Ruaumoko Manual

Induction Book

Ruaumoko Manual

Volume 5: Appendices

Author: Athol J. Carr

Civil Engineering

1

Department of Civil Engineering COMPUTER PROGRAM LIBRARY

Program name:–

RUAUMOKO

Program type:–

In-elastic Time-History Analysis

Program code:–

ANSI Fortran77

Author:–

Athol J Carr

Date:–

November 27, 2008

APPENDICES

for programs

RUAUMOKO2D, RUAUMOKO3D, HYSTERES & INSPECT

-

APPENDIX A - STRENGTH DEGRADATION

-

APPENDIX B - STIFFNESS DEGRADATIONHysteresis Loop data

Copyright \ Athol J. Carr, University of Canterbury, 1981-2007. All Rights reserved.

2

APPENDIX A

DEGRADING STRENGTH parameters (Only if ILOS > 0)

DUCT1 DUCT2 RDUCT DUCT3 RCYC

DUCT1 Ductility at which degradation begins ( > 1.0) F

DUCT2 Ductility at which degradation stops ( > DUCT1) F

RDUCT Residual Strength as a fraction of the Initial Yield Strength F

DUCT3 Ductility at 0.01 initial strength ( blank or > DUCT2) F

RCYC % reduction of strength per cycle of inelastic behaviour (ILOS = 4, 5, 6 or 7 only) F

Notes:

1. ILOS, the parameter that controls the strength degradation (see Properties tables)

ILOS = 0, No Strength Degradation.

ILOS = 1, Strength loss in each direction is a function of the ductility in that direction.

ILOS = 2, Strength loss in each direction is a function of the number of inelastic cycles.

ILOS = 3, Strength loss in each direction is a function of the maximum ductility.

ILOS = 4, As for ILOS = 1 above but strength loss is also proportional to the number of inelastic

cycles.

ILOS = 5, As for ILOS = 4 above but strength loss is also proportional to the number of inelastic cycles

and the strength due to ductility for ductilities greater than DUCT2 remains at the level of

RDUCT until the ductility reaches DUCT3 when the strength suddenly reduces to 1% of the

original strength.

ILOS = 6, As for ILOS = 3 above but strength loss is also proportional to the number of inelastic

cycles.

ILOS = 7 As for ILOS = 6 above but strength loss is also proportional to the number of inelastic cycles

and the strength due to ductility for ductilities greater than DUCT2 remains at the level of

RDUCT until the ductility reaches DUCT3 when the strength suddenly reduces to 1% of the

original strength.

2. If Strength Loss is based on cycle number rather than the ductility then DUCT1 is the cycle number that

the strength starts to reduce and DUCT2 is the cycle number at which the strength reaches the residual

value. It must be noted that the cycle number is computed as the number of times the hysteresis rule

leaves the post-yield back-bone or skeleton curve divided by 2 and this might be greater than the

number of cycles of hysteresis particularly if there has been a one sided ratchet-like behaviour of the

hysteresis. The minimum value permitted for RDUCT is 0.01. If the strength was to reduce to 0.0,

Ruaumoko would then take the member behaviour as elastic which would not be the intention of the

user.

3. If a number is provided for the variable DUCT3 above then the strength decreases linearly from

RDUCT times the initial strength at DUCT2 to 0.01 of the initial strength at ductility (cycle number)

DUCT3. If this number is omitted then the strength remains constant after DUCT2 is reached.

4. See Appendix B for information on which Hysteresis rules are able to accept strength degradation.

5. If ILOS is greater than 0 then as the strength is reduced the stiffness is reduced to match. This means

that the yield displacement, rotation or curvature remains constant as the strength decreases making

the definition of member ductility consistent. If the hysteresis loop being used has other strength

parameters such as a cracking force or moment then these are also reduced proportionally to the yield

strength. If this is not done then some of the hysteresis loops may be impossible to follow where the

yield strength would become less than the cracking strength.

6. If ILOS is supplied as a negative number, i.e. -5, the strength degradation rule would follow that for

ILOS=5 but the stiffness would not be reduced and other hysteresis rule actions would also not be

reduced. This means that the definition of ductility would be difficult to follow as the yield displacement.

rotation or curvature, which is the denominator in the expression for ductility, would decrease as the

strength decreases. Care would also be necessary insetting the levels of strength degradation to

ensure that the hysteresis loop does not become impossible to follow. See the note below.

3

Strength Reduction Variation

In earlier versions of Ruaumoko when the strength degraded the stiffness remained using its input values.

This causes problems with the definition of ductility in that as the yield force reduces and the stiffness

remains constant the yield displacement reduces and therefore for a given member deformation the apparent

ductility increases. This has shown up in that the residual strength, or the 1% strength, is reached at much

smaller displacements, or curvatures, than the user had expected. The program has now been modified

such that as the yield forces, or moments, degrade the stiffness also degrades. This means that the yield

displacements remain constant and the definitions of ductilities remain more consistant. As some hysteresis

rules have other force, or moment quantities such as cracking forces, or intercept forces (see Appendix B),

which can also cause difficulties when the yield strength degrades, such that the yield strength may reduce to

a smaller level than say the cracking moment leading to confusion within the hysteresis rule, such force

quantities are now also degraded as the yield strength degrades. This is more likely to be realistic than the

earlier operation of the strength degradation in that for most member sections the yield point is defined by the

extreme fibre yield strain and given the section properties the yield strain, or curvature, is more likely to

remain constant than is the yield force or moment. There are still some difficulties when there are different

degradations in each direction, ILOS =1, 4 or 5 as the stiffness will vary depending whether the member

displacement is positive or negative.

4

APPENDIX B

STIFFNESS DEGRADATION parameters

Hysteresis Rules for Inelastic Member Behaviour

Each of the rules is designated by the number as shown below.

0 = Elastic (default)

1 = Elasto-plastic

2 = Bi-linear

3 = Ramberg-Osgood

4 = Modified Takeda Degrading Stiffness

5 = Bi-linear with Slackness

6 = Kivell Degrading Stiffness

7 = Origin Centered Bi-linear Hysteresis

8 = SINA Degrading Stiffness

9 = Stewart Degrading Stiffness with Slackness

10 = Bi-linear Degrading Stiffness

11 = Clough Degrading Stiffness

12 = Q-HYST Degrading Stiffness

13 = Muto Tri-linear Degrading Stiffness

14 = Fukada Tri-linear Degrading Stiffness

15 = Bi-linear Elastic

16 = Non-linear Elastic (Un-Reinforced Masonry)

17 = Degrading Elastic

18 = Ring-Spring Isolator or Damper

19 = Hertzian Contact Non-linear Spring

20 = Mehran Keshavarzian's Degrading Stiffness

21 = W idodo Foundation Compliance

22 = Li Xinrong Reinforced Concrete Column Degrading Stiffness

23 = Bouc Degrading Stiffness

24 = Remennikov Out-of-plane Buckling Steel Brace

25 = Takeda with Slip Degrading Stiffness

26 = Al-Bermani Bounding Surface Hysteresis

27 = Peak Oriented Hysteresis

28 = Matsushima Strength Decay model

29 = Kato Shear model

30 = Elastomeric Damper Spring

31 = Composite Section, modified SINA model

32 = Different Stiffness in Positive and Negative directions. Modified Bi-linear rule

33 = Masonry Strut Hysteresis

34 = Hyperbolic Hysteresis

35 = Degrading Bi-linear with Gap Hysteresis

36 = Bi-linear with Differing Positive and Negative Stiffness Hysteresis

37 = Non-linear Elastic Power Rule Hysteresis

38 = Revised Origin Centred Hysteresis

39 = Dodd-Restrepo Steel Hysteresis

40 = Bounded Ramberg-Osgood Hysteresis

41 = Modified (Pyke) Ramberg-Osgood Hysteresis

42 = HERA-SHJ Steel Hinge unit.

43 = Resetting Origin Loop

44 = Pampanin Reinforced Concrete Hinge hysteresis

45 = Degrading Stiffness Ramberg-Osgood Hysteresis

46 = Dean Saunders Reinforced Concrete Column

47 = Multi-linear Elastic

48 = Isotropic/Kinematic Strain Hardening Bi-Linear

49 = Isotropic/Kinematic Strain-Hardening Ramberg-Osgood

50 = Flag-shaped Bi-linear Hysteresis

51 = Two-Four Hysteretic damper

52 = Schoettler-Restrepo Reinforced Concrete Hysteresis

53 = Rajesh Dhakal Steel Hysteresis

54 = Brian peng Concrete Hysteresis

55 = Air-column Damper

56 = Modified SINA hysteresis

5

57 = Revised TAKEDA hysteresis

58 = Shape memory Alloy Flag-shaped hysteresis

59 = Ramberg-Osgood with Alpha hysteresis

60 = IBARRA with pinching hysteresis

61 = IBARRA Peak Oriented hysteresis

62 = IBARRA Bi-linear hysteresis

63 = Bi-linear Elastic with Gap hysteresis

6

Use of Hysteresis Rules for Frame members in RUAUMOKO-2Dand RUAUMOKO-3D

IHYSTHysteresisRule

1cptbeam

R-Ccol.

Steelcol.

Gencol.

2cptbeam

VFlexbeam

0 Elastic Yes Yes Yes Yes Yes Yes

1 Elasto-Plastic Yes Yes Yes Yes Yes No

2 Bi-linear Yes Yes Yes Yes Yes Yes

3 Ramberg-Osgood Yes Yes Yes Yes Yes Yes

4 Takeda Yes Yes* Yes* Yes* Yes Yes

5 Bi-linear - Slackness Yes Yes Yes Yes Yes Yes

6 Kivell Yes Yes Yes Yes Yes Yes

7 Origin-Centered Yes Yes Yes Yes Yes Yes

8 SINA Yes Yes Yes Yes Yes Yes

9 Stewart Yes Yes Yes Yes Yes No

10 Degrading Bi-linear Yes Yes Yes Yes Yes Yes

11 Clough Yes Yes* Yes* Yes* Yes Yes

12 Q-HYST Yes Yes* Yes* Yes* Yes No

13 Muto Yes Yes Yes Yes Yes Yes

14 Fukada Yes Yes Yes Yes Yes Yes

15 Bi-linear Elastic Yes Yes Yes Yes Yes Yes

16 Non-Linear Elastic Yes Yes Yes Yes Yes Yes

17 Degrading Elastic Yes Yes Yes Yes Yes Yes

18 Ring-Spring Yes No No No No No

19 Hertzian Contact No No No No No No

20 Keshavarzian Yes Yes Yes Yes Yes No

21 Widodo Foundation Yes Yes Yes Yes Yes No

22 Li-Xinrong Column No Yes No No No No

23 Bouc Yes No No No Yes Yes

24 Remennikov Yes No Yes No No No

25 Takeda with slip Yes No No No Yes No

26 Al-Bermani Bound-Surface Yes Yes Yes Yes Yes Yes

27 Peak Oriented Yes Yes Yes Yes Yes Yes

28 Matsushima Degrading Yes Yes Yes Yes Yes Yes

29 Kato Degrading Shear Yes Yes No No Yes No

30 Elastomeric Spring No No No No No No

31 Composite Section Yes No No No Yes Yes

32 Different +/- Stiffness Yes No No No Yes Yes

33 Masonry Strut No No No No No No

34 Hyperbolic Yes Yes Yes Yes Yes Yes

35 Degrading Bi-linear Yes No No No Yes No

36 Bi-linear Differing +/- Stiffness Yes Yes Yes Yes Yes Yes

37 Non-linear Elastic Power Yes Yes Yes Yes Yes Yes

38 Revised Origin Centred Yes Yes Yes Yes Yes Yes

39 Dodd-Restrepo Steel Yes Yes Yes Yes Yes Yes

40 Bounded Ramberg-Osgood Yes Yes Yes Yes Yes Yes

41 Pyke Ramberg-Osgood Yes Yes Yes Yes Yes Yes

42 HERA-SHJ Yes No No No Yes No

43 Resetting Origin No No No No No No

7

Use of Hysteresis Rules for Frame members in RUAUMOKO-2Dand RUAUMOKO-3D

IHYSTHysteresisRule

1cptbeam

R-Ccol.

Steelcol.

Gencol.

2cptbeam

VFlexbeam

44 Pampanin Yes Yes Yes Yes Yes Yes

45 Degrading Ramberg-Osgood Yes Yes Yes Yes Yes Yes

46 Dean Saunders Conc. Column Yes Yes Yes Yes Yes Yes

47 Multi-linear Elastic Yes Yes Yes Yes Yes Yes

48 Isotropic Strain Hard. Bi-linear Yes Yes Yes Yes Yes Yes

49 Isotropic Strain Hard. Ramberg Yes Yes Yes Yes Yes Yes

50 Flag-shaped Bi-linear Yes Yes Yes Yes Yes Yes

51 Two-Four Hystertic damper No No No No No No

52 Schoettler-Restrepo Yes No No No Yes No

53 Rajesh Dhakal Steel No No No No No No

54 Brian Peng Concrete No No No No No No

55 Semi-active Air-damper No No No No No No

56 Modified SINA Yes Yes Yes Yes Yes Yes

57 Revised TAKEDA hysteresis Yes Yes Yes Yes Yes Yes

58 Shape Memory Alloy Yes Yes Yes Yes No No

59 Ramberg-Osgood with Alpha Yes Yes Yes Yes Yes Yes

60 IBARRA Pinching Yes Yes Yes Yes Yes Yes

61 IBARRA Peak-oriented Yes Yes Yes Yes Yes Yes

62 IBARRA Bi-linear Yes Yes Yes Yes Yes Yes

63 Bi-linear Elastic with Gap Yes Yes Yes Yes No No

8

Use of Hysteresis Rules for Spring members in RUAUMOKO-2D

IHYST Hysteresis Rule ITYPE = 1,3,4,5,6,7

0 Elastic Yes

1 Elasto-Plastic Yes

2 Bi-linear Yes

3 Ramberg-Osgood Yes

4 Takeda Yes

5 Bi-linear - Slackness Yes

6 Kivell Yes

7 Origin-Centered Yes

8 SINA Yes

9 Stewart Yes

10 Degrading Bi-linear Yes

11 Clough Yes

12 Q-HYST Yes

13 Muto Yes

14 Fukada Yes

15 Bi-linear Elastic Yes

16 Non-Linear Elastic Yes

17 Degrading Elastic Yes

18 Ring-Spring Yes

19 Hertzian Contact Yes

20 Keshavarzian Yes

21 Widodo Foundation Yes

22 Li-Xinrong Column No

23 Bouc Yes

24 Remennikov No

25 Takeda with slip Yes

26 Al-Bermani Bound-Surface Yes

27 Peak Oriented Yes

28 Matsushima Degrading Yes

29 Kato Degrading Shear Yes

30 Elastomeric Spring Yes

31 Composite Section No

32 Different +/- Stiffness Yes

33 Masonry Strut Hysteresis Yes

34 Hyperbolic Hysteresis Yes

35 Degrading Bi-linear Hysteresis Yes

36 Bi-linear Differing +/- Stiffness Yes

37 Non-linear Elastic Power Yes

38 Revised Origin Centred Yes

39 Dodd-Restrepo Steel Yes

40 Bounded Ramberg-Osgood Yes

41 Pyke Ramberg-Osgood Yes

42 HERA-SHJ Yes

43 Resetting Origin Yes

9


IHYST Hysteresis Rule ITYPE = 1,3 or 4

44 Pampanin Yes

45 Degrading Ramberg-Osgood Yes

46 Dean Saunders Concrete Column Yes

47 Multi-linear Elastic Yes

48 Isotropic Strain Hard Bi-linear Yes

49 Isotropic Strain Hard Ramberg Yes

50 Flag-shaped Bi-linear Yes

51 Two-Four Hysteretic Damper Yes

52 Schoettler-Restrepo Yes

53 Rajesh Dhakal Steel Yes

54 Brian Peng Concrete Yes

55 Semi-active Air-damper Yes

56 Modified SINA Yes

57 Revised TAKEDA Hysteresis Yes

58 Shape Memory Alloy Yes

59 Ramberg-Osgood with Alpha Yes

60 IBARRA Pinching Yes

61 IBARRA Peak-oriented Yes

62 IBARRA Bi-linear Yes

63 Bi-linear Elastic with Gap Yes

10



0 Elastic Yes


2 Bi-linear Yes


4 Takeda Yes

5 Bi-linear - Slackness Yes

6 Kivell Yes

7 Origin-Centered Yes

8 SINA Yes

9 Stewart Yes


11 Clough Yes

12 Q-HYST Yes

13 Muto Yes

14 Fukada Yes


16 Non-Linear Elastic Yes

17 Degrading Elastic Yes

18 Ring-Spring Yes

19 Hertzian Contact Yes

20 Keshavarzian Yes



23 Bouc Yes

24 Remennikov No

25 Takeda with slip Yes


27 Peak Oriented Yes


29 Kato Degrading Shear Yes

30 Elastomeric Spring Yes



33 Masonry Strut Hysteresis Yes






39 Dodd-Restrepo Steel Yes



42 HERA-SHJ No

43 Resetting Origin Yes

11



44 Pampanin Yes




48 Isotropic Strain Hard. Bi-lineaqr Yes

49 Isotropic Strain Hard. Ramberg Yes


51 Two-Four Hysteretic Damper Yes

52 Schoettler-Restrepo Yes

53 Rajesh Dhakal Steel Yes

54 Brian Peng Concrete Yes

55 Semi-active Air-damper Yes



58 Shape Memory Alloy Yes






12

Use of Hysteresis Rules for Foundation members in RUAUMOKO-2D and RUAUMOKO-3D

IHYST Hysteresis Rule ITYPE = 1, 2, 3, 4 or 5

0 Elastic Yes


2 Bi-linear Yes


4 Takeda Yes

5 Bi-linear - Slackness No

6 Kivell No

7 Origin-Centered No

8 SINA Yes

9 Stewart No


11 Clough Yes

12 Q-HYST Yes

13 Muto Yes

14 Fukada Yes


16 Non-Linear Elastic No

17 Degrading Elastic No

18 Ring-Spring No

19 Hertzian Contact No

20 Keshavarzian Yes



23 Bouc Yes

24 Remennikov No

25 Takeda with slip No


27 Peak Oriented No


29 Kato Degrading Shear No

30 Elastomeric Spring No



33 Masonry Strut Hysteresis No






39 Dodd-Restrepo Steel No



42 HERA-SHJ No

43 Resetting Origin No

13

Use of Hysteresis Rules for Foundation members in RUAUMOKO-2D and RUAUMOKO-3D

IHYST Hysteresis Rule ITYPE = 1, 2, 3, 4 or 5

44 Pampanin Yes




48 Isotropic Strain Hard. Bi-linear Yes

49 Isotropic Strain Hard. Ramberg Yes


51 Two-Four Hysteretic Damper No

52 Schoettler-Restrepo No

53 Rajesh Dhakal Steel No

54 Brian Peng Concrete No

55 Semi-active Air-damper No



58 Shape Memory Alloy No






14

Use of Strength Degradation and Damage Indices in RUAUMOKO-2D and RUAUMOKO-3D

IHYST Hysteresis Rule Strength Degradation Damage Indices

0 Elastic No No

1 Elasto-Plastic Yes Yes

2 Bi-linear Yes Yes

3 Ramberg-Osgood Yes Yes

4 Takeda Yes Yes

5 Bi-linear - Slackness Yes Yes

6 Kivell Yes Yes

7 Origin-Centered Yes Yes

8 SINA Yes Yes

9 Stewart Yes Yes

10 Degrading Bi-linear Yes Yes

11 Clough Yes Yes

12 Q-HYST Yes Yes

13 Muto Yes Yes

14 Fukada Yes Yes

15 Bi-linear Elastic No No

16 Non-Linear Elastic No No

17 Degrading Elastic No No

18 Ring-Spring No No

19 Hertzian Contact No No

20 Keshavarzian Yes Yes

21 Widodo Foundation No Yes

22 Li-Xinrong Column No Yes

23 Bouc No* Yes

24 Remennikov No Yes

25 Takeda with slip Yes Yes

26 Al-Bermani Bound-Surface Yes Yes

27 Peak Oriented Yes Yes

28 Matsushima Degrading No* Yes

29 Kato Degrading Shear No Yes

30 Elastomeric Spring No No

31 Composite Section Yes Yes

32 Different +/- Stiffness Yes Yes

33 Masonry Strut Hysteresis No Yes

34 Hyperbolic Hysteresis Yes Yes

35 Degrading Bi-linear Hysteresis No No

36 Bi-linear Differing +/- Stiffness Yes Yes

37 Non-linear Elastic Power No No

38 Revised Origin Centred Yes Yes

39 Dodd-Restrepo Steel No Yes

40 Bounded ramberg-Osgood Yes Yes

41 Pyke Ramberg-Osgood Yes Yes

42 HERA-SHJ Yes No

43 Resetting Origin No No

15

Use of Strength Degradation and Damage Indices in RUAUMOKO-2D and RUAUMOKO-3D

IHYST Hysteresis Rule Strength Degradation Damage Indices

44 Pampanin No No

45 Degrading Ramberg-Osgood Yes Yes

46 Dean Saunders Concrete Column Yes Yes

47 Multi-linear Elastic Yes No

48 Isotropic Strain Hard. Bi-linear No Yes

49 Isotropic Strain Hard. Ramberg No Yes

50 Flag-shaped Bi-linear Yes Yes

51 Two-Four Hysteretic Damper No No

52 Schoettler-Restrepo No No

53 Rajesh Dhakal Steel No No

54 Brian Peng Concrete No No

55 Semi-active Air-damper No No

56 Modified SINA Yes Yes

57 Revised TAKEDA Hysteresis Yes Yes

58 Shape Memory Alloy Yes No

59 Ramberg-Osgood with Alpha Yes Yes

60 IBARRA Pinching No Yes

61 IBARRA Peak-oriented No Yes

62 IBARRA Bi-linear No Yes

63 Bi-linear Elastic with Gap No Yes

16

Notes on notation for hysteresis rules:

In all of the diagrams associated with the hysteresis rules the following notation is used.

F is the force or moment in the member.

d is the deformation or curvature in the member.

0K is the initial elastic stiffness, i.e. the EI of a flexural beam or beam column, the AE/L for the axial

stiffness of a beam or beam column, or the stiffness K of a spring member.

yF is the yield force or moment.

r is the bi-linear factor or Ramberg-Osgood factor.

y y y 0: is the ductility where : = d / d where, in general, the yield displacement d = F / K

Some of these rules require further data which is described in the following pages.

Notes on the preceding tables:

If a hysteresis rule is selected and the rule is not allowed for that member then an error message is

printed in the output for the section properties and the analysis will be terminated.

Yes* implies that the hysteresis rule is now allowed for column members. However, the effects on the

small-cycle hysteresis loops of the yield moments varying with the changes in the axial force in the

member have not been studied.

If both Strength Degradation and Damage Indices are selected then the effects of Strength Degradation

on the computed Damage Indices is uncertain and a warning is printed after reading the member

properties.

If Strength Reduction or Damage Indices are not allowed for the specified Hysteresis Rule and they are

specified in the input data, the data is read and then the control parameters ILOS and/or IDAMG are

reset to zero. W arnings of these re-settings are printed in the output.

No*. The Bouc and Matsushima Hysteresis rules have their own strength degradation capability.

Damage indices for the masonry strut hysteresis only outputs the hysteretic work done.

For the Ruaumoko (2D) version Spring member ITYPE=2 is tri-linear hysteresis only. If ITYPE=3 then

the SINA hysteresis is used for the transverse (local y) direction.

For the Ruaumoko (2D) version the Frame member ITYPE=7, the four hinge beam has the allowable

hysteresis table that follow the same rules as that for the variable flexibility beam.

17

Linear Elastic Hysteresis

DEGRADING STIFFNESS parameters

IHYST = 0 Linear Elastic. - No further data required.

18

Elasto-Plastic Hysteresis

IHYST = 1 Elasto-Plastic Hysteresis. - No further data required.

Note: This rule is not available for Variable Flexibility Beam Members.

19

Bi-Linear Hysteresis

IHYST = 2 Bi-Linear Inelastic. - No further data required.

20

Ramberg-Osgood Hysteresis

IHYST = 3 RAMBERG-OSGOOD Hysteresis [Kaldjian 1967] - No further data required.

Note: The bi-linear factor in the section data is used as the Ramberg-Osgood factor r and must be

greater than or equal to 1.0

21

It must be noted that the Ramberg-Osgood loop works well when large cycle loops are exercised but an off-

set of the forces can occur in some small cycles as is shown in the diagram above. In 1984 the Ramberg-

Osgood hysteresis loop in Ruaumoko was modified to bound the forces within an envelope obtained by the

loops from the maximum and minimum displacements. In the year 2000 the Ramberg-Osgood loop reverted

to its original definition and the bounded version was moved to IHYST = 40.

22

Modified Takeda Hysteresis

IHYST = 4 Modified TAKEDA Hysteresis [Otani 1974].

Modified Takeda rule.

ALFA BETA NF KKK

ALFA Unloading stiffness (0.0 # ALFA # 0.5) F

BETA Reloading stiffness (0.0 # BETA # 0.6) F

NF Reloading stiffness power factor (NF $ 1) I

KKK =1; Unloading as in DRAIN-2D I

=2; Unloading as by Emori and Schnobrich

Note: Increasing ALFA decreases the unloading stiffness and increasing BETA increases the reloading

stiffness. The power factor NF is usually taken as 1.0

23

Bi-Linear with Slackness Hysteresis

IHYST = 5 Bi-linear with Slackness Hysteresis.

The bi-linear with slackness model can be used to represent diagonal braced systems where yield in in one

direction may stretch the members leading to slackness in the bracing system. The model allows for either

yield in compression, in say a cross-braced system, or for a simple elastic buckling in compression which

would be more appropriate in a single brace member.

Bi-linear with Slackness rule.

GAP+ GAP- IMODE RCOMP C EPS0 ILOG

GAP+ Initial slackness, positive direction. ( > 0.0) F

GAP- Initial slackness, negative direction ( < 0.0) F

IMODE = 0; Default case, normal rule holds I

= 1; Bi-linear elastic buckling in compression.

= 2; Bi-linear Elastic in Tension and Compression

RCOMP Bi-linear Factor r in Compression F

C Strain-Rate Constant (if # 0.0 Strain-rate effects are ignored) F

EPS0 Quasi-static Strain-rate (if # 0.0 then C = 0.0) F

ILOG = 0; Natural Logarithms are used for Strain-rate effects I

= 1; Base 10 Logarithms are used for Strain-rate effects

Notes:

1 Concrete and Steel Beam-column sections require GAP+ = - GAP-.

2 If no value is prescribed for RCOMP, i.e. there are less than four items on the line or RCOMP is the

word DEFAULT or D then the bi-linear factor in compression is the same as that for tension.

3 For the SPRING members the hysteresis data is the same for all actions. If different properties are

desired in the different actions then separate members should be used for the different actions. The

default bi-linear factor is that for the force components.

4 If the strain rate constant C is non zero then the positive and negative yield forces are multiplied by the

factor

where is the current strain rate and is the quasi-static strain rate EPS0.

24

Kivell Degrading Hysteresis

HYST = 6 KIVELL Degrading Stiffness [Kivell 1981].

The pinching model of Kivell was designed to represent the behaviour of the nails in steel nail-plates

connecting timber members together at the joints. The assumed cubic unloading-reloading curve is

represented by three straight lines.

Kivell Degrading rule.

ALFA


25

Origin-Centred Hysteresis

IHYST = 7 Origin-Centered Bi-linear Hysteresis - No further data is required,

On unloading and on subsequent reloading the path is on a line passing through the origin.

26

SINA Degrading Tri-linear Hysteresis

IHYST = 8 SINA Degrading Tri-linear Hysteresis. [Saiidi 1979].

SINA Degrading Tri-linear rule.

ALFA BETA FCR(i)+ FCR(i)- FCC(i)

ALFA Bi-linear factor (positive cracking to yield) F

BETA Bi-linear factor (negative cracking to yield) F

FCR(i)+ Cracking moment or force at i ( > 0.0) F

FCR(i)- Cracking moment or force at i ( < 0.0) F

FCC(i) Crack closing moment or force at i ( > 0.0) F

Notes:

1. The i refers to the different actions on the member, see the member data descriptions for the number of

actions and which action they refer to.

2. Concrete and Steel Beam-column sections require symmetry in moments and thus

FCR(i)- = - FCR(i)+ etc. and that ALFA = BETA

27

Wayne Stewart Degrading Hysteresis

IHYST = 9 W ayne Stewart Degrading Stiffness Hysteresis. [Stewart 1984].

This very general rule was initially developed by W ayne Stewart for the representation of timber framed

structural walls sheathed in plywood nailed to the framework. The model allows for initial slackness as well as

subsequent degradation of the stiffness as the nails enlarged the holes and withdrew themselves from the

framework.

Stewart Degrading with slackness rule.

FU FI PTRI PUNL GAP+ GAP- BETA ALPHA LOOP

FU Ultimate force or moment ( > 0.0) F

FI Intercept force or moment ( > 0.0) F

PTRI Tri-linear factor beyond ultimate force or moment F

PUNL Unloading stiffness factor ( > 1.0) F

GAP+ Initial slackness, positive axis ( > 0.0) F

GAP- Initial slackness, negative axis ( < 0.0) F

BETA Beta or Softening factor ( $ 1.0) F

ALPHA Reloading or Pinch power factor ( # 1.0) F

LOOP =0 Loop as defined I

=1 Modified loop

Notes: Concrete and Steel Beam-column sections require GAP+ = - GAP- and that all other

components maintain symmetry about the zero force or moment axis

This rule is not available for the Variable Flexibility or 4-Hinge Beam members.

Modified loop;

Member section yield values are taken as Fu+ and Fu- and the Fu on this line is taken as Fy+ and Fy-.

This modification allows the use of strength degradation. In the original model strength degradation

affects only the cracking moments Fy and not Fu. Also the PTRI (read as part of this data line) and r

(read as part of the member section data) are interchanged.

28

Operation of Wayne Stewart Hysteresis rule

Note: Vos = Fi etc.

29

W ayne Stewart used the following hysteresis values in his plywood wall examples [Stewart 1984].

FU = 1.5 times yield force or moment

FI = 0.25 times yield force or moment

PTRI = 0.0

PUNL = 1.45

BETA = 1.09

ALPHA = 0.38

Example: The diagram below shows the use of the modified W ayne-Stewart hysteresis loop to model a pre-

1970 reinforced concrete column hinge where plain round longitudinal reinforcement bars are

used [Liu,2001]. The two loops compare the observed experimental loop with that computed using

the program HYSTERES using the following parameters

Section Stiffness properties

K0 = 51.1 kN/mm

R = 0.001

FY+ = +58.4 kN

FY- = -58.4 kN

IHYST=9 Stiffness Degradation parameters (see preceding pages)

FRC (i.e. FU) = 27.6 kN

FI = 6.0 kN

PTRI = 0.14

PUNL = 1.1

GAP+ = 0.0

GAP- = 0.0

BETA = 1.2

ALPHA = 0.8

LOOP = 1

There is no strength degradation applied in this example.

30

Degrading Bi-linear Rule

IHYST = 10 Degrading Bi-linear Hysteresis. [Otani 1981].

This is similar to the Bi-linear rule except that the stiffness degrades with increasing inelastic deformation.

Degrading Bi-linear rule.

ALFA


31

Clough Degrading Stiffness Hysteresis

IHYST = 11 CLOUGH Degrading Hysteresis. [Otani 1981] - No further data is required.

This rule was the first degrading stiffness rules to represent reinforced concrete members. The rule is the

same as the modified TAKEDA rule with the parameters ALFA and BETA both equal to 0.0.

32

Q-HYST Degrading Stiffness Hysteresis

IHYST = 12 Q-HYST Degrading Stiffness Hysteresis. [Saiidi 1979].

This rule is the same as the Modified Takeda rule with the parameter BETA set to 0.0 and unloading as per

Emori and Schnobrich.

Q-HYST Degrading rule.

ALFA


Note: This rule is not available for the Variable Flexibility and 4-Hinge Beam Members.

33

Muto Degrading Tri-linear Hysteresis

IHYST = 13 MUTO Degrading Tri-linear Hysteresis. [Muto 1973].

After cracking the model is an Origin-Centered rule. After yield is reached the model become a Bi-linear

hysteresis with the equivalent elastic stiffness equal to the secant stiffness to the yield point.

Muto Degrading Tri-linear rule.

ALFA FCR(i)+ FCR(i)-

ALFA Bi-linear factor (cracking to yield) F



Notes:



2. Concrete and Steel Beam-column sections require symmetry in moments and therefore

FCR(i)- = - FCR(i)+ etc.

34

Fukada Degrading Tri-linear Hysteresis

IHYST = 14 FUKADA Degrading Tri-linear Hysteresis. [Fukada 1969].

Fukada Degrading Tri-linear rule.

ALFA BETA FCR(i)+ FCR(i)-

ALFA Bi-linear factor (cracking to yield) F

BETA Unloading Stiffness factor (see Takeda ALFA) F



Notes:



2. Concrete and Steel Beam-column sections require symmetry in moments and thus

FCR(i)- = - FCR(i)+ etc.

35

Bi-linear Elastic Rule

IHYST = 15 Bi-linear Elastic Hysteresis. - No further data is required.

This rule is similar to the Bi-linear hysteresis except that the rule unloads elastically down the same path which

means that no hysteretic energy is dissipated..

36

Non-linear Elastic Rule (URM)

IHYST = 16 Non-linear Elastic Hysteresis - No further data is required.

This non-linear elastic model represents the non-linear behaviour of face-loaded masonry wall units.

No hysteretic energy is dissipated.

37

Degrading Elastic Rule

IHYST = 17 Degrading Elastic Rule. - No further data is required.

The degradation of the elastic stiffness is proportional to the amount of equivalent ductility. This equivalent

ductility is equal to the displacement divided by the nominal yield deformation.

38

Ring-Spring Hysteresis

IHYST = 18 Ring-Spring Hysteresis. [Hill 1994]

This device can be used as a seismic energy dissipation device. The default model operates in the compressive

force - compressive displacement quadrant of the force-displacement plot.

Ring-Spring

RSTEEP RLOWER DXINIT KTYPE KLOOP

RSTEEP Unloading Steep Stiffness factor (usually greater than 1.0) F

RLOWER Unloading Lower Stiffness factor (usually less than the Bi-linear factor) F

DXINIT Initial Displacement F

KTYPE = 0 ; Uni-directional, = 1 ; Bi-directional I

KLOOP = 0 ; New Definition, = 1 ; Original Definition. I

Note:

1. This rule is normally only available for the Spring Members. In this case do not supply yield data as the

yield point is defined by DXINIT, see below.

2. The rule may be used for the flexural components of the Giberson one-component beam option of the

FRAME members when it would normally be expected to be used in the bi-directional mode. It may also

be used for the axial component of the Giberson beam members provided the beam has no flexural

stiffness i.e. EI is zero, representing a truss-like action. In both of these cases the appropriate yield

moments or yield forces must be provided with dummy non-zero values (the actual yield values are

computed internally by the hysteresis rule but non-zero yield forces or moments are required in order that

the member is treated as non-linear).

3. W hen initial pre-stress (FRAME members) or pre-load (SPRING members) forces are applied to the Uni-

directional Ring-spring (KTYPE = 0) they must be compressive (i.e. negative) forces.

39

Hertzian Contact Spring

IHYST = 19 Hertzian Contact Spring Hysteresis Rule. [Davis 1992]

The Hertzian contact spring is useful for modelling the contact between impacting structures. It really models

contact between spheres but this seems to be used in wider applications. It is only available for the SPRING

members and the CONTACT members.

Hertzian Contact Spring

MPP MPN PFP PFN GAP+ GAP-

MPP Stiffness Multiplier for Positive Displacement F

MPN Stiffness Multiplier for Negative Displacement F

PFP Power factor for Positive Displacement F

PFN Power factor for Negative Displacement F

GAP+ Initial slackness in Positive direction ($ 0.0) F

GAP- Initial slackness in Negative direction (# 0.0) F

40

Mehran Keshavarian Hysteresis

IHYST = 20 MEHRAN KESHAVARZIAN Degrading and Pinching Hysteresis. [Keshavarzian 1984]

Mehran Keshavarian Degrading and Pinching rule.

ALFA

ALFA Unloading stiffness (0.0 # ALFA # 0.5) (see Takeda) F

Note: This rule is not available for the Variable Flexibility and 4-Hinge Beam Members.

41

Widodo Foundation Compliance

IHYST = 21 W IDODO Foundation Compliance Model.[W idodo 1995]

These non-linear elastic rules are designed to model foundation compliance springs including the modelling of

a wall footing that can suffer partial or tip uplift. This is only appropriate to SPRING members, see section 12..

Widodo Foundation Compliance.

A(i) P(i) Q(i)

A(i) Multiplier for i th Component F

P(i) Power Factor for First part of i th Component F

Q(i) Power Factor for Second Part of i th Component F

Notes:

1. The i refers to the different actions on the member, see the member data descriptions for the number

of actions and which action they refer to.

2. This rule is not available for the Variable Flexibility Beam members.

42

Li Xinrong Reinforced Concrete Column Hysteresis

IHYST = 22 Li XINRONG Reinforced Concrete Column Hysteresis. [Li Xinrong 1994]

This rule is only available for the Reinforced Concrete column option for the FRAME members, see section 11.

The degrading rule modifies the stiffness of the member to allow for the effects due to variation of the axial force

acting in the column.

Li Xinrong Degrading Reinforced Concrete Column.

FPC RHO PB U ALFA BETA PINCH

FPC Concrete Compressive Strength f'c ( < 0.0) F

RHO Percentage Longitudinal Steel content F

PB Yield Moment - Axial Force Diagram Balance Axial Force F

U Unloading Coefficient (0.7 to 1.0) (Li Xinrong used 0.9) F

ALFA Factor for unloading (0.5 to 1.0) (Li Xinrong used 0.9) F

BETA Factor used in relocation of unloading point (0.8) F

PINCH Pinching Factor (0.7 to 0.9) F

43

IHYST = 23 BOUC Hysteresis Rule. [W en 1976]

This very general parametric hysteresis rule gives a smooth transition of the change of stiffness as the

deformation of the member changes. It has been used to represent lead-rubber bridge bearings or energy

dissipators [Bessasson 1992] and has been used for the analysis of inelastic buildings subjected to random

vibration [Baber 1981].

Bouc Degrading Stiffness.

A1 A2 A3 A4 A5 N D3 D4 D5 MODE INIT

A1 Loop Fatness parameter ( 0.1 to 0.9) F

A2 Loop Pinching parameter (-0.9 to 0.9) F

A3 Stiffness parameter (usually 1.0) F

A4 Degradation parameter (usually 1.0) F

A5 Strength parameter (usually 1.0) F

N Power Factor, Controls Abruptness (1 to 3, usually 1) I

D3 Strength Degradation parameter (0.0 to 0.1) (0.0 no degradation) F

D4 Loop Size Degradation (0.0 to 0.2) (0.0 no degradation) F

D5 Stiffness Degradation (0.0 to 0.2) (0.0 no degradation) F

MODE = 0 Constantinou Version

= 1 Baber and W en Version

INIT = 0 Normal

= 1 Bi-linear until first unloading after yielding

Note: This rule is not available for the Beam-Column Members

Bouc Hysteresis Versions

Baber and W en

Constantinou

where and are the displacement velocity, is the elastic stiffness, is the bi-linear factor and is the yield

displacement. is the yield force and is the change in displacement. Q is the force in the spring and F is

the current stiffness factor. The tangent stiffness is . and are the Bouc hysteresis control

parameters, is the power factor.

44

Bouc Hysteresis Rule

The Bouc rule is controller by the parameter , which in RUAUMOKO is initially 0.0 and is integrated step-by-step

as in the above equations. The rule is such that at the static analysis which means that the initial

stiffness is the bi-linear stiffness and the force in the member is proportional to the bi-linear stiffness and the

displacement. To over come some of these effects an option is to force the rule to be bi-linear until reversal after

the first yield. A result is that there is a marked reduction in member force at the change of rule. further work is

being done to fully understand the implications of the use of this hysteresis.

45

Steel Brace Member Hysteresis

IHYST = 24 REMENNIKOV Steel Brace Member Hysteresis.

Represents the out-of-plane buckling of a steel brace member.

REMENNIKOV Steel Brace rule.

Iminor Sminor k ALFA BETA THETA0 E1 E2 E3 E4 N SHAPE

Iminor Second Moment of Area about Minor axis F

Sminor Plastic Section Modulus about Minor axis F

effk Effective Length Parameter (L = kL) F

ALFA Strain Hardening Alpha (1.0 # " # 1.5) F

BETA Beta factor ($ > 1.0) recommended range 1.2 to 1.4 F

THETA0 Initial out-of-straightness (length units) F

E1 Effective modulus e1 (>0.0) F



E4 Effective modulus e4 F

N = 0; ALFA above used for strain hardening effects. I

= 1; Built-in strain hardening rule and ALFA is reset to 1.0

SHAPE = 1; Flanged section such as an I section. I

= 2; Circular hollow section

= 3; RHS or SHS section

Notes:

1. This hysteresis is available only for the Giberson One-component beam and the Steel beam-column

options of the FRAME member type. (see Section 11a)

2. The member only permits this hysteresis in the axial component. The member is assumed to be bi-linear

in flexure (provided the yield moments are non-zero in sections 11e or 11g).

3. The beam or beam-column cross-section area and the axial yield forces in section 11e or the yield

interaction forces and moments in section 11g must be supplied.

4. It is recommended that Iteration on Residuals, say MAXIT = 3 and FTEST = 0.001, be used with this rule

(see section 5).

5. It is recommended that a small time step be used so that the transitions within the rule may be followed.

46

Definition of Different Zones

47

Idealization Curves for Tangent Modulus History

Tangent Modulus – Axial Force Relationship

Analytical Axial Force – Plastic Hinge Rotation Curve

48

Takeda with Slip Hysteresis

IHYST = 25 TAKEDA with SLIP Hysteresis. [Kabeyasawa 1983]

This rule allows slip when the deformation reloads in the member strong direction.

TAKEDA with SLIP rule.

ALFA BETA1 BETA2 FC(i) RC(i)

ALFA Unloading Degradation parameter (see Takeda ALFA) (0.0 # ALFA # 1.0) F

BETA1 Slipping stiffness parameter F

BETA2 Re-loading stiffness parameter F

FC(i) Cracking Force for Component i ( > 0.0) F

RC(i) Cracking Displacement for Component i ( > 0.0) F

Notes:



2. The initial elastic stiffness supplied with the section properties is the secant stiffness passing through the

origin and the yield points on the member hysteresis curve.

3. The yield points in the negative direction for component i is at ( -FC(i), -RC(i) )

4. Slip only occurs when re-loading towards the stronger direction.

49

The following example show the use of the Takeda with Slip hysteresis loop to model the behaviour of a pre-1970

reinforced concrete beam which is reinforced with plain round bar reinforcement [Liu, 2001]. The loops show the

observed experimental loop and the matching loop from the program HYSTERES using the loop parameters

provided below.

Section Property parameters

K0 = 10.4 kN/mm

R = 0.05

YP+ = +93.4 kN

YP- = -61.4 kN

IHYST=25 Stiffness Degradation parameters (see previous page)

ALFA = 0.2

BETA1 = 1.2

BETA2 = 1.5

FC = 26.9 kN

RC = -0.65 mm

Strength Degradation parameters

ILOS = 1

DUCT1 = 3

DUCT2 = 8

RDUCT= 0.3

DUCT3 = 10

50

Bounding-Surface Hysteresis

IHYST = 26 AL-BERMANI Bounding-surface Hysteresis. [Zhu, 1995]

This rule allows for the Bauschinger effects in steel members by using a bounding surface rather than the more

often used but more complicated Ramberg-Osgood functions.

Bounding Surface rule.

ALFA BETA

ALFA Positive ALFA (0.0<ALFA<0.9) F

BETA Negative BETA (0.0<BETA<0.9) F

Notes:

1. The bi-linear factor p must be greater then 0.001. (see section 11c or section 12a)

This limitation on p should prevent mathematical difficulties with the logarithmic and exponential

functions.

2 The use of this rule in Concrete and Steel Beam column members will require symmetry in the actions

at each end, i.e. ALFA = BETA etc.

51

Peak Oriented Hysteresis

IHYST = 27 PEAK Oriented Hysteresis. - No further data is required.

This rule is similar to the Origin Centered rule except that on unloading the force-displacement relationship moves

along a line to the maximum force-displacement point in the opposite direction. If yield has not occurred in that

direction the opposite yield point is used as the target.

52

Matsushima Degrading Hysteresis

IHYST = 28 MATSUSHIMA Strength Reduction Hysteresis. [Matsushima 1969]

This rule represents the behaviour of short reinforced concrete columns failing in shear. The rule uses basically

a bi-linear hysteresis but that the elastic stiffness and strength degrade every time unloading takes place from

the post-yield part of the bi-linear force displacement curve.

Matsushima rule.

A B

A Stiffness Multiplier A (0.0 < A < 1.0) F

B Strength Multiplier B (0.0 < B < 1.0) F

Notes:

1. A and B are raised to then power N where N is the number of times the system unloads from the bi-linear

force-displacement hysteresis back-bone.

53

KATO Shear Hysteresis

IHYST = 29 KATO Degrading Shear Model. [Kato 1983]

Represents the behaviour of a reinforced concrete member failing in shear. A tri-linear skeleton curve with a

falling bi-linear part is used.

Kato Shear rule.

PTRI ALFA BETA GAMMA FU(i)+ FU(i)-

PTRI Tri-linear Factor (must be less than or equal to zero) F

ALFA Unloading Degrading Factor (0 < ALFA < 1) (see Takeda) F

BETA Slip Stiffness Factor (0 # BETA < 1) F

GAMMA Slip Length Factor (0 # GAMMA < 1) F

FU(i)+ Positive FU at component i (>0.0) F

FU(i)- Negative FU at component i (<0.0) F

Notes:



2. The bi-linear factor p must be less than 0.0 and less than the tri-linear factor PTRI. (see section 11c or

section 12a)

3. The actions FU should less than the appropriate yield actions FY.

4. The use of this rule in Concrete members will require symmetry in the actions at each end, i.e. FU(i)+

= -FU(i)- etc.

5. For reinforced concrete members failing in shear the recommended values are ALFA = 0.4, BETA = 0.6

and GAMMA = 0.95.

6. For reinforced concrete members flexure dominated members the recommended values are ALFA =

0.2, BETA = 0.0 and GAMMA = 0.0.

54

Elastomeric Spring Damper

IHYST = 30 Elastomeric Spring Damper Hysteresis. [Pekcan 1995]

Represents a double-acting elastomeric spring which has resistance due to both displacement and velocity. The

stiffness properties are basically bi-linear elastic.

Elastomeric Spring Damper rule.

C DMAX ALFA

C Damper Constant ($0.0) F

DMAX Maximum Damper Stroke ($0.0) F

ALFA Velocity Exponent (>0.0) F

Notes:

D1. The force F in the elastomeric damper is given by

0where the d is the displacement of the spring and where K is the initial stiffness of the device and is the

0 ylongitudinal spring stiffness of the member. The stiffness rK is the stiffness after the prestress F is

yovercome where r is the bi-linear factor for the spring member. The prestress force F is taken as the

positive longitudinal yield force of the member and if the prestress is zero the spring stiffness is taken

0as constant equal to rK .

max2. If DMAX is zero then the ratio of d to d is taken as 1.0. This has the effect of taking the exponent $

in the reference paper as zero.

3. The exponent ALFA was taken as 0.2 in the reference.

55

Composite Section Hysteresis

IHYST = 31 Composite Section. - Modified SINA Hysteresis.

This rule allows for the modelling of composite concrete-steel beams or concrete T beams where the behaviour

is different in the positive and negative flexural actions.

Composite Section rule.

BETA FCR(i) FCC(i)

BETA Stiffness Factor Cracking to Yield in Negative direction (ALFA < 1.0) F

FCR(i) Cracking Force in Negative direction at component i (FRC(i) < 0.0) F

FCC(i) Cracking Closing Force at component i (FCC(i) < 0.0) F

Notes:



2. The post cracking stiffness factor BETA must be greater than the bi-linear factor r (see member

properties sections)

56

Different Positive-Negative Stiffness Hysteresis

IHYST = 32 Different Positive and Negative Stiffness. Modified Bi-linear Hysteresis.

This rule allows for different stiffnesses in the positive and negative directions. The basic hysteresis rule is a

modification of the degrading Bi-linear rule.

Different Positive and Negative Stiffness rule.

ALFA BETA

ALFA Negative stiffness factor of the positive stiffness (ALFA > 0.1) F

BETA Unloading Degrading Factor, see TAKEDA ALFA (0.0 < BETA < 0.9) F

57

IHYST = 33 Masonry Strut Hysteresis (Crisafulli 1997).

This rule allows for the modelling of masonry panels in framed structures. If the strut model is used with the

spring members then only the longitudinal stiffness is specified for the strut member and two struts are required

to model each panel, one strut across each diagonal of the panel. The masonry strut hysteresis is also used for

the Masonry Panel Element where four struts represent the panel together with a shear spring..

Masonry Strut Hysteresis rule.

Stress-strain relationship

FC FT UC UUL UCL EMO GUN ARE

FC Compressive strength (stress units) (FC < 0.0) F

FT Tensile strength (stress units) (FT > 0.0) F

UC Strain at FC (UC < 0.0) F

UUL Ultimate strain (UUL < 0.0) (UUL # 1.5 UC) F

UCL Closing strain F

EMO Initial masonry modulus (EMO $ 2 FC/UC) F

GUN Stiffness unloading factor (GUN $ 1.0) F

ARE Strain reloading factor (ARE > 0.0) F

Strut data

AREA1 AREA2 R1 R2 IENV

AREA1 Initial strut cross-sectional area (AREA1 > 0.0) F

AREA2 Final strut cross-sectional area (AREA2 # AREA1) F

R1 Displacement at 1 (R1 < 0.0) F

R2 Displacement at 2 (R2 # R1) F

IENV = 0 ; Sargin stress-strain envelope descending branch I

= 1 ; Parabolic stress-strain envelope descending branch

58

Masonry Strut Strength Envelope

Masonry Strut Hysteresis

59

Notes:

m2FC The compressive strength f is the main parameter controlling the resistance of the strut. It must be'

noted that FC does not represent the standard compressive stress of masonry but should be adopted

taking into account the inclination of the compressive principal stresses and the mode of failure expected

in the masonry panel. See Crisafulli 1997.

tFT Tensile strength f' represents the tensile strength of the masonry or the bond strength of the panel-frame

interface, whichever is smallest. The consideration of the tensile strength has been introduced in the

model in order to gain generality. However, results obtained from different examples indicate that the

tensile strength, which is generally much smaller than the compressive strength, has no significant

influence on the overall response. Therefore, in the absence of more detailed information, the tensile

strength can be assumed to be zero.

m UC The strain at maximum stress g ' usually varies between -0.002 and -0.005 and its main effect on the

overall response of the infilled frame is the modification of the secant stiffness of the ascending branch

of the stress-strain curve.

uUUL The ultimate strain g is used to control the descending branch of the stress-strain relationship. W hen

u ma large value is adopted for, example g = 20 g ' , a smooth decrease of the compressive stress is

obtained.

clUCL The closing strain g defines the limit strain at which the cracks partially close and compressive stresses

can be developed. Values of the closing strain ranging between 0 and 0.003 lead to results which agree

cl uadequately with experimental data. If a large negative value is adopted, for example g = g , this effect

is not considered in the analysis.

moEMO The elastic modulus E represents the initial slope of the stress-strain curve and its value can exhibit

a large variation. Various expressions have been proposed for the evaluation of the elastic modulus of

masonry. It is worth noting, however, that these expressions usually define the secant modulus at a

stress level between 1/3 and 2/3 of the maximum compressive stress. In order to obtain an adequate

mo m2 mascending branch of the strength envelope it is assumed that E $ 2 f' / g ' .

unGUN The unloading stiffness factor ( controls the slope of the unloading branch. It is assumed to be greater

than or equal to 1.0 and usually ranges from 1.5 to 2.5.

reARE The reloading strain factor " defines the point where the reloading curves reach the strength envelope.

The calibration of the hysteretic model for the axial behaviour of masonry showed that good results are

obtained using values ranging between 0.2 and 0.4. However, higher values, for example 1.5, are

required to model adequately the cyclic response of the infilled frames. This is because other sources

of nonlinear behaviour, such as sliding shear, need to be indirectly considered in the response of the

masonry struts.

Four parameters are required to represent the cross-sectional area of the masonry strut. These are the initial area

ms1 ms2A = AREA1 and final area A = AREA2 and the axial displacements at which the cross-sectional area

R1 R2changes, ) = R1 and ) = R2. In a simplified model, it can be assumed that AREA1 and AREA2 are the

same using a low value of the strut area to avoid an excessive increase in the axial strength. In a more refined

analysis, a higher value of the initial area can be adopted, whereas the final area can be reduced by about 10%

m m m m mto 30%. The displacement R1 and R2 can be estimated as g ' d /5 and g ' d (where d is the length of the

masonry strut) respectively, at least until more precise information becomes available. Several empirical

expressions, which are described in section 6.2.1.3 of the reference, have been proposed for the evaluation of

the equivalent width of the masonry strut, whose value normally ranges from 0.1 to 0.25 of the diagonal length

of the infill panel.

IENV The descending branch of the stress-strain curve is usually modelled with a parabola instead of the curve

associated with Sargin's equation in order to obtain a better control of the response of the strut after the

maximum stress has been reached.

60

Hyperbolic Hysteresis

IHYST = 34 Hyperbolic Hysteresis (no further data required)

(Konduor and Zelansko (1963))

(also Duncan and Chang (1970))

This rule has been popular in representing the shear stress–shear strain relationships in soils subjected to

earthquake excitation.

.

61

Degrading Bi-linear With Gap Hysteresis

IHYST = 35 Degrading Bilinear with Gap Hysteresis

This hysteresis was initially developed to model a strain-hardening behaviour which changed with increasing cycle

number. The members were used in parallel with a member having a more conventional hysteretic behaviour

such as Bi-linear or Ramberg-Osgood. The total member force was taken as the sum of the two number forces.

Degrading Bi-linear with Gap

GAP+ GAP– PUN

GAP+ Initial gap in positive direction ($ 0.0) F

GAP– Initial gap in negative direction (# 0.0) F

PUN Unloading stiffness factor ($ 1.0) F

62

Bi-linear with Differing Positive and Negative Stiffness Hysteresis

IHYST = 36 Bi-linear with Different +/– stiffness hysteresis

The rule is to represent actions which exhibit different stiffnesses under positive or negative forces or moments.

This may be typical of reinforced concrete T sections for example.

Bi-linear with +/– stiffness

ALFA BETA GAMMA IOP

ALFA Positive stiffness is ALFA * nominal stiffness ALFA $ 0.1 F

BETA Negative stiffness is BETA * nominal stiffness BETA $ 0.1 F

GAMMA Unloading degradation factor F

see Takeda (IHYST=4) ALFA 0.5 $ GAMMA $ 0.0

IOP = 0 ; Bi-linear factor is the same in both directions I

= 1 ; Positive bi-linear factor = ALFA * bi-linear factor

Negative bi-linear factor = BETA * bi-linear factor

63

Non-linear Elastic Power Rule

IHYST = 37 Non-linear Elastic Power Rule

This rule is similar to the Bi-linear Elastic Hysteresis, IHYST = 15 except that it avoids the problems with the

sudden change of stiffness on unloading encountered due to the lack of energy dissipation in these non-linear

elastic hysteresis rules.

Non-linear Elastic Power Rule

PFP(i) PFN(i)

PFP(I) Power factor in positive direction for component i (0.01 # PFP # 3.0) F

PFN(i) Power factor in negative direction for component i (0.01 # PFN # 3.0) F

Notes:



1. Normally PFP and PFN are less than 1.0

64

Revised Origin-Centred Bi-linear Hysteresis

IHYST = 38 Revised Origin Centred Bi-linear Hysteresis (No further data is required)

On unloading the path is back to the origin. In reloading the path follows the previous unloading path on that side

of the origin.

65

Dodd-Restrepo Steel Hysteresis Rule

IHYST = 39 Dodd-Restrepo Steel Hysteresis Rule

This hysteresis rule is designed to allow for the Bauschinger effects in the steel hysteresis.

Dodd-Restrepo Steel Rule

ESH Esu Fsu OmegaF

ESH Deformation (curvature) at initiation of strain hardening F

Esu Deformation (curvature) at peak load F

Fsu Force (moment) at peak load F

OmegaF Bauschinger Effect Factor (0.6 < OmegaF < 1.3) F

(Default value = 1.0)

Reference:

Dodd, L.L. and Restrepo-Posada, J.I. Model for Predicting Cyclic Behaviour of Reinforcing Steel. J. Structural

Engineering, ASCE, Vol. 121, No. 3, Mar. 1995, pp 433–445.

66

Plot of Displacement History

Plot of Computed and Experimental Force Histories

67

IHYST = 40. Bounded Ramberg-Osgood Hysteresis. No extra data required.

This loop is similar to that for IHYST = 3 except that bounds have been applied to the forces so that off-sets to

the forces do not occur during small cycles of displacement.

68

IHYST = 41. Pyke modification to the Ramberg-Osgood Hysteresis. No extra data required.

This loop is similar to that for IHYST = 3 except that the small cycle behaviour has been modified to prevent off-

sets on the force in these small cycles. These loops were initially used to model the behaviour of soils.The small

loops indicate a greater rate of change of force with increasing displacement i.e. a greater curvature in the plots.

69

IHYST = 42. HERA - SHJ (Sliding Hinge Joint) Hysteresis.

This loop developed by the Heavy Engineering Research Association (New Zealand) is to represent the

behaviour of a sliding moment connection between steel beams with a concrete slab above them and connected

to at the joints to steel columns.

HERA-SHJ Hysteresis Rule

Cspp Cspn Ru Tdp Tdn Ispr

Cspp Positive Moment Intercept ( > 0.0 ) F

Cspn Negative Moment Intercept ( < 0.0 ) F

Ru Unloading stiffness factor ( > 0.0 ) F

Tdp Positive Design Theta ( > 0.0 ) F

Tdn Negative Design Theta ( < 0.0 ) F

Ispr = 0; W ithout Belleville Springs I

= 1; W ith Belleville Springs

70

71

IHYST = 43 Resettable Actuator Hysteresis. No extra data required.

This hysteresis is to represent the behaviour of a semi-active damper member [Hunt, 2003]. The force is

proportional to the displacement until a saturation force is attained, Fy+ or Fy- (the yield forces for the

member) when the system appears to show a perfectly plastic response. On any reversal of displacement the

force is automatically reset to zero and the origin is moved to the existing displacement and the system will

then behave as an elastic member until either saturation is achieved or the displacement again changes sign.

72

IHYST = 44 Pampanin Reinforced Concrete Beam-Column Joint Hysteresis.

Pampanin Reinforced Concrete Beam-Column Joint rule.

IOP AlfaS1 AlfaS2 AlfaU1 AlfaU2 DeltaF Beta

IOP =1; Option 1 - Reloading Power Factor I

=2; Option 2 - Reloading Slip Factor

AlfaS1 Slip Stiffness Power Factor As1 ( 1.5 # As1 # 3.0) F

AlfaS2 Option 1- Reloading Power Factor As2 ( 0.5 # As2 # 1.0) F

Option 2- Reloading Slip Factor Xi ( 1.0 # Xi # 1.5)

AlfaU1 Initial Unloading Power Factor Au1 (-1.0 # Au1 # 0.0) F

AlfaU2 Final Unloading Power Factor Au2 ( 0.3 # Au2 # 1.0) F

DeltaF Unloading Force Factor (%) Df ( 20 # DeltaF # 50) F

Beta Reloading Factor Beta (-1.0 # Beta # 0.0) F

Pampanin Hysteresis IOP=0

73

Pampanin Hysteresis IOP=1

74


IHYST = 45 Degrading Stiffness RAMBERG-OSGOOD Hysteresis.

Degrading Stiffness Ramberg-Osgood rule.

IOP ALFA BETA GAMMA RESID

IOP =1; Original Ramberg-Osgood hysteresis (see IHYST= 3) I

=2; Limited Ramberg-Osgood hysteresis (see IHYST=40)

=3; “Pyke” Ramberg-Osgood hysteresis (see IHYST=41)

ALFA % stiffness degradation per cycle (0.0 # ALFA # 10) F

BETA Ductility where stiffness starts to degrade with ductility (BETA $ 1.0 or = 0.0) F

If BETA=0.0 then there is NO degradation with ductility.

GAMMA Ductility where stiffness stops degrading with ductility (GAMMA > BETA) F

RESID Residual Stiffness when degrading with Ductility (RESID $ 0.50) F

Note: The bi-linear factor in the section data is used as the Ramberg-Osgood factor r and must be greater

than or equal to 1.0

75

IHYST = 46 DEAN SAUNDERS Reinforced Concrete Column Hysteresis.

This hysteresis is to represent the behaviour of Older Reinforced Concrete Columns where plain round

reinforcement is used.

DEAN SAUNDERS Reinforced Concrete Column rule.

DY+ DY- Funl+ Funl- ALFA BETA IOP

DY+ Positive Yield Deformation. Must be greater than (Yield Action)/Stiffness. F

DY- Negative Yield Deformation. Must be less than (Yield Action)/Stiffness. F

Funl+ Positive Threshold Action. Must be less than Positive Yield Action. F

Funl- Negative Threshold Action. Must be greater than Negative Yield Action. F

ALFA Unloading stiffness degradation factor (0.0 # ALFA # 0.5) F

BETA Reloading Factor (BETA $ 1.0) F

IOP =0; Threshold actions reduce with strength degradation. I

=1; Threshold actions remain constant when yield actions degrade.

Notes: 1. There is a 4 point Bezier curve fitted between the point (Funl+,(Funl+)/STIFF) and (YP,DY+)

with the initial slope STIFF and the final slope r*STIFF.

2. On unloading the rule is origin centered when the action is less than Funl+. W hen the action

is greater than Funl+ the unloading stiffness Ku degrades with the factor ALFA as observed

in the Degrading Bi-Linear Hysteresis (IHYST = 10).

3. The behaviour in the negative Action-Deformation quadrant is identical to that in the positive

Action-Deformation quadrant.

4. The parameter BETA only comes into action if the deformation in the other quadrant has

exceeded the yield deformation.

Dean Saunders Hysteresis Rule

76

Example of Dean Saunders Hysteresis

This example show the behaviour when strength degradation is also applied. It compares experimental results

with computational results.

In this example

Fy+ =289 kNm Fy- =-250kNm Funl =65kNm,

Ko =46.3 kNm/mm )y+ =11.35mm )y- =-11.35mm

" =0.01 $ =1.25 r =0.01

Iop =1

W ith the exception of strength degradation, which is progressive in the model, the results agree quite well for the

overall response of the observed hysteresis. The current limitation for the proposed hysteresis rule is that strength

degradation as calculated in Ruaumoko is computed outside the hysteresis rule. This results in the reloading line

converging to the previously stored force and maximum displacement coordinate before degrading down the

ductility degradation slope. To eliminate this limitation the rule will need to include a local strength degradation

feature that degrades the reloading line based on cycle number and starts from the threshold capacity.

Reference: Saunders, D.B. Seismic Performance of Pre 1970's Non-Ductile Reinforced Concrete Waffle

Slab Frame Structures Constructed with Plain Round Reinforcing Steel. Ph.D Thesis,

Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. 2004,

p184+appendices.

77

IHYST = 47 Multi-Linear Elastic Hysteresis.

Multi-Linear Elastic rule.

N F1 D1 F2 D2 F3 D3

N Number of segments beyond Bi-linear (1 # N # 3) I

F1 Fraction of stiffness in first segment beyond Bi-linear F

D1 Multiplier on Yield Displacement where F1 applies (D1 $ 1.05) F

F2 Fraction of stiffness in second segment beyond Bi-linear F

D2 Multiplier on Yield Displacement where F2 applies (D2 $ 1.05*D1) F

F3 Fraction of stiffness in third segment beyond Bi-linear F

D3 Multiplier on Yield Displacement where F3 applies (D3 $ 1.05*D2) F

Note: The F1, F2 and F3 factors should not be less than 0.0 for single degree of freedom systems.

Multi-linear Elastic Hysteresis

78

IHYST = 48 Kinematic/Isotropic Strain Hardening Bi-Linear Hysteresis.

Kinematic/Isotropic Strain Hardening Bi-Linear rule.

ALFA BETA

ALFA Stiffness Degrading Factor (See IHYST=10) (0.0 # ALFA # 0.5) F

BETA Isotropic Hardening Factor (0.0 # BETA # 1.0) F

If BETA=0.0 then there is NO Isotropic Strain Hardening.

If BETA=1.0 then there is Full Isotropic Strain Hardening.

Note: The bi-linear factor ‘r’ in the section data must be greater than 0.0 if Kinematic or Isotropic strain

hardening is to occur

79


IHYST = 49 Kinematic/Isotropic Strain Hardening RAMBERG-OSGOOD Hysteresis.

Kinematic/Isotropic Strain Hardening Ramberg-Osgood rule.

ALFA BETA IOP

ALFA Ramberg-Osgood multiplier (0.0 # ALFA) F

If ALFA=0.0 then ALFA is taken as 1.0 (Default value)

BETA Isotropic Hardening Factor (0.0 # BETA # 1.0) F

If BETA=0.0 then there is NO Isotropic Strain Hardening.

If BETA=1.0 then there is Full Isotropic Strain Hardening.

IOP =1; Original Ramberg-Osgood hysteresis (see IHYST= 3) I

=2; Limited Ramberg-Osgood hysteresis (see IHYST=40)

=3; “Pyke” Ramberg-Osgood hysteresis (see IHYST=41)

Note: The bi-linear factor in the properties section data is used as the Ramberg-Osgood factor r and must

be greater than or equal to 1.0

80

IHYST = 50 Flag-Shaped Bi-Linear Hysteresis.

Flag-shaped Bi-Linear rule.

BETA1 BETA2 ... BETAi (i = 1 to N)

BETA1 Flag Height Action 1 (0.0 # BETA1 # 1.0) F

BETA2 Flag Height Action 2 (0.0 # BETA2 # 1.0) F

BETAi Flag Height Action i (0.0 # BETAi # 1.0) F

Note: If BETA=0.0 then the loop is Bi-linear elastic.

If BETA=1.0 then the loop on unloading returns to the origin

N is the number of actions requiring data for the member. See Spring or Frame member data.

Flag-Shaped Bi-linear Hysteresis

81

IHYST = 51 Two-Four Hysteretic Damper

Two-Four Hysteretic Damper rule.

BETAi DELTAi (i = 1 to N)

BETA Initial Sticking Force1 (0.0 # BETA) F

DELTA Change Time from linear to sticking force, Seconds F

DELTA=0 implies instantaneous change

Two-Four Hysteresis

IHYST = 52 SCHOETTLER-RESTREPO Reinforced Concrete Column Hysteresis. (2 lines of data)

This hysteresis is to represent the behaviour of Reinforced Concrete Beams

SCHOETTLER-RESTREPO Reinforced Concrete Beam rule. Line 1

Kneg Rneg Fcr+ Fcr- Rho+ Rho- Dult+ Dult- IOP

Kneg Ratio of negative (compressive) stiffness to positive stiffness. (Kneg > 0.0) F

Rneg Bi-linear Factor in negative direction. (Rneg > 0.0) F

Fcr+ Ratio of positive Cracking Strength to Yield strength. (Fcr+ < 1.0) F

Fcr- Ratio of negative Cracking Strength to Yield strength. (Fcr- < 1.0) F

Rho+ Secant Stiffness Factor to Positive Yield. (Rho+ < 1.0) F

Rho- Secant Stiffness Factor to Negative Yield. (Rho- < 1.0) F

Dult+ Ultimate deformation factor. Positive direction. (Dult+ > 1.0) F

Dult- Ultimate deformation factor. Negative direction. (Dult- > 1.0) F

IOP =0; For use as reinforce concrete member I

=1; For use as a concrete strut

Notes: 1. The positive stiffness is that provided in the section properties.

2. The positive bi-linear factor is that provided in the section properties.

3. The cracking strengths are taken as fractions of the positive and negative yield strengths

provided under yield forces and moments in the section properties.

4. The Rho*Stiffness is the secant stiffness to the yield point.

5. The ultimate deformations are input as a multiplier of the respective yield displacement

and must be greater than 1.0..

6. Pinch must be greater than 0.0, if 0.0 is supplied then Pinch is reset to 1.0

SCHOETTLER-RESTREPO Reinforced Concrete Beam rule. Line 2

Alpha Beta Pinch Kappa+ Kappa- Fresid Dfactor

Alpha Unloading stiffness factor (0.0 < Alpha < 0.9) F

Beta Reloading stiffness factor . (0.0 < Beta < 5.0) F

Pinch Pinching factor (0.0 < Pinch < 1.0) F

Kappa+ Kappa factor. Positive direction . (0.0 < Kappa < 10.0) F

Kappa- Kappa factor. Negative direction . (0.0 < Kappa < 10.0) F

(Only supplied if IOP on line 1 above is 1, otherwise takes same value as Kappa+)

Fresid Residual Force factor (ratio of Cracking strength) (0.0 < Fresid < 1.0) F

Dfactor Bauschinger Effect Flag. (Dfactor = 0) I

Notes: 1. See IHYST = 4 Modified Takeda Hysteresis with the Emori and Schnobrich unloading

option for use of factor Alpha.

2. The Kappa Factor controls the rate of strength degradation.

3. The Fresid is the fraction of the Cracking strength at residual strength after degradation

has finished

4. The Bauschinger effect Dfactor parameter is reserved for future use.

5. Note Kappa- only supplied if IOP = 1

83

Figure 1: Backbone Curve

Figure 2: Uncracked – Yielded States

84

Figure 3: Cracked – Yielded States

Figure 4: Yielded – Yielded States

85

Figure 5: Yielded – Yielded States with pinching

86

IHYST = 53 DHAKAL Steel Hysteresis.

This hysteresis is to represent the behaviour of Reinforcing Steel

DHAKAL Reinforcing Steel rule. Line 1

IHARD IBCKL FYAV eHARD1 FU eU FHARD2 eHARD2 RATIO EBLKT

IHARD =1; Parabolic from initial hardening stiffness (Mander et.al. 1984) I

=2: Parabolic from intermediate point (Rodriguez et al 1999)

=3; Bilinear from intermediate point (Dhakal 2002)

IBCKL =1; Buckling included (Dhakal and Maekawa 2001) I

=0; Buckling is neglected (Tension envelope is used for compression)

FYAV Average Tensile Yield Strength, Multiplier on Yield Strength (Default= 1.0) F

eHARD1 Strain at the start of strain hardening, Multiplier of Yield Strain (Default= 10.0) F

FU Ultimate tensile strength Multiplier of Yield Strength (Default= 1.5) F

eU Strain at the ultimate point, Multiplier of Yield Strain F

(Default = 1 0.0*eHARD1)

FHARD2 Stress at intermediate point in hardening zone, Multiplier of Yield action F

(Default= FY+0.75(FU-FY))

or if (IHARD = 1) the 2nd Strain Hardening Stiffness Factor ESH2

eHARD2 Strain at intermediate point in hardening zone, Ratio of Yield Strain F

(Default = eHARD1+0.5(eU-eHARD1))

RATIO Buckling length to bar diameter ratio (Used if IBCKL=1) F

EBLKT Buckling length (if 0.0 taken as element length) (Used if IBCKL=1) F

Dhakal Reinforcing Steel Hysteresis Loop

87

IBCKL = 0

Dhakal Reinforcing Steel Hysteresis Loop

88

IHYST = 54 BRIAN PENG Concrete Hysteresis.

This hysteresis is to represent the behaviour of Concrete

BRIAN PENG Concrete rule. Line 1

TLIMIT CLIMIT BETA Fbo L TFACTOR CFACTOR eTT

TLIMIT Limit to calculate eTL (Strain at start of contact stress effect) (Default = 0.0025) F

CLIMIT Limit to calculate eCL (Strain at end of stress contact effect) (Default = 0.005)F

BETA Strain Rate factor (Static = 1.5 to 2.0; Dynamic = 1.0) (Default = 2.0) F

Fbo Residual Compressive Bond Strength (Default = 0.2) F

(Default = -0.2*FT where FT is tensile strength)

L Length factor ( = 1.0 for stress-strain) (Default =1.0) F

TFACTOR Factor for eTL (multiplying factor to magnify eTL) (Default =1.0) F

CFACTOR Factor for eCL (multiplying factor to magnify eCL) (Default =1.0) F

eTT Tensile Concrete Strain for Contact Stress Effect (Default = Ft/KC) F

Brian Peng Concrete Hysteresis Loop

89

Effect of varying Tlimit and Climit

Effect of varying Beta

90

Effect of Varying Fbo

Effect of varying Tfactor and Cfactor

91

IHYST = 55 Resettable Air-Cylinder Semi-Active Damper.

This hysteresis is to represent the behaviour of a semi-active resettable control device.

Resettable Air-cylinder Semi-Active Damper. data

IOPT AREA COEFF GAMMA FreeD+ FreeD- Fstiff Friction+ Friction-

IOPT = 1: 1-2-3-4 quadrant action (see figures below) I

= 2; 2-4 quadrant action (see figures below)


= 4; 1-2-3-4 quadrant action (see figures below, also see IHYST = 43)

= 5; 2-4 quadrant action (see figures below, also see IHYST = 51)


AREA Area of piston F

COEFF Gas Coefficient (i.e. Atmospheric constant for air = 100000 N/m ) F2

GAMMA Power Factor (i.e. Air = 1.4) (Default =1.4) F

FreeD+ Free Length Positive direction F

FreeD- Free Length Negative direction F

Fstiff Friction Stiffness factor (times nominal member stiffness) (Default =20.0) F

Friction+ Friction limit force Positive direction F

Friction- Friction limit force Negative direction F

Notes: For IOPT = 1, 2 or 3 the behaviour of the resettable device follows an isentropic compressible gas law

where

Pressure*Volume = Constant gamma

i.e.

For IOPT = 4, 5 or 6

W here Ko is the members nominal stiffness (see section properties data).

For IOPT = 2 or 5 Force = 0.0 if the displacement and the velocity have the same signs.

resetFor IOPT = 3 or 6 Force = 0.0 if the displacement and the velocity have opposing signs. d is set to

zero

In all cases, on reversal of direction of displacement the force is set to zero and the displacement resets.

If both friction force limits are equal to zero then Fstiff is set to zero and there is no friction.

If Fstiff is less than or equal to zero then there is no friction The friction follows the standard Elasto-

plastic hystersis rule (see IHYST=1).

The yield actions (forces or moments) specified for the loop are taken as limiting actions for the loop.

This means that the force (moment) cannot be greater than YP or less than YN. These must be supplied

as non-zero values if the loop is to operate.

For IOPT = 1, 2 or 3 the displacement cannot be greater or equal to FreeD+ or be less than or equal

FreeD-. If these values are reached or exceeded the analysis will terminate with an error message.

For IOPT = 4, 5, or 6 there are no displacement limits.

92

93

94

95

Resettable Air-cylinder Semi-active (1-4) Device Loop

Behaviour with air only, no friction

96

IHYST = 56 Modified SINA Degrading Tri-linear Hysteresis. [Saiidi 1979].

Modified SINA Degrading Tri-linear rule. One line for each action requiring data.

ALFA BETA GAMMA DELTA PHI FCRP FCRN FCCP FCCN IOP PMAX PMIN

ALFA Bi-linear factor (positive cracking to yield) (0.2 # ALFA # 0.9) F

BETA Bi-linear factor (negative cracking to yield) (0.2 # BETA # 0.9) F

GAMMA Unloading power factor (0.0 # GAMMA # 0.5) F

DELTA Pinching Factor (0.0 # DELTA # 0.8) F

PHI Ratio of Compression to Tensile Stiffness (0.1 # PHI #10.0) F

If PHI = 0.0 then PHI reset to 1.0

FCRP Cracking action as ratio of Positive Yield (0.3 # FCRP # 0.9) F

FCRN Cracking action as ratio of Negative Yield (0.3 # FCRN # 0.9) F

FCCP Crack closing action as ratio of Positive Yield (0.1 # FCCP # 0.7) F

FCCN Crack closing action as ratio of Negative Yield (0.1 # FCCN # 0.7) F

IOP =0; Cracking and yield deformations set at static analysis I

=1; Cracking and yield deformations set at first cracking

=2; Cracking and yield deformations set at first yield or when axial force falls

outside range of PMIN to PMAX

PMAX Maximum (Most tensile) axial force to set deformation limits. F

PMIN Minimum (Most compressive) axial force to set deformation limits. F

Notes:

1. One complete line is required for each non-linear action, i.e. 2 lines for Frame members in Ruaumoko2d

and 4 lines for Frame members in Ruaumoko3D

2. Symmetric Concrete and Steel Beam-column sections require symmetry in moments and thus

FCRP = FCRN etc. and that ALFA = BETA

3. The tri-linear factor is the bi-linear factor supplied as part of the basic member section data and is the

same in both positive and negative directions.

4. If the yield forces vary as a result of axial force-moment interaction or because of strength degradation

the stiffness in each of the positive and negative directions are adjusted to retain the same cracking and

yield deformations.

5. IOP is only relevant to columns or where the yield actions are determined from and interaction diagram

and hence may vary from call to call.

97

Modified SINA Hysteresis

98

IHYST = 57 Revised TAKEDA Degrading Tri-linear Hysteresis. [Saiidi 1979].

Revised TAKEDA Degrading Tri-linear rule. One line for each action requiring data.

ALFA BETA GAMMA DELTA PHI FCRP FCRN IOP PMAX PMIN

ALFA Bi-linear factor (positive cracking to yield) (0.2 # ALFA # 0.9) F

BETA Bi-linear factor (negative cracking to yield) (0.2 # BETA # 0.9) F


DELTA Reloading Intersection Factor (-0.1# DELTA # 0.5) F



FCRP Cracking action as ratio of Positive Yield (0.3 # FCRP # 0.9) F

FCRN Cracking action as ratio of Negative Yield (0.3 # FCRN # 0.9) F



=2; Cracking and yield deformations set at first yield or when axial force falls

outside range of PMIN to PMAX

PMAX Maximum (Most tensile) axial force to set deformation limits. F

PMIN Minimum (Most compressive) axial force to set deformation limits. F

Notes:

1. One complete line is required for each non-linear action, i.e. 2 lines for Frame members and 3 lines for

Spring members in Ruaumoko2d and 4 lines for Frame members and 6 lines for Spring members in

Ruaumoko3D


FCRP = FCRN etc. and that ALFA = BETA

3. The tri-linear factor is the bi-linear factor supplied as part of the basic member section data and is the

same in both positive and negative directions.


the stiffness in each of the positive and negative directions are adjusted to retain the same cracking

and yield deformations.

5. IOP is only relevant to columns or where the yield actions are determined from and interaction

diagram and hence may vary from call to call.

99

Revised TAKEDA Hysteresis

100

IHYST = 58 Shape-Memory-Alloy Flag Hysteresis.

Shape-Memory-Alloy Flag-shaped rule. One line for each action requiring data.

ALFA BETA DELTA PSI PHI IOP JOP KOP

ALFA Fraction of Yield at Unloading Path intersection (0.1 # ALFA # 0.9) F

BETA Stiffness Factor at Stiffening Path (Default BETA = 1.0) F

(must be larger than Bi-linear factor (see Section input data)

DELTA Multiplier of Yield Deformation where Stiffening starts (2.0 # DELTA) F

PSI Ratio of Yield Deformation to Rupture Deformation (PSI = 0.0 or PSI > 2.0) F

If PSI = 0.0 then no deformation limit

PHI Ratio of Yield Action to Rupture Action (PHI = 0.0 or PHI > 2.0) F

If PHI = 0.0 then no Action limit

IOP =0; Flag-Shaped Loop in both Positive and Negative Directions I

=1; Flag-Shaped Loop in Positive Direction and Elastic in Negative Direction

=2; Flag-Shaped Loop in Negative Direction and Elastic in Positive Direction

JOP =0; Cracking and yield deformations set at static analysis I


KOP =0; If the deformation or action limits are reached the analysis will be terminated I

=1; If the deformation or action limits are reached the analysis will continue but the

action and stiffness will be set to zero.

Notes:


Spring members in Ruaumoko2d and 4 lines for Frame members and 6 lines for Spring members in

Ruaumoko3D

2. In the direction where the Hysteresis Action is Elastic only (see IOP above) the yield action in that

direction has no effect.



and yield deformations. This will only happen once the yield deformations have been set, see JOP

above

4. If both PSI and PHI are zero then there is no limit on forces or deformations in the model. Normally

only one of PSI or PHI is set to provide the limitation. If both are prescribed then the first to occur will

terminate the action of the member.

5. If the limits set by PHI or PSI are reached then the stiffness and actions are set to zero. If KOP is zero

the analysis will be terminated.

6. JOP is only relevant to columns or where the yield actions are determined from an interaction diagram


101

SMA Flag-shape. IOP=0

SMA Flag-shape IOP=1. IOP=2 similar

102

IHYST = 59 Ramberg-Osgood with Alpha. - No further data required

Notes:

1. The Bi-linear factor in the section data is used as the Ramberg-Osgood factor r and must be greater

than or equal to 1.0.

2. The factor " is a function of the Ramberg-Osgood factor r where

3. The hysteresis bounds are set up on the same form as for the Pyke version of the Ramberg-Osgood

hysteresis model. See IHYST = 41.

Ramberg-Osgood with Alpha Hysteresis

103

IHYST = 60 IBARRA with Pinching Hysteresis. [Ibarra 2005].

IBARRA with Pinching rule. One line for each action requiring data.

ALFA GAMMA DELTA THETA PHI FC+ FC- FCC+ FCC- FRES+ FRES- DMAX+ DMAX-

DRES+ DRES- IOP

ALFA Bi-linear factor cracking to yield (0.2 # ALFA # 0.9) F


(See Takeda ALFA power factor)

DELTA Fraction of extreme axis crossing to give end of pinching

(0.1 # DELTA # 0.8) F

THETA Strength Degrading Factor (0.0 # THETA # 0.3) F



FC+ Cracking action as ratio of Positive Yield (0.3 # FC+ # 0.9) F

FC- Cracking action as ratio of Negative Yield (0.3 # FC- # 0.9) F

FCC+ Cracking Closing action as ratio of Positive Yield (0.1 # FCC+ # 0.7) F

FCC- Cracking Closing action as ratio of Negative Yield (0.1 # FCC- # 0.7) F

FRES+ Residual action as ratio of Positive Yield (0.01# FRES+ # 0.5) F

FRES- Residual action as ratio of Negative Yield (0.01# FRES- # 0.5) F

DMAX+ Ductility at Maximum Strength-Positive (1.1 # DMAX+ #30.0) F

DMAX- Ductility at Maximum Strength-Negative (1.1 # DMAX- #30.0) F

DRES+ Ductility at Residual Strength-Positive (2.0 # DRES+ #30.0) F

DRES- Ductility at Residual Strength-Positive (2.0 # DRES- #30.0) F



Notes:


Spring members in Ruaumoko2D and 4 lines for Frame members and 6 lines for Spring members in

Ruaumoko3D


FC+ = FC- etc.

3. The tri-linear factor is the bi-linear factor r supplied as part of the basic member section data and is

the same in both positive and negative directions. The value of r must be less than Alpha supplied

above.




5. IOP is only relevant to columns or where the yield actions are determined from an interaction diagram


6. This rule is still under development

104

IBARRA hysteresis back-bone

IBARRA with Pinching Hysteresis

105

IBARRA with Pinching Hysteresis

Modification when reloading deformation is to right of break point

106

IHYST = 61 IBARRA Peak-Oriented Hysteresis. [Ibarra 2005].

IBARRA Peak-Oriented rule. One line for each action requiring data

ALFA GAMMA DELTA THETA PHI FC+ FC- FRES+ FRES- DMAX+ DMAX- DRES+ DRES-

IOP




DELTA Reloading intersection on plastic path (-0.2 # DELTA # 0.5) F











DRES- Ductility at Residual Strength-Positive (2.0 # DRES- #30.0) F



Notes:



Ruaumoko3D


FC+ = FC- etc.



above..







107


IBARRA Peak-Oriented Hysteresis

108

IHYST = 62 IBARRA Bi-linear Hysteresis. [Ibarra 2005].

IBARRA Bi-linear rule. One line for each action requiring data.

ALFA GAMMA THETA PHI FC+ FC- FRES+ FRES- DMAX+ DMAX- DRES+ DRES- IOP














DRES- Ductility at Residual Strength-Negative (2.0 # DRES- #30.0) F



Notes:



Ruaumoko3D


FC+ = FC- etc.



above.







109


IBARRA Bi-linear Hysteresis

110

IHYST = 63 Bi-linear Elastic with Gap.

Bi-linear Elastic with Gap rule. One line for each action requiring data.

GAP+ GAP-

GAP+ Gap in positive direction. (0.0 # GAP+) F

GAP+ Gap in positive direction. (0.0 $ GAP-) F

Note:. One complete line is required for each non-linear action, i.e. 2 lines for Frame members and 3 lines for


Ruaumoko3D

Bi-Linear Elastic with Gap Hysteresis

111

MEMORANDA

112

MEMORANDA

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