This document summarizes calculations performed to estimate cracking and leakage rates for the VERCORS mock-up containment building. A finite element model was created to model the structure, reinforcement, and prestressing tendons. Calculations were performed to simulate dead weight, creep, shrinkage, prestressing, and internal pressure. Cracking patterns were estimated at each step. Leakage rate was calculated using permeability values corresponding to strain levels in each element. The calculated cracking lengths and leakage rate were higher than actual measured values, likely due to modeling assumptions. The study provides valuable data for benchmarking containment modeling methods.

1. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
Full paper Submission, TINCE-2016
Paris (France), September 5th
– 9th
, 2016
Estimate of cracking and leaking rate of VERCORS mock-up
Romain Vénier1
, Laure Simeoni2
, Antoine Brison3
and Romain Ragouin4
1
Project engineer and project leader, Dept. of Nuclear and Industrial structures, Tractebel Engie
– Coyne et Bellier, Lyon, France, (romain.venier@tractebel.engie.com)
2
Project engineer and project leader, Dept. of Nuclear and Industrial structures, Tractebel Engie
– Coyne et Bellier, Lyon, France (laure.simeoni@tractebel.engie.com)
3
Project engineer and project leader, Dept. of Nuclear and Industrial structures, Tractebel Engie
– Coyne et Bellier, Lyon, France (antoine.brison@tractebel.engie.com)
4
Project engineer and project leader, Dept. of Nuclear and Industrial structures, Tractebel Engie
– Coyne et Bellier, Lyon, France (romain.ragouin@tractebel.engie.com)
Introduction
The VERCORS project (French acronym for “Realistic verification of the behaviour of reac-
tor containments”) consists in designing and building a 1/3 scaled mock-up of the inner contain-
ment of a French nuclear reactor of type “P’4”. The mock-up is a prestressed-concrete building
dedicated to research purposes [MAS13]. Its overall dimensions are: height 25 m, diameter 15
m, thickness of cylinder wall 0.40 m.
Figure 1. Cross section of the VERCORS mock-up

2. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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A first benchmark was organized by EDF with three themes proposed:
1. Early-age behaviour of gusset zone of the containment;
2. Containment history: prediction of deformations, stresses and cracking history of the
whole containment during prestressing and first pressure test;
3. Leakage: estimate of air leakage during pressure test at the pressure peak of 5,2 bars
absolute
The paper deals with calculations carried out by Tractebel Engie – Coyne et Bellier for
themes 2 and 3. As regards results, we focus on cracking and leaking rate.
Model
ANSYS 11.0 software is used for modelling, calculations and post processing.
A finite element model is built, based upon construction drawings [ETP14]. Gusset, cylin-
drical wall, dome belt, dome, equipment hatch and personal hatch are modelled with solid ele-
ments. In typical zones, containment and dome are meshed with five elements in the thickness.
Global view Top view
Figure 2. Views of finite element model
Real routing of prestressing system (according to drawings) is discretised, using specific
software developed internally. Figures below show 3D plots of tendon layout.

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Gamma tendons Horizontals tendons
Verticals tendons Dome tendons
Figure 3. Views of tendon layout of VERCORS mock-up
All steel rebars are also modelled, according to construction drawings.

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Hatch view Top containment view
Figure 4. Views of steel reinforcement of VERCORS mock-up
Steel rebar and tendons discretisation are superimposed to the finite element model. In-
ternal software is used to determinate ratio of steel (rebars and tendons) going through each
concrete element. Then actual steel reinforcement pattern and prestressing tendons (cross sec-
tion and orientation) are taken into account via SOLID65 elements capabilities. There are no
specific steel rebar or prestressing element in finite element model of the VERCORS contain-
ment.
Sleeve of hatches are modelled with SHELL41 elements (3D-membrane-shell-4-node el-
ements).
The mesh is regular; the model has 39139 volume elements and 204 shell elements.
Hypotheses considered for calculations
For concrete and steel reinforcement and tendons mechanical characteristics, construction
data from site given by EDF are used (mean values). As regards concrete modulus, a creep
coefficient is taken into account, derived from creep strains studied (based on sample results
given by EDF).
Prestress losses are estimated in compliance with:
- Real routing of tendons and of course their deviations due to penetrations;
- Behaviour of material including creep and shrinkage deformations of concrete, and
steel relaxation of prestressing tendons.
Net tendon tension is computed at every point of every tendon. Thus, forces correspond-
ing to exact location of tendon discretized points are automatically computed at nearest points of
concrete volume elements, and then applied as a load to these points.
Concrete lifts are not modelled.
Gusset lower face is considered embedded in common raft: at level z = - 1.00 m, dis-
placements are set equal to zero, ux = uy = uz = 0.
No thermal computation is carried out (no thermal gradient taken into account).

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Main characteristics considered are gathered in the table below.
Table 1. Main mechanical characteristics considered
Value Unit
Concrete
Young’s modulus E 20 500 MPa
Poisson’s ratio 0,2 -
Density 2500 kg.m-3
Thermal expansion
coefficient 1,17 10-5
°K-1
28-day concrete
compressive
strength 47,3 MPa
Steel rebars and
metallic hatches
Young’s modulus E 200 000 MPa
Poisson’s ratio 0,3 -
Density 7850 kg.m-3
Thermal expansion
coefficient 1,2 10-5
°K-1

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Figure 5. Example of tension computed along two horizontal tendons
Computation methodology
Mechanical effects modelled are dead weight, creep and shrinkage at average prestress-
ing date, prestressing forces along every tendon, pressure (5,2 bar abs.). These actions are ap-
plied successively to the model, in order to remain coherent with what has been applied to the
real structure.
Cracking pattern at each step is calculated via an actual reinforced-concrete behaviour
computation, concrete behaviour law being set up at the beginning of the calculation.
Non-linear computation lasts 29 hours on a classic computer dedicated to finite element
calculations.
Main results: cracking
Stresses and strains are available at all nodes and all elements of the model, at all sub-
step required by benchmark rules (after dead weight, after creep and shrinkage [= before pre-
stressing], after prestressing [= before pressure test], at the peak of pressure test).
For theme 2, crack openings are estimated at each step as follows. Stresses are integrat-
ed in typical zones of the containment, giving normal efforts and bending moments on a typical
reinforced-concrete section. A reverse calculation leads to stresses in rebar, and via [ECO05]
formulae crack openings and spacings are computed. Total length of cracks in each zone is then
computed.
Results are gathered in tables below.
Table 2. Inner face cracks (include through cracks)
Area Total length (m) Max opening (mm) Spacing (m)
Gousset 19,32 0,061 0,173
Hatch area 0 0 0

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Cylindrical part (wall) 6,58 0,052 0,237
Dome 0 0 0
Table 3. Outer face cracks (include through cracks)
Area Total length (m) Max opening (mm) Spacing (m)
Gousset 37,48 0,113 0,184
Hatch area 109,55 0,438 0,265
Cylindrical part (wall) 13,69 0,013 0,219
Dome 0 0 0
Plots of cracking pattern after at pressure 5,2 bar absolute are as follows.
Containment view Dome view
Figure 6. Percentage of cracking for each element of finite element model, at peak of pressure
test
Main results: leaking rate
To estimate the leaking rate, main steps of the methodology we follow are detailed below.
As a first step, values of total strains are extracted for each element of the finite element
model.
Secondly the equivalent geometric permeability of each element is calculated with a for-
mula:
keq = f( ) (1)
This formula is extracted from samples studied within thesis [MIV96]. Air permeability as-
says were realised on five reinforced concrete samples submitted to traction, the results of which
are analysed in reference [DAL04]. Two laws which represent this relation are established: one
is exponential and the other is affine.

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Exponential law:
keq = a b
10-13
m², is the strain in % (2)
Linear law:
keq = (A + B) 10-13
m², is the strain in % (3)
Coefficients a, b, A and B can be deduced form interpretation of tests realised within thesis
[MIV96].
A minimal permeability kmin is taken into consideration.
Figure 7. Permeability of concrete according to strain
In our calculation we use the linear law. Coefficients A, B and kmin are determined form de-
tailed analyses of thesis [MIV96] and form our past experiences.
In a third step we estimate the leaking rate. The estimation is based on the porous medium
analytic formula, giving the mass flow by square meter of surface, in kg s-1
m-2
:
h
ppk
RT
M
Q ei
moy
22
2
(4)
With:
Dynamic viscosity (Pa.s)
k Geometric permeability (m2
)
R Perfect gas constant (J.K-1
)
M Molar mass of the fluid (kg)
h Thickness of wall (m)
pi Inner pressure (Pa)
pe Outer pressure (Pa)
Tmoy Mean temperature of fluid in the thickness of the wall h (K)
k
(m²)
(%)
kmin
k = a b
10-13k = (A + B) 10-13

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For finite elements calculation, a formal analogy is done between the equation (4) and the
heat equation by conduction in a material:
Q = T/ x (5)
To quantify the leaking rate of VERCORS mock-up, a thermal finite elements model is
used and next steps are as followed:
- calculation of mean values of principal stains: 1, 2, 3 of each volumetric element;
- determination of the principal strains 1’, 2’, 3’ for each element in local coordinated
system;
- evaluation of local permeabilities k1 1’), k2 2’), k3 3’) (with an orthotropy hypothe-
sis).
The finite element model used for this thermal calculation is geometrically the same as the
one used for mechanical calculations. SOLID65 elements are replaced by SOLID90 elements,
which are typical volume elements used for thermal calculation, and thermal conductivities of
each element are input as local permeability.
Results are given in table below (in Nm3
/h).
Table 4. Calculated leaking rate of VERCORS mock-up
Global air leakage prediction at the 5,2 bar peak (Nm3
/h) 82,7
Area
gusset 1,8
hatch area 4,1
cylindrical part (wall) 60,7
dome 17,4
Overview of experimental results and discussion
VERCORS mock-up was built between July 2014 and April 2015. Presstressing occurred
between May 2015 and August 2015, and first pressure test was carried out in November 2015.
Experimental results of the first pressure test (measured temperatures, stresses, strains,
cracking, leaking rate) were given to the benchmark participants at the beginning of 2016.
As regards cracking, we decided to use [ECO05] formulae, known to lead to overestima-
tions of cracking but widely used in civil work studies – and which use is mandatory for new pro-
jets in Europe. Other refined formulae (Model Code 2010 for example) could have led to a more
accurate estimate of total amount of cracks (see tables below).
Besides, throughout our calculation many crushed elements appear after prestressing and
at the pressure peak in the hatch area, leading to important crack lengths and openings in this
zone. This phenomenon, compared to the real behaviour of the hatch area, has not been further
studied yet.

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Table 5. Outer face cracks (include through cracks): actual and calculated values
Area Total length (m) Max opening (mm)
Gousset 37,48 0,113
Hatch area 109,55 0,438
Cylindrical part (wall) 13,69 0,013
Dome 0 0
Gousset 24,63 0,100
Hatch area 0 0
Cylindrical part (wall) 0 0
Dome 13,73 0,100
As regards leaking rate, many hypotheses can alter final results: actual material character-
istics, temperature, concrete defaults considered or not, permeability law etc. This can explain
the differences between real air leakage rate during pressure test and in our calculations (see
table below).
Table 6. Leaking rate of VERCORS mock-up: actual and calculated distributions
Global air leakage prediction at the 5,2 bar peak (Nm3
/h) 7,7 82,7
Area
gusset 56 % 1 %
hatch area 15 % 5 %
cylindrical part (wall) 25 % 73 %
dome 4 % 21 %
It also has to be noticed that we decided to set a minimal permeability kmin based on the
behaviour of a Belleville inner containment for our calculation. If we had followed Mivelaz studies
[MIV96], global leaking rate computed would have been Q = 35 Nm3
/h, closer to the actual val-
ue.
Still, we cannot imagine, with data given to the benchmark participants, how one calculator
could have managed to get the correct qualitative distribution of the leak, and especially the fact
that the gusset is the zone leaking the most.
References
[MAS13] Masson, B. and Alliard, P.-M. (2013), Objectives and design of the new experimental
program VERCORS based on a 1/3 scaled PWR containment building, Technical Innova-
tion in Nuclear Civil Engineering (TINCE 2013)
[MIV96] Mivelaz, P. (1996), Etanchéité des structures en béton armé, EPFL academic thesis no.
1539
[DAL04] Dal Pont, S. (2004), Lien entre la perméabilité et l’endommagement dans les bétons à
haute température, ENPC academic thesis
[ETP14] Eiffage construction drawings (formwork, reinforcement, prestressing)

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[ECO05] Eurocode 2 EN 1992-1-1
Please fill in the blanks at the end of this extended abstract (the additional blue lines and
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Preference: Poster X Oral
Topic: 1 - Advanced Materials 2 - Design and Hazard Assessment
3 - Civil Works Construction 4 - Long Term Operation & Maintenance
5 - Dismantling of civil works & Civil Works in Hostile Environment
6 - Geotechnical Design & Construction & Fluid Structure Interaction
X 7 - Special session VERCORS
Corresponding author: romain.venier@gdfsuez.com