A Historical Note on the Reciprocal Theorem and Theory of Statically Indeterminate Frameworks

Nature ◽  
1960 ◽  
Vol 187 (4733) ◽  
pp. 231-232 ◽  
Author(s):  
T. M. CHARLTON
Keyword(s):  
Reciprocal Theorem ◽  
Historical Note ◽  
The Reciprocal Theorem

Boundary Conditions for Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystals

2012 ◽  
Vol 580 ◽  
pp. 411-414
Author(s):  
Bao Sheng Zhao ◽  
Di Wu
Keyword(s):  
Boundary Conditions ◽  
Necessary Conditions ◽  
Reciprocal Theorem ◽  
Interior Solution ◽  
Two Dimensional ◽  
Circular Shaft ◽  
Analysis Technique ◽  
State Boundary ◽  
The Reciprocal Theorem ◽  
Layer Solution

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of torsional circular shaft in two-dimensional dodecagonal quasicrystals (2D dodecagonal QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, using the reciprocal theorem, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the torsional circular shaft in 2D dodecagonal QCs. The results of the present paper enable us to establish a set of boundary conditions.

Development of a Three-Dimensional Semi-Analytical Elastic-Plastic Contact Code

2002 ◽  
Vol 124 (4) ◽  
pp. 653-667 ◽  
Author(s):  
C. Jacq ◽  
D. Ne´lias ◽  
G. Lormand ◽  
D. Girodin
Keyword(s):  
Three Dimensional ◽  
Indentation Test ◽  
Closed Form Expression ◽  
Reciprocal Theorem ◽  
Plastic Strains ◽  
Surface Geometry ◽  
Fe Method ◽  
Elastic Plastic ◽  
Fine Mesh ◽  
The Reciprocal Theorem

A three-dimensional elastic-plastic contact code based on semi-analytical method is presented and validated. The contact is solved within a Hertz framework. The reciprocal theorem with initial strains is then introduced, to express the surface geometry as a function of contact pressure and plastic strains. The irreversible nature of plasticity leads to an incremental formulation of the elastic-plastic contact problem, and an algorithm to solve this problem is set up. Closed form expression, which give residual stresses and surface displacements from plastic strains, are obtained by integration of the reciprocal theorem. The resolution of the elastic-plastic contact using the finite element (FE) method is discussed, and the semi-analytical code presented in this paper is validated by comparing results with experimental data from the nano-indentation test. Finally, the resolution of the rolling elastic-plastic contact is presented for smooth and dented surfaces and for a vertical or rolling loading. The main advantage of this code over classical FE codes is that the calculation time makes the transient analysis of three-dimensional contact problems affordable, including when a fine mesh is required.

Reciprocal Theorem for Displacements In Inelastic Bodies

1967 ◽  
Vol 1 (2) ◽  
pp. 144-151 ◽  
Author(s):  
T.H. Lin
Keyword(s):  
Composite Materials ◽  
Strain Distribution ◽  
Elastic Strain ◽  
Body Force ◽  
Inelastic Strain ◽  
The Body ◽  
Reciprocal Theorem ◽  
Plastic Potential ◽  
The Reciprocal Theorem ◽  
Solid Bodies

Using the analogy between the inelastic strain gradient and applied body force, the reciprocal theorem for displacement in inelastic bodies is derived. The application of this theorem to find the deflection or slope at a particular point due to given in elastic strain distribution in the body is given. Using this theorem, the coincidence of yield surfaces with plastic potential is derived. This theorem may be applied to solid bodies of both homogeneous and composite materials.

scholarly journals How inertial lift affects the dynamics of a microswimmer in Poiseuille flow

2022 ◽  
Vol 5 (1) ◽  
Author(s):  
Akash Choudhary ◽  
Subhechchha Paul ◽  
Felix Rühle ◽  
Holger Stark
Keyword(s):  
Poiseuille Flow ◽  
Complex Dynamics ◽  
Fluid Flows ◽  
Reciprocal Theorem ◽  
Fluid Inertia ◽  
Active Suspensions ◽  
Active Nature ◽  
The Reciprocal Theorem ◽  
The Impact ◽  
Rich Spectrum

AbstractThe transport of motile microorganisms is strongly influenced by fluid flows that are ubiquitous in biological environments. Here we demonstrate the impact of fluid inertia. We analyze the dynamics of a microswimmer in pressure-driven Poiseuille flow, where fluid inertia is small but non-negligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a ‘swimming lift’, which we evaluate for neutral and pusher/puller-type swimmers. Accounting for fluid inertia engenders a rich spectrum of complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer’s hydrodynamic signature and goes well beyond the findings at vanishing fluid inertia. Our work will have non-trivial implications on the transport and dispersion of active suspensions in microchannels.

An Evaluation Index for Estimating Defeaturing-Induced Impacts on Finite Element Analysis

2009 ◽  
Author(s):  
Jianguo Tang ◽  
Shuming Gao ◽  
Hongwei Lin ◽  
Yusheng Liu
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Fe Analysis ◽  
Evaluation Index ◽  
Reciprocal Theorem ◽  
Element Analysis ◽  
Efficient Calculation ◽  
Practical Evaluation ◽  
The Difference ◽  
The Reciprocal Theorem

Defeaturing is a popular CAD/CAE simplification technique. However, defeaturing inevitably leads to errors that can not be effectively evaluated yet during a FE (Finite Element) analysis. In this paper, a novel evaluation index based on the reciprocal theorem is proposed to effectively and efficiently estimate defeaturing-induced impacts on FE analysis. Instead of using the difference of strain energy, the proposed evaluation index uses the difference of work to quantify the defeaturing-induced impacts on FE analysis so that it is not only effective but also can be efficiently calculated. In order for an efficient calculation of the evaluation index, a practical evaluation index is further derived from the proposed theoretical index, and its calculation method is given. The practical evaluation index has been applied to FE static computation of linear elastic structures. Therefore, we are able to adapt the simplification of a model according to a desired accuracy of the analysis results.

Ground movements above tunnels: a method for calculating volume loss

2011 ◽  
Vol 48 (3) ◽  
pp. 451-457 ◽  
Author(s):  
Andrew C. Palmer ◽  
Robert J. Mair
Keyword(s):  
Reciprocal Theorem ◽  
Volume Loss ◽  
Empirical Methods ◽  
Simple Derivation ◽  
Ground Movements ◽  
The Reciprocal Theorem

Excavation of a tunnel induces small movements at the surface of the ground above. Those movements are often estimated by empirical methods. This paper presents a simple derivation of formulas for elastic ground, based on the reciprocal theorem. It confirms that the effect of removal of the weight of the soil in the tunnel is not negligible.

Maxwell, Jenkin and Cotterill and the theory of statically-indeterminate structures

1971 ◽  
Vol 26 (2) ◽  
pp. 233-246 ◽  
Keyword(s):  
Linear Systems ◽  
General Problem ◽  
De Novo ◽  
Reciprocal Theorem ◽  
Frame Structures ◽  
German Mathematician ◽  
Metal Structures ◽  
Railway Construction ◽  
Indeterminate Structures ◽  
The Reciprocal Theorem

The decade 1860-1870 witnessed some remarkable and enduring contributions to the theory of structures in Britain by Clerk Maxwell (F.R.S. 1861), Fleeming Jenkin (F.R.S. 1865) and J. H. Cotterill (F.R.S. 1878). Indeed it was in the course of his work in this field that Maxwell formulated the reciprocal theorem for linear systems. The general problem of calculating the forces within frame structures due to applied loads was widely believed to be unresolved when Jenkin addressed himself to it in 1861. The difficulty arose when the number of structural supports and elements exceeded the number of available independent equations of equilibrium, that is, when there were supports and elements supernumerary to the needs of statics. So-called staticallyindeterminate construction of that kind was encountered frequently as the technology of economical metal structures developed partly to fulfil the needs of railway construction. Now the problem had been solved in principle by L. M. H. Navier early in the nineteenth century and published in his celebrated book (1) in France in 1826. But though that work received some attention in France, notably from Lame, Saint Venant and Levy, there seems to be no record of wider appreciation except, perhaps, in the work of the German mathematician Clebsch; certainly Maxwell approached the problem de novo at Jenkin’s instigation. Indeed Maxwell is nowadays commonly regarded as the first to solve the general problem of the statically-indeterminate framework.

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