Calculate the flexural stress distribution on a cross-section using Hooke’s law, by using electrical resistance strain gauge which are connected to the universal testing machine

UNIVERSITY OF WINDSOR
Faculty of Engineering

Mechanics of Deformable Bodies (85-218)
Laboratory #3 – Flexural Stresses in Beams

Submitted on Friday March 20, 2015

Faris, Alosaimi
Student ID: 104295764
Section 55-A

Objective:
We performed the Flexural Stresses in Beam to:-
To calculate the flexural stress distribution on a cross-section using Hooke’s law, by using electrical resistance strain gauge which are connected to the universal testing machine
To compare the experimental reading with the theoretically calculated readings.
Throughout our amusement in the experiments there will positive and negative reading with respect to the datum which means the stress at the point is a compressive and tensile respectively. As engineers we could benefit from this test by knowing the required strength of the beam to be used in the structure.

Formula to be used:-
σ = E.ε
Where
σ = experimental flexural stress
E = modulus of elasticity
ε = normal strain.

σ = -M .y/I
Where
σ = theoretical flexural stress at distance ‘y’ from neutral axis;
y = distance from neutral axis;
M = bending moment;
I = moment of inertia of cross-section about the axis of bending.

Equipment
The following equipment has been used to perform the experiment
Universal Testing Machine Tinius Olsen
Datascan Analog Measurement Processor
Dalite Datascan Configurator
Strain Gauges
Scales
Weights

Experimental Procedure:

Following measurement were verified For each beam:-
Length of the span.
Strain gauge line location.
On the gauge line from the bottom of the beam the distance of each gauge measured.

For the Steel I-beam,
To use the Universal Testing Machine For the range was zeroed

Dalite Datascan Configurator was opened at the DB2L1S.OVL file. Every second window on the monitor it was checked if the load was zero, and initial strain values for each one of the six strain gauges were recorded

Steel I-beam was loaded slowly. Load of 5kN increments were taken for five times to record the readings of the strain in each of the strain gauges. In this part the maximum load = 25kN.

After unloading the strain readings for each gauge were recorded

For the aluminum box beam,
Dalite Datascan Configurator was opened at the DB1L1A.OVL file. Initial strain values for each one of the six strain gauges were recorded

Load of 10lb increments were taken for five time to record the reading of the strain for each of the strain gauges. In this part the maximum load = 50 lb, or 222.4 N.

After unloading strain readings for each gauge were recorded.

Results:

Sample Calculations:

Steel theoretical:-
M = (P/2) x L (L = length of beam)
L= (620/2 – 100) = 210 mm
M = (0.21m) x (P/2) = (0.21m) x (5 KN/2)= 0.525 KN.m
σ =- (M/I) * y = -(0.525*10^-3N.m/ 9.3236*10^-6 m^4)*(0.074 m)= -4.167 Pa
Steel experimental:-
σ=E x ε = (200*10^3 Mpa) x (-20*10^-6)= -4 Mpa

Steel

Distance of strain gauge from neutral axis
Strain Gauge 1 2 3 4 5 6
Y, m 0.074 0.049 0.0235 -0.0235 -0.049 -0.074

Distance of Strain Gauges from bottom of beam
Distance 1 2 3 4 5 6
Y, m 0.148 0.123 0.0975 0.0505 0.025 0

Experimental Values
Load Strain Gauge Readings
(kN) Gauge 1 Gauge 2 Gauge 3 Gauge 4 Gauge 5 Gauge 6
5.000 -20.000 -18.000 -3.000 3.000 14.000 21.000
10.000 -41.000 -34.000 -11.000 7.000 27.000 45.000
15.000 -61.000 -49.000 -19.000 12.000 40.000 68.000
20.000 -50.000 -64.000 -28.000 16.000 53.000 90.000
25.000 -101.000 -81.000 -36.000 20.000 67.000 115.000

Stress (Mpa) = E x Strain
(1) (2) (3) (4) (5) (6)
-4.000 -3.600 -0.600 0.600 2.800 4.200
-8.200 -6.800 -2.200 1.400 5.400 9.000
-12.200 -9.800 -3.800 2.400 8.000 13.600
-10.000 -12.800 -5.600 3.200 10.600 18.000
-20.200 -16.200 -7.200 4.000 13.400 23.000

Theoretical Values
Load Moment Stress (Mpa) σ = – (M/I) * y
(kN) (kN.m) (1) (2) (3) (4) (5) (6)
5.000 0.525 -4.167 -2.759 -1.323 1.323 2.759 4.167
10.000 1.050 -8.334 -5.518 -2.647 2.647 5.518 8.334
15.000 1.575 -12.501 -8.277 -3.970 3.970 8.277 12.501
20.000 2.100 -16.667 -11.037 -5.293 5.293 11.037 16.667
25.000 2.625 -20.834 -13.796 -6.616 6.616 13.796 20.834

Strain ε = σ/E
(1) (2) (3) (4) (5) (6)
-20.834 -13.796 -6.616 6.616 13.796 20.834
-41.668 -27.591 -13.233 13.233 27.591 41.668
-62.503 -41.387 -19.849 19.849 41.387 62.503
-83.337 -55.183 -26.465 26.465 55.183 83.337
-104.171 -68.978 -33.081 33.081 68.978 104.171

% Error
σ ε
24.41473411 24.41473411
16.48292967 16.48292967
12.78308409 12.78308409
18.85733319 18.85733319
13.68500205 13.68500205

Aluminum

Distance of strain gauge from neutral axis
Strain Gauge 1 2 3 4 5 6
Y, m 0.0375 0.025 0.0125 -0.0125 -0.025 -0.0375

Distance of Strain Gauges from bottom of beam
Distance 1 2 3 4 5 6
Y, m 0.075 0.0625 0.050 0.025 0.0125 0

Experimental Values
Load Strain Gauge Readings
(kN) Gauge 1 Gauge 2 Gauge 3 Gauge 4 Gauge 5 Gauge 6
10 22.000 19.000 6.000 -7.000 -16.000 -26.000
20 46.000 33.000 14.000 -14.000 -38.000 -54.000
30 70.000 46.000 24.000 -21.000 -64.000 -80.000
40 97.000 61.000 34.000 -30.000 -79.000 -107.00
50 123.000 78.000 42.000 -36.000 -96.000 -138.00

Stress (Mpa) = E x Strain
(1) (2) (3) (4) (5) (6)
1.584 1.368 0.432 -0.504 -1.152 -1.872
3.312 2.376 1.008 -1.008 -2.736 -3.888
5.040 3.312 1.728 -1.512 -4.608 -5.760
6.984 4.392 2.448 -2.160 -5.688 -7.704
8.856 5.616 3.024 -2.592 -6.912 -9.936

Theoretical Values
Load Moment Stress (Mpa) σ = – (M/I) * y
(kN) (kN.m) (1) (2) (3) (4) (5) (6)
0.044 0.029 -1.961 -1.307 -0.654 0.654 1.307 1.961
0.089 0.058 -3.921 -2.614 -1.307 1.307 2.614 3.921
0.133 0.087 -5.882 -3.921 -1.961 1.961 3.921 5.882
0.178 0.116 -7.843 -5.228 -2.614 2.614 5.228 7.843
0.222 0.145 -9.803 -6.536 -3.268 3.268 6.536 9.803

Strain ε = σ/E
(1) (2) (3) (4) (5) (6)
-27.232 -18.154 -9.077 9.077 18.154 27.232
-54.463 -36.309 -18.154 18.154 36.309 54.463
-81.695 -54.463 -27.232 27.232 54.463 81.695
-108.926 -72.618 -36.309 36.309 72.618 108.93
-136.158 -90.772 -45.386 45.386 90.772 136.16

% Error
σ ε
185.3787323 185.3787323
188.8979274 188.8979274
191.8050885 191.8050885
192.7231394 192.7231394
192.5395292 192.5395292

Discussion:

While absolving the steel beam during the experiment we could found that experimental and theoretical tables in both strain and stress were negative in the beginning and then positive at the end. As will by checking the slope of the load we could notice the negative reading. From this we know that the upper half above the neutral axis of the beam is under compression; where the lower half below the neutral axis is under tension.

Moreover: while absolving the Aluminum beam during the experiment we could found that experimental tables in both strain and stress were positive in the beginning and then negative at the end; where the theoretical tables in both strain and stress were negative in the beginning and then positive at the end. As will by checking the slope of the load we could notice the negative reading. From this we know that the lower half of the beam below the neutral axis is under compression; where the upper half above the neutral axis is under tension.

Conclusion:

Using the moment, inertial and the distance of the point with respect to the normal line we could obtain the theoretical readings. Strain readings were observed and then noted down while the practical stresses calculated by using the modulus of elasticity of the respective material. There are some defects in the electrical strain gauge therefore we have some error in our readings. The initial zero error for zero loading was present, there are some other cause of error due to the purity of the material.

On the other hand; by comparing both theoretical and experimental methods, we could conclude that they are good method to calculate flexural stress distribution on a cross-section and locate natural axis.

Finally, most important point in this experiment is the natural axis, which is referenced from the bottom. All loading intersect at this point.