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BUCKLING
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BUCKLING OF COLUMNS
13.1 Critical Load
P : gaya yang menyebabkan kegagalan krn
tegangan tekan (Elastis atau Plastis)
Pcr : gaya yang menyebabkan buckling
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Mana yang akan menghasilkan Pcr paling kecil ?
Pinned ends Fixed-Free ends
Fixed ends
Fixed-Pinned ends
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13.2 Ideal Column with Pin Supports
An ideal column, meaning one that is perfectly straight before loading
The Euler formula
Pinned ends
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The two constants of integration are determined from to
boundary conditions at the ends of the column.
Since
And since
0 at
0 at
0 , then
0
, then
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The smallest value of P is obtained when n = 1, so the critical
load for the column is therefore
This load is sometimes referred to as the Euler load, named
after the Swiss mathematician Leonhard Euler, who
originally solved this problem in 1757.
The corresponding buckled shape is defined by the equation
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where
Pcr = critical or maximum axial load on the column just before it begins tu buckle.
This load must not cause the stress in the column to exceed the proportional limit
E = modulus of elasticity for the material
I = least moment of inertia for the column’s cross-sectional area
L = unsupported length of the column whose ends are pinned
For purposes od design, the above equation can also be written in a more useful
form by expressing
where A is the cross-sectional area and r is the radius gyration of the crosssectional area.
Thus,
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Here
• E = modulus of elasticity for the material
• L = unsupported length of the column, whose ends are pinned
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• The geometric ratio L/r in Eq. 13-6 is known as slenderness ratio. It is a measure of the
column’s flexibility, and as will be discussed later, it serves to classify columns as long,
intermediate, or short.
• It is possible to graph Eq. 13-6 using axes that represent the critical stress versus the
slenderness ratio. Examples of this graph for columns made of a typical structural steel
and aluminum alloy are shown in Fig. 13-7.
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• Note that the curves are hyperbolic and are valid only for critical stresses below the
material’s yield point (proportional limits), since the material must behave elastically.
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13.3 Columns Having Various Types of Supports
Pinned ends
Fixed-Free ends
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Columns Having Various Types of Supports
Fixed-Free ends
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Columns Having Various Types of Supports
The smallest critical load occurs when n = 1, so that
By comparison with Eq. 13-5, it is seen that a
column fixed-supported at its base and free at its
top will support only one-fourth critical that can be
applied to a column pin-supported at both ends.
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Effective Length
The Euler formula
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EXAMPLE
13.2
• A W6 x 15 steel column is 24 ft long and is fixed at its ends as shown in Fig. 13–11a.
• Its load-carrying capacity is increased by bracing it about
the y–y (weak) axis using struts that are assumed to be pin
connected to its midheight. Determine the load it can
support so that the column does not buckle nor the
material exceed the yield stress.
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EXAMPLE
13.2
SOLUTION
• The buckling behavior of the column will be different about the
x–x and y–y axes due to the bracing. The buckled shape for each
of these cases is shown in Figs. 13–11b and 13–11c.
29 10 ksi 29.1 in
144 in.
401.7 kip
19
!
!
!
29 10 "#$ 9.32 in
100.8 in.
262.5 kip
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NOTE : From Eq. 13-12 it can be seen that buckling will always occur about the column
axis having the largest slenderness ratio, since a large slenderness ratio will give a small
critical stress. Thus, using the data for the radius of gyration from the table in Appendix B,
we have
144 in.
2.56 in.
56.2
401.7 kip
100.8 in.
2.46 in.
!
69.0
262.5 kip
!
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144 in.
2.56 in.
56.2
!
401.7 kip
!
100.8 in.
2.46 in.
69.0
262.5 kip
4.43 in
)
401.7 kip
4.43 in
90.7 ksi
)
262.5 kip
4.43 in
59.3 ksi
Hence, y-y axis buckling will occur, which is the same conclusion reached by comparing Eqs.
1 and 2.
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F13-2. A 12-ft wooden rectangular colmn has the dimensions shown. Determine the critical
load if the ends are assumed to be pin-connected.
1.6 10 ksi . Yielding does not accur.
O
h
x
y
Iy 
b
bh 3
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13.4 The Secant Formula
• The Euler formula was derived assuming the load P is always applied through the
centroid of the column’s cross-sectional area and that the column is perfectly straight.
• This is actually quite unrealistic, since manufactured columns are never perfectly
straight, nor is the application of the load known with great accuracy.
• In reality, then, columns never suddenly buckle; instead they begin to bend, although
ever so slightly, immediately upon application of the load.
• As a result, the actual criterion for load application should be limited either to a
specified deflection of the column or by not allowing the maximum stress in the
column to exceed an allowable stress.
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13.4 The Secant Formula
Eccentric distance e
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Hence, the deflection curve, Eq. 13-14, can written as
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Maximum Deflection. Due to symmetry of loading, both the maximum deflection and
⁄2 ,
maximum stress occur at the column’s midpoint. Therefore, when
+, ,
so
Notice that if e approaches zero, then approaches zero. However, if the
terms in the brackets approach infinity as e approaches zero, then
+, will have a nonzero value.
Mathematically, this would represent the behavior of an axially loaded
column at failure when subjected to the critical load . Therefore, to find
we require
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If Eq. 13–16 is plotted as load P versus deflection +, for various values of eccentricity
e, the family of colored curves shown in Fig. 13–14 results.
Here the critical load becomes an asymptote to the curve, and of course represents the
unrealistic case of an ideal column (e = 0).
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As stated earlier, e is never zero due to imperfections in initial column straightness and load
application; however, as - → 0, the curves tend to approach the ideal case.
Furthermore, these curves are appropriate only for small deflections, since the curvature was
approximated by / ⁄/ when Eq. 13–16 was developed.
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Had a more exact analysis been performed, all these curves would tend to turn upward,
intersecting and the rising above the line
.
This, of course, indicates that a larger load P is needed to create larger column deflection.
We have not considered this analysis here, however, since most often engineering design
restricts the deflection of columns to small values.
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The Secant Formula. The maximum stress in the column can be determined by realizing
that it is caused by both the axial load and the moment, Fig. 13-15a. Maximum moment
occurs at the column’s midpoint and using Eqs. 13-13 and 13-16, it has a magnitude of
As shown in Fig. 13-15b, the maximum stress in the column is
compressive, and it has a value of
Since the radius of gyration is defined as
form called the secant formula :
⁄ , the above equation can be written in a
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Like Eq. 13-16, Eq. 13-19 indicates that there is a nonlinear relationship between the load
and the stress. Hence, the principle of superposition does not apply, and therefore the loads
have to be added before the stress is determined. Furthermore, due to this nonlinear
relationship, any factor of safety used for design purposes applies to the load and not the
stress.
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The curves in Fig. 13-16 indicate that differences in the eccentricity ratio have a marked
effect on the load-carrying capacity of columns thar have small slenderness ratios.
However, columns that have large slenderness ratios
tend to fail at or near the Euler critical load
regardless of the eccentricity ratio. When using Eq.
13-19 for design purposes, it is therefore important to
have a somewhat accurate value for the eccentricity
ratio for shorter-length columns.
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EXAMPLE
13.4
The W8 x 40 A-36 steel column shown in Fig. 13-17a is fixed at
its base and braced at the top so that it is fixed from
displacement, yet free to rotate about the y-y axis. Also it cab
sway to the side in the y-z plane. Determine the maximum
eccentric load the column can support before it either begins to
buckle or the steel yields.
SOLUTION
From the support conditions it is seen that about y-y axis the
column behaves as if it were pinned at its top and fixed at the
bottom and subjected to an axial load P, Fig. 13-17b. About the x-x
axis the column is free at the top and fixed at the bottom, and it is
subjected to both an axial load P and moment M = P (9 in), Fig.
13-17c.
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y-y Axis Buckling
From Fig. 13-10d the effective length factor is
!
0.7, so
!
0.7 12 ft
8.40 ft
100.8 in.
Using the table in Appendix B to determine ! for the W8 x 40 section and applying Eq. 13-11,
we have
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x-x Axis Yielding
From Fig. 13-10b,
2 so
2 12 ft 24 ft
Appendix B to determine
11.7 in , 3
8.25in./2
applying the secant formula, we have
)8
15
-3
sec
288 in. Again using the table in
4.125 in and
3.53 in, and
2
Substituting the data and simplifying yields
412.2
1 5 2.979 sec 0.0700
Solving for
by trial and error, noting that the argument for the
secant is in radians, we get
88.4 kip
Since this value is less than
Ans.
!
1383 kip , failure will occur about the x-x axis.
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ELASTIC BUCKLING
Centric Load
Eccentric Load
(Euler’s Formula)
(Secant Formula)
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13.5 Inelastic Buckling
In engineering practice, columns are generally classified according to the type of stresses
developed within the column at the time of failure.
Long slender columns will become unstable when the compressive stress remains elastic.
The failure that occurs is referred to as elastic instability.
Intermediate columns fail due to inelastic instability, meaning that the compressive
stress at failure is greater than the material’s proportional limit.
And short columns, sometimes called posts, do not become unstable; rather the material
simply yields or fractures.
This crane boom failed by buckling caused by an overload. Note the region of localized
collapse.
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Application of the Euler equation requires that the stress in the column remain below the
material’s yield point (actually the proportional limit) when the column buckles, and so
this equation applies only to long columns.
In practice, however, most columns are selected to have intermediate lengths. The behavior
of these columns can be studied by modifying the Euler equation so that it applies for
inelastic buckling.
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No actual column can be considered to be either perfectly straight or loaded along its
centroidal axis, as assumed here, and therefore it is indeed very difficult to develop an
expression that will provide a complete analysis of this phenomenon.
As a result, other methods of describing the inelastic buckling of columns have been
considered. One of these methods was developed by the aeronautical engineer F. R.
Shanley and is called the Shanley theory of inelastic buckling.
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Although it provides a better description of the phenomenon than the tangent modulus
theory, as explained here, experimental testing of a large number of columns, each of which
approximates the ideal column, has shown that Eq. 13–20 is reasonably accurate in
predicting the column’s critical stress. Furthermore, the tangent modulus approach to
modeling inelastic column behavior is relatively easy to apply.
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EXAMPLE
13.5
A solid rod has a diameter of 30 mm and is 600 mm long. It is made of a material that can
be modeled by the stress-strain diagram shown in Fig. 13-19. If it is used as a pinsupported column, determine the critical load.
SOLUTION
The radius of gyration is
and therefore, the slenderness ratio is
Applying Eq. 13-20 we have
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First, we will assume that the critical stress is elastic. From Fig. 13-19.
Thus, Eq. 1 becomes
Thus, Eq. 1 becomes
Since this value falls within the limits of 150 MPa and 270 MPa, it is indeed the critical stress.
The critical load on the rod is therefore
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13. 6 Design of Column for Concentric Loading
• The theory presented thus far applies to columns that are perfectly straight,
made of homogeneous material, and originally stress free.
• Practically speaking, though, as stated previously, columns are not perfectly
straight, and most have residual stresses in them, primarily due to
nonuniform cooling during manufacture.
• Also, the supports for columns are less than exact, and the points of
application and directions of loads are not known with absolute certainty.
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• In order to compensate for these effects, which actually vary from one
column to the next, many design codes specify the use of column formulas
that are empirical.
• By performing experimental tests on a large number of axially loaded
columns, the results may be plotted and a design formula developed by
curve-fitting the mean of the data.
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An example of such tests for wide-flange steel columns is shown in Fig. 13–20.
Notice the similarity between these results and those of the family of curves determined from the
secant formula, Fig. 13–16.
The reason for this similarity has to do with the influence of an “accidental” eccentricity ratio on
the column’s strength.
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• As stated in Sec. 13.4, this ratio has more of an effect on the strength of
short and intermediate-length columns than on those that are long.
• In order to account for the behavior of different-length columns, design
codes usually specify several formulas that will best fit the data within the
short, intermediate, and long column range.
• Hence, each formula will apply only for a specific range of slenderness
ratios, and so it is important that the engineer carefully observe the KL/r
limits for which a particular formula is valid.
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• Examples of design formulas for steel, aluminum, and wood columns that are
currently in use will now be discussed.
• The purpose is to give some idea as to how columns are designed in
practice. These formulas should not, however, be used for the design of
actual columns, unless the code from which they are referenced is
consulted.
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Steel Columns
• Columns made of structural steel can be designed on the basis of formulas proposed by the
Structural Stability Research Council (SSRC).
• Factors of safety have been applied to these formulas and adopted as specifications for building
construction by the American Institute of Steel Construction (AISC).
• Basically, these specifications provide two formulas for column design, each of which gives the
maximum allowable stress in the column for a specific range of slenderness ratios.
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EXAMPLE
13.6
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Aluminum Columns
Column design for structural aluminum is specified by the Aluminum
Association using three equations, each applicable for a specific range of
slenderness ratios. Since several types of aluminum alloy exist, there is a
unique set of formulas for each type. For a common alloy (2014-T6) used in
building construction, the formulas are
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These equations are plotted in Fig. 13–22. As shown, the first two represent
straight lines and are used to model the effects of columns in the short and
intermediate range. The third formula has the same form as the Euler formula
and is used for long columns.
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EXAMPLE
13.8
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EXAMPLE
13.9
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Timber Columns
Columns used in timber construction are designed on the basis of formulas published by
the National Forest Products Association (NFPA) or the American Institute of Timber
Construction (AITC). For example, the NFPA formulas for the allowable stress in short,
intermediate, and long columns having a rectangular cross section of dimensions b and d,
where d is the smallest dimension of the cross section, are
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Steel Columns
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Aluminum Columns
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Timber Columns
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13.7 Design of Columns for Eccentric Loading
Occasionally a column may be required to support a load acting either at its
edge or on an angle bracket attached to its side, such as shown in Fig. 13–28a.
The bending moment M = Pe, which is caused by the eccentric loading, must
be accounted for when the column is designed. There are several acceptable
ways in which this is done in engineering practice. We will discuss two of the
most common methods.
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Use of Available Column Formulas. The stress distribution acting over the crosssectional area of the column shown in Fig. 13–28a is determined from a superposition of
both the axial force P and the bending moment M = Pe. In particular, the maximum
compressive stress is
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The total area A for the column needed to resist both the axial load and moment requires that
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• Notice that, if the column is subjected only to an axial load, then the bending-stress
ratio in Eq. 13–31 would be equal to zero and the design will be based only on the
allowable axial stress.
• Likewise, when no axial load is present, the axial-stress ratio is zero and the stress
requirement will be based on the allowable bending stress. Hence, each stress ratio
indicates the contribution of axial load or bending moment.
• Since Eq. 13–31 shows how these loadings interact, this equation is sometimes referred to
as the interaction formula. This design approach requires a trial-and-check procedure,
where it is required that the designer pick an available column and then check to see if the
inequality is satisfied. If it is not, a larger section is then picked and the process repeated.
An economical choice is made when the left side is close to but less than 1.
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EXAMPLE
13.10
The column in Fig. 13–29 is made of aluminum alloy 2014-T6 and is used
to support an eccentric load P. Determine the maximum magnitude of P
that can be supported if the column is fixed at its base and free at its top.
Use Eq. 13–30.
SOLUTION
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The maximum compressive stress in the column is determined from the combination of axial
load and bending. We have
Assuming that this stress is uniform over the cross section, we require
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EXAMPLE
13.11
SOLUTION
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To determine the allowable compressive stress, we have
Since
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Applying the interaction Eq. 13–31 yields
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Checking the application of the interaction method for the steel section, we require
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EXAMPLE
13.12
The timber column in Fig. 13–31 is made from two boards nailed
together so that the cross section has the dimensions shown. If the
column is fixed at its base and free at its top, use Eq. 13–30 to
determine the eccentric load P that can be supported.
SOLUTION
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READY TO REJECTED
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BUCKLING INSTABILITY OF COLUMNS IN COMPRESSION
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Solution
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The value of the stress corresponding to the critical load is called the
critical stress or critical unit load and is given by
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(47) Column buckling - YouTube
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(46) SAFI ANIMATION | Steel Building Buckling Analysis YouTube
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