(hull buckling and ultimate strength)

OPen INteractive Structural Lab

Topics in Ship Structural Design(Hull Buckling and Ultimate Strength)

Lecture 4 Buckling and Ultimate Strength of Columns

Reference : Ship Structural Design Ch.09

NAOE

Jang, Beom Seon


Goal

In Lecture 02 :

Euler buckling load

Load eccentricity → Secant formula

Elastic and Inelastic column behavior

Tangent-Modulus Theory

Reduced-Modulus Theory

Shanley Theory

In this Lecture:

Residual stress effect

Load eccentricity + column geometric eccentricity (initial deflection)

→ Perry-Robertson formula

Effect of lateral load

OPen INteractive Structural LabOPen INteractive Structural Lab

Ideal Columns

The Euler buckling load PE is

(Le : Effective Length)

The Euler buckling load is the load for which an ideal column will first have

an equilibrium deflected shape.

Mathematically, it is the eigenvalue in the solution to Euler’s differential

equation

The “ultimate load” Pult is the maximum load that a column can carry.

It depends on

initial eccentricity of the column, eccentricity of the load

transverse load, end condition,

in-elastic action, residual stress.

Pult of a practical column is less than PE

Buckling - the sudden transition to a deflected shape - only occurs in the

case of "ideal" columns.

11. 1 Review of Basic Theory

2

2E

e

EIP

L

2

20

d w Pw

dx EI


Ideal Columns

If the compressive stress exceeds proportional limit even in ideal column

→ Actual buckling load < Euler load

due to the diminished slope of the stress-strain curve.

For an ideal column (no eccentricity or residual stress) buckling will occur at

the tangent modulus load, given by

(Et : tangent modulus,

the slope of the stress-stain curve corresponding to Pt/A)

Since Et depends on Pt modulusthe calculation of Pt is generally an iterative

process.

It is more convenient to deal in terms of stress rather than load

→ The effects of yielding and residual stress to be included.


2

2

tt

e

E IP

L

2

2=

( / )

EE

e

P E

A L

ρ : the radius of gyration , I= ρ2A

Le/ ρ : slenderness ratio


Ideal Columns

Load-deflection behavior for practical columns

containing residual stress and eccentricity


Slender Column (PE<PY) Stocky Column (PE>PY)

Q: Explain Pt?

2

2

tt

e

E IP

L


Ideal Columns

The ultimate strength of a column is defined as the average applied

stress at collapse

11.1 Review of Basic Theory

Stress-strain curveTangent modulus

curveTypical experiment values for straight

rolled sections and welded sections

ultult =

P

A

2

tult ideal 2

( ) =

e

E

L

due to entirely the compressive residual stress σr


Residual Stress-Rolled Sections

What is rolled section?

a metal part produced by rolling and having any one of a number of cross-

sectional shapes.


Rolled section

Welded section



The uneven cooling between Residual stresses will result after the cooling

because of the non-uniform temperature distribution through the cross

section during the cooling process at ambient temperature.

flange root : tesile residual stresses

flange tip : compressive residual stresses,

about 80 MPa for mild steel


Typical residual stresses in

rolled section



The parts of the cross section that have a compressive residual stress

may commence yielding when applied stress = σY - σr

→ particularly detrimental because it occurs in the flange tips.

The material is no longer homogeneous, and the simple tangent modulus

approach is no longer valid.

A comparatively simple solution can be achieved for the buckling strength in

the primary direction by assuming an elastic-perfectly plastic stress-strain

relationship.

The progressive loss of bending stiffens is linearly proportional to the extent

of the yielded zone → the use of an average value of tangent modulus.


Typical stub column

stress-strain curve

Structural tangent

modulus diagram

Q: What is σspl : Structural proportional limit?



Structural tangent modulus Ets by Ostenfeld-Bleich Parabola

σspl : Structural proportional limit

Substituting Ets in place of Et

For rolled wide flange section, typical value of σspl = 1/2 σY .

Johnson parabola for rolled section (essentially straight and pinned end)


)(for)(

)(Yavspl

splYspl

avYavts

E

E

E

L Ye

Y

spl

Y

spl

Y

ult

,11 2

2for,4

12

Y

ult2,

42

1,

41

2

1 222

when σult = 1/2 σspl

2

tult ideal 2

( ) =

e

E

L




Basic column curve (Johnson parabola and Perry-Robertson curve

σult = 1/2 σspl

YeYe

Ye

L

E

L

E

E

L

2

22

2

2

2

22

22

)/(

1

2

2=

( / )

EE

e

P E

A L

22

1)(

Y

ultE


Residual Stress-Welded Sections

Residual thermal stress is induced by uneven internal temperature

distribution


Residual stress close to

yield stress

in welding direction



The interaction between the different fibers results in a locked-in

tensile stress in and near the weld which is equal to approximately

equal to the yield stress.


The extent of the tension yield zone (3~6 times thickness) depends mainly

on the total heat input, the cross-sectional area of the weld deposit, the type

of welding and the welding sequence.

Typical residual stresses in welded section



From the equilibrium

ηt : width of the tension yield zone, b : total flange width


2=

2

r

Yb

t

For narrow thick sections residual stress will be high, and this will seriously

diminish the strength of the section.

It should be noted that

the effect will be much worse for open sections than for box sections due

to the tensile stress at corners of box sections.

flame-cutting creates conditions similar to welding → high tensile residual

stress is beneficial.

the effect of residual stress is somewhat diminished for higher yield

steels due to narrower tension zone.

Yr ttb 2)2(


Pinned column with an initial deflection δ(x) and addition deflection (x)

due to axial load. Bending moment is P(δ+w).

Deflection continues to grow and magnified as long as P increases.

→ no static equilibrium configuration and no sudden buckling.

Eccentricity: Magnification Factor


2

2+ ( +w)=0

d w P

dx EI

1

= sinn

n

n x

L

Let δ be represented by a Fourier series

Assume that the additional deflection

due to the bending is also a Fourier

seriesL

xnww n

sin

Eccentric columns




2 2 2

2 21

= sinn

n

d w n n xw

dx L L

The dominant term is n=1. When PE/P→1, wn /δn →∞

and the total deflection is

where ϕ is called the magnification factor.

,...3,2,1,

12

nwhere

P

Pnw

E

nn

0sin)(1

2

22

L

xnw

EI

Pw

L

n

n

nnn

0)(

2

22

nnn wEI

Pw

L

n

0)()( 2

2

22

nnEnn wPnPwL

EInP

,

1

P

Pw

E

PP

P

PP

Pww

E

E

E

T

PP

P

E

E

Load-deflection for eccentric columns



Eccentricity effects may also arise due to eccentricity of load. In this

case the magnification factor is given by

The two types of eccentricity (load application and column geometry)

can be combined linearly Δ=δ+e.

Eccentricity can be related to the slenderness ratio


=sec2 E

P

P

crP

PPeM

2secmax

Secant Formula in

Lecture 2

cA

Zr

rA

P

Z

A

A

P

Z

P

A

P

c

c

2

max

radius core

,11

2 ratioty eccentrici

c

rc

ratioty eccentrici

2

r

ec

Secant Formula in Lecture 2

resultsfor test boundlower for 0.003,L

cr

Column with eccentric load


Perry-Robertson Formula

It adopts Robertson`s value for α and assumes that the column will

collapse when the maximum compressive stress reaches the yield

stress.

It is a quadratic, and may be rewritten in a nondimensional, factored form

where R is the strength ratio of the column

η is the eccentricity ratio

λ is the column slenderness parameter

For a perfectly straight column, having η=0, reduces to the Euler curve,

, providing that λ>1.

11.2 Column Design Formulas

18

ult

ult

11

Y

E

L

ult

ult

( )1 E

Y

E

P L P

A P P

2(1 )(1 )R R R

ult

Y

R

L

2

2 2 2

1 1 1 1 11 1

2 4R

2

1R

RRultYult

Y

22 11

1,

11

E

L Y

E

Y


Accurate Design Curves

Perry-Robertson formula with different values of α being used for different

types of sections


19Column design curve for α=0.002 α=0.0035 α=0.0055

fail by yield

σult

(MPa) σult

(MPa)

σult

(MPa)

Le /ρ Le /ρ Le /ρ


Homework #1

Plot Secant curve shown below and Perry-Robertson

Formula in a graph and discuss the differences of two

approach associate with the plots.

Euler curve

Graph of the secant formula for

σmax = 250 MPa (σyield) and E = 200 GPa


Accurate Design Curves

The curves have a horizontal plateau at values

of slenderness ratio less than some threshold

value (L/ρ)0 . To achieve this the value of the

eccentricity ratio is defined

for

for

where (L/ρ)0 is given by

For welded I-and box sections, the yield

stress should be reduced by 5% to allow

for welding residual stress.


Curve selection(α) table for rolled

and welded sections

0

0c

L l

r

0 0

c

L l L L

r

0

0.2Y

L E


Use of Magnification Factor

For a pinned column subjected to lateral pressure q and P, the wmax is

The maximum bending moment

→ different for deflection, end condition, load

The simpler magnification factor which was derived for sinusoidal

initial eccentricity is used.

Max. bending moment =due to lateral load(M0)+ due to eccentricity(δ0)

11.3 Effect of Lateral Load: Beam Columns

22

4 2

max 4

5 24sec 1

384 5 2

qLw

EI

2

L P

EI

2

max 2

2(1 sec )

8

qLM

E

E

P

P P

max 0 0( )M M P max

max

MP

A Z

Homework #2 Derive wmax

and Mmax.

Central deflection of a

laterally loaded memberThe effect of axial load



To demonstrate the accuracy of this approach, if P=0.5PE, Δ=0

We assume that the column will collapse when the maximum compressive stress reaches the yield stress. Therefore

It can be expressed in terms of nondimensional parameters

where


23

max 02.030M M

2 4 2

max 0 0 02

5 50.5 2 1 2.028

384 48

EI qLM M M M

L EI

ult 0 ult 0

ult

( )

1

Y

E

P M P

A Z PZ

P

2(1 )(1 )R R R

ult

Y

R

Y Y

E

L

E

0( )A

Z

0

Y

M

Z

2

max 2

2(1 sec )

8

qLM

2

L P

EI

Homework #3 Derive left formula



The solution becomes


24

A pinned beam column subjected to a specified lateral load.

0( )A

Z

0

Y

M

Z

Initial deflection and

load eccentricity

B.M due to lateral load

2

2 2 2

1 1 1 1 11 1

2 4R




25

0( )A

Z

0

Y

M

Z

Initial deflection and

load eccentricity

B.M due to lateral load

(hull buckling and ultimate strength)

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