Diagonal Shear Failure of Beams
In the current ACI Standard 318 (Sec. 11.3), the nominal shear strength is not based on the ultimate load data but on data on the load that causes the formation of the first large cracks. The current ACI formula can be written:
where vc = VJ(bwD) nominal shear strength provided by concrete, D — effective depth of the longitudinal steel, bw — width of the beam web, /’ = compressive strength of concrete, pw = longitudinal steel reinforcement ratio, and
in which Vu — factored shear force at ultimate, Mu = factored moment at ultimate. For the case of constant shear of Fig. 10.1.3 a’ — s.
If the first diagonal shear crack were considered to be very small compared to beam depth D, no size effect would occur, as implied by Bq. (10.2.1). However, it seems that most dala refer to the formation of relatively large cracks, in which the size effect ought to occur even though it is ignored in Eq. (10.2.1).
The fact that the strength-based failure criterion used in contemporary design codes is not very realistic is, for example, confirmed by the extremely large scatter of the vast amount of test data available in the literature (Park and Paulay 1975; Bazant and Kim 1984; Bazant and Sun 1987). Moreover, in the commentary to the ACI Code (Sec. R220.127.116.11) it is acknowledged that the diagonal shear failure experiments of Капі (1966, 1967) reveal a decrease of the shear strength with the depths of the beam. These results are not considered in the code ACI 318-89, which is justified by assuming the code to be based on the load at initiation of very small cracks rather thanformalionof first large cracks or the ultimate load. For deep beams such that Lj D < 5 (L — clear span of the beam), the nominal shear strength is obtained by multiplying Bq. (10.2.1) with the factor (3.5 — 2.5s’/D), which is intended to introduce the increase of the shear strength from the first cracking load to the ultimate load in deep beams. (This is explained by assuming that the mode of shear resistance changes from flexure to arch action or the action of diagonal compression struts.)
Some revisions to the code that partially addressed some concerns stemming from fracture mechanics were proposed by ACI-ASCE Committee 426 (1973,1974, 1977) and by MacGregor and Gergely (1977),
Figure 10.2.1 Experimental data are available only for a range over which both the CEB formula and Bazanfs equation describe the results adequately, given the large experimental scatter.
but have not been incorporated into the AC1 Code (they were proposed on the basis of an exhaustive study of experimental data obtained prior to 1974). Reinhardt (1981a) analyzed some suitably. chosen test data and in 1981 found that there was a size effect and that it agreed quite well with LEFM. Later, however, the LEFM size effect was found to be too strong by Bazant and Kim (1984).
These authors, and also Bazant and Sun (1987), concluded from a statistical analysis of over 400 test scries that the code approach to design, which is not based on the maximum load, does not provide a uniform margin of safety against failure of beam of various sizes because it ignores size effect. They noted that introduction of the size effect law leads to a better agreement with the ultimate load test data compared to the current ACI Code formulas which lack the size effect (as well as an LEFM-type formula proposed by Reinhardt, in which the nominal strength decreases inversely to the sfD, which is too strong).
An empirical formula for the size effect in diagonal shear has been introduced in the CEB Model Code design recommendations (CEB 1991). It has the form vu — uo(l + л/DopD), where vo and D0 are constants. This formula, however, lias the opposite asymptotic behavior than the size effect law. For large sizes, it approaches a horizontal asymptote, and for small sizes it approaches an inclined asymptote of slope-1/2, which cannot be logically justified. The reason that this formula compared acceptably with the test data is that the data used pertained only to the middle of the size range. Due to scatter, distinguishing various laws without any theory is impossible for such limited data, as illustrated in Fig. 10.2.1.
The diagonal shear strength was also investigated using the cohesive crack model by Gustafsson(1985) and Gustafsson and Hillerborg (1988), but not with the aim to produce code formulas. Rather, their objective was to show that a size effect was theoretically predicted and to investigate how the shear strength is influenced by the fracture properties, particularly the fracture energy.
Other models have also been used to analyze the diagonal shear of beams. Jenq and Shah (1989) extended their two parameter model to describe crack growth in mixed mode and applied it to diagonal shear. A nonlocal microplane has also been used to analyze the size effect in diagonal shear of beams (Bazant, Ozbolt and Eligehausen 1994; Oz. bolt and Eligehausen 1995)
In this section, some of the most important results of the aforementioned works are summarized. In the next section, a recent modification of the classical truss model (or strut-and-tie model) is described which approximately captures the effect of energy release and explains the physical mechanism of size effect in a simple, easily understandable way.