Ductility assessment of structural

steel and composite joints

Jean-Pierre Jaspart

Adrien Corman

Jean-François Demonceau

11-13 September 2019, Prague, Czech Republic

The International Colloquium on Stability and Ductility of Steel Structures

Introduction: joint behaviour

1/9

M

z

Introduction: joint behaviour

1/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

Introduction: joint behaviour

1/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: joint behaviour

1/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: joint behaviour

1/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: joint behaviour

1/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: the component method

2/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: the component method

2/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

Tension zone

Compr. zone

Shear zone

1. Identification

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: the component method

2/9

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,iniM

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

2. Characterization

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: the component method

2/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

3. Assembly

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: the component method

2/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

3. Assembly

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Introduction: the component method

2/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

3. Assembly

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Problem: ductility assessment

3/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ductility?

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Problem: ductility assessment

3/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ductility?

Need for joint ductility:

• Design of a structure based on a plastic global analysis

Problem: ductility assessment

3/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ductility?

Need for joint ductility:

• Design of a structure based on a plastic global analysis

• Mitigation of the risk of progressive collapse under exceptional loading

Problem: ductility assessment

3/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ductility?

Need for joint ductility:

• Design of a structure based on a plastic global analysis

• Mitigation of the risk of progressive collapse under exceptional loading

• Energy dissipation under earthquakes

Problem: ductility assessment

3/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ductility?

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Problem: ductility assessment

3/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ductility?

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Φu

Estimation of Sj,st

4/9

Sj,st =Ez2

1k∗

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Estimation of Sj,st

4/9

1

k∗=

m

1

ki,m MRpl,m>MRpl,lim

+ p

1

kst,p MRpl,p≤MRpl,lim

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Sj,st =Ez2

1k∗

Estimation of Sj,st

4/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Elastic components

1

k∗=

m

1

ki,m MRpl,m>MRpl,lim

+ p

1

kst,p MRpl,p≤MRpl,lim

Sj,st =Ez2

1k∗

Estimation of Sj,st

4/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Elastic components Yielded components

1

k∗=

m

1

ki,m MRpl,m>MRpl,lim

+ p

1

kst,p MRpl,p≤MRpl,lim

Sj,st =Ez2

1k∗

Estimation of Sj,st

4/9

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

MRpl,lim = 1,65MRpl

1

k∗=

m

1

ki,m MRpl,m>MRpl,lim

+ p

1

kst,p MRpl,p≤MRpl,lim

Sj,st =Ez2

1k∗

Estimation of Sj,st

4/9

M

z

Tension zone: ki, FRpl,i, kst,i

Compr. zone: ki, FRpl,i, kst,i

Shear zone: ki, FRpl,i, kst,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

kst =

EstEki → components in tension/compression

2 1 + υ

3

EstEki → column web panel in shear

1

k∗=

m

1

ki,m MRpl,m>MRpl,lim

+ p

1

kst,p MRpl,p≤MRpl,lim

Sj,st =Ez2

1k∗

Estimation of Sj,st

4/9

1

k∗=

m

1

ki,m MRpl,m>MRpl,lim

+ p

1

kst,p MRpl,p≤MRpl,lim

Sj,st =Ez2

1k∗

M

z

Tension zone: ki, FRpl,i, kst,i

Compr. zone: ki, FRpl,i, kst,i

Shear zone: ki, FRpl,i, kst,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Estimation of MRu

5/9

MRu = min FRu,i . z

M

z

Tension zone: ki, FRpl,i, kst,i

Compr. zone: ki, FRpl,i, kst,i

Shear zone: ki, FRpl,i, kst,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Estimation of MRu

5/9

M

z

Tension zone: ki, FRpl,i, kst,i, FRu,i

Compr. zone: ki, FRpl,i, kst,i, FRu,i

Shear zone: ki, FRpl,i, kst,i, FRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

MRu = min FRu,i . z

FRpl,i (fy) ↔ FRu,i (fu)

Estimation of MRu

5/9

M

z

Tension zone: ki, FRpl,i, kst,i, FRu,i

Compr. zone: ki, FRpl,i, kst,i, FRu,i

Shear zone: ki, FRpl,i, kst,i, FRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

MRu = min FRu,i . z

Estimation of ductility

6/9

Φu =MRu −MRpl

Sj,st

M

z

Tension zone: ki, FRpl,i, kst,i, FRu,i

Compr. zone: ki, FRpl,i, kst,i, FRu,i

Shear zone: ki, FRpl,i, kst,i, FRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Ultimate rotation capacity

Estimation of ductility

6/9

Φu =MRu −MRpl

Sj,st

M

z

Tension zone: ki, FRpl,i, kst,i, FRu,i

Compr. zone: ki, FRpl,i, kst,i, FRu,i

Shear zone: ki, FRpl,i, kst,i, FRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Plastic rotation capacity

Φu −Φpl =MRu −MRpl

Sj,st−MRpl

Sj,ini/3

0

20

40

60

80

100

120

140

160

180

200

220

0 10 20 30 40 50 60 70 80 90 100

Be

nd

ing

mo

men

t [k

Nm

]

Joint rotation [mrad]

TEST T1

Experimental results

Analytical prediction

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

Ben

din

g m

om

en

t [k

Nm

]

Joint rotation [mrad]

TEST 013

Experimental results

Analytical prediction

Validation

7/9

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100

Be

nd

ing

mo

me

nt

[kN

m]

Joint rotation [mrad]

TEST 07

Experimental results

Analytical prediction

(Jaspart, 1991) (Jaspart, 1991)

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

Be

nd

ing

mo

me

nt

[kN

m]

Joint rotation [mrad]

TEST T9

Experimental results

Analytical prediction

(Zoetemeijer, 1974) (Demonceau & Jaspart, 2004)

Conclusions

8/9

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Ultimate rotation capacity: Φu =MRu−MRpl

Sj,st

• Eurocode 3, Part 1-8

• Present paper

M

z

Tension zone: ki, MRpl,i, kst,i, MRu,i

Compr. zone: ki, MRpl,i, kst,i, MRu,i

Shear zone: ki, MRpl,i, kst,i, MRu,i

MRpl

MRu

Sj,st

Φu

M

Φ

Sj,ini

Conclusions

8/9

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢

Ultimate rotation capacity: Φu =MRu−MRpl

Sj,st

• Eurocode 3, Part 1-8

• Present paper

M

z

Tension zone: ki, FRpl,i, kst,i, MRu,i

Compr. zone: ki, FRpl,i, kst,i, MRu,i

Shear zone: ki, FRpl,i, kst,i, MRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

Conclusions

8/9

M

z

Tension zone: ki, FRpl,i, kst,i, FRu,i

Compr. zone: ki, FRpl,i, kst,i, FRu,i

Shear zone: ki, FRpl,i, kst,i, FRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐅𝐑𝐮,𝐢 . 𝐳

Ultimate rotation capacity: Φu =MRu−MRpl

Sj,st

• Eurocode 3, Part 1-8

• Present paper

Conclusions

8/9

M

z

Tension zone: ki, FRpl,i, kst,i, FRu,i

Compr. zone: ki, FRpl,i, kst,i, FRu,i

Shear zone: ki, FRpl,i, kst,i, FRu,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐅𝐑𝐩𝐥,𝐢 . 𝐳

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐅𝐑𝐮,𝐢 . 𝐳

Ultimate rotation capacity: Φu =MRu−MRpl

Sj,st

• Eurocode 3, Part 1-8

• Present paper

Ductility

Thank you!

Questions?

Jean-Pierre Jaspart

Adrien Corman

Jean-François Demonceau

11-13 September 2019, Prague, Czech Republic

The International Colloquium on Stability and Ductility of Steel Structures

References

• Jaspart, J.-P., 1991. « Etude de la semi-rigidité des nœuds poutre-

colonne et son influence sur la résistance et la stabilité des ossatures en

acier », PhD dissertation , Liège University.

• Zoetemeijer, P., 1974. « A design method for the tension side of statically

loaded, bolted beam-to-column connections ». Heron, Vol. 20, N°1,

1974)

• Demonceau, J.-F. & Jaspart, J.-P., 2004. « Experimental and analytical

investigations on single-sided composite joint configuration », 5th

International PhD Symposium in Civil Engineering, Balkema, pp. 341-

349.

Introduction: the component method

M

z

Tension zone: ki, FRpl,i

Compr. zone: ki, FRpl,i

Shear zone: ki, FRpl,i

Sj,ini

MRpl

MRu

Sj,st

ΦuΦpl

M

Φ

MRelSj,ini/7

Φel

3. Assembly

4 key parameters

• Initial stiffness 𝐒𝐣,𝐢𝐧𝐢 =𝐄𝐳𝟐

𝐢𝟏/𝐤𝐢

• Plastic bending resistance 𝐌𝐑𝐩𝐥 = 𝐦𝐢𝐧 𝐌𝐑𝐩𝐥,𝐢

• Post-plastic stiffness 𝐒𝐣,𝐬𝐭 =𝐄𝐳𝟐

𝐢𝟏/𝐤∗

• Ultimate bending resistance 𝐌𝐑𝐮 = 𝐦𝐢𝐧 𝐌𝐑𝐮,𝐢