In this dissertation, the response of a slender, elastic, cantilevered beam to a simple harmonic excitation is investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. The nonlinear equations governing the motion of the beam are derived by the Lagrangian approach. The deflections are expressed as expansions in terms of the linear free-vibration modes. Galerkin’s method is used to eliminate the spatial functions from the governing equations. Three modes are used in this procedure. Approximate solutions of the temporal equations are determined by the method of multiple scales. Four first-order ordinary differential equations govern the amplitudes and phases, and predict a whirling motion under certain situations. The solutions of the modulation equations can be fixed points, limit cycles or chaotic motions. Previous studies considered whirling produced by a primary resonance. In this dissertation, secondary resonances are considered in addition to primary resonance. Previous derivations of equations of motion contain only the linear and cubic terms without consideration of the static displacement produced by the weight of the beam. As a result of this static deflection, there are quadratic terms in the governing equations which introduce the possibility of a superharmonic resonance of order two and a subharmonic resonance of order two. It is shown that out-of-plane motion is possible in every resonance when the principal moments of inertia of the beam cross-section are approximately equal. The longer the beam is, the more prominent the whirling motion becomes. If the excitation frequency is increased or decreased through a resonance, for most cases, the non-stationary response from the method of multiple scales shows good agreement with that from the original differential equations. / Ph. D.

In this study equations are developed for the direct determination of stiffness factor and carry-over factor for parabolic haunched beams. the conventional methods require long and tedious calculations, so the development of these equations, which require only the direct substitution of the variables, is useful. A digital computer was used to generate the data required to develop these general equations. A mathematical model was chosen for the general equation; then, by the statistical "Method of Least Squares," the coefficients in the general equation were determined. The equations were then checked against the generated data to determine average errors in calculated values. Illustrative examples are given to explain the use of the equations. / Master of Science

This research document focuses on prestressed PCBT girders made composite with a cast-in-place concrete deck and continuous over several spans through the use of continuity diaphragms. The current design procedure in AASHTO states that a continuity diaphragm is considered to be fully effective if a compressive stress develops in the bottom of the diaphragm when the superimposed permanent load, settlement, creep, shrinkage, 50 percent live load, and temperature gradient are summed, or if the girders are stored at least 90 days when continuity is established. It is more economical to store girders for fewer days, so it is important to know the minimum number of days that girders must be stored to satisfy AASHTO requirements. In 2005, Charles Newhouse developed the positive moment diaphragm reinforcement detail that is currently being adopted by VDOT. This thesis concludes that Newhouse's detail, four No. 6 bars bent 180° and extended into the diaphragm, is adequate for all girders except for the PCBT-77, PCBT-85, and the PCBT-93 when the girders are stored for a minimum of 90 days. It is recommended that two additional bent strands be extended into the continuity diaphragm for these three girder sizes. It was also concluded that about half of the cases result in a significant reduction in the minimum number of storage days if the designer is willing to perform a detailed analysis. The other half of the cases must be stored for 90 days because the total moment in the diaphragm will never become negative and satisfy the AASHTO requirement. In general, narrower girder spacing and higher concrete compressive strength results in shorter required storage duration. The PCA Method was used in this analysis with the updated AASHTO LRFD creep, shrinkage, and prestress loss models. A recommended quick check is to sum the thermal, composite dead load, and half of the live load restraint moments. The girder must be stored 90 days if that sum is positive, and a more detailed time-dependent analysis would result in a shorter than 90 day storage period if that sum is negative. / Master of Science

A method for determining the transient vibrational effects produced by application of a constant angular acceleration to a cantilever beam initially at rest was determined. This method is applicable to beams of uniform cross-sectional area vibrating in their planes of rotation. The governing differential equations include the effects of bending, shear deformation and rotary inertia. Coriolis' acceleration, however, is neglected. These governing equations were non-dimensionalized and solved by numerical means using a finite difference approach and dividing the beam's length into 12 sections, since their complexity made an exact solution appear impossible. This was done by the aid of a 1620 I.B.M. Computer. Application of this solution to an actual beam indicated that a wave propagation type of response becomes more clearly evident as the hub radius is increased. The numerical results also indicate that the initial displacements are a direct result of shear deformation. The effect of centrifugal force was also analyzed. At large values of time this force caused the beam to return to its undeformed axis. Solutions for a short time interval extending over 1100 steps, a large time interval extending over 275 steps, and another using the short time interval together with a reduced radius, 250 steps, were obtained. / Master of Science

As a suggestion for further work in deriving the equations by the Principle of Virtual Displacements, it might be advisable to write the equations based on the vertical displacements of the points of inflection of each column. The concentrated mass of each panel would be located at the center of the respective columns. This appears to be a better approximation than concentrating each mass at the lower joint. / Master of Science

Active control of flexural power flow in infinite, semi-infinite and finite beams by point force inputs has been analytically and experimentally studied. The systems were analyzed by assuming wave solution and then applying various terminating boundary conditions. Optimal control solutions were then obtained using a quadratic Wiener solution. The influence of system parameters such as discontinuity impedance and effects such as near fields, number and location of control actuators and error sensors is investigated and discussed. The mechanisms by which control is achieved are considered. It is demonstrated that the boundary conditions and the system configuration strongly influence the choice of optimal controller format. The experimental work is compared with the theoretical developments and found to be in good agreement. Control was achieved by a LMS Filtered-x algorithm implemented in assembler language on a TMS32020 digital signal processing chip. In general it is demonstrated that the flexural power flow in the beam types considered can be attenuated with a low number of active actuators. / Master of Science

The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method. / Master of Science

Master of Science

To study the influence of the time-dependent behavior of various materials being considered for use in orbiting precision segmented reflectors, simple sandwich beam models are developed. The beam models included layers representing face sheets, core and adhesive. The issue of time-dependency is essential because the expected life of a reflector is on the order of 20 years. Using the principle of stationary potential energy, the elastic response of three-layer and five-layer symmetric sandwich beams to mechanical and thermally-induced loads is studied. The sensitivity of the three-layer and five-layer sandwich beams to reductions of the material properties is studied. Using the correspondence principle of viscoelasticity, these elastic models are transformed to time-dependent models. Representative cases of time-dependent material properties are used to demonstrate the application of the correspondence principle and evaluate the time-dependent response of the reflector. To verify the viscoelastic models, and to obtain a better idea of the amount of time-dependency to expect from the materials, simple time-dependent experiments on candidate materials were performed. Candidate materials include a quartz-epoxy face sheet material and a glass-imide honeycomb core material. The percent increase in strain for a constant stress for the quartz-epoxy in tension and the honeycomb in shear were measured. For both, a four-parameter fluid model captured the essential characteristics of their behavior. These four-parameter fluid models were then used in the three-layer sandwich beam model to predict the time-dependent response of the beam to three-point bending. This predicted response was compared to experimental results of a sandwich beam subjected to three-point bending. / Master of Science

Thesis (M.S.) -- University of Tennessee, Knoxville, 2005. / Title from title page screen (viewed on June 30, 2005). Thesis advisor: J. Harold Deatherage. Document formatted into pages (vii, 43 p. : ill. (some col.)). Vita. Includes bibliographical references (p. 41-42).