Semester Summary
from Advanced Mechanics of Materials, 5th ed, by AP Boresi, et al, Wiley and Sons, 1993, p 5-6.

The method of mechanics of materials is based on simplified assumptions related to the geometry of deformation so that strain distributions for a cross section of the member can be determined.  A basic assumption is that plane sections before loading remain plane after loading.  The assumption can be shown to be exact for axially loaded members of uniform cross sections, for slender straight torsion members having uniform circular cross sections, and for slender straight beams of uniform cross sections subjected to pure bending.  The assumption is approximate for other beam problems.

Saint-Venant's Principle

The following was stated by Barré de Saint-Venant in Mém. savants étrangers, vol. 14, 1855.

If the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally but has a negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed. ¹

"It has been found that the boundary conditions required in the solution of most problems cannot be met exactly in practice.  Stating this a little differently, it is, in general, rarely possible to apply the edge loadings in a manner such as to satisfy the stress systems at the boundary." ²  However, all is not lost.  Saint-Venant's principle states that we can use a more practical loading technique that is statically equivalent to the theoretical loading and obtain similar results at distances greater than the largest cross-sectional dimension from the load.  "This principle is not a rigorous law of mechanics but is a common-sense observation based upon theoretical and practical experience." ³  The principle can be extended to cover most stress concentrations, such as holes and grooves.  "It should be noted, however, that this rule does not apply to every type of member and loading case.  For example, members made from thin-walled elements, and subjected to loadings that cause large deflections, may create localized stresses and deformations that have an influence a considerable distance away from the point of application of loading." ⁴

 

axial compression examples

1. Theory of Elasticity, 2nd Ed., by S. Timoshenko and J.N. Goodier, McGraw-Hill, 1934, p. 33.
2. An Introduction to Experimental Stress Analysis, by G.H. Lee, John Wiley and Sons, 1950, p. 41.
3. Mechanics of Materials, 4th Ed., by J.M. Gere and S.P. Timoshenko, PWS Publishing, 1997, pp. 135-6.
4. Mechanics of Materials, by R.C. Hibbeler, Macmillan Publishing, 1991, p. 113.