Principles of Mechanics

The response of bodies to slowly varying loads is static or quasistatic. For slowly varying loads, the sum of all forces acting on any segment of a body are in balance since there are no accelerations; i.e. for any segment of the body, the resultant of tractions on the surface of the segment is equal in magnitude but opposite in direction to any external force acting on the segment. On the other hand, rapid changes in load cause the body to accelerate; in this case the resultants of stresses acting on any part of the body are related to the product of acceleration and inertia by the laws of motion. For these two broad classes of loading the differential equations for variations in stress (or stress resultant) across an arbitrary segment of the body are obtained from either equilibrium equations or the laws of motion. These laws form the basis of several useful principles. In this chapter special forms of these principles will be developed that apply to small deflection theory for planar displacements of slender bodies.

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