A Simplified Model for Evaluating Strain Demand in a Pipeline Subjected to Longitudinal Ground Movement

2008 ◽  
Author(s):  
Nader Yoosef-Ghodsi ◽  
Joe Zhou ◽  
D. W. Murray
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Strain Hardening ◽  
Ground Movement ◽  
Simplified Model ◽  
Flow Rule ◽  
Pipe Material ◽  
Direction Parallel ◽  
Axial Forces ◽  
Sliding Zone

A simplified model was developed to calculate the maximum tensile and compressive strains due to a uniform movement of a block of soil in a direction parallel to the pipe axis using a closed-form solution of the governing differential equations. The model employs the theory of plasticity for modelling the pipe material based on normality plastic flow rule, the von Mises yield criterion, and isotropic strain hardening. While the pipe was assumed to have a bilinear, stress-strain curve with strain hardening, the pipe-soil friction was assumed to have an elastic-perfectly plastic force-deformation response. The model accounts for the initial thermal axial strains in the pipe and biaxial state of stress in the pipe due to internal pressure. The model is capable of accommodating pipe bends at the ends of the sliding zone. The relationship between the ground displacement and pipe axial force at each interface of stable and sliding zones was obtained from closed-form solutions of governing differential equations, assuming both the stable and sliding zones are infinitely long. To prevent the overestimation of the axial strains in the pipe, a limiting scenario was considered where the soil was assumed to have yielded over the entire sliding zone. Equilibrium and compatibility equations were used to calculate the pipe axial forces and strains at the two interfaces. The simplified model for longitudinal ground movement was validated against finite element solutions. The validation example presented involves a 20-inch straight pipeline subjected to longitudinal ground movement over slide lengths of 50, 100 and 200 metres, as well as a semi-infinite sliding zone case.

scholarly journals THE PARAMETRIC OSCILLATIONS OF ROTATING ELASTIC RODS UNDER THE ACTION OF THE PERIODIC AXIAL FORCES

2020 ◽  
pp. 56-64
Author(s):  
Petro Lizunov ◽  
Valentyn Nedin
Keyword(s):  
Differential Equations ◽  
Computer Program ◽  
Block Diagram ◽  
Time Integration ◽  
Numerical Differentiation ◽  
Oscillatory Motion ◽  
Elastic Rods ◽  
Axial Forces ◽  
General Block Diagram ◽  
Rotating Rods

The paper presents the results of numerical investigation of the periodic axial forces’ influence on the transverse oscillations of long rotating rods. The gyroscopic inertia forces are taken to account and space oscillating process of rotating rods is considered with account of geometric nonlinearity. The study has been done with computer program with a graphical interface that is developed by authors. The process of numerical solution of the differential equations of oscillations of rotating rods using the method of numerical differentiation of rod’s bend forms by polynomial spline-functions and the Houbolt time integration method is described. A general block diagram of the algorithm is shown. This algorithm describes the process of repeated (cyclical) solving of the system of differential equations of oscillations for every point of mechanical system in order to find the new coordinates of the positions of these points in each next point of time t+∆t. The computer program in which the shown algorithm is realized allows to monitor for the behavior of moving computer model, which demonstrates the process of oscillatory motion in rotation. Moreover, the program draws the graphics of oscillations and changes of angular speeds and accelerations in different coordinate systems. Using this program, the dynamics of a range of objects which are modeled by long elastic rods have been studied. For investigated objects is shown that on various rotational speeds and beat frequencies the oscillatory motion of the rods occurs with different character of behavior. On certain speeds with different frequencies of axial load the oscillations have definite periodicity and occur with beats of amplitude which are the result of the periodic axial force action.

scholarly journals Vibrations attenuation of a system excited by unbalance and the ground movement by an impact element

2016 ◽  
Vol 1 (2) ◽  
pp. 603-616 ◽  
Author(s):  
Marek Lampart ◽  
Jaroslav Zapoměl
Keyword(s):  
Differential Equations ◽  
Research Work ◽  
Computational Procedure ◽  
Ground Movement ◽  
Frequency Interval ◽  
Nonlinear Behaviour ◽  
Helical Springs ◽  
Impact Element ◽  
The Individual ◽  
The Impact

AbstractThis paper concentrates on the vibrations attenuation of a rotor driven by a DC motor and its frame flexibly coupled with a baseplate by linear cylindrical helical springs and damped by an element that can work either in inertia or impact regime. The system oscillation is governed by three mutually coupled second-order ordinary differential equations. The nonlinear behaviour occurs if the impact regime is adjusted. The damping element operating in inertia mode reduces efficiently the oscillations amplitude only in a narrow frequency interval. In contrast, the damping device working in impact regime attenuates vibrations of the rotor frame in a wider range of the excitation frequencies and it can be easily extended if the clearances between the rotor casing and the damping element are controlled. The development of a computational procedure for investigation of vibration of a flexibly supported rotor and for its attenuation by the inertia and impact dampers; learning more on efficiency of the individual damping regimes; finding possibilities of extension of the frequency intervals of applicability of the damping device; and obtaining more information on the character of the vibration induced by impacts are the main contributions of this research work.

scholarly journals The Krylov problem in the case of a beam on Vlasov inertial foundation

2018 ◽  
Vol 19 (6) ◽  
pp. 728-736
Author(s):  
Wacław Szcześniak ◽  
Magdalena Ataman
Keyword(s):  
Closed Form ◽  
Compressive Force ◽  
Elastic Beam ◽  
Forced Vibrations ◽  
Critical Force ◽  
Winkler Foundation ◽  
Static Deflection ◽  
Numerical Examples ◽  
Moving Force ◽  
Axial Forces

The paper deals with vibrations of the elastic beam caused by the moving force traveling with uniform speed. The function defining the pure forced vibrations (aperiodic vibrations) is presented in a closed form. Dynamic deflection of the beam caused by moving force is compared with the static deflection of the beam subjected to the force , and compressed by axial forces . Comparing equations (9) and (13), it can be concluded that the effect on the deflection of the speed of the moving force is the same as that of an additional compressive force . Selected problems of stability of the beam on the Winkler foundation and on the Vlasov inertial foundation are discussed. One can see that the critical force of the beam on Vlasov foundation is greater than in the case of Winkler's foundation. Numerical examples are presented in the paper

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

The strain hardening behaviour of particulate media

1976 ◽  
Vol 13 (3) ◽  
pp. 311-323 ◽  
Author(s):  
S. Frydman
Keyword(s):  
Strain Hardening ◽  
Three Dimensional ◽  
Flow Rule ◽  
Finite Element Program ◽  
Particulate Media ◽  
Stress Strain Relation ◽  
Element Program ◽  
Plastic Components ◽  
Associated Flow Rule ◽  
Particulate Medium

The strain increment resulting from an increment of stress applied to a particulate medium has been expressed in terms of its elastic and plastic components. The concepts of strain-hardening plasticity have been employed to develop an incremental stress–strain relation, based on a non-associated flow rule. The parameters appearing in the relation have been found using results of three-dimensional shear tests on sands and glass-microspheres. It is suggested that relations of the type developed in the paper could be beneficially incorporated into a finite-element program.

A note on the summation of divergent power series

1969 ◽  
Vol 66 (1) ◽  
pp. 109-114 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Severe Restriction ◽  
Mathematical Problems ◽  
Divergent Power Series ◽  
Euler Transformation ◽  
Direct Use

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

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