Revisiting the rocking block: closed-form solutions and similarity laws

In this paper, the dynamic response of the rocking block subjected to base excitation is revisited. The goal is to offer new closed-form solutions and original similarity laws that shed light on the fundamental aspects of the rocking block. The focus is on the transient dynamics of the rocking block under finite-duration excitations. An alternative way to describe the response of the rocking block, informative of the behaviour of rocking structures under excitations of different intensity, is offered. In the process, limitations of standard dimensional analysis, related to the orientations of the involved physical quantities, are revealed. The proposed dimensionless and orientationless groups condense the response and offer a lucid depiction of the rocking phenomenon. When expressed in the appropriate dimensionless–orientationless groups, the rocking response becomes perfectly self-similar for slender blocks (within the small rotations range) and practically self-similar for non-slender blocks (larger rotations). Using this formulation, the nonlinear and non-smooth rocking response to pulse-type ground motion can be directly determined, and need only be scaled by the intensity and frequency of the excitation.

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Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of Compressible Flow Similarity Solutions

Journal of Verification Validation and Uncertainty Quantification ◽

10.1115/1.4030929 ◽

2015 ◽

Vol 1 (2) ◽

Cited By ~ 2

Author(s):

Scott D. Ramsey ◽

Philip R. Ivancic ◽

Jennifer F. Lilieholm

Keyword(s):

Shock Wave ◽

Wave Propagation ◽

Boundary Conditions ◽

Closed Form ◽

Compressible Flow ◽

Similarity Solutions ◽

Test Problems ◽

Closed Form Solutions ◽

Self Similar ◽

Temporal Extent

This work is concerned with the use of similarity solutions of the compressible flow equations as benchmarks or verification test problems for finite-volume compressible flow simulation software. In practice, this effort can be complicated by the infinite spatial/temporal extent of many candidate solutions or “test problems.” Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, self-similar shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing self-similar shock wave propagation as a piston-driven flow in the context of various test problems featuring simple closed-form solutions of infinite spatial/temporal extent. The closed-form solutions allow for the derivation of similarly closed-form piston boundary conditions (BCs) for use in Lagrangian compressible flow solvers. The consequences of utilizing these BCs (as opposed to directly initializing the self-similar solution in a computational spatial grid) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors and global convergence rates).

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Closed-Form Solutions for a Mode-III Moving Interface Crack at the Interface of Two Bonded Dissimilar Orthotropic Elastic Layers

Mathematical Problems in Engineering ◽

10.1155/2008/930820 ◽

2008 ◽

Vol 2008 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

B. M. Singh ◽

J. Rokne ◽

R. S. Dhaliwal

Keyword(s):

Closed Form ◽

Integral Transform ◽

Practical Importance ◽

Leading Edge ◽

Shear Stresses ◽

Mode Iii ◽

Closed Form Solutions ◽

Moving Interface ◽

Elastic Layers ◽

Physical Quantities

An integral transform technique is used to solve the elastodynamic problem of a crack of fixed length propagating at a constant speed at the interface of two bonded dissimilar orthotropic layers of equal thickness. Two cases of practical importance are investigated. Firstly, the lateral boundaries of the layers are clamped and displaced in equal and opposite directions to produce antiplane shear resulting in a tearing motion along the leading edge of the crack, and secondly, the lateral boundaries of the layers are subjected to shear stresses. The analytic solution for a semi-infinite crack at the interface of two bonded dissimilar orthotropic layers has been derived. Closed-form expressions are obtained for stressing the intensity factor and other physical quantities in all cases.

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Rocking Response of Seismically-Isolated Rigid Blocks Under Simple Acceleration Pulses and Earthquake Excitations

The Open Construction and Building Technology Journal ◽

10.2174/1874836801711010217 ◽

2017 ◽

Vol 11 (1) ◽

pp. 217-236

Author(s):

Panayiotis C. Roussis ◽

Spyroulla Odysseos

Keyword(s):

Dynamic Response ◽

System Performance ◽

Base System ◽

Rigid Block ◽

Isolation System ◽

Rocking Block ◽

Pulse Type ◽

Seismically Isolated ◽

The Stability ◽

Rocking Response

Background: Although the dynamic response of rigid block-like structures standing free on a rigid foundation has been extensively studied to date, only a limited number of studies have focused on the dynamics of such systems when seismically isolated. Objective: This paper presents a comprehensive investigation on the dynamic response of base-isolated rigid blocks subjected to pulse-type base excitation, with the aim of identifying potential trends in the response and stability of the system. Method: The model adopted in this study consists of a rectangular-prismatic rigid block standing free on a seismically-isolated base, which, on the assumption of sufficiently-large friction, can be set into rocking on top of the moving base under dynamic excitation. The study examines in depth the motion of the block/base system with a large-displacement formulation that combines the nonlinear equations of motion with a rigorous model governing impact. Two isolation-system models are utilized in the analysis, a linear viscoelastic model and a bilinear hysteretic model. Results: An extensive numerical investigation was performed to calculate the rocking response of the block under simple acceleration pulses and recorded pulse-type earthquake motions of various amplitudes and frequency content. Response-regime spectra for non-isolated and isolated blocks of varying geometric characteristics have been constructed to evaluate the system performance with respect to the rocking initiation and overturning of the block. Conclusion: The study showed that, regardless of block size and excitation period, seismic isolation increases the acceleration required to initiate rocking, a benefit that increases as the isolation period increases. In regard to the stability of the rocking block, the use of isolation yields a better system performance for smaller-sized blocks both for short- and mid-period excitations, provided that the isolation system is suitably designed. On the contrary, for long-period pulses, the use of isolation is practically not beneficial in improving the stability of the rocking block, irrespective of its size.

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