Flexural Wave Solutions of Coupled Equations Representing the More Exact Theory of Bending

1953 ◽  
Vol 20 (4) ◽  
pp. 511-514
Author(s):  
Julius Miklowitz
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Shear Force ◽  
Beam Element ◽  
Transverse Load ◽  
Flexural Wave ◽  
Infinite Beam ◽  
Coupled Equations ◽  
Wave Solutions ◽  
The Moment

Abstract Presented here is a new method for deriving flexural wave solutions for the Timoshenko bending theory. The method is based on a breakdown of the total deflection into its bending and shear components. Instead of treating the full Timoshenko equation (1) an equivalent set of coupled equations, representing the rotational and translatory motions of the beam element, is solved. The advantages of this method stem from (a) the simplicity of the associated expressions for the moment and shear force, which are the elementary bending theory relations, and (b) the well-defined nature of the related boundary conditions. The latter is particularly important since it is difficult to define the proper boundary conditions associated with the full Timoshenko equation. This is evidenced in the works of Uflyand (2) and Dengler and Goland (3), both of which are concerned with wave solutions for the infinite beam under the action of a concentrated transverse load. The quoted work (3) points out the erroneous boundary conditions used in the Uflyand work (2). The present method is applied to the same case treated in the works (2, 3). Agreement is shown with the Dengler and Goland solution. The Uflyand solution is shown to have meaning when interpreted properly. The derivation of transforms for other beam cases, both finite and infinite, by the present method has also been included in this work.

Flexural Stress Waves in an Infinite Elastic Plate Due to a Suddenly Applied Concentrated Transverse Load

1960 ◽  
Vol 27 (4) ◽  
pp. 681-689 ◽  
Author(s):  
Julius Miklowitz
Keyword(s):  
Shear Force ◽  
Plate Theory ◽  
Three Dimensional ◽  
Infinite Plate ◽  
Present Theory ◽  
Transverse Load ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Flexural Stress ◽  
The Moment

The problem solved is that of an infinite plate subjected to a suddenly applied concentrated transverse shear load. The solution is derived from a plate theory that incorporates, in addition to bending, the effect of shear force and rotatory inertia on the deflection. These added effects give the present theory true wave character along with greater accuracy in the waves predicted. Numerical evaluation of the solution brings out the effects of dispersion and distortion on the moment and shear-force response of the plate. A criterion is developed for judging the accuracy of this response. It is based on a comparison, employing the stationary phase method, of the present approximate and exact (three-dimensional) theories.

Calculating Mechanics Characteristics of Single-Walled Carbon Nanotube Materials by Finite Element Method

2017 ◽  
Vol 730 ◽  
pp. 548-553
Author(s):  
Jing Ge ◽  
Hao Jiang ◽  
Zhen Yu Sun ◽  
Guo Jun Yu ◽  
Bo Su ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Finite Element ◽  
Radial Direction ◽  
Element Model ◽  
Beam Element ◽  
Beam Position ◽  
Single Walled Carbon ◽  
Finite Element Software ◽  
Calculation Results ◽  
The Moment

In this paper, we establish the mechanical property analysis of Single-walled Carbon Nanotubes (SWCNTs) modified beam element model based on the molecular structural mechanics method. Then we study the mechanical properties of their radial direction characteristics using the finite element software Abaqus. The model simulated the different bending stiffness with rectangular section beam elements C-C chemical force field. When the graphene curled into arbitrary chirality of SWCNTs spatial structure, the adjacent beam position will change the moment of inertia of the section of the beam. Compared with the original beam element model and the calculation results, we found that the established model largely reduced the overestimate of the original model of mechanical properties on the radial direction of the SWCNTs. At the same time, compared with other methods available in the literature results and the experimental data, the results can be in good agreement.

scholarly journals Impact Coefficient Analysis of Curved Box Girder Bridge Based on Vehicle-Bridge Coupling

2022 ◽  
Vol 2022 ◽  
pp. 1-11
Author(s):  
Fei Guo ◽  
Heng Cai ◽  
Huifang Li
Keyword(s):  
Shear Force ◽  
Primary Beam ◽  
Dynamic Loads ◽  
Box Girder ◽  
Impact Coefficient ◽  
Force Impact ◽  
Girder Bridge ◽  
The Moment ◽  
The Impact ◽  
Load Weight

In the current vehicle-bridge dynamics research studies, displacement impact coefficients are often used to replace the moment and shear force impact coefficients, and the vehicle model is also simplified as a moving-load model without considering the contribution of vehicle stiffness and damping to the system in some concerned research studies, which cannot really reflect the mechanical behavior of the structures under vehicle dynamic loads. This paper presents a vehicle-bridge coupling model for the prediction of dynamic responses and impact coefficient of the long-span curved bending beam bridge. The element stiffness matrix and mass matrix of a curved box girder bridge with 9 freedom degrees are directly deduced based on the principle of virtual work and dynamic finite element theory. The vibration equations of vehicle-bridge coupling are established by introducing vehicle mode with 7 freedom degrees. The Newmark-β method is adopted to solve vibration response of the system under vehicle dynamic loads, and the influences of flatness of bridge surface, vehicle speed, load weight, and primary beam stiffness on the impact coefficient are comprehensively discussed. The results indicate that the impact coefficient presents a nonlinear increment as the flatness of bridge surface changes from good to terrible. The vehicle-bridge coupling system resonates when the vehicle speeds reach 60 km/h and 100 km/h. The moment design value will maximally increase by 2.89%, and the shear force design value will maximally decrease by 34.9% when replacing moment and shear force impact coefficients with the displacement impact coefficient for the section internal force design. The load weight has a little influence on the impact coefficient; the displacement and moment impact coefficients are decreased with an increase in primary beam stiffness, while the shear force impact coefficient is increased with an increase in primary beam stiffness. The theoretical results presented in this paper agree well with the ANSYS results.

scholarly journals Analysis of beam-column elements on non-homogeneous soil using the differential transformation method

2021 ◽  
Author(s):  
Juan Sebastián Carvajal-Muñoz ◽  
Carlos Alberto Vega-Posada ◽  
Julio César Saldarriaga-Molina
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Mathematical Formulation ◽  
Transformation Method ◽  
Transverse Load ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Pasternak Elastic Foundation ◽  
Wide Range ◽  
Homogeneous Soil

This paper describes an analytical approach to conduct an analysis of beam-column elements with generalized end-boundary conditions on a homogeneous or non-homogeneous Pasternak elastic foundation. The mathematical formulation utilized herein is that presented by the senior author in a recent work. The differential equation (DE) governing the behavior of the beam-column element is solved using the differential transformation method (DTM). The DTM offers practical advantages over other conventional approaches when solving the proposed structural model. The proposed formulation provides the flexibility to account for i) combined lateral and axial load at the ends of the element, ii) homogeneous or non-homogeneous soil, iii) Pasternak elastic foundation, and iv) an external arbitrary transverse load acting on the element. The effects of various slenderness ratios, pile-soil stiffness ratios, and classical and semirigid boundary conditions can be easily studied with the proposed formulation. Examples are presented to validate the accuracy of the model and its applicability over a wide range of analyses.

Flexure of a Circular Plate With a Ring of Holes

1962 ◽  
Vol 29 (3) ◽  
pp. 489-496 ◽  
Author(s):  
H. Kraus
Keyword(s):  
Circular Plate ◽  
Shear Force ◽  
General Boundary Condition ◽  
Uniform Pressure ◽  
Simply Supported ◽  
Moment Distribution ◽  
Theory Of Plates ◽  
Circular Holes ◽  
The Moment ◽  
Kirchhoff Theory

The problem of the moment distribution resulting from a uniform pressure load acting over the surface of a circular plate containing a ring of equally spaced circular holes with, and without, a central circular hole is solved within the framework of the Poisson-Kirchhoff theory of plates. A general boundary condition is applied at the outer rim of the plate to make the solution valid for a range of conditions from the simply supported case to the clamped case. The edges of the perforations are allowed to be either free or to have a net shear force acting. Numerical results in the form of curves are given for typical cases, and the results of a photoelastic test are also presented.

Buckling Analysis of Tubular Strings in Horizontal Wells

SPE Journal ◽  
2014 ◽  
Vol 20 (02) ◽  
pp. 405-416 ◽  
Author(s):  
Wenjun Huang ◽  
Deli Gao ◽  
Fengwu Liu
Keyword(s):  
Boundary Conditions ◽  
Axial Force ◽  
Shear Force ◽  
Virtual Work ◽  
Bending Moment ◽  
Horizontal Wells ◽  
Comprehensive Description ◽  
New Model ◽  
First Category ◽  
Second Category

Summary A new buckling equation in horizontal wells is derived on the basis of the general bending and twisting theory of rods. The boundary conditions of a long tubular string are divided into two categories: the sum of the virtual work of bending moment and shear force at the ends of tubular strings is equal to zero, and the sum of the virtual work of bending moment and shear force at the ends is not equal to zero. Buckling solutions under different boundary conditions are obtained by solving the new buckling model. For the boundary conditions of the first category, the buckling solutions are identical with previous results. For the boundary conditions of the second category, the buckling solutions are different from the results under the boundary conditions of the first category. The results indicate that buckling behaviors depend on both the axial force and the boundary conditions. Compared with previous results, buckling solutions of the new model provide a more comprehensive description of tubular-buckling behaviors.

Effects of Lateral Surface Conditions in Time-Harmonic Nonsymmetric Wave Propagation in a Cylinder

1989 ◽  
Vol 56 (4) ◽  
pp. 910-917 ◽  
Author(s):  
Yoon Young Kim ◽  
Charles R. Steele
Keyword(s):  
Lateral Surface ◽  
Present Method ◽  
Lateral Wall ◽  
Computational Effort ◽  
Wave Solutions ◽  
Time Harmonic ◽  
End Conditions ◽  
Analogous Manner ◽  
Method Of Solution ◽  
Free Case

The present work is a part of the effort toward the development of an efficient method of solution to handle general nonsymmetric time-harmonic end conditions in a cylinder with a traction-free lateral surface. Previously, Kim and Steele (1989a) develop an approach for the general axisymmetric case, which utilizes the well-known uncoupled wave solutions for a mixed lateral wall condition. For the case of a traction-free lateral wall, the uncoupled wave solutions provide: (1) a convenient set of basis functions and (2) approximations for the relation between end stress and displacement which are asymptotically valid for high mode index numbers. The decay rate with the distance from the end is, however, highly dependent on the lateral wall conditions. The present objective was to demonstrate that the uncoupled solutions of the nonsymmetric waves discussed by Kim (1989), which satisfy certain mixed lateral wall conditions, can be utilized in an analogous manner for the asymptotic analysis of the traction-free case. Results for the end displacement/stress due to various end conditions, computed by the present method and by a more standard collocation method, were compared. The present method was found to reduce the computational effort by orders of magnitude.

Electromagnetic wave backscattering across a magnetic field in a bounded plasma

1979 ◽  
Vol 22 (1) ◽  
pp. 85-96
Author(s):  
Joseph E. Willett ◽  
Sinan Bilikmen ◽  
Behrooz Maraghechi
Keyword(s):  
Magnetic Field ◽  
Numerical Study ◽  
Magnetized Plasma ◽  
Absolute Instability ◽  
Moment Equations ◽  
Homogeneous Plasma ◽  
Coupled Equations ◽  
Bounded Plasma ◽  
The Moment ◽  
The Magnetic Field

The stimulated backscattering of electromagnetic ordinary waves from extraordinary waves propagating normal to a magnetic field in a plasma of finite length is studied. A pair of coupled differential equations for the amplitudes of the backscattered and scatterer waves is derived from Maxwell's equations and the moment equations for an inhomogeneous magnetized plasma. Solution of the coupled equations for a homogeneous plasma yields an expression for the growth rate of the absolute instability as a function of plasma length and damping rates of the product waves. The convective regime in which only spatial amplification occurs is discussed. A numerical study of the effects of the magnetic field on Raman and Brillouin backscattering is presented.

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