lab Manual

SECTION 2 EXPERIMENTS Experiment 1: Bending Moment Variation at the Point of Loading  This experiment examines how bending moment varies at the point of loading. Figure 3 shows the force diagram for the beam.

Check the Digital Force Display meter reads zero with no load. Place a hanger with a 100 g mass at the ‘cut’. Record the Digital Force Display reading in a table as in Table 1. Repeat using masses of 200 g, 300 g, 400 g and 500 g. Convert the mass into a load (in N) and the force reading into a bending moment (Nm). Remember; Bending moment at = Displayed force the cut (in Nm)

The equation we will use in this experiment is:

l 

0.125

Calculate the theoretical bending moment at the cut and complete Table 2.

Figure 3 Force diagram

BM (at cut) = Wa





a

Mass (g)



Load (N)

Experimental Theoretical Force bending moment bending moment (N) (Nm) (Nm)

0

l 

100

You may find the following table useful in converting the masses used in the experiments to loads.

200 300 400

Mass (Grams)

Load (Newtons)

100

0.98

200

1.96

300

2.94

400

3.92

500

4.90

500

Table 2 Results for Experiment 1 Plot a graph which compares your experimental results to those you calculated using the theory. Comment on the shape of the graph. What does it tell us about how bending moment varies at the point of  loading? Does the equation we used accurately predict the behaviour of the beam?

Table 1 Grams to Newtons conversion table

14

MM2:

Bending Moment in a Beam: Student Guide

Experiment 2: Bending Moment Variation away from the Point of Loading  This experiment examines how bending moment varies at the cut position of the beam for various loading conditions. Figure 4, Figure 5 and Figure 6 show the force diagrams.

Figure 6 Force diagram We will use the statement: “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.”

Figure 4 Force diagram

Check the Digital Force Display meter reads zero with no load. Carefully load the beam with the hangers in the  positions shown in Figure 4, using the loads indicated in Table 3. Record the Digital Force Display reading in a table as in Table 2. Convert the force reading into a bending moment (in  Nm). Remember; Bending moment at = Displayed force the cut (in Nm)



0.125

Calculate the support reactions ( RA and  RB) and calculate the theoretical bending moment at the cut. Repeat the procedure with the beam loaded as in Figure 5 and Figure 6. Comment on how the results of the experiments compare with those calculated using the theory.

Figure 5 Force diagram

Figure

W 1 (N)

W 2 (N)

4

3.92

5

1.96

3.92

6

4.91

3.92

Force (N)

Experimental bending moment (Nm)

Table 3 Results for Experiment 2 

15

R A (N)

R B (N)

Theoretical bending moment (Nm)

MM2:

Bending Moment in a Beam

NOTES:

16

MM3: Shear Force in a Beam

- Introduction and Description - Experiments 1- Shear Force Variation with Increasing Load 2- Shear Force Variation for Various Load Conditions

17

SECTION 1 INTRODUCTION AND DESCRIPTION DESCRIPTION

Figure 1 Shear forces in a beam experiment 

Introduction

How to Set Up the Equipment 

This guide describes how to set up and perform Shear  Force in a Beam experiments. It clearly demonstrates the principles involved and gives practical support to your studies.

The Shear Force in a Beam experiment fits into a Test Frame. Figure 2 shows the Shear Force of a Beam experiment assembled in the Frame. Before setting up and using the equipment, always:

Description

•

Figure 1 shows the Shear Force in a Beam experiment. It consists of a beam which is ‘cut’. To stop the beam collapsing a mechanism, (which allows movement in the shear direction only) bridges the cut on to a load cell thus reacting (and measuring) the shear force. A digital display shows the force from the load cell. A diagram on the left-hand support of the beam shows the beam geometry and hanger positions. Hanger  supports are 20 mm apart, and have a central groove which positions the hangers.

18

• •

•

Visually inspect all parts, including electrical leads, for damage or wear. Check electrical connections are correct and secure. Check all components are secured correctly and fastenings are sufficiently tight. Position the Test Frame safely. Make sure it is mounted on a solid, level surface, is steady, and easily accessible.

Never apply excessive loads to any part of the equipment.

MM3:

Shear Force in a Beam

Figure 2 Shear force of a beam experiment in the structures frame Steps 1 to 4 of the following instructions may already have been completed for you. 1. Place an assembled assembled Test Frame Frame (refer (refer to the separate instructions supplied with the Test Frame if  necessary) on a workbench. Make sure the ‘window’ of the Test Frame is easily accessible. 2. There are four securing nuts in the top member of  the frame. Slide them to approximately the positions shown in Figure 3. 3. With the right-hand right-hand end of the experiment resting on the bottom member of the Test Frame, fit the lefthand support to the top member of the frame. Push the support on to the frame to ensure that the internal bars are sitting on the frame squarely. Tighten the support in position by screwing two of  the thumbscrews provided into the securing nuts (on the front of the support only). 4. Lift the right-hand right-hand support into position and locate the two remaining thumbscrews into the securing nuts. Push the support on to the frame to ensure the

internal bars are sitting on the frame squarely. Position the support horizontally so the rolling pivot is in the middle of its travel. Tighten the thumbscrews. 5. Make sure the Digital Force Display is ‘on’. Connect the mini DIN lead from ‘Force Input 1’ on the Digital Force Display to the socket marked ‘Force Output’ on the left-hand support of the experiment. Ensure the lead does not touch the  beam. 6. Carefully zero the force meter meter using the dial on the left-hand beam of the experiment. Gently apply a small load with a finger to the centre of the beam and release. Zero the meter again if necessary. Repeat to ensure the meter returns to zero. Note: If the meter is only ±0.1 N, lightly tap the frame (there may be a little ‘stiction’ and this should overcome it).

19

SECTION 2 EXPERIMENTS Experiment 1: Shear Force Variation with an Increasing Point Load  This experiment examines how shear force varies with an increasing point load. Figure 3 shows the force diagram for the beam. W 

40 mm

a

'Cut'

R  A

Check the Digital Force Display meter reads zero with no load. Place a hanger with a 100 g mass to the left of the ‘cut’ (40 mm away). Record the Digital Force Display reading in a table as in Table 2. Repeat using masses of  200 g, 300 g, 400 g and 500 g. Convert the mass into a load (in N). Remember,

R B

Shear force at the cut = Displayed force

l 

Calculate the theoretical shear force at the cut and complete the table.

Figure 3 Force diagram The equation we will use in this experiment is: Shear force at cut, S c

=

Mass

Load

Experimental shear 

Theoretical shear 

(g)

(N)

force (N)

force (N)

W .a

0

l 

100 200

Where a is the distance to the load (not the cut) Distance a = 260 mm

300 400

You may find the following table useful in converting the masses used in the experiments to loads. Mass (Grams)

Load (Newtons)

100

0.98

200

1.96

300

2.94

400

3.92

500

4.90

500

Table 2 Results for Experiment 1 Plot a graph which compares your experimental results to those you calculated using the theory. Comment on the shape of the graph. What does it tell us about how shear force varies due to an increased load? Does the equation we used accurately predict the  behaviour of the beam?

Table 1 Grams to Newtons conversion table

20

MM3:

Shear Force in a Beam

Experiment 2: Shear Force Variation for Various Loading Conditions This experiment examines how shear force varies at the cut position of the beam for various loading conditions. Figure 4, Figure 5 and Figure 6 show the force diagrams.

Figure 6 Force diagram We will use the statement: “The Shear force at the ‘cut’ is equal to the algebraic sum of the forces acting to the left or right of the cut.”

Figure 4 Force diagram

Check the Digital Force Display meter reads zero with no load. Carefully load the beam with the hangers in the  positions shown in Figure 4, using the loads indicated in Table 2. Record the Digital Force Display reading as in Table 3. Remember, Shear force at the cut (N) = Displayed force Calculate the support reactions ( RA and  RB) and calculate the theoretical shear force at the cut. Repeat the procedure with the beam loaded as in Figure 5 and Figure 6. Comment on how the results of the experiments compare with those calculated using the theory.

Figure 5 Force diagram

Figure

W 1

W 2

Force

Experimental shear 

R A

R B

Theoretical shear 

(N)

(N)

(N)

force (N)

(N)

(N)

force (Nm)

4

3.92

5

1.96

3.92

6

4.91

3.92

Table 2 Results for Experiment 2 

21

MM3: Shear Force in a Beam: Student Guide

NOTES:

22

MM4: Deflections of Beams and Cantilevers

- Introduction and Description - Experiments 1- Deflection of a Cantilever  2- Shear Deflection of a Simply Supported Beam 3- The Shape of a Deflected Beam 4- Circular Bending

23

SECTION 1 INTRODUCTION AND DESCRIPTION

Figure 1 Deflection of Beams and Cantilevers experiment 

Introduction This guide describes how to set up and perform experiments on the deflection behaviour of beams and cantilevers. The equipment clearly demonstrates the  principles involved and gives practical support to your  studies.

Description Figure 1 shows the Deflections of Beams and Cantilevers experiment. It consists of a backboard with a digital dial test indicator. The digital dial test indicator  is on a sliding bracket which allows it to traverse accurately to any position along the test beam. Two rigid clamps mount on the backboard and can hold the  beam in any position. Two knife-edge supports also fasten anywhere along the beam. Scales printed on the  backboard allow quick and accurate positioning of the digital dial test indicator, knife-edges and loads.

Look at the reference information on the backboard. It is useful and you may need it to complete the experiments in this guide.

How to Set up the Equipment  The Deflections of Beams and Cantilevers experiment fits into a Test Frame. Figure 2 shows the Deflections of  Beams and Cantilevers experiment in the Frame. Before setting up and using the equipment, always: •

• •

•

Visually inspect all parts, including electrical leads, for damage or wear. Check electrical connections are correct and secure. Check all components are secure and fastenings are sufficiently tight. Position the Test Frame safely. Make sure it is on a solid, level surface, is steady, and easily accessible.

Never apply excessive loads to any part of the equipment.

24

MM4:

Deflections of Beams and Cantilevers

Figure 2 Deflections of Beams and Cantilevers experiment in the structures frame The following instructions may already have been completed for you.

them to roughly the positions of the thumbscrews shown in Figure 2.

1. Place an assembled Test Frame (refer to the separate instructions supplied with the Test Frame if  necessary) on a workbench. Make sure the ‘window’ of the Test Frame is easily accessible.

3. Lift the backboard into position and have an assistant secure it by threading the thumbscrews into the securing nuts. If necessary, level the backboard  by loosening the thumbscrews on one side, repositioning the backboard, and tightening the thumbscrews.

2. There are two securing nuts in each of the side members of the frame (on the inner track). Slide

25

SECTION 2: EXPERIMENTS Experiment 1: Deflection of a Cantilever  In this experiment, we will examine the deflection of a cantilever subjected to an increasing point load. We will repeat this for three different materials to see if their  deflection properties vary.

0

1 0

2 0

3 0

4 0

5 0 6 0

7 0

8 0

9 0 1 0 0

1 1 0

1 2 0 10 3 1 4 0

1 50

1 6 0 10 7 1 8 0 1 9 0 2 00

2 1 0

ZERO/ABS

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 00

1 1 0

1 2 0 10 3 1 4 0 1 5 0 1 6 0

10 7

2 3 0 20 4 2 5 0

2 60 2 7 0 2 8 0

2 90

3 0 0 30 1 3 2 0

3 3

ON/OF F

P RS EE T T

0

2 20

OL.

1 80 1 9 0 2 0 0

2 10 2 2 0 2 3 0

2 40 2 5 0 2 6 0

2 70

2 8 0 2 9 0 3 0 0

30 13 2 0 3 3

Remove any clamps and knife edges from the  backboard. Set up one of the cantilevers as shown in Figure 3. Slide the digital dial test indicator to the position on the beam shown in Figure 3, and lock it using the thumbnut at the rear. Slide a knife-edge hanger to the  position shown. Tap the frame lightly and zero the digital dial test indicator using the ‘origin’ button. Apply masses to the knife-edge hanger in the increments shown in Table 1. Tap the frame lightly each time you add the masses. Record the digital dial test indicator reading for each i ncrement of mass. Repeat the procedure for the other two materials and fill in a new table.

200 mm

Material  –2

E value: ___________ Nm I: _________________ m

4

Width b: ____________ mm Depth d: ____________ mm

W 

Figure 3 Cantilever set-up and schematic 

Mass

Actual deflection

Theoretical deflection

(g)

(mm)

(mm)

0

You may find the following table useful in converting the masses used in the experiments to loads.

100 200 300

Mass (Grams)

Load (Newtons)

100

0.98

200

1.96

300

2.94

400

3.92

500

4.90

400 500

Table 1 Results for Experiment 1 (beam 1)

Table 1 Grams to Newtons conversion table Material  –2

As well as the information given on the backboard you will need the following formula: 3

Deflection =

WL

3 EI 

E value: ___________ Nm I: _________________ m

4

Width b: ____________ mm Depth d: ____________ mm

Mass

Actual deflection

Theoretical deflection

(g)

(mm)

(mm)

0

where: W =

Load (N)  L = Distance from support to position of loading (m);  –2  E = Young’s modulus for cantilever material (Nm ); 4  I = Second moment of area of the cantilever (m ). Using a vernier gauge, measure the width and depth of  the aluminium, brass and steel test beams. Record the values next to the results tables for each material and use them to calculate the second moment of area, I .

26

100 200 300 400 500

Table 2 Results for Experiment 1 (beam 2)

MM4:

Deflections of Beams and Cantilevers

Material  –2

E value: ___________ Nm 4

I: _________________ m

Width b: ____________ mm Depth d: ____________ mm

Mass

Actual deflection

Theoretical deflection

(g)

(mm)

(mm)

0 100 200 300 400

On the same axis, plot a graph of Deflection versus Mass for all three beams. Comment on the relationship  between the mass and the beam deflection. Is there a relationship between the gradient of the line for each graph and the modulus of the material? Calculate the theoretical deflection for each beam and add the results to your table and the graph. Does the equation accurately predict the behaviour of the beam? Why is it a good idea to tap the frame each time we take a reading from the digital dial test indicator?  Name at least three practical applications of a cantilever structure.

500

Table 3 Results for Experiment 1 (beam 3)

27

MM4:

Deflections of Beams and Cantilevers

Experiment 2: Deflection of a Simply Supported Beam In this experiment, we will examine the deflection of a simply supported beam subjected to an increasing point load. We will also vary the beam length by changing the distance between the supports. This means we can find out the relationship between the deflection and the length of the beam. As well as the information given on the backboard you will need the following formula: 3

Maximum deflection =

I: _________________ m

Depth d: ____________ mm

Actual deflection

Theoretical deflection

(g)

(mm)

(mm)

0 100 200

48 EI 

300

4

Width b: ____________ mm

4

Mass

WL

where: W = Load (N);  L = Distance from support to support (m);  –2  E = Young’s modulus for cantilever material (Nm );  I =

 –2

E value: ___________ Nm

400 500

Table 4 Results for Experiment 2 (fixed beam length variable load)

Second moment of area of the cantilever (m ).

Part 1 Using a vernier gauge, measure the width and depth of  the aluminium test beam. Record the values next to the results table and use them to calculate the second moment of area, I . Remove any clamps from the backboard. Setting length between supports l  to 400 mm, set up the beam as shown in Figure 4.

Part 2 Set up the beam with the length l at 200 mm. Ensure the digital dial test indicator and load hanger are still central to the beam, as shown in Figure 5.

0

10

20 30

40

50 60

70 80

PRESET

10

20 30

40

50 60

70 80

4 0

5 0 6 0

7 0 8 0

9 0 1 0 0 10 11 2 0 1 3 0 1 4 0 1 5 0 1 6 0 10 7

1 8 0 1 9 0 2 0 2 1 0 2 2 0 2 30 2 4 0 2 5 0 2 6 0 2 7 0 2 80 2 3 9 00 03301 00 3 0 2 10 3 32 30 3 3 0 3 04

3 5 0 3 6 0 3 7 0 30 83 9 0 4 0 0 40 1 4 2 0 4 3 0 4 04

4 5 0 4 6 0 4 7 0

4 8 0 4 9 0 5 0 0 50 15 2 0 5 3 0 50 45 5 0 5 6 0 5 7 0 5 8 0 5 9 0

3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 309004410004421004432 0 4 3 0 4 4 0

45046047 0

48049 0

ON/OFF

PRESET

10

2030

40

50

60

70

80

90100110120130

14015 0

1 6 0 1 7 0 1 8109109200200201201202202 0 2 3 0 2 4 0 2 5 0 2 6 0 2 7 0 2 8 0 2 9 0

TOL.

500510520530540550560570580590

9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 2 2 0 2 3 0 2 4 0 2 5 0 2 6 0 2 7 0 2 8 0 2 9 02 9300300 03 1301 03 2 0 3 3 0 3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 3 9 0 4 0 0 4 1 0 4 2 0 4 3 0 4 4 0 4 5 0 4 6 0 4 7 0 4 8 0 4 9 0 5 0 0 5 1 0 5 2 0 5 3 0 5 4 0 5 5 0 5 6 0 5 7 0 5 8 0 5 9 0

ON/OFF

ZERO/ABS

0

2 0 3 0

ZERO/ABS

0

0

1 0

TOL.

9 0 1 0 0 111100 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 2 2 0 2 3 0 2 4 0 2 5 0 2 6 0 2 7 0 2 8 0 2 9 0 3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 3 9 0 4 0 0 4 1 0 4 2 0 4 3 0 4 4 0 4 5 0 4 6 0 4 7 0 4 8 0 4 9 0 490 5 0 0 500 5 1 0 510 520 530 540 550 560 570 580 590

l  =200 mm

l  = 400 mm 200 mm

200 mm

W 

Figure 5 Simply supported beam set-up and  schematic (fixed beam load with variable length) W 

Figure 4 Simply supported beam set-up and  schematic (fixed beam with variable load) Slide the digital dial test indicator into position on the  beam and lock it using the thumbnut at the rear. Slide a knife-edge hanger to the position shown. Tap the frame lightly and zero the digital dial test indicator using the ‘origin’ button. Apply masses to the knife-edge hanger in the increments shown in the results table. Tap the frame lightly each time, and record the digital dial test indicator reading for each increment of mass.

28

Lightly tap the frame and zero the digital dial test indicator using the ‘origin’ button. Apply a 500 g mass and record the deflection in Table 5. Repeat the  procedure for each increment of beam length. From Table 4 plot a graph of Deflection versus Applied Mass for a simply supported beam. Comment on the your graph. Inspect the ruling equation of the  beam. What is the relationship between the deflection and the beam length? Test your assumption by filling in the empty column of Table 5 with the correct variable. Plot a graph.

MM4:

Deflections of Beams and Cantilevers

Length (mm)

Deflection (mm)

 Name at least one example where this type of bending is desirable and one where it is undesirable.

200 260 320 380 440 500 560

Table 5 Results for Experiment 2 (fixed beam load  variable length)

29

MM4:

Deflections of Beams and Cantilevers

Experiment 3: The Shape of a Deflected Beam This experiment shows how the deflection of a loaded  beam varies with span.

0

1 0 2 0

3 0 4 0

5 0 66 0

0 8 1 0 0 1 01 77 0 8 00 9 9 00 1

1 2 0 1 3 0 1 4 0 10 51 6 0 1 7 0 1 8 0 1 9 0 2 00 2 1 0 2 2 0 2 3 0 2 4 0 2 50 2 6 0 2 7 0 2 8 0 2 9 0 3 0 0 30 13 2 0 3 3 0 30 43 5 0 3 6 0 3 7 0 3 8 0 3 90 4 0 0 4 1 0 4 0 24 3 0 4 4 0 4 5 0 4 6 0 4 70 4 8 0 4 9 0 5 0 5 1 0 5 2 0 5 0 35 4 0 5 5 0 50 6 5 7 0 5 8 0 5 90

PRESET

Position from

Datum

Loaded

Deflection

left (mm)

reading (mm)

reading (mm)

(mm)

0

ON/OFF

ZERO/ABS

Traverse the loaded beam with the digital dial test indicator recording the deflections.

TOL.

20 0

10

20

30

40

50

60

70

8 0 9900110000111100112200 1 3 0

140150160170180190200210220230240250260270

280290300310320330340350360370380390400410420

4 3 0 4 4 0 4 5 0 4 6 0 4 7 0 4 8500409501500502501503502 0 5 3 0 5 4 0 5 5 0 5 6 0

570580590

40 60 80 100

600 mm

150

 x 

200 250

200 mm

200 mm

300 350

W 

400

Figure 6 Simply supported beam set-up and  schematic 

450 500

Remove any clamps from the backboard and set up the  beam as shown in Figure 6. Slide the digital dial test indicator to the zero  position on the beam and, using the ‘±’ button, set it so a downward movement reads negative. Do not lock the digital dial test indicator. Slide a knife-edge hanger to the correct position on the beam. Tap the frame lightly. Roughly zero the digital dial test indicator using the ‘origin’ button. Record the actual ‘datum’ value in Table 6. Carefully slide the digital dial test indicator to the  positions shown in Table 6 (note the change in the increments after 100 mm). Remember to tap the frame each time you take a reading. Record the ‘datum’ value at each position. Apply a 500 g mass to the knife-edge hanger and return the digital dial test indicator to the zero position. Make sure the digital dial test indicator stylus passes through the gap in the knife-edge hanger.

30

550 600

Table 6 Results for Experiment 3 Work out the true deflection from the datum and loaded values. Why is it important to take datum values in this experiment? Plot a graph of deflection versus position along the  beam. What shape does the beam adopt outside the  bounds of the knife-edge supports? Why is that? Using a suitable method calculate the true deflection of the beam (within the bounds of the knife-edge supports) and add the data to the graph. Does the method you have used accurately predict the shape of  the deflected beam?

MM4:

Deflections of Beams and Cantilevers

Experiment 4: Circular Bending  In this experiment, we apply loads to a simply supported beam at its end to induce a moment and thus  produce circular bending. As well helping to establish an important relationship, this test is an accurate method for measuring Young’s modulus.

h

C 

R  = Radius of curvature (m); C  = Chord (m); h = Height of chord (m).

R  0

10

20

3040

5060

70

80

90100110120130

140150160170180190200210220230240250

26027 0

2 8 0 23900033010033120033230 3 3 0 3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 3 9 0 4 0 0 4 1 0 4 2 0

0

10

2030

40

50

60

7080

9900110000111100112200 1 3 0

140150160170180190200210220230240250260270

570580590

ON/OFF

ZERO/ABS PRESET

430440450460470480490500510520530540550560

TOL.

280290300310320330340350360370380390400410420

43044 0

45046047 0

00 50 25 01 50 35 02 0 5 3 0 5 4 0 5 5 0 5 6 0 5 7 0 5 8 0 5 9 0 4 8500409 05 1 5

Figure 8 Radius of curvature

100 mm

400 mm

W

100 mm W 

Figure 7 Circular bending set-up and schematic  In this experiment we will be using the following formula:  M 

 E  =

 I 

 R

where:  M = Applied moment (Nm);  R = Radius of curvature (m);  –2  E = Young’s modulus for cantilever material (Nm );  I =

Using a vernier, measure the width and depth of the aluminium, brass and steel test beams. For each material, record the values next to the results tables and use them to calculate the second moment of area, I . Remove any clamps from the backboard and set up the beam as shown in Figure 7. Slide the digital dial test indicator into position on the beam and lock it using the thumbnut at the rear. Slide a knife-edge hanger on to each end of the beam as shown. Tap the frame lightly and zero the digital dial test indicator using the ‘origin’ button. Tapping the frame lightly each time, apply masses to the knife-edge hangers in increments as shown in Table 7. Record the digital dial test indicator reading for  each increment of mass. Repeat the procedure for the other two specimen materials filling in a new table.

4

Second moment of area of the cantilever (m ).

You will also need to use the following mathematical relationship: 2

 R

=

C 

+

4h 2

8h

Material: _______________________ 

 –2

E value: _____ Nm

Width, b: ____ mm

Mass at each end

Deflection

Applied moment

Radius of 

(g)

(mm)

(Nm)

curvature (m)

0 100 200 300 400 500

Table 7 Results for Experiment 4 (beam 1)

31

Depth, d : ____ mm

I : ___________ m

1/R 

M   /I ( 10 )

9

4

MM4:

Material: _______________________ 

 –2

E value: _____ Nm

Width, b: ____ mm

Mass at each end

Deflection

Applied moment

Radius of 

(g)

(mm)

(Nm)

curvature (m)

Deflections of Beams and Cantilevers

Depth, d : ____ mm

I : ___________ m

1/R 

M   /I ( 10 )

4

9

0 100 200 300 400 500

Table 8 Results for Experiment 4 (beam 2)

Material: _______________________ 

 –2

E value: _____ Nm

Width, b: ____ mm

Mass at each end

Deflection

Applied moment

Radius of 

(g)

(mm)

(Nm)

curvature (m)

Depth, d : ____ mm

1/R 

I : ___________ m

4

9

M   /I ( 10 )

0 100 200 300 400 500

Table 9 Results for Experiment 4 (beam 3) From the load values calculate the applied moment in  Nm. From the deflection calculate values for the radius of curvature in m. Then complete the table by calculating 1/ R and M / I .

32

Plot a graph of   M / I  versus 1/ R. Is this a linear  relationship? If so, what is the value of the gradient.

MM4: Deflections of Beams and Cantilevers

NOTES:

33

34

MM5: Bending Stress in a Beam - Introduction and Description - Experiments 1- Bending Stress in a Beam

35

SECTION 1.0 INTRODUCTION AND DESCRIPTION

Figure 1 Bending stress in a beam experiment 

Introduction This guide describes how to set up and perform Bending Stress in a Beam experiments. The equipment clearly demonstrates the principles involved and gives practical support to your studies.

Description Figure 1 shows the Bending Stress in a Beam experiment. It consists of an inverted aluminium T- beam, with strain gauges fixed on the section (the front panel shows the exact positions). The panel assembly and Load Cell apply load to the top of the beam at two positions each side of the strain gauges. Loading the beam in this way (rather than loading the beam at just one point) has two main advantages: •

It allows a gauge to be placed on the top of the beam.

•

The constant bending moment area it creates gives better strain gauge performance and avoids stress concentration close to the gauge positions.

Strain gauges are sensors that experience a change in electrical resistance when stretched or compressed. Strain gauges are made from a metal foil formed in a zigzag pattern. They are only a few microns thick so they are mounted on a backing sheet. The backing sheet electrically insulates the zigzag element and supports it so it does not collapse when handled. The T-beam has strain gauges bonded to it. These stretch and compress the same amount as the beam, so measure strain in the beam. If you look carefully at the equipment you will notice there is another set of st rain gauges. These are called

36

MM5: Bending

Stress in a Beam

dummy gauges. The dummy gauges, and how the way they are connected in t he electrical circuit, help reduce inaccurate readings caused by temperature changes and thermal expansion. The Digital Strain Display converts the change in electrical resistance of the strain gauges to show it as displacement (strain). It shows all the strains sensed by the strain gauges, r eading in microstrain. Look at the reference information on the unit. It is useful and you may need it to complete the experiments in this guide.

How to Set up the Equipment  The Bending Stress in a Beam experiment fits into a Test Frame. Figure 2 shows the Bending Stress in a Beam experiment in the Frame. Before setting up and using the equipment, always: •

Visually inspect all parts, including electrical leads, for damage or wear.

•

Check electrical connections are correct and secure.

•

Check all components are secure and fastenings are sufficiently tight.

•

Position the Test Frame safely. Make sure it is on a solid, level surface, is steady, and easily accessible.

Never apply excessive loads to any part of the equipment.

The following instructions may already have been completed for you. 1. Place an assembled Test Frame (refer to the separate instructions supplied with the Test Frame if necessary) on a workbench. Make sure the ‘window’ of the Test Frame is easily accessible. 2. There are two securing nuts in each of the side members of the frame (on the inner track). Move one securing nut from each side to the outer track (see ST R1 instruction sheet). Slide them to about the positions shown in Figure 2. Fix the two supports on to the frame in the same position. 3. Slide two nuts into position to hold the load cell. Fix the load cell leaving the screws slightly loose. 4. Lift the beam into position and level the ends of the beam with the frame. 5. Position the load cell so the hole in the fork reaches the hole of the loading position, and it is vertical. Tighten the load cell using the 6 mm A/F hexagonal key. Secure the fork using a pin. 6. Make sure the Digital Force Display is ‘on’. Connect the mini DIN lead from ‘Force Input 1’ on the Digital Force Display to the socket marked ‘Force Output’ on the left-hand side of the load cell. 7. With no load on the load cell (the pin should turn), use the control on the front of the load cell to set the reading to around zero. 8. Make sure the Digital Strain Display is ‘on’ and set to gauge configuration 1. Matching the number on the lead to the number on the socket, connect the strain gauges to the strain display. Leave the gauges for five minutes to warm up and reach a steady state.

37

MM5:

Figure 2 Bending stress in a beam experiment in the structures frame

38

Bending Stress in a Beam

MM5: Bending

Stress in a Beam

39

SECTION 2.0 EXPERIMENTS

Experiment 1: Bending Stress in a Beam

Figure 3 Beam set-up and schematic  As well as the information given on the unit you will need the following formulae: σ

 E  = --ε

Where: = Stress (Nm-2) ε = Strain  E = Young’s modulus for the beam material (Nm –2) σ

(Typically 69 x 10 9 Nm-2 or 69 GPa) and

 M  σ ----- = -- I   y (The bending equation) where:  M = Bending moment (Nm)  I = Second moment of area of the section (m 4) -2 σ = Stress (Nm )  y = Distance from the neutral axis (m)

Ensure the beam and Load Cell are properly aligned. Turn the thumbwheel on the Load Cell to apply a positive (downward) preload to the beam of about 100 N. Zero the Load Cell using the control. Take the nine zero strain readings by choosing the number with the selector switch. Fill in Table 1 with the zero force values. Increase the load to 100 N and note all nine of the strain readings. Repeat the procedure in 100 N increments to 500 N. Finally; gradually release the load and preload. Correct the strain reading values for zero (be careful with your signs!) and convert the load to a bending moment then fill in Table 2. From your results, plot a graph of strain against bending moment for all nine gauges (on the same graph). •

What is the relationship between the bending moment and the strain at the various positions?

40

MM5: Bending

Stress in a Beam

•

What do you notice about the strain gauge readings on opposite sides of the section? Should they be identical?

•

If the readings are not identical, give two reasons why.

Gauge number

Load (N) 0

100

200

300

400

500

70

87.5

1 2 3 4 5 6 7 8 9

Table 1 Results for Experiment 1 (uncorrected)

Gauge

Bending moment (Nm)

Number

0

1

0

2

0

3

0

4

0

5

0

6

0

7

0

8

0

9

0

17.5

35

52.5

Table 2 Results for Experiment 1 (corrected)

41

MM5:

Gauge

Nominal

Actual

Number 

Vertical

Vertical

position (mm)

position (mm)

Bending Stress in a Beam

Bending moment (Nm)

0 1

0

2,3

8

4,5

23

6,7

31.7

8,9

38.1

Table 3 Averaged strain readings for Experiment 1 Calculate the average strains from the pairs of gauges and enter your results in Table 3 (disregard the zero values). Carefully measure the actual strain gauge positions and enter the values into Table 3. Plot the strain against the relative vertical position of the strain gauge pairs on the same graph for each value of bending moment. Take the top of the beam as the datum. Calculate the second moment of area and position of the neutral axis for the section (use a vernier to measure the exact size of the section) and add the position of the neutral axis to the plot. •

What is the value of strain at the neutral axis?

•

Calculate the maximum stress in the section by turning the strains into stress values (at the maximum load). Compare this to the theoretical value.

•

Does the bending equation accurately predict the stress in the beam?

42

MM5: Bending Stress in a Beam

NOTES:

43

44

MM6: Torsion of Circular Sections

- Introduction and Description - Experiments 1- Torsional Deflection of a Solid Rod 2- The Effect of Rod Length on the Torsional Deflection 3- Comparison of Solid Rod and Tube

45

SECTION 1 INTRODUCTION AND DESCRIPTION

Figure 1 Torsion of circular sections experiment 

Introduction

How to Set up the Equipment 

This guide describes how to set up and perform experiments on the torsion of circular sections. It clearly demonstrates the principles involved and gives practical support to your studies.

The Torsion of Circular Sections experiment fits into a Test Frame. Figure 2 shows the Torsion of circular  sections experiment assembled in the Frame. Before setting up and using the equipment, always:

Description

•

Figure 1 shows the Torsion of Circular Sections experiment. It consists of a backboard with chucks for  gripping the test specimen at each end. The right-hand chuck connects to a load cell using an arm to measure torque. A protractor scale on the left-hand chuck  measures rotation. A thumbwheel on the protractor  scale twists specimens. Sliding the chuck along the  backboard alters the test specimen length. The backboard has some formulae and data printed on it. Note this information – it will be useful later.

46

• •

•

Visually inspect all parts, including electrical leads, for damage or wear. Check electrical connections are correct and secure. Check all components are secured correctly and fastenings are sufficiently tight. Position the Test Frame safely. Make sure it is on a solid level surface, is steady and easily accessible.

Never apply excessive loads to any part of the equipment.

MM6:

Torsion of Circular Sections

Figure 2 Torsion of circular sections in the structures frame Steps 1 to 3 of the following instructions may already have been completed for you. 1.

Place an assembled Test Frame (refer to the separate instructions supplied with the Test Frame if necessary) on a workbench. Make sure the ‘window’ of the Test Frame is easily accessible. 2. There are two securing nuts in each of the side members of the frame (on the inner track). Move one to the outer track (see STR1 instruction sheet) then slide them to approximately the positions shown by the thumbscrews in Figure 2. 3. Lift the backboard into position and have an assistant secure the backboard with thumbscrews into the securing nuts. If necessary, level the  backboard by loosening the thumbscrews on one side and tightening when ready.

4.

Make sure the Digital Force Display is ‘on’. Connect the mini DIN lead from ‘Force Input 1’ on the Digital Force Display to the socket marked ‘Force Output’ on to the right underside of the  backboard. 5. Carefully zero the force meter using the dial. Gently apply a small torque to the left-hand chuck  and release. If necessary, zero the meter again.

47

SECTION 2 EXPERIMENTS Experiment 1: Torsional Deflection of a Solid Rod  This experiment examines the relationship between torque and angular deflection of a solid circular section. Further work will show how the properties of the material affect this relationship. With a pencil and a rule, mark the steel and brass rods with these distances from the left-hand end (note that the rubber tip is on the right-hand end):

Force

Torque, T 

Angular deflection

(N)

(Nm)

(°)

0

0

0

1 2 3 4 5

•

15 mm, • 315 mm, • 365 mm, • 415 mm, • 465 mm, • 515 mm.

Table 3 Results for a brass rod 

Wind the thumbwheel down to its stop. Position the steel rod from the right-hand side with the rubber tipped end sticking out. Line up the first mark with the lefthand chuck (note the jaws of the chuck move outward as they close!). Tighten it fully using the chuck key in the three holes. Undo the four thumbnuts which stop the chuck from sliding. Slide the chuck until the last mark (515 mm) lines up with the right-hand chuck. This procedure sets the rod length at 500 mm. Fully tighten the right-hand chuck using the chuck key in each of the three holes. Wind the thumbwheel until the force meter reads 0.3 N to 0.5 N. Zero the force meter and the angle scale using the moveable pointer arm. Wind the thumbwheel so the force meter reads 5 N and then back to zero. If  the angle reading is not zero check the tightness of the chucks and start again. Take readings of the angle every 1 N of force: you should take the reading just as the reading changes. Take readings to a maximum of 5 N of force. Enter all the readings into Table 2. To convert the load cell readings to torque multiply by the torque arm length (0.05 m). Repeat the set up and procedure for the brass rod and enter your results in Table 3. Force

Torque, T 

Angular deflection

(N)

(Nm)

(°)

0

0

0

1 2 3 4 5

Table 2 Results for steel rod 

48

From your results, on the same graph plot torque versus angle for both rods Comment on the shape of the graph. What does it tell us about how angle of deflection varies because of  an increased torque? Name at least three applications or  situations where torsional deflection would undesirable and one application where it could be desirable or of  use. Take a look at the formulas on the backboard that  predicts the behaviour of the rods. What would happen to the relative stiffness of the rod if the diameter were increased from 3 mm to 4 mm?

MM6:

Torsion of Circular Sections

Further Work  Measure the diameter of both the rods with the vernier  as accurately as you can (remember the affect of a small error in the diameter!). Calculate  J  values for each rod using the formulae on the backboard of the equipment. Fill in Tables 4 and 5 from your experimental results to establish values of  TL and  J θ. Remember you must convert your angle measurements from degrees to radians (2π radians = 360°).

Diameter of brass section, d 

 _________ mm

Polar moment of inertia, J 

 _________ × 10

m

4

0.5 m

Length L Torque (Nm)

−12

Angular deflection, (rad)

TL

J 

10

13

0 0.05 0.10

Diameter of steel section, d 

 _________ mm

Polar moment of inertia, J 

 _________ × 10− m

Length L Torque (Nm)

0.15 12

4

0.20

0.5 m

Angular deflection, (rad)

TL

0.25

J 

10

13

Table 5 Calculated values for a brass rod 

0 0.05 0.10 0.15 0.20

Plot a graph of  TL against  J θ. Examine the torsion formula and say what the value of the gradient represents. Does the value compare favourably with typical ones?

0.25

Table 4 Calculated values for a steel rod 

49

MM6:

Torsion of Circular Sections

Experiment 2: The Effect of Rod Length on Torsional Deflection This experiment examines the relationship between torsional deflection and rod length at a constant torque. If you have completed Experiment 1 you will have already completed some of the following steps. In which case you can leave the brass rod in place at 500 mm long. With a pencil and a rule, mark the steel and brass rods these distances from the left-hand end (note that the rubber tip is on the right-hand end):

the angle reading is not zero check the tightness of the chucks and start again. Wind the thumbwheel so the torque is 0.15 Nm (a reading of 3 N) and note down the angle in Table 6. Reduce the length of the rod to the next mark (450 mm) and reset. Take a reading of angle at the same torque and record. Repeat this procedure for lengths down to 300 mm. Dia. of brass rod

•

15 mm, • 315 mm, • 365 mm, • 415 mm, • 465 mm, • 515 mm.

Length (m)

 _____ mm

Torque, T 

0.15 Nm

Angular deflection (°)

0.30 0.35 0.40

Wind the thumbwheel down to its stop. Position the steel rod from the right-hand side with the rubber tipped end sticking out. Line up the first mark with the lefthand chuck (note the jaws of the chuck move outward as they close!). Tighten it fully using the chuck key in each of the three holes. Undo the four thumbnuts which stop the chuck from sliding. Slide the chuck until the last mark (515 mm) lines up with the right-hand chuck. This procedure sets the rod length at 500 mm. Fully tighten the right-hand chuck using the chuck key in each of the three holes. Wind the thumbwheel until the force meter reads 0.3 N to 0.5 N. Zero the force meter and the angle scale using the moveable pointer arm. Wind the thumbwheel so the force meter reads 5 N and then back to zero. If 

50

0.45 0.50

Table 6 Results for a brass rod 

Plot a graph of angular deflection against rod length. Comment on the shape of the plot. On most front-wheel drive vehicles have unequal length drive shafts (from side-to-side). This is because of the gearbox position being at one end of the engine. This mismatch in length causes an undesirable effect on the steering as the car accelerates (that is, as torque from the engine increases). Why is that? What could eliminate the effect?

MM6:

Torsion of Circular Sections

Experiment 3: Comparison of Solid Rod and Tube This experiment compares the torsional deflection of a solid rod and a tube with a similar diameters. With a pencil and a rule mark the brass tube and  brass rods at 15 mm and 515 mm from the left-hand end (the end without the rubber tip). Wind the angle thumbwheel down to its stop. Position the brass tube in from the right-hand side with the rubber tip end sticking out. Line up the first mark  with the left-hand chuck (note the jaws of the chuck  move outward as they close!). Tighten it fully using the chuck key in each of the three holes. Undo the four thumbnuts that stop the chuck from sliding. Slide the chuck until the last mark (515 mm) lines up with the right-hand chuck. This sets the rod length at 500 mm. Fully tighten the right-hand chuck  using the chuck key in each of t he three holes. Wind the thumbwheel until the force meter reads 0.3 N to 0.5 N. Zero the force meter and the angle scale with the moveable pointer arm. Wind the thumbwheel so the force meter reads 5 N and then back to zero. If  the angle reading is not zero check the tightness of the chucks and start again. Take readings of the angle every 1 N of force: you should take the reading just as the reading changes. Take readings to a maximum of 5 N of force. Enter all the readings into Table 7. To convert the load cell readings to torque multiply by the torque arm length (0.05 m).

If you have completed Experiment 1, enter your  results for the solid brass rod in Table 7. If not, repeat the set up and procedure for the solid brass rod. Force (N)

Torque (Nm)

Rod angular  deflection (°)

Tube angular  deflection (°)

0 1 2 3 4 5

Table 7 Results for brass rod and tube

Calculate the  J  values for the solid rod and tube. To calculate  J  for a tube, find  J  for a solid of the same diameter then subtract  J  for the missing material in the centre. Examine your results and the  J  values you have calculated and comment on the effect of the missing material. 3 Assuming a density of 8450 kgm− for brass, work  out the nominal mass per unit length of both the tube and the solid rod. Comment on the efficiency of  designing torsional members out of tube instead of solid material.

51

MM6: Torsion of Circular Sections

NOTES:

52

M12: Buckling of Struts - Introduction and Description - Experiments 1- Buckling Load of as Pinned-End Strut 2- The Effect of End Conditions on the Buckling Load

53

SECTION 1 INTRODUCTION AND DESCRIPTION

Figure 1 Buckling of struts experiment 

54

MM12:

Buckling of Struts

Introduction This guide describes how to set up and perform experiments related to the Buckling of Struts. The equipment clearly demonstrates the principles involved and gives practical support to your studies.

Description Figure 1 shows the Buckling of Struts experiment. It consists of a back plate with a load cell at one end and a device to load the struts at the top. There are five aluminium alloy struts included in a holder on the back   plate Printed on the equipment are a number of  equations and pieces of information that you will find useful while using the equipment

How to Set Up the Equipment  The Buckling of Struts experiment fits into a test frame. Figure 2 shows the Buckling of Struts experiment in the Structures Test Frame. Before setting up and using the equipment, always:

• •

• •

Visually inspect all parts (including electrical leads) for damage or wear. Replace as necessary. Check electrical connections are correct and secure. Only a competent person must carry out electrical maintenance. Check all components are secured correctly and fastenings are sufficiently tight. Position the Test Frame safely. Make sure it is on a solid, level surface, is steady, and easily accessible.

Never apply excessive loads to any part of the equipment.

The following instructions may have already been completed for you. If so, go straight to Section 2. 1. Place an assembled Test Frame (refer to the separate instructions supplied with the Test Frame if  necessary) on a workbench. Make sure the ‘window’ of the Test Frame is easily accessible.

Figure 2  Buckling of struts experiment in the structures frame

55

MM12:

2.

On the Test Frame there are securing nuts in the  bottom groove of the top member and the top g rove of the bottom member. In each member slide two of  these to approximately the positions shown in Figure 2. 3. Lift up the STR12 unit onto the frame and have an assistant secure the unit to the frame using the thumbscrews and washers provided. 4. Make sure the Digital Force Display is ‘on’. Connect the mini DIN lead from ‘Force Input 1’ on the Digital Force Display to the socket marked ‘Force Output’ on the right-hand side of the unit.

56

5.

Buckling of Struts

Carefully zero the force meter using the dial on the front panel of the experiment. Gently apply a small load with a finger to the top of the load cell mechanism and release. Zero the meter again if  necessary. Repeat to ensure the meter returns to zero.

Note: If the meter is only ±1 N, lightly tap the frame (there may be a little ‘stiction’ and this should overcome it).

SECTION 2 EXPERIMENTS Experiment 1: Buckling Load of a Pinned-End Strut  Compressive members can be seen in many structures. They can form part of a framework for instance in a roof truss, or they can stand-alone; a water tower  support is an example of this. Unlike a tension member which will generally only fail if the ultimate tensile stress is exceeded, a compressive member can fail in two ways. The first is via rupture due to the direct stress, and the second is by an elastic mode of failure called Buckling. Generally, short wide compressive members that tend to fail by the material crushing are called columns. Long thin compressive members that tend to fail by buckling are called struts. When buckling occurs the strut will no longer carry any more load it will simply continue to displace i.e. its stiffness then becomes zero and it is useless as a structural member.

Figure 3 Experimental layout (pinned ends) In this experiment we will load struts until they buckle investigating the effect of the length of the strut. To  predict the buckling load we will use the Euler buckling formulae. Critical to the use of the Euler formulae is the slenderness ratio, which is the ratio of the length of the strut to its radius of gyration (l/k ). The Euler formulae  become inaccurate for struts with a l/k  ratio of less than 125 and this should be taken into account in any design

57

work. The struts provided have an l/k  ratio of between 520 and 870 to show clearly the buckling load and the deflected shape of the struts. In practice struts with an l/k  ratio of more than 200 are of little use in real structures. We will use the Euler buckling formula for a pinned strut:  P e

= π2 EI/L2

where:  P e  E   I   L

= = = =

Euler buckling load (N); Young’s modulus (Nm−1); 4 Second moment of area (m ); Length of strut (m).

Referring to Figure 3, fit the bottom chuck to the machine and remove the top chuck (to give 2 pinned ends). Select the shortest strut, number 1, and measure the cross section using the vernier provided and calculate the second moment of area,  I , for the strut. Adjust the position of the sliding crosshead to accept the strut using the thumbnuts to lock off the slider. Ensure that there is the maximum amount of travel available on the handwheel thread to compress the strut. Finally tighten the locking screws. Carefully back off the handwheel so that the strut is resting in the notch but not transmitting any load; rezero the forcemeter using the front panel control. Carefully start to load the strut. If the strut begins to  buckle to the left, “flick” the strut to the right and vice versa (this reduces any errors associated with the straightness of the strut). Turn the handwheel until there is no further increase in load (the load may peak and then drop as it settles into the notches). Record the final load in Table 1 under ‘buckling load’. Repeat with strut numbers 2, 3, 4 and 5 adjusting the crosshead as required to fit the strut. Take more care with the shorter struts, as the difference between the  buckling load and the load needed to obtain plastic deformation is quite small. Try loading each strut several times until a consistent result for each strut is achieved. Strut

Length

Buckling load

number 

(mm)

(N)

1

320

2

370

3

420

4

470

5

520

Table 1 Results for Experiment 1 Examine the Euler buckling equation and select an appropriate parameter to establish a linear relationship

MM12:

Buckling of Struts

 between the buckling load and the length of the strut (Hint: remember π,  E  and  I  are all constants). Calculate the values and enter them into Table 1 with an appropriate title. Plot a graph to prove the relationship is linear. Compare your experimental value to those calculated from the Euler formula by entering a theoretical line onto the graph. Does the Euler formula  predict the buckling load? It would be useful at this stage to calculate the gradient of the experimental results for use in Experiment 2.

58

MM12:

Buckling of Struts

Experiment 2: The Effect of End Conditions on the Buckling Load  Follow the same basic procedure as Experiment 1, but this time remove the bottom chuck and clamp the specimen using the cap head screw and plate to make a  pinned-fixed end condition. Record your results in 2 Table 2 and calculate the values of 1/ L for the struts.  Note that the test length of the struts is shorter than in Experiment 1 due to the allowance made for clamping the specimen. 2

Strut

Length

Buckling load

1/L

number 

(mm)

(N)

(m )

1

300

2

350

3

400

4

450

5

500

-2

Table 2 Results for Experiment 2 (pinned-fixed)  Now fit the top chuck with the two cap head screws and clamp both ends of the specimen, again this will reduce the experimental length of the specimen and you will 2 have to calculate new values for 1/ L . Take care when loading the shorter struts near to the buckling load. NOTE Do not continue to load the struts after the buckling load has been reached otherwise the struts will become permanently deformed!

Figure 4 Experimental layout for pinned-fixed  conditions

Enter your results into Table 3. 2

Strut

Length

Buckling load

1/L

number 

(mm)

(N)

(m )

1

280

2

330

3

380

4

430

5

480

-2

Table 3 Results for experiment 2 (fixed-fixed) Plot separate graphs of buckling load versus 1/ L2 and calculate the gradient of each line. Establish ratios  between each end condition (taking the pinned-pinned condition as 1). Examine the Euler buckling formulae for each end condition and confirm that the experimental and theoretical ratios are similar.

Figure 5 Experimental layout for fixed-fixed  conditions

59