BEAM BENDING LABORATORY

ES196 STACTICS AND STRUCTURES PANDYA, KATHIT ID:1812049

Abstract Beams are an example of horizontal structural members that are used to carry a load, when subjected to external forces they generate internal axial forces, shear forces and bending moments. This report examines the bending moments acting on a determinant overhanging beam when different loads are applied through its length. Statically determinant structures are those in which reactions and internal forces can be determined using equilibrium equations, hence the equations of equilibrium are used to calculate the internal forces at different portions of the beam and compare these theoretical values with the recorded experimental results; the results attained by the experiment are reliable. The graph used to display the data showed a 3.12% percentage error. Percentage difference tables show the differences between the theoretical and experimental bending moment and support the accuracy and reliability of the experiment.

Introduction The laboratory is divided into three different sections to which are all focused on finding the bending moment

in different cases. In section one the aim is to observe the change in the bending moment acting on the beam when weights are added at the position of the cut of the beam. The aim of the second section is to show the changes in the bending moment at the cut position of the beam when different weights are placed at different positions of the beam. Following on, the next section of the laboratory is

about analysing the changes in bending moment when a distributed load is applied to the beam.

Theory

Beams are categorised by three main factors; geometry, supports and equilibrium. Simply supported beams are considered determinant and defined as having a pinned support at one end and a roller support at the other end. Fixed Beams have fixed supports to which cause moment resistance at either end where the support is. Cantilever Beams are supported by a Fixed Support at one end. This laboratory uses overhanging beams that have two supports, but one of the supports is not at the end of the beam.

When an external load is applied to a beam a moment is created. For the beam to remain in equilibrium a couple is induced by the internal loads called the bending moment. The resultant internal force is called the normal force or the shear force depending on where on the plane it lies. An assumption made is that newton’s first law implies that if a body isn’t rotating the moments acting on it are balanced hence there is no net force acting on an object in equilibrium.

Bending moment is a moment that acts to counter the external load in order to keep the beam in equilibrium. There are two different types of bending moments that are experienced by the beam, hogging and sagging. Hogging causes a beam to bend with the concave side upwards and sagging is when the beam bends downwards. Sagging is treated as a positive bending moment where the top surface is in compression whereas hogging is treated as a negative bending moment where the bottom surface is in compression and top surface is in tension. By calculating reactions at supports and making cuts before and after each reaction and load a bending moment diagram and shear force diagram can be created. This can show the bending moment at different points of the beam.

Apparatus and Hazards

Hazard Precaution

Attempting to move the equipment can lead to the beam testing rig to top over or person moving it to trip over.

Do not attempt to move the beam testing rig whilst carrying out the experiment.

Weights could fall off the table or the hanger, causing injury if it lands on feet.

Must wear PPE i.e. Protective (steel capped) boots.

Could get an electric shock from the wires. Do not touch bare wires and ensure all the wires are connected properly.

Possibility of hand getting caught in moving parts. Be cautious and keep hands away from the path of moving parts. Listen to the briefing on how to operate the apparatus.

The aim of the experiment is to find the bending moment acting on the determinant overhanging beam and

then compare it to the theoretical bending moment. The length of the moment arm is 125mm so that is used to work out the moment that is developed at the cut position. The experimental setup used to carry out the laboratory is shown in Fig.1, the equipment used include ‘TecQuipment’s, Bending Moments in a Beam (STR2)’, the ‘Structures Test Frame(STR1) and the ‘Digital Force Display(STRa1). Figure 1 models a beam with a roller joint on the left support and a pin joint on the right support that is fitted in an aluminium frame. Weights are added to the hanger to which is hung on the beam, these weights can cause the beam to collapse, hence, there is a moment arm that joins the rest of the beam to the cut. The bending moment force that opposes the force created by the weights when applied to the beam is measured by the force sensor which is subsequently displayed on the digital read out.

Method Experiment 1 – Bending moment change at point of loading

Before setting up the equipment make sure each part of the equipment is not loose and inspect for damages. Zero the output reading caused by the force sensor to avoid any systematic errors. After that is done set up the equipment as shown in figure 1.

Weights are then hung on the grooved hangers at the position of the cut and the beam remains horizontal relative to the weights. By doing so we observe the change in the moment acting on the beam. Figure 2 shows the free-body diagram of the beam where RA + RB =W using the law of equilibrium. Masses are hung at the cut position and the corresponding values that are displayed on the digital force display are reported in table 1. Firstly, you start with a load of mass 100g and this method is repeated with the following mass values: 200 g, 300 g, 400 g and 500 g. Table 2 is split into five columns. First column records all the mass in grams and the second column records the loads. To obtain the load values multiply the masses in the first column by the acceleration of gravity g (9.81). Furthermore, force from the moment arm is used to calculate the experimental moment by

Digital force display

Pivot

Hanger and masses

Beam

Rolling Pivot

Securing thumbscrews

Moment arm

Figure 1. Experimental set up used in this laboratory.

multiplying the force by the length of the moment arm which is 125mm. The theoretical value is calculated by calculating reactions at supports and making cuts before and after each reaction and load using the free body diagram in figure 2. Finally, the bending moment diagram for the mass 500g must be included.

Experiment 2 – Bending moment change far from point of loading The aim of this experiment is to show the changes in the bending moment at the cut position of the beam when different weights are placed at different positions of the beam. First set up equipment as shown in figure 3. A weight of 400g is hung on the left side of the beam and the moment and experimental force is recorded as done in experiment 1. Secondly, weights of 400g and 200g are hung on the beam as shown in figure 4 and the moment, experimental force and theoretical value are recorded in the table using the method from experiment 1. Thirdly, the experiment is repeated once again as presented in figure 5 where the two weights of 400g and 500g are placed on the beam. For all these cases the bending moment diagram is drawn. Experiment 3 – Bending moment change with a distributed load The aim of this experiment is to analyse the changes in bending moment when a distributed load is applied to the beam. As shown in figure 6 three weights of 200g are placed on the beam. The method from experiment 1 is once again repeated in order to get the moment and theoretical value of the bending moment from the measured experimental force. The values and are recorded in a table and the values are used to make a bending moment diagram. In addition, a point load of 300g is attached at the left side of the beam and the method from experiment 1 is repeated to work out the theoretical value of the bending moment.

Figure 2. Experimental set up 1 where weights are hung below the cut.

Figure 4. Experimental set up 2.2 where two weights are positioned at the centre.

Figure 3. Experimental set up 2.1 where one weight is hung at the left side of the beam.

Figure 5. Experimental set up 2.3 where weights are hung at the right and centre of the beam.

Figure 7. Experimental set up where weights are hung at the left side of the beam with the distributed load.

Figure 6. Experimental set up 3 where a distributed is applied to the beam.

Results

Mass (g) Load (N) Digital Force Display (N) Experimental BM (Nm) Theoretical BM (Nm)

100 0.981 0.700 0.088 0.094

200 1.962 1.500 0.1875 0.187

300 2.943 2.100 0.2675 0.281

400 3.924 3.000 0.3750 0.375

500 4.905 3.600 0.450 0.470

Table 1a. Results of experiment with the weights in the cut position.

Figure W1 (g)

W1 (N)

W2 (g)

W2 (N)

Digital Force Display (N)

Experimental BM (Nm)

RA (N) RB (N) Theoretical BM (Nm)

4 400 3.924 - - -1.400 -0.175 5.173 -1.249 -0.175

5 200 1.962 400 3.924 3.500 0.450 2.586 3.300 0.462

6 500 4.905 400 3.924 3.800 0.465 2.586 6.242 0.482

Table 2a. Results of experiment with the weights in different location.

Table 3a. Results of experiment with the distributed load.

Experimental BM (Nm)

Theoretical BM (Nm)

Percentage difference

0.088 0.094 -6.383%

0.188 0.187 0.535%

0.268 0.281 -4.626%

0.375 0.375 0.000%

0.45 0.47 -4.255%

Experimental BM (Nm)

Theoretical BM (Nm)

Percentage difference

-0.175 -0.175 0.000%

0.450 0.462 -2.597%

0.465 0.476 -2.311%

Experimental BM (Nm)

Theoretical BM (Nm)

Percentage difference

0.488 0.487 0.205%

0.350 0.356 -1.685%

Figure W1 (g/m)

W1 (N/m)

W2

(g)

W2

(N)

Digital Force Display (N)

Experimental BM (Nm)

RA (N)

RB (N)

Theoretical BM (Nm)

7 7500 73.575 - - 3.900 0.488 2.408 3.478 0.487

8 7500 73.575 300 2.943 2.800 0.350 6.287 2.542 0.356

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0.45

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0 50 100 150 200 250 300 350 400 450 500 550

Be

nd

ing

Mo

me

nt

(Nm

)

Mass (g)

The bending moment created below the cut

Experimental Bending Moment (N) Theoretical Bending Moment (N)

Table 1b. Percentage difference of experiment with the weights in the cut position.

Table 2b. Percentage difference of experiment with the weights in different location.

Table 3b. Percentage difference of experiment with the distributed load.

Figure 8. Graph of bending moment against mass.

Analysis

In experiment 1 masses are hung at the cut position and the corresponding values that are displayed on the digital force display are reported in table 1a. From this table we can identify a lot of trends in the data. The mass and the load are directly proportional to each other. To obtain the values of the load the masses were multiplied by the acceleration of gravity g so the mass of 100g has a load of 0.981. Furthermore, the table 1a also shows a positive linear relationship between the mass and experimental bending moment. This is further supported by the graph in figure 8. The blue line represents the experimental bending moment showing a linear increase. The yellow line in the graph represents the theoretical bending moment which also has a linear increase that is very similar to the experimental bending moment line. The fact that the two lines intersect and are very close supports the reliability and accuracy of the results taken in the experiment. Table 1b shows the percentage difference between the experimental and theoretical bending moment. The formula used to find the percentage difference and percentage error is listed below and is shown in figure 9. (Theoretical Value-Experimented value/Theoretical value) *100= Percentage Error (Experimental BM -Theoretical BM)/ Theoretical BM=Percentage Difference.

For the mass of 100g the difference is 6.383% which is comparatively higher than the other differences as the experimental value is 0.088 and theoretical is 0.094 the values are very small hence there is a larger percentage difference. The digital force display has a resolution of 0.1 which give san uncertainty of ±-0.01. This affects the

accuracy of the results as if there was a higher degree of accuracy there would be a lower percentage difference between the experimental and theoretical bending moment. The results would be more accurate, and the graph would have lines closer together. An example of a systematic uncertainty is the non-zero error before carrying out the experiment if the digital force display is not zeroed all measurements taken could be too big or too small. To overcome this, find the size of the uncertainty and then take it off all the measurements. The absolute uncertainty is the size of the range in which the true value lies. There could be absolute uncertainty from the instrument of ±0.1.

Figure 10 presents the BM diagram of experiment 1 which is a positive bending moment. When a mass of 200g is applied the theoretical value of the bending moment is 0.187 and the experimental value is 0.188 which supports the accuracy of the results and the experiment as the percentage difference is only 0.535% Experiment 2 shows the changes in the bending moment at the cut position of the beam when different weights are placed at different positions of the beam. When the load is applied at the free end the beam deflects upwards as the

bending moment becomes negative. As shown by the BM diagram of experiment 2 there is a drastic change in the direction of the curve. The position of the load changes the bending moment, this supports the idea that there Is relationship between different forces applied at different points of the beam. Table 2a shows a small difference between the bending moments between repeat two that had masses 200g and 400g attached and repeat 3 that has masses 500g and 400g attached. The

Figure 10. BM Diagram of experiment 1-500g.

Figure 9. Excel Formula To Work Out The Percentage Difference.

bending moment for repeat 2 was 0.450 and it was 0.465 for repeat 3. There is a 2% percentage difference between the experimental and theoretical bending moments. Finally, experiment 3 shows changes in the bending moment when a distributed load is applied to the beam. In figure 11 the bending moment starts to have parabolic characteristics as there is a point load of 300g as well as a distributed load. The BM diagram shows the right-hand side with the distributed load to be positive and the left side with the point load to being negative. It shows parabolic characteristics as it goes from positive and negative as another is added. The theoretical values are close to the experimental values. In the first part of the experiment w1 is 7500 g/m and the experimental value recorded is 0.5 Nm and the theoretical value for the bending moment is 0.499Nm. The results attained from this experiment are very accurate as they are close to the calculated theoretical bending moment. If there were a series of 6 point loads the shear force diagram would have lots of drastic changes whereas the BM diagram would remain as a parabola due to there would be no change on the theoretical values.

Conclusion The three experiments were successful due to the face that when comparing the experimental values for bending moments they were very close to the theoretical values for the bending moment. From plotting the graph of bending moment against the mass, the error was found to be 3.12%. The third experiment allowed us to see the relationship between the SF and BM diagram through the BM diagram that had the characteristics of a parabola. Experiment three showed the changes a point load and distributed load could cause to a beam and see the outcome if the forces are increased. It was very accurate as the percentage error was only 0.95%. On the other hand, there were a few uncertainties and systematic errors that could have been avoided in order to get even more accurate results. The 3.70% percentage error in experiment 2 could be decreased if apparatus were more accurate and had a higher degree of accuracy. Overall, the experiment was successful and results obtained were reliable, as measures were taken to counteract most uncertainties and errors in the results.

Reference 1) ES196 Statics and Structures, Beam Bending Laboratory Briefing Sheet - https://moodle.warwick.ac.uk/pluginfile.php/921711/mod_page/content/29/ES196%20-%20Beam%20Bending%20Briefing%202018.pdf 2) Tequipment Academia https://www.tecquipment.com/bending-moments-in-a-beam 3) Princeton https://www.princeton.edu/~humcomp/bikes/design/desi_63.htm 4)Roymech http://www.roymech.co.uk/Useful_Tables/Beams/Beam_theory.html 5)Wikipedia https://en.wikipedia.org/wiki/Beam_(structure)