scholarly journals Calculation of a beam on an inhomogeneous basis

2018 ◽  
Vol 251 ◽  
pp. 04023
Author(s):  
Nikolaj Atarov
Keyword(s):  
Numerical Calculation ◽  
Power Law ◽  
Modulus Of Elasticity ◽  
Strain Energy ◽  
Deformation Modulus ◽  
Bending Moment ◽  
System Of Differential Equations ◽  
Total Potential ◽  
Modulus Of Deformation ◽  
Symmetric Load

The questions of analytical calculating the beam on the base with the modulus of deformation (modulus of elasticity), which is changed by thickness of the layer by power law, was deals in the article. The purpose of work was receiving finite expressions for reactive pressure of the basе on a beam and internal efforts in a beam when using model of the base with two characteristics (coefficient of subgrade resistance). A system of differential equations second-order with respect to the displacements of points the surface of a layer with a continuously changing modulus of elasticity was obtained based on the minimum of the total potential strain energy. The calculation of the rigid beam on the base on the action of the symmetric load was performed and the formulas for the reactive pressures of the base were got. Numerical calculation is executed and the analysis of influence of change of the module of deformation of a layer by the amount of reactive pressure and the bending moment in a beam was given. It is shown that with increase in the module of deformation on layer thickness the basе with two characteristics on nature of work is approaching vinklerovsky.

A simplified formulation of adhesion problems with elastic plates

2009 ◽  
Vol 465 (2107) ◽  
pp. 2217-2230 ◽  
Author(s):  
Carmel Majidi ◽  
George G. Adams
Keyword(s):  
Potential Energy ◽  
Contact Area ◽  
Contact Region ◽  
Bending Moment ◽  
Total Potential Energy ◽  
Work Of Adhesion ◽  
Contact Radius ◽  
Algebraic Complexity ◽  
Elastic Plates ◽  
Total Potential

The solution of adhesion problems with elastic plates generally involves solving a boundary-value problem with an assumed contact area. The contact region is then found by minimizing the total potential energy with respect to the contact area (i.e. the contact radius for the axisymmetric case). Such a procedure can be extremely long and tedious. Here, we show that the inclusion of adhesion is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is related to the work of adhesion and flexural rigidity of the plate. Such a formulation can greatly reduce the algebraic complexity of solving these problems. It is noted that the related plate contact problems without adhesion can also be solved by minimizing the total potential energy. However, it has long been recognized that it is mathematically more efficient to find the contact area by specifying a continuous internal bending moment at the boundary of the contact region. Thus, our moment discontinuity method can be considered to be a generalization of that procedure which is applicable for problems with adhesion.

Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

1953 ◽  
Vol 20 (4) ◽  
pp. 469-474
Author(s):  
W. A. Nash
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Middle Surface ◽  
Elastic Buckling ◽  
Elastic Instability ◽  
Total Potential ◽  
External Forces ◽  
Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

scholarly journals Comparison of bending stiffness of six different colours of copolymer polypropylene

1999 ◽  
Vol 23 (1) ◽  
pp. 63-71 ◽  
Author(s):  
R. S. Ross ◽  
R. J. Greig ◽  
P. Convery
Keyword(s):  
Electron Microscope ◽  
Wall Thickness ◽  
Modulus Of Elasticity ◽  
Microscope Study ◽  
Electron Microscope Study ◽  
Bending Moment ◽  
Bending Stiffness ◽  
Ankle Foot Orthosis ◽  
Uniform Wall ◽  
Foot Orthosis

This paper compares the bending stiffness of 5 different colours of copolymer polypropylene (CCP) with that of natural copolymer polypropylene (NCP). Flesh coloured and natural sheets are supplied thicker than other pigmented sheet. The bending stiffness of a specimen may be defined as EI, i.e. the product of E, Young's modulus of elasticity and I, the 2nd moment of area. Strips of “as supplied” (AS) and “post-draped” (PD) specimen were clamped and subjected to bending to assess the effect of pigmentation on bending characteristics. The gradient of the graph of bending deflection δ versus bending moment enables EI to be estimated. The process of thermoforming polypropylene reduces EI, the bending stiffness. However, the manual draping and vacuum procedure introduces so many variables that it is difficult to quantify the effect of pigmentation. The E of a bent specimen may be estimated from the gradient of the graph of δI versus bending moment. In the case of AS sheet, the effect of pigmentation on E is inconclusive. PD specimens indicate a significant reduction in E due to thermoforming. This was verified by an electron-microscope study of AS and PD specimens. Draping an ankle-foot orthosis (AFO) results in a non-uniform wall thickness. The results of this study with respect to the effects of pigmentation on the bending stiffness of AFOs are inconclusive. More detailed studies require to be completed in order to confirm which factors are responsible for this non-uniformity in wall thickness and consequent variation in bending stiffness.

THE INFLUENCE OF SEX HORMONES ON THE BONY CHANGES OCCURRING IN PARALYSED LIMBS

1954 ◽  
Vol 11 (1) ◽  
pp. 66-70 ◽  
Author(s):  
J. A. GILLESPIE
Keyword(s):  
Young’S Modulus ◽  
Sex Hormones ◽  
Modulus Of Elasticity ◽  
Young's Modulus ◽  
Bending Moment ◽  
Breaking Point ◽  
Total Weight ◽  
Breaking Stress ◽  
Young’S Modulus Of Elasticity ◽  
Ash Weight

SUMMARY 1. Paralysis, produced in young male rats by avulsion of peripheral nerves, resulted in a highly significant reduction in the total weight, ash weight, ash percentage, X-ray density and bending moment at the breaking point, of the bones of the affected limb. The breaking stress was significantly reduced, but Young's modulus of elasticity was unaltered. 2. Certain of these changes were modified by treatment with sex hormones. Both oestradiol and testosterone significantly reduced the difference between the bones of the normal and paralysed limbs in respect of total weight, ash weight and bending moment at the breaking point. 3. Treatment with oestradiol significantly increased both the ash percentage and Young's modulus of elasticity, the increase affecting equally the bones of the normal and paralysed limbs. Testosterone treatment did not significantly affect either of these properties, and neither hormone affected the breaking stress.

scholarly journals Extension of Castigliano’s method for isotropic beams

2020 ◽  
Vol 231 (11) ◽  
pp. 4621-4640
Author(s):  
Juergen Schoeftner
Keyword(s):  
Shear Force ◽  
Strain Energy ◽  
Axial Stress ◽  
Bending Moment ◽  
Axial Direction ◽  
Present Contribution ◽  
Special Cases ◽  
Dummy Load ◽  
Complementary Strain ◽  
Castigliano’S Theorem

Abstract In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

scholarly journals Avalanches and power law behavior in aortic dissection propagation

2020 ◽  
Vol 6 (21) ◽  
pp. eaaz1173 ◽  
Author(s):  
Xunjie Yu ◽  
Béla Suki ◽  
Yanhang Zhang
Keyword(s):  
Cardiovascular Disease ◽  
Extracellular Matrix ◽  
Aortic Dissection ◽  
Power Law ◽  
Model Prediction ◽  
Strain Energy ◽  
Quantitative Agreement ◽  
High Morbidity ◽  
Failure Mechanics ◽  
Microscopic Features

Aortic dissection is a devastating cardiovascular disease known for its rapid propagation and high morbidity and mortality. The mechanisms underlying the propagation of aortic dissection are not well understood. Our study reports the discovery of avalanche-like failure of the aorta during dissection propagation that results from the local buildup of strain energy followed by a cascade failure of inhomogeneously distributed interlamellar collagen fibers. An innovative computational model was developed that successfully describes the failure mechanics of dissection propagation. Our study provides the first quantitative agreement between experiment and model prediction of the dissection propagation within the complex extracellular matrix (ECM). Our results may lead to the possibility of predicting such catastrophic events based on microscopic features of the ECM.

Subcritical crack growth along epoxy/glass interfaces

1992 ◽  
Vol 7 (9) ◽  
pp. 2621-2629 ◽  
Author(s):  
K.M. Conley ◽  
J.E. Ritter ◽  
T.J. Lardner
Keyword(s):  
Energy Release Rate ◽  
Energy Release ◽  
Crack Growth ◽  
Power Law ◽  
Release Rate ◽  
Strain Energy ◽  
Crack Velocity ◽  
Subcritical Crack Growth ◽  
Strain Energy Release Rate ◽  
Strain Energy Release

Subcritical crack growth behavior along polymer/glass interfaces was measured for various epoxy adhesives at different relative humidities. A four-point flexure apparatus coupled with an inverted microscope allowed for observation in situ of the subcritical crack growth at the polymer/glass interface. The specimens consisted of soda-lime glass plates bonded together with epoxy acrylate, epoxy (Devcon), or epoxy (Shell) adhesives. Above a threshold strain energy release rate, the subcritical crack velocity was dependent on the strain energy release rate via a power law relationship where the exponent was independent of the adhesive tested and the test humidity (n = 3). However, the multiplicative constant A in the power law relation varied by over three orders of magnitude between the various adhesives with epoxy (Shell) having the smallest value and the epoxy (Devcon) the greatest value; in addition, A was very sensitive to humidity, decreasing by over two orders of magnitude from 80% to 15% relative humidity. At high strain energy release rates, the subcritical crack velocity reached a plateau at approximately 10−6 m/s. The use of this subcritical crack velocity data in predicting thin film delamination is discussed.

The Stress State in Inhomogeneous Elastic Beam at Combined Strength

2014 ◽  
Vol 501-504 ◽  
pp. 645-648 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Elena V. Barmenkova ◽  
Alena V. Matveeva
Keyword(s):  
Inverse Problem ◽  
Stress State ◽  
Modulus Of Elasticity ◽  
Elastic Beam ◽  
Bending Moment ◽  
Optimization Method ◽  
Theory Of Elasticity ◽  
Simultaneous Action ◽  
Concentrated Force ◽  
Inhomogeneous Bodies

In paper describes a method of optimization the stress state of an elastic beam, subjected to the simultaneous action of the central application of concentrated force and bending moment. Optimization method based on solving the inverse problem of the theory of elasticity of inhomogeneous bodies, the essence of which is to determine the law of changing the modulus of elasticity on the beams height for which stress state will be given.

scholarly journals Using XFEM to Predict the Damage with Temperature of the Steel Pipe Elbows under Bending and Pressure loading

2020 ◽  
Vol 15 (55) ◽  
pp. 345-359
Author(s):  
Nourddine Hammadi ◽  
Moahmed Mokhtari ◽  
Habib Benzaama ◽  
Kouider Madani ◽  
Abdelkader Brakna ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Calculation ◽  
Internal Pressure ◽  
Structural Damage ◽  
Bending Moment ◽  
Pressure Loading ◽  
Tube System ◽  
Moment Stress ◽  
Calculation Code

The pipes, during their service, are subjected to accumulated loads such as internal pressure and that of the soil. The latter considerably accelerate their damage. In this work, the bending moment stress of API 5L X70 category steel elbows under thermo-mechanical behavior and in the presence of pressure were studied. We used FEM (finite element method) through the numerical calculation code ABAQUS and the XFEM technique for structural damage while using solid elements as a structure. Our objective is to evaluate the response and resistance capacity of the steel elbow by its location in the tube–elbow-tube system under a mixed loading of pressure and moment for all scenarios. It is based on a single standardized dimensioning of the elbow (diameter and thickness). The effect of several parameters has been studied such as the type of loading and the pressure levels, which are clearly conditioned by the level of damage. Numerical damage results are presented by moment-rotation curves. They illustrate the variation in damage as a function of these effects which act simultaneously.

scholarly journals ON THE KOLMOGOROV-WIENER FILTER FOR RANDOM PROCESSES WITH A POWER-LAW STRUCTURE FUNCTION BASED ON THE WALSH FUNCTIONS

2021 ◽  
pp. 39-47
Author(s):  
V. N. Gorev ◽  
A. Yu. Gusev ◽  
V. I. Korniienko ◽  
A. A. Safarov
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Numerical Calculation ◽  
Weight Function ◽  
Structure Function ◽  
Random Process ◽  
Power Law ◽  
Wiener Filter ◽  
Walsh Function ◽  
Better Than

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of a continuous stationary random process with a power-law structure function. Objective. The aim of the work is to develop an algorithm of obtaining an approximate solution for the weight function without recourse to numerical calculation of integrals. Method. The weight function under consideration obeys the Wiener-Hopf integral equation. A search for an exact analytical solution for the corresponding integral equation meets difficulties, so an approximate solution for the weight function is sought in the framework of the Galerkin method on the basis of a truncated Walsh function series expansion. Results. An algorithm of the weight function obtaining is developed. All the integrals are calculated analytically rather than numerically. Moreover, it is shown that the accuracy of the Walsh function approximations is significantly better than the accuracy of polynomial approximations obtained in the authors’ previous papers. The Walsh function solutions are applicable in wider range of parameters than the polynomial ones. Conclusions. An algorithm of obtaining the Kolmogorov-Wiener filter weight function for the prediction of a stationary continuous random process with a power-law structure function is developed. A truncated Walsh function expansion is the basis of the developed algorithm. In opposite to the polynomial solutions investigated in the previous papers, the developed algorithm has the following advantages. First of all, all the integrals are calculated analytically, and any numerical calculation of the integrals is not needed. Secondly, the problem of the product of very small and very large numbers is absent in the framework of the developed algorithm. In our opinion, this is the reason why the accuracy of the Walsh function solutions is better than that of the polynomial solutions for many approximations and why the Walsh function solutions are applicable in a wider range of parameters than the polynomial ones. The results of the paper may be applied, for example, to practical traffic prediction in telecommunication systems with data packet transfer.

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