Beams are structural elements designed to bear structural load (dead load) and other external loads (live loads). Bending moment of a beam denotes a reaction that a structural beam produces when an external force is applied on the beam causing its bending. In civil engineering the bending moment is referred to as the torque that keeps the beam intact and at equilibrium to the external load. Analysing beams is known to be complicated than the torsion and axially loaded members. Bending usually involves two types of loading. The applied loading usually generate internal shear forces and bending moments. When analysing beams shear force, shear force diagrams, location of centroids of cross section and calculations on second moment of area are all important when analysing beams. One most crucial function of beams is the transfer of the applied external to the supports or pillars. It’s for this reason that the concept of bending moments in beams are fundamentals that should be clearly under stood in structural engineering since it determines the final design structure. For instance from beam to columns the bending moments and the shear force decides the overall structural size and the strength required for a structural member. In design it’s important to understand the forces due to shearing in the beam because shearing forces acts maximum at the supports. Considering complete load envelope, all members will have maximum demand to capacity ratio in the case where bending shear occurs. For this reason axial forces are not important as bending moments are the most considered in structural design. Failure in beams is another aspect in analysis of bending moment in the beam. Failure in most instances is because of bending when the tensile stress exerted by outside forces is matches or exceed the ultimate strength of the beam element. However, a beam could fail due to shearing forces before failing in bending. Beams are usually subjected to bending forces as illustrated below.
Figure above shows the shear force diagram and a resultant bending moment diagram.
The main focus of the experiments is the determination of how the bending moment change at the point C of loading when different variable load are applied at the cut.
Figure shows force diagram
Theoretical determination of the bending moment at the cut, Mc is given as
Mc = Wa ((L-a)/L)
Where; W= Load (N)
a =distance of the load (m)
L = length of the beam
And the second experiment aims at the determination of how the bending moment changes in relation the positions of the loads.
This test were carried under a simply supported beam apparatus which is designed to apply and provide calculations of force across the beam.
Apparatus used
1. Bending moment apparatus
2. Digital force display
3. Loaders.
Procedure
1st
part
I. The digital force display reading was inspected to confirmed it to be zero
II. 100g mass was placed on a hanger and applied to the left side of the cut and the value of the force on the digital force display was recorded.
III. The procedure was repeated using increasing mass of 200g, 300g, 400g and 500g.
Note; the masses were converted into load in Newton (N). Bending Moment at C (N) = Recorded force * 0.125
The moment arm was 125mm in length.
2nd
experiment;
I. The digital force display was checked if its reading indicated zero when no load was applied.
II. The beam was the loaded carefully with hangers at different positions and the loads were applied as shown in fig 1.0, 2.0 and 3.0 respectively.
III. The reading were then recorded as in table 2 below.
IV. The force reading were then converted in bending moments
V. Reaction forces at a and b ends were calculated and the theoretical bending moment at the cut C determined.
Results
Table 1
Mass (g)
Load (N)
Force (N)
Experimental bending moment (Nm)
Theoretical bending moment
(Nm)
0
0
0
0
0
100
0.981
0.9
0.1125
0.09
200
1.981
1.6
0.2
0.19
300
1.962
2.4
0.3
0.31
400
2.924
3.3
0.4125
0.37
500
4.905
4.0
0.5
0.47
Table 2
No
W1(N)
W2(N)
Ra
(N)
Rb
(N)
Force
(N)
Experimental bending moment
(Nm)
Theoretical bending moment (Nm)
1
3.92
-
5.17
-1.25
-1.6
0.2
-0.17
2
1.96
3.92
2.58
3.3
3.9
0.4875
0.4662
3
4.91
3.92
2.59
-6.24
3.9
0.4875
-0.4816
Data analysis
Experiment 1: Calculating the theoretical Bending moment
Moment at the cut,
a = 0.3
L = 0.44
0.981 * 0.3 ((0.44-0.3)/0.44) = 0.09
1.962 * 0.3 ((0.44-0.3)/0.44) =0.19
2,924 * 0.3 ((0.44-0.3)/0.44) = 0.31
3.924 * 0.3 ((0.44-0.3)/0.44) =0.37
4.905 * 0.3 ((0.44-0.3)/0.44) =0.47
Experiment 2.
Calculations of reactions Ra and Rb
Fig: 1.0
∑Ma = 0
W1 =3.92
-3.92(0.14)- Rb(0.44) = 0
Rb = -1.25
∑Fy = 0
Ra + Rb = 3.92
Ra = 5.17
Fig; 2.0
∑Ma = 0
1.96 *0.22 – 3.92 x 0.26 + Rb * 0.44 = 0
Rb = (1.96 * 0.22 + 3.92 *0.26)/ 0.44 =3.3N
Ra – 1.96-3.92 -3.3 = 0
Ra = 2.58
Fig; 3.0
W1 = 4.91
W2 = 3.92
∑Ma = 0
(-Rb * 0.44) – (0.240 * 4.91) + (0.4 * 3.92) = 0
Rb = -6.24
∑Fy = 0
-6.42 + 4.91 + 3.92 = Ra
Ra = 2.59
Discussion
Derivation of the equation
Graph showing relationship between the loading and bending moment.
The graph is linear in that, when the load was increased, the bending moment also increases. From the formula bending moment = applied load * distance. Hence, an increase in load would as well increase the bending moment. The ratio at which the bending moment value increase or decrease is equivalent to the ratio at which the load that is applied increases or decreases. The graph shows a linear relationship between the bending moment and the applied load. Experimental and theoretical bending moments also show a perfect linear relationship with the applied load with little difference in their values of bending moment. The experimental values and the theoretical values in the graph also show a close relationship since their values are not far apart suggesting a minimal error percentage.
Percentage Error = (experimental shear force –theoretical shear force)/ (theoretical shear force)
PE = (0.5-0.486)/0.486 * 100
= 2.88%
The percentage of error or the errors in the experiment are likely due to instrumental errors, human errors, errors in calculation, environmental conditions such as temperature and air pressure and observation errors. This factors in one way or the other may affect values that are obtained during the experiment. In this experiment deviation between the experimental and the theoretical values is minimum, this proves that the apparatus is accurate and the skill of the observer on being keen and apt during the experiment.
From the calculations in this experiment the “bending moment at the cut is equal to the algebraic moment caused by the forces acting on the left or right of the cut.”(Newton’s law of moments) This prove the theory that for a body in equilibrium the resultant forces must be zero. This also applies to the resulting moment of the forces on the body about all points of the beam must be zero. When the beam is simply supported at each end all the downward forces are balanced by opposite upward forces. In other words the sum of clockwise moments about a point on the beam would be equal to the sum of anticlockwise forces about the same point for an element/ beam in equilibrium.
Experiment 2
On the 2nd
experiment investigating the bending moment changes for various loads of different conditions, in comparison of the results, the obtained bending moments experimentally and theoretically, for the experiment fig 1.0, 0.2 Nm, fig 2.0, 0.4875 and fig 3.0, 0.4875. Calculated theoretical values were fig 1.0, 0.17, fig 2.0, 0.4662 and fig 3.0, -0.4816.
On the calculations on percentage difference/errors
For fig 1.0:
= (0.17 – 0.2)/0.2) * 100=15%
= (0.4662-0.4875)/0.4875) * 100 =4.4%
= (0.4816-0.4875)/0.4875) * 100 =1.2%
This section of the experiment also prove that the moment at the cut is equal to the algebraic sum of the moment of force acting on the left or right of the cut. The bending moment can be calculated based on the obtained distance. This means that the distance at which a specific load acts affects the overall bending moment.
According to the bending moments obtained from the above bending moment diagrams in figure 1.0, 2.0, and 3.0. It was concluded that when load of the same magnitude is applied at different distances on the beam the bending moment will be likely to change value. Distance of application of force therefore affect the bending moment.
Conclusion
The test carried out to determine the equilibrium at the cut with it balancing the forces to its left and to its right. This has been proved in the two experiments in determining the effects of different loading on a beam at a point and the effect of changing distances of loading. In the first experiment proves through a linear graph the relationship between the bending moment and the loading to be directly proportionate and as load value increases the value of the bending moment also increases. In the second experiment also proved that change in loading distance also affects the bending moment. In determining the percentage errors were found to be minimal emphasising on the importance of eliminating obvious sources of errors. For instance human errors associated with lack of keenness and careless observation and incompetence in the laboratory should be avoided at all cost. Instrumental errors on the other hand may be inevitable but proper maintenance and handling with caution may guarantee an efficient instrument free of errors. Safety and cautionary measures in the laboratory should be also observed to prevent accidents or disturbances to other while on practical. The experiments main scope was achieved in proving the relationships of bending moment to its importance in structural design by deducing beam size, load intensity, and internal forces. Since beams are mostly used in transmission of loads to the supports, through the experiment one is able to determine points on the beams on which maximum forces act given the external forces and reactions.
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