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Saint Venant Principle
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Objective

  • To measure the average and concentrated strains on a mild steel specimen under tension due to the effect of point load by placing two strain guages at 'a/4' and average at 'a' as shown in figure 1.

       

  • By estimating the ratio of concentrated stress and average stress, compare the results with the given model graph.

Introduction

For Revision of Mechanics of solids basics, refer to the introduction section. In 1855, the French Elasticity theorist Adhemar Jean Claude Barre de Saint-Venant stated that the difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from the load.The stresses and strains in a body at points that are sufficiently remote from points of application of load depends only on the static resultant of the loads and not on the distribution of loads. That is, when a portion of a solid body is in a system of forces that are in equilibrium, the stresses produced within the body diminish with increasing distance from the point of application of load.

 

The Stress-strain diagram

 

The stress strain relationship of any material is of primary importance as it gives a good idea of the mechanical behaviour of the material in practical conditions. This is generally accomplished using the tension-compression tests. When calculating the nominal or engineering stress, we assume that the stress is constant over the entire cross section of the specimen’s central portion along the gage length. Thus,

 

Where, A0 is the area of cross-section and P is the applied load.  is the average stress distribution across the specimen’s cross section. Normally, while applying the equations for standard axial loading on specimen, we have assumed that we are adequately far from the point of application of load. That is how the distribution of normal stress is uniform.


Point loads on a surface give rise to a stress concentration near the point of application.
Stress concentration is the increase in stress along the cross-section that maybe caused by a point load or by any another discontinuity such as a hole which brings about an abrupt change in the cross sectional area. The aim of this experiment is to study the effects of stress distribution across a mild steel flat plate with uniform cross section that is subject to a tensile force within the yield strength.

Assume that a short block as shown in fig 1 is acted upon by a concentrated load at its ends. By analysing the block for stresses as a two-dimensional problem, and using methods of the theory of elasticity gives the result as shown below in fig 1. By fixing strain gages that measure the deformations at a point close to the centre of the specimen and far away from the specimen, we can understand the way stress affects points close to the location of application of load and points far away from it (the centre).



 

As per S.Timoshenko and J.N. Goodier, the concentrated stress at a distance of quarter of a width from the point of application of the load is 2.575 times the average stress. This is illustrated in the figure below.

 

 

Fig 1: Stress distribution near a concentrated force - rectangular plate.

 

Since we have already shown strain to be proportional to stress, we can get a good idea about the magnitude of normal stress by examining the normal strain in a material as it is being subjected to some loads. To allow this, we can draw lines parallel to the normal plane and see if they remain plane during load application. In each of the following cases, witness how near the discontinuity there is a non-uniform distribution in the strain (and therefore stress) field, while farther away the distribution is linear (ie. the lines remain straight).

 

The Tension test

 

In St.Venant’s Principle experiment, we fix two strain gages, one near the central portion of the specimen and one near the grips of the Universal Testing Machine’s (UTM) upper (stationary) holding chuck. The UTM is then switched on and the specimen subjected to tensile load. The respective strain values obtained from both the gages are measured and then plotted with respect to time. Since stress is proportional to strain, as per St.Venant’s principle, the stress will be concentrated near the point of application of load. Although the average stress along the uniform cross section remains constant, at the point of application of load, the stress is distributed as shown in fig.1with stress being concentrated at the load point. The further the distance from the point of application of load, the more uniform the stress is distributed across the cross section. 

Specimen details:

Material : Mild steel

Specimen test section width: 31.5 mm

Specimen thickness: 3.4 mm

 

Fig 2: Strain gages fixed on the test specimen

Fig 3: Specimen fixed on the chuck of UTM

Fig 4: Adapter used to apply point load on the specimen

 

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