The effect of panel zone (PZ) flexibility and yielding on the stiffness and strength of steel moment resisting frames (SMRF) has been the topic of numerous papers over the past thirty years. When properly detailed, the PZ is an excellent source of energy dissipation, even under large inelastic deformations. Due to these large inelastic deformations, the PZ region may also be a weak link in steel moment frame behavior. Because of the importance of PZ deformation in the behavior of steel frames, accurate modeling of this region is critical. Two of the most commonly used mathematical models for representing PZ behavior are investigated. They are referred to herein as the Krawinkler model and the Scissors model. From the literature review conducted at the beginning of this study, it was determined that there were no PZ models available that accounted for the elastic drift associated with PZ flexure which could be used in computer representations using commercial software that is currently available. This thesis details the analytical work used to establish the estimated elastic drift associated with PZ flexure and a method to include this estimated drift and the contribution of continuity plates in the Krawinkler and Scissors models. This study is initially focused on elastic deformations of individual structural subassemblages. First, formulas are derived to account for each major elastic component of drift in an individual subassemblage. The results from these derivations were implemented into a computer program named PANELS to allow for rapid calculation of the estimated drifts. Then, the properties (elastic and inelastic) for the Krawinkler and Scissors models are derived in their entirety. The Krawinkler model's results are compared to the results from PANELS, neglecting the PZ flexural component in PANELS and any inelastic contributions in the Krawinkler model. Since the Krawinkler model does not include PZ flexure, this established that the derived formulas accounted for all the remaining sources of elastic strain energy, assuming that the Krawinkler model is accurate. The results from PANELS are compared to those from finite element models developed using ABAQUS. Using the ABAQUS results, a method for determining the elastic drift associated with PZ flexure in PANELS is presented. A detailed inelastic study of the Krawinkler and Scissors models is then conducted both on the subassemblage level and on full structural frames to determine any differences associated with them. First, the two models are compared to each other on a subassemblage level to ensure that they both give the same results. Then, both PZ models are included in multiple full structural frames using various design configurations and loading conditions to ascertain their differences. Initially it was believed that there would be a large disparity between the two models. This study shows that there is actually little difference between the two models, although the kinematics of the Scissors model is still questionable. Elastic and inelastic comparisons between the PANELS formulas (elastic) and the ABAQUS models (elastic and inelastic) and data collected from tests performed at Lehigh University by Dr. James Ricles are then presented. This was done to show that the ABAQUS models and the PANELS formulas (including the PZ flexural component) give an accurate estimation of the drift of a subassemblage. The results from these comparisons show that the modeling techniques used are accurate and not including PZ flexural component of drift will cause the overall drift estimate to be unconservative. Finally, a method of including the elastic component of drift attributed to PZ flexure and continuity plates in both models is presented. The Ricles' Lehigh test data is again used in an inelastic comparison between the original Krawinkler and Scissors models and their updated counterparts. These comparisons show that including this component enables both the Krawinkler and Scissors models to more accurately estimate the total drift of an individual subassemblage. / Master of Science

<p> A research programme is presented for the analytical. evaluation of the deflections of Vierendeel trusses comprised of semi-rigid connections under panel feint loadings. The semi-rigid connections are made of two unequal width square ESS members welded at right angles. As the flexibility of the joints increases when the width ratio is less than 1.0, the joints are unable to develop the moment capacity of the web member and excessive deflections limit functional capability of the truss. Hence, several types of joint reinforcement are recommended. A yield line method is attempted to estimate the strength capacity of the joint with and without reinforcements. In addition, a plate analysis forms the basis for estimating elastic joint stiffness for evaluating anticipated deflections at mid span. </p> / Thesis / Master of Engineering (MEngr)

Timber moment resisting connections have gained considerable interest in structural design due to the numerous advantages offered by timber as a lightweight, renewable, sustainable, and aesthetically pleasing material. This research focuses on investigating the feasibility and potential benefits of hybrid timber-steel moment connections in enhancing the seismic performance and ductility of timber structures. The objective is to evaluate the response modification factors of the hybrid timbersteel moment-resisting frames to see if this type of moment connection has the ability to perform like steel moment-resisting frames in lateral loadings. The process by which the studied frames were designed was focused on preventing damage to timber elements by inducing inelastic deformations exclusively in the steel beams, while the remaining parts of the frame retain their elasticity. Nonlinear static analysis is employed to evaluate the force modification factors and nonlinear behavior of the selected structures. In this study, a total of 18 frames with different span lengths, numbers of stories, and seismicity levels were analyzed to comprehensively investigate their seismic performance. The frames were designed to represent a range of practical configurations commonly found in timber structures. The span lengths of 4, 6, and 8 meters were considered. The number of stories were 2, 4, and 6, and the frames were located in Montreal, QC, and Vancouver, BC, which are known for having varying seismic conditions. By considering a diverse set of frames, this study tried to provide a comprehensive understanding of the behavior and performance of different timber frame structures under seismic loading, taking into account the effects of span length, number of stories, and regional seismic conditions. The results of the analysis offer a preliminary understanding of the seismic performance and potential advantages of steel-timber moment connection frames. However, it should be noted that further research is needed to conduct full-scale experimental tests to validate the proposed connections and gather more accurate data. The findings from this study have the potential to contribute to the development of new seismic provisions for moment connection timber frame systems, advancing the field of timber structural design and offering potential design schemes that increase ductility and performance in timber moment resisting connections.

The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximum-entropy family of moment closures offers models described by hyperbolic systems of balance laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment model has the ability to provide accurate solutions where the Navier-Stokes equations lose physical validity due to the solution being too far from local equilibrium. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, higher-order maximum-entropy closures cannot be expressed in closed form. The only known affordable option is to propose approximations. Previous approximations to the fourteen-moment maximum-entropy model have been proposed [McDonald and Torrilhon, 2014]. Although this fourteen-moment model also predicts viscous flow with heat transfer, the necessary moments to close the system renders it more difficult to approximate accurately than the twenty-one moment model. The proposed approximation for the fourteen-moment model also has realizable states for which hyperbolicity is lost. Unfortunately, the velocity distribution function associated with the twenty-one moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in closing fluxes that can only be obtained through complex numerical methods. The goal of this work is to present a new approximation to the closing fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary results show that a proposed approximation is able to provide shock predictions that are in good agreement with the Boltzmann equation and surpassing the prediction of the Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity flows are computed using a novel approach to Knudsen layer boundary conditions. A dispersion analysis as well as an investigation of the hyperbolicity of the model is also shown. The Couette flow results are compared against Navier-Stokes and the free-molecular analytical solutions for a varying Knudsen number, for which the twenty-one moment model show good agreement over the domain. The shock-tube problem is also computed for different Knudsen numbers. The results are compared with the one obtained by directly solving the BGK equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model shows good agreement with the direct simulation Monte-Carlo (DSMC) solution.

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