous appliquons également l’approche modale à un réseau périodique bidimensionnel de nanocylindres . La comparaison avec les résultats d'une méthode numérique rigoureuse démontre un bon accord avec le calcul modal. Par rapport à des calculs entièrement rigoureux, la décomposition modale de la matrice T permet une réduction significative du temps de calcul. Comme les calculs sont analytiques une fois que les modes ont été calculés, l'approche modale est extrêmement utile pour les problèmes d'optimisation. / The presented thesis is dedicated to the development of semi-analytical accurate models for the numerical calculation of resonant nanophotonic devices. In particular, it concerns photonic crystal slabs, which can support resonances with high quality factors, and ensembles composed of several plasmonic nanoantennas, which exhibit resonances with low quality factors. The structure of the thesis is two-fold. In the first part, a semi-analytical model for the calculation of the modes supported by photonic crystal slabs (their dispersion and quality factors) is presented. Leaky modes supported by photonic crystal slabs are modeled as a transverse Fabry-Perot resonance composed of a few propagative Bloch waves bouncing back and forth vertically inside the slab. This model is applied to the study of bound states in the continuum (BICs). We show that the multimode Fabry-Perot model is perfectly suitable to predict the existence of BICs as well as their precise positions in the parameter space. We show that, regardless of the slab thickness, BICs cannot exist below a cut-off frequency, which is related to the existence of the second-order Bloch wave in the photonic crystal. Thanks to the semi-analyticity of the model, we investigate the dynamics of BICs with the slab thickness in symmetric and asymmetric photonic crystal slab. In the latter case, we investigate structures with either a broken horizontal symmetry or a broken vertical symmetry (addition of a substrate). As a result, we obtain some important insights into the nature and behavior of BICs. We evidence that, as the horizontal mirror symmetry is broken, the symmetry-protected BICs that exist in symmetric structures at the Gamma-point of the dispersion diagram are still BICs despite the absence of symmetry but change their nature. They become resonance-trapped BICs, but only for specific values of the slab thickness.The second part of the thesis is dedicated to the development of an original modal theory to model light scattering by complex structures composed of a small ensemble of plasmonic nanoantennas. The objective is to be able to model light scattering by metasurfaces from the sole knowledge of the eigenmodes of their individual constituents. For that purpose, we combine a quasi-normal mode (QNM) formalism with the multipole multiple-scattering theory based on the calculation of the so-called transition matrix (T-matrix) of a single scatterer. The T-matrix provides the relation between the incident and scattered fields in the vectorial spherical harmonics basis. It captures all the intrinsic scattering properties of the object that are due to its shape and refractive index distribution. Computation of the T-matrix is a heavy numerical burden since it requires numerous rigorous calculations of the scattered field— one for each harmonic in the basis. Using a modal expansion of the scattered field with QNMs allows us to bring both analyticity and physical understanding into the calculation. We derive a modal expansion of the T-matrix and test its accuracy on the reference case of a metallic nanosphere.Finally, we apply the modal expansion of the T-matrix to practical cases of interest in nanophotonics. From the sole knowledge of a few modes of a single plasmonic nanorod, we calculate analytically multiple light scattering by a dimer and a Yagi-Uda antenna composed of these nanorods. We apply also the modal approach to a periodic two-dimensional array of nanorods. Comparison with the results of a rigorous Maxwell’s equations solver demonstrates a good agreement with the QNM-based calculation. Compared to fully rigorous calculations, the QNM expansion of the T-matrix allows for a significant reduction of the computation time. Since the calculations are analytical once the modes have been calculated, the QNM approach is extremely useful for optimization problems.