University of Engineering and Technology

Lab Report:
Strength of Materials

Submitted to:
Sir Shiraz Ali

Submitted By :
2018-BET-MECH-08
Salman Hayat
List of Experiments

Experiment
Description
No.
To draw the load-extension curve of a metallic wire and hence to determine the
1 modulus of elasticity of the material of the wire.

To investigate the relationship between shear stress and shear strain for rubber
2 and to determine the modulus of rigidity of the material.

3 To determine the modulus of rigidity of the given material of circular shaft.

To determine the central deflection of a simply supported beam loaded by a

4 concentrated load at mid point and hence determine the modulus of elasticity of
the material of the beam.

To determine the central deflection of a fixed ended beam loaded at mid-span

5 by concentrated loads and to compare with theoretical value.

6 To determine the deflection at mid span of a propped cantilever beam and

compare with the theoretical values

To determine the deflection at three different points of a propped

cantilever beam using a brass beam and compare with the theoretical
7 values also find its comparison with aluminum beam used in lab session
6.

8 To measure the stiffness of a compression spring and compare it with

theoretical values.

9 To measure the stiffness of an Extension spring and compare it with

theoretical values.

10 To verify the relationship among load on spiral spring, number of turns

and degree of rotation of a coil spring

Page | 12
Lab Session 1

1.1 Objective:
To draw the load-extension curve of a metallic wire and hence determine the modulus of
elasticity of the material of the wire.

1.2 Apparatus:
 Young’s Modulus of Elasticity apparatus
 Hangers
 Weights
 Meter Rod
 Micrometer

Figure 1. 1Young's Modulus of Elasticity

Apparatus

1.3Summery of Theory:
The Young’s Modulus of Elasticity apparatus consists of a wire attached to a fixed support.
The lower end of the wire is attached to the hanger with the help of a metallic plate. The
extension of the wire on loading can be measured from the scale present on metallic plate.

Normal stress in a solid body is defined as:

“The internal resistance force per unit area against the applied load or external force.” It is denoted by
σ. It can be tensile or compressive.
Mathematically,
Stress = Force/Area ---------- (i)
2
Units of stress: Newton per square meter (N/m ) = Pascal (Pa) or pounds per square inch (psi)

Page | 13
Normal strain in a solid body is defined as: “Change of length per Original Length.” It is denoted by
the symbol ε.
Mathematically,
Normal Strain = Change in length/Original length ----------
(ii) Strain is measured as inch/inch.

By Hooke’s law, we know that stress is directly proportional to the strain, whenever a material is
loaded within its proportionality limit. It is denoted by E.
Mathematically,
Stress α Strain (within proportionality limit) ---------- (iii)
2
Units of E: Newton per square meter (N/m ) = Pascal (Pa) or pounds per square inch
(psi) Consider a body (wire) subjected to a tensile stress as shown in figure 1.1.

Let,
P = Load or force acting on the body
L = Length of the body
A = Cross-sectional area of the body
σ = Stress induced in the body
E = Modulus of elasticity for the material of the
body ε = Strain produced in the body δl =
Deformation of the body

From (i), (ii), and (iii)

σαε
σ=Exε
or
E = σ/ε

E = (p/ δl) (L/A)

1.4 Load-Extension Curve:

Figure 1. 2 Load Extension Curve

Page | 14
1.5 Procedure:
1. Put the initial load of 2 lb to remove wrinkles in wire.
2. Measure length of wire using meter rod.
3. Measure diameter of the wire using micrometer.
4. Adjust main scale so that zeros of two scales coincide with each other.
5. Put a load of 5 lb in the hanger and measure extension.
6. Take a set of at least five readings of increasing value of load and then take readings
on unloading.
7. Check the zeros at no load.
8. Calculate the “Young’s Modulus of Elasticity (E)” of the material of the shaft.

1.6 Observations and Calculations:

Least Count of the scale of apparatus = 0.1 mm

Least Count of micrometer = 0.001 mm
Least Count of meter rod = 1 mm
Length of wire (L) = 1011 mm
Dia of wire (d) = 1 mm
Initial Load = 13.2 N

X-area of wire (A= πd2/4) = 0.785 mm2

Table 1 Calculation of Modulus of Elasticity:

Extension-
δl Modulus of
P/δl
Elasticity
Effective Load-P (mm)
No. of Obs. (N/mm)
E=(P/δl)(L/A)
(N)
Loading Unloading Average
From Graph
(GPa)
1. 10 0.4 0.5 0.45 22.22 28
2. 20 0.6 0.7 0.65 30.76 39
3. 30 0.8 0.9 0.85 35.29 45
4. 40 1 1.1 1.05 38.09 49
5. 50 1.2 1.2 1.2 41.67 53
1.6.1 Graph:

Deformation Vs Load

60

50
50

40
40

30
30
Load

20
20

10
10

0
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Deflection

1.8 Conclusion:
The following are results can be drawn:

1.The graph between load and extension is straight line.

2. The average value of modulus of elasticity is 42.8 GPa.

1.9 Comments:
 In this experiment we experimentally calculate the value of modules of elasticity of a wire
 There was a problem in the main scale and the Vernier scale of the apparatus.
 Wire also used in experiment was not actually straight, we straight it using adding a weight in
weight pan.
Lab Session 2

2.1 Objective:
To investigate the relationship between shear stress and shear strain for rubber and to determine
the modulus of rigidity of the material.

2.2 Apparatus:
 Modulus of rigidity of rubber apparatus
 Hangers
 Weights
 Steel rule
 Dial Indicator

Rubber Block Dial Indicator

Loading Plate

Base Plate
Hanger

Back Plate Plate

Figure2.1
Figure 2.1Rubber
Rubber Block Apparatus

A rubber block 12 x 4 x 1 inch is bonded to two aluminum alloy plates. One plate is screwed to a
wall, whilst the other has a shear load applied by a loaded weight hanger. A dial gauge measures
the deflection of the block.
This equipment is part of a range designed to both demonstrate and experimentally confirm basic
engineering principles. Great care has been given to each item so as to provide wide
experimental scope without unduly complicating or compromising the design.
Each piece of apparatus is self-contained and compact. Setting up time is minimal, and all
measurements are made with the simplest possible instrumentation, so that the student
involvement is purely with the engineering principles being taught.

2.3 Summary of Theory:

The force which tends to cut off or parts off one portion of the component from the other is called
shear force. Stresses produced on the area under shear, due to shearing forces, are called shearing
stresses. Shear stress is denoted by τ.
Mathematically,
Page | 17
 Shearing stress = Shearing force/ Area under shear ------ (i)
2
Units of shear stress: Newton per square meter (N/m ) = Pascal (Pa) or pounds per square inch (psi)

Shearing strain is the angle of distortion. It can be represented by γ. ------ (ii)

The constant of proportionality relating shear stress and shear strain is modulus of rigidity. It is
represented by G.
Mathematically,
G = Shear stress/ shear strain ------ (iii)
Units of G: Newton per square meter (N/m2) = Pascal (Pa) or pounds per square inch (psi)

Let us consider the deformation of a rectangular block where the forces acting on the block are
known to be shearing stress as shown in the figure 2.2.
The change of angle at the corner of an originally rectangular element is defined as the shear
strain.

Let, w
Ps = Shearing load or force acting on the A C
body l = Length of the body
c
A = Area under shear = l x t
τ = Shear stress induced in the body l
G = Modulus of rigidity for the material of the
body γ = Shear strain produced t
δs = Deformation of the body
B D δ
d

From the figure P

Figure 2. 2 Distortion of a
Cc = Dd = δs = Shear
Rectangular Block
Deformation tanγ = Dd/BD = δs/w

For smaller angles

tanγ = γ =Shear strain = δs/w

From the information in (i), (ii), and (iii)

G=τ/γ
or
G = (Ps / δs) (w/ l.t)

Page | 18
Shear Stress-Shear Strain Curve:

Figure 2. 3Shear Stress‐Shear Strain Curve

2.4 Procedure:
1. Set the dial indicator so that its anvil rests on the top of the loading plate.
2. Set the dial indicator at zero.
3. With the hanger in position apply a load to the hanger and read the vertical displacement of
the loading plate relative to the fixing plate from the dial indicator (δs).
4. Repeat the experiment for increasing load and record the vertical displacement of the loading
plate in each case.
5. Unload and note the corresponding readings with the load decreasing.
6. Calculate the “Modulus of Rigidity (G)” of the rubber material.

2.5 Observations & Calculations:

Length of rubber block (l) = 301 mm

Width of rubber block (w) = 101 mm

Thickness of rubber block (t) = 24.5 mm
Least count of dial indicator = 0.01 mm

Page | 19
 Table 2 Calculation of Modulus of Rigidity:

Shear Deformation-δs Modulus of

Load Shear Shear
No. (mm) Stress Strain Rigidity
of Ps G =τ/γ
Obs. τ =Ps/l .t γ = δs / w
(N) Loading Unloading Average
2
(KN/m ) (MPa)
1. 10 0.09 0.09 0.10 1.3 x10 3
98 14.3
2. 20 0.20 0.20 0.21 2.6 x103 196 12.3
3. 30 0.30 0.28 0.30 3.9 x103 294 11.1
4. 40 0.41 0.43 0.41 5.3x103 392.1 13.3
5. 50 0.50 0.52 0.51 6.3 x103 502.3 12.3

2.5.1 Graph:

Shear stress vs Shear strain

600

502.3
500

392.1
400
Shear stress

294
300

196
200

98
100

0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Shear strain

2.8 Conclusion:
There are following results:
1. Plot of curve between shear stress-τ (Y-axis) and shear strain-γ (X-axis).Calculate the slope of the
graph.
2. Hand calculations showing all results requested in (5) under procedure above.
The difference between experimental value and Ideal value is due to some systematic error or random
error.

2.8 Comments:
1. Here we use the modules of rigidity apparatus to find out the shear stress and shear strain of the
rubber material.
2. We experimentally calculate the value of modulus of rigidity by the shear stress and shear strain.
3. There was some issue in the dial gauge that’s why there is some error in the readings.
Lab Session No. 3

3.1 Objective:
To determine the modulus of rigidity of the given material of circular shaft.

3.2 Apparatus:

 Torsion of shaft apparatus

 Hangers
 Weights
 Vernier Calipers
 Micrometer
 Steel rule

Figure 3. 1Torsion of shaft Apparatus

Torsion of shaft apparatus includes a shaft of circular section, two measuring scales and a pulley with
a frame.

The main purpose of the pulley with hanger is to apply some load on the circular shaft. Similarly, the
scales attached to the frame are used to measure the torsion in the circular shaft. Actually, two scales
are used, one at the front and one at the back.
The measuring arms (scales) are used to measure the magnitude of the torsion at the front and the
back of the circular shaft respectively. The front is the portion of the shaft that is near to the pulley
and the back is the portion of the shaft near the back support of the frame.
The main purpose of the frame is to support the shaft and balance the apparatus on the surface.

Page | 22
3.3 Summary of Theory:
Torsion is the engineering word used to describe the process of twisting a member about its
longitudinal axis.

Consider a solid circular shaft of radius “r” and length “L” fixed at its back face as shown in figure(b).
A line AC is marked on the shaft. If a torque “T’ is applied at its free end, line AC will acquire the
/
shape of a helix and point A will move to A .

L
Figure 3. 2Torsion of Shaft

Then from figure,

/
Angle of twist, θ = <AO A

Now consider a longitudinal fiber at distance “ρ” from the axis of the shaft.
Deformation in longitudinal fiber, δs = AA/ = ρθ
Strain in longitudinal fiber, γ = δs /L = ρθ/L
Stress in longitudinal fiber, τ = Gθ

The shearing strain is maximum on the surface of the shaft where ρ = r.

If J is the Polar moment of inertia of the shaft, then using above information the torsional
formula for a circular shaft can be written as:

T/J = τ/r = Gθ/L

` or
G = TL/ Jθ

The torsional formula describes the relation of applied torque with the angle of twist and stresses
produced in the shafts.

Page | 23
3.4 Procedure:
1. Place the apparatus on a smooth horizontal surface.
2. Measure the effective length of the shaft using steel rule.
3. Measure the diameter of the shaft using micrometer.
st nd
4. Adjust the Zeros at 1 and 2 measuring arms.
5. Put a load of ten N in the hanger.
6. Measure the 1st and 2nd angle of twist of the shaft.
7. Take a set of six readings of increasing value of load and then take readings on unloading.
8. Calculate the “Modulus of Rigidity (G)” of the material of the shaft.

3.5 Observations & Calculations:

Effective length of shaft (L) = 300 mm

Diameter of shaft (d) = 5 mm
Diameter of torque pulley (D) = 68 mm
Radius of torque pulley (R=D/2) = 34 mm

Polar Moment of Inertia of the shaft (J=πd4/32) = 61.3 mm 4

Table 3 Calculation of Modulus of Rigidity of a shaft:

st
Load Torque Angle of twist at 1 measuring arm
No.
of W WR θ1 Modulus of Rigidity
Obs. (N) (Nm) (rad) G=TL/Jθ
Loading Unloading Average (GPa)
1. 10 0.34 0.035 0.037 0.036 47
2. 20 0.68 0.061 0.063 0.062 54
3. 30 1.02 0.078 0.080 0.079 63
4. 40 1.36 0.122 0.124 0.123 54
5. 50 1.70 0.148 0.150 0.149 56
6. 60 2.04 0.165 0.165 0.165 60

Page | 24
 3.5.1 Graph:
Load vs modules of Rigidity
70
63
60
60 56
54 54

5047

40
Load

30

20

10

0
10 20 30 40 50 60
Modulas of rigidity

3.9 Comments:
1. We use the torsion on shaft apparatus to experimentally determine the value of modules
of rigidity of a shaft.
2. In the experience there some error occurred due to the lack of knowledge reading the
angle on the scale.
3. Angle reading error may occur due to corrosion on angle dial.
Lab Session 4

4.1 Objective:

To determine the central deflection of a simply supported beam loaded by a concentrated load at
mid-point and hence determine the modulus of elasticity of the material of the beam.

4.2 Apparatus:
 Deflection of beam apparatus
 Hanger
 Weights
 Meter rod
 Dial indicator
 Vernier Calipers

Figure 4. 1Simply Supported Beam

Deflection of beam apparatus contains a metal beam and two knife-edge supports upon
which the beam is supported for this experiment and hence the beam becomes of a simply –
supported type.

4.3 Summery of Theory:

Beams are structural members supporting loads applied at various points along the members. A
beam undergoes bending by the loads applied perpendicular to their axis of the structure. Beams
are of various types.

If the supports are at the ends such that one of them is pin and other is roller then such a beam is
called simply supported beam. The supports can be considered as simple wedges at the ends as shown
in figure 4.1.

Consider a simply supported beam AB of length “L” and carrying a point load “W” at the centre of
beam C as shown in figure 4.2.
The maximum deflection for simply supported beam will occur at half the distance from either
support (mid-point).

Page | 26
 δ

L L
2 2
Figure 4. 2Simply Supported Beam loaded at mid point

Let,
δ = Deflection of beam at any point along the length of the beam
δc = Central deflection of beam
x = Variable distance from end B.

From the symmetry of the figure, we find that the reaction at A is:

RA = RB = W/2

The maximum deflection (yc) at x = L/2 is given by:

3
δc = WL / 48EI
or
E= (W/ δc) (L3/ 48 I)
Where E = Modulus of elasticity for the material of beam
I = Moment of inertia of the beam

Load-Deflection Curve:

Page | 27
δ

Figure 4. 3Load‐Deflection curve

4.4 Procedure:

1. Set the Deflection of Beam apparatus on a horizontal surface.

2. Set the dial indicator at zero.
3. Apply a load of 2N and measure the deflection using dial indicator.
4. Take a set of at least five readings of increasing value of load and then take readings
on unloading.
5. Calculate the “Modulus of Elasticity (E)” of the material of the beam.

4.5 Observations and Calculations:

Least Count of the dial indicator = 0.01 mm

Least Count of vernier calipers = 0.01 mm
Effective length of beam (L) = 800 mm
Breadth of beam (b) = 25.4 mm
Height of beam (h) = 6.34 mm

Moment of inertia of the beam (I=bh3/12) = 539.41 mm 4

Table 4 Calculation of Modulus of Elasticity of S.S.Beam:

Central Deflection-δc Modulus of

Effective Load- W/ δc Elasticity
No. of W (N)
3
Obs. (N/m) E=(W/δc)(L /48I)
(N) Loading Unloading Average From Graph
(N/m2)
1. 2 0.45 0.70 0.57 3.47 68×109
2. 4 1.02 1.29 1.15 3.46 68×109
3. 6 1.66 1.80 1.73 3.45 69×109
4. 8 2.20 2.26 2.23 3.59 70×109
5. 10 2.77 2.81 2.79 3.61 71×109
6. 12 3.10 3.10 3.10 3.63 72×109
 2.5

1.5
4.5.1 Graph:
On 1graph, plot the deflection against load for the theoretical & practical results. Draw the best-fit
straight lines through the points
0.5

0
0 1 2 Deflection
3 vs4 Load 5 6 7

3.5
3.1
3 2.79

2.5
2.23

2
1.73
Load

1.5
1.15

1
0.57
0.5

0
0 2 4 6 8 10 12 14
Deflection
4.6 Industrial Applications:
 Since, beams are integral part of most of the structures in our daily life as well as in
industry.
 Almost all structures contain beams so these calculations have prime importance.
 For design purpose, we have to perform all the calculations to calculate the required
parameters.
 Load reactions should be calculated before designing supports for a bar.

4.8 Conclusions:
The following are results can be drawn:
1.The graph between load and extension is straight line.
2. The average value of modulus of elasticity is 358 (GN/m2).

4.9 Comments:
1. In the experiment we use a simply supported beam and check out the deflection at center
of the beam due to applied load.
2. Due to error in dial gauge a few errors take place in our readings.

3. Simply supported beam is also not equally straight because there is some bend in center
of beam.
Lab Session 5

5.1 Objective:

To determine the central deflection of a fixed ended beam loaded at mid-span by

concentrated loads and to compare with theoretical value.

5.2 Apparatus:

 Deflection of beam apparatus with clamps

 Hanger
 Weights
 Meter rod
 Dial indicator
 Vernier Caliper.

Figure 5. 1Fixed Ended Beam

Deflection of beam apparatus contains a metal beam and two knife-edge supports
upon which the beam is supported for this experiment. With the help of clamps
arrangement at ends it can be made fixed type of beam.

5.3Summery of Theory:

A fixed ended beam is supported by fixed supports at both ends as shown in Figure 5.1. The slope of
the beam is thus zero at each end, and a couple will have to be applied at each end and to make the
slope there have this value. The applied couples will be of opposite sign to that of bending moment,
due to loading.

Consider a beam AB of length “L” fixed at A and B and carrying a point load “W” as shown in figure
5.2.
The maximum deflection for this fixed beam will occur at center of the beam
(mid-point). Let,
δ = Actual deflection of beam at any point along the length of beam
δc = Actual central deflection of beam
yc = Theoretical central deflection of beam

Page | 31
 L L
2 2

Figure 5. 2Fixed Beam loaded at mid span

The maximum theoretical deflection (yc) at x = L/2 is given by:

yc = WL3/ 192EI

Where E = Modulus of elasticity for the material of beam

I = Moment of inertia of the beam

5.4 Procedure:

1. Set the Deflection of Beam apparatus on a horizontal surface.

2. Set the dial indicator at zero.
3. Apply a load of 0.5 lb and measure the deflection using dial indicator.
4. Take a set of at least five readings of increasing value of load and then take readings on
unloading.
5. Calculate:
a. the “Theoretical value of deflection (yc)” of beam at mid-span.
b. the %age error between theoretical and experimental values of central deflections.
5.5 Observations and Calculations:

Least Count of the dial indicator = 0.01 mm

Effective length of beam (L) = 800 mm
Breadth of beam (b) = 25.4 mm
Height of beam (h) = 6.4 mm
Modulus of elasticity of material of the beam = 70000 N/ mm2
3 = 539.41 mm 4
Moment of inertia of the beam (I=bh /12)

Page | 32
Table 5 Calculation of deflection of fixed beam:
Actual Central Deflection-δc Theoretical
Deflection W/ δc
Load- (mm)
3
No. of W yc = WL /192EI
Obs.
(N) Loading Unloading Average (mm) (N/m)

1. 5 0.19 0.49 0.34 0.34 14.7

2. 10 0.49 0.90 0.69 0.68 14.49
3. 15 0.90 1.26 1.08 1.02 13.88
4. 20 1.32 1.52 1.42 1.36 14.08
5. 25 1.74 1.74 1.74 1.70 14.36

5.5.1Graph:
Load Vs Deflection
Y-Values
1.8 1.7
30
1.6
1.36
25 1.4

1.2
1.02
20 1
Load

0.8 0.68
15
0.6

0.4 0.34
10
0.2

5 0
0 5 10 15 20 25 30
Deflection
0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Page | 33
5.8 Conclusions:
The following are results can be drawn:
1.The graph between load and extension is straight line.
2. The average value of modulus of elasticity is 358 (GN/m2)

5.9 Comments:
1. In the experiment we experimentally find out the behavior of the fixed ended beam
by applying the different load on it.
2. We got a little bit deflection in the center of the beam instead of simply supported
beam.

Page | 34
Lab Session 6

6.1 Objective:
To determine the deflection at mid span of a propped cantilever beam and compare with the
theoretical values
6.2 Apparatus:
 Propped cantilever beam apparatus
 Weights
 Dial gauge
 Vernier Caliper
 Specimen
 Hangers
 Spanner

Figure 6. 1Propped Cantilever Beam

6.3Summery of Theory:
A beam is a structural element that is capable of withstanding load primarily by resisting
bending.

6.3.1Classification of beams :
The beams may be classified in several ways, but the commonly used classification is
based on support conditions. On this basis the beams can be divided into six types:
(1) Cantilever beams (2) Simply supported beams (3) Overhanging beams
(4) Propped beams (5) Fixed beams (6) Continuous beams

6.3.1.1Cantilever beam:
A beam having one end fixed and the other end free is known as cantilever beam,
figure shows a cantilever with end ‘A’ rigidly fixed into its supports, and the other end ‘B’ is
free. The length between A and B is known as the length of cantilever.

Page | 35
Figure 6. 2Cantilever beam
6.3.1.2Simply supported beam:
A beam having both the ends freely resting on supports is called a simply supported beam.
The reaction act at the ends of effective span of the beam. Figure show simply supported
beams. For such beams the reactions at the two ends are vertical. Such a beam is free to
rotate at the ends, when it bends.

6.3.1.3Overhanging beams:
A beam for which the supports re not situated at the ends and one or both ends extend over
the supports, is called an overhanging beam. Figure represents overhanging beams.

Figure 6. 3One end overhanging beam

Figure 6. 4 Both ends overhanging beam

6.3.1.4Propped cantilever beams:

A cantilever beam for which one end is fixed and other end is provided support, in order to
resist the deflection of the beam, is called a propped cantilever bema. A propped cantilever is
a statically indeterminate beam. Such beams are also called as restrained beams, as an end is
restrained from rotation.

Figure 6. 5 Proped cantilever beam

Page | 36
6.3.1.5Fixed beams:
A beam having its both the ends rigidly fixed against rotation or built into the supporting
walls, is called a fixed beam. Such a beam has four reaction components for vertical loading
(i.e., a vertical reaction and a fixing moment at both ends) figure shows the fixed beam.

Figure 6. 6 Fixed ended beam

6.3.1.6Continuous beam:
A beam having more than two supports, is called as continuous beam. The supports at the
ends are called as the end supports, while all the other supports are called as intermediate
support. It may or may not have overhang. It is statically indeterminate beam. In these beams
there may be several spans of same or different lengths figure shows a continuous beam.

Figure 6. 7 Continuous beam

6.3.2Derivation of formula for deflection at mid span.

(Derive the formula as for experiment conditions)

6.4Procedure:

i Measure the width and depth of the beam with the help of scale to find the moment of inertia
of the beam.
ii Set the apparatus and put the required hangers at different points.
iii Measure the distances of each hanger from the reference end.
iv Set the deflection dial gauge at zero after putting the hangers.
v Take the reading of deflection after putting the loads in the hangers
vi Repeat the process for different loads
vii Find the theoretical deflection and compare with the experimental values by showing on a
graph

Page | 37
6.5 Observations and Calculations:

Width of Beam = b = 25.4 mm

Depth of beam = d = 5 mm
Moment of Inertia for rectangular metal bar = I =
3
bd /12 = 539.41 mm4
Modulus of Elasticity = E = 70000 N/mm2

Table 6.1 Variation in deflection with loads:

Obs. No LOADS
(N)

W1 W2 W3
δ δ %age
exp th Error
1 1 1 1 0.13 0.20 35
2 2 3 1 0.50 0.55 9
3 2 3 3 0.75 0.80 6
4 5 4 3 1.21 1.30 6
5 7 4 5 1.68 1.70 2

6.5.1 Graph:
On graph, plot the deflection against load for the theoretical & practical results. Draw the best-fit
straight lines through the points
Deflection vs Load
 Y-Values
1.4

1.2

0.8
Ex value

0.6

0.4

0.2

0
0 1 2 3 4 5 6 7 8 9
Load

1.4 1.3

1.2

0.8
0.8
Th value

0.6 0.55
Y-Values
0.4
3.5
0.2
0.2
3

0
2.5 0 1 2 3 4 5 6 7 8 9
Load
2

1.5

0.5

0
0.5 1 1.5 2 2.5 3

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6.6 Industrial Applications:
1- Cantilever beam is used in billboard tower.
2- Cantilever beam is widely used in traffic light towers.
3- Cantilever beam is used in parking canopies.
4- The court of basketball is the perfect example of cantilever beam.
5- Cantilever beam is used in overhang shadow roofs.

6.7 Comments:
1. In this experiment we observed the deflection in the propped cantilever beam due to three
different loads.
2. We applied three different loads and checked the deflection at the center of the proped
cantilever beam.
3. Dial is used in this experiment is not at 0.

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Lab Session 7

7.1 Objective:
To determine the deflection at three different points of a propped cantilever beam using a brass
beam and compare with the theoretical values also find its comparison with aluminum beam used
in lab session 6.
7.2 Apparatus:
 Propped cantilever beam apparatus
 Weights
 Dial gauge
 Vernier Caliper
 Specimen
 Hangers
 Spanner

Figure 7. 1 Propped Cantilever Beam

7.3Summery of Theory:
Same as for lab session#6.

7.4Procedure:
i Measure the width and depth of the beam with the help of scale to find the moment of inertia
of the beam.
ii Set the apparatus and put the required hangers at different points.
iii Measure the distances of each hanger from the reference end.
iv Set the deflection dial gauge at zero after putting the hangers.
v Take the reading of deflection after putting the loads in the hangers
vi Repeat the process for different loads
vii Find the theoretical deflection and compare with the experimental values by showing on a
graph

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7.5 Observations and Calculations:
Width of Beam = b = 11 mm
Depth of beam = d = 25.4 mm
3
Moment of Inertia for rectangular metal bar = I = bd /12= 7333.3
Modulus of Elasticity = E = 70GPa
Length of beam = 80 cm

Table 7 Variation in deflection with loads:

Obs.No LOADS Deflection δexp wL3 %age

δth= 192EI
(N) Error
Loading Un-loading (Mean) (mm)

1. 5 0.19 0.49 0.34 0.34 0

2. 10 0.49 0.90 0.69 0.68 1.4
3. 15 0.90 1.26 1.08 1.02 5.5
4. 20 1.32 1.52 1.42 1.36 4.2
5. 25 1.74 1.74 1.74 1.70 2.2

7.5.1 Graph:
On graph, plot the deflection against load for the theoretical & practical results. Draw
the best-fit straight lines through the points.

Experimental
1.8

1.6

1.4

1.2
Experimental

0.8

0.6

0.4

0.2

0
0 5 10 15 20 25 30
Load
 Theoratical
1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

0
0 5 10 15 20 25 30

7.7 Industrial Applications:

1. Cantilever beam is used in traffic light tower.
2. Cantilever beam is used in parking canopies.
3. Cantilever beam is widely used in overhang shadow roofs.
4. This also used in one sided billboards ground system.
5. The court of basketball is the perfect example of cantilever.

7.8 Comments:
1. In this experiment we observe that the cantilever beam strip is not in straight from all
surface.
2. On adding more loads we observe that the deflection in beam is increases gradually.
3. There is miner error in dial gauge used in experiment.