Project: Local Degree of Freedom Ductility Demand

Ductility demand during an earthquake has long been known to be a good measure of structural damage in buildings during earthquakes. For design purposes it is considered a priori according to some rules that may or may not apply to each building. Such ductility parameters are mostly based on the response of a single degree of freedom system to ground motions. However, it has been probed that these parameters are not accurate enough and might lead to underestimating risks to which a structure may be vulnerable to (e.g. Miranda and Bertero 1994, Esteva 1994, Santa-Ana and Miranda 2000). Application of probabilistic methods is recommended to evaluate vulnerability of structures, (ASCE 07-2010). And so, traditional probabilistic approaches (i.e. total probability theorem) are applied as part of the Performance Based Earthquake Engineering (PBEE) methodology. This project intends to evaluate the accuracy of Machine Learning algorithms when applied as an alternative to traditional methods. Linear Regression, Weighted linear regression and Neural Networks are evaluated by comparing deterministic results of nonlinear models with predictions resulting from these algorithms. Both Linear and Weighted linear regression resulted accurate enough but do not seem to capture the essence of the problem. In opposition, neural networks appear to capture the essence of the problem and further work may be promising. Introduction Earthquake resistant design is one of the most important issues to guarantee the correct behavior and serviceability of buildings and so, it is one of the major concerns for current Structural Engineering research and practice. Economy and life safety are the main goals for design, allowing inelastic behavior of materials during extreme earthquakes (i.e. design level earthquakes and maximum considered earthquake) is common for achieving these goals. However, if not designed properly, such type of behavior may lead to unstable behavior that may produce collapse. Such inelastic behavior during design process is measured (or assumed) by predefined ductility demand values pre-established in codes. Ductility is defined as the quotient of the maximum deformation of the structure during the earthquake divided by the limit of its elastic deformation. In current codes the parameter ductility is based on single degree of freedom behavior. (e.g. singlestory buildings). Commonly the reference point for measuring and understanding ductility is the top story of the building under design. However, this suggests a global measure of ductility and important information might be unnoticed by the designer. Most of the times, measuring ductility locally leads to a better understanding of the level of damage that a structure may present during an earthquake. (e.g. ductility at the first story, ductility at a certain beam or brace.) Supervised machine learning techniques are evaluated as an alternative for predicting the maximum local ductility demands. The input for the model is formed by three types of scalar variables: 1) Ground Motion Characteristics, 2) Structure properties, and 3) Variables which are a combination of both. In PBEE, the first and third kind are both grouped in the name of intensity measures. However, the writer considered important to make the distinction for this work. The result of each model is a positive real value number which defines the ductility demand at a local story. Even though probabilistic approaches are commonly used in performance based earthquake engineering for evaluating the seismic performance of different components of buildings, there is not a work known to the writer in which neural networks are used to predict the ductility demand or damage of a building for earthquake events. As part of this work, Linear Regression and Weighted Linear Regression are also applied with two purposes: 1) to improving the understanding of the behavior of machine learning techniques when applied to a problem which is more familiar to the author, and 2) to serve as benchmarks to know if a more complicated model is of use for this problem or if it would only result in “overcomplicating” the problem. The observation was that neural networks are indeed more effective for addressing this kind of problems. Datasets and Features As described above, three groups of features are considered, the ten following features were considered for the model: a) Ground Motion Characteristics: 1. Peak Ground Acceleration (PGA): is the maximum acceleration from the ground motion record. 2. Peak Ground Velocity (PGV): is the maximum velocity computed when integrating the ground motion acceleration record over time. 3. Peak Ground Displacement (PGD): is the maximum displacement computed when integrating the ground motion acceleration record over time two times. 4. Strong GM Duration (90% Arias intensity): refers to the time-length in which the middle 90% of energy of the record is contained (i.e. leaving out 5% from the beginning of the record and 5% from the end). In this energy measure, the energy is computed as the integral of the squared accelerations of the record over time. b) Building Properties: 5. Number of stories of the building 6. Fundamental period of vibration of the building (T1): is the period of vibration associated to the first mode of vibration of the building. 7. Design Ductility (μ): Is the expected ductility of the structure considering building design factors, it can be computed as the quotient of the seismic force reduction factor (R) and the overstrength factor (Ω) from the building code (ASCE 07-2010). c) Composite Properties: result of the combination of properties from both the building and the ground motion. 8. Spectral Acceleration at T1 (SaT1): the maximum acceleration response to the considered ground motion of an elastic single degree of freedom system having the same period as the fundamental period of the building. 9. Geometric Mean of Spectral accelerations at T1 (SaGmean): Geometric Mean of maximum response accelerations of single degree of freedom systems with periods in the vicinity of T1 (i.e. from 0.2T1 to 3.0 T1 spaced at 0.1 T1). 10. Average Spectral acceleration at T1 (SaAve): Mean of maximum response accelerations of single degree of freedom systems with periods in the vicinity of T1 (i.e. from 0.2T1 to 3.0 T1 spaced at 0.1 T1). Since there were not published databases of such analysis containing all these variables, simplified models’ analyses were conducted to generate the applied dataset. The variations to generate the database were as follows: Ground Motions Number of Stories μ Number of periods (T1) for each number of stories Total analyzed samples 270 1-7 1,2,3,4 5 37800 A post-process was required to evaluate the stability of the simplified model for resulting in representative values. As a result, 27 ground motions were removed and a resulting database of 34,020 samples was used to perform this project. The ground motions were considered as representative according to Eads (2013). Methods Three machine learning methods were applied for this project to improve the understanding of such methods, and to develop sensibility of how variations within these Machine Learning methods may affect the outcome when applied to this type of problems. 1. Linear Regression This is the simplest method in Machine Learning, its goal is to find a vector, the size of the number of features and predict the outcome of the problem as the result of a dot product operation of a vector containing the features’ values and the linear regression vector. The regression vector can be obtained with the following closed form formula: Where θ is the linear regression vector, X is the matrix containing the feature values of the training samples, and yԦ is the vector containing the result or outcome of each training sample. 2. Weighted Linear Regression Just a step further of linear regression, weighted linear regression intends to obtain the regression vector by making the samples closer to the intended prediction to have more influence in the outcome. This is achieved by adding weights to each sample of the training set. In this project, the weights were computed as follows: Where wii ⬚ is the weight value for the training sample i, xi is the vector containing the features’ values of the i training sample, x is the vector containing the features’ values of the intended prediction, and Σx is the covariance matrix for the features of the training data set. The regression vector is obtained with the following closed form formula: θ ≔ ሺXWXሻXWyԦ Where θ, X, and yԦ are as defined for linear regression and W is a diagonal matrix containing the weights wii . For both, linear and weighted linear regression, the prediction or hypothesis is obtained with the following equation: h ≔ θx