# LEFM Analyses of Jenq and Shah and of Karihaloo

Jenq and Shah (1989) analyzed diagonal shear fracture using LEFM. They considered the idealized diagonal crack shown in Fig. 10.2.7a and approximated the solution as the superposition of the cases

0.7

0.6

0.5

0.4

V // ‘0.3

I, J I

0. 8 1

D/lc„

Figure 10.2.6 Nondimensional shear strength vs. beam depth for various span-to-dcplh ratios and steel ratios according to Gustafsson and Hiller borg model (data from Gustafsson 1985). Tire dashed lines correspond to power law expressions of the form vu oc D~03.

shown in Figs. 10.2.7b-c. The first case corresponds to the concrete taking a load Vi: such that the stress intensity factor at the crack tip is Kjc. The second case corresponds to concrete plus reinforcement taking a load Vs computed front classical no-tension strength of material analysis (no crack singularity, neutral axis at the crack tip, linear stress distribution along the ligament). Note that in this model only the situation at ultimate load is considered, and the equations that follow cannot be used to analyze the crack growth. This means, in particular, that ac is the critical crack length (actually its vertical projection), understood as the crack length at peak load.

Because there is no closed-form expression for the precise geometry in Fig. 10.2.7b, Jenq and Shah (1989) assumed that the stress intensity factor can be approximated by the stress intensity factor of a pure bend beam with a symmetric edge notch of depth a subjected to the bending moment corresponding to the cross section at the mouth of the crack (i. e.: M — Vcx-, note that it is not clear’ why the moment should not be taken at the cross section at the tip of the crack, M* =- Vcxck). According to this, the crack growth condition is

where in the first fraction we recognize the expression for the nominal stress in bending (6M/6£>2) and k(a) is given, for example, by Srawley’s expression (5.4.8). From this we get

On the other hand, the equilibrium of moments for the case in Fig. 10.2.7c requites

V. – T{x)^~ , у – D – c (D – ac) – ~ – I – ~ – c (10.2.18)

2-е к d j D

Given x and ac (together with the initial geometry of the beam), the foregoing equations determine V if the distribution of the steel force T(x) is known. Jenq and Shah (1989) assume as a simplification that this distribution can be approximated by a power law:

T{x) -= (-)Л (10.2.19)

where 7’тах is the value of the steel 1’orce below the concentrated load. Based on test data by Ferguson and Thompson (1962, 1965), Jenq and Shah (1989) proposed a formula for7nlax. They made an intensive numerical analysis of the influence of the exponent N, from which they recommended the value N — 2.5. The recommended expression for Tnm is:

7 max = 2.509(10.2.20)

where the result is in kN if s and D are in mm and f[ in MPa. This result is strictly valid only lor beam thicknesses of 10 in (254 mm) and for a single steel bar. To obtain a dimensionally correct equation, it is better to express the force carried out by the steel at the central section as the length of the steel bars. s, times their perimeter щпПь, times the average bond shear stress (Karihaloo 1992)

7max — SUb’ixDbTb (10.2.21)

Setting now nbirDl/A — pbD and 1% oc f[j D, as deduced by Jenq and Shah from the Ferguson and Thompson data (1962, 1965), we get the result

21™ = LbsJ^fl, Lb – 2.509 m (10.2.22)

The value of Lb is determined so that this formula coincides with (10.2.20) for nb 1 and b — 254 mm.

Given x (or d) and ac, the shear strength is determined front the foregoing equations setting vu = (Vc p Vs)/bD the result is

Karihaloo analyzed this model and improved it in a series of papers (Karihaloo 1992, 1995; So and Karihaloo 1993). First he modified the way the model is applied and used it as a forensic engineering tool by using the values of x and ac as measured (optically) in a test. Using this method on two beam tests, he concluded that the Jenq-Shah model predicted shear strengths that were loo low (see exercise 10.3). So and Karihaloo (1993) extended the range of applicability of (10.2.20) to include other parameters. They reevaluated the results of Ferguson and Thompson (1962, 1965) and proposed a new formula that takes into account the bar diameter and the number of bars; for an embedment length u, the formula reads

7’max – Le s/4nbTTpbD rb

where F is a reduction factor for пь — 2 (the formula is strictly applicable only for one or two bars):

The average ultimate bond shear strength % is given by

where all the dimensions must be in millimeters and /’ in MPa. The factor F2 is given by

Although this equation substantially improves the Jenq-Shah expression for Tmax, the problem of the shape of the distribution (particularly the value of exponent N in (10.2.19)), still remains. Karihaloo (1995) further improved the treatment of the LEFM crack growth condition by explicitly considering the mixed mode condition at the crack tip. But even with this enhancement, the strength predictions of the model were too low, even for exponents N as low as 1.25.