Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

2018 ◽  
Vol 25 (3) ◽  
pp. 593-611
Author(s):  
Xiayang Zhang ◽  
Haoquan Liang ◽  
Meijuan Zhao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Impact Load ◽  
Moving Load ◽  
Numerical Results ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Load Case ◽  
Inverse Laplace Transformation ◽  
Damping Effect

This paper, taking the clamped boundary condition as an example, develops Su and Ma's fundamental solutions of the dynamic responses of a Timoshenko beam subjected to impact load. Based on that, a further extension regarding the general moving load case is also established. Kelvin–Voigt damping, whether proportionally or nonproportionally damped, is incorporated into the model, making it more comprehensive than the model of Su and Ma. Numerical inverse Laplace transformation is introduced to obtain the time-domain solution, where Durbin's formula and the corresponding convergence criteria are utilized in numerical experiments. Further, the real modal superposition method is applied at an analytical level to validate the numerical results by applying a proportionally damped condition. Total comparisons are made between the methods by sufficient case studies. The dynamic responses with and without damping effect are computed with wider slenderness to verify the correctness and effectiveness of the numerical results. Furthermore, parametric studies regarding the damping coefficients are performed to explore the nonproportional damping effect. The results show that the structural damping has significant influences on the dynamic behaviors and is especially stronger at small slender ratios. As the damping decreases the inherent frequencies and excites the low-frequency modal components more actively, a resonant phenomenon appears in high slenderness case when the beam experiences a low-speed moving load. Additionally, the computations in the moving load case indicate that the algorithm convergence is preferable when the number of grids exceeds 1000.

A New Three-Dimensional Moving Timoshenko Beam Element for Moving Load Problem Analysis

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Yan Xu ◽  
Weidong Zhu ◽  
Wei Fan ◽  
Caijing Yang ◽  
Weihua Zhang
Keyword(s):  
Dynamic Analysis ◽  
Coordinate System ◽  
Timoshenko Beam ◽  
Moving Load ◽  
Three Dimensional ◽  
Beam Element ◽  
Superposition Method ◽  
Reference Coordinate System ◽  
Current Formulation ◽  
Timoshenko Beam Element

Abstract A new three-dimensional moving Timoshenko beam element is developed for dynamic analysis of a moving load problem with a very long beam structure. The beam has small deformations and rotations, and bending, shear, and torsional deformations of the beam are considered. Since the dynamic responses of the beam are concentrated on a small region around the moving load and most of the long beam is at rest, owing to the damping effect, the beam is truncated with a finite length. A control volume that is attached to the moving load is introduced, which encloses the truncated beam, and a reference coordinate system is established on the left end of the truncated beam. The arbitrary Lagrangian–Euler method is used to describe the relationship of the position of a particle on the beam between the reference coordinate system and the global coordinate system. The truncated beam is spatially discretized using the current beam elements. Governing equations of a moving element are derived using Lagrange’s equations. While the whole beam needs to be discretized in the finite element method or modeled in the modal superposition method (MSM), only the truncated beam is discretized in the current formulation, which greatly reduces degrees-of-freedom and increases the efficiency. Furthermore, the efficiency of the present beam element is independent of the moving load speed, and the critical or supercritical speed range of the moving load can be analyzed through the present method. After the validation of the current formulation, a dynamic analysis of three-dimensional train–track interaction with a non-ballasted track is conducted. Results are in excellent agreement with those from the commercial software simpack where the MSM is used, and the calculation time of the current formulation is one-third of that of simpack. The current beam element is accurate and more efficient than the MSM for moving load problems of long three-dimensional beams. The derivation of the current beam element is straightforward, and the beam element can be easily extended for various other moving load problems.

Some Solutions of the Timoshenko Beam Equations

1955 ◽  
Vol 22 (4) ◽  
pp. 579-586
Author(s):  
B. A. Boley ◽  
C. C. Chao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Numerical Results ◽  
Beam Model ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Infinite Beam ◽  
Beam Equations

Abstract Solutions are obtained by the method of Laplace transformation for four types of loadings applied to a semi-infinite beam. Numerical results are presented for two of these, both for suddenly applied and gradually varying loads. The effects of shear deformations and rotatory inertia are taken into account according to Timoshenko’s beam model. A comparison with the corresponding results of the Bernoulli-Euler theory are briefly presented.

Numerical Simulations of a Building-Acoustic Volume Interaction System Excited by Multiple Human Footfall Impacts

2008 ◽  
Author(s):  
Jing Tang Xing ◽  
Ye Ping Xiong
Keyword(s):  
Finite Element ◽  
Variational Formulation ◽  
Impact Load ◽  
Mixed Finite Element ◽  
Moving Load ◽  
Mixed Finite Element Method ◽  
Dynamic Responses ◽  
Experimental Result ◽  
Structure Interaction ◽  
Interaction System

A mixed finite element method is used to simulate a building structure-acoustic volume interaction system subject to multiple human footfall impacts. The pressure in the acoustic volume and the displacement of the structure are chosen as the fundamental variables to describe air-structure interaction dynamics. The governing equations and the corresponding variational formulation for generalised air-structure interaction systems are presented. From the variational formulation, the finite element and substructure-subdomain equations are derived. The available experimental results of footfall impact load histories are described and discussed. Based on an experimental result, an approximate footfall load time function is proposed to model the footfall loads in two successive human foot-steps. This approximate footfall load is applied at each structure point at which a left or right foot contacts at the corresponding time instant. Therefore, this footfall load is a moving load with a speed equalling the human walking speed. Following a generalised description of the developed numerical approach and footfall loads, an example is given. In this example, three cases involving two people walking along two perpendicular directions on the top floor of the structure are simulated, respectively. The dynamic responses of the displacement of the structure and the acoustic pressure in the acoustic volume are obtained. The calculated results are compared and discussed to illustrate the developed method and to reveal the mechanism of low-frequency vibration produced by human footfall impacts. The advantages of the proposed method are summarised to provide some guidelines to house designs.

Vibration Analysis of a Flexibly Connected Double Span Beam Traversed by Uniformly Distributed Moving Load

2011 ◽  
Vol 71-78 ◽  
pp. 375-378
Author(s):  
Yu Yang ◽  
Yun Fang Yang ◽  
Qi Mao Cai
Keyword(s):  
Vibration Analysis ◽  
High Speed ◽  
Moving Load ◽  
Transportation Systems ◽  
Vertical Vibration ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Mode Superposition Method ◽  
Distributed Load ◽  
Duhamel Integral

Considering the engineering background of high speed transportation systems such as maglev, vertical vibration of a flexibly connected double span beam is studied. The beam can be taken as two Euler-Bernoulli beams flexibly connected by an elastic joint and the model of it bearing moving distributed load with constant speed is then established. Duhamel integral as well as mode superposition method is applied to solve the dynamic responses of the beam. Numerical analysis base on different parameters such as the length of the train, the span of the beam and the velocity of the train are compared. The following conclusions are made:(1) the left span and right span’s dynamic responses is different so that each span should be analyzed separately; (2)the dynamic response of the beam is tied up with the factors such as the frequency of the beam, the moving frequency of the load and the relative bending stiffness of the joint.

scholarly journals Design Optimization of Explosion-Resistant System Consisting of Steel Slab and CFRP Frame

Materials ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 2589
Author(s):  
Jung J. Kim
Keyword(s):  
Hybrid System ◽  
Impact Load ◽  
Load Condition ◽  
High Strength ◽  
Dynamic Responses ◽  
Elastoplastic Behavior ◽  
Reinforced Polymer ◽  
Steel Slab ◽  
Strength Material

This study presents an explosion-resistant hybrid system containing a steel slab and a carbon fiber-reinforced polymer (CFRP) frame. CFRP, which is a high-strength material, acts as an impact reflection part. Steel slab, which is a high-ductility material, plays a role as an impact energy absorption part. Based on the elastoplastic behavior of steel, a numerical model is proposed to simulate the dynamic responses of the hybrid system under the air pressure from an explosion. Based on this, a case study is conducted to analyze and identify the optimal design of the proposed hybrid system, which is subjected to an impact load condition. The observations from the case study show the optimal thicknesses of 8.2 and 7 mm for a steel slab and a ϕ100 mm CFRP pipe for the hybrid system, respectively. In addition, the ability of the proposed hybrid system to resist an uncertain explosion is demonstrated in the case study based on the reliability methodology.

scholarly journals FORCED RESPONSE VIBRATION OF SIMPLY SUPPORTED BEAMS WITH AN ELASTIC PASTERNAK FOUNDATION UNDER A DISTRIBUTED MOVING LOAD

2020 ◽  
Vol 4 (2) ◽  
pp. 1-7
Author(s):  
Fatai Hammed ◽  
M. A. Usman ◽  
S. A. Onitilo ◽  
F. A. Alade ◽  
K. A. Omoteso
Keyword(s):  
Shear Modulus ◽  
Moving Load ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Forced Response ◽  
Dynamic Responses ◽  
Moving Force ◽  
Pasternak Elastic Foundation

In this study, the response of two homogeneous parallel beams with two-parameter Pasternak elastic foundation subjected to a constant uniform partially distributed moving force is considered. On the basis of Euler-Bernoulli beam theory, the fourth order partial differential equations of motion describing the behavior of the beams when subjected to a moving force were formulated. In order to solve the resulting initial-boundary value problem, finite Fourier sine integral technique and differential transform scheme were employed to obtain the analytical solution. The dynamic responses of the two beams obtained was investigated under moving force conditions using MATLAB. The effects of speed of the moving force, layer parameters such as stiffness (K_0) and shear modulus (G_0 ) have been conducted for the moving force. Various values of speed of the moving load, stiffness parameters and shear modulus were considered. The results obtained indicates that response amplitudes of both the upper and lower beams increases with increase in the speed of the moving load. Increasing the stiffness parameter is observed to cause a decrease in the response amplitudes of the beams. The response amplitudes decreases with increase in the shear modulus of the linear elastic layer.

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