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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.

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 [

Mace [

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.

We will here consider the reflection of a stress pulse at the boundary of beams and strings of different diameters and materials (Figure _{0}. We denote the displacement of the reflected longitudinal wave by _{1}, the lateral displacement produced by the flexural wave by _{1} in beam 1, the longitudinal displacement in string 2 by _{2}, the lateral displacement in string 2 by _{2}, the longitudinal displacement in beam 3 by _{3}, and the lateral displacement in beam 3 by _{3}. The joint is modeled as a rigid body, and the internal forces are shown in Figure

The displacement and stress of the three-member intersection. (a) The displacement of the components. (b) The internal forces of the junction.

Continuity of the displacement of beam 1 and string 2 leads to

Continuity of the displacement of beams 1 and 3 leads to

The balance of the forces at the junction leads to_{1}_{1} and _{3}_{3} are bending stiffness of beams 1 and 3. _{1}_{1}, _{2}_{2}, and _{3}_{3} are axial stiffness of beams 1 and 3 and string 2. _{j} and _{j} are the moment of inertia and quality of the joint.

In the paper by Doyle and Kamle [

The displacements and strains can be written as_{n}, _{n}, and _{1}, _{2} are understood to be frequency dependent. If the FFT is taken of the strain history, then it can be approximated with a finite set of frequency components, that is,_{n} and _{n} at discrete frequency can be solved as follows:

This is different from that reported by Doyle and Kamle [

Finally, the incident and transmitted displacements are expanded in the forms:_{2} and _{2} are velocities of the longitudinal and flexural waves in string 2. The hypotheses that _{2} is small and perpendicular to the _{2} and _{2} can be expressed in the forms:

_{3} in (

For convenience, the joint is located at _{n} and _{n} which are given by the initial conditions. Due to the limitation of the space, the expression of the coefficients will not be given in this article.

According to the theory of Fourier transform, the arbitrary wave produced by vehicle vibration load can be explained as a superposition of the different frequency and amplitude cosine waves. In order to simplify calculation, consider a beam-string system subjected to a cosine wave. The length of the string and beam is 1 m. A square whose side length is 25 mm is used as a beam cross section, and a circle whose radius is 1 mm is used as a string cross section. The top of the string and the right of the beam are fixed. The load is applied in the left side of the beam. In order to simplify the process of calculation, a cycle of cosine wave is calculated in this article, as shown in Figure ^{2}, Poisson’s ratio = 0.3, and density = 7.8^{−9} t/mm^{3}. The string is made of the steel with Young’s modulus ^{2}, Poisson’s ratio = 0.3, and density = 7.8^{−9} t/mm^{3}.

A cycle of cosine load (a) and the finite element of the model (b).

The angle

Figures _{1}, _{2}, and _{3} are 23940 N, 405 N, and 23220 N by theoretical analysis and 20836 N, 328 N, and 20128 N by finite element analysis. It is noticed that the amplitude of the internal force by finite element analysis is slightly smaller than that by theoretical analysis. This is not surprising because of the loss in the process of the wave propagation and the error of the algorithm. It can be seen from the results of the comparison that the analytical solutions are in good agreement with finite element analysis prediction.

The internal force by two methods (

Figure _{2} exhibits a great relevance to the angle _{2} = 405, 352, and 212 N by theoretical analysis and 328, 268, and 148 N by finite element analysis. It can be concluded that the pulling force has a nonlinear relationship with the angle _{2} by theoretical analysis is close to that by finite element analysis. This can also prove the accuracy of the theoretical method on the other side.

Effect of _{1} by theoretical analysis. (b) _{1} by finite element analysis. (c) _{2} by theoretical analysis. (d) _{2} by finite element analysis. (e) _{3} by theoretical analysis. (f) _{3} by finite element analysis.

The energy density is another important research object in this paper. _{k}, _{p}, and _{2}, _{3}, _{4}, _{7}, and _{8}.

Figure _{1} has an unloading effect on the incident wave _{0}. It also can be observed that the energy density

The influence of the radius of the string on the energy density (_{0} − _{1}. (b) The energy density caused by the longitudinal wave _{2}. (c) The energy density caused by the flexural wave _{3}.

Referring to (_{1}, _{2}, and _{6} caused by the longitudinal wave _{3} are as follows:

Figure _{0} − _{1} decreases with the increase of the angle _{2} is opposite to the change in energy density _{3} is not influenced by angular change at all. It can be observed from Figures _{2} decreases with the increase of the angle. On the contrary, the energy density

The influence of _{0} − _{1}. (b) The energy density by the longitudinal wave _{2}. (c) The energy density by the longitudinal wave _{3}. (d) The energy density by the longitudinal wave _{1}. (e) The energy density by the flexural wave _{2}. (f) The energy density by the longitudinal wave _{3}.

The stress waves at a joint of a structure composed of the string and beam are investigated in this paper. The governing equations are established by displacement balance and internal force balance. The fast Fourier transform (FFT) numerical algorithm is used to deal with the incident wave of arbitrary wave type. The unknown coefficients of the reflected wave and transmitted wave are obtained by governing equations. The analytical solutions are compared with the finite element analysis by Abaqus program, and some parameters are analyzed in this paper. We can yield the following conclusions: first, the results of theoretical analysis are slightly larger than those of the finite element analysis, and the error is about 10 percent. Because the wave is dissipative in the finite element analysis, the result is reasonable, and the theoretical analysis is correct. Second, through the analysis of the influence of the angle

The authors declare that they have no conflicts of interest.

All the authors gratefully acknowledge the support of the National Natural Science Foundation of China (nos. 51408228 and 51378220).