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UEME1263 SOLID MECHANICS 1

Title: Buckling test Date of experiment: 24 January 2017 Group members: Name

Student ID

Course

Trimester

Chee Kong Meng

1504365

ME

Y1S3

Ong Juen Hau

1504845

ME

Y1S3

Tan Sheng Yee

1504425

ME

Y1S3

Chee Kit Mun

1601499

ME

Y1S3

Lim Wan Lynn

1604374

ME

Y1S3

TITLE Buckling test OBJECTIVES 1. To observe deflection shape of columns of three different configurations. 2. To prove Euler’s theory by comparing experimental values and theoretical values. 3. To determine maximum buckling force for different materials. INTRODUCTION Buckling is a mathematical instability which caused structural failure. It happens when both axial and bending forces caused great stresses in the column. The behaviour of the columns was investigated by mathematician Leonhard Euler, and he derived the Euler’s equation to predict the buckling action. The Euler’s equation is given as below.

𝜋2 𝐸𝐼 F=

𝐾𝐿2

Where F= maximum force (vertical load on column) 𝐸 = modulus elasticity of material 𝐼= area of moment of inertia of the cross section of tested specimen 𝐿= original length of column 𝐾 = column effective length factor (depends on the support condition of the column)

According to the Euler’s equation, the factors affecting the buckling behaviour are modulus elasticity of material, area of moment of inertia of the cross section of tested specimen, buckling length of column and the conditions of end of support. In this buckling test, the maximum force of a column can support before it collapse is determined and recorded. The experiment was repeated using several type of supports and different material. The experimental value is then compared to the calculated value to prove the Euler’s equation.

SETUP OF APPARATUS

A= loading mechanism B= column C= force sensor D= force meter E= Dial gauge with holder F= Test specimen block screw G=Test specimen holder plate and screw

PROCEDURE 1. The force sensor (C) cable was connected to the force meter (D). 2. The force meter (D) was connected to a 240 V AC power supply and switched on. 3. The width and length of column was measured using meter rule, the thickness of column was measured using Vernier callipers. The measurement was recorded. 4. The lower and upper bar test specimen screw were loosen. The column was placed to the lower and upper bar. (Ensure the column touches the upper bar). The screw was tighten with the Allen key provided. 5. The configuration of fixed-fixed end was set. (Ensured the screws F of upper and lower bar were tightened) 6. The max/min soft button was tared zero for three second. 7.

Load was applied to the column by rotating the loading mechanism.

8. Stop rotating once the force reached maximum point and start decreasing. 9. Hold the loading mechanism and the min/max button of the meter was pressed. The maximum force was recorded. 10. The experiment was repeated with different configuration and materials. For free end, ensured screws F was loosen.

MATERIALS Steel and Aluminium specimen APPARATUS Meter ruler, Allen key, Vernier callipers, and WP120 test stand CALCULATIONS The followings are the formulas related to the calculations in results section for this experiment: Moment of inertia,

(where b = Width of specimen and d = Depth of specimen) 𝐼=

𝑏𝑑 3 12

Euler critical load for free-free column, 𝐹𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =

𝜋 2 𝐸𝐼 𝐿2

Euler critical load for fixed-free column, 𝐹𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =

𝜋 2 𝐸𝐼 0.49𝐿2

Euler critical load for fixed-fixed column, 𝐹𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =

4𝜋2 𝐸𝐼 𝐿2

(where E = Young modulus of specimen, I = Moment of inertia and L = Length of specimen) Percentage difference =

|[𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑣𝑎𝑙𝑢𝑒]−[𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒]|

Average measurement =

[𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒]

𝑥 100%

[𝑅𝑒𝑎𝑑𝑖𝑛𝑔 1]+[𝑅𝑒𝑎𝑑𝑖𝑛𝑔 2] 2

Young Modulus used for: Steel = 200 GPa Aluminium = 69 GPa (Reference: http://www.engineeringtoolbox.com/young-modulus-d_417.html)

RESULTS Table 1: Average dimension of steel column Dimension

Reading#1

Reading 1

Reading#2

(mm)

Reading 2

Average

(mm)

(mm)

Width, b

17.80 mm

17.80

17.60 mm

17.60

17.700

Depth,d

2.51 mm

2.51

2.52 mm

2.52

2.515

Length,L

29.9 cm

299.00

30.0 cm

300.00

299.500

Test Data: Table 2: Critical load and percentage difference of each test configuration of steel column. Critical load (N) Test Configuration

Percentage

Measured

Calculated

difference (%)

Free-free column

338

516.25

34.5

Fixed-free column

443

1053.58

58.0

Fixed-fixed

564

2065.01

72.7

Moment of inertia, 𝐼 =

0.0177(0.0025153 )

Calculated load (free-free),

12

𝐹=

Calculated load (fixed-free), 𝐹 = Calculated load (fixed-fixed), 𝐹 = Percentage error (free-free)

=

Percentage error (fixed-free) = Percentage error (fixed-fixed) =

= 2.346 𝑥 10−11 𝑚 4

𝜋2 (200 𝑥109 )(2.346 𝑥 10−11 ) 0.29952 𝜋2 (200 𝑥109 )(2.346 𝑥 10−11 ) 0.49(0.29952 )

= 516.25 𝑁 = 1053.58 𝑁

4𝜋2 (200 𝑥109 )(2.346 𝑥10−11 ) 0.29952

516.25 −338 516.25 1053.58 −443 1053.58 2065.01 −564 2065.01

= 2065.01 𝑁

𝑥 100% = 34.5 𝑥 100% = 58.0 𝑥 100% = 72.7

Table 3: Average dimension of aluminium column Dimension

Reading#1

Reading 1

Reading#2

(mm)

Reading 2

Average

(mm)

(mm)

Width, b

19.20 mm

19.20

19.18 mm

19.18

19.190

Depth,d

3.20 mm

3.20

3.19 mm

3.19

3.193

Length,L

30 cm

300.00

30 cm

300.00

300.000

Test Data: Table 4: Critical load and percentage difference of each test configuration of aluminium column. Critical load (N) Test Configuration

Percentage

Measured

Calculated

difference (%)

Free-free column

314

393.91

20.3

Fixed-free column

365

803.89

54.6

Fixed-fixed

486

1575.63

69.2

Moment of inertia, 𝐼 =

0.01919(0.0031933 )

Calculated load (free-free),

12

𝐹=

Calculated load (fixed-free), 𝐹 = Calculated load (fixed-fixed), 𝐹 = Percentage error (free-free)

=

Percentage error (fixed-free) = Percentage error (fixed-fixed) =

= 5.206 𝑥 10−11 𝑚 4

𝜋2 (69 𝑥109 )(5.206 𝑥 10−11 ) 0.32 𝜋2 (69𝑥109 )(5.206 𝑥 10−11 ) 0.49(0.32 )

= 393.91 𝑁 = 803.89 𝑁

4𝜋2 (69𝑥109 )(5.206 𝑥 10−11 ) 0.32

393.91−314 393.91 803.89−365 803.89 1575.63−486 1575.63

= 1575.63 𝑁

𝑥 100% = 20.3 𝑥 100% = 54.6 𝑥 100% = 69.2

DISCUSSION Buckling is a circumstance when a column experiences at load force along its axis and bends caused by the loading force. Buckling will cause the column to bend, lose its stability and sudden failure will occur. In this experiment, two different material, steel and aluminium were analysed to enquire the maximum load on specimen before it fails in three different cases, which is free-free column, fixed-free column and fixed-fixed column. The theoretical value calculated for critical load of steel and aluminium column for the 3 different cases are 526.25 N, 1023.58 N and 2065.01 N respectively. Meanwhile, the theoretical value calculated for critical load of the aluminium column for the 3 different cases are 393.31 N, 803.89 N and 1575.63 N respectively. The theoretical values are compared with experimental values and the results are tabulated in Table 2 and Table 4. Percentage errors ranging from 20% to 73% are obtained. There are several factors that contributed to the differences in theoretical and experimental result. The percentage error calculated ranges between 20-73% which is quite large as the experiment was carried out using mechanical equipment. The results for both specimens show smaller percentage difference for free-free column configuration, the first test done, and increase in order to when the last test is done. This is because the experiment was repeated with the same specimen for each configuration and the specimen suffered fatigue as it continuously gets pressured to critical load. A new specimen of a similar dimension must be used for each different configuration to get more accurate experimental value. Another reason of deviation from experimental value is due the error which happens when the specimen is not fasten to the specimen holder properly. This will be the source of sideways force which is not directly through the axial load and will cause variance from actual critical load. Another error is hysteresis error which is visible when force meter reacts slow for the initial input of force. These two errors can be reduced by repeating the experiment several times and get an average. However, when we compare the theoretical and experimental value for this buckling analysis there will definitely be a certain amount of variance as the formulas used to calculate the theoretical value were derived based on an ideal column. This means it has been assumed that the column is perfectly uniform and loaded consistent throughout the arrangement of material. In the actual experiment, the specimens in most cases are not ideal and have uneven distribution of forces around the column.

Table 5: Experimental shape of deflection for all four configurations. Free-free configuration

Fixed-free

Fixed-fixed

configuration

configuration

Steel

Aluminium

Euler buckling cases is the possible forms of the column which depends on how the column is attached.

Free-free column

Fixed-free column

Fixed-fixed column

The critical load formula was derived from the Euler cases and the base formula, 𝐹𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =

𝜋2 𝐸𝐼 𝑠2

, (where E = Young modulus of specimen, I = Moment of inertia and s =

Buckling length.) From Table 5, the fixed-fixed configuration for both specimen shows the most significant bend caused by buckling followed by fixed-free and then free-free configuration. The specimen held fixed on both ends experiences the largest buckling effect before it loses its stability and fail. When the specimen is fixed on either end, it will be held in place and allow it to buckle more before reaching its critical load. Compared to the theoretical shape of deflection, the experimental shape is not as significant as the theoretical shape of deflection but it exhibits the correct trend of deflection. The free-free configuration for each specimen bends the least to reach its maximum axial load and the fixed-fixed configuration buckles the most among the three. The results for both specimens in all three cases prove that steel column can withstand a larger critical load compared to the aluminium column. A material with a higher Young Modulus value has a stronger resistance to buckling because the specimen is more stiff and rigid. The steel specimen with Young Modulus 200 GPa will experience maximum buckling effect only when a larger load force is applied compared to the aluminium specimen with Young Modulus 69 GPa. Buckling is not influence by the tensile strength of the material and this can be seen as aluminium has higher tensile strength compared to steel but the maximum buckling force occurs at a lower value for the aluminium specimen. Hence, the steel material with a higher Young Modulus but lower tensile strength will be preferred to prevent buckling failure. The most recommended column design among the three configurations would be the fixed-fixed column. It is the best column design as compared to the other two configurations because it can withstand the greatest load regardless which material is used for the buckling test. Referring to the results, we have proven that the critical loads for the fixed-fixed column is the highest for both steel and aluminium specimen. The value of critical load for fixed-fixed column has a large difference compared to the other two configurations. Therefore, we can say that the fixed-fixed configuration is the best column design to avoid buckling failure, in which it can resist the highest critical load before the column collapses.

CONCLUSION As conclusion, we can conclude that out of the three configurations which are the freefree column, fixed-free column, and fixed-fixed column, fixed-fixed column can withstand the largest critical load. By comparing the deflection shape of steel specimen and aluminium specimen, steel has the lesser deflection than aluminium when maximum load is applied. Hence, this indicates that steel can sustain a heavier load and it is stiffer compared to aluminium. In short, steel column with fixed-fixed configuration can say to be the best material and the best column design to support a greater load while not easily experience buckling failure. In addition, the percentage errors calculated by using the experimental values and theoretical values are ranging from 20% to 73%, indicating the data obtained is less reliable and improvements are needed for this experiment.

REFERENCES ITEM INDUSTRIECHNIK. (2015) Euler Buckling cases. [Online] 2015. Available from: http://glossar.item24.com/en/home/view/glossary/ll/en%7Cde/item/euler-buckling-cases/ [Accessed: 5th February 2017].

KASTEN, M. (2016) Strength of Aluminium vs Strength of Steel. [Online] 2016. Available from: http://www.kastenmarine.com/alumVSsteel.htm [Accessed: 5th February 2017].

MCGINTY,

B.

(2012)

Column

Buckling.

[Online]

2012.

Available

from:

http://www.continuummechanics.org/columnbuckling.html [Accessed: 5th February 2017]....