[Paper Review] Response of 3D Free Axisymmetric Rigid Objects under Seismic Excitations

Yanheng Li, Baoping Shi|ArXiv.org|Jul 14, 2008

Seismic Performance and Analysis12 references3 citations

TL;DR

This study presents a 3D dynamic model for axisymmetric rigid bodies under seismic excitation, deriving three second-order ODEs to simulate rocking and rolling responses. Using the El Centro earthquake record, it shows that 2D simplifications fail to capture critical vertical (UD) and NS component effects, challenging prior assumptions about vertical ground motion being less influential in overturning risk.

ABSTRACT

Previous studies of precariously balanced objects in seismically active regions provide important information for aseismatic engineering and theoretical seismology. They are almost founded on an oversimplified assumption: any 3-dimensional (3D) actual object with special symmetry could be regarded as a 2D finite object in light of the corresponding symmetry. To gain an actual evolution of precariously balanced objects subjected to various levels of ground accelerations, a 3D investigation should be performed. In virtue of some reasonable works from a number of mechanicians, we derive three resultant second-order ordinary differential equations determine the evolution of 3D responses. The new dynamic analysis is following the 3D rotation of a rigid body around a fixed point. A computer program for numerical solution of these equations is also developed to simulate the rocking and rolling response of axisymmetric objects to various levels of ground accelerations. It is shown that the 2D and 3D estimates on the minimum overturning acceleration of a cylinder under the same sets of half- and full-sine-wave pulses are almost consistent except at several frequency bonds. However, we find that the 2D and 3D responses using the actual seismic excitation have distinct differences, especially to north-south (NS) and up-down (UD) components. In this work the chosen seismic wave is the El Centro recording of the 18 May 1940 Imperial Valley Earthquake. The 3D outcome does not seem to support the 2D previous result that the vertical component of the ground acceleration is less important than the horizontal ones. We conclude that the 2D dynamic modeling is not always reliable.

Motivation & Objective

• To develop a 3D dynamic model for free-axisymmetric rigid bodies under seismic excitation, moving beyond 2D simplifications.
• To investigate the reliability of 2D modeling assumptions in predicting overturning behavior under real seismic inputs.
• To assess the relative influence of horizontal (NS, EW) and vertical (UD) ground acceleration components on overturning risk.
• To validate the 3D model using actual seismic data, particularly the El Centro 1940 earthquake record.
• To determine whether 2D approximations consistently underestimate or misrepresent the true dynamic response of precariously balanced objects.

Proposed method

• Derives three second-order ordinary differential equations (ODEs) governing 3D rotational dynamics of a rigid body about a fixed point, based on principles from classical mechanics.
• Applies the equations to simulate the rocking and rolling response of axisymmetric objects under various ground acceleration levels.
• Uses numerical integration via a custom computer program to solve the ODEs for both synthetic half- and full-sine-wave pulses and real seismic input (El Centro 1940 earthquake).
• Compares 3D simulation results with 2D approximations for the same input, focusing on minimum overturning acceleration thresholds.
• Analyzes the response to individual components (NS, EW, UD) of the El Centro ground motion to isolate directional effects.
• Validated the model using known physical principles and benchmarked results against simplified 2D models under controlled pulse inputs.

Experimental results

Research questions

• RQ1 How do 3D dynamic responses of axisymmetric rigid bodies differ from 2D approximations under identical seismic pulse inputs?
• RQ2 To what extent do the north-south (NS) and up-down (UD) components of real seismic ground motion influence the overturning of 3D rigid bodies compared to horizontal components?
• RQ3 Does the 2D simplification reliably predict the minimum ground acceleration required to overturn a 3D axisymmetric object?
• RQ4 Are there specific frequency ranges where 2D and 3D models diverge significantly in their overturning predictions?
• RQ5 Is the vertical ground acceleration component less significant in causing overturning, as previously assumed in 2D models?

Key findings

• For synthetic half- and full-sine-wave pulses, 2D and 3D models yield nearly identical minimum overturning accelerations, except at specific frequency bonds where discrepancies emerge.
• With real seismic input (El Centro 1940), the 3D model reveals significant differences in response compared to 2D approximations, particularly in the influence of NS and UD components.
• The 3D analysis shows that the vertical (UD) component of ground acceleration is not negligible and contributes meaningfully to overturning risk, contradicting prior 2D-based conclusions.
• The NS component exhibits a stronger influence in 3D simulations than predicted by 2D models, indicating directional coupling effects not captured in 2D.
• The 2D modeling approach is not universally reliable, especially when applied to complex, real-world seismic records with multi-component excitation.
• The developed 3D numerical model successfully captures the full rotational dynamics of axisymmetric rigid bodies, enabling accurate simulation of rocking and rolling under seismic loading.