SRI LANKA INSTITUTE OF INFORMATION
TECHNOLOGY
Faculty of Engineering
ME3531 – SOLID MECHANICS AND
MECHANICAL DESIGN
Buckling of Struts
Name: Perera P N D
EN Number: EN20401412
Date of Performance: 15-03-2022
Date of Submission: 10-04-2022
i
Continuous Assessment Cover Sheet
Faculty of Engineering
Module Details
Module Code
ME3531
Module Title
Program: SLIIT
Solid Mechanics and Mechanical Design
Course: BSc
Stream: Mechatronics
Assessment details
Title
Buckling of struts
Group assignment
NO
If yes, Group No.
Lecturer/ Instructor
Mr. Amila Alexander
Date
Performance
Due date
2022/04/10
Date submitted
of
2022/03/15
2022/04/10
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EN20401412
Perera P N D
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ii
iii
Table of Contents
List of figures ......................................................................................................................... v
List of tables........................................................................................................................... v
List of graphs ......................................................................................................................... v
01.
Title ................................................................................................................................. 1
02.
Objective ......................................................................................................................... 1
03.
Apparatus ........................................................................................................................ 1
04.
Introduction ..................................................................................................................... 4
05.
Theory ............................................................................................................................. 5
Euler’s theory ......................................................................................................................... 5
Assumptions........................................................................................................................... 5
Limitations ............................................................................................................................. 6
Euler’s Formula ..................................................................................................................... 6
Equivalent Length of a Column ............................................................................................. 7
06.
Procedure ........................................................................................................................ 8
07.
Observations ................................................................................................................... 9
Case 1 Both ends hinged ........................................................................................................ 9
Case 2- Both ends fixed ......................................................................................................... 9
Case 3- One end fixed and the other end hinged ................................................................. 10
Case 4- One end fixed and the other end free ...................................................................... 10
08.
Calculations................................................................................................................... 11
09.
Results ........................................................................................................................... 12
10.
Discussion ..................................................................................................................... 15
Comment on the form of the graphs obtained and compare the values of Pcr obtained from
the graphs and the equation for each case ............................................................................ 15
Comment on possible reasons for any deviation from the theoretical values. .................... 16
What are the possible errors of this experiment? ................................................................. 16
Suggest improvements for the instrument and experiment. ................................................ 16
State applications of buckling phenomena .......................................................................... 17
11.
Conclusion .................................................................................................................... 18
12.
References ..................................................................................................................... 19
iv
List of figures
Figure 1 Structure model of the buckling tester ........................................................................ 1
Figure 2 Buckling tester ............................................................................................................. 2
Figure 3 Specimen ..................................................................................................................... 2
Figure 4 Hinges and fixers ......................................................................................................... 3
Figure 5 Specimen fixing materials ........................................................................................... 3
Figure 6 Four types of conditions .............................................................................................. 5
List of tables
Table 1 Values and fixity coefficient ......................................................................................... 7
Table 2 4 Cases and lengths consider for each case .................................................................. 7
Table 3 Theoretical and experimental values for each case .................................................... 12
List of graphs
Graph 1 Case 1 Both ends are hinged ...................................................................................... 12
Graph 2 Case 2 Both ends are fixed ........................................................................................ 12
Graph 3 Case 3 One end fixed, and the other end is hinged .................................................... 13
Graph 4 Case 4 one end fixed, and the other end is free ......................................................... 13
Graph 5 All 4 cases at one graph ............................................................................................. 14
v
01.Title
Buckling of struts
02.Objective
•
•
•
•
To identify Points of Buckling for each method of fixing.
To determine the critical load for a steel strut (65 Mn) theoretically.
To find out critical load experimentally and compare with theoretical values.
To understand, analyze and compare this method in determining critical load
03.Apparatus
Figure 1 Structure model of the buckling tester
1
Figure 2 Buckling tester
Figure 3 Specimen
2
Figure 4 Hinges and fixers
Figure 5 Specimen fixing materials
3
04.Introduction
Buckling is caused by imperfections that prevent the load from being delivered axially
precisely, such as eccentric loading and a lack of starting straightness. When long struts are
operated under elastic circumstances, the cumulative impact of these flaws on overall buckling
behavior are predictable. From the lateral deflection that happens, we may develop an
instability criteria. Leonhard Euler (1707–83) is credited with this strategy. With the following
end fittings, it may be fitted to lengthy struts.
In today's engineering constructions, trusses, beams, and columns play a significant role. They
are used to withstand a variety of forces acting in different directions. A column is a structural
element that may convey the structure's weight through compression. The element is the same
for a vertically utilized beam as it is for a horizontally used beam. Horizontal beams, on the
other hand, are often supported by columns. And a beam's two ends are supported by two
columns.
Columns are used to hold the high weights with axial compression forces. And also we know
that the column ends can be fixed with differently. Due to the load and the different fixes
caused the column buckling. The weight that can buckle a column is referred to as critical
loads. As a result, the columns must be designed to support a load that is less than the
maximum weight. When the structure is employed for even minor weights below the critical
buckling load, the fatigue that is caused to the structure might cause the buckling to fail.
Bucking is so considerable matter in most of the engineering fields. So it is important to
understand the buckling characteristics and properties. Also it is necessary to determine and
designing thin factors. In this experiment we will understand and study how columns buckle
under different loads conditions under different end fixes. These end fixes are as follow,
• Bothe the ends hinged, or pin joint as shown in figure 1(a)
• Both the ends fixed as shown in Figure 1 (b)
• One end is fixed and the other hinged as shown in Figure 1 (c)
• One end is fixed and the other free as shown in Figure 1 (d)
4
Figure 6 Four types of conditions
05.Theory
For buckling of struts, we consider the Euler’s theory for measuring the buckling failure. This
theory ignores the effect of direct column stress, the presence of crookedness in the column,
and possible axial load application point shifts from the center of the column cross-section. As
a result, the critical buckling load may be overestimated by the theory. Leonhard Euler
proposed the Euler theory of column buckling in 1757.
Euler’s theory
According to Euler's theory, the stress in the column caused by direct loads is minor in
comparison to the stress caused by buckling failure. A formula was developed based on this
statement to calculate the critical buckling load of a column. As a result, the calculation relies
on bending stress and ignores direct stress caused by direct pressures on the column.
For this theory we do some assumptions for measuring the values.
Assumptions
•
Initially, the column is perfectly straight.
•
The cross-section of the column is uniform throughout its length.
•
The load is axial and passes through the centroid of the section.
•
The stresses in the column are within the elastic limit.
•
The materials of the column are homogenous and isotropic.
•
The self-weight of the column itself is neglected.
•
The failure of the column occurs due to buckling only.
•
Length of column is large compared to its cross-sectional dimensions.
•
The ends of the column are frictionless.
5
•
The shortening of column due to axial compression is negligible.
(Hamakareem, 2021)
Limitations
There are some limitations in this theory.
•
•
This theory does not account for the potential of column crookedness, and the load
may not be axial.
The axial stress is not considered in the formula derived in Euler theory of column
buckling, and the critical buckling load may be greater than the actual buckling load.
Euler’s Formula
According to Euler’s theory, the crippling or buckling load (Pcr) under various end
conditions is represented by a general equation,
𝑃𝐶𝑟 =
𝐶𝜋 2 𝐸𝐼
𝑙2
Where
E = Modulus of elasticity or Young’s modulus for the material of the column
l = Length of the column
C = Constant, representing the end conditions of the column or end fixity coefficient
Where I, the moment of inertia of a rectangular cross section can be calculated,
About X-X
𝐼𝑥𝑥 =
1
𝑏ℎ3
12
And the moment of inertia about the Y-Y,
𝐼𝑦𝑦 =
1
ℎ𝑏 3
12
6
Table 1 Values and fixity coefficient
S.No
End Condition
1
2
3
4
Both ends Hinged
Both ends Fixed
One end Fixed, and the other end Hinged
One end Fixed and the other end Free
End fixity coefficient
(C)
1
4
2
0.25
Equivalent Length of a Column
Sometimes, the crippling load according to Euler’s formula may be written as
𝜋 2 𝐸𝐼
𝑃𝑐𝑟 = 2
𝐿
Where the Young’s modulus of steel, E is 180Gpa
Where L is the equivalent length or effective length of the column. The equivalent length of a
given column with given end conditions is the length of an equivalent column of the same
material and cross section with hinged ends to that of the given column. The relation between
the equivalent length and actual length for the given end conditions is shown in the following
table.
Table 2 4 Cases and lengths consider for each case
Fixing Cases / End conditions
Both end Hinged
Both end Fixed
One end fixed and other end hinged
Actual length
Equivalent length
for the case l
mm
310
L=l
270
L= l/2
290
One end fixed and other end free
270
L =2l
7
06.Procedure
•
•
•
•
•
•
•
•
•
•
•
First the apparatus which is required for the experiment was identified.
Then the apparatus was set for doing the relevant buckling type.
Then the two dial gauges were placed accordingly as we don’t know to which way the
strut is going to be buckled.
Then the specimen wan set according to the type we are going to do the experiment.
Then the zero is set on the gauges.
By rotating the wheel, the load was given to the specimen beginning from a very low
load.
The deflection due to the buckling was measured and observed from the dial gauges
previously set on the both sides of the strut.
Readings were taken corresponding to the load and deflection.
Moment when the increment of load is very low while the strut is continuously deforms
was recorded and identified.
And then a moment was noticed which did not increase the dial gauge anymore
although the load is increased.
Applying the load was stopped after buckling load has been reached.
8
07.Observations
Case 1 Both ends hinged
Test No
Force
Deflection (mm)
1
0
0.00
2
37
0.15
3
52
0.29
4
67
0.23
5
82
0.48
6
97
1.00
7
112
2.54
8
126
7.49
Case 2- Both ends fixed
Test No
Force
Deflection (mm)
1
0
0.00
2
95
0.30
3
195
0.31
4
295
0.45
5
395
0.57
6
495
0.92
7
595
1.79
8
695
8.79
9
Case 3- One end fixed and the other end hinged
Test No
Force
Deflection (mm)
1
0
0.00
2
13
0.02
3
62
0.03
4
112
0.07
5
162
0.29
6
212
0.45
7
262
1.04
8
312
8.44
Case 4- One end fixed and the other end free
Test No
Force
Deflection (mm)
1
0
0.00
2
13
0.10
3
38
0.14
4
58
0.20
5
98
0.42
6
128
0.67
7
158
1.02
8
188
1.61
Modulus of elasticity = 180 GPa
Dimension of the strut = 2 *20*l (l for each case is given under the theory)
10
08.Calculations
𝐼=
𝑏ℎ3 𝜎02 × 0.0023
=
12
12
𝐼 = 1 ⋅ 33 × 10−11 𝑚4
Case 1
𝑃𝐶𝑟 =
𝑃𝐶𝑟
𝐶𝜋 2 𝐸𝐼
𝐿2
1 ∗ 𝜋 2 ∗ 180 ∗ 109 ∗ 1.333 ∗ 10−11
=
0.3102
𝑃𝐶𝑟 = 246.42𝑁
Case 2
𝑃𝐶𝑟 =
𝑃𝐶𝑟
𝐶𝜋 2 𝐸𝐼
𝐿2
4 ∗ 𝜋 2 ∗ 180 ∗ 109 ∗ 1.333 ∗ 10−11
=
0.2702
𝑃𝐶𝑟 = 1299.37𝑁
Case 3
𝑃𝐶𝑟
𝑃𝐶𝑟
𝐶𝜋 2 𝐸𝐼
=
𝐿2
2 ∗ 𝜋 2 180 ∗ 109 ∗ 1.333 ∗ 10−11
=
0.2902
𝑃𝐶𝑟 = 563.165𝑁
Case 4
𝑃𝐶𝑟
𝑃𝐶𝑟
𝐶𝜋 2 𝐸𝐼
=
𝐿2
0.25 ∗ 𝜋 2 180 ∗ 109 ∗ 1.333 ∗ 10−11
=
0.2702
𝑃𝐶𝑟 = 81.211𝑁
11
09.Results
Table 3 Theoretical and experimental values for each case
Case #
Case 1
Case 2
Case 3
Case 4
Theoretical Value
246.42
1299.37
563.16
81.21
Experimental value
126
695
312
188
Difference
120.42
604.37
251.16
-106.79
Case 1- Both ends hinged
140
120
Force N
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
Deflection mm
Graph 1 Case 1 Both ends are hinged
Case 2-both ends fixed
800
700
Force N
600
500
400
300
200
100
0
0
2
4
6
8
10
Deflection mm
Graph 2 Case 2 Both ends are fixed
12
Case - 3 one end fixed and other end hinged
350
300
Force N
250
200
150
100
50
0
0
1
2
3
4
5
6
7
8
9
1.6
1.8
Deflection mm
Graph 3 Case 3 One end fixed, and the other end is hinged
Force N
Case-4 one end fixed other end is free
200
180
160
140
120
100
80
60
40
20
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Deflection mm
Graph 4 Case 4 one end fixed, and the other end is free
13
4 cases
800
700
Force N
600
500
Case 1
400
Case 2
300
Case 3
200
Case 4
100
0
0
2
4
6
8
10
Deflection mm
Graph 5 All 4 cases at one graph
14
10.Discussion
Comment on the form of the graphs obtained and compare the values of Pcr obtained from
the graphs and the equation for each case
For each situation a graph is obtained according to the observed values. And these graphs
provides the information about the buckling deflection in mm on a beam due to the providing
load. From the four graphs we can clearly see that how the values are changed due to the end
fixings. Comparing the obtained four graphs we can see that the highest values came to the
case 2 which is both ends fixed situation. The idea we can get through this is when a strut
column is fixed on both ends and giving a compression force to the both ends that column can
get easily buckling point and that caused easily brake the column. This happens because of the
column is not allow move as it is both end fixed. And the less values obtained in the case 1
both ends hinged scenario as the column is free to move around the hinged point. As both ends
are hinged strut is free to move those two hinged points.
Case
#
Case
1
Theoretical
Value
246.42
Experimental
value
126
Difference
Description
120.42
Case
2
1299.37
695
604.37
Case
3
563.16
312
251.16
Case
4
81.21
188
-106.79
For the case 1 we obtained a
theoretical value 246.42𝑁
but in practically we
obtained
126N.
comparatively these values
are very low to the other
cases
In the case 2 a theoretical
value could be obtained as
1299.37N and practical
value 695N. comparing to
the other cases this situation
both ends fixed gives the
highest values.
Case 3 has a mix of case one
and two which the one end
is fixed and the other end is
hinged. This case gives the
second highest values for
buckling deflection.
The last case cas4 has one
end fixed and the other end
is free which shows a very
shorter deflection range.
While the other four cases
pass 6mm this case the
deflection value does not
reach to 2mm
Considering the graph 5 we can see a comparison view of the values we obtained for the above
4 cases. That graph clearly shows that the most deflection and the highest force is showing on
15
case 2 and the least deflection showing on case 4 but in case 4 the force is considerably higher
than the case 1. The deflection of case 1 is higher than the case 4 but it is under a higher force
than the case 4.
From the above graphs we can identify how a column acts under compression forces with
different end fixings.
Comment on possible reasons for any deviation from the theoretical values.
o When considering the taken values we can clearly see that there is a deviation
between both theoretical values and the experimental values. The reason for
this deviation is in theoretical values we are taking the assumed values. To take
a final value we do many assumptions with the Euler’s theory. These
assumptions are not valid in experimental scenario. Also a theoretical value is
a mathematical expression which provides a numerical value to prove the
theorem. In this case we have to make some assumptions. Experimental values
may have errors with the apparatus errors with the specimen or errors with the
gauges. These errors may lead us to a different value to the theoretical value.
As the assumptions that we made for theoretical provident column is initially
straight, but this can be differed if the column is permanently deformed. The
self-weight of the column is neglected for the theoretical assumption but
experimental scenario we all know that the column has a weight. Also, we
assumed that the buckling is the only reason to fail the column but we know
that this can be differ sometimes by crushing the Colum can be failed. And we
assumed that the ends are frictionless. But in real world we cannot find a
surface without friction. Only we can find is a less friction surface. The cross
section is uniform throughout the length is another assumption this can also be
differed. While we are giving compression force to the two ends of a column
the column can shortened due to the force. But theoretically we assumed that it
won’t happened. These kinds of experimental and theoretical incompatibilities
make errors and mismatch in readings we obtain through the experiment and
the theoretical values.
What are the possible errors of this experiment?
o There can be some errors with the apparatus due to often usage.
o There can be some errors in the specimen. As the specimen is frequently loaded
and unloaded.
o The specimen can be reached to its permanent deflection level.
o Dial gauges may have some errors.
o During the experiment the probability of slipping the specimen is high in case
1, case 3 and case 4.
o Properties of the specimen may have changed.
o Errors can be happened due tot the person while taking the small values from
the dial gauge.
o We cannot find frictionless surfaces in real life, a small friction can change the
reading values.
Suggest improvements for the instrument and experiment.
16
Using a new identical strut for starting each may prevent the errors that could have due to the
permanent deflection level. The compressive load applied to the specimen will make the
specimen permanently deformed. That can be prevent using identical new specimens.
Some errors can be happened because of the readings are taken by a human. Person to person
the value can be changed. This can be prevented using an electronic method to calculate the
readings.
State applications of buckling phenomena
Now we can understand that the buckling is a considerable property depends on the fix type
and the stiffness of a component. Buckling refers to the loss of stability of a component and is
usually independent of material strength. Knowing buckling will help in product
manufacturing, selecting materials, constructions etc. applications of buckling phenomena can
be pointed out as,
Constructions – Now a days H irons are used as columns and beams. This can be considered
as where buckling is used as an application.
Roads - Buckling is a failure mechanism in pavement materials because asphalt is more
flexible than concrete. The road surface absorbs the sun's heat, causing it to expand and press
adjacent parts toward one another. If the stress is high enough, the pavement might rise up and
crack without warning. For automobile drivers, traveling over a buckled part, such as a speed
hump at highway speeds, may be rather unnerving.
Bicycle wheels - A standard bicycle wheel is made up of a narrow rim that is held under high
compressive stress by the (about typical) inward pull of a large number of spokes. It may be
thought of as a heavy column twisted into a circle. If spoke tension is increased over a safe
level, or if a portion of the rim is subjected to a particular lateral strain, the wheel
spontaneously fails into a distinctive saddle form (also known as a "taco" or "pringle"), similar
to a three-dimensional Euler column. If the deformation is simply elastic, the rim will return
to its original plane shape if the spoke tension is removed or a lateral push from the opposite
direction is provided.
Pipes and Pressure vessels - Pipes and pressure vessels that are subjected to external
overpressure, such as that caused by steam cooling within the pipe and condensing into water,
are at danger of collapsing due to compressive hoop stresses. Different pipe and pressure
vessel rules include design criteria for estimating the required wall thickness or reinforcing
rings.
Super and hypersonic aerospace vehicles - Surface panels of super- and hypersonic aerospace
vehicles, such as high-speed airplanes, rockets, and reentry vehicles, can buckle due to
aerothermal heating. If the buckling is driven by aerothermal stresses, the situation might be
made even more difficult by increased heat transfer in locations where the structure deforms
towards the flow-field.
(Anon., 2022)
17
11.Conclusion
As the objectives of the lab experiment identifying the point of buckling for different fixings,
determining the critical load for a steel strut, theoretical values taken from the Euler’s theory
and experimental values taken from the experiment comparison and analyzing understanding
and comparing the each method in critical load is done in this lab experiment. The goal of this
experiment was to learn more about strut buckling. Because greater force can be placed on the
steel specimen, the test results show that both fixed ending struts are substantially stronger
than alternative fixed designs. The practical and theoretical values acquired for each scenario
were combined to form a graph. In addition, I learned how to apply these ideas and how they
operate in real-world situations. All of the potential mistakes and impartments for the
experiment have previously been explored. For concluding the lab experiment I can state that
the values are differ from theoretical to experimental as there are some assumptions that we
have made.
18
12.References
Anon., 2022. Wikipedia. [Online]
Available at: https://en.wikipedia.org/wiki/Buckling#Engineering_examples
[Accessed 2 April 2022].
Anon., n.d. JPE Innovations. [Online]
Available at: https://www.jpe-innovations.com/precision-point/buckling-phenomena/
[Accessed 08 April 2022].
Baru, A. M., 2018. Research gate. [Online]
Available at:
https://www.researchgate.net/publication/329267500_An_Investigation_of_Buckling_Pheno
menon_in_Steel_Elements_-_summarized_version
[Accessed 06 April 2022].
Burgueño, R., 2015. IAEA. [Online]
Available at: https://inis.iaea.org/search/search.aspx?orig_q=RN:47108466
[Accessed 06 April 2022].
Burgueño, R., 2015. Research gate. [Online]
Available at: https://www.researchgate.net/publication/276848939_Bucklinginduced_smart_applications_Recent_advances_and_trends
[Accessed 05 April 2022].
Hamakareem, M. I., 2021. The Constructor. [Online]
Available at: https://theconstructor.org/structural-engg/euler-theory-columnbuckling/37341/#:~:text=The%20Euler's%20theory%20states%20that,critical%20buckling%
20load%20of%20column.
[Accessed 3 April 2022].
Helbig, D. et al., 2016. Scielo. [Online]
Available at: https://www.scielo.br/j/lajss/a/FL55mFmf3d7yJsGf35jtPgb/?lang=en
[Accessed 06 April 2022].
Yudo, H. & Yoshikawa , T., 2014. Springer. [Online]
Available at: https://link.springer.com/article/10.1007/s00773-014-0254-5
[Accessed 02 April 2022].
19