Load Causing Cracks of Given Relative Depth

One possibility is to define the cracking load as the load that produces initial diagonal shear cracks of a depth Di representing a given percentage of beam depth D, i. e., such that the ratio D/D is agiven constant (Fig. 10.3.7a), say 0.5. We imagine an array of the initial cracks, as shown in Fig. 10.3.7a. The formation of each initial crack causes stress redistribution in triangular zones 1321 and 1341, shaded in Fig. 10.3.7a. (In contrast to Fig. 10.3.4, the stress relief zones are not strips, nor elongated triangles, because the material is not orthotropically damaged before the initial cracks form.) For the sake of simplicity, these zones may be assumed to consist of triangles with angles roughly в = 45°, each two triangles making a square. The shape of these zones and the length of the initial cracks obviously determines their spacing.

Before the initial diagonal cracks form, the vertical stress in the beam is 0, and so the stirrups have no stress, while shear force V is resisted by shear stresses in concrete taken approximately as v = V/bd. The complementary energy initially contained in the shaded square cell in Fig. 10.3.7a is Uq — (y1 /2Gc)b(ci cos 6)(cj sinff) = v2( + v)bc] sin в cos 9/Ec, where Gc = £c/2(l + tz) = elasticshear modulus of concrete, v = Poisson ratio (tz rs 0.18), and Cj is defined in Fig. 10.3.7a. After the initial cracks form, the diagonal tensile stress in the shaded square zone is reduced to 0 and the applied shear stress v is

Подпись: AU Подпись: bt^v2 /1-й bs Ec  2 "r 4nAv Подпись: E. Подпись: (10.3.32)

then carried by truss action in the cell, i. e., by tensile stress <7„ in the vertical stirrups, given by (10.3.15), and by diagonal compressive stress ac, given by (10.3.14). So the complementary energy contained in the cell after the initial cracks form is approximately calculated as Щ — (аЦ2Ес)Ь{а sinf?)(c; cos 9) -1 {vl/2Es)Av(c}/s)sin9cosO, where av — vbs ton 9/Av, ac — – i;/sin6cos0. For the sake of simplicity, we assume 0 ■= 45°. The complementary energy change per crack at constant V is AW — U* — Wq, which yields

Consider now the final infinitesimal crack length increment i5cj by which the crack size c; is reached (the shaded square zone in Fig. 10.3.7a grows with сг, and at the end of this increment, it touches the square zone corresponding to the adjacent crack). During this increment, the change of complementary energy is [d(AU*)/dci]6ci. This must be equal to the energy consumed and dissipated by the crack, which is bTZScii It is the crack resistance, which represents the critical energy release rate required for crack growth. In general, fi depends on c.;, representing an Л-curve behavior. This dependence may be approximately described as

Подпись: (10.3.33)7l = G,—^—

Co + Ci

where Co is a positive constant. For large enough c*, H = Gf = fracture energy of the material. The balance of energy during the crack length increment requires that


—-л— – Sa =– ЬШсі (10.3.34)


Load Causing Cracks of Given Relative Depth Подпись: (10.3.35)

Substituting (10.3.32) here, we obtain an equation whose solution yields, for the size effect on the applied nominal shear stress vcr at initial cracking, the following equation:

Load Causing Cracks of Given Relative Depth Load Causing Cracks of Given Relative Depth Load Causing Cracks of Given Relative Depth Подпись: 1_+ v 2~ Подпись: 1/2 Подпись: (10.3.36)

in which the following constants have been introduced:

Note that the ratio D/с. і is assumed to be a given constant by which the cracking load is defined. Equation (10.3.35) shows that the applied nominal shear stress at cracking follows again Baz. ant’s size effect law. As a special case, this equation applies to a beam without stirrups (Av — 0).