Numerical Study on Ultimate Behavior of 1/4 PCCV Model Subjected to Internal Pressure
Kenji Yonezawa l), Katsuyoshi Imoto ~), Asao Kato 2~, Masahiko Ozaki 3~, Kazuhiko Kiyohara 4), Yasuyuki Murazumi 5~, Kunihiko Sato 6)
1) Obayashi Corporation, 2) The Japan Atomic Power Company,
3) The Kansai Electric Power Co.,Inc., 4) Kyushu Electric Power Co.,Inc. 5) Taisei Corporation, 6) Mitsubishi Heavy Industries,LTD.
ABSTRACT
Static high-pressure tests on a 1/4 scale prestressed concrete containment vessel (PCCV) model were performed in September, 2000[ 1 ][2]. A Round Robin pretest analysis meeting was held among 17 organizations from 9 countries in October, 1999 in Albuquerque, USA [3].
The Japan PCCV research group participated in the pretest analysis meeting and presented our research program. Many types of global and local pretest analyses on 1/4 PCCV model were conducted to establish an analysis methodology for predicting the nonlinear behaviors of actual PCCVs subjected to internal pressure in our research program.
This paper describes global analysis methods that account for slip amount of unbonded tendon, and the analytical results in a part of our research. The influence of analytical assumptions (boundary conditions etc.) on evaluation of nonlinear behavior of PCCV is discussed here by comparing the results of the following four types of global analyses:
1) A 3D90 ° model which idealizes one quarter sector of a 1/4 PCCV model. 2) A 3D180 ° model which idealizes one half sector of a 1/4 PCCV model.
3) A 3D360 ° model which idealizes the whole 1/4 PCCV model, presented in SMiRT15[4].
4) An axisymmetric model which idealizes a general portion at 135 ° azimuth, presented in SMiRT 15[4].
Furthermore, the friction - slip relationship between unbonded tendon and RC body is investigated in these analyses to estimate the tendon behavior of 1/4 PCCV up to ultimate state by use of the friction model proposed in SMiRT15[4].
1. INTRODUCTION
Recently, an increasing number of actual PCCVs are being constructed in Japan as final barriers for hypothetical severe accidents in nuclear power plants. A JAPAN-US cooperative research project on the ultimate capacity of nuclear reactor containment vessels subjected to internal pressure was established by the Nuclear Power Engineering Corporation (NUPEC) and the Nuclear Regulatory Commission (NRC) [ 1 ] [2]. As a part of this research, ultimate capacity pressure tests were carried out on a 1/4 scale PCCV model of existing PCCVs for pressurized water reactors (PWR) in Japan in September, 2000 in Albuquerque, USA [3]. A Round Robin pretest analysis meeting was held before the test by participants among 17 organizations from 9 countries in October 1999 in Albuquerque, USA [3]. The Japan PCCV research group participated in this meeting and presented pretest analysis results obtained in our research program. Many types of global and local pretest analyses were conducted on the 1/4 PCCV model to establish an analysis methodology that can predict the nonlinear behaviors of actual PCCVs subjected to internal pressure in our research program.
The PCCV model is a uniform 1/4 scale model of actual PCCVs used in Japan. The model includes a steel liner and scaled representation of the equipment hatch (E/H), personnel airlock (A/L), and main steam (M/S) and feed water (F/W). In the region surrounding these openings, both the rebar ratios and the wall thickness are larger than those of general sections. A stretched layout of the PCCV model is shown in Fig. 1.
The 1/4 scale PCCV model contained 90 meridional and 108 horizontal prestressing tendons. It can be easi!y estimated by hand calculation that about 70% of the ultimate pressure would be borne by these tendons. Therefore, the nonlinear behavior of the PCCV is strongly influenced by the tendon behavior. In general, curved unbonded tendons have friction effect, and this should be considered in evaluating the force distribution of tendons in the analyses. Thus, in SMiRT15[4], the authors proposed a numerical model for tendon friction effects that plays an important role in determining force and relative slip distributions, and finally confirmed its applicability by applying it to the analysis of the 3D360 ° model. This idealized the whole 1/4 PCCV model with E/H, A/L, buttresses, and local concentration of rebar arrangement around M/S and F/W. This model (using an interface element) was used to evaluate the friction - slip relationship between unbonded tendon and RC body without numerical instability up to ultimate capacity.
Axisymmetric finite element methods have been widely used for a long time for evaluating the nonlinear behavior of PCCV. The 3D360 ° model might provide more accurate analytical results than the axisymmeteric model. However, it is complicated and a lot of effort and computational time is needed to process the large amounts of data, making it adequate for actual use. This
SMiRT 16, Washington DC, August 2001 Paper # 1158
paper discusses the influence of the analysis conditions on the nonlinear behavior of PCCV by comparing the analytical results obtained from the following four types of analytical model:
1) A 3D90 ° model which idealizes one quarter sector of 1/4 PCCV model. 2) A 3D 180 ° model which idealizes one half sector of 1/4 PCCV model.
3) A 3D360 ° model which idealizes the whole 1/4 PCCV model, presented in SMiRT15[4].
4) An axisymmetric model which idealizes a general portion at 135 ° azimuth, presented in SMiRT1514].
2. ANALYTICAL MODELING
The computer code used here is FINAL [5], which was developed by Obayashi Corporation.
Computational grids of two analyses are shown in Fig.2 and Fig.3. Analytical modeling methods for the axisymmetric model and 3D360 ° model are shown in detail in reference [4].
1) 3D90 ° Model
The 3D90 ° model shown in Fig.2, which idealizes a one quarter section (azimuth 180°~270 °) of a 1/4 scale PCCV model, takes into account the non-axisymmetric tendon layout of the dome, buttresses, and the local congestion of the rebars around these penetrations. This model idealizes the layout of the meridional tendons as accurately as possible. For modeling the basemat, displacements obtained from the axisymmetric model are applied to nodes at the bottom of the cylinder. The configuration of the 1/4 PCCV is not symmetrical in relation to azimuth 270 ° and 180 ° at both sides of the model due to existence of E/H and AJ L etc. However, as shown in Fig.2, the symmetric condition is assumed as a boundary condition on both sides of the model. The reinforced concrete wall is modeled using quadratic multi-layered shell elements. The meridional and hoop tendons are represented through truss elements and attached to the concrete with interface elements. The friction model (friction factor : # =0.21 [6]) presented in SMiRT15[4] is used to model the friction characteristics between concrete and tendons. The liner consists of quadratic shell elements.
2) 3D 180 ° Model
The 3D 180 ° model shown in Fig.3, which idealizes a half section (azimuth 9 0 ° ~ 2 7 0 °) of the 1/4 scale PCCV model, takes into account the non-axisymmetric meridional tendon layout of the dome, E/H, A/L and buttresses, and the local congestion of the rebars around these penetrations. The lowest end of the cylindrical wall is fixed. The configuration of the 1/4 PCCV is not symmetrical in relation to azimuth 90 ° and 270 ° at both sides of the model due to existence of E/H and A/L etc. However, as shown in Fig.3, the symmetric condition is assumed as a boundary condition on both sides of the model. Analysis modelings are the same as those in the 3D90 ° model.
3. MATERIAL CONSTITUTIVE MODEL
(1)CONCRETE
For the concrete constitutive model, the authors adopted the equivalent uniaxial strain model proposed by Darwin et al. [7]. A smeared crack model is assumed to represent cracks.
©Basic Uniaxial Stress-Strain Relationship
The modified Ahmad model [8] is used for the compression zone. Elastic behavior is assumed until cracking occurs in the tensile zone. After cracking, tension cut-off is assumed.
(2)Failure Surface
The Kupfer model [9] in the bi-axial stress state is used for the failure surfaces. (2)Shear Retention Model in Cracked Plane.
The A1-Mahaidi equation [ 10] is used to model shear retention in a cracked plane.
The strength of the concrete( o B) used in the analyses is to be that given in [3]. Young's modulus (E0), tensile strength (f t), and strain ( ~ p) at maximum strength, used in the analyses, are calculated from the following equations. These equations were developed from lots of tests.
ap = (13.97o B + 1690)/106 [11] ... (1) E 0 = (3.57x/~B + 5.71)xl 03 (kN / mm2) [11] ... (2)
ft = 0.3 9(YB °'566 ( k Y / m m 2 ) [ 12] ... (3) (2) STEEL MATERIAL
4. ANALYTICAL RESULTS
Fig.4 and Fig.5 compare the pressure - vertical and radial displacement relationships, respectively, obtained from four analyses. The four curves in Fig.4 have almost the same general features. As shown in Fig.4, the apex of the dome begins to move downward at a pressure larger than 1.5Mpa. However, this amount is fairly small in comparison with the radial displacement of the cylinder. The radial displacements of the 180 ° and 360 ° models at E/H are almost the same. In addition, those of 90 ° and 360 ° models at the mid-height (EL.6.2m) of general portions are almost the same. However, the radial displacements of the 90 ° and 360 ° models at the mid-height become larger than that of the axisymmetric model at pressures above 0.7MPa. It can be seen by comparing radial displacements at E/H and at mid-height of general potions that radial displacements of E/H become larger than those of general portions up to 1.3MPa. The sequences of non-linearity for the four analyses are found to be almost the same, and are as follows: concrete cracking at a pressure of 0.6MPa, liner yielding at 0.92MPa, rebar yielding at 0.93MPa, hoop tendons yielding around mid-height at 1.3Mpa, and finally reaching ultimate capacity of Pmax-l.55Mpa.
Fig.6 compares the relative slip between the hoop tendons and the RC body obtained from three analyses (90 °, 180 ° and 360 ° model). This figure shows the slip amount between the hoop tendon node and the concrete node at the two height levels (as shown in Fig. 1) at the four pressure steps. Tendon-A is located at the height of the A/L and E/H openings. The maximum strain in the hoop tendon at the pressure of 1.55MPa (Pmax) is found in Tendon-B at the height of 6.7m. The relative slip amounts at pressures of 0.4 and 0.8MPa are nearly zero. However, the relative slip becomes large above a pressure of 1.2Mpa, at which the rebar begins to yield. At 1.2MPa and Pmax, the relative slip becomes very large at the rebar cut-off sections around the openings where the stiffness changes, and the tendency for variance of Tendon-B is similar to that of Tendon-A, but smaller in degree. The relative slip at the center of the E/H, A/L, M/S, and buttresses are found to be nearly zero. It can be seen by comparing the analyses of three models that their slip amounts differ a little. However, the characteristics of slip amount are almost the same. Therefore, the influence of the boundary conditions assumed for the 90 ° and 180 ° models on relative tendon slip is found to be very small.
It can also be seen from Fig.6 that the friction effects between tendon and RC body can be estimated appropriately, and that the friction model discussed in this paper works reasonably well. Few past studies have tried to analyze the relative slip between RC body and unbonded tendon up to nonlinear behavior. However, the proposed model may be able to evaluate this with reasonable accuracy. It is very difficult to measure relative tendon slip experimentally, and the analysis results obtained here may be useful for PCCV design.
Fig.7 compares hoop tendon strain distributions at four pressure levels obtained from three analyses. At the pressure levels before tendon yielding, hoop tendon strains are shown to be almost the same. However, at pressure Pmax, the strains of both Tendon-A and Tendon-B in the reinforced portion are smallest at the azimuth at the center of the opening and increase with distance from the opening, peaking at the 30 °, 120 ° and 240 ° azimuth area of the general portion. In comparing the three analyses, some discontinues in strain distributions obtained in the 90 ° and 180 ° models can be seen at 90 ° and 270 o azimuth where symmetry is assumed, but no discontinues of the 360 ° model. However, the characteristics of tendon strain distributions for the three analyses are generally almost the same.
Fig.8 and Fig.9 show contours of deformation vectors obtained from the 180 ° model and the 90 ° model, respectively. The location of the maximum displacement at the pressure of 1.2MPa is seen around the E/H. However, when the pressure increases to 1.5MPa, the displacement of a general portion at about 230 ° azimuth between the buttress and M/S becomes largest. The maximum displacement appears at a different location at each pressure level.
Fig. 10 and Fig. 11 show liner von Mises strain contours obtained from the 180 ° model and the 90 ° model, respectively. It can be found by comparing the lower and higher pressure levels that the locations of the strain concentrations move to the upper portion as the pressure increases. As a result, large strain regions under 1.2MPa appear at a height of about EL.4.0m, while those at Pmax are distributed from about EL.4.0m to EL.7.0m. The liner strain concentrations occur at several portions, such as rebar cut-off sections around penetrations and buttresses, as shown in Fig. 10 and Fig. 11.
Investigation of the analyses conducted in this paper show that strain concentrations in the rebars and the liner tend to occur in the hoop direction where the stiffness is discontinuous in circumferential, such as in buttresses and rebar cut off sections around openings and penetrations.
5. CONCLUSIONS
The conclusions obtained from this study are as follows:
1) Relative tendon slip becomes very large at rebar cut-off sections around openings where the stiffness changes in the circumferential direction.
2) Few past studies have tried to analyze the relative slip between an RC body and an unbonded tendon up to nonlinear behavior. However, the proposed model can evaluate this with reasonable accuracy.
3) The nonlinear behavior of PCCV under internal pressure is strongly influenced by the non-symmetries of the structure, such as penetrations (M/S, F/W, E/H, A/L), buttresses and non-symmetric tendon layouts in the dome. The maximum displacement appears to be at the E/H under 1.3MPa. Therefore, the axisymmetrical analysis is not valid for precisely evaluating the non-linear behavior of PCCV structures.
4) As a result, the analytical results for 90 °, 180 ° and 360 ° models are almost the same. It is found by comparing analytical results that the boundary conditions assumed for 90 ° and 180 ° models are valid for evaluating its nonlinear behavior. The analyses of the 90 ° and 180 ° models are more useful and economic for design than that of the 360 ° model. This is because the 360 ° model is complicated and a lot of effort and computational time are required to process the large amounts of data.
5) In order to predict the ultimate behavior of PCCVs in detail, lots of local analyses with finer-mesh models, as listed in Ref. [ 1 ], must be performed in addition to the three dimensional global analyses here.
REFERENCES
[1 ] Dameron, R.A., Rashid Y.R., Luk V.K. and Hessheimer M.E: Preliminary analysis of a 1/4 scale prestressed concrete containment vessel model, SMiRT 14, H03/3, pp.89-96. Aug., 1997.
[2] Hessheimer, M. F., Pace D.W., Klamerus E.W., Matsumoto T. and Costello J.F.: Instrumentation and testing of a prestressed concrete containment vessel model, SMiRT 14, H03/4, pp.97-104. Aug., 1997.
[3] Pretest Round Robin Analysis of a Prestressed Concrete Containment Vessel Model, NUREG/CR-6678, SAND 00-1535
[4] Yonezawa K., Akimoto.M, Imoto.K, Watanabe Y. : Ultimate Capacity Analysis of 1/4 PCCV Model Subjected to Increasing Internal Pressure, SMiRT- 15 in Seoul, Aug., 1999.
[5] Takeda T., Yamaguchi T. and Naganuma K.: Report on Tests of Nuclear Prestressed Concrete Containment Vessels, Concrete Shear in Earthquake, Edited by T. C. C. Hsu, S. T. Mau, Elsevier Applied Science, pp. 163-172, 1992.
[6] Kashiwase, T. and Nagasaka H.: Analysis study on change of tendon tension force distribution during the pressurization process of prestressed concrete containment vessel (PCCV), Journal of structural and construction engineering (Transactions of AIJ), No.508, pp.63-70, Jun., 1998, in Japanese.
[7] Darwin, D. and Pecknold, D.A.: Nonliner Biaxial Stress-Strain Law for Concrete, J. ofEM Div., ASCE, Vol.103, No. EM2, pp.229-241, April, 1977.
[8] Naganuma, K.: Stress-Strain Relationship for Concrete under Triaxial Compression, Journal of structural and construction engineering (Transactions of AIJ), No.474, pp. 163-170, Aug., 1995, in Japanese.
[9] Kupfer, H. B.: Behavior of Concrete under Biaxial Stress, J. ofE. M. Div., ASCE, Vol.99, No.EM4,
pp.853-866,Aug.,
1973.[ 10] A1-Mahaidi, R. S. H.: Nonlinear Finite Element Analysis of Reinforced Concrete Deep Members, Report 79-1, Dep. of Structural Engineering, Cornell Univ., Jan., 1979.
[11] Amemiya A. andNoguchi H.: Development of Finite Element Analytical Program for High Strength Reinforced Concrete Members, Partl: Development of Concrete Model, Annual Convention of AIJ, Proceeding of Structures II, pp.639-640, Oct., 1990, in Japanese. [ 12] State - o f - the - Art Report on High Strength Concrete, Architectural Institute of Japan, 1991.
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