The Static Analysis of Thin-Walled Curved Beams under Inclined Eccentric Line Load

1. INTRODUCTION

Curved thin-walled beams are widely used in engineering because they offer both aesthetic appeal and a high stiffness-to-mass ratio[1]. However, their analysis is associated with two main difficulties: (i) the inherent coupling between internal forces and displacements and (ii) the dependence of the governing equations on the geometry of the cross-section. For example, the expressions for the longitudinal strain governing the behaviour of beams with doubly symmetric I-shaped, monosymmetric I-shaped, and doubly symmetric H-shaped cross-sections differ from one another, and, as a result, so do the governing equations for these beams. Furthermore, in-plane and out-of-plane responses are inherently decoupled for doubly symmetric cross-sections, whereas they are always coupled for asymmetric cross-sections. For monosymmetric cross-sections, the degree of coupling depends on the shape of the cross-section[2].

The coupling can also be caused by the type of load. When a distributed line load is inclined and eccentric, the in-plane and out-of-plane internal forces are coupled, regardless of the shape or symmetry of the cross-section. The aim of this study is to investigate these coupling effects in thin-walled curved beams with a doubly symmetric H-shaped cross-section subjected to an eccentric inclined distributed line load.

These analyses can be considerably simplified if the initial curvature of the beam is either small [TeX:] (R/b^{*} \geq 10)$(R/b^{*}\ge 10)$ or moderate [TeX:] (7 < R/b^{*} < 10)$(7<R/b^{*}<10)$, where [TeX:] b^{*}$b^{*}$ is the length of the largest cross-sectional wall and [TeX:] R$R$ is the circular arc radius of the undeformed centroid line. As far as the authors are aware, the first analysis of thin-walled curved beams with small curvature was performed by Vlasov[3], who derived the internal force-displacement relations of curved beams by substituting curvature terms into the corresponding relations for straight beams. In his formulation, Vlasov neglected the distance of the shear centre with respect to the centroid of the cross-section and thus disregarded the coupling effects that occur in monosymmetric and asymmetric cross-sections.

Nishino and Fukasawa[4] analysed thin-walled curved beams without distinguishing between the governing expressions for doubly symmetric I-shaped and H-shaped cross-sections. Their expressions were derived using Vlasov’s assumption that the shear strain in the middle surface can be neglected. These expressions were used in most subsequent studies[2], despite the drawback that this strain in the middle surface may not be zero[5]. To address this limitation, Yang and Kuo[6] proposed an approach that distinguishes between doubly symmetric I-shaped and H-shaped cross-sections and ensures that the shear strain in the middle surface is identically zero. However, their formulation neglects the membrane component of the shear stress[7]. As far as the authors are aware, Tong and Xu[5] have presented the only approach that correctly extends the Vlasov theory for straight thin-walled beams to curved thin-walled beams, regardless of cross-section type or symmetry, using the theory of shells of revolution[8]. However, in their study[5], the results were obtained using numerical procedures.

For thin-walled curved beams and similar structures, numerical methods have been developed to a much greater extent than analytical methods. Recent approaches mainly include beam finite elements[9-12], Carrera’s unified formulation[13], generalised beam theory[14-16] and isogeometric finite element analysis[17-21]. Since the authors of this paper have recently presented a comparison of approaches for analysing thin-walled curved beams[2], this paper does not further discuss existing methods. This paper also represents the latest outcome of the authors’ ongoing research on this topic.

Kustura et al.[2, 22-23] extended the approach of Yang and Kuo by including the membrane component of shear stress and then performed linear out-of-plane[2] and in-plane[22-23] static analyses. In these approaches, closed-form solutions for thin-walled curved beams with small or moderate curvature were obtained using a Maclaurin series expansion of the term [TeX:] R/(R - y)$R/(R-y)$. This simplification allows the cross-sectional properties of straight thin-walled beams to be used to analyse curved beams, as is common in most studies of thin-walled curved beams[2]. However, only one type of distributed line load was considered in each of these studies, prompting the present study to investigate the effects of an inclined eccentric distributed line load.

Chapter 2 provides a brief theoretical background necessary to understand the problem and uses the method of initial parameters to derive closed-form solutions. The out-of-plane solutions are presented in Appendix A, while Appendix B contains the in-plane solutions. It should be noted that these solutions are only valid for beams with a doubly symmetric H-shaped cross-section. Chapter 3 compares the centroid displacements and the longitudinal normal stresses for beams with small and moderate curvature and for different beam parameters, with the analytical solutions being compared with the numerical results obtained with finite shell elements. The last section summarises the main results and provides guidelines for future research.

2. THEORETICAL DEVELOPMENT

The assumptions underlying this approach[2, 22-23] are as follows: (1) strains and displacements are small; (2) the cross-section middle line remains rigid in its own plane; (3) the shear strain in the middle surface is neglected; and (4) the material is linear elastic. These assumptions are consistent with those of Vlasov[3].

The present approach is subject to the following constraints: (a) the initial curvature of the beam is small or moderate [TeX:] (R/b^{*} > 7)$(R/b^{*}>7)$; (b) only uniformly distributed line loads are considered; and (c) the cross-section is H-shaped and doubly symmetric.

The middle surface of a thin-walled curved beam with a doubly symmetric H-shaped cross-section is shown in Figure 1.a, while the cross-section geometry and applied load are shown in Figure 1.b, where [TeX:] b$b$ and [TeX:] h$h$ denote the width and height of the cross-section, respectively, and [TeX:] t_{1}$t_1$ and [TeX:] t_{0}$t_0$ denote the thickness of the flange and web. The undeformed centroid line of the curved beam is assumed to be a circular arc of radius [TeX:] R$R$ and length [TeX:] L$L$. An inclined and eccentric uniformly distributed line load of constant magnitude [TeX:] q$q$ acts at the outer web-flange junction, i.e., at point [TeX:] G$G$ in Figure 1.b, where [TeX:] \varphi$\phi$ is the inclination angle with respect to the [TeX:] O\rho\phi$O\rho \varphi$ plane. Two coordinate systems are used: the cylindrical coordinate system [TeX:] O\rho\phi Z$O\rho \varphi Z$ and the curvilinear coordinate system [TeX:] Cxyz$Cxyz$, where [TeX:] \rho = R - y$\rho =R-y$ and [TeX:] \phi\ = x/R$\varphi =x/R$. The origin [TeX:] C$C$ is at the centroid of the cross-section, while the origin [TeX:] O$O$ corresponds to the centre of the undeformed centroid line. The positive direction of the [TeX:] x$x$-axis is defined by an increase in the angle [TeX:] \phi$\varphi$, while the [TeX:] y$y$- and [TeX:] z$z$-axes coincide with the principal axes of the cross-section.

Fig. 1 a) Global and local coordinate systems[2]; b) Characteristics of an H-shaped cross-section

The adopted displacement field is given as follows[2, 5-6, 22-23]:

[TeX:] \begin{matrix} v = v_{C} - z\alpha, & w = w_{C} + y\alpha, & u = u_{C} - y\gamma + z\beta + \omega\vartheta\ R/(R - y), \end{matrix}$\begin{array}{ccc}\hfill v=v_C-z\alpha ,\hfill & \hfill w=w_C+y\alpha ,\hfill & \hfill u=u_C-y\gamma +z\beta +\omega \vartheta R/(R-y),\hfill \end{array}$ (1)

with:

[TeX:] \begin{matrix} \gamma = v_{C}^{'} + u_{C}/R, & \beta = - {w_{C}}^{'}, & \vartheta = - \left( \alpha^{'} + {w_{C}}^{'}/R \right), \end{matrix}$\begin{array}{ccc}\hfill \gamma =v_C^{\prime }+u_C/R,\hfill & \hfill \beta =-{w_C}^{\prime },\hfill & \hfill \vartheta =-({\alpha }^{\prime }+{w_C}^{\prime }/R),\hfill \end{array}$ (2)

where [TeX:] u$u$, [TeX:] v$v$ and [TeX:] w$w$ denote the displacements of any point [TeX:] S$S$ on the middle surface (Figure 1.b), while [TeX:] u_{C}$u_C$, [TeX:] v_{C}$v_C$ and [TeX:] w_{C}$w_C$ represent the displacements of the centroid [TeX:] C$C$ in the directions of the [TeX:] x$x$-, [TeX:] y$y$- and [TeX:] z$z$-axes, respectively. The term [TeX:] \omega = yz$\omega =yz$ denotes the sectorial coordinate, [TeX:] \alpha$\alpha$ is the angle of torsion (twist angle), [TeX:] \gamma$\gamma$ and [TeX:] \beta$\beta$ are the slopes of the deflection line in the [TeX:] Cxy$Cxy$ and [TeX:] Cxz$Cxz$ planes, respectively, [TeX:] \vartheta$\vartheta$ is the relative angle of torsion, and [TeX:] ()^{'} = d()/dx${\left(\right)}^{\prime }=d\left(\right)/dx$[2].

The longitudinal strain [TeX:] \varepsilon_{x}${\epsilon }_x$ is defined as follows[2, 5-6, 22-23]:

[TeX:] \begin{matrix} \varepsilon_{x} = \frac{R}{R - y}\left( \frac{\partial u}{\partial x} - \frac{v}{R} \right). \end{matrix}$\begin{array}{c}\hfill {\epsilon }_x=\frac{R}{R-y}(\frac{\partial u}{\partial x}-\frac{v}{R}).\hfill \end{array}$ (3)

Using Eqs. (1)-(3), the longitudinal normal stress [TeX:] \sigma_{x}${\sigma }_x$ is:

[TeX:] \sigma_{x} = E\left\lbrack u_{C}^{'} - \frac{v_{C}}{R} - y\left( v_{C}^{''} + \frac{u_{C}^{'}}{R} \right) - z\left( w_{C}^{''} - \frac{\alpha}{R} \right) - \frac{\omega R}{R - y}\left( \alpha^{''} + \frac{w_{C}^{''}}{R} \right) \right\rbrack\frac{R}{R - y}.${\sigma }_x=E[u_C^{\prime }-\frac{v_C}{R}-y(v_C^{″}+\frac{u_C^{\prime }}{R})-z(w_C^{″}-\frac{\alpha }{R})-\frac{\omega R}{R-y}({\alpha }^{″}+\frac{w_C^{″}}{R})]\frac{R}{R-y}.$ (4)

The internal force-normal stress relations for thin-walled curved beams with a doubly symmetric H-shaped cross-section are defined like those of straight thin-walled beams[2-3]:

[TeX:] \begin{bmatrix} N \\ M_{z} \\ M_{y} \\ B \end{bmatrix} = \int_{A}^{}{\begin{bmatrix} 1 \\ - y \\ z \\ \omega \end{bmatrix}\sigma_{x}dA} = E\left( \int_{A}^{}{\begin{bmatrix} 1 & - y & - z & - \omega\frac{R}{R - y} \\ - y & - y^{2} & \omega & y\omega\frac{R}{R - y} \\ z & - \omega & - z^{2} & - z\omega\frac{R}{R - y} \\ \omega & - y\omega & - z\omega & - \omega^{2}\frac{R}{R - y} \end{bmatrix}\frac{R}{R - y}dA} \right)\begin{bmatrix} u_{C}^{'} - \frac{v_{C}}{R} \\ v_{C}^{''} + \frac{u_{C}^{'}}{R} \\ w_{C}^{''} - \frac{\alpha}{R} \\ \frac{w_{C}^{''}}{R} + \alpha^{''} \end{bmatrix}.$\left[\begin{array}{c}\hfill N\hfill \\ \hfill M_z\hfill \\ \hfill M_y\hfill \\ \hfill B\hfill \end{array}\right]={\int }_A^{}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -y\hfill \\ \hfill z\hfill \\ \hfill \omega \hfill \end{array}\right]{\sigma }_{x}dA=E\left({\int }_A^{}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill -y\hfill & \hfill -z\hfill & \hfill -\omega \frac{R}{R-y}\hfill \\ \hfill -y\hfill & \hfill -y^2\hfill & \hfill \omega \hfill & \hfill y\omega \frac{R}{R-y}\hfill \\ \hfill z\hfill & \hfill -\omega \hfill & \hfill -z^2\hfill & \hfill -z\omega \frac{R}{R-y}\hfill \\ \hfill \omega \hfill & \hfill -y\omega \hfill & \hfill -z\omega \hfill & \hfill -{\omega }^2\frac{R}{R-y}\hfill \end{array}\right]\frac{R}{R-y}dA\right)\left[\begin{array}{c}\hfill u_C^{\prime }-\frac{v_C}{R}\hfill \\ \hfill v_C^{″}+\frac{u_C^{\prime }}{R}\hfill \\ \hfill w_C^{″}-\frac{\alpha }{R}\hfill \\ \hfill \frac{w_C^{″}}{R}+{\alpha }^{″}\hfill \end{array}\right].$ (5)

The term [TeX:] R/(R - y)$R/(R-y)$ in Eq. (1), associated with the out-of-plane displacements [TeX:] w_{C}$w_C$ and [TeX:] \alpha$\alpha$ and the sectorial coordinate [TeX:] \omega$\omega$, is a feature of the doubly symmetric H-shaped cross-section and introduces an additional curvature effect in the out-of-plane analysis. This term is absent in the out-of-plane analysis of thin-walled curved beams with a doubly symmetric I-shaped cross-section, which leads to simplified governing equations and corresponding solutions. To linearise the effect of the curvature, the following Maclaurin series expansions are used:

[TeX:] \begin{matrix} \frac{R}{R - y} \cong 1 + \frac{y}{R}, & \left( \frac{R}{R - y} \right)^{2} \cong 1 + 2\frac{y}{R}. \end{matrix}$\begin{array}{cc}\hfill \frac{R}{R-y}\cong 1+\frac{y}{R},\hfill & \hfill {\left(\frac{R}{R-y}\right)}^2\cong 1+2\frac{y}{R}.\hfill \end{array}$ (6)

The longitudinal force [TeX:] N$N$, the bending moment about the z-axis [TeX:] M_{z}$M_z$, the bending moment about the y-axis [TeX:] M_{y}$M_y$, and the bimoment [TeX:] B$B$ are defined as:

[TeX:] \begin{matrix} \begin{matrix} N = EA\left( u_{C}^{'} - \frac{v_{C}}{R} \right) - \frac{EI_{z}}{R}\left( v_{C}^{''} + \frac{u_{C}^{'}}{R} \right), \\ M_{y} = - EI_{y}\left( w_{C}^{''} - \frac{\alpha}{R} \right) - 2\frac{EI_{\omega}}{R}\left( \frac{w_{C}^{''}}{R} + \alpha^{''} \right), \end{matrix} & \begin{matrix} M_{z} = EI_{z}\left( v_{C}^{''} + \frac{v_{C}}{R^{2}} \right) \\ B = - \frac{EI_{\omega}}{R}\left( w_{C}^{''} - \frac{\alpha}{R} \right) - EI_{\omega}\left( \frac{w_{C}^{''}}{R} + \alpha^{''} \right). \end{matrix} \end{matrix}$\begin{array}{cc}\hfill \begin{array}{c}\hfill N=EA(u_C^{\prime }-\frac{v_C}{R})-\frac{E{I}_z}{R}(v_C^{″}+\frac{u_C^{\prime }}{R}),\hfill \\ \hfill M_y=-E{I}_y(w_C^{″}-\frac{\alpha }{R})-2\frac{E{I}_{\omega }}{R}(\frac{w_C^{″}}{R}+{\alpha }^{″}),\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill M_z=E{I}_z(v_C^{″}+\frac{v_C}{R^2})\hfill \\ \hfill B=-\frac{E{I}_{\omega }}{R}(w_C^{″}-\frac{\alpha }{R})-E{I}_{\omega }(\frac{w_C^{″}}{R}+{\alpha }^{″}).\hfill \end{array}\hfill \end{array}$ (7)

The in-plane internal forces ([TeX:] N$N$ and [TeX:] M_{z}$M_z$) in Eq. (7) remain valid for beams with a doubly symmetric I-shaped cross-section, since the in-plane and out-of-plane analyses are naturally decoupled, as already mentioned. Accordingly, [TeX:] u_{C}$u_C$ and [TeX:] v_{C}$v_C$ represent the in-plane centroid displacements.

Since the term [TeX:] R/(R - y)$R/(R-y)$ is linearised in the proposed approach with a Maclaurin series expansion, this method applies to beams with moderate curvature ([TeX:] 7 < R/b^{*} < 10$7<R/b^{*}<10$). In contrast, Vlasov neglected this term and restricted his analysis to beams with small curvature ([TeX:] R/b^{*} \geq 10$R/b^{*}\ge 10$). A further simplification adopted in this study is the use of the bimoment relation in Eq. (5) instead of the modified bimoment expression, [TeX:] B_{cH} = \int_{A}^{}{\omega\frac{R}{R - y}\sigma_{x}dA}$B_{cH}={\int }_A^{}\omega \frac{R}{R-y}{\sigma }_{x}dA$[2, 5]. This simplification is partially justified by Eq. (6) and constraint (a) and, consequently, results in significantly simpler closed-form solutions.

The geometric properties adopted in this study are identical to those of straight beams and consistent with most studies on thin-walled curved beams[2-7, 22-23]. This results from the constraint (a) and from the simplification of the curvature introduced in Eq. (6), i.e.:

[TeX:] \begin{matrix} \begin{matrix} A = \int_{A}^{}{dA =}bt_{0} + 2ht_{1}, \\ I_{y} = \int_{A}^{}{z^{2}dA =}2\frac{t_{1}h^{3}}{12}, \end{matrix} & \begin{matrix} I_{z} = \int_{A}^{}{y^{2}dA =}\frac{t_{0}b^{3}}{12} + 2ht_{1}\left( \frac{b}{2} \right)^{2}, \\ I_{\omega} = \int_{A}^{}{\omega^{2}dA =}ht_{1}\frac{b^{2}h^{2}}{24}. \end{matrix} \end{matrix}$\begin{array}{cc}\hfill \begin{array}{c}\hfill A={\int }_A^{}dA=b{t}_0+2h{t}_1,\hfill \\ \hfill I_y={\int }_A^{}{z}^{2}dA=2\frac{t_1{h}^3}{12},\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill I_z={\int }_A^{}{y}^{2}dA=\frac{t_0{b}^3}{12}+2h{t}_1{\left(\frac{b}{2}\right)}^2,\hfill \\ \hfill I_{\omega }={\int }_A^{}{\omega }^{2}dA=h{t}_1\frac{b^2{h}^2}{24}.\hfill \end{array}\hfill \end{array}$ (8)

The total torsional moment [TeX:] M_{x}$M_x$ is expressed as [TeX:] M_{x} = M_{\omega} + M_{t}$M_x=M_{\omega }+M_t$, where the warping moment [TeX:] M_{\omega}$M_{\omega }$[2] and the pure torsional moment [TeX:] M_{t}$M_t$ of a curved beam are defined as follows:

[TeX:] \begin{matrix} M_{t} = GI_{tc}\left( \alpha^{'} + w_{C}^{'}/R \right), & M_{\omega} = B^{'}, \end{matrix}$\begin{array}{cc}\hfill M_t=G{I}_{tc}({\alpha }^{\prime }+w_C^{\prime }/R),\hfill & \hfill M_{\omega }=B^{\prime },\hfill \end{array}$ (9)

with torsional moment of inertia for a curved thin-walled beam defined as [TeX:] I_{tc} = \frac{1}{3}\int_{A}^{}{\left( \frac{R}{R - y} \right)^{2}t^{2}dA}$I_{tc}=\frac{1}{3}{\int }_A^{}{\left(\frac{R}{R-y}\right)}^2{t}^{2}dA$. Using Eq. (6), the final simplification used in this study is [TeX:] I_{tc} \cong \frac{1}{3}\int_{A}^{}{t^{2}dA} = I_{t} = \left( bt_{0}^{3} + 2ht_{1}^{3} \right)/3$I_{tc}\cong \frac{1}{3}{\int }_A^{}{t}^{2}dA=I_t=(b{t}_0^3+2h{t}_1^3)/3$, where [TeX:] I_{t}$I_t$ represents the torsional moment of inertia for a straight thin-walled beam.

As mentioned, the in-plane and out-of-plane analyses of thin-walled curved beams with a doubly symmetric cross-section are normally decoupled, since all internal forces are defined with respect to the centroid of the cross-section. An infinitesimal segment of the curved centroid line with length [TeX:] dx = Rd\phi$dx=Rd\varphi$ is shown in Figure 2. The out-of-plane internal forces together with the corresponding line loads are shown in Figure 2.a, while the in-plane internal forces and the distributed line loads are shown in Figure 2.b.

Fig. 2 Infinitesimal segment of a curved centroid line with a) out-of-plane[2] and b) in-plane internal forces and distributed line loads

In the out-of-plane analysis, [TeX:] Q_{z}$Q_z$ and [TeX:] q_{z}$q_z$ denote the shear force and the distributed line load in the direction of the [TeX:] z$z$-axis, respectively, and [TeX:] m_{x}$m_x$ represents the distributed torsional load acting along the curved centroid line. In the in-plane analysis, [TeX:] Q_{y}$Q_y$ and [TeX:] q_{y}$q_y$ denote the shear force and the distributed line load in the direction of the [TeX:] y$y$-axis, respectively, and [TeX:] q_{x}$q_x$ corresponds to the distributed longitudinal load along the centroid line. In accordance with constraint (b), only uniformly distributed line loads of constant magnitude are considered in this study. Additionally, the uniformly distributed out-of-plane centroid line moment load [TeX:] m_{y}$m_y$ and the in-plane moment load [TeX:] m_{z}$m_z$ are neglected in this study.

The differential equilibrium equations governing the out-of-plane static analysis are derived using the free-body approach and Figure 2.a[2], and can be expressed as follows:

[TeX:] \begin{matrix} M_{x}^{'} - M_{y}/R\ \ + m_{x} = 0, & M_{y}^{'} - Q_{z} + M_{x}/R = 0, & Q_{z}^{'} + q_{z} = 0. \end{matrix}$\begin{array}{ccc}\hfill M_x^{\prime }-M_y/R+m_x=0,\hfill & \hfill M_y^{\prime }-Q_z+M_x/R=0,\hfill & \hfill Q_z^{\prime }+q_z=0.\hfill \end{array}$ (10)

Similarly, the differential equilibrium equations governing the in-plane static analysis are derived with reference to Figure 2.b[22-23], and can be written as follows:

[TeX:] \begin{matrix} N^{'} - Q_{y}/R + q_{x} = 0, & Q_{y}^{'} + N/R + q_{y} = 0, & M_{z}^{'} + Q_{y} = 0. \end{matrix}$\begin{array}{ccc}\hfill N^{\prime }-Q_y/R+q_x=0,\hfill & \hfill Q_y^{\prime }+N/R+q_y=0,\hfill & \hfill M_z^{\prime }+Q_y=0.\hfill \end{array}$ (11)

Eliminating the internal forces from the equilibrium equations yields:

[TeX:] \begin{matrix} M_{x}^{'} - M_{y}/R + m_{x} = 0, & M_{y}^{''} + M_{x}^{'}/R + q_{z} = 0, \end{matrix}$\begin{array}{cc}\hfill M_x^{\prime }-M_y/R+m_x=0,\hfill & \hfill M_y^{″}+M_x^{\prime }/R+q_z=0,\hfill \end{array}$ (12)

and

[TeX:] \begin{matrix} M_{z}^{'''} + M_{z}^{'}/R^{2} + q_{y}^{'} - q_{x}/R = 0. \end{matrix}$\begin{array}{c}\hfill M_z^{‴}+M_z^{\prime }/R^2+q_y^{\prime }-q_x/R=0.\hfill \end{array}$ (13)

By introducing the flexural-torsional constants [TeX:] k^{2} = R^{2}GI_{t}/(EI_{\omega})$k^2=R^{2}G{I}_t/\left(E{I}_{\omega }\right)$ and [TeX:] n^{2} = R^{2}I_{y}/I_{\omega}$n^2=R^2{I}_y/I_{\omega }$, and by using Eqs. (7)-(9) and (12), the governing equations for the out-of-plane analysis are defined as:

[TeX:] \begin{matrix} \alpha^{(4)} - \left( 3 + k^{2} \right)\frac{\alpha^{''}}{R^{2}} + n^{2}\frac{\alpha}{R^{4}} + 2\frac{w_{C}^{(4)}}{R} - \left( 2 + n^{2} + k^{2} \right)\frac{w_{C}^{''}}{R^{3}} - \frac{m_{x}}{EI_{\omega}} = 0, \\ 3\alpha^{(4)} - \left( 1 + n^{2} + k^{2} \right)\frac{\alpha^{''}}{R^{2}} + \left( 4 + n^{2} \right)\frac{w_{C}^{(4)}}{R} - k^{2}\frac{w_{C}^{''}}{R^{3}} - \frac{q_{z}R}{EI_{\omega}} = 0. \end{matrix}$\begin{array}{c}\hfill {\alpha }^{\left(4\right)}-(3+k^2)\frac{{\alpha }^{″}}{R^2}+n^2\frac{\alpha }{R^4}+2\frac{w_C^{\left(4\right)}}{R}-(2+n^2+k^2)\frac{w_C^{″}}{R^3}-\frac{m_x}{E{I}_{\omega }}=0,\hfill \\ \hfill 3{\alpha }^{\left(4\right)}-(1+n^2+k^2)\frac{{\alpha }^{″}}{R^2}+(4+n^2)\frac{w_C^{\left(4\right)}}{R}-k^2\frac{w_C^{″}}{R^3}-\frac{q_{z}R}{E{I}_{\omega }}=0.\hfill \end{array}$ (14)

The solutions in closed form for these equations can be found in Appendix A. Similarly, the governing equation for the in-plane analysis is obtained using Eqs. (7)-(9) and Eq. (13):

[TeX:] \begin{matrix} EI_{z}\left( {v_{C}}^{(5)} + 2\frac{{v_{C}}^{'''}}{R^{2}} + \frac{{v_{C}}^{'}}{R^{4}} \right) - q_{y}^{'} + \frac{q_{x}}{R} = 0, \end{matrix}$\begin{array}{c}\hfill E{I}_z({v_C}^{\left(5\right)}+2\frac{{v_C}^{‴}}{R^2}+\frac{{v_C}^{\prime }}{R^4})-q_y^{\prime }+\frac{q_x}{R}=0,\hfill \end{array}$ (15)

and its closed-form solution is shown in Appendix B.

A comparison of Eqs. (14) and (15) shows that the in-plane and out-of-plane displacements are generally decoupled, allowing their analyses to be performed independently of each other. For the load configuration shown in Figure 1.b, the equivalent distributed line loads acting on the centroid line are defined in the present analysis as follows:

[TeX:] \begin{matrix} q_{y} = q\frac{R + b/2}{R}\cos\varphi, & q_{z} = q\frac{R + b/2}{R}\sin\varphi, & m_{x} = - q\frac{b}{2}\frac{R + b/2}{R}\ \sin\varphi. \end{matrix}$\begin{array}{ccc}\hfill q_y=q\frac{R+b/2}{R}cos\phi ,\hfill & \hfill q_z=q\frac{R+b/2}{R}sin\phi ,\hfill & \hfill m_x=-q\frac{b}{2}\frac{R+b/2}{R}sin\phi .\hfill \end{array}$ (16)

The load case considered in this study represents a special problem in which in-plane and out-of-plane effects can be considered coupled. Thus, for a given load magnitude [TeX:] q$q$, a change in the inclination angle [TeX:] \varphi$\phi$ can significantly affect both in-plane and out-of-plane internal forces and displacements. Consequently, in-plane and out-of-plane analyses can be treated as coupled, which is the main motivation for the present study.

To derive closed-form solutions, this study uses the method of initial parameters:

[TeX:] \begin{matrix} v = Kv_{0} + I, \end{matrix}$\begin{array}{c}\hfill v=K{v}_0+I,\hfill \end{array}$ (17)

where [TeX:] \mathbf{K}$𝐊$ is the field matrix, [TeX:] \mathbf{I}$𝐈$ is the load vector and [TeX:] \mathbf{v}$𝐯$ is the state vector. The solutions of the governing equations are expressed in terms of the initial state vector [TeX:] \mathbf{v}_{0}${𝐯}_0$:

[TeX:] \left. \ v \right|_{x = 0} = v_{0} = \begin{bmatrix} v_{0}^{op} & v_{0}^{ip} \end{bmatrix}^{T},${v|}_{x=0}=v_0={\left[\begin{array}{cc}\hfill v_0^{op}\hfill & \hfill v_0^{ip}\hfill \end{array}\right]}^T,$ (18)

where the out-of-plane component [TeX:] \mathbf{v}_{0}^{op}${𝐯}_0^{op}$ and the in-plane component [TeX:] \mathbf{v}_{0}^{ip}${𝐯}_0^{ip}$ are defined as follows:

[TeX:] \begin{matrix} \left. \ v^{op} \right|_{x = 0} = v_{0}^{op} = \begin{bmatrix} Q_{z0} & M_{y0} & \beta_{0} & w_{C0} & M_{x0} & B_{0} & \vartheta_{0} & \alpha_{0} \end{bmatrix}^{T}, \\ \left. \ v^{ip} \right|_{x = 0} = v_{0}^{ip} = \begin{bmatrix} \begin{matrix} Q_{y0} & M_{z0} & \gamma_{0} \end{matrix} & \begin{matrix} v_{C0} & N_{0} & u_{C0} \end{matrix} \end{bmatrix}^{T}. \end{matrix}$\begin{array}{c}\hfill {v^{op}|}_{x=0}=v_0^{op}={\left[\begin{array}{cccccccc}\hfill Q_{z0}\hfill & \hfill M_{y0}\hfill & \hfill {\beta }_0\hfill & \hfill w_{C0}\hfill & \hfill M_{x0}\hfill & \hfill B_0\hfill & \hfill {\vartheta }_0\hfill & \hfill {\alpha }_0\hfill \end{array}\right]}^T,\hfill \\ \hfill {v^{ip}|}_{x=0}=v_0^{ip}={\left[\begin{array}{cc}\hfill \begin{array}{ccc}\hfill Q_{y0}\hfill & \hfill M_{z0}\hfill & \hfill {\gamma }_0\hfill \end{array}\hfill & \hfill \begin{array}{ccc}\hfill v_{C0}\hfill & \hfill N_0\hfill & \hfill u_{C0}\hfill \end{array}\hfill \end{array}\right]}^T.\hfill \end{array}$ (19)

The boundary conditions used in this analysis are listed in Table 1[1-2].

Table 1 Boundary conditions for the supports of thin-walled curved beams

In Table 1, the boundary conditions before the semicolon refer to the out-of-plane analysis, while the boundary conditions after the semicolon refer to the in-plane analysis.

3. COMPARATIVE ANALYSIS

The analysed beam is clamped at both ends, considering the symmetry at the beam mid-span [TeX:] (x = L/2)$(x=L/2)$ in all models, as shown in Figure 3.a for the undeformed centroid line. The geometrical parameters are defined as [TeX:] t_{1} = t_{0} = t = 5$t_1=t_0=t=5$ mm, [TeX:] h/t = 10$h/t=10$, [TeX:] b/h = 2$b/h=2$. The influence of different values of the ratios [TeX:] L/b$L/b$, representing the beam length ([TeX:] 10 \leq L/b \leq 30$10\le L/b\le 30$, in steps of 2), and [TeX:] R/b$R/b$, representing the beam curvature ([TeX:] 7 \leq R/b \leq 20$7\le R/b\le 20$, in steps of 1), is investigated, resulting in a total of 154 models. The material properties are defined by a Poisson’s ratio of [TeX:] \nu = 0.3$\nu =0.3$ and a modulus of elasticity of [TeX:] E = 73$E=73$ GPa. The loading parameters are [TeX:] q = 2$q=2$ N/mm and [TeX:] \varphi = \pi/6$\phi =\pi /6$, as shown in Figure 1.b and defined in Eq. (16). The four-node shell finite element model[24] with boundary conditions is shown in Figure 3.b. An average element size of 10 mm is used to verify the analytically obtained results. To represent the uniformly distributed torsional load [TeX:] m_{x}$m_x$, vertically distributed line loads at the inner ([TeX:] q_{i}$q_i$) and outer ([TeX:] q_{o}$q_o$) web-flange junction edges, with the corresponding values:

[TeX:] \begin{matrix} q_{i} = \frac{m_{x}}{b}\frac{R}{R - b/2}, & q_{o} = - \frac{m_{x}}{b}\frac{R}{R + b/2}, \end{matrix}$\begin{array}{cc}\hfill q_i=\frac{m_x}{b}\frac{R}{R-b/2},\hfill & \hfill q_o=-\frac{m_x}{b}\frac{R}{R+b/2},\hfill \end{array}$ (20)

are applied in the numerical model. The deformed (solid line) and undeformed (dashed line) finite element model for the parameters [TeX:] L/b = 10$L/b=10$ and [TeX:] R/b = 7$R/b=7$ is shown in Figure 3.c. The range of a centroid line central angle [TeX:] \Phi = L/R$\Phi =L/R$ is shown in Figure 3.d, and ranges from [TeX:] \Phi = 0.5$\Phi =0.5$ [TeX:] (L/b = 10$(L/b=10$ and [TeX:] R/b = 20$R/b=20$) to [TeX:] \Phi = 4.29$\Phi =4.29$ [TeX:] (L/b = 30$(L/b=30$ and [TeX:] R/b = 7$R/b=7$).

Fig. 3 Boundary conditions for: a) the curved centroid line and b) FEM model ([TeX:] L/b = 10$L/b=10$ and [TeX:] R/b = 7$R/b=7$), c) the deformed and undeformed FEM model and d) range of the centroid line central angle [TeX:] \Phi$\Phi$

In this study, the relative differences are used to compare the analytical (A) and the numerical (N) results, and are defined as:

[TeX:] \delta u = 100\ \frac{u_{C}^{A} - u_{C}^{N}}{u_{C}^{N}},\ \delta v = 100\ \frac{v_{C}^{A} - v_{C}^{N}}{v_{C}^{N}},\ \delta w = 100\ \frac{w_{C}^{A} - w_{C}^{N}}{w_{C}^{N}},\ \delta\alpha = 100\ \frac{\alpha^{A} - \alpha^{N}}{\alpha^{N}},\ \delta\sigma = 100\ \frac{\sigma_{x}^{A} - \sigma_{x}^{N}}{\sigma_{x}^{N}},$\delta u=100\frac{u_C^A-u_C^N}{u_C^N},\delta v=100\frac{v_C^A-v_C^N}{v_C^N},\delta w=100\frac{w_C^A-w_C^N}{w_C^N},\delta \alpha =100\frac{{\alpha }^A-{\alpha }^N}{{\alpha }^N},\delta \sigma =100\frac{{\sigma }_x^A-{\sigma }_x^N}{{\sigma }_x^N},$ (21)

where the numerical twist angle [TeX:] \alpha^{N}${\alpha }^N$ is defined as the average of the twist angles calculated from the numerical vertical displacements of points A, B, F, D, G, and E (Figure 1.b) and Eq. (1).

The relative differences of the maximum longitudinal stress [TeX:] \sigma_{x}${\sigma }_x$ at the beam mid-span and at [TeX:] x = b$x=b$ (close to the clamped end) are shown in Figure 4. The locations of the maximum stress were determined for the analysed cross-sections and for each analysed ratio [TeX:] L/b$L/b$ and [TeX:] R/b$R/b$ using the FEM models. The relative differences [TeX:] \delta\sigma$\delta \sigma$ were then determined analysing the analytical solutions at these locations.

Fig. 4 Relative differences [TeX:] \delta\sigma$\delta \sigma$ in the cross-section at: a) [TeX:] x = L/2$x=L/2$ and b) [TeX:] x = b$x=b$

Figure 4 shows that the results of the proposed approach are in good agreement with the numerical solutions. The deviations in the longitudinal normal stress at the mid-span increase with increasing beam length and decreasing radius. For the cross-section [TeX:] x = b$x=b$, the agreement between the analytical and numerical models is excellent, with the maximum relative difference [TeX:] \delta\sigma$\delta \sigma$ reaching 4.28 for the parameters [TeX:] L/b = 14$L/b=14$ and [TeX:] R/b = 7$R/b=7$.

Figure 5 shows the distribution of the longitudinal normal stress [TeX:] \sigma_{x}${\sigma }_x$ at the beam mid-span and in the cross-section [TeX:] x = b$x=b$, both along the inner [TeX:] (y = b/2)$(y=b/2)$ and outer [TeX:] (y = - b/2)$(y=-b/2)$ flange middle lines, for the case [TeX:] L/b = 30$L/b=30$ and [TeX:] R/b = 7$R/b=7$. This case corresponds to the largest value of [TeX:] \delta\sigma$\delta \sigma$ in Figure 4.a. In the figures, “FEM” denotes the numerical solutions, while “TP” refers to the analytical solutions obtained using the proposed approach.

The absolute maximum stress values in the section [TeX:] x = b$x=b$ are consistently higher than those in the mid-span section, regardless of the parameters analysed. Consequently, the differences between the proposed solutions for the normal stress at the beam mid-span are less significant. Moreover, in all models, the absolute maximum stress value at the beam mid-span occurs at the point [TeX:] B$B$ of the outer flange (Figure 1.b), while no general conclusion can be drawn for the section [TeX:] x = b$x=b$, since the location of the maximum stress varies between the different edge points of the two flanges. Figure 5 also shows that the proposed approach accurately reflects both the linear character of the longitudinal stress distribution and the corresponding stress values.

Fig. 5 Distribution of the longitudinal normal stress [TeX:] \sigma_{x}${\sigma }_x$ ([TeX:] L/b = 30$L/b=30$, [TeX:] R/b = 7$R/b=7$) at the cross-section flanges for: a) [TeX:] x = L/2$x=L/2$ and b) [TeX:] x = b$x=b$

The relative differences of the centroid displacements [TeX:] v_{C}$v_C$, [TeX:] w_{C}$w_C$, and in the twist angle [TeX:] \alpha$\alpha$ at the beam mid-span are shown in Figure 6, together with the relative differences in the displacement [TeX:] u_{C}$u_C$ at the beam quarter-span ([TeX:] x = L/4$x=L/4$).

Fig. 6 Relative differences: a) [TeX:] \delta u$\delta u$, b) [TeX:] \delta v$\delta v$, c) [TeX:] \delta w$\delta w$ and d) [TeX:] \delta\alpha$\delta \alpha$

For the out-of-plane centroid displacement [TeX:] w_{C}$w_C$ and the twist angle [TeX:] \alpha$\alpha$, Figure 6 shows that the deviation of the proposed solutions increases with decreasing radius, especially for moderately curved beams, while the beam length has only a minor influence. For the in-plane centroid displacements [TeX:] u_{C}$u_C$ and [TeX:] v_{C}$v_C$, the deviations are more pronounced for shorter beams with larger radii. On the other hand, for the other cases analysed, i.e., longer beams with small curvature, which are the most frequently used in practice, an excellent agreement between the proposed approach and the numerical results is found.

To further evaluate the influence of the displacement deviations on the overall response, the total displacement variable [TeX:] \Delta$\Delta$ defined as:

[TeX:] \Delta = \sqrt{u^{2} + v^{2} + w^{2}},$\Delta =\sqrt{u^2+v^2+w^2},$ (22)

is shown in Figure 7.a for the centroid line ([TeX:] y = z = 0$y=z=0$) of the beam with the parameters [TeX:] L/b = 10$L/b=10$ and [TeX:] R/b = 20$R/b=20$, which correspond to the maximum values of [TeX:] \delta u$\delta u$ and [TeX:] \delta v$\delta v$ in Figure 6. This case is used to analyse the deviations in the in-plane displacements. To study the deviations in the out-of-plane displacements, Figure 7.b shows the total displacement of the lower inner flange edge ([TeX:] y = b/2$y=b/2$, [TeX:] z = h/2$z=h/2$), passing through point D (Figure 1.b), for the beam with parameters [TeX:] L/b = 30$L/b=30$ and [TeX:] R/b = 7$R/b=7$, which correspond to very high values for [TeX:] \delta w$\delta w$ and [TeX:] \delta\alpha$\delta \alpha$.

Fig. 7 Total displacement [TeX:] \Delta$\Delta$ of the: a) centroid line ([TeX:] L/b = 10$L/b=10$, [TeX:] R/b = 20$R/b=20$) and b) lower inner edge ([TeX:] L/b = 30$L/b=30$, [TeX:] R/b = 7$R/b=7$)

Figure 7.a shows good agreement for the centroid displacement [TeX:] \Delta$\Delta$ ([TeX:] L/b = 10$L/b=10$ and [TeX:] R/b = 20$R/b=20$), with a relative difference [TeX:] \delta\Delta$\delta \Delta$ of −4.61 at the beam mid-span ([TeX:] x = 500$x=500$ mm), and −10.16 at the quarter-span ([TeX:] x = 250$x=250$ mm). Together with the results from Figure 6, these findings indicate that the out-of-plane displacements [TeX:] w_{C}$w_C$ and [TeX:] \alpha$\alpha$ contribute significantly to the total displacement [TeX:] \Delta$\Delta$, especially at the beam mid-span, where the largest value of [TeX:] \Delta$\Delta$ is observed in all models. This behaviour results from the lower stiffness of the analysed cross-section under out-of-plane loading compared to in-plane loading. Figure 7.b shows a larger deviation, with a relative difference of −12.91 at the beam mid-span and a comparable value at the beam quarter-span. These results are summarised in Table 2, which shows the relative differences [TeX:] \delta\Delta$\delta \Delta$ of the centroid and at point [TeX:] D$D$ of the cross-section, both at the beam mid-span and at the quarter-span, for beams with small and moderate curvature ([TeX:] L/b = 30$L/b=30$).

Table 2 Relative differences [TeX:] \delta\Delta$\delta \Delta$ at the centroid and at point [TeX:] D$D$ of the cross-section, evaluated at the beam mid-span and at the quarter-span, for beams with small and moderate curvature ([TeX:] L/b = 30$L/b=30$)

The results in Table 2 show that the relative difference [TeX:] \delta\Delta$\delta \Delta$ decreases with increasing radius, while the position of the analysed point has only a minor influence on its value. For beams with small curvature, the relative differences [TeX:] \delta\Delta$\delta \Delta$ are acceptable, whereas for beams with moderate curvature, especially when [TeX:] R/b = 7$R/b=7$, the relative differences [TeX:] \delta\Delta$\delta \Delta$ increase significantly as the radius decreases. Considering that the radius analysed in Figure 7.b corresponds to the lower bound of constraint (a), and that the predictions of the normal stress are also in excellent agreement, the proposed approach generally provides reliable results and can be effectively applied to beams with small and moderate curvature. Nevertheless, further research is needed to clarify the observed discrepancies in displacement, with a particular focus on a more comprehensive treatment of curvature effects.

4. CONCLUSION

This work represents the culmination of the authors’ research in which approximate closed-form solutions were developed based on the assumptions of Vlasov[3] and the linearisation of the Maclaurin series expansion. In contrast to Vlasov’s approach, which is limited to beams with small curvature, the proposed method allows the analysis of thin-walled curved beams with both small and moderate curvature, provided that the cross-section is doubly symmetric and H-shaped. In this way, the well-established cross-sectional properties of straight thin-walled beams can be used to analyse curved beams.

The selected case of an eccentric inclined distributed line load was analysed because it reflects realistic loading conditions in which several load components are typically present. For example, if the analysed beam is part of a roof structure, both in-plane loads and out-of-plane loads can occur due to wind action, where the inclination angle [TeX:] \varphi$\phi$ represents the wind direction. The results show good agreement between the proposed approach and the numerical solutions, especially with respect to the longitudinal normal stress. Although some discrepancies were observed in the displacements, these were found to be relatively insignificant.

The most important limitation of this formulation is that it only applies to thin-walled curved beams with a doubly symmetric H-shaped cross-section. For monosymmetric and asymmetric cross-sections, additional coupling effects may occur since the governing equations inherently couple the in-plane and out-of-plane responses. Moreover, the governing expressions depend on the specific geometry of the cross-section. For these reasons, further studies are needed to extend the present approach to other types of cross-sections and to investigate the influence of the Maclaurin series expansion on the accuracy of the closed-form solutions for beams with smaller radii.

5. REFERENCES

6. APPENDIX A

The closed-form solution of Eq. (14) is obtained using the method of parameters variation:

[TeX:] \begin{matrix} w_{C}(\phi) = C_{1} + C_{2}\phi + D_{3}\left\lbrack C_{3}\cosh(D_{1}\phi) + C_{4}\sinh(D_{1}\phi) \right\rbrack + \left( C_{5} + C_{6}D_{2}\phi \right)\cos\phi + \\ + \left( C_{7} + C_{8}D_{2}\phi \right)\sin\phi + RC_{9} - \frac{q_{z}R^{4}}{EI_{y}}\frac{n^{2}}{k^{2}}\frac{\phi^{2}}{2}, \\ D_{4}\left\lbrack C_{3}\cosh(D_{1}\phi) + C_{4}\sinh(D_{1}\phi) \right\rbrack - \frac{1}{R}\left( C_{8}R^{2} + C_{5} + C_{6}D_{2}\phi \right)\cos\phi \\ + \frac{1}{R}\left( C_{6}R^{2} - C_{7} - C_{8}D_{2}\phi \right)\sin\phi - C_{9}, \end{matrix}$\begin{array}{c}\hfill w_C\left(\varphi \right)=C_1+C_2\varphi +D_3[C_{3}cosh\left(D_1\varphi \right)+C_{4}sinh\left(D_1\varphi \right)]+(C_5+C_6{D}_2\varphi )cos\varphi +\hfill \\ \hfill +(C_7+C_8{D}_2\varphi )sin\varphi +R{C}_9-\frac{q_z{R}^4}{E{I}_y}\frac{n^2}{k^2}\frac{{\varphi }^2}{2},\hfill \\ \hfill D_4[C_{3}cosh\left(D_1\varphi \right)+C_{4}sinh\left(D_1\varphi \right)]-\frac{1}{R}(C_8{R}^2+C_5+C_6{D}_2\varphi )cos\varphi \hfill \\ \hfill +\frac{1}{R}(C_6{R}^2-C_7-C_8{D}_2\varphi )sin\varphi -C_9,\hfill \end{array}$ (A1)

where [TeX:] \phi = x/R$\varphi =x/R$, [TeX:] C_{1},\ \ldots,\ C_{9}$C_1,\dots ,C_9$ are integration constants, while [TeX:] D_{1},\ \ldots,\ D_{4}$D_1,\dots ,D_4$ and [TeX:] H_{1},\ \ldots,\ H_{3}$H_1,\dots ,H_3$ are additional auxiliary constants defined as follows:

[TeX:] \begin{matrix} D_{1} = \frac{kn}{\sqrt{n^{2} - 2}},D_{2} = - \frac{R^{2}}{2}\frac{k^{2} + n^{2} + 4}{n^{2} + 1},D_{3} = \frac{R^{2}}{D_{1}}\frac{H_{1}}{H_{2}},D_{4} = - \frac{R}{D_{1}}\frac{k^{2}\left( n^{2} + 2 \right)}{H_{2}}, \\ H_{1}\ = \ 2k^{2} - n^{2} + 2,H_{2}\ = \ n^{2}\left( k^{2} + 1 \right) - 2,H_{3}\ = \ k^{2}n^{2}\left( k^{2} + n^{2} + 1 \right) + 3n^{2}\left( n^{2} - 1 \right) - 6, \end{matrix}$\begin{array}{c}\hfill D_1=\frac{kn}{\sqrt{n^2-2}},D_2=-\frac{R^2}{2}\frac{k^2+n^2+4}{n^2+1},D_3=\frac{R^2}{D_1}\frac{H_1}{H_2},D_4=-\frac{R}{D_1}\frac{k^2(n^2+2)}{H_2},\hfill \\ \hfill H_1=2k^2-n^2+2,H_2=n^2(k^2+1)-2,H_3=k^2{n}^2(k^2+n^2+1)+3n^2(n^2-1)-6,\hfill \end{array}$ (A2)

and

[TeX:] \begin{matrix} C_{1} = w_{C0} + R\alpha_{0} + \frac{R}{GI_{t}}\left\lbrack M_{y0}R + B_{0} - \frac{q_{z}R^{3}}{D_{1}^{2}} + \left( q_{z}R^{3} - m_{x}R^{2} \right)\left( 1 + \frac{1}{n^{2}} \right) \right\rbrack, \\ C_{2} = \frac{R^{3}}{GI_{t}}Q_{z0}, \\ C_{3} = \frac{D_{1}}{GI_{t}}\left\lbrack - \frac{H_{1} - k^{2}}{H_{2}}M_{y0} + \frac{B_{0}}{R} + R\frac{(H_{1} - k^{2})q_{z}R - k^{2}(n^{2} + 1)m_{x}}{D_{1}^{2}H_{2}} \right\rbrack, \\ C_{4} = \vartheta_{0} + \frac{R^{2}}{EI_{\omega}H_{2}}\left\lbrack \left( 1 - \frac{H_{1}}{k^{2}} \right)Q_{z0}R + (n^{2} + 1)M_{x0} \right\rbrack, \\ C_{5} = - R\alpha_{0} + \frac{R^{3}}{EI_{\omega}H_{2}}\left\lbrack \left( \frac{H_{1} - n^{4} + 4}{H_{2}} - n^{2} \right)M_{y0}R - \left( n^{2} + 2 \right)B_{0} \right\rbrack + \\ + \frac{R^{5}}{EI_{\omega}H_{2}^{2}}\left\{ - q_{z}R\left\lbrack H_{3} + 2(H_{2} - H_{1}) + k^{2}n^{2}(n^{2} + 2) \right\rbrack \right.\ \left. \ + m_{x}\left\lbrack H_{3} + (n^{2} - 2)H_{1} \right\rbrack \right\}, \\ C_{6} = \frac{R^{2}}{EI_{\omega}}\frac{n^{2} + 1}{H_{2}}\left( - Q_{z0}R + M_{x0} \right), \\ C_{7} = - R\beta_{0} - R^{2}\frac{H_{1}}{H_{2}}\vartheta_{0} + \frac{R^{4}}{2EI_{\omega}H_{2}^{2}}\left\{ \left\lbrack H_{3} - 3H_{1} \right\rbrack M_{x0} - \right.\ \left. \ \left\lbrack 2\left( n^{2} + 2 \right)H_{2} + H_{3} - 3H_{1} \right\rbrack Q_{z0}R \right\}, \\ C_{8} = \frac{R^{2}}{EI_{\omega}}\frac{n^{2} + 1}{H_{2}}\left( M_{y0} + q_{z}R^{2} - m_{x}R \right), \\ C_{9} = \frac{R^{2}}{EI_{y}}\left( q_{z}R\frac{k^{2} + n^{2} + 2}{k^{2}} - m_{x} \right). \end{matrix}$\begin{array}{c}\hfill C_1=w_{C0}+R{\alpha }_0+\frac{R}{G{I}_t}[M_{y0}R+B_0-\frac{q_z{R}^3}{D_1^2}+(q_z{R}^3-m_x{R}^2)(1+\frac{1}{n^2})],\hfill \\ \hfill C_2=\frac{R^3}{G{I}_t}{Q}_{z0},\hfill \\ \hfill C_3=\frac{D_1}{G{I}_t}[-\frac{H_1-k^2}{H_2}{M}_{y0}+\frac{B_0}{R}+R\frac{(H_1-k^2)q_{z}R-k^2(n^2+1)m_x}{D_1^2{H}_2}],\hfill \\ \hfill C_4={\vartheta }_0+\frac{R^2}{E{I}_{\omega }{H}_2}[(1-\frac{H_1}{k^2})Q_{z0}R+(n^2+1)M_{x0}],\hfill \\ \hfill C_5=-R{\alpha }_0+\frac{R^3}{E{I}_{\omega }{H}_2}[(\frac{H_1-n^4+4}{H_2}-n^2)M_{y0}R-(n^2+2)B_0]+\hfill \\ \hfill +\frac{R^5}{E{I}_{\omega }{H}_2^2}\{-q_{z}R[H_3+2(H_2-H_1)+k^2{n}^2(n^2+2)]+m_x[H_3+(n^2-2)H_1]\},\hfill \\ \hfill C_6=\frac{R^2}{E{I}_{\omega }}\frac{n^2+1}{H_2}(-Q_{z0}R+M_{x0}),\hfill \\ \hfill C_7=-R{\beta }_0-R^2\frac{H_1}{H_2}{\vartheta }_0+\frac{R^4}{2E{I}_{\omega }{H}_2^2}\{[H_3-3H_1]M_{x0}-[2(n^2+2)H_2+H_3-3H_1]Q_{z0}R\},\hfill \\ \hfill C_8=\frac{R^2}{E{I}_{\omega }}\frac{n^2+1}{H_2}(M_{y0}+q_z{R}^2-m_{x}R),\hfill \\ \hfill C_9=\frac{R^2}{E{I}_y}(q_{z}R\frac{k^2+n^2+2}{k^2}-m_x).\hfill \end{array}$ (A3)

7. APPENDIX B

Similarly, the closed-form solution of Eq. (15) is:

[TeX:] \begin{matrix} v_{C} = C_{10}\ + \left( C_{11} + C_{12}\phi \right)\cos\phi + \left( C_{13} + C_{14}\phi \right)\sin\phi - q_{x}\frac{R^{4}}{EI_{z}}\phi \end{matrix},$\begin{array}{c}\hfill v_C=C_{10}+(C_{11}+C_{12}\varphi )cos\varphi +(C_{13}+C_{14}\varphi )sin\varphi -q_x\frac{R^4}{E{I}_z}\varphi \hfill \end{array},$ (B1)

where [TeX:] C_{10},\ \ldots,\ C_{14}$C_{10},\dots ,C_{14}$ are integration constants defined as follows:

[TeX:] \begin{matrix} C_{10} = \left( \ M_{z0} + \ N_{0}R + q_{y}R^{2}\ \right)\frac{R^{2}}{EI_{z}}, \\ C_{11} = v_{C0} - C_{10}, \\ C_{12} = \frac{R^{3}}{2EI_{z}}\left( Q_{y0}\ - \ q_{x}R \right), \\ C_{13} = u_{C0} - \gamma_{0}R - C_{12} + \frac{q_{x}R^{4}}{EI_{z}}, \\ C_{14} = - \frac{R^{3}}{2EI_{z}}\left( N_{0}\ + \ q_{y}R \right). \end{matrix}$\begin{array}{c}\hfill C_{10}=(M_{z0}+N_{0}R+q_y{R}^2)\frac{R^2}{E{I}_z},\hfill \\ \hfill C_{11}=v_{C0}-C_{10},\hfill \\ \hfill C_{12}=\frac{R^3}{2E{I}_z}(Q_{y0}-q_{x}R),\hfill \\ \hfill C_{13}=u_{C0}-{\gamma }_{0}R-C_{12}+\frac{q_x{R}^4}{E{I}_z},\hfill \\ \hfill C_{14}=-\frac{R^3}{2E{I}_z}(N_0+q_{y}R).\hfill \end{array}$ (B2)

The displacement [TeX:] u_{C}$u_C$ results from Eqs. (7), (17) and (B1), i.e.:

[TeX:] \begin{matrix} u_{C} = \ \cos\phi\left( C_{12} - \ C_{13} - \ C_{14}\phi \right) + \ \sin\phi\left( C_{11} + \ C_{14} + \ C_{12}\phi \right) + C_{15} + \\ + \frac{\begin{array}{r} \ \phi \end{array}R^{2}}{E\left( \ AR^{2} - I_{z} \right)}\left( AEC_{10} - R^{2}\left( q_{y} + \frac{AR^{2}q_{x}}{2I_{z}}\phi \right) \right), \end{matrix}$\begin{array}{c}\hfill u_C=cos\varphi (C_{12}-C_{13}-C_{14}\varphi )+sin\varphi (C_{11}+C_{14}+C_{12}\varphi )+C_{15}+\hfill \\ \hfill +\frac{\begin{array}{c}\hfill \varphi \end{array}{R}^2}{E(A{R}^2-I_z)}(AE{C}_{10}-R^2(q_y+\frac{A{R}^2{q}_x}{2I_z}\varphi )),\hfill \end{array}$ (B3)

where [TeX:] C_{15}\ $C_{15}$is the integration constant:

[TeX:] C_{15} = \gamma_{0}R - 2C_{12} + \frac{q_{x}R^{4}}{EI_{z}}.$C_{15}={\gamma }_{0}R-2C_{12}+\frac{q_x{R}^4}{E{I}_z}.$ (B4)