scholarly journals APPLICATION OF ANALYTICAL SOLUTIONS FOR BENDING BEAMS IN THE METHOD OF MOVEMENT

2021 ◽  
Vol 6 (4) ◽  
pp. 42-53
Author(s):  
Vladimir Karpov ◽  
◽  
Evgeny Kobelev ◽  
Aleksandr Panin ◽  
◽  
...  
Keyword(s):  
Analytical Solutions ◽  
Accurate Solution ◽  
Transverse Load ◽  
Bending Moments ◽  
Displacement Method ◽  
Lagrange Principle ◽  
Concentrated Force ◽  
Concentrated Load ◽  
Local Loads ◽  
Different Types

Introduction: Usually, to analyze statically indeterminate rod systems, the classical displacement method and preprepared tables for two types of rods of the main system are used. A mathematically correct representation of local loads with the use of generalized functions makes it possible to find an accurate solution of the differential equation for the equilibrium of a beam exposed to an arbitrary transverse load. Purpose of the study: We aimed to obtain analytical expressions for functions of deflection, rotation angles, transverse forces, and bending moments depending on four types of local loads for beams with different boundary conditions, so as to apply accurate solutions in the displacement method. Methods: We propose an analytical form of the displacement method to analyze rod structural models. For beams exposed to different types of transverse load (uniformly distributed force, concentrated force, or a couple of forces), accurate analytical solutions were obtained for functions of deflection, bending moments, and transverse forces at different types of beam ends’ restraint. This is possible due to the fact that concentrated load and load in the form of the moment of force can be specified by using unit column functions. By transforming Mohr’s integrals, using integration by parts, we show that the system of canonical equations of the displacement method was obtained based on the Lagrange principle. Results: Based on the analysis of a statically indeterminate frame, the effectiveness of the proposed analytical method is shown as compared with the classical displacement method.

scholarly journals Dynamic Analytical Solution of a Piezoelectric Stack Utilized in an Actuator and a Generator

2018 ◽  
Vol 8 (10) ◽  
pp. 1779 ◽  
Author(s):  
Xinnan Liu ◽  
Jianjun Wang ◽  
Weijie Li
Keyword(s):  
Analytical Solution ◽  
Dynamic Characteristics ◽  
Analytical Solutions ◽  
External Load ◽  
Piezoelectric Actuators ◽  
Displacement Method ◽  
Proposed Model ◽  
Special Cases ◽  
Piezoelectric Stacks ◽  
Comprehensive Manner

This paper presents the dynamic analytical solution of a piezoelectric stack utilized in an actuator and a generator based on the linear piezo-elasticity theory. The solutions for two different kinds of piezoelectric stacks under external load were obtained using the displacement method. The effects of load frequency and load amplitude on the dynamic characteristics of the stacks were discussed. The analytical solutions were validated using the available experimental results in special cases. The proposed model is able not only to predict the output properties of the devices, but also to reflect the inner electrical and mechanical components, which is helpful for designing piezoelectric actuators and generators in a comprehensive manner.

scholarly journals Analysis of Pure Bending Vertical Deflection of Improved Composite Box Girders with Corrugated Steel Webs

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Chi Ma ◽  
Shi-zhong Liu ◽  
Jin Di ◽  
Rui-jie Zhang
Keyword(s):  
Shear Deformation ◽  
Analytical Solutions ◽  
Analytical Calculation ◽  
Load Condition ◽  
Width Ratio ◽  
Shear Lag ◽  
Vertical Deflection ◽  
Box Girders ◽  
Concentrated Load ◽  
Span Width

Steel bottom plates are applied as replacements for the concrete bottom plates in order to reduce the dead weight of the composite box girders with corrugated steel webs and steel bottom plates (CSWSB). Due to the change in the material, the previous analytical calculation methods of vertical deflection of composite box girders with corrugated steel webs (CSWs) cannot be directly applied to the improved composite box girders. The shear lag warpage displacement function was derived based on the shear deformation laws of the upper flange and the bottom plates of the improved composite box girders. The equations for the calculation of the shear deformation and the additional deflection due to the shear lag of continuous and simply supported composite box girders with CSWSB under concentrated and uniformly distribution loads were derived by considering the double effects of the shear lag and the shear deformations of the top and the bottom plates with different elastic moduli. The analytical solutions of the vertical deflection of the improved composite box girders include the theory of the bending deflection of elementary beams, shear deformation of CSWs, and the additional deflection caused by the shear lag. Based on the theoretical derivation, an analytical solution method was established and the obtained vertical deflection analytical solutions were compared with the finite element method (FEM) calculation results and the experimental values. The analytical equations of vertical deflection under the two supporting conditions and the two load cases have verified the analyses and the comparisons. Further, the additional deflections due to the shear lag and the shear deformation are found to be less than 2% and 34% of the total deflection values, respectively. Moreover, under uniform distributed load conditions, the deflection value was found to be higher than that of the under concentrated load condition. It was also found that the ratio of the deflection caused by the shear lag or the shear deformation to the total deflection decreased gradually with the increase in the span width ratio. When the value of the span width ratio of a single box and single chamber composite box girder with CSWSB was equal to or greater than 8, the deflections caused by the shear lag and the shear deformation could be ignored.

scholarly journals Experimental Research on the Transverse Effective Bending Rigidity of Shield Tunnels

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Gang Zheng ◽  
Yawei Lei ◽  
Tao Cui ◽  
Xuesong Cheng ◽  
Yu Diao ◽  
...  
Keyword(s):  
Sandy Soil ◽  
Bending Rigidity ◽  
Concentrated Force ◽  
Concentrated Load ◽  
Ring Model ◽  
Loading Patterns ◽  
Steel Tank ◽  
Practical Engineering ◽  
Tunnel Diameter ◽  
Load Tests

The uniform rigidity ring model is commonly used to design the segmented structures of shield tunnels. However, model tests have been primarily used to study the transverse effective rigidity ratio η with a concentrated force, which is notably different from realistic loading patterns. To obtain more reasonable η values, in this study, tests were performed with a concentrated load on an experimental bench and with a realistic loading pattern in sandy soil in a rigid steel tank. Three types of segmental ring models were designed and tested: straight-jointed, stagger-jointed, and uniform rings. The test results indicated that the η values of the stagger-jointed assembly mode were clearly larger than those of the straight-jointed assembly mode under both loading patterns. η increased as the load increased under the realistic loading conditions, whereas η decreased as the load increased under the concentrated load. More importantly, the η values derived from the realistic load tests were considerably larger than those derived from the concentrated load tests for both assembly modes (i.e., 0.423–0.672 and 0.587–0.761 for the straight-jointed and stagger-jointed assembly modes, respectively), and the former should be recommended for practical engineering applications. Furthermore, formulas relating η to the ratio of the cover depth to the tunnel diameter were proposed for sandy soil.

scholarly journals Recursive Differentiation Method for Boundary Value Problems: Application to Analysis of a Beam-Column on an Elastic Foundation

2014 ◽  
Vol 44 (2) ◽  
pp. 57-70 ◽  
Author(s):  
Mohamed Taha
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Buckling Mode ◽  
Beam Columns ◽  
Differentiation Method ◽  
Adomian Decomposition ◽  
Different Types ◽  
Finite Domains

Abstract In the present work, the recursive differentiation method (RDM) is introduced and implemented to obtain analytical solutions for differential equations governing different types of boundary value prob- lems (BVP). Then, the method is applied to investigate the static behaviour of a beam-column resting on a two parameter foundation subjected to different types of lateral loading. The analytical solutions obtained using RDM and Adomian decomposition method (ADM) are found similar but the RDM requires less mathematical effort. It is indicated that the RDM is reliable, straightforward and efficient for investigation of BVP in finite domains. Several examples are solved to describe the method and the obtained results reveal that the method is convenient for solving linear, nonlinear and higher order ordinary differential equations. However, it is indicated that, in the case of beam-columns resting on foundations, the beam-column may be buckled in a higher mode rather than a lower one, then the critical load in that case is that accompanies the higher mode. This result is very important to avoid static instability as it is widely common that the buckling load of the first buckling mode is always the smaller one, which is true only in the case of the beam-columns without foundations.

Nonlinear Equilibrium and Buckling of Fixed Shallow Arches Subjected to an Arbitrary Radial Concentrated Load

2017 ◽  
Vol 17 (08) ◽  
pp. 1750082 ◽  
Author(s):  
Yong-Lin Pi ◽  
Mark Andrew Bradford ◽  
Airong Liu
Keyword(s):  
Analytical Solutions ◽  
Limit Point ◽  
Buckling Load ◽  
Equilibrium Path ◽  
Instability Mode ◽  
Concentrated Load ◽  
Shallow Arches ◽  
Buckling Behavior ◽  
Nonlinear Equilibrium ◽  
Relationship Of

This paper is concerned with an analytical study of the nonlinear in-plane equilibrium and buckling of fixed shallow circular arches that are subjected to an arbitrary radial concentrated load. The structural behavior of an arch under an arbitrary radial concentrated load is quite different from that of an arch under a central concentrated load. It is shown that a fixed arch under an arbitrary radial concentrated load can buckle in a limit point instability mode, but cannot buckle in a bifurcation mode, which is different from that of a fixed arch under a central concentrated load that can buckle in a bifurcation mode or in a limit point instability mode. Analytical solutions for the nonlinear equilibrium path and limit point buckling load of shallow circular arches under an arbitrary radial concentrated load are derived. It is found that the load position influences the buckling load significantly and the influence is much related to the modified slenderness of the arch defined in the paper. It is also found that when the modified slenderness of an arch is smaller than a specific value, the arch has no typical buckling behavior. The analytical solution for the relationship of the specific modified slenderness with the load position is also derived. Comparisons with finite element (FE) results show that the analytical solutions can accurately predict the nonlinear equilibrium and buckling load of shallow fixed arches under an arbitrary radial concentrated load.

scholarly journals Bending of a ship's skin panel loaded along the axis of symmetry

2021 ◽  
pp. 14-19
Author(s):  
Т.П. Кныш ◽  
М.В. Сухотерин ◽  
С.О. Барышников
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Double Fourier Series ◽  
Transverse Load ◽  
Bending Moments ◽  
Approximate Methods ◽  
Loaded Part ◽  
Axis Of Symmetry ◽  
Square Plate ◽  
3D Shapes

Задача изгиба прямоугольной панели обшивки от действия распределенной по оси симметрии поперечной нагрузки не имеет точного решения в конечном виде в виду сложности краевых условий и вида нагрузки. Использование другими авторами различных приближенных методов оставляет открытым вопрос о точности полученных результатов. Целью исследования является получение точного решения с помощью гиперболо-тригонометрических рядов по двум координатам. Для этого используется метод бесконечной суперпозиции указанных рядов, которые в отдельности удовлетворят лишь части граничных условий. Порождаемые ими невязки взаимно компенсируются в ходе итерационного процесса и стремятся к нулю. Частное решения представлено двойным рядом Фурье. Точное решение достигается увеличением количества членов в рядах и числа итераций. При достижении заданной точности процесс прекращается. Получены численные результаты для прогибов и изгибающих моментов для квадратной пластины при различной длине загруженной части оси пластины. Представлены 3D-формы изогнутой поверхности пластины и эпюры изгибающих моментов. The problem of bending a rectangular skin panel from the action of a transverse load distributed along the axis of symmetry does not have an exact solution in the final form due to the complexity of the boundary conditions and the type of load. The use of various approximate methods by other authors leaves open the question of the accuracy of the results obtained. The aim of the study is to obtain an exact solution using hyperbolo-trigonometric series in two coordinates. To do this, we use the method of infinite superposition of these series, which individually satisfy only part of the boundary conditions. The residuals generated by them are mutually compensated during the iterative process and tend to zero. The quotient of the solution is represented by a double Fourier series. The exact solution is achieved by increasing the number of terms in the series and the number of iterations. When the specified accuracy is reached, the process stops. Numerical results are obtained for deflections and bending moments for a square plate with different lengths of the loaded part of the plate axis. 3D shapes of the curved surface of the plate and diagrams of bending moments are presented.

scholarly journals CWENO Finite-Volume Interface Capturing Schemes for Multicomponent Flows Using Unstructured Meshes

2021 ◽  
Vol 89 (3) ◽  
Author(s):  
Panagiotis Tsoutsanis ◽  
Ebenezer Mayowa Adebayo ◽  
Adrian Carriba Merino ◽  
Agustin Perez Arjona ◽  
Martin Skote
Keyword(s):  
Finite Volume ◽  
Analytical Solutions ◽  
Unstructured Meshes ◽  
Accurate Solution ◽  
High Order ◽  
Experimental Results ◽  
Test Problems ◽  
Interface Capturing ◽  
Multicomponent Flows ◽  
Stringent Test

AbstractIn this paper we extend the application of unstructured high-order finite-volume central-weighted essentially non-oscillatory (CWENO) schemes to multicomponent flows using the interface capturing paradigm. The developed method achieves high-order accurate solution in smooth regions, while providing oscillation free solutions at discontinuous regions. The schemes are inherently compact in the sense that the central stencils employed are as compact as possible, and that the directional stencils are reduced in size, therefore simplifying their implementation. Several parameters that influence the performance of the schemes are investigated, such as reconstruction variables and their reconstruction order. The performance of the schemes is assessed under a series of stringent test problems consisting of various combinations of gases and liquids, and compared against analytical solutions, computational and experimental results available in the literature. The results obtained demonstrate the robustness of the new schemes for several applications, as well as their limitations within the present interface-capturing implementation.

Surface Vibration of a Multilayered Elastic Medium due to Harmonic Concentrated Force

2011 ◽  
Author(s):  
Akindeji Ojetola ◽  
Hamid Hamidzadeh
Keyword(s):  
Fourier Transform ◽  
Elastic Medium ◽  
Three Dimensional ◽  
Concentrated Force ◽  
Double Complex ◽  
Concentrated Load ◽  
Propagator Matrix ◽  
Displacement Vectors ◽  
Bottom Interface ◽  
Similar Information

Dynamic response of a multi-layer elastic medium subjected to harmonic surface concentrated load is considered. In development of the analytical solution, the three-dimensional theory of elasto-dynamic is utilized for derivation of the governing partial differential equations for each layer. These equations are solved in the Fourier domain by employing the Double Complex Fourier Transform technique. In the analysis, each layer of the medium is assumed to be extended infinitely in the horizontal x and z directions and has uniform depth in the y direction and is considered to be linearly elastic, homogeneous, and isotropic. Utilizing the Integral Fourier Transform, displacements and stresses at any point in each layer can be determined in terms of boundary stresses for each layer. Also, the presented solution provides the relation between stress and displacement vectors for the top and bottom of each layer in matrix notation. By satisfying the compatibility of displacements and stresses for each interface, a propagator matrix relating displacements and stresses at the top of the medium to the bottom interface will be obtained. This relates the displacement and stress vector on the top surface to the bottom interface by eliminating similar information for the other interfaces. In this study, the displacements on the surface of the layered medium are computed for the two cases where the surface of the medium is subjected to a concentrated harmonic vertical or horizontal harmonic force.

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