Boundary conditions

x = 0 or a

Simply supported (S)

$w=\text{0,}\frac{{\partial }^{2}w}{\partial x^2}=0$

Guided (G)

$\frac{\partial w}{\partial x}=\text{0,}\frac{{\partial }^{3}w}{\partial x^3}=0$

Clamped (C)

$w=0,\frac{\partial w}{\partial x}=0$

Free (F)

$\frac{{\partial }^{2}w}{\partial x^2}=0,D\frac{{\partial }^{3}w}{\partial x^3}+N_x\frac{\partial w}{\partial x}=0$