Lightly Reinforced Beams: Overview
The question of minimum reinforcement calls for answering two problems: (1) stability of a system of interacting cracks, which ensures that the cracks will remain densely spaced, and (2) avoidance of snapbaek in bending at a cross section with only one crack. The first problem controls the spacing of bending cracks in beams which in turn controls their width. A certain minimum reinforcement is required to prevent a large crack spacing, causing a large crack opening. This problem is important for serviceability under normal loads and small overloads. The second problem, which is important for preventing sudden catastrophic failure without wanting, will be discussed now.
In recent years, the analysis of lightly reinforced beams and of minimum reinforcement received considerable attention. This is probably due to the widespread feeling that this is a problem that can be handled with relative ease using fracture mechanics. In particular, lightly reinforced beams in three-point bending fail by a single crack across the central cross section, as opposed to normally reinforced beams in which multiple or distributed cracking occurs prior to collapse.
Before reviewing the various theoretical approaches to this problem, let us describe the main empirical facts. Figs. 10.4.1a-c show load-displacement curves measured for various reinforcement ratios. Fig. 10.4.1 a shows the results by Bosco, Carpintcri and Debernardi (1990b), for concrete reinforced with standard ribbed steel. Fig. 10.4. lb shows some results reported by Hedcdal and Kroon (1991) on a similar material; note that in these tests, the beams had a short notch —5% of beam depth— on the tension side of the beam, which explains the sharper peak. Fig. 10.4.1c shows very recent results of Planas, Ruiz, and Elices (1995) for lightly reinforced beams made of microconcretc. Although the materials and the test arrangements were quite different, the results are clearly similar.
From their tests and the theoretical analysis to be described later (Section 10.4.4), Ruiz, Planas and Hlices (1993, 1996) suggested that for steel with low strain hardening, the load-deflection curve can generally be sketched as shown in Fig. 10.4. Id. A linear portion OL is followed by a nonlinear zone up to the peak LM after which a li-shaped portion MNP follows, ending at point P at which the reinforcement yields (if the reinforcement is elastic-pcrfectly plastic). This is followed by a relatively long tail PT with mild softening which theoretically has no end (for ideal steel) but in practice ends by steel necking and fracture. Since the steel never follows exactly an ideal plastic behavior with a sharp transition from elastic to plastic, the actual curve may look closer to the dashed curve NP’T which rounds the corner at P due to strain-hardening.
In an ideal situation (no internal stresses due to shrinkage, no thermal gradients, nor chemical reactions) the linear limit depends only on the tensile strength of concrete, with <7n ~ ft (we do not care here about the 5% difference due to the concentrated load that was discussed in Section 9.3 with reference to the rupture modulus). After that limit, a fracture zone starts to grow towards the reinforcement across the cover, and the load-displacement curve for similar unreinforced beams is approximately followed (dashed line). When the fracture zone approaches or reaches the steel, two phenomena occur simultaneously: (1) the fracture zone is sewed Uj) by the reinforcement, which is still clastic thus requiring an extra load to continue cracking; and (2) steel pullout and slip takes place. Therefore, the peak and near postpeak in the load-displacement curve and its neighborhood is controlled by three factors: (a) The steel ratio; (b) the bond-slip properties; and (c) the steel cover.
The influence of the steel ratio on the peak load was already illustrated in Figs. 10.4. la-c. The influence of the bond is illustrated in Fig. 10.4.1c which shows the results of Planas, Ruiz, and Elices (1995) for a fixed steel ratio and for two different types of reinforcement: ribbed bars with strong bond, and smooth bars with weak bond. It is clear that the bond strength modifies substantially the response. Finally, the influence of the cover is not so evident and little experimental support is available. 1 lowcver, that the cover must p(ay a role can be inferred by the following reasoning. If the cover is large enough, the specimen load will exhibit a peak before the fracture zone reaches the reinforcement; then, alter some load decrease, the growing fracture zone will reach the reinforcement and will be arrested, thus engendering hardening followed by a second peak and further softening. Therefore, the cover must influence the response. Indeed, Ruiz and Planas (1995) have detected, both experimentally and theoretically, the existence of the double peak, as shown in Fig. 10.4. If. Certainly there also must be an indirect effect of the cover thickness and bar spacing because these variables are known to modify the bond strength, but this is a secondary influence in the usual analysis.
Several models have been proposed to describe the foregoing results; they can be classified as pertaining to three wide groups: (1) models that make use of LEFM as the basic tool; (2) models that use a simplified smeared cohesive cracks; and (3) models that use cohesive cracks, hi the following, wc describe the mean features of these three groups of models.