  We develop a rigorous mathematical framework for the weak formulation of
cracked beams and shallow arches problems. First, we discuss the crack modeling
by means of massless rotational springs. Then we introduce Hilbert spaces,
which are sufficiently wide to accommodate such representations. Our main
result is the introduction of a specially designed linear operator that
"absorbs" the boundary conditions at the cracks.
  We also provide mathematical justification and derivation of the Modified
Shifrin's method for an efficient computation of the eigenvalues and the
eigenfunctions for cracked beams.
