1Chapter 1

Introduction

1.1 Introduction

Structures and machines may fail in a variety of ways, depending upon the material, kind of

loads, and conditions of supports.

Buckling is a mathematical instability, leading to a failure mode. Theoretically, buckling is

caused by a bifurcation in the solution to the equations of static equilibrium. At a certain stage

under an increasing load, further load is able to be sustained in one of two states of

equilibrium: an un-deformed state or a laterally-deformed state. Buckling is caused by the

compressive load acting axially in the column (that is, long, slender structural members loaded

axially in compression).if such a member is slender, then instead of failing by direct

compression, it may bend and deflect laterally and we can say the column has buckled. Under

an increasing axial load, the lateral deflection increase too, and eventually the column will

collapse completely.

1.1.1 Buckling and stability

Buckling is characterized by a sudden failure of a structural member subjected to

high compressive stress, where the actual compressive stress at the point of failure is less than

the ultimate compressive stresses that the material is capable of withstanding. For example,

during earthquakes, reinforced concrete members may experience lateral deformation of the

longitudinal reinforcing bars. This mode of failure is also described as failure due to elastic

instability. Mathematical analysis of buckling makes use of an axial load eccentricity that

introduces a moment, which does not form part of the primary forces to which the member is

subjected. When load is constantly being applied on a member, such as column, it will

ultimately become large enough to cause the member to become unstable. Further load will

cause significant and somewhat unpredictable deformations, possibly leading to complete loss

of load-carrying capacity. The member is said to have buckled, to have deformed.

21.1.2 Slenderness ratio and effective length

(1) Effective length: The effective length of a given column with given end conditions is the

length of an equivalent column of the same material and section with hinged ends having the

value of crippling load equal to that of given column.

(2) Slenderness ratio (K): The ratio of the effective length of a column to the least radius of

gyration of its cross section is called the slenderness ratio (sometimes expressed with the

Greek letter lambda, ). This ratio affords a means of classifying columns. Slenderness ratio is important for design considerations.

Slenderness ratio is important for design considerations. All the following are approximate

values used for convenience.

A short steel column is one whose slenderness ratio does not exceed 50; an intermediate

length steel column has a slenderness ratio ranging from about 50 to 200, and are

dominated by the strength limit of the material, while a long steel column may be

assumed to have a slenderness ratio greater than 200.

A short concrete column is one having a ratio of unsupported length to least dimension of

the cross section not greater than 10. If the ratio is greater than 10, it is a long column

(sometimes referred to as a slender column).

The dividing line between intermediate and long columns cannot be readily evaluated. One

way of defining the lower limit of long timber columns would be to set it as the smallest

value of the ratio of length to least cross sectional area that would just exceed a certain

constant K of the material. Since K depends on the modulus of elasticity and

compressive stress parallel to the grain, it can be seen that this arbitrary limit would vary with

the species of the timber. The value of K is given in most structural handbooks.

Some rules of the buckling of columns are as follows:

(i) A column made of a ductile material like steel and whose length is more than eight

times of its least lateral dimension is likely to buckle and should be treated as a

column.

3(ii) A column made of a brittle material like cast iron and whose length is more than six

times of its least lateral dimension is likely to buckle and should be treated as a

column.

There is a basic difference between lateral deflection of a beam and buckling of columns.

The lateral deflection of the beam is gradually increased as the lateral load is increased. In

case of buckling there is no such lateral deflection till the load reaches the critical value at

this point ,there is sudden lateral deflection, which result in collapse of the column .the

failure of the buckling is ,therefore, sudden and total without any warning.

1.1.3 Important point to note

When the slenderness ratio is less than 30, there is no effect on buckling and such

components are designed on the basis of compressive stresses. Columns with slenderness

ratio greater than 30 are designed on the basis of critical load there are two namely short

and long columns that are frequently used in buckling analysis. The rules for deciding

long and short columns are follows:

(i) Cast iron columns with slenderness ratio not greater than 80 and steel columns

with a slenderness ratio not greater than 100 are considered as short columns.

(ii) Long columns are those with slenderness ratio greater than 100 for ductile

materials and greater than 80 for cast iron.

Fig1.1 Different loading conditions and their respective effective length

41.1.4 Influencing Factors

Below the influence of various characteristic values such as the E modulus, geometric

moment of inertia, length and the type of mounting on buckling behaviour will be examined

using the Euler formula.

1. E Modulus: The E modulus is a measure of the rigidity of a material. A stiff material

is sensible for high resistance to buckling. Since strength has no influence on

buckling, materials with as high an E modulus as possible should be used. For

example, in the case of buckling strength a simple constructive steel St37 with a

tensile strength of only 330 N/mm 2should be given.

2. Geometric Moment of Inertia: The geometric moment of inertia indicates the

resistance against deflection resulting from the cross-sectional shape of the rod. Since

a rod buckles in the direction of least resistance, the minimum geometric moment of

inertia is the decisive factor. The table contains the geometric moment of inertia for

several cross-sectional shapes. Here, hollow sections with small wall thickness are

more favourable at the same weight as solid cross sections.

3. Buckling Length: The length of the rod as well as the type of mounting determines

the buckling length. The influence of the length is quadratic. At twice the length the

admissible load is only one-fourth the original value.

1.2. Formulae used for finding the critical load

(i) Euler formulae

(ii) Rankine formulae

(iii) Johnson formulae

(iv) Perry Robertson method

1.2.1 Eulers formula

Mathematician LEONARD EULER derived a formula that gives the maximum axial load

that a long, slender, ideal column can carry without buckling. An ideal column is one that is

perfectly straight, homogeneous, and free from initial stress. The maximum load, sometimes

called the critical load, causes the column to be in a state of unstable equilibrium that is, the

5introduction of the slightest lateral force will cause the column to fail by buckling. The

formula derived by Euler for columns with no consideration for lateral forces is given below:.

F= 2EI(KL)2Where,

F= critical load

E= youngs modulus

I= moment of inertia

L= unsupported length of column,

K= column effective length factor, whose value depends on the conditions of end

support of the column, as follows.

Assumptions in Eulers formula

The material of the column is homogenous isotropic elastic. The section of the column is uniform throughout. The column is initially straight and is loaded axially. The column fails by buckling alone.

The self-weight of the column is negligible.

1.2.2 Rankine formula

In case of short column which fails by crushing the load at failure equals P=Fc A where Fc is

crushing stress for the column material and A is the sectional area. But for a long column

which fails by buckling, the load at failure that is the buckling load. The struts and the

columns which we come across are neither too short nor too long. The failure of the member

will be due to the combined effect of direct and buckling stress. Rankine devised a formula

for the collapse load which should cover all columns whether they are short or long.

6Let Pc be the crippling load, Rankine stated his empirical formula as:1PR = 1Pe + 1PcWhere Pe = Euler crippling load Pc = Crushing load or Yield point load in Compression PR = Actual load to cause failure or Rankine loadSince the Rankine formulae is a combination of the Euler and crushing load for a strut.1PR = 1Pe + 1PcFor a very short strut Pe is very large hence 1Pe would be large so that 1Pe can be neglected.Thus PR = Pc for very large struts, Pe is very small so 1Pe would be large and 1Pc can be neglected, hence PR = Pe.The Rankine formulae are therefore valid for extreme values of 1/k .It are also found to be

fairly accurate for the intermediate values in the range under consideration. Thus rewriting

the formula in terms of stresses, we have1? ? = 1? ? ? + 1? ? ?1? = 1? ? + 1? ?

7= ? ?1+? ?? ?

For struts with both ends pinned

= ? ?1+? ? l? ?2

The value of a' is found by conducting experiments on various materials. Theoretically, but

having a value normally found by experiment for various materials. This will take into

account other types of end conditions.

Rankine load = ? ?1 + ? ? l? ?2Rankine formula is used in design of machine components like IC engine connecting rod.

.

Fig2.1 Variation of unit load against slenderness ratio [8]

81.2.3 Johnsons parabolic formula

PA = FC g ? lk?2 For pinned struts. Where Fc = compressive yield stress

g = constant depending upon column material

E = youngs modulus

Things which we rectify by Johanssons formula:

1. The effect of direct compression has been neglected in the case of Eulers formulae.

2. The loading is not exactly applied as desired.

3. The pin joints are not practically frictionless.

4. Absolute fixation of ends is not possible.

5. The members are not perfectly straight uniform and homogenous.

1.2.4 Straight line formula

PA = FC n ?Lek ?2 Where

PA= stress corresponding to PA P = crippling load on the column

Fc=compressive yield stress

A =area of c/s of the column

Le k = slenderness ratio

n = constant whose value depends upon the material of the column

1.2.5 Perrys formulae

A column in actual cases has imperfections like slight initial curvature of longitudinal axis,

imperfect support condition and material non homogeneity .very slight unavoidable

eccentricity may also exist in the line of application of the load. Such imperfection vary

column to column .Due to such imperfection the column is subjected to not only direct load

9but also some bending moment .it is therefore reasonable to assume a centrally loaded

column with imperfections to be equivalent an ideal eccentrically loaded column. This is a

formula which is found for these cases where we have to determine the safe load that can be

applied at a column at a given eccentricity.

Professor Perrys formula? ?? 0 1? ?1 ? 0? ? ? = 1.2? ? ?? 2 Where

? = maximum permissible compressive stress. ? 0 = stress due to direct load = PA.

? ? ?? 2 is called eccentricity ratio.1.3 Note/Conclusion

These formulae give us the approximate result when compared with the actual result obtained

from the experiment due to following reasons:

a. The effect of direct compression has been neglected in case of Eulers formula.

b. The loading is not exactly applied as desired.

c. The pin joints are not practically frictionless

d. Absolute fixation of ends is not possible.

e. The members are not perfectly straight, uniform and homogenous.

So we try to use some other empirical relations for practical design. Some of them are:

Stress at critical load for structural steel PA= 367.5-2? lk? ?? ? ?

Safe working stress for mild steel =150[1-0.0038? lk? ] ?? ? ?

10

1.4 Objective

1. To first calculate the buckling load for different cross section using different theoretical

formula available (Rankine, Euler, Johnson straight line formula).

2. To observe the buckling behaviour of columns and estimate their buckling loads by varying

there length and end conditions.

3. To measure the lateral deflection of the columns during axial compression.

4. To compare the analytical and experimental values of buckling loads of columns.

5. To obtain the deflection behaviour of columns with slight initial bent shape.

6. To find the effects of material properties and end restraint on column behaviour.

1.5 Applications

1. Buckling is a very big factor for the failure of the connecting rod .we determines the cross section

of the connecting rod by buckling consideration.

Fig 3.1 Buckling of connecting rod in two planes [8]

2. To find the core and kernel of the column by buckling consideration.

3. Buckling failure is also important to select the cross section of the rods like square, triangle,

rectangle or a hexagon. Also the critical load for an equilateral triangle is 21% higher than that

of circular column so we use it instead of circular rod.

11

Chapter 2

LITERATURE REVIEW

2.1 Literature survey

[1] Jian-kang Chen (1987) presents an analytical solution for the linear elastic buckling

analysis of simply supported battened columns subjected to axial compressed loading. The

critical buckling load is derived by using the classical energy method. Unlike most of existing

work, the present approach considers not only the shear effect but also the discrete effect of

battens on the global buckling behaviour of the columns. The present analytical solution is

validated using the data obtained from the finite element analysis. The results show that the

number of battens has significant influence on the critical buckling load of battened columns,

particularly when the relative rigidity of the batten to the main member is small. It is shown

that the critical buckling load increases with the number of battens, the combined bending

and shear rigidity of battens, but decreases with the increased membrane stiffness of the two

main members, and the increased distance between the centroid of the two main members.

[3] Javad Alamatian (1999) prepares new features for applying the Dynamic Relaxation (DR)

method in buckling and post-buckling structural analyses. Here, controlling the

displacements increment is the main concept so that the minimum residual displacement and

the minimum kinetic energy criteria are designed for updating the load factor in DR

iterations. These new formulations do not affect the DRs stability and they are successfully

applied to both viscous and kinetic DR algorithms. Numerical analyses of structures with

snap-through and snap-back behaviours show that the accuracy and the convergence rate of

the suggested methods are higher than the well-known existing approaches such as the arc-

length technique, especially, if they are combined with the kinetic DR algorithm. Calculating

the buckling load is another interesting application of the proposed methods, performed here.

[2] Dongming Wei et al (2006) present analytic formulas for calculating the critical buckling

states of some plastic axial columns of constant cross-sections. The associated critical

buckling loads are calculated by Euler-type analytic formulas and the associated deformed

shapes are presented in terms of generalized trigonometric functions. The plasticity of the

material is defined by the Hollomons power-law equation. This is an extension of the Euler

12

critical buckling loads of perfect elastic columns to perfect plastic columns. In particular,

critical loads for perfect straight plastic columns with circular and rectangular cross-sections

are calculated for a list of commonly used metals. Connections and comparisons to the

classical result of the Euler-Engesser reduced-modulus loads are also presented.

[3] Safa Bozkurt Cokun (2009) variational iteration method (VIM) is applied to the problem of determination of critical buckling loads for Euler columns with constant and variable

cross-sections. VIM is a powerful method for the solution of nonlinear ordinary and partial

differential equations and integral equations. Hence it is a suitable approach for the analysis

of engineering problems where an exact solution is difficult to obtain. This study presents the

application of VIM to various buckling cases and results are produced for columns with

different support conditions and with different variation of cross-sections.

2.2 Conclusions to Literature Survey

From the above survey of the literature on various aspects of Buckling we concluded that

number of methods are applicable to find Buckling load but some of the above methods are

not applied to our experimental setup, so others methods like Eulers , Rankine , Prof. Perry

are applied to find the critical load. Eulers formula is specifically used for the long columns

and Rankine is used for both long and short column due to consideration of the crushing load

which also makes Rankine to give more accurate result than Euler approach. Perry Robertson

formula is for eccentric load. So from all these above method we find the critical load for all

loading conditions.

2.3 Problem Formulation

Values of critical load for different cross section and different end conditions are calculated

for different materials like (steel, cast iron, timber, copper) and different analytical formulae

are used to obtain the critical load under these various different conditions. These analytical

formulae also have some assumptions which cause the deviation of the theoretical critical

load value from the actual or experimental critical load. These analytical formulae give error

about 6-7%, so we will try to find out the exact value of critical buckling load by

experimental setup for buckling under its various supporting conditions.

13

Chapter 3

Methodology

3.1 Modeling of column

1. Different cross sections taken (rectangle, circular, hollow circular, I section) of specific

dimension (specified in results) for column of different material (steel, cast iron, timber,

copper). Value of moment of inertia for these different cross sections value K and other

unknowns are calculated.

2. A program is developed on C++ language for calculating the critical load for different

cross section of column using various formulae like Eulers, rankine, johanson.

3. Here the length is varied and the variation of the critical load for change in length for

different cross section using different material and different formulae is plotted in graph.

4. Different graph corresponding to various material and cross section are obtained.

Fig3.1 Screen shot for the calculation for Eulers critical load

14

Fig3.2 Screen shot for the calculation of Rankines critical load

Fig3.3 Screenshot for the calculation of the Johnsons Critical load

15

Chapter 4

4.1 Result and discussion

In the first phase of our project the following objectives are achieved The theoretical critical load

value for the different materials (steel , cast iron , timber, copper) under different end conditions

(both end hinge ,both end fixed , one hinge one fixed, one fixed and other free) and for different

cross section (rectangular,circular,hollow circular , I section) and the variation for different

length calculated and plotted in the graph.

We use different formula to find the critical buckling loads which are:

1. Eulers formula

2. Rankins formula

3. Johnson formula(Johnson straight line formula)

For Calculation following data is taken

1. Cross section taken

1. Rectangular

2. Circular

3. Hollow circular

4. I section

2. Material taken:1. Steel2. Cast iron3. Timber4. copper

3. Dimensions :1. Area (A) =12 mm2

2. Length (L) = 300 to 350 mm

16

Cross section properties

1. Radius of gyration

1. K (rectangular) = 0.29 mm

2. K (circular) = 0.98 mm

3. K (hollow circular) = 1.437

4. K (I section) = 1.08

2.Moment of inertia

1. Imin (rectangular) = 1 mm4 2. Imin (circular) = 11.47 mm4 3. Imin (hollow circular) = 26.13mm4 4. Imin (I section) = 12.986mm4

Material properties

1. Youngs modulus of elasticity

1. Esteel =200 GPA2. Ecast iron =210GPA3. Etimber = 11GPA4. Ecopper = 117GPA

2. Yield compressive stress 1. Fc (steel) = 315MPA2. Fc (cast iron) = 560MPA3. Fc (timber) = 35MPA4. Fc (copper) = 70MPA

3. Rankines constant 1. a (steel ) =0.0001332. a (cast iron) = 0.0006253. a (timber) =0.00033334. a (copper) = 0.000401

End condition to be maintained

1. both end fixed (n=4)2. both end hinged (n=1)3. one free one hinged(n=0.25)4. one fixed one hinged(n=2)

17

BY EULERS FORMULA

Table 1: Observation for hollow circular section

L (mm) Pe(steel) (N) Pe(castiron) (N) Pe(timber)(N) Pe(copper) (N)300 2294.1 2408.84 126.18 1342.07310 2148.51 2235.9 118.17 1256.88320 2016.33 2117.14 110.89 1179.55330 1895.98 1990 104.28 1109.15340 1786.09 1875.39 98.23 1044.86350 1685.48 1769.26 92.7 986.01360 1593.14 1672.8 87.62 931.99370 1508.19 1583.6 82.95 882.29380 1429.86 1501.35 78.64 836.47390 1357.47 1425.34 74.66 794.12400 1290.45 1354.97 70.97 754.91

Fig4.1 Variation of the critical buckling load wrt length for different material using Eulers

formula for hollow cross section

18

Table 2: Observation for I section

L (mm) Pe(steel)(N) Pe(castiron)(N) Pe(timber)(N) Pe(copper)(N)

300 1137.68 1194.56 62.58 665.54

310 1065.63 1118.74 58.6 625.29

320 1000.07 1049.91 55 584.95

330 940.38 987.24 51.71 550.03

340 885.87 930.02 48.72 518.16

350 835.97 877.65 45.57 488.97

360 790.18 829.56 43.46 462.18

370 748.04 785.32 41.14 437.54

380 709.12 749.53 39 414.81

390 673.29 706.84 37.03 339.81

400 640.04 671.94 35.2 374.31

Fig4.2 Variation of the critical buckling load wrt length using Eulers formula for I section

19

Table 3: Observtion for Rectangular cross section

L (mm) Pe(steel) (N) Pe(castiron) (N) Pe(timber) (N) Pe(copper) (N)

300 92.116 87.729 4.825 51.34

310 86.269 82.161 4.518 48.064

320 80.961 77.106 4.24 45.107

330 76.129 72.5 3.987 42.414

340 71.71 68.301 3.756 39.956

350 67.67 64.45 3.544 37.7

360 63.96 60.923 3.35 35.64

370 60.55 57.674 3.172 33.739

380 57.413 54.679 3 31.98

390 54.5 51.911 2.855 30.368

400 51.81 49.34 2.714 28.868

Fig4.3 Variation of critical buckling load wrt length using Eulers formula for rectangular cross

section

20

Table 4: Observation for circular cross section

L(mm) Pe(steel) (N) Pe(castiron)(N) Pe(timber)(N) Pe(copper)(N)

300 1006.26 1056.57 55.34 588.666

310 942.386 989.506 51.83 551.29

320 884.4 948.628 48.34 517.34

330 831.619 873.2 45.24 486.49

340 783.42 822.591 43.088 458.3

350 739.293 776.257 40.66 432.48

360 698.791 773.731 38.43 408.79

370 661.529 694.6 36.38 386.99

380 627.17 658.582 34.49 366.89

390 595.42 625.191 32.24 348.49

400 566.021 594.32 31.13 331.19

Fig 4.4 Variation of the critical buckling load wrt to length using Euler formula

For circular cross section

21

BY RANKINE FORMULA

Table 5: Observation for circular cross section

L (mm) Pr(steel)(N) Pr(castiron)(N) Pr(timber)(N) Pr(copper)(N)

150 962.125 429.603 47.72 135.071

155 915.167 403.973 45.015 127.275

160 871.251 380.527 42.53 120.114

165 830.149 359.028 40.23 113.525

170 791.654 339.261 38.11 107.448

Fig4.5 Variation of the critical buckling loads wrt length using Rankine formula for circular

cross section

22

Table 6: Observation for hollow circular cross section

L(mm) Pr steel (N) Pr cast iron (N) Pr timber (N) Pr copper (N)

150 1656.55 2811.12 93.72 293.16

155 1593.72 2704.49 89.03 266.09

160 1533.64 2602.54 84.65 253.95

165 1476.23 2541.06 80.57 241.71

170 1421.38 2505.11 76.75 230.25

Fig4.6 Variation of buckling load wrt load using Rankine formula for hollow cross section

23

Table 7: Observation for the rectangle cross section

L(mm) Pr(steel) (N) Pr(cast iron) (N) Pr(timber) (N) Pr( copper) (N)

300 108.25 39.94 4.66 12.966

310 101.553 37.42 4.36 12.15

320 95.455 35.133 4.1 11.4

330 89.88 33.05 3.86 10.73

340 84.78 31.14 3.68 10.11

350 80.11 29.39 3.435 9.54

Fig4.7 Variation of the critical buckling load wrt to length using Rankine formula for

the rectangle section

24

Table 8: Observation for I section

Fig 4.8 Variation of the critical buckling load wrt length using Rankines formula for the I

section

L (mm) Pr(steel)(N) Pr(castiron)(N) Pr(timber) (N) Pr(copper)(N)150 1110.62 514.693 56.576 160.726155 1058.97 484.377 53.442 151.626160 1010.97 456.602 50.549 143.729165 964.828 436 47.876 135.524170 921.927 407.626 45.401 128.388

25

BY JOHANSON FORMULA

Table 9: Observation for mild steel under variation end condition

L (mm) Pj rectangle (N) Pj circular (N) Pj hollow circular (N) Pj I section (N)300 1835.17 1801.96 1797.33 1800.67310 1836.74 1802.42 1797.64 1801.09320 1838.32 1802.89 1797.97 1801.51330 1839.89 1803.36 1798.26 1801.93340 1841.46 1803.82 1798.57 1802.36350 1843.03 1804.29 1798.88 1802.78

Fig4.9 Variation of the critical buckling load wrt different length for I cross section steel

column

26

Discussion

We plot the graphs between critical load v/s length for different materials of same cross section

and observe there variation in buckling critical load with length using different formula for

finding buckling critical load. Similarly we plot graphs for other cross sections and observe the

following:

According to Eulers formula

1. For the same area rectangular and same end condition steel is the best material having

highest buckling load.

2. For same circular cross section and end condition cast iron is the best material having highest

buckling load value.

3. For hollow circular cross section and same end condition cast iron is the best material having

highest buckling load value.

4. For the same I cross section and same end condition cast iron is the best material having

highest buckling load value.

According to Rankine formula

1. For the same rectangular cross section and end condition steel is the best material having

highest buckling load value.

2. For same circular cross section and end condition steel is the best material having highest

buckling load value.

3. For hollow circular cross section and same end condition cast iron is the best material having

highest buckling load value.

4. For same I section and same end condition steel is the best material and having the highest

buckling load value.

Second observation

According to Johnsons formula

For mild steel rectangular cross section is the best cross section having high buckling load

capacity for same end condition.

27

Limitation

1. As we know all these formula give the approximate value and they are based on so

many assumptions like material is homogenous, isentropic .also the self weight of

column is neglected in these formulas. Euler formula gives best result for long

columns.

2. In Eulers formula effect of crushing is neglected.3. During our analysis of finding the critical buckling load we came know that Johnson

formula is not applicable for the smaller cross sectional area and giving erroneous result.

4. Other formula like Perry Robertson formula, energy method, secant method is not applicable under pure axial compressive load and small cross section. They are only applicable for eccentric loads.

28

Chapter 5

Conclusion and future scope

5.1 Conclusion

From the above analysis and observation the theoretical value of critical load for different

cross section and material and there variation with length and different end condition obtain

with the help of graph .Different formula give the different value of critical load for same

material and same end condition and there variation in also observed from these theoretical

values of critical load we come to conclusion that I section is the best section and mild steel is

the best material among the chosen specimens and material. The theoretical critical load is

obtained by various different formulae and will be compared with the experimental value.

5.2 Future scope

This report has covered the theory of column buckling, the basics of preparing and

performing a column buckling experiment, and the results of this specific analysis and

simulation works. The column buckling theory and governing equations accurately predicts

the critical buckling loads for various lengths and fixities of columns. It appears to that end

fixity of a column, in addition to the length of the column; greatly affect how a compressive

structure behaves under axial compressive force. Structure will buckle under compressive

loads. Till now the theoretical analysis and study of the column for buckling under different

cross-section has been done and the experimental analysis will be done in next phase.

29

REFERENCES

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2. Dongmingwei,Alejandrosarria,MohamedElginci , Critical buckling loads of the perfect Hollomons power low column, Volume 47, January 2013, Pages 69-76.

3. JavadAlamation, Displacement- based method for calculating the buckling load and tracing the post-buckling regions with dynamic relaxation method, Volumes 114115, January 2013, Pages 84-97.

4. SafaBozkurtCokun, Mehmet Tark Atay, method for Determination of critical buckling load for elastic column of constant and variable cross section using variation iteration method, Volume 58, Issues 1112, December 2009, Pages 2260-2266.

5. S. Ramamurtham & R. Narayanam Strength of materials 17th edition 2011 Danpath rai publication.

6. Tod hunter, I., and Pearson , k., a history of the theory of elasticity and of the strength of material ,vols. I and II, Dover publication, inc., Newyork, 1960 pages 298-310.

7. Gere & Timoshenko Mechanics of material 2nd edition 2012 CBS Publication, 1972 Pages 197-278.

8. V.B Bhandari Design of machine elements 3rd edition 2011 Tata McGraw-Hill Publication page no 806.

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