###### Abstract
[28] put the research emphasis on the stability and sound and vibration control of periodic shells conveying fluid, and proposed functionally-graded-material periodic shell models to eliminate or alleviate the stress concentration induced by the material or geometrical discontinuity of periodic composite structures [29]. Recently, [PERSON] et al. [30] examined the vibration attenuation properties of a LR pipe under impact excitation, and further designed a new AM pipe presenting the LR and Bragg scattering BGs simultaneously, which improved the broadband vibration BGs [31]. The present author also investigated the vibration suppression performance of a fluid-conveying PC pipe with axially deploying motion [32]. In essence, the PC theory is a multidisciplinary subject concerned with physics, mechanics, and materials. For the analysis of dynamical properties, a variety of theoretical approaches are available in the open literature, including the transfer matrix method (TMM) [21; 27; 31], plane wave expansion method (PWEM) [33; 34], spectral element method (SEM) [35; 36], homogenization method (HM) [37; 38] and finite element method (FEM) [28; 29; 39], etc. Among these methods, the TMM is widely used in most of the studies on the periodic pipes conveying fluid. However, although the TMM can calculate the BG regions, numerical ill-conditioning often occurs when dealing with the frequency response, and the natural properties, such as natural frequency and vibration mode, can be hardly obtained using this procedure. In contrast, the SEM carried out in this paper possesses the capability of treating these problems. Actually, the SEM has been extensively applied in uniform beams [40; 41]. Some research work was also concerned with uniform pipes conveying fluid [42; 43]. For PC and AM structures, [PERSON] et al. [44] combined the SEM and PC theories to study the flexural wave attenuation of LR periodic beams. [PERSON] et al. [45] developed the SEM to investigate the BG properties of periodic rigid frames composed of Timoshenko beams. [PERSON] et al. [46] examined the effects of slowly varying spatial properties on the vibration suppression performance of PC and AM beams via the SEM. [PERSON] and [PERSON] [47] established the SE model of a PC water-filled circular cylindrical shell. Whereas, to the knowledge of the authors, the SEM has not been applied for analyzing the vibration and wave characteristics of periodic FSI structures, especially their natural properties. In terms of the foregoing review, it is found that the periodic pipe models adopted in most literatures are associated with periodically varying materials. Actually, although this type of structure can be realized theoretically, it is technically difficult to produce such thin-walled composite, as mentioned previously. Moreover, there is lack of a comprehensive study, especially on the natural properties of such periodic FSI structures. In view of this, in this paper, a fluid-conveying pipe with periodically varying cross-section is proposed, and the BG characteristics and free vibration of such PC structure are investigated. The BG distribution and frequency response are investigated, and the natural frequencies and mode shapes are studied. The effects of some critical parameters are discussed in detail.