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Biaxial behavior of plain concrete of nuclear containment building
Nuclear Engineering and Design 227 (2004) 143–153
Biaxial behavior of plain concrete of nuclear containment building Sang-Keun Lee a,∗ , Young-Chul Song b , Sang-Hoon Han c b
a B&T Engineering Company Ltd., 164-1 Anyang-7-Dong, Manan-Gu, Anyang 430-816, South Korea Korea Electric Power Research Institute, 103-16 Munji-Dong, Yuseong-Gu, Daejeon 305-380, South Korea c Department of Civil Engineering, Chungbuk National University, Chongju, South Korea
Received 5 August 2002; received in revised form 15 July 2003; accepted 2 September 2003
Abstract To provide biaxial failure behavior characteristics of concrete of a standard Korean nuclear containment building, the concrete specimens with the dimensions of 200 mm × 200 mm × 60 mm were tested under different biaxial load combinations. The specimens were subjected to biaxial load combinations covering the three regions of compression–compression, compression–tension, and tension–tension. To avoid a confining effect due to friction in the boundary surface between the concrete specimen and the loading platen, the loading platens with Teflon pads were used. The principal deformations in the specimens were recorded, and the failure modes along with each stress ratio were examined. Based on the strength data, the biaxial ultimate strength envelopes were developed and the biaxial stress–strain responses in three different biaxial loading regions were plotted. The test results indicated that the concrete strength under equal biaxial compression, f1 = f2 , is higher by about 17% on the average than that under the uniaxial compression and the concrete strength under biaxial tension is almost independent of the stress ratio and is similar to that under the uniaxial tension. © 2003 Elsevier B.V. All rights reserved.
1. Introduction From the standpoint of safety, the containment building is one of the most important components of the nuclear power plants (NPP) because it serves as the final barrier to the release of fission products to the outside environment under postulated accident conditions such as loss of coolant accident (LOCA) or main steam line break (MSLB). A major focus of operating plants, therefore, is the benchmarking of existing design criteria and the assessment of structural performance of present containment under postulated accident conditions. The latter of these can be solved through a nonlinear finite element analysis ∗ Corresponding author. Tel.: +82-31-448-8012; fax: +82-31-448-8010. E-mail address: [email protected] (S.-K. Lee).
of containment considering internal pressure that may have been caused by a severe accident. Nonlinear analysis of engineering structures, generally, requires the nonlinear constitutive models for the materials as the components of a structural body. In the case of a prestressed concrete containment for a nuclear power plant, the finite element analysis must consider the ultimate internal pressure and a biaxial model is need as one of the nonlinear concrete behavior models, because the containment under ultimate internal pressure makes nonlinear membrane behavior subjected to biaxial stress. Generally, most engineering designers have used material models not developed from failure behavior tests on applied materials for every case but based on some existing experimental results (Glomb, 1958; Kupfer and Hilsdorf, 1969; Liu et al., 1972; Nelissen, 1972; Buyukozturk et al., 1971; Tasuji et al., 1978) for similar materials with
0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2003.09.001
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slightly different physical properties. Even though it is not unreasonable to use existing nonlinear material models in the analysis of concrete structures, more reliable and accurate analysis will be achieved based on experimentally verified data from real concrete structures. Therefore, this investigation will provide experimental failure behavior characteristics under biaxial stresses for plain concrete of a standard Korean nuclear containment building. But, failure behavior tests under biaxial loading are attempted with some difficulties because they require a fine and complicated testing system which is able to solve the confining effect due to friction between a specimen and the loading platens. In the biaxial loading stage any friction forces are developed at the interface between the steel loading platens and the concrete test specimen as a result of the different lateral stiffness of the two different materials. The developed friction forces restrain the concrete specimen and may induce additional forces that add to the nominal test loads. Because of this, we used a particular method for the loading in the biaxial concrete tests and the details will be explained later. Here we used concrete specimens for two different mixtures with the design criteria strengths, 37.8 MPa (5500 psi) and 27.5 MPa (4000 psi). The first strength, 37.8 MPa (5500 psi), corresponds to the design criteria strength of wall & dome concrete of a standard Korean nuclear containment building, and the second strength, 27.5 MPa (4000 psi), corresponds to the base concrete. There are three possible combinations of biaxial loading for the concrete specimens, compression–compression, compression–tension and tension–tension, and four stress ratio levels in each biaxial combination regions which were considered. The stress ratio is defined as the ratio between normal stresses in two principal directions, f1 and f2 . The final purpose of this investigation is to provide more reliable and accurate stress–strain responses, failure modes and ultimate strength envelopes of plain concrete that is subjected to all the combinations of biaxial loading. The results of this investigation will provide the experimental material models that are essential to carry out ultimate internal pressure analysis of containment. The results also may enhance the models so that nonlinear behavior evaluation of concrete containment will be able to provide higher reliability.
2. Test program Concrete specimens with a square plate form, 200 mm × 200 mm × 60 mm (7.87 in. × 7.87 in. × 2.36 in.), were subjected to biaxial stress in the regions of compression–compression, compression–tension and tension–tension. Concrete specimens of two mixtures called ‘wall & dome concrete’ and ‘base concrete’, with an uniaxial compressive strength of 39.0 MPa (5656 psi) and 30.3 MPa (4395 psi) were tested at 28 days. Such strength values were taken from uniaxial compressive tests for plate form specimens, as they correspond to initial target design criteria strengths of 37.8 MPa (5500 psi) and 27.5 MPa (4000 psi) as explained above. Tension was taken as positive and f1 > f2 . Within each region of stress combinations four different stress ratios α (=f2 /f1 ) were chosen, and four specimens were tested for each variable. For compression–compression, the stress ratios were α = 0 (uniaxial compression), 0.2, 0.5, and 1.0; for compression–tension, the stress ratios were α = −0.05, −0.1, −0.2, and −0.3; and for tension–tension, the stress ratios were α = 1, 2, 5, and ∞ (uniaxial tension) (see Fig. 1). To maintain a constant strain rate in two principle directions of the specimen, the largest principle stress was increased with a loading velocity of approximately 1.96 MPa/min (285 psi/min) throughout the test.
Fig. 1. Stress ratios f2 /f1 considered in this investigation.
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2.1. Test specimens Type I portland cement was used. Two concrete mix designs were chosen that approximate the concrete design strength of a standard Korean nuclear containment building. The water-cement ratios for the two mixtures of concrete were 0.51 and 0.63, and the fine aggregate rates were 0.43 and 0.45 by weight, respectively. The concrete contained coarse aggregates with a maximum size of 19 mm (0.748 in.). The specimens were cast horizontally in steel molds which had been precision machined, so that no further preparation of the loaded surfaces was necessary. It was necessary to take into account the thickness variation of the specimen due to self-settlement during the initial curing after the concrete pours. Therefore the depth of the steel mold was 1 mm greater than the thickness of the specimen. All specimens were compacted by hand. Before testing the rough surface of the specimens was finished with a diamond grinder. Variations in the upper surface were less than 0.1 mm. All specimens, including the additional cylindrical specimens, were cured under water for 27 days and then stored at a temperature of 20 ◦ C (68 F) and a relative humidity of 65%. They were tested 28 days after casting in a uniaxial and biaxial loading frame. 2.2. Biaxial loading and equipment system A concrete specimen subjected to biaxial compression loads may be confined along its loaded surfaces due to friction between the solid platens of the testing machine and the concrete. It is well known that such restraint may result in an increase of the apparent strength of the test specimen (Kupfer and Hilsdorf, 1969; Sundara Raja Iyengar et al., 1965; Vile, 1965; Robinson, 1967). For example, because a specimen subjected to compression–tension loads may be confined along compressive loaded surfaces, the orthotropic principle tensile deformation may not be free. Therefore, to avoid confinement of the specimen due to friction, the solid loading platens were used with Teflon pads, which are a polyethylene series having no friction. The thickness of the Teflon pads was 0.1 mm. Fig. 2 shows an array of the loading platens with Teflon pads. As shown in Fig. 2, the width of the loading plates is 194 mm, which is 6 mm smaller than the specimen width, 200 mm. The
Fig. 2. Loading platens with Teflon pads and the geometry of concrete specimens.
reason they are smaller is to avoid another additional confinement due to contact between adjoining plates under only biaxial compression. Also, to minimize the confining effect due to the compressive force under compression–tension, two Teflon pads were laid one upon another. For the tensile loads, the concrete specimens were glued directly to the solid loading plates using epoxy adhesives. They were tested 48 h after gluing. Fig. 3 shows the loading equipment used in this investigation. Each steel frame was designed so that the
Fig. 3. The loading equipment for biaxial tests of concrete under biaxial stresses.
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Fig. 4. Hydraulic simple beam system to maintain a constant stress ratio used in this investigation.
maximum bending deformation under a prospective maximum load would be less than 1.0 × 10−5 mm. Double-acting hydraulic loading jacks mounted at the loading frames can generate maximum loads of 0.98 MN (220.462 lb) in both compression and tension. The loading platens were attached as a hinge to two orthotropic hydraulic jacks and to steel frames. The specimen was placed in the center of the two cross loading axes and then rested upon an adjustable platform which is fitted after some pre-loading. The ratio of the biaxial stresses, f2 /f1 , can be maintained constantly throughout a test by a simple beam system as shown in Fig. 4. Hydraulic Jack 1 was connected to an auto-controlled hydraulic pressure pump which applied a concentrated force to a simple beam supported by two additional hydraulic cylinders, Jacks
2 and 3. Because the pressure lines of Jacks 2 and 3 were connected to Jacks 4 and 5 which transmitted the forces to the specimen, the reaction forces from a simple beam can be transmitted equally to Jacks 4 and 5. The position of Jack 1 was adjustable along the beam and controlled the stress ratios, f2 /f1 . The control principle of the stress ratios, therefore, are represented as a pb f1 = = . b pa f2 Loads and deformations were recorded continuously using a dynamic data-acquisition system. Deformations in the plane of the specimens were measured with 60 mm (2.36 in.) electrical strain gages. For this investigation, four gages were attached to the front and back surface in order to obtain the strains
200 mm
t(=60 mm)
200 mm
3
4 2 1
strain gage (60 mm)
specimen
Fig. 5. Strain gages attached to specimen.
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in two principal directions (Fig. 5). And then loads f1 , f2 were measured using two load cells which were mounted on the bottom of hydraulic Jacks 2 and 3.
3. Test results 3.1. Ultimate strength 3.1.1. Uniaxial strength data The basic physical properties, ultimate uniaxial compressive and tensile strength, elastic modulus, and Poisson’s ratio, for the two mixtures of concrete was determined by basic physical property tests for both the 100 mm × 200 mm control cylinders and the 200 mm × 200 mm × 60 mm plates. The control cylinders were tested using a 100 kN universal testing machine in accordance with ASTM C39 and ASTM C496. For each mixture, the three cylinders were tested at 7 and 28 days, and the four plates were tested at 28 days. Table 1 provides the basic physical properties for the different mixtures called ‘wall & dome concrete’ and ‘base concrete’. In Table 1, the strength ratio means the relative ratio of compressive strength of cylinder, fcu , to the compressive strength of plate, fc at 28 days. The compressive strength of plates was evaluated and found to be slightly smaller than that for one of the cylinders. Such a difference can be attributed to the difference in geometric shapes and sizes between the two specimens and to the effect of end conditions in the two specimens. The plate specimens may have lower absorption capacity of failure energy than the cylinder specimens under uniaxial compressive test due to the
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fact that the plate specimens have four edges in the axial direction. Tensile strengths were obtained through direct tensile tests for plates and splitting tensile tests for cylinders. From these measurements the Poisson’s ratio was obtained. Since the experiments on the plate specimens were carried out using steel platens with Teflon pads, the end conditions remained the same. Thus, it was determined to use the uniaxial compressive strength, fc , of the concrete plates as the representative uniaxial compressive strength for the different types of concrete specimens. 3.1.2. Biaxial strength data The plate specimens were tested under different combinations of biaxial loading. The tests were performed under constant stress ratio controlled by a hydraulic simple beam, as shown in Fig. 4. Note that the notations for the principal stresses are f2 > f1 , and when fi is negative the specimen is in compression. All strengths are reported as fractions of the unconfined uniaxial compressive strength, fc which was obtained from the plate specimen under uniaxial loading. Table 2 provides ultimate strength normalized with respect to fc under different stress ratios for the different concrete mixtures in this investigation. Fig. 6 shows the relationship between the normalized principal stresses at failure, f1 /fc and f2 /fc , for the different concrete mixtures under combined biaxial compression, compression and tension, and biaxial tension. The graphical representation of this relationship is also referred to as the biaxial strength envelopes. In general, the ultimate strength of concrete under biaxial compression is higher than under uniaxial
Table 1 Basic physical properties for different mixtures Mixtures
Physical properties Compressive strength, MPa (psi) 7 days
28 days
Tensile strength, MPa (psi)
Strength ratio 7 days
Elastic modulus (MPa) Poisson’s ratio
28 days
Wall & dome concrete Plate specimen – 39.0 (5656) 0.982 Cylinder specimen 28.3 (4105) 39.7 (5758) 1
– −4.29 (622) 3.24a (470) 3.82a (554)
29500 30300
0.1745 –
Base concrete Plate specimen – 30.3 (4395) 0.984 Cylinder specimen 22.8 (3307) 30.8 (4467) 1
– 1.96a (284)
26200 26600
0.1764 –
a
Splitting tensile strength.
2.91 (422) 3.14a (455)
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Table 2 Relative fractions of ultimate strength with respect to fc under different stress ratios Combined regions
Stress ratios
Compression–compression
Compression–tension
Tension–tension
α =0 0.2 0.5 1.0 α = −0.05 −0.1 −0.2 −0.3 α = 00 5 2 1
Wall & dome concrete
Base concrete
fc = 39.0 MPa (5656 psi)
fc = 30.3 MPa (4395 psi)
f1 /fc
f2 /fc
f1 /fc
f2 /fc
−1 −1.25 −1.28 −1.17 −0.806 −0.588 −0.379 −0.255 0 0.0234 0.0525 0.0810
0 −0.25 −0.64 −1.17 0.0403 0.0588 0.0758 0.0765 0.11 0.117 0.105 0.081
−1 −1.27 −1.33 −1.164 −0.82 −0.477 −0.316 −0.23 0 0.0218 0.052 0.096
0 −0.254 −0.665 −1.164 0.041 0.0477 0.0632 0.069 0.096 0.10 0.104 0.096
f2/fc
0.2
α=−0.05
α=−0.1
α=−0.2
α=−0.3
0.1
f1/fc
0.0 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0. 2 -0.1 0.0 -0.1
0.1 0.2
α=0.2
-0.2
0.5 α=
-0.3 -0.4
α= 1. 0
-0.5
α= 2.0
-0.6
α=5.0
-0.7
fc = -39.0 M Pa (-5,656 psi) fc = -30.3 M Pa (-4,395 psi)
-0.8 -0.9 -1.0 -1.1α = ∞ -1.2 -1.3 -1.4
Fig. 6. Biaxial strength envelops for two different types of concrete under biaxial stress.
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compression due to the increased confinement from biaxial compression. According to Fig. 6, the relative strength increase for wall & dome concrete (fc = 39.0 MPa) under biaxial compression is slightly less than for base concrete (fc = 30.3 MPa). The relative strength increase is dependent on the biaxial stress ratio. The maximum biaxial strength occurs at a biaxial stress ratio of 0.5 for the two types of concrete tested. At this stress ratio, a strength increase of about 28% for the wall & dome concrete specimen and 33% for base concrete specimen was observed. At equal biaxial compression (α = f2 /f1 = 1.0) the relative strength increase is almost identical for the two types of concrete which were investigated. The strength increase is 17 and 16.4% for the wall & dome and base concrete specimen, respectively. It is noted herein that these trends are in good agreement with the trend that was reported by Kupfer and Hilsdorf (1969) on three types of concrete. In the range of compression–tension the relative strength decrease is almost identical for the two types of concrete. The strength of concrete under biaxial tension is almost independent of the stress ratio α and is similar to the uniaxial tensile strength. 3.2. Concrete stress–strain responses The loads and deformations in the two principal directions were measured for all the tests using the devices mentioned previously. The stress–strain responses show the relationships of normalized stress to actual strain. The conventions for the principal strains are such that ε2 > ε1 , with the tensile strain being positive. Figs. 7 and 8 show biaxial stress–strain relationships under different load combinations for the two types of concrete which were investigated. Stress–strain relationships for the specimens subjected to biaxial compression are shown in Figs. 7a and 8a. It was observed that the stress–strain curve for equal biaxial compression (f2 /f1 = −1/−1) have a trend that the initial tangent inclination is relatively steep, whereas for uniaxial compression (f2 /f1 = 0/−1) in which stress ratio is smaller than biaxial compression, it is relatively gentle. It is the reason why the stiffness of concrete increases relative to the increased confinement from biaxial compression, as the absolute value of stress ratio is large.
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The stress–strain responses for a combined compression and tension region are presented in Figs. 7b and 8b. These figures show the principal stress of the compressive direction, f1 , decreases as the absolute value of the stress ratio, f2 /f1 , increases. This is caused by the principal tensile stresses in the orthogonal direction increasing more due to the increase of absolute value of f2 /f1 and thus, the principal compressive stresses decrease. Also, the stress–strain responses of biaxial tension region as shown in Figs. 7c and 8c show an identical trend with the biaxial compression region above where the initial tangent inclination of the stress–strain curve is steep and the stress ratio is large. 3.3. Failure modes The crack patterns and failure modes that were observed in the specimens after failure were similar to those obtained in previous investigations (Kupfer and Hilsdorf, 1969; Nelissen, 1972; Tasuji et al., 1978). Also, the failure modes observed in the wall & dome concrete specimens were similar to those obtained for the base concrete specimens. There was no fundamental difference in the crack patterns and failure modes due to the increase in the compressive strength. Fig. 9 shows photographs of the failure modes that were observed for all the stress ratios considered in this investigation. In the biaxial compression region, the fracture of concrete under uniaxial compression of Fig. 9a was determined by the formation of a major crack, which is in a pattern like the alphabetic ‘H’ after many microcracks developed in the direction of the applied load. The fractures of Fig. 9b–d in the same region were determined by the formation of major cracks which are inclined at angles of 5–10, 15–20, and 40–45◦ to the direction of the applied load according to the increasing stress ratio. Simultaneously, their failure produced a large blast sound. These phenomena show that the inclined angle of a major crack under biaxial compression loading is developed largely as the stress ratio increases. Secondly, in the compression–tension region, three different patterns of the fracture mode appeared. Fig. 9e shows the first pattern which contains a major crack inclined at an angle of 45◦ to the direction of the applied compressive load, due to the relatively larger
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Fig. 7. Stress–strain relationships for wall & dome concrete mixture (fc = −39.0 MPa) under (a) biaxial compression, (b) combined compression and tension, and (c) biaxial tension.
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Fig. 8. Stress–strain relationships for base concrete mixture (fc = −30.3 MPa) under (a) biaxial compression, (b) combined compression and tension, and (c) biaxial tension.
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Fig. 9. Failure modes under different stress ratios.
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compressive load. Also many tensile cracks perpendicular to the direction of the applied tensile load were developed. Fig. 9f shows that the absolute value of the stress ratio is slightly larger versus Fig. 9e which shows another pattern where the fracture mode of the specimen is determined by a few tensile cracks perpendicular to the direction of the applied tensile load. Fig. 9g and h show that the fracture of the specimen is determined by only one apparent tensile crack. The specimen under uniaxial tension of Fig. 9i was ruptured by only one apparent tensile crack perpendicular to the direction of the applied tensile load. This was also shown in Fig. 9g and h where the crack was located near the center of the specimen. The specimens under biaxial tension loading in Fig. 9j–l show that a major crack is developed more closely to a pseudo line with a 45◦ inclination to the direction of the maximum principal stresses as the stress ratio decreases to 1.0.
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the experimental data for investigators (Chen and Chen, 1975; Chen and Saleeb, 1994; Menetrey and Willam, 1995) to develop the numerical constitutive models of the failure behaviors of concrete. In the future, to establish the complete nonlinear concrete material models of a standard Korean nuclear containment building, such additional investigations as uniaxial or triaxial failure behavior tests of concrete should be continued.
Acknowledgements This study has been performed under the support from the mid- to long-term nuclear research and development program initiated by ministry of science and technology. We acknowledge such support with a warm heart. References
4. Conclusions The biaxial failure behavior characteristics of two concrete mixtures that are representative of the concrete that is used in the standard Korean nuclear containment building were investigated. The concrete mixtures have strengths are 39.0 and 30.3 MPa. One feature of the biaxial loading method is the application of solid platens with the no friction Teflon pads instead of the existing brush loading platens (Kupfer and Hilsdorf, 1969; Liu et al., 1972; Nelissen, 1972). This approach was selected in order to remove a confining effect due to friction between a specimen and the loading platens. Although the solid platens with Teflon pads versus the brush platens are very simple the failure mode results were determined to be satisfactory and the effect was almost equal. For both of the concrete types, the ultimate strength envelopes, stress–strain responses, and failure modes according to various stress ratios under biaxial stress were developed. Such concrete biaxial behavior characteristics will further raise the reliability of the nonlinear finite element analysis of both nuclear containment buildings and the other concrete structures for which the concrete strengths are equal or similar level with those considered in this investigation. Furthermore, this investigation provides
Glomb, J., 1958. The utilization of the biaxial strength of concrete in the design of plates and shells, session 1, no. 1. In: Proceedings of the Third Congress of Prestressed Concrete, Berlin. Kupfer, H., Hilsdorf, H.K., 1969. Behavior of concrete under biaxial stress. ACI J. Proc. 66 (8), 656–666. Liu, T., Nilson, A., Slate, F., 1972. Stress–strain response and fracture of concrete in uniaxial and biaxial compression. ACI J. Proc. 69 (5), 291–295. Nelissen, L.J., 1972. Biaxial testing of normal concrete. Heron (delft) 18 (1), 603–611. Buyukozturk, O., Nilson, A.H., Slate, F.O., 1971. Stress–strain response and fracture of a concrete model in biaxial loading. ACI J. Proc. 68 (8), 590–599. Tasuji, I., Slate, F., Nilson, A., 1978. Stress–strain response and fracture of concrete in biaxial loading. ACI J. Proc. 75 (7), 306–312. Vile, G.W.D., 1965. Strength of concrete under short-time static biaxial stress. In: Proceedings of the International Conference of the Structure of Concrete, Paper F2. Sundara Raja Iyengar, K.T., Chandrashekhara, K., Krishnaswamy, K.T., 1965. Strength of concrete under biaxial compression. ACI J. Proc. 62 (2), 239–249. Robinson, G.S., 1967. Behavior of concrete in biaxial compression. Proc. ASCE 93 (ST1), 71–86. Chen, A.C.T., Chen, W.F., 1975. Constitutive relations for concrete. ASCE Proc. 101 (EM4), 465–481. Chen, W.F., Saleeb, A.F., 1994. Constitutive equations for engineering materials, vol. 1 (2), Elsevier, Amsterdam. Menetrey, Ph., Willam, K.J., 1995. Triaxial failure criterion for concrete and its generalization. ACI Struct. J. 97 (3), 311– 318.
Biaxial behavior of plain concrete of nuclear containment building Sang-Keun Lee a,∗ , Young-Chul Song b , Sang-Hoon Han c b
a B&T Engineering Company Ltd., 164-1 Anyang-7-Dong, Manan-Gu, Anyang 430-816, South Korea Korea Electric Power Research Institute, 103-16 Munji-Dong, Yuseong-Gu, Daejeon 305-380, South Korea c Department of Civil Engineering, Chungbuk National University, Chongju, South Korea
Received 5 August 2002; received in revised form 15 July 2003; accepted 2 September 2003
Abstract To provide biaxial failure behavior characteristics of concrete of a standard Korean nuclear containment building, the concrete specimens with the dimensions of 200 mm × 200 mm × 60 mm were tested under different biaxial load combinations. The specimens were subjected to biaxial load combinations covering the three regions of compression–compression, compression–tension, and tension–tension. To avoid a confining effect due to friction in the boundary surface between the concrete specimen and the loading platen, the loading platens with Teflon pads were used. The principal deformations in the specimens were recorded, and the failure modes along with each stress ratio were examined. Based on the strength data, the biaxial ultimate strength envelopes were developed and the biaxial stress–strain responses in three different biaxial loading regions were plotted. The test results indicated that the concrete strength under equal biaxial compression, f1 = f2 , is higher by about 17% on the average than that under the uniaxial compression and the concrete strength under biaxial tension is almost independent of the stress ratio and is similar to that under the uniaxial tension. © 2003 Elsevier B.V. All rights reserved.
1. Introduction From the standpoint of safety, the containment building is one of the most important components of the nuclear power plants (NPP) because it serves as the final barrier to the release of fission products to the outside environment under postulated accident conditions such as loss of coolant accident (LOCA) or main steam line break (MSLB). A major focus of operating plants, therefore, is the benchmarking of existing design criteria and the assessment of structural performance of present containment under postulated accident conditions. The latter of these can be solved through a nonlinear finite element analysis ∗ Corresponding author. Tel.: +82-31-448-8012; fax: +82-31-448-8010. E-mail address: [email protected] (S.-K. Lee).
of containment considering internal pressure that may have been caused by a severe accident. Nonlinear analysis of engineering structures, generally, requires the nonlinear constitutive models for the materials as the components of a structural body. In the case of a prestressed concrete containment for a nuclear power plant, the finite element analysis must consider the ultimate internal pressure and a biaxial model is need as one of the nonlinear concrete behavior models, because the containment under ultimate internal pressure makes nonlinear membrane behavior subjected to biaxial stress. Generally, most engineering designers have used material models not developed from failure behavior tests on applied materials for every case but based on some existing experimental results (Glomb, 1958; Kupfer and Hilsdorf, 1969; Liu et al., 1972; Nelissen, 1972; Buyukozturk et al., 1971; Tasuji et al., 1978) for similar materials with
0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2003.09.001
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slightly different physical properties. Even though it is not unreasonable to use existing nonlinear material models in the analysis of concrete structures, more reliable and accurate analysis will be achieved based on experimentally verified data from real concrete structures. Therefore, this investigation will provide experimental failure behavior characteristics under biaxial stresses for plain concrete of a standard Korean nuclear containment building. But, failure behavior tests under biaxial loading are attempted with some difficulties because they require a fine and complicated testing system which is able to solve the confining effect due to friction between a specimen and the loading platens. In the biaxial loading stage any friction forces are developed at the interface between the steel loading platens and the concrete test specimen as a result of the different lateral stiffness of the two different materials. The developed friction forces restrain the concrete specimen and may induce additional forces that add to the nominal test loads. Because of this, we used a particular method for the loading in the biaxial concrete tests and the details will be explained later. Here we used concrete specimens for two different mixtures with the design criteria strengths, 37.8 MPa (5500 psi) and 27.5 MPa (4000 psi). The first strength, 37.8 MPa (5500 psi), corresponds to the design criteria strength of wall & dome concrete of a standard Korean nuclear containment building, and the second strength, 27.5 MPa (4000 psi), corresponds to the base concrete. There are three possible combinations of biaxial loading for the concrete specimens, compression–compression, compression–tension and tension–tension, and four stress ratio levels in each biaxial combination regions which were considered. The stress ratio is defined as the ratio between normal stresses in two principal directions, f1 and f2 . The final purpose of this investigation is to provide more reliable and accurate stress–strain responses, failure modes and ultimate strength envelopes of plain concrete that is subjected to all the combinations of biaxial loading. The results of this investigation will provide the experimental material models that are essential to carry out ultimate internal pressure analysis of containment. The results also may enhance the models so that nonlinear behavior evaluation of concrete containment will be able to provide higher reliability.
2. Test program Concrete specimens with a square plate form, 200 mm × 200 mm × 60 mm (7.87 in. × 7.87 in. × 2.36 in.), were subjected to biaxial stress in the regions of compression–compression, compression–tension and tension–tension. Concrete specimens of two mixtures called ‘wall & dome concrete’ and ‘base concrete’, with an uniaxial compressive strength of 39.0 MPa (5656 psi) and 30.3 MPa (4395 psi) were tested at 28 days. Such strength values were taken from uniaxial compressive tests for plate form specimens, as they correspond to initial target design criteria strengths of 37.8 MPa (5500 psi) and 27.5 MPa (4000 psi) as explained above. Tension was taken as positive and f1 > f2 . Within each region of stress combinations four different stress ratios α (=f2 /f1 ) were chosen, and four specimens were tested for each variable. For compression–compression, the stress ratios were α = 0 (uniaxial compression), 0.2, 0.5, and 1.0; for compression–tension, the stress ratios were α = −0.05, −0.1, −0.2, and −0.3; and for tension–tension, the stress ratios were α = 1, 2, 5, and ∞ (uniaxial tension) (see Fig. 1). To maintain a constant strain rate in two principle directions of the specimen, the largest principle stress was increased with a loading velocity of approximately 1.96 MPa/min (285 psi/min) throughout the test.
Fig. 1. Stress ratios f2 /f1 considered in this investigation.
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2.1. Test specimens Type I portland cement was used. Two concrete mix designs were chosen that approximate the concrete design strength of a standard Korean nuclear containment building. The water-cement ratios for the two mixtures of concrete were 0.51 and 0.63, and the fine aggregate rates were 0.43 and 0.45 by weight, respectively. The concrete contained coarse aggregates with a maximum size of 19 mm (0.748 in.). The specimens were cast horizontally in steel molds which had been precision machined, so that no further preparation of the loaded surfaces was necessary. It was necessary to take into account the thickness variation of the specimen due to self-settlement during the initial curing after the concrete pours. Therefore the depth of the steel mold was 1 mm greater than the thickness of the specimen. All specimens were compacted by hand. Before testing the rough surface of the specimens was finished with a diamond grinder. Variations in the upper surface were less than 0.1 mm. All specimens, including the additional cylindrical specimens, were cured under water for 27 days and then stored at a temperature of 20 ◦ C (68 F) and a relative humidity of 65%. They were tested 28 days after casting in a uniaxial and biaxial loading frame. 2.2. Biaxial loading and equipment system A concrete specimen subjected to biaxial compression loads may be confined along its loaded surfaces due to friction between the solid platens of the testing machine and the concrete. It is well known that such restraint may result in an increase of the apparent strength of the test specimen (Kupfer and Hilsdorf, 1969; Sundara Raja Iyengar et al., 1965; Vile, 1965; Robinson, 1967). For example, because a specimen subjected to compression–tension loads may be confined along compressive loaded surfaces, the orthotropic principle tensile deformation may not be free. Therefore, to avoid confinement of the specimen due to friction, the solid loading platens were used with Teflon pads, which are a polyethylene series having no friction. The thickness of the Teflon pads was 0.1 mm. Fig. 2 shows an array of the loading platens with Teflon pads. As shown in Fig. 2, the width of the loading plates is 194 mm, which is 6 mm smaller than the specimen width, 200 mm. The
Fig. 2. Loading platens with Teflon pads and the geometry of concrete specimens.
reason they are smaller is to avoid another additional confinement due to contact between adjoining plates under only biaxial compression. Also, to minimize the confining effect due to the compressive force under compression–tension, two Teflon pads were laid one upon another. For the tensile loads, the concrete specimens were glued directly to the solid loading plates using epoxy adhesives. They were tested 48 h after gluing. Fig. 3 shows the loading equipment used in this investigation. Each steel frame was designed so that the
Fig. 3. The loading equipment for biaxial tests of concrete under biaxial stresses.
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Fig. 4. Hydraulic simple beam system to maintain a constant stress ratio used in this investigation.
maximum bending deformation under a prospective maximum load would be less than 1.0 × 10−5 mm. Double-acting hydraulic loading jacks mounted at the loading frames can generate maximum loads of 0.98 MN (220.462 lb) in both compression and tension. The loading platens were attached as a hinge to two orthotropic hydraulic jacks and to steel frames. The specimen was placed in the center of the two cross loading axes and then rested upon an adjustable platform which is fitted after some pre-loading. The ratio of the biaxial stresses, f2 /f1 , can be maintained constantly throughout a test by a simple beam system as shown in Fig. 4. Hydraulic Jack 1 was connected to an auto-controlled hydraulic pressure pump which applied a concentrated force to a simple beam supported by two additional hydraulic cylinders, Jacks
2 and 3. Because the pressure lines of Jacks 2 and 3 were connected to Jacks 4 and 5 which transmitted the forces to the specimen, the reaction forces from a simple beam can be transmitted equally to Jacks 4 and 5. The position of Jack 1 was adjustable along the beam and controlled the stress ratios, f2 /f1 . The control principle of the stress ratios, therefore, are represented as a pb f1 = = . b pa f2 Loads and deformations were recorded continuously using a dynamic data-acquisition system. Deformations in the plane of the specimens were measured with 60 mm (2.36 in.) electrical strain gages. For this investigation, four gages were attached to the front and back surface in order to obtain the strains
200 mm
t(=60 mm)
200 mm
3
4 2 1
strain gage (60 mm)
specimen
Fig. 5. Strain gages attached to specimen.
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in two principal directions (Fig. 5). And then loads f1 , f2 were measured using two load cells which were mounted on the bottom of hydraulic Jacks 2 and 3.
3. Test results 3.1. Ultimate strength 3.1.1. Uniaxial strength data The basic physical properties, ultimate uniaxial compressive and tensile strength, elastic modulus, and Poisson’s ratio, for the two mixtures of concrete was determined by basic physical property tests for both the 100 mm × 200 mm control cylinders and the 200 mm × 200 mm × 60 mm plates. The control cylinders were tested using a 100 kN universal testing machine in accordance with ASTM C39 and ASTM C496. For each mixture, the three cylinders were tested at 7 and 28 days, and the four plates were tested at 28 days. Table 1 provides the basic physical properties for the different mixtures called ‘wall & dome concrete’ and ‘base concrete’. In Table 1, the strength ratio means the relative ratio of compressive strength of cylinder, fcu , to the compressive strength of plate, fc at 28 days. The compressive strength of plates was evaluated and found to be slightly smaller than that for one of the cylinders. Such a difference can be attributed to the difference in geometric shapes and sizes between the two specimens and to the effect of end conditions in the two specimens. The plate specimens may have lower absorption capacity of failure energy than the cylinder specimens under uniaxial compressive test due to the
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fact that the plate specimens have four edges in the axial direction. Tensile strengths were obtained through direct tensile tests for plates and splitting tensile tests for cylinders. From these measurements the Poisson’s ratio was obtained. Since the experiments on the plate specimens were carried out using steel platens with Teflon pads, the end conditions remained the same. Thus, it was determined to use the uniaxial compressive strength, fc , of the concrete plates as the representative uniaxial compressive strength for the different types of concrete specimens. 3.1.2. Biaxial strength data The plate specimens were tested under different combinations of biaxial loading. The tests were performed under constant stress ratio controlled by a hydraulic simple beam, as shown in Fig. 4. Note that the notations for the principal stresses are f2 > f1 , and when fi is negative the specimen is in compression. All strengths are reported as fractions of the unconfined uniaxial compressive strength, fc which was obtained from the plate specimen under uniaxial loading. Table 2 provides ultimate strength normalized with respect to fc under different stress ratios for the different concrete mixtures in this investigation. Fig. 6 shows the relationship between the normalized principal stresses at failure, f1 /fc and f2 /fc , for the different concrete mixtures under combined biaxial compression, compression and tension, and biaxial tension. The graphical representation of this relationship is also referred to as the biaxial strength envelopes. In general, the ultimate strength of concrete under biaxial compression is higher than under uniaxial
Table 1 Basic physical properties for different mixtures Mixtures
Physical properties Compressive strength, MPa (psi) 7 days
28 days
Tensile strength, MPa (psi)
Strength ratio 7 days
Elastic modulus (MPa) Poisson’s ratio
28 days
Wall & dome concrete Plate specimen – 39.0 (5656) 0.982 Cylinder specimen 28.3 (4105) 39.7 (5758) 1
– −4.29 (622) 3.24a (470) 3.82a (554)
29500 30300
0.1745 –
Base concrete Plate specimen – 30.3 (4395) 0.984 Cylinder specimen 22.8 (3307) 30.8 (4467) 1
– 1.96a (284)
26200 26600
0.1764 –
a
Splitting tensile strength.
2.91 (422) 3.14a (455)
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Table 2 Relative fractions of ultimate strength with respect to fc under different stress ratios Combined regions
Stress ratios
Compression–compression
Compression–tension
Tension–tension
α =0 0.2 0.5 1.0 α = −0.05 −0.1 −0.2 −0.3 α = 00 5 2 1
Wall & dome concrete
Base concrete
fc = 39.0 MPa (5656 psi)
fc = 30.3 MPa (4395 psi)
f1 /fc
f2 /fc
f1 /fc
f2 /fc
−1 −1.25 −1.28 −1.17 −0.806 −0.588 −0.379 −0.255 0 0.0234 0.0525 0.0810
0 −0.25 −0.64 −1.17 0.0403 0.0588 0.0758 0.0765 0.11 0.117 0.105 0.081
−1 −1.27 −1.33 −1.164 −0.82 −0.477 −0.316 −0.23 0 0.0218 0.052 0.096
0 −0.254 −0.665 −1.164 0.041 0.0477 0.0632 0.069 0.096 0.10 0.104 0.096
f2/fc
0.2
α=−0.05
α=−0.1
α=−0.2
α=−0.3
0.1
f1/fc
0.0 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0. 2 -0.1 0.0 -0.1
0.1 0.2
α=0.2
-0.2
0.5 α=
-0.3 -0.4
α= 1. 0
-0.5
α= 2.0
-0.6
α=5.0
-0.7
fc = -39.0 M Pa (-5,656 psi) fc = -30.3 M Pa (-4,395 psi)
-0.8 -0.9 -1.0 -1.1α = ∞ -1.2 -1.3 -1.4
Fig. 6. Biaxial strength envelops for two different types of concrete under biaxial stress.
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compression due to the increased confinement from biaxial compression. According to Fig. 6, the relative strength increase for wall & dome concrete (fc = 39.0 MPa) under biaxial compression is slightly less than for base concrete (fc = 30.3 MPa). The relative strength increase is dependent on the biaxial stress ratio. The maximum biaxial strength occurs at a biaxial stress ratio of 0.5 for the two types of concrete tested. At this stress ratio, a strength increase of about 28% for the wall & dome concrete specimen and 33% for base concrete specimen was observed. At equal biaxial compression (α = f2 /f1 = 1.0) the relative strength increase is almost identical for the two types of concrete which were investigated. The strength increase is 17 and 16.4% for the wall & dome and base concrete specimen, respectively. It is noted herein that these trends are in good agreement with the trend that was reported by Kupfer and Hilsdorf (1969) on three types of concrete. In the range of compression–tension the relative strength decrease is almost identical for the two types of concrete. The strength of concrete under biaxial tension is almost independent of the stress ratio α and is similar to the uniaxial tensile strength. 3.2. Concrete stress–strain responses The loads and deformations in the two principal directions were measured for all the tests using the devices mentioned previously. The stress–strain responses show the relationships of normalized stress to actual strain. The conventions for the principal strains are such that ε2 > ε1 , with the tensile strain being positive. Figs. 7 and 8 show biaxial stress–strain relationships under different load combinations for the two types of concrete which were investigated. Stress–strain relationships for the specimens subjected to biaxial compression are shown in Figs. 7a and 8a. It was observed that the stress–strain curve for equal biaxial compression (f2 /f1 = −1/−1) have a trend that the initial tangent inclination is relatively steep, whereas for uniaxial compression (f2 /f1 = 0/−1) in which stress ratio is smaller than biaxial compression, it is relatively gentle. It is the reason why the stiffness of concrete increases relative to the increased confinement from biaxial compression, as the absolute value of stress ratio is large.
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The stress–strain responses for a combined compression and tension region are presented in Figs. 7b and 8b. These figures show the principal stress of the compressive direction, f1 , decreases as the absolute value of the stress ratio, f2 /f1 , increases. This is caused by the principal tensile stresses in the orthogonal direction increasing more due to the increase of absolute value of f2 /f1 and thus, the principal compressive stresses decrease. Also, the stress–strain responses of biaxial tension region as shown in Figs. 7c and 8c show an identical trend with the biaxial compression region above where the initial tangent inclination of the stress–strain curve is steep and the stress ratio is large. 3.3. Failure modes The crack patterns and failure modes that were observed in the specimens after failure were similar to those obtained in previous investigations (Kupfer and Hilsdorf, 1969; Nelissen, 1972; Tasuji et al., 1978). Also, the failure modes observed in the wall & dome concrete specimens were similar to those obtained for the base concrete specimens. There was no fundamental difference in the crack patterns and failure modes due to the increase in the compressive strength. Fig. 9 shows photographs of the failure modes that were observed for all the stress ratios considered in this investigation. In the biaxial compression region, the fracture of concrete under uniaxial compression of Fig. 9a was determined by the formation of a major crack, which is in a pattern like the alphabetic ‘H’ after many microcracks developed in the direction of the applied load. The fractures of Fig. 9b–d in the same region were determined by the formation of major cracks which are inclined at angles of 5–10, 15–20, and 40–45◦ to the direction of the applied load according to the increasing stress ratio. Simultaneously, their failure produced a large blast sound. These phenomena show that the inclined angle of a major crack under biaxial compression loading is developed largely as the stress ratio increases. Secondly, in the compression–tension region, three different patterns of the fracture mode appeared. Fig. 9e shows the first pattern which contains a major crack inclined at an angle of 45◦ to the direction of the applied compressive load, due to the relatively larger
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Fig. 7. Stress–strain relationships for wall & dome concrete mixture (fc = −39.0 MPa) under (a) biaxial compression, (b) combined compression and tension, and (c) biaxial tension.
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Fig. 8. Stress–strain relationships for base concrete mixture (fc = −30.3 MPa) under (a) biaxial compression, (b) combined compression and tension, and (c) biaxial tension.
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Fig. 9. Failure modes under different stress ratios.
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compressive load. Also many tensile cracks perpendicular to the direction of the applied tensile load were developed. Fig. 9f shows that the absolute value of the stress ratio is slightly larger versus Fig. 9e which shows another pattern where the fracture mode of the specimen is determined by a few tensile cracks perpendicular to the direction of the applied tensile load. Fig. 9g and h show that the fracture of the specimen is determined by only one apparent tensile crack. The specimen under uniaxial tension of Fig. 9i was ruptured by only one apparent tensile crack perpendicular to the direction of the applied tensile load. This was also shown in Fig. 9g and h where the crack was located near the center of the specimen. The specimens under biaxial tension loading in Fig. 9j–l show that a major crack is developed more closely to a pseudo line with a 45◦ inclination to the direction of the maximum principal stresses as the stress ratio decreases to 1.0.
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the experimental data for investigators (Chen and Chen, 1975; Chen and Saleeb, 1994; Menetrey and Willam, 1995) to develop the numerical constitutive models of the failure behaviors of concrete. In the future, to establish the complete nonlinear concrete material models of a standard Korean nuclear containment building, such additional investigations as uniaxial or triaxial failure behavior tests of concrete should be continued.
Acknowledgements This study has been performed under the support from the mid- to long-term nuclear research and development program initiated by ministry of science and technology. We acknowledge such support with a warm heart. References
4. Conclusions The biaxial failure behavior characteristics of two concrete mixtures that are representative of the concrete that is used in the standard Korean nuclear containment building were investigated. The concrete mixtures have strengths are 39.0 and 30.3 MPa. One feature of the biaxial loading method is the application of solid platens with the no friction Teflon pads instead of the existing brush loading platens (Kupfer and Hilsdorf, 1969; Liu et al., 1972; Nelissen, 1972). This approach was selected in order to remove a confining effect due to friction between a specimen and the loading platens. Although the solid platens with Teflon pads versus the brush platens are very simple the failure mode results were determined to be satisfactory and the effect was almost equal. For both of the concrete types, the ultimate strength envelopes, stress–strain responses, and failure modes according to various stress ratios under biaxial stress were developed. Such concrete biaxial behavior characteristics will further raise the reliability of the nonlinear finite element analysis of both nuclear containment buildings and the other concrete structures for which the concrete strengths are equal or similar level with those considered in this investigation. Furthermore, this investigation provides
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