# Beam Bending Lab

Urban and Civil

Nov 26, 2013 (4 years and 7 months ago)

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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

{

(

)

(

)

}

(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

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

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
,

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

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e⁯f⁉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

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

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(y潵⁷ill

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

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.