# Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams Without Stirrups

The typical pattern of cracks forming during the failure of a simply supported beam is seen in Fig. 10.3. la. Although after the failure only one final crack emerges, cracks of various orientations form during the loading process. The first cracks caused by shear loading are tensile cracks of inclination approximately 45°. On approach to the maximum load, these cracks interconnect and form a larger crack running approximately along the line connecting the application point of the load V to the support in Fig. 10.3.1a. This major crack is free of shear stresses and has approximately the direction of the maximum principal compression stress, 07/.

According to the truss model (or strut-and-tie model), we may imagine that most of the load is transferred through the shaded zone called the compression strut (in the case of distributed load, it would be a compressed arch). The normal stress in Ihe direction orthogonal to the strut is essentially zero and the material can expand freely in that direction.

The failure behavior is approximately idealized as shown in Fig. 10.3. lb for two geometrically similar beams of different size. Although for calculation purposes the compression strut is assumed to represent a one-dimensional bar connecting the point of application of V and the support, it has a finite effective width, denoted as kD (Fig. 10.3. lb) where D is the depth to the reinforcement and к is approximately a constant, independent of the beam size.

Accordingto experimental evidence, supported by finite element results, abeam(vithapositive bending moment) fails at maximum load due to compression failure of the concrete, usually near the upper end of the compression strut, provided that the longitudinal bar is anchored sufficiently so that it cannot slip against concrete near the support. Aside from the fact that the compression failure (axial splitting or compression shear crack) occurs only within a portion of the length of the strut, the basic premise of the present analysis is that the width h of the cracking z. one in the direction of the strut is lor a given concrete approximately a constant (which is probably approximately proportional to the maximum aggregate size and also depends on Irwin’s characteristic length and on other material characteristics).

The fact that h, in contrast to the length and width of the stress-relieved strip in the stmt (the white strip 56785 in Fig. 10.3.1b), is not proportional to the beam size is the cause of the size effect. If the width h of the crushing zone were proportional to the beam size, there would be no size effect. For calculation purposes, we will assume that the compression failure of the material consists of a band 12341 of splitting cracks (Fig. 10.3.1b) growing vertically across the stiut upward or downward, or both (which of these is immaterial for the present analysis). These cracks may interconnect after the peak load to produce what looks as a shear crack.

Microscopically, the compression failure may be regarded as internal buckling of an orthotropically damaged material (Bazant and Xiang 1994, 1997; see Section 9,5). The failure begins by formation of dense axial splitting microcracks in the direction of maximum compression, which greatly reduces the transverse stiffness of the material, thus causing the microslabs of the material between the microcracks to buckle laterally. The details of the process are, however, not needed for the present analysis. Neither is it important that the crushing band is pictured to propagate vertically. If it propagated across the strut in an inclined or horizontal direction, the calculation results would be about the same.

The growth of the splitting crack band, which causes the load-deflection curve to reach a maximum load and subsequently decline, relieves the compression stress from the strip 56785 shown in Fig. 10.3.1b. The reason that the boundaries of the stress relief zone, that is, the lines 16, 25, 38, and 47, are parallel to the direction of the strut is that the material is heavily weakened by cracks parallel to the strut. Otherwise, a more realistic assumption would be a triangular shape of the stress relief zone, as considered in the case of tensile failures (Section 1.4; see also the remark at the end of Section 9.5).

Now, how to make the size effect intuitively clear with minimum calculations? ‘Го this end, note that the area of the stress relief zone 56785 in Fig. 10.3.1b is proportional to ca, where c is the length of the crack band at failure. Since ca — (с/ D)(a/D)D2, and c/D and a/D are constants, independent of D, the area of the stress relief zone is proportional to D2. Because the average strain energy density in the strut is proportional to the nominal shear stress at ultimate load, v, the total energy release from the stress-relieved strip 56785 of the strut is proportional to v2D2. However, assuming the energy dissipation per unit volume of the crack band to be constant, the energy dissipation in the entire cracking band is proportional to D, because the area of the crushing band is proportional to ch — (c./D)hD. Therefore, varying the beam size D, vD2 must be proportional to D, which means that vu must be proportional to 1 /s/D. This represents a very strong size effect corresponding to LEFM.

In summary, the cause of the size effect is simply the fact that the energy release from the struc­ture is approximately proportional to v2 L)2 whereas the energy consumed by fracture is approximately proportional to D.

Let us now do the calculations in detail, following the stress relief zone approximation illustrated in Section 3.2.2. The condition that the entire shear force V must be transmitted by the compression strut yields, for the axial compression stress in the strut, the following expression; 1 V _ Vu (_f_ D bklf sin в cos 0 к D ‘ s     in which 0 is the inclination angle of the compression strut from the horizontal (note that tan в — D/s). The strain energy density in the strut is o/2Ec, where Ec is the elastic modulus of concrete. The volume of the strut is sbe (where b = beam width). Therefore, the loss of strain energy from the beam caused by stress relief during the formation of the crack band at constant load-point displacement is, approximately:     The minus sign expresses the fact that this is an energy loss rather than gain. The energy release rate due to the growth of the cracking band is obtained from (2.1.15) as  The energy dissipated by the cracking zone may be expressed on the basis of the fracture energy Gj characterizing the axial splitting microcracks in the crack band. The length of these cracks is h (width of the band), and their average spacing is denoted as sc. The number of axial splitting cracks in the band is c/sc. Thus, the total energy dissipated by the crack band is W; — (c/.4t:)bhG/. Differentiating with respect to c, we find that the energy dissipation in the crack band per unit length of the band and unit width of the beam (which we call TZ because it has the meaning of a crack growth resistance) is;

In this equation, however, it would be too simplistic to consider h to be a constant through the entire evolution of the crack band. Naturally, the crack band must initiate from a small zone of axial splitting cracks. The length of these cracks first extends in the direction of the strut until they reach a certain characteristic length ho – After that the crack band grows across the stmt at roughly constant width h — ho (see the intuitive picture of the subsequent contours of the crack zone in Fig. 10.3.1a). Such behavior may be simply described by the equation h ^ —<1— h0 w0 + c

in which h0, Wo — positive constants; ho represents the final width of the crack band. Thus, strictly speaking, our hypothesis of a constant width of the cracking zone means that the final width rather than h is a constant.

The increase. of 1Z with c, as described by (10.3.4) with (10.3.5), represents an Л-curve behavior (because 7Z represents the resistance to fracture). The Л-curvc behavior in tensile fracture is also caused by the growth of the fracture process zone size. Here, however, this growth is expressed indirectly in terms of the length of the axial splitting cracks in the cracking band.

It is also conceivable that, instead of a band of parallel splitting cracks, a shear crack would propagate in a direction inclined to the compression strut (Fig. 10.3.2). In that case

^ (!По:г ^

where Gfs — fracture energy of the shear crack and ho now characterizes the Л-curve behavior of the shear crack. This is mathematically identical to (10.3.4) if one sets Gf — GjsscJho, and so wc will not pursue it further.

The balance of energy during equilibrium propagation of the crushing band requires that Q — 1Z. Substituting here the expressions in (10.3.3)-(10.3.5), one obtains the result:

 / D~’/2 Vu ■ «Р (1 + TjJ (10.3.7) in which the following notations have been made „ I) D0 — xuq— c (10.3.8) ,, / s D ‘ vp – c„Kc -1 jJ (10.3.9) Kc = y/EcOj, CP = kJ^ V w^c S! D (10.3.10)

Here the expression for Kc is that for the fracture toughness (the critical stress intensity factor) of the axial splitting microcracks. An important point is that, because of our assumptions (constant с/D and s/D), the values of Do, vp, and cp are constant, independent of size D. The value vp is the limiting (asymptotic) value of the nominal shear strength for a very small size D.

Eq. (10.3.7) represents the size effect law discussed in Chapters 1 and 6. This law was introduced into the analysis of diagonal shear failure by Bazant and Kim (1984), however, on the basis of a more general and less transparent argument (see Section 10.2.2).

By the same calculation procedure, it can also be easily shown that if and only if, contrary to our hypothesis, the width h of the crushing band were proportional to D instead of obeying (10.3.5), there would be no size effect. If constant wo were taken as 0, one would have vu cx D~xwhich is the size effect of linear elastic fracture mechanics (LEFM), representing the strongest size effect possible. However, the experimental data exhibit a weaker size effect, which implies that the constant wo should be considered finite.

As seen in Chapters 1 and 6, the size effect curve given by (10.3.7) represents a smooth transition from a horizontal asymptote corresponding to the strength theory or plastic limit analysis to an inclined
asymptote of slope — 1/2, corresponding to LEFM. The approach to the horizontal asymptote means that the plasticity approach, that is, the truss model (or strut-and-tie model), can he used only Гог sufficiently small beam sizes D.

For very small beam sizes D, we may substitute in (10,3.1) ac — fb = compression strength of the strut, and replace vu by plastic nominal strength vp. From this we can solve:

which also shows the effect of the relative shear span s/D on the nominal shear strength. Note that fb cannot be expected to represent the uniaxial compression strength f’c of concrete because the progressively fracturing concrete in the strut is under high transverse tensile strain in the other diagonal direction and has been orthotropically damaged by cracking parallel to the strut due to previous high transverse tensile stress (Hsu 1988, 1993). So fb is a certain biaxial strength of concrete, depending both on the uniaxial compression strength /’ and the direct tensile strength //. This dependence needs to be calibrated by shear tests of beams.   It is interesting to determine the ratio to the nominal strength for bending failure, abNu. The ultimate bending moment in the cross section under the load V is Mu = Vus = abNbsD. From the moment equilibrium condition of the cross section under the load V, we also have Mu — (/ypbD)ki, D, in which fy is the yield strength of the longitudinal reinforcing bars, p is the reinforcement ratio (which means that pbD is the cross section area of the longitudinal reinforcing bars), and kyD represents the arm of the internal force couple at the ultimate load. As is well known, кь is approximately constant: Equating the expressions for Mu, we obtain abN = pfykbD/s. Considering now (10.3.12), we conclude that:

This equation shows that the ratio of the nominal bending strength to the nominal shear strength of the beam decreases when the relative shear span s/D increases, which confirms a well-known fact. It means that slender beams, for which s/D is large, fail by bending, while deep beams, for which s/D is small, fail by shear. However, as is clear from (10.3.13), the relative shear span (s/D)tr at the transition between the shear and bending failures is not constant but is larger for a larger beam siz. e D. To express it precisely, one sets Стдг — vu in (10.3.13), and needs to solve (10.3.13) for s/D, which is a cubic equation. The transitional shear span obviously exhibits a size effect.

The foregoing analysis assumes the reduction of the compressive stress crc all the way to zero. Similar to the analysis of compression fracture in Section 9.5, it could be that the compression stress ac is reduced to some small but finite residual strength ar. However, the residual stress is anyway likely to be smaller than for uniaxial compression, due to the existence of large tensile strain. A finite ar seems more realistic when we consider beams with stirrups, which provide some degree of confinement. If ar were nonzero for the present case, it would have the effect of adding a constant term to the right hand side of (10.3.7).

The tensile strength of concrete, /t’, has played no direct role in the foregoing analysis. The tensile strength is not a material parameter in LEFM, nor in the Л-curve model of nonlinear fracture. It does appear in the cohesive (fictitious) crack model or the crack band model. However, those models are too complicated for achieving a simple analytical solution. The tensile strength, of course, controls the initiation of the inclined shear cracks, however, their growth is governed by fracture energy. In the present analysis we take the view that the inclined cracks due to shear loading have already formed before the maximum load.

Does shear stress transmission across cracks due to friction and aggregate interlock play any role? It could, although according to the present analysis, it cannot be significant. As shown in Fig. 10.3.1a, only cracks rather curved within the area of the compression strut can be subjected to shear and normal loading. Their capability of shear stress transmission decreases with the crack width, and the crack width Figure 10.3.3 Involution of diagonal cracks in beam with stirrups under shear loading: (a) diagonal crack for­mation before maximum load, (b) growth of crushing band across compression strut during failure at maximum load, (c) beam at maximum load with the crushed and stress-relieved parts of the compression strut removed, (d) state of beam without crushed and stress-relieved concrete after collapse (i. e., when the load has been reduced to a small value), (e) location of the slates represented in Figs, a-d on the load deflection curve, (0 equilibrium of forces in stirrups and struts, and (g) Mohr circle of strains.

may be assumed to increase with an increasing beam size, which obviously would also introduce a size effect (this idea was proposed by Reineck 1991). The cracks are most inclined to the compression strut direction and are opened the most widely at the bottom of the beam. However, the maximum load appears to be controlled by progressive compression crushing near the major crack at the top of the beam. For this reason, the effect of crack opening on the shear stress transmission across cracks can hardly play a major role in the size effect on the maximum load.