Models Based on Cohesive Cracks
The cohesive crack model has been used by several investigators to analyze lightly reinforced beams in three-point bending. All the analyses up to now simplify the problem by assuming that a single cohesive crack forms at the central cross section while the concrete in the bulk behaves elastically and the steel is elastic-perfectly plastic. The various analyses differ in the computational method and in the way they incorporate the effect of the reinforcement.
Hawkins and Hjorsetet (1992) use a commercial finite element code to simulate the experiment of Bosco, Carpinteri and Debernardi (1990b). They use the method described in Section 7.2.3 in which the cohesive zone is modeled by an array of elastic-softening springs. Although they do not explicitly consider bond slip, they made two kinds of analysis: one standard (called P-MAX) in which perfect adherence was assumed, and another (called P-MIN) in which the cross sections were forced to remain plane. In the first case, a large strain is generated in the reinforcement as soon as the crack tip reaches it, which causes the steel to yield. Гп the second case, the strain is smeared over the element width, which is similar (although
not identical) to having a slip length equal to the element width. Fig. 10.4.7 shows the results of the computations together with the experimental results of Bosco, Carpinteri and Debernardi (1990b). Note that the P-MAX and P-MIN predictions differ appreciably, especially as far as the peak load is concerned; this indicates that the bond must play an important role in defining the minimum reinforcement.
Hcdcdal and Kroon (1991) and Ruiz et al. (Ruiz, Planas and Elices 1993, 1996; Ruiz and Planas 1994, 1995; Planas, Ruiz and Elices 1995; Ruiz 1996) use very similar computational procedures, traceable to Pelersson’s influence matrix method (see Section 7.4). The two groups consider bond slip in a very similar fashion, but use a different way to implement it numerically. They both consider the same classical load-displacement curve which is obtained for pullout from a rigid half-space, depicted in Fig. 10.4.8a. They also assume that, in the actual test, the steel displacement us is given by half the crack opening at the reinforcement, w3 (Fig. 10.4.8b); therefore, the force-crack opening displacement is given by
where Fs is the resultant tensile force in the steel at the central cross section, tc is the bond shear strength (rigid-plastic behavior assumed), ps and As are the perimeter and the area of the reinforcement, and Es, and fy are the elastic modulus and the yield limit of steel (elastic-perfectly plastic behavior is assumed).
Hededal and Kroon (1991) introduce the action of the steel on the concrete as the force Fs concentrated at the surface of the cohesive crack and treat it as a cohesive force with a load-crack opening curve as deduced from (10.4.16). Their theoretical predictions compare quite realistically with their experimental
Figure 10.4.9 Comparison of the numerical and experimental results of Hededal and Kroon (1991).
Figure 10.4.10 Approximations analyzed by Planas, Ruiz and Elices (1995): (a) concentrated forces on the crack faces; (b) concentrated forces at the center of gravity of the bond stresses; (c) distributed bond stresses. (From Planas, Ruiz and Elices 1995.)
results as shown in Fig. 10.4.9. In making the predictions, fledcdal and Kroon use material parameters determined from independent experiments, except for the bond strength which they select in each case to give a good fit of the postpeak values. The softening curve for concrete is assumed to be bilinear and is determined from tests on notched plain concrete specimens. The steel bars are threaded bars rather than conventional reinforcing steel bars. The ultimate load and the apparent elastic modulus are determined from tensile tests. Note that Hededal and Kroon use the product тср3 instead of rc to characterize the bond strength; тсря is the shear force per unit length of reinforcement.
Ruiz and Planas (1994) and Ruiz, Planas and Elices (1993) use a different numerical approach which incorporates the effect of the reinforcement by means of internal stresses. This allows considering the steel-concrete interaction to be located within the concrete rather than at the surface. They analyze the three options depicted in F’ig. 10.4. lOa-c. In their first approach, they analyze the case of perfect bond with the steel-concrete interaction represented by two forces acting on the crack faces (Fig. 10.4.10a; Ruiz, Planas and Elices 1993; Ruiz 1996).
This analysis reveals that the cohesive crack growth process follows the stages shown in Fig. 10.4.11 a-e: in stage (a), the cohesive zone extends through the cover and may go through the first peak if the cover is thick; then the cohesive crack is pinned by the steel and hardening occurs until, as shown in (b), the tensile strength is reached at points ahead of the reinforcement; from then on, two separate cracks exist at both sides of the reinforcement until the yield strength is reached in the steel as shown in (c); then a softening phase begins, with an open crack extending across the reinforcement as shown in (d).
The analysis confirms Hawkins and Hjorsetet’s (1992) conclusion that perfect adherence implies a very sharp and high peak. However, it also turns out that this peak depends strongly on the width (diameter) of the reinforcement or, if the steel force is concentrated at a node, on the width of the elements used in the computations. The reason is that, in this approach, the steel force is modeled as a nodal force, which causes that the computational procedure smears this force roughly over an element width, and thus one never deals with a concentrated force but with a distributed force; if the force were really concentrated at a
dimensionless displacement, u f(/Gp
point, the compliance would be infinite and the peak would decrease. This effect is shown in Fig. 10.4.12, where the effect of smearing the force over 1, 3, or 5 nodes is shown for equal elements 1/100 of the beam depth in size (Ruiz 1996). It is clear that the wider the reinforcement, the stiffer the response,
The foregoing problem —the effect of the element size or reinforcement width— appears whenever the steel force is concentrated at the crack faces, even if the bond slip is taken into account. To avoid introducing the width of the reinforcement as a further variable, Planas, Ruiz and Elices (1995) and Ruiz (1996) let the steel-concrete interaction occur inside the concrete as shown in Fig. 10.4. lOb-c. The simplest approach uses a concentrated force acting at the center of gravity of the shear stress distribution, much like in the approach by Bazant and Cedolin (1980), called the effective slip-length model. The location of the concentrated forces varies with the crack opening at the level of steel as follows:
where Ls is the slip length, which is readily obtained from the simple pullout model.
To solve the numerical problem in a computationally inexpensive way, Ruiz cl al. first write the actual
Reinforced Beams in Flexure and Minimum Reinforcement
Figure 10.4.13 Successive decomposition of the problem as a sum of elastic problems (from Ruiz 1996).
Figure 10.4.14 Approximate closed form solution for the internal pair of forces (from Ruiz 1996).
problem (Fig. 10.4.13a) as the superposition of the three elastic cases shown in Figs, 10.4.13b-d. The first two eases are the classical cases appearing in plain concrete. The third case—introducing crack openings, but no stresses— is handled as shown in Figs. 10.4.13e-g which involve the determination of the stresses engendered on the central cross section in an uncracked beam: this is an internal stress field which is then handled in a way similar to thermal or shrinkage stresses (Petersson 1981; Planas and Elices 1992b, 1993b). The only problem is the determination of the stresses in the auxiliary problem in Fig 10.4.13f. This is approximately solved in closed form as follows (Ruiz and Planas 1994; Ruiz 1996).
The actual problem —Fig. 10.4.14a— is considered as the elastic solution for two concentrated loads parallel to the surface of an elastic half-space (Fig. 10.4.14b) plus the elastic solution for the beam subjected to surface tractions canceling those in the previous solution (Fig. 10.4.14c). The last problem is approximately solved by replacing it with a mechanically equivalent linear stress distribution at the cross-section, as sketched in Fig. 10.4.14d-e. The complete stress distribution can thus be obtained in a closed form from Mclan’s (1932) elastic solution for a point load parallel to the surface of the half-space. The integration of the surface tractions and their moments, required to find the solution in Fig. 10.4.14e, can be performed analytically (a symbolic mathematical package was used by Ruiz to get the closed form quadratures; see Ruiz 1996 for details).
The model was further refined by Ruiz (1996) to allow distributed bond stresses to be directly used as shown in Fig. 10.4.13a. The stress distribution caused by the reinforcement can be obtained in a closed form by integrating the solution for the concentrated load. This is cumbersome but feasible if one of the modern symbolic mathematical packages is used. However, detailed comparisons showed that the differences with respect to the effective slip-length approach are negligible for most practical cases (Ruiz 1996).
This model was successfully used to describe the tests on microconcrete performed by Ruiz et al. as
illustrated in Fig 10.4.15a (Ruiz and Planas 1995; Planas, Ruiz and Flices 1995;Ruiz 1996). The important point in this comparison is that all the parameters required to make the predictions were determined by independent tests. In particular, the bond strength tc was determined form pullout tests; much better fits can be achieved if the value of rc is adequately selected for each test. Moreover, the model, conceptually simple as it is, shows that the problem is governed by four dimensionless parameters. These parameters are the following:
where l is the characteristic size based on the initial linear softening defined in (10.1.10), p is the steel ratio, /* is the relative yield strength of steel, p is a dimensionless parameter which characterizes the bond, and n = Es/Ec is the ratio of the elastic moduli of steel and concrelc.
The foregoing model was used to investigate the influence of various parameters on the behavior of lightly reinforced beams. The most important result is that a closed-form expression has been found for
the first peak of the load-displacement curve. Since this first peak occurs at the initial stages of cracking, before much softening take place, the peak load is controlled by the characteristics of the initial straight portion of the softening curve. Consequently, as discussed in Section 7.2.4, the size effect is controlled by l, rather than by lch; see (7.1.17). Moreover, since the steel remains elastic at this stage, the yield strength of steel cannot influence the value of this first peak. This means that the nominal stress at the first peak о/vc can be written as
where ф is a dimensionless function and cs is the steel cover. Numerical simulations showed that this function may be, in a crude approximation, expressed as
vkc – />■ + pf’tP 6 (l – Ф
where fr is the rupture modulus and ф a factor depending on the beam depth and cover thickness, approximately given by
where the last inequality defines the range of application of the formula.
Note that the modulus of rupture in the foregoing formulas is itself size-dependent and can be approximated by the formula (9.3.12) due to Planas, Guinea and Elices (1995).