An Efficient and Robust Nonlinear Dynamic Analysis Method for Framed Structures Using the Rigid Body Rule

In this paper, by implanting the rigid body rule (RBR)-based strategy for static nonlinear problems into the implicit direct integration procedure, an efficient and robustness nonlinear dynamic analysis method for the response of framed structures with large deflections and rotations is proposed. The implicit integration method proposed by Newmark is improved by inserting an intermediate time into the time step and by adding the 3-point backward difference in the second substep so as to preserve the momentum conservation and to maintain the stability of the direct integration method. To solve the equivalent incremental equations of motion, the RBR is built in to deal with the rigid rotations and the resulting additional nodal forces of element. During the increment-iterative procedure, the use of RBR-qualified geometric stiffness in the predictor reduces the numbers of iterations, while the elastic stiffness alone in the corrector to update the element nodal forces makes the computation efficiency and convergence with no virtual forces caused by the ill geometric stiffness. The proposed algorithm is advanced in the applications of several framed structures with highly nonlinear behavior in the dynamic response by its simplicity, efficient and robustness.

Determination of the first natural frequency of vibration of a steel pole, under the effect of geometric nonlinearity, using optimization techniques / Determinação da primeira frequência natural de vibração de um pólo de aço, sob o efeito da não linearidade geométrica, utilizando técnicas de optimização

This work aims to find a procedure to obtain an alternative formulation that represents the first mode of vibration of slender steel poles considering the effect of geometric non-linearity, using the Reyleigh-Ritz method, trigonometric formulations with optimization techniques and a finite element mathematical model. The application was on an existing polygonal steel pole. In order to consider the geometric non-linearity in the calculation of the natural frequencies of the respective structure, the concept of initial stiffness, geometric stiffness and effective stiffness, computed by the Rayleigh method for vibration problems in mechanical systems, was used. So, to optimize the computational time to obtain the modal response in dynamic analysis of the described structure, without neglecting the precision of the results of a rigorous analysis with sophisticated methodologies, alternative formulations to those described in NBR 6123 (1988) will bepresented in this work.

Highly Sensitive Nonlinear Identification To Track Early Fatigue Signs in Flexible Structures

This study describes a physics-based and data-driven nonlinear system identification approach for detecting early fatigue damage due to vibratory loads. The approach also allows for tracking the evolution of damage in real-time. Nonlinear parameters such as geometric stiffness, cubic damping and phase angle shift can be estimated as a function of fatigue cycles, which are demonstrated experimentally using flexible aluminum 7075-T6 structures exposed to vibration. Nonlinear system identification is utilized to create and update nonlinear frequency response functions, backbone curves and phase traces to visualize and estimate the structural health. Findings show that the dynamic phase is more sensitive to the evolution of early fatigue damage than nonlinear parameters such as the geometric stiffness and cubic damping parameters. A modifed Carrella-Ewins method is introduced to calculate the backbone from the nonlinear signal response, which is in good agreement with the numerical and harmonic balance results. The phase tracing method is presented, which appears to detect damage after approximately 40% of fatigue life, while the geometric stiffness and cubic damping parameters are capable of detecting fatigue damage after approximately 50% of the life-cycle.

A Superconvergent Isogeometric Method with Refined Quadrature for Buckling Analysis of Thin Beams and Plates

A superconvergent isogeometric method is developed for the buckling analysis of thin beams and plates, in which the quadratic basis functions are particularly considered. This method is formulated through refining the quadrature rules used for the numerical integration of geometric and material stiffness matrices. The criterion for the quadrature refinement is the optimization of the buckling load accuracy under the assumption of harmonic buckling modes for thin beams and plates. The method development starts with the thin beam buckling analysis, where the material stiffness matrix with quadratic basis functions does not involve numerical integration and thus the refined quadrature rule for geometric stiffness matrix can be obtained in a relatively easy way. Subsequently, this refined quadrature rule for thin beam geometric stiffness matrix is conveniently generalized to the thin plate geometric stiffness matrix via the tensor product operation. Meanwhile, the refined quadrature rule for the thin plate material stiffness matrix is derived by minimizing the buckling load error. It turns out that the refined quadrature rule for the thin plate material stiffness matrix generally depends on the wave numbers of buckling modes. A theoretical error analysis for the buckling loads evinces that the isogeometric method with refined quadrature rules offers a fourth-order accurate superconvergent algorithm for buckling load computation, which is two orders higher than the standard isogeometric analysis approach. Numerical results well demonstrate the superconvergence of the proposed method for the buckling loads corresponding to harmonic buckling modes, and for those related with non-harmonic modes, the buckling loads given by the proposed method are also much more accurate than their counterparts produced by the conventional isogeometric analysis.

Consistent co-rotational framework for Euler-Bernoulli and Timoshenko beam-column elements under distributed member loads

Conventional co-rotational formulations for geometrically nonlinear analysis are based on the assumption that the finite element is only subjected to nodal loads and as a result, they are not accurate for the elements under distributed member loads. The magnitude and direction of member loads are treated as constant in the global coordinate system, but they are essentially varying in the local coordinate system for the element undergoing a large rigid body rotation, leading to the change of nodal moments at element ends. Thus, there is a need to improve the co-rotational formulations to allow for the effect. This paper proposes a new consistent co-rotational formulation for both Euler-Bernoulli and Timoshenko two-dimensional beam-column elements subjected to distributed member loads. It is found that the equivalent nodal moments are affected by the element geometric change and consequently contribute to a part of geometric stiffness matrix. From this study, the results of both eigenvalue buckling and second-order direct analyses will be significantly improved. Several examples are used to verify the proposed formulation with comparison of the traditional method, which demonstrate the accuracy and reliability of the proposed method in buckling analysis of frame structures under distributed member loads using a single element per member.

Historical Trends in Alpine Ski Design: How Skis Have Evolved Over the Past Century

Alpine skis have changed dramatically in the last century. Long and straight wood skis have evolved into shorter lengths and now contain a plethora of modern materials. Shaped skis have become the norm. Today’s skis also offer a variety of waist widths and shapes to cater to specific uses. By studying how skis have evolved, it is possible to gain insight into how the design of alpine skis has progressed. To do so, the mechanical properties of 1016 skis, from the 1920s to 2019, were measured with a machine developed at the University of Sherbrooke. The resulting data are used to calculate various geometric, stiffness and performance parameters. The evolution of these parameters over the years is analyzed. This analysis provides a better understanding of the evolution of ski design and shows when the introduction of new materials and shaping concepts has changed the way skis are designed.