Analysis and Design of Compression-Member - Ch 05 from xxx.pdf
35
http://syaifulsipil96.blogspot.com/ [email protected] 5 - 1 5.1 INTRODUCTION Compression member/column is a structural steel member subjected to axial compressive load. Compression member occur usually at roof trusses, truss bridge, column in building has axial compressive load combined with the bending moment. This chapter describes the analysis and design procedure of compression member such as strength of compression member when local buckling is not occur, strength of compression member when there is local buckling, effective length etc. 5.2 BASIC THEORY OF BUCKLING 5.2.1 GENERAL Buckling load is defined as limit of load to produce the column buckling. If the column is loaded with axial load the column will be deflect laterally and reach buckling if the load is increased continuously. When the applied load less than buckling load and the load is removed the column will return to its original position. Leonhard Euler proposed the buckling theory in 1744. The basic assumption of his theory are : Elastic Column, column is behaves in elastic manner until axial load reach buckling load. Prismatic, the column section is prismatic over the length. Straight, column is straight. Homogen Material, column consists of homogen material. Pin Ended Support, the end support is pin support. 5.2.2 EULER BUCKLING LOAD OF ELASTIC COLUMN The mathematical solution for a differential equation for buckling analysis is : 2 2 2 cr L EI n P π = [5.1] where : P cr = buckling load n = number of half-sine waves in the length of column EI = flexural stiffness of column L = length of column CHAPTER 05 ANALYSIS AND DESIGN OF COMPRESSION MEMBER
Transcript of Analysis and Design of Compression-Member - Ch 05 from xxx.pdf
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5 - 1
5.1 INTRODUCTION Compression member/column is a structural steel member subjected to axial compressive load.
Compression member occur usually at roof trusses, truss bridge, column in building has axial
compressive load combined with the bending moment.
This chapter describes the analysis and design procedure of compression member such as strength of
compression member when local buckling is not occur, strength of compression member when there is
local buckling, effective length etc.
5.2 BASIC THEORY OF BUCKLING 5.2.1 GENERAL Buckling load is defined as limit of load to produce the column buckling. If the column is loaded with
axial load the column will be deflect laterally and reach buckling if the load is increased continuously.
When the applied load less than buckling load and the load is removed the column will return to its
original position.
Leonhard Euler proposed the buckling theory in 1744.
The basic assumption of his theory are :
Elastic Column, column is behaves in elastic manner until axial load reach buckling load.
Prismatic, the column section is prismatic over the length.
Straight, column is straight.
Homogen Material, column consists of homogen material.
Pin Ended Support, the end support is pin support.
5.2.2 EULER BUCKLING LOAD OF ELASTIC COLUMN
The mathematical solution for a differential equation for buckling analysis is :
2
22
crL
EInP π= [5.1]
where :
Pcr = buckling load
n = number of half-sine waves in the length of column
EI = flexural stiffness of column
L = length of column
CHAPTER
05 ANALYSIS AND DESIGN OF COMPRESSION MEMBER
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5 - 2
The buckling load / critical load of elastic column proposed by Euler occurs when n=1.0 results the
smallest buckling load it is called Euler Buckling Load.
The Euler buckling load is :
2
2
crLEIP π
= [5.2]
The Euler buckling load is defined as the smallest buckling load result from one half-sine wave. If
the column were able to buckle with n>1.0 then its buckling load is increase. In other word the
buckling load of column with n=1.0 is smaller than buckling load of column with n>1.0.
Axial load less than the Euler buckling load (Pu<Pcr) will not produce buckling and axial load equal
and greater than Euler buckling load (Pu≥Pc) will produce buckling.
n also represent the buckling modes, mode 1, mode2 and so on. The buckling load is reached at the
first mode (mode 1) n=1.0.
The figure below shows the definition of half-sine wave.
FIGURE 5.1 DEFINITION OF HALF-SIN WAVE TABLE 5.1 HALF-SIN WAVE
n=1.0 n=2.0 N=3.0
2
22
cL
EI1P π=
2
22
cL
EI2P π=
2
22
cL
EI3P π=
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The Euler buckling load was generally ignored for design purpose because it is different with the test
result, usually column will fail by inelastic buckling rather than elastic buckling.
5.2.3 CRITICAL BUCKLING LOAD & CRITICAL STRESS
In steel structure engineering the Euler buckling load usually named with critical buckling load and its
stress is named with critical buckling stress.
If 2ArI = then critical buckling load for pin-ended column is :
( )( )2
2
2
22
2
2
crrLEA
LArE
LEIP π
=π
=π
= [5.3]
Critical buckling stress for pin-ended column is :
( )22
crcr
rLE
APF π
== [5.4]
where :
Pcr = critical buckling load
Fcr = critical buckling stress
E = modulus of elasticity
A = section area
L = length of column
I = minimum moment of inertia
r = minimum radius of gyration
rL = largest slenderness ratio
If the minor axis of the section is y then minimum moment of inertia is Iy and minimum radius of gyration is ry.
For column with other end support the critical buckling load and critical buckling stress becomes :
( )22
crrKL
EAP π=
( )22
crrKLEF π
= [5.5]
5.3 EFFECTIVE LENGTH 5.3.1 GENERAL
Effective length is defined as column length of an auxiliary pin-ended column with has the same
Euler buckling load value or the distance between zero moment value (point of contra flexure / point
of inflection of the column). Effective length is calculated using the effective length factor K. The value
of K is lies between 0.5 and 2.0 depended to the end support condition.
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When a column is supported with other than pin support there is rotation restraint at the end of
column and the buckling length is less than if the end support is pin. In this case we use the
effective length concept KL.
5.3.2 PINNED – FIXED SUPPORT CONDITION
The effective length factor K of single column with typical support condition is tabulated in the
table below :
TABLE 5.2 EFFECTIVE LENGTH FACTOR – SINGLE COLUMN
THEORETICAL K
K=1.0 K=1.0 K=0.5 K=0.7 K=2.0
Pinned Fixed Fixed Pinned Free
Δ
Δ
Δ
Δ
Pinned Fixed Fixed Fixed Fixed
K=1.0 K=1.2 K=0.65 K=0.8 K=2.0
DESIGN K
The condition of pure pinned or pure fixed support is cannot be achieved in the actual structure.
The actual condition of end support is lies between pinned support (no rotation restraint) and fixed
support (rotation restraint). 5.3.3 BRACED FRAME / NON-SWAY FRAME
Braced frame structure is defined by LRFD code as a structure which the lateral stability is
provided by diagonal bracing, shear wall or equivalent means, the vertical column is braced frame
structure will not has laterally movement of its top relative to its bottom.
The effective length factor of braced compression member is always less than 1.0 K<1.0 for design
purpose it can be taken conservatively K=1.0. The effective length factor K can be computed using
nomograph from LRFD or approximate equation from France design rules.
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FIGURE 5.2 EFFECTIVE LENGTH FACTOR – BRACED FRAME
The effective length factor of braced compression member / braced frame structure is :
( )( ) 28.1GG0.2GG3
64.0GG4.1GG3KBABA
BABA++++++
= [5.6]
where :
K = effective length factor
GA = ratio of flexural stiffness of end A of column
GB = ratio of flexural stiffness end of end B of column
5.3.4 UN-BRACED FRAME / SWAY FRAME
Un-braced frame structure is defined by LRFD code as a structure which the lateral stability is
provided by the bending stiffness of the connected beam and column, the vertical column is
braced frame structure will has laterally movement of its top relative to its bottom.
The effective length factor of braced compression member is always greater than 1.0 K>1.0 minimum value of K is 1.0. The effective length factor K can be computed using nomograph from LRFD or
approximate equation from France design rules.
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FIGURE 5.3 EFFECTIVE LENGTH FACTOR – UN -BRACED FRAME
The effective length factor of un-braced compression member / un-braced frame structure is :
( )5.7GG
5.7GG0.4GG6.1KBA
BABA++
+++=
( ) ( )αα
=+−α
tanGG636GG
BA
BA2
απ
=K
[5.7]
where :
K = effective length factor
GA = ratio of flexural stiffness of end A of column
GB = ratio of flexural stiffness end of end B of column
5.3.5 ELASTIC FLEXURAL STIFFNESS RATIO The elastic flexural stiffness ratio of column to beam is defined as :
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
=
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
=
b
b
c
c
b
bb
c
cc
elastic
LILI
LIE
LIE
G [5.8]
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where :
Ec,b = elastic modulus of elasticity of column and beam
Ic = moment of inertia of column
Ib = moment of inertia of beam
Lc = height of column
Lb = length of beam
This flexural stiffness ratio is calculated at two ends of column, these are Gi and Gj.
For bottom level of structure which is the support can be pinned or fixed the value of G is :
TABLE 5.3 FLEXURAL STIFFNESS RATIO – PINNED AND FIXED SUPPORT
G FIXED
SUPPORT
PINNED SUPPORT
Theoretical 0.0G = =≈G
Design 0.1G = 10G =
FIGURE 5.4 G VALUE FOR SPECIAL CONDITIONS
The elastic G is used if the behavior of the compression member is elastic, if the behavior is
inelastic the inelastic G must be used, it’s depended to the slenderness parameter λc. The value of
inelastic G is less than elastic G because inelastic modulus of elasticity is less than elastic modulus of elasticity. The slenderness parameter will be explained in the next section when
analysis of compression member is described.
5.3.6 INELASTIC FLEXURAL STIFFNESS RATIO
When the column is inelastic and the beam is elastic then the inelastic G must be used to calculate
the effective length factor K.
As described previously the elastic ratio of flexural stiffness of beam and column is :
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∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
=
b
bc
c
cc
elastic
LIE
LIE
G [5.9]
where :
Ec,b = elastic modulus of elasticity of column and beam
For inelastic column the equation becomes :
elastict
b
b
c
c
t
b
b
c
ct
inelastic GEE
LILI
EE
LEIL
IE
G ⎟⎠
⎞⎜⎝
⎛=
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎠
⎞⎜⎝
⎛=
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
= [5.10]
where :
Et = inelastic modulus of elasticity of column
E = elastic modulus of elasticity of beam
Ratio of inelastic E and elastic E is obtained by divide the critical buckling stress for inelastic and
elastic behavior. This parameter is explained detailed in the next section, but to determine the
inelastic G the equation is firstly written.
The critical buckling stress of inelastic column is :
ycr F685.0F2c ⎟⎠⎞⎜
⎝⎛= λ [5.11]
The critical buckling stress of elastic column is :
y2c
cr F877.0F ⎟⎟⎠
⎞⎜⎜⎝
⎛
λ= [5.12]
The ratio of inelastic E and elastic E is :
⎟⎠⎞⎜
⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ λ== λ2
c685.0877.0F
FEE 2
c
elastic,cr
inelastic,crt [5.13]
where :
λc = slenderness parameter
The equation above is depended to the slenderness ratio of column which is not known until the
design is finish, so it is need trial and error calculation.
The equation above is simplified by Yura – Disque as follows :
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( )( ) ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛
λ
φ=
φφ
≈φ
φ≈
2c
gu
glong,cr
n
glong,cr
gshort,crt
877.0
APAF
PAFAF
EE
[5.14]
Or inelastic critical buckling stress is approximately as follows :
g
uyinelastic,cr A
PF685.0F2c
φ=⎟
⎠⎞
⎜⎝⎛= λ [5.15]
From the relation above the λc is can be computed, and the elastic critical buckling stress is
computed using equation 5.12.
The ratio of inelastic E and elastic E then can be calculated.
5.4 ANALYSIS OF COMPRESSION MEMBER – COMPACT SECTION 5.4.1 GENERAL
The calculation of compression member strength is based on the gross section area, with a reason that
the maximum compressive load at the net section is always less than in the gross section.
When column is loaded with compression load the column will be buckle. There are two buckle types,
as follows :
Column Buckling, when the critical buckling load is achieved the column will be buckle with half
sine wave shape, occurs in compact section.
Local Buckling, if the flange or web plate is too thin they will be buckle before the column
buckling, occurs in non-compact section.
This section describes the analysis of compression member for a column with column buckling
without local buckling occurs (compact and non-compact section). To ensure there is no local
buckling at the section plate the width – thickness ratio of plate must be achieved.
There are three column buckling modes depended to the cross section configuration, as follows :
Flexural Buckling, the column bends about the minor principal axis or largest slenderness
ratio (smallest radius of gyration). The flexural buckling can happen to any type of column
cross section.
Torsional Buckling, the column twist about the longitudinal axis. The torsional buckling can
happen to doubly symmetric cross section or built up member from thin plate.
Flexural-Torsional Buckling, the column is bends and twist simultaneously. This failure is
combination of bending and twisting, combination of flexural buckling and torsional buckling.
The flexural-torsional buckling can happen to single symmetric cross section or unsymmetric
section.
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5.4.2 LIMITATION OF WIDTH – THICKNESS RATIO
To prevent local buckling occurs before buckling the width-thickness ratio λ must be checked for
plate element in the steel section. If the plate element of steel section is too thin, due to axial
compression load the plate will be buckle and local buckling occurs and the compressive strength
is reduced.
There are two types of element must be checked, these are :
Unstiffened Element, plate element that is supported at one edge of the plate such as flange of I
shape.
Stiffened Element, plate element that is supported at two edges of the plate such as web of I shape.
FIGURE 5.5 UNSTIFFENED & STIFFENED ELEMENT
To ensure there is no local buckling the width-thickness ratio λ must not greater than λr as noted in
LRFD code.
rλ≤λ [5.18]
where :
λ = width-thickness ratio
λr = limits of width-thickness ratio for slender column
Width-thickness ratio is defined as ratio of width of plate to thickness of plate, as follows :
tb
=λ
th
=λ [5.16]
where :
λ = width-thickness ratio
b = width of plate
h = height of plate
t = thickness of plate
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For I shape becomes :
f
ft2
b=λ
wth
=λ [5.17]
where :
bf = width of flange
h = height of web
tf = thickness of flange
tw = thickness of web
The figure below shows the definition of width-thickness ratio of many types of steel section.
FIGURE 5.6 DEFINITION OF WIDTH-THICKNESS OF PLATE
The limitation of width-thickness ratio for compact, non compact and slender section is tabulated in
the table below. The definition of compact section and non compact section will be explained in the
next chapter for analysis and design of flexure member / beam.
For compression member if the width-thickness ratio λ greater than λr as noted in the table then
local buckling occurs and design procedure in follows in the next section, there is a reduction factor
Q if local buckling is occurs. The λr value is maximum limits of non compact section.
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TABLE 5.4 LIMITATION OF WIDTH-THICKNESS RATIO
λp (COMPACT) λr (NON COMPACT) TYPE λ
ksi MPa ksi MPa
ftb (R)
yF65
yFE38.0
10F141
y −
70FE83.0
y −
ftb (W)
yF65
yFE38.0 ( ) cy k/5.16F
162−
( ) cy k/115FE95.0
−
125.0PP yBu ≤φ
⎟⎟⎠
⎞⎜⎜⎝
⎛
φ−
yB
u
y PP75.21
F640
125.0PP yBu ≤φ
⎟⎟⎠
⎞
⎜⎜⎝
⎛
φ−
yB
u
y PP75.21
FE76.3 WF
wth 125.0PP yBu >φ
⎟⎟⎠
⎞⎜⎜⎝
⎛
φ−
yB
u
y PP33.2
F191
yF253
≥
125.0PP yBu >φ
⎟⎟⎠
⎞
⎜⎜⎝
⎛
φ−
yB
u
y PP33.2
FE126.1
yFE49.1≥
⎟⎟⎠
⎞⎜⎜⎝
⎛
φ−
yB
u
y PP74.01
F970
⎟⎟⎠
⎞
⎜⎜⎝
⎛
φ−
yB
u
y PP74.01
FE70.5
ftb
yF190
yFE12.1
yF238
yFE40.1
BOX
wth As WF As WF As WF As WF
f
ftb As WF As WF As WF As WF
C
wth As WF As WF As WF As WF
ftb As WF As WF As WF As WF
T
wtd NA NA
yF127
yFE75.0
L tb NA NA
yF76
yFE45.0
LL tb NA NA
yF76
yFE45.0
NA C NA
C yF/3190 C
( )yF/E11.0 C
PIPE tD
yF/2070 F yF/2070 F yF/8970 F ( )yF/E31.0 F
where :
Fy = yield strength ksi or MPa
E = modulus of elasticity
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The residual stress Fr is taken as :
TABLE 5.5 RESIDUAL STRESS
RESIDUAL STRESS SECTION ksi MPa
Rolled 10 70
Welded 16.5 115
kc in the table is defined as :
w
c
th4k =
[5.18]
where :
h = height of web
tw = web thickness
5.4.3 FLEXURAL BUCKLING Flexural buckling occurs when compression member is buckle about minor principle axis or axis
with minimum radius of gyration (largest slenderness ratio). The flexural buckling can occurs to
any shape.
The nominal strength of compression member based on the gross section when flexural buckling
governs is :
crgn FAP = [5.19]
where :
Pn = strength of compression member
Ag = gross section area
Fcr = critical buckling stress of flexural buckling column
The design strength of compression member then becomes :
( )crgCnC FAP φ=φ
85.0C =φ [5.20]
where :
φC = strength reduction factor of compression member
The critical buckling stress Fcr is a function of slenderness ratio KL/r, but LRFD use slenderness
parameter λc as follows :
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EF
rKL y
c π=λ [5.21]
where :
λc = slenderness parameter of flexural buckling column
K = effective length factor
L = length of member
Fy = yield strength
E = modulus of elasticity
And Fcr is defined as :
( )22
y2c
crr/KL
EF1F π=
λ=
cr
y2c F
F=λ
[5.21]
To fabricate pure straight steel member is cannot be achieved, there is must be a crookedness as
initial curvature. The maximum of crookedness is 1000Le ≤ and for rolled section is 1500e . The
effect of crookedness is significant in a range of slenderness ratio 135rL50 << .
where :
e = initial vertical camber
L = length of member
FIGURE 5.7 CROOKEDNESS OF COMPRESSION MEMBER
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FIGURE 5.8 EFFECT OF CROOKEDNESS TO STRENGTH
The figure above shows the effect of crookedness at compression member. When axial compression
load is zero P=0 the member is already bend and when the axial compression load is applied the
lateral deflection is increase and the strength is reduced. It is shown in the graph that the critical
buckling load Pcr is reduced due to crookedness by a factor 0.877.
AISC LRFD consider the effect of crookedness by reduce the critical buckling stress, LRFD assumed
initial crookedness is e=L/1500 where L is the length of member.
FIGURE 5.9 ELASTIC & INELASTIC BUCKLING OF COLUMN
The strength of compression member is depended to the type of buckling occurs, as follows :
Elastic Buckling, when buckling occurs none of fiber is yields. Elastic buckling happens in long
compression members.
Inelastic Buckling, when some of fiber is reach yield then the behavior is becomes inelastic into
the strain-hardening range. When portion of section is yields then the yield part then buckle.
Inelastic buckling happens in short compression members.
The parameter to of elastic buckling and inelastic buckling is use slenderness parameter λc, the
boundary of elastic and inelastic column is λc=1.5.
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The critical buckling stress of elastic and inelastic column is :
TABLE 5.6 CRITICAL BUCKLING STRESS – FLEXURAL BUCKLING COLUMN
BUCKLING λc Fcr
Elastic 5.1c >λ y2c
cr F877.0Fλ
=
Inelastic 5.1c ≤λ ycr F658.0F2c ⎟⎠⎞⎜
⎝⎛= λ
where :
λc = slenderness parameter
Fy = yield strength
Fcr = critical buckling stress of flexural buckling column
From the equation above it can be known that elastic column occur when the 1.5 x critical buckling
stress is below the yield strength (none of fiber is yields) and inelastic column occurs when the
1.5 x critical buckling stress is above the yield strength (part of fiber yields).
To analyze flexural buckling column we use maximum slenderness ratio of the section or minimum
radius of gyration. The slenderness parameter also uses the maximum slenderness ratio.
The following table shows the minimum radius gyration used to analyze a flexural buckling column.
TABLE 5.7 MINIMUM RADIUS OF GYRATION – FLEXURAL BUCKLING COLUMN
SECTION rmin
ry
rz
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5.4.4 FLEXURAL – TORSIONAL BUCKLING
Flexural – torsional buckling occurs when compression member is bend and twist simultaneously.
The flexural – torsional buckling can occur to single symmetric section (channel, tee, double angle,
angle with equal legs) and un-symmetric section (angle with unequal legs).
The calculation procedure of flexural-torsional buckling column is similar as flexural buckling column,
the difference is at the equation of critical buckling stress Fcr.
The nominal strength of compression member based on the gross section when flexural – torsional
buckling governs is :
crftgn FAP = [5.22]
where :
Pn = strength of compression member
Ag = gross section area
Fcrft = critical buckling stress of flexural-torsional buckling column
The design strength of compression member then becomes :
( )crftgCnC FAP φ=φ
85.0C =φ [5.23]
where :
φC = strength reduction factor of compression member
The critical buckling stress Fcrft is a function of slenderness ratio (KL/r)ft, but LRFD use slenderness
parameter λcft as follows :
crft
ycft F
F=λ [5.24]
where :
λcft = slenderness parameter of flexural-torsional buckling column
Fy = yield strength
Fcrft = critical buckling stress of flexural-torsional buckling column
And Fcrft is defined as :
TABLE 5.8 CRITICAL BUCKLING STRESS – FLEXURAL-TORSIONAL BUCKLING COLUMN
SECTION Fcrft
Single Symmetric
(Double angle,
channel, tee,
angle with
equal legs)
( ) ⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+−−
+= 2
crzcry
crzcrycrzcrycrft
FF
HFF411
H2FF
F
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SECTION Fcrft
Un-Symmetric
(Angle with
unequal legs)
( )( )( ) ( )
( ) 0ryFFF
rxFFFFFFFFF
2
0
0crxcrft
2crft
2
0
0crycrft
2crftcrzcrftcrycrftcrxcrft
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−−−−−
take the smallest root
The critical buckling stress about x, y, z axis are :
( )2xx
2
crxrLKEF π
=
2
yy
2
cryrLK
EF⎟⎠⎞⎜
⎝⎛
π=
( ) 20
2z
w2
crzAr1GJ
LKECF
⎟⎟⎠
⎞⎜⎜⎝
⎛+
π=
[5.25]
where :
Fcrx = Fcr about X axis, bend perpendicular X axis
Fcry = Fcr about Y axis, bend perpendicular Y axis, minor axis of symmetry
Fcrz = Fcr about Z axis, twist about Z axis
Cw = warping constant
G = shear modulus of elasticity
J = torsional constant
A = gross section area
The un-braced length for torsion is the member length about minor axis.
The other variables of equation above are :
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−= 2
0
20
20
ryx1H
⎟⎟⎠
⎞⎜⎜⎝
⎛ +++=
AII
yxr yx20
20
20
[5.26]
where :
x0, y0 = coordinates of shear center from center of gravity
The shear center is defined as the point when the load is applied through this point there is no
bending or twisting occurs. For some section the shear center is coincides with the center of gravity
such as I section, hollow tube, pipe section. Section such as tee, channel, angle and double angle the
shear center is not coincides with the center of gravity.
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5 - 19
The following figure shows the location of shear center for any cross section.
FIGURE 5.10 LOCATION OF SHEAR CENTER
The axis of symmetry to calculate the Fcry is not always about Y axis but depended to the shape, as
follows :
Tee, Y axis is the axis of symmetry
Channel, X axis is the axis of symmetry
Angle, shown in the figure above
Double Angle, Y axis is the axis of symmetry
The flexural – torsional buckling is occurs at the axis of symmetry (X axis for C) and flexural
buckling is occurs at the other axis/minor axis (Y axis for C). For single symmetric section there are
two possible strengths must be checked, the strength of flexural buckling and strength of flexural-torsional buckling the minimum value is governs.
The critical buckling stress of elastic and inelastic column is similar as for flexural buckling column
:
TABLE 5.9 CRITICAL BUCKLING STRESS – FLEXURAL-TORSIONAL BUCKLING COLUMN
BUCKLING λcft Fcrft
Elastic 5.1cft >λ y2cft
crft F877.0Fλ
=
Inelastic 5.1cft ≤λ ycrft F658.0F2
cft ⎟⎠⎞⎜
⎝⎛= λ
where :
λc = slenderness parameter
Fy = yield strength
Fcr = critical buckling stress of flexural-torsional buckling column
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5 - 20
5.4.5 TORSIONAL BUCKLING Torsional Buckling occurs when compression member is twist about longitudinal axis. The
torsional buckling can occur to double symmetric section (box, pipe, I, rectangular).
The calculation procedure of flexural-torsional buckling column is similar as flexural buckling column,
the difference is at the equation of critical buckling stress Fcr.
The nominal strength of compression member based on the gross section when flexural – torsional buckling governs is :
crtgn FAP = [5.27]
where :
Pn = strength of compression member
Ag = gross section area
Fcrf = critical buckling stress of torsional buckling column
The design strength of compression member then becomes :
( )crftgCnC FAP φ=φ
85.0C =φ [5.28]
where :
φC = strength reduction factor of compression member
The critical buckling stress Fcft is a function of slenderness ratio (KL/r)ft, but LRFD use slenderness
parameter λct as follows :
crt
yct F
F=λ [5.29]
where :
λct = slenderness parameter of torsional buckling column
Fy = yield strength
Fcrf = critical buckling stress of torsional buckling column
And Fcrt is defined as :
TABLE 5.10 CRITICAL BUCKLING STRESS –TORSIONAL BUCKLING COLUMN
SECTION Fcrt
Double Symmetric
(Box,Pipe,
I, Rectangular ) ( ) yx
2z
w2
crt II1
LKECF
+⎟⎟⎠
⎞⎜⎜⎝
⎛ π=
where :
Fcrt = critical buckling stress of torsional buckling column
Cw = warping constant
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5 - 21
The un-braced length for torsion is the member length about minor axis.
The critical buckling stress of elastic and inelastic column is similar as for flexural buckling column
:
TABLE 5.11 CRITICAL BUCKLING STRESS –TORSIONAL BUCKLING COLUMN
BUCKLING λct Fcrt
Elastic 5.1ct >λ y2ct
crt F877.0Fλ
=
Inelastic 5.1ct ≤λ ycrt F658.0F2ct ⎟⎠⎞⎜
⎝⎛= λ
5.5 ANALYSIS OF COMPRESSION MEMBER – NON COMPACT SECTION
5.4.6 GENERAL
When the section is too thin the plate element may be buckle it’s called local buckling occurs in non
compact section. When local buckling occurs then the section will be no longer effective and the
maximum compressive strength cannot be achieved. The local buckling is controlled use the width-
thickness ratio of the plate element as already explained in the previous section. When the width-
thickness ratio of the section λ is greater than λr (upper limit of slender section) then local
buckling will be occurs. The limitation of width-thickness ratio is shown in the table in the section 5.4.
To account the local buckling effect LRFD use the reduction factor Q to calculate the compressive
strength.
This section describes the analysis procedure of compression member when local buckling occurs.
FIGURE 5.11 LOCAL BUCKLING
5.4.7 COMPRESSIVE STRENGTH
The nominal strength of slender compression member based on the gross section when flexural
buckling governs is :
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5 - 22
crgn FAP = [5.30]
where :
Pn = strength of compression member
Ag = gross section area
Fcr = critical buckling stress of flexural buckling column
The design strength of compression member then becomes :
( )crgCnC FAP φ=φ
85.0C =φ [5.31]
where :
φC = strength reduction factor of compression member
The critical buckling stress of elastic and inelastic column is :
TABLE 5.12 CRITICAL BUCKLING STRESS – FLEXURAL BUCKLING COLUMN
BUCKLING λc Fcr NOTES
Elastic 5.1Qc >λ y2c
cr F877.0Fλ
= cry F5.1F >
Inelastic 5.1Qc ≤λ yQ
cr F658.0QF2c ⎟⎠⎞
⎜⎝⎛= λ cry F5.1F ≤
where :
λc = slenderness parameter
Fy = yield strength
Fcr = critical buckling stress of flexural buckling column
Q = reduction factor
The reduction factor Q for compact and non-compact section (no local buckling) taken 1.0 and for
slender section is taken as :
asQQQ = [5.32]
where :
Q = reduction factor of slender element
Qs = reduction factor for un-stiffened slender element
Qa = reduction factor of stiffened slender element
For other type if buckling mode such as flexural-torsional buckling and torsional buckling the equation
is similar as already explained in the previous section, the difference just to compute the reduction
factor Qs for un-stiffened element and Qa for stiffened element.
There are three possible conditions to analyze a slender section, as follows :
The column section only contains of un-stiffened element (angle, tee section). Strength reduction
factor Q=Qs is used to compute the compressive strength.
The column section only contains of stiffened element (box section). Strength reduction factor
Q=Qa is used to compute the compressive strength.
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5 - 23
The column section is contains of un-stiffened and stiffened element (I section). Strength
reduction factor Q=QsQa is used to compute the compressive strength.
5.4.8 REDUCTION FACTOR QA – STIFFENED ELEMENT
The reduction factor Qa for stiffened element is :
g
ea A
AQ = [5.33]
where :
Qa = reduction factor for stiffened element
Ae = effective area of stiffened element
Ag = gross section area
∑−= ige AAA
( )∑ −−= tbbAA ege [5.34]
where :
Ai = ineffective area of stiffened element
b = width of stiffened element
be = effective width of stiffened element
t = thickness of stiffened element
The effective width is used because when local buckling occurs the stiffened element is buckle in the
middle and this area is not provide more compressive strength any more, the area which is
provide strength is the effective area.
b and be can also noted as h and he if the position of the stiffened element is vertical as in I, box and
channel section.
The figure below shows the effective area of stiffened element of box and I section.
FIGURE 5.12 EFFECTIVE AREA OF STIFFENED ELEMENT
For rectangular hollow section there are two type of stiffened element must be checked these are the flange and the web, in I section there is only one stiffened element that is the web.
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5 - 24
The effective width of stiffened element is tabulated in the table below :
TABLE 5.13 EFFECTIVE WIDTH FOR STIFFENED ELEMENT
EFFECTIVE WIDTH
(psi)
EFFECTIVE WIDTH
(MPa) TYPE λ he λ he
f253
th
w≤ hhe =
fE49.1
th
w≤ hhe =
WF
f253
th
w>
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
fth2.571
ft326h
w
we
fE49.1
th
w< [ ] ⎟
⎟⎠
⎞⎜⎜⎝
⎛−=
fE
th34.01
fEt91.1h
wwe
f253
th
w≤ hhe =
fE49.1
th
w≤ hhe =
f253
th
w>
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
fth2.571
ft326h
w
we
fE49.1
th
w< [ ] ⎟
⎟⎠
⎞⎜⎜⎝
⎛−=
fE
th34.01
fEt91.1h
wwe
f238
tb
f≤ bbe =
fE40.1
tb
f≤ bbe =
BOX
f238
th
w>
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
ftb9.641
ft326h
f
fe
fE40.1
tb
f> [ ] ⎟
⎟⎠
⎞⎜⎜⎝
⎛−=
fE
tb38.01
fEt91.1h
ffe
f253
th
w≤ hhe =
fE49.1
th
w≤ hhe =
C
f253
th
w>
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
fth2.571
ft326h
w
we
fE49.1
th
w< [ ] ⎟
⎟⎠
⎞⎜⎜⎝
⎛−=
fE
th34.01
fEt91.1h
wwe
T bbe =
LL bbe =
L bbe =
yF3300
tD
≤ 0.1Qa = yF
E11.0tD
≤ 0.1Qa =
PIPE
yF3300
tD
> ( ) 3
2FtD
1100Qy
a += yF
E11.0tD
> ( ) 3
2FE
tD038.0Q
ya +⎟
⎟⎠
⎞⎜⎜⎝
⎛=
gAPf = [5.35]
where :
f = axial compressive stress
P = axial compressive load
Ag = area of gross section
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5 - 25
5.4.9 REDUCTION FACTOR QS – UN-STIFFENED ELEMENT
The reduction factor for un-stiffened element is shown in the table.
TABLE 5.14 REDUCTION FACTOR FOR UN-STIFFENED ELEMENT – KSI UNITS
REDUCTION FACTOR
(psi) TYPE λ Qs
yf
f
F95
t2b
≤ 0.1Qs =
yf
f
y F176
t2b
F95
<< ( ) yffs Ft2b00437.0415.1Q −= WF (R)
yf
f
F176
t2b
≥ ( ) y
2ff
sFt2b
20000Q =
cyf
f
kF109
t2b
≤ 0.1Qs =
cyf
f
cy kF200
t2b
kF109
<< ( ) cyffs kFt2b00381.0415.1Q −=
WF (W)
cyf
f
kF200
t2b
≥ ( )( ) y
2ff
cs
Ft2bk26200Q =
BOX 0.1Qs =
C As for WF, but f
ft2
breplaced with
f
ftb
For flanges as for WF, for web see below
yw F127
td
≤ 0.1Qs =
ywy F176
td
F127
<< ( ) yws Ftd00715.0908.1Q −= T
yw F176
td
≥ ( ) y
2w
sFtd
20000Q =
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5 - 26
TABLE 5.14 REDUCTION FACTOR FOR UN-STIFFENED ELEMENT – KSI UNITS
REDUCTION FACTOR
(psi) TYPE λ Qs
yF76
tb≤ 0.1Qs =
yy F155
tb
F76
<< ( ) ys Ftb00447.0340.1Q −= LL
yF155
tb≥
( ) y2sFtb
15500Q =
EF
446.0tb y≤ 0.1Qs =
EF
910.0tb
EF
446.0 yy << ( ) EFtb761.034.1Q ys −= L
EF
910.0tb y≥ ( ) ( )EFtb
534.0Qy
2s =
PIPE 0.1Qs =
TABLE 5.15 REDUCTION FACTOR FOR UN-STIFFENED ELEMENT – MPA UNITS
REDUCTION FACTOR
(MPa) TYPE λ Qs
yf
fFE56
t2b
≤ 0.1Qs =
yf
f
y FE03.1
t2b
FE56 << ( )
EF
t2b74.0415.1Q yffs −= WF (R)
yf
fFE03.1
t2b
≥ ( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
y2
ffs F
Et2b
69.0Q
( )cyf
fkF
E64.0t2
b≤ 0.1Qs =
( ) ( )cyf
f
cy kFE17.1
t2b
kFE64.0 << ( )
EkF
t2b64.0415.1Qc
yffs −= WF (W)
( )cyf
fkF
E17.1t2
b≥
( )( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
y2
ff
cs F
Et2bk90.0Q
BOX 0.1Qs =
C As for WF, but f
ft2
breplaced with
f
ftb
T For flanges as for WF, for web see below
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5 - 27
REDUCTION FACTOR
(MPa) TYPE λ Qs
yw FE75.0
td
≤ 0.1Qs =
ywy FE03.1
td
FE75.0 <<
( )EF
td22.1908.1Q yws −=
yw FE03.1
td
≥ ( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
y2
ws F
Etd69.0Q
yFE45.0
tb≤ 0.1Qs =
yy FE910.0
tb
FE45.0 << ( ) EFtb761.034.1Q ys −=
LL
yFE910.0
tb≥
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛=
y2s F
Etb
534.0Q
yFE45.0
tb≤ 0.1Qs =
yy FE910.0
tb
FE45.0 << ( ) EFtb761.034.1Q ys −=
L
yFE910.0
tb≥
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛=
y2s F
Etb
534.0Q
PIPE 0.1Qs =
The value of kc is :
TABLE 5.16 VALUE OF KC
WF OTHERS
w
c
th4k =
763.0k35.0 c ≤≤
763.0kc =
where :
h = depth of the web
tw = thickness of the web
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5 - 28
5.6 ANALYSIS OF COMPRESSION MEMBER – BUILT UP MEMBER 5.6.1 GENERAL
Built up member is defined as member assembled from more than one steel section, such as double
angle, double channel etc. The analysis procedure of built up member is similar as analysis of single
section member. The difference of analysis of built up member is at the modification of slenderness
ratio.
5.6.2 BEHAVIOR OF BUILT UP MEMBER The basic concept of built up member is to ensure that the built up section is can acts as a one unit
section. To ensure this behavior the connector plate must be installed at a specified distance.
The following figure shows the behavior of built up angle section, if the member is buckle about the X-axis the connector plate is not subjected to shear force but if buckle about Y-axis the connector
plate is subjected to shear force so the slenderness ratio must be modified. When buckling about
X-axis the built up section is acts as a units, but buckling about Y-axis and there is no connector
plate the member will act alone.
There is no modification of slenderness ratio about X-axis and there is modification of slenderness
ratio about Y-Axis (connector plate is subjected to shear) greater than unmodified slenderness
ratio.
FIGURE 5.13 BUILT UP MEMBER
5.6.3 LOCATION OF CONNECTOR PLATE
The location and number of connector plate is provided to follows the condition below :
⎟⎠
⎞⎜⎝
⎛≤r
KL43
ra
i [5.36]
where :
⎟⎟⎠
⎞⎜⎜⎝
⎛
ira = largest slenderness ratio of one component
a = spacing of connector plate
ri = minimum radius of gyration of one component
⎟⎠
⎞⎜⎝
⎛r
KL = maximum slenderness ratio of built up member
ri for angle section is refer to rz as explained previously.
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5 - 29
5.6.4 MODIFIED SLENDERNESS RATIO
The modification slenderness ratio is compute for the axis that produce shear force at the connector plate, for example above is the Y-axis :
TABLE 5.17 MODIFICATION OF SLENDERNESS RATIO
SNUG TIGHT BOLT WELDED OR
FULLY TENSIONED BOLT
2
i
2
0m ra
rKL
rKL
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠
⎞⎜⎝
⎛=⎟⎠
⎞⎜⎝
⎛
2
ib2
22
0m ra
182.0
rKL
rKL
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
α+
α+⎟
⎠
⎞⎜⎝
⎛=⎟⎠
⎞⎜⎝
⎛
ibr2h
=α
where :
mrKL
⎟⎠
⎞⎜⎝
⎛ = modified (increased) slenderness ratio of built up member
0rKL
⎟⎟⎠
⎞⎜⎜⎝
⎛ = unmodified slenderness ratio of built up member
⎟⎟⎠
⎞⎜⎜⎝
⎛
ira = largest slenderness ratio of one component
⎟⎟⎠
⎞⎜⎜⎝
⎛
ibra = slenderness ratio of one component about axis parallel to Y-axis
a = spacing of connector plate
ri = minimum radius of gyration of one component
rib = radius of gyration of one component about axis parallel to Y-axis
h = distance between components perpendicular to Y-axis
The Y-axis means the buckling axis that produce shear force at the connector plate.
TABLE 5.18 BUILT UP DOUBLE ANGLE
SECTION ri rib h
rz ry1 As shown
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5.7 DESIGN OF COMPRESSION MEMBER 5.7.1 GENERAL
Design is a state of the art (SOA) rather than a science. In a design process a structural engineer
combine the ability of analysis, engineering judgment, experience, construction method, economic
design etc.
The design procedure is similar as analysis, we try to find the required steel section for a compression
member. Design cannot be done if engineer do not know the basic concept of analysis.
There are two consideration when a compression member is designed, as follows :
Strength, the compression member must adequate to resist the ultimate axial compressive load.
Stiffness, the compression member must not fail due to serviceability requirements.
5.7.2 STRENGTH CONSIDERATION
The basic equation of compression member design is :
unC PP ≥φ 85.0C =φ
[5.37]
where :
Pn = nominal strength of compression member
Pu = ultimate axial compressive load
φC = strength reduction factor of compression member
The design of compression member must follows the condition below :
crC
ug F
PAφ
≥ [5.38]
5.7.3 STIFFNESS CONSIDERATION If the compression member is too slender the compressive strength is reduced because of the
local buckling of the flange or the web. When the local buckling occurs before buckling then the
section cannot development its maximum compressive strength. To control the situation above we
must limits the slenderness ratio of the section.
The maximum slenderness ratio of compression member is :
200r
KL≤ [5.39]
where :
K = effective length factor
L = length of tension member
r = minimum radius of gyration
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5.8 STEP – BY – STEP PROCEDURES The followings are step-by-step procedure can be used as a guide for design of compression
member, as follows :
Determine the ultimate axial compressive load Pu from elastic structural analysis.
Choose the steel section, calculate the gross section area Ag.
Check the width-thickness ratio of un-stiffened and stiffened element to investigate is there
any local buckling occurs. If local buckling occurs then the compressive strength reduced by
reduction factor Q=QaQs.
Calculate the nominal and design compressive strength based on the flexural buckling, flexural-
torsional buckling and torsional buckling.
CONDITION NOMINAL
STRENGTH φ
Flexural
Buckling crgn FAP = 0.85
Flexural-
Torsional
Buckling crftgn FAP = 0.85
Torsional
Buckling crtgn FAP = 0.85
Check for the stiffness consideration, this is the maximum slenderness ratio.
200r
KL≤
Repeat the design process until the basic equation is achieved.
unC PP ≥φ
5.9 ANALYSIS & DESIGN OF LATERAL BRACING 5.9.1 GENERAL
As previously explained the major part of compression member is the slenderness effect that is
related to critical buckling load. For short column the critical buckling load is increase and for long
column the critical buckling load is decrease. Based on buckling theory we can increase the critical
buckling load by reduce the length of the column (effective length of the column). To reduce
length of column we can provide sufficient lateral bracing.
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5.9.2 POINT BRACING OF ELASTIC COLUMN – ONE MODE
FIGURE 5.14 POINT BRACING – ONE MODE
If we model the hinged support becomes spring, we need large spring stiffness (very stiff bracing)
to prevent the side sway of the column. The idea is to find the bracing stiffness to prevent the side
sway.
The equilibrium equation is :
( )LkQLP Δ==Δ [5.40]
where :
P = compressive axial load
k = axial stiffness of bracing
L = length of column
Δ = side sway of column
If ( ) Δ<Δ PLk side sway occurs, if ( ) Δ>Δ PLk no side sway occur and the column is considered
braced. The ideal lateral brace is a brace that have enough axial stiffness to prevent movement.
The stiffness of lateral bracing is defined as :
LPk =
LPk cr
ideal = [5.41]
where :
kideal = axial stiffness of lateral bracing
L = length of column
Pcr = critical buckling load of column
Braced system occurs when idealkk ≥ , Pcr is reach and column buckle without end side sway.
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Un Braced system occurs when idealkk < , side sway occurs.
After we find the ideal axial stiffness of bracing we must find the required section area of bracing, as
follows :
bracingbracing L
EAk = [5.41]
kbracing = axial stiffness of lateral bracing
E = modulus of elasticity
A = section area of lateral bracing
Lbracing = length of lateral bracing
5.9.3 POINT BRACING OF ELASTIC COLUMN – TWO MODES
FIGURE 5.15 POINT BRACING – TWO MODES
The equilibrium equation is :
L2
kL2QP ⎟
⎠
⎞⎜⎝
⎛ Δ=⎟
⎠
⎞⎜⎝
⎛=Δ [5.42]
The axial stiffness of lateral bracing is defined as :
LP2k cr
ideal = [5.43]
5.9.4 POINT BRACING OF ELASTIC COLUMN – GENERAL
Based on the previous explanation the axial stiffness of lateral bracing for any modes of buckling is :
LPk cr
idealβ
= [5.44]
The value of β is lies from 1.0 to 4.0.
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5 - 34
The following table shows the value of β, as follows :
TABLE 5.19 β VALUE
EQUAL SPAN β
1 1.0
2 2.0
3 3.0
4 3.4
5 3.75
6 3.8
7 3.9
5.9.5 POINT BRACING WITH INITIAL CROOKEDNESS It is cannot be ensure that the column is perfectly straight, perfectly vertical and perfectly loaded as
assumed in calculation there is always a initial crookedness. In other word before the load is applied
the column is already have a initial lateral deflection.
FIGURE 5.16 POINT BRACING WITH INITIAL CROOKEDNESS
Assume there is a initial crookedness Δ0, then the equilibrium becomes :
( ) ( )LkP 0 Δ=Δ+Δ [5.45]
⎟⎟⎠
⎞⎜⎜⎝
⎛ΔΔ
+=0
crreq 1
LPk
⎟⎟⎠
⎞⎜⎜⎝
⎛ΔΔ
+=0
idealreq 1kk [5.46]
The initial crookedness is les between 500L to
1000L , AISC suggest
500L as initial crookedness is
acceptable.
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5 - 35
5.9.6 LRFD DESIGN OF LATERAL BRACING
LRFD consider assume that Δ = Δ0 = 500L .
The required axial stiffness of lateral bracing is :
( )idealreq k2k = [5.47]
where :
kreq = required axial stiffness of lateral bracing
kideal = ideal axial stiffness of lateral bracing
