Civil engineering tends to propose solutions increasingly safe and at the same time designed to optimize materials. In the field of seismic design, Peak Response method offers a new approach for seismic linear design using response spectra. Traditional methods such as CQC+Newmark or CQC+SRSS lead sometimes to conservative solutions because of lack of information related to concomitance of sign of responses. Therefore structural solutions using these methods can be overdesigned. Peak Response method yields an interaction ellipsoidal envelop which defines the concomitance domain of a linear combination of two, or more, Normal random variables. This paper aims at presenting the results obtained by applying the Peak Response method to the Finite Element Model for seismic analyses of EPR’s HNX (Nuclear Auxiliary Building). Then steel reinforcement quantities will be compared to those obtained by traditional method (CQC+SRSS) to highlight the differences between both methods.

1. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
Full paper Submission, TINCE-2016
Paris (France), September 5th
– 9th
, 2016
Response Spectrum Design using Peak Response method and Application to the UK-EPR
Jordi Alerany Canut1
, Amanda Parisis2
1
Former intern in Tractebel Engie / Coyne et Bellier, Nuclear and Industry, 5 rue du 19 mars 1962, 92622
Gennevilliers – France (jordi.alerany-canut@tpi.setec.fr)
2
Tractebel Engie / Coyne et Bellier, Nuclear and Industry, 5 rue du 19 mars 1962, 92622 Gennevilliers –
France (amanda.parisis@tractebel.engie.com)
1-Introduction
For several years, civil engineering proposes solutions increasingly safe and at the same
time designed to optimize materials. Currently, seismic design has a major importance due to
the construction of new infrastructures in seismic zones. One of the most common methods to
perform linear seismic analysis is the response spectrum design that allows estimating the seis-
mic forces applied to the structure for each vibration mode. Then each modal peak response is
combined using, for example, the complete quadratic combination (CQC) method that provides
the absolute value for each response. Finally in order to combine the different seismic directions,
most standards propose the SRSS or the Newmark’s method. Nevertheless during the process
there is a lack of information concerning the concomitance of responses and their sign. As a
consequence, it must be assumed that the maximum responses do occur simultaneously with
any possible signs and hence resulting efforts are overestimated which leads to overdesign
structures. The simultaneous peak responses’ occurrence was studied by A.K.Gupta [GUP90],
L.Leblond [LEB80] and many others, leading to the definition of an interaction ellipsoid envelop.
The Peak Response method yields an interaction ellipsoidal envelop and it’s the basis of the
following article.
In the first part of the paper, a brief introduction to this method is proposed by presenting
the major points of the method and the discretization of an ellipsoid envelop. Thus, this part will
permit to underline the main problems related to the method and the use of shell elements (Nx,
Ny, Nxy, Mx, My, Mxy, Vx, Vy) in FEM. In the second part a solution to those problems is proposed
and applied to a FEM of the EPR’s HNX (Nuclear Auxiliary Building). Then, results obtained by
the traditional method (CQC+SRSS) are compared to those obtained by Peak Response meth-
od in terms of peak and total steel reinforcement. The final part will concern convex hulls, a new

2. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
to 9th
September
approach aiming to reduce the number of load cases that should be considered when combining
results from Peak Response method and pseudo-static loads and/or concomitant static or pseu-
do-static loads.
2-Main features of the Peak Response method
The Peak Response method is widely explained in [LEB80], [NGU12] and [ALE15] thus
the following is simply a reminder of method’s major points.
In linear seismic analyses it is usual to consider that seismic action can be likened to a
normal random variable. Then when finding the maximum value “P” of a linear combination of 2
normal random variables (NRV) “x” and “y” (with Xi and Yi the elementary maximum responses
of NRV xi and yi, with  ixx and  iyy so  jiij XXX  and  jiij YYY 
), Eqs (2.1a) and (2.1b) can be deduced:
1 yx   (2.1a)
21 2222
YYXX
i
iiij    (2.1b)
Eqs (2.1a) and (2.1b) are in fact the tangential equations to the ellipse of Eq(2.2):
12
2
222
2
222 














YX
YX
Y
y
yx
YX
YX
X
x jiijjiij 
(2.2)
Therefore the concomitant peak values describe an interaction ellipse in 2D (Eq(2.2)) that
will be considered as an interaction ellipsoid in N dimensions (N > 2). This elliptical domain may
be enveloped by an order 2 polyhedron defined by p x 2p
points when combining a set of “p”
parameters. This discretization can be carried out using an “α” coefficient
Figure 1. Discretization of the elliptical domain for p=2, 8 points (η, ξ: local axis) (a-b) and for
p=3 , 24 points (1/8 of the polyhedron) (c)
(c)(a) (b)

3. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
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September
The value of “α” is always 12  for an order 2 polyhedron and it corresponds to the loca-
tion of a vertex referring to semi-major axes (λ1, λ2) of the interaction ellipse (in local axes). The
following table presents the total number of points needed to define the polyhedron envelope
depending on the number of parameters:
Table 1: Number of vertices depending on the number of parameters.
Number of parameters p 2 3 4 5 6 7 8 9 10
Number of vertices p.2p
8 24 64 160 384 896 2048 4608 10240
By applying Peak Response method to shell elements with 8 efforts (Nx, Ny, Nxy, Mx, My,
Mxy, Vx, Vy) the total number of points is 2048 which is large. Consequently these seismic load
cases must be combined with static and pseudo-static concomitant load cases, so the total
amount of load cases will easily reach more than 200.000. This number is not negligible and
should be determined for over 80.000 elements (in Nuclear Plants’ FEM).
The first way to reduce load cases is to divide the 8 efforts in two groups, the first one
(Nx,Ny,Nxy,Mx,My,Mxy) will lead to the determination of the longitudinal steel reinforcement and the
second one (Nx,Ny,Nxy,Vx,Vy) will lead to the determination of the shear steel reinforcement.
Consequently the total number of points is now 384 + 160 = 544 that is quite lower than 2048.
The steel reinforcement design for EPR’s HNX (Nuclear Auxiliary Building) performed in (3), will
take into account this division.
3-Steel reinforcement design of EPR’s HNX
The Nuclear Auxiliary Building is a reinforced concrete building 32m large, 36,5m long and
46,7m high (Fig.2). The Finite Element Model used in this part is a preliminary version of EPR’s
HNX (Nuclear Auxiliary Building). This FEM is made using ANSYS and contains mainly “Sol-
id45”, “Shell43” and “Mass21” elements, the number of shell elements is 14352. In order to sim-
plify the structure, all nodes on the base have been embedded (these nodes should be linked to
springs for modeling soil-structure interaction for a design calculation).
Steel reinforcement is determined using the traditional method CQC+SRSS and the Peak
Response method. Results taken into account are mainly peak steel reinforcements in both di-
rections and on both faces, shear reinforcements and the theoretical steel mass that should be
implemented for the load case combination analyzed.

4. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
to 9th
September
Figure 2. Top view of the HNX Building (a) and 3D view of the FEM (b).
Loads applied to the model are the self-weight (G) using a density of 2500kg/m3
and the
response spectra presented in Fig.3 considering a damping ratio of 7% for concrete parts. Note
that in this case, response spectra have been established using Eurocode 8 (FR version) even
though Nuclear Plants should be calculated using EUR’s response spectra. Therefore the load
combination is:
EdAGaULS :. (3.1)
Steel reinforcement design is based on ETC-C AFCEN standards and it’s performed us-
ing Tractebel Engie’s internal program “Ferrail” which applies Capra-Maury method for shell’s
steel reinforcement. Concrete characteristics used for design are, fck=30MPa, γc=1,2 ; αcc=1;
steel characteristics are, fyk=500MPa, γs=1,0 ;.
Figure 3. Response spectra (EC-8; zone 3; soil class C; St=1,0; ξ=0,07).
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
0 0.5 1 1.5 2 2.5 3 3.5 4
Se(m/s²)
T(s)
Response Spectra
Horizontal spectrum
Vertical spectrum
(a) (b)

5. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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to 9th
September
 Results :
As mentionnedd, the following results correspond to longitudinal reinforcement in both di-
rections and shear reinforcement calculated applying a traditional method (CQC+SRSS) and
Peak Response method. To facilitate the understanding of the results, Fig.4 presents axes crite-
ria for steel reinforcement design. Note that steel reinforcement is determined using local axes of
shell elements.
Figure 4. Local and global axes for shell elements (a); upper and lower face depending on local
axes for shell elements (b).
(a) (b)
A1X
CQC
(cm²/m)
A1X
Peak
Response
(cm²/m)
A2X
Peak
Response
(cm²/m)
A2X
CQC
(cm²/m)

6. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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Figure 5. Steel reinforcement depending on local axes and on shell faces (CQC vs Peak Re-
sponse).
It should be emphasized that steel reinforcements presented above correspond to theo-
retical quantities, no minimal steel reinforcement quantity has been considered and no smooth-
ing or actual installed reinforcement analysis have been performed. Then using these results,
AT
Peak
Response
(cm²/m²)
A1Y
CQC
(cm²/m)
A2Y
CQC
(cm²/m)
AT
CQC
(cm²/m²)
A1Y
Peak
Response
(cm²/m)
A2Y
Peak
Response
(cm²/m)

7. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
to 9th
September
the theoretical total amount of steel reinforcement has been determined. Fig.6 shows the differ-
ence between peak reinforcements (in cm²/m & cm²/m²) and total amount of steel (in tons).
Figure 6. Comparison of peak steel reinforcement results (a) and comparison of total mass of steel
(right) (CQC vs Peak Response).
As shown in Fig.6, Peak Response method provides a lower quantity of steel reinforce-
ment than CQC+SRSS method. In this case, peak reinforcements decreased on average 22%
while the total mass of steel is reduced by 24% using Peak Response method. However this
reduction depends on the structure and on the seismic excitation hence it is not possible to an-
ticipate the reduction percentage. The results also will be mitigated by smoothing and installed
reinforcement analysis when exploiting the reinforcement charts.
4-Convex hull approach
Usually, in linear seismic analysis some loads due to seismic excitation (hydrodynamic
pressure, dynamic earth pressure) are treated apart from dynamic analysis. These also called
pseudo-static loads are applied to FEMs as a static load for each earthquake’s direction and
then can be combined with Newmark method resulting on 24 load combinations. Afterwards
pseudo-static efforts are combined with seismic analysis efforts.
Furthermore the Peak Response results must be combined with those 24 pseudo-static
load cases so the amount of load cases to be post-processed increases dramatically, see Tab.2.
Table 2: Load cases to be considered after combining Peak Response and pseudo-static results
Parameters 2 3 4 5 6 7 8
Load cases (Peak Response) 8 24 64 160 384 896 2048
Load cases (Peak Response +Pseudo-static) 192 576 1536 3840 9216 21504 49152
0
20
40
60
80
100
120
A1x (cm²/ml) A1y (cm²/ml) A2x (cm²/ml) A2y (cm²/ml) At (cm²/m²)
Peak reinforcement ULS-a (CQC vs Peak response)
-21% -34% -15% -28%
-12%
0
100
200
300
400
500
600
700
Mtot,As (t)
Total Mass of steel reinforcement ULS-a (CQC vs
Peak response)
CQC
Peak response
-24%
(a) (b)

8. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
to 9th
September
In addition, if variable load cases are considered, the total amount of load cases will easi-
ly reach more than 200.000. It’s therefore necessary to apply a criterion in order to reduce the
number of load cases while saving those that are dimensioning. The new approach proposed in
this paper is the convex hull (see Fig.7).
Figure 7. Convex hull applied to 384 load cases for a couple of forces (F5, F6)
The convex hull approach is currently used in many field such as architectural, medical,
and entertainment. The first algorithms appeared in the early 1970s and some permit to create
convex envelopes in 3D, nevertheless the following concerns only the 2D approach. The princi-
ple of this method is quite simple: it consists in choosing points that form a convex envelope
according to a couple of axes. Since shell efforts are processed in groups of 6 or 5 efforts, see
(2), it’s necessary to repeat this envelope according to several couple of efforts. The couple of
axes proposed for applying the convex hull are, MX-NX ; MY-NY ; MXY-MX ; NX-NXY ; NXY-MXY for 6
efforts’ group (used for longitudinal reinforcement determination) and NX-VX ; NY-VY ; VX-VY for 5
efforts’ group (used for shear reinforcement determination).
To show how accurate can this approach be, it will be applied to EPR’s HNX model ana-
lyzed in (3). Given that peak reinforcements and total mass of steel have already been calculat-
ed using all the load cases, a comparison of these results with those obtained by the convex hull
approach shall be carried out.
 Results :
Once applied convex hull Tractebel Engie’s internal program “Hybrid”, only 101 load cases
are retained from 384 for 6 efforts’ group and 59 out of 160 for 5 efforts’ group. Note that is not
possible to anticipate the final number of retained load cases. Reinforcement axes definition are
presented in Fig.4.

9. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
to 9th
September
A1X
Peak
Response
(cm²/m)
A2X
Peak
Response
(cm²/m)
A1X
Peak
Response
+
Conv.Hull
(cm²/m)
A2X
Peak
Response
+
Conv.Hull
(cm²/m)
A1Y
Peak
Response
(cm²/m)
A1Y
Peak
Response
+
Conv.Hull
(cm²/m)
A2Y
Peak
Response
(cm²/m)
A2Y
Peak
Response
+
Conv.Hull
(cm²/m)

10. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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to 9th
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Figure 8. Steel reinforcement depending on local axes and on shell faces (Peak Response vs
Peak Response +Conv.hull).
As said, the results above are calculated using the internal program “Ferrail” thus these
steel reinforcement values are theoretical. After comparing the results obtained by post-
processing all load cases (384 for As,long/transv and 160 for As,Shear) and those obtained applying
convex hull (101 load cases for As,long/transv and 59 load cases for As,Shear), we can observe that
both are almost identical. Peak reinforcement values are the same in both sides so to realize
what the differences are, the total mass of steel reinforcement is calculated using the results
shown in Fig.8. A summary of the results is presented in Fig.9, please note that minimal steel
reinforcement has not been considered in this paper.
Figure 9. Comparison of peak steel reinforcement results (a) and comparison of total mass of steel
(b) (Peak Response vs Peak Response +Conv.hull).
AT
Peak
Response
(cm²/m²)
AT
Peak
Response
+
Conv.Hull
(cm²/m²)
0
20
40
60
80
100
120
A1x (cm²/ml) A1y (cm²/ml) A2x (cm²/ml) A2y (cm²/ml) At (cm²/m²)
Peak reinforcement ULS-a (Peak response vs Peak
response+Conv.Hull)
0%
0%
0%
0%
0%
0.0
100.0
200.0
300.0
400.0
500.0
600.0
Mtot,As (t)
Total Mass of steel reinforcement ULS-a (Peak
response vs Peak response+Conv.Hull)
Peak response
Peak response+Conv.Hull
-0,4%
(a) (b)

11. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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Notice that convex hull provides the dimensioning load case for most part of elements ex-
cept for some of them. Nevertheless the selected load cases for these elements are closer to the
dimensioning ones so the variation of the total mass of steel reinforcement is just -0.4% (see
Fig.9-b). It must be said that it is not possible to anticipate steel reinforcement variation however
the results of this paper and in [ALE15], show that convex hull approach leads to a variation of
the total mass of steel around -0.5% which is quite low. In other words, convex hull approach
provides dimensioning load cases by the time it reduces considerably the number of load cases
that must be post-processed. For other examples of this approach, see [ALE15].
Another criterion used to choose dimensioning load cases is the max-min approach. In this ap-
proach, for shell elements, there are 26 load cases that are selected to calculate the steel rein-
forcement. Load cases are selected using the following criteria, the maximum and minimum val-
ue of NX, NY, NXY, MX, MY, MXY, VX, VY, NP, MP, VP, with:
    4/2/ 22
P XYYXYX NNNNNN  ,     4/2/ 22
P XYYXYX MMMMMM 
and 22
P YX VVV  . Then the last criteria are σ(+)max, σ(+)min, σ(-)max, σ(-)min, with
    5,02/)( 22
YXXYYX   where σX, σY, σXY, are stress values calculated on
the upper fiber; σ(-) can be determined as σ(+) but with σX, σY, σXY, as stress values calculated
on the lower fiber. The following presents a comparison of the total mass of steel reinforcement
obtained by applying the convex hull and the max-min approach. Then we have estimated the
amount of elements for which the dimensioning load case is retained.
Figure 10. Comparison of total mass of steel (a) and Comparison of dimensioning load cases
retained (b) (Peak Response vs Peak Response +Conv.hull vs Max-Min).
0
2000
4000
6000
8000
10000
12000
14000
16000
Dimensioning Load cases
retained for long-transv.
reinforcement
Dimensioning Load cases
retained for shear.
reinforcement
Dimensioning Load cases retained for long-transv. and shear
reinforcements
ULS Peak response
ULS Peak response Conv. Hull
ULS Peak response MAX-MIN
62%
100%
6,5%
100% 100% 97%
0.0
100.0
200.0
300.0
400.0
500.0
600.0
Mtot,As (t)
Total Mass of steel reinforcement ULS-a
100% 99,6% 95,8%
(a) (b)

12. 3rd
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As shown in Fig.10, even if the number of load cases retained using max-min approach is
very low (6,5%, Fig.10-b) the value of steel reinforcement calculated is very similar to the dimen-
sioning load case (95,8%, Fig.10-a). Anyway, for this example, the convex hull approach is more
precise than the max-min approach even if both provide accurate results.
Conclusions
In this paper CQC+SRSS and Peak Response methods are compared by analyzing the
reinforced concrete HNX building. According to the results presented, the application of Peak
Response method in seismic linear analysis using response spectra can lead to an important
reduction of steel reinforcement in concrete shell elements. However this reduction depends on
the structure and the seismic excitation thus general reduction factors cannot be proposed be-
forehand. The main issue of this method concerns the huge number of seismic load cases re-
sulting from Peak Response method. In this paper, a way to overcome this problem has been
presented by using convex hull, a novel approach in civil engineering that can be applied for
static and dynamic analysis. In the last part of this paper, convex hull is used to select the load
cases that will provide a very similar steel reinforcement amount of the dimensioning ones.
References
[LEB80] Leblond, L. (1980), “Calcul sismique par la méthode modale. Utilisation des réponses
pour le dimensionnement”, Théories et méthodes de calcul N°380, 119-127 (In French).
[GUP90] Gupta, A.K. (1990), “Response Spectrum Method In Seismic Analysis and Design
Structures”, Blackwell Scientific Publication, 1990,Chapter 4, 68-82.
[NGU12] Nguyen, Q.S. and Erlicher, S. and Martin, F. (2012), “Comparison of several variants
of the response spectrum method and definition of equivalent static loads from peak re-
sponse envelopes”, 15th World Conference on Earthquake Engineering, Lisbon, Portugal,
2012, 10p.
[SMI03] Smid, M. (2003), “Computing the convex hull of a planar point set”, Lecture notes, Car-
leton University, Ottawa (Canada), 33p.
[ALE15] Alerany Canut, J., Parisis, A. and Pecker, A. (2015), “Dimensionnement sismique des
structures par la méthode des Ellipses. Application à l’Ilot Nucléaire EPR-UK”, Final
Project, Ecole des Ponts ParisTech at Paris, France (In French), 108p.

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Preference: � Poster  Oral
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Corresponding author: Jordi.alerany-canut@tpi.setec.fr