Raffaele Barretta

Non-linear mechanics of elastic Bernoulli - Euler curved beams is investigated within 4-D spacetime geometric framework. Large displacements of such structures are addressed by novel rate elasticity methodology, avoiding finite deformation approaches and linearization of nonlinear kinematic relations common in literature. The variational rate virtual power principle (RVPP) is formulated and ensuing notion of effective stress rate leveraged. A rate form of nonlinear elasticity for planar curved beams is consequently established, and application of the RVPP yields the consistent elastic stiffness operator governing the structural evolution problem. Straightforward and computationally efficient incremental procedure is implemented in the brand new automatic program named REBUV. An enhanced version of the Riks - Crisfield arc-length method is developed by exploiting the derived expression of the stiffness operator and is successfully applied to the investigated highly nonlinear benchmark problems. Comparison with analytical finite deformation approaches and established computational methods is carried out, supporting effectiveness of the presented approach. This paper provides significant extension to curved beams of a methodology proved capable of solving problems and limitations affecting conventional analytical and computational approaches.

Non-Linear Mechanics of elastic Bernoulli-Euler curved beams is investigated within 4-D spacetime geometric framework. The concept of natural stress state is introduced for the considered structural model and the Virtual Power Principle (VPP) formulated. The associated Rate Virtual Power Principle (RVPP) is leveraged and ensuing notion of effective stress rate established, thus extending to slender curved beams special contributions in literature. Obtained theoretical outcomes are exploited to formulate nonlinear elasticity of planar Bernoulli-Euler structures by rate approach. Time and spatial discretizations are carried out to solve geometrically nonlinear beam problems as sequence of linear ones naturally derived by RVPP. Conveniently, no finite deformation measures are involved nor linearization needed in the proposed treatment. Computationally efficient incremental procedure is implemented in the automatic program referred to as REBUV. Emblematic case-problems are investigated by enhanced version of Riks-Crisfield arc-length method where necessary. Comparison with established computational approaches is carried out for purely elastic structures, supporting effectiveness of the presented approach. Outcomes highlight potential of the proposed methodology to solve nonlinear elasto-visco-plastic beams, thus by-passing known difficulties and limitations of finite deformation strategies.