Determining Critical Excitation of MDOF Systems for Displacement Objective Function and Energy Constraint

Considering the increasing use of dynamic analysis methods in structural design, the selection of appropriate design earthquake has been an important part of the design procedure. For intermediate and low importance structures, the design earthquake is typically provided by the seismic codes as a design spectrum. However, for important structures for which time-history analysis should be performed, the use of recorded ground motions as input is inevitable. On the other hand, the existing ground motions only show a small part of the reality. Experiences of past earthquakes indicate that sole reliance on existing data will never resolve all issues, and new damage problems have occurred recently. In order to overcome this problem, a new paradigm has to be used. The concept of “critical excitation” and the structural design based on this concept can become one of such new paradigms. In the present paper, the probabilistic critical excitation method is used to determine the critical excitations for three shear building models, which are modeled as MDF system. Selection of appropriate constraints is the main problem when using this method. It is shown that the upper bound of earthquake input energy per unit mass can be considered as suitable constraint for the critical excitation. This bound can be a reasonable benchmark to estimate the allowable range of possible energy in similar earthquake. Considering this bound as constraint, the method is used to determine the critical PSD functions and generating synthetic accelerograms. In order to demonstrate the effectiveness of the method, three real accelerograms are selected as benchmarks and linear dynamic analysis was conducted using these accelerograms and the generated critical excitations, and the key parameters of response including maximum displacement, drift and acceleration of stories are compared. Input base acceleration is defined as the product of an envelope function and a stationary Gaussian process with zero mean. Considering the sum of the mean-square interstory drift of the system as the objective function, the critical excitation problem is defined as follows: Given the mass, stiffness and the viscous damping matrix of a linear elastic MDOF system, as well as the envelope function, find the critical PSDF, so that the objective function is maximized under specific constraints. In order to solve this problem, first the constraints have to be selected. According to the concept of the constraints, there is no straight way to recommend a specific value of them. Thus that is very difficult to estimate that what value of one constraint is sufficient to make the final nonstationary input critical. Nevertheless, at least a reasonable assumption can be made for the ratio of constraints. Therefore, the method can be used to locate the rectangular function. Takewaki [1] by setting constraints on the acceleration and velocity time history of the ground motion determined an upper bound of total input energy per unit mass of a record for a damped linear elastic system. This bound is well defined by two curves that perfectly bound the actual input energy curve in the range of short and long periods. Investigations on the time-history of various ground motions indicate that even for the same level of energy bound for acceleration constraint, the maximum amount of actual energy is not constant. Moreover, this maximum value may occur in different periods. The upper bound of the total input energy for acceleration constraint can be used to estimate the possible range of energy in similar ground motions. This bound is only related to the area of the PSDF of excitation and the maximum value of it. These two parameters can define a class of ground motions that existing record is just one realization of them. As a result it is desirable to fix the upper bound of input energy and let the excitation to change in amplitude and frequency content in such a way that the objective function is maximized. By this mean, using a sample of ground motion, a synthetic accelerogram can be generated with the same upper bound but different energy content. In order to examine the effect of the critical excitations on the structures, a numerical simulation was carried out using the three typical shear building of 8, 14 and 20 stories. These buildings were designed as steel moment frames, using the ETABS commercial software, and then the mass and stiffness of stories were determined and the corresponding MDOF models were used for linear dynamic analysis. This study shows that the upper bound of the total input energy for acceleration constraint can be a reasonable benchmark to estimate the possible range of energy in similar earthquakes. This bound is only related to the area of the PSDF of excitation and the maximum value of it. By controlling these two parameters, the upper bound of input energy can be fixed and then by changing the amplitude and frequency content using the critical excitation method required accelerograms are determine so that the objective function is maximized. Comparison of linear dynamic analysis of three designed models under critical excitations and the corresponding actual records, which have been selected in such a way that the energy content of the record is the maximum in the natural frequency of the fundamental mode of structures, shows that the synthetic accelerograms can reasonably estimate the behavior of structures, including the maximum displacement, interstory drift and absolute acceleration of stories.