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Solving a system of linear or non-linear equations is required to analyze any kind of structures. There are many ways to solve a system of equations. They can be classified as implicit and explicit techniques. The explicit methods eliminate round-off errors and use less memory. The dynamic relaxation method (DRM) is one of the powerful and simple explicit processes. The important point is that the DRM does not require to storage the global stiffness matrix. It just uses the residual loads vector.
Utilizing the virtual masses, damping and time steps, the DRM convert a system of static equations to dynamic ones. The process is started by assuming an initial solution. The next steps are done in such a way that the residual forces are decreased. The proper value of fictitious mass and time step guarantees the convergence of the proposed DR procedure. On the other hand, the convergence rate is dependent on value of damping factor, which is calculated using the lowest eigenvalue of artificial dynamic system in the common dynamic relaxation method. It is evidence; the dynamic system oscillates when damping is zero. The convergence of DRM with zero damping factors is achieved utilizing kinetic damping or -damping. In the kinetic dynamic relaxation process, the velocities of the joints are set to zero when a fall in the level of total kinetic energy of the structure occurs. However, it is difficult to calculate the extreme point of kinetic energy. Topping suggested assuming the peak point at the mid-point of the previous time-step, when a fall down in kinetic energy is occurred. The factor  in the -damping method is time step ratio of two sequence steps. The time-step ratio can be calculated in such a way that the responses converge to exact solutions.
In this paper, a comprehensive review of dynamic relaxation algorithms is presented. Of these, the popular and kinetic damping DR methods are described in detail. Then, the new dynamic relaxation algorithm is proposed. In this procedure, the artificial mass and time steps are similar to the DR methods that have been recently introduced. However, the damping factor is different with these methods. Damping factor is calculated in some specified steps. In other words, damping is zero in the most step of DR algorithm. Therefore, the total number of calculations is reduced. The concentrated damping is imposed, when the value of total kinetic energy of system is at its peak point. Utilizing the proper values of concentrated damping factors, the kinetic energy converges to zero. The presented formulation shows the relation between common and kinetic dynamic relaxation processes, too. It should be noted; the procedures of minimizing the kinetic energy of proposed method and Topping algorithm are different. The kinetic technique is required more calculations. Finally, some benchmark problems of truss and frame structures are selected. The linear and geometric nonlinear analyses are performed. The numerical results also show that the convergence rate of the new DRM increases in the majority of cases with respect to kinetic damping and also popular damping.

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The high ductile of steel moment-resisting frames (SMRFs) during earthquakes has been challenged due to the brittle fractures of their welded (rigid) beam to column connections. Consequently, SMRFs have suffered severe damages and have produced collapse in main structural members (such as beams and columns). During previous years, energy dissipative devices in connections have been developed by researchers to resolve the ductility problem in rigid beam to column connections of SMRFs. Slit steel damper (SSD) as one of these devices contains a plate or a standard section with a number of slits in its web. The damper can dissipate the seismic input energy with inelastic deformation absorption and also prevent seismic energy transmission to the main structural members (such as beam and column). Due to the uniform strut width of SSD, stress concentration at the end parts of the damper struts is produced and unbalanced distribution of von-Mises stresses along the struts is shown. Furthermore, slit dampers are commonly fractured in the end parts of its struts. The low participation of the middle parts of struts in the energy dissipation is caused. Henec, finding the best shape of slits has been attracted by researchers. In this study, new geometry shape of SSD was proposed for improving rigid beam to column connections of steel structures. For investigating the performance of the proposed damper, the behavior of a rigid connection with the common and proposed SSD was assessed subjected to monotonic and cyclic loads in ABAQUS software. The proposed SSD has the same weight in comparison with that of the common SSD. The results of assessment was shown that in the proposed SSD reducing the width of damper slits in two its ends and increasing its middle parts improved its seismic performance in comparison with that of the common SSD. The proposed damper in comparison with common one subjected to shear load can effectively contribute to about 41% of the total dissipated energy. Furthermore, using the proposed damper in a rigid beam to column connection subjected to cyclic loading can effectively contribute to about 51.8% of the total dissipated energy. The performance of the proposed SSD shows that first, the middle part of strip treat like fuse and the suitable ductility provide. Then, the maximum stresses transfers to the top and bottom of strips. Due to the distribution of stresses in more area, the strength of the proposed damper increases. Therefore, withstanding a large number of loading cycles until the failure in this proposed damper, it can be used instead of welded connection in SMRFs.

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The high ductile of steel moment-resisting frames (SMRFs) during earthquakes has been challenged due to the brittle fractures of their welded (rigid) beam to column connections. Consequently, SMRFs have suffered severe damages and have produced collapse in main structural members (such as beams and columns). During previous years, energy dissipative devices in connections have been developed by researchers to resolve the ductility problem in rigid beam to column connections of SMRFs. Circular pipe steel damper (CPSD) proposed as a type of steel damper can indicate and dissipate seismic energy mainly through inelastic deformation. Among steel dampers such as shear panel damper, the advantage of CPSD is to resiste applied load in all direction. Under cyclic loading the circular shape of CPSD can change to elliptical shape which causes an extra energy in its absorption capacity. The previous study indicated that the stress concentration was high at both ends in the loading direction. The maximum stress was also observed at lower ends in the direction of loading. Henec, finding the best shape of cross section can enhance the behaviour of pipe steel damper (PSD). In this study, ellipse PSD (EPSD) was proposed for improving rigid beam to column connections of steel structures. For investigating the performance of the proposed EPSD, the behavior of a rigid connection with the common slit steel damper (SSD) SSD was assessed subjected to cyclic load in ABAQUS software. The proposed EPSD has the same weight in comparison with that of the common CPSD. The results of assessment were shown that in the energy dissipation of the proposed EPSD and CPSD subjected to cyclic load is equal to 11.11 kJ and 9.11 kJ, respectively. Thus, the proposed damper in comparison with CPSD can effectively contribute to about 22% of the total dissipated energy. The distribution of stress in the proposed EPSD in comparison with that of CPSD was also uniformly caused in the hight of EPSD. Furthermore, the performance of a rigid beam to column connection equipped with the proposed EPSD and SSD in subjected to cyclic loading was compared. The results revealed that EPSD in the rigid connection increased to about 63% of the total dissipated energy. Due to the distribution of stresses in more area, the strength of the proposed damper increases. Finally, the performance of a rigid beam to column connection equipped with the proposed EPSD and the welded connection in subjected to cyclic loading effectively was compared. The results demonstrated that the connection equipped with the proposed EPSD colud withstand a large number of loading cycles until the failure. Therefore, the proposed EPSD can be used instead of welded connection in SMRFs.

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One of the forming pipes methods is the rotary draw bending process. Today, bending of thin-walled pipes with low radius of curvature is widely used in the automotive, military and aerospace industries, which is used to bend high-strength pipes. In this paper, at first the necessary models were created to simulate the bending process of the rotary pipe, and then the necessary mechanical and physical properties for stainless steel 304 and elastomers were determined. Then, experimental and numerical study of the forming force and changes in pipe wall thickness were performed. The process simulation was analytically performed using polyurethane elastomeric mandrels and nitrile rubber based on ABAQUS finite element software on 304 steel. The results show a good agreement between simulation and experimental results. Finally, the effects of process parameters including mandrel type, pipe diameter and bending radius were analyzed on the maximum forming force by factorial analysis. The results showed that the maximum forming force for both types of mandrel materials is obtained for pipes with small diameter and high curvature radius. Also, the bending forces increase 5 times by 30%increasing the bending radius, for pipes with smaller diameters. In addition, in equal diameter and radius of bending, the bending forces in the case of using polyurethane mandrel are 25% more than nitrile mandrel.