BARC/2003/E/023

GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION

BASE ISOLATION STRATEGIES FOR STRUCTURES AND COMPONENTS

byVeto Varma, G. R. Reddy, K. K. Vaze and H. S. Kushwaha

Reactor Safety Division

BHABHA ATOMIC RESEARCH CENTREMUMBAI, INDIA

2003

BARC

/200

3/E/

023

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/2003/E/023

07 Part No. or Volume No. :

08 Contract No. :

10 Title and subtitle : Base isolation strategies for structures and components

11 Collation : 36 p., 21 figs., 4 tabs., 2 ills.

13 Project No. :

20 Personal author(s) : Veto Varma; G.R. Reddy; K.K. Vaze; H.S. Kushwaha

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/2003/E/023

BARC/2003/E/023

30 Date of submission : July 2003

31 Publication/Issue date : August 2003

40 Publisher/Distributor : Head, Library and Information Services 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 : 17 refs.

53 Gives data on :

60

70 Keywords/Descriptors : PHWR TYPE REACTORS; FINITE ELEMENT METHOD; RUBBERS;EARTHQUAKES; CONTAINMENT BUILDINGS; VALIDATION; STRESSES; STRAINS;GROUND MOTION; NUCLEAR POWER PLANTS

71 INIS Subject Category : S21

99 Supplementary elements :

-ii-

Abstract : In the present report the effect of Laminated Rubber Bearing (LRB) system on thedynamic response of the structure was studied. A LRB system was designed and tested in thelaboratory for its dynamic characteristics. Finite element analysis was also performed and basedon this analysis, isolator for PHWR nuclear power plant was designed. Analysis of the buildingwas performed with and without isolator. Comparison of responses was made in terms of

frequencies, accelerations, and displacements and floor response spectra.

Base Isolation Strategies For Structures

And Components

Veto Varma

G R Reddy

K K Vaze

H S Kushwaha

Reactor Safety Division

Bhabha Atomic Research Centre

i

Abstract

In the present report the effect of Laminated Rubber Bearing (LRB)

system on the dynamic response of the structure was studied. A LRB

system was designed and tested in the laboratory for its dynamic

characteristics. Finite element analysis was also performed and based on

this analysis, isolator for PHWR nuclear power plant was designed.

Analysis of the building was performed with and without isolator.

Comparison of responses was made in terms of frequencies,

accelerations, and displacements and floor response spectra.

ii

Contents

S.N. Title Page No.

Abstract i

Contents ii

List of Figures iii

Notations v

1. Introduction 1

2. Design of LRB System For Laboratory Testing 3

3. Analytical Studies 5

4. LRB System For A PHWR Building 7

5. Analysis of PHWR Building 10

6. Results And Discussions 13

7. References 16

Appendix A 18

Appendix B 20

iii

List of Figures

S.N. Title Page no.

1. Laminated Rubber Bearing (Test Model) 3

2. One story Steel frame supported on LRB system 4

3. Finite element model of LRB test model 5

4. Deflected shape of LRB Test Model under

vertical and Horizontal Load

5

5. Comparison of analytical and experimental

deformation of

LRB (Test Model)

6

6. Load deformation behavior of LRB (test model)

under horizontal and vertical load obtained

analytically

6

7. FE implemented curve-fitting procedure 8

8. Stress Strain Behaviour of rubber 8

9. Stress against stretch ratio 11

10. 500 tones capacity Laminated Rubber bearing 12

11. Load deformation behaviour of Prototype LRB

under vertical and horizontal Load

13

12. Finite Element Model of Prototype LRB

for PHWR building

14

13. Deformed shape of prototype LRB 14

14. Response spectra for 5% damping 15

15. A schematic Diagram of PHWR Building with

LRB system

16

16. Beam Model of PHWR Building 16

17. Deflection vs Time of LRB under reactor

building

17

18. Comparison of response spectra at various

location in Reactor building with and without

LRB

19

iv

S.N. Title Page no.

19. Comparison Of Acceleration History At the Top

Node of Reactor Building

20

20. Comparison of Displacement History at the Top

node of Reactor Building

20

21. Comparison of Acceleration History at Node 21 20

v

Notations

Ac Area under compression

Af Force free surface area

As Area under shear

∇a Allowed range of horizontal deformation

c Damping

d Density

D Diameter

Eb1

Apparent compression modulus

for bending without considering

the bulk Modulus

Eb Apparent compression modulus

for bending considering the bulk

modulus

Ec’ Apparent modulus of elasticity

For compression without

considering the bulk Modulus

Eϕ Bulk modulus of elasticity

G Shear Modulus

Kc Vertical stiffness of the bearing

Kc’ Vertical stiffness of individual

layer of rubber

Kf Form factor

Kh Horizontal stiffness of the bearing

Km Multiplication factor

l Total thickness of rubber bearing

n number of rubber layers

P Compressive Load

Pcr Critical buckling load

S Shape factor

Sb Bending stiffness of bearing

SS Shear stiffness of bearing

vi

t Thickness of rubber

tr Thickness of individual rubber

layer

ts Steel plate thickness

x Numerical factor depending on

amount of carbon black filler

Xc Displacement

Pc Critical buckling load

S.F. Safety Factor

σc Stress

ε Strain

εb Elongation at break

1

1. INTRODUCTION

Excitation of structures by earthquake ground motions includes inertia forces, the

intensity of which depends on the dynamic properties of the soil structure system and

characteristics of ground excitations. There is basically two design approaches for

reducing earthquake damage to structures: Seismic-Design (Earthquake resistance based)

and Aseismic-Design (Earthquake isolation based) approaches.

One of the effective control methods of seismic vibration is to modify the vibration

transmission path from source to structure. For the above purpose various methods have

been introduced for control of seismic vibrations. The four major classes of control

systems are passive, active, semi active and hybrid. A passive control system may be

defined as a system which does not require an external power source for operation and

utilizes the notion of the structure to develop the control forces. Seismic isolation system

represents one of the form of passive control systems.

In the traditional Seismic-Design approach, both strength and energy absorbing capacity

of the structure is generally increased. If earthquake forces are high, energy is absorbed

by inelastic deformations. In earthquake isolation based aeismic approach, a

comparatively novel approach, the structure is essentially de-coupled from earthquake

ground motions by providing separate isolation devices between the base of the structure

and its foundation as shown in the following figure.

Behavior of Base isolated and fixed base building under earthquake loading

2

There are currently many types of seismic base isolation systems in practice e.g.

Laminated rubber bearing, Lead rubber bearing (New Zealand Type), friction bearings

etc. Derham and Kelly (1985) have shown that a building on rubber bearing will be

simultaneously protected from unwanted vibration and from earthquake excitation. The

main purpose of base isolation devices is to attenuate the horizontal acceleration

transmitted to the superstructure. All the base isolation systems have certain features in

common; the most important of one is the flexibility and energy dissipation capability.

The main concept of base isolation is shifting the fundamental period of the structure out

of the range of dominant earthquake energy and increasing energy absorbing capability.

The first mode of isolated structure involves deformation only in the isolation system and

the superstructure above remains almost rigid. Thus, the high energy in the ground

motion at the higher mode frequencies is deflected. In this way, the isolation becomes a

very attractive approach where protection of expensive sensitive equipment and internal

non-structural component is needed. The structural design, therefore, can be done

independent of the design basis earthquake (DBE) that may occur during its lifetime.

Base isolated building has fundamental frequency lower than both its fixed base

frequency and the dominant frequencies of ground motion. The structure designed using

base isolation is likely to see damage only in the isolation system, which can be replaced.

Base isolation is therefore recommended for life saving and life threatening structures

such as hospitals, nuclear power plants and in the overall sense it can be more economical

than seismic design.

Laminated Rubber Bearings is most commonly used base isolation system and consists of

alternating layers of rubber and steel with rubber being vulcanized to the steel plates. The

dominant feature of this system is the parallel action of springs and dashpots. Generally,

an LRB system exhibits high damping capacity, horizontal flexibility and high vertical

stiffness. The seismic response of isolated structures in the horizontal direction is

strongly influenced by low frequency components (about 0.5 Hz) of earthquake waves.

Thus, the LRB is required to have a large horizontal displacement capacity.

3

2. DESIGN OF LRB SYSTEM FOR LABORATORY TESTING

The isolator system designed for the present study is shown in the Fig 1. The important

points considered while designing the LRB system are given below

1. The Height to width ratio of the LRB system ≤1

2. Rubber thickness between two plates < 2 cms.

3. Frequency of isolator considering structure as rigid ≈ 0.5 Hz. (Due to limitation in

testing it is considered approximately 5 Hz.)

4. Maximum vertical load to be supported ≈ 100 kg. (Limitation due to testing facility).

The various properties of the isolator are listed in table1. The properties are calculated

according to formulation given in Appendix A. This test LRB’s are manufactured and

tested on one story frame as shown in Fig.2 .The frequency of superstructure without

isolator was 20 Hz which came down to 6.59 Hz after introducing LRB isolation. The

sine loading of 0.2g amplitude is applied on the model. The acceleration experienced

has come down from 5.8g to 0.2g because of LRB. This shows the effectiveness of

the isolator.

110

50

30 60

61.5 36

12

All dimensions are in mm.Fig 1. Laminated Rubber Bearing (Test Model)

4

Table 1. Properties of Isolator For Laboratory Testing

Horizontal frequency 6.26 Hz.

Vertical frequency 24.80 Hz.

Rocking Frequency 618.67 Hz.

Torsional Frequency 6.2 Hz.

Horizontal Stiffness 40.02 Kgf/cms

Vertical Stiffness 618.67 Kgf/cms

mb = 52 Kg

LRB

m = 52 Kg

Fig 2. One story Steel frame supported on LRB system

5

3. ANALYTICAL STUDIES

A Finite element model of the laminated rubber bearing system is prepared as shown in

Fig 3. The Steel plate is modeled as linear system while the rubber layers are modeled as

non-linear elements. Model consists of eight-nodded plain strain element. The Mooney-

Rivlin model is used to model the rubber behavior. The details of Monney Rivlin model

and various constants are given in Appendix B.

Deflected shape of LRB is shown in Fig 4. The Horizontal load deflection curve of the

LRB is obtained which is compared with the test data as shown in Fig 5 and 6. The

testing has shown a 5% damping of the isolator (Ref 11).

In Finite Element analysis rubber is generally described as hyper elastic material; this

means that there is a strain energy density function, which when differentiated with

respect to a strain measure yields the corresponding stress measure. The mechanical

response of a material is defined by giving the required parameters in the chosen strain

Fig 3. Finite element model of LRB test model

Fig 4 . Deflected shape of LRB Test Model under vertical and Horizontal Load

6

energy potential. A least squares fit to the data is then performed to calculate the material

parameters.

Fig 5. Comparison of analytical and experimental deformation of LRB (Test Model)

0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200

220

240

Analytical Data

Test Data

Load

in K

g

Displacements in mms

Fig 6. Load deformation behavior of LRB (test model) under horizontal and vertical load obtained analytically

0 50 100 150 200 250

0

2000

4000

6000

8000

10000

12000

14000

16000

Load

in K

g

Displacements in mms

7

4.0 LRB system for a PHWR building

The LRB system designed for PHWR building is shown in Fig 10. This LRB has the

capacity of 500 tonnes. The various properties of this LRB are listed in Table 3. The

properties of rubber used in calculations (Appendix A) are given in Table 4. Finite

element model of the prototype Laminated Rubber bearing is prepared. Eight nodded

plain strain elements are used for the modeling. The Mooney-Rivlin model (Appendix B)

is used to simulate the rubber behavior.

Fig 10. 500 tones capacity Laminated Rubber bearing

1500 1400

243.2

Rubber 8 mm x 22 Steel 3.2 mm x 21 All dimensions are in mm

323.2

8

Table 3: Properties of the Laminated Rubber Bearing Used

Rated Vertical Load (MN) 4.9

Horizontal Natural Frequency (Hz) 0.5

Vertical Natural Frequency (Hz) 19.8

Horizontal Stiffness (MN/m) 4.8

Vertical Stiffness (MN/m) 7700

Allowed range of horizontal deformation (m) 0.50

Fig. 11. Load deformation behaviour of Prototype LRB under vertical and

horizontal Load

0 10 20 30 40 50 60 70 80 90 100-50

0

50

100

150

200

250

300

350

400

450

500

550

Formulae Calculation

FE Analysis

Load

in T

onne

s

Displacements in cms

9

Table 4: Properties of the Rubber Used G (MN/m2) 0.58

EΨ (MN/m2) 2030

εb 6

Hardness (IRHD) 40

x 0.85

Fig 12 shows the FE model of the prototype LRB. The deformed shape of the prototype

LRB is shown in Fig. 13. The load deflection characteristic of the LRB is obtained as

shown in Fig. 11.

Fig 12. Finite Element Model of Prototype LRB for PHWR building

Fig. 13. Deformed shape of prototype LRB

10

5.0 Analysis of PHWR Building

The Reactor building consists of four substructures namely outer containment wall

(OCW), Inner containment wall (ICW), Internal structure (IS) and calandria vault (CV).

The OCW, ICW and IS are supported on a common foundation raft. The CV structure is

connected to the internal structure at 93 m. elevation. The outer containment wall

consists of an RCC cylindrical wall. It is capped by torispherical dome in RCC. The

substructure is concentric with the raft in Plan. The inner containment wall and the dome

are of pre-stressed concrete cylindrical wall. It is capped by torispherical dome of pre-

stressed concrete. The internal structure consists of RCC slabs; shear walls,

miscellaneous panel walls and pre-stressed pressure walls etc. The structure centerline is

concentric with the raft in plan. All the systems and components in the reactor are

mounted on the IS. The calandria vault is a rectangular box of RCC. It is connected to the

internal structure. The foundation raft is massive pedestal, circular in plan and is founded

on rock. In the mathematical model each of these substructure is represented by an

0 5 10 15 20 25 30 35 400.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

Acce

lerati

on in

g

Frequency In Hz

Fig. 16 Response spectra for 5% damping

A- Acceleration of structure without Base isolation B- Acceleration of structure with isolator

A

B

11

equivalent stick formed by beam element. Each of these sticks represents OCW, IS, CV

where the top of IS is connected horizontally by a steel link to the ICW at that level. It is

free in vertical direction. This model is used for the present study. A study is made on the

responses of the structure. Based on total weight of reactor building 300 numbers of

isolation were used for the analysis. Five percent damping is used in the analysis. The

response spectra is generated at different levels and compared with the structure.

Analysis shows that the static deflection is more then the total deflection due to seismic

rotation and vertical deflection hence there is no loss of contact.

Fig 17. Deflection vs Time of LRB under reactor building

0 5 10 15 20 25 30 35 40

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

Defle

ction

in m

.

Time in Seconds

12

Fig 15. Beam Model of PHWR Building

Fig 14. A schematic Diagram of PHWR Building with LRB system

13

6.0 Results And Discussions

1. Experimental study was made on a LRB isolator system designed for a low load

capacity. It was found that within the testing limits the response of the isolator was linear.

2. Analytical calculations were made using Finite Element technique to study the

deformation behavior of the isolator. It is found that the analytical results are matching

the experimental results. It was found that up to about 200% of the shear strain the

behavior is linear.

3. With the above experience prototype isolator of 500 tones capacity is designed given

in appendix A. This LRB isolator is used to analytically study the response behavior of

the PHWR building. Analysis was performed in the Linear range as it is clear from the

Fig. 11 that under the vertical load present the behaviour of the isolator is linear.

4. The deflection behavior of the isolator under the Reactor building is shown in Fig 17.

It is clear that the maximum deflection is within the linear range of isolator as shown in

Fig 11. The shear deformation is approximately 40%.

5. A response spectrum at various locations in the Reactor Building is generated. It is

clear from the Fig. 18, that the acceleration experienced by the structure at different

frequencies is greatly reduced.

6. From Figure 19 –21 it can be observed that isolator has filtered out many frequencies

passing to the structure. The acceleration experienced by the structure is reduced while

the displacements have increased in the form of rigid body motion of the superstructure.

14

Fig 18 Comparison of response spectra at various location in Reactor building with and without LRB

-5 0 5 10 15 20 25 30 35-505

101520253035404550

Response with LRB

Response without LRB

NODE 60Connecting node of SGand IS

Acc

eler

atio

n in

m/s

ec2

Frequency in Hz-5 0 5 10 15 20 25 30 35

0

5

10

15

20

25

30

Response without LRB

Response with LRB

NODE 37Connecting node of pump and IS

Acc

eler

atio

n in

m/s

ec2

Frequency in Hz

0 5 10 15 20 25 30 35

0

510

1520

2530

3540

45

50

Response without LRB

Response with LRB

NODE 1Top of OCW

Acc

eler

atio

n in

m/s

ec2

Frequency in Hz

0 5 10 15 20 25 30

02468

10121416182022

Respose with LRB

Response without LRB

Node21Connecting node of CV and IS

Acc

eler

atio

n In

m/s

ec2

Frequency In Hz

15

Without Isolator With Isolator

Fig 19. Comparison Of Acceleration History At the Top Node of Reactor Building

0 5 10 15 20 25 30 35 40-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Acc

eler

atio

n in

m/se

c2

Time in Second0 5 10 15 20 25 30 35 40

-8

-6

-4

-2

0

2

4

6

8

Acc

eler

atio

n in

m/se

c2

Time in Second

Fig 20. Comparison of Displacement History at the Top node of Reactor Building

0 5 10 15 20 25 30 35 40

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Disp

lace

men

ts in

met

ers

Time in Second0 5 10 15 20 25 30 35 40

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008D

ispla

cem

ents

in m

eter

s

Time in Second

Fig 21. Comparison of Acceleration History at Node 21

0 5 10 15 20 25 30 35 40

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Acc

eler

atio

n in

m/se

c2

Time in Second0 5 10 15 20 25 30 35 40

-8

-6

-4

-2

0

2

4

6

8

Acc

eler

atio

n in

m/se

c2

Time in Second

16

REFERENCES 1. B.C. Lin et. al, “Response of base isolated buildings to random excitation described

by the Clough- Penzien spectral model”, Earthquake engineering and structural

dynamics, vol.18,49-62,1989.

2. Christos J. Younis et al. , “Response of sliding rigid structures to base excitation”,

Journal of engineering mechanics, vol. 110,no. 3, March 1984

3. G.R.Reddy, V. Varma, Y.M.parulekar, K.K.Vaze, H.S.Kushwaha, “Base Isolation

Strategies For Aseismic Design of Structures and Components”, BRNS Project report ,

BARC, Mumbai 1999.

4. J.M.Kelly, Earthquake Resistance Design with Rubber, П edition, Springer-Verlag

London ,1997.

5. E.F.Gobel, A.M. Brichta, Rubber Spring Design, English Translation Butterworth and

Co.; London1974

6. Faruq M.A. Siddiqui et.al., ”Simplified analysis method for multistorey base isolated

structures on viscoelastic halfspace”, Earthquake engineering and structural dynamics,

vol.18,63-77,1989.

7. G.R.Reddy, V. Varma, Y.M.parulekar, K.K.Vaze, H.S.Kushwaha, “Base Isolation

Strategies For Aseismic Design of Structures and Components”, BRNS Project report ,

BARC, Mumbai 1999.

8. J.M.Kelly, Earthquake Resistance Design with Rubber, П edition, Springer-Verlag

London ,1997.

9. Jong She Lee and Jong Who Ho, “ Stability of Rubber bearing For seismic isolation”,

Smirt-15, Seoul, Korea, 1999, pp IX 375.

10. M.Kurihara, K. Nishimoto, M. Shigeta, “A Study on Responses during Large

Deformations in a Seismic Isolation System of Nuclear Island Buildings”, JSME

International Journal, Series Ш, vol..33, no.3, (1990)

11. R.S. Jangid, T.K. Dutta, “Seismic Behaviour of Base isolated Buildings; a state-of-

the-art review”, Proc. of Institution of Civil. Engineers. Structures and Buildings,

1995,110,May,186-203

12. R.S. Jangid, “Base isolation strategies for aseismic design of structures and

components”, Project report for TPDM 2000, BARC, January 2000.

17

13. R.W.Clough, J.Penzien, Dynamics of Structures, П edition, McGraw-Hill

International Edition, 1993

14. T. Fujita, S. Fujita, S. Tazaki, T. Yoshizawa, S. Suzuki,” Research Development and

Implementation of Rubber Bearings for Seismic Isolation”, JSME International Journal,

Series Ш, vol.33, no.3, (1990).

15. Takafumi Fujita etc., “ Seismic isolation of industrial facilities using Lead-rubber

Bearings”, JSME Series III,vol. 33, no. 3,1990.

16. Leslie R. Treloar, (1974), “The mechanics of rubber elasticity”, J. Polymer Sci.:

Polymer symposium no. 48, 107-123.

17. D Boast and V A Coveney (1999), “Finite element analysis of Elastomers”,

Professional Engineering Publishing Limited.

18

Appendix A

Design Calculations of LRB

Horizontal Stiffness

As = π/4 D 2 = 1.539 m2

I = π/64 D4 = 0.18857 m4

l = n tr+ (n-1)ts = 0.2432 m

S = Ac /Af = 43.75

E b= 3G(1+2xS2/3)= 2222.0525 MN/m2

E b’= E b Eψ/( E b+ Eψ)

= 1060.8445 Mn/m2

Ss = GAs(tr+ts)/tr = 1.2497 MN

Sb= Eb’ I(tr+ ts )/ tr = 280.06 MN-m2

q =√{ P/Sb ( 1+P/Ss)} = 0.2934 m-1

KH = P2/[2qSb Tan(ql/2) – Pl]

= 5.1 MN/m

Horizontal Frequency

Natural Frequency Along horizontal direction = 1/2π (KH g / P)1/2

= 0.51 Hz

Value specified = 0.5 Hz

Rubber Sandwich Calculations

Ec’ = Eψ [1- { Eψ / 12GS2 }1/2]

= 1237.57 MN/m2

Kc’ = Ac Ec

’/tr = 238077.53 MN/m

Kc = Kc’/n = 10821.706 MN/m

Value specified = 7700 MN/m

Xc = Pc/Kc = 4.53 x 10-4 m.

σc = P/Ac = 3.18 MN/m2

ε = {Xc / (22 tr) } 100 = 0.257%

if there is no spacer

19

Kf = Ac/ Af

=1.4

Km = 20

Ec =KmG =11.6 MN/m2

Kc = AEc/t = 102.037 MN/m

Increase in the vertical stiffness due to steel layers =10821.706/102.037

= 106.056

Vertical Frequency

Natural Frequency along vertical direction = 1/2π (Kcg/P) ½

=23 Hz (approximately)

21

Appendix B

Mooney- Rivlin Material Model The Mooney-Rivlin form is a more general form of the of the strain energy potential. It

uses a linear relationship between stress and strain in simple shear.

Assumptions of the Mooney’s Theory:

1. Rubber is incompressible and isotropic in the unstrained state

2. Hooke’s law is obeyed in simple shear or in simple shear superposed in a plane at

right angles to a prior simple extension or uniaxial compression.

Rivlin examined the general form of the strain energy function W for a rubber, which is

isotropic in the unrestrained state, and concluded that it must be expressible in terms of

the three quantities 1I , 2I , 3I termed strain invariants, which are even powered

functions of the principal extension ratios, i.e.

23

22

213

21

23

23

22

22

212

231

212

211

λλλ

λλλλλλ

λλλ

=

++=

++=

III

for an incompressible rubber

13 ≡I

hence

2

32

22

12−−− ++= λλλI

The most general form of strains energy function for an incompressible rubber consistent

with this formulation

22

ji

jiij IICW )3()3( 2

0,01 −−= ∑

∞

==

in which ijC are independent elastic constants.

The most general first order expression in 1I and 2I is

( ) ( )33 2211 −+−= ICICW

Stress-Strain Relations

( )

∂∂+∂

∂−=2

22

1

23

211 2 I

wI

wt λλλ

For a particular case of a simple extension

t2 = t3 = 0

these yields

( )

∂∂+∂

∂−=211

211

112 Iw

Iwt λλλ

which is similar to

( ) ( )21211

112 CCt λλλ +−=

These equations provides basis for the experimental examination of the form of W in a

pure homogeneous strain of the most general type. By stretching a sheet of rubber in two

perpendicular directions, and measuring the principal stresses t1 and t2 corresponding to

given values of 1λ and 21λ the necessary data are obtained to enable these equations to

be solved simultaneously.

( )

∂∂+∂

∂−=2

21

1

23

222 2 I

wI

wt λλλ

23

Fitting Procedures in Finite element analysis

The parameters of material models are usually determined in curve fitting procedures.

The conventional finite element fitting procedure determines all necessary coefficient in

an iterative non-linear fit.

Material Model Selection

Experimental Stress/strain

data

Model evaluation in stress/strain

Iterative Nonlinear fit

Minimization Fitted Parameter

FE implemented curve-fitting procedure

24

Regression Analysis Procedure

The reason for regression analysis is to determine correlation between physical test data

and the material model within a given set of parameters. The function is generally

satisfied if the deviation between test and analysis is within a small limit. It is important

to investigate whether the material model has only one minimum in the least square fit.

Moreover, the quality of fit is no more important than determining the absolute minimum

for the physical condition described.

It is require to fit m data points (yj,exp, x1j, x2j ,x3j ,…..xrj ) with (j=1…n)as measured

independents in a mathematical model that has q adjustable parameters cs with ( s=1…q).

The mathematical model predicts a functional relationship between the measured

independent (x1j,….. xrj) and dependent variable yj,exp with a number of parameters (C1

…. Cq ) independent from each other.

yj,exp = f (x1j,….. xrj, C1 …. Cq)

The guiding function of determining material parameters is based on the least square fit;

the minimization of:

Stress Strain Behaviour of rubber

0

0.015

0.03

0.045

0.06

0.075

0.09

0.105

0.12

0.135

0.15

0.165

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6

Strain

Stress

25

( )∑=

−n

jjcalj yy

1

2exp,,

for finite element curve fit procedures it is assumed that the least squares compeutation is

additionally weighted with a function gj, so that

( )2exp,,1

jcalj

n

jj yyg −∑

=

is minimized. The minimization takes place in varying the value of the parameters to

determine the least square fit.

Verification of model

The uniaxial test data as shown in Fig:2 is used to calculate the stress strain behaviour

with the help of Mooney-Rivlin material model and the same data is fed in to Finite

element software (COSMOS/M) and again the stress strain behaviour is plotted, which is

shown in Fig.9 The constants calculated by finite element software (COSMOS/M) and

By hand calculations are given in the Table 2.

Constants calculated

Constants COSMOS Formulae

(Hand Calculation)

C1 -1.94 x 10-5 -1.281 x 10-5

C2 7.13 x 10-3 6.4423 x 10-3

26

Stretch Ratio

Stress

Stress against stretch ratio