This paper presents an intuitive method for deriving the asymptotic solution of all detectable waves reflected at the free surface due to the incidence of a line P source, i.e., P- and S-disturbances that propagate horizontally at their respective velocities with a decrease rate of x_3/2 (where x denotes the horizontal distance). At first, the line P source is regarded as a combination of plane-wave components incident with real angles and those incident with imaginary angles. Based on the theory of plane-wave reflection with an extension to the range of imaginary incidences, the two transmissible plane-wave incidences of small real angle and of finite imaginary angle which directly lead to the actual P- and S-disturbances at large distances, respectively, are identified. Consequently, the geometric ray representation for near-field reflection associated with the S-disturbance is clarified after a modification upon the conventional theory. In analogy with the classical plane-wave reflection theory, the reflected waves are obtained by multiplying the fully reflected waves due to the incident cylindrical wave by the corresponding reflection coefficients, which are determined by the stress-free condition at the surface. Hence, the present solution can capture all the reflected waves with a decrease rate of x_3/2, unlike the conventional asymptotic solution. This intuitive method not only simplifies the calculation of asymptotic solutions using the method of steepest descent or stationary phase, but is physically more meaningful via establishment of a direct link between the reflection of plane waves and that of cylindrical waves. Results show that the S-disturbance due to a shallow or slowly oscillating line P source becomes significant especially for the responses beneath the surface. In contrast, the Sdisturbance was recognized as a minor response and ignored in many existing studies.

Seismic response of seabeds with irregular terrain (i.e., canyons and hills) under oblique incident P- or SVwaves is analyzed by the finite/infinite element method (FIEM). Initially, based on the exact solution for a corresponding free field, the equivalent seismic forces for the solid-liquid site are calculated. Then, the solid-liquid coupling for the terrain slope boundaries is implemented with key numerical parameters selected for the FIEM. Furthermore, the adequacy of the solid and liquid meshes each was verified by comparing the solutions with existing ones for two extreme cases. Finally, the effect of irregular terrain on the seabed seismic responses was assessed for P- and SV-waves based on the 1999 Chi-Chi Earthquake. Key conclusions include: (1) The difference in the vertical acceleration aymaxfor SV-waves is the largest for the irregular seabed with one canyon and one hill under super-critical conditions. (2) The canyon shows more prominent amplification for the seabed responses near the seismic source for both P- and & sdot;SV-waves.& sdot;(3) Deeper liquid drastically amplify the difference in horizontal accelerations on two sides of irregular local terrain for sub-critical P-waves and near-critical SVwaves & sdot; Additional observations made for the numerical analysis are given inside the text.

Angle cross-section members are widely used in various transmission towers owing to their convenient connections, easy availability, simple production processes, and high efficiency. Such members are very susceptible to nonlinear behavior when subjected to abnormal or extreme loads. In this work, a new geometric stiffness matrix for thin-walled steel members with an angle cross-section is derived based on the principle of virtual displacement, rigid body rule and updated Lagrangian equations. The generalized displacement control (GDC) method is employed to deal with the issues of non-convergence at the extreme value point and rebound point in the geometric nonlinear analysis, which offers clear physical meaning and can adjust the loading direction. The geometric and the traditional elastic stiffness matrices derived in this work are applied during the prediction phase, while only the traditional elastic stiffness matrix is used in the correction phase. The comparisons of the results obtained by the proposed method with those in the previous literature and ANSYS numerical solutions show that the proposed method is sufficient to ensure the accuracy and applicability for addressing the structural geometrically nonlinear issues.

A novel formula is derived for determining the bridge damping ratio from the correlation of the front and rear wheels of a test vehicle using the modal components extracted by the wavelet transform (WT). Firstly, closed-form solutions are derived for the dynamic responses of the bridge under a moving two-axle vehicle. Next, to remove the masking effect by vehicle's frequencies, the wheel-bridge contact responses back-calculated from the vehicle responses are used in analysis. Then, the WT is employed to obtain the instantaneous amplitudes of the component responses of the two contact points. Finally, using the correlation between the two contact points, a novel formula is derived for the bridge damping ratio. The above derivations are validated by numerical simulations. Through the theoretical analysis and numerical simulations, the following conclusions are made: (1) the spatial correlation of the front and rear contact points is utilized to derive the bridge damping formula using the WT; (2) for multi-span bridges, the RANdom SAmple Consensus (RANSAC) can beneficially downplay the data points with large deviations near the internal supports in fitting the bridge damping ratio; (3) the first span can be reliably used to calculate the damping ratio of multi-span bridges, especially in the presence of pavement roughness; and (4) the formula has been attested to be robust against various levels of vehicle and bridge damping.

Mode shapes recovered for bridges are often distorted by bridge damping when using the vehicle scanning method (VSM). To remedy such an effect, a novel recursive formula utilizing the spatial correlation between the front and rear contact points of a two-axle scanning vehicle is theoretically established. For a bridge subjected to a two-axle moving test vehicle, the dynamic response of the system is first derived in closed form. The vehicle-bridge contact response is used as a substitute for the vehicle response in signal processing, since it is free of the masking effect posed by vehicle frequencies. Using either the Hilbert transform (HT) or wavelet transform (WT), instantaneous amplitudes are extracted for the component response of the bridge of concern. Further, recursive formula that accounts for the spatial correlation between the two contact points is established to recover the undistorted bridge mode shapes. The following are the conclusions: (1) The distortion effect of damping on bridge mode shapes can be effectively removed by the recursive formula against various factors; (2) the WT is more effective in recovering undistorted bridge mode shapes than the HT; and (3) the proposed formula works well for restoring the mode shapes of bridges with rough pavement, by adding an accompanying truck during the vehicle scanning.

Lateral-distortional buckling may occur in frames composed of non-aligned I-members, for which the buckling resistance may be overestimated if the distortional deformation of the cross-section was ignored. In this regard, the geometric stiffness matrix for a novel straight beam element with nine DOFs per node that incorporates the effect of angling (or concentrated transverse stresses induced by in-plane bending moments) between two non-aligned members via the symmetric and anti-symmetric distortion modes is derived. Specifically, the virtual work done by the induced moments by in-plane bending moments near the angled joints when, in buckling, undergoing the angle of twist and the anti-symmetric distortion, is consistently included in the virtual work formulation. Although the local geometric stiffness matrices appear to be asymmetric, the symmetry of the structural stiffness matrix is ensured in the global assembly process, provided that the compatibility and equilibrium conditions at joints in the deformed configuration are satisfied. The reliability of the present distortional beam element is verified by comparing the present solutions with those by the Abaqus shell element in several well-designed examples, by which the superiority of the present element to the non-distortional beam elements is also demonstrated. Comparison study shows that the effect of distortional deformation is extremely significant when the angled frame is composed of short members, or the arch is of high curvature, and when rigid flanges or flexible webs are adopted. Moreover, the distortional beam element excluding the angling effect will produce significant errors in some particular cases.

The seismic behavior of underwater half-space under arbitrary 3D incidence of P- or SV-waves is analyzed by the 2.5D finite/infinite element method. Firstly, the 3D responses of solid and liquid in free field are newly derived exactly, which cover both the sub- and super-critical conditions. Then, the equivalent seismic forces acting on the solid boundary of the 2.5D finite/infinite element method are merely computed from the free-field displacement for given earthquakes. Subsequently, the equation of motions for the underwater half-space under arbitrary 3D incident seismic waves are established in the 2.5D finite/infinite element system, with the key parameters specified. The proposed method was verified against the previous 2D solutions and inversely computed free-field solutions in frequency domain. Major conclusions drawn from the present analysis include: (1) The effects of liquid, concerning enlargement, reduction or disturbance, on the seabed responses become stronger for increasing liquid depth under sub-critical condition; (2) The seabed interior responses are less sensitive to the variation in liquid depth under super-critical condition; and (3) Under the super- and sub-critical conditions, respectively, more attention should be given to the interior response for P-waves and to the seabed surface response for SV-waves.

This paper investigates the effect of damping on the torsional-flexural frequencies of monosymmetric thin-walled beams via the scanning by a single-axle test vehicle. A bi-directional damping model is adopted to account for the vertical and torsional-flexural motions of the beam, as they are mechanically uncoupled. To start, the closed-form solutions are derived for the vehicle, the beam and the vehicle-bridge contact responses. Based on the hypothesis of rigid cross sections, the rocking contact response is derived, which enables the torsional-flexural frequencies of the beam to be separated from the vertical ones. Both uniform and bi-directional damping properties are considered for the beam. The pollution effect of pavement roughness is overcome by using the residual contact response generated by two connected single-axle test vehicles. Through the parametric analysis, it is confirmed that: (1) the rocking contact response enables the first few torsional-flexural frequencies to be separately retrieved; (2) the damping ratio in each direction only affects the detectability of the frequencies, especially those of the high modes, in that direction for monosymmetric cross sections; (3) the residual contact response exhibits some robustness in identifying the frequencies of thin-walled beams with surface roughness and environmental noise effect; (4) a test vehicle moving in the side lane (with larger eccentricity from bridge's centerline) at a speed of 10 m/s (36 km/h) is recommended for the field test.

Warping-distortion coupling is common in frames and curved beams made of I-sections, which tends to decrease significantly the lateral and torsional resistance of structures. Traditionally, it has been difficult to consider such effects in structural analysis, because the warping and distortional degrees of freedom (DOFs) of two connected elements at a common joint cannot be easily transformed to a common coordinate system for global stiffness assembly. The purpose of this paper is to conquer such a problem. By introducing the symmetric and anti-symmetric distortion modes, in addition to the warping and conventional six DOFs, this paper develops a new theory that consists of nine DOFs per node for the two-node I-beam element. The three deformational DOFs of the cross-section, i.e., the symmetric distortion, anti-symmetric distortion and warping, can be regarded as three mechanical couples relating to the twisting, shearing and bending, respectively, of the two flanges in the opposite sense. This allows all the DOFs of the connected elements at a common joint to be easily transformed to the global coordinates for stiffness assembly. As a result, the warping-distortion compatibility problem that occurs in frames and curved beams is resolved. In the exemplar studies, the present beam element has been demonstrated to be capable of producing results that are in excellent agreement with those of the shell element for the lateral deformation of angled frames with unstiffened and stiffened joints and of curved beams with various boundary conditions. It is also observed that the cross-sectional distortion effect becomes more manifest in curved I-beams of high curvature or of high flange-to-web rigidity ratio.

Continuous rails are often modeled as infinite beams. In this paper, closed-form solution is newly derived by the residue theorem and Green’s function for the dynamic response of an infinite beam resting on viscoelastic foundation under a harmonic moving load. The most suitable span length will be determined for the analogic finite beam (which can be more conveniently used in practice) to best represent the deflection of the infinite beam for practical reasons. Starting from the equations of motion, closed-form solution is firstly derived for the infinite beam. Then, it is verified by the finite element method (FEM) using semi-infinite elements to simulate the infinite boundary conditions. The deflection curves of the infinite beam and analogic finite beam are compared for various parameters. It is concluded that: (1) a larger span length should be adopted for the finite beam under higher load speeds or with foundations of softer stiffness or larger damping; typically, for load speeds [Formula: see text][Formula: see text]m/s (324[Formula: see text]km/h); (2) typically, a span length of 50 m can be adopted for the analogic beam for UIC rails resting on foundations with stiffness [Formula: see text] and damping coefficient [Formula: see text]; (3) the peak deflection of the infinite and analogic finite beams occurs after the moving load for foundations with larger damping; and (4) the shape and amplitude of the deflection curve vary as the load frequency and amplitude varies, but the difference between the two beams is insignificant for moving loads with high frequencies.

With the three dimensional (3D) oblique incident waves exactly determined for the free field, the soil seismic responses in both frequency and time domains are studied by the 2.5 dimension (2.5D) finite/infinite element method. First, the free-field responses in frequency domain are solved exactly for 3D arbitrary incident P and SV waves, which requires no coordinate conversion or extra effort for SV waves with super-critical incident angles. Next, the earthquake spectra are incorporated by the concept of equivalent seismic forces on the near-field boundary, based only on the displacements input derived for unit ground accelerations of each frequency using the 2.5D approach. For the asymmetric 2.5D finite/infinite element model adopted, the procedure for soil seismic analysis is presented. The solutions computed by the proposed method are verified against those of Wolf’s and de Barros and Luco’s and for inversely calculated ground motions. Of interest is that abrupt variation in soil response occurs around the critical angle on the wave propagation plane for SV waves. In addition, the horizontal displacements attenuate with increasing horizontal incident angle, while the longitudinal ones increase inversely for 3D incident P and SV waves.

In this paper, a simple formula is derived for the modal damping ratio of the bridge using the correlation between the instantaneous amplitudes of the related front and rear contact responses of a two-axle test vehicle by the Hilbert transform (HT). To start, closed-form solutions were derived for the dynamic response of the damped bridge and vehicle-bridge contact responses. Next, the HT was employed to generate the instantaneous amplitudes of the two contact points. Based on their correlation, a simple formula is derived for the bridge damping ratio. Finally, the reliability of the derived formula was verified in the numerical study. It was demonstrated that the proposed formula can be successfully used to determine the first bridge damping ratio, even in the presence of rough pavement, but with the aid of random traffic.

This paper presents for the first time a top-down procedure for calculating the multi contact responses from the response of a multi DOF vehicle (containing two bogies and four wheelsets), which can be measured or simulated. Two key steps are involved. First, the nodal distribution method was devised for distributing the vehicle’s responses to those of the four wheelsets. Then by treating the wheelset as a single-DOF system, the contact response is calculated exactly by the enhanced integration algorithm for each time step. Two scenarios are studied. In Scenario 1, the vehicle is kept stationary on the rails, but excited differently via the contact points, by which the transmission of vibration from the wheelsets up to the vehicle components and down again by the present method is validated. In Scenario 2, the vehicle is set to move over a simple bridge, by which the contact responses are shown to outperform the vehicle’s responses in extracting the bridge frequencies, due to the fact that the vehicle’s frequencies were totally eliminated. In addition, the present method has been demonstrated to be of excellent accuracy, efficiency and robustness in each application.

This paper investigates comprehensively the resonance and cancellation conditions for the free vibration of elastically-supported (ES) beams subjected to successive moving loads. Focus is placed on application of the cancellation condition to minimize bridge vibrations, considering particularly the effect of elastic supports. In terms of the modal amplitude R of free vibration of the ES beam, both resonance and cancellation conditions are identified. This paper is featured by the fact that the cancellations are classified into two types as the external (load-related) and internal (structure-related) ones. Through the (internal) cancellation function, the criterion for selecting the optimal support stiffness ratio (SSR) is derived for the first time for suppressing the resonance of short to medium-span railway bridges. It depends solely on the bridge/vehicle length ratio L/d, and can be utilized to achieve near-perfect cancellation. The theoretical findings are validated by the finite element method (FEM) for various parameters. The results reveal that for beams with lengths in the ranges of (0.5d, d] and (1.5d, 2d], an SSR closer to the lower bound of the acceptable range should be selected to achieve the best effect. And for beams with lengths in the range of (d, 1.5d], the SSR should be selected as close to the optimal value as possible. Besides, it was found that damping in the beam and supports contributes to further suppression of vibration for bridges designed with optimal SSR.

Conventional beam elements ignoring distortion may overestimate the lateral resistance of frames and curved beams made of monosymmetric I-sections. This paper introduces two new distortional modes represented by mechanical couples relative to twisting and shearing of the two flanges that are opposite in directions but unequal in magnitudes. A straight beam element with nine degrees of freedom (DOFs) per node, including the conventional three translations, three rotations, warping and the new two distortions, is newly derived. This allows all the DOFs of the connected elements at a common joint to be easily transformed to the global coordinates for stiffness assembly. As a result, the warping–distortion compatibility problem that occurs in frames and curved beams is resolved. In the numerical examples, the results produced by the present beam element is demonstrated to agree excellently with the shell-element solutions for the lateral-distortional deformation of the angled frame and curved beam. It is observed that the cross-sectional distortion effect becomes extremely significant for angled frames of short unbraced length and for curved beams of high curvature.

Dual-function amplifiers are proposed for the first time herein for enhancing the capability of a scanning test vehicle for bridges. To start, closed-form solutions are derived for the dynamic responses of the amplifier-vehicle-bridge system with a moving test vehicle. Then, the dynamic amplification factors of the amplifier and vehicle are presented for assessing the bridge/vehicle and vehicle/amplifier transmissibility. It was known that the spectrum of the vehicle may be hindered for extracting the bridge frequencies because of vehicle frequency and rough pavement. Two differentially tuned amplifiers are called on to tackle the problem: one (i.e., the vehicle damper) is to suppress the effect of vehicle's frequency, acting like the tuned mass damper (TMD), and the other (i.e., the bridge amplifier) is to enlarge the amplitude of bridge frequency of concern. Based on the parametric study, it is concluded that (1) the bridge amplifier performs better than the vehicle one in extracting bridge frequencies by increasing their visibility in the spectrum; (2) the effect of vehicle's frequency can be suppressed by tuning the vehicle damper such that it functions like a TMD of the vehicle; (3) by tuning the bridge amplifier to any of the first few bridge frequencies, the latter can be well detected even for rough pavement.

The two support bearings of an elastically supported (ES) bridge may be unequal in stiffness due to aging or other factors. This paper proposed an effective technique of using the contact residual response (CRR) generated by a two-axle test vehicle in its round-trip movement to detect the weak end and frequencies of the bridge. First, the round-trip CRR is presented in closed form. It is featured by three facts: (1) the CRR is given in terms of the vehicle-bridge "contact" response and thus is free of the annoying vehicle's frequency; (2) being created as the "residual" of the responses of the two axles, the CRR is immune to the disturbance of the surface roughness; and (3) by letting the vehicle move in "round trip," the weak end of the bridge can be detected, while all the bridge frequencies are enhanced. A procedure is presented for calculating the CRR considering the discrete nature of field measured data recorded by the test vehicle. In the numerical study, the closed-form solution is validated using the finite-element method (FEM). The results from a realistic example demonstrated that weak-end amplification can be clearly recognized for the CRR (both temporal and spectral) for the vehicle moving in a round trip. Such a feature enables the weak end of the bridge to be easily detected. The other advantage is that more bridge frequencies of higher orders can be identified using the CRR generated by the vehicle moving from the weak end.

The advent of railways and especially highspeed railways marks great strides in human transportation history. To guarantee exclusive right-of-way, highspeed railways are often built on equal simply supported beams resting on piers. In this paper, a historical review will be given of the resonance and cancellation phenomena observed for simply supported beams traveled by a set of moving loads, as they are typical of highspeed railways. The phenomenon of resonance was observed by early investigators including Timoshenko, Bolotin, Frýba, Matsuura, etc. However, the phenomenon of cancellation was noted lately in 1997 by Yang et al. By letting the conditions of resonance and cancellation coincident, they proposed the optimal span length for suppressing the resonance of simple beams, which is equal to 1.5 times the car length. This 1.5 times rule has been verified and adopted in the design of some highspeed railways. In this article, the theoretical solution for the problem will be revisited for unveiling the key parameters such as the resonance speed (in temporal domain) and resonance wavelength (in spatial domain). Then a rather in-depth review will be given of existing works on the resonance and cancellation of railway bridges from the waves perspective. Some new developments along these lines will also be identified.

Artificial boundaries are crucial for simulating the dynamic response of semi-infinite problems. In this paper, a novel hybrid artificial boundary for the simulation of the far field is proposed by combining the perfectly matched layer (PML) and infinite element method (IEM), while the near field is simulated by the finite element method (FEM). The new hybrid artificial boundary (PML-IEM) is devised such that it can enjoy the relative advantage of each method. To start, a brief description is given on the fundamentals of the PML, IEM and the proposed PML-IEM. Then, the procedure for implementing the FEM-PML (using finite elements) and the IEM (using infinite elements) in the stretched coordinates is derived. In the parametric analysis, the factors affecting the accuracy of the PML, IEM, and the proposed PML-IEM are assessed in details. Based on this, optimal parameters are selected for implementing the proposed PML-IEM. Through the examples, the accuracy and reliability of the hybrid method proposed for analysing the semi-infinite problems under the surface line loads are demonstrated to be superior to those by the pure PML or IEM.

This paper presents, for the first time, the consideration of three-dimensional (3D) oblique incident P and SV waves in calculating the 3D seismic response of a lined tunnel embedded in a half-space by the 2.5D finite/infinite element method (FIEM). Firstly, the applicability of the 2.5D FIEM for 3D seismic analysis is summarized. With the exact solutions obtained for the free field in the Appendix, the equivalent seismic forces are rationally computed for the near-field boundary, considering the horizontal and vertical excitations of the Chi-Chi Earthquake. By performing seismic analysis of the half space embedded with a tunnel using the 2.5D FIEM, the time-domain responses of the tunnel are obtained. The accuracy of the present solutions is verified against those of de Barros and Luco. Conclusions drawn from the parametric study include: (1) Stress concentration for the principal stress under oblique incident seismic waves occurs at the polar angles of 0° (vault), 90°, 180° (inverted arch), and 270° of the lining wall. (2) The vault and inverted arch are the weakest parts of the tunnel during earthquakes. (3) The accelerations of the tunnel during earthquakes can be regarded as of the rigid body type. (4) The responses of the tunnel lining caused by SV waves of an earthquake are much more critical than those by P waves. (5) For arbitrary seismic waves, the maximum longitudinal acceleration azmax is of the same order of magnitude as the maximum horizontal acceleration axmax.