Yield Criterion for Concrete Material

In case analyzing concrete structure, since the confined effects of concrete material can be clearly appeared, it is generally pointed out that Drucker-Prager’s yield criterion should be used. How­ever, it has been seen by authors that elasto-plastic impact response behavior for small-scale RC beams under falling-weight impact loading can be better simulated using von Mises yield criterion (Kishi et al., 2005). In this study, an applicability of both yield criterion on impact response ana­lysis for prototype RC girder was investigated with the parameters listed in Table 5 by comparing with the experimental results. The Drucker-Prager type yield criterion can be written as shown in Equation (1):

f(I1,J2) = aI1+Jh-k = 0, (1)

where I1 is the first invariant of stress tensor, J2 is the second invariant of deviatoric stress tensor, a is a material constant, and k is a yield strength under pure shear. Coefficients a and k were determined referring to the book by Chen (1982) as a = 0.472 and k = 3.19 MPa assuming that the tensile strength of concrete is 1/10th of compressive strength which is ft = 3.04 MPa.

Figure 8 shows the comparisons of the impact response waves of the girder obtained using both yield criterion with the experimental results. From Figures 8(a) and 8(b), it is observed that time increment of the first dominant wave obtained using Drucker-Prager yield criterion is greater than that obtained using von Mises one. It is supposed that yield strength of concrete near impacted area is estimated so as to be upgraded due to three-dimensional confined effects in case using Drucker-Prager’s yield criterion. However, even though Drucker-Prager’s yield criterion is used, time increment of the wave cannot be increased up to that of the experimental results and the maximum amplitude is also a little smaller than experimental one.

From the comparisons of second dominant wave shown in Figure 8(b), it is observed that: (1) in case using von Mises yield criterion, impact force wave has not been excited during 65 ms after first dominant wave being excited, and then a half sine wave with high amplitude was excited. This wave configuration is greatly different from that of experimental results; and (2) on the other hand, in case using Drucker-Prager’s yield criterion, four half-sine waves with a few ms duration time were excited with the time intervals of 10 to 15 ms and then the wave has been decreased to zero level. The wave configuration is some different from that of experimental results but is more similar to the experimental one than that obtained using von Mises yield criterion.

From Figure 8(c), it is seen that reaction force waves for one supporting point obtained using both yield criterion are almost the same to each other. The numerically estimated period for free vibration excited during a heavy weight being rebounded is shorter than that obtained from the experimental results in spite of yield criterion of concrete material. From Figure 8(d) of enlarged wave configuration in the beginning of impact, configurations of the first dominant wave obtained using both yield criterion are almost the same to each other. The maximum amplitude for those wave configurations is similar to that of experimental results but time increment of the wave at the

beginning of impact is larger than that of the experimental results and is contradictory to the case of impact force wave.

From Figures 8(e) and 8(f) for displacement waves at the points D — 1/2, it is confirmed that numerical response wave during the impact load surcharging to the RC girder is similar to that of the experimental results irrespective of yield criterion of concrete material considered here. From Figure 8(f), it is observed that numerically estimated period for free vibration during a heavy weight being rebounded is shorter than that from the experimental results as well as reaction force wave. Residual displacement obtained using Drucker-Prager yield criterion is almost the same to the experimental result but that obtained using von Mises yield criterion is estimated larger than the experimental result. This implies that the local stress concentration and confined effects of concrete near impacted area of the girder can be better evaluated by using Drucker-Prager’s yield criterion. Then, in case of spherical head of a heavy weight impacting for the prototype RC girder, impact response behavior of the girder may be better analyzed using Drucker-Prager yield criterion for concrete material.