Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations

Abstract

This paper presents exact solutions for the nonlinear bending problem, the buckling loads, and postbuckling configurations of a perfect and an imperfect bioinspired helicoidal composite beam with a linear rotation angle. The beam is embedded on an elastic medium, which is modeled by two elastic foundation parameters. The nonlinear integro-differential governing equation of the system is derived based on the Euler–Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial curvature. The Laplace transform and its inversion are directly applied to solve the nonlinear integro-differential governing equations. The nonlinear bending deflections under point and uniform loads are derived. Closed-form formulas of critical buckling loads, as well as nonlinear postbuckling responses of perfect and imperfect beams are deduced in detail. The proposed model is validated with previous works. In the numerical results section, the effects of the rotation angle, amplitude of initial imperfection, elastic foundation constants, and boundary conditions on the nonlinear bending, critical buckling loads, and postbuckling configurations are discussed. The proposed model can be utilized in the analysis of bio-inspired beam structures that are used in many energy-absorption applications.

1. Introduction

Due to their exceptional strength and stiffness properties, the usage of laminated composite (LC) structures in engineering applications such as automotive, aerospace, spacecraft, energy harvester, and marine structures has increased intensely [1]. In nature, there are many examples of helicoidal structures, such as the osteons in mammalian bones, certain plant cell walls, various insect cuticles, and DNA structure [2,3]. Motivated by these examples, several authors explored the applicability of twisted laminae arrangements rather than the conventional design of laminates [4,5]. Using the finite element (FE) method, Morozov et al. [6] studied the buckling of anisotropic grid composite lattice cylindrical shells under different loading conditions. Kacar and Yildirim [7] investigated the buckling and natural frequencies of noncylindrical unidirectional composite helical springs subjected to initial static axial force and moment. Ginzburg et al. [3] studied damage tolerance of bioinspired helicoidal composites under low-velocity impact. Sabah et al. [8,9] illustrated the dynamic response and failure mode maps of bioinspired conventional sandwich composite beams under low-velocity impact. By using a refined beam model, She et al. [10] presented thermal buckling/postbuckling of functionally graded (FG) tubes resting on elastic foundations under a uniform temperature distribution. She et al. [11,12] studied analytically the postbuckling and resonance of porous FG curved nonlocal strain gradient nanobeams. Eltaher et al. [13] studied numerically the buckling stability of LC beams under varying axial in-plane loads by using unified beam theory. Qian et al. [14] examined the bioinspired bistable piezoelectric harvester for broadband vibration energy harvesting. Kueh and Siaw [15] explored computationally the impact resistance of the proposed bioinspired sandwich curved beam. Fani and Behrooz [16] investigated analytically the thermal buckling/postbuckling of LC beams reinforced with shape memory alloy via the Reddy Bickford theory. Lu et al. [17] explored analytically the postbuckling of geometrically imperfect graphene-reinforced composite microtubes in the frame of modified couple stress theory. Hosseini and Arvin [18] presented the buckling and postbuckling of rotating FG microbeams in thermal environment based on the Euler–Bernoulli beam assumption. Sojobi and Liew [19] performed multi-objective optimization to predict the high performance of bio-inspired prefabricated composites for sustainable and resilient construction. Melaibari et al. [20] studied free vibration of FG shell and nanoshell with different geometries using the analytical Galerkin method.

Even though beams as a structural element have gained interest due to their applications in aerospace, marine, civil engineering, and MEMS/NEMS, they may suffer from geometrical imperfections and curvature through manufacture or prior to usage in the application. Therefore, it is necessary to consider this point in order to obtain the accurate design and response. As one of the earliest endeavors for the vibration analysis of curved structures, Petyt and Fleischer [21] studied the free radial vibration of a curved beam using FE models. Wang and Abdalla [22] numerically explored the global and local buckling analysis of grid-stiffened composite panels. Li and Qiao [23] studied buckling and postbuckling behavior of shear deformable anisotropic laminated beams with initial geometric (sine, local, and global type) imperfections subjected to axial compression by Newton’s iterative and Galerkin’s methods. Wu et al. [24] investigated the sensitivity of imperfections on postbuckling behavior of FG-CNT beams based on the first-order shear deformation beam theory with a von Kármán geometric nonlinearity. Wu et al. [25] illustrated the impact of thermoelectro-mechanical properties on the postbuckling of piezoelectric FG-CNTRC imperfect beams via the differential quadrature method. Wu et al. [26] studied the postbuckling response of FG-CNT imperfect beams under a parabolic variation of thermal load considering different modes of imperfection. Mohamed et al. [27] investigated the nonlinear free and forced vibrations of a curved beam in the vicinity of postbuckling configuration by using the numerical approach. Eltaher et al. [28] studied the effects of periodic sine/cosine and nonperiodic imperfection modes on buckling, postbuckling, and dynamics of isotropic beam rested on nonlinear elastic foundations. Mohamed et al. [29] and Eltaher et al. [30] exploited the energy equivalent model in the analysis of buckling and postbuckling of an imperfect carbon nanotube (CNTs) installed on a nonlinear elastic foundation. Dabbagh et al. [31] studied postbuckling of multi-scale hybrid nanocomposite beams installed on a nonlinear stiff substrate within the framework of the Galerkin’s method. Emam and Lacarbonara [32] analytically analyzed the buckling and postbuckling response of extensible, shear deformable beams with different boundary conditions. Zhu et al. [33] presented analytical solutions for nonlinear stability and dynamic behaviors of fluid conveying FG imperfect pipes. Xu et al. [34] developed an exact model to show the effect of nanovoids distribution on forced vibration of FG curved nanobeams. In the framework of higher-order shear theory, Mohamed et al. [35] studied buckling and postbuckling behaviors of CNTs via an energy-equivalent model. Golewski [36] presented a comprehensive review of the different types of dynamic loads acting on concrete structures and procedures to diagnose their effects. Zhang et al. [37] studied bistable characteristics of hybrid composite laminates embedded with bimetallic strips.

With the aid of the Laplace transform, Zhang and Qing [38] analyzed the nonlinear bending behavior of Timoshenko curved beams based on a modified nonlocal strain gradient model. Li et al. [39] solved the problems of linear bending, buckling, and free vibrations of an axially loaded Timoshenko beam. They applied the Laplace transform to solve the bending problem. Nazmul and Devnath [40] analyzed the bending of 2D FG nanobeams based on Eringen’s nonlocal elasticity theory by utilizing the Laplace transformation. The Laplace transform and its inversion were used to solve the problems of elastic buckling and free vibration response of the FG material Timoshenko beam, based on the nonlocal strain gradient integral model by Tang and Qing [41].

Based on a review of previous works, it is concluded that the analytical investigation of the buckling/postbuckling and nonlinear bending of bioinspired helicoidal composite beam installed on a linear substrate has not been considered elsewhere. Therefore, this article aims to discuss this topic comprehensively. The rest of this article is organized as follows. The problem formulation and governing equation is discussed in Section 2. In Section 3, the Laplace transform method is employed to deduce the exact solution of the nonlinear bending deflection. In Section 4, the buckling problem is considered and exact solutions of critical buckling load and postbuckling response are derived. Through Section 5, numerical calculations are performed to investigate the effects of different parameters on the bending and buckling responses. Finally, some conclusions are drawn in Section 6.

2. Problem Formulation

A laminated composite simply supported beam of $N_L$ layers with uniform total thickness $h$, length $L$, and width $b$ has a helicoidal lamination scheme as shown in Figure 1.

The Euler–Bernoulli beam theory is adopted to describe the displacement field of composite beam structures as [42]:

$$U(\widehat{x},\widehat{z},\widehat{t})=\widehat{u}(\widehat{x},\widehat{t})-\widehat{z}\left[\frac{\partial \widehat{w}(\widehat{x},\widehat{t})}{\partial \widehat{x}}-\frac{d{\widehat{w}}_0(\widehat{x},\widehat{t})}{d\widehat{x}}\right]$$

(1a)

$$W(\widehat{x},\widehat{z},\widehat{t})=\widehat{w}(\widehat{x},\widehat{t})$$

(1b)

in which $U$ and $W$ are the axial and lateral displacements at any generic point of the beam domain. $\widehat{u}$ and $\widehat{w}$ are the displacements at the neutral axis and ${\widehat{w}}_0$ denotes the beam’s initial rise. The strain due to deformation is given by Boutahar et al. [43]:

$${\epsilon }_x={\epsilon }_0-\widehat{z}{\varkappa }_0$$

(2)

in which ${\epsilon }_0$ is the axial strain and ${\varkappa }_0$ is curvature in the x–z plane, which are defined as [42]:

$${\epsilon }_0=\frac{d\widehat{u}}{d\widehat{x}} + \frac{1}{2}\left[{\left(\frac{\partial \widehat{w}\left(\widehat{x}\right)}{d\widehat{x}}\right)}^2-{\left(\frac{d{\widehat{w}}_0\left(\widehat{x}\right)}{d\widehat{x}}\right)}^2\right]$$

(3a)

$${\varkappa }_0=\frac{d^2\widehat{w}(\widehat{x},\widehat{t})}{d{\widehat{x}}^2}-\frac{d^2{\widehat{w}}_0(\widehat{x})}{d{\widehat{x}}^2}$$

(3b)

The resultant axial force $N$ and bending moment $M$ can be defined as [27]:

$$N=b\left(A_{11}{\epsilon }_0+B_{11}{\varkappa }_0\right)$$

(4a)

$$M=b\left(B_{11}{\epsilon }_0+D_{11}{\varkappa }_0\right)$$

(4b)

The laminated in-plane rigidities $A_{ij}$, $B_{ij}$, and $D_{ij}$ are described by:

$$\left(A_{ij},B_{ij}, D_{ij}\right)=\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{\overline{Q}}_{ij}\left[1,\widehat{z},{\widehat{z}}^2\right]d\widehat{z}, \left(i, j=1, 2, 6\right)$$

(5)

The transformed reduced stiffness material constants at any fiber orientation angle $\theta$, can be identified as [13]:

$${\overline{Q}}_{11}=Q_{11}{\cos }^4\left(\theta \right)+\left(Q_{12}+2Q_{66}\right){\sin }^2\left(2\theta \right)+Q_{22}{\sin }^4\left(\theta \right)$$

(6a)

$${\overline{Q}}_{12}=\frac{1}{2}\left(Q_{11}+Q_{22}-4Q_{66}\right){\sin }^2\left(2\theta \right)+Q_{12}\left({\sin }^4\left(\theta \right)+{\cos }^4\left(\theta \right)\right)$$

(6b)

$${\overline{Q}}_{22}=Q_{11}{\sin }^4\left(\theta \right)+\left(Q_{12}+2Q_{66}\right){\sin }^2\left(2\theta \right)+Q_{22}{\cos }^4\left(\theta \right)$$

(6c)

$${\overline{Q}}_{16}=\frac{1}{2}\left[\left(Q_{11}-Q_{12}-2Q_{66}\right){\cos }^2\left(\theta \right)+\left(Q_{12}-Q_{22}+2Q_{66}\right){\sin }^2\left(\theta \right)\right]\sin \left(2\theta \right)$$

(6d)

$${\overline{Q}}_{26}=\frac{1}{2}\left[\left(Q_{11}-Q_{12}-2Q_{66}\right){\sin }^2\left(\theta \right)+\left(Q_{12}-Q_{22}+2Q_{66}\right){\cos }^2\left(\theta \right)\right]\sin \left(2\theta \right)$$

(6e)

$${\overline{Q}}_{66}=\frac{1}{2}\left(Q_{11}+Q_{22}-2Q_{12}-2Q_{66}\right){\sin }^2\left(2\theta \right)+Q_{66}\left({\sin }^4\left(\theta \right)+{\cos }^4\left(\theta \right)\right)$$

(6f)

The plane stress-reduced stiffnesses $Q_{ij}$ are given by:

$$Q_{11}=\frac{E_1}{1-{\nu }_{12}{\nu }_{21}}, Q_{12}=\frac{{\nu }_{12}{E}_2}{1-{\nu }_{12}{\nu }_{21}}, Q_{22}=\frac{E_2}{1-{\nu }_{12}{\nu }_{21}}, Q_{66}=G_{12}$$

(7)

where $E_1, E_2, {\nu }_{12}$, and $G_{12}$ are four independent material constants. The equations of motion of the helicoidal composite beam accounting for mid-plane stretching can be given as:

$$m\frac{{\partial }^2\widehat{u}}{\partial {\widehat{t}}^2}+{\widehat{\mu }}_0\frac{\partial \widehat{u}}{\partial \widehat{t}}-\frac{\partial N}{\partial \widehat{x}}={\widehat{F}}_u$$

(8a)

$$m\frac{{\partial }^2\widehat{w}}{\partial {\widehat{t}}^2}+{\widehat{\mu }}_1\frac{\partial \widehat{w}}{\partial \widehat{t}}-\frac{{\partial }^{2}M}{\partial {\widehat{x}}^2}-N\frac{{\partial }^2\widehat{w}}{\partial {\widehat{x}}^2}={\widehat{F}}_w$$

(8b)

in which $m$ is the mas per unit length, ${\widehat{F}}_u$ is the axial distributed force along the $\widehat{x}$-axis, ${\widehat{F}}_w$ is the transverse force, and ${\widehat{\mu }}_0 \mathrm{and} {\widehat{\mu }}_1$ are damping coefficients in axial and transverse directions, respectively. To reduce the governing equations into a single equation, the inplane inertia and damping are negligible and the distributed axial force is zero. The result is:

$$\begin{array}{cc}\hfill m\frac{{\partial }^2\widehat{w}(\widehat{x},\widehat{t})}{\partial {\widehat{t}}^2}+& \widehat{\mu }\frac{\partial \widehat{w}(\widehat{x},\widehat{t})}{\partial \widehat{t}}+b\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)\left(\frac{{\partial }^4\widehat{w}\left(\widehat{x},\widehat{t}\right)}{\partial {\widehat{x}}^4}-\frac{d^4{\widehat{w}}_0\left(\widehat{x}\right)}{d{\widehat{x}}^4}\right)\hfill \\ & +(\widehat{P}-{\widehat{k}}_s+\frac{b}{L}{B}_{11}\left(\frac{\partial \widehat{w}\left(L,\widehat{t}\right)}{\partial \widehat{x}}-\frac{\partial \widehat{w}\left(0,\widehat{t}\right)}{\partial \widehat{x}}-\frac{d{\widehat{w}}_0\left(L\right)}{d\widehat{x}}+\frac{d{\widehat{w}}_0\left(0\right)}{d\widehat{x}}\right)\hfill \\ & -\frac{b}{2L}{A}_{11}\underset{0}{\overset{L}{{\displaystyle \int }}}\left[{\left(\frac{\partial \widehat{w}\left(\widehat{x},\widehat{t}\right)}{\partial \widehat{x}}\right)}^2-{\left(\frac{d{\widehat{w}}_0\left(\widehat{x}\right)}{d\widehat{x}}\right)}^2\right]d\widehat{x})\frac{{\partial }^2\widehat{w}\left(\widehat{x},\widehat{t}\right)}{\partial {\widehat{x}}^2}+{\widehat{k}}_L\widehat{w}\left(\widehat{x},\widehat{t}\right)=\widehat{q}\left(\widehat{x}\right)+\widehat{F}\cos \left(\widehat{\Omega }\widehat{t}\right)\hfill \end{array}$$

(9)

To simplify the mathematical treatment of the governing equations, the following nondimensional parameters are defined [28]:

$$x=\frac{\widehat{x}}{L}, w=\frac{\widehat{w}}{r}, w_0=\frac{{\widehat{w}}_0}{r}, r=\sqrt{\frac{I}{A}}, t=\widehat{t}\sqrt{\frac{b\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}{m{L}^4}}$$

(10)

where $\widehat{P}$ is the axial imposed force, ${\widehat{k}}_s$ is the elastic shear stiffness of the foundation, ${\widehat{k}}_L$ is the elastic stiffness of the foundation, $\widehat{q}$ and $\widehat{F}$ are the distributed transverse and axial loads along the beam length.

As a result, the nondimensional governing equation can be expressed as [23]:

$$\begin{array}{l}\ddot{w}+\mu \dot{w}+w^{iv}+\left(P-k_s+\gamma \left(w^{\prime }\left(1,t\right)-w^{\prime }\left(0,t\right)-w_0^{\prime }\left(1\right)+w_0^{\prime }\left(0\right)\right)-\frac{1}{2}\alpha \underset{0}{\overset{1}{{\displaystyle \int }}}\left({w^{\prime }}^2-w_0^{\prime }{}^2\right)dx\right)w^{″}+k_{L}w\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-w_0^{iv}=q\left(x\right)+Fcos\left(\Omega t\right)\end{array}$$

(11)

in which over-dot is a derivative with respect to time $t$, and prime stands for a derivative with respect to spatial coordinates $x$ [28]:

$$\begin{gathered} \alpha =\frac{A_{11}{r}^2}{\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, \gamma =\frac{B_{11}r}{\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, \Omega =\widehat{\Omega }\sqrt{\frac{m{L}^4}{b\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right) }}, \mu =\frac{\widehat{\mu }{L}^2}{\sqrt{mb\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}}, q\left(x\right)=\frac{\widehat{q}\left(\widehat{x}\right)L^4}{rb\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, \\ P=\frac{\widehat{P}{L}^2}{b\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, F=\frac{\widehat{F}{L}^4}{rb\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, k_s=\frac{{\widehat{k}}_s{L}^2}{b\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, k_L=\frac{{\widehat{k}}_L{L}^4}{b\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}, \end{gathered}$$

(12)

Note that for a symmetric laminate, the coupling stiffness $B_{11}$ vanishes ($\gamma =0$). Besides, the boundary conditions in dimensionless form are [35]:

$$w=0 \mathrm{and} w^{″}=0 \mathrm{at} x=0, 1$$

(13a)

$$w=0 \mathrm{and} w^{\prime }=0 \mathrm{at} x=0, 1$$

(13b)

For S–S and C–C beams, respectively, the initial configuration of geometrically imperfect composite beams is supposed to have the following form [27]:

$$w_0\left(x\right)=A_0\sin \left(\sigma x\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{S}\text{–}\mathrm{S}$$

(14a)

$$w_0\left(x\right)=\frac{1}{2}{A}_0\left(1-\cos \left(\sigma x\right)\right)\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{C}\text{–}\mathrm{C}$$

(14b)

where $A_0$ is the amplitude of initial imperfection. The corresponding values of $\sigma$ for S–S and C–C are $\pi$ and $2\pi$, respectively.

3. Bending Problem

It is well known that the implementation of point loads represents a challenging task, since a strong discontinuity has to be inserted in the structural model. In this section, the nonlinear bending problem of helicoidal composite perfect and imperfect beams subjected to point and uniform loads is exactly solved.

For the static bending problem, all time derivatives and the external axial load $P$ are set to zero and the following equilibrium equation is obtained:

$$w_q^{iv}-\left(k_s+\frac{1}{2}\alpha \underset{0}{\overset{1}{{\displaystyle \int }}}\left(w_q^{\prime }{}^2-w_0^{\prime }{}^2\right)dx\right)w_q^{″}+k_L{w}_q-w_0^{iv}=q\left(x\right)$$

(15)

where $w_q$ is the bending deflection due to the transverse load $q$. The applied transverse load $q\left(x\right)$ can be expressed as:

$$\begin{gathered} q\left(x\right)=q_0\delta \left(x-x_0\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Point}\mathrm{load} \\ q\left(x\right)=q_0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Uniform}\mathrm{load} \end{gathered}$$

(16)

in which $q_0$ is the intensity of the load, $\delta \left(.\right)$ is the Dirac delta function and $x_0$ is the application position of the point load. Equation (15) is a fourth order nonlinear nonhomogeneous ordinary differential equation and its analytical solution is unavailable. Herein, we can find the exact solution of Equation (15). Furthermore, we can rewrite Equation (15) as:

$$w_q^{iv}-{\eta }_q^2{w}_q^{″}+k_L{w}_q=q\left(x\right)+w_0^{iv}$$

(17a)

in which

$${\eta }_q^2=k_s+\frac{1}{2}\alpha \underset{0}{\overset{1}{{\displaystyle \int }}}\left(w_q^{\prime }{}^2-w_0^{\prime }{}^2\right)dx$$

(17b)

Applying Laplace transform to Equation (17a), the result is:

$$\begin{array}{l}\mathcal{L}\left\{w_q\left(x\right)\right\}=\frac{1}{\left(s^2-m_1^2\right)\left(s^2-m_2^2\right)}(\mathcal{L}\left\{q\right\}+\mathcal{L}\left\{w_0^{iv}\right\}+\left(s^3-{\eta }_q^{2}s\right)w_q\left(0\right)+\left(s^2-{\eta }_q^2\right)w_q^{\prime }\left(0\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+s{w}_q^{″}\left(0\right)+w_q^{‴}\left(0\right))\end{array}$$

(18)

where $\mathcal{L}\left\{\right\}$ stands for the Laplace transform operator, the symbol $s$ represents a complex Laplace variable, and $w_q\left(0\right)$, $w_q^{\prime }\left(0\right)$, $w_q^{″}\left(0\right)$, and $w_q^{‴}\left(0\right)$ are undetermined constants and:

$$m_{1,2}=\sqrt{\frac{{\eta }_q^2}{2}\pm \sqrt{\frac{{\eta }_q^4}{4}-k_L}}$$

(19)

Note that

$$\mathcal{L}\left\{q\right\}=\left\{\begin{array}{c}{q}_0{e}^{-s{x}_0} \mathrm{Point}\mathrm{load}\\ \frac{q_0}{s} \mathrm{Uniform}\mathrm{load}\end{array}\right.$$

(20)

and

$$\left\{w_0^{iv}\right\}=\left\{\begin{array}{lllll}\frac{A_0{\sigma }^5}{s^2+{\sigma }^2}& & & & \mathrm{S}\text{–}\mathrm{S}\\ \frac{A_0{\sigma }^{4}s}{2\left(s^2+{\sigma }^2\right)}& & & & \mathrm{C}\text{–}\mathrm{C}\end{array}\right.$$

(21)

Then, utilizing the inverse Laplace transform leads to:

$$w_q\left(x\right)=w\left(0\right){\Psi }_0\left(x\right)+w^{\prime }\left(0\right){\Psi }_1\left(x\right)+w^{″}\left(0\right){\Psi }_2\left(x\right)+w^{‴}\left(0\right){\Psi }_3\left(x\right)+q_0{\Psi }_4\left(x\right)+A_0{\Psi }_5\left(x\right)$$

(22)

where the functions ${\Psi }_i\left(x\right), i=1,2\dots 5$ are defined as:

$${\Psi }_0\left(x\right)=\frac{1}{m_2^2-m_1^2}\left(m_2^2\cosh \left(m_{1}x\right)-m_1^2\text{cosh}\left(m_{2}x\right)\right)$$

(23a)

$${\Psi }_1\left(x\right)=\frac{1}{\left(m_2^2-m_1^2\right)}\left(\frac{m_2^2}{m_1}\text{sinh}\left(m_{1}x\right)-\frac{m_1^2}{m_2}\text{sinh}\left(m_{2}x\right)\right)$$

(23b)

$${\Psi }_2\left(x\right)=\frac{1}{m_2^2-m_1^2}\left(\cosh \left(m_{2}x\right)-\text{cosh}\left(m_{1}x\right)\right)$$

(23c)

$${\Psi }_3\left(x\right)=\frac{1}{m_2^2-m_1^2}\left(\frac{1}{m_2}\sinh \left(m_{2}x\right)-\frac{1}{m_1}\text{sinh}\left(m_{1}x\right)\right)$$

(23d)

$${\psi }_4\left(x\right)=\left\{\begin{array}{c}H\left(x-x_0\right){\Psi }_3\left(x-x_0\right) \mathrm{Point}\mathrm{load}\\ \underset{0}{\overset{x}{{\displaystyle \int }}}{\Psi }_3\left(x\right)dx \mathrm{Uniform}\mathrm{load}\end{array}\right.$$

(23e)

$$\begin{array}{l}{\Psi }_5\left(x\right)={\sigma }^4(\frac{-1}{\left(m_1^2+{\sigma }^2\right)\left(m_2^2+{\sigma }^2\right)}{\zeta }_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{\left(m_2^2-m_1^2\right)}\left(\frac{1}{\left(m_1^2+{\sigma }^2\right)} {\zeta }_2-\frac{1}{\left(m_2^2-{\sigma }^2\right)}{\zeta }_3\right))\\ \left[{\zeta }_1, {\zeta }_2, {\zeta }_3\right]=\left\{\begin{array}{c}\left[\sin \left(\sigma x\right),\frac{\sigma }{m_1}\sinh \left(m_{1}x\right),\frac{\sigma }{m_2}\sinh \left(m_{2}x\right)\right] \text{For} \mathrm{S}\text{–}\mathrm{S} \text{beam}\\ \frac{1}{2}\left[\cos \left(\sigma x\right),-\cosh \left(m_{1}x\right), \cosh \left(m_{2}x\right)\right] \text{For} \mathrm{C}\text{–}\mathrm{C} \text{beam}\end{array}\right.\end{array}$$

(23f)

in which $H\left(x\right)$ is the Heaviside function. In the absence of Linear Elastic foundation constant ($k_L=0$), the functions ${\Psi }_i\left(x\right),i=1,2\dots 5$ are defined as:

$${\Psi }_0\left(x\right)=1$$

(24a)

$${\Psi }_1\left(x\right)=x$$

(24b)

$${\Psi }_2\left(x\right)=\frac{1}{{\eta }_q^2}\left(\cosh \left({\eta }_{q}x\right)-1\right)$$

(24c)

$${\Psi }_3\left(x\right)=\frac{1}{{\eta }_q^2}\left(\frac{1}{{\eta }_q}\sinh \left({\eta }_{q}x\right)-x\right)$$

(24d)

$${\psi }_4\left(x\right)=\left\{\begin{array}{c}H\left(x-x_0\right){\Psi }_3\left(x-x_0\right) \mathrm{Point}\mathrm{load}\\ \underset{0}{\overset{x}{{\displaystyle \int }}}{\Psi }_3\left(x\right)dx \mathrm{Uniform}\mathrm{load}\end{array}\right.$$

(24e)

$$\begin{gathered} {\Psi }_5\left(x\right)=\frac{{\sigma }^2}{{\eta }_q^2+{\sigma }^2}\left({\zeta }_2-\frac{{\sigma }^2}{{\eta }_q^2}{\zeta }_1\right) \\ \left[{\zeta }_1, {\zeta }_2\right]=\left\{\begin{array}{c}\left[\sigma \left(x-\frac{1}{{\eta }_q}\sinh \left({\eta }_{q}x\right)\right),\left(\sin \left(\sigma x\right)-\sigma x\right)\right] \text{For} \mathrm{S}\text{–}\mathrm{S} \text{beam}\\ \frac{1}{2}\left[\left(1-\cosh \left({\eta }_{q}x\right)\right), \left(1-\cos \left(\sigma x\right)\right)\right] \text{For} \mathrm{C}\text{–}\mathrm{C} \text{beam}\end{array}\right. \end{gathered}$$

(24f)

Herein, the unknown constants are needed to be determined by the boundary conditions, given in Equation (13a,b).

(a)

S–S boundary conditions

Substituting Equation (13a) into Equation (22), we can therefore obtain the constants $w^{\prime }\left(0\right)$ and $w^{‴}\left(0\right)$ from:

$$\left[\begin{array}{cc}{\Psi }_1\left(1\right)& {\Psi }_3\left(1\right)\\ {\Psi }_1^{″}\left(1\right)& {\Psi }_3^{″}\left(1\right)\end{array}\right]\left[\begin{array}{c}{w}_q^{\prime }\left(0\right)\\ w_q^{‴}\left(0\right)\end{array}\right]=-\left[\begin{array}{c}{q}_0{\Psi }_4\left(1\right)+A_0{\Psi }_5\left(1\right)\\ q_0{\Psi }_4^{″}\left(1\right)+A_0{\Psi }_5^{″}\left(1\right)\end{array}\right]$$

(25)

By solving Equation (25), one can obtain:

$$w_q^{\prime }\left(0\right)=\frac{A_0\left({\Psi }_5\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_5^{″}\left(1\right)\right)+q_0\left({\Psi }_4\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_4^{″}\left(1\right)\right)}{{\Psi }_3\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_3^{″}\left(1\right)}$$

(26a)

$$w_q^{‴}\left(0\right)=\frac{A_0\left({\Psi }_5\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_5^{″}\left(1\right)\right)+q_0\left({\Psi }_4\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_4^{″}\left(1\right)\right)}{{\Psi }_1\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_1^{″}\left(1\right)}$$

(26b)

Substituting Equations (13a) and (26a,b) into Equation (22), the following exact solution of S–S imperfect Helicoidal composite beam can be obtained as:

$$\begin{gathered} w_q\left(x\right)=\left[\frac{{\Psi }_4\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_4^{″}\left(1\right)}{{\Psi }_3\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_3^{″}\left(1\right)}{\Psi }_1\left(x\right)+\frac{{\Psi }_4\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_4^{″}\left(1\right)}{{\Psi }_1\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_1^{″}\left(1\right)}{\Psi }_3\left(x\right)+{\Psi }_4\left(x\right)\right]q_0 + \\ \left[\frac{{\Psi }_5\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_5^{″}\left(1\right)}{{\Psi }_3\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_3^{″}\left(1\right)}{\Psi }_1\left(x\right)+\frac{{\Psi }_5\left(1\right){\Psi }_1^{″}\left(1\right)-{\Psi }_1\left(1\right){\Psi }_5^{″}\left(1\right)}{{\Psi }_1\left(1\right){\Psi }_3^{″}\left(1\right)-{\Psi }_3\left(1\right){\Psi }_1^{″}\left(1\right)}{\Psi }_3\left(x\right)+{\Psi }_5\left(x\right)\right]A_0 \end{gathered}$$

(27)

(b)

C–C boundary conditions

Substituting Equation (13b) into Equation (22), subsequently, the following equation can be evaluated:

$$\left[\begin{array}{cc}{\Psi }_2\left(1\right)& {\Psi }_3\left(1\right)\\ {\Psi }_2^{\prime }\left(1\right)& {\Psi }_3^{\prime }\left(1\right)\end{array}\right]\left[\begin{array}{c}{w}_q^{″}\left(0\right)\\ w_q^{‴}\left(0\right)\end{array}\right]=-\left[\begin{array}{c}{q}_0{\Psi }_4\left(1\right)+A_0{\Psi }_5\left(1\right)\\ q_0{\Psi }_4^{\prime }\left(1\right)+A_0{\Psi }_5^{\prime }\left(1\right)\end{array}\right]$$

(28)

From Equation (28), the constants $w^{″}\left(0\right)$ and $w^{‴}\left(0\right)$ can be determined as:

$$w_q^{″}\left(0\right)=\frac{A_0\left({\Psi }_5\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_5^{\prime }\left(1\right)\right)+q_0\left({\Psi }_4\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_4^{\prime }\left(1\right)\right)}{{\Psi }_3\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_3^{\prime }\left(1\right)}$$

(29a)

$$w_q^{‴}\left(0\right)=\frac{A_0\left({\Psi }_5\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_5^{\prime }\left(1\right)\right)+q_0\left({\Psi }_4\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_4^{\prime }\left(1\right)\right)}{{\Psi }_2\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_2^{\prime }\left(1\right)}$$

(29b)

As a result, the solution of C–C imperfect Helicoidal composite beam can be derived as:

$$\begin{array}{l}{w}_q\left(x\right)=\left[\frac{{\Psi }_4\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_4^{\prime }\left(1\right)}{{\Psi }_3\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_3^{\prime }\left(1\right)}{\Psi }_2\left(x\right)+\frac{{\Psi }_4\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_4^{\prime }\left(1\right)}{{\Psi }_2\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_2^{\prime }\left(1\right)}{\Psi }_3\left(x\right)+{\Psi }_4\left(x\right)\right]q_0\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[\frac{{\Psi }_5\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_5^{\prime }\left(1\right)}{{\Psi }_3\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_3^{\prime }\left(1\right)}{\Psi }_2\left(x\right)+\frac{{\Psi }_5\left(1\right){\Psi }_2^{\prime }\left(1\right)-{\Psi }_2\left(1\right){\Psi }_5^{\prime }\left(1\right)}{{\Psi }_2\left(1\right){\Psi }_3^{\prime }\left(1\right)-{\Psi }_3\left(1\right){\Psi }_2^{\prime }\left(1\right)}{\Psi }_3\left(x\right)+{\Psi }_5\left(x\right)\right]A_0\end{array}$$

(30)

In order to compute the constant ${\eta }_q$, which is necessary to obtain the explicit solutions with various boundary conditions, the solutions given in Equations (27) and (30) and the initial shape of imperfections defined in Equation (14a,b) are enforced in Equation (17b). The result is a complex algebraic equation, which is solved numerically.

Hint: The solutions for the bending problem of perfect beams can be obtained by setting $A_0=0$ in Equations (27) and (30) for S–S and C–C beams, respectively.

4. Buckling Problem

To obtain the critical buckling load and postbuckling configurations, all time derivatives and the external transverse load $q\left(x\right)$ should be set to zero. Thus, one obtains:

$$w_p^{iv}+\left(P-k_s-\frac{1}{2}\alpha \underset{0}{\overset{1}{{\displaystyle \int }}}\left(w_p^{\prime }{}^2-w_0^{\prime }{}^2\right)dx\right)w_p^{″}+k_L{w}_p=w_0^{iv}$$

(31)

in which $w_p$ is the static deflection due to the applied axial load $P$. Equation (31) can be rewritten as:

$$w_p^{iv}+{\eta }_p^2{w}_p^{″}+k_L{w}_p=w_0^{iv}$$

(32a)

where

$${\eta }_p^2=P-k_s-\frac{1}{2}\alpha \underset{0}{\overset{1}{{\displaystyle \int }}}\left(w_p^{\prime }{}^2-w_0^{\prime }{}^2\right)dx$$

(32b)

4.1. Buckling of Perfect Beam

Equations governing the buckling problem of perfect composite beams are obtained by setting $w_0=0$ in Equation (32a,b), that is:

$$w_p^{iv}+{\eta }_p^2{w}_p^{″}+k_L{w}_p=0$$

(33a)

$${\eta }_p^2=P-k_s-\frac{1}{2}\alpha \underset{0}{\overset{1}{{\displaystyle \int }}}{w}_p^{\prime }{}^{2}dx$$

(33b)

Performing Laplace transform on Equation (33a), the result is:

$$\mathcal{L}\left\{w_p\left(x\right)\right\}=\frac{1}{\left(s^2+n_1^2\right)\left(s^2+n_2^2\right)}\left(\left(s^3+{\eta }_p^{2}s\right)w_p\left(0\right)+\left(s^2+{\eta }_p^2\right)w_p^{\prime }\left(0\right)+s{w}_p^{″}\left(0\right)+w_p^{‴}\left(0\right)\right)$$

(34)

where

$$n_{1,2}=\sqrt{\frac{{\eta }_p^2}{2}\pm \sqrt{\frac{{\eta }_p^4}{4}-k_L}}$$

(35)

Talking the inverse Laplace leads to:

$$w_p\left(x\right)=w_p\left(0\right){\Phi }_0\left(x\right)+w_p^{\prime }\left(0\right){\Phi }_1\left(x\right)+w_p^{″}\left(0\right){\Phi }_2\left(x\right)+w_p^{‴}\left(0\right){\Phi }_3\left(x\right)$$

(36)

where the functions ${\Phi }_i\left(x\right), i=1, 2, 3$ are defined as

$${\Phi }_0\left(x\right)=\frac{1}{n_2^2-n_1^2}\left(n_2^2\cos \left(n_{1}x\right)-n_1^2\cos \left(n_{2}x\right)\right)$$

(37a)

$${\Phi }_1\left(x\right)=\frac{1}{n_2^2-n_1^2}\left(\frac{n_2^2}{n_1}\sin \left(n_{1}x\right)-\frac{n_1^2}{n_2}\sin \left(n_{2}x\right)\right)$$

(37b)

$${\Phi }_2\left(x\right)=\frac{1}{n_2^2-n_1^2}\left(\cos \left(n_{1}x\right)-\cos \left(n_{2}x\right)\right)$$

(37c)

$${\Phi }_3\left(x\right)=\frac{1}{n_2^2-n_1^2}\left(\frac{1}{n_1}\sin \left(n_{1}x\right)-\frac{1}{n_2}\sin \left(n_{2}x\right)\right)$$

(37d)

The constants $w_p\left(0\right)$, $w_p^{\prime }\left(0\right)$, $w_p^{″}\left(0\right)$, and $w_p^{‴}\left(0\right)$ can be fixed by the boundary conditions.

(a)

S–S boundary conditions

Substituting Equation (13a) into Equation (36) and using Equation (37), one can obtain the following:

$$\left[\begin{array}{cc}\frac{n_2^2}{n_1}\sin \left(n_1\right)-\frac{n_1^2}{n_2}\sin \left(n_2\right)& \frac{1}{n_1}\sin \left(n_1\right)-\frac{1}{n_2}\sin \left(n_2\right)\\ n_1{n}_2\left(n_1^2\sin \left(n_2\right)-n_2^2\sin \left(n_1\right)\right)& n_1\sin \left(n_1\right)-n_2\sin \left(n_2\right)\end{array}\right]\left[\begin{array}{c}{w}_p^{\prime }\left(0\right)\\ w_p^{‴}\left(0\right)\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

(38)

To obtain a nontrivial solution to Equation (38), the determinant of the coefficient matrix should vanish. This condition yields the following characteristic equation:

$$\sin \left(n_1\right)\sin \left(n_2\right)=0\to n_1=k\pi , k=1, 2, 3, \dots$$

(39)

As a result, the buckling mode shapes can be obtained as:

$$w_P\left(x\right)=a \sin \left(k\pi x\right)$$

(40)

The postbuckling amplitude can be computed by substituting Equation (40) into Equation (33b) and using Equation (39), which yields:

$$a=\pm \frac{2}{k\pi }\sqrt{\frac{1}{\alpha }\left(P-k_s-{\eta }_p^2\right)}$$

(41)

and the critical buckling loads are

$$P_{cr}=k_s+\frac{1}{k^2{\pi }^2}\left(k_L+k^4{\pi }^4\right)$$

(42)

(b)

C–C boundary conditions

Substituting the boundary conditions, Equation (13b), into the general solution given in Equation (36) and using Equation (37):

$$\left[\begin{array}{cc}\cos \left(n_1\right)-\cos \left(n_2\right)& \frac{1}{n_1}\sin \left(n_1\right)-\frac{1}{n_2}\sin \left(n_2\right)\\ n_2\sin \left(n_2\right)-n_1\sin \left(n_1\right)& \cos \left(n_1\right)-\cos \left(n_2\right)\end{array}\right]\left[\begin{array}{c}{w}_p^{″}\left(0\right)\\ w_p^{‴}\left(0\right)\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

(43)

The determinant of the coefficient matrix is set to zero, the result is:

$$2n_1{n}_2-2n_1{n}_2\cos \left(n_1\right)\cos \left(n_2\right)-\left(n_1^2+n_2^2\right)\sin \left(n_1\right)\sin \left(n_2\right)=0$$

(44)

The corresponding mode shapes are:

$$\begin{gathered} w_P\left(x\right)=a\left(\cos \left(n_{1}x\right)-\cos \left(n_{2}x\right)+\frac{n_1\sin \left(n_1\right)-n_2\sin \left(n_2\right)}{n_1\left(\cos \left(n_1\right)-\cos \left(n_2\right)\right)}\sin \left(n_{1}x\right) \\ -\frac{n_1\sin \left(n_1\right)-n_2\sin \left(n_2\right)}{n_2\left(\cos \left(n_1\right)-\cos \left(n_2\right)\right)}\sin \left(n_{2}x\right)\right) \end{gathered}$$

(45)

Using trigonometric identities, Equation (44) can be manipulated as follows:

$$\left(n_1\sin \left(\frac{n_1}{2}\right)\cos \left(\frac{n_2}{2}\right)-n_2\sin \left(\frac{n_2}{2}\right)\cos \left(\frac{n_1}{2}\right)\right)\left(n_2\sin \left(\frac{n_1}{2}\right)\cos \left(\frac{n_2}{2}\right)-n_1\sin \left(\frac{n_2}{2}\right)\cos \left(\frac{n_1}{2}\right)\right)=0$$

(46)

It is noticed from Equation (46) that there are two cases. First:

$$n_1\tan \left(\frac{n_1}{2}\right)-n_2\tan \left(\frac{n_2}{2}\right)=0$$

(47)

which yields the symmetric mode shapes

$$w_p\left(x\right)=a\left(\cos \left(n_{1}x\right)-\cos \left(n_{2}x\right)+\tan \left(\frac{n_1}{2}\right)\sin \left(n_{1}x\right)-\tan \left(\frac{n_2}{2}\right)\sin \left(n_{2}x\right)\right)$$

(48)

Second

$$n_2\tan \left(\frac{n_1}{2}\right)-n_1\tan \left(\frac{n_2}{2}\right)=0$$

(49)

and yields the asymmetric mode shapes

$$w_p\left(x\right)=a\left(\cos \left(n_{1}x\right)-\cos \left(n_{2}x\right)-\cot \left(\frac{n_1}{2}\right)\sin \left(n_{1}x\right)+\cot \left(\frac{n_2}{2}\right)\sin \left(n_{2}x\right)\right)$$

(50)

Substituting the mode shapes, Equations (48) and (50), into Equation (33b) solves the postbuckling amplitude. For symmetric mode shapes, the postbuckling amplitude is:

$$a=\pm \frac{2}{n_1}\sqrt{\frac{1}{\alpha }\frac{P-k_s-{\eta }_p^2}{1+\frac{n_2^2}{n_1^2}+2\tan \left(\frac{n_1}{2}\right)\left(\tan \left(\frac{n_1}{2}\right)+\frac{2}{n_1}\right)}}$$

(51)

and the postbuckling amplitude of asymmetric mode shapes is

$$a=\pm \frac{2}{n_1}\sqrt{\frac{1}{\alpha }\frac{P-k_s-{\eta }_p^2}{1+\frac{n_2^2}{n_1^2}+2\cot \left(\frac{n_1}{2}\right)\left(\cot \left(\frac{n_1}{2}\right)-\frac{2}{n_1}\right)}}$$

(52)

and the critical buckling loads are

$$P_{cr}=k_s+\frac{1}{n_1^2}\left(k_L+n_1^4\right)$$

(53)

4.2. Buckling of Imperfect Beam

Similarly, applying the Laplace transform and its inversion on Equation (32a), the result is:

$$w_p\left(x\right)=w_p\left(0\right){\Phi }_0\left(x\right)+w_p^{\prime }\left(0\right){\Phi }_1\left(x\right)+w_p^{″}\left(0\right){\Phi }_2\left(x\right)+w_p^{‴}\left(0\right){\Phi }_3\left(x\right)+A_0{\Phi }_4\left(x\right)$$

(54)

where the functions ${\Phi }_i\left(x\right), i=1, 2, 3$ are given in Equation (37) and ${\Phi }_4\left(x\right)$ is defined as

$$\begin{gathered} {\Phi }_4\left(x\right)={\sigma }^4\left(\frac{1}{\left(n_1^2-{\sigma }^2\right)\left(n_2^2-{\sigma }^2\right)}{\xi }_1-\frac{1}{\left(n_2^2-n_1^2\right)} \left(\frac{1}{\left(n_1^2-{\sigma }^2\right)} {\xi }_2-\frac{1}{\left(n_2^2-{\sigma }^2\right)}{\mathsf{\xi }}_3\right)\right) \\ \left[{\xi }_1, {\xi }_2, {\mathsf{\xi }}_3\right]=\left\{\begin{array}{c}\left[\sin \left(\sigma x\right),\frac{\sigma }{n_1}\sin \left(n_{1}x\right),\frac{\sigma }{n_2}\sin \left(n_{2}x\right)\right] \text{For} \mathrm{S}\text{–}\mathrm{S} \text{beam}\\ \frac{-1}{2}\left[\cos \left(\sigma x\right), \cos \left(n_{1}x\right),-\cos \left(n_{2}x\right)\right] \text{For} \mathrm{C}\text{–}\mathrm{C} \text{beam}\end{array}\right. \end{gathered}$$

(55)

Substituting the corresponding boundary conditions, the unknown constants can be determined.

(c)

S–S boundary conditions

Substituting Equation (13a) into Equation (54), and using Equations (37) and (55), the first buckling mode of S–S imperfect helicoidal composite beam is computed as:

$$w_p\left(x\right)=\frac{A_0{\sigma }^4}{\left(n_1^2-{\sigma }^2\right)\left(n_2^2-{\sigma }^2\right)}\sin \left(\sigma x\right)$$

(56)

Substituting Equation (56) into Equation (32b) leads to a cubic polynomial with respect to ${\eta }_P^2$:

$${\left({\eta }_P^2\right)}^3-a_2{\left({\eta }_P^2\right)}^2+a_1{\eta }_P^2-a_0=0$$

(57)

in which

$$a_0=\frac{1}{{\pi }^4}\left(\frac{1}{4}\alpha g^2{\pi }^2+P-k_s\right)k_L^2+2\left(\frac{1}{4}{g}^2{\pi }^2\alpha +P-k_s\right)k_L+{\pi }^4\left(P-ks\right)$$

(58a)

$$a_1=\left({\pi }^4+k_L\right)\left(\frac{1}{2}\left(\alpha g^2+2\right)+\frac{2}{{\pi }^2} \left(P-ks\right)+\frac{1}{{\pi }^4}{k}_L\right)$$

(58b)

$$a_2=\left(\frac{1}{4}\alpha g^2+2\right){\pi }^2+\left(P-k_s\right)+\frac{1}{{\pi }^2}{k}_L$$

(58c)

Equation (57) has at least one real root. Hence, the first critical buckling load occurs when the discriminant of Equation (57) is equal to zero:

$$P_{cr}={\pi }^2\left(1-\frac{1}{4}\alpha g^2{\pi }^2+\frac{3}{\sqrt[3]{16}}{\alpha }^{\frac{1}{3}}{g}^{\frac{2}{3}}\right)+ k_s+\frac{1}{{\pi }^2}{k}_L$$

(59)

(d)

C–C boundary conditions

Correspondingly, the boundary conditions, Equation (13b), are substituted into the general solution given in Equation (54). The corresponding first buckling configuration is:

$$\begin{gathered} w_p\left(x\right)=\frac{1}{2}\frac{A_0{\sigma }^4}{\left(n_1^2-{\sigma }^2\right)\left(n_2^2-{\sigma }^2\right)}\left(\frac{1}{n_2\tan \left(\frac{n_2}{2}\right)-n_1\tan \left(\frac{n_1}{2}\right)}\left[n_2\tan \left(\frac{n_2}{2}\right)\cos \left(n_{1}x\right)-n_1\tan \left(\frac{n_1}{2}\right)\cos \left(n_{2}x\right)\right] \\ +\frac{\tan \left(\frac{n_1}{2}\right)\tan \left(\frac{n_2}{2}\right)}{n_2\tan \left(\frac{n_1}{2}\right)-n_1\tan \left(\frac{n_2}{2}\right)}\left[n_2\sin \left(n_{1}x\right)-n_1\sin \left(n_{2}x\right)\right]-\cos \left(\sigma x\right)\right) \end{gathered}$$

(60)

Similarly, computing the constant, ${\eta }_P$, is achieved by enforcing Equation (60) into Equation (32b). This leads to a nonlinear algebraic equation, which will be solved numerically. Herein, the critical buckling load is computed by solving the nonlinear algebraic equation at a definite axial load with various initial guesses and then checking if the iteration converges to a unique solution or multiple solutions.

In the absence of a linear elastic foundation constant ($k_L=0$), the functions ${\Phi }_i\left(x\right),i=1,2,3,4$ are given as:

$${\Phi }_0\left(x\right)=1$$

(61a)

$${\Phi }_1\left(x\right)=x$$

(61b)

$${\Phi }_2\left(x\right)=\frac{1}{{\eta }_p^2}\left(1-\cos \left({\eta }_{p}x\right)\right)$$

(61c)

$${\Phi }_3\left(x\right)=\frac{1}{{\eta }_p^2}\left(x-\frac{1}{{\eta }_p}\sin \left({\eta }_{p}x\right)\right)$$

(61d)

$$\begin{gathered} {\Phi }_4\left(x\right)=\frac{{\sigma }^2}{{\eta }_p^2-{\sigma }^2}\left(\frac{{\sigma }^2}{{\eta }_p^2}{\xi }_1-{\xi }_2\right) \\ \left[{\xi }_1,{\xi }_2\right]=\left\{\begin{array}{c}\left[\left(\frac{\sigma }{{\eta }_p}\sin \left({\eta }_{p}x\right)-\sigma x\right),\left(\sin \left(\sigma x\right)-\sigma x\right)\right] \text{For} \mathrm{S}\text{–}\mathrm{S} \text{beam}\\ \frac{1}{2}\left[\left(1-\cos \left({\eta }_{p}x\right)\right),\left(1-\cos \left(\sigma x\right)\right)\right] \text{For} \mathrm{C}\text{–}\mathrm{C} \text{beam}\end{array}\right. \end{gathered}$$

(61e)

Following a similar procedure, one can obtain the nonlinear characteristic equations for buckling loads and buckling modes of perfect and imperfect helicoidal composite beams. For more details, see Eltaher et al. [28].

5. Numerical Analysis

This section presents the numerical results of nonlinear bending and buckling problems of helicoidal composite beams. Different laminate specifications are considered. The laminate configurations are shown in

Table 1. Specifications of the chosen layup Configurations.

Designation	Stacking Sequence	Description
UD	${\left[0\right]}_{32}$	Unidirectional
CP	${\left[0∕90\right]}_{8s}$	Cross ply-symmetric
QI	${\left[45∕-45∕0∕90 \right]}_{4s}$	Quasi isotropic-symmetric
3H	${\left[0/3\dots ∕45\right]}_s$	Helicoidal (3°)-symmetric
6H	${\left[0/6\dots ∕90\right]}_s$	Helicoidal (6°)-symmetric
12H	${\left[0/12\dots ∕180\right]}_s$	Helicoidal (12°)-symmetric

. The beam is composed of $32$ layers and the following material properties are used [3]:

$$E_1=160.5 \mathrm{GPa}, E_2=12.5 \mathrm{GPa}, G_{12}=4.6 \mathrm{GPa}, \nu =0.303$$

It is also assumed:

$$h=4 \mathrm{mm}, b=4 \mathrm{mm}, L=100h$$

5.1. Results of Nonlinear Bending

In this section, the numerical results are presented to clarify the effects of load position, helicoidal rotation angle, amplitude of initial imperfection, and elastic foundation parameters on the nonlinear bending behavior of composite beams.

The load parameter $q_0$ of uniform and point loads is normalized respectively as:

$$q_0=\frac{{\widehat{q}}_0{L}^4}{rb\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}\mathrm{and}{q}_0=\frac{{\widehat{q}}_0{L}^3}{rb\left(D_{11}-\frac{B_{11}^2}{A_{11}}\right)}$$

To show the influence of point load position on the nonlinear bending response, the nonlinear bending deflections of perfect S–S and C–C helicoidal composite beams with layup sequence 3H subjected to point load exerted on different positions with load intensity ${\widehat{q}}_0=1 N$ are displayed in Figure 2. Among the different values of $x_0$ considered in this figure, the deflection is the largest for the point load applied at the midpoint of the beam ($x_0=0.5$) whereas it is the smallest for the load applied near boundaries ($x_0=0.1 \mathrm{and} 0.9$).

Figure 3 and Figure 4 show the effect of rotation angle on the nonlinear bending response of perfect and imperfect composite beams subjected to uniform load with load intensity ${\widehat{q}}_0=5 N/m$, respectively. The results were obtained for the S–S and C–C boundary conditions. One can see that the UD beam has the lowest, while the QI beam has the greatest deflection amongst the six chosen layup configurations. It is also noted that the deflection of 3H is slightly higher than the deflection of UD especially for the C–C beam.

Figure 5 illustrates the influence of the elastic foundation constants on the nonlinear bending characteristics of perfect beams with layup sequence 3H subjected to point load with load intensity ${\widehat{q}}_0=5 N$ exerted on the midpoint of the beam. As expected, the deflection of the beam gets larger by decreasing the values of these parameters since structural responses are softening for decreasing the values of foundation parameters. Secondly, one can observe that the effect of elastic foundations is more pronounced on the S–S boundary conditions. Furthermore, it is noticed that effect of the shear parameter is more noticeable than the effect of the linear foundation constant.

5.2. Buckling Analysis

In this section, the influence of rotation angle, foundation constants, and amplitude of imperfection on critical buckling loads and postbuckling behaviors of helicoidal composite beams are examined. In

Table 2. First three critical buckling loads ($N$) of perfect S–S and C–C composite beams for different layups ($k_s=0, L=100h$).

Laminate	${\mathit{k}}_{\mathit{L}}=20$			${\mathit{k}}_{\mathit{L}}=50$
	${\mathit{P}}_{\mathit{cr1}}$	${\mathit{P}}_{\mathit{cr2}}$	${\mathit{P}}_{\mathit{cr3}}$	${\mathit{P}}_{\mathit{cr1}}$	${\mathit{P}}_{\mathit{cr2}}$	${\mathit{P}}_{\mathit{cr3}}$
(a) S–S
UD	256.4084	861.8419	1919.4286	321.9250	878.2210	1926.7082
CP	149.2720	501.7343	1117.4243	187.4135	511.2696	1121.6623
QI	108.1414	363.4857	809.5277	135.7733	370.3937	812.5979
3H	232.0188	779.8635	1736.8525	291.3035	794.6846	1743.4397
6H	188.9145	634.9810	1414.1813	237.1854	647.0487	1419.5447
12H	137.8104	463.2093	1031.6245	173.0232	472.0125	1035.5370
(b) C–C
UD	883.6214	1749.6677	3411.9220	932.4433	1763.0029	3424.4339
CP	514.4136	1018.5955	1986.3019	542.8360	1026.3588	1993.5860
QI	372.6714	737.9303	1438.9936	393.2622	743.5545	1444.2706
3H	799.5713	1583.2392	3087.3799	843.7493	1595.3060	3098.7017
6H	651.0276	1289.1062	2513.8087	686.9981	1298.9312	2523.0271
12H	474.9151	940.3841	1833.7866	501.1551	947.5513	1840.5113

, the numerical values for the first three buckling loads of perfect composite beams with different layups and various values of linear foundation constant are listed. It observed that the layup configurations affected the critical buckling loads.

The first buckling load of S–S and C–C imperfect composite beams with different laminate specifications and various values of imperfection amplitude are tabulated in

Table 3. First critical buckling loads ($N$) of imperfect S–S composite beams for different layups ($k_L=0,k_s=0, L=100h$).

${\mathit{A}}_0$	UD	CP	QI	3H	6H	12H
$0.5$	358.9829	207.2039	151.5052	316.3949	249.3464	189.6495
$1$	412.8145	238.8817	174.1865	366.2812	289.9060	219.1237
$2$	402.0357	237.3041	169.3567	376.7940	312.7668	221.6148
$3$	260.9025	164.7909	109.2545	291.3420	274.9440	162.9004
$4$	0	27.3479	−1.6466	118.7747	182.9625	48.3833

and

Table 4. First critical buckling loads ($N$) of imperfect C–C composite beams for different layups ($k_L=0,k_s=0, L=100h$).

${\mathit{A}}_0$	UD	CP	QI	3H	6H	12H
$0.5$	1239.6624	716.3208	523.1438	1096.6105	869.0241	656.4225
$2$	1651.2582	955.5267	696.7461	1465.1250	1159.6239	876.4950
$4$	1608.1426	949.2164	677.4268	1507.1759	1251.0673	886.4591
$6$	1043.6099	659.1638	437.0182	1165.3680	1099.7761	651.6016
$8$	0	109.3918	−6.5864	475.0987	731.8499	193.5332

, respectively. In the range of small values of initial imperfection, the UD layup results in higher buckling loads compared with the helicoidal composite ones. However, the opposite is exact for larger values of imperfection amplitude, in which the helicoidal composite beams layups improve the critical buckling load compared with the UD layup.

To account for the effect of the elastic foundation constants on the critical buckling load, the critical buckling load values are listed in

Table 5. First critical buckling loads ($N$) of perfect and imperfect composite beams for different layups, $L=100h$.

	${\mathit{k}}_{\mathit{s}}=5$			${\mathit{k}}_{\mathit{s}}=10$
	${\mathit{k}}_{\mathit{L}}=0$	${\mathit{k}}_{\mathit{L}}=20$	${\mathit{k}}_{\mathit{L}}=50$	${\mathit{k}}_{\mathit{L}}=0$	${\mathit{k}}_{\mathit{L}}=20$	${\mathit{k}}_{\mathit{L}}=50$
(a) A0 = 0
UD	320.5012	364.1790	429.6956	428.2718	471.9496	537.4662
CP	186.5846	212.0123	250.1538	249.3249	274.7526	312.8941
QI	135.1728	153.5941	181.2261	180.6256	199.0469	226.6788
3H	290.0151	329.5383	388.8230	387.5346	427.0577	486.3424
6H	236.1363	268.3169	316.5877	315.5387	347.7192	395.9901
12H	172.2580	195.7332	230.9461	230.1808	253.6561	288.8689
(b) A0 = 4
UD	107.7706	151.4484	216.9650	215.5412	259.2189	324.7356
CP	90.0882	115.5159	153.6574	152.8285	178.2562	216.3977
QI	43.8061	62.2274	89.8594	89.2589	107.6802	135.3121
3H	216.2941	255.8173	315.1020	313.8136	353.3367	412.6214
6H	262.3648	294.5454	342.8162	341.7672	373.9477	422.2186
12H	106.3061	129.7814	164.9943	164.2290	187.7042	222.9171

for various layup configurations of perfect and imperfect S–S composite beams. Increasing the values of elastic foundation constants leads to an increase in the stiffness of the beam, and consequently leads to an increase in the values of critical buckling loads. Similar to bending results, Figure 5, the effect of the shear foundation constant is more pronounced than the linear foundation parameter.

Figure 6 displays the load-deflection curve related to first and second postbuckling configurations of S–S and C–C perfect composite beams. It is observed that the layup configurations have great influence on the postbuckling response of the beams. It is also noted that the perfect beam buckles through a pitchfork bifurcation.

Figure 7 presents the nonlinear load-deflection curves of the S–S imperfect helicoidal composite beams with different layup configurations. In contrast with a perfect beam, the imperfect beam buckles through a saddle-node bifurcation. It is noted that the imperfection amplitude as well as the layup configurations can be used to optimize the response of the composite beams.

6. Concluding Remarks

In this paper, based on the Laplace transform and its inversion, exact solutions were deduced for nonlinear bending and buckling behaviors of helicoidal composite perfect and imperfect beams embedded in a linear elastic medium. Parametric studies were carried out to show the effects of layup configurations, load type, elastic foundation constants, amplitude of initial imperfection, and boundary conditions on the nonlinear responses. The findings can be summarized as follows:

• The nonlinear bending deflection due to point load is highly dependent on the application position of the load.
• The hardening structural responses are associated with increasing the elastic foundation constants.
• The buckling strength is improved with an increase in the amplitude of initial imperfection; next, the critical values are continuously decreased with an increase in the initial imperfection amplitude.
• For larger values of amplitude of imperfection, the helicoidal composite beams can enhance the critical buckling loads.
• The layup configurations have a great influence on the nonlinear bending and buckling responses of perfect and imperfect beams.
• The proposed model can be exploited broadly in various engineering applications, such as airplane wings, helicopter blades, wind turbine blades, as well as many others in the aerospace, mechanical, and civil industries.