PULS Course content - Fakultet for ingeniørvitenskap - IV ... - NTNU.… · PULS Course content...

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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

2


PULS Course Part 1 - Overview

PULS course objective

Motivation for ultimate strength assessment

PULS areas of application

PULS features


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


PULS – Panel Ultimate Limit State

ls

PULS is a code for bucklingand ULS assessments

of stiffened and unstiffenedpanels

4


PULS - Element library

Unstiffened plate element (U3): (non-linear)

Stiffened plate element (S3): (non-linear)

Stiffened plate element (T1):(linear, non-regular geometry)


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

5


PULS - Advanced Viewer


PULS - Advanced Viewer – OutputOutput options: ULS-loads, deflections,

stress distribution, interaction curves

6


PULS - Excel Application


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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Plate buckling

PULS principles

Theoretical basis

PULS solution method


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

8


Buckling response curves

Linearized buckling theory

Load

Deformation

Eigenvalue




Load

Eigenvalue

Non-linear geometry

Deformation

9




Load

Eigenvalue

Non-linear geometryNo buckling

Deformation




Load

Eigenvalue

Non-linear geometryNo buckling

Non-linear material

Deformation

10


Buckling concepts

Load

Deformation

Elastic BucklingYielding

Ultimate strength

Pre-buckling

Post-buckling Post-collapse


Slenderness variation

0

1

Slenderness (-)

Load

(-)

Elastic bucklingUltimate strengthSquash yield

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PULS: Detailed results

0

1

Slenderness (-)

Load

(-)

”Stocky”

design


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=


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

13


Effect of imperfection shape

Py

Px


Effect of imperfection shape

Capacity envelope

= minimum value

Py

Px

14


Effect of imperfection magnitude

Increasing imperfection magnitude

Load

Deflection


Effect of imperfection magnitude

Load

DeflectionWmax

15


Effect of boundary conditions

Rotational boundary conditions

(linear effect)

In-plane boundary conditions

(nonlinear effect)

Should represent the effect ofsurrounding structure


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


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


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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∑∑

∑∑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


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



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)



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)


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


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


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


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


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


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


∑∑

∑∑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


PULS T1 example, bi-axial load space

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PULS Corrugated panel

Element under development

Axial load

In-planeshear

Lateral pressure




Axial loading In-plane shear loading

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Example validation

Axial load - in-plane shearcombinatins



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


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)


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


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


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


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 demonstration

Exercises


PULS demonstration

PULS Advanced viewer

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