The Internet has fundamentally changed the way of modern communication. Current trends indicate that high-capacity demands are not going to be saturated anytime soon. From Shannon's theory, we know that information capacity is a logarithmic function of signal-to-noise ratio (SNR), but a linear function of the number of dimensions. Ideally, we can increase the capacity by increasing the launch power, however, due to the nonlinear characteristics of silica optical fibers that imposes a constraint on the maximum achievable optical-signal-to-noise ratio (OSNR). So there exists a nonlinear capacity limit on the standard single mode fiber (SSMF). In order to satisfy never ending capacity demands, there are several attempts to employ additional degrees of freedom in transmission system, such as few-mode fibers (FMFs), which can dramatically improve the spectral efficiency. On the other hand, for the given physical links and network equipment, an effective solution to relax the OSNR requirement is based on forward error correction (FEC), as the response to the demands of high speed reliable transmission. In this dissertation, we first discuss the model of FMF with nonlinear effects considered. Secondly, we simulate the FMF based OFDM system with various compensation and modulation schemes. Thirdly, we propose tandem-turbo-product nonbinary byte-interleaved coded modulation (BICM) for next-generation high-speed optical transmission systems. Fourthly, we study the Q factor and mutual information as threshold in BICM scheme. Lastly, an experimental study of the limits of nonlinearity compensation with digital signal processing has been conducted.

In this study, seismic interferometry is used to analyze dynamic processes in the Earth’s shallow subsurface caused by environmental processes and ground shaking. In the first part of the thesis, the feasibility of a passive monitoring with ambient seismic noise at the pilot site for CO2 injection in Ketzin is investigated. Monitoring the expansion of the CO2 plume is essential for the characterization of the reservoir as well as the detection of potential leakage. From June 2008 until August 2013, more than 67000 tons of CO2 were injected into a saline aquifer at a depth of about 650 m. Passive seismic data recorded at a seismic network around the injection site was cross-correlated in a frequency range of 0.5-4.5 Hz over a period of 4 years. The frequency band of 0.5-0.9 Hz, in which surface waves exhibit a high sensitivity at the depth of the reservoir, is not suitable for monitoring purposes as it is only weakly excited. In a frequency range of 1.5-3 Hz, periodic velocity variations with a period of approximately one year are found that cannot be caused by the CO2 injection. The prominent propagation direction of the noise wave field indicates a wind farm as the dominant source providing the temporally stable noise field. This spacial stability excludes variations of the noise source distribution as a spurious cause of velocity variations. Based on an amplitude decrease associated with time windows towards later parts of the coda, the variations must be generated in the shallow subsurface. A comparison to groundwater level data reveals a direct correlation between depth of the groundwater level and the seismic velocity. The influence of ground frost on the seismic velocities is documented by a sharp increase of velocity when the maximum daily temperature stays below 0 C. Although the observed periodic changes and the changes due to ground frost affect only the shallow subsurface, they mask potential signals of material changes from the reservoir depths. To investigate temporal seismic velocity changes due to earthquake-related processes and environmental forcing in northern Chile, 8 years of ambient seismic noise recorded by the Integrated Plate Boundary Observatory Chile (IPOC) are analyzed. By autocorrelating the ambient seismic noise field, approximations of the Green’s functions are retrieved and velocity changes are measured with Coda Wave Interferometry. At station PATCX, seasonal changes of seismic velocity caused by thermal stress as well as transient velocity reductions are observed in the frequency range of 4-6 Hz. Sudden velocity drops occur at times of mostly earthquake-induced ground describing the seismic velocity variations based on continuous observations of the local ground acceleration. The model assumes that not only the shaking of large earthquakes causes velocity drops, but any small vibrations continuously induce minor velocity variations that are immediately compensated by healing in the steady state. The shaking effect is accumulated over time and best described by the integrated envelope of the ground acceleration over one day, which is the temporal resolution of the velocity measurements. In the model, the amplitude of the velocity reduction as well as the recovery time are proportional to the strength of the excitation. The increase of coseismic velocity change and recovery time with increasing excitation is confirmed by laboratory tests with ultrasound. Despite having only two free scaling parameters, the model fits the data of the shaking-induced velocity variation in remarkable detail. Additionally, a linear trend is observed that might be related to a recovery process from one or more earthquakes before the measurement period. A clear relationship between ground shaking and induced velocity reductions is not visible at other stations. The outstanding sensitivity of PATCX to ground shaking and thermal stress can be attributed to the special geological setting of the station, where the subsurface material consists of a relatively loose conglomerate with high pore volume leading to stronger nonlinearity compared to the other IPOC stations. / In dieser Studie werden mit Hilfe von seismischer Interferometrie kleinste dynamische Prozesse in der Erdkruste beobachtet, welche beispielsweise durch umweltbedingte oder anthropogene Einflüsse sowie Bodenerschütterungen hervorgerufen werden können. Im ersten Teil der Arbeit werden Änderungen in der seismischen Geschwindigkeit am Pilotstandort für CO2-Speicherung in Ketzin untersucht. In einer Tiefe von 650m wurden dort zwischen Juni 2008 und August 2013 über 67000 Tonnen CO2 eingelagert. In einem Frequenzbereich vom 0,05-4,5 Hz wurden Kreuzkorrelationen des seismischen Hintergrundrauschens an einem kleinräumigen Netzwerk über einen Zeitraum von 4 Jahren berechnet. Der Frequenzbereich zwischen 0,5 und 0,9 Hz weist eine hohe Sensitivität von Oberflächenwellen in der Tiefe des Reservoirs auf, ist aber nur sehr schwach angeregt und eignet sich deswegen nicht für die Analyse. In einem Frequenzbereich von 1,5-3 Hz zeigen sich periodische Geschwindigkeitsänderungen mit einer Periode von einem Jahr, welche nicht durch die Einlagerung von CO2 erzeugt werden können. Eine Analyse des seismischen Hintergrundrauschens zeigt, dass dieses über den gesamten Zeitraum hinweg hauptsächlich aus der Richtung eines Windparks kommt. Durch die Stabilität des Wellenfeldes können Änderungen in den Quellpositionen, welche sich in scheinbaren Geschwindigkeitsänderungen zeigen können, ausgeschlossen werden. Eine Amplitudenabnahme der Geschwindigkeitsänderungen hin zu späteren Zeitfenstern in der Coda lässt auf oberflächennahe Prozesse als Ursache schließen. Ein Vergleich zwischen den jährlichen Geschwindigkeitsänderungen mit Schwankungen im Grundwasserspiegel zeigt eine direkte Korrelation. Ein sprunghafter Anstieg in der Geschwindigkeit zeigt sich im Winter, wenn die Tageshöchsttemperaturen unter den Gefrierpunkt sinken und der Boden zufriert. Obwohl Bodenfrost und Änderungen im Grundwasserspiegel nur einen sehr oberflächennahen Bereich betreffen, so überdecken sie dennoch mögliche Signale durch die Einlagerung von CO2. Im zweiten Teil der Arbeit werden Geschwindigkeitsänderungen in Nordchile untersucht, welche durch erdbebeninduzierte Prozesse und umweltbedingte Einflüsse hervorgerufen werden. Dazu wurden über einen Zeitraum von 8 Jahren Autokorrelationen des seismischen Hintergrundrauschens des IPOC Netzwerkes (Integrated Plate Boundary Observatory Chile) berechnet und mit seismischer Interferometrie ausgewertet. An der Station PATCX können in einem Frequenzbereich von 4-6 Hz periodische Geschwindigkeitsänderungen beobachet werden, welche durch thermisch induzierte Dehnung hervorgerufen werden. Außerdem treten transiente Geschwindigkeitsabnamen nach Bodenerschütterungen auf, welche hauptsächlich von Erdbeben verursacht werden. Die seismische Geschwindigkeit kehrt daraufhin langsam wieder auf ihr vorheriges Niveau zurück. Für die Geschwindigkeitsänderungen wurde ein empirisches Modell entwickelt, welches auf Messungen der lokalen Bodenerschütterung basiert. Dabei wird angenommen, dass nicht nur große erdbebeninduzierte, sondern auch kleinste Bodenerschütterungen einen Abfall der Geschwindigkeit erzeugen, welche wiederum innerhalb kürzester Zeit durch Heilung in den Gleichgewichtszustand zurückkehrt. Dabei summieren sich die Effekte durch die Bodenerschütterungen mit der Zeit auf und werden am besten mit dem Integral der lokalen Bodenbeschleunigung über die Messwerte eines Tages beschrieben. Die Diskretisierung von einem Tag entspricht der zeitlichen Auflösung in der Messung der Geschwindigkeitsänderungen. Sowohl die Amplitude der Geschwindigkeitsabnahme als auch die Zeit bis der Gleichgewichtszustand wieder erreicht ist (Heilungszeit) werden im Modell als proportinal zur Größe der Anregung angenommen. Eine Korrelation der Heilungszeit und der Amplitude der koseismischen Geschwindigkeitsabnahme mit der Größe der Anregung konnte mit Hilfe von Laboruntersuchungen mit Ultraschall bestätigt werden. Mit nur zwei Parametern beschreibt das Modell die transienten Geschwindigkeitsänderungen in bemerkenswerter Genauigkeit. Desweiteren beinhaltet das Modell einen linearen Verlauf in den Geschwindigkeitsänderungen, welcher vermutlich durch einen Heilungsprozess hervorgerufen wird, der auf ein oder mehrere Erdbeben vor dem Messzeitraum folgte. Eine Beziehung zwischen Bodenerschütterung und Geschwindigkeitsänderung ist an anderen Stationen des IPOC Netzwerkes nicht erkennbar. Die herausragende Sensitivität von PATCX im Hinblick auf Bodenerschütterung und thermische Dehnung kann den speziellen geologischen Gegebenheiten an der Station zugeschrieben werden. Bei dem dort vorliegenden Material handelt es sich um ein relativ loses Konglomerat mit großem Porenvolumen, welches ein starkes nichtlineares Verhalten aufweist, was an anderen IPOC Stationen nicht zu erwarten ist.

The superficial tangential zone (STZ) plays a significant role in normal articular cartilage’s ability to support loads and retain fluids. To date, tissue engineering efforts have not replicated normal STZ function in cartilage repairs. Finite element models were developed to examine the STZ’s role in normal and repaired articular surfaces under different contact conditions. Models were developed by incrementally adding improvements which culminated in contact loading of curved models by permeable and impermeable rigid surfaces and a normal cartilage layer. In the normal STZ, permeability was strain-dependent on volumetric strain; tension-compression nonlinearity modeled collagen behavior. Nonlinear geometry accounted for finite deformation. Results showed that STZ properties of sufficient quality maybe critical for the survival of transplanted constructs in vivo. As compared to rigid surfaces, loading via normal cartilage provided more physiologic results. These models can provide guidance in identifying critical features for the design of tissue engineered articular cartilage constructs.

Nos últimos anos grande número de modelos computacionais tem sido propostos com o intuito de descrever o movimento de solutos no perfil do solo, apesar disso, o que se observa é que existe grande dificuldade em se modelar esses fenômenos, para que o modelo possa predizer o processo de deslocamento e retenção dos solutos na natureza. Para tanto, o objetivo deste trabalho foi utilizar um modelo estatístico para descrever o transporte dos solutos no perfil do solo. Dessa forma, foi realizado um experimento em laboratório e observado os níveis de potássio e nitrato ao longo do perfil dos solos Latossolo Vermelho Amarelo e Nitossolo Vermelho. Para inferir sobre essas variáveis foram consideradas duas abordagens. Para a primeira abordagem foi utilizado um modelo de regressão não linear para cada uma das variáveis, cujos parâmetros do modelo apresentam uma interpretação prática, na área de solos. Para esse modelo foi realizado um esboço sobre a não linearidade do mesmo para verificar as propriedades assintóticas dos estimadores dos parâmetros. Para o método de estimação foi considerado, o método de mínimos quadrados e o método de bootstrap. Além disso, foi realizada uma análise de diagnóstico para verificar a adequação do modelo, bem como identificar pontos discrepantes. Por outro lado, para outra abordagem, foi utilizado um modelo multiresposta para analisar o comportamento das variáveis nitrato e potássio ao longo do perfil dos solos, conjuntamente. Para esse modelo foi utilizado o método da máxima verossimilhança para encontrar as estimativas dos parâmetros do modelo. Em ambas as situações, observou-se a adequação dos modelos para descrever o comportamento dos solutos nos solos, sendo uma alternativa para os pesquisadores que trabalham com estudo de solos. O modelo logístico com quatro parâmetros se destacou por apresentar melhores propriedades, como medidas de não linearidade e boa qualidade de ajuste. / In the last years, several computational models have been proposed to describe the movement of solutes in the soil profile, but what is observed is that there is great difficulty in model these phenomena, so that model can predict the displacement process and retention of solutes in nature. Thus, the aim of this study was to use a statistical model to describe the transport of solutes in the soil profile. Therefore, an experiment was conducted in the laboratory and observed levels of potassium and nitrate along the depth of soil Oxisol (Haplustox) and Hapludox,. To make inferences about these variables were considered two approaches. For the first approach was utilized a non-linear regression model for each variable and the model parameters have a practical interpretation on soil. For this model we performed a sketch on the nonlinearity of the model to check the asymptotic properties of parameter estimators. To estimate the parameters were considered the least squares method and the bootstrap method. In addition, we performed a diagnostic analysis to verify the adequacy of the model and identify outliers. In the second approach considered was using a multi-response model to analyze the behavior of the variables nitrate and potassium throughout the soil profile together. For this model we used the maximum likelihood method to estimate the model parameters. In both cases, we observed the suitability of the models to describe the behavior of solutes in soils, being an alternative for researchers working on the study of soils. The logistic model with four parameters stood out with better properties, such as non-linearity and good fit.

Este trabalho trata da evolução natural do modelo de cálculo em regime elástico linear, largamente empregado no dimensionamento de pavimentos de concreto armado, para um modelo com maior capacidade de representação. A consideração da não-linearidade física do material concreto armado é incorporada a um sistema computacional em elementos finitos segundo modelos não-estratificados, através da generalização das formulações para o campo dos esforços. O elemento de barra é abordado em campo uniaxial, verificando-se o escoamento produzido pela flexão. Para o elemento de placa, estabelece-se o critério de escoamento de Von Mises com leis associativas particularizado ao estado plano de tensões (EPT). Essas tensões são integradas ao longo da espessura do elemento, permitindo escrever o critério em função do terno de valores de momentos atuantes. Por fim, os modelos de barra e placa são integrados ao sistema computacional, resultando um sistema de análise não-linear de pavimentos de concreto armado. A caracterização do comportamento físico da seção transversal é feita segundo um diagrama momento-curvatura trilinear. A aplicação do modelo proposto a elementos estruturais isolados, e a um pavimento convencional, confirma a melhoria do sistema computacional, e os tempos de processamento requeridos evidenciam a viabilidade do emprego do modelo não-linear físico em projetos usuais de pavimentos de concreto armado. / This work deals with the natural evolution of the design model based on linear elasticity, widely employed on reinforced concrete slabs designs, to a improved design model. The consideration of physical nonlinearity of the reinforced concrete is introduced into a finite element computational system by nonlayered models with generalization of the formulation to the bending moment field. The beam element is treated in a uniaxial field, with bending moment yielding. In the plate element it is established the Von Mises yield criterion with associative laws particularized to the plane stress. These stresses are integrated along the element depth, enabling to write the criterion with plate bending moments. At last, the beam and the plate models are incorporated into a computational system, resulting in a reinforced concrete slabs analysis system. The characterization of the cross section physical behavior is made by a trilinear bending moment-curvature diagram. The application of the proposed model to isolate structural elements and to a conventional slab confirms the improvement of the finite element system and the required computational analysis times show the feasibility of the nonlinear model application into usual reinforced concrete slab design.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-17T10:55:51Z No. of bitstreams: 2 Tese - Erivelton de Oliveira Alves - 2015.pdf: 5869721 bytes, checksum: 7ad25bf25a87b400628abe44e6c6f340 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-17T10:56:06Z (GMT) No. of bitstreams: 2 Tese - Erivelton de Oliveira Alves - 2015.pdf: 5869721 bytes, checksum: 7ad25bf25a87b400628abe44e6c6f340 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-01-17T10:56:06Z (GMT). No. of bitstreams: 2 Tese - Erivelton de Oliveira Alves - 2015.pdf: 5869721 bytes, checksum: 7ad25bf25a87b400628abe44e6c6f340 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-10-16 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work the influence of a saturable nonlinearity on the modulation instability (MI) in the contexts oppositely directed coupler and optical fibers in the presence of high-order effects are investigated. The instability gain is attained by using standard linear stability analysis. In particular, we study the combination of a saturable nonlinearity with self-steepening or intrapulse Raman scattering effects on MI for both normal and anomalous group velocity dispersion regimes. In the case of the directional couplers we investigated how the gain of modulational instability is affected by the saturation model and self-steepening or intrapulse Raman scattering effects. Our results show that instability gain exhibits significant changes due to the effects of saturable nonlinearity. When we analyze self-steepening effect, we show that its effective influence on the gain of the modulational instability depends on the algebraic sum of the parameters in each channel. Analyzing the intrapulse Raman scattering, we observe a symmetry break in the gain regions when the Raman scattering parameter with opposite signals in each channel is considered. Finally, in the context of optical fibers we show how the reduction of the Of the gain frequency of the IM, induced by saturation, can drastically limit the formation of soliton trains. / Neste trabalho estudamos a influência dos efeitos de não linearidades de ordem cúbica e quını́tica saturável sobre o espectro de ganho da instabilidade modulacional (MI) em acopladores direcionais e fibras ópticas. O ganho de instabilidade foi obtido usando a análise de estabilidade linear padrão. Em particular, estudou-se a combinação de efeitos de nãolinearidade saturável, autoinclinação ou espalhamento sobre MI para regimes de dispersão de velocidade de grupo normal e anômalo. No caso dos acopladores direcionais investigamos como o ganho da instabilidade modulacional é afetado pelo modelo de saturação e pelos efeitos de autoinclinação e espalhamento Raman. Nossos resultados mostram que ganho de instabilidade exibe mudanças significativas devido aos efeitos da não linearidade saturável. Quando analisamos o efeito de autoinclinação, mostramos que sua influência efetiva sobre o ganho da instabilidade modulacional depende da soma algébrica dos parâmetros em cada canal. Analisando o espalhamento Raman, observamos uma quebra de simetria nas regiões de ganho quando se considera o parâmetro do espalhamento de Raman com sinais contrários em cada canal. Finalmente, no contexto de fibras ópticas mostramos como a redução da faixa de frequência do ganho da IM, induzida pela saturação, pode limitar drasticamente a formação de trens de sólitons.

Esta dissertação tem como objetivo principal realizar uma investigação sobre técnicas de identificação e caracterização de não linearidades em dinâmica estrutural. Em particular, busca-se aplicar as transformadas de Hilbert e Hilbert-Huang na identificação e caracterização de um sistema estrutural composto de uma viga metálica em balanço com uma massa concentrada em sua extremidade livre e apresentando uma não linearidade cúbica. Este efeito não linear é obtido através de um fio metálico tracionado que é fixo à extremidade livre da viga em balanço. O sistema é modelado através de ferramentas analíticas convencionais e o modelo do sistema é usado em simulações numéricas. Ensaios experimentais de vibração livre são realizados e os sinais medidos são usados como dados de entradas nas transformadas afim de se investigar o fenômeno não linear presente na estrutura. Os resultados simulados numéricamente apresentaram uma excelente correlação com os resultados experimentais no que se refere à identificação e caracterização do efeito não linear cubico presente no sistema. Algumas distorções entre resultados simulados e experimentais foram observados no que se refere à quantificação do parâmetro não linear, mas de forma geral, as técnicas empregadas produziram resultados bastante satisfatórios. / The major goal of this dissertation is to investigate currently available techniques to the identification and characterization of nonlinearities in structural dynamics. Particular attention is paid to the application of the Hilbert and Hilbert-Huang transforms in the nonlinearity identification and characterization process. For that purpose, a structural system composed of a steel cantilever beam carrying a lumped mass at its free end is used. The nonlinear effect is introduced by attaching the lumped mass to a thin steel wire that is positioned transversely to the beam\'s longitudinal axis. By varying the traction force on this wire different levels of nonlinearity can be observed on the beam\'s bending motion when it undergoes free vibrations. The system analytical model is obtained by employing standard modeling techniques and this model is used in numerical simulations. An experimental survey is carried out on an actual prototype in order to provide a comparison basis for the numerically simulated results. The output vibration signals resulting from either the numerical simulations and experimental tests are then used with the transform methods studied and the results are compared. A good correlation is observed between numerical and experimental data, what is a clear indication of the robustness of the Hilbert and Hilbert-Huang transforms.

All three essays in this thesis are concerned with the interrelation of family, gender and labour market outcomes. The first paper investigates family earnings mobility between parents and sons, and parents and daughters, highlighting the role of assortative mating. The results suggest that daughters are more mobile than sons. I also find that Sweden has a higher degree of mobility compared to the U.S., and that assortative mating is an important underlying channel for earnings transmission. The difference in mobility between the two countries does not inherently depend on factors affecting the marriage match. Moreover, adult economic outcomes are more dependent on family background for those at the lower end of the earnings distribution. The second study analyses the long-run effects of an increase in family size on the 1980-2005 labour market outcomes of Swedish men and women. The decision to have (more) children is dependent on current and future labour market prospects. I use the exogenous variations in the sex composition of the first two children to overcome this endogeneity problem. My findings suggest that having an additional child has a stronger negative impact on earnings than on participation. However, mothers experience a substantial but not complete long-term recovery in earnings. The third paper illustrates the difficulty in disentangling the underlying channels of intergenerational earnings persistence using a path analysis model. On closer examination, such a model has a potential shortcoming since the covariates are correlated to other unobserved factors. The results suggest that education is the most influential mechanism in the earnings transmission process, while IQ, mental ability and BMI are of secondary importance. However, education is sensitive to the inclusion of other covariates and the order in which these are entered into the equation. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript.

In practice, steel frames are analyzed and designed by assuming all beam to column connections as either rigid or simple. In real life, there are no such idealizations as rigid or simple and all connections would actually belong to a group of connections named as semi rigid connections. Various difficulties exist in modeling an accurate non-linear behavior of a steel structure, where one of these challenges is the modeling of semi-rigid behavior of connections. A detailed finite element model would take into account the complex interaction between all surfaces due to contact, friction and bolt pretension besides the material and geometrical nonlinearity effects. All these nonlinearity effects could be simply lumped as a moment-rotation type model at the connection region. Such a methodology is followed in this thesis and the main aim is to study the lumped nonlinear behavior of steel semi-rigid connections on the overall structural responses of steel Special Moment Frames. In this thesis three, nine and fifteen story steel Special Moment Frames are analyzed and designed as rigid frames first, and then the frames are reanalyzed considering non-linear effects due to semi-rigid connections. Changes in the ductility and overstrength reduction factors obtained from pushover curves are compared between the rigid and semi rigid modeling alternatives.

During shearing of stiff clays, plate-shaped clay particles are parallel-oriented in the direction of shear reaching the minimum resistance of &ldquo / residual shear strength&rdquo / . The residual shear strength envelopes of stiff clays are curved, but for practical purposes represented by linear envelopes. This study investigates the nonlinearity of the residual shear strength envelope using experimental evidence (i) from laboratory reversal direct shear tests on two stiff clays (Ankara clay and kaolinite) at 25 to 900 kPa effective normal stresses and (ii) from laboratory data collected from literature. To evaluate the importance of nonlinearity of the envelope for geotechnical engineering practice, by limit equilibrium method, (a) case histories of reactivated landslides are analyzed and (b) a parametric study is carried out. Conclusions of this study are: (1) The residual shear strength envelopes of both Ankara clay and kaolinite are nonlinear, and can be represented by a power function (cohesion is zero). (2) At least 3 reversals or cumulative 20 mm shear displacement of direct shear box is recommended to reach residual condition. (3) Empirical relations between plasticity index and residual friction angle can accurately estimate the residual strength of stiff clays. (4) Nonlinearity is especially important for landslides where average effective normal stress on the shear plane is less than 50 kPa, both for translational and rotational failures. For such slopes using a linear strength envelope overestimates the factor of safety (more significantly for the case of high pore pressures). (5) As the plasticity index increases, the power &ldquo / b&rdquo / of the nonlinear shear strength envelope decreases, indicating more significant nonlinearity. For less plastic materials, using linear and nonlinear shear strength envelopes does not affect the factor of safety.