  Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring,
$M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this
paper we study the graded structure of the $i$-th local cohomology module of
$M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More
precisely, we discuss finiteness property and vanishing of the graded
components $H^{i}_{R_{+},J}(M)_{n}$.
  Also, we study the Artinian property and tameness of certain submodules and
quotient modules of $H^{i}_{R_{+},J}(M)$.
