lab Manual

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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.

Figure 3 Force diagram

The equation we will use in this experiment is:





l a l Wa 

=

cut)

(at

BM

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

Table 1 Grams to Newtons conversion table

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

the cut (in Nm) = Displayed force 0.125

Calculate the theoretical bending moment at the cut and complete Table 2. Mass (g) Load (N) Force (N) Experimental bending moment (Nm) Theoretical bending moment (Nm) 0 100 200 300 400 500

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?

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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.”

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

the cut (in Nm) = Displayed force 0.125

Calculate the support reactions ( R_A _and R_B_{) 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.

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MM2:Bending Moment in a Beam

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MM3: Shear Force in a Beam

--

Introduction and Description

--

Experiments

1-1- Shear Force Variation with Increasing Load

Shear Force Variation with Increasing Load

2-2- Shear Force Variation for Various

Shear Force Variation for Various Load

Load

Conditions

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SECTION 1 SECTION 1 INTRODUCTION

INTRODUCTION AND DES

AND DESCRIPTION

CRIPTION

Figure 1

Figure 1 Shear forces in Shear forces in a beam a beam experiment experiment

Introduction

How

How to

to Set

Set Up

Up the

the Equipment

Equipment

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

your studies.

The Shear Force in a Beam experiment fits into a Test The Shear Force in a Beam experiment fits into a Test Frame. Figure 2 shows the Shear Force of a Beam Frame. Figure 2 shows the Shear Force of a Beam experiment assembled in the

experiment assembled in the Frame.Frame.

Before setting up and using the equipment,

Before setting up and using the equipment,always:always:

Description

•• Visually inspect all parts, including electrical leads,Visually inspect all parts, including electrical leads,

for damage or wear. for damage or wear. Figure 1 shows the Shear Force in a Beam experiment.

Figure 1 shows the Shear Force in a Beam experiment. It consists of a beam which is ‘cut’. To stop the beam It consists of a beam which is ‘cut’. To stop the beam collapsing a mechanism, (which allows movement in collapsing a mechanism, (which allows movement in the shear direction only) bridges the cut on to a load cell the shear direction only) bridges the cut on to a load cell thus reacting (and measuring) the shear force. A digital thus reacting (and measuring) the shear force. A digital display shows the force from the load cell.

display shows the force from the load cell.

•

• Check electrical connections are correct and secure.Check electrical connections are correct and secure.

•

• Check all components are secured correctly andCheck all components are secured correctly and

fastenings are sufficiently tight. fastenings are sufficiently tight.

•

• Position the Test Frame safely. Make sure it isPosition the Test Frame safely. Make sure it is

mounted on a solid, level surface, is steady, and mounted on a solid, level surface, is steady, and easily accessible.

easily accessible. A diagram on the left-hand support of the beam

A diagram on the left-hand support of the beam shows the beam geometry and hanger positions. Hanger shows the beam geometry and hanger positions. Hanger supports are 20

supports are 20 mm apart, and have mm apart, and have a central groovea central groove

which positions the hangers. which positions the hangers.

Never

Never apply excessive loads to any part of theapply excessive loads to any part of the equipment.

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Figure 2

Figure 2 Shear force of Shear force of a beam a beam experiment in the structures frameexperiment in the structures frame

Steps 1 to 4 of the following instructions may already Steps 1 to 4 of the following instructions may already have been completed for you.

have been completed for you. 1.

1. Place an assePlace an assembled Test Fmbled Test Frame (reframe (refer to the er to the separateseparate

instructions supplied with the Test Frame if instructions supplied with the Test Frame if necessary) on a workbench. Make sure

necessary) on a workbench. Make sure the ‘window’the ‘window’

of the Test Frame is easily accessible. of the Test Frame is easily accessible. 2.

2. There are four There are four securing nuts in the securing nuts in the top member top member of of

the frame. Slide them to approximately the positions the frame. Slide them to approximately the positions

internal bars are sitting on the frame squarely. internal bars are sitting on the frame squarely. Position the support horizontally so the rolling pivot Position the support horizontally so the rolling pivot is in the middle of its travel. Tighten the is in the middle of its travel. Tighten the thumbscrews.

thumbscrews.

5. Make sure the Digital Force Display is ‘on’. 5. Make sure the Digital Force Display is ‘on’. Connect the mini DIN lead from ‘Force Input 1’ on Connect the mini DIN lead from ‘Force Input 1’ on the Digital Force Display to the socket marked the Digital Force Display to the socket marked ‘Force Output’ on the left-hand support of the ‘Force Output’ on the left-hand support of the

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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 R _B R _A l a 'Cut' 40 mm

The equation we will use in this experiment is:

Shear force at cut,

l a W S _c ₌

.

Wherea _{is the distance to the load (not the cut)}

Distancea_{= 260 mm}

100 0.98

200 1.96

300 2.94

400 3.92

500 4.90

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,

Shear force at the cut = Displayed force

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

Mass (g) Load (N) Experimental shear force (N) Theoretical shear force (N) 0 100 200 300 400 500

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?

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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.”

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 ( R_A _and R_B_{) 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.

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MM3: Shear Force in a Beam: Student Guide

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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

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SECTION 1 INTRODUCTION AND DESCRIPTION

Figure 1 Deflection of Beams and Cantilevers experiment

Introduction

Look at the reference information on the backboard.

It is useful and you may need it to complete the experiments in this guide.

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.

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.

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.

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.

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Figure 2 Deflections of Beams and Cantilevers experiment in the structures frame 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). Slide

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

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.

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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.

Figure 3 Cantilever set-up and schematic

100 0.98

200 1.96

300 2.94

400 3.92

500 4.90

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

Deflection = EI WL 3 3 where: W _{= Load (N)}

L _{= Distance from support to position of loading}

(m);

E _{= Young’s modulus for cantilever material (Nm} –2_);

I _{= Second moment of area of the cantilever (m}4_).

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 _.

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. 0 1 02 0 3 0 4 05 06 0 7 08 09 0 1 001 101 2 0 1 30 1 4 0 1 5 0 1 6 0 1 701 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 3 10 3 2 0 3 3 0 1 02 03 0 4 05 0 6 0 7 0 8 09 0 1 0 0 1 101 2 0 1 30 1 4 0 1 501 6 0 1 70 1 8 0 1 9 0 2 00 2 1 0 2 202 3 0 2 40 2 5 0 2 60 2 7 0 2 8 0 2 903 0 0 3 10 3 2 0 3 3 P R ES E T TOL. Z ERO/ABS ON/OF F W 200 mm

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 increment of mass.

Repeat the procedure for the other two materials and fill in a new table.

Material E value: ___________ Nm –2 Width b: ____________ mm I: _________________ m4 Depth d: ____________ mm Mass (g) Actual deflection (mm) Theoretical deflection (mm) 0 100 200 300 400 500

Table 1 Results for Experiment 1 (beam 1)

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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.

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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:

Maximum deflection = EI WL 48 3 where: W _{= Load (N);}

L_{= Distance from support to support (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. 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 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 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 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 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 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 02 9 0 3 0 0 3 1 0 PRESET TOL. ZERO/ABS ON/OFF W 200 mm l _{= 400 mm} 200 mm 490 500 510 9 0 1 0 0 1 1 0

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.

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.

E value: ___________ Nm –2 Width b: ____________ mm I: _________________ m4 Depth d: ____________ mm Mass (g) Actual deflection (mm) Theoretical deflection (mm) 0 100 200 300 400 500

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

Part 2

Set up the beam with the lengthl _{at 200 mm. Ensure the}

digital dial test indicator and load hanger are still central to the beam, as shown in Figure 5.

01 02 0 3 04 05 06 07 08 09 0 1 0 0 1 1 0 1 2 0 1 3 01 4 0 1 501 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 03 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 04 5 0 4 6 0 4 704 8 0 4 905 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 01 02 0 3 04 05 0 6 07 0 8 09 0 1 0 0 1 10 1 2 0 1 3 0 1 40 1 5 0 1 6 0 1 701 80 1 9 0 2 00 2 1 0 2 2 0 2 30 2 4 0 2 50 2 6 0 2 7 0 2 80 2 9 0 3 0 0 3 10 3 2 0 3 3 0 3 403 0 0 3 1 0 3 2 0 3 3 3 50 3 6 0 3 7 0 3 80 3 9 0 4 0 0 4 10 4 2 0 4 3 0 4 404 50 4 6 0 4 704 80 4 9 0 5 0 0 5 10 5 2 0 5 3 0 5 40 5 5 0 5 6 0 5 70 5 8 0 5 9 0 PRESET TOL. ZERO/ABS ON/OFF W l _{=200 mm} 0 0 4 1 0 4 2 0 4 3 1 9 0 2 0 0 2 1 0 2 2 0

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

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Length (mm) Deflection (mm) 200 260 320 380 440 500 560

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

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

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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.

01 02 03 04 05 06 07 08 09 0 1 0 0 1 1 0 1 2 0 1 3 01 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 02 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 04 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 05 7 0 5 8 0 5 9 0 01 0 2 03 0 4 05 0 6 067 0 8 0 9 0 17 0 8 09 0 1 0 0 1 101 20 1 3 0 1 4 0 1 50 1 6 0 1 7 0 1 80 1 9 0 2 00 2 1 0 2 2 0 2 30 2 4 0 2 50 2 6 0 2 7 0 2 80 2 9 0 3 0 0 3 10 3 2 0 3 3 0 3 40 3 5 0 3 6 0 3 70 3 8 0 3 90 4 0 0 4 1 0 4 20 4 3 0 4 4 0 4 50 4 6 0 4 70 4 8 0 4 9 0 5 00 5 1 0 5 2 0 5 30 5 4 0 5 5 0 5 60 5 7 0 5 8 0 5 90 PRESET TOL. ZERO/ABS ON/OFF W 200 mm 200 mm 5 0 0 5 1 0 5 2 0 5 3 0 9 0 1 0 0 1 1 0 1 2 0 x 600 mm

Figure 6 Simply supported beam set-up and schematic

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. Donot_{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.

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

Position from left (mm) Datum reading (mm) Loaded reading (mm) Deflection (mm) 0 20 40 60 80 100 150 200 250 300 350 400 450 500 550 600

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?

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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.

01 02 0 3 04 05 06 07 0 8 09 0 1 0 0 1 1 0 1 2 0 1 3 01 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 02 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 04 3 0 4 404 5 0 4 6 0 4 704 8 0 4 905 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 01 02 03 0 4 05 0 6 07 08 09 0 1 0 0 1 1 0 1 2 0 1 3 01 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 02 6 0 2 702 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 03 0 0 3 1 0 3 2 0 3 3 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 05 7 0 5 8 0 5 9 0 PRESET TOL. ZERO/ABS ON/OFF W W 400 mm 100 mm 100 mm 5 0 0 5 1 0 5 2 0 5 3 0 9 0 1 0 0 1 1 0 1 2 0

Figure 7 Circular bending set-up and schematic In this experiment we will be using the following formula: R E I M = where: M _{= Applied moment (Nm);} R_{= Radius of curvature (m);}

You will also need to use the following mathematical relationship: R ₌ h h C 8 4 2 2 + R C h = Radius of curvature (m); = Chord (m); = Height of chord (m). h C R

Figure 8 Radius of curvature

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.

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.

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Material: _______________________ E value: _____ Nm –2 Width, b: ____ mm Depth, d : ____ mm I : ___________ m4

Mass at each end (g) Deflection (mm) Applied moment (Nm) Radius of curvature (m) 1/R M /I ( 10 9 ) 0 100 200 300 400 500

Material: _______________________ E value: _____ Nm –2 Width, b: ____ mm Depth, d : ____ mm I : ___________ m4

Mass at each end (g) Deflection (mm) Applied moment (Nm) Radius of curvature (m) 1/R M /I ( 10 9 ) 0 100 200 300 400 500

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 _.

Plot a graph of M _/ I _{versus 1/} R_{. Is this a linear}

relationship? If so, what is the value of the gradient.

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MM5: Bending Stress in a Beam

-

Introduction and Description

-

Experiments

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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

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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 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.

Neverapply 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.

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MM5:_{Bending Stress in a Beam}

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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:

Where:

σ= Stress (Nm-2)

ε= Strain

E = Young’s modulus for the beam material (Nm –2)

(Typically 69 x 109Nm-2or 69 GPa)

and

(The bending equation) where:

M = Bending moment (Nm)

I = Second moment of area of the section (m4)

σ= Stress (Nm-2)

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 (down-ward) 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?

E

σ ε

---=

M

I

---

σ

y

---=

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• 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.

Table 1 Results for Experiment 1 (uncorrected)

Gauge Load (N) number 0 100 200 300 400 500 1 2 3 4 5 6 7 8 9

Gauge Bending moment (Nm)

Number 0 17.5 35 52.5 70 87.5 1 0 2 0 3 0 4 0 5 0 6 0

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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?

Gauge Number Nominal Vertical position (mm) Actual Vertical position (mm) Bending moment (Nm) 0 1 0 2,3 8 4,5 23 6,7 31.7 8,9 38.1

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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

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SECTION 1 INTRODUCTION AND DESCRIPTION

Figure 1 Torsion of circular sections experiment

Introduction

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.

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.

How to Set up the Equipment

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:

• Visually inspect all parts, including electrical leads,

for damage or wear.

• 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.

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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)

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.

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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.

Force (N) Torque, T (Nm) Angular deflection (°) 0 0 0 1 2 3 4 5

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):

•_{15 mm,}

Table 3 Results for a brass rod

•_{315 mm,}

•_{365 mm,} _{From your results, on the same graph plot torque versus}

angle for both rods

•_{415 mm,}

•_{465 mm,} _{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.

•_{515 mm.}

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 left-hand chuck (note the jaws of the chuck move outward as they close!). Tighten it fully using the chuck key in the three holes.

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?

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 (N) Torque, T (Nm) Angular deflection (°) 0 0 0 1 2 3 4 5

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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 steel section, d _________ mm

Polar moment of inertia, J _________ ×10−12 m4 Length L 0.5 m Torque (Nm) Angular deflection, (rad) TL J 10 0 0.05 0.10 0.15 0.20 0.25

Table 4 Calculated values for a steel rod

Diameter of brass section, d _________ mm

Polar moment of inertia, J _________ ×10−12m4

Length L 0.5 m Torque (Nm) Angular deflection, (rad) TL J 10 0 0.05 0.10 0.15 0.20 0.25

Table 5 Calculated values for a brass rod

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?

13

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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):

•_{15 mm,} •_{315 mm,} •_{365 mm,} •_{415 mm,} •_{465 mm,} •_{515 mm.}

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 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 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.

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 _____ mm Torque, T 0.15 Nm

Length (m) Angular deflection (°)

0.30 0.35 0.40 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?

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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.

Assuming a density of 8450 kgm−3 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.

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MM6: Torsion of Circular Sections

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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

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SECTION 1 INTRODUCTION AND DESCRIPTION

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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.

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.

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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.

5. 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).

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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.

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 _{= Euler buckling load (N);}

E _{= Young’s modulus (Nm}−1_);

I _{= Second moment of area (m}4_);

L _{= 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.

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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.

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Experiment 2: The Effect of End Conditions on the Buckling Load

Figure 4 Experimental layout for pinned-fixed conditions

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

Table 2 and calculate the values of 1/ L2 _{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. Strut number Length (mm) Buckling load (N) 1/L2 (m-2) 1 300 2 350 3 400 4 450 5 500

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

have to calculate new values for 1/ L2_{. 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!

Enter your results into Table 3.

Strut number Length (mm) Buckling load (N) 1/L2 (m-2) 1 280 2 330 3 380 4 430 5 480

Table 3 Results for experiment 2 (fixed-fixed)

Plot separate graphs of buckling load versus 1/ L2 _and