Resistance of columns and beam- columns - IV - · PDF fileResistance of columns and...

Resistance of columns and beam-columns

TMR41205 Buckling and ultimate load analysis of marine structures Jørgen Amdahl

Dept. Marine structures

Eaxmple of column buckling due to excessive jacking during installation.

Element

STRUCTURE

Beam Column

M

l

Idealized

effl

N

Idealisering av stavelementer i et rammeverk til en isolert bjelkesøyle

CM

Column buckling

The characteristic axial compressive resistance based Johnson-Ostenfeldt expression with 10% reduction :

c2

y

f 0.9 for 1.34f

λλ

= >

21.0 0.28 for 1.34c

y

ff

λ λ= − ≤

( )2

2,y EE

f EIff k A

πλ = =l

If local buckling strength fcl < fy replace fy by fcl

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

Euler

ISO (1-0.28 l², 0.9/l²)

Chen & Ross - fabricated

Smith, Somerville & Swan - fabricated

Smith, Somerville & Swan - seamless

Steinmann & Vojta - ERW

Yeomans - ERW

Yeomans - seamless

λ = (Fyc/Fe)0.5

fc/Fyc

Bias 1.057 COV = 0.041 n = 84

Figure A.13.2-1 Comparison of Test Data with Column Buckling Design Equation forFabricated Cylinders Subjected to Axial Compression

ISO column buckling curve Comparison with test data

Effect of axial local buckling

2

1 0 0 412

1 047 0 274 0 412 1 382

1 382

cl

y

cl

y

cl ce

f . .ff . . . .f

f f .

= λ ≤

= − λ ≤ λ ≤

= ≤ λ

λ =

f y

fxe

λ2 =f xef y

fcl

f y

= 1.0 λ ≤ 0.412

fcl

f y

= 1.047 − 0.274λ2 0.412 ≤ λ ≤1.382

fcl = fce 1.382 ≤ λ

fxe = 0.3E

tr

Elastic buckling

If fcl < fy replace fy by fcl in column buckling check

Yield stress D/t local buckling limit 235 91 355 60 420 51

Local buckling strength versus D/t-ratio

0"

0.2"

0.4"

0.6"

0.8"

1"

1.2"

0" 20" 40" 60" 80" 100" 120" 140"

Norm

alise

d"local"buckling"stress"

Diameter/thickness"ra=o"

Cri=cal""local"buckling"stress"

Yield stress achieved for D/t < 60

Capacity in bending

Typical normalized moment-rotation curves for cylinders for various D/t-ratios (MPS is plastic bending moment)

Capacity in bending m

y

f Zf W

= 0.0517yf DEt

≤

1 13 2 58 ym

y

f Df Z. .f Et W

⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠0.0517 0.1034yf D

Et< ≤

0 94 0 76 ym

y

f Df Z. .f Et W

⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠0.1034 120y yf D f

Et E< ≤

4 4( 2 )32

D D tD

π ⎡ ⎤− −⎣ ⎦

3 31 ( 2 )6D D t⎡ ⎤− −⎣ ⎦

W = elastic section modulus

Z = plastic section modulus

Allowable bending stress for tubular members ISO/DIS 19902/NNORSOK N_004

Plastic thin-walled

Yield

Yield stress D/t plas>c limit D/t "Elas>c" 235 46 92 355 31 61 420 26 52

Allowable bending stress versus D/t-ratio (yield stress 355 MPa)

•  Fully plastic bending moment achieved only for D/t < 30

0"

0.2"

0.4"

0.6"

0.8"

1"

1.2"

1.4"

1.6"

0" 20" 40" 60" 80" 100" 120"

Allowab

le(ben

ding(stress(fm

/fY%

Diameter/thickness(ra6o(D/t(

Interaction equation for tubular beam-columns subjected to axial force and

bending moment (ISO, NORSOK)

0.1

NN1

MC

NN1

MCM1

NN

5.02

Ez

Sd

Sdz,mz

2

Ey

Sd

Sdy,my

RdRdc,

Sd ≤

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+

Table 6-2 Effective length and moment reduction factors for member strengthcheckingStructural element k Cm

(1)

Superstructure legs - Braced 1.0 (a) - Portal (unbraced) k(2) (a)Jacket legs and piling - Grouted composite section 1.0 (c) - Ungrouted jacket legs 1.0 (c) - Ungrouted piling between shim points 1.0 (b)Jacket braces - Primary diagonals and horizontals 0.7 (b) or (c) - K-braces(3) 0.7 (c) - Longer segment length of X-braces(3) 0.8 (c)Secondary horizontals 0.7 (c)

Notes:1. Cm values for the cases defined in Table 6-2 are as follows:

(a) 0.85(b) for members with no transverse loading,Cm = 0.6 - 0.4 M1,Sd/M2,Sdwhere M1,Sd/M2,Sd is the ratio of smaller to larger moments at the ends of that portion of the memberunbraced in the plane of bending under consideration. M1,Sd/M2,Sd is positive when the number is bentin reverse curvature, negative when bent in single curvature.(c) for members with transverse loading,Cm = 1.0 - 0.4 Nc,Sd/NE, or 0.85, whichever is less, and NE = NEy or NEz as appropriate.

2. Use Effective Length Alignment Chart in Commentary.

3. At least one pair of members framing into the a K- or X-joint must be in tension if the joint is notbraced out-of-plane. For X-braces, when all members are in compression, the k-factor should bedetermined using the procedures given in the Commentary.

4. The effective length and Cm factors given in Table 6-2 do not apply to cantilever members and themember ends are assumed to be rotationally restrained in both planes of bending.

Buckling coefficient for X-braces dependeing on force condition and rotational end support

Capacity checks of deep water tubular members

Stresses in circular cross-sections for external hydrostatic pressure

Hoop stress from external hydrostatic pressure

h=pr/2t h Axial stress from capped end forces x=0.5 h q,Sd

p

Hoop buckling •  General formula

–  see buckling of cylindrical shells

Hoop buckling strength of fabricated cylinders subjected to hydrostatic pressure- test data compared with design equation (from ISO19902)

y

he

ff

λ =

fhe = 2ChEtD

⎛⎝⎜

⎞⎠⎟

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0 1 2 3 4 5 6 7

ISO

Miller & Kinra - fabr.+ rings

Miller, Kinra & Marlow - fabr.+rings

Miller, Kinra & Marlow - fabr. unstif.

Eder et al - fabr.+ rings

Eder et al- ERW + rings

Kiziltug et al - ERW unstif.

Steinmann & Vojta - ERW unstif.

Fy/Fhe

fh/Fy

Hoop buckling resistance versus D/t-ratio

•  Notice the rapid decrease of buckling strength with increasing D/t-ratio •  For D/t > 30 buckling is predominantly elastic (yield stress 355 MPa)

0"

0.2"

0.4"

0.6"

0.8"

1"

0" 20" 40" 60" 80" 100" 120" 140"

Norm

alise

d+ho

pp+buckling+s

tress+

Diameter6thickness+ra7o+

Hoop+buckling+stress+versus+diameter6thickness+ra7o+

Elas-c"buckling"stress"f_he/f_Y"

Cri-cal"buckling"stress"f_h/f_Y"

Interaction between column buckling,local buckling and hydrostatic pressure

fch

fcl

⎛

⎝⎜⎞

⎠⎟

2

−fc

fcl

−2σ x

fcl

⎛

⎝⎜⎞

⎠⎟fch

fcl

+σ x

fcl

σ x

fcl

−1⎛

⎝⎜⎞

⎠⎟= 0, λ < 1.34 (1−

2σ x

fcl

)⎡

⎣⎢

⎤

⎦⎥

−1

fch

fcl

= 12

[ξ −2σ x

fcl

+ ξ 2 +1.12λ 2 σ x

fcl

], λ <1.34 (1−2σ x

fcl

)⎡

⎣⎢

⎤

⎦⎥

−1

Interaction local buckling –hydrostatic pressure

0

50

100

150

200

250

300

350

400

450

500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Wat

er d

epth

[m]

f_ch

/f_cl

Sigma_x/f_cl

Interaction between hydrostatic pressure and column buckling

lamda=0.5

lamda=0.7

lamda=0.9

lamda=1.1

lamda=1.3

lamda=1.5

D/t = 30

D/t = 40

D/t = 50

D/t = 60

Interaction equations 1) Combined tension and hydrostatic pressure 2) Combined bending and hydrostatic pressure (shown below)

Characteristic bending stress modified for hydrostatic pressure

Bending resistance with no external hydrostatic pressure

Design hoop buckling resistance

fmh

fm

= 1+ 0.09σh

fh

⎛

⎝⎜⎞

⎠⎟

2

−σh

fh

⎛

⎝⎜⎞

⎠⎟

2η

− 0.3σh

fh

⎛

⎝⎜⎞

⎠⎟

fmhfm

⎛

⎝⎜⎞

⎠⎟

2

+ 2vfmhfm

⎛

⎝⎜⎞

⎠⎟σhfh

⎛

⎝⎜⎞

⎠⎟+

σhfh

⎛

⎝⎜⎞

⎠⎟

2η

= 0 η = 5 − 4fhf y

Interaction between bending and hydrostatic pressure

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

f_m

h/f_

m

Sigma_x/f_h

f_h/f_y =0.2

f_h/f_y =0.4

f_h/f_y =0.6

f_h/f_y =0.8

f_h/f_y =1.0

fmh

fm

⎛

⎝⎜⎞

⎠⎟

2

+ 2vfmh

fm

⎛

⎝⎜⎞

⎠⎟σh

fh

⎛

⎝⎜⎞

⎠⎟+

σh

fh

⎛

⎝⎜⎞

⎠⎟

2η

= 0 η = 5 − 4fh

f y

fmh

fm

= 1+ 0.09σh

fh

⎛

⎝⎜⎞

⎠⎟

2

−σh

fh

⎛

⎝⎜⎞

⎠⎟

2η

− 0.3σh

fh

⎛

⎝⎜⎞

⎠⎟

Reduction of characteristic bending stress versus water depth

0

50

100

150

200

250

300

20 30 40 50 60 70 80 90 100

Diameter/thickness

Wat

erde

pth

(m)

100 %10 %5 %

2 %

Reduction in bending strength as a function of water depth and D/t-ratio

Interaction equation for beam-columns with external hydrostatic

pressure

•  Two formulations given, depending on capped end forces included or not

•  Here, only the formulations for capped end forces NOT included are shown

σ a

fch+ 1fmh

Cmyσmy

1− σ a

fEy

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

2

+ Cmzσmz

1− σ a

fEz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

2⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

0.5

≤1.0

Interaction equation for tubular beam-columns subjected to axial compression, bending moment and external hydrostatic pressure

(capped end forces NOT included).

Axial compressive strength in the presence of external hydrostatic pressure

Bending resistance in the presence of external hydrostatic pressure

Yield check for tubular beam-columns subjected to axial compression, bending moment

and external hydrostatic pressure (capped end forces NOT included).

σ a +σ x

fcl+

σ my2 +σ mz

2

fmh≤1.0

Local buckling strength not affected by external hydrostatic pressure


Yield check for tubular beam-columns subjected to axial tension, bending moment and

external hydrostatic pressure (capped end forces NOT included).

σ a −σ x

fth+

σmy2 +σmz,

2

fmh≤1.0

Axial tensile resistance in the presence of external hydrostatic pressure


σ ac ≥σ x (net axial tension)

Capacity check for tubular beam-columns subjected to axial tension, bending moment and external hydrostatic pressure

(capped end forces NOT included).

σ a −σ x

fcl+

σ my2 +σ mz

2

fmh≤1.0

Local buckling resistance


σ a,Sd <σ q,Sd (net axial compression)

The material factor for beam-column checks depends on the utilisation wrt local buckling and

hoop buckling

0.1λfor1.45γ

1.0λ0.5forλ0.600.85γ

0.5λfor 1.15γ

sM

ssM

sM

>=

≤≤+=

<=

NORSOK standard N-004 Rev. 3, February 2013

NORSOK standard Page 19 of 264

0.1Ȝfor1.45Ȗ

1.0Ȝ0.5forȜ0.600.85Ȗ

0.5Ȝfor 1.15Ȗ

sM

ssM

sM

!

dd�

�

( 6.22)

where

h

2

h

Sdp,c

c

Sdc,s Ȝ

fı

Ȝf

ıȜ �¸

¸¹

·¨¨©

§��

l

( 6.23)

where fcl is calculated from Equation ( 6.6) or Equation ( 6.7) whichever is appropriate and fh from Equation ( 6.17), Equation ( 6.18), or Equation ( 6.19) whichever is appropriate.

ec

yc f

fȜ

l

, and he

yh f

fȜ

( 6.24)

fcle and fhe is obtained from Equation ( 6.8), and Equation ( 6.20) respectively. Vp,Sd is obtained from Equation ( 6.16) and

W

MM

ANı

2Sdz,

2Sdy,Sd

Sdc,

��

( 6.25)

NSd is negative if in tension.

6.3.8 Tubular members subjected to combined loads without hydrostatic pressure

6.3.8.1 Axial tension and bending Tubular members subjected to combined axial tension and bending loads should be designed to satisfy the following condition at all cross sections along their length:

0.1M

MM

NN

Rd

2Sdz,

2Sdy,

75.1

Rdt,

Sd d�

�¸¸¹

·¨¨©

§

( 6.26)

where

My,Sd = design bending moment about member y-axis (in-plane) Mz,Sd = design bending moment about member z-axis (out-of-plane) NSd = design axial tensile force

If shear or torsion is of importance, the bending capacity MRd needs to be substituted with MRed,Rd calculated according to subclause 6.3.8.3 or 6.3.8.4.

6.3.8.2 Axial compression and bending

Tubular members subjected to combined axial compression and bending should be designed to satisfy the following condition accounting for possible variations in cross-section, axial load and bending moment according to appropriate engineering principles:

0.1

NN

1

MC

NN

1

MC

M1

NN

5.02

Ez

Sd

Sdz,mz

2

Ey

Sd

Sdy,my

RdRdc,

Sd d

°°

¿

°°

¾

½

°°

¯

°°

®

»»»»

¼

º

««««

¬

ª

��

»»»»

¼

º

««««

¬

ª

��

( 6.27)

and at all cross sections along their length:



0.1Ȝfor1.45Ȗ

1.0Ȝ0.5forȜ0.600.85Ȗ

0.5Ȝfor 1.15Ȗ

sM

ssM

sM

!

dd�

�

( 6.22)

where

h

2

h

Sdp,c

c

Sdc,s Ȝ

fı

Ȝf

ıȜ �¸

¸¹

·¨¨©

§��

l

( 6.23)


ec

yc f

fȜ

l

, and he

yh f

fȜ

( 6.24)


W

MM

ANı

2Sdz,

2Sdy,Sd

Sdc,

��

( 6.25)




0.1M

MM

NN

Rd

2Sdz,

2Sdy,

75.1

Rdt,

Sd d�

�¸¸¹

·¨¨©

§

( 6.26)

where





0.1

NN

1

MC

NN

1

MC

M1

NN

5.02

Ez

Sd

Sdz,mz

2

Ey

Sd

Sdy,my

RdRdc,

Sd d

°°

¿

°°

¾

½

°°

¯

°°

®

»»»»

¼

º

««««

¬

ª

��

»»»»

¼

º

««««

¬

ª

��

( 6.27)




0.1Ȝfor1.45Ȗ

1.0Ȝ0.5forȜ0.600.85Ȗ

0.5Ȝfor 1.15Ȗ

sM

ssM

sM

!

dd�

�

( 6.22)

where

h

2

h

Sdp,c

c

Sdc,s Ȝ

fı

Ȝf

ıȜ �¸

¸¹

·¨¨©

§��

l

( 6.23)


ec

yc f

fȜ

l

, and he

yh f

fȜ

( 6.24)


W

MM

ANı

2Sdz,

2Sdy,Sd

Sdc,

��

( 6.25)




0.1M

MM

NN

Rd

2Sdz,

2Sdy,

75.1

Rdt,

Sd d�

�¸¸¹

·¨¨©

§

( 6.26)

where





0.1

NN

1

MC

NN

1

MC

M1

NN

5.02

Ez

Sd

Sdz,mz

2

Ey

Sd

Sdy,my

RdRdc,

Sd d

°°

¿

°°

¾

½

°°

¯

°°

®

»»»»

¼

º

««««

¬

ª

��

»»»»

¼

º

««««

¬

ª

��

( 6.27)




0.1Ȝfor1.45Ȗ

1.0Ȝ0.5forȜ0.600.85Ȗ

0.5Ȝfor 1.15Ȗ

sM

ssM

sM

!

dd�

�

( 6.22)

where

h

2

h

Sdp,c

c

Sdc,s Ȝ

fı

Ȝf

ıȜ �¸

¸¹

·¨¨©

§��

l

( 6.23)


ec

yc f

fȜ

l

, and he

yh f

fȜ

( 6.24)


W

MM

ANı

2Sdz,

2Sdy,Sd

Sdc,

��

( 6.25)




0.1M

MM

NN

Rd

2Sdz,

2Sdy,

75.1

Rdt,

Sd d�

�¸¸¹

·¨¨©

§

( 6.26)

where





0.1

NN

1

MC

NN

1

MC

M1

NN

5.02

Ez

Sd

Sdz,mz

2

Ey

Sd

Sdy,my

RdRdc,

Sd d

°°

¿

°°

¾

½

°°

¯

°°

®

»»»»

¼

º

««««

¬

ª

��

»»»»

¼

º

««««

¬

ª

��

( 6.27)


Local buckling Hoop buckling



M

yRdSd

32

fAVV

J d

( 6.13)

where VSd = design shear force fy = yield strength A = cross sectional area JM = 1.15

Tubular members subjected to shear from torsional moment should be designed to satisfy the following condition:

M

ypRdT,SdT,

3D

f2IMM

J d

( 6.14)

where MT,Sd = design torsional moment

Ip = polar moment of inertia = > @44 )t2D(D32

��S

6.3.6 Hydrostatic pressure

6.3.6.1 Hoop buckling Tubular members subjected to external pressure should be designed to satisfy the following condition:

M

hRdh,Sdp,

ff

JV d

( 6.15)

t2DpSd

Sdp, V

( 6.16)

where fh = characteristic hoop buckling strength Vp,Sd = design hoop stress due to hydrostatic pressure (compression positive) pSd = design hydrostatic pressure JM = see 6.3.7

If out-of-roundness tolerances do not meet the requirements given in NORSOK M-101, guidance on calculating reduced strength is given in Clause 12.

,ff yh yhe f44.2ffor ! ( 6.17)

4.0

y

heh f

ff7.0f

»»¼

º

««¬

ª y for yhey f55.0f2.44f !t

( 6.18)

,ff heh yhe f55.0ffor d ( 6.19)

The elastic hoop buckling strength, fhe, is determined from the following equation:

DtEC2f hhe

( 6.20)

where Ch = 0.44 t/D for P t 1.6D/t = 0.44 t/D + 0.21 (D/t)3/P4 for 0.825D/t d P < 1.6D/t = 0.737/(P - 0.579) for 1.5 d P < 0.825D/t

Resistance of columns and beam- columns - IV - · PDF fileResistance of columns and...

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