11. Let $\triangle A B C$ be inscribed in a circle $\odot O$ with radius $R$, and $A B=A C, A D$ be the altitude from $A$ to the base $B C$. Then the maximum value of $A D + B C$ is $\qquad$ .
11. $R+\sqrt{5} R$.

If finally 3, let $\angle O B D=\alpha$,
then
$$
\begin{array}{l}
A D=R+R \sin \alpha, \\
\frac{1}{2} B C=B D=R \cos \alpha, \\
B C=2 R \cos \alpha . \\
\text { Therefore, } A D+B C \\
=R+R \sin \alpha+2 R \cos \alpha \\
=R+\sqrt{5} R \sin (\alpha+\varphi),
\end{array}
$$

where $\tan \varphi=2$.
Thus, the maximum value of $A D+B C$ is $R+\sqrt{5} R$.