A continuum approach to modeling the vibrations within a block subjected to free rocking
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Seismic events are unpredictable and in locations susceptible to these events, buildings and bridges must be designed to minimize societal impacts and structural damage. Accurately modeling structures subjected to seismic loads is important for improving the design techniques for structures in active seismic areas. Employing rocking motions in structural design can help minimize the effect of seismic loads on these structures by allowing critical structural elements to move with the earthquake motion. The rocking movement dissipates the energy imparted to the structure by the earthquake. This reduces the amount of structural deformation within the structure’s members. Mathematical modeling techniques have the ability to capture the internal structural deformation of rocking structures. The models created in this research project yield an in-depth understanding of rocking structures by providing deeper insight into the characteristics of rocking dynamics.
The rocking system investigated in this research project is composed of a rectangular block rocking on a planar foundation. Traditional rocking models idealize this rocking block system by assuming the rocking block and foundation are infinitely rigid. Rocking models that consider the block and foundation infinitely rigid are referred to as rigid rocking models. In the rigid rocking model the energy dissipation characterized by rocking motion is concentrated at the interface between the block and foundation. In reality, structural systems are finitely rigid and there are many locations throughout the structure that can contribute to the total energy dissipation of the system. In particular, energy can be dissipated within the block or radiated through the foundation to the surrounding media, such as soil. Considering multiple locations for the energy dissipation complicates the mathematical representation of the rocking system. In the rigid rocking model, a single location that generalizes the energy dissipation is sufficient. The location of the energy dissipation is chosen according to the component of the system that is being investigated.
The focus of this study is the vibration energy within the block and the contribution of this vibration energy to the overall rocking motion. Rocking models that describe a rocking block system where the vibration energy is generated are generally referred to as flexible rocking block models. The rocking model in this study assumes the vibration energy in the block is the only contributor to the energy dissipation that occurs during rocking motion and that the only energy lost between impacts is the vibration energy developed within the block during each impact. The foundation is assumed to be perfectly rigid to simplify the impact dynamics and to further isolate the vibration energy in the block from other possible sources. The block vibration energy can be dissipated by being transferred to the foundation or to nonlinear phenomena such as internal friction, heat energy, and sound energy. Once this energy transfer occurs, the energy is assumed never to return to the block. This energy loss in the flexible rocking block model can account for as much as 3% of the equivalent amount of energy introduced into the rigid rocking block model, indicating the possibility that the vibration energy in the block contributes significantly to the energy loss of the rocking system. The model presented in this project is the result of finding the variational representation of the law of conservation of energy for the considered modes of deformation. This method is a modification of the derivation of the Mindlin-Timoshenko plate bending model detailed in Lagnese and Lions. The model created in this research project is equivalent to accepted rigid models in the absence of internal deformations and represents a higher degree of nonlinearity in rocking dynamics when vibrations are included.
The characteristics of the rocking block system that cause an increased amount of block vibration energy depend on the flexibility of the block and the initial displacement of the block. The mathematical model described herein substantiates the significance of the contribution of block vibration energy to rocking systems. The internal flexural, axial and shear deformations, in addition to the angular rocking displacement, are incorporated in the model. The minimum amount of block vibration energy in the rocking response is defined and based on the contribution of the block vibration energy, a rocking criterion is established. This criterion defines the initial displacement that will cause a block to rock on a rigid foundation, given the block’s flexibility. The rocking criterion also defines the completion of the rocking response. These assertions can help influence experimental design and further investigation into the nonlinear motions that occur during rocking motion.