In engineering and other fields related to it, bending moment is defined as a measure of the reaction produced by an element such as a beam when an external force or moment is applied to a structural element, thereby causing the element to bend (Gere and Timoshenko, 1996). Simply put, it is the induced reaction in a structural element after application of an outside force. A beam is a structural element that is designed and used to bear high loads of a structure and other loads (Atta, 2018). For a section through an element, bending is defined as the algebraic sum of all moments of all external forces acting about the section. In engineering today, the beam is the simplest and most common structural element to be subjected to bending moments. Consider the application of an external load to the beam, should the beam be displaced from its recent axis as per newton’s first law, it would have undergone the process that is bending. Alternatively, the bending moment of a beam is the product of the force applied on it and the distance between the point of application of the force and the fixed end of the beam (Atta, 2018). The concept of bending moment is very important in various engineering fields, most especially mechanical and civil engineering.
For equilibrium to be maintained on the section of the beam, the algebraic sum of clockwise bending moments (which result in sagging if taken negative) must be equal to the anticlockwise moments (which result in hogging if taken as positive). In simpler terms, the clockwise moments must equal the anticlockwise moments for equilibrium to occur and be maintained. Taking into consideration the cross-section of the structural elements, its internal reaction loads can be resolved into a resultant force and a resultant couple. Shear force is the resultant internal force while the bending moment is the resultant internal couple. The bending moment of a beam is usually influenced by three factors, tensile stresses, compressive stresses and cross-sectional shape of the beam. This bending moment is usually called a Plastic Hinge.
If, for a beam, the applied loading is increased beyond a critical value, a crack forms across the beam cross-section (Levy, 1983). A failure in bending for an element thereby occurs when the bending moment is enough to set off tensile stresses larger than yield stresses of the element. For a beam, if not well restrained, a bending force may result into
rotation about a pivot-point.
Figure 1: Sketch showing the free body diagram (FBD) of a beam under bending moment
Bending moment is the product of all the external forces acting on a component such as a beam and the corresponding distances orthogonal to the direction of the force. The diagram above (Figure 1) shows the force diagram for a beam subjected to bending moment. Theoretically, bending moment, Mc is given by the following formula:
Where: W=Load (N)
a=distance to the load (m)
l=length of the beam (m)
The aim of this experiment was to determine the effect of forces applied by varying loads on the bending moment of a beam. Simply put, the experiment’s aim was to offer an understanding on the effects of the magnitude of the force on the bending moment of the beam.
Objectives
The experiment had many learning outcomes which included:
i. To understand the difference between a variable action and a permanent action.
ii. To understand the basic theory and visualize the bending moment in structures.
iii. To understand the way that structures support loads.
iv. To understand the relationship between bending moment and a deflected shape.
Procedure
During the entire experiment, caution was taken not to apply excessive loads to any part of the equipment.
Test One
i. The digital force display was checked to ensure it had been turned on and was reading zero without any load.
ii. A load hanger was placed at the position of the cut and a one-hundred-gram mass was applied at the point.
iii. The digital force display reading was recorded in a table of results, shown below, that had been drawn to collect the data.
Mass (g)
Load (N)
Bending Moment (Nm)
Experimental
Theoretical
0
0
0
0
Table 1: Table of Results for Collection of Test One Data
iv. Procedure ii and iii were repeated while varying the mass in increments of a hundred grams per test.
v. The procedure was stopped after testing masses up to five hundred grams.
vi. The moment arm length for the experiment was measured and found to be 125 mm.
The bending moment at the cut (Nm)=digital force displayed x 0.0125
Test Two
i. The digital force display was checked to ensure that it was on and was reading zero in the absence of the load.
ii. The beam was loaded with hangers positioned as shown in the figure above (Figure 1), the equivalent mass was then applied in order to give the stated loads.
iii. The digital force reading was recorded in the table of results below.
iv. The procedure was repeated for the 3 loading arrangements.
Loading condition
W1
(N)
W2
(N)
RA
(N)
RB
(N)
Bending Moment (Nm)
Experimental
Theoretical
Figure 1.0
Figure 2.0
Figure 3.0
Table 2: Table of Results for Collection of Test Two Data
Results
In the first part of the experiment, Test One, the following data was collected after increasing the mass at the point of loading from zero to five hundred in 100-gram increments. The table below shows the results obtained from part one of the experiment.
Mass (g)
Load (N)
Bending Moment (Nm)
Experimental
Reading and the Reading + 0.125
Theoretical
0
0
0
0
0
100
0.981
0.9
0.1125
0.0936
200
1.962
1.7
0.2125
0.1873
300
2.943
2.4
0.3000
0.2509
400
3.924
3.3
0.4125
0.3746
500
4.905
4.0
0.5000
0.4682
Table 3: Table with the actual test one data
After careful and precise work, the second test, test two, produced the following results. The table below shows the results that were obtained from the second part of the experiment.
Loading Condition
W1 (N)
W2 (N)
RA (N)
RB (N)
Bending Moment (Nm)
Experimental and Experimental*0.125
Theoretical
Figure 1.0
3.92
/
5.17
-1.25
-1.7
-0.2125
-0.1750
Figure 2.0
1.96
3.92
2.58
3.29
4.0
0.5
0.4606
Figure 3.0
4.91
3.92
2.59
6.24
4.0
0.5
0.4816
Table 4: Table with the actual test two data
Discussion of Results
All the values of the bending stresses that were obtained from the experiment were recorded in their suitable tables. The tables were arranged basing on the cell that produced the specific bending moment. Upon completion of data collection in the respective table of results, an analysis of the results was conducted in an effort of understanding what the experiment was trying to tell us. The bending moment was calculated using the equation below:
Where: W=Load (N)
a=distance to the load (m)
l=length of the beam (m)
In theory, the following statement determines how bending moment for as beam may be determined; the bending moment at the cut is equal to the algebraic sum of the moments caused by the forces at the right or left of the cut.
A graph showing the relationship between the variable load and bending moment for both experimental and theoretical moments of bending for Test One were plotted using excel and the graph shown below was the result.
From the graph, it can be concluded that, a constant varying load increment during the experiment resulted in the uniform change in bending moment, this can be seen by the type of line that the graph becomes. The graph may be interpreted to mean that an increase in the applied load will result in an increase in the bending moment of the structural element under investigation. The reverse is also true, which means that a decrease in the applied load will result in a decrease in the corresponding bending moment.
When constructing the variable load diagram, it is paramount to obtain reactions at the ends of the beam (RA
and RB). The vertical components of forces are then added from the left end of the beam in an effort to preserve the mathematical conventional signs used. We recall that the algebraic sum of forces in the y direction is zero since the beam is static and has zero rotation. Also, the algebraic sum of moments about any chosen point is zero.
Usually, any error is calculated from subtracting the actual value test value from the theoretical value. In our case, the experimental error was calculated by obtaining the difference between the experimental value and the theoretical value of the bending moments. This difference was then expressed as a percentage of the experimental value of bending moment. A sample calculation is done from test two.
For example, from procedure 2:
Percentage Error (PE) = (Experimental value - Theoretical value) / (Experimental value) × 100%
Percentage Error = (0.5-0.4606) / (0.5) ×100%=7.88%
Comparing this with the experimental findings of those of prediction of bending moment as in test one, considering that a load of three hundred grams is applied,
PE= (0.300-0.2509) / (0.300)×100%=16.37%
An error of less than ten percent, as is the case for this experiment, falls within the allowable error limit and therefore the experiment may be considered a success. For the theoretical bending moment graph, the error on the graph could have resulted from the measurement drift of the digital force display.
The advantage of plotting the graph of bending moment against variable load is that it helps us to determine the maximum absolute value of variable load as well as the bending moment.
Further research into the topic discovered that for a beam whose end is free or pinned, the algebraic sum of moments is equal to zero since it is in equilibrium. If this end is in equilibrium (as in the case of a cantilever), the resultant moment is equal to the moment calculated for the reaction.
Conclusion
In theory, bending of a beam results when an external load applied on the beam or other component is large enough to displace the latter from its current axis. Bending moment, on the other hand is the resultant of multiplying the force applied on the beam and the distance between the point of application of the force and the fixed end of the beam. This experiment was aimed at studying the effect of varying forces on the bending moment for the beam, and from the experiment, it is noted that there is a linear relationship between bending moment and applied variable loads. This means that the bending moment is directly proportional to the applied variable loads.
In experiments, errors are likely to be obtained as a result of various factors. Some of these errors are either systematic, random or personal errors. Systematic errors result from, reading measurements in parallax or measurement drift for electronic measurements as the instruments warm up. Random errors result from variations in the experimental conditions. Taking additional data may reduce the effect of random errors as opposed to systematic data where additional data does not reduce the error. Personal errors, not usually categorized under common errors may be a result of the student being skeptical before even attempting the experiment. The experiment achieved its objectives and thus successful.
References
Anne H. (2018). Sources of Error in Science Experiments. Retrieved from; https://googleweblight.com/i?u=https://sciencenotes.org/error-in-science/"hl=en-KE
Beer F, Johnston E. R (1984). Vector Mechanics for Engineers: Statics, McGraw Hill. Pgs. 62-76.
C. Levy (1982): Effect of Bending moment on the dynamic fracture of a beam or plate under tensile loading. Engineering Fracture Mechanisms, Vol 18, Issue 1(1983). Pgs. 39-43.
Gere, J.M.; Timoshenko, S.P. (1996). Mechanisms of Materials: 4th Edition Nelson Engineering, ISBN 053 4934293.
Green Mechanic: Bending Moment in a Beam Lab Report. Retrieved from; https://www.green-mechanic.com/2017/01/bending-moment-in-beam--lab-report -pdf.html?m=1
International Journal of Fracture Mechanisms, Vol 5, Issue 4(1969). Pgs. 269-286.
Procedure for drawing shear force and bending moment diagram: Lecture 23 and 24. Retrieved from; https://nptel.ac.in/courses/Webcourse-contents/IIT- ROORKEE/strength%20of%20materials/lects%20"%20picts/image/lec23%20and%2 024/lecture%2023%20and%2024.htm
Shear Force and Bending Moment Diagrams. Retrieved from; https://en.m.wikiversity.org/wiki/Shear_Force_and_Bending_Moment_Diagrams.
Shear force and Bending Moment: An Introduction to Shear force and bending moments in beams. Available at: https://www.codecogs.com/library/engineering/materials/shear- force-and-bending-moment.php
