# Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams
## 4 Calculation of Fixed-End Moments  
### Fixed Beams with Both Ends Subjected to the Distributed Loads  
The positive directions for the end moments are different in MIV and SDM (see Figure 3).  
Figure 3: (**a**) Positive directions in MIV; (**b**) positive directions in SDM.  
Figure 2: (**a**) Beam subjected to singular force; (**b**) both ends of a fixed beam subjected to a distributed load.  
A fixed beam with both ends subjected to the distributed load is shown in Figure 2b. In this case, two of the initial values and end conditions can be written as  
\[v_{0}=0,\ \varphi_{0}=0\ ;\ v(L)=0,\ \varphi(L)=0 \tag{28}\]  
The other two initial values are found with the help of Equations (6) and (7) as follows.  
\[\begin{array}{l}M_{0}=\Big{(}\int_{0}^{L}\frac{\alpha}{E(\alpha)I(\alpha)}d \alpha\Big{)}\times\int_{0}^{L}p(\tau)\Big{(}\int_{0}^{L-\tau}-\frac{\alpha(a -L+\tau)}{E(\alpha)I(a)}d\alpha+\frac{\gamma^{2}(\tau-L)}{E(\tau)I(\tau)}\Big{)} \ d\tau\\ -\Big{(}\int_{0}^{L}\frac{\alpha(L-\alpha)}{E(\alpha)I(a)}d\alpha\Big{)} \times\int_{0}^{L}p(\tau)\Big{(}\int_{0}^{L-\tau}\frac{\alpha}{E(\alpha)I(a)} d\alpha-\frac{\gamma^{2}}{E(\tau)I(\tau)}\Big{)}d\tau\\ /\Big{(}\Big{(}\int_{0}^{L}\frac{L-\alpha}{E(\alpha)I(a)}d\alpha\Big{)}\ \int_{0}^{L}\frac{\alpha}{E(\alpha)I(a)}d\alpha-\Big{(}\int_{0}^{L}\frac{1}{E( \alpha)I(a)}d\alpha\Big{)}\int_{0}^{L}\frac{\alpha(L-a)}{E(\alpha)I(a)}d\alpha \Big{)}\end{array} \tag{29}\]  
\[\begin{array}{l}T_{0}=\Big{(}\Big{(}\int_{0}^{L}\frac{L-\alpha}{E(\alpha)I(a )}d\alpha\Big{)}\int_{0}^{L}p(\tau)\Big{(}\int_{0}^{L-\tau}\frac{E(\alpha)I(a )}{E(\alpha)I(a)}d\alpha-\frac{\gamma^{2}}{E(\tau)I(\tau)}\Big{)}d\tau\\ -\Big{(}\int_{0}^{L}\frac{1}{E(\alpha)I(a)}d\alpha\Big{)}\times\int_{0}^{L}p( \tau)\Big{(}\int_{0}^{L-\tau}-\frac{\alpha(a-L+\tau)}{E(\alpha)I(a)}d\alpha+ \frac{\gamma^{2}(\tau-L)}{E(\tau)I(\tau)}\Big{)}d\tau\Big{)}\\ /\Big{(}\Big{(}\int_{0}^{L}\frac{L-\alpha}{E(\alpha)I(a)}d\alpha\Big{)}\ \int_{0}^{L}\frac{\alpha}{E(\alpha)I(a)}d\alpha-\Big{(}\int_{0}^{L}\frac{1}{E( \alpha)I(a)}d\alpha\Big{)}\int_{0}^{L}\frac{\alpha(L-a)}{E(\alpha)I(a)}d\alpha \Big{)}\end{array} \tag{30}\]  
Special cases  
1. Uniform beam with constant cross section subjected to the uniform load (see Figure 4a). In this case, \[p(z)=p,\ E(z)=E,\ I(z)=I\] (31) Using Equations (29) and (30), the initial values \(M_{0}\) and \(T_{0}\) are found as follows. \[M_{0}=-\frac{1}{12}p(12\gamma^{2}+L^{2}),\ T_{0}=\frac{Lp}{2}\] (32) The variation of the initial moment (\(M_{0}\)) in classical and nonlocal theory with respect to the \(\gamma/L\) ratio is given in Figure 5a. 2. Uniform beam with a constant cross section subjected to the triangular load (see Figure 4b). In this case, \[p(z)=\frac{p}{L}z,\ E(z)=E,\ I(z)=I\] (33) Using Equations (29) and (30), the initial values \(M_{0}\) and \(T_{0}\) are found as follows. \[M_{0}=-\frac{L^{2}p}{30},\ T_{0}=\frac{3 Lp}{20}-\frac{\gamma^{2}p}{L}\] (34) The end moments are found using Equation (8) as below. \[M(L)=-\frac{1}{20}p(20\gamma^{2}+L^{2}),\ T(L)=-\frac{\gamma^{2}p}{L}-\frac{ 7 Lp}{20}\] (35) The variation of the end moment (\(M_{L}\)) in classical and nonlocal theory with respect to the \(\frac{\gamma}{L}\) ratio is given in Figure 5b.  
Figure 4: (**a**) Beam subjected to the uniform load; (**b**) beam subjected to the triangular load.