Ductility assessment of structural steel and composite joints€¦ · 2/9 M Rpl M Ru S t Φ u M-S...
Transcript of Ductility assessment of structural steel and composite joints€¦ · 2/9 M Rpl M Ru S t Φ u M-S...
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
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