# Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams
## 3 Applications of the Initial Values Method
### Beam Subjected to Two Different Distributed Loads  
In this case, Equation (4) can be written for both regions as follows (see Figure 1b).  
\[\overrightarrow{y}_{1}(z)=TM(z).\overrightarrow{y}_{0}+\int_{0}^{z}TM(z-\tau). \overrightarrow{f}_{1}(\tau)d\tau,\ 0\leq z\leq\eta L \tag{9}\]  
\[\overrightarrow{y}_{2}(z)=TM(z).\overrightarrow{y}_{1}(\eta L)+\int_{0}^{z}TM( z-\tau).\overrightarrow{f}_{2}(\tau)d\tau,\ 0\leq z\leq(1-\eta)L \tag{10}\]  
where  
\[\overrightarrow{f}_{1}(z)=\begin{pmatrix}0\\ -\frac{\gamma^{2}p_{1}(z)}{E(z)I(z)}\\ 0\end{pmatrix},\ \overrightarrow{f}_{2}(z)=\begin{pmatrix}0\\ -\frac{\gamma^{2}p_{2}(z)}{E(z)I(z)}\\ 0\\ -p_{2}(z)\end{pmatrix} \tag{11}\]  
The unknowns of a beam subjected to two distributed loads are calculated using Equations (9) and (10) as below (see Figure 1b).  
\[\begin{array}{l}v_{1}(z)=-M_{0}\Big{(}\int_{0}^{z}\frac{z-a}{E(a)I(a)}da \Big{)}-T_{0}\Big{(}\int_{0}^{z}\frac{\alpha(z-a)}{E(a)I(a)}da\Big{)}+v_{0}+ \varphi_{0}z\\ +\int_{0}^{z}\frac{p_{1}(\tau)\big{(}E(\tau)I(\tau)\int_{0}^{z-\tau}-\frac{ \alpha(a+\tau-i)}{E(a)I(a)}da+\gamma^{2}(\tau-z)\big{)}}{E(\tau)I(\tau)}d\tau \\ \\ \varphi_{1}(z)=-M_{0}\Big{(}\int_{0}^{z}\frac{1}{E(a)I(a)}da\Big{)}-T_{0}\Big{(} \int_{0}^{z}\frac{\alpha}{E(a)I(a)}da\Big{)}+\varphi_{0}\\ +\int_{0}^{z}p_{1}(\tau)\Big{(}\int_{0}^{z-\tau}\frac{\alpha}{E(a)I(a)}da- \frac{\gamma^{2}}{E(\tau)I(\tau)}\Big{)}d\tau\\ \\ M_{1}(z)=M_{0}+\int_{0}^{z}p_{1}(\tau)\ (\tau-z)\ d\tau+T_{0}z,\ T_{1}(z)=\int_{0}^{z}-p_{1}( \tau)\ d\tau+T_{0}\end{array} \tag{14}\]  
\[\begin{array}{l}v_{2}(z)=-M_{0}\Big{(}\int_{0}^{\eta L}\frac{\eta L-a}{E(a)I (a)}da\Big{)}-\Big{(}\int_{0}^{z}\frac{(z-a)}{E(a)I(a)}da\Big{)}\left(\int_{0} ^{\eta L}p_{1}(\tau)(\tau-\eta L)d\tau+\eta LT_{0}+M_{0}\right)\\ -M_{0}z\Big{(}\int_{0}^{\eta L}\frac{1}{E(a)I(a)}da\Big{)}+\int_{0}^{\eta L} \frac{p_{1}(\tau)\big{(}E(\tau)I(\tau)\int_{0}^{\eta L-\tau}-\frac{\alpha(a- \eta L+\tau)}{E(a)I(a)}da+\gamma^{2}(\tau-\eta L)\big{)}}{E(\tau)I(\tau)}d\tau \\ -\Big{(}\int_{0}^{z}\frac{\alpha(z-a)}{E(a)I(a)}da\Big{)}\Big{(}\int_{0}^{\eta L }-p_{1}(\tau)d\tau+T_{0}\Big{)}+z\int_{0}^{\eta L}p_{1}(\tau)\Big{(}\int_{0}^ {\eta L-\tau}\frac{\alpha}{E(a)I(a)}da-\frac{\gamma^{2}}{E(\tau)I(\tau)}\Big{)} d\tau\\ -T_{0}\Big{(}\int_{0}^{\eta L}\frac{\alpha(\eta L-a)}{E(a)I(a)}da+z\int_{0}^{ \eta L}\frac{\alpha}{E(a)I(a)}da\Big{)}+\int_{0}^{z}\frac{p_{2}(\tau)\big{(}E( \tau)I(\tau)\int_{0}^{z-\tau}-\frac{\alpha(a+\tau-i)}{E(a)I(a)}da+\gamma^{2}( \tau-z)\big{)}}{E(\tau)I(\tau)}d\tau\\ +\varphi_{0}(\eta L+z)+\eta_{0}\\ \\ \varphi_{2}(z)=-M_{0}\Big{(}\int_{0}^{\eta L}\frac{1}{E(a)I(a)}da\Big{)}-\Big{(} \int_{0}^{z}\frac{1}{E(a)I(a)}da\Big{)}\left(\int_{0}^{\eta L}p_{1}(\tau)( \tau-\eta L)d\tau+\eta LT_{0}+M_{0}\right)\\ +\int_{0}^{\eta L}p_{1}(\tau)\Big{(}\int_{0}^{\eta L-\tau}\frac{\alpha}{E(a)I(a) }da-\frac{\gamma^{2}}{E(\tau)I(\tau)}\Big{)}d\tau-\Big{(}\int_{0}^{z}\frac{ \alpha}{E(a)I(a)}da\Big{)}\left(\int_{0}^{\eta L}-p_{1}(\tau)d\tau+T_{0}\right) \\ -T_{0}\Big{(}\int_{0}^{\eta L}\frac{\alpha}{E(a)I(a)}da\Big{)}+\int_{0}^{z}p_{2 }(\tau)\Big{(}\int_{0}^{z-\tau}\frac{\alpha}{E(a)I(a)}da-\frac{\gamma^{2}}{E( \tau)I(\tau)}\Big{)}d\tau+\varphi_{0}\\ \\ M_{2}(z)=\int_{0}^{\eta L}p_{1}(\tau)(\tau-\eta L)d\tau+z\int_{0}^{\eta L}-p_{1}( \tau)d\tau+T_{0}(\eta L+z)+M_{0}\\ +\int_{0}^{z}p_{2}(\tau)(\tau-z)d\tau\end{array} \tag{17}\]  
Figure 1: (**a**) A beam subjected to a distributed load; (**b**) a beam subjected to two different loads.  
\[T_{2}(z)=\int_{0}^{\eta L}-p_{1}(\tau)d\tau+\int_{0}^{z}-p_{2}(\tau)d\tau+T_{0} \tag{18}\]