# Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams
## 4 Examples  
We consider three example problems corresponding to the three types of boundary conditions examined in the previous section, and for each of them, we find in closed form the transverse displacement (deflection) \(w(x)\) for two different types of transverse distributed loads \(q(x)\). It is noted that in all instances except the case of classical (local) theory, the solutions are generally large algebraic expressions. Let a nanobeam have length \(L\), height \(h\), width \(b\), Young's modulus \(E\), and a load intensity parameter \(q_{0}\), as shown in Table 1[31]. The same table also has the intervals for the values of the nonlocal material constant \(\tau\) and the parameter \(\xi_{1}\) (\(\xi_{1}+\xi_{2}=1\)). It is remarked that [PERSON], [PERSON] and [PERSON] [28] stated that the nonlocal effect is noticeable when the length of the structure is less than 20 nm and recommended \(e_{0}a<2.1\) nm, while [PERSON] [10] suggested a value of parameter \(e_{0}\) to be 0.39. First, we study the bending behavior of a cantilever beam (CF) for which the boundary conditions are as in (20) loaded by a transverse distributed load  
\[q(x)=q_{0}\quad\text{or}\quad q(x)=-q_{0}\sin(n\tau\frac{x}{L}),\]  
where \(n\) is a positive integer. For the case of uniformly distributed load \(q(x)=q_{0}\), the deflection \(w(x)\) throughout the beam according to local (\(\xi_{1}=1\)) and nonlocal (\(\xi_{1}=0.1\)) elasticity for various values of the nonlocal parameter \(\tau=e_{0}a\) is depicted in Figure 1. Figure 2 shows the deflection \(w(x)\) for \(\tau=2\) and several values of the parameter \(\xi_{1}\) that controls the influence of local and nonlocal integral models in the constitutive relation. In the case of a variable distributed load \(q(x)\) with \(n=3\), the deflection \(w(x)\) is sketched in Figure 3 for different values of the nonlocal parameter \(\tau\). Next, we consider the case of a clamped pinned beam (CP) with the boundary conditions as in (34). For the case of uniformly distributed load \(q(x)=q_{0}\), the deflection \(w(x)\) for the whole beam in both local (\(\xi_{1}=1\)) and nonlocal (\(\xi_{1}=0.1\)) elasticity for several values of the nonlocal parameter \(\tau=e_{0}a\) is outlined in Figure 4. In Figure 5, we give the deflection \(w(x)\) for \(\tau=2\) and various values of the control parameter \(\xi_{1}\). The shape of deformation of \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(L\) (nm) & \(b\) (nm) & \(h\) (nm) & \(q_{0}\) (nN/nm) & \(E\) (TPa) & \(\tau=e_{0}a\) (nm) & \(\xi_{1}\) \\ \hline
10 & 1 & 1 & 10\({}^{-4}\) & 5.5 & \([1.0,2.0]\) & \([0.1,1]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry, loading, and material parameters of the nanobeam. Figure 1: Deflection of cantilever beam (CF) under uniform load and various values of \(\tau\). the beam loaded by a variable distributed load \(q(x)\) of the above type with \(n=3\) is shown in Figure 6 for different values of \(\tau\). As a third example, we take the case of a clamped beam (CC) for which the boundary conditions are given in (42). In the case of a uniformly distributed load, the deflection \(w(x)\) across the beam in both local (\(\xi_{1}=1\)) and nonlocal (\(\xi_{1}=0.1\)) theory for different values of \(\tau=e_{0}a\) is given in Figure 7, while Figure 8 shows how the deflection changes as \(\xi_{1}\) varies. In the case of the above variable distributed load \(q(x)\) with \(n=3\), the beam deforms as shown in Figure 9 in a local (\(\xi_{1}=1\)) and nonlocal (\(\xi_{1}=0.2\)) model for different values of the nonlocal parameter \(\tau\). Figure 3: Deflection of cantilever beam (CF) under variable load and different values of \(\tau\). Figure 2: Deflection of cantilever beam (CF) under uniform load and several values of \(\xi_{1}\). From the results presented, it can be concluded that in all three cases of boundary conditions and loading cases, the solutions obtained are characterized by the softening effect that the nonlocal theory has on the beam deformation.