This paper aims to present the exact closed-form solutions for the free vibration of double-beam systems composed of two parallel beams connected by an arbitrary number of discrete elastic supports. The general solutions of the mode shapes of the double-beam system are derived employing the Laplace transform method from a perspective of the entire domain of beams without enforcement of any segmentation. A unified strategy applied to various boundary conditions is proposed to determine the independent constants involved in the general solutions, as well as the frequency equation. Numerical calculations are performed to verify the present solutions by comparing the results from the previous literature and finite element simulation, and to discuss the effects of support parameters (stiffness, location, and number) on the modal characteristics of the double-beam system in detail. Outcomes show that the support location plays a pivotal role in regulating the modal characteristics of the double-beam system; for each-order mode, there are one or more potential optimal positions to maximize the effect of the elastic support. The mode veering phenomenon is detected as the support parameters change. It is highlighted that, by introducing an amplitude similarity index, the proximity degree for the mode shapes of the two beams influenced by the support parameters can be evaluated quantitatively. The present analysis is greatly helpful to the optimal design, health monitoring, and vibration control of the double-beam system. 

Exact closed-form solutions for free vibration of double-beam systems interconnected by elastic supports under axial forces

Adjacent tunnel construction and environmental disturbances can lead to longitudinal deformation in pipe-jacking tunnels. The longitudinal deformation of the tunnel is closely related to the occurrence of joint dislocation, joint opening, and other defects. In view of the difficulty of obtaining 3D longitudinal deformation curves, a method is proposed to obtain 3D longitudinal deformation curves based on a large number of 3D point cloud data with high spatial resolution and large spatial dimensions. Combined with the mechanism of defects occurrence, a theoretical basis for tunnel defects assessment based on tunnel longitudinal deformation is proposed. Taking one pipe jacking tunnel as an example, the longitudinal settlement curve and the 3D longitudinal deformation curve are compared. The correlation between the 3D longitudinal deformation curve and defects such as mud leakage, cracks, and differential deformation is illustrated from the perspective of three indexes: deformation amount, bending deformation, and shearing deformation. The accuracy and reliability of the 3D longitudinal deformation curve in tunnel defects detection and assessment are verified.

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A Novel Detection and Assessment Method for Operational Defects of Pipe Jacking Tunnel Based on 3D Longitudinal Deformation Curve: A Case Study

In this paper, an analytical solution is proposed for longitudinal dynamic responses of long tunnels under arbitrary excitations. For the derivation, the tunnel is assumed as a Timoshenko beam rest...

Pan Li

Papers

Analytical Solution for Longitudinal Dynamic Responses of Long Tunnels under Arbitrary Excitations