This dissertation addresses the ultimate strength analysis of nonlinear beam-columns under axial compression, the sensitivity of the ultimate strength, structural optimization and reliability analysis using ultimate strength analysis, and Reliability-Based Design Optimization (RBDO) of the nonlinear beam-columns. The ultimate strength analysis is based on nonlinear beam theory with material and geometric nonlinearities. Nonlinear constitutive law is developed for elastic-perfectly-plastic beam cross-section consisting of base plate and T-bar stiffener. The analysis method is validated using commercial nonlinear finite element analysis. A new direct solving method is developed, which combines the original governing equations with their derivatives with respect to deformation matric and solves for the ultimate strength directly. Structural optimization and reliability analysis use a gradient-based algorithm and need accurate sensitivities of the ultimate strength to design variables. Semi-analytic sensitivity of the ultimate strength is calculated from a linear set of analytical sensitivity equations which use the Jacobian matrix of the direct solving method. The derivatives of the structural residual equations in the sensitivity equation set are calculated using complex step method. The semi-analytic sensitivity is more robust and efficient as compared to finite difference sensitivity. The design variables are the cross-sectional geometric parameters. Random variables include material properties, geometric parameters, initial deflection and nondeterministic load. Failure probabilities calculated by ultimate strength reliability analysis are validated by Monte Carlo Simulation. Double-loop RBDO minimizes structural weight with reliability index constraint. The sensitivity of reliability index with respect to design variables is calculated from the gradient of limit state function at the solution of reliability analysis. By using the ultimate strength direct solving method, semi-analytic sensitivity and gradient-based optimization algorithm, the RBDO method is found to be robust and efficient for nonlinear beam-columns. The ultimate strength direct solving method, semi-analytic sensitivity, structural optimization, reliability analysis, and RBDO method can be applied to more complicated engineering structures including stiffened panels and aerospace/ocean structures. / Ph. D.

Thesis (MScEng)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Higher grade steels are being rolled in South Africa by suppliers and results in structural members having an increased axial and bending moment capacity due to an increased yield stress. Structural elements used in designs are stronger and therefore lighter sections with sufficient axial and bending moment capacity are used. Displacements of structural elements are calculated using the stiffness and Young’s modulus of a profile. These values are not affected by the increased yield stress in higher steel grades and therefore have a negative effect on the displacements of the structure. The potential of these higher grade structural elements are not utilized through serviceability limit state criteria, since the displacement determination does not account for the increased capacities of higher grade steels, but only stiffness and elasticity of the members. Structural analysis of portal frames does not account for the real behaviour of steel connections and column bases. It is assumed that connections and bases are either fully rigid or perfectly pinned. This assumption is used in the analysis and design of the structure. Although it is assumed that connections and bases are either rigid or pinned, the real behaviour is in between these two extremes. Rigid connections exhibit a certain flexibility under loading whereas pinned bases provide a certain restraint under loading. The real behaviour of connections and bases are referred to as the moment-rotation behaviour of the connection. For a certain applied moment to the connection or base, the connection exhibits a certain rotation. The focus of this study is placed on the accuracy and feasibility of modelling the real behaviour of connections and bases in a structural analysis of a portal frame. A connection stiffness is determined from the connection’s moment-rotation behaviour, and is assigned to a rotational spring of zero length in a structural analysis. An experimental investigation was conducted to obtain the real displacement data of a portal frame subject to loads for two different support conditions, i.e. a perfect hinge and grouted-support. A perfect hinge support was used to isolate the moment-rotation response of the ridge and eaves connection. The experimental results were used to compared to the results obtained from a structural analysis to determine the accuracy of the numerical results. A real design case was investigated with load combinations imposed on the frame in accordance with SANS 10160:2011. Three methods of modelling connections and bases in an analysis were considered. Firstly modelling connections as rigid and bases as pinned, secondly modelling connections as linear rotational springs and bases as pinned. Lastly was to model connections as linear rotational springs and bases as non-linear rotational springs. The outcome of the research was that more accurate displacements of a portal frame could be obtained by modelling the real behaviour of rigid connections as rotational springs, but this is not the case with grouted column bases. It is thus not feasible to model the real behaviour of connections and bases in a structural analysis as the current method of modelling connections as rigid and bases as pinned provides reliable and accurate displacement results. / AFRIKAANSE OPSOMMING: Hoë graad staal word tans in Suid Afrika gerol deur verskaffers en lei daartoe dat strukturele elemente oor ’n groter aksiale- en buigmomentkapasiteit het as gevolg van ’n groter vloeispanning. Strukturele elemente in ontwerpe is sterker en gevolglik het ligter elemente die benodigde aksiale- en buigmoment-kapasiteit. Verplasings van strukturele elemente word bepaal vanaf die styfheid en Young modulus van die element. Hierdie waardes word nie beïnvloed deur die groter vloeispanning van hoë graad staal nie, en het dus ’n negatiewe uitwerking op die verplasings van die struktuur. Die potensiaal van die gebruik van hoë graad staal word nie benut in die geval van voldoening aan diensbaarheids kriterium nie, aangesien verplasings bepaal word vanaf die styfheid en elastisiteit van die elemente, en nie vloeispanning nie. Strukturele analise van portaalrame neem nie die ware gedrag van konneksies en kolomvoetstukke in ag nie. Die aanname word gemaak in analises en ontwerpe dat konneksies en voetstukke óf rigied óf geskarnierd is. Hierdie is slegs ’n aanname en in die werklikheid lê die ware gedrag van konneksies en voetstukke tussen hierdie grense. Rigiede konneksies toon ’n sekere buigbaarheid tydens belasting en geskarnierde voetstukke toon ’n sekere beperking teen rotasies. Die ware gedrag van konneksies en voetstukke word gedefinieer as moment-rotasie gedrag. Vir ’n spesifieke aangewende moment, ondergaan die konneksie of voetstuk ’n sekere rotasie. Hierdie studie fokus op die akkuraatheid en uitvoerbaarheid van die modellering van die ware gedrag van konneksies en voetstukke in ’n strukturele analise van portaalrame. Die styfheid van ’n konneksie word bepaal vanaf sy unieke moment-rotasie gedrag, en word ingevoer as ’n styfheid van ’n rotasieveer in ’n strukturele analise. ’n Eksperimentele ondersoek was gedoen om verplasingswaardes van ’n portaalraam onder belastings te bepaal. Twee ondersteunings is ondersoek in die eksperimentele program, naamlik ’n geskarnierde ondersteuning asook ’n breivul ondersteuning. Die gebruik van die geskarnierde ondersteuning isoleer die moment-rotasie gedrag van die nok en dakrand konneksies. Die eksperimentele resultate was gebruik om die akkuraatheid van die resultate vanaf die strukturele analise te ondersoek. Laastens was ’n ontwerpsprobleem ondersoek deur laskombinasies, soos bepaal volgens die riglyne van SANS 10160:2011, op ’n portaalraam aan te wend. Drie gevalle van modellering van konneksies in ’n strukturele analise is ondersoek. Eerstens om konneksies as rigied en voetstukke as geskarnierd te beskou. Tweedens was die konneksies as linieêre rotasievere gemodelleer en voetstukke as geskarnierd te beskou. Laastens was om konneksies as linieêre rotasievere te modeleer en voetstukke as nie-linieêre rotasievere. Die navorsing het getoon dat meer akkurate verplasings van portaalrame bepaal kan word deur rigiede konneksies te modelleer as rotasievere, maar dit is nie die geval met breivul ondersteunings nie. Die gevolg is dat die uitvoerbaarheid van die modellering van konneksies en voetstukke as rotasievere nie effektief is nie, aangesien die huidige metode van die modellering van konneksies as rigied en voetstukke as geskarnierd akkurate en betroubare resultate lewer.

This research introduces and develops a counter-intuitive seismic retrofit strategy, referred to as “Selective Weakening” (SW), for pre-1970s reinforced concrete (RC) frames with a particular emphasis on the upgrading of exterior beam-column joints. By focusing on increasing the displacement and ductility capacities of the beam-column joints, simple retrofit interventions such as selective weakening of the beam and external post-tensioning of the joint can change the local inelastic mechanism and result in improved global lateral and energy dissipation capacities. The thesis first presents an extensive review of the seismic vulnerability and assessment of pre-1970s RC frames. Following a review of the concepts of performance-based seismic retrofit and existing seismic retrofit solutions, a thorough conceptual development of the SW retrofit strategy and techniques is presented. A “local-to-global” design procedure for the design of SW retrofit is proposed. Based on the evaluation of the hierarchy of strength at a subassembly level, a capacity-design retrofit outcome can be achieved using various combinations of levels of beam-weakening and joint post-tensioning. Analytical tools for the assessment and design of the SW-retrofitted beam-column joints are developed and compared with the test results. Nine 2/3-scaled exterior joint subassemblies were tested under quasi-static cyclic loading to demonstrate the feasibility and effectiveness of SW retrofit for non-ductile unreinforced beam-column connections. Parameters considered in the tests included the presence of column lap-splice, slab and transverse beams, levels of post-tensioning forces and location of beam weakening. Extensive instrumentation and a rigorous testing regime allowed for a detailed experimental insight into the seismic behaviour of these as-built and retrofitted joints. Experimental-analytical comparisons highlighted some limitations of existing seismic assessment procedures and helped in developing and validating the SW retrofit design expressions. Interesting insights into the bond behaviour of the plain-round bars, joint shear cracking and post-tensioned joints were made based on the experimental results. To complement the experimental investigation, refined fracture-mechanic finite-element (FE) modelling of the beam-column joint subassemblies and non-linear dynamic time-history analyses of RC frames were carried out. Both the experimental and numerical results have shown the potential of SW retrofit to be a simple and structurally efficient structural rehabilitation strategy for non-ductile RC frames.

Propõe-se um modelo analítico para determinação da resistência ao cisalhamento de nós de pórtico externos em estruturas de concreto armado. Faz-se um estudo de modelos analíticos propostos por pesquisadores e sua aplicação em uma extensa base de dados com resultados experimentais de diferentes ligações, a fim de verificar a eficiência de cada modelo analítico. Em paralelo realiza-se uma análise paramétrica por meio de simulação numérica utilizando o software DIANA® de maneira a compreender o comportamento do nó externo analisando a influência da geometria, nível de tensões no pilar, taxa de armadura da viga e taxa de estribos, e assim propõe-se um modelo de cálculo para a resistência ao cisalhamento com base nos resultados desta análise paramétrica. Por fim o modelo analítico proposto é aplicado na base de dados apresentando bons resultados e comprovando sua eficácia tanto para nós de pórtico externos sem estribo como com estribos. / A proposal of an analytical model for determining the shear strength of exterior reinforced concrete beam-column joints is made in this work. A study of analytical models proposed by researchers and their application in a wide database with experimental results is done in order to verify the effectiveness of each analytical model. In parallel a parametric analysis by numerical simulation using the software DIANA® is performed in order to understand the behavior of the joint by analyzing the influence of the geometry, stress level in the column, longitudinal reinforcement ratio of beam and stirrup ratio and thus, it is proposed a model for predicting the shear strength based on the results of this parametric analysis. Finally the analytical design model proposed is applied to the database and presents good results proving its effectiveness for both external joint with and without stirrups.

Este trabalho consiste em um estudo do comportamento de nós de pórtico de ligações viga-pilar de concreto armado submetidos a ações cíclicas. O trabalho é composto por duas etapas: uma de investigação experimental e outra de cunho teórico. A etapa experimental teve como base os ensaios realizados em cinco modelos físicos de ligações viga-pilar. Os resultados desses ensaios permitiram a observação de diversos aspectos do comportamento estrutural dos nós de pórtico frente às ações cíclicas e a avaliação da influência de algumas variáveis. Neste trabalho, as variáveis escolhidas foram a variação do carregamento aplicado e o detalhamento da armadura transversal na região do nó. Os principais efeitos das ações cíclicas sobre as cinco ligações ensaiadas foram: aumento generalizado dos deslocamentos globais, perda de resistência e de rigidez dos elementos estruturais, propagação e aumento da abertura das fissuras inicialmente formadas, tendência de uniformização das deformações dos estribos do nó ao longo dos ciclos e maior deterioração do nó de pórtico em relação aos elementos viga e pilar. Na etapa teórica, foram empregados essencialmente dois tipos de modelos: i) modelos derivados das idealizações dos modelos clássicos de biela e tirante. ii) modelos simplificados que empregam conceitos da Mecânica do Dano e da Fratura, para a avaliação dos efeitos de perda de rigidez (dano) e das deformações permanentes (plasticidade) na resposta global da ligação viga-pilar. Por fim, foram feitas comparações entre os resultados experimentais e teóricos, chegando-se a importantes conclusões sobre o comportamento mecânico do nó frente às cargas cíclicas e sobre a eficiência dos modelos teóricos em simular a resposta do nó de pórtico e da ligação como um todo. As comparações mostraram que, dentro de certos limites, os modelos teóricos fornecem resultados satisfatórios frente aos resultados experimentais / The present thesis studies the behavior of frame joints in reinforced concrete beam-column joints subjected to cyclic loads. The thesis is divided into two parts: an experimental study and a theoretical-based study. The experimental study is based on laboratory tests on five physical models of beam-column joints. The results of this experimental study permitted the observation of various aspects of the structural behavior of frame joint subjected to cyclic load and estimating the influence of certain variables the affect the behavior of such joints. In the present thesis, the variables studied are the applied load and reinforcement detailing at the joint location. The effects of cyclic loads on the tested joints were: an increase in the overall deformation, loss of strength and stiffness of structural elements, crack propagation and increase in crack width of previously formed cracks, a tendency towards a uniform strain distribution in the transverse reinforcement at the joint section and a more rapid deterioration of frame joints compared to beam/column joints. At the theoretical level, two types of models were used to study the behavior of the joint: i) a model derived from the idealization of the classical strut-tie model, and ii) a simplified damage and fracture mechanics-based model for estimating the effects of the loss of stiffness (damage) and permanent strains (plasticity) on the overall response of the beam column joint. To reach any conclusion on the mechanical behavior and the efficiency of existing theoretical models employed in the design of reinforced beam-column joints and other joints subjected to cyclic loading, a comparative study between the theoretical and experimental results was carried out. The results show that within certain limits, the theoretical models employed are in good agreement with experimental results

Este trabalho abordou o estudo de ligação viga de aço e pilar misto preenchido com concreto a partir da aplicação de modelos numérico para representar o comportamento de duas tipologias de ligação: uma ligação parafusada com aderência entre os parafusos passantes e o núcleo de concreto e uma ligação soldada, validadas com base em resultados experimentais existentes. Os resultados obtidos nessa etapa foram considerados satisfatórios, dentro da proposta de modelagem, apesar das simplificações adotadas. Após a validação do modelo numérico foi desenvolvido um estudo paramétrico dos modelos de ligação para entender a influência de alguns parâmetros no comportamento global da estrutura, que foram: resistência à compressão do concreto e espessura do perfil tubular e perfil da viga, com posterior discussão da relação momento-rotação das duas tipologias de ligação estudadas. A mudança do perfil da viga foi o parâmetro que mais influenciou no comportamento global da ligação parafusada; enquanto que, para a ligação soldada, as variações de todos os parâmetros influenciaram no comportamento da ligação, com destaque para a variação do perfil da viga. / This research studied connections typology between steel beams to column filled with concrete from the application of numerical models that reproduce the behavior of two connections: bolted beam-column connection and welded beam-column connection, which were calibrated using existing experimental results. At this stage, the results were considered satisfactory, within the proposed model, despite the adopted simplifications. After the calibration, a parametric study was developed with the aim of verifying the influence of some parameters on the overall behavior of the structure, such as the concrete compression strength, the thickness of the tube wall and the beam properties, with discussions of bending moment-rotation of the studied connections. The beam properties were the parameters that most influenced on the overall behavior of the bolted beam-column connection, while, for the welded connection, all parameters influenced on the overall behavior of the connection, specially the beam properties.

The main objective of the thesis is evaluating and comparing Second-Order Elastic Analysis Methods defined in two different specifications, AISC 2005 and TS648 (1980). There are many theoretical approaches that can provide exact solution for the problem. However, approximate methods are still needed for design purposes. Simple formulations for code applications were developed, and they are valid as acceptable results can be obtained within admissible error limits. Within the content of the thesis, firstly background information related to second-order effects will be presented. The emphasis will be on the definition of geometric non-linearity, also called as P-&amp / #948 / and P-&amp / #916 / effects. In addition, the approximate methods defined in AISC 2005 (B1 &ndash / B2 Method), and TS648 (1980) will be discussed in detail. Then, example problems will be solved for the demonstration of theoretical formulations for members with and without end translation cases. Also, the results obtained from the structural analysis software, SAP2000, will be compared with the results acquired from the exact and the approximate methods. Finally, conclusions related to the study will be stated.

Este trabalho abordou o estudo de ligação viga de aço e pilar misto preenchido com concreto a partir da aplicação de modelos numérico para representar o comportamento de duas tipologias de ligação: uma ligação parafusada com aderência entre os parafusos passantes e o núcleo de concreto e uma ligação soldada, validadas com base em resultados experimentais existentes. Os resultados obtidos nessa etapa foram considerados satisfatórios, dentro da proposta de modelagem, apesar das simplificações adotadas. Após a validação do modelo numérico foi desenvolvido um estudo paramétrico dos modelos de ligação para entender a influência de alguns parâmetros no comportamento global da estrutura, que foram: resistência à compressão do concreto e espessura do perfil tubular e perfil da viga, com posterior discussão da relação momento-rotação das duas tipologias de ligação estudadas. A mudança do perfil da viga foi o parâmetro que mais influenciou no comportamento global da ligação parafusada; enquanto que, para a ligação soldada, as variações de todos os parâmetros influenciaram no comportamento da ligação, com destaque para a variação do perfil da viga. / This research studied connections typology between steel beams to column filled with concrete from the application of numerical models that reproduce the behavior of two connections: bolted beam-column connection and welded beam-column connection, which were calibrated using existing experimental results. At this stage, the results were considered satisfactory, within the proposed model, despite the adopted simplifications. After the calibration, a parametric study was developed with the aim of verifying the influence of some parameters on the overall behavior of the structure, such as the concrete compression strength, the thickness of the tube wall and the beam properties, with discussions of bending moment-rotation of the studied connections. The beam properties were the parameters that most influenced on the overall behavior of the bolted beam-column connection, while, for the welded connection, all parameters influenced on the overall behavior of the connection, specially the beam properties.

Este trabalho consiste em um estudo do comportamento de nós de pórtico de ligações viga-pilar de concreto armado submetidos a ações cíclicas. O trabalho é composto por duas etapas: uma de investigação experimental e outra de cunho teórico. A etapa experimental teve como base os ensaios realizados em cinco modelos físicos de ligações viga-pilar. Os resultados desses ensaios permitiram a observação de diversos aspectos do comportamento estrutural dos nós de pórtico frente às ações cíclicas e a avaliação da influência de algumas variáveis. Neste trabalho, as variáveis escolhidas foram a variação do carregamento aplicado e o detalhamento da armadura transversal na região do nó. Os principais efeitos das ações cíclicas sobre as cinco ligações ensaiadas foram: aumento generalizado dos deslocamentos globais, perda de resistência e de rigidez dos elementos estruturais, propagação e aumento da abertura das fissuras inicialmente formadas, tendência de uniformização das deformações dos estribos do nó ao longo dos ciclos e maior deterioração do nó de pórtico em relação aos elementos viga e pilar. Na etapa teórica, foram empregados essencialmente dois tipos de modelos: i) modelos derivados das idealizações dos modelos clássicos de biela e tirante. ii) modelos simplificados que empregam conceitos da Mecânica do Dano e da Fratura, para a avaliação dos efeitos de perda de rigidez (dano) e das deformações permanentes (plasticidade) na resposta global da ligação viga-pilar. Por fim, foram feitas comparações entre os resultados experimentais e teóricos, chegando-se a importantes conclusões sobre o comportamento mecânico do nó frente às cargas cíclicas e sobre a eficiência dos modelos teóricos em simular a resposta do nó de pórtico e da ligação como um todo. As comparações mostraram que, dentro de certos limites, os modelos teóricos fornecem resultados satisfatórios frente aos resultados experimentais / The present thesis studies the behavior of frame joints in reinforced concrete beam-column joints subjected to cyclic loads. The thesis is divided into two parts: an experimental study and a theoretical-based study. The experimental study is based on laboratory tests on five physical models of beam-column joints. The results of this experimental study permitted the observation of various aspects of the structural behavior of frame joint subjected to cyclic load and estimating the influence of certain variables the affect the behavior of such joints. In the present thesis, the variables studied are the applied load and reinforcement detailing at the joint location. The effects of cyclic loads on the tested joints were: an increase in the overall deformation, loss of strength and stiffness of structural elements, crack propagation and increase in crack width of previously formed cracks, a tendency towards a uniform strain distribution in the transverse reinforcement at the joint section and a more rapid deterioration of frame joints compared to beam/column joints. At the theoretical level, two types of models were used to study the behavior of the joint: i) a model derived from the idealization of the classical strut-tie model, and ii) a simplified damage and fracture mechanics-based model for estimating the effects of the loss of stiffness (damage) and permanent strains (plasticity) on the overall response of the beam column joint. To reach any conclusion on the mechanical behavior and the efficiency of existing theoretical models employed in the design of reinforced beam-column joints and other joints subjected to cyclic loading, a comparative study between the theoretical and experimental results was carried out. The results show that within certain limits, the theoretical models employed are in good agreement with experimental results

Propõe-se um modelo analítico para determinação da resistência ao cisalhamento de nós de pórtico externos em estruturas de concreto armado. Faz-se um estudo de modelos analíticos propostos por pesquisadores e sua aplicação em uma extensa base de dados com resultados experimentais de diferentes ligações, a fim de verificar a eficiência de cada modelo analítico. Em paralelo realiza-se uma análise paramétrica por meio de simulação numérica utilizando o software DIANA® de maneira a compreender o comportamento do nó externo analisando a influência da geometria, nível de tensões no pilar, taxa de armadura da viga e taxa de estribos, e assim propõe-se um modelo de cálculo para a resistência ao cisalhamento com base nos resultados desta análise paramétrica. Por fim o modelo analítico proposto é aplicado na base de dados apresentando bons resultados e comprovando sua eficácia tanto para nós de pórtico externos sem estribo como com estribos. / A proposal of an analytical model for determining the shear strength of exterior reinforced concrete beam-column joints is made in this work. A study of analytical models proposed by researchers and their application in a wide database with experimental results is done in order to verify the effectiveness of each analytical model. In parallel a parametric analysis by numerical simulation using the software DIANA® is performed in order to understand the behavior of the joint by analyzing the influence of the geometry, stress level in the column, longitudinal reinforcement ratio of beam and stirrup ratio and thus, it is proposed a model for predicting the shear strength based on the results of this parametric analysis. Finally the analytical design model proposed is applied to the database and presents good results proving its effectiveness for both external joint with and without stirrups.