Bazant-Kim-Sun Formulas

Bazant and Kim (1984) and Bazant and Sun (1987) developed a set of phenomenological equations to describe the dependence of the diagonal shear strength on the size, shape, and steel ratios of beams failing in diagonal shear. The work of Bazant and Kim has three essential ingredients. The first one is the general structure of the formula which is based on the approach described in Section 10.1.5, and thus takes the form (10.1.13). The second is the development of a rather general expression for avNu derived by analyzing the arch action and the composite beam action and summing their contributions. The combination of these



Bazant-Kim-Sun Formulas



і (


Х. П-





h— X——

pH——— s




Figure 10.2.2 Diagonal shear strength analysis of Bazant and Kim (1984)’. (a) geometry; (b) actions at intermediate cross section; (c) decomposition of normal forces.


Bazant-Kim-Sun Formulas

M(x) = T(x)j(x)D (10.2.5)

The overall equilibrium equation for the beam requires that V = dM{x)/dx and thus

V = Vi+V2, V, = d^-j{x)D, V2 – ~T(z)£ (10.2.6)

where Vj and V% are known as the composite beam action and arch action, respectively. Bazant and Kim empirically approximated j(x) by a power law function:

j(x) = kp~m 0)Г ^~T{x)D (10.2.7)

The value of dT/dx is obtained from the equilibrium condition along the reinforcement which requires dT/dx — ПьттОьтъ, where щ is the number of bars, TrDj, their perimeter, and r;, the shear bond stress. Since the perimeter of the bars is proportional to the square root of their area —hence, proportional to y/p— and the ultimate bond strength is roughly proportional to f ‘c4 with q — 1/2, Bazant and Kim were able to write V at the critical section x — s as

V] = kopx/2-mfc4bD (10.2.8)

where ко is a constant. Next, using (10.2.7) and assuming that the critical cross section for arch action is at x — D, and that, at failure, the steel stress is a constant, they found

V2 = Copl~m bD




Bazant-Kim-Sun FormulasBazant-Kim-Sun FormulasBazant-Kim-Sun Formulas

Bazant-Kim-Sun Formulas

Bazant-Kim-Sun Formulas

Figure 10.2.3 Size-effect plot for Bazant-Kiin-Sun formula, compared to 461 available data points for beams without stirrups (data from Bazant and Sun 1987).



Substituting the last two expressions into the first of (10.2.6) and rearranging leads to Hq. (10.2.4) for V? (where the dummy stress cri has been introduced for dimensional compatibility).

Bazant and Kim compared this formula to a large number of tests from the literature in order to get average values for the foregoing empirical parameters. It was shown that Bazant’s size effect law was able to describe the size dependence of the classical data by Капі (1966, 1967) and Walraven (1978). As shown in Section 1.5, this finding was further supported by the tests by Bazant and Kazemi (1991); Fig. 1.5.7, series K1 and K2.

Comparison of equation (10.2.3) to the results from seven classical data series was used by Bazant and Kim to optimize the parameters in that equation. The values of the parameters so determined were as follows :

p — – , q – , r — ~ , k ~ 10, кг ~ 3000, Do 25d„ (10.2.10)

With this formula, Bazant and Kim were able to fit 296 experimental data points with a coefficient of variation of 30%, much better than the ACI formula.

Later, Bazant and Sun (1987) further improved Eq. (10.2.4) by introducing the effect of maximum aggregate size da. This led to the replacement of the value 10 for the factor k in (10.2.10) with the expression

k — 6.5 ^1 1- /co/da’j, со = 0.2 in–5.1 mm (10.2.11)

Bazant and Sun also collected and tabulated a still larger set of data than Bazant and Kim (1984), involving 461 test data, and showed that the improved formula gives still better results, reducing the coefficient of variation to 25%. Fig. 10.2.3 shows the size effect plot for the 461 data points.

Подпись: vt + vl’ Bazant-Kim-Sun Formulas Подпись: 1/2 Подпись: (10.2.12)

Bazant and Sun further introduced in the formula the influence of the stirrups that the ACI code neglects. Although in the original work the approach was completely empirical, a theoretical background is now provided by the analysis in Section 10.1.6. The equations (10.1.22) and (10.1.23) introduce the modification of the size effect due to the stirrups. Thus, the final formula taking all factors into account is

Подпись: V', Подпись: pvfyuisina + cos a) Подпись: (10.2.13)

in which?;J, u£c and Do are given by

1.00 «- = 0.90 < 0.80 0.70

Подпись:Подпись:t 0.60

— 0.50 0.40

Bazant-Kim-Sun Formulas Подпись: а і = 1 psi — 6.895 kPa (10.2.14) (10.2.15)


in which p„ is the steel ratio of stirrups, fyv the yield strength of the stirrups, c. — 6.5, and m is given by (10.1.23). The foregoing value of c. i is adequate to obtain the best fit on average. For design, tq is reduced to C| = 4.5.

Fig. 10.2.4 shows the size effect plot for Bazant-Kim-Sun formula and 87 available data points for beams of a rectangular cross section with stirrups from the test data listed by Bazant and Sun (1987). Although the scatter is large, the experimental results lie relatively close to the Bazant-Kim-Sun formula, closer than to other expressions including the ACI formulas.