Experimental and Numerical Study of Damaged Cantilever

The introduction of a crack in a steel structure will cause a local change in the stiffness and damping capacity. The change in stiffness will lead to a change of some of the natural frequencies of the structure and a discontinuity in the associated mode shapes. This paper contains a presentation of the results from experimental and numerical tests with hollow section cantilevers containing fatigue cracks. Two different finite-element (FE) models have been used to estimate the modal parameters numerically. The first FE model consists of beam elements. The second FE model consists of traditional rectangular shell elements and one rectangular shell element with a transverse, internal, open crack. The analytical results from the numerical models are compared with data obtained from experimental tests. The numerical models give good agreements with the experimental data. The beam model takes into account only the first mode of the crack evaluation. In the shell model all three modes of the crack growth are taken into account. Nevertheless, the results obtained for both models are satisfactory because the beam is subjected to bending. It can be concluded that it is sufficient to use crack models for calculating natural frequencies in bending, taking into account the first mode of the crack extension only.     

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.     

Vibration Studies of Tsing Ma Suspension Bridge

A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and towers. Significant coupling between torsional and lateral modes is also observed. The numerical results are in excellent agreement with the measured first 18 natural frequencies and mode shapes. The established dynamic model and computed dynamic characteristics can serve further studies on a long-term monitoring system and aerodynamic analysis of the bridge.     

The Most Degrading Postbuckling Mode of Pin-Jointed Structures

This paper describes a numerical method which is used to determine the most unstable postbuckling mode capable of developing at the ultimate critical state of complex pin-jointed structures. In this method, the set of independent variables are the flexural shortenings of those members that are a subset of the critically loaded members in the lattice structure. This choice of independent variables greatly simplifies the analysis and promotes the evaluation of the most degrading mode under both equilibrium and collapse conditions. In this Technical Note, this method is used to evaluate these two modes for some rather simple structures. The deformed modes are plotted in each case and differences between the most degrading mode under equilibrium and collapse conditions are noted.     

Assignment of Geometrical and Physical Parameters for the Confinement of Vibrations in Flexible Structures

A strategy for confinement of flexural vibrations in flexible structures by proper selection of their geometrical and physical parameters is proposed. We first show that the problem of vibration confinement can be formulated as an inverse eigenvalue problem (IEP) where the mode shapes and/or natural frequencies are assumed and the geometrical and physical properties are unknown functions of the space variables. It is required that the assumed modes form a complete and independent set of spatial functions that satisfy the boundary conditions and guarantee confinement within the desired spatial subdomain(s) of the structure. Using simple spatial functions, such as polynomials and exponentials, we determined approximate solutions of the geometrical and physical parameters by applying the orthogonality of the mode shapes with respect to the stiffness and mass density. The order of the selected polynomials or exponentials depends on the number of modes retained in the discretized model. Numerical simulations are presented on a beam and then on a plate to examine convergence of the solution to the IEP. We show that convergence is attained with few assumed mode shapes. The approximated parameters are finally substituted into the forward eigenvalue problem to confirm confinement at the desired locations.     

圆锯片振动模态的准确解

通过对Bessel函数的精确数值计算,对普通圆锯片在不考虑离心惯性力效应时的频率方程和振型函数分别进行了精确的求解和计算,得到了锯片振动模态的准确解,为了检验计算结果的正确性,又用有限单元法计算了1例,两种方法的计算结果吻合.另外,本文还通过计算证明了钢材泊松比的变化对锯片振动模态的影响很小.其计算结果可供锯片设计时直接查取.     

Stability of triple-helical poly(dT)-poly(dA)-poly(dT) DNA with counterions

Structural conformation of triple-helical poly(dT)-poly(dA)-poly(dT) has been a very controversial issue recently. Earlier investigations, based on fiber diffraction data and molecular modeling, indicated an A-form conformation with C'3-endo sugar pucker. On the other hand, Raman, solution infrared spectral, and NMR studies show a B-form structure with C'2-endo sugars. In accordance with these experimental results, a theoretical model with B-form, C'2-endo sugars was proposed in 1993. In the present work we investigate the dynamics and stability of the two conformations within the effective local field approach applied to the normal mode calculations for the system. The presence of counterions was explicitly taken into account. Stable equilibrium positions for the counterions were calculated by analyzing the normal mode dynamics and free energy of the system. The breathing modes of the triple helix are shifted to higher frequencies over those of the double helix by 4-16 cm-1. The characteristic marker band for the B conformation at 835 cm-1 is split up into two marker bands at 830 and 835 cm-1. A detailed comparison of the normal modes and the free energies indicates that the B-form structure, with C'2-endo sugar pucker, is more stable than the A-form structure. The normal modes and the corresponding dipole moments are found to be in close agreement with recent spectroscopic findings.     

Failure of Web-Flange Junction in Postbuckled Pultruded I-Beams

A numerical procedure for analyzing a common and catastrophic failure mode in pultruded composite material I-beams is presented in this paper. Pultruded wide-flange profiles (often referred to as I-beams) exhibit a number of different failure modes when loaded in flexure or axial compression. The particular failure mode of interest to this paper is that due to the local separation of the flange from the web of the profile following local buckling of the flange. A node-separation technique is used to simulate the progressive failure of the joint between the flange and the web of the wide-flange beam in the postbuckled regime. The procedure has been implemented in NIKE3D, a multipurpose nonlinear implicit finite-element code. The fundamentals of the separation algorithm and the mechanics of the implementation in NIKE3D are described. The results of simulations using the proposed procedure are compared with experimental observations.     

Realistic Bond Strength of FRP Rebars in NSC from Beam Specimens

The bond strength of reinforcing bars in concrete is a prerequisite for the evaluation of the development length in reinforced concrete structures. This study concerns these phenomena for fiber reinforced polymer (FRP) rebars in normal strength concrete (NSC). Three different types of rebars were tested using the beam specimen: Carbon, glass, and steel. This involved a total of 26 beam specimens containing 10, 16, and 19?mm rebars. The test embedment lengths were 10, 15, and 20 times the rebar diameter (db). For each rebar tested, the results concern load deflection curves, bond stress-slip responses, and the mode of failure. The results showed that the bond strength of a FRP rebar is, generally, lower than that of steel rebar. Based on this and previous research, proposals for the average bond strength and for the development length of straight FRP rebars under tension in NSC are made.     

Assessment of Soil Stiffness Properties by Dynamic Tests on Bridges

This paper presents a method to determine soil stiffness properties using measured structural modes of bridges. Normally, the identified mode shapes have to be smoothed. The mode shapes are approximated using functions describing the transverse vibration of distributed–parameter systems. Artificial coefficients are introduced into this solution in order to sum up the error contributions of displacements and its derivatives up to second order. Then, a pier-soil model based on normalized mechanical impedance functions is used. Applying this method along with more than one vertical mode shape leads to acceptable and more accurate results. The amplitudes of pier bottom vibrations are chosen as the suitable weights for the averaging procedure. For the Warth Bridge situated near Vienna, shear wave velocities and shear moduli at the pier foundations have been estimated. The results correspond quite well to the geological investigation.     

Nonclassical Modes of Unrestrained Shear Beams

This paper considers the effect that a rotational motion has on the normal modes of a shear beam that is free to rotate, either because it is free in space or it is pivoted at one end. It is shown that the classical solutions for these two cases violate the principle of conservation of angular momentum, and that this is true even when the rotational inertia of the beam vanishes or is neglected.     

Prediction of laser transformation hardening depth using a line source model

Laser treatments have been used for metal and alloy surface modification. In these applications, the processing is mainly controlled by the laser parameters such as the laser power, the specimen velocity, the beam width and the beam mode (energy distribution). It is very difficult to predict the temperature field and the case depth when complex laser modes are involved. Based on previous work, a practicable model, considering a laser source as a “line source” and using the superposition of a number of Gaussian line sources, is established in the present work, and found to have a good agreement with the experimental data obtained from a medium carbon steel. The present model is believed to be applicable to various laser modes including a non-stationary beam, so long as the energy distribution is known.     

Global and Local Buckling of a Sandwich Beam

A two-dimensional mechanical model is developed to predict the global and local buckling of a sandwich beam, using classical elasticity. The face sheet and the core are assumed as linear elastic isotropic continua in a state of planar deformation. The core is assumed to have two deformation modes: antisymmetrical and symmetrical with respect to the core geometric midplane. Characteristics of the two deformation modes and the corresponding buckling behavior are shown and it appears that they are identical when the buckling wavelength is short. The present analysis is compared with various previous analytical studies and corresponding experimental results. On the basis of the model developed here, validation and accuracy of several previous theories are discussed for different geometric and material properties of a sandwich beam. The results presented in this paper, verified through finite-element analysis and experiment, are an accurate prediction of the overall buckling behavior of a sandwich beam, for a wide range of material and geometric parameters.     

Failure Mode Analyses of Reinforced Concrete Beams Strengthened in Flexure with Externally Bonded Fiber-Reinforced Polymers

As existing structures age or are required to meet the changing demands on our civil infrastructure, poststrengthening and retrofitting are inevitable. A relatively recent technique to strengthen reinforced concrete (RC) beams in flexure uses fiber-reinforced polymer (FRP) strips or sheets glued to the tension side of the beam. A number of researchers have reported that the failure mode of an FRP-strengthened RC beam can change from the desired ductile mode of an underreinforced beam to a brittle one. This paper analyzes the effects of this strengthening technique on the response and failure modes of a reference RC beam. A nonlinear RC beam element model with bond-slip between the concrete and the FRP plate is used to study how the failure mechanism of simply supported strengthened RC beams is affected by the following parameters: plate length, plate width, plate stiffness, and loading type. The beam geometry is kept constant. The parametric studies confirm the experimentally observed results according to which the most commonly observed failure modes due to loss of composite actions are affected by the plate geometric and material properties. In addition, distributed loads (difficult to apply in an experimental test) may not be as sensitive to plate debonding in the region of maximum bending moment as are beams subjected to point loads.     

Research on mode localization of reticulated shell structures

Reticulated shell structures (RSSs) are characterized as cyclically periodic structures. Mistuning of RSSs will induce structural mode localization. Mode localization has the following two features: some modal vectors of the structure change remarkably when the values of its physical parameters (mass or stiffness) have a slight change; and the vibration of some modes is mainly restricted in some local areas of the structure. In this paper, two quantitative assessment indexes are introduced that correspond to these two features. The first feature is studied through a numerical example of a RSS, and its induced causes are analyzed by using the perturbation theory. The analysis showed that internally, mode localization is closely related to structural frequencies and externally, slight changes of the physical parameters of the structure cause instability to the RSS. A scaled model experiment to examine mode localization was carried out on a Kiewit single-layer spherical RSS,and both features of mode localization are studied. Eight tests that measured the changes of the physical parameters were carried out in the experiment. Since many modes make their contribution in structural dynamic response, six strong vibration modes were tested at random in the experimental analysis. The change and localization of the six modes are analyzed for each test. The results show that slight changes to the physical parameters are likely to induce remarkable changes and localization of some modal vectors in the RSSs.     

Free Vibrations of Cylindrical Storage Tanks: Finite-Element Analysis and Experiments

This note summarizes a theoretical and experimental study undertaken to provide a deeper understanding of the effect of different parameters on the coupled modal characteristics of circular cylindrical tanks. First, the most common case of clamped-free tanks resting on rigid foundations is investigated by using finite-element (FE) modeling and holographic experiments. A good agreement between experimental and numerical results is a basis to draw a number of conclusions. For both tank geometries investigated, the frequencies for modes of circumferential parameter n = 1 (the “beam” modes) are found to be reduced most significantly by the presence of liquid. Very significant dependence of the radial shell mode shapes on the filling ratio is confirmed both by the FE and experimental results. In addition, nonclassical vibration patterns for radial shell modes were extracted numerically and recorded experimentally. Special attention is paid to the pairs of shell modes. Second, the effects of a flexible foundation and axial compression are investigated using holographic interferometry. The modal responses of this shell–liquid system are found to be different from those of the existing theoretical models.     

Debonding Failures of RC Beams Strengthened with Near Surface Mounted CFRP Strips

This paper presents the results of a series of tests conducted on reinforced concrete (RC) beams strengthened in flexure with near surface mounted (NSM) carbon fiber-reinforced polymer (CFRP) strips. As the main focus of the research is on debonding failure mechanisms, the only test variable investigated was the embedment length of the NSM strip and the NSM strip was extensively strain-gauged to monitor its bond behavior. Load-deflection curves, failure modes, strain distributions in the CFRP strip, and local bond stresses at the CFRP–epoxy interface from the tests are all examined in detail and compared with the predictions of a simple analytical model where appropriate. Of the four embedment lengths investigated, all but the shortest one led to a notable increase in the load-carrying capacity and, to a lesser extent, in the postcracking stiffness of the beam. Debonding was found to be the primary failure mode in all cases except for the beam with the longest embedment length. Also reported in this paper are results from preliminary bond tests used to characterize the local bond-slip behavior of the NSM system. Apart from gaining a better understanding of debonding failures in RC beams with NSM FRP strips, the test results reported in the paper should be useful for future verification of numerical and analytical models.     

Simple General Solution for Interfacial Stresses in Plated Beams

The flexural strength of a reinforced concrete, metallic or timber beam can be increased by bonding a thin plate, made of steel or fiber-reinforced polymer, to its tension face. A main failure mode of such plated beams involves debonding of the plate end from the beam and such plate-end debonding depends strongly on the interfacial stresses between the beam and the plate. Consequently, many analytical solutions have been developed for the interfacial stresses of specific plated beam problems, with almost all of them being for simply supported plated straight beams of constant section subjected to simple loadings. The existing analytical solutions are therefore neither general enough nor simple enough for direct exploitation in assessing the risk of plate-end debonding failure. This paper corrects this deficiency by presenting a simple, accurate yet general solution for interfacial stresses. The solution is applicable to plated beams of all geometric (e.g., curved beams), sectional (e.g., tapered beams), loading (e.g., a linearly varying distributed load), and boundary conditions (e.g., continuous beams). The accuracy of the solution is demonstrated through comparisons with finite element results. The paper also presents simple and accurate approximations for the peak values of interfacial shear and normal stresses at the plate end. In these approximate expressions, only the sectional forces and properties of the plate end section are involved, which greatly facilitates their direct exploitation in predicting debonding failure.     

Production and propagation of Hermite-sinusoidal-Gaussian laser beams

Hermite-sinusoidal-Gaussian solutions to the wave equation have recently been obtained. In the limit of large Hermite-Gaussian beam size, the sinusoidal factors are dominant and reduce to the conventional modes of a rectangular waveguide. In the opposite limit the beams reduce to the familiar Hermite-Gaussian form. The propagation of these beams is examined in detail, and resonators are designed that will produce them. As an example, a special resonator is designed to produce hyperbolic-sine-Gaussian beams. This ring resonator contains a hyperbolic-cosine-Gaussian apodized aperture. The beam mode has finite energy and is perturbation stable.     

Investigation of Microstructure, Natural Frequencies and Vibration Modes of Dragonfly Wing

In the present work, a thorough investigation on the microstructural and morphological aspects of dragonfly wings was carried out using scanning electron microscope. Then, based on this study and the previous reports, a precise three-dimensional numerical model was developed and natural frequencies and vibration modes of dragonfly forewing were determined by finite element method. The results shown that dragonfly wings are made of a series of adaptive materials, which form a very complex composite structure. This bio-composite fabrication has some unique features and potential benefits. Furthermore, the numerical results show that the first natural frequency of dragonfly wings is about 168 Hz and bending is the predominant deformation mode in this stage. The accuracy of the present analysis is verified by comparison of calculated results with experimental data.This paper may be helpful for micro aerial vehicle design concerning dynamic response.