Plastic hinge properties play a crucial role in predicting the NONLINEAR response of structural elements. The plastic hinge region of reinforced concrete normal beams has been previously studied experimentally and analytically. The main objective of this research is to evaluate the behavior of the plastic hinge region of reinforced concrete deep beams and its comparison with normal beams through finite element simulation. To do so, ten beams contain six deep beams, and four normal beams, under concentrated and uniformly distributed loading, are investigated. Lengths in the plastic hinge region involving curvature localization, rebar yielding, and concrete crushing zones are studied. The results indicate that the curvature localization zone is not suitable for the prediction of plastic hinge length in reinforced concrete deep beams. Based on the results it can be stated that in simply supported normal beams the concrete crushing zone is focused on the middle span, but in simply supported deep beams by creating a compression strut between loading place and support, the concrete crushing zone spreads along the compression trajectory. The rebar yielding zone of simply supported beams increases as the loading type is changed from the concentrated load at the middle to the uniformly distributed load.

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.

Our theorems are on ordered cone metric spaces which are not necessarily normal. In the end, we describe the application of the main results in the integral equation. Although Du in [W. S. Du, A note on cone metric fixed point theory and its equivalence, NONLINEAR ANALYSIS, 72(2010) 2259-2261. ], showed that the fixed point results in the setting of cone metric spaces in which linear contractive conditions appear, can be reduced to their respective results in the metric setting, but it has been shown in the papers [W. S. Du, New cone fixed point theorems for NONLINEAR multivalued maps with their applications, Applied Mathematic Letters, 24(2011) 172-178. ] and [S. Jankovic, Z. Kadelburg, S. Radenovic, On cone metric spaces: A survey, NONLINEAR ANALYSIS, 74(2011) 2591-2601. ] when, cone metric spaces are non-normal, the results may not hold. The results from our paper belong to this category.

Analyzing the NONLINEAR vibration of beams is one of the important issues in structural engineering. According to this, an impressive analytical method which is called modified iteration perturbation method (MIPM) is used to obtain the behavior and frequency of a cantilever beam with geometric NONLINEAR. This new method is combined by the Mickens and iteration methods. Moreover, this method does not require small parameter in the equation which is difficult to be found for NONLINEAR oscillation. The accuracy of the solution that is obtained by the use of MIPM has been shown graphically and compared with exact solution. Comparison shows that good adaptation is obtained and MIPM as a powerful method for solving the vibrational behavior of structures analytically.

In this paper, the effect of angle between predictor and corrector surfaces on the structural ANALYSIS is investigated. Two objective functions are formulated based on this angle and also the load factor. Optimizing these functions, and using the structural equilibrium path’s geometry, lead to two new constraints for the NONLINEAR solver. Besides, one more formula is achieved, which was previously found by other researchers, via a different mathematical process. Several benchmark structures, which have geometric NONLINEAR behavior, are analyzed with the proposed methods. The finite element method is utilized to analyze these problems.The abilities of suggested schemes are evaluated in tracing the complex equilibrium paths.Moreover, comparison study for the required number of increments and iterations is performed. Results reflect the robustness of the authors’ formulations.

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.