St Venant’s Principle

However, in practice, as we shall show in the next chapter it is not possible to apply such a uniform traction on the ends of a bar and the stress must diffuse into the bar. This causes an end effect. This was first examined for isotropic beams by St. Venant5) who considered the cases of uniaxial tension, of torsion and of bending. He proposed that the assumed uniform distribution of forces over any section within the specimen was a limiting state to which the forces in the real specimen approached, the further from the extremities of the specimen. This proposal has become known as St. Venant s principle and is often interpreted as implying that local eccentricities are not felt at distances... 

Exact solutions are, however, not easy to find nor are they usually manageable by non-mathematicians, and in consequence St. Venant s principle has led to a rule of thumb that if a specimen is about 10 diameters in length the stress under any form of end loading can be considered as uniform. There are several objections to the rule of thumb being taken as general practice. 

St. Venant touched on these in his original work and the validity of St. Venant s principle is normally assumed in tests using beams. Again, however, it should be stressed that in precise work the effects of the points of support should be assessed, since St. Venant s principle is not quantitative nor, as we have pointed out, is it valid in its conventionally stated form, when high anisotropy is present. 

St. Venant s principle See statistical equivalent loading system. 

Figure 4.1 A complex loading configuration can be simplified at large distances from the (shaded) contact region by St Venant s Principle.

The S5rmmetry of the single period of the configuration shown in Figure 3.9 has another important consequence. From overall equilibrium of the segment, it follows that the resultant force on any plane 2 = constant is also zero. In particular, the shear force vanishes identically on any such plane and, by St. Venant s principle, it follows that the local shear stress becomes negligibly small for depths beyond about 2p into the substrate. The absence of shear traction on a section of the substrate is one of the hallmarks of pure flexural deformation. This reasoning supports the use of cylindrical or spherical substrate curvature concepts to study the behavior of patterned surface Aims. 

For anisotropic materials, it is important to recognise that it is generally necessary to measure samples of very high aspect ratio to minimise end effects . These end effects arise from non-uniform stress conditions near the clamps that are much more severe than would be anticipated on the basis of St Venant s principle. The situation has been discussed in detail by Horgan [4,5] and by Folkes and Arridge [6]. 

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