There are general concentration results for convex bodies that apply (see [this answer to another question][1]), but for the simplex you can get away with calculating higher moments directly and using Markov's inequality.  For example see equation (19) in [this paper][2] (or equation (12) in the published version of that paper), which, after some renormalizing, states that
$$
\mathbb{E} x_1^{r_1} \cdots x_n^{r_n} = \frac{(n-1)! r_1! \cdots r_n!}{(r+n-1)!},
$$
where $r_i \ge 0$ and $r = r_1 + \cdots r_n$.  So in particular
$$
\mathbb{E} ||x||^2 = 
$$

Other relevant results are in [this famous paper][3] of Diaconis and Freedman.


  [1]: http://mathoverflow.net/questions/67521/concentration-of-measure-for-arbitrary-convex-bodies/67770#67770
  [2]: http://arxiv.org/abs/math/0505618
  [3]: http://stat-reports.lib.berkeley.edu/accessPages/85.html