# Exercises

1.1 In fracture mechanics manuals, it is customary to use for crjv the maximum tensile stress computed elastically for an unnotchcd specimen. Express on for a beam in terms of the maximum bending moment A/, the beam depth D, and the central moment of inertia of the cross-section [. (Answer: on — M D/2I)

1.2 Determine on as in the previous exercise for a hollow cylindrical bar of outer diameter U and inner diameter aD (a < 1) subjected to a torsional moment Mr. (Answer: on 16Л/т/[тгІ)‘,(1 —a4)])

1.3 . With the same criteria as inthe previous exercises, determine o. n for a circular bar of diameter D subjected to simultaneous tension and torsion; let F be the tensile force and Mt — PPD the torque, where p is some dimensionless constant. Give the coefficient cn corresponding to Eq. (1.4.1). Hint: use Mohr’s circle to find the maximum tensile stress.

1.4 Results from the literature were analyzed by the authors using a characteristic specimen dimension D and a nominal stress оn defined with c, v -= 1. The results for the best fit of Bf’t and Do were 1.15 MPa and 322 mm, respectively. To compare with other results, you want to use a nominal stress defined using the same characteristic size D, but a constant c. v = 2.5. What would the values of the best-fit constants (say В f[ and Do) in this case. (Answer: Вf’t = 2.88 MPa, Do – 322 mm.)

1.5 Generalize the previous exercise and prove that if, for a particular selection of c. v and D, the size effect parameters arc В ft and Do, for a different selection c. v and D (where D/D — constant), their value is Bfl •= (cN/cN)Bfi and Do = (J0/D)D0.

1.6 Find the relationship between hj and cj that make identical Eqs. (1.4.12) and (1.4.14).

 Series Material Specimen type" ao/D S/D b (mm) CN Reference АІ-А6 concrete SEN-TPB 1/3 4 76.2 6 Walsh 1972 B1 concrete SEN-TPB 1/6 2.5 38.1 3.75 Bazant and Pfeiffer 1987 B2 concrete DEN-F. C 1/6 — 38.1 1 Ibid. B3 concrete DEN-T 1/6 — 19.1 1 Ibid. B4 concrete DEN-S 1/6 — 38.1 1 Bazant and Pfeiffer 1986 Cl mortar SEN-TPB 1/6 2.5 38.1 3.75 Bazant and Pfeiffer 1987 C2 mortar DEN-F. C 1/6 — 38.1 1 Ibid. C3 mortar DEN-T 1/6 . ——— 19.1 1 Ibid. C4 mortar DEN-S 1/6 — 38.1 1 Bazant and Pfeiffer 1986 D1 HSC SEN-TPB 1/3 2.5 38.1 3.75 Gettu, Bazant and Karr 1990 El marble SEN-TPB .0.5 4 30 6 Fatby 1992 E2 granite SEN-TPB 0.5 4 30 6 Ibid. FI limestone SEN-TPB 0.4 4 – 13 6 Bazant, Gettu and Kazemi 1991 G1 Si02 – SEN-TPB 0.2b 4 = D 6 McKinney and Rice 1981° G2 SiCCN-137 SEN-TPB 0.2 4 = D 6 Ibid. G3 SiC CN-163 SEN-TPB 0.2 4 = D 6 Ibid. Hl-2 concrete DP ■— — — 0.4 Marti 1989 11 microcon. BPO —– —– — — 4/тг Bazant and §ener 1988 J1 mortar UPT — — — 0.75 Bazant, §ener and Prat 1988 J2 R. mortar RPT —– —– — — 0.75 Ibid. K1 R. mortar LRB-UB — — — 0.5 Bazant and Kazemi 1991 K2 R. mortar LRB-AB — — — 0.5 Ibid. LI microcon. PS — — — 1 / 7Г Bazant and Cao 1987
 “See specimen types in Fig. 1.5.1. ^Variable. Results have been reduced to a fixed ao/D — 0.2. cSee also Bazant and Kazetni 1990b.