Abstract

Buckling is a failure mode which occurs in long slender structural members, before a

plastic deformation, such as yielding or crushing can happen. This report deals with

a series of experiments on struts of different lengths, but the same cross-sectional

area. It will show that the Euler Buckling load is affected by the slenderness ratio,

radius of gyration, second moment of inertia, and support conditions of a member. It

is also concluded that Euler’s theorem is unsuitable for design purposes as buckling

occurs at a far lower weight than the ideal value given by the theorem.

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Contents

Contents .......................................................................................................................................................... 2

1. Introduction ............................................................................................................................................ 3

2. Procedure: ............................................................................................................................................... 3

I. Experiment .............................................................................................................................................. 3

3. Recorded Results ..................................................................................................................................... 4

4. Calculations and Graphs .......................................................................................................................... 4

Assumptions in Theory ....................................................................................................................................... 4

4.1. Theoretical Results ................................................................................................................................ 4

4.1.1. The second moment of area (I) ......................................................................................................... 5

4.1.2. Radius of Gyration ............................................................................................................................ 5

4.1.3. The effective lengths of the struts ..................................................................................................... 5

4.1.4. Slenderness Ratio .............................................................................................................................. 5

4.1.5. Euler’s Buckling formulae ................................................................................................................. 5

4.1.6. Derivation ......................................................................................................................................... 6

4.1.7. Example ............................................................................................................................................ 6

5. Observations and Conclusions ............................................................................................................... 10

5.1. The Material ........................................................................................................................................ 10

5.2. Definition of Buckling Load ................................................................................................................. 11

5.3. Euler’s Theorem ................................................................................................................................... 11

5.4. Analysis of Graphs ............................................................................................................................... 11

5.5. Difference between Theoretical and Measured Results ...................................................................... 12

5.6. Effect of I-value and Radius of Gyration on the Euler Load ................................................................. 14

5.7. Practical examples of buckling ............................................................................................................ 14

6. References ............................................................................................................................................ 15

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1. Introduction

Compressive members are found in most structures, and as they can fail in two ways, unlike

tension members, it is important to know the limits of the materials used. Buckling is the

elastic mode of failure investigated in this report. Buckling can be defined as an axially

loaded member becoming unstable as load increases till it falls out of equilibrium and

elastically collapses.

2. Procedure:

Blockley (2005) defines a strut as a structural member that resists axial forces and states

that the difference between short/ stocky struts and long/ slender struts is the crushing

failure of the first one and the buckling failure of the latter.

The objectives of this experiment were to investigate the relationship between buckling

load, strut length, boundary conditions and the actual deflected shape which was then

compared with the theoretical deflected shape (sine wave).

I. Experiment

Each of the 3 struts were tested in a pinned-pinned, pinned-fixed and fixed-fixed positions.

The struts varied in length, but were of the same composition depth and width.

The length, modulus of elasticity (which was constant throughout), buckling load and the

deflected shape of the strut were recorded throughout, these results can be seen in below.

The exact method and equipment can be seen in the appendix, which comprises a section of

the laboratory instructional handout.

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3. Recorded Results

For all struts the Second Moment of Area (I) is 1.3 x 10-11 and the Modulus of Elasticity is 69

GPa (i.e. 69 kN/m²)

Figure. 1 Recorded Results

Pin-Pin

Pin-Fixed Fixed-Fixed

Strut length Buckling load Strut length Buckling load Strut length Buckling load

m N m N m N

0.520 -19 0.300 -130 0.480 -124

0.420 -49 0.400 -82 0.380 -209

0.320 -67 0.500 -60 0.280 -360

4. Calculations and Graphs

Assumptions in Theory

For the calculations the aluminium struts were (like most construction materials) assumed

to behave in a linear-elastic way, i.e. the struts deflected under the axial load, but recovered

their original shape after the load was removed, because the limit of proportionality was not

reached. (Hulse & Cain, 2000)

4.1. Theoretical Results

The dimensions, of the cross-area of the three struts are shown below.

0.02m

0.002m

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4.1.1. The second moment of area (I)

I = = = 1.3

4.1.2. Radius of Gyration

= = = 5.7

4.1.3. The effective lengths of the struts

Pin-pin = L

Pin-Fixed = 0.7L

Fixed-Fixed = 0.5L

4.1.4. Slenderness Ratio

The slenderness ratio is calculated by:

Sl =

For pin-pin test 1. Sl = = 91.22

4.1.5. Euler’s Buckling formulae

Euler’s Bucking formulae varies for each occasion due to the whether the ends were pinned

or fixed.

A pin-pin strut would use

A pin-fixed strut would use

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A fixed-fixed strut would use

4.1.6. Derivation

The original equation for = The theoretical values of effective length, taken from

(Hulse, R and Cain, J) are substituted into this equation. For example for the fixed-fixed end

strut = 0.5L.

This gives = =

4.1.7. Example

The pin-pin ended strut.

Young’s modulus of elasticity (E) for aluminium is 69Gpa. This equals 69 x N/

The second moment of area (I) is 1.3 .

Therefore using the appropriate equation and the values of the strut lengths a theoretical

value of buckling can be calculated.

The first strut used was 0.520 metres in length.

Using = = 33.59 (negative)

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Figure.2 Table of Calculated Results

Type of end Length of

Strut/m

Effective

length/m

Slenderness

ratio

Pin-Pin L / r

Where r =

0.00057

1 0.520 0.520 912.3 -33.59

2 0.420 0.420 736.8 -51.49

3 0.320 0.320 561.4 -88.70

Pin-Fixed 0.7L

1 0.500 0.35 614.0 -72.68

2 0.400 0.28 491.2 -113.56

3 0.300 0.21 368.4 -201.89

Fixed-Fixed 0.5L

1 0.480 0.24 421.1 -157.68

2 0.380 0.19 333.3 -251.59

3 0.280 0.14 245.6 -463.39

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Figure. 3 Errors Between Theoretical and Measured Results

(theoretical) (actual) Percentage error % Average % error

Pin-Pin

-32.7 -18

82

37.5 -50.2 -49

2.45

-86.4 -67

28

Pin-Fixed

-68.4 -59

15.93

19.6 -106.9 -82

30.37

-190 -169

12.43

Fixed-Fixed

-153.6 -125

22.88

18.9 -245.2 -209

8.85

-451.5 -361

25.07

Table 2, shows the percentage difference between the theoretical and actual buckling loads.

Graph 1.1 (figure. 4) shows the theoretical and the actual buckling loads against strut

lengths. This allows the errors between each point to be seen.

Graph 1.2 (figure. 5) shows the buckling loads for struts with pin-fixed ending.

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Graph 1.3 (figure. 6) shows the buckling loads for struts with fixed-fixed endings.

Graph 1.4 (figure. 7) Shows how the slenderness ration affects the maximum stress.

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5. Observations and Conclusions

5.1. The Material

Aluminium has a weight of about 2 770 kg/m³ in comparison to 7 848 kg/m³ for steel

(Baden-Powell, 2001) but the yield strength for aluminium lies between 3.5 x 104 and 4.5 x

104 psi (i.e. 241 x 106 to 310 x 106 N/m²) whereas that of stainless steel is 4.0 x 104 to 5.0 x

104 psi (i.e. 275 x 106 to 345 x 106 N/m²) (Engineers Edge, 2000-2010). Due to this high

strength-to-weight ratio (Blockley, 2005) aluminium is useful in practice and for this

experiment.

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5.2. Definition of Buckling Load

The buckling load of a structure is the load for which an ideal structure will be disturbed

from its ideal position of equilibrium and buckle.

5.3. Euler’s Theorem

Euler’s theorem is based upon Euler-Bernoulli beam theory, which ignores the effects of

transverse shear deformation. This effect is apparent in non-slender members, and means

Euler’s is only appropriate for slender members. The definition of when the slenderness

ratio is low enough that a beam is assumed to be non-slender is demonstrated by the

graphs figure 7 and figure 8. Due to this Euler’s theorem is only valid for elastic deformation

of beams i.e. after a load is applied and the material experienced bending, the object gets

back into its original shape unless it has reached the yield stress (Hulse & Cain, 2000).

5.4. Analysis of Graphs

From the figures 4-6 it is very obvious that as the length of the strut increases the buckling

load decreases, therefore they are inversely proportional. This can also be seen in equation

V.II.IV which shows that for struts of equal cross sectional area; as the length increases the

slenderness ratio increases, and from figure 7 as the slenderness ratio increases the buckling

load decreases. It is also easy to see by comparing these graphs that the type of constraint

has a major effect on the results, with fixed ends taking the largest buckling load and the

pinned ends taking the least buckling load. From this we can deduce that it is the effective

length of the strut rather than the actual length that determines the buckling load. Two

struts of equal length will have vastly different effective lengths determined by the

boundary conditions, this is demonstrated below.

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Also, plotted alongside our measured results in figures 4-7 are the theoretical values

calculated using Euler’s method. The theoretical values seem to vary inversely with the

length completely linearly, whereas our measured results show a significant divergence

from the calculated values.

5.5. Difference between Theoretical and Measured Results

Evidently there are pronounced errors present between our calculated buckling values and

the values we read from the experiment. In table (insert) in the appendix the percentage

errors of the results for each set of constraints are shown highlighting this point.

Euler’s theorem is an ideal solution that assumes a perfect sine wave deformation, a

constant temperature, perfectly elastic bending and no inherent flaws in the evaluated

section. Obviously this is completely unreasonable and hence a large percentage error

between the recorded and calculated values is to be expected.

Effective Lengths

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In terms of our experiment, the most likely causes of the difference between our measured

and calculated results is inelastic deformation of the strut due to the residual stresses and

any retained plastic deformation because of its repetitive prior use. Also, we had no control

in the experiment over whether the modulus of elasticity we used in the calculations

corresponds perfectly with the grade and quality of aluminium that we used. Consequently

our observed deflections showed far more of an exaggerated point of contraflexure at the

apex of the strut than could be expected of an exact half sine wave deformation.

Euler’s Method In Practice . The design values are used to increase the safety of the

structural member.

In practice Euler’s method is regarded as a fairly basic demonstration of buckling, and isn’t

used at all for any design purposes thus, the effective length constants are different in

practice and theory. However, the theory values are useful for structural analysis they are a

very good way of seeing the linear elastic buckling load of a strut and can be used to quickly

give a broadly accurate picture of how a strut will eventually fail.

Calvert (2007) states that Euler’s theory is a good method for columns with a length = 30 x

width. For short columns the failure (e.g. yielding of steel or crushing of concrete) can be

determined by the ultimate compressive stress, for intermediate columns in practice the

Tangent-Modulus Theory (which can be simplified to the reduced-modulus theory) is often

in use. (eFunda, 2009). The tangent theory replaces the Young’s modulus of elasticity with

the local tangent value Et to get the critical load and the critical stress

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5.6. Effect of I-value and Radius of Gyration on the Euler Load

As described earlier the Euler buckling load Pe [N] was calculated as follows:

For the 0.52m strut with pin-pin-ended boundary conditions thus

If the strut was turned about its central axis, the cross-sectional area would stay the same,

but the width (b) and the height (h) would be exchanged and the second moment of area (I)

would become 1.3 x 10-9 resulting in a buckling load of:

If instead of a rectangular cross-section 0.002m x 0.02m (area = 0.00004 m²) the tested strut

had a squared cross-sectional area with sides of 0.006325m length, the area would stay the

same, but the I-value would increase from 1.3 x 10-11 to 1.3 x 10-10 and the buckling load

would become:

In this case for the same cross-sectional areas the exchange of width and height had a larger

effect on the Euler load than the change of the geometry of the cross-section. Hulse & Cain

(2000) claim that both the geometry and the radius of gyration (which is affected by the

second moment of area I) affect the slenderness ratio, which is the key factor in the

behaviour of the structural member.

5.7. Practical examples of buckling

Practical examples of potential buckling can be seen everywhere, from steel framed

buildings to buckling under self-weight of lamp-posts.

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6. References

Baden-Powell, C. (2001). Architect's Pocket Book (2nd ed.). Oxford: Architectural Press.

Blockley, D. I. (2005). New Dictionary of Civil Engineering. London: Penguin Books Ltd.

Duncan, I., Liddell, W., & Williams, C. (1982). Current Trends in the Treatment of Safety. In R.

Narayanan (Ed.), Axially compressed Structures (pp. 41-50). Barking: Applied Science

Publishers Ltd.

Engineers Edge. (2000-2010). Yield Strength - Strength (Mechanics) of Materials. Retrieved

December 3, 2010, from Engineers Edge:

http://www.engineersedge.com/material_science/yield_strength.htm

Hulse, R., & Cain, J. (2000). Structural Mechanics (2nd ed.). New York: Palgrave Macmillan.

Tall, L. (1982). Centrally Compressed Members. In R. Narayanan (Ed.), Axially Compressed

Structures (pp. 1-40). Barking: Applied Science Publishers Ltd.

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Appendix A – Lab sheet