  In this article, we consider eigenfunctions $u$ of the bi-harmonic operator,
i.e.,
  $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary
conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a
$C^{\infty}$ bounded domain, $\partial\Omega$ is piecewise analytic and
$\partial\Omega$ is analytic except a set $\Gamma\subseteq\partial\Omega$ which
is a finite union of some compact $(n-2)$ dimensional submanifolds of
$\partial\Omega$. The main result of this paper is that the measure upper
bounds of the nodal sets of the eigenfunctions is controlled by
$\sqrt{\lambda}$. We first define a frequency function and a doubling index
related to these eigenfunctions. With the help of establishing the monotonicity
formula, doubling conditions and various a priori estimates, we obtain that the
$(n-1)$ dimensional Hausdorff measures of nodal sets of these eigenfunctions in
a ball are controlled by the frequency function and $\sqrt{\lambda}$. In order
to further control the frequency function with $\sqrt{\lambda}$, we first
establish the relationship between the frequency function and the doubling
index, and then separate the domain $\Omega$ into two parts: a domain away from
$\Gamma$ and a domain near $\Gamma$, and develop iteration arguments to deal
with the two cases respectively.
