This study describes the use of the double integration approach to determine beam deflection. This technique involves the use of integration tools to create an equation that may be used to calculate beam deflection for beams with various cross-sectional areas. When using this procedure, keep the modulus of elasticity of the beam material and the cross-section of the beam in test in mind. The cross-sectional area of the beam is important since it allows an individual to precisely compute the second moment of area that will be used for estimating the deflection utilizing the deflection method. The technique of double integration one of the most utilised method for analysing beam deflection due to its simplicity. Thus, whether in an industrial setting or in the field or in an academic institution, one is able to utilise it in determining the deflection of a beam provided that the modulus of elasticity is known. Other than determining deflection, this technique can also be exercised when resolving the bending stress and the maximum bending stress of a beam.
This experimentation, therefore, explores the application of the double integration technique in determining deflection in a 3-point bending, 4-point bending and in a cantilever when subjected to a bending load. At the culmination of this report, it shall be clear that there is a difference in the level of deflection for 3-point and 4-point bending tests with the same quantity of load.
Introduction
The objective of this experimentation is to examine and contrast the 3-point and 4-point bend tests. In addition, the experimentation shall be suitable in determining the behaviour of a cantilever that is subjected to incremental point load. In both cases, there shall be a comparison of the theoretical and the tentative results. This shall involve the determination of uncertainty and error.
Background information
Beams have a variety of applications in the design and fabrication of machines and various structures. These beams, at the point of application, are customarily subjected to different kinds of loads. These loads usually cause the beam to deflect. This deflection of the beam is a sign that the beam is experiencing a stress and undergoing deformation. The stresses experienced by a beam comprise of the shear, bending and the normal stresses . Normal stress develops when a beam, a structural member, is loaded axially. A typical structural member that experiences normal stress is the column.
Bending stress alternatively develops when a beam is loaded perpendicular to its length as shown in figure 1, next page. The beam can be fixed either at both ends or at one end and be loaded at any point along its length. There are various loading techniques that can set a beam to undertake the bending stress. These techniques encompass uniformly distributed loading, point loading, and unevenly distributed loading . In all these loading conditions, it is anticipated that the beam deflection will be maximum at the point that experiences the utmost loading force. In addition, the beam that is in bending stress experiences tension and compression on its fibres when loaded.
Shear stress, unlike the bending and normal stresses, transpires when the load is subjected parallel to an area on the beam. Therefore, the shear stress varies throughout the cross-sectional area.
These stresses are essential when performing calculations for the design of different sorts of beams. These stresses determine the material to be utilised in designing beams for diverse applications and loadings. It is these stresses that are used in determining the safety factor of a given beam. Thus, they are very vital in ensuring that designers design structures that are safe. In addition, they use the stress analysis of these stresses in setting the secure load limit for a particular design of beams. Thus, bending stress and beam deflections is a vital and valuable topic in determining and testing of the strengths of materials in engineering materials.
Figure 1 below represents the beam that undergoes a bending stress when loaded.
Reaction A
Reaction B
Figure 1: Beam diagram for a point load
Point load P
Tension in bottom fibres
Compression in top fibres
Unloaded Beam
Bending beam
Bending Moment, Shear Force and Bending Stress in beams
Bending moments and shear along a beam are usually conveyed using diagrams. The shear diagram indicates the shear throughout the length of the beam, whilst the moment diagram indicates the bending moment through the length of that exact beam. Often, they are customarily portrayed arranged in a stacked manner with one on top of the other. When combined, these two diagrams result to the shear-moment diagram.
The shear and the bending moment at any point lateral to the beam are respectively given by
V= , and
M=
Where M is the bending moment on a beam at a specific distance, x, on the beam, and V is the shear force.
The bending stress at any location on the cross-section of the beam is calculated using the bending moment, M, at that specific location. Based on the flexure formula, the bending moment is found to vary over the height of the beam’s cross-section. The flexure formula is represented as follows;
σb = - ,
where σb is the bending stress, M is the bending moment at the position of interest along the length of the beam, Ic is the second moment of area, and y is the deflection of the beams from the neutral axis to the location of interest lengthwise the height of the beam’s cross-section.
Thus, the bending stress can be determined when the deflection is known and the bending moment and the second moment of area have been calculated. Besides, in order to find the maximum bending stress, the second moment of area is replaced with the sectional modulus. The section modulus combines the second moment of area, Ic, and the centroidal distance.
Experimental method
3-point and 4-point bending
The rigs were configured to ascertain a fair comparison of the tests. The distance amid the supports, L0 was set to 500mm. The load hangers and the digimatic indicators were aptly positioned in the middle for 3-point test whereas for the 4-point they were positioned 200mm from both ends respectively. The theoretical elastic modulus of the material was recorded. The height and width of the cantilever were measured using the Vernier calliper and recorded accordingly. These were later utilized in the calculation of the second moment of area and were recorded in table 2.
The dial gauge was zeroed and the frame of the rig was tapped lightly to eliminate stiction before loading. The rig was loaded using a mass hanger, a knife edge, and washers. At every incremental load, the frame was lightly tapped and the procedure replicated for three trials. The values obtained were utilised in completing table 1 for both 4-point and the 3-point bending.
Cantilever
The cantilever was set as illustrated in the laboratory manual. The knife edge and the dial gauge were set at the 200mm length. The height and width of the cantilever were measured using the Vernier calliper and recorded accordingly. These were later utilized in the calculation of the second moment of area and were recorded in table 3. The frame of the rig was lightly tapped and the dial gauge zeroed. The rig was loaded with a mass hanger, knife edge, and washers. The frame was lightly tapped at each incremental load and the process repeated for three trials. The results obtained were used in completing table 4.
Results
The results comprise of the records made for the two experimentations and are in four tables.
3-point and 4-point bending
Table 1 indicates the results of the measured deflections against their respective loads.
Table 2 displays the elastic modulus, width, height, and the second moment of area.
Figure 1 gives the graphical comparison of the average deflection (mm) against the load
Cantilever
Table 3 represents the trend of deflection (in mm) of the cantilever at various loads
Table 4 shows the elastic modulus, width, height, the second moment of area of the cantilever, and the analysis of the uncertainty and the error
Figure 2 expresses the trend of deflection when loading a cantilever with incremental loads.
Discussion
Theoretically, it is expected that the 3-point bend will have a higher deflection than the 4-point bend even for the same load. This is because the 4-point bend has a relatively a larger second moment of area as compared to the 3-point bend. Thus, by applying the deflection formula, the 4-point bend has to have a smaller deflection. Therefore, for the 4-point and 3-point bend experiment, the results obtained are coherent to the anticipated theoretical results. In fact, figure 1 gives a better representation of this expectation. The shear force and bending moment are the same for both the 3 and 4-point bending test. Though they have dissimilarly shaped shear-moment diagrams. In addition, these two could also have a Mohr diagram that is similar.
The second moment of area for the 3-point, 4-point and cantilever bending was calculated using the formula for calculating the second moment of area of a beam with rectangular cross-section.
I =
Where b and h are the width and the height of the cross-section of the beam.
The trend of deflection presented in figure 2 indicates that a cantilever behaves like an elastic spring that is within the elastic region. The cantilever experiment signifies that the material it is made of exhibits elastic properties. This is true because mild steel is indeed a metal alloy that has elastic properties and is even used in making vehicle suspension, especially the coiled spring varieties.
Ultimately, the results in table 4 indicate that the experimental deflection is beyond the acceptable range of expected theoretical deflection allowance. The error is at 31%. The source of this error could be as a result of positioning the knife-edge slightly beyond the 200mm position. In addition, the error could have resulted due to the sliding of the hanger when the cantilever deflects, thus causing the cantilever to deflect even further and cause the error. The error was determined after taking into account the extent of uncertainty. The uncertainty was obtained using the expression below
Δσ = |
Conclusion
In conclusion, the double integration technique used in this experimentation to determine the deflection of a beam is a quick, simple and acceptably accurate method. The other techniques that are suitable for establishing the beam deflection include the area-moment system, the tactic of superposition, the conjugate beam technique and the strain-energy system, alternatively recognised as the Castigliano’s method.
Bibliography
[1] Bertram, Albrecht. 2016. Solid Mechanics. [S.L.]: Springer International Pu. 223-241
[2] Holman, J.P. 2012. Experimental Methods for Engineers. New York: McGraw-Hill. 19-22
[3] Hunt, John F., Houjiang Zhang, and Yan Huang. 2015. "Analysis of Cantilever-Beam Bending Stress Relaxation Properties of Thin Wood Composites". Bioresources 10 (2): 15-18. doi:10.15376/biores.10.2.3131-3145.
[4] Philpot, Timothy A. 2014. Mechanics of Materials. Singapore: Wiley.
[5] Ratcliffe, Colin P. 2016. Doubt-Free Uncertainty in Measurement. [Place of publication not identified]: Springer International Pu.
Appendix
Cantilever computations
Computation of the second moment of area for the cantilever
I= == 57.4 mm4
Computation of the theoretical deflection
σ = = = 1.1mm
Computation of uncertainty
Δσ = |
Δσ = |
Δσ = 0.39mm
Computation of % error.
True value = theoretical deflection + uncertainty
σ = 1.1mm + 0.39mm = 1.49mm
Error = = = 31%
Bending Moment and Shear Force
3-point bending test
RA+RB = P, considering forces acting along y. The forces acting along x are zero.
Taking moments about A,
Σ MA=P×0.5L – RB ×L= 0
Σ MA=4.905×0.5(0.5) – RB×0.5 = 0
0.5RB = 1.22625
RB = 2.4525N
RA = P – RB = 4.905 – 2.4525 = 2.4525N
Thus, Bending Moment = RA×0.5L = 2.4525×0.5(0.5) = 0.613125Nm
P = 4.905N
SF
BMD
0.613125Nm
Figure 3: 3-point bending shear- moment diagram
A
B
0
2.4525N
-2.4525N
0
0
0
4-point bending test
RC+RD = 0.5P+0.5P = P, considering forces acting along y. The forces acting along x are zero.
Taking moments about C,
Σ MC=0.5(P×a+P(a+0.1))- RD × L = 0,
where a = 200mm = 0.2m, and L = 0.5m
Σ MC= 0.5(4.905 × 0.2 + 4.905(0.2+0.1)) – RD × 0.5 = 0
0.5RB = 1.22625
RD = 2.4525N
RC = P – RD = 4.905 – 2.4525 = 2.4525N
Thus, Bending Moment = RC×0.5L = 2.4525×0.5(0.5) = 0.613125Nm
1/2P
1/2P
0
0
0
SF
BMD
0.613125Nm
2.4525N
-2.4525N
Figure 4: 4-Point moment -shear diagram
