MS15.3 - Nonlinear Dynamics and Dynamic Stability

Frictional joints are present in a plethora of applications and fields, such as the aerospace, automotive, and building industries. Therefore, an ever-important challenge in the analysis of engineering systems is the understanding of friction damping in structural dynamics. Due to its nonlinear and non-smooth nature, the available approaches, including proposing alternative constitutive laws and validating models based on experimental data, cannot deal with the identification of friction forces. One way around this is to take advantage of the rapid increase of data availability through measurements for complex engineering systems, and newly developed identification techniques of the underlying differential equations of physical problems based on noisy measurements. A promising framework called Sparse Identification of Nonlinear Dynamics (SINDy) was developed for this purpose, aiming to derive parsimonious solutions for nonlinear systems. This framework has been recently further developed by combining the principles of dictionary-based learning, which is a key concept in SINDy, with numerical analysis tools, and more specifically the 4th-order Runge-Kutta integration scheme. This approach, the so-called RK4-SINDy was proven to be more efficient when dealing with noisy and sparsely collected data, not exploring though the incorporation of physics in discovering nonlinear models. In the current work, the incorporation of physics in RK4-SINDy is investigated for identifying the governing equations of a Single-Degree-of-Freedom (SDOF) oscillator under harmonic excitation, including Coulomb friction damping. In the proposed methodology, part of the system’s equation of motion is assumed known during the identification of the vector field of the global response, incorporating in this way part of the known physics. This simple, yet representative case study, is examined using both artificially generated noisy data, as well as data obtained from an experimental setup. It is shown that this approach can lead to accurate results even for significant noise levels, while maintaining a parsimonious solution to avoid overfitting the noisy data.

Large amplitude vibrations in slender structures tend to introduce nonlinearities in the structural response due to coupled bending and membrane stretching effect. This is generally observed as a hardening or a softening type nonlinearity which results in amplitude dependent shifts in modal frequencies [1]. These effects cannot be captured using the traditional eigenvalue analyses approach due to linearization approximations. Therefore, new methods are required to be developed in order to account for the nonlinear effects. For generality in application to various types of structures, it is convenient to have finite element (FE)-based formulations. Recently, an approach for model order reduction technique using a momentum subspace formulation [2] has been developed which is applicable to FE models. The geometric nonlinearity is introduced in the system using the von Karman kinematic model where higher order displacement derivatives are accounted for. The stiffness tensors are subsequently computed as up to fourth order derivatives of the total strain energy in the system. This results in an equation of motion (EOM) with quadratic and cubic nonlinearities. To obtain the reduced order model (ROM), an adaptation of the Koiter-Newton method [3] for post-buckling analyses is utilized. A quadratic mapping is used to correlate the real displacements to the reduced subspace. Eigenmode shapes obtained using the linear eigenvalue analysis are utilized in the reduction basis matrix to obtain the ROM. The EOM in the reduction subspace is transformed to a set of first order differential equations using the Hamiltonian formulation. A solution in the time domain is subsequently obtainable using numerical integration techniques. Alternatively, a frequency response curve is obtained using continuation algorithms [4]. Studies conducted using this approach on simple structures, such as rectangular plates, have demonstrated high accuracy and efficiency. Furthermore, experiments conducted on a stiffened plate to measure the nonlinear frequency response have been used to validate the numerical results. Comparisons show that for the chosen test cases, accurate results are obtainable using up to 2-DOF reduced order models. References: [1] Alijani, F., & Amabili, M. (2014). Non-linear vibrations of shells: A literature review from 2003 to 2013. International journal of non-linear mechanics, 58, 233-257. [2] Sinha, K., Singh, N. K., Abdalla, M. M., De Breuker, R., & Alijani, F. (2020). A momentum subspace method for the model-order reduction in nonlinear structural dynamics: Theory and experiments. International Journal of Non-Linear Mechanics, 119, 103314. [3] Liang, K., Abdalla, M., & Gürdal, Z. (2013). A Koiter‐Newton approach for nonlinear structural analysis. International journal for numerical methods in engineering, 96(12), 763-786. [4] Krauskopf, B., Osinga, H. M., & Galán-Vioque, J. (2007). Numerical continuation methods for dynamical systems (Vol. 2). Berlin: Springer.

In recent years there has been an increasing demand for housing in developing countries such as Colombia. Therefore, a significant number of mid-rise residential buildings with thin RC shear walls have been built for low-cost construction requirements. These buildings have been constructed in areas of high and intermediate seismic risk such as Bogotá, the country’s capital city, where they represent 32% of its building stock. The main characteristic of this system is the reduced thickness of the walls, ranging from 100 to 150 mm, which results in high slenderness and low reinforcement ratios, reflected in reduced construction cost. These led to the use of welded wire mesh (WWM), as the main reinforcement in the core of the wall, which is a material with low ductility. Experimental studies have shown that currently available WWM do not meet the requirements to be used as reinforcement in concrete structures requiring intermediate or special energy dissipation. The main concerns are related to the lack of ductility, insufficient experimental information, scarce evidence on its behavior during earthquakes and lack of clarity of design guidelines in modern seismic-resistant codes that could lead to inadequate behavior of these buildings. This research aims to provide evidence by analyzing the structural behavior of a 6-story prototype building with reinforced concrete walls, located in Bogotá, Colombia and designed according to the requirements of the Colombian code. The modeling consists in a three-dimensional computational representation of the building in OpenSees using the Multiple-Vertical-Line-Element-Model (MVLEM) where the walls are connected through a multi-point constraint between floor nodes. This approach allows the simulation of flexure-dominated RC wall behavior. The performance is numerically evaluated by pushover and nonlinear dynamic analyses; the structural parameters are obtained from available information of the prototype and Colombian materials. In the dynamic analysis, records were selected with acceleration spectrum that reproduce the seismic hazard for the city of Armenia, considering different hazard levels and with occurrence rates assigned using the Conditional Scenario Spectrum (CSS) methodology. Different performance levels of the structure were analyzed with the results of the static analysis and compared with those available in the literature. The results show that the performance of the buildings is adequate in terms of system over-strength, which reaches a value of 2.6. On the other hand, once the maximum capacity of the structure is reached, its capacity decreases rapidly due to mesh rupture, which shows the low ductility of this type of material.