and                                                     (12.16)

Equation (12.15) is the equation of motion (EOM) and equation (12.16) are natural boundary conditions. In the EOM the first and the second terms are corresponding to the inertia and the elasticity, respectively and terms in right hand side are external forces. In boundary conditions [i.e. equation (12.16)] the first term contains the bending moment and the slope terms and the second term contains the shear force and the deflection. It should be noted that either the bending moment or the slope must be zero to satisfy the condition of equation (12.16). Similarly, the shear force or the deflection must be zero. For example, the cantilever beam has the deflection and the slope zero at fixed end however the shear force and the bending moment are not zero. For the free end the deflection and the slope are not zero however the shear force and the bending moments are zero.