In determining the reactions of a statically indeterminate beam, several methods can be used. This project aims at assessing the reaction of beams at the supports when loads are subjected. To achieve this, the project uses a support frame, reaction support piers, beam specimen, a meter ruler, and a set of weights. Varying weights are placed at the center of the beam specimen, and the resulting reactions read from the indicators. From doing this, it is seen that the reactions at the supports of the beam specimen are equal. The support at the center of the beam offers the highest load support. As such, the beam can take up more load at the middle than at the end supports. Conclusively, the project recommends that the indicator readings should always be zero before placing a load and that all screws are tight enough to avoid the load hanger from moving.
Introduction
When placed under stress, beams react in numerous ways. For instance, when loaded, bending of beams is a frequent phenomenon experience in practice. When a slender member is subjected to a transverse load, the member is termed as a beam under bending. At any cross-section of the beam, the transverse loading generates bending m0mnets and shear to maintain an equili9brium. This bending causes the beam to change its curvature, and at the same time induces compressive and tensile stresses in the cross-section of the beam. The maximum stresses are achieved in layers that are furthest from the neutral axis, at the point where there is no strain (Banthia, Mindess, and Bentur, 1987)
This experiment focused on the analysis of stresses in beam bending and was conducted by loading different loads to beam. In doing this, this project aims at showing the relationship between the load on a beam and the reactions at its supports.
Furthermore, the project also aims at showing the moments theory may help in predicting the reaction forces due to the positioning of a load at different points along a beam. To achieve these objectives, the experiments are done stages. The first part of the experimental procedure is placing point loads at the center of the beam. The point load is changed gradually as the resulting reaction s are recorded. The loads vary from 100, 200, 300, 400, and 500g loads. The second part, a loading of 500g is placed along the beam at different lengths of 0.36 m. Finally, the loads are uniformly distributed along the beam at even spaces.
This experiment is particularly important, as it is applicable in numerous real-life applications. For instance, a uniformly distributed load is applied in the design of a building, a lorry trailer, and an airplane’s cargo hold.
Theory
This project tends to explain the theory of ultimate load applied to a simply supported beam. In any simply supported beam, there are three types of loadings. The point load, the uniformly distributed load, and the uniformly varying loads all of which are covered in this experiment. In the simplified engineering bending theory, the following assumptions are made:
1. The beam is initially unstressed and straight.
2. The beam is assumed to be internally statically indeterminate
3. Materials of the beam are homogenous and isotropic.
4. The Young’s Modulus of Elasticity value is the same in both compression and tension.
5. The transverse sections that were plane before the loading and bending remain plane even after the bending and loading (Lee et al., 2018).
The self-weight of the building cannot be neglected in any way of the cases. It is usually considered an extra UDL load such that it spreads evenly rather than being applied at one point.
To compute the value of bending stresses in loaded beams, consider two cross-sections of a beam HE and GF.
Considering fiber AB in the material, at a distance y from the Neutral Axis, when the beam bends it stretches to A’B’.
Since CD and C’D’ are o the neutral axis, it is assumed that the stress on the neutral axis is zero. Thus, there is no strain on the neutral axis.
Thus, the term becomes the property of the material and is referred to as the second moment of area of the cross-section denotated as l.
Therefore,
This equation is referred to as the Bending Theory Equation. The equation gives the distribution of stresses, which are normal to the cross-section in the x-direction.
Experiment methods
Apparatus
1. A Beam specimen
2. A set of weights
3. A meter ruler for measuring the span of the beam
4. A support frame
The procedure below is used to investigate the relationship between the loads on a beam and the reaction of the beam. Moreover, the second procedure helps in predicting the reaction of the beam by placing the loads at different points on the beam.
1. The weight hangers are lifted from the beam while a hook at the center is left on the beam.
2. Both spring balance supporting the beam are zeroed. The balance stubs are then adjusted to make the beams level.
3. Nine 10g masses are added to the weight hanger thereby giving a total weight hanger of 100 g, which is attached to the hook.
4. The reaction forces measured by each spring are then recorded. On recording the results, the figures were rounded off to the nearest three decimal places to increase the accuracy.
Results
The data collected are in the tables below and the relationship between the variable represented on the graphs.
Load (g)
Load (N)
R 1 (N)
R 2 (N)
R1 +R2 (N)
100
0.98
0.5
0.5
1.0
200
1.96
0.95
1.25
2.1
300
2.943
1.45
1.65
3.4
400
3.924
1.95
2.15
4.4
500
4.905
2.4
2.6
5
Table 1: Point load at the center results
Figure 2: Graph 1; Experiment 1 results
Load 1 (g)
Load 2 (g)
Load 3 (g)
Load 4 (g)
Load 5 (g)
Total Load (g)
Total Load (N)
R1 (N)
R2 (N)
R1 +R2 (N)
50
50
50
50
50
250
2.45
1.25
1.25
2.5
100
100
100
100
100
500
4.905
2.5
2.5
5
150
150
150
150
150
750
7.3595
3.75
3.75
7.5
200
200
200
200
200
1000
9.8
5
5
10
250
250
250
250
250
1250
12.25
6.25
6.25
12.5
Table 2: Uniformly distributed load results
Figure 3: Graph 2; Experiment 2 results
Discussion
Table 1 shows the values of the reactions of the beam when subjected to a point load. When a load is placed at the center of the beam, the sum of the opposite reactions is equal to the load placed, i.e., R1+R2= P. As such, the reactions of the beam should be same, i.e., R1=R2. So that R1=R2=P/2. However, from the figures obtained from experiment 1, apart from the 100 g load, the reactions were not equal to each other with the subsequent loads. This indicates an error in the experiment. The errors could have arisen from numerous factors. For one, the load was not placed at the center of the beam. Secondly, the spring balances could have had zero errors and lastly, the balance stub was not balanced properly before taking the reading.
Table 2 on the other hand, indicate the results of experiment 2 which is placing a point load along the beam. In the same light, as in the first experiment, the reactions should be equal. From table 2, the reaction figures obtained are equal to each other. Similarly, graph 2 having a straight line indicate this relationship.
Conclusion and Recommendations
Ultimately, from the experiment conducted it is conclusive that the value of the reactions at the left and the right supports are the same, and the value of the reaction at the center is the highest. This goes to show that the middle support will support more loading relative to the right and the left supports. Additionally, the total load supported by the beam is equal to the total reactions at the three supports the beam is in equilibrium (Saadatmanesh, and Ehsani, 1991)
Recommendations
1. Ensure that all screws are tight enough to avoid the load hanger and the beam from moving.
2. Take readings from the indicators only when the indicators stop are stable.
3. Ensure that the load is on a static state when taking the reading.
4. Ensure that the reading on the indicator is always zero before placing the load.
5. To prevent damage to the indicator, ensure that the loads are placed slowly at the load hanger.
References
Banthia, N.P., Mindess, S. and Bentur, A., 1987. Impact behaviour of concrete beams. Materials and Structures, 20(4), pp.293-302.
Lee, J.Y., Shin, H.O., Yoo, D.Y. and Yoon, Y.S., 2018. Structural response of steel-fiber-reinforced concrete beams under various loading rates. Engineering Structures, 156, pp.271-283.
Saadatmanesh, H. and Ehsani, M.R., 1991. RC beams strengthened with GFRP plates. I: Experimental study. Journal of structural engineering, 117(11), pp.3417-3433.
