Advances in Civil Engineering

Volume 2018 (2018), Article ID 4863503, 10 pages

https://doi.org/10.1155/2018/4863503

## The Analysis of Stress Waves at a Junction of Beam and String

^{1}School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China^{2}School of Civil Engineering, Shijiazhuang Tiedao University, Shjiazhuang 050043, China

Correspondence should be addressed to Niujing Ma

Received 3 September 2017; Revised 14 November 2017; Accepted 19 December 2017; Published 7 March 2018

Academic Editor: Tayfun Dede

Copyright © 2018 Mu Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the bridge engineering, there are some problems about the dynamics that traditional theory cannot solve. So, the theory about stress waves is introduced to solve the related problems. This is a new attempt that the mechanic theory is applied to practical engineering. The stress wave at a junction of the structure composed of beams and strings is investigated in this paper. The structure is studied because the existence of a soft rope makes the transmission of the force in the bridge structure different from the traditional theory, and it is the basis for further research. The equilibrium equations of the displacement and the internal force are built based on the hypothesis. The fast Fourier transform (FFT) numerical algorithm is used to express an incident pulse of arbitrary shape. The analytical solutions are substantiated by comparing with the finite element programs. The conclusion that if the cross section of the string is relatively small, then the energy density of the structure is relatively large, which is disadvantageous to the structure, can be obtained from this paper.

#### 1. Introduction

Complex structures, such as arch bridge, suspension bridge, and cable-stayed bridge, are used increasingly in the world. Different from the traditional theory, the vehicle load is regarded as the vibration load in the bridge structure. The theory about stress waves is introduced to study the influence of the vibration load. The propagation speed of longitudinal waves and transverse waves in the concrete and steel is in the range of 1000 m/s to 5000 m/s, and in those large structures, the time scale of vehicle load and stress wave in balance are in the same order of magnitude. So, different from the traditional theory of structural dynamics that ignores stress wave propagation and interaction before the static equilibrium, the spread of the stress process is what we concern. Investigation of stress waves at a junction composed of beams and strings is the most important and foundational work.

Lee and Kolsky [1] investigate the generation of stress pulses at the junction of two noncollinear rods. They used the simple one-dimensional theory to describe the propagation of longitudinal waves and used the Timoshenko equation for flexural propagation. The analytic relations between the incident pulse and the four generated pulses are derived both for an incident longitudinal pulse and an incident flexural pulse. Atkins and Hunter [2] study the propagation of longitudinal elastic waves around right-angled corners in rods of square cross section. In their work, the rod is assumed to behave as a rigid body, and it was demonstrated that longitudinal elastic waves in rods of square cross section are transmitted in part around right-angled corners. Doyle [3] used the fast Fourier transform algorithm that allows the force history to be determined to establish a strain relation for the transverse impact of beams.

Mace [4] studied wave reflection and transmission in beams. In his paper, it is seen that incident near fields can give rise to substantial propagation components. Doyle and Kamle [5, 6] had taken an experimental study of the reflection and transmission of flexural waves at an arbitrary T-joint. It could be obtained from their study that the dynamic response is not sensitive to particular values of the joint model, and it depends primarily on the member arrangement and only secondarily on the particular shape and mass of the joint. Horner and White [7] predicted vibrational power transmission through bends and joints in beam-like structures. They built some models which determine the wave type which carries most power in each section of the system. By establishing the wave type, it is then possible to apply the most suitable vibration control technique. Sack [8] studied transverse oscillations in travelling strings. He found that the frequency response to a simple harmonic disturbance and the envelope of the oscillations are similar to those of a clamped string, but the phase of oscillation varies from point to point so that the instantaneous configuration of the string is not a sine curve. Little and Heywoodb [9] had described the axial stress waves in an elastic-perfectly-plastic bar. Different from the classical elastic solution, the load is greater than half the yield load in their study. The paper describes and verifies the model and presents the predictions that it gives for various levels of load. Chen [10] had thoroughly reviewed the research of transverse vibrations of axially moving strings and their control. Chen presents the governing equations with large amplitude and reviews progress on discretized or direct approximate analytical analyses and numerical approaches based on the Galerkin method or the finite difference method. Guo et al. [11] investigated the elastic wave and energy propagation in angled beams. Experimental and numerical methods were taken. It could demonstrate that, after several reflections from and transmission across the bends, energy is progressively smeared throughout the entire beam and does not concentrate at any particular segment. Li et al. [12] developed a fractal damage joint model based on the fractal damage theory to investigate the transmission and reflection of stress waves across joints. The fractal geometrical characteristics of joint surfaces are investigated by using a laser profilometer to scan the joint surfaces in their study. Lancioni et al. [13] studied the scattering of seismic waves induced by the presence of underground cavities in homogeneous soils by using the numerical method. Singh and Goya [14] established the governing equations of a transversely isotropic microstretch material, which are specialized in *x*-*z* plane. Plane wave solutions of these governing equations result in a biquadratic velocity equation. They found that, for a specific material, numerical simulation in the presence as well as in the absence of microstretch shows that the coupled longitudinal displacement (CLD) wave is the fastest wave and the coupled transverse microrotational (CTM) wave is the observed slowest wave. Piccolroaza et al. [15] presented a novel analysis of Floquet–Bloch flexural waves in a periodic lattice-like structure consisting of flexural beam ligaments. A special feature of this structure is in the presence of the rotational inertia, which is commonly neglected in conventional models of the Euler–Bernoulli type. Numerical examples were taken to illustrate directional localization, negative refraction, localization at an interface, and neutrality for propagating plane waves across a structured interface for a frequency range corresponding to a Dirac cone.

The previous studies focused on the distribution of the stress wave in the junction of the beams. According to the need of practical engineering, the distribution of the stress wave that was produced by transverse impact in the junction of the beam and string is the focus of the study in this paper.

#### 2. Theoretical Analysis

##### 2.1. Wave Propagation through the Junction

We will here consider the reflection of a stress pulse at the boundary of beams and strings of different diameters and materials (Figure 1(a)). There is an angle *β* between two beams and *α* between beam 1 and string 2. We call the flexural displacement due to the incident wave *v*_{0}. We denote the displacement of the reflected longitudinal wave by *u*_{1}, the lateral displacement produced by the flexural wave by *v*_{1} in beam 1, the longitudinal displacement in string 2 by *u*_{2}, the lateral displacement in string 2 by *v*_{2}, the longitudinal displacement in beam 3 by *u*_{3}, and the lateral displacement in beam 3 by *v*_{3}. The joint is modeled as a rigid body, and the internal forces are shown in Figure 1(b).