Experimental Analysis into the Buckling Failure Condition of a Range of Aluminium Test Samples PDF

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Summary

Aluminium test struts of different dimensions were tested until buckling, to outline the important quantities in Euler's Buckling Equation. Assumptions, test configuration, experimental data and analysis are all included in this report...

Description

Buckling Lab Report

Name: Steven Wright

Lab Date: 14th October 2014

Group 20

Dr. Rendall

1. Abstract The objective of this experiment is to compare the relationship between the buckling load of an aluminium alloy strut and the length and end-fixityconditions imposed upon the member. Using a column buckling machine the member was tested by measuring the compressive load required to deflect the material by 2mm intervals until the member displaced 16mm (the point at which further testing would have induced plastic deformation). The load was recorded for both forced and natural displacement. Results were compared and analysed against the theoretical buckling load given by Euler’s Critical Load equation with the anticipation that the two figures would align closely. The conclusions drawn from this experiment were that the critical load does adhere to the inverse relationship Euler defined in the theoretical buckling load equation, and that the member can withstand more load if one, or both, of the member ends are fixed.

2. Introduction Many modern engineering structures are a combination of trusses, beams and struts designed to withstand large forces acting through many different planes of motion. An individual and independent member within an engineering structure may experience either a tensile or compressive axial force, perpendicular to the normal plane of the strut, potentially in conjunction with shear force and resulting moments.

Subject to a high, compressive axial force, a member in some state of fixity will remain straight until the critical load of the material has been reached, at which point “the introduction of the slightest lateral force will cause the column to fail by buckling.”1 In 1757, Swiss Mathematician, Leonard Euler derived and equation demonstrating the critical load to be dependent “on the stiffness constant of the material, on the way the rod is supported at either end, and inversely proportional to the square of the length of the rod.”2 Once a member has undergone buckling it becomes redundant in carrying any further load.

𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙

𝑘𝜋 2 𝐸𝐼 = 𝐿2

Figure 1: Euler's Theoretical Buckling Load Equation

As stated above, it is important that the fixity conditions imposed throughout the experiment are accounted for, with respect to the value of k, (see figure 2). This effect is observed because when the strut is fixed, there are more reaction forces acting along the horizontal axis, counteracting the resulting forces from the applied load that displace the member in the lateral direction. The proportional relationship between fixity conditions and Euler’s equation are considered in the value of k.)

1 2

Wikipedia contributors, Buckling, Wikipedia: The Free Encyclopedia, October 2016 Gautschi, Walter: Leonhard Euler: His Life, The Man and His Works, 2008, SIAM Review

Pinned-Pinned Pinned-Fixed Fixed-Fixed

K=1 K=2 K=4

Figure 2: Table to Show How the Value of K Changes with Configuration

The aims of this experiment are to verify the authenticity of Euler’s equation for critical load through comparison of the theoretical and experimental buckling loads collated throughout and to investigate the effect of the support conditions and length of member on this load.

“Euler’s theory is recognised as providing a reliable upper limit to the buckling load for long struts,”3 however detailed mechanics requires in depth calculation that cannot be fully satisfied by an assumptive equation such as Euler’s. Throughout this experiment, it was assumed that the strut was of a constant cross-sectional area, load was applied directly through the centre of the strut and that the material was linearly elastic “perfectly straight, homogeneous, and free from initial stress.”4

3

Rees, David, The Mechanics of Engineering Structures , World Scientific Publishing Co Inc, Nov 2014 Ye Jianqiao, Structural and Stress Analysis: Theories, Tutorials and Examples, Second Edition ,CRC Press, 2 Dec 2015

4

3. Procedure 3.1 Equipment TecQuipment STR12 [Buckling Of Struts] Aluminium Alloy test struts, (20mm wide, 2mm thick, Range of Lengths; 320mm, 370mm, 420mm, 470mm, 480mm, 500mm, 520mm) Magnetic Ruler (mm precision) Scientific Calculator Loading Handwheel Height Adjustment Bar Pinned Configuration

Test Strut

Magnetic Ruler

Digital Force Display

Pinned Configuration

Figure 3: Buckling of Struts [STR12] Testing Apparatus

3.2 Experiment Pinned-Pinned The loading assembly was set to pinned-pinned configuration by removing the restraining clamps, revealing a v-shaped indent designed such as to hold the aluminium alloy strut in place, whilst giving it enough of a degree of horizontal freedom as to eliminate the reactions forces a fixed-fixed strut would exhibit. The loading handwheel was turned clockwise so it touched the tip of the strut, applying negligible force yet holding it in place. Next, the force meter was set to zero by lightly adjusting the dial on the front panel of the experiment. It was ensured that the position of the magnetic ruler was aligned on the face of the buckling apparatus such that the deflection of the strut was set to zero. Using a calculator, Euler’s theoretical buckling load equation was concluded for each strut’s length so that the expected buckling load was known; this was to ensure it wasn’t overloaded causing the member plastic deformation. Then a load was applied by turning the loading handwheel clockwise, until a 2mm deflection could be seen by reading carefully from the magnetic ruler. Here the reading was taken from the force meter and denoted as the natural buckling load. The strut was then pushed horizontally until it displaced in the opposite horizontal direction, giving the load required to make the strut buckle in the forced direction. A strut will buckle naturally in a given direction due to molecular inconsistencies and weaknesses within the material itself, therefore the relationship between natural and forced buckling loads should be that the forced buckling load is greater. Data collected regarding both ‘natural’ and ‘forced’ buckling loads were inputted into a table that compared these buckling forces to the member deflection. The experiment, collating readings for natural and forced loads, was repeated for deflections at 2mm intervals until the member had deflected 16mm. At this point the buckling load was only changing incrementally because it had exceeded the critical load, the value at which a member buckles and becomes redundant in taking any further force. Continued loading would only have

served to plastically deform the material. The experiment was repeated for struts of lengths, 320mm, 370mm, 420mm, 470mm and 520mm. The assembly had to be adjusted using the height adjustment bar, (labelled in figure 3) to test the members of different lengths.

Pinned-fixed For the pinned-fixed experiment the strut was secured at the bottom of the apparatus, (as illustrated in figure 4) and a strut of length 500mm, of the same material, was tested. Once the height of the assembly was adjusted to the length of the strut the same procedure as described above was repeated and the results recorded in a table mapping the ‘natural’ and ‘forced’ buckling loads against the deflection of the member, as so for the pinned-pinned configuration. Figure 4: The Strut Was Secured at The Bottom of The Apparatus

Fixed-Fixed For the fixed-fixed experiment a member of length 480mm was tested, so the height adjustment bar had to be adjusted accordingly. For this instance, both restraining clamps were applied to the top and bottom of the member, completely securing it and ensuring it had no degree of lateral freedom where the clamps were acting. Figure 5 illustrates how the strut was secured at the top of the apparatus. The same Figure 5: The Strut Was Secured at The Top of The Apparatus

procedure was repeated as in both above experiments and the results recorded in the table.

3.3 Data Processing Once all the data had been collated into the table it was processed into different graphs to give a visual representation of the experiment. Two of the graphs mapped the relationship for the pinned-pinned configuration, one mapping ‘Load’ against ‘Deflection’ and the other representing ‘Buckling Load’ against

1 𝐿𝑒𝑛𝑔𝑡ℎ 2

. Two more graphs presented the relationship between ‘Load’

and ‘Deflection’ for the pinned-fixed and fixed-fixed experiments.

3.4 Hazards and Safety It was ensured that for the pinned configurations the strut was secured in the V-shaped indent and for the fixed instances the clamping screws were tightened securely to prevent the strut from flying out of the apparatus when under compression. The Height Adjusting Bar had to be held when loosening the screws to stop it from dropping, potentially damaging an individual or the equipment. Goggles were worn to protect against the possibility of the strut failing and snapping. Care was also taken to ensure the buckling load was not exceeded.

4. Results Experimental Configuration

Length L (mm)

PinnedPinned

320

PinnedPinned

370

PinnedPinned

420

PinnedPinned

470

PinnedPinned

0.00 2.00

Theoretical

Mid Strut Deflection (mm) 4.00 6.00 8.00 10.00 12.00 14.00 16.00 Resulting Strut Load (N) 81 86 84 95 86

Natural

0

57

Forced

0

76

84

106

104

109

Natural Forced Natural

0 0 0

41 18 30

51 90 39

55 92 43

58 91 47

61 80 49

62 77 51

62 76 51

64 75 54

Forced

0

59

78

80

73

73

67

66

64

Natural

0

23

30

33

36

37

37

38

38

Forced

0

3

33

54

54

52

50

50

48

520

Natural

0

18

25

27

30

31

32

32

PinnedFixed

500

Forced Natural

0 0

47 38

50 52

49 51

46 62

44 64

41 66

Forced

0

97

93

108

98

93

Fixed-Fixed

480

Natural

0

104

122

128

133

137

Forced

0 167 169 163 159 157 154 Figure 6: Table of Results for All Experiments

Buckling Theoretical Load Buckling (N) Load (N) 86 88 69.5

66

59

51

44

41

33

37

34

40 68

42 69

77

72

88

86

84

140

143

144

148

157

154

154

A Graphical Illustration of the Relationship Between the Buckling Load and Deflection of Pinned-Pinned Members of Different Lengths 100 90 80

Load (N)

70 60 50 40 30 20 10 0 0

2

4

6

8

10

12

14

16

Deflection (mm) Length 320mm

Length 370mm

Length 420mm

Length 470mm

Length 520mm

Figure 7: The Trendline Gives an Insight into the Buckling for non-recorded deflections

18

A Graphical Representation of the Relationship Between the Buckling Loads and Deflection of a Pinned-Fixed Member of Length 500mm 80 70 60

Load (N)

50 40 30 20 10 0 -10

0

2

4

6

8

10

12

14

16

18

Deflection (mm)

Figure 8: The Trendline Gives an Insight into the Relationship Between the Buckling Load and Deflection for Non-Recorded Members Within the Pinned-Fixed Experiment

A Graphical Representation of the Relationship Between the Buckling Load and Deflection of a Fixed-Fixed Member of Length 480mm 160 140

Load (N)

120 100 80 60 40 20 0 0

2

4

6

8

10

12

14

16

18

Deflection (mm)

Figure 9: The Trendline Gives an Insight into the Relationship Between the Buckling Load and Deflection for Non-Recorded Members Within the Pinned-Fixed

A Bar Chart to Illustrate the Difference Between Theoretical and Experimental Buckling Loads 180 160 140 120 100 80 60 40 20 0

Buckling Load (N)

Theoretical Buckling Load (N)

Figure 10: A Graphical Representation of the Deviation Between Theoretical and Bucking Loads.

5. Discussion The results of this experiment do provide supportive evidence for Euler’s theoretical buckling load equation as the most inaccurate result was within 8.3% of the figure predicted by Euler’s equation. (See figure 6: Euler’s theoretical buckling load equation predicts a load of 157n to cause the member to buckle, however, the experimental results show the buckling load was 144N. The difference between the experimental load and the theoretical load is 8.28%). This inaccuracy, however, is possibly due to systematic error, (refer to ‘5.2 Errors’ for detail). If this value is dismissed due to the presence of systematic error, then it is found that the highest level of inaccuracy is in fact 7.32%, (found from the difference in theoretical and experimental load when examining the pinned-pinned member of length, 470mm, see figure 6). From figure 7 it can be concluded that for pinned-pinned members, there is an inverse relationship between the amount of load a material can withstand before buckling, and its’ length. Euler’s equation also confirms this, going on to state that the relationship is in fact square inverse, (refer to figure 1, Euler's Theoretical Buckling Load Equation). As predicted earlier in the report and in Euler’s equation, there is also a strong relationship between the fixity conditions of the member and the critical load. It is easiest to spot this relationship by examining figure 6. The pinned-pinned strut of length 320mm is the only non-fixed member with a higher buckling load than the pinned-fixed strut and this is simply because the pinned-fixed member has a length of 500mm, which in accordance with Euler’s inverse square law, greatly reduces its critical load. The relationship is further reinforced by figure 6, as the fixed-fixed member has twice the ‘natural’ or ‘forced’ buckling load of any length of the pinned-pinned members. Newton provides an explanation for this phenomenon stating that a fixed member will produce a reaction force to any horizontal force applied. This cancels out more of the applied force, and so the buckling load of a fixed-fixed member will be larger than that of a pinned-pinned. Another thing to note is the fact that the forced buckling load is always greater than the natural buckling load for any length of member, or fixity conditions imposed upon it. From analysing figure 6 we can see that the forced load is between 9.3% and 22% greater than the natural buckling load. (Data calculated by taking the smallest difference between ‘forced’ and ‘natural’ critical loads

exhibited by the fixed-fixed members, and the largest difference between ‘forced’ and ‘natural’ loads exhibited by the pinned-pinned strut of length 520mm.) This is due to molecular weaknesses and inconsistencies in the member at atomic level, causing it to deflect to one side with more ease than the other, meaning that the force required to deflect it in the opposite direction is greater. This observation highlights one of the limitations in using Euler’s equation, as it assumes that the member is perfectly engineered, (consistent and perfectly aligned at molecular level) whereas, in reality, this is not the case. Other fixity conditions such as fixed-free or fixed-fixed-fixed were not considered during this experiment, however, the theory and data collected could be applied to predicting how a member would react under these conditions. If a member was imposed with fixed-free fixity it could be predicted from the results of this buckling experiment that it would have a smaller buckling load than the pinned-pinned members. This is because there is less restraint, resulting in less reaction force to counter the horizontal forces the load causes. Furthermore, the force would concentrate directly in the centre of the strut, as it would deflect in a parabolic manner, (see figure 10) causing increased stress and eventually fracture in the centre.

Figure 11: The Force Would Concentrate Directly in the Centre of the Strut as it Would Deflect in a Parabolic Manner

For a fixed-fixed-fixed strut the data in this experiment would predict that the critical load should be much larger as k, (refer to figure 1) increases as the fixity increases. The buckling load is proportional to k, as seen in Euler’s equation.

Each member was only tested up to 16mm deflection, (apart from the pinnedpinned member of length 320mm, which was tested up to 12mm deflection, as it was extremely close to plastic deformation) because the experimental buckling load did not vary much between 12-16mm deflection, insinuating that the buckling load had already been reached, (see figure 6)

5.1 Errors As mentioned before, there was systematic error in the experiment for the fixed-fixed pin. The clamping restraint did not tighten completely as some of thread had been stripped due to wear. (refer to figure 11) This prevented the strut from being held securely and may have given it some lateral freedom, possibly accounting for the experimental buckling load being so far below from the theoretical. Other systematic error included the precision of the magnetic ruler, which measured to the nearest millimetre. This gives an uncertainty of ±0.5mm in the readings. The measurements for load were also taken to the nearest Newton, giving an uncertainty of ±0.5N. The measurements from the magnetic ruler also had to be read by eye and, of course, appear different depending on the position of the reader relative to the ruler. Therefore, it was vital to stand exactly face on to the strut and the ruler, however, there may have been some error in this reading. Furthermore, it is unlikely each strut was aligned exactly as the one before it, either in the V-shaped indent or the clasp. This leads to a small, yet notable, amount of random, human error in the experiment.

6. Conclusion The experiment was successful in providing evidence for the relationships addressed in Euler’s equation, and the results had a relatively small percentage of inaccuracy when compared with the theoretical buckling load. Random and systematic error within the experiment and limitations/assumptions, (as explained in 6.2) made by the equation account for the difference between these loads. Completion of the experiment has led to a greater understanding of the horizontal and vertical components of force, and how they act independently. It has also...