DYNAMIC BEHAVIORS OF BEAM AND PLATE STRUCTURES SUBJECTED TO VARIOUS COMPLICATING EFFECTS USING ARTIFICIAL SPRING MODELS

Abstract

This thesis, on the dynamic behaviors of beam and plate structures subjected to various complicating effects using artificial spring models, is divided into six chapters. Chapters 1 and 6 are the introduction and conclusion respectively. Formulations and numerical results of the various case studies are presented and discussed in chapters 2 to 5. In chapter 2, the free vibration of beam structures subjected to various complicating effects was studied. The cases discussed include vibration of a stepped beam, in-plane vibration of planar frame structures, and the flexural vibration of a cracked beam. Natural frequencies of a stepped simply-supported beam were determined using two different algorithms based on the Rayleigh-Ritz method. The first algorithm, which assumed a series of assumed functions tha.t satisfied only the external boundary conditions and ignored the presence of the step, was found to be only suitable for beams with moderate to small thickness variations across the step. The second algorithm, which used artificial linear and torsional springs of large stiffness to model the geometric continuities at the step, considered the beam as two sub-beams divided by the step. Two different sets of admissible functions which satisfied the respective geometric boundary conditions of each fictitious sub-beam were assumed. This second method was found to be better suited for stepped beams with large thickness variations across the step. Artificial linear and torsional springs of large stiffness were also employed for the in-plane vibrations of planar frame structures. The springs were used to enforce the continuity conditions due to the axial rigidity of the members and the rigidity of the joints. The deformation for each member of the of the frame structures was described by a different set of admissible functions. Numerical results obtained for the portal frame were found to be in good agreement with exact results. The flexural vibration of a beam with intermediate open cracks was analyzed with the crack being modeled by an artificial linear spring of large stiffness and a torsional spring of equivalent spring constant depending on the crack. Numerical results obtained for a simply-supported beam with a pair of double-sided cracks at mid-span showed good agreement with the published results. In chapter 3, the free vibration of plate structures subjected to various complicating effects was investigated. Of interest were the vibration of symmetrically laminated rectangular composite plates reinforced by intermediate stiffeners (parallel to the edges of the plate), the vibration of isotropic rectangular plates reinforced by an intermediate stiffener oriented at different angles, and the vibration and critical speeds of a spinning annular disk of varying thickness. The free vibration of symmetrically laminated rectangular plates with beam stiffeners parallel to the edges of the plate was examined based on the Rayleigh-Ritz method. Effects of the location of the stiffeners and their relative stiffness to the flexural rigidities of the plate on the natural frequencies and modes were investigated. The effects of material properties and stacking sequence of the laminated plate were also examined. It was found that the natural frequencies do not necessarily show significant increases even if there was a significant increase in the modulus of the stiffeners. The results indicated that the relative positions of the nodal lines to the stiffeners' location strongly affect the natural frequencies. The free vibrations of isotropic rectangular plates with stiffeners oriented at different angles relative to the plate were next examined. Numerical results otained once again indicated that relative positions of the nodal lines to the stiffener location affected the natural frequencies more significantly than the modulus of the stiffeners. The importance of including torsional effects of the beam was also reflected in the studies. The natural frequencies and and critical speeds of a spinning isotropic annular plate of variable thickness were determined based on Hamilton's principle and the assumed mode method. The effects of tapering in the thickness on the critical speeds of the spinning disk were examined. Numerical results obtained for the critical speeds compared well with published results. The results also suggest that critical speeds may be increased by tapering the annular plate inwards towards the rotating axis. Chapter 4 presents the dynamic analysis of plate and beam structures under external excitations. The cases examined include the dynamic response of a. cracked beam subjected to a moving load, and the effects of support configuration on the parametric excitation of a plate. The effects of the speed of the moving load and the crack size on the deflection under the load of a cracked beam were examined and compared with the ideaI case where the beam has no crack. The necessary continuity conditions at the crack were enforced using an artificial linear spring of very large stiffness and an artificial torsional spring of equivalent spring constant depending on the crack size. For a relatively slow moving load, the presence of a crack was found to increase drastically the deflection under it when compared with the case of an uncracked beam under the same load. For a relatively fast moving load, the presence of a crack caused much less significant increase in deflection under the moving load when compared with the uncracked case. The equations of motions of a plate moving over multiple line supports were formulated based on Hamilton's principle and the assumed mode method. The supports were assumed to be frictionless line supports spanning the entire or partial width of the plate, with the plate being pushed or pulled over them. These line supports were modeled as linear line springs of very large stiffness. The effects of support length and eccentricity in the locations of the support were examined for various prescribed longitudinal excitations of the plate. For in-plane sinusoidal excitation of the plate with a prescribed amplitude of excitation over the line supports, the stability of the plate was found to be dependent on the frequency of excitation and also the support configuration. Chapter 5 looks into the dynamic stability of plate structures subjected to various in-plane loadings. Bolotin's method was employed to generate the instability regions and the equations of motion for the dynamic behavior of plate structures were formulated based on Hamilton's principle and the assumed mode method. For a rectangular plate on multiple line or point supports under conservative in-plane loads, the effects of sinusoidal excitation for in-plane loads on the stability of the plate were investigated. The instability regions were found to be shifted to the left having lower frequencies of excitation when the average magnitude of the in-plane compressive loads was increased. The multiple intermediate point and line supports were again modelled as linear springs (point and line respectively) of very large stiffness. The positions of the instability regions were also found to be strongly influenced by the support configurations. The dynamic stability of a moving rectangular plates with four free edges and two opposite edges subjected to in-plane acceleration and force excitation was also examined. The effects of sinusoidal excitation with respect to the in-plane acceleration and the external loads were investigated. The instability regions were found to be shifted to the left when magnitudes of in-plane acceleration and in-plane force excitation were increased. The sizes of the instability regions were also found to be very much influenced by the magnitudes of the in-plane acceleration and the external loads. The axisymmetric dynamic stability of an annular plate subjected to pulsating conservative radial loads was also discussed and the instability regions for various combinations of in-plane loads and boundary conditions were compared. Instability regions for various combinations of in-plane loads and boundary conditions were obtained and the instability regions were observed to be shifted to the left having lower frequencies of excitation when the average magnitude of the in-plane radial compressive loads was increased. It was also observed that the sizes of the instability regions were significantly smaller for the case where the annular plate was loaded only along the inner edge as compared with the cases where the plate was loaded along the outer edge or along both edges.