Objective:
Part 1: To study the relationship between load, span, width, height and the
deflection of a beam that placed on two bearers and affected load at the
center.
Part 2: To distinguish the coefficient of elasticity for the steel and brass.
Introduction
Young Modulus of a material is defined as the ratio of longitudinal stress to
strain, where
Young Modulus, E
¿
stress
strain
Force
cross sectional area
¿
extension
original length
Dimension of Young Modulus
=
=
Dimension of stress
=
=
dimension of stress
dimensionof strain
dimension of force
dimensionof cross sectional area
MLT −2
L2
−1
ML T
−2
Since strain has no dimension because it is the ratio of two lengths, hence,
dimension of Young Modulus =
−1
ML T
−2
.
The SI unit for Young modulus is Nm-2.
When force is applied to the material used, assuming the before elastic limit,
the extension in length of material increase but the cross-sectional area of
material doesn’t change with force applied, within the limit of proportionality,
the extension is directly proportional to the force, the material obeys the
Hooke’s Law.
In this experiment, we measure the deflection of material caused by the force
to determine the Young Modulus and other quantities that affect the Young
Modulus are constant values, i.e. its length and cross sectional area. The
formula is modified in term of Force, Deflection, Length, Moment of Inertia.
Graph of Force against Deflection is constructed, the gradient of graph is
determined =
Force
Deflection
.
Apparatus and Materials:
1) Shear forces apparatus :- 1 set of 80mm x 50mm x 38mm aluminium
section with 2 adjustable span support.
2) 1 unit of shear force dynamometer.
3) 2 sets of weight hangers.
4) 1 set of weights.
5) 2m measuring tape.
Procedure:
Part I
A) One fixed end and one simple support end.
1) The clamping length was adjusted to 800mm.
2) The width and height(depth) of the test specimen measured by using a
caliper and the value was recorded.
3) The test specimen was placed on the bearers.
4) One end was set as fixed end by tightening the screw.
5) The load(F) hanger was mounted on the center of the test specimen.
6) The dial gauge was moved to the center of the test specimen. The height
of the gauge was adjusted to ensure the needle touches the test specimen.
The initial reading of gauge is recorded.
7) The load of 2N weight was loaded onto the weight hanger and the dial
gauge reading is recorded.
8) The procedure (7) was repeated by using load of 4N, 6N, 8N and 10 N. All
the gauge readings must be recorded.
9) Remove all the loads after the results have been taken.
10) Get the average deflection value by repeated the experiment once again.
11) The graph of force versus deflection was plotted.
12) The experimental Young’s Modulus for respective beam/material was
calculated and compared with the theoretical(published) value.
13) The experiment was repeated by using different beam material (steel and
brass).
Part II
B) Two simple supports end.
1) The clamping length was adjusted to 600mm.
2) The width and height(depth) of the test specimen measured by using a
caliper and the value was recorded.
3) The test specimen was placed on the bearers.
4) The screw did not tightened since both ends are simple supports.
5) The load(F) hanger was mounted on the center of the test specimen.
6) The dial gauge was moved to the center of the test specimen. The height
of the gauge was adjusted to ensure the needle touches the test specimen.
The initial reading of gauge is recorded.
7) The load of 2N weight was loaded onto the weight hanger and the dial
gauge reading is recorded.
8) The procedure (7) was repeated by using load of 4N, 6N, 8N and 10 N. All
the gauge readings must be recorded.
9) Remove all the loads after the results have been taken.
10) Get the average deflection value by repeated the experiment once again.
11) The graph of force versus deflection was plotted.
12) The experimental Young’s Modulus for respective beam/material was
calculated and compared with the theoretical(published) value.
13) The experiment was repeated by using different beam material (steel and
brass).
Part III
C) Two simple supports end.
1) The steel beam was selected in the experiment
2) The clamping length was adjusted to 500mm.
3) The width and height(depth) of the test specimen measured by using a
caliper and the value was recorded.
4) The test specimen was placed on the bearers.
5) The screw did not tightened since both ends are simple supports.
6) The load(F) hanger was mounted on the center of the test specimen.
7) The dial gauge was moved to the center of the test specimen. The height
of the gauge was adjusted to ensure the needle touches the test specimen.
The initial reading of gauge is recorded.
8) The load of 2N weight was loaded onto the weight hanger and the dial
gauge reading is recorded.
9) The procedure (8) was repeated by using load of 4N, 6N, 8N and 10 N. All
the gauge readings must be recorded.
10) Remove all the loads after the results have been taken.
11) Get the average deflection value by repeated the experiment once again.
12) The graph of force versus deflection was plotted.
13) The experimental Young’s Modulus for respective beam/material was
calculated and compared with the theoretical(published) value.
Calculation:
−3
−3 3
b h 3 (26.0∗10 )(6.53∗10 )
Ifor brass=
=
=6.033∗10−10 m4
12
12
−3 3
−3
I for steel=
3
b h (24.5∗10 )(3.1∗10 )
−11 4
=
=6 .0823∗10 m
12
12
Part 1:
Brass:
y = 0.0539x + 0.9889
Gradient, m =
( )(
F
E=
δ
F
δ
=
0.0539
=
53.9
N
mm
N
m
3
3.5 ( 0.8 )
3.5 L3 (
= 53.9 )
384 I
384 ( 6.033∗10−10 )
)
(
)
¿
416929112.1 Pa
¿
4.2* 10
¿
0.42GPa
8
Pa
Theoretical value of young modulus for brass is 100GPa.
Percentage error =
¿ E−100∨ ¿ ∗100
100
¿
= 99.583%
Stainless Steel:
y = 0.0184x + 2.4578
Gradient, m =
E=
( )(
F
δ
F
δ
=
0.0184
=
18.4
N
mm
N
m
3
(
3
3.5 ( 0.8 )
3.5 L
= (18.4 )
384 I
384 ( 6.0823∗10−11 )
)
¿
)
1411746653 Pa
¿
9
1.41* 10
¿
1.41GPa
Pa
Theoretical value of young modulus for stainless steel is 200GPa.
Percentage error =
¿ E−200∨ ¿ ∗100
200
¿
= 99.294%
Part 2:
Brass:
y = 0.0529x + 1.3604
gradient , m =
( )(
F
E=
δ
F
δ
=
0.0529
=
52.9
N
mm
N
m
( 0.8 )3
3.5 L3 (
)
= 52.9
384 I
48 ( 6.033∗10−10 )
(
)
¿
)
935300292.8 Pa
9
¿
0.94* 10
¿
0.94GPa
Pa
Theoretical value of young modulus for brass is 100GPa.
¿ E−100∨ ¿ ∗100
100
¿
Percentage error =
= 99.0647%
Stainless Steel:
y = 0.0079x + 0.5343
Gradient, m =
( )(
F
E=
δ
F
δ
=
0.0079
=
7.9
N
mm
N
m
( 0.8 )3
3.5 L3 ( )
= 7.9
384 I
48 ( 6.0823∗10−11 )
)
(
¿
)
1385440815Pa
9
¿
1.38* 10
¿
1.38GPa
Pa
Theoretical value of young modulus for stainless steel is 200GPa.
Percentage error =
¿ E−200∨ ¿ ∗100
200
¿
= 99.307%
Discussion (Quiz):
Q1. What is stiffness and young modulus? What is the relationship between
stiffness and young modulus?
Answer:
Stiffness is a structural property, classically corresponds to a ratio between
force and length change, commonly associated to the properties of a
structure, influenced by the geometry of the specimens as well as the
material of which it is comprised. Under consideration of specimen geometry,
it can be determined from force-displacement curve.
Stiffness,
k=
F
e
F = Force Applied
e = extension of material
Young modulus is a material property, is calculated with ratio of stress to
strain, is intrinsic to the material, is not influenced by specimen geometry. It
can be determined from the slope of the stress-strain curve in the elastic
region.
Young modulus,
E=
FL
eA
Where
L = length of material
A = Cross-sectional area
Young modulus,
E=
¿
FL
eA
( Fe ) LA
=k
=
(
L
)
A
stiffness(
length
)
area
Q2: Do you agree with this statement “Stiffness is an important property in
the design of structures which are only allowed to deflect certain amount”?
Discuss.
Answer:
Yes. Stiffness is the rigidity of an object to resists deformation in response to
an applied force. It is very important because objects can quickly regain their
original shape after being deformed by a force, with the molecules or atoms
of their material returning to the initial state of stable equilibrium. High
stiffness of a material indicates it can absorb larger force and with little
extension and therefore gives smaller deflection.
For example, stiffness is dominant in the design of bicycle frame. This is
because the bicycle frame has to support the weight of the biker. It also has
to bear the additional force that is generated from a sudden direction
changes and riding over a rock or pothole and etc. When these forces are too
large and the stiffness of the bicycle frame is low, it will exceed its elastic
limit. It will begin to deform and might even break eventually.
Discussion (Experiment):
There is a relationship between deflection of beam and mass of the load.
Referring to the graph of Load vs. Deflection in the result section, we know
that the mass of load and the deflection of the beam are directly proportional
to each other. An increase in mass will lead to a corresponding increase in
deflection whenever the end of the beams is fixed or both are just simple
support.
Besides, the deflection of the beam depends on the length of the materials. A
shorter length of the beam will result in a smaller deflection. According to the
table, deflection of 500mm steel bar in part 3 is much more smaller than
deflection of 800mm steel bar in part 2, regarding any mass of the load.
Based on our calculation, the experimental value deviates widely from
theoretical value.
In part 1 and part 2, the experimental Young’s Modulus, E obtained for brass
are 0.42GPa and 0.94GPa respectively. However, the real theoretical value for
Young’s Modulus of brass is 100GPa. The percentage errors are up to 99%.
For the steel in part 1 and part 2, the experimental Young’s Modulus, E
obtained are 1.41GPa and 1.38GPa respectively. The real theoretical value for
Young’s Modulus of steel is 200GPa. The percentage errors are also up to
99%.
One of the errors affecting the result is the load hanger is not being mounted
correctly. The loads should be hanged in the middle of beam to prevent
inaccuracy.
Besides, the non-uniform cross-sectional area of the beam might influence
the value of Young Modulus. During the experiment, we had measured the
height and width of beam at its top end, middle part and bottom end
respectively. It results in the different figures obtained for the dimensions.
The cross-sectional area of beam is inconsistent and force is not exerted
exactly on the center of gravity of the beam. These errors will alter the values
of Young Modulus.
Another error could have been creeping. Creeping is the tendency of a solid
material to move slowly or deform permanently under the influence of
mechanical stresses, such as bending or applying load to it. It can happen as
a result of long-term exposure to high levels of stress, which are still below
the yield strength of the material. Depending on the magnitude and duration
of the applied stress, the deformation might become so large that a
component can no longer perform its function.
Other errors that could have taken place include the errors in the equipment
and the other random systematic errors that can occur.
Conclusion:
From the results we obtained, we can conclude that the length of
deflection of beam is directly proportional to the magnitude of forces applied
on the beam no matter the ends of beam are fixed or both of the ends are
simply support. Besides that, the length of deflection of beam is inversely
proportional to the Young’s modulus. In addition, the stainless steel is stiffer
than the brass because the value of Young’s modulus of stainless steel is
higher than the brass. Moreover, the width, b and the height, h of a
rectangular section beam is directly proportional to its moment of Inertia.
Reference
1.
2.
3.
4.
5.
http://en.wikipedia.org/wiki/Stiffness
http://en.wikipedia.org/wiki/Young's_modulus
http://www.engineeringtoolbox.com/young-modulus-d_417.html
http://www.engineeringtoolbox.com/young-modulus-d_773.html
http://www.skf.com/in/industry-solutions/racing/requirements/highstiffness/index.html
6. http://www.homebuiltairplanes.com/forums/aircraft-designaerodynamics-new-technology/11076-importance-stiffness-howget.html
7. http://www.researchgate.net/post/What_is_the_relation_between_stiffne
ss_and_youngs_modulus2