  We consider a single living semi-flexible filament with persistence length
l_p in chemical equilibrium with a solution of free monomers at fixed monomer
chemical potential mu_1 and fixed temperature T. While one end of the filament
is chemically active with single monomer (de)polymerization steps, the other
end is grafted normally to a rigid wall to mimick a rigid network from which
the filament under consideration emerges. A second rigid wall, parallel to the
grafting wall, is fixed at distance L<<l_p from the filament seed. In
supercritical conditions the filament tends to grow and impinges onto the
second surface which, in suitable conditions (non-escaping filament regime)
stops the filament growth. We first establish the grand-potential and derive
some general properties, in particular the filament size distribution and the
force exerted by the living filament on the obstacle wall. We apply this
formalism to the semi-flexible, living, discrete Wormlike chain (d-WLC) model
with step size d and persistence length l_p, hitting a hard wall. By original
Monte-Carlo calculations we justify the use of the weak bending universal
expressions of Gholami et al. (Phys.Rev.E. 74,(2006), 041803) over the whole
non escaping filament regime. Employing this universal form for living
filaments, we find that the average force exerted by a living filament on a
wall at distance L is in practice L independent and very close to the value
predicted by Hill, his expression being strictly valid in the rigid filament
limit. The average filament force results from the product of the cumulative
size fraction x, where the filament is in contact with the wall, times the
buckling force on a filament of size L_c ~ L. We discuss several consequences
of the L independence of the stalling force for our specific filament model.
