ME2114-2 Combined Bending & Torsion
Transcript of ME2114-2 Combined Bending & Torsion
-
8/10/2019 ME2114-2 Combined Bending & Torsion
1/11
ME2114 Mechanics of Materials II
1
ME2114-2 Combined Bending & Torsion (D2)
Formal Report
Matriculation Number: A0101934
Name : Ong Wei Quan
Subgroup : 2K1
-
8/10/2019 ME2114-2 Combined Bending & Torsion
2/11
ME2114 Mechanics of Materials II
2
Objectives
To analyse the stresses at the surface of shaft subjected to combined bending and twisting
using strain gauge technique.
To compare the experimental results with theoretical results.
Introduction
Shafts subjected to both bending and twisting are frequently encountered in engineering,
applications. By applying St. Venant's principle and the principle of superposition, the stresses at the
surface of the shaft may be analysed. The main purpose of this experiment is to analyse problems of
this kind using, the strain gauge technique and to compare the experimental results with theoretical
results. As the strain gauge technique enables only the determination of states of strain at about a
point. Hooke's law equations are used to calculate the stress components. In this experiment, the
elastic constants of the test material are first determined.
Experimental Procedures
Determination of elastic constants
1. Measure the diameter of the tensile test piece and mount it on the tensometer.
2. Use a quarter bridge configuration and for each tensile load applied to the testpiece, record
the longitudinal and transverse strains in order to evaluate the Young's modulus and
Poisson's ratio.
A. Combined bending and torsion test
1. Measure the dimensions of a and b.
2. Connect the strain gauges to the strain-meter using, a quarter bridge configuration and
balance all the gauges.
3. For each loading, on the shaft record the strain readings.
4. From the strain readings compute the stresses.
5. Using, a full bridge configuration in a manner illustrated in Figures (3a) & (3b) record the
strain-meter reading for each applied load.
-
8/10/2019 ME2114-2 Combined Bending & Torsion
3/11
ME2114 Mechanics of Materials II
3
Sample Calculations
Table 1 (Using load of 200N)
Direct Stress MPa88.210940.6
200
AreaSectionalCross
loadTensile5
Table 3 (Using load of 0.5kg for Quarter Bridge Configuration)
6
4321
6
3241
1027)1225()1024()()(
1071)2510()1224()()(
b
a
Table 4 (Using load of 0.5kg)
MPaEalExperiment
MPaD
aPlTheoretica
MPaE
alExperiment
MPaD
bPlTheoretica
xy
xy
x
x
654.0)3182.01(2
10)]10(24[(107.50
)1(2
)(
952.0)0159.0(
)81.95.0)(15.0(1616
892.03182.01
10)1224(107.50
1
)(
26.1)0158.0(
)81.95.0)(9945.0(3232
6921
33
69
41
33
-
8/10/2019 ME2114-2 Combined Bending & Torsion
4/11
ME2114 Mechanics of Materials II
4
Results
A. Determination of elastic constants
Diameter of Tensile Test Piece (mm) Cross Sectional Area (mm2
)D1 D2 Daverage
69.49.37 9.42 9.40
mmN
DDDaverage 40.9
2
42.937.921
Cross Sectional Area 2522
10940.64
)0094.0(
4m
d
Table 1Load (N) Direct Stress, x(MPa) Longitudinal Strain, x(10
-
6)
Transverse Strain, y(10-6
)
200 2.88 61 -19
400 5.76 120 -37
600 8.65 177 -55
800 11.53 232 -73
1000 14.41 282 -91
1200 17.29 334 -107
Youngs modulus
x
xE
= Gradient of Graph 1
GPa7.50
y = 0.0507x
0
2
4
6
8
10
12
14
16
18
20
0 50 100 150 200 250 300 350 400
DirectStress(MPa)
Longitudinal Strain (x10-6)
Graph 1: Direct Stress vsLongitudinal
Strain
-
8/10/2019 ME2114-2 Combined Bending & Torsion
5/11
ME2114 Mechanics of Materials II
5
Poissons ratiox
y
=Gradient of Graph 23182.0)3182.0(
B. Combined Bending and Torsion Test
Table 2
Load P (kg)Strain (10
-6) [Quarter Bridge Configuration]
1 2 3 4
0.00 0 0 0
0.524 -10 -25 12
1.0 45 -20 -48 26
1.568 -30 -73 38
2.088 -40 -98 50
2.5111 -50 -122 62
3.0132 -60 -147 75
Table 3: Comparing Quarter Bridge Configuration with Full Bridge Configuration
Load P (kg)Quarter Bridge Configuration Full Bridge Configuration
a (10-6
) b (10-6
) a (10-6
) b (10-6
)
0.0 0 0 0 0
0.5 71 27 69 24
1.0 139 47 138 48
1.5 209 73 205 72
2.0 276 96 274 95
2.5 345 121 340 120
3.0 414 144 410 142
y = -0.3182x
-120
-100
-80
-60
-40
-20
0
0 50 100 150 200 250 300 350 400
TransverseStrain(x10
-6)
Longitudinal Strain (x10-6)
Graph 2: Transverse Strain vsLongitudinal
Strain
-
8/10/2019 ME2114-2 Combined Bending & Torsion
6/11
ME2114 Mechanics of Materials II
6
a b D (diameter of the shaft)
0.150m 0.995m 0.0158m
y = 0.0072x
y = 0.0073x
0
0.5
1
1.5
2
2.5
3
0 50 100 150 200 250 300 350 400 450
Load(kg)
a(x10-6)
Graph 3: Load vs a
Quarter Bridge
Full Bridge
y = 0.0207x
y = 0.021x
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100 120 140 160
Load(kg)
b(x10-6)
Graph 4: Load vs b
Quarter Bridge
Full Bridge
-
8/10/2019 ME2114-2 Combined Bending & Torsion
7/11
ME2114 Mechanics of Materials II
7
Table 4
Load P (kg)Bending Stress, x(MPa) Shear Stress, xy(MPa)
Theoretical Experimental Theoretical Experimental
0.0 0.00 0.000 0.00 0.000
0.5 1.23 0.918 0.93 0.796
1.0 2.47 1.938 1.85 1.546
1.5 3.70 3.162 2.78 2.319
2.0 4.93 4.284 3.70 3.138
2.5 6.17 5.406 4.63 3.958
3.0 7.40 6.324 5.55 4.731
y = 2.5194x
y = 1.4366x
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3
BendingStress(MPa)
Load (kg)
Graph 5: Bending Stress vs Load
Theoretical
Experimental
y = 1.904x
y = 1.2371x
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
ShearStress(MPa)
Load (kg)
Graph 6: Shear Stress vs Load
Theoretical
Experimental
-
8/10/2019 ME2114-2 Combined Bending & Torsion
8/11
ME2114 Mechanics of Materials II
8
Discussion
1. Compare the theoretical stresses with the experimental values. Discuss possible reasons for
the deviation if any, in the results obtained.
The experimental results are much smaller than the theoretical results. This may be due to
the following:
The metal used in the tensile test piece may have different properties (youngs
modulus) to the one used in the hanger causing the theoretical and experimental
graphs to deviate for example in graphs 5 and 6.
The orientation of each rosette is such that the axis of any one gauge makes an angle
of 45 with the axis of the shaft. The angle may not be exactly 45 degrees as the strain
gauges are taped using scotch tape and the orientation of the gauges may change
over time.
When the hanger is loaded, there is some oscillation which contributes to the
inaccuracy of the readings.
2. From the results of step (B5), deduce the type of strain the strain-meter readings represent.
The shear strain, xy, by transformation of axes, is given by
2cos2
2sin22
3443
xy
Where the subscripts 3 and 4 refer to the direction of the gauge numbers 3 and 4 respectively.
Since = -45o,
2134 xy
Hence, xya 2)()()()( 34213241
by transformation of axes,at point B,
224
xy
B
y
B
x
or
xB xy
2
14
Where the superscript B refers to the point B.
Thus, the bending strain at point A is then given by
12
)( 41B
x
A
x
x
So,
)1(41 x and similarly, )1(32 x
Hence, )1(2)()()()( 32414321 xb
Therefore, configuration bmeasures the product of strain in x-direction with the term 2(1-).
-
8/10/2019 ME2114-2 Combined Bending & Torsion
9/11
ME2114 Mechanics of Materials II
9
3. Apart from the uniaxial tension method used in this experiment, how can the elastic
constants be determined.
Poisson's Ratio, can be determined by using the expressionG E
v
2 1( ) To determine Young's modulus, E, a cantilever beam with weights hung from the beam at a
few locations along the beam. The resulting deflections v wL
EI
3
3caused by the weights
can be
measured using a dial-gauge, where I is the second moment of area. By varying v with L, a
graph of v against L3could be plotted and the gradient obtained. The value of the youngs
modulus can be obtained from the gradient.
To determine Shear modulus, G, the torsion test could be used. The angle of twist is given by
TL
GI. By varying the torque T with the angle , a graph of against T could be plotted and
the gradient obtained. Since L and I are known by measuring the dimensions of the bar, G can
be obtained.
4. Instead of using Equations (3) and (8) for strains, develop alternative equations to enable
the determination of strains from the four gauges readings.
Since the full bridge configuration is quite similar to the Wheatstone bridge, 2 expressions can
be derived
xya 2)()()()( 34213241
)1(2)()()()( 32414321 xb
where 1, 2, 34, are the four gauge readings and a, bare the resulting strains.
5. Develop stress equations for combined bending, and twisting, of hollow shafts with K as the
ratio of inside to outside diameter.
-
8/10/2019 ME2114-2 Combined Bending & Torsion
10/11
ME2114 Mechanics of Materials II
10
appliediswherepoint toonefromdistancewhere
1
32=
164
2
164
16464
=areaofmomentSecond=
4344
44
4
4444
PbKD
bP
KD
DbP
KDD
dDdDI
I
yM
x
z
z
xy
x
armtorsiontheoflengthwhere
1
16=
132
2
132
13232
=areaofmomentsecondPolar=
4344
444
4
444
aKD
aP
KD
DaP
KDDdDdDJ
J
rT
xy
x
x
xxxy
D = outer diameter of shaft; d = inner diameter of shaft;D
dK
6.
In certain installations shafts may be subjected to an axial load F in addition to torsionaland bending loads. Would the strain gauge arrangement for this experiment be acceptable
to the determination of stresses?
Give reasons for your answer. For simplicity, a solid shaft may be considered.
The shaft will experience a higher axial strain and axial stress when an additional axial load F
is applied. This additional strain caused by the load can still be measured by the strain gauge
and the readings show the resultant axial and shear strain due to the combined bending and
torsion and axial load (principle of superposition). Therefore, the strain gauge arrangement
for this experiment is acceptable for the determination of both the axial and shear strain.
Conclusion
The stresses and strains due bending and twisting were determined using the strain gauge
technique.
-
8/10/2019 ME2114-2 Combined Bending & Torsion
11/11
ME2114 Mechanics of Materials II
11
The deviation of experimental results to theory may be due to the different elastic properties
of the test piece and the experimental setup.
The Youngs modulus, E and the Poissons ratio, were determined to be 50.7GPa and 0.32
respectively.