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Laboratory Report

Strut Buckling

Introduction

Buckling is a mathematical instability, leading to a failure mode. It is caused by a bifurcation in the solution to the equations of static equilibrium. At a certainstage under an increasing load, further load is able to be sustained in one of two statesof equilibrium: an undeformed state, or a laterally-deformed state. In practice,

 buckling is characterized by a sudden failure of a structural member subjected to highcompressive stress, where the actual compressive stress at the point of failure is lessthan the ultimate compressive stresses that the material is capable of withstanding.

For example, during earthquakes, reinforced concrete members mayexperience 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, whichdoes not form part of the primary forces to which the member is subjected. When loadis constantly being applied on a member, such as column, it will ultimately becomelarge enough to cause the member to become unstable. Further load will causesignificant and somewhat unpredictable deformations, possibly leading to completeloss of load-carrying capacity. The member is said to have buckled.

Objective

To determine the buckling load for a:

1. Pinned and Strut2. Fixed end Strut

Required Equipment and Components

Name Quantity Remarks

Strut Buckling 1 Apply Compress Load to Specimen

Micro Meter 1Record The Gauge of the Length

which been Bended.

Digital Indicator 1 Record Apply LoadPinned-end Specimen 1

Fixed-end Specimen 1

Strut Buckling

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Micro Meter 

Digital Indicator 

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Pinned-end Specimen

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Fixed-end Specimen

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Procedures

1. We fix the pinned ended specimen to the strut apparatus.

2. Note the reading on the digital indicator.

3. We required checking the position of the dial gauge to ensure that is at mid-

span. Beside, set the dial gauge reading to zero.4. The specimen should be load at suitable increments by moving the screw jack 

slowly in the up wards direction.

5. For each load increment, we required to record the load and the corresponding

mid-span deflection.

6. Well all were done, unload the specimen.

7. Fix the fixed end specimen to the strut apparatus and repeat step 2 to 6.

The following is the step to be taken toward performing the experiment:1) To increase the clear height between the platen:

a) Unscrew the top nuts upwards to the desired position, approximately theheight of the specimen. b) Turned the bottom nut so that it moves upwards with the platen.c) When the top surface of the platen touches the top nuts tighten all the nuts and

insert the specimen.2) To decrease the clear height between the platen

a) Unscrew the bottom nuts downwards to the desired position, approximatelythe height of the specimen.

 b) Turned the top nut so that it moves downwards to the top surface of the platen.c) When it touches the platen tightened all the nuts and inserts the specimen.

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3%1)To mount a fixed end specimen.a) Lower the screw jack shaft.

 b) Position the top platen such that the clear distance between the platen isslightly longer than the length of the specimen.

c) Slide the specimen into the top and bottom groove at the fixture for the end

condition.d) At the top end, push the specimen so that its end touches the base of the

groove.i) Tighten the screw at the side of the fixture.ii) Base of the grooveiii) End of specimeniv) Correct position of knife-edge placedv) Knife edge being wrongly

e) At the bottom end, raise the shaft of the screw jack so that the base of the

specimen just touches the base of the groove. Check that the specimen is

straight. If the specimen is slightly bent, lower the screw jack shaft.

f) Tightened the screw at the side of the fixture. The specimen is now ready for testing.4) To mount a pinned end specimen.

a) Lower the screw jack shaft. b) Position the top platen such that the clear distance between the platen is

slightly longer than the length of the specimen.c) Slide the knife edge attachment into the top and bottom groove at the fixture

for the end condition.d) Push the knife edge attachment so that its end touches the base of the groove.

Tighten the screw at the side of the fixture.Fixed the V groove on each end of the specimen.

f. Place one end of the specimen on the lower knife edge.e) Raise the screw jack shaft so that the top end of the specimen just touches the

knife edge. Check that the specimen is straight. If the specimen is slightly

 bent, lower the screw jack shaft slightly.f) The specimen is now ready for testing.

Result

Buckling of a Pinned-Ended Strut

Length = 600 mm Width = 25 mm Thickness = 3 mm

Elastic Modulus = 200000 N/mm2 Moment of Inertia = 56.25 mm4

K=1.0

Pinned-End Condition

Load, P Mid-Span Deflection, d d/P

 N div mm mm/N

30 24 0.24 0.008

60 58.5 0.585 0.00975

90 107 1.07 0.011889120 171.2 1.712 0.014267

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150 269.5 2.695 0.017967

180 448 4.48 0.024889

210 915 9.15 0.043571

223 1520 15.2 0.068161

From the experimental plot the Euler load = slope of the graph

Therefore Pcr (experimental) = 250.43N

Theoretical Euler Load = π2EI(KL)2 

= π2 ×200000Nmm2 ×56.25 mm4 (1 ×600mm)2

= 308.43N

Percentage error = 308.43-250.43N308.43N ×100%

= 18.80%

Comment: There was a high percentage error in this pinned-ended structure

experiment. Because the specimen had been test a lot of times, the internal bond of 

the specimen would be weak.

Buckling of a Fixed-Ended Strut

Length = 600 mm Width = 25 mm Thickness = 3 mm

Elastic Modulus = 200000 N/mm2 Moment of Inertia = 56.25 mm4

K=0.5

Fixed-End Condition

Load, P Mid-Span Deflection, d d/P

 N div mm mm/N

100 15 0.15 0.0015

200 30.5 0.305 0.001525

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300 53.5 0.535 0.001783

400 82 0.82 0.00205

500 124 1.24 0.00248

600 182 1.82 0.003033

700 279 2.79 0.003986

800 428.5 4.285 0.005356

900 725.5 7.255 0.008061

970 1458.2 14.582 0.015033

From the experimental plot the Euler load = slope of the graph

Therefore Pcr (experimental) = 1063.1N

Theoretical Euler Load = π2EI(KL)2 

= π2 ×200000Nmm2 ×56.25 mm4 (0.5 ×600mm)2

= 1233.70N

Percentage error = 1233.70-1063.1N1233.70N ×100%

= 13.83%

Comment: There was a high percentage error in this fixed-ended structure experiment

 but less than the pinned-ended structure experiment. Except the reason from pinned-

ended structure experiment, it also because the fixed-ended structure can lead the

specimen more strength to resist the deform.

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From above diagram we observed that the slope of fixed-end is more than the slope of 

 pinned-end which mean that fixed-end can apply more load than the pinned-end to

deform.

Conclusion

The lab has given the main understanding of measuring the deform of pinned-ended

strut and fixed-ended strut. But the experiment has not indeed been successful due to

the specimens had been test a lot of times, the Elastic Modulus of the specimens

maybe not as high as the new specimen. If we try to find out the actual elastic

modulus of the specimen which been test by tensile test, the theoretical Euler load

would be calculated accuracy, then the percentage would decrease.

References

Analysis of Structures book written by V.N Vazirani (M.Sc.Structures) andM.M.Ratwani (Ph.D.Structures) from page 217 – 272 and page 368 – 399.

http://www.wikipedia.com

Theory

In 1757, mathematician Leonhard Euler derived a formula that gives themaximum axial load that a long, slender, ideal column can carry without buckling. Anideal column is one that is perfectly straight, homogeneous, and free from initialstress. The maximum load, sometimes called the critical load, causes the column to bein a state of unstable equilibrium. Any increase in the load, or the introduction of theslightest 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.

However, if lateral forces are taken into consideration the value of critical loadremains approximately the same.

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Fixed -

Pinned -

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where

 F = maximum or critical force (vertical load on column), E = modulus of elasticity, I = area 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.For both ends pinned (hinged, free to rotate),  K = 1.0.For both ends fixed,  K = 0.50.For one end fixed and the other end pinned,  K = 0.699....For one end fixed and the other end free to move laterally,  K = 2.0. KL is the effective length of the column.

Examination of this formula reveals the following interesting facts with regard to theload-bearing ability of slender columns.

1. Elasticity and not compressive strength of the materials of the columndetermines the critical load.

2. The critical load is directly proportional to the second moment of area of thecross section.

3. The boundary conditions have a considerable effect on the critical load of slender columns. The boundary conditions determine the mode of bending andthe distance between inflection points on the deflected column. The closer together the inflection points are, the higher the resulting capacity of the

column.

A demonstration model illustrating the different "Euler" buckling modes. The modelshows how the boundary conditions affect the critical load of a slender column.

 Notice that each of the columns is identical, apart from the boundary conditions.

The strength of a column may therefore be increased by distributing the material so asto increase the moment of inertia. This can be done without increasing the weight of the column by distributing the material as far from the principal axes of the cross

section as possible, while keeping the material thick enough to prevent local buckling.

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This bears out the well-known fact that a tubular section is much more efficient than asolid section for column service.

Another bit of information that may be gleaned from this equation is the effect of length on critical load. For a given size column, doubling the unsupported lengthquarters the allowable load. The restraint offered by the end connections of a column

also affects the critical load. If the connections are perfectly rigid, the critical load will be four times that for a similar column where there is no resistance to rotation (hingedat the ends).

Since the moment of inertia of a surface is its area multiplied by the square of a lengthcalled the radius of gyration, the above formula may be rearranged as follows. Usingthe Euler formula for hinged ends, and substituting A·r 2 for I, the following formularesults.

where  F  /  A is the allowable stress of the column, and l  / r  is the slenderness ratio.

Since structural columns are commonly of intermediate length, and it is impossible toobtain an ideal column, the Euler formula on its own has little practical application for ordinary design. Issues that cause deviation from the pure Euler strut behaviour include imperfections in geometry in combination with plasticity/non-linear stressstrain behaviour of the column's material. Consequently, a number of empiricalcolumn formulae have been developed to agree with test data, all of which embodythe slenderness ratio. For design, appropriate safety factors are introduced into theseformulae.

Limit point vs bifurcation buckling

Bifurcation buckling is sometimes called Euler buckling even when applied tostructures other than Euler columns. As the applied load is increased by a smallamount beyond the critical load, the structure deforms into a buckled configurationwhich is adjacent to the original configuration. For example, the Euler column

 pictured will start to bow when loaded slightly above its critical load, but will notsuddenly collapse.

In structures experiencing limit point instability, if the load is increasedinfinitesimally beyond the critical load, the structure undergoes a large deformation

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into a different stable configuration which is not adjacent to the originalconfiguration. An example of this type of buckling is a toggle frame (pictured) which'snaps' into its buckled configuration.

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