  We show that a contact $(+1)$-surgery along a Legendrian sphere in a flexibly
fillable contact manifold ($c_1=0$ if not subcritical) yields a contact
manifold that is algebraically overtwisted if the Legendrian's homology class
is not annihilated in the filling. Our construction can also be implemented in
more general contact manifolds yielding algebraically overtwisted manifolds
through $(+1)$-surgeries. This gives new proof of the vanishing of contact
homology for overtwisted contact manifolds. Our result can be viewed as the
symplectic field theory analog in any dimension of the vanishing of contact
Ozsv\'ath-Szab\'o invariant for $(+1)$-surgeries on two-component Legendrian
links proved by Ding, Li, and Wu.
