Lab4.Bending Moment

Faculty of Engineering and Applied Science ENGR 2420U – Solid Mechanics Winter 2012 Course Instructor Dr. Ahmad Barari Laboratory Instruction Manual Lab 4: The Bending Moment Experiment

W

P C

W M

C V

Bending Moment 1.1 Objective In this experiment the concept of bending moment and its calculation along a straight beam is investigated. Measurements are performed to improve the understanding of the concept and to gain experience in the use of modern bending moment testing machine. The bending moment test is performed to study how the bending moment varies with variation of the o Load magnitude o Load distribution It also helps students to find out how the bending moment can be measured in a point along the beam. 1.2 Introduction and Theoretical Background W Members that are slender and support loadings that are P applied perpendicular to their longitudinal axis are C called beams. In general beams are long, straight bars having a constant cross-sectional area. Because of the applied loadings, beams develop an internal force W called shear force and an internal moment known as M C bending moment. In general both shear force and bending moment vary from point to point along the V beam, as shown in Fig. 1. They are calculated at each point using the following procedure. 1- Determine all the reactive forces and couple Fig. 1: A beam under various loadings moments acting on the beam. 2- Section the beam perpendicular to its length at the desired point and draw the free body diagram of one segment. Make sure that V (Shear Force) and M (Bending Moment) are shown acting in their positive sense, in accordance with the sign convention given in Fig. 2. 3- The shear force is obtained by summing the forces perpendicular to the beam’s axis and the bending moment is obtained by summing the moments about the sectioned end of the Fig. 2: Beam sign conventions segment.

In order to properly design a beam, it is important to know the variations of the shear force and the bending moment along its axis in order to find the points in which these values are maxima. Both of these important parameters at each point vary by changing the magnitude and the distribution of the applied loads. This test is being preformed to study the effects of the magnitude and distribution of the applied loads on the bending moment value at a specified point.

1.3 Apparatus The bending moment machine, shown in Fig. 3, used in this laboratory consists of a beam fitted into the structure test frame. The structure test frame is a sturdy aluminum frame, which stands on a workbench. Loads are applied to experiments using hangers, which hold various masses. The Digital Force Display electronically measures and displays forces during experiments. It conveniently fixes to the test frame. All the equipment connects to a computer by means of an Automatic Data Acquisition Unit and software (STR2000). The beam is ‘cut’ by a pivot. To stop the beam collapsing a moment arm bridges the cut on to a load cell thus reacting (and measuring) the bending moment force. A digital display shows the force from the load cell. The beam geometry and hanger positions are shown in Fig. 4. Hanger supports are 20 mm apart, and have a centre slot that positions the hangers. The moment arm is 125 mm long. The beam hangs from the top member of the test frame, rather than sitting on the bottom, so the supports do not interfere with loading positions.

‘Set Zero’ Control

Force sensor conditioning Force sensor

Support Moment arm

Beam Rolling pivot Grooved hangers

‘Cut’ position Support pivot Fig. 3: Bending moment in a beam apparatus

60

Rolling pivot

Beam

380

60

Securing thumbscrews Moment arm

Hangers & masses

Pivot

Digital force display

Fig. 4: Beam geometry and hangers positions Since the load cell has almost zero deflection, no Hook compensation in the level of the beam is needed for an increase in force. Thus, the beam remains perfectly horizontal regardless of load. All pivots run on sealed ball races, the left support allows rotational and horizontal movement while the right support allows rotation only. The masses supplied with the equipment give maximum flexibility and ease of use. Fig. 5 shows Securing clip a hanger loaded with masses. Use the clips provided to hold the masses on the hangers. Masses There are one-hundred-and-fifty 10-gram masses and five 10-gram weight hangers. This allows any load, in increments of 10-grams, to be made up to Base plate maximum 500-grams. Alternatively, one hanger can be made up into 100-grams, 200-grams, 300-grams, Fig. 5: Hanger loaded with masses 400-grams or 500-grams.

1.4 Safety Instructions • • • •

There is a risk of electric shock. Always unplug first. During test do not touch any parts of the test apparatus except the designated handles IMPORTANT! Never attempt any form of machine maintenance. IMPORTANT! Never attempt to apply any excessive load over than the designed loads; it may cause damage to the load cell and also plastic deformation in the beam.

1.5 Test Procedure In order to find out how the bending moment varies with the variation of the magnitude of the applied load, and its distribution along the beam, two experiments are designed. The steps of both experiments are illustrated in the following. 1.5.1 Bending moment variation at the point of loading In this experiment the bending moment is measured at ‘Cut’ position when the loading is applied in the same position (‘Cut’), as shown in Fig. 6. The steps of the experiment are as follows: 1. 2. 3. 4. 5.

Make sure that the beam is load less and in its horizontal position. Turn on the load digital force displayer. Make sure that the digital force displayer displays zero force. Hang the load hanger exactly in the ‘Cut’ position (according to Fig. 6). Put proper masses to get loads according to the first column of the following Table 1.

6. Read the force displayed by the digital force displayer and write it, in the third column of Table 1, for each case. 7. When you finished recording the data, depart the hanger and prepare for the next experiment. Table 1: Mass (g) 0.00 100 200 300 400 500

Load (N) 0.00 0.98 1.96 2.94 3.92 4.90

Force (N)

Experimental Bending Moment (Nm)

Theoretical Bending Moment (Nm)

W (Load) Cut

RB

RA

Fig. 6: Force diagram in experiment one.

1.5.2 Bending moment variation away from the point of loading In this experiment the bending moment is measured at ‘Cut’ position and loadings are applied in some other points along the beam. Three load cases according to Figures 7, 8, and 9 are considered. The steps of the experiment are as following: 1. 2. 3. 4. 5. 6.

Make sure that beam is load less and in its horizontal position. Turn on the load digital force displayer if it is off. Make sure that the digital force displayer displays zero force. Hang the load hanger exactly according to the position shown in Fig. 7. Put proper masses to get load according to the load shown in Fig. 7. Read the force displayed by the ‘digital force displayer’ and write it in the first row forth column of Table 2. 7. Remove the load hanger. 8. Hang the load hangers exactly according to the positions shown in Fig. 8. 9. Put proper masses to get loads according to the loads shown in Fig. 8. 10. Read the force displayed from the ‘digital force displayer’ and write it in the second row forth column of Table 2. 11. Remove the hanger W1 and read the force and then hang W1 and remove the hanger W2. Check if the summation of these two cases is equal to the force you read in step 10. 12. Remove the load hangers and hang them according to Fig. 9 with the shown loads. 13. Read the force displayed from the ‘digital force displayer’ and write it in the third row forth column of Table 2. 14. Repeat the step 11 similarly.

Table 2: Figure

W1 (N)

7 8 9

3.92 1.96 4.90

W2 (N)

Force (N)

RA (N)

Experimental Bending Moment (Nm)

RB (N)

Theoretical Bending Moment (Nm)

3.92 3.92

Cut

W1

RA

RB

W1 = 3.92 N (400 g) Fig. 7: First load case force diagram in experiment two.

Cut W1 = 1.96 N (200 g) W2 = 3.92 N (400 g)

RB

W1 W2

RA

Fig. 8: Second load case force diagram in experiment two.

Cut W1 = 4.90 N (500 g) W2 = 3.92 N (400 g)

1.6 Analysis

RB

RA

W1 W2 Fig. 9: Third load case force diagram in experiment two.

1.6.1 Bending moment variation at the point of loading 1- Using the obtained readings calculate experimental bending moment by the following formula: Experimental bending moment = Force X Moment arm length (1) 2- Using the above formula calculate the experimental bending moment for each case and fill in the forth column of Table 1. 3- UsingG the equilibrium equations, i.e. G (2) F ∑ G = OG ∑M = O (3) Determine the values of RA and RB for each case. 4- Using the method discussed in the section 1.2 calculate the bending moment for each case and fill in the fifth column of Table 1. 5- Draw the load (vertical axis) vs. experimental and theoretical bending moments (horizontal axis) as in Fig. 10 drawn for some typical results. 1.6.2 Bending moment variation away from the point of loading 1- Using the obtained readings calculate the experimental bending moment from Eq. 1. 2- Using the above formula calculate the experimental bending moment for each load case and fill in the fifth column of Table 2. 3- Using the equilibrium equations determine RA and RB for each load case and fill in the sixth and seventh column of Table 2.

Fig. 10: Load vs. theoretical and experimental bending moment.

4- Using the method discussed in the section 1.2 calculate the bending moment for each case and fill in the eighth column of Table 2. 1.7 Analysis Report: A full report would usually require a brief outline of the steps taken in performing the experiment and of precautions taken to minimize errors. Your report should contain the following items: 1. 2. 3. 4. a)

A record of all measurements made on the test specimens Tabulated needed values of Tables 1 and 2 for each load case A figure showing variation of the load vs. bending moment for experiment 1. Answer of two following questions: What are the probable sources of difference between the theoretical and experimental values of the bending moment? b) Concerning to your experience done in experiment 2, what is your comment about “The bending moment at the ‘cut’ is equal to the algebraic sum of the moments caused by the forces acting to the left or right of the cut.”? References: Hibbeler, R. C., Mechanics of Materials (6th Edition), Prentice Hall: New Jersey, 2004. Testing Machine User Manual, TecQuipment Education and Training Limited 2004.