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

Abstract

The present research is focused on the linear analysis of a spatial Bernoulli-Euler beam. Metrics of the reference and deformed configurations are rigorously defined with respect to the convective coordinate frame of reference. No higher order terms are neglected which makes the formulation ideally suited for analysis of arbitrarily curved spatial beams in the frame of finite (but small) strain theory. The well-known issue of nonorthogonality of local coordinate system at an arbitrary point of a spatial beam is solved by the introduction of a new coordinate line that is orthogonal to the normal plane of the beam axis at each point. Generalized coordinates of the present model are translations of the beam axis and the angle of twist of a cross section. Two different parameterizations of this angle are discussed and implemented. Both geometry and kinematics are described with the same set of NURBS functions, in line with the isogeometric approach. Numerical analysis proved that the theoretical considerations are correct. An in-depth analysis of convergence properties has confirmed the fact that models with the highest interelement continuity have an improved accuracy per DOF, for problems that result in a smooth structural response.

On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli–Euler beam

“…The analogous convention is used for the designation of the base vectors which constitute the original basis and its reciprocal counterpart. For a more thorough treatment, the paper [2] is recommended.…”

“…The Christoffel symbols of the second kind associate the partial derivative of base vectors with the same set of these vectors: , , k i j ij k = Γ g g (6) and they are often represented in matrix form, [2]:…”

“…For spatial beams with variable curvature, this definition is more complex. One option is to use the value Kmaxhmax, where Kmax is the maximum value of the curvature of the beam axis while hmax is the maximum dimension of the cross section in the planes parallel to the osculating plane at the position of Kmax, [2]. In this way, Kmaxhmax∈(0,2) and it is referred here to as the (maximum local) curviness of a beam.…”

“…All the other cases belong to the category of big-curvature beams. The lack of benchmark test results for the recent developments of big-curvature beam model in the frame of isogeometric analysis (IGA), [2], [3], [4], and [5], was one of the main motives for the present research.…”

“…All these considerations are related to the big-curvature beam model, where no higher-order terms of the beam metric are neglected. These models are rarely examined within the analytic context and a few isogeometric numeric approaches have emerged lately, [2], [3], and [5]. Generally, IGA enables the high-accuracy calculation of curved beams and analytical results can serve for preliminary validation of these formulations, [36], [37], [38], [39], [40], [41].…”