Progressive collapse studies generally assess the performance of the structure under gravity and blast loads, while earthquakes may also lead to the progressive collapse of a damaged structure. In this study, the progressive collapse response of concentrically braced dual systems with steel moment-resisting frames was assessed under seismic loads through pushover analysis using triangular and uniform lateral load patterns. Two different bracing types (X and inverted V braces) were considered, and their performances were compared under different lateral load patterns using the Nonlinear static alternate path method recommended in the Unified Facilities Criteria (UFC) guideline. Eventually, the seismic progressive collapse resistance of models was compared to their progressive collapse response under gravity loads. These studies showed that models under the seismic progressive collapse loads satisfied UFC acceptance criteria and limited rehabilitation objective. The structures had better performance under seismic progressive collapse than models under gravity loads because of more resistance, ductility, suitable load redistribution, and more structural elements that participated in load redistribution. Furthermore, despite studies on progressive collapse under gravity loads, the dual system with X braces showed better progressive collapse performance (more resistance, residual reserve strength ratio and ductility) under seismic loads than the model with inverted V braces.

Introduction: Exchange rate is a measure of the equality of a country's currency against the currencies of other countries. It indicates the measurement of that country's economic situation in comparison with other countries. In the framework of conventional economic theory, the exchange rate system refers to the mechanism of determining the exchange rate through market forces exerted on the supply and demand. The purpose of this study is to understand the dynamics governing the exchange rate behavior using Nonlinear models. By understanding the exchange rate dynamics, one can recognize its unusual and worrying behavior over time and apply the necessary policies accordingly. methodology: The baseline linear model used in this study is a finite-order autoregressive (AR) model with relation (1): (1) In the real logarithm of the real exchange rate, the interrupted polynomials are placed in the roots of ϕ (z) = 0 on or outside a single circle. The roots outside the unit circle mean that PPP remains stable in the long run. Regime change regression Two-mode TAR (2) models are defined as follows: (2) In the above relation, there is a sequence of white noises with mean zero and variance 1. In this model, it is assumed that the variance in each regime is different from that in the other regimes. In order to complete the above definition, the R1 and R2 regimes need to be described more precisely. It depends on how each regime changes over time. Introducing the Bayesian model In order to estimate the Bayesian SETAR, we assume that the exchange rate variable has a normal distribution. (3) In the SETAR model, the regime change is defined as a discrete variable as follows: In the next step, in order to have a Bayesian estimate, we need to specify the backgrounds of the model coefficients and the other parameters. An appropriate assumption in this regard is to assume that the anterior distribution r is a continuous uniform distribution whose boundaries include the minimum and maximum time series data as follows: In the next step, we will define the backgrounds for the coefficients. According to the objectives of this study, we will use the background for the ignorance of the normal part as follows: Bayesian estimation method The basis of Bayesian inferences is Bayesian theorem. According to this theorem, the posterior probability of an event varies according to the product of the previous probability in the logarithm of the orthogonality. In mathematical terms, Bayes' theorem is as follows: Results and Discussion: Table 1 reports the results of the SETAR model estimation for the exchange rate return (Rials against the dollar with the monthly rotation in the period from 2004 to December 2020). The validity intervals of the coefficients are adjusted and do not include zero, which, like the classical case, is based on the significance of these coefficients. Table 1. Bayesian SETAR model coefficients (1) for the dollar exchange rate Coefficients Posterior average Posterior standard deviation 95% confidence interval 0, 0079 0, 0023 (0, 0034, 0, 0123) 0, 3179 0, 0713 (0, 1773, 0, 4567) 0, 0147 0, 0161 (-0, 0172, 0, 0465) 0, 4338 0, 1468 (0, 1427, 0, 7244) 0, 0223 0. 0006 (0, 0208, 0, 0230) Table 2. Variances of the two regimes in the Bayesian SETAR (1) model for the dollar exchange rate Coefficients Posterior average Posterior standard deviation 95% confidence interval 0/0007 0/0001 (0, 0006, 0, 0009) 0/0094 0/0019 (0, 0064, 0, 0137) According to the results of Table (2), when the exchange rate is lower than the latter value of thresholds, its variability is much less than when the exchange rate is higher than the threshold value (variance in regime 2 is greater than that in regime 1) Figure (1) shows the autocorrelation of the simulated values in the latter estimation of the model parameters in the two regimes. Figure 1. Autocorrelation of the posterior coefficients in both regimes 1 and The results in Figure 1 show that the correlation of the simulated values for all the model parameters rapidly decreases to zero. Therefore, we are faced with a suitable sample of values to simulate the posterior distribution of the parameters. There is also no need to increase the simulation volume. Figure (2) shows the effect curves of all the later parameters of the model used in this research: Figure 2. Effect diagrams for the SETAR pattern parameters (1) Based on the findings about the curve of the effect related to all the parameters, no regular pattern exists in the simulated values ​​of the parameters. Therefore, the stability of these coefficients is confirmed, and the results of the Bayesian model SETAR (1) used in this study are statistically valid. Conclusion: The exchange rate as a price variable plays a very important role in the performance of an economy. The results of this study indicate that there are two exchange regimes in which the exchange rate adjustment parameter will be in equilibrium with a 95% probability in regime 1 (0. 1773, 0. 4567). This is very small due to the deviation of the latter standard of this coefficient (S. Dev = 0. 0713). Also, the same parameter with a 95% probability in regime 2 will be at the distance (0. 1427, 0. 7244), which is a relatively long distance. The results showed that this is due to the high volatility of the exchange rate in regime 2. In addition, the adjustment in the first regime to the long-term equilibrium path is much safer than the adjustment to the long-term path in the second regime because the variability in the conditions of the exchange rate increase is very high. Finally, the results of this study showed that the expansionary exchange rate regime (regime 1) has a deviation from higher regime standards than the mild exchange rate regime (regime 2), which indicates high currency fluctuations in this regime and more uncertainty. So, using regime 1 provides the conditions for a proper economic growth in the future.

Ever-increasing demands of the modern world and the growth of industry requirements toward more accurate analyses have made the engineering community develop the first fundamental step and meet the needs by extending the previous prevalent linear methods into Nonlinear areas. In this regard, the improvement of linear modes as one of the most pervasive and widespread analytical methods opens a new window to analyses with more closeness to reality. In this paper, after the deep identification of Nonlinear Normal Modes, an approach is proposed to analyze the multidegree-of-freedom structures with Nonlinear material under undamped free vibration. Afterward, through an in-depth investigation of the calculation methods, a novel algorithm for identifying Nonlinear Normal Modes was proposed, and by expanding this algorithm to the existing invariable motion equation used in free vibration analysis, the possibility of the extraction of all Nonlinear Normal Modes has emerged. After that, to investigate the functionality of the proposed approach, the Finite Elements method-based Model of a 2-story steel structure was developed and, after verification, it was used to form the invariable differential equations. Finally, after verifying the Independent Periodic method, pseudo-continuous masses of Nonlinear Normal Modes and FrequencyEnergy curves of the mentioned structure were calculated. It is worth noting that the independency of resulted response to previous points, the possibility of capturing Nonlinear Normal Modes with different frequencies in each degree-of-freedom, the potential of capturing all internal resonances, the expendability of Finite Elements Model to a set of invariable motion equations, and considering material Nonlinearity are among achievements of the current paper.

In this paper, Nonlinear Klein-Gordon equation with quadratic term is solved by means of an analytic technique, namely the Homotopy analysis method (HAM).Comparisons are made between the Adomian decomposition method (ADM), the exact solution and homotopy analysis method. The results reveal that the proposed method is very effective and simple.

In this paper, we propose a new approximate method, namely homotopy analysis sumudu transform method (HASTM) to solve various linear and Nonlinear Fokker-Planck equations.The homotopy analysis sumudu transform method is a combined form of the sumudu transform method and the homotopy analysis method. The proposed technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The results obtained by the proposed method show that the approach is very efficient, simple and can be applied to other Nonlinear problems.

In this paper, we present a new iterative method with order of convergence sixth for solving Nonlinear equations. This method is developed by extending a fourth order method of Ostrowski. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. A general error analysis providing the sixth order of convergence is given. Several numerical examples are given to illustrate the efficiency and performance of the new method.