FACULTY OF ENGINEERING AND THE BUILT
IN ENVIRONMENT
DEPARTMENT OF ELECTRICAL POWER
ENGINEERING
STRENGTH OF MATERIALS II
STMT201
Bending Stress in a Beam
Group 7
Surname and Initials
Student Number
Mbewu S
Mkhize MC
Mncube SN
Subbadu K
21801509
21801740
21804080
21803030
Abstract
Beams are important structural members used in infrastructure projects. It should be observed
that beams can be made of a variety of materials. Some beams are made of steel, while others
may be made of reinforced concrete. The use of beams in construction is dependent on their
purpose and strength because they must bear both dead and live loads. When shear forces and
bending moments develop in a beam as a result of external forces, the beam creates internal
resistance to these forces, which is referred to as resisting shearing and bending stresses [1].
Bending stresses are the stresses caused by bending moments. It is critical to calculate shear
and bending stresses at various locations of a beam for beam design purposes. Bending stress
ranges from zero at the neutral axis to a maximum on the beam's tensile and compressive sides.
The bending stress experiment was conducted at the Strengths of Materials laboratory to
determine and understand the bending stresses at various positions of a beam section. Load is
applied to two different locations in this experiment using a special mechanism attached to the
apparatus. Strain gauges are attached to the beam at various locations and are used to collect
data on the stress produced in the beam. The experiment is to be done according to the
instructions and procedures given and results obtained are clearly tabulated under the results
section. All calculations are done and in discussion, the theoretical results and experimental
results will be compared to account for errors deviation of results.
i
Table of Contents
Abstract ....................................................................................................................................... i
List of figures, tables and equations ........................................................................................ iii
List of figures ....................................................................................................................... iii
List of tables ......................................................................................................................... iii
List of equations ................................................................................................................... iii
Chapter One: Introduction ......................................................................................................... 1
Simple bending stress ............................................................................................................ 1
Pure bending stress ................................................................................................................ 1
Aim ........................................................................................................................................ 2
Objective ................................................................................................................................ 2
Methodology .............................................................................................................................. 3
List of Apparatus.................................................................................................................... 4
Experimental Procedure ......................................................................................................... 5
Experiment Precautions ......................................................................................................... 5
Data Sheet .............................................................................................................................. 6
Calculation of the Model ....................................................................................................... 7
Results & Discussion ............................................................................................................... 14
Conclusion ............................................................................................................................... 18
Recommendations ................................................................................................................ 18
References ................................................................................................................................ 19
ii
List of figures, tables and equations
List of figures
Figure 1 - Simple Bending ......................................................................................................... 1
Figure 2 - Pure Bending ............................................................................................................. 1
Figure 3 - STR8A Load Cell [4] ................................................................................................ 3
Figure 4: The laboratory test frame ........................................................................................... 4
Figure 5: Digital force display unit ............................................................................................ 4
Figure 6: Digital strain display unit ........................................................................................... 5
Figure 7: Beam setup for the experiment ................................................................................... 7
Figure 8: Section 1 of the beam ................................................................................................. 7
Figure 9: Section 2 of the beam ................................................................................................. 8
Figure 10: Section three of the beam ......................................................................................... 9
Figure 11: SFD and BMD of the beam .................................................................................... 10
Figure 12: Cross section of the T-beam ................................................................................... 11
Figure 13: T-beam cross-section with Gauge positions ........................................................... 12
Figure 14: The graph of strain vs bending moment ................................................................. 15
Figure 15 - Bending Moment vs Strain.................................................................................... 16
List of tables
Table 1 - Experiment Results (Uncorrected) ............................................................................. 6
Table 2 - Experiment Results (Corrected) ................................................................................. 6
Table 3: Beam geometric properties calculated ....................................................................... 11
Table 4: Experiment bending moment results ......................................................................... 14
Table 5 - Strain in all gauges with the use of various bending moments ................................ 14
Table 6 - Average Data ............................................................................................................ 15
List of equations
Equation 1 - Beam Formula ....................................................................................................... 2
Equation 2 - Bending Formula ................................................................................................. 12
Equation 3 - Hooke's Law ........................................................................................................ 13
iii
Chapter One: Introduction
If a constant bending moment (zero shear force) acts on a length of beam, the stress set up on
any cross section must be a pure couple equal to and opposite to the bending moment. As a
result, one part of the cross section is in tension, while the other is in compression. It is obvious
that the inside edge of an initially straight beam will be in compression and the outside edge
will be in tension [2]. There will be an intermediate surface with zero stress, which is known
as the neutral axis. The bending stress can be separated into two parts, simple bending stress
and pure bending stress.
Simple bending stress
When bending occurs due to beam self-load plus external load, it is referred to as simple
bending. This sort of bending is also referred to as ordinary bending, because it causes both
shear and normal stresses in the beam.
Figure 1 - Simple Bending
Pure bending stress
When bending happens only due to coupling on one end, it is referred to as pure bending. There
is no risk of shear stress in the beam in this case. Normal stress, on the other hand, will
propagate in the beam as a result of this. Because it does not cause any damage to the beam,
normal stress is used [3].
Figure 2 - Pure Bending
1
Aim
The purpose of this experiment was to see how accurate the beam equation is in a bending
stress device when applying various loads and measuring the elongation, stress, and bending
moment of the corresponding.
Objective
The main objective for this experiment is conducting the test on the test frame and obtain results
for the strain which can be used to find the bending moment and to do all necessary calculations
to bring about a solid conclusion for the experiment. Other objectives are:
 Setting up the equipment for the experiment
 Studying and following the procedure given to carry out the experiment successfully
 Carrying out the experiment according to the procedure
 Considering all the precautionary measures for the experiment
Due to the occurrence of bending moments in a beam, these occurrences give rise to stresses
known as bending stresses. In solving problems related to a beam, the beam equation is capable
of being altered. The loaded diagram of the beam is as follows.
The beam formula can be expressed as:
Equation 1 - Beam Formula
𝑀 𝜎
=
𝐼
𝑦
Where:
M = Bending Moment (N-m).
I = Moment of Inertia about the neutral axis (m4).
𝜎 = Bending Stress (Pa) at a given distance y(m) from the neutral axis.
y= distance of the element from the neutral axis (m).
When adjusting the load, different strains can be determined at different strain gauge points.
You can then compare these to the theoretical strain calculation from equation one, beam
equation.
2
Methodology
The experiment was conducted at the Strengths of Materials laboratory with the help of the
laboratory technician in charge. All apparatus that were used were fixed in the test frame
designed by TecQuipments provided in the laboratory. The test frame is shown in the figure
below which symbolises the materials that were used to complete the experiment.
Figure 3 - STR8A Load Cell [4]
From the figure above, you can use the STR8A load cell to control the load on the T-beam on
the roller support. By using strain gauges and digital strain displays connected to your PC, you
can read loads at nine different measuring gauge positions. From the measured values, the
relationship between bending moment, elongation and bending moment can be observed. Only
one individual can operate the test frame at a time to maintain accuracy of results obtained to
maintain the order or sequence of the experiment. The actual test frame used in the laboratory
is shown below and is identical to the figure above.
3
Figure 4: The laboratory test frame
List of Apparatus
1) Test Frame
2) Inverted Aluminium T-Beam, with strain gauges fixed on the section.
3) Loading Frame & STR8A Load Cell.
4) Digital Force Display Unit.
Figure 5: Digital force display unit
4
5) Digital Strain Display Unit.
Figure 6: Digital strain display unit
Experimental Procedure
1) Make sure that the beam and load cell are properly aligned. Turn the knurled wheel on
the load cell which results in an addition of pretension (positive downward) to a beam
of about 100N. Zero the load cell by utilizing the control unit provided.
2) Note down a total of nine zero strain readings by choosing the number with the selector
switch. Using the values collected, fill in Table 1 with zero force values.
3) The load is increased to 100N, and the values are taken down for a total of nine of strain
readings. The process is then repeated, for a range of 100N to 500N. Finally; gradually
release the load and preload.
4) The strain reading values are then corrected zero, once completed, convert the load to
a bending moment then fill in Table 2.
5) From your results, plot a graph of strain against bending moment for all nine strain
gauges (on the same graph).
Experiment Precautions
1) An excessive load must never be applied to any portion of the test equipment.
2) Inspect the test equipment, which includes the transmission (electrical) wires for any
damage or wear.
3) Check and ensure all connections of the test equipment are tight and secure.
4) Visually inspect all components to ensure it is fastened and secure to a certain extent.
5) The positioning of the test frame must be done safely. This is done by ensuring it is
placed on a solid surface which does not have any inclines, the frame must be steady
and must be easily accessible.
5
Data Sheet
Table 1 - Experiment Results (Uncorrected)
Gauge
Number
Load (N)
0
100
200
300
400
500
1
-15.0
-77.5
-145.4
-231.3
-320.6
-423.8
2
-9.4
-48.9
-89.6
-146.4
-210
-280.7
3
-7.7
-51.3
-98.9
-156.3
-219.1
-289.1
4
-5.7
-3.5
-1.2
-7.5
-15.8
-23.7
5
-0.4
1.9
4.0
5.6
7.2
9.1
6
-0.6
30.4
64.7
91.4
118.4
145.3
7
12.2
22.0
39.5
56.3
70.6
88.8
8
8.3
45.6
48.8
51.2
53.3
56.2
9
14.9
38
70.8
104.4
137.5
175.2
Table 2 - Experiment Results (Corrected)
Gauge
Number
Bending moment (Nm)
0
17.5
35
52.5
70
87.5
1
0
-62.5
-130.4
-216.3
-305.6
-408.8
2
0
-39.5
-80.2
-137.0
-200.6
-271.3
3
0
-43.6
-91.2
-148.6
-211.4
-281.4
4
0
2.2
4.5
-1.8
-10.1
-18
5
0
2.3
4.4
6.0
7.6
9.5
6
0
31.0
65.3
92.0
119.0
145.9
7
0
9.8
27.3
44.1
58.4
76.6
8
0
37.3
40.5
42.9
45.0
47.9
9
0
23.1
55.9
89.5
122.6
160.3
6
Calculation of the Model
Figure 7: Beam setup for the experiment
∑𝐹𝑌 = 0,
𝐴𝑦 + 𝐵𝑦 −
𝑊 𝑊
− =0
2
2
𝐴𝑦 = 𝑊 − 𝐵𝑦
∑𝑀𝐴 = 0,
𝑊
𝑊
− ( ) ( 0.35𝑚) − ( ) (0.485𝑚) + (0.835𝑚)𝐵𝑦 = 0
2
2
𝐵𝑦 =
(0.4175𝑚)𝑊
0.835𝑚
𝐵𝑦 = 0.5𝑊
𝐴𝑦 = 0.5𝑊
SECTION 1: 0 ≤ 𝑥 ≤ 0.35𝑚
Figure 8: Section 1 of the beam
7
∑𝐹𝑦 = 0,
𝑊
2
𝑉=
∑𝑀𝑁𝐴 = 0,
𝑀−
𝑊
(𝑥) = 0
2
𝑀=
𝑊𝑥
2
At x = 0,
𝑉=
𝑊
&𝑀 =0
2
At x = 0.35m,
𝑉=
𝑊
& 𝑀 = 0.175𝑊
2
SECTION 2: 0.35𝑚 ≤ 𝑥 ≤ 0.485𝑚
Figure 9: Section 2 of the beam
∑𝐹𝑦 = 0,
𝑉=0
∑𝑀𝑁𝐴 = 0,
8
𝑊
𝑊
𝑀 − ( ) (𝑥) + ( ) (𝑥 − 0.35) = 0
2
2
𝑀=
𝑊𝑥 𝑊𝑥
−
+ 0.175𝑊
2
2
𝑀 = 0.175𝑊
At x = 0.35m
𝑉 = 0 & 𝑀 = 0.175𝑊
At x = 0.485m
𝑉 = 0 & 𝑀 = 0.175𝑊
SECTION 3: 0.485𝑚 ≤ 𝑥 ≤ 0.835𝑚
Figure 10: Section three of the beam
∑𝐹𝑦 = 0,
𝑉=−
∑𝑀𝑁𝐴 = 0,
9
𝑊
2
𝑊
𝑊
𝑊
𝑀 − ( ) 𝑥 + ( ) (𝑥 − 0.35) + ( ) (𝑥 − 0.485)
2
2
2
𝑀=
𝑊𝑥 𝑊𝑥
𝑊𝑥
−
+ 0.175𝑊 −
+ 0.2425𝑊
2
2
2
𝑀=−
𝑊𝑥
+ 0.4175𝑊
2
At x = 0.485m
𝑉=−
𝑊
& 𝑀 = −0.2425𝑊 + 0.4175𝑊 = 0.175𝑊
2
At x = 0.835m
𝑉=−
𝑊
& 𝑀 = −0.4175𝑊 + 0.4175𝑊 = 0
2
Figure 11: SFD and BMD of the beam
10
Cross-Section Analysis of T-Beam:
Centroid:
𝑦=
𝑦=
∑𝑦𝑖 𝐴𝑖
∑𝐴𝑖
[(22.25 ∗ 202.88) + (3.2 ∗ 243.84)]
202.88 + 243.84
𝑦 = 11.851𝑚𝑚
Figure 12: Cross section of the T-beam
Table 3: Beam geometric properties calculated
Area
y
dy
31.7 ∗ 6.4
22.25𝑚𝑚
22.25 − 11.851
Segment
Ⅰ
= 202.88𝑚𝑚2
= 10.399𝑚𝑚
𝐼𝑥𝑥 =
(38.1 ∗ 6.4)
3.2𝑚𝑚
= 243.84𝑚𝑚2
3.2 − 11.851
= −8.651𝑚𝑚
A∗dy2
[(6.4 ∗ 31.73 )]
12
= 1.6989
ⅠⅠ
𝑏ℎ3
12
(202.88 ∗ 10.3992 )
= 21.94 ∗ 103 𝑚𝑚4
∗ 104 𝑚𝑚4
[(38.1 ∗ 6.43 )]
12
= 832.310𝑚𝑚
(243.84 ∗ −8.6512 )
= 18.248 ∗ 103 𝑚𝑚4
4
Moment of Inertia about Neutral Axis:
𝐼𝑥𝑥 = 𝐼𝑥 𝑠𝑒𝑔𝑚𝑒𝑛𝑡1 + 𝐼𝑥 𝑠𝑒𝑔𝑚𝑒𝑛𝑡2
= [(1.6989 ∗ 104 ) + (21.94 ∗ 103 )] + [(832.310) + (18.248 ∗ 103 )]
11
= 38929 + 19080.31
= 58009.31𝑚𝑚4 = 58.00 ∗ 10−9 𝑚4
𝐼𝑦𝑦 = 𝐼𝑦 𝑠𝑒𝑔𝑚𝑒𝑛𝑡1 + 𝐼𝑦 𝑠𝑒𝑔𝑚𝑒𝑛𝑡2
= 1.6989 ∗ 104 + 832.310
= 1.78216 ∗ 10−8 𝑚4
The y distance from neutral axis to strain gauges:
Figure 13: T-beam cross-section with Gauge positions
y1 = [38.1 – 11.85] mm = 26.25mm
y2,3 = [30.1 – 11.85] mm = 18.25mm
y4,5 = [15.1- 11.85] mm = 3.25mm
y6,7 = [11.85 – 6.4] mm = 5.45mm
y8,9 =11.85 mm
Flexural Bending Equation:
The stress due to bending at a cross-section at a distance y from the neutral axis is provided by,
Equation 2 - Bending Formula
𝜎=
𝑀𝑦
𝐼
12
Where:
𝜎 – Bending Stress
M – Bending Moment
I – Second Moment of Area about the neutral axis
Hooke’s Law:
Equation 3 - Hooke's Law
𝜎 = 𝐸𝜀
Therefore, the bending equation can be written as,
𝑀𝑦
𝐼
𝐸𝜀 =
And
𝜀=
𝑀𝑦
𝐼𝐸
Where:
𝜀 – Strain
E – Young’s Modulus (Aluminium = 69 GPa)
13
Results & Discussion
Table 4: Experiment bending moment results
Gauge
Bending Moment (Nm)
y- position
(m)
0
17.5
35
52.5
70
87.5
1
0
-1.640625
-3.423
-5.677875
-8.022
-10.731
0.02625
2
0
-0.720875
-1.46365
-2.50025
-3.66095
-4.951225
0.01825
3
0
-0.7957
-1.6644
-2.71195
-3.85805
-5.13555
0.01825
4
0
0.00715
0.014625
-0.00585
-0.032825
-0.0585
0.00325
5
0
0.007475
0.0143
0.0195
0.0247
0.030875
0.00325
6
0
0.16895
0.355885
0.5014
0.64855
0.795155
0.00545
7
0
0.05341
0.148785
0.240345
0.31828
0.41747
0.00545
8
0
0.442005
0.479925
0.508365
0.53325
0.567615
0.01185
9
0
0.273735
0.662415
1.060575
1.45281
1.899555
0.01185
Table 5 - Strain in all gauges with the use of various bending moments
Gauge
Bending Moment (N-m)
0
17.5
35
52.5
70
87.5
Strain
1
0
-1.07612 x10-5
-2.24522 x10-5
-3.72424 x10-5
-5.26181 x10-5
-7.0387 x10-5
2
0
-3.28735 x10-6
-6.67457 x10-6
-1.14017 x10-5
-1.66947 x10-5
-2.2579 x10-5
3
0
-3.62857 x10-6
-7.59003 x10-6
-1.23671 x10-5
-1.75936 x10-5
-2.3419 x10-5
4
0
5.80647 x10-9
1.18769 x10-8
-4.75075 x10-9
-2.6657 x10-8
-4.7507 x10-8
5
0
6.0704 x10-9
1.16129 x10-8
1.58358 x10-8
2.00587 x10-8
2.50734 x10-8
6
0
2.30079 x10-7
4.84651 x10-7
6.82816 x10-7
8.83208 x10-7
1.08286 x10-6
7
0
7.27348 x10-8
2.02618 x10-7
3.27306 x10-7
4.3344 x10-7
5.68519x10-7
8
0
1.30879 x10-6
1.42107 x10-6
1.50528 x10-6
1.57896 x10-6
1.68072 x10-6
9
0
8.10535x10-7
1.96142 x10-6
3.14038 x10-6
4.3018 x10-6
5.62462 x10-6
14
Table 6 - Average Data
Gauge
y position (mm) Actual y position (mm)
Bending Moment
from neutral axis.
0
17.5
35
52.5
70
87.5
1
0
-1.07612x10-5
-2.24522 x10-5
Strain
-3.72424 x10-5
-5.26181 x10-5
-7.0387 x10-5
0
26.25
2,3
0
-3.45796 x10-6
-7.1323 x10-6
-1.18844 x10-5
-1.71441 x10-5
-2.2999 x10-5
8
18.25
4,5
0
5.93844 x10-9
1.17449 x10-8
5.54254 x10-9
-3.29913 x10-9
-1.1217 x10-9
23
3.25
6,7
0
1.51407 x10-7
3.43635 x10-7
5.05061 x10-7
6.58324 x10-7 8.25688 x10-7
31.7
5.45
8,9
0
1.05966 x10-6
1.69125 x10-6
2.32283 x10-6
2.94038 x10-6 3.65267 x10-6
38.1
11.85
From the results obtained, a graph of Strain vs Bending moment was plotted and the graph is
shown in the figure below.
Figure 14: The graph of strain vs bending moment
15
1. What is the relationship between bending moment and strain at various positions?
Figure 15 - Bending Moment vs Strain
As the moment increases at different positions, the value of the strain also increases
(proportional relationship).
2. What do you notice about the strain gauge reading on the opposite sides of the section?
Should they be identical?
According to the theory taught, the opposite sides should be identical. However, experimentally
demonstrated by tables 1 and 2, it is evident that the opposite sides are not identical.
3. If the readings are not identical give two reasons why?
a. In the theory, we assume that the load distribution is uniform. However, because the
opposing sides of the experiment have different measurements, the load distribution is
not as uniform as it should be.
b. The instrument does not provide precise and accurate readings. For example, when the
thumbwheel reads 100N, the computer reads either little more or slightly less than
100N, which has a detrimental impact on the measurements and computations.
16
4. What is the value of the strain reading at neutral axis?
Strain is zero.
5. Calculate the maximum stress in the section by turning the strains into stress values
(at maximum load). Compare this to theoretical value.
Experimentally:
𝜎 = 𝐸𝜀
= 69 ∗ 109 ∗ −7.0387 𝑥10−5
= − 4.857 x106 Pa
Theoretically:
σ=
=
𝑀𝑦
𝐼
(−10.731)( 0.02625)
(58 ∗ 10−9 )
= − 4.857 𝑥106 𝑃𝑎
Experiment value and theoretical value are the same.
6. Does the bending equation accurately predict the stress in the beam?
Yes, the bending equation predicts accurately the stress in the beam.
17
Conclusion
The beam is the most common structural element subject to bending moments, as it can bend
at any point along its length when loaded. Bending failure can occur when the tensile stress
applied by a force is equal to or greater than the element's ultimate strength (or yield stress).
This is an important concept in structural engineering because it can be used to calculate where
and how much bending will occur when forces are applied. The study of beams has enabled
engineers to carefully construct support beams for various huge projects where safety is a very
crucial aspect of a successful project. This experiment was conducted successfully and both
the students and the laboratory technician were happy with the results. It is important to
understand how the different types of beams react to applied forces and use that knowledge to
improve the technology and infrastructure for the future. The bending moment is directly
proportional to the strain. As the bending moment increases, the strain increases as well. This
was proven in the experiment by studying the tabulated results and the relationship between
strain and bending moment in the plotted graph.
Recommendations
To improve the reliability of the results obtained in the laboratory, the experiment should be
done more than one time to compare the results and perhaps take the average of the results
obtained from at least two repeated experiments. Also, a simulation software can be introduced
whereby the beam structure and all forces applied can be drawn and simulated to get another
set of results to compare to. Simulations have been trusted to give accurate results when all
inputs are inputted correctly. This is because there is very small chance of error when a
simulation is running as compared to physical experiments. Human error can occur and the
simplest mistake can cause a significant change in the results obtained and therefore making
them less accurate and unreliable.
18
References
[1] Admin,
“What
Is
A
Beam,”
SkyCiv,
[Online].
https://skyciv.com/docs/tutorials/beam-tutorials/what-is-a-beam/.
Available:
[Accessed
29
November 2021].
[2] Admin, “Bending Stress,” Corrosionpedia, 04 November 2018. [Online]. Available:
https://www.corrosionpedia.com/definition/6270/bendingstress#:~:text=Bending%20stress%20is%20the%20normal,subjected%20to%20a%20ten
sile%20load.. [Accessed 04 December 2021].
[3] J. Afsar, “Types Of Bending Stresses,” ENGINEERING INTRO, 05 August 2015.
[Online]. Available: https://www.engineeringintro.com/mechanics-of-structures/supportreactions/types-of-bending-stresses/. [Accessed 04 December 2021].
[4] TecQuipment, “Bending Stress In A Beam,” Durban University of Technology, Durban,
2008.
19