Saint-Venant

Saint-Venant stated that two different loadings that are statically equivalent produce the same stresses and deformations at a distance sufficiently far removed from the area of application of the loadings. Thus, if two statically equivalent loadings are applied and the observation point is near the end where the loading is applied, then the stresses and deformations will be different for each loading. Hence the name Saint-Venant end effects. 

The hydraulic performance of sewer pipes can be described at different levels. In the case of nonstationary, nonuniform flow, the Saint Venant Equations should be applied. However, under dry-weather conditions, the Manning Equation is an adequate description of the wastewater flow in a gravity sewer pipe when considering the prediction of wastewater quality changes under transport. There are no grounds for using advanced hydraulic models because of the uncertainties in the prediction of the microbial transformations of the wastewater. 

Sandorf P.E. (1980). Saint-Venant effects in an orthotropic beam. J. Composite Mater. 14, 199-212... 

Saint-Venant, B. M6moires de l academie des sciences des savants Strangers 14, 233 (1855)... 

The experimental methods used to measure residual stresses are essentially the same for metals and polymers. The most widespread are mechanical methods, which can be destructive, non-destructive, and semi-destructive146. Destructive methods involve cutting off part of a sample the residual part reacts to this procedure by deformations or displacements proportional to the inherent stresses. This approach is based on the Saint-Venant principle the response in the residual part does not depend on the stress distribution in the cut part of a sample. After measuring the distribution of deformations, residual stresses in the initial sample can be calculated. 

A.J.C. Barre de Saint-Venant in C.L.M.H. Navier, Resume des Lecons sur 1 Application de la Mechanique (Dunod, Paris, 1864). 

In accordance with the Saint-Venant and Wantzel formula... 

Navier-Stokes equation Navier-Saint-Venant... 

The equations that form the theoretical foundation for the whole science of fluid mechanics were derived more than one century ago by Navier (1827) and Poisson (1831) on the basis of molecular hypotheses. Later the same equations were derived by de Saint Venant (1843) and Stokes (1845) without using such hypotheses. These equations are commonly referred to as the Navier-Stokes equations. Despite the fact that these equations have been known of for more than a century, no general analytical solution of the Navier-Stokes equations is known. This state of the art is due to the complex mathematical (i.e., nonlinearity) nature of these equations. 

According to the Saint Venant principle, the stresses of two statically equivalent applied loads are closely similar except near the point of application of the load. This principle, which can be empirically tested, allows the boundary conditions to be expressed in terms of a resultant force rather than an exact distribution of stresses. Obviously, this principle is of great significance in many practical problems. 

Note that the condition expressed by Eq. (16.147) is an integral instead of a z = 0 this fact represents the application of the Saint Venant principle to the z face of the cylinder. For this reason the stress state in the cylinder is only precise enough for points far from the ends of the cylinder. 

The analysis of the stresses and strains in beams and thin rods is a subject of great interest with many practical applications in the study of the strength of materials. The geometry associated with problems of this type determines the specific type of solution. There are cases where small strains are accompanied by large displacements, flexion and torsion in relatively simple structures being the most relevant examples. Problems of this type were solved for the elastic case by Saint Venant in the nineteenth century. The flexion of viscoelastic beams and the torsion of viscoelastic rods are studied in this chapter. 

It is convenient to describe these properties in terms of the following mechanical models [396] the Hooke body (an elastic spring), the Saint-Venant body modeling dry friction (a bar on a solid surface), and the Newton body (a piston in a vessel filled with a viscous fluid). By using various combinations of these elementary models (connected in parallel and/or in series), one can describe situations which are rather complex from the rheological viewpoint. 

It is well known [38, 118, 125, 280, 379] that for foam there exists a yield stress ro that classifies the types of rheological behavior of foam as follows for r < to, the foam is a solid-shaped substance, and for t > to, it is fluid-shaped. For this reason, mechanical models of foam must include the Saint-Venant body. One of the simplest macrorheological models of the foam body is shown in Figure 7.3. 

In 1871, French mathematician and engineer Barre de Saint-Venant (1797-1886) wrote a paper on elasto-plastic analysis of partly plastic problems, such as the twisting of rods, bending of rectangular beams and pressurizing of hollow cylinders. Sain-Venant considered the following assumptions ... 

Saint-Venant is famous for his principle in the strength of material that states that except in the immediate vicinity of the points of application of the load, the stress distribution may be assumed independent of the acmal mode of application of the load as long as loadings are statically equivalent. This principle is conveniendy used to find out the stresses far away from the load. In the immediate vicinity of the load, the stresses can be determined using advanced theoretical or experimental methods (Beer et al. 2004). Von Karman carried out compression test on marble under high pressure and results were published in 1911. 

Saint-Venant (1797-1886) was one of the foremost elasticians of the period. As a student at the Ecole Polytechnique he was disliked by his contemporaries for refusing to fight for Napolean and defend Paris when the students were... 

It is not possible in this text to cover comprehensively the relevant scientific developments over this period to do this reference should be made to, for example, Straub [47], Heyman [48], Timoshenko [49]. As an indication of the main stream of developments, brief mentions of the work of the prominent figures such as Coulomb, Navier, Cauchy, Saint Venant, Culmann, Mohr and Castigliano will be made. 

The 1973 ASCE paper presents a conceptual model to alleviate flood damages due to overtopping failures of future small earthfill dams including the erosion pattern. The potential reduction in the reservoir release due to the proposed erosion retarding layer is also investigated. A method to determine the optimum layer location is provided so as to minimize the maximum possible reservoir release due to a gradually-breached earth dam. The transient reservoir flow is simulated by a numerical model based on the solution of the one-dimensional Saint-Venant equations, which are solved by the method of characteristics subjected to appropriate boimdaiy conditions. The numerical simulation provides the reduction in release discharge in terms of various parameters. 

In 1843, Adhemar-Jean-Claude Barre de Saint Venant developed the most general form of the differential equations describing the motion of fluids, known as the Saint Venant equations. They are sometimes called Navier-Stokes equations after Claude-Louis Navier and Sir George Gabriel Stokes, who were working on them around the same time. 

The sHde block (Saint-Venant body) simulates ideal plastic behavior with no strain at aU below a critical yield stress 0 (Figure 2.10), although at and above the critical yield stress the strain increases without Hmit This behavior is frequently invoked in problems of slope stabihty for wet fine-grained materials such as silt and clay, as the sHde block wiU not move due to friction until sufficient stress is applied. 

Figure 2.10 Model elements for viscoelastic simulations (from left linear spring element, linear dash pot element, nonlinear spring element, nonlinear dash pot element, sliding block (Saint-Venant body) with yield stress 9 (after Hennicke, 1978).

Figure 2.14 Nonlinear rheological models, (a) Saint-Venant model (b) PrandtI-Reuss model (c) Bingham model.

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