Lecture 6. Instability of members 2021_MP_new.pdf

Design of offshore structures and foundations – Prof. C. Mazzotti, Dr. Palermo
1
Claudio Mazzotti
DICAM
University of Bologna
INSTABILITY PROBLEMS OF MEMBER
ELEMENTS

2
Buckling Phenomena
Buckling is a mode of failure generally resulting from structural instability due to
compressive action on the structural member or element involved.

3
Buckling of beam in the jacket.
Buckling in the member for K bracing.
Buckling of the tubular joint.

4
Buckling Phenomena
1. Global buckling:
- Flexural;
- Torsional;
- Torsional-flexural
2. Local buckling of thin-walled members;

5
Column Buckling: flexural buckling
P<
P<
 Buckling occurs when a straight,
homogeneous, centrally loaded column
subjected to axial compression
suddenly undergoes bending.
 Buckling is identified as a failure limit-
state for columns.
 The value of P at which a straight
column becomes unstable is called the
Critical Load.
 When column bends at critical load, it
is said to have buckled.
 Therefore, critical load is also called
the buckling load.

6
 The column will go back to its original straight position. Just as the ball returns to the bottom of
the container.
 Gravity tends to restore the ball to its original position while in columns elasticity of column itself
acts as a restoring force.
 This action constitutes STABLE EQUILIBRIUM.
 The same procedure can be repeated with increased load until some critical value is reached.
CASE 1: P<PCR
1. Axial load P is applied to the column;
2. The column is then given a small deflection by giving a small force F;
3. If the force P is sufficiently small, when the force F is removed, the column will go back to its original
straight position.

7
 The amount of deflection depends on amount of force F.
 The column can be in equilibrium in an infinite number of bent position =>
NEUTRAL EQUILIBRIUM
CASE 2: P=PCR

8
 The elastic restoring force was not enough to prevent small disturbance growing into
an excessively large deflection.
 Depending on magnitude of load P, column either remain in bent position, or will
completely collapse or fracture.
 This type of behavior indicates that for axial loads greater than Pcr the straight
position of column is one of UNSTABLE EQUILIBRIUM in that a small disturbance
will tend to grow into an excessive deformation.
CASE 3: P>PCR

9

10

11

12
IDEAL COLUMN WHIT PIN SUPPORTS
- An ideal column is perfectly straight
before loading, made of
homogeneous material, and upon
which the load is applied through the
centroid of the X- section.
- The slope of the elastic curve is small
and deflections occur only in
bending.
- The material behaves in a linear
elastic manner and the column
buckles or bends in a single plane

13
Euler’s Formula for Pin-ended Columns
(Eigen-value Problem)
e
i M
M 
0
2
2

 v
EI
P
dx
v
d
kx
C
kx
C
v cos
sin 2
1 

0
0
)
0
( 2 


 C
x
v

.....
3
,
2
,
1
,
2
2
2

 n
L
EI
n
P

2
2
cr
L
EI
P



0
sin
0
)
( 1 


 kL
C
L
x
v

0
sin
)
'
,
0
1 

 kL
flexion
(no
solution
trivial
s
it
then
C
If
Smallest value of P is obtained for n=1:
Which is satisfied if:
x
L
C
x
v


 sin
)
( 1
BUCKLED SHAPE
EULER LOAD
EI
Pv
EI
M
dx
v
d



2
2
v
P
v
I
E 



 '
'
L
n
k
n
kL






14
2
2
cr
L
EI
P


Euler’s Formula for Pin-ended Columns
Buckling equation for a pin-supported long slender column:
Pcr = critical or maximum axial load on column just before it begins to buckle. This
load must not cause the stress in column to exceed proportional limit.
E = modulus of elasticity of material
I= least modulus of inertia for column’s x-sectional area
L= unsupported lenght of pinned-end columns

15
Critical Stress in Pin-ended Columns
2
2
cr
cr
AL
EI
A
P 



A
I
,
y 2
2


 
r
where
Ar
dA
I
  2
2
2
2
2
2
2
cr
/ 







E
r
L
E
L
Er
= Radius of Gyration
The tendency of column to buckle is usually measured by its Slenderness Ratio
λ=L/r. It is a measure of the column flexibility and will be used to classify columns as
long, intermediate or short.
EULER’S
HYPERBOLE
λ>λc
λ<λc
=>σy< σcr
SLENDER COLUMN
y
c
f
E





*λc=76÷94
S355 S235

16
Slenderness Ratio:
200
150
250
200








Dynamic loads
Static loads
MAIN STRUCTURES
SECONDARY STRUCTURES
MAIN STRUCTURES
SECONDARY STRUCTURES

17
• A column will buckle about the principal axis of the x-section
having the least moment of inertia (weakest axis)
• For example, the meter stick shown will buckle about the a-a
axis and not b-b axis.
• Thus circular tubes made axcellent columns!!!
Consideration in Section Property
2
2
cr
cr
AL
EI
A
P 
 

In the formula, J is a minimum on the
cross-section. Therefore, the buckling of
columns has a selective direction.
I is isotropic Iy = Iz Iy < Iz

18
Extension of Euler’s Formula
  2
2
2
2
2
2
cr
4
1
2 L
EI
L
EI
L
EI
P
e






   2
2
2
2
cr
/
4
/ r
L
E
r
L
E
e


 

• A column with one fixed and one free end, will
behave as the upper-half of a pin-connected
column
• Euler’s formula:
Le = column effective length = distance between the zero moment points
Le=K*L
K = effective length factor
K=2

19
Euler’s Formula for Clamp-ended Columns
  2
2
2
2
2
2
cr
4
2
/ L
EI
L
EI
L
EI
P
e






K=0,5

20
Effective Length of Various Columns
Le = column effective length =
distance between the zero
moment points
Le=K*L
K = effective length factor
The factor K for some idealised
boundary conditions are given in
the table.
Because idealized boundary
conditions seldom are attained for
actual structures, values
recommended for design are
slightly conservative.
2
2
cr
e
e
L
EI
P
P



Pe = Euler
buckling load

21
Effective lenghts in different directions:

22
EXAMPLE
A 7,2 m long A-36 steel tube having the x-section shown is
to be used as pin-ended column.
Determinate the maximum allowable axial load the column
can support so that it does not buckle.
Solution:
Est= 200 GPa
Since σcr<σy=250 MPa, application of Euler’s equation is appropriate.
 
kN
m
mm
m
m
kN
L
EI
P 2
.
228
2
.
7
)
1000
/
1
(
)
70
(
4
1
]
/
)
10
(
200
[
2
4
4
2
6
2
2
2
cr 






MPa
mm
N
mm
kN
N
kN
A
Pcr
100
/
2
.
100
]
)
70
(
)
75
(
[
)
/
1000
(
2
.
228 2
2
2
2
cr 







754-704

23
The stress in the column cross-section can be calculated as:
= P/A
where,  is assumed to be uniform over the entire cross-section => IDEAL STATE
This ideal state is never reached.
THE STRESS-STATE WILL BE NON-UNIFORM DUE TO:
1. ACCIDENTAL ECCENTRICITY of loading with respect to the centroid;
2. Member out-of –straightness => GEOMETRIC IMPERFECTIONS
3. RESIDUAL STRESSES in the member cross- section due to fabrication
processes.
Effect of material Imperfections and Flaws
Slight imperfections in tension members can be safely disregarded as they are of little
consequence. On the other hand, slight defects in columns are of great significance.

24
1. Eccentric Loading:
2
2
max
2
2
1
2
sec
e
e
e L
EI
P
P
P
e
y
EI
e
P
y
P
dx
y
d


















 







• Eccentric loading is equivalent to a centric
load and a bending moment.
• Bending occurs for any non-zero eccentricity.
Question of buckling becomes important
when the resulting deflection is excessive.
• The deflection becomes infinite when P = Pcr
M(x) =P·e - P·y(x)

25
 
























 



r
L
EA
P
r
c
e
A
P
r
c
e
y
A
P e
2
1
sec
1
1 2
2
max
max
THE SECANT
FORMULA
Maximum stress
c= distance from neutral axis to
outer fiber of column where
maximum compressive stress
occurs.
1. Eccentric Loading:
y
ccentricit
ratio of e
r
c
e


2
0÷3; usually taken ≤ 1
y=250 Mpa
E= 200 GPa
λ=Le/r
P/A
(MPa)
As the slenderness ratio λ
increases, eccentrically loaded
columns tend to fail near the Euler
buckling load

26
2. Effect of geometric imperfections
 The initial out-of-straightness is also termed "initial
crookedness" or "initial curvature".
 It causes a secondary bending moment as soon as
any compression load is applied, which in turn
leads to further bending deflection and a growth in
the amplitude of the lever arm of the external end
compression forces.
 A stable deflected shape is possible as long as the
external moment, i.e. the product of the load and
the lateral deflection, does not exceed the internal
moment resistance of any section.
 When straight column buckles, it assumes a
stable, bent equilibrium, but with slightly larger
load.
 In Crooked column deflection increases from
beginning of loading and column is in unstable
condition when it reaches to maximum load.
δ0 = maximum initial deflection,whitout load.
v0(z)= initial deformation=δ0*sin(πz/L)

27
0
)
(
'
' 0 

 v
v
P
v
EIx
)
( 0
v
v
P
Mx 

0
'
' 0
2
2


 v
k
v
k
v
0
sin
'
' 0
2
2






 



L
z
k
v
k
v
…some calculations





 




L
z
A
v
v
v F
tot sin
0
0
0
'
' 0





x
x EI
v
P
EI
v
P
v
)
(
sin
0
crooked
initial
to
due
moment
A
M
L
z
P
A
M
F
x
F
x







 







 


L
z
v sin
0
0
2
2
e
e
L
EI
P


1
1
1




e
F
P
P
factor
on
amplifiati
A

28
Of longitudinal axis Of trasversal axis
frequence

29

30
3. Effect of Residual Stresses
 Complete yielding of x-section did not occur until applied strain equals the
yield strain of base material.
 The residual stresses do not affect the load corresponding to full yield of x-
section.
Effect of residual stresses:

31
If the maximum stress σn reaches the yield stress fy, yielding
begins to occur in the cross-section. The effective area able
to resist the axial load is, therefore, reduced.
3. Effect of Residual Stresses
R
y
n f 



σn

32
Previous analyses assumed stresses below the proportional limit σ<σy and initially
straight, homogeneous columns
Experimental data demonstrate:
 LONG COLUMN:
for large Le/r, cr follows
Euler’s formula and depends
upon E but not Y.
 SHORT COLUMN:
for small Le/r, cr is
determined by the yield
strength Y and not E.
 INTERMEDIATE
COLUMN:
mixed behavior.
IDEAL COLUMN vs REAL COLUMN

33

34
BUCKLING RESISTANCE OF MEMBERS
ACCORDING TO EC3

35
Class 1 cross-sections are those which can
form a plastic hinge with the rotation capacity
required from plastic analysis without reduction
of the resistance.
Class 2 cross-sections are those which can
develop their plastic moment resistance, but
have limited rotation capacity because of local
buckling.
Class 3 cross-sections are those in which the
stress in the extreme compression fibre of the
steel member assuming an elastic distribution
of stresses can reach the yield strength, but
local buckling is liable to prevent development
of the plastic moment resistance.
Class 4 cross-sections are those in which local
buckling will occur before the attainment of
yield stress in one or more parts of the cross-
section.
Cross section classification

36

37

38
Uniform members in compression (clause 6.3.1 of EC3-1-1)
Buckling resistance
A compression member should be verified against buckling as follows
where
Ned is the design value of the compression force;
Nb,Rd is the design buckling resistance of the compression member.
where c is the reduction factor for the relevant buckling mode.

39
Buckling curves
α is an imperfection factor corresponding to the appropriate buckling curve
(Table 6.1 and 6.2);
Ncr is the elastic critical force for the relevant buckling mode based on the gross
cross sectional properties.
(See Figure 6.4)

40
Figure 6.4: Buckling curves

41
Lateral-Torsional Buckling
 Instability phenomenon characterized by the occurrence of large transversal
displacements and rotation about the member axis, under bending moment about the
major axis (y axis).
 This instability phenomenon involves lateral bending (about z axis) and torsion of cross
section.
 Beams with sufficient restraint to the compression flange are not susceptible to lateral-
torsional buckling. In addition, beams with certain types of cross-sections, such as
square or circular hollow sections, fabricated circular tubes or square box sections are
not susceptible to lateral-torsional buckling.

42
 In the study of lateral-torsional buckling of beams, the Elastic Critical Moment
Mcr plays a fundamental role; this quantity is defined as the maximum value of
bending moment supported by a beam, free from any type of imperfections.
 For a simple supported beam with a double symmetric section, with
supports prevent lateral displacements and rotation around member axis (twist
rotations), but allowing warping and rotations around cross section axis (y and
z), submitted to a uniform bending moment My (“standard case”), the elastic
critical moment is given by:

43

44
Uniform members in bending (clause 6.3.2 of EC3-1-1)
Buckling resistance
A laterally unrestrained member subject to major axis bending should be verified against
lateral torsional buckling as follows:
In determining Wy holes for fasteners at the beam end need not to be taken into account.

45
Mcr is the elastic critical moment for lateral-torsional buckling and it is based on gross cross
sectional properties and takes into account the loading conditions, the real moment
distribution and the lateral restraints.
αLT imperfection factor corresponding to the appropriate buckling curve
Lateral torsional buckling curves for rolled sections or equivalent welded sections
f takes into account the moment distribution between the lateral restraints of members:

46
Recommended values for lateral torsional
buckling curves for cross sections
Figure 6.4: Buckling curves

47
This occurs when some part or parts of x-section of a column are so thin that
they buckle locally in compression before other modes of buckling can occur
Local Buckling=> related to plate buckling
 When thin plates are used to carry compressive
stresses they are particularly susceptible to
buckling about their weak axis due small
moment of Inertia.
 If local buckling of the individual plate elements
occurs, then the column may not be able to
develop its global buckling strength.
 Each plate element must be stocky enough, i.e.,
have a b/t ratio that prevents local buckling from
governing the column strength.
Local buckling of the compression flange

48
 The coefficient k has a minimum value of 4
for a/b=1,2,3 etc.
 The error in using k =4 decreases with
increasing a/b and for a/b= 10 or more, it is
extremely small.
Local Buckling
a
N N
2
2
a
D
K
Nxcr 

2
2
2
)
1
(
12











a
t
E
K
xcr
)
2
1
(
12
3





t
E
D
D bending stiffness of the plate;
t thickness of the plate.
K = Constant depends on
 How edges are supported
 Ratio of plate length to plate width
 Nature of loading
2
2
4
a
D
Nxcr 


49
Numerical solution: example
Result of buckling analysis:
modes shapes

50
t
b
f
t
b
P eff
d
y
eff
Rd
c



 ,
max
,

b
beff 
 
fy,d= fy /gM1
gM1=1,1.
673
,
0

P

673
,
0

P

- =1
- 2
22
,
0
P
P





cr
y
P
f

 

51
Plate with imperfections for (geometrically) non linear buckling theory
Ingrandimento nell'intorno di P/Pc=1
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
Spostamento w (mm)
P/Pcr Imperfezione
0,05 mm
Imperfezione
0,5 mm
Imperfezione
0,75 mm
Imperfezione
1 mm
w[mm]

52
BUCKLING OF TUBULAR MEMBERS
ACCORDING TO ISO19902
Range of validity of the verification formula:
- unstiffened and ring stiffened cylindrical tubulars having a thickness t > 6 mm
- diameter to thickness ratio D / t < 120
- material meeting the general requirements of ISO
- yield strengths shall be less than 500 Mpa
- ratio of yield strength to ultimate tensile strength shall not exceed 0,90

53
STRENGTH REQUIREMENTS FOR MEMBER ELEMENTS
ISO19902:2020

54
ISO19902:2020

55
WITHOUT HYDROSTATIC PRESSURE

56
AXIAL COMPRESSION
σc axial compressive stress due to forces from factored actions;
fc representative axial compressive strength, in stress units;
gR,c partial resistance factor for axial tensile strength, gR,c = 1,10;
Um Utilization of member under axial compression.

57
COLUMN BUCKLING
In the absence of hydrostatic pressure, the representative axial compressive strength for
tubular members should be the smaller of the in-plane and the out-of-plane buckling
strengths determined from the following equations:
where:
fc Representative axial compressive strength, in stress units;
fyc Representative local buckling strength, in stress units;
λ Column slenderness parameter;
fe Smaller of the Euler buckling strengths in the y- and z-
direction, in stress units;
E Young’s modulus of elasticity;
K Effective length factor;
L Unbraced length in the y- or z-direction;
r Radius of gyration
I = moment of inertia of the cross-section;
A = cross-sectional area.

58
AXIAL COMPRESSION + IN PLANE BENDING + OUT OF PLANE BENDING
1
2
AND
ONE MUST VERIFY BOTH THE CONDITIONS !!!!
Expression 1 set a strength verification;
Expression 2 set a buckling verification
c axial compressive stress due to forces from factored actions;
fc Representative axial compressive strength, in stress units;
σby bending stress about the member’s y-axis (in-plane) due to forces from factored actions;
σbz bending stress about the member’s z-axis (out-of-plane) due to forces from factored actions.
fb representative bending strength, in stress units

59
COLUMN BUCKLING
Annex of Clause 13 gives some further details on the derivation of the empirical equations
and limitations:

60
Comparison of test data with representative COLUMN BUCKLING strength
equations for fabricated CYLINDERS subjected to axial compression
The calibrated equations are representative of lower bound of the tested strength

Cm,y and Cm,z moment reduction factors corresponding to the y- and z-axes, respectively (see
Table 13.5.1);
fey and fez Euler buckling strengths corresponding to the y- and z-axes respectively, in
stress units.
61
Ky and Kz effective length factors for the y- and z-directions, respectively (see
Table 13.5.1);
Ly and Lz unbraced lengths in the y- and z-directions, respectively.
ry and rz radius of gyration
Where:
I is the moment of inertia of the cross-section
A is the area of the cross-section
2
AXIAL COMPRESSION + IN PLANE BENDING + OUT OF PLANE BENDING

62
Utilization of a member, Um, under axial compression and bending shall be the larger
value calculated from Equations:
Utilization of a member

65
EFFECTIVE LENGTHS AND MOMENT REDUCTION FACTORS
The effective lengths and moment reduction factors may be determined using an analysis
that includes joint flexibility.
Lengths to which the effective length factors K are applied are normally measured from
centerline to centerline of the end joints.
Safety values of effective length factors (K) and moment reduction factors (Cm) may be
taken from following Table 3.2.

66
1
2
3

67
Local Buckling
Local buckling of pipe line
Local buckling, with breaking, of a pipe line

68
Cylinder buckling:

69
Schematic postbuckling response of an axially compressed cylinder.
Figure shows the load-
deformation curve of an axially
compressed cylindrical shell,
which exhibits a postbuckling
response with multiple bifurcation
points due to changes in the
deformed geometry after each
critical point.
Cylindrical shells can attain their
postbuckling regime due to the
natural transverse deformation
restraint provided by their
geometry.

70
Load-displacement curves

71
fyc representative local buckling strength, in stress units;
fxe representative elastic local buckling strength, in stress units;
Cx critical elastic buckling coefficient;
E Young’s modulus of elasticity;
D outside diameter of the member;
t wall thickness of the member.
Cx = 0.6 for an ideal tubular member
Cx = 0.3 to account for the effect of initial geometric imperfections within the tolerance limits
A reduced value of Cx = 0.3 is also implicit in the limits for fy /fxe
Local Buckling
and
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400
fxe/fy (fy=300 MPa)
1

72
Local Buckling
and limitations:

73
ISO 19902:2007(E)
Comparison of test data and the representative LOCAL BUCKLING strength
equations for CYLINDERS subjected to axial compression

74
WITH HYDROSTATIC PRESSURE
ACCORDING TO ISO19902
• A tubular member below the water line is subjected to hydrostatic pressure unless it
has been flooded in the installation procedure.
• Platform legs are normally flooded in order to assist in their upending and placement
and for pile installation.
• Remember that, even if members are flooded in the in-place condition, they can be
subjected to hydrostatic pressures during launch and installation.

75
 For analyses using factored actions that include capped-end actions:
 For analyses using factored actions that do not include the capped-end actions:
σt axial stress resulting from the analysis without capped-end actions
σq compressive axial stress due to the capped-end hydrostatic actions:
σh hoop stress due to forces from factored hydrostatic pressure
CAPPED-END ACTIONS: axial components of hydrostatic pressure on each member
APPROXIMATION

76
Hoop Buckling
Closed, unstiffened cylinders subjected to hydrostatic pressure will experience compressive
stresses in the axial and ring direction, respectively.
Simple equilibrium considerations show that the ring stress is twice the magnitude of the axial
stress.
Stresses in closed, unstiffened circular cross-sections
for external hydrostatic pressure

77
σh hoop stress due to forces from factored hydrostatic pressure;
p factored hydrostatic pressure
D outside diameter of the member;
t wall thickness of the member;
γR,h partial resistance factor for hoop buckling strength: γR,h = 1.25;
fh representative hoop buckling strength, in stress units;
fy representative yield strength, in stress units:
fhe elastic hoop buckling strength, in stress units.
Hoop Buckling

78
Ch critical elastic hoop buckling coefficient:
μ geometric parameter:
Lr length of tubular between stiffening rings, diaphragms or end connections.
MEMBERS THAT VIOLATE THE ALLOWABLE TOLERANCE:
Dmax and Dmin are the maximum and minimum values of any measured outside diameter
at a cross-section and Dn is the nominal diameter.
Members that have out-of-roundness greater then 1% and less than 3%, a reduced elastic
hoop buckling stress, fhe,red, is applicable:
α=geometric
imperfection factor
out-of-roundness (%)

Ch critical elastic hoop buckling coefficient:
μ geometric parameter:
Lr length of tubular between stiffening rings, diaphragms or end connections.
0
1
2
3
4
5
6
7
0 100 200 300 400
0
20
40
60
80
100
120
0 100 200 300 400
D/t
fxe/fy (fy=300 MPa)
D/t
fxe/fy (fy=300 MPa)

80
Comparison of test data with representative hoop buckling strength equations
for fabricated cylinders subjected to hydrostatic pressure

81
Ring stiffener design
For m≥1,6 D/t, the elastic critical hoop
buckling stress is approximately equal to
that of a long unstiffened tubular.
Hence, to be effective, stiffening rings,
if required, should be spaced such that:
CIRCUMFERENTIAL STIFFENING RING SIZE
Ic is the required moment of inertia for the composite ring section;
Lr is the ring spacing;
D is the outside diameter of the member;
Dr is the diameter of the centroid of the composite ring section;
These formulae for
determining the required
moment of inertia of the
stiffening rings, provide
sufficient strength to resist
buckling of the ring and shell
even after the shell has
buckled between stiffeners.

82

83
and limitations:

84
AXIAL COMPRESSION, BENDING AND HYDROSTATIC PRESSURE
fc,h representative axial compressive strength in the presence of external hydrostatic
pressure, in stress units
ONE MUST VERIFY BOTH THE CONDITIONS !!!!
1
2
 is the column slenderness parameter

85
If the maximum combined compressive stress σx = σ b + σc,c and the elastic local buckling
strength fxe exceeds following the limits:
fhe elastic hoop buckling strength;
fxe representative elastic local buckling strength
then the next equation should also be satisfied:
3

86
The utilization of a member, Um, under axial compression, bending and hydrostatic
pressure shall be the largest value calculated from the following Equations:
Utilization of a member
Eq. 1
Eq. 2
Eq. 3

87
Dented tubular members
Dent depth, h: h ≤0,3 D & h ≤ 10 t

88
Axial compression
σc,d is the axial compressive stress due to forces from factored actions on the
undamaged cross-section;
P is the member axial compressive force;
fc,d is the representative axial compressive strength of dented members, in
stress units;
gc,d = 1.18 is the partial resistance factor for axial compressive strength of
dented members.

89
Column buckling
Where:
Dy is the maximum out-of-straightness of the dented member;
L is the unbraced member length, in the plane of buckling which coincides
with the plane of y;
fc,d,o is the representative axial compressive strength of dented members when
y/L≤ 0,001, in stress units

90
fyc is the representative local buckling strength, in stress units;
fb is the representative bending strength, in stress units;
Ze is the elastic section modulus of the undamaged member;
fe,d is the Euler buckling strength of the dented member, in stress units;
λd is the slenderness parameter of the dented member;
rd is the radius of gyration of the dented member;
Id is the effective moment of inertia of the dented cross-section;
Ad is the effective cross-sectional area of the dented section

91
A is the cross-sectional area of the undamaged section,

92
ISO 19902:2007(E)

93
ISO 19902:2007(E)

Types of
analyses
Elastic
I order
II order
Elasto-
plastic
I order
II order
picture taken from “Ballio and Bernuzzi,
Hoepli Ed., 2004”
Amplification due to the
presence of compressive forces
Linear elastic behaviour
Collapse
analysis
Both mechanical and geoemtric non-
linearities are taken into account
Buckling of Frames

Buckling of
frames
&
II-order
analysis
methods
Elastic
Inelastic
Simple
frames
Multi-storey
frames
In closed-form (starting from basic
cases, e.g. contained in Belluzzi)
Intuitive considerations
Use of geometric stiffness matrix
(iterative procedures)
Simplified methods (P-D method,
charts...)
Merchant-Rankine approximated
formula
Numerical methods
...
Newmark formula
Buckling of Frames

96
Buckling of Frames:
simplified methods based on alignment charts
GA
GR
K

97
Unbraced frames
The system analysis is reduced to a determination of the effective buckling lengths of the
individual components
Within the system, the individual components are distinguished between the compression
members (columns) and the bending members providing restraint, and denoted,
respectively, by the subscripts c and b. Then, the following factor, which measures the
relative stiffness between the compression and the restraining members, is defined.
The factors are easily derived from simple beam theory accounting for actual
boundary conditions
Σ indicates a summation of all members rigidly
connected to that joint and lying in the plane in
which buckling of the column is being
considered;
I c moment of inertia and l the unsupported
length of the column section;
I b is the moment of inertia and lb the
unsupported length of a girder or other
restraining member.

98

99
Depending upon the geometry of the joint, all restraining beams may not be considered as
perfectly rigid. Accordingly the stiffness of these members will be reduced.
The stiffness of the resisting beams, Cb, and the joint, Cj, can be considered as two
springs in series.
The β-factor, which represents the
softening of the beam stiffness due to
joint flexibility, is introduced into the G-
factor.
Joint flexibility
The joint rotational stiffness Cj is given by parametric equations.

100
In-plane bending:
Out-plane bending :
These expressions are valid for
For other joints it is conservative to use the above relations to assess local shell stiffness
for buckling control
T-joint

101
In case of connecting by more than one
member, the buckling factor K can be
obtained from the ALIGNMENT CHART .
The subscripts A and R refer to the joints at
the two ends of the column section being
considered.
K
G-factor end a G-factor end b
The determination of the effective buckling
length is based upon the G-factor calculated
at each end of the considered member.
Then a straight line is drawn between the
values on the g-factor
axis. Then the effective buckling length
factor can be read as the intersection of this
line with the K-axis.
Effective buckling length factor K: ALIGNMENT CHART

102
1 in= 2.54 cm
1 foot=30.48 cm
1 kips=4.44 kN
K
Example

103
Buckling of X-braces.
X-braces are commonly used in jackets and jack-up platforms. The braces carry most of the lateral force
on the platform.
For a platform with single X-braces the lateral force caused by WAVES, CURRENT AND WIND will be
carried by compression in one of the braces and tension in the other. These forces are of equal
magnitude. In addition there will be a contribution in compression due to compression of the legs from the
topside weight. Consequently the total force in the compressive brace will be larger than the force in the
tension brace.
If the environmental forces are
assumed to come in from the left,
force N1 will be in compression and
force N2 will be in tension
Out-of-plane buckling is critical.
There is interaction between the
two braces because the
tension brace acts as a support
for the compression brace

104
THE COMPRESSION BRACE MAY FAIL IN TWO MODES AS SHOWN IN FIGURE:
• Symmetric buckling
• Asymmetric buckling.
Support is modelled as an equivalent spring with stiffness KN2.
In addition, the legs and any other braces framing into the ends of the X-braces provide a
rotational restraint, which is modelled by a rotational spring with stiffness C. For simplicity it is
assumed equal at all four supports.
If the tension brace is not sufficiently strong the symmetric buckling mode governs.

105
Symmetric buckling:
Critical
load
1

106
Asymmetric buckling:
Critical
load
2
The critical loads as given by Equations 1 and 2 may conveniently be solved
by means of an iterative procedure, e.g. bi-section method.

107
The left axis with = 0
corresponds to pinned
ends, the right axis with
100/  = 0 corresponds
to fixed ends.
Graphic solutions for symmetric and asymmetric buckling as a function of the non-
dimensional rotation stiffness
The symmetric mode
depends upon the axial
force in the tension
brace, so that N2/N1 is
introduced as a
parameter
Some results are easily recognized.
Es.: for pinned ends (= 0 ) and N2/N1= 1 both braces are in compression and there is no
stiffening effect from N2. The curve applies to a single brace, hence K = 1 for pinned
conditions and K = 0.5 for fixed ends.
N2/N1=1
0.5
1.0
Rotational end restrain C rotational
stiffness

108
0.5-0.8 L
Q/P=1
ALIGNMENT CHART ISO 19902

109

110

111
and limitations:

Lecture 6. Instability of members 2021_MP_new.pdf

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