Nucl Eng Technol xxx (2015): 1-13

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Original Article

MECHANICAL PROPERTIES OF TWO-WAY DIFFERENT CONFIGURATIONS OF PRESTRESSED CONCRETE MEMBERS SUBJECTED TO AXIAL LOADING

CHAOBI ZHANG*, JIANYUN CHEN, QIANG XU, and JING LI

School of Civil and Hydraulic Engineering, Dalian University of Technology, Number 2 Linggong Road, Ganjingzi District, Dalian 116024, People's Republic of China

ARTICLE INFO

ABSTRACT

Article history:

Received 29 December 2014 Received in revised form 14 May 2015 Accepted 15 May 2015 Available online xxx

Keywords: Axial loading Biaxial

Concrete damaged plasticity Finite element method Mechanical property Nuclear power plant Numerical simulation Prestressed concrete

In order to analyze the mechanical properties of two-way different configurations of pre-stressed concrete members subjected to axial loading, a finite element model based on the nuclear power plant containments is demonstrated. This model takes into account the influences of different principal stress directions, the uniaxial or biaxial loading, and biaxial loading ratio. The displacement-controlled load is applied to obtain the stress —strain response. The simulated results indicate that the differences of principal stress axes have great effects on the stress—strain response under uniaxial loading. When the specimens are subjected to biaxial loading, the change trend of stress with the increase of loading ratio is obviously different along different layout directions. In addition, correlation experiments and finite element analyses were conducted to verify the validity and reliability of the analysis in this study.

Copyright © 2015, Published by Elsevier Korea LLC on behalf of Korean Nuclear Society.

1. Introduction

Nuclear containments are unique structures that are constantly exposed to harsh environmental conditions, including high temperature, high pressure, and nuclear radiation [1]. Under these conditions, the configurations of

prestressed tendons and reinforced steel in the containment are complex, and the concrete damaged property and the ultimate capacity are generally difficult to evaluate [2-4]. Furthermore, the containment is subjected to biaxial loading in extreme environments [5,6]. Using the ultimate stress under uniaxial loading to design and analyze the structure

* Corresponding author. E-mail address: zhangchaobi@aliyun.com (C. Zhang).

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http:// creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. http://dx.doi.org/10.1016/j.net.2015.05.005

1738-5733/Copyright © 2015, Published by Elsevier Korea LLC on behalf of Korean Nuclear Society.

2 Nucl Eng Technol xxx (2015) : 1-13

will result in inevitable overvalue prediction of the capacity causing the waste of materials, or underestimation of the capacity bringing hidden dangers [7-9].

Many experimental researches have been conducted on the mechanical behavior of reinforced concrete members. Williams [10] prepared a technical report about the stress— strain response of normal-strength concrete panels under direct tension. Comparisons among the results obtained from the experimental program, the existing design code, and other formulae for the tension stiffness were discussed in the report. Gilbert and Warner [11] explained in detail the mechanism of tension stiffening at different loading stages before yielding reinforcement steel in reinforced concrete. Experiments examined the influence of the concrete strength, and a further model for predicting the average tensile stress of concrete after cracking was proposed based on the experiments [12]. Also, thin reinforced concrete slabs and thick reinforced concrete plates were researched via experiments [13-16], in which many variables including the strength of concrete, the reinforcement ratio, the reinforcement bar distribution, the thickness of concrete cover, reinforcing spacing, uniaxial and biaxial loading, and the loading ratio under biaxial loading were considered. The cracking behavior under the influence of transverse reinforcement by the testing programs of reinforced concrete members has been discussed [17-20]. The tension stiffening of the cracked concrete with a test investigation has also been studied, and a model to introduce the average tensile stress-strain for concrete has been developed [21].

Along with the experimental method to explore the mechanical behavior of reinforced concrete, the finite element (FE) codes were also used to study the property. The tension stiffening capacity of describing the cracking response of reinforced concrete under tensile stress (uniaxial and biaxial) was built by Choi and Cheung [22]. Belarbi and Hsu [23] and Hsu and Zhang [24] investigated the in-plane behavior of reinforced concrete membrane elements and developed expressions relating the average principal tensile stress to the average tensile strain of the panel based on the numerical results. Noh [25] used the smeared crack model to simulate the ultimate behavior of large scale reinforced concrete shell structures. Christiansen and Nielsen [26] presented a simple model for predicting the plane stress behavior of reinforced concrete through determining stress, strain, and crack width. Several numerical models which can implement the tension-stiffening effect into the stress-strain relationship of concrete have been proposed [27-29]. Cho et al [30,31] conducted the tensile tests of six half-thickness concrete wall elements as part of the Korea Atomic Energy Research Institute program and developed the constitutive model of concrete panels. Besides, the American Concrete Institute committee 224 [32] and CEB-FIP [33] predict, in an empirical manner, the average stress-strain curves of reinforced concrete elements subjected to biaxial loading.

Most of the current research focuses on the tension stiffness of reinforced concrete members. However, the compressive stresses, especially under biaxial loading, are rarely explored. Besides, existing research only pays attention to the common reinforced concrete members, but most nuclear containments that are under operation or being built are

prestressed concrete structure. What is more important is that the previous studies are mainly based on the similar reinforcement layouts along different directions, but different reinforcement layouts along two directions are frequently encountered in the practical structure. Thus, it is necessary to explore the stress-strain whole constitutive model of pre-stressed concrete members under two-way different reinforcement ratios. An FE model was recently developed, and the details of the sizes and reinforcement layout patterns are completely based on a practical nuclear containment cylinder structure which is widely built and operated in China.

Herein, we focus on the mechanical properties for pre-stressed concrete structure under axial loading. The main variables considered in the numerical program include the different principal stress axes under uniaxial loading, the tension-tension behavior, and compression-compression behavior under biaxial loading, and the different loading ratios under biaxial loading. The FE code ABAQUS was employed with slow load application to ensure a quasi-static solution. Elastic-plastic material model was used for all steel components and the concrete damaged plasticity model was used for the concrete element. Furthermore, a qualitative experiment was tested for a deeper understanding of the different stress strengths along different reinforcement configurations directions. Extensive validities of the analysis were demonstrated by comparing the analytical predictions with experimental results of reinforced concrete members including uniaxial tension specimen, uniaxial compression column, and biaxial tension plates.

2. FE model

In this paper, the selected model is based on a practical nuclear power cylinder structure, the sizes are 900 mm x 900 mm x 900 mm, the distribution of prestressed tendons and reinforcement steel are entirely the same with the nuclear power structure, as shown in Fig. 1. The reinforcement ratios along the hoop and vertical directions are 5.76% and 1.84%, respectively.

The initial strain was imposed to reflect the prestressing effect. Subsequently, the load was applied to obtain the stress-strain response by displacement. For the case of uniaxial tension, the loads were exerted along the hoop and vertical directions, respectively. And for the biaxial case, the loads were simultaneously applied along each direction using various loading ratios (1:0.2,1:0.4, and 1:0.6).

Fig. 2 illustrates the specimens' names, where the first index indicates the type of the member, and the next indices are loading type, stress direction, and mechanical properties in sequence. The tested members are presented in Table 1.

3. Material properties

The general-purpose FE program ABAQUS version 6.10 (Hibbit, Karlsson and Sorensen Inc. USA, 2010) was used in this study to build a FE model for reinforced concrete cube. A reduced

Nucl Eng Technol xxx (2015): 1-13 3

Fig. 1 - Layouts of the prestressed concrete.

integration eight-node solid brick element was chosen to model the concrete. This type of element is capable of modeling simple linear as well as complex nonlinear analysis aspect of contact, plastic behavior, and large deformation. The reinforcement bar is modeled by the T3D2 element which is the three-dimensional (3D) stress/displacement two-node linear displacement truss element.

In addition, the perfect bond and the displacement compatibility between concrete and steel material allow treating the steel as an integral part of the 3D finite element. The steel stiffness matrix is added to that of the concrete, and thus the total stiffness can be obtained as Equation 1:

K — Kc + Ks

CY-B1-HO-T

Mechanical properties T: tension C: compression W: tension & compression -Stress direction

HO: hoop direction of cylinder VE: vertical direction of cylinder

N: not considered the direction Test method

U: uniaxial load B1 : applied load ratio (1:0.2) B2: applied load raio (1:0.4) B3: applied load ratio (1:0.6) Specimen's type CY: cylinder SI: simple concrete

Fig. 2 - Specimens' identification.

3.1. Constitutive model of concrete

In order to describe the complex mechanical behaviors of concrete material under uniaxial and biaxial loading, a number of constitutive models have been developed, including the isotropic damage models [34—37] and anisotropic damage models [38,39]. In this section, we adopted a basic constitutive model developed by Lubliner et al [35] and modified by Lee and Fenves [36]. This model provides a general capability for modeling plain concrete and reinforced concrete subjected to monotonic and cyclic loading under low confining pressures. In this model, the uniaxial strength functions are factorized into two parts to represent the permanent (plastic) deformation and degradation of stiffness (degradation damage). It is assumed that there are mainly two failure mechanisms of the concrete material: one for tensile cracking and the other for compressive crushing.

3.1.1. Damaged evolution

In the incremental theory of plasticity, the total strain tensor e is decomposed into elastic part ee and the plastic part ep for linear elasticity as Equation 2.

£ — t + £p

The scalar damaged elasticity equation is adopted as Equation 3.

s. = (1 - dj ej - eS) (3)

Where s. is the stress tensor, e. and e? are the strain tensor and the plastic strain tensor, respectively, Diejkl is the initial (undamaged) elastic matrix and d is the scalar stiffness

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Table 1 - Summary of specimen program.

No. Specimen symbol Type Loading ratio Stress direction Mechanical properties

1 SI-U-N-W Simple concrete 1:0 Not considered Tension & compression

2 CY-U-HO-W Cylinder 1:0 Hoop Tension & compression

3 CY-U-VE-W Cylinder 1:0 Vertical Tension & compression

4 CY-B1-HO-T Cylinder 1:0.2 Hoop Tension

5 CY-B2-HO-T Cylinder 1:0.4 Hoop Tension

6 CY-B3-HO-T Cylinder 1:0.6 Hoop Tension

7 CY-B1-VE-T Cylinder 1:0.2 Vertical Tension

8 CY-B2-VE-T Cylinder 1:0.4 Vertical Tension

9 CY-B3-VE-T Cylinder 1:0.6 Vertical Tension

10 CY-B1-HO-C Cylinder 1:0.2 Hoop Compression

11 CY-B2-HO-C Cylinder 1:0.4 Hoop Compression

12 CY-B1-VE-C Cylinder 1:0.2 Vertical Compression

13 CY-B2-VE-C Cylinder 1:0.4 Vertical Compression

degradation variable which can take values from 0 (undamaged material) to 1 (fully damaged material). The damage associated with the failure mechanisms of the concrete (cracking and crushing), therefore, it results in a reduction in the elastic stiffness which is assumed to be a function of a set of the internal variable k consisting of tensile and compressive damage variables, i.e., k = {kt, kc}. Damage functions in tension dt and in compression dc are nonlinear functions calculated by comparing the uniaxial response with experimental data.

3.1.2. Yield criterion

In terms of effective stress, the yield function takes the form of Equation 4.

potential G used for the mo del is th e Drucker-Prager hyperbolic function, G = 2 st0 tan j)2 + q - ptan j, where j is the dilation angle measured in the p-q plane at high confining pressure, st0 is the uniaxial tensile stress at failure and 2 is a parameter (referred to as the eccentricity) that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). In this work, the dilation angle j = |f is adopted, where f is the internal-friction angle as a critical parameter of the Mohr-Coulomb failure criterion model and can be measured from the triaxial compression test. The flow potential, which is continuous and smooth, ensures that the flow direction is uniquely defined.

' = (q - 3ap + b(?') (Smax> - g( - w» - Sc (g?) = 0 (4)

a= (Sb0=Sc0)-1 ; oK a< 0.5

2(sbo/sco) -1'

3.2. Constitutive model of steel

The constitutive law of steel is represented by the bilinear law shown in Fig. 5 or in analytical form by the function of Equation 5.

b = ^^ (1 - a)-(1 + a)

g =■

3(1 - Kc) 2Kc - 1

a and g: dimensionless material constants; ffb0/ffc0: the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress; sc(èpl): the effective tensile cohesion stress; st(èpl): the effective compressive cohesion

stress;

the maximum principal effective stress;

p = —1 trace(s): the hydrostatic pressure stress; q = yj§ (S : S):is the von Mises equivalent effective stress; S: the effective stress deviator; and Kc: the strength ratio of concrete under equal biaxial compressive to triaxial compressive.

Typical yield surfaces in the deviatoric plane are shown in Fig. 3, and Fig. 4 shows the initial shape of the yield surface in the principal plane stress space.

3.1.3. Flow rule

Plastic flow is governed by a flow potential function G(s) according to nonassociated flow rule depl = dAdG(s)/dff. The flow

Fig. 3 - Yield surfaces in the deviatoric plane.

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Fig. 4 — Initial yield function in plane stress space.

Uniaxial loading

In the above formula Es denotes the Young's modulus of steel while esy and ssy are in turn the steel strain and stress at yielding. The value ssy is obtained as ssy = fsk/gs where fsk is the characteristic yielding strength of steel and gs is the relevant partial safety factor. Furthermore,

esy — Ssy/Es.

Due to the different reinforcement ratios along hoop and vertical directions, the stress-strain may be different. Different strains were applied along hoop and vertical directions. Fig. 6 presents the stress-strain curves along the hoop and vertical directions under uniaxial loading (compressive stress is positive).

Tables 2 and 3 summarize the tensile and compressive stress obtained from the numerical model under uniaxial and biaxial loading, respectively.

£su < £ < £sy

Ss(£j — Es£ £sy < £ < £sy

sy < £ < £

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Table 2 - Tensile behavior under axial loading.

Specimen Stress/MPa

Cracking Yield

CY-B1-HO-T 3.73 31.2

CY-B2-HO-T 3.73 31.2

CY-B3-HO-T 3.69 31.2

CY-B1-VE-T 1.45 3.11

CY-B2-VE-T 1.58 5.72

CY-B3-VE-T 2.17 7.54

CY-U-VE-W 3.04 9.05

CY-U-HO-W 3.82 31.2

5. Biaxial loading

As the hoop stiffness response is obviously higher than the vertical one, the hoop direction is chosen as the principal stress axes. More specifically, the hoop direction applies similar displacement as the uniaxial loading, and the vertical direction applies 0.2 times, 0.4 times, or 0.6 times of the displacement under uniaxial loading.

5.1. Tension-tension behavior

4.1. Hoop direction

Compared to CY-U-HO-W and SI-U-N-W, the relative improvement of compression stiffening effect is 62.7% when concrete lose the contribution after the ultimate strength, and the compression stiffening response decreases gradually up to the yield of prestressed tendons and reinforced steels. Meanwhile, for the tension stiffening behavior, the improvement of the strength increases gradually up to 68.4% at the concrete damaged stage. Afterwards, the bearing capacity of concrete decreases and the prestressed tendons and reinforced steels mainly bear the strength. When steels have achieved the yield strength, the ultimate stress of specimen CY-U-HO-W is 13.77 times of that of SI-U-N-W.

4.2. Vertical direction

As indicated in Fig. 6, the ultimate compressive stress of CY-U-VE-W is improved by 18.15% than SI-U-N-W. However, after the ultimate stress, the decreasing amplitude of the compression stiffening is obvious during the cracking stage. For the tensile capacity, the load-bearing capacity is higher when the concrete attains the ultimate capacity. Subsequently, the strength is mainly borne by the prestressed tendons and reinforced steels, and in the end the ultimate stress when the reinforced steels yielded is found to be 3.4 times of that of SI-U-N-W.

Compared with the stress-strain relationship between CY-U-HO-W and CY-U-VE-W, the unlimited capacities of compressive behavior, or tensile behavior of CY-U-VE-W are lower than that of CY-U-HO-W. Moreover, in the cracking stage, the compression stiffening effect of specimen CY-U-VE-W is even lower than the specimen SI-U-N-W, and the main reason is the less reinforcement distribution along the vertical direction.

Table 3 - Compressive property under axial loading.

Specimen Stress/MPa

CY-B1-HO-C 63.30

CY-B2-HO-C 62.51

CY-B1-VE-C 27.78

CY-B2-VE-C 34.24

CY-U-VE-W 35.99

CY-U-HO-W 49.56

Comparisons for the stress—strain curves between hoop (CY-B1-HO-T, CY-B2-HO-T, and CY-B3-HO-T) and vertical (CY-B1-VE-T, CY-B2-VE-T and CY-B3-VE-T) directions are presented in Fig. 7, and the numerical results also are shown in detail in Table 2. When concrete loses capacity, the hoop stress is slightly lower than the corresponding one of SI-U-N-W. Moreover, the ultimate stress is equal to the uniaxial ultimate tensile strength when the reinforcement steels yield, which indicates that biaxial loading has little effect on the hoop stress when the hoop direction is used as the principal stress axe.

It is noted that the stress with higher loading ratio develops a higher cracking stress at the vertical direction, as shown in Fig. 7. However, after the cracking stage, the capacity has a small decrease because concrete stiffness decreases rapidly as the stiffness of steels gradually increases. Until the concrete diminishes to zero, the prestress tendons and reinforcement steels bear all stiffness construction.

In comparison with the stress along hoop direction, the capacities under vertical direction are obviously lower, which means that the structures will result in failure along vertical direction.

Compression-compression behavior

Fig. 8 presents a comparison of stress-strain curves between hoop and vertical direction under different biaxial loading ratio. The ultimate capacities along hoop direction (CY-B1-HO-C and CY-B2-HO-C) are much higher than that of specimen SI-U-N-W, the increased amplitude is approximately

I 30. 25 20 15105

CY-B1-HO-T CY-B1-VE-T CY-B2-HO-T CY-B2-VE-T CY-B3-HO-T CY-B3-VE-T

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

e/10 3

Fig. 7 - Stress-strain relationships under biaxial tension.

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1 6050-

CY-B1-HO-C CY-B1-VE-C CY-B2-HO-C ¿r- CY-B2-VE-C

e/10 "

Fig. 8 - Stress-strain relationship under biaxial compression.

-«— P-N -•— O-S — O-D

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

Fig. 10 - Average stress-strain curves for specimens.

26.9% (see Table 3). Furthermore, the ultimate stresses are hardly affected by the change of the loading ratio. The vertical ultimate capacity increases as the loading ratio increased, and specimen CY-B2-VE-C is close to the stress of specimen CY-U-VE-W. In addition, the elastic module of the specimens decreases as the loading ratio increases.

6. Experimental verification

A qualitative experimental investigation was carried out to validate the proposed model and enhance the understanding of the stress-strain effect along different reinforcement configurations directions.

Fig. 9 - Typical configuration and dimension of the specimens.

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Fig. 11 - Square reinforced concrete model.

A special material consisting of water, cement, ore powder, barite power, and barite sand in the ratios by mass of 8.0%, 1.5%, 10.5%, 30%, and 50% respectively, was developed for the nuclear power plant model. The elasticity and brittleness of this new material are similar to those of mass concrete, and its density and Poisson's ratio also approach those of concrete [40,41]. The characteristics of the material agree well with those of mass concrete. Its tensile strength and modulus of elasticity are much lower than for mass concrete and are controlled by its age. Iron wires were used to replace the steel bars.

Fig. 9 shows the typical irons configurations and specimens' sizes. It takes 24 hours to cure it in a suitable laboratory environment, which consists of a temperature range of

Fig. 12 - Load-displacement relationship.

15-18°C and a humidity range of 50-70%. The experiment was conducted using a computer-control and using the SANS hydraulic pressure universal testing machine (manufactured by the SANS INC. Shenzhen, China). In this test, the displacement-control compression loads were applying along the X or Y directions, and the corresponding stress-strain curves were automatically recorded by the computer.

The average compressive strength curves were presented in Fig. 10. Herein, P-N means no iron wires specimens, O-S is X direction specimens, and O-D is Y directions specimens. The ultimate strength stresses of O-S and O-D are 0.649 MPa and 0.909 MPa, respectively, and the corresponding increasing amplitudes are 45.8% and 104.3% compared with P-N. This indicates that stresses of specimens are affected by different reinforcement configurations, and this result is consistent with the conclusion analyzed by the FE code.

7. FE verification

To establish more sufficient validity in simulating the mechanical properties, three related experimental results were given by ABAQUS code and compared with numerical results. It should be reminded that the element styles and material constructive laws of concrete and reinforce steel elements are completely the same as the above description in this section.

Uniaxial tension test

A reinforced concrete prism with square cross section based on the numerical code RFPA3D (realistic failure process

Fig. 13 - Crack pattern of uniaxial tension specimen. (A) The RFPA3D result (B) The ABAQUS result.

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Fig. 14 - Specimen for reinforced concrete columns.

analysis) that was similar to the international cooperative experiment of RILEMTC 147-FMB [19] has been carried out by Zhang [42]. The displacement-controlled loading scheme was used to simulate failure process under uniaxial tension. The geometric characteristics of tension member are: concrete section 720 mm x 30 mm x 30 mm, reinforcement with the diameter of 6 mm was placed at the center of the square cross section along the transverse direction (see Fig. 11). The material properties of concrete and steel used in this analysis are: fc — 80MPa, ft — 0.395(fC)aBBMPa, Ec — 25 GPa, fy — 435 MPa, and Es — 210 MPa, Poisson's ratio vc — 0.18, vs — 0.30.

As shown in Figs. 12 and 13, the stress-strain curve simulated is very close to the result of RFPA3D, the simulated failure location is very similar to the RFPA3D result with the phenomenon of same fracture spacing. Although the load by the ABAQUS analysis is lower than that of RFPA3D at the stage of concrete failure, the subsequent curves were concordant, thus the results of the simulations can be acceptable.

Table 4 - Material properties used in columns.

Concrete

fc/MPa 65.8

Reinforced steel

f6 Aj/mm2 28.3

fy/MPa 541

As/mm2 113.1

fy/MPa 475

As/mm2 153.9

fy/MPa 376

7.2. Uniaxial compression test

A set of axial compression tests of large-size high-strength reinforced concrete columns with different geometric dimensions and stirrup ratios have been conducted to reveal the size effect law and axial compressive performance by Fu et al [43]. In this study, two completely identical parametric columns, NM400-1 and NM400-2, were selected. The geometry and cross-section dimensions of the columns are presented in Fig. 14, and the material properties are summarized in Table 4.

Fig. 15 presents the comparison of the stress-strain curves between simulated and experimental results. The experimental ultimate stresses of NM400-1 and NM400-2 are 55.59 MPa and 62.98 MPa, respectively, and the ABAQUS analytical result is 59.31MPa, lies between that of NM400-1 and NM400-2. Failure phenomena after the tests are shown in Fig.16A—C. It is shown that the failure area is mainly concentrated on the mid-column, the compressive stress causes longitudinal steel bars to buckle and the corresponding concrete cover to spall. Tensile failure and obvious necking behavior of stirrups appeared at the corner.

Fig. 16D shows that the simulated concrete damaged value at the mid-column is 0.947, which indicates that the

Fig. 15 - Stress-strain curves of column.

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Fig. 16 - The final destruction of the specimens. (A) The concrete spall. (B) The longitudinal steels buckle. (C) The necking region of stirrups. (D) The concrete damaged. (E) The steel's stress.

Fig. 17 - Layout and dimensions of biaxial tension specimens.

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T 1 I 1 I ' I 1 I 1 I ' I

0.0 0.5 1.0 1.5 2.0 ,2.5 3.0

Fig. 18 — Average steel stress—strain curves of biaxial loading panel.

corresponding concrete loses strength. The longitudinal steels and stirrups were buckling and reached the yield strength (see Fig. 16E). In a word, the simulated curves and crack location are satisfactory when compared with experimental results.

7.3. Biaxial tension test

In order to check the validity of the model mechanical behavior under biaxial loading, a model was tested to predict the experimental behavior of previously tested slabs loaded transversely [15].

The specimen dimensions are 600 mm x 600 mm x 190 mm, as shown in Fig. 17, reinforcements are orthogonally placed, and the biaxial tension loads directly act to the reinforcement. The biaxial loading ratio is 1:1, and the average concrete compressive strength f is found to be equal to 35 MPa, and the steel bar diameter is 15 mm with a yield stress and ultimate tensile strength of 410 MPa and 650 MPa,

respectively. The steel bars had an elastic modulus of 200 GPa.

The analytical prediction is compared with the experimental average steel stress-strain relations. As shown in Fig. 18, the cracking stress and failure stress from the numerical analyses agree fairly well with those of the experimental results. The cracks of the numerical analysis and experimental result are illustrated in Fig. 19. Also, the numerical result is close to the experimental result. In fact, the revealed cracks of numerical model are impossible to be completely identical to the experimental result because of the experimental complexity.

8. Conclusion

This paper presents a numerical model of prestressed concrete members used for nuclear power stations, and predicts the stress—strain response subjected to uniaxial and biaxial loading. The influences of loading direction under uniaxial loading and the loading ratio under biaxial loading are investigated.

Due to the lower reinforcement ratio along the vertical direction, the stress stiffness is obviously lower than the corresponding one of hoop direction subjected to uniaxial loading. The vertical tensile and compressive stresses are 29.0% and 72.6% times of the corresponding values of the hoop direction.

Under biaxial loading, the effect of tension stiffness along the hoop direction with the change of loading ratio is indistinctive, but the improvement of the compression stiffness as the increase of the loading ratio is obvious. The stresses of tension and compression along vertical direction increase as the increase of loading ratio, but are lower than the corresponding ones under uniaxial loading.

Besides, the reliability of the analytical results was validated by a simple reinforced concrete experiment and several FE simulations analyses in this paper. The results can provide a reference for designing and analyzing containment structures for nuclear power plants.

Fig. 19 — Final crack patterns of the panel. (A) The experimental result. (B) The simulated result.

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Conflicts of interest

The authors have no conflicts of interest. Acknowledgments

The research reported in this paper is granted by the State Key Program of National Natural Science of China (Grant No. 51138001) and the financial support from the Specialized Research Fund for Doctoral Program of Higher Education Institutions, China (Grant No. 20110041110012). The authors would like to express their gratitude to these organizations for their financial support.

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