CEE370L_Test5_Report_Vergara, Kailah Reign
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CEE 370 Mechanics of Materials Lab
Beam Bending Theory
Kailah Reign Vergara
University of Hawaii at Manoa
CEE 370L Mechanics of Materials
November 14, 2023
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
EXECUTIVE SUMMARY
During the laboratory experiment, a large steel beam specimen with simple support
was exposed to a transverse load. The steel beam was equipped with multiple electrical
resistance strain gauges at different locations to track the strain in the steel. These
gauges were employed to assess the specimen's response under various loads. By
calculating the theoretical stresses based on the measured microstrains, we could
observe the behavior of the plot. The outcomes of the test will be utilized to validate the
correlation between bending moment and bending stresses, commonly referred to as
the beam bending theory.
Page
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
TABLE OF CONTENTS EXECUTIVE SUMMARY………………...……………………………………………………………...i
1
INTRODUCTION
.............................................................................................................................
1
1.1
B
ACKGROUND
..............................................................................................................................
1
1.2
R
EASON
FOR
E
XPERIMENT
...........................................................................................................
1
1.3
T
HEORY
........................................................................................................................................
1
1.4
O
BJECTIVE
...................................................................................................................................
1
2
APPROACH
......................................................................................................................................
1
2.1
T
EST
S
ETUP
AND
I
NSTRUMENTATION
...........................................................................................
1
2.2
T
EST
S
PECIMENS
..........................................................................................................................
2
2.3
T
EST
P
ROCEDURE
.........................................................................................................................
3
3
RESULTS
...........................................................................................................................................
3
3.1
...............................................................................................................................................................
3
3.2
...............................................................................................................................................................
3
4
ANALYSIS
.........................................................................................................................................
3
4.1
...............................................................................................................................................................
4
4.2
...............................................................................................................................................................
4
5
CONCLUSIONS/RECOMMENDATIONS
.....................................................................................
4
6
REFERENCES
..................................................................................................................................
5
APPENDIX
TABLE OF FIGURES
FIGURE 1.3.1: DIAGRAM FOR A TYPICAL STEEL BEAM SUBJECTED TO AN EXTERNAL LOAD
FIGURE 2-2: DIAGRAM OF THE BEAM WITH THE DISTANCES AND LAYOUT OF THE VARIOUS GAGES
FIGURE 2-3: CROSS SECTION OF THE BEAM
Figure 2-4: BEAM BENDING THEORY TEST SETUP
Figure 3.1: Stress vs Load Relationship of Top and Bottom Gages
Figure 3.2: Average Stress vs Distance Relationship of Front and Back Gages
Figure 3.3: Stress vs Distance Relationship from Center of Beam
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CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
1
Introduction
1.1
Background
Beam bending arises when external forces act on a beam, inducing deformation.
This phenomenon is prevalent in the construction of diverse structures. To assess a
beam's ability to bear loads, shear force and bending moment diagrams are employed
to pinpoint areas of peak shear force and moment. Steel, widely utilized for beams,
exhibits proficiency in both tension and compression. Its versatility in being shaped into
various forms with distinct capabilities further enhances its appeal. By scrutinizing stress
distribution across the cross-section, the highest stress occurs farthest from the neutral
axis. Additionally, when assessing stress variation along the beam span, stress peaks
align with maximum bending moments. 1.2
Reason for Experiment
The objective of conducting this laboratory experiment was to aid a civil engineering
professional in the design phase by identifying the most suitable variation of a specific
material for a project. The stress-strain curve generated through the experiment
provides essential characteristics that play a vital role in the material selection process.
This careful consideration is crucial for ensuring the safety and stability of the structure
being worked on. The broader public stands to gain valuable insights from the outcomes
of this experiment. Knowledge of how various materials behave and respond to applied
forces can instill confidence in the public regarding the safety and structural integrity of
their surrounding environment.
1.3
Theory
Page 1
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
The fundamental principle of beam bending theory, also known as Bernoulli Euler
beam theory, plays a crucial role in mechanical engineering. It elucidates the behavior
of a beam when subjected to external loads. As a beam undergoes deformation or
bending under loading, we can compute the resulting stresses and deflections. Notably,
any cross-section of the beam that is initially perpendicular to the neutral axis will
maintain this perpendicular orientation throughout the deformation process. The core
tenet of beam bending theory asserts that a beam will deform in a manner that
minimizes its potential energy under the given external loads. This principle holds as
long as the cross-section remains constant during bending, and the material exhibits
consistent mechanical properties in all directions.
Figure 1.3.1: Diagram for a typical steel beam subjected to an external load
1.4
Objective
The goal of this laboratory experiment was to confirm the correlation between
bending moment and bending stresses. Furthermore, the validity of the assumption that
plane sections remain flat will be assessed.
Page 2
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
2
Approach
2.1
Test Setup and Instrumentation
This test used a steel beam and electrical gages to identify the strain in the beam. Figure 2-2: Diagram of the beam with the distances and layout of the various gages. Figure 2-3: Cross section of the beam. Page 3
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CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
Figure 2-4: Beam Bending Theory Test Setup
Test frame: Simple supports (reaction anchors), Hydraulic jack with two points of contact
Test Instrument: Electrical resistance strain gages Test materials: Steel tube
2.2
Test Specimens
Table 2.1: Materials and their properties
Table 2.1: Materials and their properties
Material Span
length (in)
Height
(in)
Depth
(in)
E
(ksi)
I (in^4)
Vertical Strain Gage Distance (in)
Horizontal Strain Gage Distance (in)
Steel Beam
96
6
3 29000
13.4
6
1
2.3
Test Procedure
Initially, the sensors were calibrated to ensure accurate data collection. The first
load of 0 kips was subsequently applied to the beam, and shortly thereafter, the
Page 4
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
microstrain from the top sensors was read aloud. This step was then repeated for loads
of 1, 2, 1, and 0 kips. In total, 5 strains were recorded for each of the 5 loads. This entire
process was then repeated three more times for each of the strain gage positions
(bottom, front, and back).
Page 5
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
3
Results
3.1
Stress vs. Load
Load
T1
U1
0
-0.087
0
1
2.552
-2.581
2
5.278
-5.075
1
2.61
-2.726
0
-0.145
-0.145
0
0.5
1
1.5
2
2.5
-6
-4
-2
0
2
4
6
0
-2.58
-5.08
-2.73
-0.15
-0.09
2.55
5.28
2.61
-0.15
f(x) = − 2.51 x − 0.1
f(x) = 2.7 x − 0.12
T1
Linear (T1)
U1
Linear (U1)
Load (in kips)
Stress (in ksi)
Figure 3.1: Stress vs Load Relationship of Top and Bottom Gages
3.2
Stress vs Distance From Neutral Axis
Table 3.2: Experimental Stress
Load (kips)
Experimental Stress
Average
T1
T2
T3
T4
T5
0
-0.087 -0.058 -0.058 -0.058
0
-0.0522
1
2.552
2.61
2.61
2.407
1.972
2.4302
2
5.278
5.336
5.336
4.872
4.031
4.9706
1
2.61
2.697
2.668
2.436
2.059
2.494
0
-0.145 -0.058 -0.087 -0.087 0.058
-0.0638
Page 6
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CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
U1
U2
U3
U4
U5
0
0
0
0
0
-0.116
-0.0232
1
-2.581 -2.552 -2.581 -2.146 -1.972
-2.3664
2
-5.075 -4.988
-4.93
-5.046 -3.799
-4.7676
1
-2.726
-2.61
-2.407 -2.146 -1.972
-2.3722
0
-0.145
0
0.319
0.087
0
0.0522
F1
F2
F3
F4
F5
0
-0.145 -0.087 -0.087 -0.087
0
-0.0812
1
1.74
0.986
0
-0.928
-1.74
0.0116
2
3.451
1.711 -0.087 -1.856 -3.509
-0.058
1
1.682
0.812 -0.145 -1.044 -1.885
-0.116
0
-0.174 -0.174 -0.116 -0.145 -0.087
-0.1392
B1
B2
B3
B4
B5
0
-0.116 -0.058
0
-0.145 0.029
-0.058
1
1.653
0.841 -0.087 -1.044 -1.798
-0.087
2
3.451
1.943 -0.029 -1.885 -3.451
0.0058
1
1.711
0.87
-0.087 -1.102 -1.885
-0.0986
0
-0.174 -0.116 -0.087 -0.174 -0.087
-0.1276
Distance Ex. Stress
Theo.
Stress
3
5.278
7.164
2
3.451
4.776
1
1.74
2.388
0
-0.058
0
-1
-1.8705
-2.388
-2
-3.48
-4.776
-3
-5.075
-7.164
Page 7
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
-4
-3
-2
-1
0
1
2
3
4
-8
-6
-4
-2
0
2
4
6
8
7.164
4.776
2.388
0.000
-2.388
-4.776
-7.164
5.28
3.45
1.74
-0.06
-1.87
-3.48
-5.08
f(x) = 2.39 x − 0
f(x) = 1.73 x − 0
Experimental Stress
Linear (Experimental Stress)
Theoretical Stress
Linear (Theoretical Stress)
Y-Distance from Origin Axis (in inches)
Stress (in ksi)
Figure 3.2: Average Stress vs Distance Relationship of Front and Back Gages
3.3
Stress vs Distance from Center Of Beam
X-Dist.
Ex T1-T5
Ex U1-U5
Theo T1-
T5
Theo U1-
U5
0
5.278
-5.075
7.164
-7.164
6
5.336
-4.988
7.164
-7.164
12
5.336
-4.93
7.164
-7.164
18
4.872
-4.292
6.716
-6.716
24
4.031
-3.799
5.373
-5.373
Page 8
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
0
5
10
15
20
25
30
-10
-7.164
-7.164
-7.164
-6.716
-5.373
7.164
7.164
7.164
6.716
5.373
-5.08
-4.99
-4.93
-4.29
-3.8
5.28
5.34
5.34
4.87
4.03
f(x) = 0.01 x² − 0.08 x − 7.07
f(x) = − 0.01 x² + 0.08 x + 7.07
f(x) = 0 x² − 0.01 x − 5.07
f(x) = − 0 x² + 0.06 x + 5.24
Experimental T1-T5
Logarithmic (Experimental T1-T5)
Power (Experimental T1-T5)
Power (Experimental T1-T5)
Power (Experimental T1-T5)
Polynomial (Experimental T1-T5)
Experimental U1-U5
Polynomial (Experimental U1-U5)
Theoretical T1-T5
Polynomial (Theoretical T1-T5)
Theoretical U1-U5
Polynomial (Theoretical U1-U5)
X-Distance from midspan (in inches)
Stress (in ksi)
Figure 3.3: Stress vs Distance Relationship from Center of Beam
Page 9
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CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
4
Analysis
Observing trends becomes possible by plotting both theoretical and measured data
results obtained from the strain gauges. Linear behavior is evident in the stress vs. load
relationship for both the top and bottom gauges at mid-span. At maximum load, the
average stress plot across the height of the section, derived from the front and back
gauges, displays an increasing linear trend. As height increases, so does the average
stress. Analysis of stress vs. distance from the center of the beam indicates a decrease
in stress as the distance from the center load increases. Notably, in the constant
moment section of a beam, plane sections deviate from their original flatness. The
conducted tests validate beam bending theory, accounting for the interplay between
bending moment, curvature, and material properties. Within the constant moment
segment, where the bending moment remains constant along the beam's length, cross-
sectional deformation is non-uniform. This results in an alteration of the angle between
the normal to the original plane section and the normal to the deformed plane section,
as evident in the measured results. The stress distribution aligns with beam bending
theory, assuming linear elasticity and small deformations. The maximum normal stress
occurs at the furthest distance from the neutral axis, specifically at the top and bottom of
the beam cross-section. In summary, experimental measurements of stress distribution
in the beam corroborate theoretical predictions from beam bending theory. Nonetheless,
some deviations from these predictions arise due to factors such as non-uniform
material properties, loading conditions, and geometric imperfections in the cross-
section.
Page 10
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
Page 11
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
5
Conclusions/Recommendations
This laboratory experiment focused on applying a load to a steel tube, utilizing electrical
strain gauges to validate the connection between bending moment and bending
stresses. The assessment also addressed the assumption regarding the persistence of
plane sections. - Linear behavior was evident in the stress vs. load relationship for the top and
bottom gauges at mid-span. - Under maximum load, the plot of average stress from the front and back across
the section height displayed a progressively linear trend. Increasing height correlated
with an increase in average stress. - Analysis of stress vs. distance from the center of the beam indicated a reduction in
stress as the distance from the center load increased. - In the constant moment segment of a beam, plane sections failed to maintain their
planarity. This segment, where the bending moment remains constant along the beam's
length, resulted in non-uniform deformation of the cross-section. This led to a change in
the angle between the normal to the original plane section and the normal to the
deformed plane section, as observed in the measurements. - The stress distribution in the beam aligned with beam bending theory, assuming
linear elasticity and small deformations. - Deviations from theoretical predictions might be attributed to factors such as non-
uniform material properties, loading conditions, and geometric imperfections in the
cross-section.
Page 12
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CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
6
References
Gere, James M., 2001. Mechanics of Materials 5
th
Edition
. Pacific Grove, California:
Brooks/Cole.
“CEE 370L - Mechanics of Solids Laboratory Stress-Strain Relationships for Various
Materials Tension Test.” University of Hawaii at Manoa Dept. of Civil & Environmental
Engineering.
Page 13
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
Appendix
A-1 Equations and Sample Calculations
Experimental stress:
?
= E*ε_x
Theoretical stress:
?
=-MY/I
Moment of Inertia:
I=(bh^3)/12
Page 14
CEE 370L Mechanics of Materials
Fall 2020
Laboratory (No.)
A-2 Raw Data
RAW DATA
Load
(kips)
Strain (macrostrain)(10^-6)
T1
T2
T3
T4
T5
0
-3
-2
-2
-2
0
1
88
90
90
83
68
2
182
184
184
168
139
1
90
93
92
84
71
0
-5
-2
-3
-3
2
U1
U2
U3
U4
U5
0
0
0
0
0
-4
1
-89
-88
-89
-74
-68
2
-175 -172 -170 -174 -131
1
-94
-90
-83
-74
-68
0
-5
0
11
3
0
F1
F2
F3
F4
F5
0
-5
-3
-3
-3
0
1
60
34
0
-32
-60
2
119
59
-3
-64
-121
1
58
28
-5
-36
-65
0
-6
-6
-4
-5
-3
B1
B2
B3
B4
B5
0
-4
-2
0
-5
1
1
57
29
-3
-36
-62
2
119
67
-1
-65
-119
1
59
30
-3
-38
-65
0
-6
-4
-3
-6
-3
Page 15
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Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
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- Problem 1: (50 points) In Avengers 2, Captain America's improved shield is made from unobtainium, a new material that will soon be available in a store near you. Unobtainium has the normal stress- strain diagram shown. The proportional limit, the elastic limit and the yield point are identical in this material. o [MPa] 2001 175 150 125 100 Fig.1 Normal stress-strain 75 50 25 0 0. 0.05 0.1 0.15 0.2 0.25 Unobtainium has a Poisson's ratio of 0.3. a. [6pts] Identify the yield point oy, the ultimate stress ou and the fracture stress of. Include units. c. [25pts] A bar of unobtainium has a length of 1.5 m, a width of 100mm and a height of 50 mm, as shown. The cross-sectional area is 50mm 100mm = 5 x 10-³m². The bar is subjected to an axial force of 375 kN. Find the normal stress in a cross-section of the bar. b. [5pts] How do you find Young's modulus from this graph? Find its value with units. whesta fins dus al hoss P = 375 KN 1.5 m 0.3 50 mm 100 mm & [mm/mm] Is the material within its…arrow_forwardQ1 pleasearrow_forwardHi can you please help me with the attached question?arrow_forward
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