Structural Response of 34‐m Darrieus Rotor to Turbulent Winds

The results of a mathematical investigation of the dynamic response of the Sandia/Department of Energy/U.S. Department of Agriculture 34‐m‐diameter Darrieus rotor wind turbine erected at Bushland, Tex., are presented. The formulation used a double multiple stream‐tube aerodynamic model combined with a turbulent airflow and included the effects of (linear) aeroelastic forces. The structural analysis was carried out using previously established procedures with the computer program MSC/NASTRAN. A number of alternative expressions for the spectrum of turbulent wind were used. The modal loading represented by each did not differ significantly; a more significant difference was caused by imposing full lateral coherence of the turbulent flow. Spectra of the predicted stresses at various locations show that without aeroelastic forces, very severe resonance is likely to occur at certain natural frequencies. Inclusion of aeroelastic effects greatly attenuates this stochastic response, especially in modes involving in‐plane blade bending. Comparison is made with preliminary field data collected at medium‐low wind speeds (11.3 m/s) from critical locations on the blade. This comparison shows generally good agreement between measured stress spectra and model predictions based on 10% turbulence intensity and including aeroelastic forces.     

Large Deflection Stability of Slender Beam-Columns with Semirigid Connections: Elastica Approach

The large-deflection elastic analysis of slender beam-columns of symmetrical cross sections with semirigid connections under end loads (forces and moments) including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions are utilized in the proposed method which can be used in the large-deflection stability analysis of slender beam-columns with rigid, semirigid, and simple connections under any combination of end loads (conservative and nonconservative). The proposed method consisting of a closed-form solution of the Elastica can also be utilized in the large deflection analysis of beam-columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The main limitation of the Elastica is that only flexural strains are considered (the effects of axial and shear strains are neglected). Therefore results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for plane beam-columns under bending actions. The large-deflection analysis of a beam-column with flexible connections at both ends becomes a complex problem requiring the simultaneous solution of at least two highly nonlinear equations with elliptical integrals. The solution of this problem becomes even more complex when the end connections are nonlinear or the direction of the applied end load changes (like “follower” loads). The validity and effectiveness of the proposed method and equations are verified against available solutions of very large deflection elastic analysis of beam-columns. Four comprehensive examples are included for verification and easy reference.     

Vibro-Impact Behavior of Two Orthogonal Beams

Structural interaction between two beam-like structures is a situation that occurs in piping systems, among other applications relevant to the nuclear, petroleum, biomedical, and automotive industries, for instance. This paper analytically investigates the repetitive impact dynamics of two orthogonal pinned–pinned beams subjected to base excitation at specified frequency and acceleration. The orthogonal beam configuration restricts the contact to a single point, and the contact interface is modeled by a spring. Although many approaches have been developed for multibody dynamics, the constraint and modal mapping method is efficiently applied herein to obtain the forced response through modal analysis. The vibration is described in a piecewise fashion as switching between the linear in-contact and not-in-contact states, and compatibility conditions are applied at their junctions. The development of the conjoined mode shapes and their orthogonality is derived in detail. The contact impulse is used to describe the structures’ complex interacting behavior through repetitive impact frequency-response functions. In order to determine major response factors, parameter studies are performed on contact stiffness, relative beam stiffness, contact location, modal damping, and stand-off gap.     

Transient Response of Supported Beams to Moving Forces with Sinusoidal Time Variation

Moving forces are a common loading pattern for flexible beams, found in many applications in both civil and mechanical engineering. These forces give rise to a transient response, the nature of which depends on the time variation of the amplitude of the force and its position along the beam. In addition to the possibility of numerical evaluation, closed form solutions of the beam response are beneficial for their simplicity of use, and because they allow an understanding of the system behavior. On the other hand, these prove to be rather complicated in most cases, and only a limited number of cases are available in the literature. This paper studies the simple but common case of a supported beam loaded by a force with sinusoidal time variation moving at a constant speed. Simple equations are presented for the approximated responses at and away from resonance, and their accuracy is discussed. Transient frequency response functions are also shown. Finally, as an example, the results are applied to an evaluation of the response of a beam footbridge to the action of a walker, and compared to code specifications.     

Peak Response of a Nonlinear Beam

A model for estimating the peak dynamic response distribution of a nonlinear beam, based on a special class of non-Gaussian stochastic processes, is proposed in this paper. It is shown that the stochastic response of a cantilever beam with geometrically nonlinear behavior can be accurately calibrated with translation processes. Different models to describe the significant bimodal features in the marginal probability density functions of the response time histories are proposed. Finally, two of these models are used to estimate the response peak value distributions and the results are compared. This comparison demonstrates the effects of inaccurate models for the parent response processes on the peaks estimation.     

Nonlinear Dynamic Response of Piles under Horizontal Excitation

This paper presents test results from cast-in situ reinforced concrete single and group piles subjected to strong horizontal excitation. The tests were conducted for different eccentric moments simulating different excitation levels to obtain the frequency-amplitude response of the pile. Moderate nonlinear behavior is observed in both horizontal and rocking components of vibration. The experimental results were compared with dynamic interaction factor approach using nonlinear solutions. The accuracy of the nonlinear analysis in predicting the dynamic response depends on the choice of parameters that best characterize the response of boundary zone around the pile and the realistic length of pile separation. It is shown in this study that by allowing for boundary zone and separation between pile and soil, close agreement between theoretical predictions and measured response curves can be achieved.     

Response of a Buckled Beam Constrained by a Tensionless Elastic Foundation

In this paper, we study the deformation and stability of a pinned buckled beam under a point force. The buckled beam is constrained by a tensionless elastic foundation, which is flat before deformation. From static analysis, we found a total of five different deformation patterns: (1)?noncontact, (2)?full contact, (3)?one-sided contact, (4)?isolated contact in the middle, and (5)?two-sided contact. For a specified set of parameters, there may coexist multiple equilibria. To predict the response of the buckled beam foundation system as the point force moves from one end to the other, we have to determine the stability of these equilibrium configurations. To achieve this, a vibration method is adopted to calculate the natural frequencies of the system, taking into account the slight variation of the contact range between the buckled beam and the tensionless foundation during vibration. It is concluded that among all five deformation patterns, Deformations 1, 2, 3, and 4 may become stable for certain loading parameters. In the extreme case in which the foundation is rigid, on the other hand, only Deformations 1 and 3 are stable.     

Mathematical Modeling and Analysis of Flutter in Bending-Torsion Coupled Beams, Rotating Blades, and Hard Disk Drives

In this study I present a review of several research directions in the area of mathematical analysis of flutter phenomenon. Flutter is known as a structural dynamical instability that occurs in a solid elastic structure interacting with a flow of gas or fluid, and consists of violent vibrations of the structure with rapidly increasing amplitudes. The focus of this review is a collection of models of fluid-structure interaction, for which precise mathematical formulations are available. The main objects of interest are analytical results on such models, which can be used for flutter explanation, its qualitative and even quantitative treatments. This paper does not pretend to be a comprehensive review of the enormous amount of engineering literature on analytical, computational, and experimental aspects of the flutter problem. I present a brief exposition of the results obtained in several selected papers or groups of papers on the following topics: (1) bending-torsion vibrations of coupled beams; (2) flutter in transmission lines; (3) flutter in rotating blades; (4) flutter in hard disk drives; (5) flutter in suspension bridges; and (6) flutter of blood vessel walls. Finally, I concentrate on the most well-known case of flutter, i.e., flutter in aeroelasticity. The last two sections of this review are devoted to the precise analytical results obtained in my several recent papers on a specific aircraft wing model in a subsonic, inviscid, incompressible airflow.     

Dynamic Response of Closed-Loop System with Uncertain Parameters Using Interval Finite-Element Method

Using the interval finite-element method, the vibration control problem of structures with interval parameters is discussed, which is approximated by a deterministic one. Based on the first-order Taylor expansion, a method to solve the interval dynamic response of the closed-loop system is presented. The expressions of the interval stiffness and interval mass matrix are developed directly with the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of dynamic responses is developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The effect of the different levels of uncertainties of interval parameters on responses is discussed. The comparison of the present method with the classical random perturbation is given, and the numerical results show that the present method is valid when the parameter uncertainties are small compared with the corresponding mean values.     

Identification of Cracking in Beam Structures Using Timoshenko and Euler Formulations

Timoshenko and Euler beam formulations, using energy approach, have been used to estimate the influence of crack size and location on the natural frequencies of cracked beams. Fracture mechanics approach has been used to consider the effect of cracking on the dynamic response of the beam. Galerkin’s approach has been used to solve the problem numerically. It is shown that for slender beams the deep beam influence is felt only when the [(basic?bending?length)/h] ratio of the fundamental sinusoid of a beam becomes very small for higher modes. When the (l/h) ratio becomes small (<10), the influence of shear rotation and rotary inertia effects become dominant; the inclusion of these effects makes the beam less stiff than a Euler beam. The crack influence on Euler and Timoshenko beams are similar for beams with l/h>10; but when l/h<10, the results of cracked Euler and Timoshenko beams slowly become different and diverge. The frequency contour method identifies the crack size and location properly, using the lower order frequencies. When structural symmetry gives an ambiguity regarding the crack location, the vibration behavior of the same beam with an asymmetrically placed mass, in conjunction with the frequency contour method, would uniquely identify the crack size and location.     

Long-Term Response Prediction of Integral Abutment Bridges

The importance of long-term behavior in integral abutment (IA) bridges has long been recognized. This paper presents an analytical, long-term, response prediction methodology using finite-element (FE) models and compares results to measured response. Three instrumented Pennsylvania IA bridges have been continuously monitored since November 2002, November 2003, and September 2004 to capture bridge response. An evaluation of measured responses indicates that bridge movement progresses year to year with long-term response being significant with respect to static predictions. Both two-dimensional and three-dimensional FE models were developed using ANSYS to determine an efficient and accurate analysis level. Seasonal cyclic ambient temperature and equivalent temperature derived from time-dependent strains using the age adjusted effective modulus method were employed as major loads in all FE models. The elastoplastic p-y curve method, classical earth pressure theory, and moment-rotation relationships with parallel unloading paths were used to model hysteretic behavior of soil-pile interaction, soil-abutment interaction, and abutment-to-backwall connection. Predicted soil pressures obtained from all FE models are similar to the measured response. Predicted abutment displacements and corresponding design forces and moments at the end of the analytically simulated 100-year period indicate the significance of long-term behavior that should be considered in IA bridge design.     

Time‐Delay Effect on Dynamic Response of Actively Controlled Structures

The effect of time‐delayed control signals on the stability and the dynamic response of actively controlled structures is investigated. The necessary optimal control law that takes the effect into account is derived using the variational approach. With the help of an example of a moment‐controlled simply supported beam subjected to a concentrated load, it is shown how time delay cannot only degrade the system response, but can also destabilize actively controlled structures. A gain‐compensating technique is proposed, and its use is illustrated with the help of the example. The results indicate that the effect of smaller values of time delay can be compensated for by applying additional energy, but the effects of higher values may not be compensated for, because of the prohibitively larger amounts of energy required in those situations.     

Time Domain Analysis on the Dynamic Response of a Flexible Floating Structure to Waves

A time-domain numerical method is developed to analyze the hydroelastic responses of flexible floating structures to waves; in which, the boundary element method is applied to evaluate the fluid motion and the finite-element method to analyze the elastic deformation of structure. The dynamic wave-structure interaction is simulated by prescribing the conditions on a wave generation boundary for each time step and by satisfying the continuity of the pressure and displacement on the fluid-structure interface. A time-domain solution is obtained in a predictor-corrector scheme and through a time-stepping computation. The effect of space and time discretizations on the convergence and stability of solution for regular, random and solitary waves is discussed by comparing among numerical solutions. The validity of the present method is verified by comparing it with the experimental results for the three kinds of waves mentioned. Further, the fission of a solitary wave under a flexible floating structure is observed both in numerical analysis and experiments.     

Dynamic Response of Footings Resting on a Sand Layer of Finite Thickness

Dynamic response of foundations depends on several factors, namely, size and shape of the foundations, depth of embedment, soil profile and properties, frequency of loading, and mode of vibration. An attempt was made in this Technical Note to investigate the dynamic behavior of foundations resting on a sand layer underlain by a rigid layer. Model block vibration tests were carried out in a pit of size 2.0?m×2.0?m×1.9?m using a concrete footing of size 0.4?m×0.4?m×0.1?m and a vertically acting rotating-mass type mechanical oscillator. Using locally available river sand, a sand layer of six different thicknesses was prepared, and, for each thickness, tests were carried out for two different static weights and three different dynamic loadings. It was observed that the resonant frequency decreases with an increase in layer thickness and it nearly equals that of the half-space when the thickness of the layer is more than three times the width of the footing. It was also observed that the radiation damping of the sand layer was affected by the presence of a rigid layer at bottom. Inclusion of rigid layer causes a 9.8% reduction with respect to homogeneous sand condition even for a sand layer of thickness four times the width of the footing.     

Nonlinear Response of Deep Immersed Tunnel to Strong Seismic Shaking

Critical for the seismic safety of immersed tunnels is the magnitude of deformations developing in the segment joints, as a result of the combined longitudinal and lateral vibrations. Analysis and design against such vibrations is the main focus of this paper, with reference to a proposed 70?m-deep immersed tunnel in a highly seismic region, in Greece. The multisegment tunnel is modeled as a beam connected to the ground through properly calibrated interaction springs, dashpots, and sliders. Actual records of significant directivity-affected ground motions, downscaled to 0.24 g peak acceleration, form the basis of the basement excitation. Free-field acceleration time histories are computed from these records through one-dimensional wave propagation equivalent-linear and nonlinear analyses of parametrically different soil profiles along the tunnel; they are then applied as excitation at the support of the springs, with a suitable time lag to conservatively approximate wave passage effects. The joints between the tunnel segments are modeled realistically with special nonlinear hyperelastic elements, while their longitudinal prestressing due to the great (7?bar) water pressure is also considered. Nonlinear dynamic transient analysis of the tunnel is performed without ignoring the inertia of the thick-walled tunnel, and the influence of segment length and joint properties is investigated parametrically. It is shown that despite ground excitation with acceleration levels exceeding 0.50 g and velocity of about 80?cm/s at the base of the tunnel, net tension and excessive compression between the segments can be avoided with a suitable design of joint gaskets and a selection of relatively small segment lengths. Although this research was prompted by the needs of a specific project, the dynamic analysis methods, the proposed design concepts, and many of the conclusions of the study are sufficiently general and may apply in other immersed tunneling projects.     

Stability and Delay Compensation for Real-Time Substructure Testing

Real-time substructure testing is a method for establishing the dynamic behavior of structural systems. The method separates a complex structure into physical and numerically modeled substructures, which interact in real-time allowing time-dependent nonlinear behavior of the physical specimen to be accurately represented. Displacements are applied to the physical specimen using hydraulic actuators and the resulting measured forces are fed back to the numerical substructure. This feedback loop is implemented as a time-stepping routine. One of the key factors in obtaining reliable results using this method is the accurate compensation of the delayed response of the actuator. If this is not accounted for, instability of the feedback loop is likely to occur. This paper presents a method for estimating the delay while a test is in progress and accurately compensating for it during the test. The stability of both linear and nonlinear single-actuator systems is examined and the behavior of twin-actuator systems controlling two degrees-of-freedom at the substructure interface is presented. The effectiveness of the method is clearly demonstrated by comparisons between experimental and theoretical behavior.     

Stochastic Response and Stability of a Nonintegrable System

For determining the stochastic response and stability of a strongly nonlinear single-degree-of-freedom system using the stochastic averaging technique, the size of excitations should be small such that the response of the system converges weakly to a Markov process. This condition is not often met with practical problems, and therefore, application of this method for obtaining their responses becomes difficult. Further, for systems with nonlinearities that cannot be integrated in closed form, stability analysis by examining the conditions of the two boundaries of the problem is not possible. A semianalytical method along with a weighted residual technique is presented here to circumvent these difficulties and to determine the response and stability of a strongly nonlinear system subjected to sizable stochastic excitation. The weighted residual technique is employed to correct the errors in averaged drift and diffusion coefficients resulting due to the size of the stochastic excitation. Two example problems are solved as illustrations of the method.     

Frequency Spectrum Analysis of Impact-Echo Waveforms for T-Beams

Numerical and experimental studies were performed to assess the transient impact response of 11 T-beams with various dimensions and aspect ratios. Numerical modeling was performed using a three stage finite-element modeling procedure which included modal analysis, resonant analysis, and three-dimensional transient dynamic analysis. The response at impact locations on both the top centerline of the flange and the bottom centerline of the web was investigated. Physical models of three of the beams were constructed in the laboratory to determine the physical response of the beams when subjected to a transient impact and to verify the numerical results. Relationships between the fundamental frequencies and the frequencies of higher cross-sectional modes of vibration were established for the various aspect ratios. Shape factors were derived from the numerical and experimental results. The practical significance of the results is demonstrated for a project where impact-echo testing was used to nondestructively assess the condition of a decommissioned concrete T-shaped girder.     

Lateral–Torsional Buckling of Singly Symmetric Tapered Beams: Theory and Applications

A general variational formulation to analyze the elastic lateral–torsional buckling (LTB) behavior of singly symmetric thin-walled tapered beams is presented, numerically implemented, validated and illustrated. It (1) begins with a precise geometrical definition of a tapered beam; (2) extends the kinematical assumptions traditionally adopted to study the LTB of prismatic beams; (3) includes a careful derivation of the beam total potential energy; and (4) employs Trefftz’s criterion to ensure the beam adjacent equilibrium. In order to validate and illustrate the application and capabilities of the proposed formulation, several numerical results are presented, discussed and, when possible, also compared with values reported by other authors. These results (1) are obtained by means of the Rayleigh–Ritz method, using trigonometric functions to approximate the beam critical buckling mode, and (2) concern the critical moments of doubly and singly symmetric web-tapered I-section simply supported beams and cantilevers acted by point loads. In particular, one shows that modeling a tapered beam as an assembly of prismatic beam segments is conceptually inconsistent and may lead to rather inaccurate (safe or unsafe) results. Finally, it is worth mentioning that the paper includes a state-of-the-art review concerning one-dimensional analytical formulations for the LTB behavior of tapered beams.