# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 1 Introduction  
Among novel structural systems in civil engineering, coupled beams have increasingly been used in the last decades for highways, subgrades, and other facilities. This kind of system consists of beams with a straight or curved geometry, connected by an elastic layer, with a linear, non-linear, and plastic behavior. Despite the large amount of studies from the literature focusing on the structural behavior of single beams [1; 2; 3; 4], some static and dynamic aspects for coupled beam systems have already been explored in the pioneering works by [PERSON] and [PERSON] [5] and [PERSON] and [PERSON] [6], drawing increased attention from researchers in the recent years. These studies focused on static and dynamic behavior [7; 8; 9], transverse vibration [10], and forced vibration [11], using viscously damped interlayer [8; 12] among others. Among the most recent works, we can find studies focusing on the dynamic response of coupled curved beams, including both singly curved [13; 14; 15; 16] and doubly curved systems [17; 18; 19]. The earliest research on the vibrational behavior of beams resting on an elastic foundation was performed by [PERSON] [20; 21] and [PERSON] [22], involving one or two foundation parameters. Several researchers have recently been concerned with obtaining the natural frequencies of a curved beam with different geometries, loading, boundary conditions, and elastic foundations [23; 24; 25; 26; 27; 28; 29; 30]. Although few studies have assessed the vibration of homogeneous coupled beams [31; 32; 33; 34; 35; 36], research has focused on the effect of nanomaterials on the frequency behavior of these systems. Recently, [PERSON] et al. [37] have presented an exact solution to define the dynamic characteristics of double-beam systems using an improved Wittrick-Williams algorithm. The vibrational behavior of a double-beam systemconnected to a viscoelastic layer was also studied [38]. Additionally, numerical studies have been performed to determine the damping properties of a system using two beams [39]. Advanced nanomaterials and their application have also motivated many studies to investigate their possible application in civil (structural and geotechnical) engineering, to create more efficient and sustainable materials [40; 41]. Among them, single-layered hollow tubes are largely used as single-walled carbon nanotubes with diameter 0.4 to 2 mm [42]. Functionally graded materials (FGMs) have been also proposed for their high performance, capacity, and low weight with an overall improvement of the mechanical behavior of the structure due to a non-uniform distribution of the material along different directions. Among them, functionally graded carbon nanotubes (FG-CNTs) represent the most common type of nanocomposite used in many structural members, such as beams, plates, and shells [43; 44; 45; 46; 47; 48; 49; 50; 51]. [PERSON] et al. [52] investigated the dynamic response of double-beam systems in which CNTs were used as the reinforcing phase in the structure with different distribution patterns. In the further work [53], the authors focused on the natural frequency of FG-CNT reinforced double beam systems in a thermal environment, where FG-CNTs stand for the reinforcing phase. The dynamic behavior of nanocomposite coupled beam systems is still being considered by different researchers. In such a context, many scientists have continued to investigate the dynamic responses of structures over the past few decades, accounting for different analytical solutions such as GDQ and DSC [54; 55; 56; 57; 58]. The GDQ approach is a semi-analytical solution method with a fast convergence rate, as largely demonstrated in many applications of beams, plates, and shells [59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72] for different curved geometries, both in a static and dynamic sense. It is crucial to satisfy the boundary conditions accurately in order to obtain reliable and accurate solutions.