مجله علوم و مهندسی زلزله
سال سوم شماره 4 (پیاپی 9، زمستان 1395)

Peak ground acceleration is one of the most important factors that needs to be investigated in order to predict the devastation potential resulting from earthquakes in reconstruction sites. Besides, the maximum level of shaking control is subjected criteria that can be worth considering. In this research, a training algorithm based on gradient descent and Levenberg-Marquart (Train LM) were developed and employed by using strong ground motion records. The Artificial Neural Networks (ANN) algorithm indicated that the fitting between the predicted PGA values by the networks and the observed PGA values were able to yield high correlation coefficients of 0.78 for PGA.
From a deterministic point of view, the determination of the strongest level of shaking that is expected at a site has long been a significant consideration in earthquake engineering. Besides, knowledge of the maximum physically possible ground motions allows a meaningful truncation of the distribution of ground motion residuals, and as a result, leads to falling of the values computed in probabilistic seismic hazard analysis (Strasser and Bommer, 2009).
The peak ground acceleration parameter is often estimated by the attenuation of relationships and by using regression analysis. PGA is one of the most important parameters, often analyzed in studies related to damages caused by earthquakes (Gullo and Ercelebi, 2007). It is mostly estimated by the attenuation of equations and is developed by a regression analysis of powerful motion data.
Kerh and Chaw (2002) used software calculation techniques to remove the lack of certainties in declining relations. They used the mixed gradient training algorithm of Fletcher-Reeves’ back propagation error (Fletcher and Reeves, 1964). They applied three neural network models with different inputs including epicentric distance, focal depth and magnitude of the earthquakes. These records were trained and then the output results were compared with available nonlinear regression analysis.
In this article, to estimate strong ground motion acceleration component in an area, four artificial neural networks with different algorithms were used, including General Regression Neural Network (GRNN), Nonlinear Auto Regression neural network (NARX), Feed-Forward Back-Propagation error (FFBP) and General Feed-Forward Neural Network (GFFNN). Input vectors of neural networks include four parameters, which have key effects in occurrence of an earthquake in an area. The parameters include magnitude of moment, rupture distance of earthquake center, mechanism of faults, and ranking of site. Output vector has only one component: maximum peak ground acceleration for an earthquake in an area is used as a target output.
After different tests, GRNN network has maximum output correlation coefficient (0.87) and General Feed-forward Back-Propagation error neural network (FFBP) has the least (0.41). Besides, GRNN network had the least mean square error (0/014), and Back-Propagation network had 0.125. In this research, GRNN neural network is the best neural network, which can estimate possible peak acceleration more than 1g in an area.
Artificial neural networks are a set of non-linear optimizer methods which do not need certain mathematical models in order to solve problems. In regression analysis, PGA is calculated as a function of earthquake magnitude, distance from the source of the earthquake to the site under study, local condition of the site and other characteristics that are linked to the earthquake source such as slippery length and reverse, normal or wave propagation. In non-linear regression methods, non-linear relations that exist between input and output parameters are expressed as estimations, through statistical calculations within a specified relationship (Douglas, 2003).

Gravity walls are commonly used as earth retaining systems supporting fill slopes adjacent to roads and residential areas, especially to protect the transportation facilities and/or nearby structures in regions prone to earthquakes. Analysis of retaining walls behavior against earthquake is an important task for geotechnical engineers for reasons such as soil complex seismic behavior and inefficiency of quasi-static analyses. Seismic analysis and design of earth retaining walls is a difficult task, which traditionally requires the determination of the dynamic soil pressures induced by the soil seismic motion on the wall.
Understanding the performance of a retaining wall during an earthquake is very important for an economical design and reducing the damages caused by large earthquakes. Calculated displacement of retaining walls has a key role in the optimal performance design of these structures under seismic loadings. The efficiency of a wall after an earthquake depends on its seismic displacement. Excessive displacements may not only cause the wall to collapse, but also cause to damage the adjacent structures. There have been numerous examples of this type of failure in recent earthquakes. Though the quasi static method for rational design methods of retaining structures has been performed for several decades, deformations ranging from slight displacement to catastrophic failure have been observed in many earth retaining structures during the recent major earthquakes.
Many researchers have developed design methods for retaining walls during earthquakes by using different approaches. In this paper, an algorithm for calculation of permanent displacements of retaining walls in seismic conditions is presented. Formulation of this algorithm is based on the upper bound limit analysis. Displacement of the wall is calculated by obtaining its yield acceleration by limit analysis, and then combination of the proposed method with Newmark method. Effect of various parameters on the displacement of the walls is studied.
For the upper bound theorem to be valid, the velocity field in the failure mechanism must conform to the normality flow rule (associated with the yield condition). The term normality rule originates from the geometric property of the potential law where the deformation rate vector is perpendicular (normal) to the yield surface.
When dense sand is subjected to shear, it simultaneously exhibits volumetric changes (dilatancy). These changes, when described by the flow rule associated with the Mohr–Coulomb yield condition, tend to overestimate the true dilatancy. There are two distinct issues that need to be addressed: (1) How does the departure from the normality rule affect the yield acceleration of the structure; and (2) what flow rule should be used to obtain a reasonable estimation of the true displacements of a structure subjected to seismic excitation?
The first question was addressed earlier by recent researchers who indicated that the yield acceleration of a soil structure built of “nonstandard" soil (“nonstandard” soil is one with deformation governed by the non-associative flow rule) can be obtained with sufficient precision by the kinematic approach if internal friction angle and cohesion of the soil is modified. For the second issue, the deformation description is described by the true dilatancy angle to conform the true material behavior and for prediction of the true (finite) displacements,
Effect of various parameters on yield acceleration and the displacement of the walls is studied. Internal friction and dilatancy angles of the soils have the most important influence on the results.

In recent years, interests in utilizing performance-based design to achieve earthquake-resistant structures have grown. One of a robust procedure in this category, which is presented in 1993 by Prof. Priestley, is Direct Displacement-Based Design (DDBD) method. Extensive and developed researches have shown that DDBD has a great potential to overcome existing shortcomings of the force-based design method. During last decade, DDBD, which was initially proposed for designing RC buildings and bridge piers, are developed for steel structures. DDBD has two main factors: hysteretic damping capacity and yield displacement of the building. Accurate estimation of these parameters is very important to determine proper value of design base shear of the building under consideration. These factors were firstly estimated experimentally for concrete structures but now, they are intensely studied analytically by many researchers for various forms of steel structures. In this regard, this paper attempts to consider more realistic estimation of equivalent viscous damper capacity of moment resisting steel frame structures and its influence on determined base shear values.
In this paper, 30 different moment-resisting frames with various numbers of stories (3, 6, 9, 12 and 15) and spans (3 and 6) are studied. For each frame, different analysis methods are carried out: (1) nonlinear static or pushover analysis, (2) nonlinear time history analysis employing synthetic accelerograms, (3) nonlinear time history analysis employing two sinusoidal protocols with different excitation frequencies: initial and effective frequencies, (5) nonlinear static cyclic analysis using an incremental sinusoidal displacement protocol, (6) simple linear analysis of an equivalent single-degree-of-freedom (SDOF) model of the structure subjected to a sinusoidal load, and finally (7) the proposed relation in the Model Code for the Displacement-Based Seismic Design of Structures, DBD12. Comparing the results shows that the equivalent damping ratio obtained using DBD12 relation for life-safety (LS) level is significantly lower than the values obtained by the analyses conducted in this study. This means that the determined base shear for designing such steel building is much more than values for a safe building. In other words, steel buildings using relations of DBD12 tend to be stiffer and stronger than needed. Hence, a new relation is derived to determine the hysteretic damping of MR steel frame structures in the LS performance level as a function of a ductility coefficient. Furthermore, using the relationship between the initial and effective period mentioned in ATC40, another practical relationship is proposed as a ratio of the effective period over the initial period of the considered building.

In this paper, a method is presented based on approximating the objective function and constrains in optimization problems in conjunction with Lagrange multiplier method. Besides, an algorithm is developed in this relation. Instead of linear or parabola terms employed in Taylor expansion to proceed cautiously with short step lengths, in the method presented here, an arc with constant curvature is used that makes it possible to proceed with relatively longer step lengths. For an n-dimensional optimization problem, the spheres are n-dimensional too. The radius of curvature and center of spheres can be determined at the tangent point between each function and its corresponding sphere. For the objective function, the parameters of sphere are determined at the reference point obtained by Lagrange equations, but for the constraints, first the reference point is returned to the surface of all the active constraints, then at the points on the constraints, the approximate parameters are calculated. Hence every computational step includes two parts: the determination of the reference point and returning it to the surface of active constraints. The criterion for returning to the active constraint is based on the shortest distance of the reference point from each of the active constraint, because the reference point is the output of optimization represented by Lagrange equations and so is the basis of the calculations. For returning the reference point to the active constraint, only one scalar variable is involved in the calculations. The introduction of the n-dimensional spheres both reduces the number of and simplifies the form of equations that need to be solved simultaneously to determine the optimum point and Lagrange multipliers at each optimization step, because the unknowns are now the Lagrange multipliers. This results in a significant reduction in computation time. Separating the design variables from Lagrange equations, the time of calculations may be saved for the loops of time-history analysis in the optimal seismic design. The method is applied to the optimization of two major parts of the lateral resistance systems, and the results are compared with those from penalty method. Considerable reduction of solution time is observed.
The following remarks and conclusions are pertinent with regard to the formulation and the results presented in the paper:(1) The structural examples solved by the method presented here have also been solved by the exterior penalty method where both methods have provided exactly the same optimum solutions.
(2) The proposed method does not depend on the convexity or the concavity of the constraints or the objective function, because the radius of sphere that indicates the curvature is directly utilized at each computational step.
(3) Similar to the other optimization methods, the convergence behaviour and success of the proposed method depends on the starting point.
(4) Separating the design variables from Lagrange equations, the time of calculations may be saved for the loops of time-history analysis in the optimal seismic design.
(5) In this method, the criterion for the returning to the active constraint was utilized that is based on the shortest distance of the reference point from each of the active constraint (residual error); however, the methods based on Taylor expansion do not consider the minimization of residual error in every computational step.
(6) Lagrange multipliers related to the constraints of lower and upper bound in the structural optimization problems can be decoupled from the others by the proposal method.
(7) Though of the time of computation to converge, the final solution is of great importance and should be discussed in detail. The space limitation does not let a proper comparison of convergence behaviour between the presented method and the exterior penalty method. Hence this issue has been postponed to a follow-up paper, but just qualitatively, the presented method has shown the convergence faster.

Installation of stone columns is one of the proper and known methods for the improvement of weak soils. Stone column construction are employed to improve the bearing capacity, slope stability, and drainage rate, as well as reducing the settlement and liquefaction potential of the soft soil. In geotechnical earthquake engineering, stone columns are generally used to control the liquefaction potential of loose granular soils. However, the seismic performance of these inclusions has been partially studied and requires more researches. On the other hand, it is important to estimate a fundamental frequency of site for the seismic design of buildings and infrastructures and considers the basis of site classifications in seismic codes.
In this paper, the effects of stone column construction on the fundamental frequency of the sites are studied numerically. Finite element analysis was performed using ABAQUS. The analysis is a modal analysis through the calculation of eigenvalues. Analyses was carried out in 3D and 2D in some cases. According to the modal analysis of the problem, the behavior of the soil and stone column are considered linear elastic. Additionally, the shear wave velocity and density of the soil and stone columns are assumed constant in depth. The results demonstrated that stone columns construction can increase the fundamental frequency of the site to four times. The fundamental frequency amplification factor of the site (α) can be defined according to the dimensionless parameters including stone column to soil shear wave velocity, height to diameter, distance to diameter, and stone column arrangements.
The results indicated that α decreased with a rise in the ratio of the stone column height to diameter. Stone column arrangements are either square or triangle. When the triangle and square arrangements is used, zones of influences by each column as a regular hexagon and square, respectively. A comparison of the stone column arrangements demonstrated that, in triangle arrangement, α was greater than the corresponding value in square arrangement. The reason behind this is that in triangle arrangements, the zones of influence of each column is greater than the similar value in the square arrangement. Depending on the height of the column and depth of the bedrock, stone columns can be constructed as end bearing with their end on the bedrock or as floating with free end in the soil. The results indicate that, in floating stone columns, the effects of stone columns on α with respect to the condition where the stone column was end bearing, was considerably insignificant. In the following, tri-variant relation was determined for α. This relation was achieved using the Evolutionary Polynomial Regression (EPR). This method utilizes multi-objective genetic programming to derive regression equations by constructing symbolic models. Two-thirds of the data chosen to operate as training data and the other was used as testing data. The statistical parameters showed the good correlation and high accuracy of the derived relation for training and testing data. In the following, the problem is done in plane strain condition (2D). For this purpose, stone columns which were in a row, were assumed as equivalent strips and these strips were supposed as a set of considerable rigid retaining walls in the soil profile. Similar to the 3D case, α can be presented by the values of dimensionless parameters. Finally, a 2D equivalent method for simplification of the 3D actual problem will be presented by examining the various cases. The results suggest that in the case the inertial moment of stone columns in 3D equal to 2D, relatively good approximation exists between the actual 3D and the equivalent 2D results.