Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.

Study on transient beam loading compensation for China ADS proton linac injectorⅡ

Significant transient beam loading effects were observed during beam commissioning tests of prototype II of the injector for the accelerator driven sub-critical （ADS） system, which took place at the Institute of Modern Physics, Chinese Academy of Sciences, between October and December 2014. During these tests experiments were performed with continuous wave （CW） operation of the cavities with pulsed beam current, and the system was configured to make use of a prototype digital low level radio frequency （LLRF） controller. The system was originally operated in pulsed mode with a simple proportional plus integral and deviation （PID） feedback control algorithm, which was not able to maintain the desired gradient regulation during pulsed 10 mA beam operations. A unique simple transient beam loading compensation method which made use of a combination of proportional and integral （PI） feedback and feedforward control algorithm was implemented in order to significantly reduce the beam induced transient effect in the cavity gradients. The superconducting cavity field variation was reduced to less than 1.7% after turning on this control algorithm. The design and experimental results of this system are presented in this paper.

Study of the beam loading effect in the CSNS/RCS

The China Spallation Neutron Source/Rapid Cycling Synchrotron （CSNS/RCS） accelerates a high-intensity proton beam from 80 MeV to 1.6 GeV. Since the beam current and beam power is high, the beam loading is a severe problem for the stability of the circulating beam in the RCS. To study the beam loading effect in the CSNS/RCS theoretically, the RLC circuit model of the rf cavity, the method of Fast Fourier Transform and the method of Laplace transform have been employed to obtain the impedance of the rf system, the beam spectrum and the beam-induced voltage, respectively. Based on these physical models, the beam dynamics equations have been revised and a beam loading model has been constructed in the simulation code ORIENT. By using the code, the beam loading effect on the rf system of the CSNS/RCS has been investigated. Some simulation results have been obtained and conclusions have been drawn.

Analytical solution to the transient beam loading effects of a superconducting cavity

Transient beam loading is one of the key issues in any high beam current intensity superconducting accelerators, and needs to be carefully investigated. The core problem in the analysis is to obtain the time evolution of effective cavity voltage under transient beam loading. To simplify the problem, the second order ordinary differential equation describing the behavior of the effective cavity voltage is intuitively simplified to a first order one, with the aid of two critical approximations which lack proof of their validity. In this paper, the validity is examined mathematically in some specific cases, resulting in a criterion for the simplification. It is popular to solve the approximate equation for the effective cavity voltage numerically, while this paper shows that it can also be solved analytically under the step function approximation for the driven term. With the analytical solution to the effective cavity voltage, the transient reflected power from the cavity and the energy gain of the central particle in the bunch can also be calculated analytically. The validity of the step function approximation for the driven term is examined by direct evaluations.After that, the analytical results are compared with the numerical ones.

Prediction of Load Carrying Capacity of Corroded Reinforced Concrete Beam

A novel method for prediction of the load carrying capacity of a corroded reinforced concrete beam (CRCB) is presented in the paper. Nine reinforced concrete beams, which had been working in an aggressive environment for more than 10 years, were tested in the laboratory. Comprehensive tests, including flexural test, strength test for corroded concrete and rusty rebar, and pullout test for bond strength between concrete and rebar, were conducted. ne flexural test results of CRCBs reveal that the distribution of surface cracks on the beams shows a fractal behavior. The relationship between the fractal dimensions and mechanical properties of CRCBs is then studied. A prediction model based on artificial neural network (ANN) is established by the use of the fractal dimension as the corrosion index, together with the basic information of the beam. The validity of the prediction model is demonstrated through the experimental data, and satisfactory results are achieved.

Seismic performance of interior precast concrete beam-column connections with T-section steel inserts under cyclic loading

An experimental investigation was conducted to study the performance of precast beam-column concrete connections using T-section steel inserts into the concrete beam and joint core,under reversed cyclic loading.Six 2/3-scale interior beam-column subassemblies,one monolithic concrete specimen and five precast concrete specimens were tested.One precast specimen was a simple connection for a gravity load resistant design.Other precast specimens were developed with different attributes to improve their seismic performance.The test results showed that the performance of the monolithic specimen M1 represented ductile seismic behavior.Failure of columns and joints could be prevented,and the failure of the frame occurred at the flexural plastic hinge formation at the beam ends,close to the column faces.For the precast specimens,the splitting crack along the longitudinal lapped splice was a major failure.The precast P5 specimen with double steel T-section inserts showed better seismic performance compared to the other precast models.However,the dowel bars connected to the steel inserts were too short to develop a bond.The design of the precast concrete beams with lap splice is needed for longer lap lengths and should be done at the beam mid span or at the low flexural stress region.

Experimental Study of RC Beams Strengthened with CFRP Sheets Under Sustaining Loads

Seven reinforced concrete （ RC ） beams strengthened in flexure using carbon fiber reinforced polymer （CFRP） sheets subjected to different sustaining loads were tested. The effects of initial load and load history on the ultimate strength of strengthened RC beams were examined by externally bonded CFRP sheets. The main experimental parameters included different levels of sustaining load at the time of strengthening, and load history. Experimental results show that sustaining load levels at the time of strengthening have important influences on the ultimate strength of strengthened RC beams. If the initial load is the same, the ultimate strength of RC beams strengthened with CFRP sheets is almost the same regardless of load history at the time of strengthening.

Transient response of a beam on viscoelastic foundation under an impact load during nondestructive testing

Beam responses under an impact NDT load are studied using the Fourier and Laplace transforms. Numerical computation is performed for a parametric study of beam and load parameters. It is shown that under an impact load, the time duration for displacement to vanish is ten times longer than that for velocity and acceleration. The maximum response is achieved first in acceleration, followed by velocity, and finally displacement, all of which occur before the impact load is removed. At the moment that the impact load is removed from the beam, there is a discontinuity in the velocity and acceleration responses, but not in displacement response. The effect of K and ph on beam response is much less significant than that of EI and C, which have similar effect on beam response. As El increases, peak values of displacement, velocity and acceleration response decrease, The effect of K on beam response only becomes appreciable after the impact load is removed. While the peak of displacement remain almost unchanged large K values than for small K values. as K increases, the displacement response decays to zero faster for

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slen- der, non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed con- tinuous load and a concentrated load at the free end of the beam. The material of the cantilever is assumed to be non- linearly elastic. Different nonlinear relations between stress and strain in tensile and compressive domain are considered. The accuracy of numerical solutions is evaluated by com- paring them with results from previous studies and with a laboratory experiment.

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.

Dynamic response of a beam on a Pasternak foundation and under a moving load

The dynamic response of an infinite beam placed on a Pasternak foundation when the system was subjected to a moving load was investigated.We used the double Fourier transform and its inversion to solve the formulations of the problem.A closed form analytic solution of the beam was obtained by the theorem of residues.We selected a numerical example to illustrate the dynamic response of the beam on Pasternak and Winkler foundations,respectively.We discuss the effect of the moving load velocity on the dynamic displacement response of the beam.The maximum deflection of the beam increases slightly with increased load velocity but increases significantly with reduced shear modulus of subgrade at a given velocity.The maximum deflection of a beam resting on a Pasternak foundation is much smaller than that of a beam on a Winkler foundation.

Codimension-two bifurcation of axial loaded beam bridge subjected to an infinite series of moving loads

A novel model is proposed which comprises of a beam bridge subjected to an axial load and an infinite series of moving loads. The moving loads, whose distance between the neighbouring ones is the length of the beam bridge, coupled with the axial force can lead the vibration of the beam bridge to codimension-two bifurcation. Of particular concern is a parameter regime where non-persistence set regions undergo a transition to persistence regions. The boundary of each stripe represents a bifurcation which can drive the system off a kind of dynamics and jump to another one, causing damage due to the resulting amplitude jumps. The Galerkin method, averaging method, invertible linear transformation, and near identity nonlinear transformations are used to obtain the universal unfolding for the codimension-two bifurcation of the mid-span deflection. The efficiency of the theoretical analysis obtained in this paper is verified via numerical simulations.

Tensile Strength Characteristics of GFRP Bars in Concrete Beams with Work Cracks under Sustained Loading and Severe Environments

To investigate the effect of different environmental conditions of GFRP bars in concrete beams with work cracks subjected to sustained loads, the beams were exposed in indoor, freeze/thaw cycles and immersed in alkaline solution at elevated temperature. The bars were carefully extracted from the beams and tested in order to evaluate residual tensile properties. The results show that the tensile strength decreased significantly in the highly aggressive conditions but not in the natural conditions. The effect of GFRP bars casting in concrete beams demonstrated approximately 2.5% decrease of tensile strength caused by pore water environment in concrete beams on basis of those of the original bars. The effect of sustained loading plus work cracks demonstrated about 10.5% tensile strength decrease on basis of those of the bars only casted in concrete beams. The effect of environments under sustained loading plus work cracks demonstrated about 17% tensile strength decrease caused by a saturated solution of Ca（OH）2 and 60-2 ℃ tap water （pH=12-13） and about 8% tensile strength decrease caused by freezing and thawing cycle （F/T）, both on basis of those of the bars of the indoor beams only under sustained loading plus work cracks. The results demonstrate the effects of the tensile strengths under different environmental conditions of GFRP bars in concrete beams with work cracks subjected to sustained loads.

RC beam strengthened with pre-stressed CFP under the secondary load

Feasibility of using pre-stressed carbon fiber plates to strengthen reinforced concrete beams was studied. Based on the characteristics of carbon fiber plates, we developed a pre-stress clamp and a device for applying the pre-stress. Contrast tests were conducted between ordinary carbon fiber plates and a pre-stressed carbon fiber plate and between secondary loaded carbon fiber plates and a concrete beam strengthened with a secondary loaded carbon fiber plate. On this basis, we analyzed the failure pattern, the width of cracks and their distribution, the cracking load, the yield load, the limit load and the relation between load and deflec- tion. The results indicate that using pre-stressed carbon fiber plates to strengthen concrete beams is feasible and effective.

AN ANALYSIS oF THE SHEAR FAILURE OF RIGID-LINEAR HARDENING BEAMS UNDER IMPULSIVE LOADING

A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which tabes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with Nonaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams.

Geometrically Nonlinear Bending Analysis of Metal-Ceramic Composite Beams under Thermomechanical Loading

A new method is developed to derive equilibrium equations of Metal-Ceramic beams based on first order shear deformation plate theory which is named first order shear deformation beam theory2(FSDBT2). Equilibrium equations obtained from conventional method (FSDBT1) is compared with FSDBT2 and the case of cylindrical bending of Metal-Ceramic composite plates for non-linear thermomechanical deformations and various loadings and boundary conditions. These equations are solved by using three different methods (analytical, perturbation technique and finite element solution). The through-thickness variation of the volume fraction of the ceramic phase in a Metal-Ceramic beam is assumed to be given by a power-law type function. The non-linear strain-displacement relations in the von-Kármán sense are used to study the effect of geometric non-linearity. Also, four other representative averaging estimation methods, the linear rule, Mori-Tanaka, Self-Consistent and Wakashima-Tsukamoto schemes, by comparing with the power-law type function are also investigated. Temperature distribution through the thickness of the beams in thermal loadings is obtained by solving the one-dimensional heat transfer equation. Finally it is concluded that for Metal-Ceramic composites, these two theories result in identical static responses. Also the displacement field and equilibrium equations in the case of cylindrical bending of Metal-Ceramic plates are the same as those supposed in FSDBT2.

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half- space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of （sub-） hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of （sub-） soft beam and hard half-space but are similar to each other for the system of （sub-） hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity （the shear wave velocity of the beam for the system of soft beam and hard half-space）. The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave racliation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.

Dynamic Response of Curved Beam under Moving Load

In this paper an integral transform method is used to analyze the dynamic response of simple supported curved beam under single moving load with constant speed, and some parameters are defined. These parameters, such as radius of curvature, ratio of stiffness, velocity, warping stiffness, which may influence the response, are also discussed.

CALCULATION ANALYSIS OF SHEARING SLIP FOR STEEL-CONCRETE COMPOSITE BEAM UNDER CONCENTRATED LOAD

The strain difference of steel and concrete under vertical concentrated load was analyzed on the basis of elastic theory, and was compared with ideal solution of steel and concrete under vertical uniform load. The results indicate that the computing formula concluded from the paper is believable. The practical structure usually bears concentrated load, so it can be used in the practical engineering.

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube （SWCNT） subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.