where p is the longitudinal load and q is the transverse load. Via the Hamilton’s principle, we have the equations of motion u$_{xx}$ ( c$_{11}$ − d 2 31 a$_{33}$ ) + u$_{zz}$ ( c$_{55}$ − d 2 15 a$_{11}$ ) + w$_{xz}$ ( c$_{13}$ + c$_{55}$ − d 2 15 a$_{11}$ ) − ϕ$_{xz}$ ( d$_{31}$ a$_{33}$ + d$_{15}$ a$_{11}$ ) + w$_{xxx}$ ( d$_{15}$ a$_{11}$ f$_{1111}$ − d$_{31}$ a$_{33}$ f$_{1313}$ ) + f$_{1331}$ a$_{11}$ d$_{15}$u$_{zzz}$ − w$_{xzz}$ ( d$_{31}$ a$_{33}$ f$_{3333}$ + d$_{15}$ a$_{11}$ f$_{1331}$ ) f$_{1111}$ a$_{11}$ [ ϕ$_{xxx}$ + f$_{1111}$u$_{xxxx}$ + f$_{1331}$ ( u$_{xxzz}$ + w$_{xxzz}$ ) + f$_{3311}$w$_{xxxz}$ ] f$_{1331}$ a$_{11}$ [ ϕ$_{xzz}$ + f$_{1111}$u$_{xxzz}$ + f$_{1331}$ ( u$_{zzzz}$ + w$_{xzzz}$ ) + f$_{3311}$w$_{xzzz}$ ] f$_{1133}$ + f$_{1313}$ a$_{33}$ [ ϕ$_{xzz}$ + f$_{1133}$u$_{xxzz}$ + f$_{1313}$ ( u$_{xxzz}$ + w$_{xxxz}$ ) + f$_{3333}$w$_{xzzz}$ ] − p = ϱ ¨ u, w$_{xx}$ ( c$_{55}$ − d 2 15 a$_{11}$ ) + w$_{zz}$c$_{33}$ + u$_{xz}$ ( c$_{13}$ + c$_{55}$ − d 2 15 a$_{11}$ ) (23) − d$_{15}$ a$_{11}$ ϕ$_{xx}$ − d$_{15}$ a$_{11}$ f$_{3311}$w$_{xxz}$ + u$_{xxx}$ ( d$_{31}$ a$_{33}$ f$_{1313}$ − d$_{15}$ a$_{11}$ f$_{1111}$ ) − f$_{1313}$ a$_{33}$ [ ϕ$_{xxz}$ + f$_{1133}$u$_{xxxz}$ + f$_{1313}$ ( u$_{xxxz}$ + w$_{xxxx}$ ) + f$_{3333}$w$_{xxzz}$ ] + f$_{1331}$ a$_{11}$ [ ϕ$_{xxz}$ + f$_{1111}$u$_{xxxz}$ + f$_{1331}$ ( u$_{xzzz}$ + w$_{xxzz}$ ) + f$_{3311}$w$_{xxzz}$ ] f$_{3333}$ a$_{33}$ [ ϕ$_{zzz}$ + f$_{1133}$u$_{xzzz}$ + f$_{1313}$ ( u$_{xzzz}$ + w$_{xxzz}$ ) + f$_{3333}$w$_{zzzz}$ ] − q = ϱ ¨ w, ϕ$_{xx}$ ( 3 a$_{11}$ − ε$_{0}$ ) + ϕ$_{zz}$ ( 3 a$_{33}$ − ε$_{0}$ ) − u$_{xz}$ ( d$_{15}$ a$_{11}$ + d$_{31}$ a$_{33}$ ) − d$_{15}$ a$_{11}$ w$_{xx}$ − f$_{1111}$ a$_{11}$ u$_{xxx}$ − f$_{3333}$ a$_{33}$ w$_{zzz}$ − u$_{xzz}$ ( f$_{1331}$ a$_{11}$ + f$_{1313}$ a$_{33}$ ) + w$_{xxz}$ ( f$_{1331}$ a$_{11}$ + f$_{3311}$ a$_{11}$ + f$_{1313}$ a$_{33}$ ) = 0 u = u ′ x = w = w ′ x = ϕ = 0 , (24) ∂H ∂u$_{x}$ − ∂ ∂x ( ∂H ∂u$_{xx}$ ) − 1 2 ∂ ∂z ∂H ∂u$_{xz}$ = P$_{∗}$, ∂H ∂u$_{xx}$ = 0 , ∂H ∂w$_{x}$ − ∂ ∂x ( ∂H ∂w$_{xx}$ ) − 1 2 ∂ ∂z ∂H ∂w$_{xz}$ = 0 , (25) ∂H ∂w$_{xx}$ = 0 , ∂H ∂ϕ$_{x}$ = 0 , with the boundary conditions presented for the brevity sake in the form of partial derivatives of the electric enthalpy function as: at x = 0 at x = l 13