Influence of Prestressing on Diagonal Shear Strength
The effect of longitudinal prestressing on the diagonal shear of longitudinally reinforce beams was addressed by Bazant and Cao (1986) and also by Gustafsson( 1985). Similar to the procedure of Bazant, Kim and Sun, Bazant and Cao first used simple equilibrium considerations to get an approximate expression for г>£ and then applied the size effect correction (10.1.13). Their result is:
where Do, ci, and c.2 are empirical constants, s is the shear span, and acc the uniaxial stress due to prestressing; cti = 1 psi = 6.895 MPa is introduced for dimensional compatibility. From the analysis of
235 test results from the literature, Ba&mt and Cao proposed the following values of the constants:
D0 – 25da, c, – 4.9 , c2 – 0.54 (10.2.29)
These values yield the average shear strength; for design, they proposed the values Cj — 4, c2 —■ 0.4.
It must be realized that although the analysis of data showed a clear size effect, the scatter was so large that the values of the parameters are merely roughly indicative. Nevertheless, the study of Bazant and Cao also shows that the proposed formula provides a better agreement with test results than other formulas found in the literature such as the ACI formula or the formula proposed by Sozen, Zwoyer and Siess (1958).
In contrast to nonprestressed beams, the arch action was not considered in the derivation of the foregoing formula. However, in prestressed beams it is difficult to distinguish the arch action from the effect of prestress. To some extent, the separate consideration of shear force c2occD/s associated with prestress substitutes for the consideration of arch action. However, improvements might be in order.
The foregoing formula did not anchor the size effect into a complete plasticity solution, which should be applicable for an infinitely small size. However, according to the size effect data, the plasticity solution would be applicable, in theory, for beam depths smaller than the aggregate size, and thus it is not clear whether the application of plasticity is permitted. It calls for further research to determine whether this might be so and, if it would, then one could draw on various elegant plasticity solutions for diagonal shear (for example, the recent developments in truss analogy, see Section 10.3).
10.1 Karihaloo (1992) reported tests on two reinforced concrete beams tested in three-point bending. The dimensions of the beams were as follows (refer to Fig 10.2.7 for notation): s — B00 mm, D — 150 mm, b — 100 mnt, c = 25 mm. Beam number 1 was reinforced with one ribbed bar 12 mm in diameter and beam number 2 with two ribbed bars of the same characteristics, giving steel ratios of 0.0075 and 0.015, respectively. The steel had a yield strength fy — 463 MPa. The concrete mix had the following characterislics: da = 20 mm, /(. = 38 MPa, E = 30 GPa, /, — 3.4 MPa. The fracture toughness was estimated (from tests on similar mixes) as Kic = 1.27 MPayTn. Beam number 1 failed in bending with a main crack close to the central cross-section, while beam number 2 failed in diagonal shear. The failure loads were approximately equal to 24 kN and 33 kN, respectively (note that this is the total load P — 2V). (a) Determine the expected strength of the beams according to the Bazant, Kim and Sun’s formula whenever applicable, (b) Determine the design strength according to the Bazant, Kim and Sun’s model, and determine the actual safety factor for the beam that failed in diagonal shear.
10.2 Consider the beams in the previous example, (a) Make an estimate of the values of 6V and 4a to be used in the Gustafsson-Hillerborg model, (b) Plot the experimental results on a copy of Fig. 10.2.6. (c) Give at least two reasons for each beam (not necessarily the same) why the results ofllillerborg and Petersson cannot be applied directly.
10.3 Consider again the beams in the previous examples, (a) Determine the shear strength of beam number 2 using the Jenq-Shah model with N — 2.5, with the following assumptions (based on the observed failure, taken from Karihaloo 1992): x яз 250 mm, ac яз 125 mm, 0 яз 45°. Compare the result to the experimental value, (b) Determine the value of exponent N that should be used to make the model deliver the observed strength.