Dynamic Deflection Curve Research Articles

Two span bridge under moving load

The dynamic effect of the moving load on the bridge construction is the subject of the solution in this article. The bridge is modeled as two span continuous beam with two degrees of freedom. The assumption describing the dynamic deflection curve and the assumption describing the load distribution on individual lumped masses are adopted at the creation of bridge computational model. The plane computational model of heavy vehicle with five degrees of freedom is adopted. The problem is described by ordinary differential equations which are solved numerically by using MATLAB. The results are presented by graphical and numerical form.

Discussion of “New Method for High-Speed Railway Bridge Dynamic Deflection Measurement” by Xianlong He, Xueshan Yang, and Lizhen Zhao

Discussion of “New Method for High-Speed Railway Bridge Dynamic Deflection Measurement” by Xianlong He, Xueshan Yang, and Lizhen Zhao

Deflection Estimation of Bending Beam Structures Using Fiber Bragg Grating Strain Sensors

This paper uses strains measured by fiber Bragg grating (FBG) sensors to estimate the static or dynamic deflection curve of bending beam structures. The deflection estimation method is only based upon the geometric equations of a beam structure without knowing the material information. At each cross section of a beam structure, two FBG strain sensors are installed, and the curvature at the cross section can be estimated by using the two measured strains from the geometric equation. Then the curvature function is assumed as a polynomial function and the coefficients can be estimated using least squares method. Finally, the deflection curve is estimated by integrating the curvature function twice. For dynamic deflection estimation, since only the geometric equations are used, and at each time step, the geometric equations can be also used for a dynamic system. Therefore, at each time step, the deflection can be estimated using the proposed method and the dynamic deflection can be finally obtained. The method is verified by the simulations of a continuous beam under static loads and experimental tests of a simply supported beam under various static loads. A simply support beam under moving loads is also simulated to verify the method for dynamic deformation estimation. All the numerical and test results show that the method can estimate the static and dynamic deflection curve of beam structures with a high accuracy.

A Simplified Numerical Method for Blast Induced Structural Response Analysis

Efficiently and accurately predicting structural dynamic response and damage to external blast loading is a big challenge to both structural engineers and researchers. The conventional numerical treatment to this problem is proved being able to give reliable predictions, however at the cost of enormous computational time and resource. Simplified SDOF approach is popularly used in design as it is straightforward to use and also gives good structural response predictions if the response is governed by a global response mode (shear or bending) and the accurate dynamic deflection curve is available, but it cannot predict the detailed local structural damage. In this study, a new numerical approach that combines the recently proposed two-step method and the static condensation method is proposed to analyze structure response and collapse to blast loads. The two-step method divides the structural response into two phases, i.e. forced vibration phase (blast loading duration) and free vibration phase. Single- Degree-of-Freedom system approach is adopted to solve the structural element responses at the end of the forced vibration phase, and the structural free vibration simulation is carried out using the hydro-code LS-DYNA to calculate the detailed structural response and damage. The static condensation technique is utilized to condense structural components that are relatively away from the explosion center to further reduce the computational effort. To demonstrate the proposed method, the structural responses of a three story RC frame to blast loads are calculated by four approaches, i.e. the traditional detailed FE simulation, the two-step method, the model condensation method, and the new combined two-step and dynamic condensation method. Through the results comparison, the efficiency and accuracy of the proposed combined approach are demonstrated.

New Method for High-Speed Railway Bridge Dynamic Deflection Measurement

The measurement of dynamic deflection is an important component of bridge loading experiments for high-speed railway bridges, but it is difficult to do directly for some long high-speed railway bridges. The deflection curve of the bridge in theory can be obtained by integrating the inclination curve of the bridge. This may be a good indirect method for dynamic deflection measurements of high-speed railway bridges if it meets two conditions: first, it must use a high-precision inclinometer to get a high-precision inclination curve of the bridge, and second, it must have a reliable integration method to calculate the deflection curve from the inclination curve. Based on this theory, the authors propose a new indirect dynamic deflection measurement method that uses precision inclinometers and a reliable integration method to measure the dynamic deflection curves of high-speed railway bridges. First, the authors designed a precision inclinometer called the QY inclinometer; second, the authors propose an inclinometer placement optimization method and a reliable integration method; last, two comparison experiments are conducted to prove this method on the Baima River Bridge. Test results indicate that this method can achieve the same accuracy as the method that uses cable-displacement sensors to test deflection and can achieve 0.01-mm accuracy. These experiments also prove indirectly that QY inclinometers have high precision and a response that is rapid enough to produce a dynamic intimation curve of a beam bridge when a train runs across it at high speed; the integration method proposed in this paper is also reliable. Compared with other methods of dynamic deflection measurement, this indirect measurement method is very easy to execute and does not need a static reference point to install inclinometers, and it can measure the static and dynamic deflection of any long- or short-beam bridge.

Rapid Assessment Method of Girder Bridge Carrying Capacity Based on Traffic Running Test

The dynamic response under running vehicle of the bridge includes the static response. The frequency components of simple supported beam dynamic response under the uniform constant force and the harmonic force were discussed. A method using the low pass filter to extract the static displacement from dynamic response curves was proposed and extended to the continuous girder bridge. Simulation analysis describes the process of extracting static displacement. The results show that the method is effective to separate the static and dynamic compositions from the vibration deflection. The dynamic deflection curves of an actual bridge was studied to get the generalized influence lines and then vehicles were laid on them as the static load test program to estimate the static deflection. The results of research show the estimative deflection has high precision and meets the requirements of the bridge inspection. This method can replace the static load test for rapid assessment of the girder bridge.

A large deflection light-induced bending model for liquid crystal elastomers under uniform or non-uniform illumination

In this paper, we propose a large deflection model for the light-induced bending of liquid crystal elastomers, in which we consider the in-plane membrane force and the geometrical nonlinearity. Based on the Hamilton principle, we derive the in-plane force balance equation and the dynamic deflection curve differential equation. The effect of light on the bending is defined as an effective optical bending moment, which is generated by the inhomogeneous light-induced strain and the membrane force. By coupling solving these two equations, we can obtain the deflection curves of LCEs under any boundary conditions and illumination. As examples, we solve the equations by an analytical-numerical method and simulate the bending under uniform illuminations. Then we consider the situations of non-uniform laser illumination and solve the equations by finite difference method. The deflection of the LCE sample can be varied and controlled by changing the illumination position, the illumination direction, the light intensity and the distribution half width of the electric field. As a result, when the LCE sample is constrained at the two boundaries, the small deflection bending theory is no longer applicable, and we need to use the large deflection theory.

A gradient model of light-induced bending in photochromic liquid crystal elastomer and its nonlinear behaviors

Photochromic liquid crystal elastomer was recently reported to be able to deform largely and bend under illumination. In this paper, considering the opto-chemical process and the nematic–isotropic phase transition, we introduce the light and temperature into the constitutive relation of the liquid crystal elastomers, and propose a model for the light-induced bending. The dynamic deflection curve equation of the light-induced bending is derived based on the Hamilton principle. In the equation, the effect of light is introduced as an effective optical bending moment, which is caused by the inhomogeneous light-induced strain and Young's modulus. Several simulation examples are given to show the light-induced bending under different boundary conditions and various illumination or temperature controlling. Under the condition of deep nematic phase and weak enough illumination, the approximate analytical expression of the effective moment and the stress distribution can be obtained. Rich nonlinear behaviors are found in this model. The effective moment is a non-monotonic function of time, thickness ratio, and light intensity when the thickness ratio is not very large. The stress distribution through the thickness is nonlinear with two or three zero-stress planes.

High speed dynamic weighing of unit loads

1. In order to save time and also to reduce the weighing error it is recommended to read the signal at a time that corresponds to the point of intersection between the curve of dynamic deflection and the line of static equilibrium of the weighing device. 2. The formulas that are derived give the theoretical calculation of this time for transient process whenω0>λ,ω0=λ; they also determine the error of weighing. 3. Calculations show that when the mass measured is within acceptance limits corresponding to 3–5% of the measured mass, the dynamic deflection curve intersects the static equilibrium line at practically one and the same instant of time. The error of measurement is least during the transient process whenω0>λ.

Coupled Bending-Bending Vibrations of Pre-Twisted Cantilever Blading Treated by the Rayleigh-Ritz Energy Method

The Rayleigh-Ritz method is used to determine the natural frequencies and mode shapes of vibration of pre-twisted rectangular cross-section beams. The method is dependent upon a good choice of approximating functions for the dynamic deflection curves. In the present analysis, series of the characteristic functions representing the normal modes of vibration are taken as the approximating functions for the bending displacements in the directions of the co-ordinate axes. The choice of this particular series leads to a considerable reduction in the number of elements in the final matrix equation and also considerably reduces the computing time. The natural frequencies of vibration are obtained for various width-to-thickness ratio beams with pre-twist angle in the range 0-90°, and the mode shapes of vibration are presented for one particular width to thickness ratio beam. The results are compared to results obtained by other methods and to experimental results, and good agreement is shown to exist.