### Governing equation of motion for a curved beam

The equation of motion for a curved beam’s post-buckled state, taking into account the effects of shear deformation and rotary inertia, as well as, the extension of the neutral axis, can be written as (Jianbei et al., 2014):

$$ N\left(\frac{1}{R}-\frac{\partial \varphi }{\partial S}\right)-\frac{\partial Q}{\partial S}=-\gamma A\ddot{U} $$

(1)

$$ Q\left(\frac{1}{R}-\frac{\partial \varphi }{\partial S}\right)+\frac{\partial N}{\partial S}=\gamma A\ddot{W} $$

(2)

$$ \frac{\partial M}{\partial S}-N\frac{Q}{kAG}+Q\left(\frac{N}{EA}+1\right)=\gamma A\ddot{\varphi} $$

(3)

$$ \frac{N}{EA}=\left(\left(\frac{\partial U}{\partial S}+\frac{W}{R}+1\right)\cos \left(\varphi \right)+\left(\frac{\partial W}{\partial S}-\frac{U}{R}\right)\sin \left(\varphi \right)-1\right) $$

(4)

$$ \frac{Q}{kAG}=\left(-\left(\frac{\partial U}{\partial S}+\frac{W}{R}+1\right)\mathit{\sin}\left(\varphi \right)+\left(\frac{\partial W}{\partial S}-\frac{U}{R}\right)\mathit{\cos}\left(\varphi \right)\right) $$

(5)

$$ M= EI\frac{\partial \varphi }{\partial S}, $$

(6)

where dot means the derivative with respect to time. As shown in Fig. 1, which represents an element of a curved beam, *W*, *U*, and *φ* denote the radial and tangential displacements, and the angle of rotation. Parameters *M*, *N*, and *Q* show the bending moment, normal and shear forces respectively. Moreover, *A*, *I*, *γ*, *G*, *E*, and *k* are the cross-section area of the beam, area moment of inertia of the cross-section, mass density per unit volume of the beam material, shear and Young’s modulus of elasticity, and shear factor of the cross-section, respectively.

In what follows, the equations of motion are solved by the DQE method which is one of the efficient numerical methods for fast solving linear and nonlinear differential equations.

### DQEM

The differential quadrature element method (DQEM) is a new and efficient numerical method for rapidly solving linear and nonlinear differential equations. The method is based on the differential quadrature (DQ) method which is an approximate method for expressing partial derivatives of a function at a point located in the domain of the function, as the weighted linear sum of the values of the variable function at all the defined precision points in the derivation direction. Equation (7) is the mathematical representation of the DQ expansion (Chen, 2005):

$$ {\left.\kern0.50em \frac{df}{dx}\right|}_{x={x}_i}=\sum \limits_{j=1}^N{C}_{ij}^{(1)}{f}_j,i,j=1,2,3,\dots, N $$

(7)

where *f* is the desired function, *N* is the number of precision points, *x*_{i} is the precision associated with the *i*th point of the function domain and also represents the weighting coefficients used to find the first derivative of the function at the *i*th precision point of the function domain.

In our case study, the beam is discretized to *m* elements and each element itself is divided into *N* nodes (see Fig. 2). These nodes are the aforementioned precision points. In order to use Eq. (7), the physical coordinate*θ* is transformed to the natural coordinate *x* with such the following relation (Chen, 2005):

$$ \theta =\left(1-x\right){\theta}_1+x{\theta}_N $$

(8)

Where *θ*_{1} and *θ*_{N} are the angular coordinates of the first and *N*th nodes of the element respectively. Note that *x* is a value in [0, 1] domain, i.e., (0 <*x*<1). In fact, applying this relation, the differentiation with respect to angular coordinate *θ* can be obtained using Eq. (8). According to Eqs. (7) and (8), there are two important factors in applying the DQ method; firstly, calculation of DQ weighting coefficients and, secondly, selection of the precision points. In this article, the Lagrangian functions were used to compute the weighted coefficients, and the Gauss–Lobatto Chebyshev polynomial was used to select the precision points. Finally, it is to be noted that to reach overall consistency, the continuity conditions at inter-element boundaries of two adjacent elements and the boundary conditions of the whole beam, as well as the governing equations on each element, must be satisfied.

As indicated previously, the beam is discretized to *m* elements and each element itself is divided into *N* nodes (see Fig. 2). Assuming a harmonic excitation of *ω* frequency for *U*, *W*, and *φ* in Eqs. (1–3), i.e., \( \ddot{U}=-{\omega}^2U \), \( \ddot{W}=-{\omega}^2W \), and \( \ddot{\varphi}=-{\omega}^2\varphi \), then applying the DQ discretization to the equations of motion at an interior node *m*^{i} of the element *i*, anybody can reach to such the following discrete equations (Jianbei et al., 2014):

$$ {N}_{m^i}^i\left(\frac{1}{R}-\frac{\partial {\varphi}_{m^i}^i}{\partial S}\right)-\frac{\partial {Q}_{m^i}^i}{\partial S}={\omega}^2{\gamma}^i{A}^i{U}_{m^i}^i, $$

(9)

$$ {Q}_{m^i}^i\left(\frac{1}{R}-\frac{\partial {\varphi}_{m^i}^i}{\partial S}\right)+\frac{\partial {N}_{m^i}^i}{\partial S}=-{\omega}^2{\gamma}^i{A}^i{W}_{m^i}^i, $$

(10)

$$ \frac{\partial {M}_{m^i}^i}{\partial S}-{N}_{m^i}^i\frac{Q_{m^i}^i}{KAG}+{Q}_{m^i}^i\left(\frac{N_{m^i}^i}{EA}+1\right)=-{\omega}^2{\gamma}^i{A}^i{\varphi}_{m^i}^i, $$

(11)

$$ {N}_{m^i}^i= EA\left(\left(\frac{\partial {U}_{m^i}^i}{\partial S}+\frac{W_{m^i}^i}{R}+1\right)\cos \left({\varphi}_{m^i}^i\right)+\left(\frac{\partial {W}_{m^i}^i}{\partial S}-\frac{U_{m^i}^i}{R}\right)\sin \left({\varphi}_{m^i}^i\right)-1\right), $$

(12)

$$ {Q}_{m^i}^i= kAG\left(-\left(\frac{\partial {U}_{m^i}^i}{\partial S}+\frac{W_{m^i}^i}{R}+1\right)\mathit{\sin}\left({\varphi}_{m^i}^i\right)+\left(\frac{\partial {W}_{m^i}^i}{\partial S}-\frac{U_{m^i}^i}{R}\right)\mathit{\cos}\left({\varphi}_{m^i}^i\right)\right), $$

(13)

$$ {M}_{m^i}^i=E{I}^i\frac{\partial {\varphi}_{m^i}^i}{\partial S}. $$

(14)

To solve the nonlinear vibration of the post-buckled curved beam (Eqs. (9–14)), first, the system of equations is solved statically to determine what the equilibrium shape is. As a result, considering small harmonic variations around the post-buckled equilibrium configuration, the solution can be written as the sum of the equilibrium and harmonic parts in the following form:

$$ {U}_{m^i}^i={\left({U}_{m^i}^i\right)}_e+{\left({U}_{m^i}^i\right)}_d\;\sin \left(\omega t\right), $$

(15)

$$ {W}_{m^i}^i={\left({W}_{m^i}^i\right)}_e+{\left({W}_{m^i}^i\right)}_d\;\sin \left(\omega t\right), $$

(16)

$$ {\varphi}_{m^i}^i={\left({\varphi}_{m^i}^i\right)}_e+{\left({\varphi}_{m^i}^i\right)}_d\;\sin \left(\omega t\right), $$

(17)

$$ {N}_{m^i}^i={\left({N}_{m^i}^i\right)}_e+{\left({N}_{m^i}^i\right)}_d\;\sin \left(\omega t\right), $$

(18)

$$ {Q}_{m^i}^i={\left({Q}_{m^i}^i\right)}_e+{\left({Q}_{m^i}^i\right)}_d\;\sin \left(\omega t\right), $$

(19)

$$ {M}_{m^i}^i={\left({M}_{m^i}^i\right)}_e+{\left({M}_{m^i}^i\right)}_d\;\sin \left(\omega t\right). $$

(20)

Eliminating the time-dependent terms in Eqs. (9–11), the following equilibrium equations are obtained:

$$ {\left({N}_{m^i}^i\right)}_e\left(\frac{1}{R}-\frac{\partial {\left({\varphi}_{m^i}^i\right)}_e}{\partial S}\right)-\frac{\partial {\left({Q}_{m^i}^i\right)}_e}{\partial S}=0, $$

(21)

$$ {\left({Q}_{m^i}^i\right)}_e\left(\frac{1}{R}-\frac{\partial {\left({\varphi}_{m^i}^i\right)}_e}{\partial S}\right)+\frac{\partial {\left({N}_{m^i}^i\right)}_e}{\partial S}=0, $$

(22)

$$ \frac{\partial {\left({M}_{m^i}^i\right)}_e}{\partial S}-{\left({N}_{m^i}^i\right)}_e\frac{{\left({Q}_{m^i}^i\right)}_e}{KAG}+{\left({Q}_{m^i}^i\right)}_e\left(\frac{{\left({N}_{m^i}^i\right)}_e}{EA}+1\right)=0, $$

(23)

$$ \frac{{\left({N}_{m^i}^i\right)}_e}{EA}-\left(\left(\frac{\partial {\left({U}_{m^i}^i\right)}_e}{\partial S}+\frac{{\left({W}_{m^i}^i\right)}_e}{R}+1\right)\cos \left({\left({\varphi}_{m^i}^i\right)}_e\right)+\left(\frac{\partial {\left({W}_{m^i}^i\right)}_e}{\partial S}-\frac{{\left({U}_{m^i}^i\right)}_e}{R}\right)\sin \left({\left({\varphi}_{m^i}^i\right)}_e\right)-1\right)=0, $$

(24)

$$ \frac{{\left({Q}_{m^i}^i\right)}_e}{kAG}-\left(-\left(\frac{\partial {\left({U}_{m^i}^i\right)}_e}{\partial S}+\frac{{\left({W}_{m^i}^i\right)}_e}{R}+1\right)\mathit{\sin}\left({\left({\varphi}_{m^i}^i\right)}_e\right)+\left(\frac{\partial {\left({W}_{m^i}^i\right)}_e}{\partial S}-\frac{{\left({U}_{m^i}^i\right)}_e}{R}\right)\mathit{\cos}\left({\left({\varphi}_{m^i}^i\right)}_e\right)\right)=0, $$

(25)

$$ {\left({M}_{m^i}^i\right)}_e-E{I}^i\frac{\partial {\left({\varphi}_{m^i}^i\right)}_e}{\partial S}=0. $$

(26)

Now, small harmonic variations are considered around the post-buckled static equilibrium state. Substituting Eqs. (15–20) into (9–14), and removing the nonlinear terms, the linear dynamic equations of motion can be obtained as

$$ {\left({N^i}_{m^i}\right)}_e\left(-\frac{\partial {\left({\varphi^i}_{m^i}\right)}_d}{\partial S}\right)+{\left({N^i}_{m^i}\right)}_d\left(\frac{1}{R}-\frac{\partial {\left({\varphi^i}_{m^i}\right)}_e}{\partial S}\right)-\frac{\partial {\left({Q^i}_{m^i}\right)}_d}{}=-{\omega}^2{\gamma}^i{I}^i{\left({U}_{m^i}^i\right)}_d, $$

(27)

$$ {\left({Q}_{m^i}^i\right)}_e\left(-\frac{\partial {\left({\varphi}_{m^i}^i\right)}_d}{\partial S}\right)+{\left({Q}_{m^i}^i\right)}_d\left(\frac{1}{R}-\frac{\partial {\left({\varphi}_{m^i}^i\right)}_e}{\partial S}\right)+\frac{\partial {\left({N}_{m^i}^i\right)}_d}{\partial S}={\omega}^2{\gamma}^i{I}^i{\left({W}_{m^i}^i\right)}_d, $$

(28)

$$ {\displaystyle \begin{array}{l}\frac{\partial {\left({M}_{m^i}^i\right)}_d}{\partial S}+{\left({Q}_{m^i}^i\right)}_d{\left({N}_{m^i}^i\right)}_e\left(\frac{1}{EA}-\frac{1}{KAG}\right)+{\left({N}_{m^i}^i\right)}_d{\left({Q}_{m^i}^i\right)}_e\left(\frac{1}{EA}-\frac{1}{KAG}\right)+{\left({Q}_{m^i}^i\right)}_d\\ {}\kern19.80001em ={\omega}^2{\gamma}^i{I}^i{\left({\varphi}_{m^i}^i\right)}_d,\end{array}} $$

(29)

$$ {\displaystyle \begin{array}{l}\frac{{\left({N}_{m^i}^i\right)}_d}{EA}-{\left({\varphi}_{m^i}^i\right)}_d\left(\frac{{\left({Q}_{m^i}^i\right)}_e}{kAG}\right)-\left(\frac{\partial {\left({W}_{m^i}^i\right)}_d}{\partial S}-\frac{{\left({U}_{m^i}^i\right)}_d}{R}\right)\sin \left({\left({\varphi}_{m^i}^i\right)}_e\right)\\ {}-\left(\frac{\partial {\left({U}_{m^i}^i\right)}_d}{\partial S}+\frac{{\left({W}_{m^i}^i\right)}_d}{R}\right)\cos \left({\left({\varphi}_{m^i}^i\right)}_e\right)=0,\end{array}} $$

(30)

$$ {\displaystyle \begin{array}{l}\frac{{\left({Q}_{m^i}^i\right)}_d}{kAG}+{\left({\varphi}_{m^i}^i\right)}_d\left(\frac{{\left({N}_{m^i}^i\right)}_e}{EA}+1\right)+\left(\frac{\partial {\left({U}_{m^i}^i\right)}_d}{\partial S}+\frac{{\left({W}_{m^i}^i\right)}_d}{R}\right)\sin \left({\left({\varphi}_{m^i}^i\right)}_e\right)\\ {}-\left(\frac{\partial {\left({W}_{m^i}^i\right)}_d}{\partial S}-\frac{{\left({U}_{m^i}^i\right)}_d}{R}\right)\cos \left({\left({\varphi}_{m^i}^i\right)}_e\right)=0,\end{array}} $$

(31)

$$ {\left({M}_{m^i}^i\right)}_d-E{I}^i\frac{\partial {\left({\varphi}_{m^i}^i\right)}_d}{\partial S}=0. $$

(32)

Continuity conditions must also be applied at the interface of the beam segments. To have a better look at the equilibrium condition and the corresponding variation, the beam under a concentrated load (*P*), and its deformed centerline is shown in Fig. 3. In this figure, the solid arc is the initial shape of the beam and the dotted curve is the deformed shape of the centerline.

The radial and tangential displacements and the angular rotation continuity conditions at the inter-element boundaries of two adjacent elements *i* and *i +* 1, except for the peak of the beam, are expressed as

$$ {W}_{N^i}^i={W}_1^{i+1},\kern0.36em {U}_{N^i}^i={U}_1^{i+1},\kern0.36em {\varphi}_{N^i}^i={\varphi}_1^{i+1} $$

(33)

Based on the action-reaction rule, to satisfy the continuity conditions at the inter-elements boundary of two adjacent elements, the normal and shear forces and the bending moment of the last node of segment *i* must be equal to corresponding loads of the first node of segment *i* + 1. In fact, with reference to Eqs. (12–14), the normal and shear forces and the bending moment continuity conditions at the inter-element boundary of two adjacent elements *i* and *i +* 1, can be expressed, respectively, as

$$ {\displaystyle \begin{array}{l}\ {E}^i{A}^i\left(\left(\frac{\partial {u}_{N^i}^i}{\partial S}+\frac{w_{N^i}^i}{R}+1\right)\cos \left({\varphi}_{N^i}^i\right)+\left(\frac{\partial {w}_{N^i}^i}{\partial S}-\frac{u_{N^i}^i}{R}\right)\sin \left({\varphi}_{N^i}^i\right)-1\right)=\\ {}{E}^{i+1}{A}^{i+1}\left(\left(\frac{\partial {u}_1^{i+1}}{\partial S}+\frac{w_1^{i+1}}{R}+1\right)\cos \left({\varphi}_1^{i+1}\right)+\left(\frac{\partial {w}_1^{i+1}}{\partial S}-\frac{u_1^{i+1}}{R}\right)\sin \left({\varphi}_1^{i+1}\right)-1\right),\end{array}} $$

(34)

$$ {\displaystyle \begin{array}{l}\ {k}^i{A}^i{G}^i\left(-\left(\frac{\partial {u}_{N^i}^i}{\partial S}+\frac{w_{N^i}^i}{R}+1\right)\mathit{\sin}\left({\varphi}_{N^i}^i\right)+\left(\frac{\partial {w}_{N^i}^i}{\partial S}-\frac{u_{N^i}^i}{R}\right)\mathit{\cos}\left({\varphi}_{N^i}^i\right)\right)=\\ {}{k}^{i+1}{A}^{i+1}{G}^{i+1}\left(-\left(\frac{\partial {u}_1^{i+1}}{\partial S}+\frac{w_1^{i+1}}{R}+1\right)\mathit{\sin}\left({\varphi}_1^{i+1}\right)+\left(\frac{\partial {w}_1^{i+1}}{\partial S}-\frac{u_1^{i+1}}{R}\right)\mathit{\cos}\left({\varphi}_1^{i+1}\right)\right),\end{array}} $$

(35)

$$ {E}^i{I}^i\frac{\partial {\varphi}_{N^i}^i}{\partial {S}_{N^i}^i}={E}^{i+1}{I}^{i+1}\frac{\partial {\varphi}_1^{i+1}}{\partial {S}_1^i}. $$

(36)

The continuity conditions at the peak of the arch in the static state are

$$ {\displaystyle \begin{array}{l}\ {U}_{N^i}^i={U}_1^{i+1},{\varphi}_{N^i}^i={\varphi}_1^{i+1},{W}_{N^i}^i={w}_0,\kern0.36em {W}_1^{i+1}={w}_0\\ {}\frac{\partial {u}_{N^i}^i}{\partial S}=\frac{\partial {u}_1^{i+1}}{\partial S},\kern0.36em \frac{\partial {\varphi}_{N^i}^i}{\partial S}=\frac{\partial {\varphi}_1^{i+1}}{\partial S},\kern0.48em i=\frac{m}{2},\end{array}} $$

(37)

where *w*_{0} is the radial displacement of the middle point of the beam (see Fig. 3).

The continuity conditions at the peak for the dynamic state are similar to Eq. (37) except for *w*_{0} which is considered to be 0.

To make the system of equations solvable, three boundary conditions must be considered at each endpoint of the beam. Boundary conditions for a beam clamped at both ends in static and dynamic states are

$$ {W}_1^1=0\kern0.75em {\varphi}_1^1=0\kern1.25em {U}_1^1=0, $$

(38)

$$ {W}_{N^i}^m=0\kern1em {\varphi}_{N^i}^m=0\kern1.25em {U}_{N^i}^m=0\kern0.5em . $$

(39)

Noting that for a perfect circular arch without any imperfection, the arch only deforms under a fully symmetric mode, so the beam is assumed to have some geometrical imperfections. In this study, the geometric imperfection functions are represented in terms of the first modal shape of the beam with an arbitrary amplitude of 0.001.

Applying the differential quadrature element method to the equations of motion of the beam, as well as, the continuity and boundary conditions, they are transformed into an algebraic system of eigenvalue problem that must be solved in terms of natural frequencies and mode shapes.

### Formula for the buckling load

Regarding Fig. 3, if the load *P* is increased to a point which makes the beam unstable (the buckling load), the beam as a part of a detailed structure undergoes a large displacement with a small excitation. Since this phenomenon can fail the whole structure, predicting this force is an important part of a design in this respect. To model the application of buckling load, the radial displacement of the peak of the beam (*w*_{0}) is defined as the input of the arc length strategy meeting the converge criterion; so, the buckling load (*P*_{c}) is obtained as (Jianbei et al., 2014)

$$ {\displaystyle \begin{array}{l}\ {F}_1={N}_N^i\;\sin \left({\varphi}_N^i\right)+{Q}_N^i\;\cos \left({\varphi}_N^i\right),\\ {}{F}_2={N}_1^{i+1}\;\sin \left({\varphi}_1^{i+1}\right)+{Q}_1^{i+1}\;\cos \left({\varphi}_1^{i+1}\right),\\ {}{P}_c=\mid {F}_1-{F}_2\mid, \kern0.48em i=\frac{m}{2}.\end{array}} $$

(40)

In the static analysis, this force can be equalized to the radial displacement in the continuity conditions at the peak of the arch as

$$ {\displaystyle \begin{array}{l}{U}_{N^i}^i={U}_1^{i+1},{\varphi}_{N^i}^i={\varphi}_1^{i+1},{W}_{N^i}^i={W}_1^{i+1}\\ {}\frac{\partial {u}_{N^i}^i}{\partial S}=\frac{\partial {u}_1^{i+1}}{\partial S},\kern0.36em \frac{\partial {\varphi}_{N^i}^i}{\partial S}=\frac{\partial {\varphi}_1^{i+1}}{\partial S}\\ {}{F}_1={N}_N^i\;\sin \left({\varphi}_N^i\right)+{Q}_N^i\;\cos \left({\varphi}_N^i\right),{F}_2={N}_1^{i+1}\;\sin \left({\varphi}_1^{i+1}\right)+{Q}_1^{i+1}\;\cos \left({\varphi}_1^{i+1}\right)\\ {}P=\mid {F}_1-{F}_2\mid, \kern0.48em i=\frac{m}{2}\kern1.44em \end{array}} $$

(41)