Lab Report: To Study the Buckling of Struts

 Abstract

Buckling of struts is an important phenomenon in engineering. In this experiment buckling of struts for different materials has been studied. Materials used in this study were aluminum, steel and brass. Ten samples of circular struts of similar diameter but of different length were taken for each material type. Tensile testing machine was used to perform the experiment. Two different methods were used to find the buckling load and stress theoretically. Euler formula which is usually used for long struts was used to calculate the amount of buckling load and stress in the samples. Rankine’s formula was also used to calculate the load and stress in all the samples of different materials. The data for experimental, Euler’s formula and Rankine’s formula was then gathered, manipulated and put in tabular form to make it eye-catching and easily understandable. Graph for all materials show that value of stress is very close to experimental values at high slenderness ratio while the difference in the value of stress is quite large at low slenderness values.

Aim

“To Study the Buckling of Struts”

The aim of this experiment is to study different techniques to measure buckling of struts made up of different engineering materials. Two methods from theory are also used to calculate the amount of buckling load and stress. The materials included in this study are brass, steel and aluminum. 

Objectives

In order to complete the experiment of buckling of struts, the following pieces of work will be completed in the given order.

Gathering data from the apparatus to complete the test log sheet

Calculation of buckling load using Euler’s theory

Calculation of buckling load using Rankine’s theory

Comparison of experimental and theoretical value for each material

Discussion on the results 

Theory

Basics of buckling and struts

Struts are an important engineering component found in different engineering applications usually installed to bear compressive loads. They can also be used to take tension loads. In theory, there are two theories for studying buckling in the struts. One of which is Euler’’s theory and the second is Rankine’ formula. Formulas and important parameters used in both theories are discussed one by one.

Euler Formula for Buckling

Euler Buckling Theory is a classical theory to find the critical buckling load for the column of any cross-section. It is usually adopted to calculate the buckling load in long columns. The derivation of Euler’s formula for buckling starts from noting that bending moment in a loaded column and buckled column is ‘Py’. Where ‘P’ is the load applied and ‘y’ is the deflection of strut or column. This expression is then put into beam deflection equation and appropriate boundary conditions are then applied which leads to the final formula of Euler’s theory of buckling.

Euler’s formula for calculating the buckling load is given as

P_E=(π^2 EI_min)/(L_e^2 )

Where 

‘P_E’ is Euler’s load (Buckling or Crippling) or sometimes known as critical load. It is called critical because increasing load beyond this value will cause the strut or column to buckle. ‘E’ is the Young’s Modulus of column’s material. This value will change for all materials. ‘I_min’ is the minimum moment of inertia for a column’s cross section. Suppose if the column is rectangular, the moment of inertia for the smaller side will be put into the formula. Because column bends in the direction of smaller dimension. But in our study, we are using a circular column, so there will be only one moment of inertia. L_e is the effective length of the column that actually takes part in bending. Sometimes, it may happen that complete length of the column does cause bending. Euler’s stress can then be calculated by simply dividing P_E by the cross-sectional area of the strut or column.

Rankine’s Formula for Buckling

Rankine’s theory for finding buckling load is used to calculate buckling load in the columns of short, average and large length. The column is decided to be short, medium or long based on the l/k ratio. The derivation of Rankine’s formula starts from the statement 

1/P_R =1/P_E +1/P_c 

Where P_R is Rankine’s crippling or buckling load, P_E is Euler Buckling load and P_c is crushing or compressive load. For the small sized struts or columns, 1/P_E  becomes almost equal to zero so the Rankine’s buckling load will be equal to P_c. Similarly, for the long struts or columns 1/P_c  becomes almost equal to zero and the Rankine’s buckling load will be equal to P_E. The mathematical manipulation of above equation leads to the following form of Rankine formula,

P_R=(σ_c A_c)/(1+a〖(Le/k_min )〗^2 )

Where σ_c compressive load, a is the Rankine’s constant whose value will change depending upon the material of the column.

Procedure

The experiment to find the buckling load is performed on tensile testing machine. Following set of steps are to be performed to complete the experiment.

First of all, length of column is measured using micrometer or simple steel ruler and value is recorded in mm.

Cross-sectional area of the each specimen is then calculated from the radius of specimen.

After that, specimen is placed between the jaws of the machine as per the instructions provided and the buckling test is performed.

The value of load is printed for each person in the group.

The value of load is recorded in N (Newton).

Experimental Results

 

Aluminum

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling load, Pb (N)

Actual Stress (MPa)

225

5.00

180

918

47

200

5.00

160

816

41

175

5.00

140

357

18

150

5.00

120

1193

61

125

5.00

100

1718

87

100

5.00

80

2126

108

85

5.00

68

3199

162

75

5.00

60

2915

148

60

5.00

48

3626

184

50

5.00

40

4483

228

 

Steel

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling load, Pb (N)

Actual Stress (MPa)

225

5.00

180

1219

62

200

5.00

160

1093

56

175

5.00

140

876

44

150

5.00

120

1346

68

125

5.00

100

2075

105

100

5.00

80

5412

275

85

5.00

68

8228

418

75

5.00

60

7549

383

60

5.00

48

7864

399

50

5.00

40

9749

495

 

Brass

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling load, Pb (N)

Actual Stress (MPa)

225

5.00

180

870

44

200

5.00

160

821

42

175

5.00

140

1262

64

150

5.00

120

716

36

125

5.00

100

2378

121

100

5.00

80

2736

139

85

5.00

68

4280

217

75

5.00

60

3965

201

60

5.00

48

5015

255

50

5.00

40

4950

251

Calculated results Using Euler Formula:

 

Aluminum

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling Load (Euler’s Formula)

Euler’s Stress (MPa)

225

5.00

180

455

23

200

5.00

160

577

29

175

5.00

140

753

38

150

5.00

120

1025

52

125

5.00

100

1477

75

100

5.00

80

2308

117

85

5.00

68

3194

162

75

5.00

60

4103

208

60

5.00

48

6411

326

50

5.00

40

9232

469

 

Steel

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling Load (Theoretical)

Euler’s Stress (MPa)

225

5.00

180

1259

64

200

5.00

160

1594

81

175

5.00

140

2082

106

150

5.00

120

2834

144

125

5.00

100

4041

205

100

5.00

80

6377

324

85

5.00

68

8827

448

75

5.00

60

11338

576

60

5.00

48

17716

900

50

5.00

40

25511

1295

 

Brass

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling Load (Theoretical)

Euler’s Stress (MPa)

225

5.00

180

599

30

200

5.00

160

759

39

175

5.00

140

991

50

150

5.00

120

1349

69

125

5.00

100

1943

99

100

5.00

80

3037

154

85

5.00

68

4203

213

75

5.00

60

5399

274

60

5.00

48

8436

428

50

5.00

40

12148

617

Calculated results Using Rankine’s Formula:

 

Aluminum

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling Load (Rankine’s Formula)

Rankine’s Stress (MPa)

225

5.00

180

305

15

200

5.00

160

338

17

175

5.00

140

242

12

150

5.00

120

552

28

125

5.00

100

794

40

100

5.00

80

1107

56

85

5.00

68

1598

81

75

5.00

60

1704

87

60

5.00

48

2316

118

50

5.00

40

3018

153

 

Steel

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling Load (Rankine’s Formula)

Rankine’s Stress (MPa)

225

5.00

180

620

31

200

5.00

160

648

33

175

5.00

140

617

31

150

5.00

120

913

46

125

5.00

100

1376

70

100

5.00

80

2928

149

85

5.00

68

4258

216

75

5.00

60

4532

230

60

5.00

48

5446

277

50

5.00

40

7053

358

 

Brass

Length (l), mm

Diameter, d (mm)

Slenderness ratio, l/k

Buckling Load (Rankine’s Formula)

Rankine’s Stress (MPa)

225

5.00

180

355

18

200

5.00

160

394

20

175

5.00

140

555

28

150

5.00

120

468

24

125

5.00

100

1069

54

100

5.00

80

1439

73

85

5.00

68

2121

108

75

5.00

60

2286

116

60

5.00

48

3145

160

50

5.00

40

3517

179

 


Discussion:
Actual stress, stress using Euler’s formula and Rankine’s formula for Aluminum, steel and brass respectively has been found in this experiment. Figure-1 shows the graph between slenderness ratio and buckling stress for aluminum. The results of rankine’s theory are closer to the experimental values at higher slenderness values but initially the error is quite high as initial stress value for Rankine Formula is very low as compared to that of the experimental value. While results of Euler’s theory are closer to experimental values at high slenderness ratio but the error is quite high at low values of slenderness ratio. The result provided by the Euler’s theory area non uniform as the graph show sudden increase and decrease in the stress value for increasing value of slenderness ratio. This sudden increase and decrease can only be explained in way that either stress production depends highly on material properties like strength, hardness and ductility or the experimental setup has limitations for this method. It also shows that with increasing value of slenderness ratio the stress produce in the truss start to decrease. The graph pattern of stress and slenderness ratio is non-linear which show that it depends a lot on other material properties rather than just on the specimen dimensions or slenderness ratio. 
 


In the figure-2 for steel, the pattern is quite similar as in the figure-1 where the results of rankine’s theory are closer to the experimental values at higher slenderness values but initially the error is quite high as initial stress value for Rankine Formula is very low as compared to that of the experimental value.


 

Similar pattern in the figure-3 can also be observed for the third type of material which is brass. It can be further concluded that Euler’s theory demands extra strength for material which can boost the costs while while Rankine’s theory underestimates the material’s strength which can cause serious problem in real life structures.
 


Figure 3- Stress in Brass
Sources of Errors
Any slippage in grip can distort the measurement.
Axial misalignment of machine jaws can cause error.
The large error at low values of slenderness can be due to fact that the grip weight might be ignored

Conclusion
The aim of this experiment was to study different techniques to measure buckling of struts made up of different engineering materials. Two methods from theory were used to calculate the amount of buckling load and stress. The materials included in this study were brass, steel and aluminum. The results of rankine’s theory are closer to the experimental values at higher slenderness values but Euler’s theory are closer to experimental values at high slenderness ratio but the error is quite high at low values of slenderness ratio. It was concluded that Euler’s theory demands extra strength for material which can boost the costs while Rankine’s theory underestimates the material’s strength which can cause serious problem in real life structures.

No comments:

Post a Comment