Beam Bending Lab
–
Preparation
Page
1
Beam Bending Lab
Preparation
1. Overview of the Beam Bending Lab
The Beam Bending Lab introduces basic concepts that
engineers
use
to
design structures.
In th
e Beam Bending L
ab,
you
will
:
1.
Investigate and apply the concepts of
stress, strain and Young's modulus for structural materials
.
2.
A
pply the stress

strain equation to calculate
how applied forces
deform
structure
s
.
3.
Ca
lculate the moment of inertia of various beam geometries and determine how beam geometry
affects t
he st
iffness
and strength
of beams.
4.
Use
dial caliper
s
and dial indicator
s
to make accurate measurements of the dimensions and deflections
of
structural
beams.
5.
A
ppl
y
forces
to cantilever beams and measure beam deflections
.
2
.
Engineering
Structures
This lab explores the engineering applications of materials and structures. Engineers choose
suitable
materials
–
metal, plastic, glass, concrete, etc.
–
to design useful, economical and safe structures. The
se
structures range from very small
–
nanotechnology devices and computer chips
–
to very large
–
airplanes,
bridges and tall
buildings.
Engineers have to be certain that
the structures
they
design will be stable and safe under the most adverse
operating conditions.
But first,
let's
inve
stigate how structural materials behave when they are pushed and pulled by forces.
2.1
Force and deflection for a simple spring
You learned in physics that a
n
applied force
F
makes a spring
stretch (or compress)
by
the
distance
x
.
The
linear relationship
between
F
and
x
is
Hooke's Law:
where
k
is called the
spring constant
. Note that for very large values of the spring constant, the spring
stretches
only a
very small amount in response to a
n
applied force.
In
SI units
,
the force
F
is in
Newtons
,
distance
x
is in
meters
,
and
the
spring constant has units of
N/m
.
In
English units
,
F
is in
lbf
(pounds

force)
,
x
is in
inches
,
and
the spring constant has units of
lbf
/
in
.
,
(1)
Beam Bending Lab
–
Preparation
Page
2
L
L+ L
F
F
F
F
Figure 1b.
Aluminum Bar, Force F
Area, A
L
Figure 1a.
Aluminum Bar, no Force
2.
2
Simpl
ified
model of structural materials
It turns
out th
at solid materials also exhibit
an
elastic, spring

like response to applied forces.
Figure 1a (below) shows an aluminum rod with cross

s
ection area
A
and
initial length
L
when
no force
is
applied. The atoms in the rod are all connected to their nearest neighbors by electromagnetic forces that
are "
spring

like"
–
here
shown
as
atoms connected by springs
.
Figure 1b shows the same aluminum rod, but with
two
opposing forces
F
applied at each end. In
response to the forces, the
length of the
aluminum bar
increases
(
stretches
)
by an amount
Δ
L
.
When the
forces
pull outward
, as
shown
in Figure 1b, the bar is said to be in "
tension
".
If
both
forces
we
re
reversed to
push inward
on the ends of the bar, the length of the bar
decrease
s
by the
same
amount
Δ
L
and
we would say that
the bar
is
in "
compression
".
2.
3
Stress, strain and Young's modulus
For Figure 1, the "
spring

like equation
" for the aluminum bar is
The proportionality constant E is called
Young's modulus
, or the
modulus of elasticity
,
and has units of
N/m
2
(Pascals)
or
lbf/in
2
(pounds per square inch, or
simply
psi)
.
The
ratio "
F/A
" is called "
stress
"
and is given the Greek letter symbol σ:
The physical interpretation of
stress
is
:
the
internal pressure
in the aluminum bar
caused by
external
ly
applied
forces
.
(㈩
却Se獳
{
(
)
(
)
}
(㌩
Beam Bending Lab
–
Preparation
Page
3
The
ratio "
Δ
L/L"
is called
"strain"
(
note that
it has no units)
and is given the Greek letter symbol
ε
:
Therefore,
strain
is the
fractional elongation
(or compression) of a structural member caused by applied
forces. For instance, a strain of
ε
= 0.02
5
would correspond to a
2.5%
increase
in the length of the bar.
Substituting Equations (3) and (4) into (2) gives the fundamental "
stress

strain
" equation
for materials
:
For
most structural materials (steel, etc.),
Young's modulus is very large, so large forces
cause only small
deformations. Have you ever
been in a car on a bridge and felt the
slight
up

and

down motion of the
bridge
caused by
heavy trucks
moving across
the bridge?
Metric and English values of Young's modulus
for the materials we will use in the Beam
Bending Lab
are shown in
T
able
1
.
F
rom the table, you can see that steel is
almost
three
times stronger than aluminum
.
2.
4
Stress

strain graphs and structural failure
Figure 2 shows the graph of stress versus strain for a
typical structural material.
The slope of the
linear
portion of the curve
is equal to Young's modulus.
The linear portion of the curve
(up to point 2)
is
called the
elastic
region
,
because
every time
the
force is removed,
the
beam returns to its original
length.
Point 2 is called the Proportional Limit.
After
Point 2,
however,
the curve
becomes
non

linear and the beam
will be
permanently deformed.
The graph also shows that i
f
the beam is stressed to
Point
4
and the force is removed, then
the beam
retains a permanent 0.2% elongation.
If the beam is
stretched to the end of the
curve
(Point
X
)
,
the beam
fails
completely and
breaks apart
.
E
ngineers design structures
so
that the maximum
stress
in
building
s
or airplane
s
will
never exceed the
Proportional L
imit (Point 2)
,
even under the most adverse conditions.
Professional design standards,
building codes and governmental regulations
also
require
structural engineers to
use safety factors
in
the
de
sign of bridges, buildings, airplanes and other critical structures.
Strain
{
(
)
}
(
4
)
{
(
)
(
)
}
(㔩
T慢l攠ㄮ†1潵湧'猠m潤畬畳潲 t桥⁂h慭⁂敮摩湧⁌慢
Y潵湧o猠m潤畬畳
Ⱐ
E
Material
GPa (10
9
N/m
2
)
psi (lbf/in
2
)
Aluminum
70
10,000,000
Steel
186
27,000,000
Polystyrene
3
435,000
Basswood
35
5,000,000
Figure 2. Graph of Stress versus Strain.
ΔL
x
Beam Bending Lab
–
Preparation
Page
4
A = 10 sq mm
Aluminum Bar
m = 100 kg
L = 1.0 m
F
g
2.
5
Sample
calculation
of
stress
and
strain
Let's
use
t
he
stress

strain
equation
to solve a sample problem.
Consider
a
round aluminum bar
(shown in Figure 3)
that has
a
n
initial
length
and a cross

section
al
area
.
I
ts
upper end
is
fixed to
the ceiling and
the weight
(
m
=
1
0
0
kg
)
hangs on the free end
, as shown in
the
Figure.
Calculate:
(1) the stress σ in the bar, (2) the strain
ε
, and (3)
the elongation
Δ
L
.
(1)
Stress in the bar
The f
orce
of
gravity on the mass is
(
)
(
)
Thi猠f潲捥
灵ll猠
摯d渠
潮oth攠e湤f⁴桥hr⸠.Th攠
異uar搬d
潰灯oi湧潲c攠
i猠灲潶id敤e批⁴桥eili湧⸠⁔h敲敦or攬et桥
i湴敲湡l
Stre獳
i渠n桥hr
(
)
is
(2)
Strain in the bar
Since
Young's modulus for aluminum
is
(3)
Elongation
The strain
ε
is equal to the fractional elongation
,
ε
=
ΔL/L
= 0.0014
.
And
since we
know the initial length of the
bar,
L = 1 m
,
then
the
elongation
ΔL
is
2.
6
Student exercise
Us
ing
the
method
from
the example above
,
find stress, strain and elongation for a
suspended
aluminum
ba
r
(
L = 5 ft
,
A = 0.01 in
2
)
that holds a weight
of
mass
m = 100 lbm
. Do your calculations in English
units. Note that the force of gravity on a
100 lbm
mass is
F
g
= 100 lbf
.
Check
the
correct answer
,
ΔL =
(A)
0.006 m ___ (B) 0.060 ft ___ (C) 0.060 in ___ (D) 1.5 mm ___
Figure 3. Suspended aluminum bar.
Stress
Strain
Elongation
(
)
(
)
Beam Bending Lab
–
Preparation
Page
5
3
.
Cantilever Beams
3
.1
Horizontal cantilever beam
A c
antilever beam
is a structure
with one
end
firmly anchored
and
the
other end free to move
.
Figure 4 shows a cantilever
beam
with the beam
oriented in a horizontal plane.
The free end of the beam will move
down if
an
e
xternal force
F
is applied
to
the end.
T
he
deflection of the free end of t
he beam
due to
the
applied force
F
depends on:
(1)
the dimensions of
the beam (length
L
, width
w
, and thickness
t
)
,
(2)
Young's modulus
(
E
)
for the beam material
, and
(3)
a geometry factor called the
Moment of Inertia
.
Examples of horizontal cantilever
s
are: airplane
wings, diving boards, and the overhanging
section
of the upper level deck
in
Ohio Stadium.
T
he
force
F
causes the end of the beam to deflect downward by an amount
δ
. T
he equation to calculate
the deflection
δ
is:
Deflection
{
}
(㘩
T桥
䵯M敮ef⁉湥ntiaⰠ
I
, is a geometry factor that depends on
ly on
the cross

sectional
dimensions
(width
w
and thickness
t
)
of the beam. For the rectangular beam shown in Figure 4, the moment of inertia is
3
.
2
Sample calculation of deflection
Given
a rectangular aluminum beam with
L = 1 m, w = 5 cm, t = 1 cm
and
F = 10
0
N
, first calculate
I
:
Then calculate
the
deflection
at the end of the beam (
Young's modulus for aluminum
is in
Table
1
)
:
Figure 4. A horizontal cantilever beam.
Moment of
Inertia
{
}
(㜩
D敦le捴i潮
(
)
(
)
(
⁄
)
(
)
䵯M敮e ⁉湥ntia
(
)
(
)
Beam Bending Lab
–
Preparation
Page
6
3
.
3
Vertic
al cantilever
beam
In the Beam Bending Lab, you will test vertical
cantilever beam
s, similar to
th
e beam
in Figure 5.
In th
e
figure, th
e force is horizontal
and
the beam bends
to the right.
A few
familiar
examples of vertical cantilevers
are:
Trees
S
top sign
s
Tall buildings
W
ind turbine
tower
s
On a windy day,
the
force
of the wind
is
distributed
over
exposed
surfaces
and
cause
s
the
se
structures
to bend
.
The Sears Tower in Chicago (now called the Willis
Tower
) is 110 stories and 1450 feet
tall
. A 60 mph wind
causes the
building to bend and the
top of the tower
to
move laterally by 6 inches.
The tower
's
structure is designed to safely
withstand the
largest wind
speed
ever
expected
in Chicago.
Figure 5. A vertical cantilever
beam.
Beam Bending Lab
–
Preparation
Page
7
4
.
Beam Bending Lab Apparatus
4.1
Major components of the Beam Bending Lab apparatus
Clamp
Vertical
Cantilever
Beam
Dial Indicator
Pulley
Weight
Holder
Extra
Weights
Figure 6. Beam Bending Lab apparatus.
The
major components of the Beam Bending Lab apparatus
are
shown in Figure 6
:
Clamp
–
used to securely hold the fixed end
of
the vertical cantilever beam
Vertical Cantilever Beam
–
made of different
sizes
,
shapes,
and materials,
including:
o
aluminum
o
steel
o
copper
Dial Indicator
–
a precision instrument used to measure beam deflection
Pulley
–
transmits the
vertical
force of the we
ights to
a horizontal
force
on
the beam
Weight Holder
–
holds up to ten weights
Extra Weights
–
ten weights are supplied with each lab kit (50 gm
each
)
During the lab, you will:
1.
Select a beam (from one of the five beam materials listed above)
2.
Securely
fasten the beam in the Clamp
3.
Place
the
weights
, one at a time, on
the Weight Holder and observe the beam
deflection
4.
Record beam deflection
s
as indicated on the face of the Dial Indicator.
Beam Bending Lab
–
Preparation
Page
8
4.
2 Theoretical model of the Beam Bending Apparatus
In the beam bending apparatus, the force
F
is applied at
a location that is a distance
L = 8.75"
from the fixed end
of the beam, as shown in Figure 7.
Also shown, the deflection of the beam is measured (by
the Dial Indicator) at the location
S = 7.5"
from the
fixed end
of the beam.
The equation that gives deflection as a function of the
applied force and beam properties is:
Deflection
(at location S)
(
)
(㠩
䱥L'猠摯 m灬攠e慬c畬atio渠n潲 愠at敥l 慭
(y潵⁷ill
畳u⁴桩s敡m渠n桥⁂敡h⁂敮摩湧⁌慢
t桡t慳
t桥h
f潬l潷i湧⁰ o灥pti敳:
L = 8.75"
S = 7.5"
w =
0.50"
t = 0.125"
E = 27,000,000 lbf / in
2
F = 1.0 lbf
Use Eq. 8 to
calculate the deflection.
First, calculate the Moment of Inertia using Eq. 7:
Then calculate deflection:
Deflection
(at location S)
(
)
(
)
(
)
(
⁄
)
(
)
(
(
)
)
y
x
F
L = 8.75"
S = 7.5"
Deflection
䙩F畲攠㜮†Th敯e整i捡l潤敬 t桥h扥bm
扥湤b湧灰慲at畳.
䵯M敮e ⁉湥ntia
(
)
(
)
Beam Bending Lab
–
Preparation
Page
9
4.
2 Moment of Inertia for other beam geometries
In actual practice, a beam with a solid rectangular cross

s
ection is
not always the strongest or the most economical for use in
structures.
For instance, the vertical support tower for a wind turbine is
usually made of steel and has a
tapered,
hollow
,
circular cross

s
ection.
And because of its circular symmetry,
it has the same
strength regardless of which way the wind is blowing.
Many beams used in bridges and buildings have
the
cross

s
ection
of the
block letter "I" and for that reason are called "I

beams".
For specific applications, b
eams with
many different
shapes and
geometries are designed to be light, strong and as economical as
possible.
Figure
9
shows the cross

s
ection
geometries
of a
hollow
circular beam, an
I

b
eam and a box

beam.
(1)
(2)
(3)
Figure
9
. Cross

s
ection geometries of (1)
hollow
circular beam, (2) I

beam, and (3) a box

beam.
5.
Further preparation assignment
In order to finish your preparation for the Beam Bending Lab, watch the video on how to use a Dial
Caliper and take the Beam Bending Lab quiz on Carmen.
Figure 8
. Wind turbine tower.
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