Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups

Consider now a beam with stirrups (Fig. 10.3.3a). The stirrups cause the diagonal cracks due to shear to be more densely distributed. The first hairline cracks, shown by the dashed lines in Fig. 10.3.3a, form near the neutral axis, with inclination about 45° before the maximum load. These cracks later interconnect and form continuous major cracks at inclination angle 0 with the horizontal (Fig. 10.3.3a). These cracks run in the direction of the maximum principal compressive stress an, transmitting no shear stresses. They are, of course, cohesive cracks transmitting tensile bridging stresses. These stresses will probably be less than one half of the tensile strength, /t’, while the compression stresses in the truss will be equal or nearly equal to the compressive strength of concrete, /’. So, it is safe to assume that the tensile principal stress is negligible (sigma, ~ 0) compared to the compressive principal stress, which justifies treating the beam approximately as a truss. This makes the truss statically determinate. It is this circumstance that makes the well-known simple analysis of the truss model (or strut-and-tie model) possible. If oy were not negligible, the truss model would be invalid.

The failure at maximum load is assumed to be caused by the progressive crushing of concrete in the compression struts between the major inclined cracks. Similar to beams without stirrups, a crack band which consists of dense axial splitting microcracks first widens to its full width h and then propagates sideways as shown in Fig. 10.3.3b. For the case of a positive bending moment, this crack band probably forms near the top of the beam and may be assumed to propagate horizontally, to the left or to the right,

Подпись: Figure 10.3.4 Stress relief zones caused by a crack band propagating across the compression strut in beams of different sizes with stirrups. The beam in (b) is similar to the beam in (a) except that the crack band width h is the same.

or both. The location and direction of the propagation of the crack band is actually not important for the present analysis, and the same results would be obtained if the band propagated at other inclinations to the compression strut. An important point, however, is that the final length ho of the axial splitting cracks, that is, the final width of the band, is a material property, independent of the size of the beam. If the width ho of the band were proportional to beam depth D, there would be no size effect. Since it is less than proportional to D, there must be size effect.

Thus, the cause of the size effect is the localization of the compression failure of the strut into a crack band of a fixed width, and the growth of this band across the strut.

An important point is that the stirrups as well as the longitudinal steel bars are not necessarily yielding during the failure at maximum load. They might not have yielded before the crushing of the strut began, or they may have yielded and unloaded. There is no reason why the yielding of steel should occur simultaneously with the progressive compression crushing.

The formation of the crack band 12341 (Fig. 10.3.3b) may again be assumed to relieve the compression stress from the entire length of the compression struts in the region 12561 (Fig. 10.3.3b). This causes a release of strain energy from the compression struts, which is then available to drive the propagation of the crack band. This represents the mechanism of failure at maximum load.

With the stress relieved from the compression struts, the beam acts essentially as shown in Fig. 10.3.3c, as if there were a gap in concrete (provided the residual strength of crushed concrete is neglected). However, since the steel is not in general yielding, this does not represent a failure mechanism. A failure mechanism can be created only when a sufficient number of compression struts fail as shown in Fig. 10.3.3d, in which case even nonyielding bars permit free movement because the bending resistance of the bars is negligible. However, this type of collapse mechanism corresponds to a postpeak state at which the load is already reduced to a very small value (such as slate D in Fig. Ю. З.Зе). Thus, the stress relief at maximum load does not imply the structure has become a mechanism.

First, let us explain the size effect mechanism in the simplest possible terms. The area of the compression struts from which the compression stress is relieved, that is, area 12561 in Fig. 10.3.4, is proportional to cD, which is equal to (c/D)D2. But since the failure is assumed to be geometrically similar for beams of different sizes (shown in Fig. 10.3.4), c/D is a constant, and so the area of the stress relief zone is proportional D2. The strain energy density before the stress relief is proportional to v2/2Ec, and so the total energy release is proportional to v2D2. The area of the crack band is proportional to ch = (c/D)hD. Since both h and c/D are constant for beams of different sizes, the area of the crack band is proportional to D, and so is the energy dissipated in the crack band. So, considering the failures of geometrically similar beams of different sizes, v2D2 must be proportional to D, which means that vu must be proportional to 1 /s/D. Again, same as for the beam without stirrups, we thus obtain a size effect, and it is the strong size effect of LEFM. In practice, the size effect for smaller beam sizes is weaker because of the Л-curve behavior of the crack band 12341.

Подпись: Fc = vubD 1 bD cos 0 sinf) bDcosD Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups Подпись: (10.3.14)

We assume the stirrups to be uniformly distributed (smeared). Equilibrium on a vertical cross section of the beam (Figs. 10.3.4 and 10.3.3Ґ) requires that

in which 0 is the inclination of the compression struts, Fc — compression force in the strut per length D, and oc is the compression stress transmitted by the strut (which, in general, is not equal to the standard
compression strength /’ of concrete and depends on the size of the beam in a manner to be determined). Equilibrium on an inclined cross section of the beam parallel to the compression struts further requires that

_ cr„ – (VsvjAvD)lmB = vusvblanB/Av (10.3.15)

in which Au = cross section of the stirrups, sv — spacing of the stirrups, and cr„ — tensile stress in the stiiTups, which, in general, is not equal to the yield stress. The stress in the longitudinal bars is obtained from the moment equilibrium condition in a cross section and is as — M/Ayk^D, in which M — bending moment, As = cross section area of the longitudinal bars, and k/,D = arm of the internal force couple in the cross section.

We do not attempt to determine the angle в of the diagonal cracks and the struts by fracture analysis. The diagonal cracks delineating the struts start to form before the maximum load, and not during failure. For the sake of simplicity, we assume the orientation of the major diagonal cracks not to rotate and adopt the method introduced into the truss model by Mitchell and Collins (1974) in their compression field theory, in which they used the compatibility condition for the average strains in the truss in a similar way as Wagner (1929) used the compatibility condition for approximate analysis of the shear buckling of the webs of steel beams. The average strains of the truss are defined as the strains of a homogeneously deforming continuum that is attached to the joints of the truss at the nodes (tops and bottoms of the stirrups). According to the Mohr circle shown in Fig. 10.3.3g (in which є denotes the strains, ande^ is the strain in the longitudinal bars), the overall compatibility of the average strains of the struts, the stirrups, and the longitudinal bars requires that

Подпись: (10.3.16)£y ~ £c ^ (cr„/£3) – /(ffc)

£/i – £c {<Ts[Es) – /Ос)

Here the strains have been expressed in terms of the stresses assuming the steel not to be yielding and denoting by f(<Jc) the stress-strain diagram of concrete. (For the precise method in which the strains entering (10.3.16) are calculated, see Mitchell and Collins 1974.) The foregoing calculation, of course, requires that the diagonal cracks and the struts be aligned with the direction of the compressive principal strain, which coincides with the direction of the compressive principal stress.

Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups Подпись: (10.3.17)

Fracture analysis begins by expressing the strain energy change (Fig. 10.3.4) caused by the formation of the crack band of length c at constant load-point displacement:

The minus sign reflects the fact that this is an energy loss rather than gain.

The stress oy in the foregoing equation represents the residual compression strength of the crack band of concrete. In this study, the residual compression strength o> is considered to be an empirical property. However, it can be mathematically expressed on the basis of the concept of internal buckling of a material heavily damaged by axial splitting microcracks, as proposed in Bazant (1994a) and Bazant and Xiang (1997); see Section 9.5.

Подпись: Q Подпись: J. 8AUC' _ (ac - a,.)2 b be _u 2 Ec Подпись: (10.3.18)

The energy release rate may be calculated as:

The energy dissipation rate (fracture resistance) of the crack band is again given by (10.3.4), i. e., 77. — G fh/sc, in which the width of the crack band may be assumed to evolve again according to (10.3.6), i. c.,

h – h0c/(w0 + c).

Substituting now (10.3.14) and (10.3.15) into (10.3.18), and using the fracture propagation criterion Q — 77, we obtain an equation which can be easily solved for vu. This provides the result:

Figure 10.3.5 Size effect in shear failure of concrete beam in terms of the logarithm of either vu or vu — vr – in which we introduced the notations:

Подпись: (10.3.20)

Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups

n D

D0 — w0—

Подпись: sin 20 <T 2 Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups Подпись: (10.3.21)

c

The size effect described by (10.3.19) is plotted in Fig. 1.0.3.5 in two ways, in terms of Iog(nu — vf) and in terms of log o,,. By virtue of the residual compression strength, the nominal shear strength of the beam tends at infinite size to a finite value. An equation of the form of (10.3.19) was proposed on the basis of general considerations in Bazant (1987a).

The question whether the confinement of concrete by stirrups suffices to cause the residual compression strength o>, and thus the residual nominal strength vr, to be nonzero needs to be studied further. It is on the safe side to take vr = 0, in which case, the effect of stirrups on the residual nominal strength provided by concrete is neglected.