# Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load
## 3 Static Analysis  
The equation governing the static response of initially curved, multilayer nanobeams clamped at both ends can be obtained from Equation (16) by dropping the time-dependent terms. The result is  
\[\left[1-\overline{\mu}\;\left(\overline{P}-\frac{1}{2}\alpha\int\limits_{0}^{ 1}\Big{(}{w_{s}^{\prime}}^{2}-{w_{0}^{\prime}}^{2}\Big{)}dx\right)\right]\;w _{s}^{i\
u}+\left(\overline{P}-\frac{1}{2}\alpha\int\limits_{0}^{1}\Big{(}{w_ {s}^{\prime}}^{2}-{w_{0}^{\prime}}^{2}\Big{)}dx\right)w_{s}^{\prime\prime}=w_{0 }^{i\
u} \tag{17}\]  
where \(w_{s}(x)\) is the lateral static deflection of the beam. Again, \(\overline{P}\) is the nondimensional initial pre-stress. Because geometric imperfection is present, Equation (17) governs all equilibrium positions that may exist at a given pre-load, \(\overline{P}\). This is not the classical buckling problem that is basically an eigenvalue problem. Indeed, when the beam exhibits some imperfection, and for any nonzero load, there exists a static equilibrium position that grows gradually as the load is increased. The problem with imperfection, as given by Equation (17), is a boundary-value problem. This is evident by the fact that the right-hand-side of Equation (17) is nonzero. The question is can we still find a buckling load in this case. The answer is yes. The point of buckling, in this case, will not refer to Euler buckling or bifurcation buckling, but it refers to a limit point, which may define a buckling load that is less or larger than the Euler buckling load. The structural dynamics community use the terms, bifurcation pointsand limit points, to refer to Euler buckling of initially straight columns and initially curved columns, respectively [68]. In the meantime, the nonlinear dynamics community use the terms, pitch-fork bifurcation and saddle-node bifurcations, to refer to these two points, respectively [69]. To gain insight, we define the effective compressive force that acts at the beam ends after buckling as  
\[\Lambda=\left(\overline{P}-\frac{1}{2}\alpha\int\limits_{0}^{1}\left({w^{ \prime}}^{2}-{w_{0}^{\prime}}^{2}\right)dx\right) \tag{18}\]  
This is simply the initial pre-stress load minus the induced tensile midplane stretching due to buckling. Therefore, Equation (17) can be expressed as  
\[\left(1-\overline{\mu}\ \Lambda\right)w_{\mathrm{s}}^{\mathrm{i}p}+\Lambda\ w_{ \mathrm{s}}^{\prime\prime}=w_{0}^{\mathrm{i}p} \tag{19}\]  
As such, the size effect is about reducing the bending stiffness, which will result in reduced buckling loads and an increased postbuckling response, as will be shown next. The initial configuration of a clamped-clamped imperfect beam is assumed to have the form [17]  
\[w_{0}(x)=\frac{1}{2}a\ (1-\cos\ 2\ \pi x) \tag{20}\]  
where \(a\) is the beam's midspan initial rise, which is used as a control parameter representing the nondimensional imperfection amplitude. [PERSON] [17] found that a general solution for the static deflection, \(w_{s}(x)\), of a perfect or imperfect beam, can be expressed as follows  
\[w_{s}(x)=\frac{1}{2}h(1-\cos\ 2\ \pi x) \tag{21}\]  
where \(h\) is the buckling amplitude. Substituting Equations (20) and (21) into Equation (17) yields  
\[\Big{(}1+4\pi^{2}\Big{)}\alpha h^{3}+\bigg{[}16-a^{2}\alpha\Big{(}1+4\ \pi^{2} \overline{\mu}\ \Big{)}-\frac{4\overline{P}}{\pi^{2}}\Big{(}1+4\ \pi^{2}\overline{\mu}\ \Big{)}\bigg{]}h-16\ a=0 \tag{22}\]  
Equation (22) governs the buckling amplitude, \(h\), for a given pre-stress load, \(\overline{P}\), initial imperfection amplitude, \(a\), laminate layup coefficient, \(\alpha\), and size parameter, \(\overline{\mu}\). Buckling occurs when the two roots of Equation (22) coalesce. This condition is satisfied if the discriminant of Equation (22) is zero. Equation (22) takes the form  
\[c_{3}h^{3}+c_{1}h+c_{0}=0 \tag{23}\]  
where the constants, \(c_{i}\), can be defined in light of Equation (22). In the prebuckling state, Equation (22) yields only one real solution that corresponds to the stable equilibrium position.