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INSTRUCTION MANUAL

HST3/2 Deflection of Trusses

Prepared by :- Dr. N.P.Roberts

Validated by :- Nigel R. Hart

Issue No. :- 10

Date :- 06 April 2001

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LABORATORY TECHNIQUE

Safety in the Laboratory

The principal hazards in using apparatus that demonstrates the static and dynamic performance of associated theorems and the assumptions involved are where rotary or linear motion occurs and where the handling of loose heavy items, for example weights, is part of the procedure.

Generally, rotating parts where the speed is more than a few revolutions per minute are shielded. Nevertheless a thoughtful approach to those sorts of experiments is a necessary part of the procedure and learning process.

Of the loose items the heavier weights must be regarded as the most dangerous objects. Should one of these fall onto the feet of those around the apparatus the potential for damage is present. Hence it is recommended that cast iron weights be handled carefully and when moving and placing the heavier ones (say 10 N upward) on load hangers this should be regarded as a two handed operation. It is surprisingly easy to spill a complete stack of weights off a hanger when adding a further one.

In addition to weights there are some heavy parts that have to be interchanged during some experiments and a similar approach using two hands where required is suggested. It may also be both sensible and necessary for two people to take part in changes to the apparatus

Success in the Laboratory

Work in the laboratory depends on understanding, observation and skill. In the first place a good understanding of the performance, and limitations, of experimental models is needed. To know about the theory involved is useful but not essential. In the second place keen observation leads to better results and avoidance of mechanical mistakes. Lastly, the way in which students handle the apparatus can influence the accuracy and speed of the work.

To help students gain experience and improve their experimental technique a range of information is offered in the following notes. The important points are highlighted in bold. Bear in mind that in the world of real civil engineering it is often necessary to check the behaviour of materials and structures using the methods and instruments of laboratory experiments.

Design of Experimental Models

The purpose of each experiment is to illustrate an item of structural theory, or to show how well simplifying assumptions in the applied mathematics correspond to actual behaviour. This often requires the model to exaggerate the behaviour of a real structure.

In order to achieve specific objectives each experiment has a particular arrangement best suited to the theoretical requirement. These arrangements of the apparatus are described in the Construction Appendix, where included, of each experimental Instruction Manual. Before starting an experiment students should read through the Instruction Manual and be prepared to follow the recommended procedure.

Increased deflections are usually achieved by using very flexible models. The stiffness depends on EI or EA so a change of material from steel (E = 205 kN/mm2) to aluminium (E about 1/3 E for steel) or a plastic (E about 1/80 E for steel) is a solution. The alternative is to use thin steel beams with a low I.

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One disadvantage experimentally is that friction in bearings may affect displacements and force measurements. The other is that large changes in dimension (geometry) of models must be accommodated if possible.

Results can be improved by using stiffer models and larger loads, but this reduces visual effects such as curvature of beams.

Sources of Resistance

A frictionless pin or bearing can be simulated by a knife edge, but horizontal movements demand ball bearings. These are packed with grease and fitted with shields to keep out dust and grit. Hence ball bearings have some torsional restraint which affects forces in the order of magnitude 1 N. This shows up as a difference in readings for loading and unloading.

Pin joints in trusses are also subject to friction which increases in proportion to the loading.

Dial gauges have springs which push the spindle outward. The spring force is in the range 1-2 N. If a dial gauge is moved during an experiment it may affect displacements of flexible beams.

Repeatability of Readings

The ability to obtain accurate and repeatable experimental results is generally a matter of care and technique. Of course it helps to know the sources of error and to recognise when the apparatus contributes to the variability of readings.

Frictional variation can be minimised by using vibration. The extent of the friction can be observed by first increasing and then decreasing an applied load by hand to get the difference in readings. Banging the HST1 frame in which the experiment is mounted will reduce the variation. A dial gauge may be tapped on the front with a pencil.

Cast iron weights for loading must always be applied gently. A load suddenly added will instantaneously apply twice its static value. Although weights are hand finished there is a manufacturing tolerance of ½%. This may affect linearity in purely elastic experimental readings.

Deflection gauges with analogue dials are used because they indicate the rate of change between successive readings visually. Although the outer circular scale can be rotated through 3600 of arc it is bad practice to adjust the scale zero position by more than 200 from its normal position (when the revolution counting pointer is on a particular mark). Take care to note which way the pointers move when a change occurs. Take care not to read just 2xx, for example, when the revolution counter moves from 22 through 0 to 2 but record the fact because the real reading is then 27xx. Also remember that when the revolution counter is on, for example, 4 and the big pointer is on 83 the true reading is 383 (NOT 483). In some experiments dial gauges are mounted on special brackets in order to avoid errors due to elastic movement of the HST1 frame.

Use of Computers

The numerical readings taken during an experiment can be processed by a computer and then printed in tables and/or graphs. The selection of a system and software is based on universal availability by potential users and hence depends on Lotus 1-2-3 and the IBM XT/AT or compatible computer. Dedicated macros are available for the experiments. The procedure for entering and processing results is the same for all experiments, and is explained in detail in a manual. It makes sense to study the manual on a once-for-all basis before starting on the laboratory work, as the manual has been written for a class demonstration purpose.

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DEFLECTIONS OF TRUSSES

INTRODUCTION

Plane trusses in their various forms constitute one of the most widely used structural components. They occur as bridges and roof support structures, they form a vital part of most cranes, and find important applications in the air frames of aircraft and spacecraft, to name but a few. Moreover in a large number of these applications it is common to use these trusses in a form in which the internal member forces are dependent only upon the externally imposed loads and truss geometry itself. This form of statically determinate truss provides considerable practical advantages. It allows for easier fabrication and removes the uncertainties of loading that would otherwise arise from temperature differentials or the thermal effects caused by welding, and it means that the force distribution is not sensitive to small movements of foundation supports.

It is now feasible to use a computer to analyse a truss with rigid joints. However, experience shows that the secondary stresses due to the rigid joints in place of pinned joints are small enough to be ignored. In effect this assumes that the overall factor of safety used in designing includes an allowance for these secondary stresses.

This experiment therefore acts as a general introduction to the functional design of a plane truss. It can be used to compare joint displacements with those predicted by the simple pin jointed theory. Being based on strain energy it shows which members have the greatest influence on

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HST3/2

deflection.TEXT BOOK

The method of finding the force in one member in the middle of a truss is explained in the reference :-

Structural Mechanics by Durka, Morgan/Williams

The deflection of a truss is explained fully in

Understanding Structural Mechanics by RobertsLIST OF PARTS

See Packing List at back of Instruction Manual.

APPARATUS

The experimental equipment provides for the construction of three regular plane trusses cantilevered from two wall springings by using a selection of members as in the following table.

Truss jointcentres(mm)

Actual Overalllength of

member (mm)

Quantity Number usedon truss

A B C361.30 319.30 1 - - 1300.00 258.00 3 3 3 1256.30 214.30 3 - - 3240.00 198.00 5 5 2 3180.00 138.00 5 2 5 1 90.00 48.00 1 - - 1

The members are made of steel and have a gross sectional area of 20 mm2 . Trusses A and B are based on a 3 : 4 : 5 triangle while C has a top member inclined at 20.50 as shown below.

The joints are constructed by attaching the members to sub-assemblies of swivelling lugs which simulate a pinned joint (albeit not entirely frictionless). It will be noted that the lugs are in aluminium alloy. The joints should be lubricated with oil containing silicon or PTFE

Deflections are measured by a dial gauge mounted on an adjustable clamp fixed on a two member sub-frame. To provide true deflections relative to the springings of the truss the sub-frame is attached to the springings, and can be inverted as required.

Load is applied to the joint L3 by a hanger and a link suspended from the central attachment plate. To prevent a horizontal buckling failure of the bottom compression boom a steady bar is

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provided to restrain it with adjustable forks.

EXPERIMENT

OBJECT

The object of this experiment is to compare the joint deflections of a model truss with the theoretical values assuming a pin-jointed truss.

PROCEDURE

It is anticipated that one of the trusses A, B or C will have been assembled before the experiment begins. (See the Construction Appendix for details).

Set up the dial gauge truly vertically over joint L3 using the top of the load attachment plate. Add the load hanger and a 20 N load. Regard this as the “no load” state.

Before taking readings put 100 N load on and off the hanger several times to stabilise the truss.

Add 100 N in 20 N increments recording the gauge reading at each load in Table 1.

Repeat the above procedure for all the other joints using the central collar on the joints. It will be found necessary to invert the dial gauge sub-frame for some joints.

Table 1Vertical deflections of the joints

Joint U1 x U2 + L1 L2 L3 xLoad

(N)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

RESULTS

Plot the joint deflections against load on a graph and draw the best fit straight lines through the points ignoring the zero load origin. Use the graph to assess the deflection at each joint for 100 N and enter the derived values in Table 2. Calculate the deflections of all the joints by the unit load method taking E = 204 kN/mm2 and add the values to Table 2. It is necessary to allow for the elasticity of the joint assemblies which are made of aluminium of various shapes and fittings of stainless steel. The nominal E value for each assembly can be taken as

This can be taken into account by using a modified value of E for the combined members as shown below.

The effective value of E for a member of constant cross section with a 50 mm length of lugs and

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50 K of steel leads to the formula

Note that Eeff varies with the length of the member.

Table 2Vertical deflections for 100 N at L3

Joint ExperimentalDeflection

(mm)

TheoreticalDeflection

(mm)

OBSERVATIONS

Was the truss behaving in a linear elastic manner?

How well did the theoretical and experimental deflections agree?

To reduce the deflections which are the best members to stiffen?

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THEORY APPENDIX

The deflection of any joint of a truss can be found by using Castigliano’s first theorem with the unit load method. The form of the solution is

where NW = force in each member due to given loadingn = force in each member due to a unit load at the position and in the direction

of the required deflectionL = length of each memberA = cross sectional area of each memberE = effective modulus of elasticity of each member

Member LengthL

AreaA

ElasticityEeff

Load ForcesNW

Unit Forcesn

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CONSTRUCTION APPENDIX

HST1

HST100 and HST100b

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The kit of parts supplied is intended for making up one of three "pin-jointed" trusses.

Note: Overall length of the steel members is 42 mm less than truss joint centres.

Before starting construction it is essential to identify the parts. As the joint lugs have the fixing screws set on a 50 mm pitch circle diameter it follows that the overall length of members is 42 mm less (that gives 50 mm less for the hole centres) than the dimensions of the trusses shown above. There are four identical joints with four lugs and a central collar, one joint with four lugs and a load attachment plate instead of the collar, and two sets of pairs of lugs which fit on the springings. Note that each joint has an M4 screw which acts as the pin on which the rest of the parts is assembled and retained by the M4 nut. Also note that the plane of the members of the frame is central to the joints.

The trusses are to be built off the left hand side of the HST1 frame. They should be assembled on a flat bench as described below and fixed to the frame to which the steady bar has already been attached.

Assembling a truss

1. The joints along the top or upper boom of the truss are designated U0 (springing), U1 and U2; along the bottom or lower boom the joints are L0 (springing), L1, L2 and L3 . This identifies the position and direction of any member - for example the member U0L1 is a diagonal downward from the top springing.

2. In describing the assembly, the parts of the joints will be defined as shown in the following diagram

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When laid out on the bench the back will be underneath and the front on top.

3. Take the pairs of lugs off the two springings and lay them on the bench to the left. Place the four identical joints at their positions and the load carrying joint at the right. Align the lugs to suit the truss being built in accordance with the system shown here. Pick the required members and lay them in their correct positions between the joints.

4. Take the joints apart carefully and lay each lug in its correct orientation.

5. Starting from the right (loading) end bolt the members to the lugs ensuring that the back lugs are behind the member and the front lugs are in front. This means the frame will be in a plane central to the joints when they are assembled. If there is difficulty in aligning the lugs with the member try putting the screw and nut on finger tight and then placing the member and lug in a vice with the nut underneath (it will be held by the vice) and using the hexagon wrench to tighten the fastening.

6. Working from the right assemble the joints and add the M4 retaining nuts as you proceed. Do the nuts up finger tight and then turn them back not more than a quarter turn.

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7. Fix the springings to the HST1 frame or the side extension HST100b on the HST100 bench mounted frame (see the HST100 Instruction Manual) at the positions shown. Lift the truss into position and put the springing lugs onto the pins. The upper springing will be ready for the retaining nut. There is a loose lug to fit on the lower springing before the retaining nut can be added.

8. The steady bar should be fixed on the front of the HST1 or the HST100 frame at 70 mm above the base. Insert the adjustable forks in the appropriate holes. Before setting the forks to restrain the truss ensure that the truss is hanging in the vertical plane of the centre line between the twin members of the frame (view it through one end of the frame). If it does not then insert card packing between one side of the springing and the frame as shown below. Finally set the forks to restrain the load attachment plate at the end and the bottom members close to the two joints but not directly under a joint.

9. Suspend the load hanger from the load attachment plate and hanger link.

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Attaching or inverting the dial gauge sub-frame

10. The sub-frame is attached to the rear of the springing brackets using the M6 threaded holes that hold the springing plates. Two long screws and tubular spacers are provided. The sub-frame can be inverted as shown by the dashed lines for measuring deflections of upper joints.

11. To move the dial gauge clamp from the inclined to horizontal rods of the sub-frame undo the screw at the apex of the frame and slide the clamp off the one rod and replace it on the other.

12. When fixing the dial over or under a joint make sure that the flat anvil is truly horizontal transversely so that sidesway of the truss does not give a false deflection reading.

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RESULTS APPENDIX

The following results were obtained by an experienced demonstrator.

Truss A



(N)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm) 0 0.01 0.04 0.00 0.01 0.00 20 0.12 0.11 0.36 0.32 0.11 0.11 0.31 0.30 0.55 0.55 40 0.23 0.22 0.67 0.63 0.21 0.21 0.62 0.61 1.11 1.11 60 0.34 0.33 0.98 0.94 0.31 0.31 0.93 0.92 1.65 1.65 80 0.46 0.45 1.31 1.27 0.41 0.41 1.23 1.22 2.21 2.21100 0.58 0.57 1.64 1.60 0.51 0.51 1.56 1.55 2.75 2.75



(mm)


(mm)U1 0.56 0.55L1 0.50 0.49U2 1.56 1.52L2 1.52 1.46L3 2.75 2.74

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Truss BTable 1

Vertical deflections of the joints


(N)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm) 0 0.00 -0.01 0.02 -0.01 0.16 20 0.05 0.05 0.11 0.12 0.06 0.04 0.11 0.12 0.39 0.23 40 0.10 0.10 0.24 0.25 0.11 0.09 0.23 0.24 0.62 0.46 60 0.15 0.15 0.36 0.37 0.16 0.14 0.35 0.36 0.83 0.67 80 0.20 0.20 0.49 0.50 0.21 0.19 0.47 0.48 1.05 0.89100 0.25 0.25 0.62 0.63 0.26 0.24 0.59 0.60 1.26 1.10



(mm)


(mm)U1 0.25 0.35L1 0.24 0.29U2 0.63 0.63L2 0.60 0.57L3 1.10 1.04

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Truss C



(N)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm)

DialRdg.(mm)

Defn.

(mm) 0 0.03 0.01 -0.01 0.02 0.06 20 0.05 0.02 0.15 0.14 0.02 0.03 0.16 0.14 0.61 0.55 40 0.08 0.05 0.28 0.27 0.05 0.06 0.31 0.29 1.15 1.09 60 0.11 0.08 0.42 0.41 0.08 0.09 0.46 0.44 1.68 1.62 80 0.13 0.10 0.55 0.54 0.10 0.11 0.59 0.57 2.22 2.16100 0.17 0.14 0.69 0.68 0.13 0.14 0.71 0.69 2.74 2.68



(mm)


(mm)U1 0.15 0.14L1 0.14 0.14U2 0.68 0.74L2 0.69 0.74L3 2.65 2.73

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Appendix to ResultsJoint Elasticity

Loading several joints with 100 N produced an average extension per joint of 0.055 mm on the overall length of 50 mm.

Taking these values with a fictitious area of 20 mm2 (the steel member c.s.a.) leads to an apparent modulus of elasticity

Hence

And were r = 45

Table 3Effective E of Members

LengthL

mmk

Overall Modulus

Eeff

KN/mm2

90.0 0.8 8.0180.0 2.6 15.4240.0 3.8 20.1256.3 4.13 21.3300.0 5.0 24.5361.3 6.23 28.8

MAINTENANCE APPENDIX

HST3/2. .

Generally speaking HI-PLAN equipment needs little maintenance since so far as possible materials and finishes are corrosion proof and long lasting. However, dial gauges are precision instruments and require correct treatment as explained below. Should a dial gauge fail (typically by accidental misuse) a replacement can be ordered by quoting the experiment and part number of the item on which the gauge is mounted. If re-calibration is involved (for example a load measuring pier or the 5 kN loading device) the complete item should be returned to Hi-Tech Limited if possible.

Dial Gauges

The moving spindle of a dial gauge is a honed fit in its bearings so the spindle must be kept completely clean and no lubricant is to be used. Dust or finger marks should be wiped off with a clean, dry lint-free cloth (like a cotton handkerchief). There will always be minute stiction in the gauge mechanism, and a light tap on the yellow face with a pencil will help to bring the gauge to its true reading.

Electrical Resistance Strain Gauges

Although gauges fixed in production are given protection against handling, they must be treated with care as the gauge itself is fragile. The flexible leads are not directly attached to gauges to reduce accidental damage. Spare gauges can be supplied, but not always with the same gauge factor. If re-calibration is involved the complete item should be returned to Hi-Tech Limited if possible.

Load Indicating Meters

Unless a customer has the expertise to identify the failed component in a meter, the whole unit must be returned to Hi-Tech Limited for servicing.

Replacement and Spares

These can be ordered using the experiment and part number plus a description.

Test Specimens

Most of the experiments are within the linear elastic range of the test specimen. In the case of plastic bending, batches of new beams or portals are available from Hi-Tech Limited. It is, of course, feasible for elastically deformed specimens to be bent back to shape by heating and working them. If this is done, be sure to finish by annealing the part worked on to restore the typical elastic yield of the black mild steel.

HST3/2, DEFLECTION OF TRUSSES

HST3/2. .

PACKING LIST

Comprises:

1 HST1617 Dial Gauge Sub Frame

1 HST1618 Steady Bar Assembly

1 HTB2 Text Book

1 HST3/2 Instruction Manual

1 HAC102 Dial Gauge 0.01 mm/division with Flat Anvil

1 Small Parts :-

2 Adjustable Springing Bracket Assembly 4 Assembled Joints 1 Assembled Load Hanger Joint 1 Member 319 mm 3 Member 258 mm 3 Member 214.3 mm 5 Member 198 mm 5 Member 138 mm 1 Member 48 mm 1 2 N Load Hanger 24 M4 x 16 mm Socket Head Cap Screws 24 M4 Hex. Nuts 1 7 mm Spanner 1 6 mm Hex. Wrench 1 3 mm Hex. Wrench 1 2.5 mm Hex. Wrench 1 10 mm Spanner

HST302W Set of Weights

6 20 N

ORDER No. SERIAL No.

SIGNED: CHECKED: Date:

HST3/2. .

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