Spatial free plastic forming of slender parts — fundamental relations and first example

Since plane air bending of slender parts, controlled only by the motion of both end supports has been established as a feasible metal forming process from the theoretical, experimental and technological point of view the fundamentals shall now be elaborated also for the corresponding spatial forming process. After presenting the basic kinematic relations the resultant forces and moments, acting on the material cross sections are introduced as generalized stresses producing work against the associated generalized strain rates. The equilibrium and the incremental compatibility conditions are formulated mutually with a generalized elastic–plastic constitutive law. Limitations to the forming process due to hurting the yield condition are also discussed. As a special event the deformation of the Bernoulli beam is examined, particularly due to moving yield hinges. The general theory is illustrated as the example of forming a helicyclic spiral.