Dominance of Shear Stresses in Early Stages of Impulsive Motion of Beams

1960 ◽  
Vol 27 (1) ◽  
pp. 132-138 ◽  
Author(s):  
H. H. Bleich ◽  
R. Shaw
Keyword(s):  
Differential Equations ◽  
Timoshenko Beam ◽  
Shear Stresses ◽  
Impulsive Motion ◽  
Early Stages ◽  
Plastic Analysis ◽  
Bending Stresses ◽  
Impulsive Forces

In order to compare the magnitude of bending stresses and shear stresses in beams under the action of impulsive forces, the values of these stresses are determined from the known differential equations for the Timoshenko beam. It is found that in the early stages, soon after the initiation of the motion, the shear stresses are of much larger magnitude than the bending stresses. This result indicates that for sufficiently large initial velocities first yielding will be in shear, a matter of consequence in plastic analysis.

A Combined Fourier Series–Galerkin Method for the Analysis of Functionally Graded Beams

2004 ◽  
Vol 71 (3) ◽  
pp. 421-424 ◽  
Author(s):  
H. Zhu ◽  
B. V. Sankar
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Galerkin Method ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Functionally Graded Beam ◽  
Fourier Series Method ◽  
Bending Stresses ◽  
The Galerkin Method ◽  
Graded Structures

The method of Fourier analysis is combined with the Galerkin method for solving the two-dimensional elasticity equations for a functionally graded beam subjected to transverse loads. The variation of the Young’s modulus through the thickness is given by a polynomial in the thickness coordinate and the Poisson’s ratio is assumed to be constant. The Fourier series method is used to reduce the partial differential equations to a pair of ordinary differential equations, which are solved using the Galerkin method. Results for bending stresses and transverse shear stresses in various beams show excellent agreement with available exact solutions. The method will be useful in analyzing functionally graded structures with arbitrary variation of properties.

Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
J. Awrejcewicz ◽  
A. V. Krysko ◽  
S. P. Pavlov ◽  
M. V. Zhigalov ◽  
V. A. Krysko
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Stability Loss ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Timoshenko Beams ◽  
Size Dependent ◽  
Rigid Layer ◽  
Graded Material

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

scholarly journals USE OF ATANGANA–BALEANU FRACTIONAL DERIVATIVE IN HELICAL FLOW OF A CIRCULAR PIPE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040049 ◽  
Author(s):  
KASHIF ALI ABRO ◽  
ILYAS KHAN ◽  
KOTTAKKARAN SOOPPY NISAR
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rheological Parameters ◽  
Shear Stresses ◽  
Fractional Differentiation ◽  
Retardation Time ◽  
Governing Equations ◽  
Partial Differential ◽  
Gamma Functions ◽  
Helical Pipe

There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana–Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox- H function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.

The Distribution of Shear Stresses in an I Section Beam

1961 ◽  
Vol 65 (603) ◽  
pp. 198-201
Author(s):  
W. G. Wood
Keyword(s):  
Shear Stiffness ◽  
Bending Stiffness ◽  
Combined Effect ◽  
Transverse Load ◽  
Substantial Part ◽  
Shear Stresses ◽  
Short Beam ◽  
Bending Stresses

When a Beam is subjected to a static transverse load acting through its flexural centre, the resulting deflection may be considered as the combined effect of the deflection due to bending stresses and the deflection due to shear stresses. In a majority of cases, where the span of the beam is large, or where the shear stiffness is large compared with the bending stiffness, the bending component only need be considered. However, for a short beam, the shear deflection can account for a substantial part of the total deflection. This is especially so in the case of a deep I section beam with a thin web, where the bending stiffness is large compared with the shear stiffness.

Plane Waves in an Elastic-Plastic Half-Space Due to Combined Surface Pressure and Shear

1966 ◽  
Vol 33 (1) ◽  
pp. 149-158 ◽  
Author(s):  
H. H. Bleich ◽  
Ivan Nelson
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Half Space ◽  
Nonlinear Differential Equations ◽  
Plane Waves ◽  
Yield Condition ◽  
Shear Stresses ◽  
Von Mises ◽  
Plane Wave Propagation ◽  
Normal And Shear Stresses

The most general case of plane wave propagation, when normal and shear stresses occur simultaneously, is considered in a material obeying the von Mises yield condition. The resulting nonlinear differential equations have not been solved previously for any boundary-value problem, except for special situations where the differential equations degenerate into linear ones. In the present paper, the stresses in a half-space, due to a uniformly distributed step load of pressure and shear on the surface, are obtained in closed form.

Stresses and deformations in overlapped diesel engine crankshafts, Part 2: Evaluation of results

1998 ◽  
Vol 212 (4) ◽  
pp. 255-270 ◽  
Author(s):  
I Bickley ◽  
V D'Olier ◽  
H Fessler ◽  
T. H. Hyde ◽  
N. A. Warrior
Keyword(s):  
Diesel Engine ◽  
Stress Concentration ◽  
Cross Sections ◽  
Shear Stresses ◽  
Previous Issue ◽  
Rectangular Section ◽  
Bending Stresses ◽  
Concentration Factors ◽  
Stress Concentration Factors ◽  
The Web

The extensive results presented in Part 1 (in the previous issue*) have been supplemented and analysed further. A large number of cross-sections which could be reasonably used to calculate nominal stresses has been listed and evaluated. An inclined (flat, rectangular) section through the web is shown to be the best to calculate nominal stresses due to torsion, pure radial bending and bending due to crankpin forces; its width is h (see Fig. 1), the length of the shortest line joining crankpin and journal fillets in the plane of symmetry. Stress concentration factors based on these nominal stresses show only modest scatter from single curves for crankpin and journal fillets for torsion and radial bending. Predictions using the most commonly used method underestimate shear stresses due to torsion and overestimate bending stresses.

scholarly journals Numerical Study of Lorentz Force Interaction with Micro Structure in Channel Flow

Energies ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 4286
Author(s):  
Shabbir Ahmad ◽  
Kashif Ali ◽  
Sohail Ahmad ◽  
Jianchao Cai
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Viscous Dissipation ◽  
Numerical Study ◽  
Nonlinear Differential Equations ◽  
Thermal Control ◽  
Shear Stresses ◽  
Fluid Temperature ◽  
X Rays ◽  
Micro Particles

The heat transfer Magnetohydrodynamics flows have been potentially used to enhance the thermal characteristics of several systems such as heat exchangers, electromagnetic casting, adjusting blood flow, X-rays, magnetic drug treatment, cooling of nuclear reactors, and magnetic devices for cell separation. Our concern in this article is to numerically investigate the flow of an incompressible Magnetohydrodynamics micropolar fluid with heat transportation through a channel having porous walls. By employing the suitable dimensionless coordinates, the flow model equations are converted into a nonlinear system of dimensionless ordinary differential equations, which are then numerically treated for different preeminent parameters with the help of quasi-linearization. The system of complex nonlinear differential equations can efficiently be solved using this technique. Impact of the problem parameters for microrotation, temperature, and velocity are interpreted and discussed through tables and graphs. The present numerical results are compared with those presented in previous literature and examined to be in good contact with them. It has been noted that the imposed magnetic field acts as a frictional force which not only increases the shear stresses and heat transfer rates at the channel walls, but also tends to rotate the micro particles in the fluid more rapidly. Furthermore, viscous dissipation may raise fluid temperature to such a level that the possibility of thermal reversal exists, at the geometric boundaries of the domain. It is therefore recommended that external magnetic fields and viscous dissipation effects may be considered with caution in applications where thermal control is required.

Multiply Loaded Timoshenko Beam on a Stressed Orthotropic Half-Plane via a Thin Elastic Layer

1993 ◽  
Vol 60 (2) ◽  
pp. 541-547 ◽  
Author(s):  
H. Bjarnehed
Keyword(s):  
Half Plane ◽  
Timoshenko Beam ◽  
Elastic Layer ◽  
Beam Theory ◽  
Singular Integral Equations ◽  
Uniaxial Stress ◽  
Shear Stresses ◽  
Fe Model ◽  
Tangential Load ◽  
Thin Elastic Layer

The problem of bonded contact between a uniform finite Timoshenko beam and an orthotropic half-plane via a thin elastic layer is considered in this paper. The beam is loaded by distributions of normal and tangential forces, and a uniaxial stress load is applied to the half-plane. The Timoshenko beam theory is extended in such a way that the tangential load is included when the shear contribution to the beam central line deflection is calculated. The layer is formulated as a generalized Winkler cushion including also shear stresses and strains. Governing singular integral equations are stated and numerically solved for the unknown interface stresses. A comparison with a corresponding FE-model is also performed.

scholarly journals Variational Principles for Bending and Vibration of Partially Composite Timoshenko Beams

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Halil Özer
Keyword(s):  
Differential Equations ◽  
Timoshenko Beam ◽  
Inverse Method ◽  
Variational Principles ◽  
Timoshenko Beams ◽  
Governing Equations ◽  
Partially Composite ◽  
Composite Timoshenko Beam

Variational principles are established for the partially composite Timoshenko beam using the semi-inverse method. The principles are derived directly from governing differential equations for bending and vibration of the beam considered. It is concluded that the semi-inverse method is a powerful tool for searching for variational principles directly from the governing equations. Comparison between our results and the results reported in literature is given.

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