INSTRUCTION MANUAL
HST3/1
Deflection Of Beams
Prepared by :-
Dr. N. P. Roberts
Validated by :-
Mr. Nigel R. Hart
Issue No. :-
10
Date :-
29 March 2001
HST3/1. Page 1.
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.
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.
HST3/1. Page 2.
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.
HST3/1. Page 3.
HST3/1
DEFLECTION OF BEAMS
INTRODUCTION
In designing structures it is often necessary to calculate deflections of beams. A numerical
method of doing this can be developed from a theorem stated by Castigliano, namely, for a
linear elastic structure the displacement.
 
δU
δP
where U = total strain energy due to the applied load
P = force at the point and in the direction of the required displacement
It can be shown that it is generally possible to evaluate the partial differential by the use of
Simpson's rule for calculating the area under the graph of the function that expresses the
solution.
This, then, is the third method of obtaining beam deflections and rotations, and it is the most
powerful and simple of all. In addition, it is entirely suitable for working with a computer.
Furthermore it leads to one general method for analysing redundant (indeterminate)
structures.
HST3/1. Page 4.
In carrying out this experiment to compare experimental displacements with calculated
values, the opportunity has been taken to introduce another theorem on reciprocity. Clerk
Maxwell stated that for any determinate or indeterminate structure the component of
displacement in a particular direction (1) at a point A due to a unit force acting at point B in
direction (2) is equal to the displacement at B in direction (2) due to a unit force acting at A in
direction (1). This principle of reciprocal displacements can be written mathematically as
AB = BA
or
AB = BA
where AB is the deflection at A due to a unit moment at B
and BA is the rotation at B due to a unit load at A.
The conditions attached to the reciprocity are that no external work is done at the support
reactions and the structure is in a linear elastic state.
TEXT BOOK
The use of strain energy for calculating displacements of all types of structure by the
application of Castigliano's first theorem is dealt with in the following reference:
Understanding Structural Mechanics by Roberts
LIST OF PARTS
See Packing List at back of Instruction Manual.
APPARATUS
An elastic steel beam 25 x 3 mm nominal cross section is provided with two special supports.
One end support has a pivoted bearing to which the beam or cantilever is clamped: the other
end support is similar except that the beam clamp permits the beam free longitudinal travel.
On both a moment arm from the underside allows tangential force to be applied at a radius of
150 mm and rotation to be measured at 100 mm by the dial gauge attached.
The tangential force is provided in either direction by horizontal cords passing over pulleys to
vertical load hangers. To counterbalance the arm and the dial gauge an ochre coloured load
hanger heavier than the standard hangers must always be used on the same side as the dial
gauge. A steel rod can be inserted to lock the moment arm vertically.
To provide an upward load or reaction a double pulley bracket and a cord assembly is used.
A clamp-on moment application fixture with a counterbalancing system is provided so that a
pure moment can be applied. The clamp-on part is free to deflect as the beam moves, while
its rotation is derived from two dial gauge readings. The moment is developed by equal up
and down forces acting through cords on a pair of 75 mm diameter torque wheels, one each
side of the beam.
HST3/1. Page 5.
EXPERIMENT
OBJECT
The objects of this experiment are to investigate Castigliano's method of calculating beam
displacements and to observe the reciprocity of displacements as stated by Maxwell.
PROCEDURE
Part 1
Set up the apparatus to provide a simply supported beam of 960 mm span by withdrawing the
locking rods in the end supports. Fix a hanger clamp at 240 mm from the right hand end (The
moment application device should be temporarily moved to the left and detached from the
beam as described in the Construction Appendix). Put load hangers on the two clamps and
initially set up the dial gauge over the non-central clamp to read about 1600. Level the beam
by placing 5 N weights on the inner hangers of the two end supports. By hand depress and
release the beam to ensure that it will slide horizontally in the roller fixture of the left hand
end support.
Read the deflection dial gauge and the support rotation dial gauges as a record of the "no
load" values. Apply 10 N to the mid-span hanger and again read the three dial gauges.
Record these
or loading 1. Remove the 10 N from mid span and apply it to the left hand (ochre coloured)
hanger at the right end support. Read and record the dial gauges for loading 2. Remove the
10 N.
Now move the beam deflection gauge to midspan and read the "no load" values of the three
dial gauges. Apply 10 N to the non-central hanger and note the readings of the three gauges
for loading 3. Remove the 10 N and
HST3/1. Page 6.
apply it to the ochre hanger at the right end again. Note the readings for this repeat of loading
2, (defined as loading 4) plus the beam deflection δCB .
4
Table 1
Deflections and Rotations of an 960 mm span beam
Loading
LHE
Dial
Rdg.
(0.01 mm)
¾ Span
Midspan
Rotation
(radians)
Dial
Rdg.
(0.01 mm)
Deflection
(mm)
Dial Rdg.
(0.01 mm)
RHE
Deflection
(mm)
Dial Rdg.
(0.01 mm)
Rotation
(radians)
(1) Zero
10 N Mid.
(2) Zero
10 N RHE
(3) Zero
10 N ¾
(4) Zero
10 N RHE
Part 2
With the beam set up as at the end of Part 1, insert the locking pin in the support at the left
end A. With no applied load note the beam deflection and right end rotation dial gauges.
Read and record them again when a 10 N load is applied at midspan. Remove the load. Use
table 2.
Insert a locking pin in the right end B so the beam is fully fixed. Note the "no load" dial
gauge reading at mid span. Apply the 10 N load and record the dial gauge reading. Add
another 10 N load to obtain a further set of readings.
Table 2
Deflections of beams with end fixing
Loading
(1) Zero
10 N Mid.
(2) Zero
10 N Mid.
20 N Mid.
LHE
Dial Rdg.
Rotation
(0.01 mm)
(radians)
LOCKED
LOCKED
Midspan
Dial Rdg.
(0.01 mm)
Deflection
(mm)
RHE
Dial Rdg.
(0.01 mm)
Rotation
(radians)
LOCKED
HST3/1. Page 7.
Part 3
Rearrange the beam with its 800 mm span and with the moment application unit clamped to
the beam at mid span. The end supports must be free to rotate, thus providing a simply
supported beam with a moment at mid-span. Set up the beam deflection dial gauge at a
quarter span position. Check that the beam is free to deflect. Read and record the five dial
gauge values, namely one for beam deflection, two for end rotations and the pair on the
moment applicator in Table 3.
Apply a moment by adding a pair of 5 N weights to the load hangers simultaneously, and read
the dial gauges for the deflection and the end rotations. Also at each load read the two dial
gauges on the moment application fixture. Repeat this by increments of 5 N up to a total of
20 N on each hanger. Take further gauge readings as the load is decreased in 5 N steps back
to zero.
Table 3
Slope and Deflection of Beam with Midspan
Load
2 x (N)
LHE
Dial Rdg Rotation
(.01mm) (.01 rad)
Dial Rdg
(.01mm)
Midspan
Dial Rdg
(.01mm)
Rotation
(.01 rad)
¾ Span
Dial Rdg Deflection
(.01mm)
(mm)
RHE
Dial Rdg Rotation
(.01mm)
(.01 rad)
RESULTS
Tabulate on pairs of diagrams as under the differences between the "no load" and "loaded"
dial gauge readings for Part 1. Calculate the rotations in radians (divide by 100 mm). For
loading 2 derive the displacements for an applied moment of 10 N.mm (divide the
experimental values by 150).
Loading 1
Loading 3
Note that the correlation between the pair of diagrams is the interchange of loading and
deflection positions. Compare the two deflections to test Maxwell's reciprocal theorem that
DC = CD
HST3/1. Page 8.
The next pair of diagrams is
Loading 1
Loading 2
(repeated)
There is one more pair of diagrams. Together they test the reciprocities.
BC = CD
BD = DB
In addition take two or three of the experimental displacements and check them by using
Castigliano's first theorem and the unit load method.
The results of Part 2 should be checked by calculation (see the Theory Appendix for a
shortened version of the unit load method).
Tabulate the results for Part 3 and plot a graph of the rotations and deflection against applied
moment. Draw the best fit straight lines to obtain average values of /M and /M. Compare
these with calculated values, and also the anti-symmetry of the rotations.
OBSERVATIONS
How well did the experimental results compare with the calculated values? Comment on the
various methods for calculation of displacements that have been explained in all the beam
theory work.
The testing for reciprocity was not the primary object of this experiment. How could the
results have been improved? For the first test set out the load and unit load diagrams and
comment on the feasibility of the reciprocal theorem.
CONCLUSIONS
Does Castigliano's theorem and the unit load method offer a universal way of calculating
displacements of beams?
HST3/1. Page 9.
CONSTRUCTION APPENDIX
HST1
HST100 and HST100a (4A)
HST3/1. Page 10.
Several of the items for this experiment can be permanently fixed in the HST1 or HST100
frame as follows. However, as part of the experiment it is necessary to move and replace the
moment application assembly, so instructions for that are given also. If the HST100 frame is
used ensure that the higher end is at the right hand side.
1.
Fix the right hand end support HSA114 (the one with the clamp plate for gripping the
beam) with its centre 120 mm from the right hand side of the frame.
2.
Fix the left hand end support HSA115 (this one has needle rollers in the beam holding
system) with its centre 960 mm from the right hand support.
3.
Anchor the dial gauge bracket 480 mm from the right hand side of the frame.
4.
Fix the double pulley bracket to the top of the frame so that the centre of the right
hand pulley is 625 mm from the right side of the frame.
or
4A.
When using the HST100 fix the HST100a Overhead Pulley Bracket at 640 mm from
the right side of the frame before attaching the double pulley bracket as in 4 above.
The bottom and top bars of the 100a must rotate in opposite directions to offset the
double pulley bracket to the left.
5.
Attach the moment application stand temporarily with the centre about 200 mm from
the left hand support. Unscrew the counterbalance cord attachment and lay the cord
and counterbalance aside. Lay the beam moment applicator on the base of the frame
as shown with the short cord and spacer under it. Put a hanger link on the spacer.
Drape the long cord over the top of the pulleys on the stand and fit a hanger link to the
spacer.
6. Remove the beam clamp plate from the right hand support and slide the beam into
position through the rollers of the left hand support. Press firmly on the top plate of
the roller system and finger tighten the four holding down screws. Replace the clamp
plate at the right hand support and use the Allen key to tighten the four screws evenly.
Depress the beam at mid-span to check that it can move horizontally through the
needle rollers at the left-hand support.
7.
Fix two hanger clamps on the beam at three-quarter and mid-span as shown
8.
The apparatus is now ready for Parts 1 & 2 of the experiment.
9.
For Part 3 of the experiment the mid-span hanger clamp and load hanger are removed
and replaced by the moment applicator as shown.
HST3/1. Page 11.
Replacing the moment application assembly
10.
Starting from position A (clause 5), where the cord and its attachment have been laid
aside, take out the two cap head screws and remove the beam clamp of the moment
applicator. Lift up the applicator turning it through 290 anti-clockwise, taking care to
keep the cords in the peripheral grooves of the torque wheels. The applicator will then
have the spindle between the torque wheels touching the underside of the beam. Refit
the clamp and the fixing screws so that the applicator can slide along the beam
(position B).
HST3/1. Page 12.
11.
Loosen the anchorage under the moment application stand and slide the applicator and
stand to the right until the applicator is at mid-span of the beam and the stand centre is
672 mm from the right hand side of the HST1 frame. Tighten the beam clamp.
12.
Take the counterbalance cord and screw the attachment into the beam clamp. Then
lay the cord over the double pulleys on the bracket already fixed to the top of the
frame (clause 4) and add the counterbalance.
Removing the moment application assembly
13.
This is the exact reverse of the above procedure, namely
(1)
Remove and lay aside the counterbalance and its cord and attachment
(2)
Loosen the beam clamp of the moment applicator
(3)
Loosen the moment stand anchorage
(4)
Slide the moment stand and applicator 165 mm leftward (position B)
(5)
Undo the beam clamp and remove it
(6)
Immediately following on (5) lower the applicator while turning it 290
clockwise
(7)
Park the applicator on the base of the HST1 frame (position A)
HST3/1. Page 13.
THEORY APPENDIX
The use of Castigliano's theorem with the unit load method for calculating displacements can
be simplified when working with statically indeterminate structures. As an example take a
12 m fully fixed beam with a uniformly distributed load of 20 kN/m and a flexural rigidity of
EI. Find the central deflection.
The first stage is to produce the bending moment diagram due to the load alone. This is a
standard case where the fixing moments are 2/3 of the mid-span bending moment due to the
load on a simply supported beam. (This was an area-moment proposition to make the net area
of the BMD zero)
Applying Castigliano's first theorem
C 

M W mdx
EI
It can be shown that the unit load can be applied to any convenient statically determinate form
of the proper structure. The possibilities for a fixed beam are
(1) a simply supported beam, or
(2) a cantilever
HST3/1. Page 14.
In this case (1) is better because of the symmetry.
The ordinate of the MW curve at quarter span is
MW
x = 3
MW
=
MSS - 240
=
20 x 2 

120x  - 240
2 

=
360 - 90 - 240 = 30
Using Simpson's rule
C  2 x
=
6
0  4 x 1.5 x 30  3 x 120
6 EI
1080
EI
(This can be compared with the area-moment calculation in 'Understanding Structural
Mechanics'.)
HST3/1. Page 15.
RESULTS APPENDIX
The following results were obtained by an experienced demonstrator.
Part 1
Table 1
Deflections and Rotations of an 960 mm Span Beam
Loading
(5) Zero
10 N Mid.
(6) Zero
10 N RHE
(7) Zero
10 N ¾
(8) Zero
10 N RHE
LHE
Dial Rdg.
Rotation
(0.01 mm) (radians)
1446
1983
0.054
1446
1228
-0.022
1459
1784
0.033
1459
1245
-0.021
Midspan
Dial Rdg. Deflection
(0.01 mm)
(mm)
1287
142
1287
2064
11.45
-7.77
¾ Span
Dial Rdg. Deflection
(0.01 mm)
(mm)
1421
233
11.88
1421
2109
-6.88
RHE
Dial Rdg.
Rotation
(0.01 mm) (radians)
1294
753
-0.054
1294
1716
0.042
1293
835
-0.046
1293
17101
0.042
HST3/1. Page 16.
Part 2
Table 2
Deflection of Beams with End Fixing
Loading
(3) Zero
10 N Mid.
(4) Zero
10 N Mid.
20 N Mid.
LHE
Dial Rdg.
Rotation
(0.01 mm)
(radians)
LOCKED
LOCKED
Midspan
Dial Rdg.
Deflection
(0.01 mm)
(mm)
1407
680
7.27
1390
979
4.11
583
8.07
RHE
Dial Rdg.
Rotation
(0.01 mm)
(radians)
1325
1068
-0.026
LOCKED
Theoretical values for 10N are
(1)
 = 6.98 mm
(2)
 = 3.93 mm
 = 0.024
HST3/1. Page 17.
Part 3
Table 3
Slope and Deflection of Beam with Midspan Moment
Load
2 x (N)
0
5
10
15
20
15
10
5
0
LHE
Dial Rdg Rotation
(.01mm) (.01 rad)
1430
0.00
1415
0.15
1399
0.31
1384
0.46
1367
0.63
1383
0.47
1398
0.32
1415
0.15
1429
0.01
Dial Rdg
(.01mm)
1194
1195
1194
1194
1193
1193
1192
1194
1194
Midspan
Dial Rdg
(.01mm)
1210
1242
1268
1298
1328
1299
1269
1243
1212
Rotation
(.01 rad)
0.00
0.31
0.58
0.88
1.19
0.90
0.61
0.33
0.02
¾ Span
Dial Rdg Deflection
(.01mm)
(mm)
1480
0.00
1451
0.29
1428
0.52
1405
0.75
1385
0.95
1407
0.73
1428
0.52
14490
0.31
1475
0.05
RHE
Dial Rdg Rotation
(.01mm)
(.01 rad)
1321
0.00
1304
0.17
1293
0.28
1280
0.41
1268
0.53
1280
0.41
1293
0.28
1304
0.17
1319
0.02
Theoretical values for 2 x 20N loading are
A = B = -0.0062 rad
C = 0.011 rad
D = 1.11 mm
HST3/1. Page 18.
MAINTENANCE APPENDIX
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 HiTech 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.
HST3/1. Page 19.
HST3/1, DEFLECTION OF BEAMS
PACKING LIST
Comprises:



1
HAC6m
Dial Gauge Assembly
1
HSA106
Double Pulley Bracket

2
HSA109
Hanger Clamp

1
HSA114
End Support with Clamp complete with Dial
Gauge, 0.5 N Hanger and Counter Balance Hanger

1
HSA115
End Support with Roller Fixture complete with Dial
Gauge, 0.5 N Hanger and Counter Balance Hanger

1
HST603
Moment Application Unit including Centre Support
Bracket and Two Special Dial Gauges

1
HST605
Cord Assembly

2
HST607
Hanger Link

1
HST610
Counterbalance Hanger







HT-B4
HWH2
HWH4
3 mm
6 mm
HTB2
HST3/1
1
2
2
1
1
1
1
25 mm x 3 mm Mild Steel Beam
Hanger 0.5 N
Hanger 1 N
Hex. Wrench
Hex. Wrench
Text Book
Instruction Manual
HST301W Set of Weights




2
4
4
2
1N
2N
5N
10 N
ORDER No.
SIGNED:
SERIAL No.
CHECKED:
Date:
HST3/1. Page 20.