Elastic Stability of Concrete Beam-Columns

2017 ◽  
Vol 17 (01) ◽  
pp. 1750095 ◽  
Author(s):  
Mamta R. Sharma ◽  
Arbind K. Singh ◽  
Gurmail S. Benipal
Keyword(s):  
Dynamic Stability ◽  
Concrete Beam ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Damping Ratio ◽  
Elastic Stability ◽  
Lumped Mass ◽  
Nonlinear Dynamical ◽  
Beam Columns ◽  
Concrete Beam Columns

In the first part of this paper, elastostatic stability of cracked conservative flanged concrete beam-columns has been analyzed. Using the derived expression for the lateral stiffness under constant axial force, their elastodynamic stability is investigated in this second part. As expected, the instantaneous values of the stiffness and the damping coefficients of the lumped-mass underdamped SDOF nonlinear structures are found to depend upon the vibration amplitude. The natural frequency has been found to vanish at the two critical axial loads defined in the first part. For axial load exceeding the second critical value, the concrete beam-columns in the second equilibrium state are shown to exhibit loss of dynamic stability by divergence. Depending upon the initial conditions, the phase plane has been partitioned into dynamically stable and unstable regions. Under harmonic excitations, the nonlinear dynamical systems exhibit subharmonic resonances and jump phenomena. Loss of dynamic stability has been predicted for some ranges of damping ratio as well as of peak sinusoidal force and forcing frequency. Sensitivity of dynamic stability to the initial conditions and the sense of the peak sinusoidal force have also been predicted. The theoretical significance and the methodology adopted in this paper are also discussed.

Elastic Stability of Concrete Beam-Columns

2017 ◽  
Vol 17 (01) ◽  
pp. 1750094
Author(s):  
Mamta R. Sharma ◽  
Arbind K. Singh ◽  
Gurmail S. Benipal
Keyword(s):  
Dynamic Stability ◽  
Concrete Beam ◽  
Reference Frames ◽  
Lateral Displacement ◽  
Critical Value ◽  
Critical Values ◽  
Buckling Mode ◽  
Beam Columns ◽  
Lateral Forces ◽  
Concrete Beam Columns

Following the empirical-computational methodology, the contemporary investigations deal with inelastic stability and dynamics of concrete beam-columns. Even under service loads, the concrete structures exhibit physical nonlinearity due to presence of axio-flexural cracks. The objective of the present paper is to analyze the static and dynamic stability of conservative physically nonlinear fully cracked flanged concrete beam–columns. In this paper, using proper reference frames, analytical expressions are developed for the lateral displacement and stiffness of a flanged concrete cantilever under axial compressive and lateral forces. Two critical values of both the axial and lateral loads are identified. For constant lateral force smaller than its first critical value, the concrete beam–columns exhibit brittle buckling mode. Higher lateral forces lesser than the second critical value introduce alternate stable and unstable domains with increase in axial force. The lateral stiffness is predicted to vanish when the axial loads reach the critical values and when the limiting displacement is reached for axial load exceeding its second critical value. The load-space is partitioned into stable and unstable regions. Accessibility of these equilibrium states in the load space has been investigated. Such distinguishing aspects of the predicted behavior of elastic concrete beam–columns are discussed. Their dynamic stability is investigated in second part of the paper.

scholarly journals Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

2017 ◽  
Author(s):  
Geoff Boeing
Keyword(s):  
Dynamical Systems ◽  
Visual Analysis ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Systems Of Nonlinear Equations ◽  
Self Similarity ◽  
Nonlinear Dynamical ◽  
Qualitative Approaches ◽  
Divergent Behavior ◽  
Behavior Systems

Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior.

scholarly journals iLQR-VAE : control-based learning of input-driven dynamics with applications to neural data

2021 ◽  
Author(s):  
Marine Schimel ◽  
Ta-Chu Kao ◽  
Kristopher T. Jensen ◽  
Guillaume Hennequin
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Linear Quadratic Regulator ◽  
External Input ◽  
Learning Approaches ◽  
Linear Quadratic ◽  
Nonlinear Dynamical ◽  
Neural Data ◽  
Recognition Model ◽  
Variational Autoencoder

Understanding how neural dynamics give rise to behaviour is one of the most fundamental questions in systems neuroscience. To achieve this, a common approach is to record neural populations in behaving animals, and model these data as emanating from a latent dynamical system whose state trajectories can then be related back to behavioural observations via some form of decoding. As recordings are typically performed in localized circuits that form only a part of the wider implicated network, it is important to simultaneously learn the local dynamics and infer any unobserved external input that might drive them. Here, we introduce iLQR-VAE, a novel control-based approach to variational inference in nonlinear dynamical systems, capable of learning both latent dynamics, initial conditions, and ongoing external inputs. As in recent deep learning approaches, our method is based on an input-driven sequential variational autoencoder (VAE). The main novelty lies in the use of the powerful iterative linear quadratic regulator algorithm (iLQR) in the recognition model. Optimization of the standard evidence lower-bound requires differentiating through iLQR solutions, which is made possible by recent advances in differentiable control. Importantly, having the recognition model implicitly defined by the generative model greatly reduces the number of free parameters and allows for flexible, high-quality inference. This makes it possible for instance to evaluate the model on a single long trial after training on smaller chunks. We demonstrate the effectiveness of iLQR-VAE on a range of synthetic systems, with autonomous as well as input-driven dynamics. We further show state-of-the-art performance on neural and behavioural recordings in non-human primates during two different reaching tasks.

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