\textbf{Question 6}
\begin{center}
Nous savons que: $ x_1(\zeta, t) = \frac{d^2 w}{d^2\zeta}(\zeta,t)$
$\Rightarrow w(\zeta,t)=\int\int_0^L x_1(\mu , t)d\mu $\\
comme $x_1(\mu,t)=\Phi^T(\mu)x_{1d}(t)$\\
d'où 
\begin{equation}
w(\zeta,t) = 
\begin{bmatrix}
\int\int_0^\zeta \Phi(\mu)d\mu & 0_{{\rm I\!R}_{1\times4}}
\end{bmatrix}x_d(t)
\end{equation}
\end{center}