Herrmann’s Approximate Method to Obtain Q by Beam Theory
A remarkably simple method for close approximation of Q in notched beams was discovered by Kienzler and Herrmann (1986) and Herrmann and Sosa (1986). The method was derived from a certain unproven hypothesis (postulate) regarding the energy release when the thickness of the fracture band is increased. Bazant found a different derivation of this method (Bazant 1990a) which is simpler and at the same time indicates that the hypothesis used by Herrmann et al. might not be exact but merely a good approximation. Also, Herrmann’s method relies on more sophisticated concepts (material forces) which are elegant but seem more complicated than necessary to obtain the result. An even a simpler method of deriving Bazant’s and Herrmann’s results has been recently developed by Planas and Eliccs (199Id). This last will be presented now.
The method consists of approximating the cracked beam by a triangularly notched beam as shown in Fig. 3.2.3, and calculating its energy in the frame of the strength of materials theory (bent beam of variable inertia.)
Let к be the slope of the sides of the triangular notch, to be determined empirically. Let b and D be, respectively, the thickness, and depth of the unnotched beam, and let M be the constant bending moment over a central portion of the beam of length 2L > 2ka. With the axis shown in Fig. 3.2.3, the complementary energy of the central portion is
where El і is the bending stiffness of the unnotched beam and EI(c) is the bending stiffness when the
(: — D -• a d-
where, since 1(D) — 11, the second and third terms cancel out. Moreover, from Eq. (3.2.9), dc/da =– —kdc/dx. Integration now delivers Herrmann’s result
where Eh is the bending stiffness of the central (notched) section.
When the remote flexural stress a/ = MD/2I is used as a measure of the load, and the expressions of the inertia moments for rectangular cross-section are substituted, the previous equation reads
According to Kienzler and Herrmann (1986, Pigs. 3 and 4), this compares (for к = 1) very well with accurate solutions from handbooks. However, it appears that the agreement would be even better for some value к ф 1 (Bazant 1990a). For very shallow notches, this method requires rather large A;-value (about 4) to accurately fit the results. But in this case, the approximation by a beam of variable thickness is poor.
The results of Herrmann and Sosa (1986) for double-cdge-notched and center-cracked specimens may also be obtained using the Planas and Elices expedient of approximating the crack by a triangular notch (coinciding with the shaded areas in Pig. 3.2.2a for the center cracked panel), and performing a classical analysis with the assumption that the cross-sections remain plane.