One End Hinged and One End Fixed

BUCKLING OF COLUMN WHEN ONE END HINGED AND ONE END FIXED AIM:

To final buckling load of column using column test setup arrangement under one end hinged and one end fixed .

APPARATUS REQUIRED: Column test, load indicator, specimen rod, two ends supported.

EULER’S COLUMN THEORY: As per Euler’s equation for buckling load of long column based on  bending stress, the effect of direct stress is neglected. This may be justified with the statement; the direct stress induced in a long column is negligible as compared to the bending stress. General Euler’s equation

π2 EI ρ = ---------- ℓ2

ρ  Crippling load E Young’s modulus I  moment of inertia  ℓ  Actual length (varies with end conditions) THEORY:Euler’s column theory deals with perfect columns only. I.e. columns with no imperfections. In practice, no column is perfect. Imperfections always exist in column and may manifest in anyone of the following ways 1. The column column is not not initially initially straight. straight. It is curved initiall initially. y.

2. The column column materia materiall behavior behavior may may not be be perfectly perfectly linear. linear. The column is not slender. It may be short, thick column

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ASSUMPTION: • Initially the column is perfectly straight and the load applied is truly axial. • The cross section of column is uniform throughout its length. • The column material is perfectly elastic homogenous and isotropic and they Obey’s Hooks law. • The length of column is very large as compared to its cross-sectional dimensional and the failure occurs due to buckling load. PROCEDURE: • Consider a column AB of length ‘ ℓ’ with one end fixed other end hinged. • The column cannot be rotated because it is one end fixed end and other  end hinged. • It is positioned to have a complete supports. • The load is gradually applied by rotating the loading wheel connected to load cell intern to the digital meter. • The load indicator is connected with column test setup. • The load indicator is viewed simultaneously from the display of digital load indicator. •  Now the column just starts buckling. • Till the column deflection of column occurs as shown in fig. meanwhile applied load value approximately coincides with the theoretical value. MODEL CALCULATION : (ONE END FIXED AND OTHERS ENDS ARE HINGED)

16π2EI ρ = ------------; [where ℓ =3 ℓ/4] 9 ℓ2 for stainless steel E=2

105 N/mn2

π I = ---- × 54 mm4 where 5 mm is the dia of shaft. 64

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 ℓ = 1000mm 2π2 × 2 × 105 × π × 54 ρ = --------------------------------10002 × 9.81 × 64 = 10.97 Kg

Euler’s theoretical buckling load

Note: Similarly calculate for 4 mm & 6 mm Diameter rods BUCKLING OF COLUMNS: When one end is hinged and other end’s fixed before loading.

S. No.

1) 2) 3)

Specimen

Stainless steel Stainless steel Stainless steel

Young’s modulus N/mm2 2 × 105 2 × 105

2 × 105

Length ‘mm’

Dia in mm

Crippling load Kg

1000 1000 1000

4 5 6

4.49 10.97 22.75

Loading column is stopped at crippling load. This load to known as  buckling load of column.

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TABLE OF READINGS & CALCULATION (ONLY FOR GUIDANCE)

Hinged-Fixed Condition Ratio, [ ( /P)]/ [ ( )]

Sl. No.

Diameter,d 'mm'

Load 'Kg'

Load, P 'N'

Deflection, 'mm'

( )

Deflection/Loa d, P

1

4

1

9.81

2

-4

0.204

( / P) 0.10 2

2

4

2

19.62

6

-2

0.306

0.03 4

-0.017

39.24 0 58.86 0

0.08 5

0.014

70.632

0.025

39.240

0.014

73.575

0.005

188.35 2

0.001

1177.2 00

0.010

98.100

3 4

4 4

3 4

29.43 39.24

8 14

-6 14

0.272 0.357

5

5

2

19.62

0

-2

0.000

6

5

4

39.24

2

-5

0.051

7

5

6

58.86

7

-4

0.119

8

5

8

78.48

11

-3

0.140

9

5

10

98.1

14

14

0.143

0.05 1 0.06 8 0.02 1 0.00 3 0.14 3

0.025

Invers e

5

10

6

4

39.24

0

-2

0.000

11

6

8

78.48

2

-4

0.025

12

6

12

117.72

6

-3

0.051

13

6

16

156.96

9

-2

0.05733945

14

6

20

196.2

11

11

0.05606524

0.02 5 0.02 5 0.00 6 0.00 1 0.05 6

0.013

78.480

0.006

156.96 0

0.002

-0.001 0.005

470.88 0 1569.6 00 196.20 0

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GRAPHS:

RESULT: Thus the buckling load of column as found and compared with its theoretical value.

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CALCULATION (ONLY FOR GUIDANCE):

∆(δ)= 2-6= -4 ∆(δ)= 6-8 = -2

Load p= 1×9.81 =9.81 N δ/P = Deflection /load = 2/9.81 = 0.204 ∆(δ/P)= 0.204 – 0.306 = -0.102 ∆(δ/P)= 0.308-0.272 = 0.034 ∆(δ/P)

-0.102 Ratio = -------- = ---------- = 0.025 ∆(δ) -4 1 1 Inverse= --------- = ----------- = 39.240 Ratio 0.025

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