The equipment describedin this manual is
manufacturedand distributed by
TBCQUIPMENT LlMITBD
Suppliers of technologicallaboratory
equipment designedfor teaching.
Tel: (0115)9722611 : Pax: (0115)9731520
@ TecQuipment Limited
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mechanical, including photocopy, recording or any
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permission of TecQuipment Limited. Exception to this
restriction is given to bona fide customers in educational or
training establishments in the normal pursuit of their
teaching duties.
Whilst all due care has been taken to ensure that the contents
of this manual are accurate and up to date, errors or
omissions may occur from time to time. If any errors are
discovered in this manual please inform TecQuipment Ltd.
so the problem may be rectified.
A Packing Contents List is supplied with the equipment and
it is recommended that the contents of the package(s) are
carefully checked against the list to ensure that no items are
missing, damaged or discarded with the packing materials.
In the event that any items are missing or damaged, contact
your local TecQuipment agent or TecQuipment direct as
soon as possible.
CONTENTS
Specification
1.
Introduction
2.
Unpacking
3.
lacking
of
4.
Forces
due
5.
load
6.
Cell
1
and Assembly
load
to
2
4
Cells
Dial
7
Calibration
9
Experiments
6.1
Support reactions
simply-supported
6.2
Variation
of
beam with
load,
Verification
Influence
Simply
beam
9
of
a simply-supported
beam thickness
of
the
theory
for
pure
fixed
18
21
beam with
28
supports
central
prop
a cantilever
Propped
cantilever
(with
rigid
6.10
Propped cantilever
(with
elastic
-
about
a principal
beams
31
33
6.9
of
bending
25
of
Bending
14
theory
Deflection
Sheet
material
deflection
beam with
supported
and
of
of reciprocal
line
Continuous
for a point-loaded,
deflection
Demonstration
formula
5
Gauges
36
prop)
prop)
axis
40
43
r
SPECIFICATION
Load
Cells
3
off,
0-40
and locking
Dial
Gauges
.3 off,
N capacity
screw
0-25
for
mm in
with
adjustable
converting
0.1
to
divisions
knife
rigid
mounted
edges
supports.
on magnetic
carriers.
Spring
Balance
1 off,
0-10
Weight
Hangers
4 off,
fitted
6 off,
10 N; 4 off,
Weights
Cantilever
Support
1
off,
able
Beams
3
Services
Required
Space Required
with
rigid
for
nwn
cursors.
5 N; 4 off,
pillar
all
steel,
1350
N capacity.
test
1
with
2N.
clamping
arrangement
suit-
beams.
1 aluminium,
brass,
mmwide and
all19
long.
NOr..l:.
Bench
the
area
approximately
apparatus
2 x 0.6
can span two small
m.
Alternatively,
benches
of
the
same
height.
Ancillaries
The
SMlO4a
tributed
cells,
supplied
Specimen
loads,
dial
to
Beams, SMI04b uniformally
SMIO4c moving
gauges,
order.
hangers
Quote serial
loads,
and/or
additional
weights
number of
disload
can be
equipment
when ordering.
NOTE
SPBOMBNS
The dimensions stated for specimensare for reference only. To ensure the
accuracyof experimental results and calculations,it is recommendedthat all
specimensare measuredprior to experiments.
c
-1-
1.
INTROOl£TION
The
SMl04 Mk III
Beam Apparatus
of
experiments
to
the bending of beams.
to cover virtually
ing beams on simple
the
beam.
with
hangers
allow
allow
supplied
Three
6
alloy
Beam
the
whole
and all
fourth
for
the
,
clamping one end of
the basic
up to four
unit.
deflections
length
Apparatus
almost
any experiment
bending
under
of
along
action
dial
at three
the beam.
rour
gauges
points.
measurement
Five beams are
of these are 19 mmwide and 1350 mm long.
one each of thicknesses
(Note:
with
involving
of
These,
supports.
and three
beam is made of brass 6 mmthick
6 mm thick.
the
with
relating
for mount-
gauge along the upper cross-member allows
standard
mm; a
facilities
at up to four points
of
requirements
and also for
provide
of the beams are mild steel,
aluminium
The
loading
along
as
provides
are supplied
unit,
which extend the range
coursework
supports
measurement
a dial
deflection
and
the clamping
simultaneous
Traversing
of
or sinking
point
all
The basic unit
Three load cells
together
has many features
point
3 rom, 4.5 mm
and the fifth
of
these are nominal dimensions).
its
ancillary
the
investigation
loads,
equipment
distributed
can be used for
of beams subjected
loads,
moving
to
loads
and couples.
Additional
hangers,
equipment
dial
TecQuipment.
gauges,
such as load
special
cells,
beams
clamped
and
loads
supports,
are
weights
available
and
from
f
-2-
2.
UNPACKING AND ASSEMBLY
2.1
l.h1packing
1.
Remove all
to
of the components from their
discard
any packing material
packing,
until
taking
care not
the Packing Contents list
has been checked.
2.
all
Unwrap
bench.
amongst
accounted
The
main
of
the
and transport
The
packing
the
costs.
tee
material
with
face
for
sny
tighten
One
the
the
of
which
the
the
will
two
rear
lower
head.
until
other
carry
lower
the
two
the
the
be
acale
upper
components
Use washers
whole
frame
not
and the
use.
(the
the
face
back of
with
under
all
This
member is
are
at
holes,
one
each
side
Again
front
tighten
up the
six
nuts.
and
the
upright
fitt-
and do not
legs.
when all
four
attached
the
rear
the back of
tee
and,
forward
has two brackets
two members to the
the
the
nuts
these
fron
for
a washer
Bolt
through
be bolted
brackets
front
brackets
the
as follows:
should
member.
of
to minimise
has been assembled.
cross-members
in
on it
should
to
the
order
proceed
holes
bolted
face
in
frame
The scale
beams not
members
the
graduated
head should
bolt
nuts
dismantled
inwards).
should
The bolt
below
the
through
cross-member
members.
supplied
To assemble
legs
members
the
ed
SMlO4 is
cross-member
lower
3.
the
the Packing Contents List.
for.
frame
to
2.
components and lay them out on a suitable
Assemly
packing
1.
the
Check the components against
Check
2.2
of
legs
nuts
through
pass
are
of
the
the
the
bolts
in
place,
~
-3members in
be
clamped
washer
clamp
4.
The
the
the
the
tee
the
6.
The
or
stored
cells.
plastic
plastic
Before
fitting
but
the
do
The load
cells
should
knob down and placing
knob should
legs.
not
can
These
whichever
clamp
can
top
the
apparatus
weights
until
load
a 'C'
now be wound up to
cell.
apparatus,
beam
The
run on the
from
loading
the
gauges
magnets
The
winding
it.
load
dial
removed
5.
by
above
magnets.
the
same way as the
dial
discard
the
can
is
and are held
position
remove the
them.
Whenever the
mounted
be fixed
either
the
convenient.
more
keepers
by
from
gauges
are
must be replaced.
on rods
at
in
gauges
keepers
be stored
be attached
required.
rail
one
end
in
of
front
the
on the
of
or
feet
of
behind
beam apparatus
m
-4'"-
J.
LOCKING (T LOAD CELkS
The
knife
the
locating
cell
body
locating
NOTE:
t--
edges of
screw
load
knife
load
cells
cells
supplied
(as shown in rig
the
The
the
can be locked
through
0.1).
Note
in
position
the OIl
rest
plate
the OIl
should
edge.
will
by screwing
into
the
load
be removed before
n"
,;
be locked
for
shipment.
r~
l
rig
0.1
-5-
4.
In
fORCES
the
charts
the
load
cell
dial
gauge.
Zero
the
the
the flat
SlCMly
dicated
gauges
GAU(l:S
.
it
(fig
dial
gauge is
dial
Ensure
spring
raise
that
marker.
on the
balance.
Repeat
A
graph
of
be
linear.
force
crocodile
(say
of
allow
top
of
the
the
crocodile
gauge (fig
procedure,
against
clear
the
gauge
this
is
with
balsnce
done
to the
will
Attach
by
top
the
using
load
exert-
calibration
clip
cross
member or the
obstructions.
the
required
unit
clip
to
Zero
This
until
the
the
movement.
attached.
balance
it
is
is
knurled
done
level
part
of
0.3).
until
s convenient
5 rom) and note
increasing
deflection
include
follows:
properly
the
to
best
O-ring
spring
dial
is
the
of the dial
the
as
gauge contact
balance
contact
necessary
This
fitted
or unscrewing
zero
is
0.2).
may be constructed
and the
screwing
with
DIAl
experiments
dial
which
Ensure
by
TO
load-sensitive
ed by
~
Dl£
should
the
the
deflection
reading
deflection
be drawn.
on the
is
in-
spring
each time.
The graph
should
-6-
J
[
r
Fig 0.2
Dial
Gauge
-urig 0.3
Spring
Balance
Attachment
L
l
fig 0.4 Calibration
of Load Cells
-7-
5.
LOAD CELL CALIBRATION
The
three
beam
A
load
as shown in
locked
this
to
read
knife
set
zero.
span
(do not
found
using
using
the
ensure
cell
The
Tap
the
on the
value
(e.g.
dial
the
cell
deflect
with
when the
the
the
always
load
due to
and adjust
dial
gently
cell
B so that
and take
at B is
°10 N) and then
Plot
W/2.
edge A and adjust
load
the
beam at mid-
hanger,
height
of
is
load
to
a calibration
which
can be
knife
edge B
to zero.
This
is
to
perpendicular
cell
Increase
decrease
beam above the
gauge again
load
the
loaded;
dial
gauge returns
and the
cell
the
the
the
load
beam is
on the
read zero.
horizontal
frame
load
of
at B to
balance)
until
gauge to rest
height
include
at each step.
should
identify
curve
load
requires
is
not
beam horizontal
supported
reading.
the
zero,
to
load
The
to some
taking
the
the
particular
curve
for
other
load
load
B.
to
typical
to
spring
beam
procedure
care
load
thumb wheel
reading
load cell
The
the
exerted
maximumm
will
the
up a simply
gauge on the beam above knife
adjust
forget
beam.
force
and
by setting
set
Now move the
Set the
the
rirst
support
up a dial
B
reads
0.4.
the
zero.
edge
can be calibrated
rig
so that
done,
the
cells
is
each
shown
cells
a
then
force
can be seen by the
are
of
in
be repeated
calibration
fig
the
curve
with
its
cells,
own load
taking
cell.
A
0.5.
calibrated
2 N.
position
for
so that
1 mm of
downward displacement
The number of revolutions
of
marks below the main load cell
the
rest
plate
relative
body, as shown in rig
of
the
dial
to
the
indication
0.6.
gauge
-9~
6.
It
EX~~J~~I~
is
possible,
TecQuipment
bending
ing
especially
is
of
obtained,
beams
experiments
1.
2.
and
can
ed in detail
to
cantilevers
and
Verification
of
Demonstration
for
Simulation
of
a rolling
Continuous
beam with
9.
Deflection
a point-loaded
supported
of
of
fixed
beam with
Propped
cantilever
with
elastic
As
stated
quoted
6.1
Support Reactions
6.1.1
Introduction
---simple
for
but
themselves
by
beam supported
starting
the
it
at
mid-span
the
thickness
the
dimensions
manual
of
student
set
the
experiment
the
specimens
using
suitable
Simply-supported
which
apparatus
1/4-span
measure
beam.
specimens
For
accurate
should
be
measuring
and
Beam
allows
its
students
sensitivity
up as shown diagrammatically
l/4-span
of the
the
supplied).
experiment
the
of
are nominal.
a Point-loaded,
is
and at the
and width
(not
with
at
prop
the
useful
The apparatus
Before
prop
dimensions
instruments
the
load,
prop
this
the
measured
with
beam with
bending
earlier,
in
results
accuracy.
supported
a cantilever
11.
familiarise
pure
central
rigid
a
beam
supports
with
is
describ-
load
cantilever
This
are
follow-
deflection
Propped
Note:
and
the
theory
10.
Experimental
unit
The
simply-supported
a simply
of
reciprocal
lines
Simply
involving
apparatus.
basic
from
pages.
theory
Influence
8.
experiment
SMlO4
the
available
beam material
the
of
equipment
any
the
using
deflection
beam thickness
almost
out
for
of
cdditional
using
be carried
reactions
Variation
the
perform
in the following
Support
3.
4.
5.
6.
7.
if
to
and
in Fig
1.1
points.
the
points
length
for
of
the
beam and mark
easy reference.
Measure
r
-10-
r
Arrangement of Supports
Fig 1.1
6.1.2
EQuipment
Two load
the
and Loads
ReQuired
cells,
one
dial
gauge,
two
load
hangers,
the
weights
and
one
of
beams.
6.1.3
Procedure
1.
Choose
the
a suitable
reading
mid-span
of
on the
the
beam.
upper
scale
of
the
apparatus
for
(One of the 10 cm markers is most
convenient).
2.
Set
up
of
of
3.
one
the
any
of
the
marker
offset
in
the
position
in
4.
Place
5.
Position
Lock
the
the
beam in
load
position
may press
so
step
that
1.
it
(Do
is
not
1/4-span
forget
to
the
left
to
take
account
of
the mid-span
cursor).
cell
knife
two hangers
(The cursors
cells
chosen
Set up the second
reading.
load
1/4-span
to
the
right
edge.
with
equidistant
lightly
1.
1/4-span
from
against
overhang
at
the
mid point
the
scale).
either
end.
of
beam.
the
L
-11-
6.
Place
a dial
the
ball
rests
on the
centre-line
of
end
the
bottom
dial
gauge to
the
dial
very
read
the
the
cell
6.1.4
Theory
to
the
2
to the
read
readings
1
lock
the
above the
and take
Resolving
cross
the
moments
about
1.2
R2
gauge reads
knife
a systeinatic
of the
graphs
load
to
edges.
manner,
Move
check
adjust
the
zero.
Adjust
the
tap
the
beam
cells.
illustrate
the
theory.
2
. . . . 1.1
R2 :
+
WZ(!£ - b)
+ W18 - W2b
. . . . 1.2
1
in
support,
the
+ W
!(W1 + W2)
R
positinn.
and
vertically;
RiR. = Wi (!R. + a)
Therefore
Adjust
member, then
dial
both
vertical
zero.
in
plot
bezel
right-hand
WI + Wz - Rl
R2 =
Therefore:
stem is
in
hangers
and
beam immediately
and then
gauge and unlock
results
the
member so that
stem.
edge so that
to
the
cross
has been moved down the
the
= w
R + R
Substitute
Check that
parallel
indicators
to Fig 1.2.
1
is
dial
gently
Process
zero
knife
loads
9.
a-ring
beam
of
Apply
support.
gauge to a position
the
Remove
Taking
upper
the
load
Refer
on the
left-hand
height
8.
position
above the
that
7
gauge in
1.1:
=
!(W1 + WZ)
-
W1a/.Q, + Wzb/.Q,
. . . . 1.3
-13-
6.1.6 {::omparison
Substituting
following
the
table
is
with
Theory
value
of
R.,
a and
b in
equations
obtained:
Fig 1.2
Theoretical
arrangement
1.2
and
1.3,
the
-14-
6.2
Variation
of
Deflection
Beam Thickness
6.2.1
of,a
theory
of
book
and
shows
only
in
the
pure
bending
that
of
a beam can be found
when a beam is
plane
of
the
applied
the
bending
I
the
second
is
moment
of
Modulus
R is
the
radius
a is
the
bending
stress
at
y is
the
distance
from
the
close
approximation,
is
deflection
of
of
by
the
Using
equation
2.2
jected
to direct
loading
the
of
a is
a constant
of
the
and
.
.
.2.1
beam section
inertia)
it
y from
beam at
the
neutral
axis
axis
curvature
of
derivative
distance
x from
a beam l/R
of
the
is
given,
deflection.
a chosen
origin
= 1 = M
R
can be shown that
can always
to
a
If
z
the
be expressed
deflection
in
the
of
form:
B-~'
value
. . . . 2.3
El
depends
load
acting
on the
beam
upon
the
type
[
a beam sub-
deflection
whose
[
then:
. . . . 2.2
EI
supports
W is
the
neutral
second
z =
the
distribution
.
distance
the
dx'
z is
bends
curvature
~
where
it
Elasticity
shown that
of
stress
R
area
the
be
a way that
text
moment
E is
also
the
such
any standard
by
(moment
can
in
moment,
y
I
H is
loaded
in
= -a = ~
H
the
Beam with load
and Material
curvature of the beam are related
It
orted
Theory
The
where
Sim 1
of
loading
and
l.
-15l
is
[
and
I
is
the
relationship
.2
the
span
are
defined
Equipment
load
above
investigated
in
this
experiment
Required
cells,
one dial
gauge,
one hanger,
set
of weights,
all
of .the
beams.
6.2.3
1- 4
5.
Procedure
As for
experiment
1.
Place
the
at
hanger
centre-line
6.
7.
of
Exactly
as
Place
read
8.
dial
zero
Apply
the
Do this
the
deflection
at
Repeat
12.
For
each
the
gradient
For
the
step
Taking
N/m2
and
a graph
at
each
the
the
five
is
on the
the
ball
end
Adjust
the
of
the
dial
to
the
beam deflection
dimensions
on the
dial
on the
gauge
are
new dial
gauge
reading
(deflect-
times.
the
for
same steps
all
a graph
as 9 and
record
the
beam
of
beams
l/d'
(d
the
beams.
deflection
plot
is
N/m2,
values
gradient
of
against
load.
Determine
graph.
21 X 1010
the
point
step.
steel
12 against
of
that
setscrew.
record
scale
record
by
each
three
using
so
the
and
the
least
load
of
=
of
hanger
experiment
ES
span
loading
bezel.
and
beam plot
the
the
m.
load
Decrease
mid
centre
the
the
10
11.
14.
to
that
1.
at
Note:-
mm i.e.
Increase
in
lock
a load
ion).
13.
on the
and
so
beam.
gauge
gauge.
0.1
sp~n
experiment
rests
dial
9.
for
the
plunger
the
mid
the
a graph
thickness
EB = 10.5
obtained
obtained
of
for
in
step
the
of
x 1010
the
gradient
the
beam).
N/m2,
6 mm thick
12 against
obtained
EA = 7.6
beams,
l/E.
x 10
plot
-16-
6.2.4
Analysis
of Results
[£FlECTION
z (mm)
LOADW
(N)
STEEL
6 mm
STEEL
STEEL
BRASS
4.5mm
3 mm
6 mm
5
10
15
20
25
30
Table 1
rig
2.1
results
Steel
shows
in tables
the
and
Brass
typical
Table 2
Beams
shape
of
graphs
obtained
Aluminium
by
plotting
Beam
the
1 and 2.
Steel
-EE
-E
-u
0
.-
"OJ,
\4Q,
C
-
c
0
.-
OJ
Q'
[.
-E
c.
/Steeli..5mm
/~.Bross 6 mm
u
QI
-
"
/",'" .~,,'"
/,,'".
/~
.{/
.
Steel 6 mm
/'
"""""'"
,,~
10
20
30
W(N)
-17-
6.2.5
Conwnents
The
graphs
The
transverse
of
the
For
as in
of
lines
the
proportional
the
rig 2.2
accordance
to
straight
lines
for
all
beams can be determined
stiffness
in
obtained
nn]
from
five
from
beams.
the
slopes
x 10-').
fig
2.1
can be recorded
below.
6 mm beams,
in
are
= W + [deflection
beams the
table
the
stiffness
stiffness/(thickness)'
three
lIE
of
(stiffness
Theoretically
i.e.
vs load
stiffness
five
the
deflection
l'/I.
z/W v lIE
the
slope
with
of
a beam is
=
constant.
of
the
the
lines
theory.
pro.portional
of
to
(thickness)',
fig
2.2 shows that,
fig
2.1
The
slope
are
of
for
proportional
this
line
the
to
is
-186.3
Verification
6.3.1
Introduction
-
The
theory
for
Experiment
that
equation
3.1,
the
Theory
of
Pure
(section
is
bending
is
of
The object
measured
by the
obtained
h(2R - h)
=
82
2Rh - h2
a2 in
2.2:
1
Therefore
Two
the
load
this
at
experiment
the
start
is
to
of
show
method.
from
which
the
h2 is
Referring
to
relationship:
usually
very
small
1 = ~
82
=
~
h
and M = ~
[I
R
Equipment
outlined
Sagital
is
R
rig
of
curvature
Therefore
Also from Equation
was
valid.
curvature
radius
to
6.2).
ie
6.3.2
r'
Bending
[
2.2
of
the
a beam subjected
2
The radius
rig
of
=
. . . . 3.1
Wa1b/2EI
rig 3.2
3.1
l
Required
cells,
beams preferably
three
dial
gauges,
two hangers,
set
of weights,
one of
L
6 mm thick.
l
L
-19-
6.3.3
Procedure
To
set
up
constant
the
apparatus
bending
so
moment over
that
the
the middle
beam is
subjected
of
length
half
its
to a
follow
steps 1 to 4 of experiment 1.
5.
one hanger near the left-hand
Place
load is applied
on the centre-line
hanger at the right-hand
the same distance
6.
Place
one
dial
of the beam.
at
mid-span
and
side of it
7.
Adjust
the
8.
Apply equal loads to the two hangers
9.
Gently
tap
as shown in fig
the
gauges
beam near
to
its
(See fig
the
other
the hangers are
3.2).
two
equidistant
on
3.2.
either
dial
Place the second
end of thhe beam so that
from the supports.
gauge
three
end of the beam so that the
read
zero.
mid-span
and
take
the
readings
of
the dial gauges, h , h , and h .
Increase
11.
the
values
of W.
Process
the
load
results
and repeat
as shown
in
steps
the
8 and
following
9 for
example
at
least
five
-20-
r
. .
6.3.4
Analysis
of
Results
Beam - Brass,
Y =
!(h..
6 mm thick
1350 mm long
[
+ h3)
r
w
(mm)
300
100
300
100
300
100
300
100
300
100
5
300
150
10
300
150
15
300
150
20
300
150
25
300
150
5
300
200
10
300
200
15
300
200
20
300
200
25
300
200
Test 1
Test 3
a
(mm)
(N)
Test 2
b
hI
h2
h3
(mm)
(mm)
(mm)
y
(mm)
h 2-Y
(mm)
[
[
1.
6.'.5
~
The
Calculations
straight
equation
~
since
line
3.1)
from
graphs
against
this
obtained
by
W show that
equation
the
plotting
equation
gradient
is:
3.1
h2 - Y (this
is
of
the
is
correct
h in
form,
~
2EI
~
Thus when b is 300 mm ( =
.3m) the
gradient
= ~1!:
[1
~
Therefore
for
constant
against
a2 should
flexural
rigidity
The
second
moment
b
(in
be linear
of
the
of
area
this
and its
case
0.3
gradient
will
the
above
give
of
gradient
a measure
of
the
beam.
of
the
beam
= 19 x 6' = 3.42 x 10-10
12 x 10
'I
~
Using
m) a graph
EI =
results:
m~
.15a2
gradient
Therefore:
~.
=
E
Grd.
6.4
Demonstration of Reciprocal
6.4.1
Theory
to
Referring
simply
N/m2
.1582
x I
Theory
fig 4.1a, it can be shown that the deflection
beam AB with
supported
Zo
point
load W at C is given by:
= Web (L2 - 82 - b2)
. . . . 4.1
6l
It
is
seen
i.e.
changed,
ion
""-
at
that
r
is
the
if
the
the
at D of the
value
loading
same as the
of
z will
is
as shown in
deflection
not
at D.
change
rig
if
4.lb,
a and bare
then
the
interdeflect-
r-22If
the
will
beam is
be
reciprocal
theory
the
same as the
theory
should
now loaded
which
is
be referred
as
shown
in
deflection
rig
at
demonstrated
to
in
a standard
4.lc
D.
in
then
This
this
is
the
deflection
a particular
experiment.
at.G
case
The general
textbook.
of
r
(a)
A
[
rig
6.4.2
Equipment
Two load
cells,
4.1
Reciprocal
Theory
Required
three
dial
gauges,
three
hangers,
weights,
one beam.
-23-
L
6.4.3
Procedure
1-4
Exactly
5.
'"
as in Experiment 1.
Place load hangers and dial gauges as shown in fig 4.2.
and b need not be equal but are equal in the example below.)
6.
Use
one of the dial
Return the dial
7.
8.
9.
10.
Apply
a
three
dial
Move the
gauges to go through
gauge to its
suitable
original
record
the readings
of all
gauges.
load
to
the other
of all
three
Various
combinations
dial
of
two hangers in turn
loads can be applied
the readings
of all
Use
results
demonstrate
to
three
dial
the
and inter-changed
gauges.
the reciprocal
,,~~...2I4Q~..1.1..
'-'-'
and record
gauges.
recording
..'"
~I
~~:~.~.'Wl."'\1/.
.. L/4- . t- -Lf4- ~
fig 4.2 Demonstration of Reciprocal
,~
1.
position.
load to any hanger,
readings
the
step 6 of experiment
below~
"
(Note: a
Theory
theory
as shown
-24-
6.4.4
Analysis
=
a
W
h
=
=
of
Results
=
200
b
Load
(N)
deflection
WI
J.J.
1.1.1.
iv
(mm)
a)
WI
(mm)
(mm)
(N)
5
0
0
0
10
0
5
0
W2
(mm)
h2
W3
(mm)
(N)
h3
(N)
5
0
5
0
0
15
0
0
10
iv
h3
Wa
0
10
ii
.5
(mm)
TWO POINT LOADS
b)
Examination of R~sults
Comparison
of
reciprocal
theory
to
points
1
the
deflection
is
of
part
that
h2
SINGLE POINT LOAD
hi
(N)
6.4.5
Wz
(N)
0
5
0
0
i
ii
(mm)
hI
(N)
1
mm
Examination
results
and
the
a(i),
applies
3.
a(ii),
to points
Comparison
doubled
b results
and
a(iii),
1 and 2,
of
results
when the
will
applied
verify
will
to points
a(i)
the
is
that
2 and 3 and also
and a(iv)
load
verify
will
doubled.
following:
verify
L
-25-
from
a)
(i)
the
These
(ii)
additional
From
b)
and
(i)
ing
the
two
values
the
additional
10 N at
and (iii)
the
additional
x
point
deflections
x
additional
deflection
to
applying
at
1, due to apply-
2 = ~.
a g ree
should
2
2 due
1 = Xl"
10 N at point
1 and
at
to
within
a
very
small
be clear
that
percent-
age.
results
From
c)
applies
theory
it
will
6.5
Influence
6.5.1
Introduction
The
the
position
of
ion
of
the
obtained
by
the
the
for
point
the
between
due
the
which
and 3 and from
apply
to points
deflection
of a unit
applying
of
1
should
reciprocal
results
b(iii)
and
2 and 3.
Deflection
for
application
it
be seen to
relationship
example
to
the
the
load
the
acting
of
line
a point
deflection
application
principle
influence
of
on a beam is
of
on the
that
point
beam.
of any load
superposition
of
and the
The deflectto the
after
by the magnitude
a line
beam is
multiplying
the
load.
(See
follows).
Equipment
Two load
points
also
line
ordinate
6.5.2
and b(iv)
to
line
influence
showing
b(ii)
Required
cells,
dial
6.5.3
Procedure
1-4
As for
5.
Set
gauge,
experiment
up
the
hanger,
weights,
one
beam.
shown in
fig
5.1
1.
equipment
as
with
the
as for
experiment
1.
values
of
load
cells
locked.
6.
Carry
out
7
Vary
the
record
the
the
levelling
value
dial
operation
.of W for
chosen
gauge readings.
a and b (fig
5.1)
and
r
-26-
8.
for
each
increment
9
Plot
of
(see
the
results).
influence
line
the
defined
10.
set
Apply
given
as the
a load
by
readings
incremental
system
determine
for
load.
mean deflection
The 'unit'
per
may be
load.
and compare
calculation
'unit'
the
based
the
measured
on superposition
deflection
and the
with
that
influence
line.
Fig 5.1
Schematic
of
Experimental
Set-up
l
Fig 5.2
Influence
lines
for
W = 5 & W = 10 N
6.5.4
Analysis
of
a = 200 mm h
k
=
at
=
/5
deflection
[(h
Results
W=5 -
deflection
N
(mm)
(mm)
h at
W=0)+5]+[(h
at
W=lO -
h at
W=5)+5]
...etc
.
These results
can be presel,ted
6.5.5
Example
1.
10 N applied
in
of Application
at b
to
of the
Deflectio~
due
Deflection
due to 5 N
Measured
obtained
10 N
It
Influence
as in
Line
with
=
2 x k at
450
=
k at 250
(mm)
for
fig
5.2.
W= 5
5 N at b
250 mm
=
(mm)
= 2 x k at 450 + k at 250 (mm)
Deflection
above.
form
450 mm together
=
Total Deflection
The
graphical
is
then
should
compared
be found
that
with
the
the
total
two figures
deflection
are
in
close
agreement
2.
5 N applied
at b
=
400 mm together
Deflection
due to
5 N
Deflection
due to
10 N
Total
Again
the
total
deflection.
Deflection
measured
=
deflection
with
10 N at b
=
200 mm
k at 400 (mm)
=
=
2 x k at 200 (mm)
k at 400 + 2 x k at 200 (mm)
should
be in
very
close
agreement
with
the
t
r
-28-
6.6
Continuous
6.6.1
Introduction
A
great
from
the
amount
this
of
backlash.
It
of
care
experiment.
heights
difficulty
Beam with
in
is
knife
errors
as shown in
the
fig
~2
W
Rl
6.1,
-
the
reading
the
of
load
the
be obtained
adjustment
load
cells
cells
can lead
of
for
to
results.
for
a continuous
beam carried
on rigid
that:
3 (S2
+ b2)
16
+ b)
(s
. . . . 6.1
. . . . 6.2
o. 5 - .t!2
. . . . 6.3
~
=
2 - ftl- ft,
W
W
R.
. . . . 6.4
w
~~~
6.1
to
to accurate
Wa
B.,
w
rig
of
are
= 0.5 - ~2
W
.B.2
good results
must be paid
can be shown from the theory
supports,
if
edges and correction
in
interpreting
Supports
required
Attention
the
Small
rixed
Ra
R.
Continuous Beam on 3 Supports
-29-
:1:
6.6.2
Three
3
1.
Equipment
load
Required
cells,
three
Set
up the
load
the
Place
the
3.
Place
a hanger
4.
Set
5.
Place
the
a
two hangers,
the
O-ring
Move
dial
upper
weights,
so
at
equal
of
each
one beam.
of
from
the
overhang
part
of
at
the
will
either
end,
span.
If
stem it
requires
the
Set the
Rl
they
span.
above Rl.
stem so that
beam.
1/3
so that
zero
position
part
that
positions
and
to
down the
gauge
edge
middle
pointers
the
O-ring.
ends
with
gauge in
move the
knife
convenient
its
the
cell
made. with
the
at
position
at
dial
on
Move the
be
beam in
load
O-ring
cells
beam near
2.
6.
gauges,
Procedure
support
.L
dial
to
zero,
R2 and
dial
it
gauge
the
allows
lock
adjuust
just
gauge has no
one fitting.
contact
just
to
the
bezel,
do NOT
the
height
of
contacts
the
the
beam,
no
further.
7.
Repeat
8.
Repeat
that
9.
just
'U
~
5,
the
Place
dial
~~t"
6 with
6,
load
gauges
dial
cells
then
the
dial
place
the
displace
at
the
indicate
on the
contacts
just
gauge
and 7 until
O-rings
to
Place
the
part
gauges
the
(There
load
loads
on the
of
at Rl'
cell
gauges.
equal
beam is
horizontal,
making
sure
zero.
upper
beam.
R,.
hangers.
the
stems of
the
R2 and R, so that
should
be sufficient
indicator.)
Set the
remaining
the
ball
pressure
zeros
of
-3011.
Adjust
the
regained
height
with
the
12.
Repeat
10 and
6.6.4
Analysis
of
of
dial
the
knife
gauges
11 until
edges so that
then
sufficient
take
the
results
contact
load
have
cell
been
is
just
readings.
obtained
Results
BEAM Steel
6 mm thick
ARRANGEMENT Refer
t~.
to fig 7.1
400
8
mm
b
800
RIll
.
6.6~5
Calculated
Substituting
following
the
results
Value~
values
are
of
and
(8)
in
equations
6.1
to
6.4
the
obtained:
!1
W
=
0.5 - 125 =
400
!,
=
0.5- ~
=
0.1875
0.3438
800
w
fl.2
(b)
= 2
-
0.1875
-
0.3438
=
1.4687
W
Comparing
be seen that
the
above values
there
is
close
with
those
agreement.
obtained
by experiment
it
should
-31-
Simply
theory
ferred
7.1
Beam with
C!!!tral
Prop
Theory
.}
The
Supported
to in
it
beams supported
for
the
can
standard
by rigid
textbook.
be shown that
the
or elastic
For
force
the
props
should
configuration
induced
in
the
be re-
shown in
rigid
prop
is
Fig
given
by:
1.375 W
p
It
can
also
.
be shown
that
-11 x
Wl'
the
deflection
at
mid-span
6.7.2
Three
6.7.3
Equipment
load
.7.1
. . . . 7.2
Simply
Supported
Beam with
Central
Prop
Re~ired
cells,
dial
gauge,
two hangers,
weights,
one
beam.
Procedure
Carry
not
out
Arrange
6.
lower
lock
Place
the
essential
5.
7.
.
is:
[1
Fig 7~1
.
to
the
the
the
usual
other
a dial
support
hangers
knife
setting
as shown
edge
two
gauge
the
at
knife
at
up procedure
beam at
in
mid-span
fig
so
the
but
1/4
in
this
case
and
is
points.
7.1.
that
it
is
clear
of
edges.
mid-span
it
set
it
to
read
zero.
the
beam.
t
-Jl-
8.
9.
Apply
10.
Measure
Set
the
~
loads
to
to
the
zero.
12.
Alter
6.7.4
Analysis
the
two
with
down with
the
middle
the
edge
Record
load
of
at
a finger
knife
the
load
cell
to
zero.
hangers.
deflection
contact
Adjust
11.
on the
the
in
edge
Steel
pointer
mid-span.
knife
edge.
whilst
the
by
screwing
load
cell
Make sure
If
necessary
deflection
up until
press
is
the
the
beam is
the
knife
measured.
dial
gauge
returns
reading.
9 through
repeating
that
12 as many times
as required.
Results
beam 6 mm thick;
19 mm wide;
W (N)
span 1200 rom; 1=342
Deflection
z (mm)
P' (N)
very
with
(mm)4.
P/W
5
10
15
20
25
6.7.5
Comments
1
P/W
=
2}
From
equation
should
agree
closely
E
7.2:
=
11
384
Therefore:
E
11
384
x
5 x
equation
7.1.
[
x ~'
Iz
1200'
342 x z
= 72.368 x 1& N/rnrn2
z
l.
where W = 5 N
L
-33-
of a Cantilever
Deflection
6.8.1
Introduction
Reference
to
the
for a cantilever
textbook
will
show that
the
deflection
under
the
load
loaded at the free end is given by:
z = Wl'
. . . . 8.1
3EI
If
EI
and
l
are
maintained
constant
then:
z
= k .W
1
. . . . 8.2
z
=
. . . . 8.3
where k 1 is constant
Similarly
if
[I
and Ware
likewise z = ~,
E
J
It
follows
and z = ~
if
constant:
k2l'
E and I respectively
are made
I
the
that
therefore
between
relationship
maintained
experiments
the
deflection
illustrated
below
can be carried
and each of
the
variables.
out
showing
quantities
the
E,
I,
l
and W.
In the
experiment
Note
Since
large,
the
edge
due to
of
the
cantilever
a load
the
dial
cell
Clamped
support,
6.8.3
Procedure
2.
is
of
the
Land
upwards.
for
high
be taken
Ware
cantilever
made to deflect
gauge should
Equipment
..
deflection
only
upwards
accuracy
into
varied.
due to self
weight
by screwing
the
the
loading
of
is
knife
the
beam
account.
Required
one
load
(Refer
Set
up
a load
the
frame.
Set
up the
clamp
cell,
one
dial
gauge,
one
beam or
more.
to Fig 8.1.)
cell
to
at
give
a convenient
a cantilever
position
of
near
convenient
to
one
length.
side
of
-34-
3.
Pass
one
on the
end
of
load
cell.
assembly).
using
4.
Set
5.
the
Adjust
the
load
6.
cell.
Adjust
the
recording
7.
edge;
slacken
more
convenient
8.
Repeat
9.
In
it
so that
of
load
the
edge
knife
the
order
the
and
load
vary
and
lock
tie
rest
the
up the
dial
and
the
clamp
than
moving
for
E use
to
to
clamp
end
of
the
the
knife
other
edge
free
end
of
set
the
zero.
end
during
the
beam
dial
give
cantilever,
gauge
Hove
unlock
returns
a convenient
dial
give
to
the
zero.
gauge
reading
on the
reading.
a number
of
load
increments
readings.
its
initial
and move
the
load
several
the
the
and
zero.
the
gauge
edge
the
to
to
upwards
experiment
to
cell
upwards
edge
loads
the
to
adjust
knife
Return
near
free
Record
to
and
the
knife
clamp
string.
to
and
pointer
clamp
of
the
convenient
gauge
gauge
edge
is
the
dial
dial
knife
(It
piece
the
the
beam through
Tighten
a short
Place
the
position;
it
to
lock
a new position
the
(this
knife
is
cell).
lengths
Aluminium,
of
Brass
cantilever.
and
Steel
beams
6 mm
thick.
10.
In
order
to
vary
I
use the steel
beams 3 mm, 4.5 mm and 6 mm
thick.
rig
8.1
Schematic
of
Experimental
Set-up
r
-36-
6.8.4
Typical
Results
BEAM Steel
l
r~
6 nIn thick
(mm)
200
W (N)
400
300
z (mm) z (mm)
500
600
z (mm) z (mm)
z (mm)
2
4
.6
'8
10
6.8.5 Comments
The
each
L'
of
graphs
length
z v W verify
that
equation
given by the respective
the
a
graph
gradients
obtained
of
z
L'
=
graphs
is
a measure
equation
8.2 is correct,
with
with
k1 for
The graph of z v
of the graph.
8.3 is correct,
gradient
the
v
give
in
8.1:
of
x/W
will
substituted
rrom
equation
being given by the gradient
verifies
If
that
k2 for
each load being
of the graph.
for
z/W are
plotted,
the
of the
obtained
gradient
flexural
for
of
rigidity
each length
the
of
straight
the
and
line
beam when
8.1.
W-'
3£1
JEI
= ~'
z
since
I
=
342 mm4 then:-
E
=
inverse
of
Qradient
3 x 342
N/mm2
-376.9
Propped' C"antilever
6.9.1
Introduction
The
theory
standard
for
a
ProD)
propped beams and cantilevers
cantilever
free
end it
the
loaded
can be referred
to in most
as shown in rig
9.1
with
a rigid
the
prop
is
prop
deflection
at
the
free
end,
2
6
if
absent,
is
given
. . . . 9.1
prop the force in the prop is:
P=~
w
. . . . 9.2
16
the
its
z =
for a rigid
Since
at
can be shown that:
by:
b)
Riaid
textbooks.
ror
a)
(with
deflection
of
z =
a cantilever
loaded
at
its
free
end is
given
by
PL'
3[1
then
P, Wand a are related
PL'
=
WL'
by the equation:
(3cl
. . . . 10.3
- C2
where c
In
this
experiment
the
relationships
given
in
=
all
equations
9.2
and 9.3
investigated.
w
!.z J.
T!'c:;
fig
9.1. Loaded Cantilever
L
~
Supported by Rigid Prop at End
are
r
-38-
r
rig
6.9.2
load
Equipment
cell,
beam*,
*
end
due
6.9.3
1.
to
of
self
Set
the
alIen
screws
one
in
of
aluminium
the
free
cantilevers.
so that
knife
a cantilever
edge.
of
beam in
knife
edge.
member of
the
clamped.
clamp,
(See fig
gauge on the
position
lock
and replace
zero.
short
the
the
members.
read
the
of
the
cross
the
of
deflection
Remove the
clamp
to
use
the
position.
tightly
a dial
but
approximate
on the
lightly
with
and
it
6 mm thick
9.2)
its
upper
clamp
only
cross
Clamp,
a convenient
at
rests
the
contact
and set
the
and support
the
is
Place
rig
at
clamp
to
required
lower
to
cell
end
parallel
beam
may be used
can be accommodated.
locate
top
Weights,
necessitates
(Refer
up a load
that
J.
material
loading
Procedure
Hanger,
edge.
other
I m long
2.
Gauge,
straight
A beam
Set-up
Experimental
ReQuired
Dial
steel
9.2
four
the
the
clamp
so
that
the
frame
then
replace
the
screws
so that
the
alIen
Now, with
set
the
Ensure
the
a rule
length
of
beam and lock
held
the
beam is
cantilever
the
the
bezel.
to
the
clamp
l
vertically
clamp
as
to
the
9.3.)
beam as close
lock
recess
four
as possible
t
4.
the
dial
Unlock
the
knife
edge
gauge
reads
zero.
(Note:
pointer
is
Remove
gauge and place it
on the beam above the knife
edge.
5.
6.
zero
slide
the
dial
watch
the
pointer).
Place
the
dial
load
7.
8.
the
cell
Place
Adjust
its
that
the whole
position
height
the
and not +100 or -100
gauge in
so that
0 - indicated
divisions.
length
above the
the
of
If
the
knife
dial
by the
necessary,
cantilever
and
edge and set
the
to zero.
hanger
the
adjust
Be sure
gauge along
pointer
the
and
in
height
position.
of
the
knife
edge so that
the
dial
gauge reads
zero.
9.
10.
Record
Add
the
a
obtain
load
load
at
cell
to
least
the
Remove the weights
12.
Place
the
apply
a suitable
Adjust
zero
14.
hanger
the
Now place
the
and repeat
at 0.2
load
and repeat
of the
(e.g.
of the
the
load
steps
B, 9 and 10 so as to
results.
and hanger
height
and record
hanger
five
11.
}).
reading.
load
at 0.4
step
6.
span (measured
from the
clamp)
and
8 N).
knife
cell
edge so that
the
dial
gauge reads
reading.
of the
span and repeat
for
0.5,0.6
and
0.8.
15.
Either
the
plot
results
graphs
in
tabular
to verify
form
equations
as shown in
Fig 9.3
10.2
the
and 10.3
example
or arrange
below.
r
-40-
6.9.4
Analyais
of
Results
[
=
Part 2:
L
Note:
from
i.e.
w =
= 1000 mm w
equation
PIc.
9.7
9.3:
= 0.5
500 nwn c
N
iW(3 - c)
PIc!
I (1.5 - O.5c)
I w
-iii
~
c
P
0.2
O.J
0.4
0.5
0.6
0.8
1.4
1.35
1.3
1.25
1.2
1.1
PIc!
1.5-0.5c
W
Error ~
6.9.5
Conwnents
The average values of ~ from part 1 agree with equation 9.2,
~W
C
The
results
experimental
also
that
of
Part
errors
the
2 verify
occur
load
cell
equation
reading
number then the reading
c
are highly
magnified
o
when the
larger
= 0.2
p
5
J..e. - = ->
W
16
load
for
for c
9.3
cell
and show that
is
c = 0.2
=
0.8;
compared with
lightly
the
loaded.
are multiplied
therefore
the same errors
greatest
Notice
by a much
small
errors
at c = 0.8.
at
L
-41(With
ProPped Cantilever
6.10
Prop)
Elaatic
6.10.1 Introduction
Referring
to
applies
in
equations
the
introduction
the
present
applying
zw =
W
[1
~
b)
If
prop
force
i)
the
experiment
case and is
to a cantilever
a)
the
to
9,
restated
loaded
here
equation
as the
as shown in
fig
is
P then:
of
=
the
cantilever
due to the
prop
deflection
obvious
difference
between
i.e.
and if
that
a =
the
of
=
the
prop
due to
the
prop
force
cL
. .
deflections
-
p
w
=
3c2
of
given
the
prop must be equal
by 10.1
to
and 10.2.
Zp
(3c2
6EI
whence
10.3
of the prop.
deflection
w..'
then:
is:
f.
the
zw
~
is:
. . . . lC
wher~ k is the stiffness
is
force
-PL'
K
It
the
8'
deflection
~
of
10.1:
XI
the
first
still
6""
2
zP
ii)
9.1
-
c~
2 + (6EI/kl')
- C')
= f.
k
JE1
Pt.'
= constant
.
.
.
.10.4
the
r
-42-
6.10.3 EQuipment
Load
cell,
beam, steel
ReQuired
dial
gauge,
hanger,
weights,
clamp,
6 mm thick
aluminium
straight-edge.
6.10.4 Procedure
1-9
10.
Exactly
Add
the
same as for
successive
loads
deflection
and load
11.
Remove the
load
12.
Compare
the
experiment
6.10.5 Analysis
The
of
stiffness
of
to the
cell
hanger
recording
and repeat
of the
that
9.
the
values
of
load,
reading.
and hanger
value
with
experiment
Modulus
obtained
in
experiment
B (see
of Elasticity
step
obtained
note).
in
this
11.
Results
the
load
cell
(elastic
prop)
is
easily
determined
(see
page 3).
The experimental
The results
set-up
can
was shown in rig
be presented
in
tabular
10.2.
form
as
shown
below.
6.10.6 Ca~!J_lations
From equation
11.4
L.
p
w
=
i~_~__Q~21)-- O.~'
(6EI/Id.')
+ 2
. . . . 10.5
~
-43-
0.625
(6EI/~')
J.
From
the
above
~= ~
table
6W
I,
for
the
equations
beam used in
10.5
and 10.6
~
+ 2
5
the
.
experiment,
is
and substituting
E =
D.625
.
x
5
values
x
;;J;
.!
.l
~
..t
.J
using
k and l:
x
1.46199 x Iff
N/mm2
~p
result
8) will
'c:t:
I,
10.6
6 x 342
l...ll1 - 2
The
~
of
.
3 x 1000'
~p
J
.
342 (mm)~, therefore,
the
- 2
.
in close
from
give
step
another
11 of
value
agreement with
for
the
procedure
the
Modulus
the value obtained
(i.e.
carrying
of Elasticity
above.
out
which
experiment
should
be
-44-
BENDI~- OF BE.A1'1S
ABOU1A PRINC1~LAXIS
~
L
~
DEFLEalONEND
AT 'X'
SLOPE
RfACll0 N
MOMENT
MF
~
.w
WL3
IW
Wa2b2
3EIL
Wab(L+b)
6EIL
r
SwL4
384EI
-'ilL3
24E1
[
~ a t:
MF
b
~
~F .
Wa3b3
3'fj"'[3"-
Wab2
wlL.
384[1
wL2
[L-
MF
~
WL2
}W
Wl
L
w
4r"""""-""""
~x
wLL.
wL3
ru
wl2
ill
2
~
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,.
r
~
25481
~
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~
-.
25482 -
25486-
-
TQD~l
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Sheetf~ Acti~
25487-=-'
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lliSl-=
25552-'
15
16
17
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CHECK
I C~evef-supriiirt
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1
1
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--
,...'
j
13
01159722611
1159722611
01159731520
~1159731520
~
Clay_I>esl~t
1
6mm-A/F A11~nKey
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1
i
1
-
,
57812 57877 --
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I
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