BACHELOR IN MECHANICAL ENGINEERING

Buckling Test

MOHAMED Al Askari

SUKD1702067

EME3431

Mr Amares

Group 5
Contents:
Abstract...................................................................................................................................................... 3
Objective .................................................................................................................................................... 3
Introduction ............................................................................................................................................ 3-4
Apparatus ................................................................................................................................................... 6
Procedure ................................................................................................................................................... 7
Results .................................................................................................................................................. 8-13
Discussion ................................................................................................................................................ 13
Conclusion ............................................................................................................................................... 13
References ............................................................................................................................................... 14
Abstract:
Buckling is an unpredictability that leads to a structural failure. Once buckling commence this
uncertainty can lead to a failure. Therefore, in this experiment we determined the critical buckling
loads for a steel bar, in which one was under the connection of a pin and pin, the other case was
under the connection of a pin and fixed from the other side. The device used was the WP120. Both
forces and deflections were recorded in a table and a graph was made to show the outcome of the
experiment. Also, the critical force was calculated. Euler’s theory of buckling was also examined
in this report.

Objective:
 To determine the critical buckling loads for columns with supports.
 To examine the Euler theory of buckling and plot a graph of force against deflection.

Introduction:
Whenever a member is designed, it is necessary that it satisfies specific strength, deflection, and
stability requirements. Some members, however, may be subjected to compressive loadings, and
if these members are long and slender, the loading may be large enough to cause the member to
deflect laterally or sideways, i.e. causing deflection in the mid-span for most cases. To be specific,
long slender members subjected to an axial compressive force are called columns, or strut, as stated
in the objective above, and the lateral deflection that occurs is called buckling.
An ideal column is one which is perfectly straight before any loading is applied, made of
homogenous material, and which the loading is applied through the material cross section.
Quite often the buckling of a column can lead to a sudden and dramatic failure of a structure or
mechanism, and as a result, absolute attention must be given to the design of columns so that they
can support their intended loadings without buckling.
The maximum axial load that a column can support when it is on the verge of buckling is called
the critical load. Any additional loading will cause the column to buckle or deflect laterally.
In this experiment, focus is drawn to only two cases. The experiment is to be conducted on a fixed
to pinned and pinned to pinned connections.

Case 1(Pinned from both sides) Case 2 (pinned and fixed)

 Euler formula:

lk

i
(equation 1)
where, l k = characteristic length of bar that takes both the actual length
of the bar and the mounting conditions into consideration.

Figure B: Euler cases of buckling

For example, clamping the ends of the odds causes rigidly. The buckling
length decisive for slenderness is shorter than the actual length of the bar. Altogether a
differentiation is made between four types of mountings, each having a different
buckling length.

The influence of diameter in the slenderness ratio is expressed by the inertial radius, i. it
is calculated using the minimum geometrical moment of inertial, I y and the cross-
sectional area, A.
i  Iy / A
(equation 2)

The influence of material is taken into consideration by the longitudinal rigidity of the
rod EA. Here, E is the modulus of elasticity of the respective material and A is cross-
sectional area. The influence of various factors on the critical load are summarized in
Euler Formula.

EA
Fcrit   2 (equation 3)
2
Or expressed in a different form where it changes according to the connection type:
 EI y
Fcrit   2 (equation 4)
l2
In order to determine whether a rod has failed due to exceedingly the
admissible compressive strain or by buckling, the normal compressive strain
in the rod, which is part of the critical load must be calculated.
F E
6k  k   2 2
A 
(equation 5)
In this normal compressive strain is lower than the admissible compressive
strain, the rod will fail due to buckling. If the admission compressive strain is
used as the normal compressive strain, the critical slenderness ratio, λcrit at
which buckling occurs can be calculated:
E
crit   2
p
(equation 6)
The buckling force can be determined according to the Euler formula:
EI y
Fcrit   2 2
l
(equation 7)
And moment inertia, I y is calculated as the following for a square cross
section:
bh 3
Iy 
12
(equation 8)
Apparatus:
a. Load spindle
b. Load nut
c. Load cross bar
d. Clamping screws
e. Guide columns
f. Force gauge
g. Attachment socket
h. Basic frame
- Force measuring device
- A specimen made of flat steel bar
- Measurement apparatus (ruler, caliper, divider, etc).

Figure 3: WP120 Buckling Test device.

Procedure:
 Euler Case 1 has been chosen to run the test on buckling of the specimen and the maximum
force is 450 N.
 The thrust piece was inserted with V notch into attachment socket and fasten with clamping
screw as shown in Figure E.
 Long thrust piece was inserted with V notch into the guide bush of the load cross-bar and
Hold it firmly as shown in Figure D.
 The specimen was inserted with edges in the V notch.
 The load cross-bar was clamped on the guide column in such a manner that there is still
approx. 5mm for the top thrust piece to move.
 The specimen was Aligned in such a manner that its buckling direction points in the
direction of the lateral guide columns. Here, the edges must be perpendicular to the load
cross-bar.
 Pre-tightened the specimen with low, non-measurable force.
 Aligned the measuring gauge to the middle of the rod specimen using the support clamps.
The measuring gauge has been set at a right angle to the direction of buckling.
 Pre-tightened the measuring gauge to 10 mm deflection with the adjustable support.
 Slowly subjected the specimen load using the load nut.
 The deflection was recorded from the measuring gauge. Read and record the deflection
every 100N.
 The result was tabulated and repeated the experiment twice for each connection.
Results:
Table 1: Pin to Pin connection
Pin to Pin connection
First Second
trial Trial
Buckling Force, Buckling Force, Deflection,
F(N) Deflection, d(mm) F(N) d(mm)
0 0 0 0
100 0.250 100 0.200
200 0.535 200 0.470
300 1.040 300 1.000
400 2.555 400 2.560
Average
Buckling Force,
F(N) Deflection, d(mm)
0 0
100 0.225
200 0.5025
300 1.020
400 2.5575
Table 2: Pin to fixed connection
2 Pin connection
First Second
trial Trial
Buckling Force, Buckling Force, Deflection,
F(N) Deflection, d(mm) F(N) d(mm)
0 0 0 0
100 -0.135 100 -0.125
200 -0.380 200 -0.375
300 -0.760 300 -0.715
400 -1.155 400 -1.085

Average
Buckling Force,
F(N) Deflection, d(mm)
0 0
100 -0.130
200 -0.3775
300 -0.7375
400 -1.120
Graphs:

(Pin to Pin connection)

450
400
350
Buckling Force(N)

300
250
200
150
100
50
0
0 0.5 1 1.5 2 2.5 3
Deflection (mm)

Graph 1 (Pin to Pin connection)

(Pin to Fixed connection)

500

400
Buckling Force(N)

300

200

100

0
0 0.2 0.4 0.6 0.8 1 1.2
Deflection (mm)

Graph 2 (Pin to fixed connection)

Bar Dimensions:
Length: 0.65m
Width: 0.02m
Thickness: 0.004m

Calculations:

Modulus of Elasticity (E): 210Gpa

𝑏ℎ3 1
Moment of inertia: Ixx = = (0.02)(0.004)3 = 1.0667 𝑋 10−10 𝑚4
12 12

EA 210 𝑋 109 𝑋 1.0667 𝑋 10−10

Critical Force: Fcrit   2 = 𝜋2 2 = 1067.9N
2 0.455

Pin to Pin connection:

𝐸𝐼𝑑𝑣 2 𝐹𝑥 𝐸𝐼𝑑𝑣 𝑓𝑥 2 𝑓𝑥 3
= = +𝑐 𝐸𝐼𝑣 + + 𝑐𝑥 + 𝑐2
𝑑𝑥 2 2 𝑑𝑥 4 12

𝐿 2
𝑑𝑣 𝐿 𝐿 2 𝐹( )
2
= 0 𝑥 = 2, 0 = 𝐹 (2) + 𝐶 𝐶=−
𝑑𝑥 4

𝑋 = 0, 𝑣 = 0
𝑋 = 𝐿, 𝑣 = 0

𝐿 2 𝐿 3 𝐿 3
1 𝐹𝑥 3 𝐹( ) 𝑋 𝐹( ) 𝐹( )
2 2 2
𝑣 = 𝐸𝐼 [ 12 − ] −
4 12 4

100(0.325)3 100(0.325)3 0.225

− = 0.5721mm 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = 1 − 0.5721 (100) = 60.66%
12 4
 Table 3 (Percentage error of pin to pin connection)

Buckling Force, F (N) Deflection, δ (mm) Theoretical Value, Percentage Error, %

v(mm)
0 0 0 0

100 0.225 0.57 60.6

200 0.5025 1.44 65.1

300 1.020 1.72 40.5

400 2.5575 2.29 11.7

Pin and Fixed Connection

Σ𝐹𝑦 = 0

𝐴𝑦 + 𝐵𝑦 − 𝐹1 = 0
𝐴𝑦 + 𝐵𝑦 = 100
Σ𝑀𝐵 + = 0
0.65𝐴𝑦 − 0.325(100) − 𝑀𝐴 =0
0.65𝐴𝑦 = 𝑀𝐴 +32.5 eq (1)

Using Macaulay’s Theory

𝐴𝑦 (𝑥) − 𝑀𝐴 − 100(𝑥 − 0.325)-M(1)=0

M(1)= 𝐴𝑦 (𝑥) − 𝑀𝐴 − 100(𝑥 − 0.325)
𝑑2 𝑣
EI 2 = 𝑀
𝑑𝑥
𝑑2 𝑣
EI 2 = 𝐴𝑦 (𝑥) − 𝑀𝐴 − 100(𝑥 − 0.325)
𝑑𝑥
𝑑𝑣 𝐴𝑦 𝑥 2 100(𝑥 − 0.325)2
EI = − 𝑀𝐴 (𝑥) − + 𝐶1
𝑑𝑥 2 2
𝐴𝑦 𝑥 3 𝑀𝐴 𝑥 2 100(𝑥 − 0.325)3
EI𝑣 = − − + 𝐶1 (𝑥) + 𝐶2
6 2 6
At
x=0
v=0
𝐶2 = 0
At x=0
𝑑𝑣
=0
𝑑𝑥
𝐶1 = 0
At
x=0.65
v=0
𝑀𝐴 0.652 100(0.65 − 0.325)3
− − =0
2 6
0.0457𝐴𝑦 − 0.2113𝑀𝐴 − 𝟎. 𝟓721=0
𝑀𝐴 = 0.2163𝐴𝑦 -2.7075 eq (2)
Substituting eq 2 to 1
0.65𝐴𝑦 = 𝑀𝐴 +32.5
0.65𝐴𝑦 = 0.2163𝐴𝑦 -2.7075+32.5
0.4337𝐴𝑦 = 29.7925
𝐴𝑦 = 68.69 𝑁
𝐵𝑦 = 31.3 𝑁
𝑀𝐴 = 12.15 𝑁𝑚
x=0.325

68.69(0.325)3 12.15(0.325)2 100(0.325 − 0.325)3

EI𝑣 = − −
6 2 6

EI𝑣 = −0.2488

−0.2488
v = (210 x 109)(2.667×10−9 )

v= 0.444 mm
Percentage Error
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
Percentage of error, (%) = × 100%
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
0.444−0.130
Percentage of error, (%) = 0.444 × 100%
Percentage of error, (%) = 70.7%
 Table 3 (Percentage error of pin to fixed connection)

Buckling Force, F (N) Deflection, δ (mm) Theoretical Value, Percentage Error, %

v(mm)
0 0 0 0

100 0.130 0.444 70.7

200 0.3775 1.21 68.8

300 0.7375 2.825 73.89

400 1.120 2.734 59.0

Discussion:
After conducting the experiment its seen from the graphs made that both (Pinned and Pin to fixed
connection) increase linearly when buckling was made, however, a percentage error was found
that went up to 73%. These errors could’ve been due to error in the placement of the sensor on
the steel column or even human errors other errors could’ve been vibration on the apparatus by
placing the hands on the apparatus. This kind of error can be reduced by repeating the
experiment multiple times to ensure the stability of the data collected. The critical force was also
calculated which was 1067.9N this means the steel bar column can withstand this force before
breaking.

Conclusion:

We can conclude that whenever the force increased the deflection increased respectively and this
means that the correlation between the force and deflection is directly proportional the
percentage error was high in lower force and decreased when the force increased. Errors can be
avoided by repeating the experiment more.
 References

 Berham, P. P., Crawford, R. J., Armstrong, C. G. 1996, Mechanism of

Engineering Materials, 2nd Edition, Pearson Education Limited, China.
 Hornbostel, C., 1991. Construction Materials. 1st ed. New York: John Wiley & Sons.
 Hibbeler, R. C. 2005, Mechanics of Materials, 6th Edition, Prentice Hall, Singapore.