A new ground motion intensity measure, peak filtered acceleration (PFA), to estimate collapse vulnerability of buildings in earthquakes

In this thesis, we develop an efficient collapse prediction model, the PFA (Peak Filtered Acceleration) model, for buildings subjected to different types of ground motions.

For the structural system, the PFA model covers modern steel and reinforced concrete moment-resisting frame buildings (potentially reinforced concrete shear wall buildings). For ground motions, the PFA model covers ramp-pulse-like ground motions, long-period ground motions, and short-period ground motions.

To predict whether a building will collapse in response to a given ground motion, we first extract long-period components from the ground motion using a Butterworth low-pass filter with suggested order and cutoff frequency. The order depends on the type of ground motion, and the cutoff frequency depends on the building’s natural frequency and ductility. We then compare the filtered acceleration time history with the capacity of the building. The capacity of the building is a constant for 2-dimentional buildings and a limit domain for 3-dimentional buildings. If the filtered acceleration exceeds the building’s capacity, the building is predicted to collapse. Otherwise, it is expected to survive the ground motion.

The parameters used in PFA model, which include fundamental period, global ductility and lateral capacity, can be obtained either from numerical analysis or interpolation based on the reference building system proposed in this thesis.

The PFA collapse prediction model greatly reduces computational complexity while archiving good accuracy. It is verified by FEM simulations of 13 frame building models and 150 ground motion records.

Based on the developed collapse prediction model, we propose to use PFA (Peak Filtered Acceleration) as a new ground motion intensity measure for collapse prediction. We compare PFA with traditional intensity measures PGA, PGV, PGD, and Sa in collapse prediction and find that PFA has the best performance among all the intensity measures.

We also provide a close form in term of a vector intensity measure (PGV, PGD) of the PFA collapse prediction model for practical collapse risk assessment.

Random propagation in complex systems : nonlinear matrix recursions and epidemic spread

This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.