AXISYMMETRIC GLOBAL STRUCTURAL ANALYSIS OF BARC ...

2008

BARC/2008/E/020B

AR

C/2008/E

/020

AXISYMMETRIC GLOBAL STRUCTURAL ANALYSIS OF BARCPRESTRESSED CONCRETE CONTAINMENT MODEL FOR BEYOND DESIGN PRESSURE

byTarvinder Singh, R.K. Singh and A.K. Ghosh

Reactor Safety Division

BARC/2008/E/020BA

RC/2

008/

E/02

0

GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION

BHABHA ATOMIC RESEARCH CENTREMUMBAI, INDIA

2008

AXISYMMETRIC GLOBAL STRUCTURAL ANALYSIS OF BARCPRESTRESSED CONCRETE CONTAINMENT MODEL FOR BEYOND

DESIGN PRESSUREby

Tarvinder Singh, R.K. Singh and A.K. GhoshReactor Safety Division

BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT(as per IS : 9400 - 1980)

01 Security classification : Unclassified

02 Distribution : External

03 Report status : New

04 Series : BARC External

05 Report type : Technical Report

06 Report No. : BARC/2008/E/020

07 Part No. or Volume No. :

08 Contract No. :

10 Title and subtitle : Axisymmetric global structural analysis of BARC prestressed concretecontainment model for beyond design pressure

11 Collation : 40 p., 13 figs., 2 tabs.

13 Project No. :

20 Personal author(s) : Tarvinder Singh; R.K. Singh; A.K. Ghosh

21 Affiliation of author(s) : Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai

22 Corporate author(s) : Bhabha Atomic Research Centre,Mumbai-400 085

23 Originating unit : Reactor Safety Division,BARC, Mumbai

24 Sponsor(s) Name : Department of Atomic Energy

Type : Government

Contd...

BARC/2008/E/020

BARC/2008/E/020

30 Date of submission : September 2008

31 Publication/Issue date : October 2008

40 Publisher/Distributor : Associate Director, Knowledge Management Group andHead, Scientific Information Resource Division,Bhabha Atomic Research Centre, Mumbai

42 Form of distribution : Hard copy

50 Language of text : English

51 Language of summary : English, Hindi

52 No. of references : 5 refs.

53 Gives data on :

60

70 Keywords/Descriptors : TARAPUR-3 REACTOR; TARAPUR-4 REACTOR;CONTAINMENT; PRESTRESSED CONCRETE; ULTIMATE STRENGTH; CONCRETE BLOCKS;FINITE ELEMENT METHOD; COMPUTERIZED SIMULATION; RISK ASSESSMENT

71 INIS Subject Category : S21

99 Supplementary elements :

Abstract : In order to check the adequacy of the Indian Pressurized Heavy Water Reactor (PHWR)containment structure to withstand severe accident induced internal pressure load, the ultimate loadcapacity assessment is required. Reactor Safety Division (RSD) of Bhabha Atomic Research Centre (BARC)has initiated an experimental program at BARC Tarapur Containment Test Facility to evaluate the ultimateload capacity of Indian PHWR containment. For this study, BARC Containment Model (BARCOM),which is 1:4 scale representation of Tarapur Atomic Power Station (TAPS) unit-3&4 540 MWe PHWRInner Containment of Pre-stressed Concrete has been constructed. The model includes all the importantmajor design features of the prototype containment and simulates Main Air Lock (MAL), Steam Generator(SG), Emergency Air Lock (EAL) and Fueling Machine Air Lock (FMAL) openings. The design pressure(Pd) of BARCOM is 1.44kg/cm2 (g), which is same as the prototype. The pretest analysis of BARCOM hasbeen performed with finite element axi-symmetric modeling. The objective of this simulation was tounderstand the behavior of containment model under internal pressure and find out the various failuremodes and critical locations important for instrumentation during the experiment. The structural responseof the containment model is assessed in terms of wall and dome displacement; cracking of concrete,longitudinal and hoop strains and stresses. Another objective of the analysis was to predict the variousfailure modes of BARCOM with regard to the concrete cracking, reinforcement yielding and tendoninelastic behavior along with the estimation of the ultimate load capacity of the containment model. It isnoted that the BARCOM has an ultimate load capacity factor of 3.54Pd. However, further analysis isneeded to quantify the factor of safety with detail 3D model, which should account for the local structuralbehavior due to various openings. Meanwhile, this preliminary simplified analysis helps to understandthe general overall structural behavior of the BARCOM and the results presented in this report will behelpful to identify the critical locations for instrumentation.

iii

CONTENTS

ABSTRACT iv1. INTRODUCTION 12. STRUCTURAL DETAILS OF BARCOM 23. FEM MODEL 2 3.1 Numerical Model 2 3.2 Boundary condition 5 3.3 Material constitutive models 6 3.3.1 Concrete Model 6 3.3.2 Steel Model 12 3.4 Type of load considered 12 3.5 Convergence criteria 12

4. ANALYSIS RESULTS AND DISCUSSION 12 4.1 Deformation Response 12 4.2 Strain Response 20 4.3 Crack Development in BARCOM 20 4.4 Inelastic Response of Steel Members in BARCOM 325. CONCLUSIONS 336. REFERENCE 34

iv

ABSTRACT

In order to check the adequacy of the Indian Pressurized Heavy Water Reactor (PHWR) containment structure to withstand severe accident induced internal pressure load, the ultimate load capacity assessment is required. Reactor Safety Division (RSD) of Bhabha Atomic Research Centre (BARC) has initiated an experimental program at BARC Tarapur Containment Test Facility to evaluate the ultimate load capacity of Indian PHWR containment. For this study, BARC Containment Model (BARCOM), which is 1:4 scale representation of Tarapur Atomic Power Station (TAPS) unit-3&4 540 MWe PHWR Inner Containment of Pre-stressed Concrete has been constructed. The model includes all the important major design features of the prototype containment and simulates Main Air Lock (MAL), Steam Generator (SG), Emergency Air Lock (EAL) and Fueling Machine Air Lock (FMAL) openings. The design pressure (Pd) of BARCOM is 1.44kg/cm2 (g), which is same as the prototype. The pretest analysis of BARCOM has been performed with finite element axi-symmetric modeling. The objective of this simulation was to understand the behavior of containment model under internal pressure and find out the various failure modes and critical locations important for instrumentation during the experiment. The structural response of the containment model is assessed in terms of wall and dome displacement; cracking of concrete, longitudinal and hoop strains and stresses. Another objective of the analysis was to predict the various failure modes of BARCOM with regard to the concrete cracking, reinforcement yielding and tendon inelastic behavior along with the estimation of the ultimate load capacity of the containment model. It is noted that the BARCOM has an ultimate load capacity factor of 3.54Pd. However, further analysis is needed to quantify the factor of safety with detail 3D model, which should account for the local structural behavior due to various openings. Meanwhile, this preliminary simplified analysis helps to understand the general overall structural behavior of the BARCOM and the results presented in this report will be helpful to identify the critical locations for instrumentation.

1

1. INTRODUCTION

Reactor Safety Division (RSD) of Bhabha Atomic Research Centre (BARC) has initiated an experimental program at BARC Tarapur Containment Test Facility to evaluate the ultimate load capacity of Indian PHWR containment. For this study, BARC Containment Model (BARCOM), which is 1:4 scale representation of Tarapur Atomic Power Station (TAPS) unit-3&4 540 MWe PHWR Inner Containment of Pre-stressed Concrete has been constructed. The BARCOM shown schematically in Fig 1 includes all the main features of the prototype containment structure such as the pre-stressed concrete cylindrical wall structure with a tori-spherical dome, two steam generator (SG) openings in the dome along with main air lock (MAL), fuelling machine airlock (FMAL) and emergency air lock barrel (EAL) openings in the cylindrical wall. BARCOM has been designed for an internal design pressure (Pd) of 1.44 kg/cm2 (g), which is same as the prototype containment structure.

The objective of the present numerical simulation was to understand the behavior of containment model under internal pressure and find out the various failure modes and critical locations important for instrumentation during the experiment. The structural response of the containment model was assessed in terms of wall and dome displacement, cracking of concrete, longitudinal and hoop stresses and strains. Another objective of the analysis was to predict the various failure modes of BARCOM with regard to the concrete cracking, reinforcement yielding and tendon inelastic behavior along with the estimation of the ultimate load capacity of the containment model.

To predict the ultimate load carrying capacity of the complex structure like inner containment structure of PHWR containment structure, nonlinear analysis has been actively pursued in RSD. For the present study, the analysis was performed using 2D axi-symmetric finite element model representing the 157.5 degree azimuth. This free field region is free from local stresses due to discontinuity of buttresses or penetrations and has been developed to predict the structural response for static over-pressurization load. Nonlinear properties of the concrete and steel members are taken into consideration to trace the failure sequence in terms of first appearance of cracking, containment wall and dome through thickness cracking, reinforcement and tendon yielding. For modeling the behavior of concrete after cracking and to consider the interaction between concrete and reinforcement, tension stiffening parameters are used. The reinforcement and tendon are modeled as embedded membrane elements in concrete.

At the design pressure (1.0 Pd) stresses are within elastic limit. The first through wall cracks are observered at 2.27Pd near the mid height of IC wall. At over-pressure of 2.83Pd the yielding is initiated in hoop reinforcements near IC wall mid height. The Hoop pre-stressing cables start yielding at pressure of 3.52Pd at nearly the same location of the IC wall. Finally, the ultimate load capacity of the model is predicted as 3.54Pd, with a failure mode of cylindrical wall bursting. However, further analysis is needed to quantify the factor of safety more accurately with detail 3D model, which can account for the local structural behavior due to various

2

openings. This preliminary simplified analysis helps to understand the general overall structural behavior of the BARCOM and the results presented in this report will be helpful to identify the critical locations for instrumentation. 2. STRUCTURAL DETAILS OF INNER CONTAINMENT:

The 1:4 scale PHWR containment model BARCOM consists of a prestressed concrete cylindrical wall capped with a pre-stressed concrete tori-spherical dome. The details of the model geometry are shown in Fig 1. It includes cylindrical and dome portion along with ring beam and raft. The internal radius of containment model is 6.188m with the cylindrical wall thickness of 188mm with the height of 15.750m. The dome has mean radius of 9.794m and average thickness of 164mm with higher thickness at ring beam and opening locations. The thickness of concrete raft is 3000 mm and the size of stressing gallery is 1500 X 1800 mm. All the five major openings namely, two steam generator (SG) openings in the dome along with main air lock (MAL), fuelling machine airlock (FMAL) and emergency air lock barrel (EAL) openings in the cylindrical wall have been located in the model. The local areas around openings are thickened to account for the discontinuity effects. BARCOM has been designed for an internal design pressure (Pd) of 1.44 kg/cm2 (g), which is same as the prototype containment structure.

The model is pre-stressed by post-tensioning 176 vertical tendons and 108 hoop tendons. A typical vertical tendon is anchored at stressing gallery and ring beam. The hoop tendon is C- cable which is anchored at buttresses and covers the full circumference of the containment wall. The buttresses are located at 0o, 90o, 180o, and 270o. Dome is pre-stressed by 95 J- cables alternately in each direction which continue from raft to the other end of the ring beam. All the cable profiles are presented in Fig 1(b). The model raft foundation supports the structure and transfers its load to the sub soil.

3. BARCOM FEM MODEL

The finite element model, the boundary conditions along with the material model used in this numerical analysis are presented in this section. The geometrical and structural simplifications made to represent the various structural members are described so that meaningful results are obtained from the present numerical model.

3.1 Numerical Model

A simplified numerical analysis procedure is selected for the analysis of BARCOM with axi-symmetric model. With this assumption on the model geometry, the various openings are not considered for the present model. This numerical model thus predicts the average free field response of the BARCOM and the influence of the discontinuities due to the raft wall junction and the wall ring beam and dome junctions are also accounted in the model.

3

Fig.1. Outline sketch of BARC Containment Model (BARCOM)

4

Fig 2 shows the FEM model details that have been used for the present numerical analysis. The concrete structure is modeled by 8-node continuum axi-symmetric elements. In IC wall and dome the element size is governed by spacing of pre-stressing cable and the cover around the reinforcement. In raft the element size is governed by reinforcement spacing. Total 1810 element are used to model the concrete structure. The hoop reinforcements and tendons are modeled as rebar elements, which are represented as steel layers of equivalent smeared thickness in a particular continuum axi-symmetric element. These rebar elements have uni-axial behavior resisting only the axial force in the bar direction, which is the hoop direction in the present model. A total of 470 steel rebar elements for hoop reinforcement members and 235 number of rebar elements for hoop tendons have been included in the present model. The longitudinal reinforcements and tendons were modeled as embedded axi-symmetric membrane elements with orthotropic material property so that the all the bars carry stress only along their individual axial directions. The thicknesses of the steel layers have been calculated so that it represents the BARCOM reinforcement and pre-stressing tendons in the axi-symmetric model. The number of longitudinal embedded steel elements is 301 for the longitudinal reinforcement members and 147 for the longitudinal tendons. A constant pre-stress of 1090 MPa (equivalent to 15.5-Ton force) has been applied as initial stress in the hoop rebar elements and embedded longitudinal tendon elements. In present study, it is assumed that there is no slip between the concrete and steel rebar /embedded members. To consider the effect of the reinforcement, the tension stiffening is used in concrete material model.

(a)

r

z

5

(b)

(c)

(a) Concrete FE model , (b) & (c) Longitudinal Tendon and Reinforcement FE Models

Fig.2. FE Model Details

3.2 Boundary conditions:

At the base of raft fix boundary conditions are applied (ur=0 and uz=0) and on the axis of symmetry the axisymmetric boundary conditions are applied (ur=0)

6

3.3 Material constitutive models: To simulate the inelastic behavior up to the ultimate load capacity for the BARCOM, the composite structural response needs to be computed. This has been realized with the inelastic concrete and steel material models as described below.

3.3.1 Concrete Model

For the present analysis, smeared crack model for concrete [1] was used the reinforcement was modeled by combining elements, with rebar elements—rods, defined singly or embedded in oriented surfaces. One-dimensional strain theory was used to model the reinforcement.

The modeling approach was to consider concrete behavior independently of the rebar. Effects like bond slip and dowel action, were not considered. Tension stiffening was used to simulate load transfer across cracks through the rebar.

When the principal stress components are dominantly compressive, the response of the concrete was modeled by an elastic-plastic theory, using simple form of yield surface defined in terms of the first two stress invariants. Flow rule and isotropic hardening were considered. The inelastic response of concrete when subjected to very high pressure stress was not modeled.

The cracking and compression responses of concrete used in the finite element model are illustrated by the uniaxial response as shown in Fig 3

Figure 3 Uniaxial behavior of plain concrete.

Stress

Strain

Idealized unloading/reloading

Elastic Limit

Tension softening

Cracking

7

The initial response of concrete is elastic in compression. As stress increases, the response of the material softens due to inelastic straining and the total stain is decomposed in elastic and plastic stains as shown in Eq 1 & Eq2.

pl

cel ddd εεε += (1)

Where εd is the total mechanical strain, eldε is the elastic strain (which includes crack detection strains), and pl

cdε is the plastic strain associated with the “compression” surface. Assuming small strain theory, the equation can be integrated as

plc

el εεε += (2)

After that ultimate stress condition is reached and the material softens until it can no longer carry any stress. When a uniaxial specimen is loaded into tension, it responds elastically until, at a stress that is typically 5–10% of the ultimate compressive stress, cracks generates very quickly. To consider this effect it was assumed that the material loses strength through a softening mechanism. Any permanent strain associated with cracking was not modeled. The concept of surfaces of failure and of ultimate strength in stress space was used. These surfaces are defined below and are fitted to the experimental data. Typical surfaces are shown in Fig 4 and Fig 5.

Figure 4 Concrete failure surfaces in plane stress.

8

Figure 5 Concrete failure surfaces in the (p–q) plane.

Concrete Plasticity The inelastic response when the principal stresses are dominantly compressive is modeled as isotropic hardening compressive yield surface as shown in Eq 3.

033 0 =−−= cc paqf τ (3)

Where p is the effective pressure stress, defined as

)(trace31 σ−=p (4)

And q is the Mises equivalent deviatoric stress:

S:S23

=q (5)

Where IS p+= σ are the deviatoric stress components;

0a is a constant, which was chosen as the ratio of the ultimate stress reached in biaxial compression to the ultimate stress, reached in uniaxial compression;

)( cc λτ is a hardening parameter ( cτ is the size of the yield surface on the q-axis at 0=p , so that cτ is the yield stress in a state of pure shear stress when all components ofσ are zero except cτσσ == 2112 ).

Crack detecting surface

Compression surface

Uc

qσ

Uc

pσ

9

The hardening is measured by the value of cλ

The model simulates the concrete behavior for relatively monotonic loadings under fairly low confining pressures. Cracking was assumed to occur when the stresses reach a failure surface i.e. crack detection surface. The failure surface was assumed as a simple Coulomb line defined in terms of the first and second stress invariants, p and q, that are explained above by Eq 4 & Eq 5.

The model calculations were performed independently at each integration point of the finite element model, and the presence of cracks was considered into these calculations by the way the cracks affect the stress and material stiffness associated with the integration point.

The value of 0a is calculated as follows. In uniaxial compression cp σ31

= and cq σ= , where

cσ is the stress magnitude. Therefore, on, 0=cf

⎟⎠

⎞⎜⎝

⎛ −=33

1 0ac

c

στ (6)

In biaxial compression bcp σ3/2= and bcq σ= , where bcσ is the magnitude of each nonzero principal stress. Therefore, on 0=cf ,

⎟⎠

⎞⎜⎝

⎛ −=3

23

1 0abc

c

στ (7)

The value of σσσ bcuc

ubc r=/ was specified as part of the failure surface data. 0a was

calculated from Eq-(6) and Eq-(7) as

σ

σ

bc

bc

rra

21130 −−

=

The compression surface is shown in Fig 4 and Fig5.

Hardening in Concrete

The hardening was defined by supplying the value of the stress, 11σ in a uniaxial compression

test as a function of the inelastic strain magnitude, 11ε . These data were used to define the

)( cc λτ relationship. In uniaxial compression cp σ31

= and, cq σ= , where cσ is the stress

magnitude. During active plastic loading 0=cf , cτ was obtained by

10

cca στ ⎟⎠

⎞⎜⎝

⎛ −=33

1 0 (8)

Flow Rule The flow rule was modeled as if 0=cf and 0>cdλ ,

σσλε

∂∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+= c

cc

plc

fpcdd2

01 (9)

Otherwise, 0=plcdε .

In Eq-(9) 0c is a constant that was calculated as the ratio of pl11ε in a monotonically loaded

biaxial compression test to pl11ε in a monotonically loaded uniaxial compression test.

The gradient of the flow potential for the compressive surface is

Since

I31

−=∂∂σp and

qq S

23

=∂∂σ

,

then

I3

S23 0a

qfc +=∂∂σ

In uniaxial compression cp σ31

= , cq σ= and cS σ32

11 −= , so Eq-(9) defines

( ) ⎟⎠

⎞⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛ += 1

391 00

11acdd c

cplc λε (10)

This equation was integrated to give

( ) ⎟⎠

⎞⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛ += 1

391 00

11ac

ccpl

c λε (11)

So cλ is known from ( )cplc 11ε and the constants 0a and 0c .Eq-(8) and Eq-(11), therefore, define the

( )cc λτ relationship from the concrete model input data once 0c is known.

The constant 0c is calculated from εbcr , the ratio of ( )bcpl

c 11ε to ( )cplc 11ε ; the total plastic strain

components that would occur in monotonically loaded biaxial and uniaxial compression tests. In biaxial compression when both nonzero principal stresses have the

magnitude bcσ , cbcbc rp σσ σ

32

32

== , cbcbc rq σσ σ== and cbcrS σσ31

11 −= , so the flow rule gives

σσσ ∂∂

−∂∂

=∂∂ paqfc

03

11

( ) ( ) ⎟⎠

⎞⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛ +=

21

3941 0

02

11acrdd bcc

cplc

σλε

Using this equation and Eq-(10) then defines 0c from σbcr and the other constants as

( ) ( )( ) ( ) ( )0

20

000

43232339

araraarc

bcbc

bc

−+−

−+−=

εε

ε

Crack detection When cracking occurs during tension the orientation of the cracks is stored and oriented, damaged elasticity was then used to model the existing cracks and crack formation was assumed in orthogonal directions at a point. Rankine criterion was used to detect crack initiation, which states that a crack forms when the maximum principal tensile stress exceeds the tensile strength of the brittle material.

Tension stiffening: Due to bond effect concrete section between the cracks carries a certain amount of tensile force normal to the cracked plane. The concrete adheres to the reinforcing bars and contributes to the overall stiffness of the structure. Several approaches based on experimental results have been employed to simulate the tension-stiffening behavior. A gradual release of the concrete stress component normal to the cracked plane (Fig. 6) is adopted. Unloading and reloading behavior as shown in the Fig. 7 with fictitious elasticity modulus (Ei) is given by

( ) imi

'ti 1 εεε−= fE (12)

Where ft and εm are tensile stiffening parameters and εi is the value reached by the tensile strain at the point considered.

Fig.6: Loading and unloading of cracked concrete illustrating tension stiffening behavior

StrainCompression

tf

Stress

σi

εt εi εm

tensionEi

12

To summarize the present concrete material model consists of a compressive yield/flow surface to model the concrete response in predominantly compressive states of stress, together with damaged elasticity to represent cracks that have occurred at a material calculation point, the occurrence of cracks being defined by a “crack detection” failure surface that is considered to be part of the elasticity.

3.3.2 Steel Model:

The reinforcement bar and pre-stress cable were considered as steel layers of equivalent smeared thickness in the present model. Each steel layer has uniaxial behavior resisting only the axial force in the bar direction. A bilinear elastic-perfect plastic stress-strain model is used for steel. The thicknesses of steel layer were calculated to represent the average reinforcement and pre-stressing in the axi-symmetric model.

3.4 Type of loads for numerical analysis

The containment structure is designed to resist internal pressure due to design basis accident condition. To estimate the margin on the design pressure the load under consideration are the self weight and pre-stressing forces in cables and the internal pressure at the inner surface of the containment structure. The self-weight and pre-stressing forces are constant loads whereas internal pressure is incremented in different steps to simulate the over-pressurization induced response of the model beyond the design basis pressure.

3.5 Convergence criteria:

For the numerical solution Newton-Raphon’s method was used and force/displacement convergence parameters were used. For this analysis, the tolerance limit for residual force is taken as 5% of the average force of the complete model at a particular time step. Similarly the tolerance limit for displacement is taken as 5%. These limits are defined as the ratio of largest residual force / displacement to the average force /displacement for all the degrees of freedom.

4. Analysis Results and Discussions:

The nonlinear analysis is carried out with the present axi-symmetric model using incremental load stepping technique. In the initial first step the dead weight and pre-stressing forces were considered for analysis. Subsequently the design pressure of 1.44 Kg/sq cm is applied, which was incremented in steps to arrive at the ultimate capacity of the model. Displacements, stresses, strains and the inelastic response in terms of concrete cracking, reinforcement and tendon yielding are studied at different load steps. The results obtained are described below.

4.1 Deformation Response:-

As shown in the Fig (7a), for pre-stress and dead weight load the inward deformation of -3.35

13

mm at dome apex and -1.28mm at mid height of IC wall of the model is observed mainly due to pre-stressing in the cables. At design pressure, (1.0 Pd), the dome apex and mid height radial displacements are -1.10 and -0.515 mm respectively as presented in Fig (7b). Hence, radial outward displacement of 2.25 mm at the dome apex and 0.765 mm in cylindrical wall is expected at 1.0 Pd if the reference configuration of the model is taken after the pre-stressing. At 2.0 Pd, the maximum displacements in the dome apex and wall are 1.137mm and 0.243 mm respectively as shown in Fig 7(c).

(a)

(b)

14

(c)

(d)

15

(e)

(f)

Fig.7. Deformation profiles for internal pressure (a) 0.0 Pd, (b) 1.0Pd, (c) 2.0Pd, (d) 3.0Pd (e) 3.25Pd and (f) 3.54Pd with constant dead weight and pre-stressing force in all the cases

The inward radial displacement in cylindrical membrane portion of the containment structure is recovered due to over pressurization at pressure of 1.46Pd. At P=3.0Pd fig (7d) the cylindrical portion begins to deform significant in outward direction. Finally, large outward deflection in the

16

cylindrical wall is observed at P=3.54Pd fig (7f). The maximum radial displacement in the cylindrical wall and dome apex are 11.51mm and 70.5 mm respectively at the ultimate load factor of 3.54Pd. Four locations are selected for monitoring the deflections as shown in Fig. 8. Fig. 9 shows the radial displacement versus internal pressure at different locations. Initially at all the locations negative radial deflection was observed due to the pre-stressing force and the maximum radial inward deflection of 1.14mm is observed at location 2 near the mid height of cylinder. With the increase in pressure the deflection increases linearly at all the locations initially and at higher pressure the deflection increases in a nonlinear manner. It is observed that at location-2 deflection is increasing at a relatively higher rate. Locations 1 and 3 have lower deflection values because they are near to raft and ring beam respectively. At location 2 the deflection increases rapidly after 2.30 Pd because after this point, concrete cracked in longitudinal direction. The rate of deflection increases with pressure and at 3.54 Pd the deflection is very large due to full cracking of containment concrete and yielding of reinforcement and pre-stresseing cables with the peak value of 70.5 mm at wall mid height. Fig. 10 shows the vertical displacement versus internal pressure at these different locations. Initially at all the locations inward deflection due to pre-stressing is noticed with the maximum value of 3.35 mm at the dome apex. Initially the vertical deflection increases linearly with pressure. It becomes nonlinear with respect to the pressure due to concrete cracking in cylindrical portion of the containment structure. Maximum deflection observed at the dome apex (location 4) is 11.51 mm. At raft (location 1) the deflection is the minimum (0.52 mm).

Fig.8: Location Under consideration

17

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Dis

plac

emen

t (m

m)

Pressure Factor

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-10

0

10

20

30

40

50

60

70

80

Pressure Factor

Dis

plac

emen

t (m

m)

(b)

18

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Pressure Factor

Dis

plac

emen

t (m

m)

(c)

Fig.9: Radial Deflection vs Internal Pressure Factor (P/Pd) (a) At Location 1 (b) At Location 2 (c) At Location 3

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pressure Factor

Dis

plac

emen

t (m

m)

(a)

19

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Pressure Factor

Dis

plac

emen

t (m

m)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-2

-1

0

1

2

3

4

5

Pressure Factor

Dis

plac

emen

t (m

m)

(c)

20

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-4

-2

0

2

4

6

8

10

12

Pressure Factor

Dis

plac

emen

t (m

m)

(d)

Fig.10: Vertical Deflection vs Internal Pressure Factor (P/Pd) (a) At Location 1 (b) At Location 2 (c) At Location 3 (d) At Location 4

4.2 Strain Response in Concrete:

Fig. 11 shows hoop strain versus pressure at the above mentioned four different locations. Initially concrete is in compression due to pre-stressing in cable. The Hoop stain at location 2 (near the mid height of IC wall) increases linearly up to 2.30 Pd after that it increases rapidly due to the initiation of longitudinal cracking with the maximum concrete strain of 1.12 %. Strain near the raft at location 1 is almost zero due to the stiff raft. The strain response is low (0.006 %) at location 3 due to the stiffer ring beam. There is a drastic change in strain at location 4 after 3.4Pd (~ 0.0082%) because of cracking of concrete at the dome apex. 4.3 Crack Development in BARCOM

The stresses in concrete wall are approximately uniform in IC wall except near the ring beam or raft because of discontinuity, up to 2.0 Pd as shown in Fig. 12 (a). At 2.27Pd, cracks are initiated in the concrete because at this pressure, the tensile stresses approach the tensile strength (2.75 MPa) of the concrete (Fig. 12(b)). At pressure 2.30Pd, there is a through thickness cracking in the IC wall near the mid height. After the cracking, the hoop stresses in concrete in the IC wall start decreasing as shown in Fig. 12(c). The cracks propagate throughout the wall height at 2.5 Pd (Fig 12 (d)). Through the thickness cracking with a crack width of 0.12mm is postulated at 2.9 Pd near the wall mid height and the stresses in concrete of the IC wall become nearly zero

21

due to stress relaxation, as shown in Fig. 12(e). At the ultimate load factor of 3.54 Pd, the IC wall cracks completely in longitudinal direction and due to this the hoop stresses are fully relaxed thought the IC wall, as shown in Fig. 12 (f). Fig 13 (a) presents the hoop stress profile at the 2.0Pd along the longitudinal direction from the dome apex to the ring beam. In this case, the hoop stresses are compressive throughout the dome (-4.9 MPa at dome and -10.0 MPa at ring beam dome junction). With further increase in pressure there is reduction in the compressive stress near the dome apex (Fig. 13(b)). At 3.0Pd tensile stress are developed in the dome up to 25 degree from dome apex (Fig13 (c)). At 3.25 Pd, tension cracking is observed in dome at ~23 degree (Fig13 (d)). At the ultimate load of 3.54 Pd, as presented in Fig 13(e), the uniform tensile stress level of ~ 2 MPa in the dome membrane region is noticed while the compression zone of the ring beam dome junction has the maximum compressive stress level of 13.9 MPa. The concrete stress versus strain plot at the dome apex shown in Fig 13(f) shows that the concrete cracking is initiated at this location at 3.42 Pd. Further, detail analysis is needed to study the dome cracking profile in the presence of steam generator openings.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-1.0x10-5

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

Hoo

p St

rain

Pressure Factor

(a)

22

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-2.0x10-3

0.0

2.0x10-3

4.0x10-3

6.0x10-3

8.0x10-3

1.0x10-2

1.2x10-2

Hoo

p St

rain

Pressure Factor

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-1.2x10-4

-1.0x10-4

-8.0x10-5

-6.0x10-5

-4.0x10-5

-2.0x10-5

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

Hoo

p St

rain

Pressure Factor

(c)

23

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-4.0x10-4

-3.0x10-4

-2.0x10-4

-1.0x10-4

0.0

1.0x10-4

2.0x10-4

3.0x10-4

Hoo

p S

train

Pressure Factor

(d)

Fig.11: Hoop Strain vs Internal Pressure Factor (P/Pd) (a) At Location 1 (b) At Location 2 (c) At Location 3 (d) At Location 4

-2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

Stress ft

Hoo

p S

tress

(N/m

m2 )

Elevation Level(m)

(a)

24

-2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

Stress ft

Hoo

p S

tress

(N/m

m2 )

Elevation Level(m)

(b)

-2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

Stress ft

Hoo

p St

ress

(N/m

m2 )

Elevation Level(m)

(c)

Crack initiated Crack initiated

Through crack

25

-2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

Stress ft

Hoo

p S

tress

(N/m

m2 )

Elevation Level(m)

(d)

-2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

Stress ft

Hoo

p S

tress

(N/m

m2 )

Elevation Level(m)

(e)

Cracked region

Cracked region

Crack width > 0.12mm

26

-2 0 2 4 6 8 10-12

-10

-8

-6

-4

-2

0

2

4

Stress ft

Hoo

p St

ress

(N/m

m2 )

Elevation Level(m)

(f)

Fig.12: Hoop Stress Profile in IC wall at (a) 2.0Pd (b) 2.27Pd (c) 2.30Pd (d)2.50Pd (e)2.90Pd (f)3.54Pd

0 5 10 15 20 25 30 35 40-12

-10

-8

-6

-4

-2

0

2

4

Stre

ss (N

/mm

2 )

Angle from Verticle Axies (in Theta)

(a)

Fully cracked region

27

0 5 10 15 20 25 30 35 40-12

-10

-8

-6

-4

-2

0

2

4

Stre

ss (N

/mm

2 )


(b)

0 5 10 15 20 25 30 35 40-14

-12

-10

-8

-6

-4

-2

0

2

4

Stre

ss (N

/mm

2 )


(c)

28

0 5 10 15 20 25 30 35 40-14

-12

-10

-8

-6

-4

-2

0

2

4

Stre

ss (N

/mm

2 )


(d)

0 5 10 15 20 25 30 35 40-14

-12

-10

-8

-6

-4

-2

0

2

4

Stre

ss (N

/mm

2 )


(e)

Fig.13 Hoop Stress Profile in dome at (a) 2.0Pd (b) 2.5Pd (c) 3.0Pd (d) 3.25Pd (e) 3.54Pd

Crack Initiated

29

-3.0x10-4 -2.0x10-4 -1.0x10-4 0.0 1.0x10-4 2.0x10-4-1.2x107

-1.0x107

-8.0x106

-6.0x106

-4.0x106

-2.0x106

0.0

2.0x106

4.0x106

H

oop

Stre

ss (N

/mm

2 )

Strain

Fig 13 (f) Stress-Strain curve in concrete at dome apex

Fig. 14 shows the different locations of cracks formed during the pressurization of BARCOM. First cover crack is formed at the junction of raft and IC wall (Fig 14(a)) due to pre-stressing. However, these cracks close on the application of the internal pressure and are stable up to 1.35 Pd as the stresses remain compressive. As pressure increases some surface cracks are formed at the inner surface of dome near the dome ring beam junction at 1.6Pd (Fig 14(b)), which propagate further at pressure of 2.0Pd (Fig 14(c)). The major through thickness cracks are formed near the mid height of IC wall in hoop direction at 2.37Pd (Fig 14(d)), which propagate throughout the IC wall at 2.5Pd (Fig 14(e)). Subsequently the cracks spread towards the raft thickened wall junction at 3.25Pd (Fig 14(f)). New cracks also appear on the raft wall junction inner surface and stressing gallery inner corner at 2.5 Pd (Fig 14(e)), on the outer corner of stressing gallery at 3.25 Pd (Fig 14(f)) and all these cracks spread further at the ultimate load factor of 3.54 Pd (Fig 14(g)) in these regions. The dome cracking is noticed at 3.25 Pd (Fig 14(f)) near dome ring beam junction at an angle of 25 degrees from the dome apex and spread towards the apex, at 3.54 Pd (Fig 14(g)).

30

(a) Location of crack at P=0 (b)Location of crack at P=1.6Pd

(c) Location of crack at P=2.0 Pd

(d) Location of crack at P=2.37 Pd

31

(e) Location of crack at P=2.5Pd

(f) Location of crack at P=3.25Pd

(g) Location of crack at P= 3.5 Pd

Fig. 13: Location of crack at different Pressure

32

4.4 Inelastic Response of Steel Members in BARCOM

At the pressure of 2.83Pd the outer hoop reinforcement show yielding behavior. The pre-stressing tendons are subjected to higher tensile strains with increase in pressure and the Hoop pre-stressing tendons start yielding at pressure 3.52Pd near the mid height of IC wall. The maximum strains in the hoop rebar and the tendons are 1.13 % and 1.12 % respectively at the ultimate load of 3.54 Pd. The stresses / strains in reinforcement and cable at various locations with pressure are listed below in the Table 1(a-b). Table 1(a): Stresses (MPa) in the Reinforcement and Tendons at various locations

Dead Load + Pre-stress

Dead Load + Pre-stress +1.0Pd

Dead Load + Pre-stress + 2.0Pd


Dead Load + Prestress + 3.54 Pd

Inner Longitudinal -45.97 -21.87 2.24 63.04 129.46Inner Hoop -1.22 -0.19 0.831 4.86578 12.37812outer Longitudinal 6.36 -3.25544 -12.87 -51.36 -78.73outer Hoop -0.43 0.11 0.65 2.90 8.81Longitudinal Cable 1226.42 1238.34 1250.25 1315.78 1395.26Lo

catio

n1

Hoop Cable 966.51 966.98 967.46 969.92 974.32Inner Longitudinal -31.21 -22.21 -13.21 -20.18 -19.53Inner Hoop -36.87 -12.102 12.69 415 415outer Longitudinal -26.48 -17.66 -8.85 38.16 415outer Hoop -35.74 -11.68 12.40 415 415Longitudinal Cable 1225.47 1234.37 1243.26 1267.51 1453.42Lo

catio

n2

Hoop Cable 930.69 955.10 979.53 1466.36 1674Inner Longitudinal -52.16 -28.73 -5.68 151.09 377.06Inner Hoop -23.11 -21.42 -19.56 -4.25 13.16outer Longitudinal -7.57 -5.50 -3.13 -29.31 -22.24outer Hoop -22.37 -20.78 -19.04 -4.18 12.79Longitudinal Cable 1226.22 1238.36 1250.46 1310.9 1406.09Lo

catio

n3

Hoop Cable 943.87 945.57 947.43 962.74 980.14Inner Longitudinal -54.57 -35.19 -15.76 3.316 61.73Inner Hoop -53.50 -34.12 -14.67 4.42 45.56outer Longitudinal -52.08 -33.88 -15.61 2.89 59.51outer Hoop -52.55 -34.37 -16.10 2.38 54.81Longitudinal Cable 1036.11 1054.90 1073.74 1092.53 1150.74Lo

catio

n4

Hoop Cable 1035.15 1053.82 1072.71 1091.54 1141.72

33

Table 1(b): Micro-strains in the Reinforcement and Tendons at various locations

Dead Load + Pre-stress

Dead Load +Pre-stress +1.0Pd



Dead Load + Prestress + 3.54 Pd

Inner Longitudinal ‐229.863 ‐109.352 11.1892 315.277 639.183

Inner Hoop ‐2.41502 ‐0.202865 2.01696 9.11074 22.9597

outer Longitudinal 31.8284 ‐16.2613 ‐64.3521 ‐256.829 ‐393.671

outer Hoop ‐2.14662 0.542858 3.23851 14.5187 44.0741

Longitudinal Cable ‐139.275 ‐79.6964 ‐20.12 307.515 704.919Loca

tion1

Hoop Cable ‐2.39764 ‐0.0161289 2.37298 14.6634 36.6573

Inner Longitudinal ‐156.049 ‐111.064 ‐66.0842 ‐100.93 ‐97.6472

Inner Hoop ‐184.392 ‐60.5123 63.4715 2525.09 11289.8

outer Longitudinal ‐132.436 ‐88.3318 ‐44.2376 190.812 2078.27

outer Hoop ‐178.709 ‐58.3949 62.0215 2469.19 11042.2


tion2

Hoop Cable ‐181.489 ‐59.4225 62.7472 2496.91 11164.7

Inner Longitudinal ‐260.817 ‐143.678 ‐28.4113 755.394 1885.29

Inner Hoop ‐115.553 ‐107.083 ‐97.7815 ‐21.2429 65.7748

outer Longitudinal ‐37.83 ‐27.4979 ‐15.6462 ‐146.561 ‐111.166

outer Hoop ‐111.834 ‐103.915 ‐95.1898 ‐20.8983 63.9262


tion3

Hoop Cable ‐113.56 ‐105.417 ‐96.4546 ‐21.011 65.0326

Inner Longitudinal ‐272.881 ‐176.004 ‐78.8606 16.5421 308.61

Inner Hoop ‐269.261 ‐172.317 ‐75.1155 20.3424 283.458

outer Longitudinal ‐260.451 ‐169.44 ‐78.0714 14.4143 297.51

outer Hoop ‐262.774 ‐171.83 ‐80.5208 11.9117 274.054


tion4

Hoop Cable ‐268.781 ‐175.443 ‐80.9929 13.1668 264.056

5.0 CONCLUSIONS

The free field behavior of the BARCOM wall and dome along with the response at discontinuities due to wall raft junction and ring beam has been presented in this report. The simplified axi-symmetric model predicts the crack initiation, through thickness cracking and steel member yielding with the ultimate load factor of 3.54 Pd. Our detail observations on the inelastic behavior of the BARCOM test model is presented below in Table 2, which have been inferred from the present analysis results. The various milestones indicating the non-linear behavior and failure patterns are summarized in the remarks columns. The results of the present

34

analysis shall be used to locate the surface type sensors at the critical locations in addition to the embedded sensors placed in BARCOM. Detail analysis with 3D shell and solid elements are further required to obtain the BATCOM inelastic response at the identified sensor locations and the ultimate load factor. Table 2:

MILESTONES PRESSURE REMARKS First appearance of crack with depth above 50%

2.0Pd Near the ring beam

First appearance of through thickness crack

2.3Pd Near the mid height of IC wall (between 1m to 2m EL)

Nonlinear deformation behavior 2.30Pd Near the mid height of IC wall (between 1m to 2m EL)

Reinforcement yielding 2.83Pd In Hoop Direction near the mid height of IC wall (between 1m to 2m EL)

First appearance of crack with crack width more than 0.12mm

2.90Pd Vertical Crack near the mid height of IC wall.

Dome cracking 3.0Pd At 25 degree angle form apex of dome

Pre-stress Cable yielding 3.52Pd In Hoop Direction near the mid height of IC wall (between 1m to 2m EL)

Whole IC wall cracked with width more than 0.12mm(Structural failure)

3.54 Full IC wall cracked

6.0 Reference

[1] BARC/X Plan/5.06A1/Round Robin Analysis/ Model Document/version r0

[2]Crisfield, M. A., “A Fast Incremental/Iteration Solution Procedure that Handles `Snap-Through',” Computers and Structures, vol. 13, pp. 55–62, 1981

[3] DRG. NO. BARCOM - NPCIL-01 to BARCOM - NPCIL-36

[4] Hillerborg, A., M. Modeer, and P. E. Petersson, “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements,” Cement and Concrete Research, vol. 6, pp. 773–782, 1976.

[5] Kupfer, H.B., Gerstle, K.K., 1973. Behaviour of concrete under biaxial stresses. ASCE J. Engy. Mech.Div. 99 (EM4).

AXISYMMETRIC GLOBAL STRUCTURAL ANALYSIS OF BARC ...

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