Ultimate/Accidental Limit State Analysis and Design
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
1
Outline
Introduction Plastic hinge concept Pl ti methods Plastic th d for f beams b andd plates l t Brief on mechanism analysis Stiffness of beams including geometry effect Tensile fracture NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
2
Introduction wind, wave, current loads oads
Bracing configuration
piles, f foundation d ti NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
3
Characteristic Ch t i ti response - global l b l load l d versus global displacement for an offshore structure load level
wind, wave, current loads
member fracture ? limit load
global post-collapse member instability first plastic hinge first yield
Members
Connections
load redistribution
piles, foundation deck displacement NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
4
Can reserve systems effects be utilized? For ULS design - ductility requirements must be complied with : fracture, local buckling, cyclic effects etc. Accidental A id l actions i - systems effects must be considered Ship collision collision, Explosions Explosions, Fires Fires, Dropped objects, Specified damages.. NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
5
Definitions of ultimate resistance in intact and damaged conditions global load capacity,intact structure
reserve strength
reserve strength
100-year load level response,damaged structure
residual strength
global displacement NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
6
Nonlinear analysis of offshore structures challenges Effects: • Nonlinear material, geometry, load, Local and global failure modes: • Yielding, buckling,fracture, Modeling • Beam columns, shell, solids, springs • St Stress-strain t i representations, t ti stresst resultant approach Load control procedures NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
7
Design against accidental actions according to e.g. NORSOK Step1 Plastic
Damage due to accidental actions
Step 2
Elastic
Plastic
Resistance of damaged structure to design environmental loads Partial safety f y factors f = 1,0
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
8
Material modelling for plastic analysis stress strain relationship Rigid-perfectly plastic
y
Elastic-perfectly plastic y ~0.001
NUS July 10-12, 2006
p ~20 p
u ~ 100 p
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
9
Plastic hinge concept
Plastic hinge
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
10
Plastic hinge concept Elastic
Elastic-plastic
Fully plastic Note distribution of plasticity NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
11
Plastic hinge concept Plastic moment rectangular g cross-section
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
12
Plastic hinge concept Plastic moment circular cross-section
M p f y d 2t
NUS July 10-12, 2006
Thin-walled tube: d >>t
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
13
Normalised moment
Pl ti hinge Plastic hi conceptt Plastic
Elastic
Normalised max. strain NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
14
Plastic hinge concept
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
15
Plastic collapse resistance Kinematic analysis Pc
q
q
2q w External virtual work
Internal virtual work
We M p 2
We Pc w Kinematics
Plastic collapse load NUS July 10-12, 2006
w
2
Pc
4M p L
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
16
Elastic-plastic collapse analysis of clamped beam Total resistance q
l q1l 2 M = Mp = 12
+
q1l 2 M =24
=
q2l M=8
q1 L2 Mp Beam end: 12 q1 L2 q2 L2 In the middle: M p 24 8 q2 L2 q1 L2 1 q2 q1 8 24 3
Total ota resistance es sta ce
2
M = Mp
16 M p 4 qc q1 q2 q1 3 L2
M = Mp NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
17
Elastic-plastic collapse analysis of clamped beam Comparison with usfos Fy D t Wp Wp p Area Area I I L qc qc disp 1 yield disp2
330 0.5 0.02 0.004608 0.004611 0.030159 0.03016 0.000869 8.70E-04 10 0.243461 240000 2.60E-02 6.94E-02
plastic rotation
1.39E-02 calcuated
calculated mean diameter usfos calcuated mean diameter usfos calculated mean diameter usfos calculated usfos referencew load 0.1E+6 first yeild hinge calculated 3 hinges calculated
Plastic rotation vs displacement Load factor vs displacement
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
18
Elastic-plastic collapse analysis of clamped beam Load versus mid span deformation q1
w1
2 q1 L4 1 M p L q1 w1 384 EI 24 EI qc
q2
q w1 qc M p L2 / 24 EI
w2
Deformation in step 1. 1 q / qc
2 5q2 L4 5 M p L q2 w2 384 EI 24 EI qc
q w1 2 qc M p L / 24 EI
Deformation in step 2. k=1
k = 0.2
1 Hi Hinge att mid id span 0.75
0
Hinges at ends
0
0.75 1
2
Load - mid span deformation NUS July 10-12, 2006
w M l 2/ 24 EI p
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
19
Elastic-plastic collapse analysis of clamped beam Plastic rotation analysis M1
M0 1
1/2Mp
Plastic rotation at ends
p 1 2
M 0 M1 1 2 1 1 M p dx M p 1 EI 2 3 2 EI 6 EI
For rectangular cross cross-section section fy h 2 / 4 y p 6 E h3 /12 2h A il bl ffor lilinear elastic-perfectly Available l ti f tl plastic l ti material t i l p y / 2h 0.75 ep y 4 / 2 / 3h
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
20
Strains in elasto-plastic p region g Cantilever beam h 2
1 y 2 4 y0 2 M 2 y dy M p 1 M p 1 3 h 0 3 0 x M M p 1 1
0 1 y x 3
1
2 y 1/ 3 1 41 y 2 0 d dx d dx 3h h h 1 x 1 3 1 1/ 3
ep NUS July 10-12, 2006
y 1 max
41 y 3h
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
21
Compactness requirements for various cross sections cross-sections TVERRSNITT ELLER
OG / ELLER
TVERRSNITTSDEL
TVERRSNITTSKLASSE
TRYKKKRAFT MOMENT
b TRYKK
Lokal knekning
Flytning ytterste fiber
Siste flyteledd
b £ 1.0 t
E fy
b £ 1.2 t
E fy
b £ 1.3 t
E fy
b £ 2.0 t
E fy
b £ 2.6 t
E fy
b £ 3.3 t
E fy
Full plastisk
TVERRSNITT LIVPLATE
TVERRSNITTSKLASSE
TRYKKKRAFT OG
MOMENT
Fl t i Flytning ytterste fiber
L k l Lokal knekning
F ll Full plastisk
Si t Siste flyteledd
t 1/2d
d
E fy
0 33 b £ 0.33 t a
E fy
0 43 b £ 0.43 t a
E fy
MOMENT OG
TVERRSNITTS-
TRYKKRAFT
b b £ 1.1 t
E fy
b £ 1.25 t
E fy
b £ 1.5 t
E fy
t
t d
E d £ 0.056 t fy
E d £ 0.078 fy t
E d £ 0.112 fy t
TVERRSNITTSKLASSE 3
t1
b1
t1 t1 b2
NUS July 10-12, 2006
b £ 0.4 t
E fy
KLASSE 1 OG 2
N d £ 4.20(1 - 0.59 9 ) t Np
b
KAPASITETEN KAN BESTEMMES EETTER PKT 5.6
03 b £ 0.3 t a
t d2
E fy
a· b
t
d1
N = s ·d·t
E fy
N £ 0.10, Np
b £ 0.43 t
0£
E fy
N d £ 3.80(1 - 0.55 5 ) Np t
t
E fy
E fy
TRYKK
b £ 0.30 t
E fy
MOMENT
N d £ 2.20(1 - 0.20 ) Np t
b
d £ 4.2 t
N = s ·d·t
b £ 0.33 t
TRYKKRAFT
E fy
N £ 0.10, Np
E fy
d £ 3.8 t
0£
13 b £1.3 t a
N £ 1.0 Np
E fy
0.15 £
12 b £1.2 t a
Np = fd·d·t
E fy
N £ 0.15, Np
MOMENT OG
b £ 1 t a
E fy
E fy
a· b t b
d £ 2.5 t
t
0£
t
KAPASITETEN KAN BESTEMMES ETTER PKKT 5.6
MOMENT
N d £ 2.50(1 - 0.93 3 ) t Np
t
E fy
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
22
Bending moment axial axial–force force interaction Mechanism analysis works well for beam and frames where the resistance is governedd by b bending b di In many structures the resistance contribution from axial force important, important either initially (truss-works) or during force redistribution (beams under finite deformations)
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
23
Pl ti hinge Plastic hi conceptt Bending moment- axial force interaction Generalised yield criteria M F sin Mp 2
Tube Compression
Bending
NUS July 10-12, 2006
1
N 1 0 Np
Tube 2
M N F 1 0 M p N p
Rectangular cross.
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
24
Plastic resistance for beam with concentrated l d att midspan load id (1) P
D,t
w
Pipe section
E q u i l ib r iu m N Np
R = 8
M
2N
w /2
B e n d in g m o m e n t – a x ia l f o r c e in te r a c t io n
F
N M cos 0 2 N p Mp
N N (w )?
NUS July 10-12, 2006
U nknow n
M M pAnalysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
25
Plastic resistance for beam with concentrated load at midspan (2) P w
Kinematics Plastic elongation in each hinge 2 1 1 w2 2 u w ~ u 2 2 2 2 Plastic rotation in each hinge
w /2 NUS July 10-12, 2006
w w
w /2
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
26
Plastic resistance for beam with concentrated load at midspan (2) P w
Kinematics Plastic elongation in each hinge 2 1 N 1 w2 N 2 u w ~ 2 2 2 2k 2 2k Plastic rotation in each hinge
NUS July 10-12, 2006
w /2
u
w /2
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
27
Plastic resistance for beam with concentrated load at midspan (3) P w N Np
vp M Mp
Plastic flow - normality criterion F M v p F u N
1 w Mp 2 N N w M 1 w F cos 0 sin i 2 N p Mp 2 N p 2 N p Analysis and Design for Robustness of Offshore Structures NUS July 10-12, 2006 NUS – Keppel Short Course
28
Plastic resistance for beam with concentrated l d at midspan load id (4) P w
R e su lts o f a n a ly sis w N 2 Np D
w 1 D
W h e n w /D > 1
N N NUS July 10-12, 2006
p
M 0
w 1 D
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
29
Plastic resistance curve for beam with concentrated load at midspan (5) P w Collapse model for beam with fixed ends
R u = 1 - ( w 2 + w arcsin w ) D D D Ro
w 1 D
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
30
Plastic resistance curve for beam with concentrated load at midspan (6) P w
8
R/R0
6
Transition from bending & mebrane to pure tension at w/D =1 R/R0 = /2
4 2
The displacment at this transition is denoted characteristic displacment wc
Bending only
0 0 NUS July 10-12, 2006
1
2 3 Deformation w/D
4
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
31
Plastic resistance curve for beam with concentrated d lload d at midspan id (7) P
K
K
w Kinematics- Plastic elongation in each hinge 2 1 2 N 1 w2 N u w ~ 2 2 2 2k 2 2K
w N u w 2K
In real structures beam ends ends are not fully fixed. The axial flexibility of the adjacent structure may be represented by a linear spring with stiffness K. This affects the kinematic relationship for plastic axial elongation. Closed form solution is no longer possible, but simple incremental equation may be solved l d numerically i ll NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
32
Elastic-plastic p resistance curve for tubular beam with conc. Load at midspan Factor c includes the effect of elastic flexibility at ends 6,5 6
5 4,5
0.2
4 0
3,5
R/
3
Bending & membrane Membrane only F-R k k w
Rigid plastic Rigid-plastic
5,5
0,3
0.1
05 0.5 1
2,5
c
2
0.05
4c Kw c c 1 f y A
1,5 1 0,5
2
0 0
0,5
1
1,5
2
2,5
Deformation
NUS July 10-12, 2006
3
3,5
4
w
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
33
Tensile Fracture According to plastic theory no limitation to resistance and energy gy dissipation p in beams with axial restraint Ultimately the member will undergo fracture due to excessive straining In order to predict fracture a strain model for the plastic hinges must be developed NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
34
Strain hardening paradox In plastic analysis the stress-strain curve is assumed rigid-plastic or linear-elastic perfectly plastic If the material behavior is really like this, the member b bbehaves h bbrittle i l in i a global l b l sense andd plastic theory cannot be applied St i hardening Strain h d i is i crucial i l in i distributing di t ib ti plastic l ti strains axially in the member, so that significant energy dissipation can be achieved NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
35
Y
max
Y h
Y h
M
M
Stress distribution
Strain
Approximate stress distribution
S Stress-strain d distribution b - bilinear bl materiall 50
45 40
Hardening parameter H = 0.005
Strain S
35
Maximum strain cr/Y = 50 = 40 = 20
30 25 20
P x
15
No hardening
10 5 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x/
Axial variation of maximum strain for a cantilever beam with circular cross-section NUS July 10-12, 2006
Assumption:
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course Bilinear stress-strain relationship
36
Tensile Fract Fracture re • The critical strain in parent material depends upon:
stress gradients dimensions of the cross section presence of strain concentrations material yield to tensile strength ratio material ductility
• Critical strain (NLFEM or plastic analysis)
cr
t 0.02 0.65 ,
NUS July 10-12, 2006
5t : length g of pplasticzone
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
37
Critical deformation for tensile fracture in yield hinges c w 1 d c 2c f displacement factor
c1 c l W WP
1 4c
w c f ε cr
1 2 W εY κ c w c lp 1 c lp 41 c1 3 WP ε cr d cr
plastic l i zone length l h factor f
ε cr 1 W H ε W y P c lp ε cr W 1 H 1 ε W y P
axial flexibility factor
c c f 1 c
non-dim. plastic stiffness
H
Ep E
=
2
for clamped ends
=
1
for pinned ends
cr
non-dimensional spring stiffness
εy
=
/ c1 1
= fy
2
2
1 f cr f y E ε cr ε y critical strain for rupture
yield strain E 0.5l the smaller distance from location of collision load fy = yield strength fcr = strength corresponding to cr to adjacent joint dc = D diameter of tubular beams = elastic section modulus = 2hw twice the web height for stiffened plates Analysis and Design for Robustness of Offshore Structures 38 NUS 10-12, 2006 = h height of cross-section for symmetric I-profiles = July plastic section modulus NUS – Keppel Short Course
Deformation at rupture for a fully clamped beam as a function of the axial flexibility factor c 5 4.5 4 35 3.5
w/D
3 2.5 2 1.5
/D = 30
/D = 20
c= 0 = 0.05 = 0.5 = 1000
c= = = =
0 0.05 0.5 1000
1 0.5 0 0
20
40
60
80
100
120
cry NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
39
Tensile fracture in yield hinges D t Determination i ti off H A1= A2 A2
fcr
HE
A2
fcr
A1
A1
E
HE
E
cr cr Determination of plastic stiffness f HE
Use true yyield stress
Even if the stress strain curve lies below the true relationship such that the energy dissipation for the fiber is smaller,, the hardening g exaggeration gg may give too large energy dissipation in the member as a whole
Erroneous determination of plastic stiffness NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
40
Tensile fracture in yield hinges • Recommended values for cr and H for g different steel grades Steel grade S 235 S 355 S 460
NUS July 10-12, 2006
cr 20 % 15 % 10 %
H 0 0022 0.0022 0.0034 0.0034
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
41
Plastic hinge concept Bending moment –axial force history
M,P
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
42
Stiffness matrix for beam with axial force w QB QA
B
MA
N
w(x)
wA
MB
N
wB
X=x
X
EIw, xx M A Q AX N w w A 0
Differential equilibrium equation NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
43
Stiffness matrix for beam with axial force Q A M A = QB M B
6EI 12EI 6EI 12EI w A 3 5 2 2 3 5 2 2 6EI 2EI 4EI 3 2 4 A 2 12EI 6EI 2 wB 5 3 2 symmetry 4EI 3 B
NUS July 10-12, 2006
N
2
NE
NE
1
1
tan
1 2 3 1 1 2
tanh
1 2 3 1 1 2
1 4
3 4
1 4
3 4
3 1 2 3 1 2 1 2
3 2
1
3
4 1 2 4 2 1 2 2 5 12
5 12
2 EI
l
2
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
44
Stiffness matrix for beam with axial force 5
4 3 2
-value
1
0 -4
-3
-2
-1
0
1
2
3
-1
4
-2
N
2
NE
-3 -4
-5 -6 Axial force
NUS July 10-12, 2006
E
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
45
Buckling of column – Example E l 1 N
A
B
N
l
2EI 2 3 4 K 4 2 3
Critical force
NUS July 10-12, 2006
K 0,
4 0 ; 2 3
2 4
1 3 4 2
N NE
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
46
Buckling of column – Example 2 N
A
B l
4EI 3 K
K 0
N 2NE
1 0.7 Buckling length k 2
NUS July 10-12, 2006
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
47
Buckling of column – Example 3 N
A
B l
2EI K 3
6 5 3 2 - 3 2 2 3 2
Critical force
NUS July 10-12, 2006
K 0 12l 235 9l 222 0
1235 922 N 0.25 N E
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
48
The stiffness matrix for beam with axial force contains all ll information i f i needed d d to predict di the h exact buckling b kli load for beams subjected to end forces Q A M A = QB M B
6EI 12EI 6EI 12EI w A 3 5 2 2 3 5 2 2 6EI 2EI 4EI 3 2 4 A 2 12EI 6EI 2 wB 5 3 2 symmetry 4EI 3 B
NUS July 10-12, 2006
N
2
NE
NE
1
1
tan
1 2 3 1 1 2
tanh
1 2 3 1 1 2
1 4
3 4
1 4
3 4
3 1 2 3 1 2 1 2
3 2
1
3
4 1 2 4 2 1 2 2 5 12
5 12
2 EI
l
2
Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course
49
