course_10_buckling.pdf

TRANSILVANIA UNIVERSITY OF BRASOV MECHANICAL ENGINEERING FACULTY

DEPARTMENT OF MECHANICAL ENGINEERING

ONLY FOR STUDENTS

STRENGTH OF MATERIALS - PART II Prof.dr.ing. Ioan Calin ROSCA

1

Course 10

BUKLING

10.1. Introduction

Structural members that carry compressive loads can be divided into two types depending

on their relative lengths and cross-sectional dimensions:

a) short and thick members, defined as columns that usually fail by crushing when the

yield stress of the material in compression is exceeded;

b) long and slender columns or struts that fail by buckling some time before the yield

stress in compression is reached.

The buckling occurs owing to one or more of the following reasons:

(a) the strut may not be perfectly straight initially;

(b) the load may not be applied exactly along the axis of the strut;

(c) one part of the material may yield in compression more readily than others owing to

some lack of uniformity in the material properties throughout the strut.

At the buckling load the strut is said to be in a state of neutral equilibrium, and

theoretically it should then be possible to gently deflect the strut into a simple sine wave provided

that the amplitude of the wave is kept small. This can be demonstrated quite simply using long thin

strips of metal, e.g. a metal rule, and gentle application of compressive loads.

Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with

loads exceeding the buckling load, any slight lateral disturbance then causing failure by buckling;

this condition is never achieved in practice under static load conditions.

Buckling occurs immediately at the point where the buckling load is reached owing to

the reasons stated earlier.

The above comments and the contents of this chapter refer to the elastic stability of struts

only. It must also be remembered that struts can also fail plastically, and in this case the failure is

irreversible.

When the axial force reach the critical force of buckling CRP then the strut loss the stable

equilibrium. To critical buckling force corresponds a critical buckling stress:

A

PCR

CR ,

where A is the cross section area.

At values of load P below the buckling load a strut will be in stable equilibrium where the

displacement caused by any lateral disturbance will be totally recovered when the disturbance is

removed. Between the two forces can be written the following relationship:

b

CR

c

PP ,



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where bc is the safety buckling coefficient 1bc .

There are two types of equilibrium:

a) stable equilibrium, when CRPP ;

b) unstable equilibrium, when CRPP .

10.2. Euler’s theory

Consider the axially loaded strut shown in Figure 10.1 subjected to the crippling load cP ,

producing a deflection v at a distance x from one end. Assume that the ends are either pin-jointed or

rounded so that there is no moment at either end. It is assumed that it is in the displaced state of

neutral equilibrium associated with buckling so that the compressive axial load has reached the

value CRP .

PCR PCR

l

x

v

v

x

Figure 10.1

The cross section of the beam is constant and then, the stiffness modulus .constEI The

bending moments developed at a considered distance x, referred to the axes Oxv, are positive and equal

with:

vPM CRb , (10.1)

and the differential equation of the deformed medium axe is:

EI

M

dx

vd b2

2

. (10.2)

If is introduced relationship (10.1) in (10.2) one can obtain the differential equation:

02

2

EI

vP

dx

vd CR , (10.3)

or, 02

2

vdx

vd



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where: EI

PCR . (10.4)

The solution of the equation (10.4) is obtained by integration and is:

xBxAv cossin , (10.5)

where A and B are constants that can be found out from boundary conditions:

.0

;00

vlx

vx (10.6)

Considering the above boundary conditions, one can obtain the following values of the

constant quantities A and B:

from the first condition in (10.6) 0B ; (10.7)

from the second condition in (10.6) 0sin lA . (10.8)

The value of constant A must be different of zero 0A otherwise there is no buckling. So

results that it is necessary to have the condition:

0sin l , (10.9)

that leads to: nl , (10.10)

where n is a positive integer number.

Considering both relationships (10.4) and (10.10) results the crippling critical force as:

2

22

l

EInPCR

. (10.11)

For the situation described in Figure 10.1, the value of integer is 1n . The other solutions

...,4,3,2n leads to greater values of crippling force that are not achieved after the struts reach

the first critical force CRP .



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Observation: The higher values of buckling load

correspond to more complex buckling modes.

Theoretically these different modes could be

produced by applying external restraints to a

slender column at the points of contraflexure to

prevent lateral movement (Figure 10.2).

However, in practice, the lowest value is never

exceeded since high stresses develop at this load

and failure of the column ensues. We are not

therefore concerned with buckling loads higher

than this.

In case of identical boundary conditions

around all cross section central axes then it has

to be considered the minimum axial central

moment because the buckling is done around the

central axe.

PCR

PCR

l

l / 3

l / 2 l / 2

l / 3 l / 3

Figure 10.2

So, the Euler’s relationship becomes:

2

min

2

l

EIPCR

. (10.12)

In the fundamental case, the buckling state corresponds to a half wave, having the length

approximate equal with the struts.

Definition:

The length that corresponds to a halfwave (the distance between two consecutive inflexion

points of the buckling state) is called buckling length (equivalent length).

The above presented case is called the fundamental case because the equivalent length is

equal with the struts length:

lle . (10.13)

10.3. Buckling load for a column with one end fixed and one end free

In this configuration the upper end of the column is free to move laterally and also to rotate as

shown in Figure 10.3. At any section x the bending moment bM is given by (10.1):

vPM CRb ,

and the differential equation of the deformed medium axe is (10.2):

EI

M

dx

vd b2

2

.



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l

x

v

v

PCR

x

le

Figure 10.3

If is introduced relationship (10.1) in (10.2) one can obtain the differential equation (10.3):

02

2

EI

vP

dx

vd CR ,

or, 02

2

vdx

vd

The solution of the equation is obtained by integration and is:

xBxAv cossin ,

where A and B are constants that can be found out from boundary conditions:

.0

;00

dx

dvlx

vx

(10.14)

Considering the above boundary conditions, one can obtain the following values of the

constant quantities A and B:

from the first condition in (10.14) 0B ; (10.15)

from the second condition in (10.14) 0cos lA . (10.16)

The value of constant A must be different of zero 0A otherwise there is no buckling. So

results that it is necessary to have the condition:

0cos l , (10.17)



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that leads to:

2

12

nl , (10.18)

where n is a positive integer number.

Considering both relationships (10.4) and (10.18) results the crippling critical force, for

1n , as:

2

2

4l

EIPCR

. (10.19)

In this case the equivalent length is: lle 2 . (10.20)

10.4. Buckling of a column with one end fixed, the other pinned

It is considered the strut from Figure 10.4. In this case, the equivalent length is equal with:

lle 7.0 , (10.21)

and the crippling force, given by (10.12) becomes:

2

22

l

EIPCR

. (10.22)

PCR

le 0,5le

l

Figure 10.4



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PCR

0,5le

l

0,5le 0,5le 0,5le

Figure 10.5

10.5. Buckling load for a column with fixed ends

In practice, columns usually have their ends restrained against rotation so that they are, in

effect, fixed. Figure 10.5 shows a column having its ends fixed and subjected to an axial

compressive load that has reached the critical value, CRP , so that the column is in a state of neutral

equilibrium. In this case the ends of the column are subjected to fixing moments, M , in addition to

axial load.

The equivalent length is:

lle2

1 , (10.23)

and the crippling force is: 2

24

l

EIPCR

. (10.24)

10.6. Conclusion

Comparing the values of crippling forces and equivalent lengths can be done the following

table (Table 10.1).

Table 10.1

Quantity Case

I II III IV

Crippling force

2

min

2

l

EIPCR

2

2

4l

EIPCR

2

22

l

EIPCR

2

24

l

EIPCR

Equivalent length lle lle 2 lle 7.0 lle

2

1



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Comparing the crippling force values for all four cases one can conclude that:

IVCRIIICRIICRICR PPPP ,., .

The relationship of Euler can be generalised as:

min

2

2

e

CRl

IEP . (10.25)

10.7. Limitations of the Euler theory

For a column of cross-sectional area A the critical stress, CR , from equation (10.25) results:

2

min

22

min

2

min

2

2

E

l

iE

l

I

A

E

A

P

ee

CR

CR

, (10.26)

where: i

le , (10.27)

is defined as slenderness ratio, and i is is the radius of gyration of the cross-section.

Clearly if the column is long and slender ile is large and CR is small; conversely, for a

short column having a comparatively large area of cross-section, ile is small and CR is high.

The maximum value of the for slenderness ratio for a given cross section is:

minmax

maxi

l

i

l ee

. (10.28)

A graph of CR against ile for a particular material has the form shown in Figure 10.5. For

values of ile less than some particular value, which depends upon the material, a column will fail

in compression rather than by buckling so that CR as predicted by the Euler theory is no longer

valid. Thus in Figure 10.6 the actual failure stress follows the dotted curve rather than the full line.



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cr

le /i

Euler’s theory

Actual failure stress

Figure 10.6

From relationship (10.26) one can conclude

that the critical stress depends on both

material type (by Young’s modulus E) and

slenderness ratio.

The Euler’s relationship is valid only

if the deformation is done in the elastically

domain (the Hooke’s law is valid).

The Euler’s relationship can be used only if

the critical stress is smaller that the stress

that corresponds to the proportionately limit

of the material p :

pCR

E

2

max

2

. (10.29)

Based on relationship (10.29) one can calculate the slenderness ratio 0 that bound the

Euler’s relationship validity domain:

p

E

2

0max . (10.30)

There are defined two cases:

a) 0max , then the buckling is defined as an elastic one. The buckling is done based

on Euler’s relationship;

b) 0max , then the buckling is done in the plastic domain and the Euler’s

relationship is not a valid one.

The buckling calculus in the plastic domain is done using some different relationships that

were developed by different scientists (F. Iasinski, L. Tetmajer, M. Rankine, J. B. Johnson, etc.).

Generally these relationships have the following forms:

,

;

2

CBA

ba

CR

CR (10.31)

where CBAba ,,,, are constant values that depend on the material.

In Figure 10.7 is shown the resume of the critical crippling stress CR domains.



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Right line of Tetmajer, Iasinski

Eulers’s parabola

CR

Elastic buckling

Plastic

buckling

0 1

Pure

compre-

ssion

0

CR

y

yield stress

p

proportional

stress

all

allowed

stress

Figure 10.7

10.8. Problems solving procedure

The buckling problems are focused on verification and dimensioning.

Verification problems – it is checked if the buckling safety factor bc is greater than the prescript

one bpc .

It is checked the buckling domain (elastic or plastic). So, is calculate the slenderness ratio

and can be two situations:

a) 0 , when is used the Euler’s relationship

min

2

2

e

CRl

IEP and the strut is a stable

one if bp

CR

b cP

Pc ;

b) 01 , when are used the relationships (10.31), the crippling force is AP CRCR

and it is necessary to verify the condition bp

CR

b cP

Pc ;

c) 10 , is the case of the compression problems.

Dimensioning problems Step 1 - from the beginning it is considered that the buckling is done in the elastic domain. So, first

is calculate the second moment:

E

clPI beCR

ned 2

2

.



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Step 2 – there are chose the cross sectional dimensions and is calculate the slenderness ratio .

Step 3 – it is compared the slenderness ratio with 0 . There are the following situations:

a) 0 , the buckling phenomenon is in the elastic domain and the dimensioning problem

is considered to be finished and can be used the cross sectional obtained dimensions;

b) 01 , it is necessary to be checked the safety coefficient (comparison with the

prescript safety coefficient). If bpb cc than it is necessary to amplify the cross section

dimensions till the safety condition is fulfilled;

c) 10 , the calculation is done for compression problems.

10.10. The method of allowed stress

This method is used mainly for verification problems and for finding the allowed force CRP .

According with this method the stability condition is:

allb

CR

A

P, , (10.32)

where the buckling allowed stress allb, is:

allallb , , (10.33)

is the buckling coefficient, and all is the allowed stress of the material.

The buckling coefficient depends on the material and is given in a tabular shape.

course_10_buckling.pdf

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