TINCE2016 - Response Spectrum Design using Peak Response method and Application to the UK-EPR - J.A. Canut & A. Parisis

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 ([email protected]) 2 Tractebel Engie / Coyne et Bellier, Nuclear and Industry, 5 rue du 19 mars 1962, 92622 Gennevilliers –

France ([email protected])

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


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 YYXXi

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

yyx

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)



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.



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 spectrumVertical spectrum

(a) (b)



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)



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)



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)



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.



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)



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)



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 casesretained for long-transv.

reinforcement

Dimensioning Load casesretained 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)



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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