The authors are to be complimented for the valuable experimental work described in the paper. As pointed out very little experimental works exist on continuous deep beams. Equally valuable is the demonstration of the fact that the upper bound plasticity approach can be used to predict the ultimate strength of this type of structures.
When applying the theory of plasticity to structural concrete a topic of frequent discussions among researchers is the effectiveness factor.
The authors use an effectiveness factor νc given by Equation 7 in the paper. In this formula the factor Kc depends on the principal strain ratio ε1/ε3. To quantify this ratio, the authors make use of the expressions for the plastic strains ε1 and ε3 in the yield line and thereby substituting −ε1/ε3 in Kc with 1 + sinα/1 − sinα. At first glance, this approach appears rather elegant and appealing. However, the following comments should be made.
(a) It should be noticed that ε1/ε3 as appeared in Equation 7 is a total strain ratio. On the other hand ε1/ε3 in the yield zone (yield line) must be interpreted as incremental plastic strain rate. These two definitions cannot be equalised without additional assumptions.
(b) The approach of the authors has the consequence that the angle α between the yield line and the relative displacement δ enters the formula for νc. When calculating νc, it seems that the authors have used the angle α as defined in Figure 12 and in Equation 2. This is inconsistent. As pointed out by the authors, α in Equation 2 is to be understood as the angle between the relative displacement at the mid-point of the chord and the yield line chord. This angle is not the same as the angle between the yield line and the displacement. In fact, when using the failure mechanism shown in Figure 12 the angle between yield line and relative displacement varies along the path of the yield line. Thus, if the approach of the authors should be followed consistently, the effectiveness factor will vary along the yield line and numerical integration with small steps and with many different values for νc should be used in order to calculate the internal work.
(c) The hyperbolic yield line, which is the basis for calculations in the paper, is only an optimum yield line if the effective compressive strength is constant and independent of the path of the yield line. This is not the case if νc is a function of the angle α between the yield line and the displacement δ.
(d) The assumed failure mechanism is only possible if the reinforcement is yielding. This is, however, not the case as stated in the paper (reinforcement stresses were below 400 MPa at ultimate). To carry out the calculations the authors use fy = 420 MPa. This approach is problematic as it may lead to different predicted failure loads for similar beams. The argument is as follows: Imagine that a similar test series is conducted with reinforcement having an actual yield stress fy = 420 MPa instead of 562 MPa. Reinforcement areas are assumed to be 34% larger than the areas listed in Table 1 (562/420 = 1·34). For this test series experimental failure loads similar to those reported in the paper are expected. However, calculations using the actual yield stress, 420 MPa, and the increased steel areas will lead to different theoretical strength compared with the calculations reported in the paper.
Authors' reply
The authors would like to thank the discussers for their valuable comments. As pointed out by the discussers, the determination of concrete effectiveness factor in plasticity analysis is a major challenge. This factor is introduced in the analysis to account for the limited ductility of concrete and to absorb other shortcomings of applying the plasticity theory to concrete. In the literature, different techniques were suggested for evaluating the effectiveness factor of concrete (references 16, 17, 18, 19 of the original paper). Based on various concrete panel tests, it was shown (Collins and Mitchell, 1986; Pang and Hsu, 1995) that the angle of failure plane nearly followed the principal strain angle, and can be assumed to be the same through the failure plane during the propagation of failure cracks. As a result, shear design of concrete beams using truss models such as modified compression field theory uses average principal strains (−ε1 and ε3 ) assumed to be constant along the diagonal failure crack of beam web to explain the softening effect of concrete (Collins et al., 1996). One principal aim of the mechanism analysis presented in the paper was the simplicity. Therefore the authors used one value of the angle α between the relative displacement at the mid-point of the chord and the yield line chord as a reasonable approximation.
Shear failure of reinforced concrete deep beams is mainly governed by concrete and yielding of longitudinal reinforcement does not usually occur at failure. Therefore the contribution of main longitudinal steel reinforcement should be limited to a certain value beyond which there is no increase in the shear capacity of reinforced concrete beams. The problem of limiting yield strength of steel bars identified by the discusser is a valid comment. However, as the reinforcement did not yield in the actual tests, the use of the yield strength of steel reinforcement in the mechanism analysis would give a higher prediction and, therefore, a reasonably selected value of yield strength is proposed in the paper. For example, the load capacity of deep beams reinforced with longitudinal reinforcement having yield strength of 820 MPa was highly overestimated by the mechanism approach, because the actual stress of longitudinal reinforcement at beam failure remained in the elastic range below 450 MPa (reference 13 of the original paper). Another way to overcome this problem is to accommodate all shortcomings of applying the plasticity theory to shear failure of reinforced concrete deep beams in the effectiveness factor of concrete. In a previous investigation, Ashour and Morley (reference 16 of the original paper) introduced a similar mechanism analysis for two-span continuous deep beams to that presented in the paper. They calibrated the mechanism analysis prediction against experimental results of continuous deep beams available in the literature and hence obtained a mean best value of the effectiveness factor as low as 0·28 when all steel reinforcements were included in the prediction.
