Thin Extensional Beams under Large Deformations
Under the usual kinematical assumptions and plane loading the basic equations for an extensional beam including shearing deformation are established for the case of a semilinear material. The corresponding mixed variational formulation allows independent variations of six field variables. If the force equilibrium is satisfied a priori, there results a mixed stationarity principle with only two field variables: The cross section rotation angle and the couple. From this one complementary variational principles are derived and conditions for the existence of global bounds are given. Generally the solution is not unique. The following cases are observed: (a) JI = min, JII = max (stable equilibrium, global bounds); (b) JI = min, JII = stat (stable equilibrium, no global bounds); (c) JI = stat, JII = stat (no stable equilibrium, no global bounds). Here JI means the total potential energy and JII the total complementary potential. Some illustrative examples are worked out numerically using finite elements.
KeywordsStable Equilibrium Total Potential Energy Stiffness Ratio Dead Weight Extremum Principle
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