St. Venant’s Principle

St. Venant’s Principle

One of the most useful arguments in solid mechanics is one for which there is no precise analytic statement. St. Venant,** in discussion of beams, notes that:

“If a system of forces acting over ‘small’ areas of a solid are replaced by a statically equivalent system (same force and moment) then the stresses change ‘significantly’ only in the ‘neighborhood’ of the loaded region.”

One should appreciate that this ill-defined St. Venant principle is quite pro-found.* It is a physical statement of a fundamental property of the governing field equations of elasticity and the proper specification of boundary condi-tions for solution. It is certainly important since in the hands of an engineer “with good judgement,” it can be used to great effect to simplify and give insight to design. St. Venant’s principle, for example, is the reason why sim-ple strength-of-materials type analysis and design is so powerful in practice, and conversely, why more advanced elasticity solutions are often just local-ized. We will refer to the St. Venant principle often and the student can see it in evidence even more frequently if on the lookout for it.