# Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework
## 5 Examples  
### Peristatic Solutions  
The derived closed-form solutions can be applied to a large class of beam problems for various boundary conditions and loading cases, including distributed and concentrated loads. Below, we present plots of derived solutions for the deflection function for several examples. In addition, well-known analytical solutions to the classical beam theory will be plotted in order to demonstrate the validation of both the PD beam theory and the derived results. Consider a beam with a rectangular cross-section with high \(h=0.01\) m and width \(b=0.01\) m. The length of the beam is assumed to be \(L=1\) m, and [PERSON]'s modulus of the beam material is \(E=200\) GPa. conditions. The PD horizon size is chosen as \(\delta=0.001\) m. Figure 13 illustrates the plots of closed-form series solutions developed in this study. Different cases of loadings, including concentrated moment applied at different points of the beam, uniformly distributed load over the beam length, concentrated force at different points of the beam, and the load distributed over a part of the beam are considered. Furthermore, different examples of boundary conditions, including simply supported-simply supported, clamped-clamped, simply supported-roller-clamped, clamped-roller-clamped, and clamped-free, are analyzed. For comparison, solutions according to the classical beam theory for the same loadings and boundary conditions are presented. A very good agreement between the results for the selected value of the horizon size can be observed. Let us note that the horizon size is an important parameter in PD, providing a length-scale influence and the non-locality of the system. If the horizon size tends to zero, the PD solution converges to that of the classical beam theory in the case of elastic material behavior. The horizon size enters the derived analytical solutions in the explicit form. Plots of deflection vs. horizon size for a simply supported beam are presented in [27]. Let us summarize the closed-form analytical solutions for the six cases presented in Figure 13 as follows:  
1. Simply supported-simply supported \[w(x)=\frac{2}{cL}\sum\limits_{n=1}^{\infty}\frac{-\frac{n\pi}{L}M\text{cos} \big{(}\frac{n\pi}{2}\big{)}}{\left[\int\limits_{-\delta}^{\delta}\frac{1}{ \xi^{2}}\left(1-\cos\frac{n\pi\xi}{L}\right)d\xi\right]^{2}}\text{sin}\frac{n \pi x}{L}\] based on PD and \[w(x)=-\frac{M}{24 LE1}\Big{\{}\Big{[}-3L^{3}+12L^{2}x-12 Lx^{2}\Big{]}H(2x-L)-L^{ 2}x+4x^{3}\Big{\}}\] based on CBT. 2. Clamped-simply supported \[w(x)=-\frac{2}{cL}\sum\limits_{n=1}^{\infty}\frac{\frac{\int\limits_{-\delta}^{ L}p\text{sin}\frac{n\pi x}{L}dx-\frac{n\pi}{L}\frac{p}{8}L^{2}}{\left[\int\limits_{- \delta}^{\delta}\frac{1}{\xi^{2}}\left(1-\cos\frac{  
based on PD and \[w(x)=\frac{F}{162 EI}\Big{[}4x^{2}(3L-5x)-(L-3x)^{3}H(3x-L)\Big{]}\] based on CBT. 4. Simply supported-roller-clamped \[w(x)=-\frac{2p}{cL}\sum_{n=1}^{\infty}\frac{\frac{\int\limits_{ \frac{1}{4}}^{\delta}\sin\frac{(2n-1)\pi x}{2L}d\xi}}{\int\limits_{-\delta}^{ \delta}\frac{1}{\xi^{2}}\Big{(}1-\cos\frac{(2n-1)\pi\xi}{2L}\Big{)}d\xi}\Big{]}^ {2}\sin\frac{(2n-1)\pi x}{2L}\] based on PD and \[w(x)=-\frac{p}{6144 EI}\Bigg{\{} \begin{array}{l}\big{[}L^{4}-16L^{3}x+96L^{2}x ^{2}-256 Lx^{3}+256x^{4}\big{]}H(L-4x)\\ +\big{[}-81L^{4}+432L^{3}x-864L^{2}x^{2}-256x^{4}+768 Lx^{3}\big{]}H(3L-4x)\\ -L^{3}(L-16x)+27L^{3}(3L-16x)+768L^{2}x^{2}\end{array} \Bigg{\}}\] based on CBT. 5. Clamped-roller-clamped \[w(x)=-\frac{2p}{cL}\sum_{n=1}^{\infty}\frac{\sin\frac{(2n-1)\pi}{4}-\frac{3}{ \xi^{2}}(2n-1)\pi}{\left[\int\limits_{-\delta}^{\delta}\frac{1}{\xi^{2}}\Big{(} 1-\cos\frac{(2n-1)\pi\xi}{2L}\Big{)}d\xi\right]^{2}\sin\frac{(2n-1)\pi x}{2L}}\] based on PD and \[w(x)=\frac{F}{48 EI}\Big{[}x^{2}(9L-8x)-(L-2x)^{3}H(2x-L)\Big{]}\] based on CBT 6.