SM1005 User Guide_0314.pdf

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© TecQuipment Ltd 2014

Do not reproduce or transmit this document in any form or byany means, electronic or mechanical, including photocopy,recording or any information storage and retrieval systemwithout the express permission of TecQuipment Limited.

TecQuipment has taken care to make the contents of this

manual accurate and up to date. However, if you find any errors,please let us know so we can rectify the problem.

TecQuipment supplies a Packing Contents List (PCL) with theequipment. Carefully check the contents of the package(s)against the list. If any items are missing or damaged, contactTecQuipment or the local agent.

SM1005 Loading and Buckling of

Struts User Guide

DB/AD/bs/0314

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8/17/2019 SM1005 User Guide_0314.pdf


TecQuipment Ltd User Guide

Contents

SM1005 Loading and Buckling of Struts

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

The Main Parts and the Load Meter (Display) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3The Struts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The Deflection Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Eccentric End Fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Weights and Hangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Adjustable and Removable Fixings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Versatile Data Acquisition System (VDAS®) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Young’s Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Noise Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Installation and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Location and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Electrical Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Connections (including VDAS®) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Notation, Useful Equations and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Useful Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Theory of Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Useful Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Experiment 1 - Deflection of a Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials . . . . . . . . . . . . . . . . . . . . 31Further Experiments with Beams - Beam Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Experiment 3 - The Deflected Shape of a Strut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Experiment 4 - The Euler Buckling Load using Pinned-end Struts . . . . . . . . . . . . . . . . . . . . 39Experiment 5 - Comparing Buckling loads with End Conditions . . . . . . . . . . . . . . . . . . . . . . 41Experiment 6 - The Southwell Plot and the Buckling Load . . . . . . . . . . . . . . . . . . . . . . . . . . 43Experiment 7 - The Southwell Plot and Eccentricity of Loading . . . . . . . . . . . . . . . . . . . . . . 45Further Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Experiment 1 - Deflection of a Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . 49Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials . . . . . . . . . . . . . . . 50Experiment 3 - The Deflected Shape of a Strut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Experiment 4 - The Euler Buckling Load using Pinned-end Struts . . . . . . . . . . . . . . . . 54Experiment 5 - Comparing Buckling loads with End Conditions . . . . . . . . . . . . . . . . . 55Experiment 6 - The Southwell Plot and the Buckling Load . . . . . . . . . . . . . . . . . . . . . . . 55Experiment 7 - The Southwell Plot and Eccentricity of Loading . . . . . . . . . . . . . . . . . . 56

Useful Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Maintenance, Spare Parts and Customer Care . . . . . . . . . . . . . . . . . . . . . . . 61

Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Spare Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Customer Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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TecQuipment Ltd User Guide


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TecQuipment Ltd 1 User Guide

SM1005 Loading and

Buckling of Struts

User Guide Introduction

Figure 1 Loading and Buckling of Struts (SM1005)

Engineers learning about structures need to know how to predict the effects of compression forces onstruts. They can use this information to decide the right type and thickness of materials for their own

designs. TecQuipment’s Loading and Buckling of Struts shows students how struts of different sizes,materials and cross-section deflect and buckle under load. This mimics struts in real applications, suchas roof supports in buildings or parts of frameworks in a structure.

The equipment includes ten struts of different metals, lengths and cross-sections. You can also buy theadditional pack of struts (SM1005a) for more experiment with struts of different materials, includingwood and glass fibre.

The apparatus teaches students about the most important factors that affect how well a strut can resista buckling load. These include:

• Different end conditions (how you hold or clamp the ends of a strut).• Eccentricities of loading.• The material and dimensions of the strut.

It also includes parts to allow basic beam bending tests, to help introduce students to bending theory.

This product works with VDAS ®

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User Guide 2 TecQuipment Ltd


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Description

Figure 2 Parts of the Loading and Buckling of Struts (SM1005)

The Main Parts and the Load Meter (Display)

The main part of the Loading and Buckling of Struts (SM1005) is a precision frame with adjustable feet.The frame has slots to hold the loading end and the load measuring end. The slots allow you to adjustthe distance between the two ends to fit different size struts. The frame also has slots on its front andback, each above a scale. These slots allow you to fit two adjustable knife-edge supports and thedeflection indicator, for simple beam bending tests.

The loading end has a hand wheel that turns a thread to give a compression force on the end of a strut.

The load measuring end has a load sensor connected with a unique mechanism. This mechanism allowsthe sensor to measure the axial force (buckling load) on the strut, but ignores any bending (rotating) forces.

The load sensor connects to a separate Load Meter (Display) that shows the axial force on the strut. TheLoad Meter has a socket for connection to TecQuipment’s optional VDAS®. VDAS® allows dataacquisition from this equipment, with the use of a suitable computer (not supplied).

The Load Display has two buttons. Press one button to zero the load display before you take anyreadings. The other button sets the display to hold a peak value of the force. This is useful to help findthe maximum buckling load of a strut. To set the peak hold, press and hold this button for a few seconds.

A small symbol appears in the display to show that the peak hold is working. The display now shows tworeadings, one is actual load, the other is the peak (maximum) value that the display has measured during

your experiment. Press and hold the peak hold button to cancel the peak hold.

Weights and parts to apply loadsfor beam bending and side loads

Load Meter

Deflectionindicator Load measuring end Loading endwith Hand

wheel

Standard set of different lengthstruts (supplied).

Standard set of different material andcross-section struts (supplied).

Knife-edgesupport

Knife-edgesupport

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The Struts

Figure 3 Each Standard Strut Has a Number Stamped Near Its End and Some have Extra Holes

Included as standard are ten struts. Each standard strut has a number stamped near one end (see Figure3). Some struts also have extra holes to accept the special Eccentric End Fittings. Six struts are the samematerial, thickness and width, but different lengths, to compare the effect of strut length. The longerstruts also allow for length ‘lost’ in the end fixings, so you can compare them with shorter struts. Fourstruts are of different materials, thickness and length.

The standard struts are all solid cross-section metal struts. For extra experiments you can also buy theSM1005A pack of optional struts. This includes struts of different material and cross-section, includingwooden struts, a compound strut and struts with angle and channel cross-section. See TechnicalDetails on page 9 for more details.

The Deflection Indicator

Figure 4 The Deflection Indicator Fits on Two Different Holders for Different Experiments

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A digital deflection indicator measures how much the struts deflect (bend). It mounts on an L-shapeholder for strut experiment on the top of the base. It also mounts on a flat holder on the side of the base

for beam bending experiments. Supplied with the equipment is a cable to connect the indicator toTecQuipment’s optional VDAS®.

An adjustable weight hanger, knife edge hanger and pulley allow students to apply a light biasing load

or side load to the strut under test. This is good for side load tests, and to prove any theory about strutsthat are already curved. Students also use the weight hanger and knife edge hanger to apply a load forbeam bending tests.

Eccentric End Fittings

Figure 5 The Eccentric End Fittings

Also included are two special end fittings to help apply out-of-centre loads to the struts, for tests oneccentric loading. They have two sides, to allow two different out-of-centre (eccentric) loading distanceson a standard strut (see Figure 6).

Figure 6 Out of Centre Loading with the Eccentric End Fittings

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Weights and Hangers

Figure 7 Weights and Weight Hangers

Included with the equipment are some extra parts. These are:

• A pulley assembly that fits in one of the deflection indicator holders. You use this with thecord and knife edge weight hanger to apply load for side load tests.

• A Weight Hanger, Weights and a second Knife Edge Weight Hanger. You use these to apply

a load for beam tests and for side load tests.

Adjustable and Removable Fixings

Figure 8 To Remove Fixings

TecQuipment put the fixings in the slots of the frame to suit a standard arrangement. However, if youneed to move the deflection holder or the beam supports to the opposite side of the frame, you can usea steel rule to remove the fixings (see Figure 8). Carefully insert them back into the slots where you needthem.

Pulley assembly

Knife EdgeWeight Hanger

WeightHanger andWeights

Cord andKnife Edge

WeightHanger

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Versatile Data Acquisition System (VDAS ® )

Figure 9 The VDAS® Hardware and Software

TecQuipment’s VDAS® is an optional extra for the Loading and Buckling of Struts. It is a two-part

product (Hardware and Software) that will:

• automatically log data from your experiments

• automatically calculate data for you

• save you time

• reduce errors

• create charts and tables of your data

• export your data for processing in other software

NOTE You will need a suitable computer (not supplied) to use TecQuipment’s VDAS®.

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Technical Details

Item Details

Nett Dimensions Main Unit: 1350 mm long x 500 mm front to back x 500 mm highDigital Load Display: 170 mm x 60 mm x 200 mm

Nett Weight Main Unit: 24 kgDigital Load Display: 1.4 kg

Electrical Supply (for theDigital Load Display PowerSupply

Input 90 VAC to 264 VAC50 Hz to 60 Hz at 1A Output 12 VDC at 5 A Centre Positive

Fuse No fuses fitted.

Operating Environment Indoor (laboratory) Altitude up to 2000 mOvervoltage category 2 (as specified in EN61010-1).Pollution degree 2 (as specified in EN61010-1).

Maximum Load Capacity ofLoad Measurement Unit

2000 N

Maximum strut lengthallowable

Approximately 800 mm in fixed - fixed ends.

Eccentric End Fittings Offset the loading centre of a 3 mm strut by 5 mm and 7.5 mm.

Standard StrutsNominal Dimensions

Strut 1 - Steel, 20 mm x 3 mm x 750 mm






Strut 7 - Brass, 19 mm x 4.8 mm x 750 mm

Strut 8 - Aluminium, 19 mm x 4.8 mm x 750 mm



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Young’s Modulus

Noise Levels

The noise levels recorded at this apparatus are less than 70 dB (A).

Optional StrutsNominal Dimensions

Strut A - Hardwood (Mahogany), 20 mm x 6 mm x 550 mm with steel knifeedge inserts.

Strut B - Plywood (Marine Ply), 20 mm x 6 mm x 550 mm with steel knife edgeinserts.

Strut C - Glass Fibre, 20 mm x 5.5 mm x 550 mm with steel knife edge inserts.

Strut D - Moulded Brass (Extruded brass), D shape, 19 mm x 4.5 mm x 550 mm

Strut E - Steel Compound, Two steel struts, bolted together. Both 13 mm x3 mm. One longer (650 mm) with knife edge ends, the other 640 mm.

Strut F - Aluminium channel, 13 mm x 13 mm and 1.5 mm thick wall x 750 mmwith steel end fittings.

Strut G - Aluminium angle, 13 mm x 13 mm and 1.5 mm thick wall x 750 mmwith steel end fittings.

Strut H - Aluminium angle, 13 mm x 13 mm and 1.5 mm thick wall x 750 mmwith steel end fittings.

Strut I - Steel Rectangular, 13 mm x 6.4 mm x 650 mm.

Strut J - Steel Round, 6.4 mm diameter x 650 mm.

Item Details

MaterialYoung’s Modulus

(Nominal)

Steel 207 GN.m -2 (207 GPa)

Aluminium 69 GN.m -2 (69 GPa)

Brass 105 GN.m -2 (105 GPa)

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Installation and Assembly

The terms left , right, front and rear of the apparatus refer to the operators’ position, facing the unit.

Location and Assembly

Use the Loading and Buckling of Struts in a clean, well-lit laboratory or classroom type area. Put it onthe top of a solid, level workbench.

The Loading and Buckling of Struts uses a bench area of 1350 mm x 500 mm. If you are to use theoptional VDAS®, allow room nearby for a computer.

Obey the manufacturer’s instructions (supplied) to fit the batteries to the digital deflection indicator.

Assembly 1. Fit the four legs to the main part of the frame, then fit the scale bar to the legs, underneath the base

unit (see Figure 10 ).

Figure 10 Fit the Legs to the Frame, then fit the Scale Bar

2. Adjust the feet on the legs until the frame is level.

3. Connect the power supply for the Load Display to the electrical supply, then connect the sensorcable from the load measuring end to the socket at the back of the Load Display.

4. Refer to the experiment for more assembly details.

NOTE• A wax coating may have been applied to parts of this apparatus to

prevent corrosion during transport. Remove the wax coating by usingparaffin or white spirit, applied with either a soft brush or a cloth .

• Follow any regulations that affect the installation, operation andmaintenance of this apparatus in the country where it is to be used.

WARNINGWhen assembled, the equipment weighs more than 20 kg. Ask anassistant to help you move it by holding its support legs.

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Electrical Connection

Use the cable supplied to connect the Load Display power supply to a single-phase electrical supply.

These are the colours of each individual conductor:

Connections (including VDAS ® )

If you are to use the optional VDAS® with the Loading and Buckling of Struts, read the VDAS® UserGuide and connect the Loading and Buckling of Struts to the VDAS-B Interface and computer as shownin Figure 11 .

Included with the SM1005 is the SPC cable that connects the digital deflection indicator to the optional VDAS-B Interface.

Figure 11 Connection to VDAS®

WARNING

Connect the apparatus to the supply through a plug and socket. The

apparatus must be connected to earth.The mains supply connector at the Power Supply is its mains disconnectdevice. Make sure it is always easily accessible.

GREEN AND YELLOW: EARTH E OR

BROWN: LIVE

BLUE: NEUTRAL

To mainselectrical supply

Force Sensor Load Cell

Digital Load Meter Optional VDAS Hardware

DIGITALINPUTS

POWER COMMSRS232

OR

HUB BOARD

DIGITALOUTPUT

DTI INPUT BOARD

1

2

3

4

DIGITALOUTPUT

ANALOGUE INPUT BOARD

INPUT SOCKET 1

INPUT SOCKET 2

VERSATILE DATAACQUISITION SYSTEM

V

Mains to low voltagepower supply

Digital Indicator

To mainselectrical supply

Mains to low voltagepower supply

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Notation, Useful Equations and Theory

This section only gives the basic information needed to do the experiments. For full theory, refer to thetextbooks listed in Maintenance, Spare Parts and Customer Care on page 61 .

Notation

Convers ions

Second Moment of Area: To convert mm 4 into m 4, multiply by 10 -12 (1 mm 4 = 1 x 10 -12 m 4)

Young’s Modulus: 1 GPa = 1 GN.m -2 = 1 kN.mm -2

Symbol Definition Units

F or P Force or a ‘Prop Force’ N

l ‘Effective Length’ m (or mm where stated)

L Total length m (or mm where stated)

A Area of cross-section m 2 (or mm 2 where stated)

σ Normal Stress N.m -2

ε Direct Strain Strain or micro-strain

K Column effective length factor -

I ‘Area moment of inertia’ or ‘Second moment of area’ m 4 (or mm 4 where stated)

E Young’s Modulus Pa (or N.m -2 or kN.mm -2 where stated)

x Distance along a strut m (or mm where stated)

y Displacement or deflection at a position along a strut m (or mm where stated)

yo Initial displacement or deflection at a position along astrut

m (or mm where stated)

yc Displacement or deflection at the centre of (half-wayalong) a strut


yco Initial displacement or deflection at the centre of (half-way along) a strut


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Useful Equations

Forc e (F), Prop Fo rce (P) and L oad

You use a set of weights to apply a load for beam tests, but in your calculations, you must use the forcecaused by the weights that make the load.

From Newton’s Law of F = ma :

Force (N) = Load (kg) x g (9.81 m.s -2)

Many standard equations use the letter F for force. This works for the beam experiments, but for strutexperiments, the force is a prop force , applied to the end of the strut. So, this theory uses the letter ( P )

for this force.

Youn g’s Modulus ( Ε )

This is a ratio of the stress divided by the strain on a material. An English physicist - Thomas Youngdiscovered it. It is a measure of the stiffness of a material (a stiffer material has a higher value of Young’sModulus). It is found by the equation:

Second Mo ment of Area

The second moment of area for a rectangular cross-section beam is:

(1)

Figure 12 Cross-section of a Beam

NOTE This is the minimum second moment of area for the strut or beam with a loadapplied so that it bends in its weakest direction.

E σε---=

I bd 3

12--------=

b

d

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Flexural Rigidity ( EI )

The second moment of area ( I ) of a beam or strut links to its dimensional strength.

The Young’s Modulus ( E ) of a beam or strut links to the strength of its material (while being used in itselastic region).

The product of these two values ( E x I ) gives a measure of the beam’s flexural rigidity. A higher flexuralrigidity means that you need more force to bend or flex a beam of a known length than an identicalbeam, that has a lower flexural rigidity. The length of the beam and other factors decide the actual forceneed to bend the beam.

Beam Bending Theory

Deflection of a Beam on Simple Supports

Figure 13 Beam on Two Supports

Figure 13 shows a simply supported beam on two supports, loaded at exactly the mid-point. For thisarrangement, the theoretical deflection at the mid-point ( yc) is:

(2)

Stiffness of a beam and Young’s Modulus

As shown later in this guide, struts are not usually perfect. They may not be perfectly straight. They maynot have a constant cross-sectional area along their length, or may not have a textbook value Young’sModulus. This can give errors in your results. To reduce these errors, you can do accurate stiffness testson each strut to find its true Young’s Modulus. To make things simple, you can use the simply supportedbeam test and its theory.

To do this, you can rearrange Equation 2 to give:

The right hand side of this equation relates to the beam’s stiffness. You need more force to bend a stifferbeam. This equation is also in the form y = mx. From this, a chart of F against 48 yc I / L

3 will give a straightline, with the gradient E . This will be a good average measurement of the Young’s Modulus along thebeam’s test length.

c FL

3

48 EI ------------=

F E 48 yc I

L3

-------------×=

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Figure 14 How to Use Your Results to Find Young’s Modulus

Beam Bending Moment

Figure 15 Beam Bending Theory - Bending Moment and Deflection

Figure 15 shows a cantilever beam. This is similar to a strut with one fixed end and one free end.

Bending Moment ( M ) = - F x ( L - x)

Also, from the differential equation of bending:

(3) M EI d 2 y

dx2

--------=

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Theory of Buckling

Fai lure of Lon g and Shor t Colum ns

Figure 16 Long and Short Columns

Columns are parts of structures that resist compressive axial loads (usually vertical loads). Stanchionsare upright (usually metal) columns in buildings. Struts are the smaller parts (members) that resistcompression in trusses and frames.

Columns can be either long ( slender ) or short (fat ) (see Figure 16 ). When compressed by too muchaxial load, long, slender columns fail by suddenly bending out of line (for example - a plastic ruler). Theybecome ‘ unstable’ and ‘ buckle’ at a maximum or ‘ critical (buckling ) load’ . Short, fat columns fail in severalways, mostly determined by the material they are made from (for example - concrete crushes and mildsteel yields).

In reality, most columns are ‘ intermediate’ - they fail by a combination of effects, where bending starts amaterial failure. However, this equipment examines a single effect - how slender struts or columns failby buckling.

A Long Column

A Short Column

AxialCompressive Load




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The Fou r Main Factors that Affect Bu cklin g

The words ‘long’ and ‘slender’ suggest the length ( L) and cross-section of a strut affect its buckling load.However, the cross-sectional shape also affects the buckling load. The ‘ second moment of area ’ ( I ) is ameasure of both the cross-sectional area and shape. It affects the stiffness of the strut.

In addition, the elastic bending of a strut depends on the Young’s Modulus ( E ) of the material it ismade from. So, the E value affects the buckling load.

Finally, and perhaps slightly less obviously, the buckling load of a slender column depends on its endconditions . Firmly clamped or ‘fixed ends’ help the column to withstand higher buckling loads than acolumn that has end fixings that are less rigid.

Eulers Maximu m (Crit ical) Ax ial Bu cklin g Lo ad and ‘EffectiveLength’

A Swiss mathematician - Leonhard Euler, created a formula that predicts the maximum (critical) axialbuckling load ( P cr ) of a strut.

(4)

Where K is an ‘effective length factor’ - determined by how you fix the ends of the strut. It is the ratioof the ‘effective length’ ( l ) between two points, to the overall length ( L) of the strut.

The equation again shows that the Young’s Modulus and cross-sectional dimensions (second momentof area) affect the maximum buckling load. It also shows that the buckling load varies linearly with these

quantities. This allows you to see that, for example, a steel strut with an E value of 200 GPa shouldbuckle at twenty times the load of an equivalent wooden strut, if the wood has an E value of only10 GPa.

Figure 17 Length Squared Against Buckling Load

The equation also shows that buckling load is inversely proportional to the square of a column’s length.

A chart of 1/ L2 against buckling load will be linear (see Figure 17 ). This proves that longer columns havelower buckling loads, but also shows that buckling load is sensitive to column length (doubling the

length will quarter the buckling load).

P cr π

2 EI

KL( )2

--------------=

1/ L2

Buckling Load

X

X

X

XResults for struts of different lengths

Longer struts

Shorter struts

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Figure 18 shows that the way you fix a strut decides its effective length. A strut with one fixed end hasan effective length of 0.7 of its total length. A strut with two fixed ends has an effective length of 0.5 ofits total length. This assumes that you fix the ends firmly - any movement in the ends will affect yourcalculations.

Figure 18 Eulers Equations for Different Strut End Conditions

Shape and Deflection of Bu ckled Struts

Figure 19 Shape of a Pin-ended Strut Under Load

P cr P cr

Buckled shape= half a complete sine wave= effective length

x

y

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Figure 19 shows the buckled shape of a Pin-ended strut that is initially straight . The shape issymmetrical (half a sine wave), and its bending moment:

(at a distance x and a deflection y)

Also, from the standard differential equation of bending:

Figure 20 Shape of a Fixed end Strut Under Load

Figure 19 shows that a pinned-end strut under load buckles so that it forms a symmetrical curve (halfcycle sine wave). Its effective length is the full length of the strut. Figure 20 shows that a fixed-endedstrut buckles so that it forms a full sine wave, but its effective length (that corresponds with the pinned-end strut) is only half its entire length. So, we can consider a fixed-end strut to have half the effectivelength of the pinned-end strut . Figure 18 gives the Euler equation for the fixed end strut. It shows thata fixed-end strut has four times the buckling load of an equivalent pin-ended strut.

Figure 21 Fixed and Pin-ended Strut

For a strut that is fixed at one end but pinned at the other (see Figure 21 ), it is not possible to predictits effective length precisely by looking at it, but bending theory shows mathematically that it isapproximately 0.7 L , so that its buckling load is slightly greater than twice (2.04 times, more accurately)the buckling load of the pinned-end strut. It’s shape is approximately 1/2 of a sine wave.

M P cr y× – =

M EI d 2 y

dx2

--------=

l = 0.5 L

L

= one complete c y cle of a sine wave

= half a complete c y cle of a sine wave= effective length

P cr P cr

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From this theory, if the pinned ends condition has a buckling load of 1 kN, the fixed and pinned endcondition has a buckling load twice this (2 kN). The fixed ends condition has a buckling load of fourtimes this (4 kN).

Figure 22 Buckled Shape of an Initially Curved Strut

Figure 22 shows the buckled shape of a strut that already has a displacement at its central position. Thiscurve is symmetrical (a half cycle of a sine wave). The deflection equation for any point along the initialcurve is:

(5)

The bending equation ( 3) becomes:

OR

The solution to this equation is:

Where

At the conditions y = 0, x = 0, L gives A = 0 and B = 0, so:

OR

So

(6)

yo ycoπ x L------ sin=

EI d 2 y

dx

2-------- P y y o

+( ) – = d 2 y

dx

2-------- P

EI ----- - y+ P

EI ------ yco

π x L------sin – =

A µ xcos B µ xsinµ

2 yco

π2

L2

----- µ2

–

----------------- π x L------sin+ +=

µ2 P

EI ----- -=

µ2 yco

π2

L2

----- µ2

–

----------------- π x L------sin= yc

yco

π2 EI

PL2

----------- 1 –

---------------------=

yc yco

P cr

P ------- 1 –

----------------=

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Slenderness Ratio and Bu ckling Stress

The radius of gyration, r , of a section is the distance from its centroid at which its area may be effectivelyconsidered to be concentrated.

As stated earlier, the second moment of area ( I ) links with the cross-sectional shape and area of a beamor strut. It also links with the area and the radius of gyration, so that:

Substituting this in the Euler buckling equation gives:

(7)

As shown earlier, stress ( σ ) is the force divided by the area ( F/A ). From this, the stress ( σ cr ) at the bucklingload is:

(8)

Substituting with Equation 7, we get:

(9)

and:

(10)

KL/r (or l/r in this theory) is the slenderness ratio of the strut. It is a measure of buckling resistance.Equation ( 10 ) shows that buckling stress is inversely related to the square of the ratio. From this, you

should always use the minimum dimensions of your strut (for example - if you have a non-symmetricstrut), to calculate the slenderness ratio. This is because a strut always buckles in the direction thatmatches the weakest dimension of the strut. However, of course the end fixings affect this as well.

I Ar 2

=

P cr π

2 EI

KL( )2

-------------- π2 EAr

KL( )2

----------------= =

σ cr P cr

A-------=

σ cr π

2 Er

2

KL( )2

--------------=

σ cr π

2 E

KLr

------- 2

---------------=

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Effects of Imperfection s

Figure 23 Comparison of Straight and Curved Strut

Equation ( 4) depends on three assumptions:

• The strut has constant values of E and I , so that it is homogeneous (has constant materialproperties).

• The strut is prismatic (has constant cross-section and therefore I value).

• The strut is perfectly straight.

In reality, none of these assumptions can be perfectly true, especially straightness. This is important.Remember that Figure 22 (an already curved strut) and its theory shows a strut with a known lack ofinitial straightness. It also shows that deflection increases rapidly as the compressive loadapproaches its critical value .

The Southwel l Plot and ‘Eccentr ic i ty of L oading’

Figure 23 shows a comparison of the deflection under load of initially straight and curved struts. Withthe initially straight strut under perfect conditions, it only deflects when the load reaches the criticalvalue. This gives a clear visual display of the point of buckling. For the initially curved strut, the gradualincrease in deflection makes it difficult to see the point of buckling. In reality, struts are not perfect and

most will gradually deflect as you apply load.

To help with this, you can use a Southwell Plot . A rearranged version of Equation 6 gives this plot, so:

(11)

Initially straight strut

Initially curved strut

Deflection

Load

Critical (buckling) Load

yc P cr yc

P ---- yco – =

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Figure 24 Southwell Plot

Equation 6 shows that a plot of yc against yc/ P will be linear (see Figure 24 ) and that its slope gives anapproximate value of the critical load. Also, the intercept on the y-axis gives an approximate value forthe original central out-of-straightness or ‘ eccentricity of loading’ ( yco). This helps you to see howimperfect the strut is.

The Southwell Plot and Testing Struts

During the experiments, you will find two important factors that make the Southwell Plot a useful tool

for predicting the properties of a strut.1. You do not need to test a real strut to its critical (buckling) load to create the Southwell Plot. This

allows you to do tests without risk of damaging the struts.

2. For accurate results in some experiment, you need to do two tests and work out the averagebuckling load (due to the struts natural buckling direction). This is not necessary for the SouthwellPlot.

Gradient = P cr

Intercept = yco

yc

yc

P

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Experiments

Safety

Useful Notes

• Do these experiments in order, the results and procedures of the earlier experiments help youto understand the later experiments.

• The Euler theory assumes that the struts are perfectly straight, this will never be what happensin real life, and some of these experiments compare the theory and reality. It is impossible tomake a perfect strut, but TecQuipment take care to make sure the struts supplied arereasonably straight.

• The struts will give good results unless someone accidently bends them past their elastic limit.The experiments only use the struts within their elastic limit, so you can reuse them. However,if you do not use the equipment correctly you may bend the struts too far, so take care as youreach the buckling load.

• A strut is still useable, even if it has a slight curve. It is not useable if it has a sharp bend or‘kink’. You may straighten a slight bend in a strut. To do this, carefully bend it back in theopposite direction by hand, or use a set of rollers (if you have them). This may affect some

properties of the strut, but will not affect its bending and buckling properties.

WARNINGNever try to release a strut from its end fixings when it is under load.

Always reduce the load force to zero before you change or adjust a strut.

WARNING If you do not use the equipment as described in these instructions, its protective parts may not work correctly.

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Experiment 1 - Deflection of a Simply Supported Beam

A i m s

To verify the simple beam bending equation for a beam on two supports and show the background forthe Euler formula.

Procedure

1. Create a blank results table, similar to Table 1. If you have VDAS®, select ‘Beam Experiments’. Thesoftware will create a table for you automatically when you start taking readings.

Table 1 Blank Results Table

2. Loosen the fixings of the measuring end if necessary and move it to the end of the base (see Figure25 ).

3. Fit the two knife edge supports to the front side of the main base, above the measurement scale,so they are exactly 600 mm apart (for example - set them to 200 mm and 800 mm). Make sure thesharp edge is upwards. If necessary, use a screwdriver or the thumbscrews to slide the fixings alongin the slots so you can fit the thumbscrews (see Figure 26 ).

Beam Material:

Distance between supports ( L):

Second Moment of Area ( I ):

Young’s Modulus ( E ):

Load (g) Force (N)Measured

Deflection (mm)Measured

Deflection (m) FL 3Theoretical

Deflection ( y)

0 0

100

200

300

400

500

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Figure 25 Loosen the Fixings and Move the Measuring End Along to the End of the Base

Figure 26 Fit the Knife Edge Supports

4. Fit the flat plate digital deflection indicator holder to the front of the base, so that it is exactly mid-way between the two knife edge supports (for example - at 500 mm).

5. Find the 750 mm long steel specimen strut (number 1). Use an accurate vernier or micrometer andcarefully measure the dimensions of the strut. Use these to calculate the second moment of area forthe strut. If you have VDAS®, enter this value into the software. Also enter the material type and itsnominal Young’s modulus (see Technical Details on page 9 ).

6. Put the beam onto the knife edge supports.

NOTE From this point on, the strut works as a beam, so we will call it a beam, to makethings clearer.

Thumbscrew

Use ascrewdriver or athumbscrew toslide the fixingsalong.

Sharp edge upwards

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7. Fit the digital indicator into its holder, so its display faces forward. The beam will bend downwardsless than 10 mm in this experiment, so move the beam backwards temporarily. Adjust the deflectionindicator in its holder to allow it to measure approximately 10 mm of downward beam deflection(see Figure 27 ).

Figure 27 Temporarily Push the Beam Behind the Tip and Adjust the Deflection Indicator for 10 mmDownward movement

8. Adjust the beam to be central across the Knife Edge Supports (so that an equal amount of beam‘sticks out’ over the Knife Edge Supports) (see Figure 28 ). Or, use a pencil to mark the beam at itscentre (375 mm) and adjust it so that the pencil mark is just under the tip of the Deflection Indicator.

Figure 28 Adjust the Beam So that an Equal Amount Sticks Out (Overhangs) Each End

10 mm ofdownwardmovement

888

Equal‘overhang’

Beam

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9. Hook the Knife Edge Weight Hanger onto the beam at the mid position, just under the tip of theDeflection Indicator (see Figure 29 ).

Figure 29 Fit the Knife Edge Weight Hanger to the Beam, Just Under the Tip of the Deflection Indicator

10. Zero the deflection indicator. The deflection reading from this point onwards ignores any initialbend in the beam and any bend caused by the small weight of the Knife Edge Weight Hanger.

11. If you have VDAS®, enter the distances between supports and the deflection indicator position.

12. Fit the Hooked Weight Hanger to the bottom of the Knife Edge Weight Hanger. Add 9 x 10 gweights to give a total of 100 g load*. Gently tap the frame to reduce the effect of friction. Recordthe reading of the deflection indicator. If you are using VDAS®, enter the load value and click onthe ‘Record Data Values’ button.

13. Increase the load to 200 g, 300 g, 400 g and 500 g. At each increase, record the deflection.

Resul ts An alys is

Convert your load into force and if necessary, convert your deflection into metres (you can use mm ormetres, but the correct SI unit is metres). Plot a curve of force (vertical axis) against deflection (horizontalaxis) for the beam. What does your curve suggest about the behaviour of the beam?

Use your measurements of the beam dimensions to calculate the second moment of area for the beam.Refer to the Technical Details on page 9 to find the Young’s Modulus for the material that makes yourbeam. Use the simple beam bending equation in the theory section of this guide to find the theoreticaldeflection of the beam for each load. Add your theoretical curve to the same chart as the actual curveand compare the two curves.

Does the theory accurately predict the deflection of the beam? What are the possible sources of error-if any?

NOTE *The hooked Weight Hanger weighs 10 g, so you must allow for this.

Knife EdgeWeight Hanger

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Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials

A i m s

To test the struts and use your results to find their actual stiffness, and from this, find the Young’sModulus for the material that makes the strut.

Notes

In experiment 1 you use a nominal value of Young's Modulus for your calculations, which can causeerrors, as the metal that makes your struts may not have a consistent Young’s Modulus. Accurate testswith this equipment will give you more useful and accurate values for the Young’s Modulus for yourstruts. You can then use these values for more accurate results in later experiments.

Procedure

1. Create a blank table of results, similar to Table 2.


2. Repeat the procedure for experiment 1, using the 750 mm long steel, aluminium and brass struts.

Resul ts An alys is

Convert your load into force and if necessary, convert your deflection into metres (you can use mm ormetres, but the correct SI unit is metres). For each strut, create a chart of force (vertical axis) against48 yI/L3. Find the gradient of each curve to find the true Young’s Modulus for the metal used to makethe strut.

Compare your results to the nominal (textbook) values given in Young’s Modulus on page 10 . Do theycompare well?

Beam Material:Beam Width:Beam Thickness:Distance between supports ( L): Second Moment of Area ( I ):

Load (g) Force (N)Measured

Deflection (mm)Measured

Deflection (m) 48yI/L 3

0 0

100

200

300

400

500

Calculated Young’s Modulus ( E ):

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Further Experiments with Beams - Beam Deflection

The Loading and Buckling and Struts apparatus works well to prove theory for simple beams on twosupports. You can adjust the knife-edge supports to different positions and apply the load at differentpositions to create different beam experiments. You could also use the deflection indicator to measuredeflection at different positions along the beam, to find the deflection profile of the beam.

Sugg ested Procedure to Measure Deflection Profi le of a Beam

1. Choose your beam.

2. Set the knife edge supports to a suitable distance apart.

3. Check that your beam looks straight. Rest it on the supports.

4. Loosen the fixings for the deflection indicator and slide it to be as near as possible to one of thesupports. Tighten its fixings and set it to zero. This gives you a zero datum.

5. Loosen, move and re-tighten the deflection indicator along the base in 25 mm steps. At each step,record the deflection and position. This gives you a datum profile for the unloaded beam.

6. Add the load to the beam. Do not reset the deflection indicator. Move it to the same 25 mm spacedpositions you used in the last step and again measure the deflection. This gives you a profile for theloaded beam. Subtract the (datum) unloaded profile from the loaded profile to get a true deflectionprofile for the beam.

NOTE

Take care how you adjust and slide the deflection indicator holder along tomeasure deflection. Always re-tighten the fixings before you take a reading. Donot leave the fixings loose and slide the holder along as you take readings,because you will get unreliable results.

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Experiment 3 - The Deflected Shape of a Strut

A im

To prove the theory about the sinusoidal shape of buckled struts, for each end condition (fixing).

Notes

You will see immediately the sinusoidal shape of the strut as it buckles. Its amplitude will grow as theload increases towards the buckling load. So, in this experiment you add enough load to create areasonable and measurable deflection. The actual load value is not important.

You need two people to do this experiment correctly. One person to load the strut and record the positions for the deflection indicator, the other person to move the deflection indicator and record itsdeflection readings.

Procedure

1. Find the strut you need for your test and make a pencil mark at its mid-point (for example - makea pencil mark at 375 mm along a 750 mm strut). For reference, measure the thickness and widthof the strut.

2. Connect and switch on the Load Display. Allow a few minutes for the display and the load cell ofthe measuring end to warm up. Tap the load measuring end to remove any effects of friction, thenzero the display.

3. Turn the hand wheel of the loading end to give 5 to 10 mm gap behind its chuck (see Figure 30 ).

Figure 30 Turn the Hand Wheel to Give 5 to 10 mm Gap Behind the Chuck

NOTEThis procedure works for all struts and all ends conditions. TecQuipmentrecommend that you start with pinned - pinned ends condition and the 600 mmsteel strut.

5 to 10 mm

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4. Use the hexagon tool supplied, to loosen the four screws securing the loading end and slide it alongthe base until your strut fits into each chuck for the end condition you need, as shown in Figure 31 .Re-tighten the four screws.

Figure 31 Different End Fixings

NOTE

* Note that for the fixed-pinned ends condition, you may need to use the 1 mm

or 3 mm spacers in the chucks to keep the strut loading accurately aligned. Thishelps to prevent an eccentric fixing.

Pinned - Pinned Ends

Fixed - Fixed Ends

Fixed - Pinned Ends

*

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5. Fit the deflection indicator on its L-shaped holder, to the top of the base (see Figure 32 ).

6. Adjust it so its tip touches your pencil mark, half-way along the strut.

7. In this experiment, your strut will only bend by a maximum of 10 mm away from the DeflectionIndicator, so adjust it in its holder, so that its tip will extend at least 10 mm when the strut bends.

Figure 32 Fit the Deflection Indicator to its L-shape Holder and Fit it to the Top of the Base

8. Create a table of results, similar to Table 3. If you are to use VDAS®, select the Strut Experiments.The software will create your results table for you automatically.

9. Use the large hand wheel to apply a small force to the strut. Check that its bends away from thedeflection indicator. If not, reduce the force and turn the strut around.

10. Use the large hand wheel and carefully add a small load to the strut (less than 5 N). This helps tocheck the end fixings are holding the strut securely, especially if you are testing with the pinnedends condition.

11. Zero the deflection indicator reading.

NOTE There are two holders that will hold the deflection indicator. You must use the L-shaped holder for the strut experiments.

NOTE If the indicator has a +/- button, set it to show a positive number when it extends.

Pencil mark at halfway

along the strut.

Adjust the Indicator to allow 10 mm ofmovement, this way (outwards).

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Beam Material:Beam Length:Beam Dimensions:

End Fixing Conditions:Load:

DeflectionPosition

(25 mm steps)

DeflectionReading(Datum)

DeflectionReading(Loaded)

ActualDeflection(Loaded -Datum)

0 (Mid Point) 0

Right(Positive)

+25 mm

+50 mm

+75 mm

+100 mm

+125 mm

+150 mm

+175 mm

+ 200 mm

+ 225 mm

+ 250 mm

+ 275 mm

Left(Negative)

-25 mm

-50 mm

-75 mm

-100 mm

-125 mm

-150 mm

-175 mm

-200 mm

-225 mm

-250 mm

-275 mm

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12. Move the indicator along the strut, from the halfway pencil mark towards the right in 25 mm steps. At each step, re-tighten the deflection indicator fixings before you take a deflection reading. If youare to use VDAS®, remember to record the deflection indicator position and click the ‘Record Data

Values’ button.

13. When you have reached as far right as you can go, set the indicator back to the halfway pencil marksand move in 25 mm steps to the left of the strut, recording the deflection results as negative values(as shown in the results table).

14. Move the deflection indicator back to the halfway point. Use the hand wheel to load the strut untilthe central deflection reaches approximately 6 mm. Record the load for reference.

15. Repeat steps 12 and 13 , recording deflected readings for the loaded strut.

16. Repeat the experiment with fixed - fixed end conditions and strut number 3.

17. Repeat the experiment with fixed - pinned end conditions and strut number 4. Use the loading endas the fixed end.

Resul ts An alys is

As shown in the results table for each test, subtract your unloaded (datum) results from the loadedresults to get the actual deflection. Be careful with your signs when doing this.

Plot a graph of deflection (vertical axis) against position along the strut (horizontal axis). Make sure yourhorizontal axis has an equal negative and positive scale.

For the pinned ends condition, over your results, draw a half sine wave of the same amplitude and cyclelength. For the fixed ends condition, draw a full sine wave over your results. For the fixed - pinned endscondition, draw a 3/4 sine wave over your results.

Do your results match the theory for sine wave shapes?

NOTE Just as in the test with struts as beams, you must now find the unloaded (datum)shape of the strut and subtract this from the loaded shape to find the actualshape due to the load.

NOTE Your last positions may not be an exact 25 mm, because of the tip of thedeflection indicator, so just note the actual position.

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Experiment 4 - The Euler Buckling Load using Pinned-end Struts

A i m s

To compare theoretical buckling load with actual buckling loads of pinned end struts from experimentsand prove the theory and show its limits.

Procedure

1. Create a blank results table, similar to Table 4. If you are to use VDAS®, select the Strut Experiments.The software will create a table of results for you automatically.



3. Find the 750 mm steel strut. Use a micrometer or vernier and carefully measure its dimensions, andcalculate its second moment of area.

4. Fit the strut in the pinned ends condition as described in Experiment 3 - The Deflected Shapeof a Strut , but remove the deflection indicator.

5. Use the large hand wheel to load the strut slowly. As you turn the hand wheel, watch the loadreading and the deflection of the strut. When you see that the load does not increase, but the strut

is still deflecting, the strut has buckled. Record the ‘peak load’, shown in the Load Display. Releasethe load.

Strut Details

Peak(Buckling)

Load 1

Peak(Buckling)

Load 2

Average Peak(Buckling)

LoadTheoretical

Buckling LoadLength Material

SecondMoment of

Area

NOTE

You must now buckle the strut, then buckle it again, in the opposite direction. Thisgives you two test results, and you find the average peak (buckling) load to geta good result.

The first time the strut buckles, it buckles in its ‘natural’ direction.

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6. Apply a light load, and gently push the strut to make it buckle the opposite way to your last test.Increase the load until the strut buckles, and record the peak load.

7. Repeat the test for other struts of the same cross-section and second moment of area , butdifferent lengths.

Resul ts An alys is

For each strut, calculate the average peak (buckling) load.

Plot a curve of the length (vertical axis) against the buckling load. Use the second moment of area tocalculate the theoretical buckling load for each length and plot it on the same chart.

Does the Euler theory predict the buckling load well? You will notice that when the strut buckles in it'snatural direction that the load is lower. Why is that? (Hint - see to the assumptions made in the Eulertheory).

Your curve will be non-linear, so it is difficult to see errors. The Euler Buckling Formula shows that youcan plot 1/ L2 against buckling load to give a linear plot that makes it easier to compare results. This alsoshows that bucking load is proportional to 1/ L2.

NOTE As with all your results, use suitable units. SI units are best, but can give numberswith many decimal places.

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Experiment 5 - Comparing Buckling loads with End Conditions

A i m s

• To test a strut fixed with all three end conditions and prove the relationship between thebuckling load and the end conditions.

• To help show the ‘effective length’ principle.

Procedure

1. Create a blank results table, similar to Table 5. If you are to use VDAS®, select the Strut Experiments.The software will create a table of results for you automatically.



3. Find the 600 mm steel strut (number 5). Measure its dimensions accurately and find its secondmoment of area. Fit it in the pinned ends condition as described in Experiment 3 - TheDeflected Shape of a Strut , but remove the deflection indicator.

4. Use the large hand wheel to load the strut slowly. As you turn the hand wheel, watch the loadreading and the deflection of the strut. When you see that the load does not increase, but the strutis still deflecting, the strut has buckled. Record the ‘peak load’, shown in the Load Display. Releasethe load.

5. Apply a light load, and gently push the strut to make it buckle the opposite way to your last test.Increase the load until the strut buckles, and record the peak load.

6. Release the load and remove the strut.

Second Moment of Area for the Strut:

FixingCondition Strut

Peak(Buckling)

Load 1

Peak(Buckling)

Load 2

Average Peak(Buckling)

Load

Pinned - Pinned 5

Pinned - Fixed 4

Fixed - Fixed 3

NOTE

You must now buckle the strut, then buckle it again, in the opposite direction. Thisgives you two test results, and you find the average peak (buckling) load to get

a good result.The first time the strut buckles, it buckles in its ‘natural’ direction.

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7. Now find and fit strut number 4 in the fixed - pinned condition. Use the loading end as the fixedend. This strut is 25 mm longer than strut number 3, to allow for the length lost in one fixing. Thisgives a constant test length for correct comparisons.

8. Repeat the test and record the loads.

9. Repeat the test for strut number 3 in the fixed-fixed end condition. Again, the length of this strutallows for the length lost in the fixings, to give a fair comparison.

Resul ts An alys is

For each strut, calculate the average peak buckling load. Do the loads for each fixing condition followthe theory (fixed-fixed buckles at four times the load of pinned-pinned, and fixed-pinned buckles attwice the load of pinned-pinned).

Your results for the fixed-fixed condition may be lower than you expect. Can you explain why? Thinkabout the load on the fixings (chucks) as it buckles, and the effect it has on the end conditions.

Add your results from this experiment to those of the last experiment (if you have done it). What do youthink about the effective length idea?

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Experiment 6 - The Southwell Plot and the Buckling Load

A i m s

To show how to use the Southwell plot to find the buckling load of a strut, and prove its usefulness.

Procedure




3. Find the 600 mm steel strut (strut number 5).

4. Fit the strut in the pinned ends condition as described in Experiment 3 - The Deflected Shapeof a Strut .

5. Use the large hand wheel to load the strut slowly to get a deflection of 0.5 mm. As you turn thehand wheel, gently tap the base to help remove any friction in the deflection indicator. Watch theload reading and the deflection of the strut. Record the deflection and load at approximately0.5 mm intervals until you reach 4 mm deflection. Release the load.

Resul ts An alys is

Divide your central deflection ( yc) results by the load ( P ) at each deflection to complete your table.

Create a Southwell Plot of central deflection ( yc) against ( yc /P ), in fundamental units. From this, notethe gradient, to give you the buckling load of the strut. If you have done the earlier experiments, doesthe gradient agree with the theoretical and actual buckling loads for this strut?

Load (N)Deflection

(mm)Deflection

/Load

0 0

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Experiment 7 - The Southwell Plot and Eccentricity of Loading

Notes

The last experiment showed that the gradient of the Southwell plot gave us the Euler buckling load andcompared well with theory and actual experiment results. This plot can also show the effectiveeccentricity of loading, or out-of-straightness. As shown in the theory, imperfections in the strut causethe eccentricity of loading (also, the mechanical tolerances in the equipment may have a small effect).

From this, if you know the eccentricity of each of a set of otherwise identical struts, the Southwell plots for each should have the same gradient, but the intercept should match the known eccentricity plus orminus the other imperfections.

A i m s

To show how to use the Southwell Plot to identify the eccentricity of loading.

Notes

This test uses the longest strut (750 mm long) as the reference, with zero eccentricity. It then uses thenext longest strut fitted with the special end fittings to mimic a strut with a known eccentricity. The end

fittings make this second strut the same length as the first, for a fair comparison. You test the secondstrut with the minimum and maximum eccentricity that the end fittings will allow.

The deflection of each strut near to its ends will not be the same, due to the special end fittings. Butbecause you use relatively long struts, the actual deflection at the ends is relatively small (compared tothe deflection at the mid point of the strut), so the error is small.

Procedure


2. As shown in earlier experiments, set up the 750 mm long steel specimen strut (number 1) as apinned - pinned strut.

3. As in Experiment 4 - The Euler Buckling Load using Pinned-end Struts , test this strut inboth directions. Increase the load in steps of 0.5 mm deflection until it buckles (the load stops risingas fast, but deflection continues).

4. Fit the Eccentric End fittings to strut number 2, with both fittings set to give the smallest eccentricity(5 mm) (see Figure 6).

5. Fit strut number 2 and readjust the position of the deflection indicator to allow for the offset.

6. Repeat the test with this strut.

7. Reverse the end fittings at both ends to give the larger eccentricity, adjust the indicator again andrepeat the tests.

NOTEFor the longest strut, the maximum deflection will be less than 15 mm, so do notbend your struts more than this.

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Eccentricity (0, 5 mm or 7.5 mm):

Deflection (mm) Load 1(N) Load 2 (N) Average LoadDeflection/

Average Load

0 (0) 0 00.51.01.52.02.53.03.54.04.55.05.56.06.5

7.07.58.08.59.09.5

10.010.511.011.512.012.513.0

13.514.014.515.015.516.016.517.017.518.018.519.019.5

20.020.521.021.522.022.523.023.524.024.525.025.526.026.527.0

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Resul ts An alys is

For each line of results, find the average load and deflection/load.

Create one chart of load (vertical axis) against deflection. Add to this chart your results from each strut.

Create one Southwell Plot chart. Add to this chart your results from each strut.

Compare the results on the load against deflection chart. What effect does the offset have?

Compare the Southwell plots, especially the gradients. Are the gradients similar? Has the eccentricityaffected the buckling load?

Compare the eccentricities, (the intercept on the y axis) to the actual known offset made by the specialend fitting (5 mm and 7.5 mm). Do the results confirm what we expect (plus or minus the other non-ideal effects)?

Other than the effects of the end fittings, examine the equipment and see if you can find an important

part of its design that might affect eccentricity of loading on the struts.

NOTEThe way you draw a line through your results on the Southwell Plot affects theintercept greatly. Take care to draw a straight line through only the most linearpart of your results.

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Further Experiments

The Loading and Buckling of Struts (SM1005) allows many more experiments than those suggested inthis guide. For example - it has parts that allow more advanced study into the effects of additional lateral(side) loads.

To apply a lateral (side) load, see Figure 33 :

1. Fit your strut as described in Experiment 3 - The Deflected Shape of a Strut .

2. Fit the pulley into the flat holder to the side of the base.

3. Hook the knife edge weight hanger with cord around the strut. Lay the cord across the pulley.

4. You can now add a load to the cord to apply a side load to the strut.

Figure 33 How to Set Up a Lateral (side) Load.

Also, the extra specimens pack (SM1005A) will allow students to investigate:

1) The flexural rigidity and buckling loads for a further range of materials.

2) Tests on different engineering sections.

3) The effect of flexibility of end fittings.

4) The case of a compound (composite) Strut with imperfect shearing connections between the twocomponents.

Contact TecQuipment or your local agent for details.

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Results

Note: These results are sample results only, actual results may be slightly different.

Experiment 1 - Deflection of a Simply Supported Beam

Figure 34 Typical Results For Experiment 1

Your results should show that the theory predicts the deflection accurately, while you deflect the beamwithin the elastic limits of its material. As shown in the theory, errors can be due to manufacturingtolerances in the material - its Young’s Modulus may not be accurate. Also, the second moment of areamay only be accurate for part of the beam, as its cross-sectional dimensions may change slightly alongits length.

Steel Strut (Beam) Deflection

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5

Deflection (mm)

F o r c e

( N )

Actual Deflection

Theory Deflection

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Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials

Figure 35 Typical Results For Experiment 2 - Steel Strut

Figure 36 Typical Results For Experiment 2 - Aluminium Strut

Steel Strut (Beam) Force Against 48 yI / L 3

Slope = approx 200 GPa

0

1

2

3

4

5

6

0 5E-12 1E-11 1.5E-11 2E-11 2.5E-11 3E-11

48 yI / L 3

F o r c e

( N )

Aluminium Strut (Beam) Force Against 48 yI / L 3


0

1

2

3

4

5

6

0 1E-11 2E-11 3E-11 4E-11 5E-11 6E-11 7E-11 8E-11

48 yI / L 3

F o r c e

( N )

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Figure 37 Typical Results For Experiment 2 - Brass Strut

Your results should be similar to the nominal values, but will be more useful for your later experiments.

For reference, use a pencil to write your correct value for Young’s Modulus onto each of the beams youtest.

Brass Strut (Beam) Force Against 48 yI / L 3


0

1

2

3

4

5

6

0 1E-11 2E-11 3E-11 4E-11 5E-11 6E-11

48 yI / L 3

F o r c e

( N )

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Experiment 3 - The Deflected Shape of a Strut

Figure 38 Typical Results for a Pinned-Pinned End Strut

Figure 39 Typical Results for a Fixed-Fixed End Strut

Deflection of Pinned - Pinned Strut against a Half Sine Wave

0

1

2

3

4

5

6

7

-350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350

Position Along Strut (mm)

D

e f l e c

t i o n

( m m

)

Deflection of Fixed - Fixed Strut against a Full Sine Wave

0

1

2

3

4

5

6

7

-350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350

Position (mm)

D e

f l e c

t i o n

( m m

)

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Figure 40 Typical Results for a Fixed-Pinned End Strut

Your results should show that the theory accurately predicts the deflected shape of your struts.

Deflection of Fixed - Pinned Strut against 1/2 Sine Wave

0

1

2

3

4

5

6

7

-350 -250 -150 -50 50 150 250 350

Position Along Strut (mm)

D e

f l e c

t i o n

( m m

)

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Experiment 4 - The Euler Buckling Load using Pinned-end Struts

Figure 41 Typical Strut Length against Load Results

Figure 42 Typical Reciprocal Strut Length Squared against Load Results

Your results should be similar to the theory. The 1/ L2 chart should show the linearity of the lengthagainst load relationship.

Strut Length against Buckling Load

500

550

600

650

700

750

800

150 200 250 300 350

Buckling Load (N)

L e n g

t h o

f s

t r u t ( m m

) Test Results

Theoretical Results

1/ L Squared against Load

0

0.5

1

1.5

2

2.5

3

3.5

0 50 100 150 200 250 300 350

Load (N)

1 / L

S q u a r e

d ( m - 2 )

1/L Squared against actual results

1/L Squared against theory

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Experiment 5 - Comparing Buckling loads with End Conditions

Table 8 Typical Results for Experiment 5

Your results should show the pinned - fixed buckling load to be twice that of the pinned - pinnedbuckling load. The fixed - fixed buckling load should be four times that of the pinned-pinned buckling

load. However, the end fixings are not perfect, as they only use a simple mechanical clamp, so the endsof the strut might move slightly in the fixed-fixed condition (you would need to weld the fixings to thestrut for a better fixing method). This will give a slightly lower than expected buckling load for the fixed-

fixed ends condition.

Experiment 6 - The Southwell Plot and the Buckling Load

Figure 43 Typical Southwell Plot for the 600 mm Steel Strut with Pinned-Pinned Ends (I = 4.54 x 10 -11 )

This Southwell Plot gave a slope of 257 for a strut with a second moment area = 4.54 x 10 -11 . This is

very similar to the average buckling load of 254.5 N for this strut, found from earlier experiments, andalso the theoretical buckling load of 261 N found in Experiment 4 - The Euler Buckling Load usingPinned-end Struts .

FixingCondition Strut

Peak

(Buckling)Load 1

Peak

(Buckling)Load 2

Average Peak

(Buckling)Load Ratio

Pinned - Pinned 5 248 261 254.5 1

Pinned - Fixed 4 505 515 510 2

Fixed - Fixed 3 943 988 965.5 3.8

Southwell Plot for 600 mm Steel Pinned - Pinned Strut

Slope = 257

-2

-1

0

1

2

3

4

5

-0.005 0 0.005 0.01 0.015 0.02

Deflection/Load (mm/N)

D e f l e c

t i o n a

t c e n

t r e ( m m

)

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Experiment 7 - The Southwell Plot and Eccentricity of Loading

Figure 44 Typical Load against Deflection for the Three Eccentricities

Figure 45 Typical Southwell Plots for the Three Eccentricities

Test on Struts with Three Different Eccentricities

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30

Deflection (mm)

L o a

d ( N )

No fittings

Fittings set f or 5 mm eccentricity

Fittings set f or 7.5 mm eccentric ity

Peak (buckling) load = approx 150 N

Peak (buckling) load = approx 130 N

Peak (buckling) load= approx 110 N

Southwell Plots for Three Different Eccentricities

-10

-5

0

5

10

15

20

25

30

-0.05 0 0.05 0.1 0.15 0.2 0.25

Deflection/load (mm/N)

D e

f l e c t i o n

( m m

)

No Fittings

Fittings set for 5 mm eccentric ity

Fittings set for 7.5 mm eccentricity

All slopes = approx 155

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The load against deflection chart shows that the standard strut (no eccentricity fittings) is ‘stiffer’,because its initial rate of deflection is lower than the other two results. It also accepts more load than theother struts before it reaches its buckling load (when the curves become almost level). This shows thatincreasing eccentricity lowers the actual buckling load.

The three gradients of the Southwell Plot show that the Euler buckling load is similar (the gradients are

the similar) even though the eccentricity increases. The intercepts on the Southwell Plot are interesting.They show a ‘base line’ eccentricity of approximately 2 mm for the strut with no fittings. This baselineeccentricity is also present and in addition to the known eccentricities of the other two struts, to giveintercepts of approximately 7 mm and 10.5 mm.

This shows that the Southwell Plot is a reliable tool to help find the Euler buckling load of struts, evenwith large eccentricities. It also allows you to test a strut without danger of applying too much load anddamaging the strut, as you do not need to test it to the full buckling load.

Figure 46 shows the possible alignment errors (exaggerated for clarity) that might be present in theequipment, before or during loading, allowing for manufacturing tolerances.

Figure 46 Possible Alignment Errors (Exaggerated)

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Useful Textbooks

Structural and Stress Analysis

By T H G Megson

Published by John Wiley & Sons

ISBN 0 470 23563 2

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Maintenance, Spare Parts and Customer Care

Maintenance

General

Regularly check all parts of the equipment for damage, renew if necessary.

When not in use, store the equipment in a dry dust free area, preferably covered with a plastic sheet.

If the equipment becomes dirty, wipe the surfaces with a damp, clean cloth. Do not use abrasivecleaners.

Regularly check all fixings and fastenings for tightness; adjust where necessary.

Electr ical

You cannot repair the power supply for the Load Display. If it is faulty, replace it with a new and identicalpower supply.

Fuse Location

There are no user replaceable fuses on this equipment.

NOTE Renew or replace faulty or damaged parts or detachable cables with anequivalent item of the same type or rating.

WARNING Do not open the power supply.

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Spare Parts

Check the Packing Contents List to see what spare parts we send with the apparatus.

If you need technical help or spares, please contact your local TecQuipment agent, or contactTecQuipment direct.

When you ask for spares, please tell us:

• Your name

• The full name and address of your college, company or institution

• Your email address

• The TecQuipment product name and product reference

• The TecQuipment part number (if you know it)

• The serial number

• The year it was bought (if you know it)

Please give us as much detail as possible about the parts you need and check the details carefully before you contact us.

If the product is out of warranty, TecQuipment will let you know the price of the spare parts.

Customer Care

We hope you like our products and manuals. If you have any questions, please contact our CustomerCare department:

Telephone: +44 115 9722611

Fax: +44 115 973 1520

Email: [email protected]

For information about all TecQuipment products visit:

www.tecquipment.com

SM1005 User Guide_0314.pdf

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