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PULS Course 2007
Department of Marine Technology, NTNU
Lars Brubak27.04.2007
Version Slide 230 May 2007
PULS Course contentPart 1: General introduction
Part 2: Theory and principles
Part 3: PULS elements
Part 4: Comparison with FEM and buckling-codes
Part 5: PULS demonstration and exercises
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PULS Course Part 1 - Overview
PULS course objective
Motivation for ultimate strength assessment
PULS areas of application
PULS features
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PULS Course objective
To gain:
Knowledge and skills related to PULS
General knowledge about buckling and ultimate strength
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Motivation
• Prevent ship hull collapse disasters
• Increased control of available safety margins for ship operations
• Safeguard life, properties and the environment
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PULS – Panel Ultimate Limit State
ls
PULS is a code for bucklingand ULS assessments
of stiffened and unstiffenedpanels
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PULS - Element library
Unstiffened plate element (U3): (non-linear)
Stiffened plate element (S3): (non-linear)
Stiffened plate element (T1):(linear, non-regular geometry)
Version Slide 830 May 2007
PULS - Software implementation
Stand alone software:Excel spreadsheet
Advanced Viewer (commercial code)
Nauticus Hull rule package (Ship Rules):Section Scantling - longitudinal strength check
Automatic Buckling Check (ABC); Rule check of FE mid-ship model
Nauticus Hull FPSO rule package (RP-C201):Section Scantling, longitudinal strength check
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PULS - Advanced Viewer
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PULS - Advanced Viewer – OutputOutput options: ULS-loads, deflections,
stress distribution, interaction curves
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PULS - Excel Application
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PULS 2.06 download and installation
Go to the internet page:
www.dnv.com/software/nauticus/nauticushull/bucklingassessment.asp
Click on ”Download PULS”
Unpack .zip-file
Install (setup.exe)
Execute PULS from the Start-meny
Included in download:- Installation instructions- User’ manual
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PULS Course Part 2 - Overview
Plate buckling
PULS principles
Theoretical basis
PULS solution method
Version Slide 1430 May 2007
Plate buckling
Buckling deflections tend to be regular and periodic
Representation by trigonometricseries:
- Need very few degrees of freedomcompared to FEM
- Any shape can be represented by applying sufficiently many termsDeflection due to
axial compression
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Buckling response curves
Linearized buckling theory
Load
Deformation
Eigenvalue
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Buckling response curves
Linearized buckling theory
Load
Eigenvalue
Non-linear geometry
Deformation
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Buckling response curves
Linearized buckling theory
Load
Eigenvalue
Non-linear geometryNo buckling
Deformation
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Buckling response curves
Linearized buckling theory
Load
Eigenvalue
Non-linear geometryNo buckling
Non-linear material
Deformation
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Buckling concepts
Load
Deformation
Elastic BucklingYielding
Ultimate strength
Pre-buckling
Post-buckling Post-collapse
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Slenderness variation
0
1
Slenderness (-)
Load
(-)
Elastic bucklingUltimate strengthSquash yield
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PULS: Detailed results
0
1
Slenderness (-)
Load
(-)
”Stocky”
design
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0
1
Slenderness (-)
Load
(-)
PULS: Detailed results
”Slender”
design
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Combined loads – capacity surface
Combined loads – load history in load space
Capacity boundary/surface
in load space:
Sig1
Sig2
Sig3
Proportional load history:
Sig1
Sig2
tau = 0
tau = fixedSig1E
Sig2E
303202
101
SigUSigSigUSigSigUSig
U
U
U
Λ=Λ=Λ=
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ImperfectionsImperfections:
- Geometrical imperfections (initial deformation)- Material imperfections (residual stress)
In real life: Imperfections introduced during fabrication (welding) and operation
In calculation model: Initial deformations are introduced to account for geometrical and material deformations
Initial deflections characterized by:- Deflection shape- Deflection magnitude
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Effect of imperfection shape
Py
Px
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Effect of imperfection shape
Capacity envelope
= minimum value
Py
Px
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Effect of imperfection magnitude
Increasing imperfection magnitude
Load
Deflection
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Effect of imperfection magnitude
Load
DeflectionWmax
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Effect of boundary conditions
Rotational boundary conditions
(linear effect)
In-plane boundary conditions
(nonlinear effect)
Should represent the effect ofsurrounding structure
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Simply supported
Clamped
Effect of rotational supportPy
Px
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PULS principles
PULS design principles for shipstructures
Extreme loads
Accepts elasticbuckling deflections
Do not acceptpermanent sets/buckles
in plates
Ensure strongstiffeners
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PULS – Theoretical basis
von Karman and Marguerre’s geometric non-linear plate theory
Establish non-linear elastic equilibrium equations - Energy methods, virtual work/stationary potential energy
- Raleigh-Ritz discretization of deflections (Fourier series)
Solves non-linear elastic equilibrium equations: - Incremental perturbation procedure with arc length control
- Stepping along equilibrium curve
Solve local stress limit state functions - Trace redistributed stresses in plate and stiffeners
- Check of material yield in internal critical “hot spot” positions
Moderate large deflections
Load
Deflection
Ultimate load
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Nonlinear plate theory
Geometrical non-linearity
Membrane strain-displacement relation (kinematic relations)
)wwww(21)ww(
21)uu(
21
ww w21 u
ww w21 u
1,02,2,01,2,1,1,22,112
2,02,2
2,2,222
1,01,2
1,1,111
++++=ε
++=ε
++=ε
von Karman, 1930
Perfect plate
Marguerre, 1938
Imperfect plate
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Energy methodsPrinciple of stationary potential energy:
Intuitively: The structure adjusts itself to the shape that requires theleast energy
P
P
0TU =δ+δ=Πδ
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PULS solution method
∑∑
∑∑ππ
==
ππ==
m21
nmn
021i0i0
m21
nmn21ii
)xb
nsin()xa
msin( A)x,x(fqw
)xb
nsin()xa
msin( A)x,x(fqwDeflections:
Potential energy:
Stationary pot energy, equilibrium equations:
w
Non-linear equilib. eq., cubic in Amn
0)P,....,A,A(fVA
0)P,....,A,A(fVA
12111212
12111111
==∂
∂
==∂
∂
)Ainquartic(;)P,.....,A,A(V
uPdV21V
1211
ijij
=
Δ−εσ= ∫∫∫mn
Version Slide 3630 May 2007
PULS theory
ηΔw,Aij
w
Solves incrementally
Load, P
Equil. eq. on incremental form (linearization) perturbation (Taylor) expansion
KA + GΛ = 0
As+1 = As + Α Δη +...
Λs+1 = Λs + Λ Δη + ...
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Combined loads - staging
Assume proportional loading(piecewise linear load path)
Reduce number of loadparameters to one
)PP(P)(P 1-si
si
1-sii −Λ+=Λ
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PULS solution method
Ultimate capacity assessment:Stops load incrementation at first von Mises yield in ”hot spot” stress location
Deflection
Load
Elastic buckling load
Ultimate load
Overcritical strength
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Summary of the model theoryRepresent deflections by shape functions
Nonlinear plate theory (elastic deflections accepted)
Principle of minimum potential energy
Incremental solution procedure
Hot spot stress control (plastic deformations not accepted)
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PULS Course Part 3 - Overview
PULS U3-element
PULS S3-element
PULS T1-element
PULS Advanced Viewer (AV)
PULS Excel
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PULS U3-element
U3 element usage:
For plates with sufficient lateral support at all edges
Validity range: Geometric requirements with respect to aspect ratio and slenderness
Aspect ratio limit: L1/L2 < 20 for L1 > L2 (or equivalent L2/L1 < 20 for L1 < L2) Plate slenderness ratio: Li/tp < 200 (Li = minimum of L1 and L2)
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PULS U3-element
Typical buckling modes in unstiffened plates:
a) Axial compression b) Transverse compresson
c) shear c) Axial bending c) Transverse bending
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PULS rectangular plate, bi-axial load space
a) b)
c) d) Fixed geometry
Variable shear prestress
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PULS square plate – bi axial load space
10 mm plate, ULS capacity curves 50 mm plate ≅ Von Mises yield
Bi-axial load space,
Variable Shear pre-stress
”Slender”
design
”Stocky”
design
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PULS S3 element
S3 element usage: For regularly stiffened plates, supported by frames or bulkheads
Validity limits:Web slenderness for flat bar stiffeners: 35Web slenderness for L or T profiles: 80Free flange for L or T profiles: 10t/f ff <
Plate between stiffeners: 200t/s <Aspect ration of plate between stiffeners 0.25 < L1 /s <10
/h ww <t/h ww
<t
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PULS S3-element
1) Local plate/stiffener deflection
2) Global/lateral panel deflection
Coupled by reducedstiffness approach
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Local buckling model
∑∑==
ππ−+
π=
Ms
1mm2
Ms
1mm1 a
xmVh2z1
axmV
hzxv )sin())cos(()sin()(
Local plate/stiffener deflection
∑∑
= =
=Ms
m
Ns
n
smnl b
yna
xmAyxw1 1
)sin()sin(),( ππ
v1
v2
Version Slide 4830 May 2007
PULS S3-element
Rotational continuity
Longitudinal continuity
0y0z yw
zv
== ∂∂
−=∂∂
∫∫ =a
sx
a
px dxudxu ,,
P
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Global buckling model
Global/lateral panel deflection
∑∑∑∑= == =
−+=Mc
m
Nc
n
cmn
Ms
m
Ns
n
smng B
yna
xmAb
yna
xmAyxw1 11 1
)sin())2cos(1(2
)sin()sin(),( ππππ
Sx
Sx
p
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PULS S3-element
Failure criteria:
4
5
6 1 3
2
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Typical buckling modes
a) Weak/thin plate - strong stiffener sideways: thin plate/wide stiffenerflange
b) Weak stiffener sideways/torsional: High stiffener/small flange
a) + b) effekt interactingc) Weak stiffener out-of-plane: Low stiffenerheight/long span/small flange: prevented by PULS design principles
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Capacity curve
ULS Capacity Curve
Linear elastic buckling
Elastic buckling region - plate returns to original shape after unloading
Stress redistributionfrom plate to stiffeners
σ1
σ2
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PULS T1-elementComplex geometry,
simplified theory:
Analysis of stiffened panels witharbitrary oriented stiffeners
Based on linear theory
Ultimate limit state estimates basedon hot spot stress control
Capacity limited by linear elasticeigenvalue
Version Slide 5430 May 2007
∑∑
∑∑ππ
==
ππ==
m21
nmn
021i0i0
m21
nmn21ii
)xb
nsin()xa
msin( A)x,x(fqw
)xb
nsin()xa
msin( A)x,x(fqwDeflections:
Potential energy:
w
)Amn in quadratic( ; )P,.....,A,A(V
uPdV21V
1211
ijij
=
Δ−εσ= ∫∫∫
neglect nonlinear membrane strains
PULS T1-element
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Load-deflection response curves
Linearized buckling theory, T1 element
Load
Eigenvalue
Non-linear geometry, U3 +S3 elementsNo buckling
Non-linear material
Deformation
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PULS T1 example, bi-axial load space
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PULS Corrugated panel
Element under development
Axial load
In-planeshear
Lateral pressure
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PULS Corrugated panel
Element under development
Axial loading In-plane shear loading
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PULS Corrugated panel
Element under development
Example validation
Axial load - in-plane shearcombinatins
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PULS Course Part 4 - Overview
Validation of method:- Nonlinear FEM analysis- Experimental data
Comparison with buckling codes:- DNV Rules- CSR Bulk
Demonstration
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Nonlinear FEM
Model extent
Mesh
Element type
Boundary conditions
Imperfections
Frame
Frame
Frame
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Membrane stress – bulk panel case
von Mises stress plot calculated in ABAQUS for 12mm panelsubjected to pure transverse load
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Biaxial compression, tanker bottom panel
0
20
40
60
80
100
120
140
0 50 100 150 200 250 300
σ1 (MPa)
σ2 (
MPa
)
Abaqus
PULS
PULS (LEB)
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Shear/transverse compression, bulk side panel
0
50
100
150
200
250
300
350
0 20 40 60 80 100 120 140
σ2 (MPa)
τ12 (
MPa
)
Abaqus
PULS
PULS (LEB)
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Transv compression / lateral load, tanker bottom panel
0
50
100
150
200
250
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6p (MPa)
σ2 (
MPa
)Abaqus PULS
Version Slide 6630 May 2007
Validation against GL lab test
GL experimental test model-2 "as measured yield stresses" in PULS analysis
PULS ; bi-axial capacity curves
-500
-300
-100
100
300
-500
-400
-300
-200
-100
0 100 200 300
Axial stress σ1 [MPa]
trans
vers
e st
ress
σ2
[MPa
]
PULS UC with p = 0.0
PULS UC with p = 5.30 Wh
Local eigenvalues (PULSLEB)GL test (190,43) and p = 5.30Wh)
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Comparison with buckling codes
Relevant codes for comparison:
DNV Rules
CSR Bulk
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Biaxial compression, bottom shell panel
0
20
40
60
80
100
120
140
160
180
200
0 50 100 150 200 250 300 350
σ1 (MPa)
σ2 (
MPa
)
PULS DNV Rules
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Axial compression, tanker case
Aframax
0
50
100
150
200
250
300
350
Bot1 Bot7Bot1
3Bot2
0
Side26
Side34
Side41
Deck21
Deck15
Deck9Deck
3InB
ot4
InBot1
0
InBot1
6
InSide2
7
InSide3
5
InSide4
2Gird
4Str6
Lbhd
9
Lbhd
15
Lbhd
21
Panel id.
Capa
city
(MPa
)
PULSCSR Bulk
Version Slide 7030 May 2007
Summary of the comparisons
Good agreement between PULS and nonlinear FEM, as well as withavailable experimental data
PULS vs DNV Rules:- Increased capacity for axial and biaxial load, especially for thin plates- Increased capacity for in-plane compression combined with shear load- Effect of lateral pressure on buckling strength important
PULS vs CSR Bulk:- Significantly lower capacity for pure transverse load- Increased capacity for axial load for thin plates- Increased capacity for in-plane compression combined with shear load
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PULS Course Part 5 - Overview
PULS demonstration
Exercises
Version Slide 7230 May 2007
PULS demonstration
PULS Advanced viewer
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Version Slide 7330 May 2007