SOLUTION 4 & 5

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NTNU

Faculty of
Engineering Science and Technology

Department of Marine
Technology

SOLUTION
4

& 5

TMR 4195
DESIGN OF OFFSHORE STRUCTURES

Problem 1


In the ULS control

it is necessary to
consider the load cases given

in

Table
1
.




Functional & permanent,

F

Environmental,

E

ULS
-
a

1.3

0.7

ULS
-
b

1.0

1.3

Table
1
: ULS load factors

1.15
  

  
V V E E m
m
R
S S


a)

The axial force in the brace can be calculated from simple considerations of the
trusswork. We assume that the shear force in the main girder is carried
by

axial forces in
the vertical and inclined braces. Nearby the column (at brace A), some of the shear force
will in reality be carried by the upper and lower chord. We conservatively neglect this
effect and calculate the axial force
S
A

in brace A
by
,


1
2 sin45
2
 
A
S qL

(2 braces in
the trusswork carry shear force
)

V F E E
q q q
 
 


We assume that the
functional and permanent loads are related to weight of equipment
and
weight of deck structure, such that i
nertia forces
caused by the

vertical
wave
-
induced
acceleration will be evenly distributed along the main girder

with intensity

given by
,


2.5
9.81
E F
q q



ULS
-
a can be found to give the most critical load condition

with the following axial force
in brace A,


0.523
A F
S q L
 


The brace must be checked for
beam
-
column
buckling failure, i.e.
i
nteraction
between
axial compressive force and

bending moments
should be included in the assessment
. The
bending moments cannot be fo
und by simple hand calculations, and for
simplicity we
neglect the effect of bending moments and do a column buckling check instead. This is
non
-
conservative and should be avoided in a more accurate assessment.



The procedure
in the DNV Classification Notes 30.1, which is
used to calculate the

c
haracteristic column
buckling resistance
,

is outlined in the following
.
The input
parameter in this

procedure is the reduced slenderness ratio,


2 2
2
 



Y
A l
EI
,


where A is the cross
-
sectional area, β is the buckling length factor,
l

is the
bra
ce
length
and
I

is the 2.area moment.


Based on the boundary conditions at the ends of brace A
,

it is reasonable to assume that

β=
0.7.
However, a
ccording to
the
DNV Classification note

30.1

a design value of
β=
0.8
should be

used instead
.


The internal widt
h of the cross
-
section is denoted
b
i

and the thickness as
t
. Area and
2.area moment is then expressed as,


4

i
A bt



4
4
1
2
12
 
  
 
i i
I b t b


The cross
-
section is
assumed to be “hot finished” and therefore
assi
gned buckling curve a,
see

DNV C
lassification
N
ote
s

30.1. Then the characteristic buckling stress can be
expressed as,







2
2 2 2
2
1 0.2 0.2 1 0.2 0.2 4
2
CR Y
    
 

 
       
 

 
 
 


A cross
-
section width,
b
i
, which fulfills the following
relationship

must be selected,




0.131 1.15
F
A m CR i A m
i
q L
b
bt
   
   


The equations above

are included in EXCEL to determine the required cross
-
section
width. Here, it was found that a
n internal

width/height

of
4
50 mm will give suf
ficient
capacity against column
buckling. See
Table
2
.


450 mm
i
b






b
i

[mm]

I

[mm
4
]

A
[mm
2
]

λ

[
-
]

σ
cr

[Mpa]

σ
a

*
γ
m

[Mpa]

σ
a

*
γ
m

/

σ
cr

[
-
]

t
[mm]

30


50

1.17E07

6000

4.03

20.8

2169.4

153.53

σ
Y

[MPa]

355


100

4.63E07

12000

2.86

40.4

1084.7

36.05

l
[mm]

16971


150

1.2E08

18000

2.18

67.9

723.1

13.66

E

[MPa]

2.10E05


200

2.47E08

24000

1.75

101.5

542.3

6.63

β
[
-
]

0.8


250

4.44E08

30000

1.46

139.3

433.9

3.75

L
[m]

72


300

7.25E08

36000

1.25

178.3

361.6

2.35




350

1.1E09

42000

1.10

214.6

309.9

1.60




400

1.6E09

48000

0.97

245.1

271.2

1.16




450

2.22E09

54000

0.88

268.4

241.0

0.89




500

2.99E09

60000

0.80

285.5

216.9

0.72

Table
2
: Determination of cross
-
section width
/height

for brace A



b)

It is not obvious which ULS load condition
that will be the critical one. Both ULS
-
a and
ULS
-
b must

hence

be checked. The

stress resultants
and external pressure
in the two
ULS

conditions are

calculated as
,


F
p gh



F F E E
N N N
 
 

F F E E
M M M
 
 

F F E E
Q Q Q
 
 


The pressure is calculated at a depth corresponding to 20 m.
With the load factors in
Table
1

the two load conditions are given by,



p

[Pa]

N

[MN]

M

[MN]

Q

[MN]

ULS
-
a

2.61
E
5

79

76

26.5

ULS
-
b

2.01
E
5

76

127

28

Table
3
: ULS stress resultant in buckling check


The stresses in the cylindrical shell structure must be
determined. According to the DNV
-
RP
-
C202 code, it is usu
ally permissible to account for the longitudinal stiffeners
by an
equivalent
shell
thickness when
axial

membrane

stresses are calculated,


S
eq
A
t t
s
 

eq eq
A Dt




3
8
eq eq
I D t




Here
t

is the shell thickness,
A
s

is the cross
-
sectional area of the stiffener
without

shell
plating
,
D

is column diameter

and
s

is the stiffener spacing.


The axial
stress
and bending stress

at “outer fiber”

can then be calculated as,


a
eq
N
A



max
2
b
eq
MD
I




The shear stress will be zero at the location where maximum bending stress occurs. At the
“neutral axis” of the column
, where the bending stress is zero
, maximum shear stre
ss
is
given by,


max
2
Q
Dt





The hoop
(circumferential)
stress is calculated a
s
,



2
h
pD
t






According to the DNV
-
RP
-
C202 code it is necessary to consider 3 buckling modes for
longitudinally stiffened shells.




Shell buck
ling



Panel stiffener buckling



C
olumn buckling







Shell buckling


Panel stiffener buckling

Column buckling




S
hell

buckling
can
be trigged by th
ree different stress components.





A
xial membrane stress

caused by

axial
compressive force
and

compressive
bending stress in the column.



S
hear stresses from
torsion or
shear force

in the column.



H
oop stresses c
aused by net external pressure.


The
three
elastic shell buckling stress
es

are given by formulas valid from curved she
ll
theory. i.e.
we focus on
the shell plating
enclosed by two
longitud
inal stiffeners
.

The
formulas are expressed in the following form,



2
2
2
,,
12(1 )
Ex E Eh
E t
C
s


  

 

 

 







1
C



 


Here
E

is Young’s modulus,
ν

is Poisson’s ratio,
s

is the stiffener spacing and
t

is the shell
thickness.
The formula for C is obtained by an elliptic interpolation of the asymptotic
solutions for a flat plate and a curved shell. The parameter ψ accounts for the flat plate

solution
, while ξ accounts for t
he curved shell

solution
.
Cylindrical s
hells are sensitive for
initial
imperfections and will never reach the elastic buckling stress

for an ideal cylinder
.
Therefore a knock
-
down factor

of typically 0.5
-
0.6
, expressed by ρ, is applied to the shell
part of

the solution.

The parameters ψ, ξ and ρ depend on geometrical quantities

of the
shell, s
ee the DNV
-
RP
-
C202 code
Sec 3.3
included in the exercise appendix for further
details
.


Panel stiffener buckling is caused by the following effects




Compressive
membrane axial stress due to bending and axial force in the column



Shear stresses due to torsion or shear force in the column



Lateral load due to net external pressure


Panel stiffener
b
uckling stress
es

in the DNV
-
RP
-
C202 code
are expressed
by formulas
bas
ed on ort
h
otr
op
ic shell theory where the stiffener
s

are

s
meared over the shell
thickness.

Orthotropic shell theory is in principle only valid if the stiffeners are “light”
and if the spacing is small (Compendium in “TMR4205
-

Buckling and Ultimate Strength
Analysis of Marine Structures”).
T
he elastic buckling stresses are given by formulas on
a

similar

form as for
the
shell buckling

problem
,


2
2
2
,,
12(1 )
Ex E Eh
E t
C
l


  

 

 

 





1
C



 


Here
l

is the length of the stiffener, and
C

accounts for the same effects as described for
shell buckling. Since the stiffeners are smeared over

the shell thickness,
the parameters
involved in
C depends
also
on the stiffener effective 2.area moment,
I
eff
, which is
defined

by the

effective shell flange,
s
e
.

See

Figure
1
.



Figure
1
: Effective shell flange


In this problem the effective shell flange was set equal to 275 mm.
The distance
e
z
,
stiffener area,
A
s

and stiffener 2.area mom
ent,
I
s
, are

found

in

Fi
gure A in the exercise
appendix.

I
eff

can then be calculated by,



e
z
e S
s t
d
s t A





2
3 2
1
0.5
12
eff e z z e s z s
I s t e t d s t I d A
     




In the following the procedure
recommended

in the DNV
-
RP
-
C202
for buckling strength
assessment

is outlined.
This calculation procedure must be performed separately for both
the shell buckling check and the panel stiffener buckling check.


There will be interaction between the stress components which cause buckling. This
interaction is

taken into account
in
a

for
mula for the equivalent reduced slenderness,


2
a b h
Y
eq
j Ex Ex E Eh

  



    
 
   
 
 
,


w
here
σ
Y

is the yield strength,
σ
a

is axial compressive stress from axial force, σ
b

is
compressive bending stress, τ is shear stress and σ
h

is hoop stress from net external
pressure. If any of the
normal
stress components are tensile, they must be set equal to zero
in the equation for the
equivalent
reduced slenderness.


P
lasticity is accounted for by the equivalent stress, σ
j
,









2 2 2
2
3
j b a h b a h
       
         


Note that
the normal stress components are not set equal to zero if they are tensile

in the
equivalent stress equation
.



Thereafter,
the DNV
-
RP
-
C202 code
calculate
the design buckling strength

as
,


4
1
Y
ksd
m eq


 




Where the mater
ial factor
γ
m

depends on the
equivalent
reduced slenderness,


1.15 for 0.5
0.85 0.60 for 0.5 < 1.0
1.45 for 1.0
eq
m eq eq
eq

  





  







Finally, the buckling capacity is found acceptable if the equivalent stress is less than the
design buckling strength,


j ksd
 




The shell
buckling check and panel stiffener buckling check must be performed for both
ULS
-
a and ULS
-
b.
T
wo locations of the column are
checked in the buckling assessment
.




T
he
“outer column fibre”
where
maximum compressive

bending stress
occurs.



At the neutral axis

where

maximum shear stress is found.



In the following pages

results from the buckling control
are

presented.

The key
observation
from the analysis
can be summarized as,




Shell buckling at neutral axis comes out to be the critical point
, and

here

scantlings of
the L
-
stiffeners
give

minor effect on
the
shell buckling strength.
The
buckling control at this location therefore determines the m
inimum shell thickness
which can be used.



If the stiffener scantlings are increased, a
pronounced

increase of e
quivalent shell
thickness occurs. This is negative for material costs and structural weight.



The utilization
of buckling capacities

in ULS
-
a and ULS
-
b
are

almost equal.


Based on this, the design which gives optimal material costs must be based on the L200
-
stiffeners and a shell thickness of 21 mm.

See results in
the
next

pages.

Stress analysis
based on
equivalent shell thickness


Stiffener

A
S

[mm2]

t
eq

[mm]

I
eq

[mm4]

σ
a

[Mpa]

σ
bmax

[Mpa]

τ
max

[Mpa]

σ
h

[Mpa]

L200

3.51E+03

26.4

1.04E+13

95.3

36.7

80.3

62.1

L300

4.90E+03

28.5

1.12E+13

88.1

33.9

80.3

62.1

L400

7.75E+03

32.9

1.29E+13

76.4

29.4

80.3

62.1

Table
4
:
Stress analysis ULS
-
a




Shell buckling control

















stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

162.4

1.01

1.45

172.4

L300

159.7

1.00

1.45

173.3

L400

155.9

0.98

1.44

175.8

Table
7
: Buckling control at neutral axis of column, ULS
-
a

stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

114.3

1.16

1.45

159.6

L300

105.7

1.19

1.45

157.2

L400

92.1

1.25

1.45

153.0

Table
8
:
Buckling control at

outer fibre of column
, ULS
-
a

stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

167.1

0.85

1.36

198.6

L300

164.4

0.83

1.35

201.9

L400

160.6

0.81

1.33

207.4

Table
9
:
Buckling control
at neutral axis of column, ULS
-
b

Stiffener

A
S

[mm2]

t
eq

[mm]

I
eq

[mm4]

σ
a

[Mpa]

σ
bmax

[Mpa]

τ
max

[Mpa]

σ
h

[Mpa]

L200

3.51E+03

26.4

1.04E+13

91.6

61.3

84.9

47.9

L300

4.90E+03

28.5

1.12E+13

84.8

56.7

84.9

47.9

L400

7.75E+03

32.9

1.29E+13

73.5

49.1

84.9

47.9

Table
5
: Stress analysis ULS
-
b

failure
by

ψ
[
-
]

ρ

[
-
]

Z
s

[
-
]

ξ

[
-
]

C
[
-
]

σ
E

[Mpa]

axial stress

4

0.50

3.84

2.69

4.22

836.2

Shear stress

5.53

0.60

3.84

1.09

5.57

1102.8

Hoop stress

1.10

0.60

3.84

0.44

1.13

223.4

Table
6
: Shell elastic buckling stress


stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

135.5

1.04

1.45

169.6

L300

124.6

1.09

1.45

165.3

L400

107.0

1.20

1.45

157.0

Table
10
:
Buckling control

at outer fibre of column, ULS
-
b



Panel stiffener buckling control













Table
12
: Elastic
panel L3
00
-
stiffener buckling stresses






Table
13
: Elastic panel L4
00
-
stiffener buckling stresses



stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

162.4

1.09

1.45

165.5

L300

159.7

0.86

1.37

196.7

L400

155.9

0.70

1.27

228.8

Table
14
:
Buckling control
at neutral axis of column, ULS
-
a

stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

114.3

1.22

1.45

155.2

L300

105.7

0.95

1.42

181.0

L400

92.1

0.78

1.32

211.4

Table
15
:
Buckling control

at outer fibre of column, ULS
-
a

failure
by

α
C

[
-
]

Z
t

[
-
]

ψ

[
-
]

ξ

[
-
]

ρ
[
-
]

C [
-
]

σ
ksd

[Mpa]

axial stress

106.9

81.8

67.1

57.4

0.5

73.0

679.0

Shear stress

106.9

81.8

71.7

23.3

0.6

73.1

679.6

Lateral pressure

106.9

81.8

22.8

9.4

0.6

23.5

218.2


Table
11
: Elastic panel L200
-
stiffener buckling stresses

failure
by

α
C

[
-
]

Z
t

[
-
]

ψ

[
-
]

ξ

[
-
]

ρ
[
-
]

C [
-
]

σ
ksd

[Mpa]

axial stress

281.6

81.8

152.9

57.4

0.5

155.5

1446.5

Shear stress

281.6

81.8

97.0

23.3

0.6

98.0

911.5

Lateral pressure

281.6

81.8

35.6

9.4

0.6

36.1

335.4

failure
by

α
C

[
-
]

Z
t

[
-
]

ψ

[
-
]

ξ

[
-
]

ρ
[
-
]

C [
-
]

σ
ksd

[Mpa]

axial stress

692.8

81.8

296.2

57.4

0.5

297.6

2768.0

Shear stress

692.8

81.8

129.1

23.3

0.6

129.8

1207.6

Lateral pressure

692.8

81.8

54.7

9.4

0.6

55.0

511.2


stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

167.1

1.01

1.45

172.3

L300

164.4

0.80

1.33

209.0

L400

160.6

0.65

1.24

240.3

Table
16
:
Buckling control at neutral axis of column,
ULS
-
b


stiffener

σ
j

[Mpa]



[
-
]

γ
m

[
-
]

σ
ksd

[Mpa]

L200

135.5

1.08

1.45

166.4

L300

124.6

0.83

1.35

203.1

L400

107.0

0.68

1.26

234.2

Table
17
:
Buckling control at outer fibre of column, ULS
-
b






Column buckling should according to

the

DNV
-
RP
-
C202

code

also be checked if,


2
2.5
Y
column
A E
kL
I

 

 
 
 


Here
k

is the buckling length factor,
A

is the cross
-
sectional area, 2.area moment

is
denoted
I
,
and
L

is
the column length
. Since we have no information of the structural
design of the hull and the platform deck, we c
onservatively assume that all 4 columns will
buckle simultaneously in a side
-
sway

mechanism.
The buckling length factor is
therefore
set equa
l to 2
,

and

w
ith the L200
-
stiffeners
and a shell thickness of 21 mm
this gives,


2
2
3
3
13 4
26.4 10 10
2 35 10 392
1.04 10
column
A mm mm
kL mm
I mm

 
 
  
    
 
 
 
 

 
 


3
210 10
2.5 2.5 1479
355
Y
E MPa
MPa


  


Therefore it is

not necessary to assess the column
buckling mode.








c)

Brace A has a plate thickness of 30 mm and the service temperature is 0 °C.
The brace

is

considered to be

part of the primary structure

since it transfer

global forces from the deck
loads, and is t
herefore

assigned steel grade
B

a
ccor
ding to
Table
D1, D2 and
D3 in DNV
-
OS
-
C101 Sec. 4.


The calculated column shell has thickness 21 mm

and is a primary member.

S
teel grade
B

can
therefore
be used here.



d)

Accordin
g to Sec. 3.3.12 in DNV
-
RP
-
C203

the stress concentration facto
r,
SCF
,
for the
joint can be set equal to 2.9 since we have a gusset plate with favourable geometry
.
The
brace cross
-
sectional area was found to be 54

000 mm
2

in problem 1a).
The largest

stress
range in the joint during 20 years is then equal to,

3
20
2
2.9 1200 10
2 2 129
54000
brace
yr
brace
SCF N
N
MPa
A mm


 
     


A type F1

SN
-
curve should be used
. There is no fatigue limit since the environment is
corrosive, and the SN
-
curve is therefore one
-
sloped with m=3.0,

0.25
30
log log11.699 3 log
25
mm
N
mm

 
 
   
 
 
 
 
 






The parameters needed in the fatigue calculations

are

then
,



log log11.640 3log log 11.640 , 3.
0
N a m

     


The fatigue damage is given by,

1
m
D
Z
T
q m
D
T a h
 
  
 
 




Here
T
D

[s]

is the design life and
T
Z

[s]
is the zero up
-
crossing period.
As described in
DNV
-
RP
-
C203 the q
-
parameter is calculated by,


1/1.1
129
9.126
3600 24 365 20
ln
6.3
q
 
    
 
 
 
 
 


The
accumulated
fatigue damage can then be calculated,

3
11.64
3600 24 365 25 9.126 3 1.0
1 0.94<
6.3 10 1.1
D
DFF
  
 
     
 
 


The accumulated fatigue damage
,
D
,

multiplied with a design fatigue factor,
DFF
, must
be less than 1.0. According to DNV
-
OS
-
C101 Sec. 6, the
DFF

is equal to 1.0 for an
external structural member that is accessible for inspection and repair in dry and clean
conditions. Therefore, the fatigue life is sufficient for a service life of 25 years.

e)

The yield stress
in brace A
is increased by
50

% and the utilization with respect to
buckling shall

remain unchanged. Utilization with respect to
buckling
calculated

in
problem

1a) is found in
Table
2
,


0.89
m a
cr



 

By inserting a yield stress of 532.5 MPa in the
Excel

work
book, and thereafter adjusting
the thickness, it
can be

found that a plate thickness of 25 mm gives the same utilization
ratio with respect to buckling

as in problem 1a).





b
i

[mm]

I



[mm
4
]

A
[mm
2
]

λ

[
-
]

σ
cr

[Mpa]

σ
a

*
γ
m

[Mpa]

σ
a

*
γ
m

/

σ
cr

[
-
]

t
[mm]

25


50

7.81E+06

5000

5.51

17.0

2603.2

153.53

σ
Y

[MPa]

532.5


100

3.39E+07

10000

3.74

36.1

1301.6

36.05

l
[mm]

16971


150

91145833

15000

2.79

63.5

867.7

13.66

E

[MPa]

2.10E05


200

1.92E+08

20000

2.22

98.2

650.8

6.63

β
[
-
]

0.8


250

3.49E+08

25000

1.84

139.0

520.6

3.75

L
[m]

72


300

5.76E+08

30000

1.57

184.4

433.9

2.35




350

8.83E+08

35000

1.37

232.3

371.9

1.60




400

1.28E+09

40000

1.21

279.8

325.4

1.16




450

1.79E+09

45000

1.09

323.8

289.2

0.89




500

2.42E+09

50000

0.99

361.7

260.3

0.72

Table
18
: Modified plate thickness of brace A


It is assumed that

the
change in

stiffness of
members in
the deck structure members
does
not change
the

global
force transfer

of the wave loads
. The largest stress range in the joint
during 20 years is then equal to,

3
20
2
2.9 1200 10
2 2 155
45000
brace
yr
brace
SCF N
N
MPa
A mm


 
     


The
new
q
-
parameter
and the fatigue damage is equal to
,

1/1.1
155
10.965
3600 24 365 20
ln
6.3
q
 
    
 
 
 
 
 

3
11.699
3600 24 365 25 10.965 3
1 1.42
6.3 10 1.1
D
  
 
     
 
 


The accumulated fatigue damage is larger than 1.0, and the fatigue life of the joint must
thus
be improved. This can be done by applying weld improvement methods such as
grinding, heat relief

methods
,
peening, re
dressing of welds etc. These methods are
exp
ensive, and the best option is
usually

to change the joint design
.


In general, a large

increase

of

yield stress
should

never be accepted
without reassessment
of the

fatigue life.