Analytical Solution for Longitudinal Dynamic Responses of Long Tunnels under Arbitrary Excitations

2021 ◽  
pp. 2150174
Author(s):  
Haitao Yu ◽  
Xizhuo Chen ◽  
Weile Chen ◽  
Pan Li
Keyword(s):  
Analytical Solution ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Differential Forms ◽  
Beam Theory ◽  
Dynamic Responses ◽  
Pasternak Foundation ◽  
Longitudinal Dynamic ◽  
Shear Distortion ◽  
Algebraic Forms

In this paper, an analytical solution is proposed for longitudinal dynamic responses of long tunnels under arbitrary excitations. For the derivation, the tunnel is assumed as a Timoshenko beam resting on a visco-Pasternak foundation. The Timoshenko beam theory is employed to consider both effects of the shear distortion as well as the rotary inertia of the tunnel, which are neglected by the Euler–Bernoulli beam. The visco-Pasternak foundation is applied to represent the viscoelastic compressive and shear resistance of soil. The governing equations of motion are transformed from partial differential forms into algebraic forms through integral transformations, and thus the solutions are conveniently obtained. The analytical solutions of the tunnel under several specific dynamic loads, including impulsive loads, moving line loads as well as traveling loads, are presented in detail and verified by comparing to the known degraded solution in literature and finite element results. Several examples are also conducted to investigate the influence of the relative stiffness ratio between the soil and the tunnel structure on the tunnel responses.

Analytical Solution for Vibrations of a Modified Timoshenko Beam on Visco-Pasternak Foundation Under Arbitrary Excitations

2021 ◽  
Author(s):  
Haitao Yu ◽  
Xizhuo Chen ◽  
Pan Li
Keyword(s):  
Analytical Solution ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Differential Forms ◽  
Infinite Length ◽  
Beam Model ◽  
Pasternak Foundation ◽  
Timoshenko Beam Model ◽  
The Time Domain ◽  
Wave Passage

An analytical solution is derived for dynamic response of a modified Timoshenko beam with an infinite length resting on visco-Pasternak foundation subjected to arbitrary excitations. The modified Timoshenko beam model is employed to further consider the rotary inertia caused by the shear deformation of a beam, which is usually neglected by the traditional Timoshenko beam model. By using Fourier and Laplace transforms, the governing equations of motion are transformed from partial differential forms into algebraic forms in the Laplace domain. The analytical solution is then converted into the time domain by applying inverse transforms and convolution theorem. Some widely used loading cases, including moving line loads for nondestructive testing, travelling loads for seismic wave passage, and impulsive load for impact vibration, are also discussed in this paper. The proposed generic solutions are verified by comparing their degraded results to the known solutions in other literature. Several examples are performed to further investigate the differences of the beam responses obtained from the modified and the traditional Timoshenko beam models. Results show that the modified Timoshenko beam simulates the beam responses more accurately than the traditional model, especially under the dynamic loads with a high frequency. The analytical solutions proposed in this paper can be conveniently used for design and applied as an effective tool for practitioners.

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.

scholarly journals Comparison of 1D and 2D solutions for a beam under transverse impact

2018 ◽  
Vol 148 ◽  
pp. 05005 ◽  
Author(s):  
Vítězslav Adámek
Keyword(s):  
Analytical Solution ◽  
Timoshenko Beam ◽  
Elastic Beam ◽  
Beam Theory ◽  
Two Dimensional ◽  
Transverse Impact ◽  
Dimensional Theory ◽  
Beam Geometry ◽  
Stationary Vibration ◽  
Main Factors

The problem of non-stationary vibration of an elastic beam caused by a transverse impact loading is studied in this work. In particular, two different approaches to the derivation of analytical solution of the problem are compared. The first one is based on the Timoshenko beam theory, the latter one follows the exact two-dimensional theory. Both mentioned methods are used for finding the response of an infinite homogeneous isotropic beam. The obtained analytical results are then compared and their agreement is discussed in relation to main factors, i.e. the beam geometry, the character of loading and times and points at which the beams responses are studied.

A NEW NONLOCAL BEAM THEORY WITH THICKNESS STRETCHING EFFECT FOR NANOBEAMS

2013 ◽  
Vol 12 (04) ◽  
pp. 1350025 ◽  
Author(s):  
ABDELOUAHED TOUNSI ◽  
SOUMIA BENGUEDIAB ◽  
MOHAMMED SID AHMED HOUARI ◽  
ABDELWAHED SEMMAH
Keyword(s):  
Shear Deformation ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Dynamic Responses ◽  
Theoretical Development ◽  
Small Scale ◽  
Small Scale Effect ◽  
Nonlocal Beam

This paper presents a new nonlocal thickness-stretching sinusoidal shear deformation beam theory for the static and vibration of nanobeams. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and it accounts for both shear deformation and thickness stretching effects by a sinusoidal variation of all displacements through the thickness without using shear correction factor. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanobeam are derived using Hamilton's principle. The effects of nonlocal parameter, aspect ratio and the thickness stretching on the static and dynamic responses of the nanobeam are discussed. The theoretical development presented herein may serve as a reference for nonlocal theories as applied to the bending and dynamic behaviors of complex-nanobeam-system such as complex carbon nanotube system.

Vibration of infinite Timoshenko beam on Pasternak foundation under vehicular load

2017 ◽  
Vol 20 (5) ◽  
pp. 694-703
Author(s):  
Weili Luo ◽  
Yong Xia
Keyword(s):  
Timoshenko Beam ◽  
Current System ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Surface Irregularity ◽  
Radius Of Gyration ◽  
Winkler Foundation ◽  
Shear Rigidity ◽  
Maximum Displacement ◽  
Line Load

The vibration of beams on foundations under a vehicular load has attracted wide attention for decades. The problem has numerous applications in several fields such as highway structures. However, most of analytical or semi-analytical studies simplify the vehicular load as a concentrated point or distributed line load with the constant or harmonically varying amplitude, and neglect the presence of the vehicle and the road irregularity. This article carries out an analytical study of vibration on an infinite Pasternak-supported Timoshenko beam under vehicular load which is generated by the passage of a quarter car on a road with harmonic surface irregularity. The governing equations of motion are derived based on Hamilton’s principle and Timoshenko beam theory and then are solved in the frequency–wavenumber domain with a moving coordinate system. The analytical solutions are expressed in a general form of Cauchy’s residue theorem. The results are validated by the case of an Euler–Bernoulli beam on a Winkler foundation, which is a special case of the current system and has an explicit form of solution. Finally, a numerical example is employed to investigate the influence of properties of the beam (the radius of gyration and the shear rigidity) and the foundation (the shear viscosity, rocking, and normal stiffness) on the deflected shape, maximum displacement, critical frequency, and critical velocity of the system.

Vibration analysis of stiffened multi-plate structure based on a modified variational principle

2015 ◽  
Vol 23 (17) ◽  
pp. 2767-2781 ◽  
Author(s):  
Guo-qing Yuan ◽  
Wei-Kang Jiang
Keyword(s):  
Variational Principle ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Energy Functional ◽  
Dynamic Responses ◽  
Published Data ◽  
Discrete Elements ◽  
Displacement Fields ◽  
Analysis And Design

In order to study the vibration characteristics of flow-induced open cavity structures, the dynamic model of stiffened multi-plate is established. The first-order shear deformable plate theory and the Timoshenko beam theory are used to model the displacement fields of isotropic plates and stiffeners, respectively. A modified variational principle combined with a multi-segment partitioning procedure is employed to formulate the discretized equations of motion. The stiffeners are considered as discrete elements, and the energy contributions are included into the system energy functional by using the displacement compatibility conditions. The displacement and rotation components of each plate segment are expanded by a duplicate series of Chebyshev orthogonal polynomials of first kind. The convergence and accuracy of the present results for isotropic stiffened plates with different boundary conditions have been validated using comparisons with the published data and those obtained from the finite element analyses. Free vibration and dynamic responses of stiffened multi-plates with either longitudinal or orthogonally oriented stiffeners are discussed. The mathematical model and methodology presented in this paper may be used as an appropriate numerical tool in the analysis and design of stiffened multi-plate structures.

Ritz Series Analysis of Rotating Machinery Incorporating Timoshenko Beam Theory

2001 ◽  
Author(s):  
Nicole L. Zirkelback ◽  
Jerry H. Ginsberg
Keyword(s):  
Timoshenko Beam ◽  
Space Form ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Cross Sectional ◽  
Energy Functionals ◽  
Orthogonality Conditions ◽  
Rigid Disks ◽  
Gyroscopic Coupling

A shaft with attached rigid disks is modeled as a rotating Timoshenko beam supported by general compliant, nonconservative bearing supports. The continuous shaft-disk system is described with kinetic and potential energy functionals that fully account for transverse shear, translational and rotatory inertia, and gyroscopic coupling. Ritz series expansions are used to describe the flexural displacements and cross-sectional rotations about orthogonal fixed axes. The equations of motion are derived from Lagrange’s equations and placed in a state-space form that preserves the skew-symmetric gyroscopic matrix, as well as the cross-coupling displacement and velocity coefficient matrices describing the effects of bearings. Both the general and adjoint eigenproblems for the nonsymmetric equations are solved. Bi-orthogonality conditions lead to the ability to evaluate dynamic response via modal analysis. Two examples, which show close agreement with prior analyses of critical speeds, demonstrate the ease with which the method may be applied.

Analysis of sandwich Timoshenko porous beam with temperature-dependent material properties: Magneto-electro-elastic vibration and buckling solution

2019 ◽  
Vol 25 (23-24) ◽  
pp. 2875-2893 ◽  
Author(s):  
M. Bamdad ◽  
M. Mohammadimehr ◽  
K. Alambeigi
Keyword(s):  
Timoshenko Beam ◽  
Material Properties ◽  
Vinylidene Fluoride ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Distribution Patterns ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Temperature Dependent

Vibration and buckling analysis of a magneto-electro-elastic sandwich Timoshenko beam with a porous core and poly-vinylidene fluoride (PVDF) matrix reinforced by carbon nanotubes (CNTs) is considered as face layers and material properties of CNTs and PVDF are assumed to be temperature-dependent. Different CNT distribution patterns including uniform distribution, AV (which top and bottom face sheets have functionally graded-A (FG-A) and functionally graded-V (FG-V) CNT distribution patterns, respectively) and VA patterns are employed. The governing equations of motion are derived based on Timoshenko beam theory, and Navier's solution is used to solve these equations. The sandwich beam resting on a Pasternak foundation and face layers are subjected to electric and magnetic potentials. The effect of different parameters such as porosity coefficient, electric and magnetic potential, parameters of foundation, and geometrical parameters are investigated on vibration and buckling behavior of the sandwich beam. Numerical results of this paper show that porosity distribution has a significant effect on the stiffness of the sandwich beam. The results can be used for future analysis of magneto-electro-mechanical sandwich systems as actuators and sensors.

scholarly journals A study on dynamic response of functionally graded sandwich beams under different dynamic loadings

2018 ◽  
Vol 192 ◽  
pp. 02011
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Functionally Graded ◽  
Transverse Shear Deformation ◽  
Sandwich Beams ◽  
Governing Equations ◽  
Dynamic Loadings ◽  
Parametric Studies

In this research, free and forced vibration of functionally graded sandwich beams is considered using Timoshenko beam theory which takes into account the significant effects of transverse shear deformation and rotary inertia. The governing equations of motion are formulated from Lagrange's equations and they are solved by using The Ritz and Newmark methods. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, etc. on natural frequencies and dynamic deflections of the beams. According to the numerical results, all parametric studies considered in this research have significant impact on free and forced behaviour of the beams; for example, the frequency is low and the dynamic deflection is large for the beams which are hinged at both ends.

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