Considering the vulnerability of Double-layer grid space structures to progressive collapse phenomenon, it is necessary to pay special attention to this phenomenon in the design process. Alternate path method is one of the most appropriate and accepted methods for progressive collapse resistant design of structures. Alternate Path Method permits local failure to occur but provides alternate paths around the damaged area so that the structure is able to absorb the applied loads without overall collapse. Following the sudden initial local failure event, severe dynamic effects may arise which should be taken into account in determining the realistic collapse behavior of the structure. In this paper, a new methodology based on alternate path method is presented to apply dynamic effects of initial local failure. The method is called nonlinear dynamic alternate path method. Due to its capability to take account of dynamic nature of the failure, this method can be used to evaluate realistic collapse behavior of the structure and to investigate the vulnerability of the structure to progressive collapse phenomenon.

High stiffness to weight ratio, ease and speed of handling, as well as having favorable architectural appearance cause that, Double layer grids with ball joint system are widely used to cover large spans. A Double-layer grid has a complex behavior due to a large number of elements and a particular type of joints; hence, structural identification of this type of structure is an important issue, which refers to the determination of natural frequencies, mode shapes, and damping ratios. These results are necessary to complete the structural health monitoring, finite element model updating and damage detection. Due to the limitations of input-output methods, modal parameters of civil engineering structures such as bridges, dams, tall buildings, and Double layer grids are determined mainly by output-only modal identification. In output-only methods, the vibration parameters are determined based on the information acquired only from the structure’s output. In this work, physical model of a ball jointed Double-layer grid with dimensions of 2.8 m at 2.8 m, which is supported on four steel pipes in four corners was made in the laboratory. The grid consists of 32 members connected together with 13 balls, each having ten threaded holes at different angles. each member consists of a middle pipe and connecting parts including conical piece, sleeve and high strength bolt at both ends of the pipe. The middle pipe has the nominal length, diameter and thickness of 120 cm, 7.64 cm and 0.35 cm, respectively. The horizontal center to center distance of adjacent balls in each layer of the grid is 1.414 m and the total height of the structure includes the column length (1.3 m) and the distance between the top and bottom layers (1 m), which is equal to 2.3 m in total. The approximate weight of the structure is 3532 N. All the members and the balls used in the grid are identical. After all the members of the grid have been assembled, the bolt at each joint is tightened in a series of steps by twisting the corresponding sleeve. Exciting the grid, its acceleration response was measured. The modal parameters were obtained using four output-only modal identification techniques; namely enhanced frequency decomposition (EFDD), curve-fit frequency domain decomposition (CFDD), data-driven stochastic subspace identification (SSI-DD) and covariance-driven stochastic subspace identification (SSI-Cov). Two types of excitations were used in output-only modal tests, namely direct and indirect excitations. Since the modal parameters obtained via input-output modal analysis have less uncertainty compared to the output-only modal analysis techniques, an input-output modal test was also performed and the results are considered as reference values. It deduced that parameters identified in the direct excitation, were more accurate compared to indirect excitation. The results showed that the natural frequencies and mode shapes of the Double-layer grid were estimated with a high accuracy via the four methods. The greatest relative difference between the natural frequencies belonged to the second mode and equaled 2.07%. The dispersion of estimated damping was much higher compared to natural frequencies and mode shapes. The results indicated that identified damping in the direct excitation was lower than indirect one. Among the 4 methods, SSI-Cov had the least error in damping estimation of the Double-layer grid. The values of estimated modal damping ratios were relatively low (fraction of 1%). The mean relative error of the identified parameters showed that the time-domain methods estimated the damping ratios with less error; While the frequency-domain methods identified natural frequencies and mode shapes with higher accuracy.

Modal parameters of large civil engineering structures such as Modal Damping Ratios (MDRs) are determined mainly through output-only modal identi cation. In this paper, MDRs of a Double-layer grid were obtained using output-only modal identi cation. For this purpose, a Double-layer grid constructed of a ball-joint system was tested. Through some random tapping on the structure, the acceleration response in multiple locations was measured. The acquired data were processed using output-only modal identi cation to arrive at MDRs. The MDRs corresponding to the  rst eight modes of the grid were extracted by  ve output-only modal identi cation techniques, namely Enhanced Frequency Domain Decomposition (EFDD), Curve- t Frequency Domain Decomposition (CFDD), and three di erent methods of data-driven stochastic subspace identi cation. To determine the appropriate model order used in SSI methods, sensitivity analysis was carried out and the resulting number of orders was 200. The proper frequency resolution of 1600 was determined to estimate the MDRs of the grid by EFDD and CFDD. The results showed that the MDRs of the grid obtained from di erent methods were in good agreement with each other. The grid has very low MDRs because the MDRs of the modes measured using di erent methods varied from 0. 06% to 0. 11%.

Large-scale spatial skeletal structures belong to a special kind of 3D structures widely used in exhibition centers, supermarkets, sport stadiums, airports, etc., to cover large surfaces without intermediate columns. Space structures are often categorized as grids, domes and barrel vaults. Double layer grid structures are classical instances of prefabricated space structures and also the most popular forms which are frequently used nowadays. Topology optimization of large-scale skeletal structures has been recognized as one of the most challenging tasks in structural design. In topology optimization of these structures with discrete cross-sectional areas, the performance of meta-heuristic optimization algorithms can be increased if they are combined with continuous-based topology optimization methods. In this article, a hybrid methodology combining evolutionary structural optimization (ESO) and harmony search algorithm (HSA) methods is proposed for topologyoptimization of Double layer grid structures subject to vertical load. In the present methodology, which is called ESO-HSA method, the size optimization of Double layer grid structures is first performed by the ESO. Then, the outcomes of the ESO are used to improve the HSA. In fact, a sensitivity analysis is carried out using an optimization method (ESO) to determine more important members based on the cross-sectional areas of members. Then, the obtained optimum cross-sectional areas of members are used to enhance the HSA through a modification. Structural weight is minimized against constraints on the displacements of nodes, internal stresses and element slenderness ratio. In topology optimization of Double layer grid structures, the geometry of the structure, support locations and coordinates of nodes are fixed and this structure is assumed as a ground structure. Presence/absence of bottom nodes, and element cross-sectional areas are selected as design variables. In topology optimization of the ground structure, tabulating of nodes is carried out based on structural symmetry: this leads to reduce complexity of design space and nodes are removed in groups of 8, 4 or 1. Also, to further reduce the search space, the membersare grouped as mentioned in literature. The presence or absence of each node group is determined by a variable (topology variable) which takes the value of 1 and 0 for the two cases, respectively. The ground structure is assumed to be supported at the perimeter nodes of the bottom grid. Therefore, these supported nodes will not be removed from the ground structure. In order to achieve a practical structure, the existence of nodes in the top grid will not be considered as a variable. This causes the load bearing areas of top layer nodes to remain constant. Also, discrete variables are used to optimize the cross-sectional area of structural members. These variables are selected from pipe sections with specified thickness and outer diameter. The proposed approach is successfully tested in topology optimization problem of Double layer grid structure. In particular, ESO-HSAis very competitive with other metaheuristic methods recently published in literature and can always find the best design overall. Also, it is determined that HSA method can find better answer in the topology optimization of large-scale skeletal structures, in comparison to optimum structures attained by the GSA and ICA.

Although natural frequencies and mode shapes can be accurately measured by dynamic tests, error bounds in values of damping estimation can be large. Since damping strongly influences the design and control of structures, efforts are made to find the damping that has the least error. The random and bias errors are the two important types of error in damping estimation of structures via frequency domain methods. These errors can be reduced by choosing appropriate frequency spacing. In this study, the frequency spacing leading to the least bias and random errors for estimation of modal damping is determined. The complexity of the damping phenomenon on the one hand and complexity of the structural behavior of a Double layer grid, on the other hand, led to this study. For this purpose, a Double layer grid constructed from ball joint system was tested. the modal damping ratios related to the first 6 modes of a Double layer grid with the ball jointed system were identified for different frequency spacing via two output only modal identification techniques; namely enhanced frequency domain decomposition (EFDD), curve fit frequency domain decomposition (CFDD). The modal damping ratio estimations identified through the two methods were then compared with the results of the input output identification method of Ibrahim time domain (ITD) as the reference value. The results showed that there is an almost linear relationship between the modal damping ratio estimations and the frequency spacing in each mode. At the frequency spacing of 0. 0625 Hz, the modal damping ratios obtained from the output only methods showed the least difference (between 0 to 21. 43%) with the reference values. At this frequency spacing, the root mean square deviation of modal damping estimations between enhanced frequency domain decomposition (EFDD) and curve-fit frequency domain decomposition (CFDD) with the corresponding reference values was 0. 02.

In the present work, a feasibility study has been carried out in order to investigate the suitability of the use of skeletal Double layer grid systems as the ceilings and walls of residential, educational, hospital, commercial and administrative buildings in seismic areas by a series of the nonlinear static as well as dynamic time history analyses of representative models. In the case of the sample single story buildings designed in accordance with the common Allowable Stress Design methods, the pushover analysis results have indicated that a lighter structure with higher stiffness and lateral load carrying capacity can be achieved compared with a similar building composed of the moment resisting frame system. It has been demonstrated that with the imposition of appropriate limitations on member slenderness ratios for a few critical members, the ductility of the structure can be improved considerably. Also, the influence of the slope of the post buckling curve representing the member behavior has been found to be remarkable. A comparison of the response of the structure under the influence of three translational components of ground motion with single component excitation has demonstrated the significance of the consideration of the vertical component of ground motion in the analysis and design of such building structures. With due consideration of the lightweight and high indeterminacy of the structural system under consideration, the preliminary investigation carried out here has revealed that the proposed system is a potential candidate for the mass production of common types of building structures in seismic areas and hence deserves further theoretical and experimental investigations.

In large civil engineering structures, the output-only modal identification is the most applicable technique for estimating the modal parameters such as damping. However, due to no measurement and control of excitation force, the identified parameters obtained by output-only technique have more uncertainty than those derived from the input-output technique. Given the different nature and uncertainties of the two modal identification techniques, in the present study, the damping related to the first 12 modes of a Double-layer grid developed from the ball joint system were identified via the two techniques and compared with each other. For this purpose, a Double-layer grid was constructed by pipes and balls with free-free boundary conditions provided for both input-output and output-only experiments. Exciting the grid, its acceleration response was measured at appropriate degrees of freedom. Then, by using these data and performing modal analysis, involving four different methods of input-output and five different methods of output-only, the natural frequencies and damping ratios of the desired modes were extracted. The results indicated that despite the good agreement between the modal damping of the grid, as identified by different methods of input-output together and by different methods of output-only together, the results of input-output and output-only methods were different with each other. The damping values through the input-output modal identification methods were on average 65% higher than the corresponding values of the output-only modal identification methods.

Changes in modal strain energy of elements before and after damage is a robust damage detection index. However, in structures like Double layer grids which have large number of elements, this method has some problems. First, In large structures this method needs more mode shapes to detect damage through modal strain energy method which in practice is difficult to determine. Second, this method introduce some healthy elements as damaged element. To overcome these problems, in this paper a two stage damage detection technique based on modal strain energy method is presented for detecting damage in Double layer grids. First, the Modal Strain Energy Based Index (MSEBI) for each mode shape is determined. Then a data fusion technique based on Bayesian theory is used to combine MSEBI values obtained from each mode shape to find damaged elements. Then Charged System Search (CSS) optimization method which is a powerful optimization method is employed to optimize an objective function based on natural frequency to determine damage severity of damaged elements. To demonstrate the performance of the proposed method, a large Double layer grid with 1536 elements and different single and multiple damage cases is considered. Numerical results show that the proposed method can successfully find damaged elements and their severities using only few first numbers of mode shapes and frequencies.

Due to the some uncertainties during manufacture and assembly of Double layer grids with ball joint system, the behavior of this jointing system in the structure is different from the behavior of individual and discrete joints. In the present work, the behavior of a ball joint system has been determined in true conditions of its operation in a Double layer grid by the general framework of inverse problem. A full scale Double layer grid was constructed using ball joint system. A series of modal testing were performed on the Double layer grid in free-free support conditions and its FRFs (Frequency Response Functions) were measured in appropriate degrees of freedom. Using the measured FRFs, natural frequencies of the eight vibration modes of the grid were extracted. A beam element was used at each end of members in the finite element model of the grid in order to simulate the ball joint behavior. Through modal analysis of the finite element model, natural frequencies of the eight vibration modes of the grid were calculated for different section properties of the joint beam element. With the finite element model updating of the grid through reduction of difference between its experimental and analytical natural frequencies, section properties of the beam element representing behavior of the ball joint system in the model were obtained. Whilst the updated model resulted in a very good approximation of the natural frequencies of the Double layer grid, a flexible behavior in different degrees of freedom was obtained for the ball joint system. Also, the analytical frequency response functions that were calculated using the obtained section properties for the joint beam element, had a good correlation with the experimental ones.

Analysis and design of space structures are normally time consuming since a large number of members is involved. The problem becomes more involved when optimization is performed, where numerous analyses should be carried out. In this paper, neural network and genetic algorithm are applied for optimization of Double steel layer grids. 180 models with three topologies, spans between 10 and 75 m and height between I and 2.5 m are analyzed and designed for optimum weight. The results have been used for training and testing of proposed neural networks system to predict the weight of the structures. The genetic algorithm based on neural networks is used for design optimization. Thus, for a site plan with arbitrary length and width, the output of the neurongenetic system is the selection of a Double steel grid with optimized shape based on three available topologies with height, distance between columns and length of horizontal members.