CE 384
DESIGN FUNDAMENTALS FOR
STEEL STRUCTURES
Chapter 5
COMPRESSION MEMBERS
Main Reference:
William T. Segui 2007 and 2013.
«Steel Design», CENGAGE Learning,
4th and 5th Edition.
Prof. Dr. Nesrin YARDIMCI TİRYAKİOĞLU
5.1 INTRODUCTION
 Members that are subjected to axial (concentric) loads.
 Compression members: struts (serve as bracing), posts
or pillars, truss web and chord members.
 Shapes: Most of the rolled shapes.
 For larger loads built-up cross sections (allow a designer
to tailor to specific needs). Lacing bars or perforated
cover plates make the section as a single unit.
 The problem of stability is of great importance; they are
very sensitive to factors that may tend to cause lateral
displacements or buckling.
5.2 LIMIT STATES OF BUCKLING
There are two general modes by which axially loaded steel
columns can fail.
 Local buckling: when some part or parts of the cross section of a
column are so thin.
 Global (member) buckling: Which are: Flexural buckling,
torsional buckling and flexural-torsional buckling.
 Flexural buckling: This type of buckling is caused by bending
or flexure about the axis corresponding to the largest
slenderness ratio (this is usually minor principal axis).
 Torsional buckling: This type of buckling is caused by
twisting about the longitudinal axis of the member. It can only
occur with doubly symmetrical cross-sections with very
slender cross-sectional elements. Standard hot-rolled shapes
are not susceptible to torsional buckling.
 Flexural-torsional buckling: This type of buckling is caused
by a combination of flexural and torsional buckling. It can
occur only with unsymmetrical cross-sections (one axis of
symmetry or no axis of symmetry).
Flexural Buckling
Torsional buckling
Torsional-flexural buckling
5.3 ELASTIC (EULER) BUCKLING
Flexural buckling
Column is,
 Pin-ended,
 prismatic and doubly symmetrical
 perfectly straight.
 Compressive load is applied along the centroidal axis.
 There are no transverse loads.
 The material is homogeneous and obeys Hooke’s Law.
 Plane sections before deformation remain plane after
deformation.
 Deformations of the member are small,
 influence of shear on deformations is neglected and
 no twisting or distortion of the section occurs.
Critical buckling load, Pcr:
Pcr  PE 
 2 EI
L2
Critical buckling stress, Fcr:
Pcr PE  2 EI
 2E
Fcr 



2
A
A
AL
( L / r )2
I:
Moment of inertia
A: Cross-sectional area
r : Radius of gyration
(L / r): Slenderness ratio
I  Ar 2
Considering typical elastic flexural buckling of member with I-shape,
axial load – member length relationship is shown in the following
figure.
About strong axis
P
PEx 
 2 EI x
L2
PEy 
L
 2 EI y
L2
About weak axis
L
L  Lx  L y
PEx and PEy
E
Ix and Iy
L
:
:
:
:
Elastic buckling load
Modulus of elasticity
Principal moments of inertia about x and y axis
Pin-ended column of length
5.4. INELASTIC BUCKLING
If the stress at which buckling occurs is greater than the proportional
limit of the material, the relation between stress and strain is not
linear, and the modulus of elasticity E can no longer be used.
Instead, variable modulus of elasticity, ET (tangent modulus) must
be used.
π 2 Et
Fcr 
(L/r)2
5.5 BEHAVIOUR OF
COMPRESSION MEMBERS
Slender compression members (Long column):
If the axial load is slowly applied and increased,
the member becomes unstable and is said to
have buckled: Critical buckling load
If the member is stockier (Intermediate column),
the larger load will be required to bring the
member to the point of instability.
For extremely short members (Short column)
failure may occure by yielding.
 Slender column: It will fail by elastic buckling.
 Short column: It will crush owing to yielding. Compressive
stresses are in inelastic range.
 Intermediate column: Falls between. It is analyzed and
designed by using empirical formulas obtained by test
results.
5.6 EFFECTIVE LENGTH (KL)
Dealing with boundary conditions the critical loads for
elastic and inelastic buckling loads are:
π 2E t I
π 2 EI
Pcr 
...and...Pcr 
2
KL 
KL 2
Critical buckling stresses are:
π 2E t
π 2E
Fcr 
...and...Fcr 
2
2
L/r 
L/r 
L : Actual length of the column
K : Effective length factor
(due to end conditions)
KL: Effective length of the column
Slenderness ratio: KL/r= λ
Failure stresses versus KL/r ratios determined by testing.
Right of line A-A: Failure stresses closely
predicted by Euler’s formula.
Elastic buckling occurs at a stress
less than the proportional limit.
Left of line A-A: Fail by inelastic
buckling, yielding occurs.
• (a) and (b): Very little
resistance to end
rotations.
• (c), (d) and (e):
Sufficient resistance to
end rotations.
• To apply Euler’s formula
to columns having other
than pin ends, effective
lengths are used.
Column effective length
It behaves as a pin ended column of length L/2.
Effective length for frame columns
Restraint factors of GA and GB, defined for either end of a
column can be obtained as:
G

 EI 
 
 L columns
 EI 
 
 L  girders

I /L

G
 I /L
c
c
g
g
 Ic /L c
=Sum of the stiffness of all
columns at the end of column
under consideration
 Ig / Lg
=Sum of the stiffness of all
girders at the end of column
under consideration
If the column is: pin ended : G=10,
fixed ended : G=1.0
After GA and GB, effective length factor of K can be determined using
appropriate alignment chart.
For fixed connection to the base, G = 0. AISC recommends that G = 1.
For pinned connection to the base, G = . AISC recommends that G =
The boundary condition of the end of a girder away from the joint being
considered must be taken into account in determination of G. Accordingly,
the terms in G equation that contain girder rigities (I / L) must be
multiplied by the following values.
In braced frames;
for pinned end connection 1.5,
for fixed end connection 2.0,
In unbraced frames;
for pinned end connection 0.5,
for fixed end (encastre) connection 0.67.
For braced frames (side sway is prevented), expected
effective length factors K is:
0.5  K  1.0
Idealized Braced Frame
Shear
Wall
Braced Frame
For unbraced frames (Sidesway is unprevented), expected
effective length factors K is:
1.0  K  
Unbraced Frame
The alignment charts based upon assumptions of idealized conditions
that seldom are completely satisfied in real structures. The assumptions
are as follows:

The structure consists of regular rectangular frames,

Material behavior is linear elastic,

All members have constant cross section,

All girder-to-column connections are rigid connections,

For braced frames, at the onset of buckling, the girders exhibit
symmetric single-curvature bending,

For unbraced frames, at the onset of buckling, the girders exhibit
reverse-curvature bending,

All columns reach their buckling loads simultaneously,

No significant axial compressive force exists in the girders.
GA and GB can be computed as :
GA 
GB 
 g1
g3
I c I c1

Lc Lc1
I g1
Ig2
 g2
Lg1
Lg 2
Ic Ic2

Lc Lc 2
I g3
Ig4
 g4
Lg 3
Lg 4
5.7 DESIGN OF COMPRESSION
MEMBERS
Nominal compressive strength
Allowable compressive strength
: Pn = Fcr Ag
: Pd = Pn / Ωc
Pn
Pa  Pd 
c
Pa : Required compressive strength
Ωc : Safety factor for compression = 1.67
If allowable stress formulation is used
fa ≤ Fa = Fcr/Ωc = Fcr/1.67 = 0.6 Fcr
fa : Computed axial compressive stress = Pa/ Ag
Fa : Allowable compressive strength
Flexural buckling of members without slender elements
 Elastic buckling load:
 Compressive strength
is in inelastic range:
 Compressive strength
is in elastic range:
 K xLx
K yLy 
KL

 max
...or...


r
r
r
x
y


K yLy
K xLx
λx 
...and...λ y 
rx
ry
Pe
π 2E
Fe  
A  KL  2


 r 
Fy
KL
E
 4.71
......or.....  2.25
r
Fy
Fe
Fy

Fcr   0.658 Fe



F
 y

Fy
KL
E
 4.71
......or.....  2.25
r
Fy
Fe
Fcr  0.877 Fe
No buckling
(Short column)
Inelastic range
(Intermediate column)
Elastic range
(Slender column)
5.8 LOCAL STABILITY
 This type of instability is a localized buckling or wrinkling at
an isolated location. If it occurs the cross-section is not fully
effective (I and H sections with thin flanges and webs).
(The compressive strength must be reduced.)
 Unstiffened elements: Unsupported along one edge parallel
to the direction of load (flanges of I and H sections).
 Stiffened elements: Supported along both edges (webs of I
and H sections).
 Limiting values of width-thickness values are give in
specifications.
Cross-sectional shapes are classified as:
Compact
Noncompact
Slender
λ=width / thickness
If λ ≥ λr : Slender
Flange
Web
b b f /2 b f
λf  

t tf
2t f
h
λw 
tw
λr : Upper limit
E
λ rf  0.56
Fy
λ rw
E
 1.49
Fy
For slender columns these requirements can not be satissfied and
such members may not be permitted to carry the same load as
compact members.
b: Half of full nominal width
h: The distance between
root of flanges
If a cross-sectional shape does not satisfy the width-thickness
ratio requirements, the strength could be reduced because of local
buckling. The reduction factor (for slender columns) is
Q=QaQs
Qs is for unstiffened and Qa is for stiffened elements.
 If the shape has only stiffened elements
: Qs=1.0, Q=Qa
 If the shape has only unstiffened elements
: Qa=1.0, Q=Qs
 If the shape stiffened and unstiffened elements: Q=QaQs
Many of the shapes commonly used as column are not slender
(most of the I and H sections). However a large number of hollow
structural shapes (HSS), double angles and tees have slender
elements.
Computation Qa and Qs are given in AISC Specification.
The nominal compressive strength, Pn
Pn  Fcr A g
Critical stress, Fcr
Fe 
 2E
 KL 


r


2
With slender elements
QFy
KL
E
 4.71
......or.....
 2.25
r
QFy
Fe
QFy

Fcr  Q 0.658 Fe



F
 y

QFy
KL
E
 4.71
......or.....
 2.25
r
QFy
Fe
Fcr  0.877 Fe
5.9 BUILT-UP COLUMNS
Each plate act as a
separate column.
If two plates connected
sufficiently to prevent
slippage on each other
they will act as a unit.
Ends will deform
together.
As a compression member bends, a shearing component of
the axial force arise. Shear in columns is caused by:
“Lateral load”, “Slope with respect to the line of thrust”, “End
eccentricity of load”.
Latticed columns: The lacing or batten plates used to tie
together compression elements.
Compression members may be constructed with two or more
shapes built up into a single member. For instance, they may
consist of parts in near contact with each other, such as pairs of
angles or of channels. Its analysis is the same as for any other
compression member, provided the component parts of cross
section are properly connected.
Built-up Compression
Members in nearly contact
each other
For long columns, it may be convenient to use built-up sections
where the parts of the cross sections are spread out or widely
separated from each other. Before heavy I-shapes were made
avaliable, such sections were very commonly used for crane booms
and for the compression members of various kinds of towers. The
widely spaced parts of these types of built-up members must be
carefully laced or tied together.
Built-up Compression
Members with the parts
widely spaced
Slenderness of built-up sections,
Modified slenderness ratio, (KL/r)m must be used instead of
(KL/r) in calculation of the nominal compressive strength. Accordingly,
(a) For intermediate connectors that are bolted snug-tight:
 KL 
 KL   a 

  
  
 r m
 r 0  ri 
2
2
(b) For intermediate connectors that are welded or are connected by
means of pretensioned bolts:
a
 KL 
 KL 
 40....




ri
 r  m  r 0
a
 KL 
 KL   K i a 
 40....
  
 
ri
 r m
 r 0  ri 
2
Ka 3 KL

ri
4 r
2
Ki = 0.50 for angles back-to-back
0.75 for channels back-to-back
0.86 for all other cases
a : distance between connectors
ri : minimum radius of gyration of
individual component
Slenderness ratio of built-up member
 KL 

 : acting as a unit in the buckling direction
r

o beaing considered
5.10 TORSIONAL AND
FLEXURAL-TORSIONAL BUCKLING
 Four doubly symmetrical shapes (torsional buckling)
 Four singly symmetrical shapes (flexural-torsional
buckling)
 For shapes no axis of symmetry (flexural-torsional
buckling)
2
 x0 
 y0 
2


( Fe  Fex )( Fe  Fey )( Fe  Fez )  Fe ( Fe  Fey )   Fe ( Fe  Fex ) 
 r0 
 r0 
2
2
xo, yo : Coordinates of shear center
with respect to centroid
ho : the distance between flange
centroids
Cw :Warping constant
G : Shear modulus
J : Torsional constant
Kz : Effective length factor for
torsional buckling
Ix : Moment of inertia about x axis
Iy : Moment of inertia about y axis
H : Flexural constant
The nominal compressive strength, Pn
Pn  Fcr A g
Critical stress, Fcr
Without slender elements,
Fy

 Fy

Fe


2.25

F

0.658


cr

F
 e


 Fy

 2.25   Fcr  0.877Fe

 Fe


 Fy


With slender elements,
QFy

 QFy

 2.25   Fcr  Q  0.658 Fe


 Fe


 QFy

 2.25   Fcr  0.877Fe

 Fe


 Fy

