Geometrically exact static isogeometric analysis of arbitrarily curved plane Bernoulli-Euler beam

Abstract

We present a geometrically exact nonlinear analysis of elastic in-plane beams in the context of finite but small strain theory. The formulation utilizes the full beam metric and obtains the complete analytic elastic constitutive model by employing the exact relation between the reference and equidistant strains. Thus, we account for the nonlinear strain distribution over the thickness of a beam. In addition to the full analytical constitutive model, four simplified ones are presented. Their comparison provides a thorough examination of the influence of a beam's metric on the structural response. We show that the appropriate formulation depends on the curviness of a beam at all configurations. Furthermore, the nonlinear distribution of strain along the thickness of strongly curved beams must be considered to obtain a complete and accurate response.

Geometrically exact static isogeometric analysis of an arbitrarily curved spatial Bernoulli-Euler beam

“…If the small-curvature beam model is utilized, the axial strain of the beam axis must also be zero. However, an accurate model, such as the one presented here, results in the dilatation of the beam axis, in a manner similar to that in [52]. The distribution of the axial strain and the normal force at the final configuration are given in Fig.…”

“…As the curviness increases, its influence becomes noticeable and a more involved model is required. A simple yet effective solution to improve the accuracy of small-curvature formulations should include the exact nonlinear distribution of strain and stress in the post-processing phase [52].…”

“…The boundary conditions are imposed in a well-known manner and the rotations are treated with special care since two components are not utilized as DOFs [45]. Components of external moments with respect to the principal axes of inertia are applied as force couples which must be updated at each iteration [52]. Standard Gauss quadrature with p + 1 integration points per element are used here.…”

“…On the other hand, the D 1 model results in zero dilatation and non-zero normal force. It should be noted that this model would result in zero normal force if the reduced constitutive matrix is used for the post-processing of the section forces [52]. Here, the full constitutive relation is utilized for the calculation of section forces [45].…”

“…Although known for a long time, it is only recently that the strongly curved beams have been scrutinized in the framework of the modern numerical methods. Linear analysis of strongly curved plane beams is given in [49,50,51], while nonlinear analysis is considered in [52]. The linear response of spatial beams is analyzed in [45].…”