## 5 Results and Discussion  
Using the validated method, this section focuses on investigating the effects of non-uniformly distributed loads, inhomogeneous foundations, and axially functionally graded materials on the dynamic response of beams. Consider the square AFG beam resting on an inhomogeneous viscoelastic foundation and simply supported, as shown in Figure 7, with length \(L=2\) m, and width and height \(b=h=0.1\) m. Different types of FGMs, foundations, and loads are considered in order to gain a deeper insight into these effects. They are given by the following formulations.  
Figure 5: Dynamic response of the non-uniform AFG beam. (**a**) Sinusoidal load, \(x=L/4\); (**b**) Pulse load, \(x=L/4\); (**c**) Sinusoidal load, \(x=L/2\); (**d**) Pulse load, \(x=L/2\).  
Two types of axially functionally graded materials are taken into account, including:
* Power: \[\rho(x)=\rho_{1}+(\rho_{2}-\rho_{1})\Big{(}\frac{x}{L}\Big{)}^{n},\quad E(x)=E_{1 }+(E_{2}-E_{1})\Big{(}\frac{x}{L}\Big{)}^{n},\quad G(x)=\frac{E(x)}{2(1+\mu)}\] (53)
* Exponential: \[\rho(x)=\rho_{1}\exp\Big{[}\ln\Big{(}\frac{\rho_{2}}{\rho_{1}}\Big{)}\Big{(} \frac{x}{L}\Big{)}^{n}\Big{]},\,E(x)=E_{1}\exp\Big{[}\ln\Big{(}\frac{E_{2}}{E_{ 1}}\Big{)}\Big{(}\frac{x}{L}\Big{)}^{n}\Big{]},\,G(x)=\frac{E(x)}{2(1+\mu)}\] (54) where the subscripts \(1\) and \(2\) represent the aluminum and zirconia components, respectively. Their properties \(\rho_{1}=5700\) kg/m\({}^{3}\), \(\rho_{2}=2702\) kg/m\({}^{3}\), \(E_{1}=200\) GPa, \(E_{2}=70\) GPa, the Poisson's ration \(\mu=1/3\). The shear correction factor \(\kappa=2/3\). \(n\) is the material gradient index, which determines the variation of the material properties along the \(x\) direction, as shown in Figure 8.  
Two kinds of inhomogeneous viscoelastic foundations are considered, whose foundation parameters are given by the following:  
* Linear: \[k_{f}(x)=2k_{f1}\Big{(}1-\lambda_{1}\frac{L-x}{L}\Big{)},\quad c_{f}(x)=2c_{f1 }\Big{(}1-\lambda_{2}\frac{L-x}{L}\Big{)}\] (55)
* Parabolic: \[k_{f}(x)=2k_{f1}\Bigg{[}1-\lambda_{1}\Big{(}\frac{L-x}{L}\Big{)}^{2}\Bigg{]}, \quad c_{f}(x)=2c_{f1}\Bigg{[}1-\lambda_{2}\Big{(}\frac{L-x}{L}\Big{)}^{2} \Bigg{]}\] (56)  
Figure 8: Variations of the material properties with material gradient index \(n\). (**a**) Young’s modulus; (**b**) Material density.  
Figure 7: An AFG beam on an inhomogeneous viscoelastic foundation subjected to a non-uniformly distributed moving load.  
where \(\lambda_{1},\lambda_{2}\) are the variation coefficients of the foundation, and  
\[k_{f1}=\bar{k}_{f}E_{1}I/L^{4},\quad c_{f1}=\bar{c}_{f}\sqrt{\rho_{1}AE_{1}I}/L^{ 2}, \tag{57}\]  
the dimensionless parameters \(\bar{k}_{f}=0.6\pi^{4}\), \(\bar{c}_{f}=1\). The moment of inertia \(I=bh^{3}/12\), and the area of the cross-section \(A=bh\). The variation of the foundation properties with variation coefficients \(\lambda_{1}\) and \(\lambda_{2}\) is shown in Figure 9.  
To model the moving load, three load models are employed, including the realistic non-uniformly distributed load, the idealized uniformly distributed load, and the idealized concentrated load, as shown in Figure 10.  
The non-uniformly distributed moving load can be expressed in the moving coordinate system \(\bar{\sigma}=\bar{\sigma}\bar{\sigma}\) as  
\[q(\bar{x})=\frac{q_{0}}{\sqrt{2\pi\sigma}}e^{-(\bar{x}-\bar{I}/2)^{2}/(2\sigma ^{2})} \tag{58}\]  
where \(\bar{I}\) is the span of the distributed load, \(\sigma=\bar{I}/6\). Both the concentrated load model and the uniformly distributed load model are commonly used as simplified models for simulating loads that are actually non-uniformly distributed. The relationships between them can be expressed as  
\[\int_{0}^{\bar{I}}q(\bar{x})\mathrm{d}\bar{x}=q_{u}\bar{I}=q_{c}=q_{0}=10\; \mathrm{kN} \tag{59}\]  
The errors resulting from the simplification of the concentrated and uniformly distributed load models are investigated in order to confirm their available situations.