Accumulated Damage In Nonlinear Cyclic Static And Dynamic Analysis Of Reinforced Concrete Structures Through 3D Detailed Modeling

Accurate nonlinear cyclic static and dynamic analysis of reinforced concrete structures is necessary when trying to capture the behavior of concrete structures during earthquake excitations. The development of an objective and robust 3D constitutive modeling approach that will be able to account for the accumulated material damage during the cyclic loading of concrete structures is of great importance in order to realistically describe the physical failure mechanisms [1]. The proposed method is based on the experimental results and the concrete modelling of Kotsovos and Pavlovic [2] as modified by Markou and Papadrakakis [3]. The objective of this research work is to propose a computationally efficient modeling method that accounts for the accumulated damage developed in both concrete and steel materials during cyclic static and dynamic excitations.Two new damage factors are proposed herein that take into account the number of openings and closures of cracks during the nonlinear cyclic analysis, thus provide with the ability to account for the accumulated damage in both steel and concrete materials. Furthermore, a solution strategy that describes the behavior of concrete during the cyclic static and dynamic analysis is also presented.The proposed numerical method is validated by comparing its numerical response with the corresponding experimental data of a beam-column frame joint and a two-storey reinforced concrete frame, which were tested under cyclic static and dynamic loading conditions, respectively. Based on the numerical findings, the proposed algorithm manages to accurately capture the experimental results, while the simulation of the understudy models was performed with computational robustness and efficiency. This numerical outcome demonstrates the potential of the proposed 3D detailed modeling approach to be implemented for the seismic assessment of full-scale reinforced concrete structures through nonlinear cyclic static and dynamic analysis

Computationally Efficient and Robust Nonlinear 3D Cyclic Modeling of RC Structures Through a Hybrid Finite Element Model (HYMOD)

A computationally efficient and robust simulation method is presented in this work, for the cyclic modeling of reinforced concrete (RC) structures. The proposed hybrid modeling (HYMOD) approach alleviates numerical limitations regarding the excessive computational cost during the cyclic analysis and provides a tool for the detailed simulation of the 3D cyclic nonlinear behavior of full-scale RC structures. The simplified HYMOD approach is integrated in this work with a computationally efficient cyclic concrete material model so as to investigate its numerical performance under extreme cyclic loading conditions. The proposed approach adopts a hybrid modeling concept that combines hexahedral and beam-column finite elements (FEs), in which the coupling between them is achieved through the implementation of kinematic constraints. A parametric investigation is performed through the use of the Del Toro Rivera frame joint and two RC frames with a shear wall. The proposed modeling method managed to decrease the computational cost in all numerical tests performed in this work, while it induced additional numerical stability during the cyclic analysis, in which the required number of internal iterations per displacement increment was found to be always smaller compared with the unreduced (hexahedral) model. The HYMOD provides for the first time with the required 3D detailed FE solution tools in order to simulate the nonlinear cyclic response of full-scale RC structures without hindering the numerical accuracy of the derived model nor the need of developing computationally expensive models that practically cannot be solved through the use of standard computer systems

Accurate and computationally efficient nonlinear static and dynamic analysis of reinforced concrete structures considering damage factors

Accurate nonlinear dynamic analysis of reinforced concrete structures is necessary for estimating the behavior of concrete structures during an earthquake. A realistic modeling approach to assess their strength and their ability to carry the expected seismic forces is of great importance. Although a number of constitutive models and modeling approaches have been proposed in order to capture the behavior of reinforced concrete structures under static loading conditions, only a few of these numerical models have been extended to dynamic problems.The objective of this paper is to integrate a computationally efficient 3D detailed modelling of concrete structures with damage factors that take into account the opening and closing of cracks, as well as, damage factors for steel reinforcement considering the surrounding concrete damage level, in order to capture the level of damage and stiffness degradation of structures undergoing many loading cycles.In the adopted numerical model, the concrete domain is discretized with 8-noded isoparametric hexahedral finite elements, which treat cracking with the smeared crack approach, while the steel reinforcement is modeled as embedded beam elements inside the hexahedral mesh. The validity of the proposed method is demonstrated by comparing the numerical response with the corresponding experimental results of various reinforced concrete structural members and structures. Based on the numerical investigation, it was found that the proposed integration of the damage factors with computationally efficient concrete and steel material models can efficiently predict both static and dynamic nonlinear behavior of concrete structures, with the ability to capture the complicated phenomenon of the pinching effect

Accurate and computationally efficient nonlinear static and dynamic analysis of reinforced concrete structures considering damage factors

Accurate nonlinear dynamic analysis of reinforced concrete structures is necessary for estimating the behavior of concrete structures during an earthquake. A realistic modeling approach to assess their strength and their ability to carry the expected seismic forces is of great importance. Although a number of constitutive models and modeling approaches have been proposed in order to capture the behavior of reinforced concrete structures under static loading conditions, only a few of these numerical models have been extended to dynamic problems.The objective of this paper is to integrate a computationally efficient 3D detailed modelling of concrete structures with damage factors that take into account the opening and closing of cracks, as well as, damage factors for steel reinforcement considering the surrounding concrete damage level, in order to capture the level of damage and stiffness degradation of structures undergoing many loading cycles.In the adopted numerical model, the concrete domain is discretized with 8-noded isoparametric hexahedral finite elements, which treat cracking with the smeared crack approach, while the steel reinforcement is modeled as embedded beam elements inside the hexahedral mesh. The validity of the proposed method is demonstrated by comparing the numerical response with the corresponding experimental results of various reinforced concrete structural members and structures. Based on the numerical investigation, it was found that the proposed integration of the damage factors with computationally efficient concrete and steel material models can efficiently predict both static and dynamic nonlinear behavior of concrete structures, with the ability to capture the complicated phenomenon of the pinching effect

Inverse mass matrix via the method of localized lagrange multipliers

An efficient method for generating the mass matrix inverse is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces, and constructs the free-free inverse mass matrix employing the standard FEM procedure. The various boundary conditions are realized by the method of localized Lagrange multipliers. Numerical experiments with the proposed inverse mass matrix method are carried out to validate the effectiveness proposed technique when applied to vibration analysis of bars and beams. A perfect agreement is found between the exact inverse of the mass matrix and its direct inverse computed through biorthogonal basis functions

Gradient and relaxation nonlinear techniques for the analysis of cable supported structures

The purpose of this work is to investigate the efficiency of numerical nonlinear solution procedures when applied to the static analysis of cable supported structures. Gradient and relaxation methods are developed and compared with existing nonlinear solution techniques. In order to obtain a more general picture of the performances of the above methods, stiffness methods with Newton Raphson iterative schemes have also been included in the comparative study. Chapter 1 examines the behaviour and characteristics of cable supported structures and investigates the analytical requirements for static analysis. A state of the art of numerical solution techniques used to analyse these structures is presented. An extensive review of published work in relation to the analysis of single unstiffened cables, dual cables and cable networks is also presented. Chapter 2 approaches the solution of the structural problem through total energy formulations. Three basic energy formulations are discussed with particular emphasis given to the total potential energy formulation. The principles of the unconstrained minimization method are considered and different search techniques for approximating the minimum are discussed. Expressions for the gradient vector of the total potential energy are obtained and the tangent stiffness matrix is evaluated as the matrix of the second partial derivatives of the total potential energy formulation. Different scaling techniques are reviewed and the effects of the termination criterion used, for different methods of analysis, on the final accuracy of the methods is also discussed. In Chapter 3 there is an extensive theoretical treatment of gradient methods for the nonlinear solution of structural problems. Particular emphasis is given to the conjugate gradient algorithm and the modifications proposed by various investigators since it first appeared in 1952. A number of one dimensional linear searches are studied which approximate the minimum along the p direction and determine the scalar parameter a for the next iteration. And extensions of the conjugate gradient algorithm for the evaluation of the scalar parameter e, as proposed by Sorenson and Polak and Ribiere are discussed, Finally, the memory gradient method which employs a two dimensional linear search for a simultaneously evaluation of a and e is also presented. Chapter 4 examines the efficiency of the methods discussed in Chapter 3 when applied to the nonlinear solution of a number of test problems. The problems are selected to have varying numbers of degrees of freedom and the respective stiffness matrices to have differing condition numbers in order to study the response of the methods for different structural characteristics. The Fletcher and Reeves method with Davidon's linear search with a cubic equation to approximate the minimum, Stanton's algorithm for bracketing the solution and the regular falsi-bisection algorithm to approximate the minimum, a combined algorithm of Davidon and Stanton's techniques, Buchholdt's method, Polak and Ribiere's algorithm, Sorenson's version, the memory gradient method and a number of linearized conjugate gradient algorithms are developed and their convergence characteristics are compared. The effects of scaling and reinitialization are also studied. In Chapter 5 there is a theoretical investigation of relaxation methods and in particular the dynamic relaxation and the successive overelaxation methods. A rigorous examination of the characteristic properties of dynamic relaxation is carried out. The method is treated as a standard eigenvalue problem for error vectors and expressions for the iteration parameters are developed with respect to the minimum and maximum eigenvalue of the current stiffness matrix. A theoretical comparison of a number of pure iterative methods is performed and relationships between the iteration or scalar parameters of the conjugate gradient method, the dynamic relaxation method, the jacobi semi-iterative method, and the Tchebycheff methods, are established. This suggests that all these methods in fact belong to the same family of methods called "three term recursion formulae". A combined conjugate gradient and Tchebycheff type method is also studied. A method for the automatic evaluation of the dynamic relaxation parameters is developed by the author which can guarantee convergence for almost any arbitrary initial estimate of the minimum and maximum eigenvalues of the current stiffness matrix. The concept of using kinetic energy damping instead of viscous damping in the dynamic relaxation iterative process is also examined. Finally, the successive overelaxation method is modified to be applicable to the nonlinear analysis of structural problems, and two ongoing processes for automatic evaluation of the optimum overelaxation parameter w , proposed by Carre and Hageman, are also examined. Chapter 6 is devoted to a theoretical and numerical investigation of the problem of finding the minimum and maximum eigenvalues of a symmetric matrix. The power method, the steepest descent method, the conjugate gradient method, and the coordinate relaxation method, are among the techniques examined and compared in this Chapter. Several other modifications to the initial conjugate gradient algorithm are also studied, including the modification proposed by Fried for the evaluation of the scalar parameters and the one proposed by Geradin. An orthogonalization process is also applied to alleviate the dependency of the convergence of the method on the initial approximation for the final eigenvector. In Chapter 7 numerical studies of the relaxation methods discussed in Chapter 5 are performed. Alternative forms of the dynamic relaxation methods with an "a priori" evaluation of the iteration parameters (using one of the methods discussed in Chapter 6), with automatic adjustment of the relaxation parameters based on the method developed in Chapter 5, and with the incorporation of kinetic damping, are applied for different test problems. Techniques to avoid the occurrence of instability of the method, when the current maximum eigenvalue of the iteration matrix becomes greater than the estimated maximum eigenvalue, are also developed and compared. Finally, the efficiency of the successive overelaxation method, with both constant and adjustable relaxation parameters is examined and compared with the efficiency of the dynamic relaxation method. In Chapter 8 a review of methods operating through the formulation of the overall stiffness matrix is carried out. The efficiency of these methods is dependent on both the method employed to perform the linear solution when this is necessary and the nonlinear technique used to approximate the nonlinear equilibrium position in each iteration. A compact store elimination scheme, proposed by Jennings, is studied in conjunction with the Gaussian elimination procedure. Three different classes of nonlinear techniques are discussed together with the area in which each one has proved to be more suitable. Chapter 9 performs a general comparative study of the convergence characteristics of the best methods from each classification {gradient, relaxation and stiffness methods}, and examines the advantages and disadvantages involved in the application of the methods to the nonlinear elastic analysis of cable supported structures with members being allowed to slacken. The computer time required to obtain a certain degree of accuracy, the storage requirements and the cost involved are all examined and compared in an effort to select the most suitable method for each particular class of problem. In Chapter 10 the ultimate load carrying capacity of cable structures is studied, with members being allowed to slacken and with the inclusion of nonlinear stress-strain relationships. Two different solution procedures are employed: the stiffness method with or without the compact store elimination scheme in conjunction with Newton Raphson iteration, and Stanton's conjugate gradient algorithm. The convergence of the methods are tested for different values of the termination parameters and load increments. A continuous stress-strain curve as proposed by Jonatowski is used and provision for the cable members to reload following a different path is also included. Finally, Chapter 11 reviews the general conclusions resulting from the experience gained from the theoretical and numerical treatment of the methods discussed in this work, together with suggestions for further research

A new damage factor for seismic assessment of deficient bare and FRP-retrofitted RC structures

The seismic assessment of reinforced concrete (RC) structures before and after retrofitting is a challenging task, mainly because existing numerical tools cannot accurately model the evolution of concrete damage. This article proposes an innovative numerical method suitable to model and assess the ultimate carrying capacity of RC structures. The modelling approach proposes a steel constitutive material model with a damage factor that accounts for accumulated damage within the surrounding concrete domain, which effectively captures bar slippage. The proposed method is validated with experimental results from full-scale cyclic tests on deficient bare and CFRP-retrofitted RC joints tested previously by the authors. The results indicate that the proposed simulation method captures the extreme nonlinearities observed in the tested RC joints, with acceptable accuracy and computational robustness. The results of this study are expected to contribute towards the development of more reliable numerical tools and design guidelines for efficient seismic assessment of RC structures before and after earthquakes

A new damage factor for seismic assessment of deficient bare and FRP-retrofitted RC structures

The seismic assessment of reinforced concrete (RC) structures before and after retrofitting is a challenging task, mainly because existing numerical tools cannot accurately model the evolution of concrete damage. This article proposes an innovative numerical method suitable to model and assess the ultimate carrying capacity of RC structures. The modelling approach proposes a steel constitutive material model with a damage factor that accounts for accumulated damage within the surrounding concrete domain, which effectively captures bar slippage. The proposed method is validated with experimental results from full-scale cyclic tests on deficient bare and CFRP-retrofitted RC joints tested previously by the authors. The results indicate that the proposed simulation method captures the extreme nonlinearities observed in the tested RC joints, with acceptable accuracy and computational robustness. The results of this study are expected to contribute towards the development of more reliable numerical tools and design guidelines for efficient seismic assessment of RC structures before and after earthquakes

Vulnerability analysis of large concrete dams using the continuum strong discontinuity approach and neural networks

Probabilistic analysis is an emerging field of structural engineering which is very significant in structures of great importance like dams, nuclear reactors etc. In this work a Neural Networks (NN) based Monte Carlo Simulation (MCS) procedure is proposed for the vulnerability analysis of large concrete dams, in conjunction with a non-linear finite element analysis for the prediction of the bearing capacity of the Dam using the Continuum Strong Discontinuity Approach. The use of NN was motivated by the approximate concepts inherent in vulnerability analysis and the time consuming repeated analyses required for MCS. The Rprop algorithm is implemented for training the NN utilizing available information generated from selected non-linear analyses. The trained NN is then used in the context of a MCS procedure to compute the peak load of the structure due to different sets of basic random variables leading to close prediction of the probability of failure. This way it is made possible to obtain rigorous estimates of the probability of failure and the fragility curves for the Scalere (Italy) dam for various predefined damage levels and various flood scenarios. The uncertain properties (modeled as random variables) considered, for both test examples, are the Young’s modulus, the Poisson’s ratio, the tensile strength and the specific fracture energy of the concrete

Fire analysis of steel frames with the use of artificial neural networks

The paper presents an alternative approach to the modelling of the mechanical behaviour of steel frame material when exposed to the high temperatures expected in fires. Based on a series of stress-strain curves obtained experimentally for various temperature levels, an artificial neural network (ANN) is employed in the material modelling of steel. Geometrically and materially, a non-linear analysis of plane frame structures subjected to fire is performed by FEM. The numerical results of a simply supported beam are compared with our measurements, and show a good agreement, although the temperature-displacement curves exhibit rather irregular shapes. It can be concluded that ANN is an efficient tool for modelling the material properties of steel frames in fire engineering design studies. (c) 2007 Elsevier Ltd. All rights reserved