Development of Direct Integration Algorithms for Structural Dynamics Using Discrete Control Theory

In structural dynamics, integration algorithms are often used to obtain the solution of temporally discretized equations of motion at selected time steps. Various time integration algorithms have been developed in the time domain using different methods. In order for an integration algorithm to be reliable it must be stable and accurate. A discrete transfer function is used to study the properties of integration algorithms. A pole mapping rule from control theory in conjunction with a discrete transfer function is used to develop new integration algorithms for obtaining solutions to structural dynamics problems. A new explicit integration algorithm, called the CR (Chen and Ricles) algorithm, is subsequently developed based on the proposed method. The properties of the algorithm are investigated and compared with other well established algorithms such as the Newmark family of integration algorithms. By assigning proper stable poles to the discrete transfer function the newly developed CR explicit algorithm is unconditionally stable and has the same accuracy as the Newmark method with constant acceleration. In addition, the CR algorithm is based on expressions for displacement and velocity that are both explicit in form, making it an appealing integration algorithm for solving structural dynamics problems.     

Human Jumping and Bobbing Forces on Flexible Structures: Effect of Structural Properties

The behavior of humans jumping and bobbing on flexible structures has become a matter of some concern for both structural integrity and human tolerance. The issue is of great importance for a number of structure types including stadia terraces. A unique test rig has been developed for exploring the forces, accelerations, and displacements that occur when a human subject jumps or bobs on a flexible structure where motion can be perceived. In tests reported earlier, it was found that the subject is able to generate near resonant structural response but it is extremely difficult, if not impossible, to jump or bob at or very near to the natural frequency of the structure when its vertical motion is significant. Also, under such near-resonant conditions, the force developed by the subject was found to drop significantly. In this paper, the effect of altering the subject-to-structure mass ratio and the damping ratio of the structure on these phenomena is presented. As would be expected, it is shown that as the structure becomes more massive and more highly damped it moves less for nominally the same excitation. In this situation, it becomes easier to jump and bob near to resonance and the degree of force dropout reduces, although it is still significant for even the most massive and highly damped case considered. A method for including these effects of human-structure interaction in a load model for dynamic response calculations is then proposed.     

Two-Stepped Evolutionary Algorithm and Its Application to Stability Analysis of Slopes

Based on genetic algorithm and genetic programming, a new evolutionary algorithm is developed to evolve mathematical models for predicting the behavior of complex systems. The input variables of the models are the property parameters of the systems, which include the geometry, the deformation, the strength parameters, etc. On the other hand, the output variables are the system responses, such as displacement, stress, factor of safety, etc. To improve the efficiency of the evolution process, a two-stepped approach is adopted; the two steps are the structure evolution and parameter optimization steps. In the structure evolution step, a family of model structures is generated by genetic programming. Each model structure is a polynomial function of the input variables. An interpreter is then used to construct the mathematical expression for the model through simplification, regularization, and rationalization. Furthermore, necessary internal model parameters are added to the model structures automatically. For each model structure, a genetic algorithm is then used to search for the best values of the internal model parameters in the parameter optimization step. The two steps are repeated until the best model is evolved. The slope stability problem is used to demonstrate that the present method can efficiently generate mathematical models for predicting the behavior of complex engineering systems.     

Identification of Structural Systems Using an Evolutionary Strategy

The problem of system identification is an inverse problem of difficult solution. Currently, difficulties lie in the development of algorithms that use measured data from the system to characterize it without significant a priori knowledge of the system. In this paper, a parameter estimation technique based on an evolution strategy (an optimization algorithm inspired by natural evolution) is presented to overcome some of the difficulties encountered in the field. Using this method, a set of direct problems is solved instead of directly tackling the inverse problem. If the uniqueness of the identification solution is guaranteed for the assumed model and the available data, this heuristic method is able to find a solution without incurring restrictions of other classical optimization methods, like the need for reliable initial estimates and convergence to local optima. Some results obtained with this algorithm are presented for the identification of 3 degrees of freedom (DOF) and a 10?DOF structural system under conditions including limited input/output data, noise polluted signals, and no prior knowledge of mass, damping, or stiffness of the system.     

Computer Modeling for the Complex Response Analysis of Nonstandard Structural Dynamics Problems

Over the past several decades, two intriguing classes of problems, having a wide range of applications in engineering, have been of interest to many researchers: (1) coupled dynamics of a distributed parameter system traversed by one or more moving oscillators; and (2) transient dynamic analysis of axially moving media (and associated phenomena of parametric resonances). Bridge vehicle interaction falls into the first class of problems, and the analysis of flexible appendages deployed from a satellite or a spacecraft is typical of the second class. Mathematically, these two problems are dual to each other, and they often are highly nonlinear in nature and typically involve large overall motion in space with complex effects of convective inertia terms in their governing equations of dynamic equilibrium. The “nonstandard” analytical nature of these problems stems from the fact that we are dealing with one or more of the following peculiarities: (1) variable problem domain; (2) varying spatial distribution of forces over the time duration of the analysis; and/or (3) changing location and type of constraints. Many researchers are trying to formulate the response of these problems, each having a different approach, but applicable only to certain specific details. Moreover, few researchers have concluded that these problems are beyond the scope of the present commercial finite-element (FE) software packages. However, we believe that if the nature and details of these problems are studied properly and carefully, it is immediately possible to simulate these problems in available commercial FE programs. An added advantage would also be the avoidance of many unrealistic simplifying assumptions that are often introduced to reduce the mathematical complexity; e.g., neglecting possible separation (after periods of prior contacts) in beam-moving vehicle problems, assuming linear behavior of suspension systems, and restriction to beam configuration only, among many others. For demonstration, we use ABAQUS in a large number of test cases to be presented. The results are compared with those presented in literatures.     

Energy–Momentum Conserving Algorithm for Nonlinear Dynamics of Laminated Shells Based on a Third-Order Shear Deformation Theory

The paper describes an energy–momentum conserving time stepping algorithm for nonlinear dynamic analysis of laminated shell type structures undergoing finite rotations and large overall motion. The shell model is based on a third order shear deformation theory and falls within the class of geometrically exact shell theories. This algorithm is based on a general methodology for the design of exact energy-momentum conserving algorithms proposed recently by Simo and Tarnow. It is second-order accurate, unconditionally stable, and preserves exactly, by design, the fundamental constants of the shell motion such as the total linear momentum, the total angular momentum, and the total energy in case the system is Hamiltonian. The finite element discretization of the present shell model is closely related to a recent work by the authors dealing with the static case. Particular attention is devoted to the consistent linearization of the weak form of the fully discretized initial boundary value problem in order to achieve quadratic rate of convergence typical of the Newton–Raphson solution procedure. A range of numerical examples is presented to demonstrate the performance of the proposed formulation.     

Fatigue Damage of Steel Bridges Due to Dynamic Vehicle Loads

This paper focuses on the fatigue damage caused in steel bridge girders by the dynamic tire forces that occur during the crossing of heavy transport vehicles. This work quantifies the difference in fatigue life of a short-span and a medium-span bridge due to successive passages of either a steel-sprung or an air-sprung vehicle. The bridges are modeled as beams to obtain their modal properties, and air-sprung and nonlinear steel-sprung vehicle models are used. Bridge responses are predicted using a convolution method by combining bridge modal properties with vehicle wheel forces. A linear elastic fracture mechanics model is employed to predict crack growth. For the short-span bridge, the steel-sprung vehicle caused fatigue failure up to 6.5 times faster than the air-sprung vehicle. For the medium-span bridge, the steel-sprung vehicle caused fatigue failure up to 277 times faster than the air-sprung vehicle.     

Domain-by-Domain Algorithm for Nonlinear Finite-Element Analysis of Structures

Parallel and distributed computers have been shown to provide the necessary computational power to solve large-scale engineering problems. However, in order for this computation style to be effectively used, efficient computational algorithms must be devised. In this work, a domain-by-domain algorithm is developed for the parallel solution of nonlinear structural dynamics problems. In the proposed algorithm, the original structure is partitioned into a number of subdomains. Each subdomain is solved independently and, therefore, concurrently using a traditional direct-solution method. Finally, the solution for the interface degrees of freedom between neighboring subdomains is obtained by enforcing compatibility and equilibrium using an iterative procedure. The nonlinear version of the algorithm involves two iterations: The nonlinear dynamic equilibrium iteration and the interface equilibrium and compatibility iteration. The integration of these two iterations is investigated and two strategies are developed. It is found that the strategy in which the two iterations are isolated is the most efficient. As a demonstration, the fully nonlinear transient analysis of a 20-story model building is carried out. Excellent accuracy in the solution and significant speed up values are obtained.     

Contribution of Nonlinear Finite-Element Analysis to Evaluation of Two Structural Concrete Failures

The failure of two reinforced concrete structures is recounted, one involving a warehouse structure and the other an offshore platform base structure. Design details and factors leading to the collapses are identified and discussed. The structures were subsequently analyzed using nonlinear finite-element analysis procedures, taking into account relevant second-order behavior models. The analyses provided an accurate assessment of the load capacities and failure modes observed, as well as meaningful insights into the underlying behavior mechanisms and factors leading to the failures. This paper supports the view that nonlinear analysis techniques have become useful everyday tools for design office applications, particularly in forensic work, and also gives evidence suggesting that errors in the design of modern structures can be potentially more catastrophic than in the past, and that advanced assessment techniques will assume increased importance as a result.     

On the Comparison of Timoshenko and Shear Models in Beam Dynamics

The classical Timoshenko beam model and the shear beam model are often used to model shear building behavior both for stability or dynamic analysis. This technical note questions the theoretical relationship between both models for large values of bending to shear stiffness parameter. The simply supported beam is analytically studied for both models. Asymptotic solutions are obtained for large values of bending to shear stiffness parameter. In the general case, it is proven that the shear beam model cannot be deduced from the Timoshenko model, by considering large values of bending to shear stiffness parameter. This is only achieved for specific geometrical parameter in the present example. As a conclusion, the capability of the shear model to approximate Timoshenko model for large values of bending to shear stiffness parameter is firmly dependent on the material and geometrical characteristics of the beam section and on the boundary conditions.     

Variational Basis of Nonlinear Flexibility Methods for Structural Analysis of Frames

There have been a number of contributions to the literature on a class of structural analysis methods referred to as nonlinear flexibility methods. These methods appear to perform very well compared to classical stiffness approaches for problems with constitutive nonlinearities. Although most of these methods appeal to variational principles, the exact variational basis of these methods has not been entirely clear. Some of them even seem not to be variationally consistent. We show in this paper that, because the equations of equilibrium and kinematics are directly integrable, a nonlinear flexibility method (in the spirit of those presented in the literature) can be derived without appeal to variational principles. The method does not involve interpolation of the displacement field and the accuracy of the method is limited only by the numerical scheme used to perform element integrals. There is no need for h refinement to improve accuracy. Further, we show that this nonlinear flexibility method is essentially identical, with some subtle algorithmic differences, to a two-field (Hellinger-Reissner) variational principle when the stress interpolation is exact (which is possible for this class of problems). We demonstrate the utility of the nonlinear flexibility method by applying it to a problem involving cyclic inelastic loading wherein the strain fields evolve into functions that are difficult to capture through interpolation.     

Structural Dynamics and a Virtual Prototype Simulator of the HAB System for Oil Sands Production

Crude oil and gas are strategic commodities for U.S. national energy security. The demand for fossil fuel energy will continue to grow amid dwindling global supply. Other energy sources, such as oil sands and heavy oil production, are expected to provide a significant portion of the anticipated shortfall with diversity. The drive towards economic oil sands recovery has become a focal point in the pursuit of low-cost, bulk production methods for competitive oil sands production. The hopper-attached belt-wagon (HAB) system is one of the potential economic haulage systems being pursued towards this strategic goal. The use of this technology will introduce unique dynamic, design, and implementation problems that must be articulated, modeled, and solved via advanced research. The HAB system kinematics and dynamics have been modeled as a planar mechanism of six-bar linkages using the Newton-Euler formulation. A virtual HAB prototype model has been developed and simulated in the automatic dynamic analysis of mechanical systems software environment. The computational simulation results show that the required joint torques vectors vary with time and the HAB system has good operating characteristics. The main novelty of this research is the application of the Newton-Euler-Kutzbach formulation and virtual engineering for creating efficient equipment-formation models toward economic haulage of materials.     

Dynamic Analysis of Nonlinear Frames by Prandtl Neural Networks

A new type of activation function, based on the use of the Prandtl–Ishlinskii operator, has been developed and used in the feed forward neural networks in order to improve their capabilities in learning to identify and analyze nonlinear structures subject to dynamic loading. The genetic algorithm has been used in its training. The neural network, which is referred to as the Prandtl neural network here, has been trained and used in the analysis of two shear frames, a single degree of freedom (SDOF) and a 3DOF, both subjected to earthquake excitations. To assess the capabilities of the Prandtl neural network under ideal situations, the data on the response of the frames have been obtained through the integration of their governing nonlinear equations of motion. The training has been based on the white noise while the strong earthquakes of 200% El Centro in 1940 and Gilroy have been used for testing. Results have shown the high precision of the Prandtl neural network in solving highly hysteretic problems. The issue is important for two main applications in structural dynamics and control: (1) analysis of highly nonlinear structures where it is desired to train a neural network to directly learn the behavior of a structure from experimental data; and (2) intelligent active control of structures where neural network emulators are designed to provide as precise predictions about the future response of the structures as possible, in order to be used in the determination of the required control forces.     

Exact Solution of Nonlinear Hysteretic Responses Using Complex Mode Superposition Method and its Application to Base-Isolated Structures

In this paper, the responses of the structures subjected to arbitrary ground motions are evaluated. With the problem formulated in a state-space form, an exact solution scheme capable of dealing with a variety of material cases is proposed, including cases when postyield stiffness exhibits strain-hardening, strain-softening, and elastic-perfectly plastic properties, respectively. The proposed method can provide much higher accuracy, and requires less computational effort than the traditional step-by-step integration solution technique. The reason for these advantages is discussed and the related formulas are provided. In addition, a new efficient approach is provided for evaluating dynamic response of nonlinear base-isolated structures by taking advantage of their characteristics. Two buildings excited by real earthquake and harmonic ground motions are considered in numerical examples to demonstrate the efficiency of the proposed method.     

Overview of Vibrational Structural Health Monitoring with Representative Case Studies

In the field of nondestructive evaluation and damage detection, there is continued interest in the utilization of vibrational techniques. Structural damage will result in permanent changes in the distribution of structural stiffness. These changes may be detected through structural monitoring. Because of the direct relationship of stiffness, mass, and damping of a multi-degree-of-freedom system to the natural frequencies, mode shapes, and modal damping values, many studies have been directed at using these dynamic properties for the purpose of structural health monitoring. The use of vibrational monitoring is a developing field of structural analysis and is capable of assisting in both detecting and locating structural damage. Vibrational data have been shown to be most useful when used in conjunction with other monitoring systems if a remote and robust damage detection scheme is desired. This paper includes a literature review that summarizes the basic approaches to vibrational monitoring, suggested guidelines for sensor selection and monitoring, and concludes with three example case studies.     

Probability Density and Excursions of Structural Response to Imperfectly Periodic Excitation

A single-degree-of-freedom system under periodic excitation with random phase modulation is considered. Probability density functions (PDF) of the response are obtained numerically using the path integration method. Basic results are presented in the form of expected number of excursions over a given displacement level. They clearly illustrate the transformation of the response PDF with increasing excitation/system bandwidth ratio from one corresponding to the sinusoid with random phase at small values to asymptotically Gaussian PDF at high values of the above ratio. While this qualitative trend is known from previous analysis of the response excess factor by the method of moments, the present quantitative results may be of direct use for reliability predictions. An analytical study is also made for reduced stochastic differential equations of motion, as obtained by stochastic averaging, by the method of moments resulting in a simple explicit expression for the mean square amplitude.     

Proper Orthogonal Decomposition-Based Modeling, Analysis, and Simulation of Dynamic Wind Load Effects on Structures

Multicorrelated stationary random processes/fields can be decomposed into a set of subprocesses by diagonalizing their covariance or cross power spectral density (XPSD) matrices through the eigenvector/modal decomposition. This proper orthogonal decomposition (POD) technique offers physically meaningful insight into the process as each eigenmode may be characterized on the basis of its spatial distribution. It also facilitates characterization and compression of a large number of multicorrelated random processes by ignoring some of the higher eigenmodes associated with smaller eigenvalues. In this paper, the theoretical background of the POD technique based on the decomposition of the covariance and XPSD matrices is presented. A physically meaningful linkage between the wind loads and the attendant background and resonant response of structures in the POD framework is established. This helps in better understanding how structures respond to the spatiotemporally varying dynamic loads. Utilizing the POD-based modal representation, schemes for simulation and state-space modeling of random fields are presented. Finally, the accuracy and effectiveness of the reduced-order modeling in representing local and global wind loads and their effects on a wind-excited building are investigated.     

Identification of a Distribution of Stiffness Reduction in Reinforced Concrete Slab Bridges Subjected to Moving Loads

The method for identifying arbitrary stiffness reduction in damaged reinforced concrete slab bridges under moving loads is proposed and dynamic signals measured at several points are used as response data to reflect the properties of the moving loads sensitivity. In particular, the change in stiffness in each element before and after damage, based on the system identification method, is described and discussed by using a modified bivariate Gaussian distribution function. The proposed method in this work is more feasible than the conventional element-based damage detection method from the computational efficiency because the procedure of finite-element analysis coupled with microgenetic algorithm using six unknown parameters irrespective of the number of elements are considered. The validity of the technique is numerically verified using a set of dynamic data obtained from a simulation of the actual bridge modeled with a three-dimensional solid element. The numerical calculations show that the proposed technique is a feasible and practical method that can prove the exact location of a damaged region as well as inspect the complex distribution of deteriorated stiffness, although there is a modeling error between actual bridge results and numerical model results as well as a measurement error like uncertain noise in the response data.     

Control of Suspension Bridge Nonlinear Vibrations due to Moving Loads

The flexibility and low damping of the long-span suspended cables in the suspension bridges make them prone to vibrations due to wind and moving loads, which affect the dynamic response of the suspended cables and the bridge deck. This paper shows the design of two control schemes to control the nonlinear vibrations in the suspended cable and the bridge deck due to a vertical load moving on the bridge deck with a constant speed. The first control scheme is an optimal state feedback controller. The second control scheme is a robust state feedback controller, whose design is based on the design of optimal controllers. The proposed controllers, whose design is based on Lyapunov theory, guarantee the asymptotic stability of the system. A vertical cable between the bridge deck and the suspended cable is used to install a hydraulic actuator able to generate the active control force on the bridge deck. The MATLAB software is used to simulate the performance of the system with the designed controllers. The simulation results indicate that the proposed controllers are capable of significantly reducing the nonlinear oscillations of the system. In addition, the performance of the system with the proposed controllers is compared to the performance of the system controlled with a velocity feedback controller. It is found that the system with the proposed controllers can provide better performance than the system with the velocity feedback controller.     

Analysis of Dynamic Response of Architectural Glazing Subject to Blast Loading

The effect of blast loading on civilian structures has received much attention over the past several years. The behavior of architectural glazing is of particular interest owing to the disproportionate amount of damage often associated with the failure of this component in a blast situation. This paper presents the development of a simple yet accurate finite element-based tool for the analysis of architectural glazing subjected to blast loading. This has been achieved through the creation of a user-friendly computer program employing the explicit finite-element method to solve for the displacements and stresses in a pane of glass. Both monolithic and laminated panes have been considered, in single and insulated unit configurations, and employing several types of glass. In all cases, the pane of glass has been modeled as a plate supported by an array of boundary conditions that include spring supports, and two failure criteria are employed. Furthermore, the program is designed to predict the hazard level, given a particular glazing configuration and blast load.