One critical consideration in the design of buildings constructed in the western United States is the weight of settled snow on the roof of the structure. Engineers are tasked with selecting a design snow load that ensures that the building is safe and reliable, without making the construction overly expensive. Western states use historical snow records at weather stations scattered throughout the region to estimate appropriate design snow loads. Various mapping techniques are then used to predict design snow loads between the weather stations. Each state uses different mapping techniques to create their snow load requirements, yet these different techniques have never been compared. In addition, none of the current mapping techniques can account for the uncertainty in the design snow load estimates. We address both issues by formally comparing the existing mapping techniques, as well as creating a new mapping technique that allows the estimated design snow loads to be represented as an interval of values, rather than a single value. In the process, we have improved upon existing methods for creating design snow load requirements and have produced a new tool capable of handling uncertain climate data.

The objective of this research is to develop new stochastic methods based on most probable points (MPPs) for general reliability analysis and reliability-based design optimization of complex engineering systems. The current efforts involves: (1) univariate method with simulation for reliability analysis; (2) univariate method with numerical integration for reliability analysis; (3) multi-point univariate for reliability analysis involving multiple MPPs; and (4) univariate method for design sensitivity analysis and reliability-based design optimization. Two MPP-based univariate decomposition methods were developed for component reliability analysis with highly nonlinear performance functions. Both methods involve novel function decomposition at MPP that facilitates higher-order univariate approximations of a performance function in the rotated Gaussian space. The first method entails Lagrange interpolation of univariate component functions that leads to an explicit performance function and subsequent Monte Carlo simulation. Based on linear or quadratic approximations of the univariate component function in the direction of the MPP, the second method formulates the performance function in a form amenable to an efficient reliability analysis by multiple one-dimensional integrations. Although both methods have comparable computational efficiency, the second method can be extended to derive analytical sensitivity of failure probability for design optimization. For reliability problems entailing multiple MPPs, a multi-point univariate decomposition method was also developed. In addition to the effort of identifying the MPP, the univariate methods require a small number of exact or numerical function evaluations at selected input. Numerical results indicate that the MPP-based univariate methods provide accurate and/or computationally efficient estimates of failure probability than existing methods. Finally, a new univariate decomposition method was developed for design sensitivity analysis and reliability-based design optimization subject to uncertain performance functions in constraints. The method involves a novel univariate approximation of a general multivariate function in the rotated Gaussian space; analytical sensitivity of failure probability with respect to design variables; and standard gradient-based optimization algorithms. In both reliability and sensitivity analyses, the proposed effort has been reduced to performing multiple one-dimensional integrations. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of the sensitivity of failure probability and leads to accurate design optimization of uncertain mechanical systems.

The reliability of structures subjected to time-invariant or time-variant random loads is considered herein. This is an important field of engineering, as it provides the framework for assessing whether newly designed or existing structural systems meet their design requirements in a given lifetime, or whether they experience what is termed “structural failure”. An important aspect of reliability analysis is the study of structures subjected to multiple time-varying loads. For this class of systems, it is well-known that by modelling the loads as (time-variant) random processes, the reliability may be evaluated by considering the outcrossing of a vector process out of a safe domain. However, due to the possibility that the loads may not be fully-dependent, all loads may not necessarily contribute to structural failure. To account for this the treatment of vector-outcrossings may need to allow for the possibility of outcrossings being caused by individual loads, as distinct from combinations of all loads. The procedure used to analyse combinations of loads depends on the stochastic process model used to represent the loads. Two well-known load models have been presented in the literature—they are referred to herein as the ‘on-off’ model and the ‘standard’ model. The ‘on-off’ model typically assumes loads are non-negative, and are either ‘on’ (eg their value is non-zero) or ‘off’ (eg their value is strictly zero). They can contribute to failure only when they are ‘on’. This model is represented by a somewhat artificial ‘composite’ probability distribution, obtained by modifying the original load probability density function (pdf) so that a ‘finite’ non-zero probability represents explicitly the possibility that the load is ‘off’. To implement this model in time-variant analysis, it is necessary to consider all possible combinations of loads being ‘on’ and ‘off’. In contrast, the ‘standard’ model (which is the more commonly used) typically allows loads to be negative; it is also typically represented solely by the original load pdf, and therefore effectively assumes each load is always ‘on’. To allow for the possibility of one or more loads not to cause failure, herein the value of such loads is held ‘constant’ at the time of failure, when the value of all loads actually causing failure is allowed to change. Use of the ‘standard’ model is examined herein. The “Directional Simulation in the Load Space (DS-LS)” approach is a tool used to perform reliability analysis. It is particularly suitable for time-variant analysis, as it allows loads to be represented as random processes, and to be modelled properly. DS-LS has so far been shown to work well for relatively simple structures subjected to one or more time-invariant random loads, and has been used to examine vector outcrossings in systems comprising either discrete or continuous loads. To enable the proper consideration of load combinations, and to provide some improvements in the formulation of the technique, a generalisation of the DS-LS approach is proposed herein. The generalisation is achieved in two stages. The first involves modifying the time-invariant and time-variant DS-LS formulation to allow for the possibility of positioning the origin of DS-LS not only in the ‘safe’ region of the load space (which the formulation currently requires) but in the ‘failure’ region, or even ‘exactly’ on the boundary separating the safe and failure regions. The modifications are necessary because for even simple structures, the ‘exact’ location of the safe and failure region is not always known explicitly ‘a priori’. The second involves developing the time-variant DS-LS formulation to consider explicitly outcrossings caused by combinations of one or more loads, during analysis of systems comprising stationary continuous gaussian loads. To do this, the direction of the load process vector is ‘fixed’ at each point of outcrossing, to physically represent the particular combination of loads causing the outcrossing. By considering each possible load combination, all loads not causing an outcrossing are then held constant during radial integration, thereby modelling those that do not contribute to each outcrossing. The proposed formulation differs from most load combination analysis techniques (which, evidently, simplify the analysis) as it is analytically ‘exact’, and it considers explicitly all possible combinations of loads. The concepts and formulations proposed herein may provide further understanding of reliability analysis performed by DS-LS (or other techniques) and may aid their future development. / PhD Doctorate

Performance-based earthquake engineering is an emerging field of study that complements the prescriptive methods that the design codes provide to ensure adequate seismic performance of structures. Accounting for uncertainties in the performance assessments forms an important component in this area. In this context, the present study focuses on two broad themes; first, treatment of uncertainties and the application of the first-order reliability method (FORM) in finite-element reliability analysis, and second, the seismic risk assessment of reinforced concrete structures for performance states such as, collapse and monetary loss. In the first area, the uncertainties arising from inherent randomness (“aleatory uncertainty”) and due to the lack of knowledge (“epistemic uncertainty”) are identified. A framework for the separation of these uncertainties is proposed. Following this, the applicability of FORM to the linear and nonlinear finite-element structural models under static and dynamic loading is investigated. The case studies indicate that FORM is applicable for linear and nonlinear static problems. Strategies are proposed to circumvent and remedy potential challenges to FORM. In the case of dynamic problems, the application of FORM is studied with an emphasis on cumulative response measures. The limit-state surface is shown to have a closed and nonlinear geometric shape. Solution methods are proposed to obtain probability bounds based on the FORM results. In the application-oriented second area of research, at first, the probability of collapse of a reinforced concrete frame is assessed with nonlinear static analysis. By modelling the post-failure behaviour of individual structural members, the global response of the structure is estimated beyond the component failures. The final application is the probabilistic assessment of monetary loss for a high-rise shear wall building due to the seismic hazard in the Cascadia subduction zone. A 3-dimensional finite-element model of the structure with nonlinear material models is subjected to stochastic ground motions in the reliability analysis. The parameters for the stochastic ground motion model are developed for Vancouver, Canada. Monetary losses due to the damage of structural and non-structural components are included.

A practical engineering design problem is used to examine the over-conservativeness of designs obtained using deterministic optimization with worst-case parameter assumptions and a safety factor. Additionally, an attempted application of reliability-based design optimization (RBDO) demonstrates the limits of RBDO for practical problems. The design problem considered here is TESCO's Internal Casing Drive System (ICDS), which is used in feeding pipeline, or casing, into predrilled holes. After developing a finite element model of the ICDS, experimental data is used to successfully validate modeling methods and assumptions. The validated model is then subjected to multiple analyses to determine an appropriate design configuration to be used as the starting point for optimization. Worst-case, safety factor-based design optimization (SFBDO) is then applied considering two and three design variables, and is successful in increasing the critical load of the ICDS, Pcrit, by 35% and 45%, respectively. An efficient and recognized RBDO method, Sequential Optimization and Reliability Assessment, is selected for application to the design problem to determine an optimum design based on reliability. Due to the optimization formulation, however, SORA cannot be applied. The ICDS design problem represents a practical example that demonstrates the difficulties and limits in applying RBDO to practical engineering design problems. To evaluate the over-conservativeness of worst-case SFBDO, structural reliability analysis is performed on the deterministic optimum designs. It is found that the value of Pcrit for both the two and three variable optimum designs can be increased by 53% while maintaining acceptable probability of failure, demonstrating the over-conservativeness of the worst-case SFBDO. / Thesis (Master, Mechanical and Materials Engineering) -- Queen's University, 2008-09-05 10:51:26.273

Performance-based earthquake engineering is an emerging field of study that complements the prescriptive methods that the design codes provide to ensure adequate seismic performance of structures. Accounting for uncertainties in the performance assessments forms an important component in this area. In this context, the present study focuses on two broad themes; first, treatment of uncertainties and the application of the first-order reliability method (FORM) in finite-element reliability analysis, and second, the seismic risk assessment of reinforced concrete structures for performance states such as, collapse and monetary loss. In the first area, the uncertainties arising from inherent randomness (“aleatory uncertainty”) and due to the lack of knowledge (“epistemic uncertainty”) are identified. A framework for the separation of these uncertainties is proposed. Following this, the applicability of FORM to the linear and nonlinear finite-element structural models under static and dynamic loading is investigated. The case studies indicate that FORM is applicable for linear and nonlinear static problems. Strategies are proposed to circumvent and remedy potential challenges to FORM. In the case of dynamic problems, the application of FORM is studied with an emphasis on cumulative response measures. The limit-state surface is shown to have a closed and nonlinear geometric shape. Solution methods are proposed to obtain probability bounds based on the FORM results. In the application-oriented second area of research, at first, the probability of collapse of a reinforced concrete frame is assessed with nonlinear static analysis. By modelling the post-failure behaviour of individual structural members, the global response of the structure is estimated beyond the component failures. The final application is the probabilistic assessment of monetary loss for a high-rise shear wall building due to the seismic hazard in the Cascadia subduction zone. A 3-dimensional finite-element model of the structure with nonlinear material models is subjected to stochastic ground motions in the reliability analysis. The parameters for the stochastic ground motion model are developed for Vancouver, Canada. Monetary losses due to the damage of structural and non-structural components are included.

Performance-based earthquake engineering is an emerging field of study that complements the prescriptive methods that the design codes provide to ensure adequate seismic performance of structures. Accounting for uncertainties in the performance assessments forms an important component in this area. In this context, the present study focuses on two broad themes; first, treatment of uncertainties and the application of the first-order reliability method (FORM) in finite-element reliability analysis, and second, the seismic risk assessment of reinforced concrete structures for performance states such as, collapse and monetary loss. In the first area, the uncertainties arising from inherent randomness (“aleatory uncertainty”) and due to the lack of knowledge (“epistemic uncertainty”) are identified. A framework for the separation of these uncertainties is proposed. Following this, the applicability of FORM to the linear and nonlinear finite-element structural models under static and dynamic loading is investigated. The case studies indicate that FORM is applicable for linear and nonlinear static problems. Strategies are proposed to circumvent and remedy potential challenges to FORM. In the case of dynamic problems, the application of FORM is studied with an emphasis on cumulative response measures. The limit-state surface is shown to have a closed and nonlinear geometric shape. Solution methods are proposed to obtain probability bounds based on the FORM results. In the application-oriented second area of research, at first, the probability of collapse of a reinforced concrete frame is assessed with nonlinear static analysis. By modelling the post-failure behaviour of individual structural members, the global response of the structure is estimated beyond the component failures. The final application is the probabilistic assessment of monetary loss for a high-rise shear wall building due to the seismic hazard in the Cascadia subduction zone. A 3-dimensional finite-element model of the structure with nonlinear material models is subjected to stochastic ground motions in the reliability analysis. The parameters for the stochastic ground motion model are developed for Vancouver, Canada. Monetary losses due to the damage of structural and non-structural components are included. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

This thesis presents the results of an investigation into the evaluation of a selected test bridge using instrumentation to obtain site-specific factors contributing to the evaluation, with the ultimate objective of improving the estimate of the bridges reliability in order to assess allowable loading more accurately. The experimental portion of the research program involved instrumenting the test bridge with strain gauges, and recording field measurements using two forms of loading. The analytical portion of the research program involved the analysis of the bridge in the as-designed state, based on the design drawings and specification, followed by a re-analysis of the bridge using the site-specific factors measured on-site. The bridge was evaluated using methods outlined in the Canadian Highway Bridge Design Code CAN/CSA-S6-00 (CSA 2000). <p>The test bridge is located near the community of Hudson Bay, Saskatchewan. The bridge is constructed of steel-reinforced concrete, and there are three, three-span arch-shaped girders. There are also external steel bars added after initial construction to increase the midspan bending moment resistance. In total, 45 strain gauges were placed on the middle spans of the three girders to record strain induced by two forms of loading: controlled loading, in which a truck of known weight and dimensions was driven over the bridge in a number of pre-determined configurations, and in-situ loading, in which normal truck traffic was used. The current allowable loading on the bridge is a gross vehicle weight of 62.5 t, although increasing the allowable loading to 110 t has been proposed, along with two strengthening alternatives to make this increased loading feasible. <p>To provide a base-line analysis for comparison purposes, the bridge was first evaluated based strictly on information taken from the design drawings and specifications. The evaluation was performed using the load and resistance factor method, in which load and resistance factors were used to account for uncertainty, as well as by the mean load method, in which statistical properties of the variables parameters included in the design were used to account for uncertainty. The result of the load and resistance factor method was a live load capacity factor, indicating the overall rating of the bridge. In addition to the live load capacity factor, the mean load method was also used to determine the reliability index. The results of the as-designed analysis showed that the mean load method gave more conservative estimates of the bridge capacity. Furthermore, it was determined that, based on these assessments, the bridge would not have sufficient capacity to carry the proposed 110 t truck loads.<p>The bridge was re-evaluated using site-specific factors with the mean load method. Using the measured strains, statistical parameters were determined for live load effects, distribution factors, dynamic load allowance, and resistance. Statistical parameters that could not be obtained readily through testing were obtained from the literature. The results indicated that code-predicted estimates of a number of factors were highly conservative. Flexural and shear load effects in the girders were found to be less than 15% of the theoretical predictions, as a result of apparent arching action in the girders, generating significant axial forces. For this arching action to occur, horizontal restraint was required at the supports, either through unanticipated restraint in the bearings, or tension tie action of the tensile girder reinforcement. Furthermore, the dynamic amplification was found to be less than 1.0. The resulting reliability indices indicated that the bridge would be safe under the proposed increased allowable loading (110 t). <p>Finite element models were used to confirm the dynamic amplification observations and examine the effects of different degrees of bearing restraint. The model showed results similar to those measured for dynamic amplification. It was found that if the bearings were to become completely fixed against horizontal translation, the bridge would become overloaded as a result of increased shear effects, demonstrating the need for proper bearing maintenance. <p>An analysis of relative costs was completed to determine the most cost-effective solution for hauling logs. Assumptions were made regarding truck and maintenance and operating costs. The results indicated that the most economic solution was to use the method outlined in the research to increase the allowable loading on the bridge to 110 t, over the strengthening alternatives and simply leaving the bridge in the current state.

Overheight vehicle impact to bridge decks is a major problem in the transportation networks in the United States. An important factor that causes this problem is inadequate vertical clearance of bridges. Using steel pedestals to elevate bridge decks is an efficient and cost-effective solution for this problem. So far, steel pedestals have been used in the low seismic regions of the United States and therefore, their design has been based on providing enough strength to carry vertical loads and the lateral behavior of bridges elevated with pedestals have not been a major concern. But even in low seismic zones the seismic hazard should not be completely ignored. Also there might be some bridges in medium or high seismic regions that need to be elevated because of the lack of enough vertical clearance and using steel pedestals can be considered as an option for elevating those bridges. To address the mentioned needs, this dissertation proposes a framework to determine the structural reliability of bridges elevated with steel pedestals by developing probabilistic capacity and demand models for the slab-on-girder bridges subjected to lateral loads. This study first compares the behavior of previously tested pedestals with the behavior of elastomeric bearings in low seismic regions using statistical tests. Then, to provide a general framework, which can be applied to all bridges that are elevated with steel pedestals, this dissertation develops probabilistic capacity and demand models for steel pedestals considering all the aleatory and epistemic uncertainties of the problem. Using the developed probabilistic models along with the available models for other components of bridges, seismic fragility curves for elevated bridges are obtained and used to determine the structural reliability. Finally, this study uses the developed framework in a decision analysis that helps the engineering community and decision makers to check if the installation of steel pedestals on a specific bridge has financial justification or not. Results show that for a typical two-span slab-on-girder bridge, the use of steel pedestals has financial justification only in low seismic regions and if the societal benefits of elevating the bridge can at least cover the installation cost of pedestals.

This thesis presents the results of an investigation into the evaluation of a selected test bridge using instrumentation to obtain site-specific factors contributing to the evaluation, with the ultimate objective of improving the estimate of the bridges reliability in order to assess allowable loading more accurately. The experimental portion of the research program involved instrumenting the test bridge with strain gauges, and recording field measurements using two forms of loading. The analytical portion of the research program involved the analysis of the bridge in the as-designed state, based on the design drawings and specification, followed by a re-analysis of the bridge using the site-specific factors measured on-site. The bridge was evaluated using methods outlined in the Canadian Highway Bridge Design Code CAN/CSA-S6-00 (CSA 2000). <p>The test bridge is located near the community of Hudson Bay, Saskatchewan. The bridge is constructed of steel-reinforced concrete, and there are three, three-span arch-shaped girders. There are also external steel bars added after initial construction to increase the midspan bending moment resistance. In total, 45 strain gauges were placed on the middle spans of the three girders to record strain induced by two forms of loading: controlled loading, in which a truck of known weight and dimensions was driven over the bridge in a number of pre-determined configurations, and in-situ loading, in which normal truck traffic was used. The current allowable loading on the bridge is a gross vehicle weight of 62.5 t, although increasing the allowable loading to 110 t has been proposed, along with two strengthening alternatives to make this increased loading feasible. <p>To provide a base-line analysis for comparison purposes, the bridge was first evaluated based strictly on information taken from the design drawings and specifications. The evaluation was performed using the load and resistance factor method, in which load and resistance factors were used to account for uncertainty, as well as by the mean load method, in which statistical properties of the variables parameters included in the design were used to account for uncertainty. The result of the load and resistance factor method was a live load capacity factor, indicating the overall rating of the bridge. In addition to the live load capacity factor, the mean load method was also used to determine the reliability index. The results of the as-designed analysis showed that the mean load method gave more conservative estimates of the bridge capacity. Furthermore, it was determined that, based on these assessments, the bridge would not have sufficient capacity to carry the proposed 110 t truck loads.<p>The bridge was re-evaluated using site-specific factors with the mean load method. Using the measured strains, statistical parameters were determined for live load effects, distribution factors, dynamic load allowance, and resistance. Statistical parameters that could not be obtained readily through testing were obtained from the literature. The results indicated that code-predicted estimates of a number of factors were highly conservative. Flexural and shear load effects in the girders were found to be less than 15% of the theoretical predictions, as a result of apparent arching action in the girders, generating significant axial forces. For this arching action to occur, horizontal restraint was required at the supports, either through unanticipated restraint in the bearings, or tension tie action of the tensile girder reinforcement. Furthermore, the dynamic amplification was found to be less than 1.0. The resulting reliability indices indicated that the bridge would be safe under the proposed increased allowable loading (110 t). <p>Finite element models were used to confirm the dynamic amplification observations and examine the effects of different degrees of bearing restraint. The model showed results similar to those measured for dynamic amplification. It was found that if the bearings were to become completely fixed against horizontal translation, the bridge would become overloaded as a result of increased shear effects, demonstrating the need for proper bearing maintenance. <p>An analysis of relative costs was completed to determine the most cost-effective solution for hauling logs. Assumptions were made regarding truck and maintenance and operating costs. The results indicated that the most economic solution was to use the method outlined in the research to increase the allowable loading on the bridge to 110 t, over the strengthening alternatives and simply leaving the bridge in the current state.