  Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on
$[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not
contain $d$ sets with union of size at most $s$ and empty intersection.
Answering a question of Frankl, we present some structural results for families
that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families
that are $(k,2k)$-conditionally intersecting. As applications of our structural
results, we present some new proofs to the upper bounds for the size of the
following $k$-uniform families on $[n]$.
  (a) $(d,2k+d-3)$-conditionally intersecting families with $n\ge 3k^5$.
  (b) $(k,2k)$-conditionally intersecting families with $n\ge k^2/(k-1)$.
  (c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\ge
3k\binom{2k}{k}$.
  Our results for $(c)$ confirms a conjecture of Mammoliti and Britz for the
case $d=3$.
