Using rulers and protractors, we will analyze scale factors when dilating shapes. Participants will double and triple various triangles. Our discussion and activity will focus on the mathematics of similar figures including angle measures, scale factors, and algebraic rules that can be used to predict how the figures are affected.

Participants will investigate the results of doubling & tripling the dimensions of triangles. Mathematical foci include measurement, area, perimeter, and similarity & congruence.

This research focuses on the effect of several convex and concave nose shapes on cavity formation for both hydrophilic and hydrophobic projectiles. It specifically investigates the effect of convex shape on the threshold velocity for cavity formation as well as the effect of concave shapes on cavity formation in terms of impact velocity, geometry of the concave shape and wettability of the projectile. For the convex cases, the streamlined axisymmetric shape significantly increases the threshold velocity when cavities form and is most pronounced for the ogive and cone. The study demonstrates that measuring the wetting angle and impact velocity is not enough to predict cavity behavior, rather the roughness and nose shape must also be taken into consideration for convex projectiles. For the concave cases, the cavities formed are highly influenced by impact speed and nose shape. Wetting angle did not have any visible effect on the cavity formed at higher impact speeds (7 m/s). The dynamics of the cavity formation are dominated by the pocket of trapped air formed when the concave projectiles impact the water. At low impact speeds (~0-1 m/s) the trapped air can separate the flow from the leading edge of the projectile nose when venting out and cause a large cavity to form, depending on the specific concave shape and speed. At moderate impact speeds (1-4 m/s) the trapped air will vent completely underwater forming a small ring-shaped cavity. At high impact speeds (4-10 m/s) the trapped pocket of air compresses tremendously and causes an unsteady pressure pulse, which can result in the formation of a bubble and jet in front of the cavity. The jet is formed by water passing behind the pocket of trapped air along the walls of the concave nose and converging into a jet at the top of the concave shape and entraining the trapped air as it descends.

This paper focuses on illustrating the effectiveness of the new material model, 𝜏𝐵𝑇𝑅 in comparison with the Specifications for Structural Steel Buildings (2016) material model, 𝜏𝐴𝐼𝑆𝐶 , against a detailed finite element model to determine the accuracy of modeling the inelastic behavior of steel W-Shapes. A total of seven steel columns were analyzed, using a W8x31 section, and eleven benchmark frames to compare the performance of the two material models. An ultimate strength study was conducted using the following slenderness ratios, L/r, of 40, 60, 80, 100, 120, 160, and 200 and oriented such that minor-axis bending occurs. The benchmark frames were modeled under a limit load analysis to illustrate the magnitude of stiffness reduction considering both major and minor-axis bending. Lateral displacements were recorded and compared for the eleven frames up to the collapse condition. Additional information is provided discussing the capabilities of the two material models and their performance when compared to a detailed finite element model. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

ABSTRACT Forced Vibration Testing and Analysis of Pre- and Post- Retrofit Buildings Erica Dawn Jacobsen The primary goal of the thesis was to detect the retrofit through vibration testing of both buildings. The secondary goal focused on correctly identifying the behavior of the building through FVT, comparing that behavior to computational model predictions, and determining the necessary level of detail to include in the computational modeling. Forced vibration testing (FVT) of two stiff-wall/flexible-diaphragm buildings yielded natural frequencies and mode shapes for the two buildings. The buildings were nearly identical with the exception that one had been retrofitted. Both buildings were comprised of concrete shearwalls and steel moment frames in the north/south direction and moment frames in the east/west direction. The retrofit strengthened the moment connections and added braces to the perimeter walls in the east/west direction. The natural frequencies were found through FVT by setting a 30-lb shaker on the roof of both buildings and sweeping through a range of frequencies in both the east/west and north/south directions. Accelerometers were placed on the building to detect the accelerations. The peaks on the Fast Fourier Transform (FFT) graphs indicated the frequencies at which the structure resonated. Mode shapes were tested for by placing the shaker in a position ideal for exciting the mode and setting the shaker to the natural frequency detected from the FFT graphs. The accelerometers were placed around the roof of the building to record the mode shape. After testing, computational models were created to determine if the models could accurately predict the frequencies and mode shapes of the buildings as well as the effect of the retrofit. A series of increasingly complex computational models, ranging from hand calculations to 3D models, were created to determine the level of detail necessary to predict the building behavior. Natural frequencies were the primary criteria used to determine whether the model accurately predicted the building behavior. The mid-diaphragm deflection and base shear from spectral analysis were the final criteria used to compare these select models. It was determined that in order to properly capture the modal behavior of the building, the sawtooth framing, major beams, and the lateral-force-resisting-system (LFRS) must be modeled. Though the mode shape of the building is dominated by the flexible diaphragm, the LFRS is necessary to model to accurately predict both the natural frequency of the building as well as the diaphragm deflection.

Encoding the shape of objects within the visual environment is one of the important roles of the visual system. This thesis investigates the proposition that human sensitivity to a broad range of closed-contour shapes is underpinned by multiple shape channels (Loffler, Wilson, & Wilkinson, 2003). Radial frequency (RF) contours are a novel type of stimulus that can be used to represent simple and complex shapes; they are created by sinusoidally modulating the radius of a circle, where the number of cycles of modulation defines the RF number (Wilkinson, Wilson, & Habak, 1998). This thesis uses RF contours to enhance our understanding of the visual processes which support shape perception. The first part of the thesis combines low and high RF components, which Loffler et al. have suggested are detected by separate global and local processes respectively, onto a single contour and shows that, even when combined, the components are detected independently at threshold. The second part of the thesis combines low RF components from across the range where global processing has been demonstrated (up to approximately RF10) onto a single contour in order to test for interactions between them. The resulting data reveal that multiple narrow-band contour shape channels are required to account for performance, and also indicate that these shape channels have inhibitory connections between them. The third part of the thesis examines the local characteristics which are used to represent shape information within these channels. The results show that both the breadth (polar angle subtended) of individual curvature features, and their relative angular positions (in relation to object centre) are important for representing RF shapes; however, processing is IV not tuned for object size, or for modulation amplitude. In addition, we show that luminance and contrast cues are effectively combined at the level where these patterns are detected, indicating a single later processing stage is adequate to explain performance for these pattern characteristics. Overall the findings show that narrow-band shape channels are a useful way to explain sensitivity to a broad range of closed-contour shapes. Modifications to the current RF detection model (Poirier & Wilson, 2006) are required to incorporate inhibitory connections between shape channels and also, to accommodate the effective integration of luminance and contrast cues.

<strong>Objective:</strong> This study was designed to inform the development of a web tool to simplify access to and analysis of available evidence for youth tobacco control. This tool will support planning, evaluation and research related to youth tobacco control. Primary objectives were to examine useful functions and characteristics of potential end-users. <br /> <strong>Methods:</strong> A pilot-tested, web-based questionnaire was administered to potential end-users (N=43). A purposeful sample of participants was selected to represent individuals working in youth tobacco control from research, health, and education sectors in Western, Central, and Eastern Regions of Canada. Data were analyzed using frequency calculations and cross-tabulations by sector. Results guided the creation of a prototype. Interviews were conducted with a subset of participants (N=6) to validate the prototype and identify further functions. Data were analyzed using content analysis. <br /> <strong>Results:</strong> Useful functions of the prototype identified varied slightly across sectors. Research participants felt that access to raw data was a useful function. Health sector practitioners were interested in accessing data and creating summary reports of data. Educators were mostly interested in obtaining summary information from data, through reports. Further, health and research participants felt the information resources function was redundant. This research has provided important insights that will guide the development of a functional web tool. <br /> <strong>Implications:</strong> This tool can enable users to identify effective interventions, track the progress of school or health regions relative to benchmarks, and identify high risk schools or communities to target intervention efforts. This tool is an innovative way to maximize the use of available resources to link research, policy, and practice.

This work develops optimal assembly sequences for modular building blocks. The underlying concept is that an automated device could take a virtual shape such as a CAD file, and automatically decide how to physically build the shape using simple, identical building blocks. This entails deciding where to place blocks inside the shape and generating an efficient assembly sequence that a robot could use to build the shape. The blocks are defined in a general, parameterized manner such that the model can be easily modified in the future. The primary focus of this work is the development of methods for generating assembly sequences in a time-feasible manner that ensure static stability at each step of the assembly. Most existing research focuses on complete enumeration of every possible assembly sequence and evaluation of many possible sequences. This, however, is not practical for systems with a large number of parts for two reasons: (1) the number of possible assembly sequences is exponential in the number of parts, and (2) each static stability test is very time-consuming. The approach proposed here is to develop a multi-hierarchical rule-based approach to assembly sequences. This is accomplished by formalizing and justifying both high-level and mid-level assembly rules based on static considerations. Application of these rules helps develop assembly sequences rapidly. The assembly sequence is developed in a time-feasible manner according to the geometry of the structure, rather than evaluating statics along the way. This work only evaluates the static stability of each step of the assembly once. The behavior of the various rules is observed both numerically and through theory, and guidelines are developed to suggest which rules to apply. A secondary focus of this work is to introduce methods by which the inside of the structure can be optimized. This structure optimization research is implemented by genetic algorithms that solve the multi-objective optimization problem in two dimensions, and can be extended to three dimensions.

This study investigates the actual glaciers and the major properties of glacial landscapes (valleys, cirques and lakes) located over the Eastern Black Sea mountain chain using Geographic Information Systems (GIS) and Remote Sensing (RS) technologies. A database is created for each glacial feature that includes fundamental properties of each landscape. Data layers used in the study include digital and analog topographic maps, satellite images, geological maps and drainage maps. The studies carried out yielded identification of 93 glacial valleys (30 main, 63 tributary), 1222 cirques and 685 lakes. Several properties (length, size, aspect, elevation, slope, orientation, roundness, elongation) of each glacial landscape are investigated for the northern and southern parts separately. The frequency of each landscape is found to be more in the northern part of the area. Total area of the actual glacier is found as 0.43 to 0.53 km2 by two methods of remote sensing applications.

Many technologically important alloys such as Ni base superalloys and Ti-Al base alloys benefit from the precipitation of an ordered β phase from a disordered α matrix. When the crystallographic symmetry of the β phase is a subgroup of that of the disordered α phase, the microstructure may contain multiple orientational variants of the β phase, each with its own (anisotropic, crystallographically equivalent) misfit (lattice parameter mismatch) with the matrix phase. Examples include orthorhombic precipitates in a hexagonal matrix in Ti-Al-Nb alloys, and tetragonal precipitates in a cubic matrix in ZrO2-Y2O3. We have studied two-phase microstructures containing multiple variants of the precipitate phase. In particular, we have used phase field simulations to study the effect of elastic stresses in a two dimensional system containing a disordered matrix and three different orientational variants of the precipitate phase, with a view to elucidate the effect of different levels of anisotropy in misfit. We consider a two dimensional, elastically homogeneous and isotropic model system in which the matrix (α) and precipitate (β) phases have hexagonal and rectangular symmetries, respectively, giving rise to three orientational variants of the β phase. Therefore, our phase field model has composition (c) and three order parameters (η1, η2, η3) as the field variables.Due to the difference in crystallographic symmetry, the precipitate-matrix misfit strain tensor, ε*, can be anisotropic. ε*maybe represented in its principal form as ε *= (ε xx 0 ) 0 εyy where ε xx and ε yy are the principal components of the misfit tensor. We define t= εyy/εxx as the parameter representing anisotropy in the misfit. In this thesis, we report the results of our systematic study of microstructural evolution in systems with different values of t, representing different levels of anisotropy in misfit: •Case A: t=1 (dilatational or isotropic misfit) • Case B: 0 <t<1 (principal misfit components are unequal but have the same sign) • Case C: t=0 (the principal misfit along the y direction is zero) • Case D: -1 <t<0 (principal misfit components have opposite signs and unequal magnitudes) • Case E: t= -1 (principal misfit components are equal, but with opposite signs; pure shear) In Cases D and E, there is an invariant line along which the normal misfit is zero. In Case D, this invariant line is at ±54.72◦, and in Case E, it is at ±45◦, with respect to the x-axis. Our simulations of microstructural evolution in this system are based on numerical integration of the Cahn-Hilliard and Cahn-Allen equations which govern the evolution of composition and order parameter fields, respectively. In each case, we have studied two different situations: isolated particle (single variant) and many interacting particles (multivariant). Dynamical growth shape of an isolated precipitate In systems with an isotropic misfit (Case A), the precipitate shape remains circular at all sizes. In Cases B and C, the precipitate shape is elongated along the y-axis, which is also the direction in which the magnitude of the misfit strain is lower. In all these cases, the symmetry of the particle shape remains unaltered at all sizes. In contrast, in Cases D and E, the particle shape exhibits a symmetry-breaking transition. In Case D, the precipitate elongates initially along the y direction (i.e. the direction of lower absolute misfit), before undergoing a transition in which the mirror symmetry normal to x and yaxes is lost. In Case E, the particle has an initial square-like shape (with its sides normal to the 11directions) before losing its four-fold rotation axis to become rectangle-like with its long axis along one of the the 11directions. The critical precipitate size at which the symmetry-breaking shape transition occurs is obtained using bifurcation diagrams. In both Cases D and E, the critical size for the dynamical growth shapes is larger than those for equilibrium shapes[1].This critical size is larger when the matrix supersaturation is higher or shear modulus is lower. Microstructural Evolution In all the five cases, the elastic stresses have a common effect: they lead to microstructures in which the precipitate volume fraction is lower than that in a system with no misfit. This observation is consistent with the results from the thermodynamics of stressed solids that show that a precipitate-matrix misfit increases the interfacial composition in both the matrix and the precipitate phase. In systems with isotropic misfit (Case A), the microstructure consists of isolated circular domains of the precipitate phase that retain their circular shape during growth and subsequent coarsening. In Cases B and C with anisotropic misfit with t≥0, the three orientational variants of the precipitate phase are elongated along the directions of lower misfit (y-axis and ±120◦to y-axis). At a given size, particles in Case C (in which one of the principal misfits is zero) are more elongated than those in Case B. Systems with a higher shear modulus enhance the effect of misfit stresses, and therefore, lead to thinner and longer precipitates. When the precipitate volume fraction is increased, these elongated precipitates interact with (and impinge against) one another to a greater extent, and acquire a more jagged appearance. For Cases D and E, each orientation domain is associated with an invariant line along which the normal misfit is zero. Thus, in Case D, early stage microstructures show particles elongated along directions of lower absolute misfit (y-axis and ±120°to y-axis). At the later stages, the domains of the precipitate phase tend to orient along the invariant lines; this leads some of the particles to acquire a ‘Z’ shape before they completely re-orient themselves along the invariant line. In Case E, each variant grows as a thin plate elongating along the invariant line. The growth and impingement of these thin plates leads to a microstructure exhibiting complex multi-domain patterns such as stars, wedges, triangles, and checkerboard. These patterns have been compared (and are in good agreement) with experimental observations in Ti-Al-Nb alloys containing the precipitate (O) and matrix (α2)phases[2]. Since in Case E the sum of misfit strains of the three variants is zero, elastic energy considerations point to the possibility of compact, self-accommodating clusters of the three variants, sharing antiphase boundaries (APBs). Thus, if the APB energy is sufficiently low, the microstructure may consist of such compact clusters. Our simulations with such low APB energy do show triangle shaped clusters with six separate particles (two of each variant)in a self-accommodating pattern. (Refer PDF file)