Objective:
To study the buckling of a column under axially load with various boundary conditions and compare
the experimental buckling loads with the Euler buckling formulae.
Apparatus:
Figure below shows the details of the buckling of Struts apparatus. It consists of a back plate with a
load cell at one end and a device to load the struts at the top. The bottom check fixes to an
articulated parallelogram mechanism, which prevents rotation but allows movement in the vertical
direction against the ring load cell. The mechanism reacts to the considerable side thrust produced
by the strut under buckling conditions, with little friction in the vertical direction. It also has a digital
force display.
Figure: Buckling of strut experimental setup
Theory:
The general Euler buckling formula for struts is
2
π EI
Pcr = 2
Le
Where,
a) Pcr is the Euler buckling load (N)
b) E is Young’s modulus (GPa)
c) I, is the minimum second moment of area (
4
mm ¿(Based on the cross−section chosen , I =I xx ∨I yy )
d) Le is the effective length of the strut (mm );
Depending on the clamping condition, effective length is as follows:
i. Hinged- Hinged Le = L
L
ii. Hinged-Fixed Le =
√2
L
iii. Fixed-Fixed Le = .
2
Young’s modulus of Steel = 210 GPa.
Second moment of area for the strut is given by:
3
bd
I xx =
12
3
db
I yy =
12
Procedure:
Turn on the digital force display. Connect the mini-DIN lead from ‘Force Input 1’ on the Digital Force
Display to socket marked ‘Force Output’ on the right-hand side of the unit. Bring the force meter to
zero using the dial on the front panel of the instrument. Gently apply a small load with a finger to the
top of the load cell mechanism and release. Zero the meter again if necessary. Repeat to ensure that
the meter returns to zero. It the meter is only +/- 1N, lightly tap the frame (there may be a little
‘stiction’ and this should overcome it)
I. Buckling Load of a Pinned-Pinned End Strut:
a) Fit the bottom chuck to the machine and remove the top chuck.
b) Use the strut given. Measure the cross section using the vernier.
c) Adjust the position of the sliding crosshead to accept the strut using the thumbnuts
to lock off the slider. Ensure that there is maximum amount of travel available on the
hand wheel thread to compress the strut. Finally tighten the locking screws.
d) Carefully back off the hand wheel so that the strut is resting in the notch but not
transmitting any load; re-zero the force meter using the front panel control.
e) Carefully start the loading of the strut.
f) Turn the hand wheel until there is no further increase in load. Record the final load
as ‘buckling load’.
II. Buckling load of a Pinned-Fixed End Strut:
a) Remove the bottom chuck and clamp the specimen using the cap head screw and
plate to make a pinned-fixed end condition.
b) Use the strut given. Measure the cross section using the vernier.
c) The length of the strut, L = Original length – 20mm.
d) Adjust the position of the sliding crosshead to accept the strut using the thumbnuts
to lock off the slider. Ensure that there is maximum amount of travel available on the
hand wheel thread to compress the strut. Finally tighten the locking screws.
e) Carefully back off the hand wheel so that the strut is resting in the notch but not
transmitting any load; re-zero the force meter using the front panel control.
f) Carefully start the loading of the strut.
g) Turn the hand wheel until there is no further increase in load. Record the final load
as ‘buckling load’.
Observations:
I. Buckling load of a Pinned-Pinned End Strut:
Length Thickness Breadth I xx I yy Buckling load ( Theoretical % Error
Strut
(L) (d) (b) Pcr ¿
No. (mm )
4
(mm )
4 Buckling Load
(mm) (mm) (mm) (N)
(N)
1. 370 1 30 2.5000 2250.000 -28 -37.85 26.02%
2. 470 1.1 30 3.3275 2475.000 -22 -31.22 29.53%
3. 470 1 25 2.0833 1302.083 -15 -19.55 23.27%
II. Buckling load of a Pinned-Fixed End Strut:
Length Thickness Breadth I xx I yy Buckling load ( Theoretical % Error
Strut
(L) (d) (b) Pcr ¿
No. (mm )
4
(mm )
4 Buckling Load
(mm) (mm) (mm) (N)
(N)
1. 350 1 30 2.5000 2250.000 -80 -84.60 5.43%
2. 450 1.1 30 3.3275 2475.000 -53 -68.11 22.18%
3. 450 1 25 2.0833 1302.083 -38 -42.64 10.88%
*The values in the observation table of loadings are negative to indicate compression.
Calculations
Following are the calculations of I xx , I yy, Pcr for Pinned-Pinned case with L = 370 mm
Calculating I xx ∧I yy :
3
b d3 30∗(1)
I xx = = =2.5000 mm4
12 12
3
d b3 1∗( 30 )
I yy = = =2 250 mm4
12 12
In the Euler buckling formula, the value of Second Moment of Inertia is the minimum of I xx , I yy,
4
which in this case is I xx =2.5000 mm .
Calculating Pcr :
2
π ∗E∗I min
Pcr = 2
N
Le
2 9 −12
π ∗210∗10 ∗2.5000∗10
Pcr = N
( 370 2 )∗10−6
Pcr =37.85 N
Graph
Buckling Load vs Strut length for Hinged-Hinged Boundary Condition
Buckling Load vs Strut length for Fixed-Hinged Boundary Condition
Result
a) Buckling of a column was studied with varying clamping conditions, Pinned-Pinned
and Pinned-Fixed.
b) Both the Theoretical and Experimental observations are recorded in the observation
table.
Conclusion
a) The Critical Load, both the theoretical and experimental values are inversely
proportional to the Slender Ratio.
L
b) When one end was fixed the effective length became which is apparent in the
√2
increased Pcr values. In Pinned-Fixed case, the experimentally observed values are
approximately double the values of the Pinned-Pinned case, consistent with the
theory.
c) The error observed between the experimental and theoretical values could be due to
several reasons:
i. Error in the measuring instruments
ii. Due to fatigue in the material caused by constant bending
iii. Material may not be perfectly straight before loading
iv. Error could be due to over-loading beyond the Pc r
