Although the nonlinear dynamics of hanging cantilevered pipe conveying fluid ha ve been extensively scrutinized, a few investigations exist about the nonlinear behavior of standing one . Hence, the objective of this study is to examine the geometrically exact nonlinear static and dynamic responses of standing cantilevered pipe conveying fluid. The geometrically exact rotation -based model, along with the shooting method and Galerkin technique is applied to assess the nonlinear static behavior of system and its stability characteristics. Moreover , to compute the nonlinear dynamics of system, the geometrically exact quaternion -based model , together with the Galerkin technique is employed. It is revealed that the system may undergo buckling via a supercritical or subcritical pitchfork bifurcation depending on the gravity parameter , which may give rise to extremely large -amplitude responses . The system may also experience flutter instability via a supercritical Hopf bifurcation, which bring s about self-excited perio dic oscillation s. The generic behavior of system for a specific range of the gravity parameter is investigated for four distinct scenarios depending on the gravity parameter and mass ratio , which only one of them is similar to that comes to pass for the hanging case . Keywords Standing c antilevered pipe conveying fluid; Geometrically exact nonlinear analysis; Subcritical and supercritical pitchfork bifurcation ; Supercritical Hopf bifurcation ; Extremely large amplitude ; Self-excited periodic oscillation. 1. Introduction Pipes conveying fluid as a kind of fluid -solid interaction system have drawn noteworthy research interest due to their applications in numerous engineering systems [1-6]. It is well -known that fluid -conveying pipes supported at both ends undergo buckling instability at a sufficiently high flow velocity [3, 7] and their nonlinear dynamic behavior has been widely analyzed [8-16]. According to the existing literature, the t heoretical and experimental studies on the flutter instability of cantilevered pipe s conveying fluid first appeared in the papers by Bourrières [17], Benjamin [18, 19] , and Païdoussis and Gregory [20, 21] . Baja et al. [22, 23] developed a nonlinear mathematical model to study the bifurcation analysis of cantilevered pipe conveying fluid and some interesting features of system were reported in their studies . However, their nonlinear 1 Email addresses: a.m.dehrouye@ut.ac.ir , a.m.dehrouye@gmail.com (A.M. Dehrouyeh -Semnani) 2 mathematical model has some incomplete terms which are discussed in detail in Ref. [3]. Semler et al. [24] established a third -order approximation nonlinear mathematical model for planner motion of cantilevered pipe conveying fluid , and Wadham -Gagnon et al. [25] developed it for three -dimensional motion . In m any studies related to the cantilevered pipe conveying fluid , the third -order approximation model or its modified versions (due to added elements , imperfect support , materi al properties, and so on ) have been utilized to study the nonlinear dynamics of system [26-35]. Since the third -order approximation model was derived [24] based on the inextensibility assumption, Ghayesh et al. [36] developed a new approximation model, which the third -order approximation model can be recovered from it, to investigate the role of extensibility in the nonlinear dynamics of cantilevered pipe conveying fluid. Recently, the geometrically exact modeling and analysis of cantilevered pipe conveying fluid have drawn notable research interest due to the limitation of the third -order approximation model in the simulation of large -amplitude responses of system . The geometrically exact models for dynamic analysis have been developed based on four different approaches . In the first approach, a geometrically exact model based on the Cosserat rod model and applying balance laws for the linear and angular momentum using an intrinsic orthogonal framework [37]. In the second approach, a geometrically exact model based on the total moment of the fluid and pipe was proposed [38]. The fully nonlinear differential governing equations of system were established according to the momentum balance of pipe and fluid in floating non -inertial frames. The third approach is related to developing a geometrically exact model in terms of the rotation angle based on Hamilton’s principle where the third -order approximation model can be recovered from this model [39-45]. The fourth approach is associated with the derivation of a geometrically exact model based on the quaternion formulation , along with Hamilton’s principle [46]. It was indicated that the rotational -based model developed in Ref. [39] can be recovered from the newly developed model. A major difference between this model and that developed in Ref. [39] is that its governing equations are expressed in terms of integro -partial differential algebraic equations while the governing equation of the model presented in Ref. [39] is an integro -partial differential equation . The comparison studies indicated that the quaternion -based model is capable of capturing the nonlinear dynamics of system with a remarkable reduction in computational cost compared to the rotation -based model . The nonlinear dynamics of standing cantilevered pipe conveying fluid have been investigated in a few studies. Li and Paidoussis [47] examined the nonlinear dynamics of standing cantilevered pipe conveying fluid via the third -order approximation model, together with the Galerkin technique with two modes. Prior to applying Galerkin’s technique, the non -linear inertia terms were replaced by equivalent ones through a perturbation procedure. The focus of this study was on the stability and double degeneracy , where the subcritical pitchfork bifurcation and Hopf bifurcation take place simultaneously , as well as on the chaotic motion of system . Wang and Ni [48] studied the stability and chaotic motions of standing cantilevered pipe conveying fluid with elastic support and motion -limiting constraints . In the mathematical model, the source of 3 nonlinearity was the support and constraints, and a linear model was adopted for the pipe conveying fluid. Similar to the previous reference , Galerkin’s technique with two modes was used to analyze the system. Bou-Rabee et al. [49] studied the stability characteristics of standing cantilevered pipe conveying fluid using a simple geometrically exact model . To construct the mathematical model, they supposed that the fluid -structure interaction induces a concentrated force and a point source of damping, respectively, tangent and normal to the free end. A long, thin conduit with a nozzle at its free end can be an appropriate representation for this model. Chen et al. [50] analyzed the nonlinear behavior of standing cantilevered pipe conveying fluid based on the geometrically exact rotation -based model for a specific case (see Appendix A ). The review of the papers related to the nonlinear analysis of cantilevered pipes conveying fluid shows that the nonlinear dynamics of standing system have not been disclosed appropriately, unlike the hanging and horizontal system s. Therefore, the focus of this paper is on exploring the geometrically exact nonlinear static and dynamic behavior of standing cantilever ed pipe conveying fluid. In Section 2 , two different mathematical models utilized to acquire the geometrically exact nonlinear responses of system are explained in detail . As well, the solution strategies for determining the static, stability, and dynamic behavior of system are elucidated. In Section 3 , the available results in the existing literature related to the geometrically exact nonlinear dynamics of standing cantilevered pipe conveying fluid are employed to verify the mathematical model and solution strategy used in this paper for dynamic analys is. A convergence study on the verified results is also presented in this section . In Section 4 , a geometrically exact nonlinear analysis o f the generic behavior of system is carried out. Firstly, the stability of undeformed configuration of system without flow velocity due to gravity , along with its stable positions is studied. Subsequently, the influence of flow velocity, together with the gravity parameter on new static responses and their stability are examined. The role of flow velocity, along with the gravity parameter and mass ratio in the dynamic instability and correspon ding motion characteristics is also investigated. A generic behavior for four distinct scenarios is discussed in detail. Additionally, the i mpact of initial conditions on the final response of system is explored. Eventually, the summary of findings is presented in Section 5 . 2. On m athematical model s and solution strateg ies A schematic representation of a standing cantilevered pipe conveying fluid is depicted in Figure 1. The system under investigation includes a pipe of length 𝐿, cross -sectional area 𝐴, flexural rigidity 𝐸𝐼, mass per unit length 𝑚, conveying a fluid of mass per unit length 𝑀 with flow velocity 𝑈̅. 4 Figure 1: Schematic representation of a standing cantilevered pipe conveying fluid To assess the geometrically exact static responses of system, the geometrically exact rotation - based model (GERM) is used. The reason for choosing this model is that it can easily capture the static responses of system in conjunction with the shooting scheme. However, their stability is determined by Galerkin’s technique. T he geometrically exact quaternion -based model (GEQM) , along with the Galerkin technique is implemented for the dynamic analysis . The advantage of this model compared to the former model i s its low computational cost. It should be pointed out that the mathematical model s of hanging and standing cantilever ed pipe conveying fluid are identical except for the gravity parameter. In the case of the hanging pipe, the gravity parameter is positive , but this parameter is negative for the standing pipe. Thus , the geometrically exact mathematical models of hanging cantilevered pipe conveying fluid developed in the previous studies can be utilized for the standing one. Static and stability analysis via GERM 2.1. Static and stability analyses via GERM The dimensionless form of geometrically exact equation of motion and corresponding boundary conditions of cantilevered pipe conveying fluid in terms of rotation angle, 𝜃, were established as follows [39, 40, 50] 5 ()() ()() ()()()() ( ) ()()()() ( ) ()()()() ( )2 ˆ 2 10 ˆ 2 10 111 sin sin 1 ˆ cos sin cos ˆ sin cos sin ˆˆ 2 sin sin cos cos 0,ss ss sssU ds ds ds ds U ds ds                       + − − + − +− −+ − + =   (1) ()() 0 0, 1 0, == (2) where the dimensionless parameters are ()()()1/2 * *3 4, , = , , U = U, = , ,s EI M M M ms t gLL M m L EI M m EI    + == ++  (3) in which 𝑠 is dimensionless distance along the pipe from the clamped end , 𝜁 and 𝜂, respectively, are dimensionless axial and transverse displacements , 𝜏 is dimensionless time, 𝛼 is dimensionless damping coefficient, 𝑈 is dimensionless flow velocity, 𝛽 is mass parameter, and 𝛾 is gravity parameter. Moreover, the prime and dot symbols denote derivative s with respect to 𝑠 and 𝜏, respectively. The dimensionless axial and transverse displacements, respectively, can be obtained by [39] ()() ()()00ˆ ˆ ˆ ˆ cos 1, sin .ss uvs ds s dsLL   = = − = = (4) The geometrically exact rotation -based static model can be achieved by dropping the time - dependent terms in Eq. (1) [50]. ()() ()()21 sin sin 1 0,s s s s sU    + − + − = (5) in which 𝜃𝑠 stands for static rotation angle. Similar to Ref. [51], the shooting scheme is used to evaluate the geometrically exact responses of system based on the static equation given in Eq. (5). To determine their stability, the linearized dynamic model around the static responses is employed. Substituting 𝜃+𝜃𝑠 instead of 𝜃 in Eq. (1), incorporating Eq. (5), and neglecting nonlinear terms in the resulting equation , one can obtain the following linear dynamic equation [50]. ()() ()()()() () () ()() ()()()() ( )2 1 0 1 0 111 sin 1 cos 1 ˆ cos cos sin sin ˆˆ 2 sin sin cos cos 0,s s s s s s ss s s s sss s s ssU ds ds ds ds U ds ds                      + − − + − −     + − −       − + =     (6) In view of Galerkin’s technique, the rotation angle is approximated as follows 1,N nn np == (7) 6 in which 𝑁, 𝜑𝑛 and 𝑝𝑛 stand for number of approximation function s, 𝑛th approximation function , and 𝑛th generalized coordinate, respectively. Similar to Ref. [50], 𝜑𝑛=𝑠𝑖𝑛((2𝑛−1)𝜋𝑠2⁄). It should be mentioned that to calculate the stability characteristics of system around its undeformed configuration, it is enough to set 𝜃𝑠=0 in Eq. (6). 2.2. Dynamic analysis via GEQM The geometrically exact mathematical model of cantilevered pipe conveying fluid in the quaternion system was derived in Ref. [46]. The quaternion elements 𝑒0 and 𝑒3 can be written in terms of the rotation angle as follows [46]: 𝑒0=𝑐𝑜𝑠(𝜃2⁄) and 𝑒3=𝑠𝑖𝑛(𝜃2⁄), along with the following constraint: Φ=𝑒02+𝑒32=1. The resulting dimensionless form of this mathematical model is [46] () ( )()() ()() ( ) ( ) () ()2 2 0 3 0 3 3 0 3 0 3 3 0 ˆˆ 22 0 0 0 0 3 3 3 3 0 3 0 3 3 01 0 1 0 0 0 0 3 3 3 3 0 3 0114 2 2 2 1 ˆˆ 2 2 2 2 2 ˆˆ 2 2 2 2 2s s s s sse e e e e e e e e e e s e e e e e e e dsds e e e e e e e dsds U e e e e e ds e e e e e ds      − − + + −    + + − − + + +          + − + +           ()()( ) ( )2 2 2 0 0 3 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 3 0 3 0 3 3 0 3 3 3 0 3 3 0 3 0 3 3 0 3 02 (1) (1) 2 2 (1) (1) 8 4 2 0, U e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e          − − + − − + − − −               + + + − − − − − = (8) () ( )()() ()() ( ) ( ) () ()2 2 0 3 0 0 3 0 3 0 0 3 3 ˆˆ 22 3 0 0 0 3 3 3 0 0 3 0 3 3 01 0 1 0 3 0 0 3 3 0 3 0 3 0114 2 2 2 1 ˆˆ 2 2 2 2 2 ˆˆ 2 2 2 2 2s s s s sse e e e e e e e e e e s e e e e e e e dsds e e e e e e e dsds U e e e e e ds e e e e e ds      − − + − −    − + − − + + +          − − − +           ()()( ) ( )2 2 2 3 0 3 0 0 3 0 3 0 0 0 3 0 3 0 0 0 3 0 3 0 0 3 0 0 0 3 0 0 3 0 3 3 0 3 0 0 0 3 0 0 3 32 (1) (1) 2 2 (1) (1) 8 4 2 0, U e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e          − − − + − + − − +               + + + − − − − − = (9) 23 00 1, ee+= (10) 0 3 0 3(0) 1, (0) 0, (1) 0, (1) 0,e e e e  = = = = (11) in which 𝜆 is dimensionless Lagrangian multiplier function. The dimensionless axial and transverse displacements in terms of the quaternion elements, respectively, can be written [46] ()22 0 3 0 300ˆˆ1, 2 .sse e ds e e ds= − − = (12) The mathematical model of system in terms of the rotation angle is an integro -partial differential equation (Eqs. (1)-(2)), but it is an integro -partial differential algebraic equation in terms of the quaternion elements (Eqs. (8)-(11)). In Ref. [46], it was proved that Eqs. (1)-(2) can be recovered 7 from Eqs. (8)-(11). Furthermore, it was discussed that the quaternion -based model significantly reduces the computational cost compared to the rotation -based model. In view of Galerkin’s technique, the quaternion elements and the dimensionless Lagrangian multiplier function are approximated as follows 03 03 1 1 11 , , eeNN N n n n n n n n n ne p e q r     = = == + = =   (13) where 𝑁𝑒0, 𝑁𝑒3, and 𝑁𝜆 are number of approximation functions for 𝑒0, 𝑒3, and 𝜆, respectively ; 𝜙𝑛, 𝜓𝑛, and 𝜒𝑛 are 𝑛th approximation function for 𝑒0, 𝑒3, and 𝜆, respectively; 𝑝𝑛, 𝑞𝑛, and 𝑟𝑛 are 𝑛th generalized coordinate of 𝑒0, 𝑒3, and 𝜆, respectively. Similar to Ref. [46], 𝑁𝑒0=𝑁𝑒3=𝑁𝜆=𝑁, 𝜙𝑛=𝜓𝑛=𝑠𝑖𝑛((2𝑛−1)𝜋𝑠2⁄) and 𝜒𝑛= 𝑠𝑖𝑛(𝑛𝜋(𝑠+0.5)4⁄). Similar to Ref. [46], to solve the resulting ordinary differential algebraic equations , the ode15s solver in MATLAB software is utilized . To reduce the index of resulting equation s from three to one, Eq. (10) is replaced with the following equation [46] 22 0 3 0 0 3 3 0. e e e e e e+ + + = (14) which is the second derivative of Eq. (10) with respect to 𝜏. 3. Verification study In Ref. [46], the quaternion -based model was utilized to verify the geometrically exact nonlinear behavior of hanging cantilevered pipe conveying fluid. Therefore, in this section, this model is verified for the standing one. Chen et al. [50] investigated the nonlinear responses of a standing pipe conveying fluid when 𝛽=0.142, 𝛾=−18.9, and 𝛼=0.005 via the geometrically exact rotation -based model with 𝑁=4. To verify the results reported by the aforementioned reference, the geometrically exact nonlinear responses of quaternion -based model with 𝑁=4 and 5 are calculated. Comparison studies are presented in Appendix A. The comparison studies in Figures A1-A3 show that the results of quaternion -based model with 𝑁=4 are in good agreement with those obtained by Chen et al. [50] via the rotation -based model. Additionally, the plots depicted in these figures indicate that the results with 𝑁=5 can better describe the geometrically exact nonlinear dynamics of system. 4. Nonlinear geometrically exact analysis For a hanging pipe with 𝑈=0, the undeformed configuration of system is stable. But for a corresponding standing one, the stability of undeformed configuration is dependent on the gravity parameter. The geometrically exact nonlinear static responses and their stability characteristics for standing pipe s when −100 ≤𝛾≤0 are plotted in Figure 2. The plots show that the undeformed configuration of system is stable if the gravity parameter is large enough. However, the undeformed configuration of system becomes unstable and the system undergoes buckling instability at a critical gravity parameter 𝛾𝑐𝑟=−7.83. In other words, the system buckles due to 8 its weight (self-buckling) . The plots indicate that the system undergoes extremely large deformation when the gravity parameter is low enough . It should be stressed that the simple model proposed by Bou-Rabee et al. [49] is capable of capturing the geometrically exact static behavior of system. It is due to the fact that simplifying in their model does not affect the static instability of system. The results plotted in Figure 2a were presented in their work for −60≤𝛾≤0 based on a dynamic analysis. There is a good agreement between the present results and those reported in the aforementioned