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CEE 370 Mechanics of Materials Lab
BEAM BENDING THEORY
He Shi
University of Hawaii at Manoa
CEE 370L Mechanics of Materials
Nov 26, 2023
1
CEE 370L Mechanics of Materials
Laboratory 1
Fall 2023
EXECUTIVE SUMMARY
During an experiment, an aluminum beam was examined to validate the principles of
beam bending. The theory suggests that a beam made of flexible & homogeneous material with
an equivalent modulus of elasticity in tension and compression will deform equally under the
same stress. The moment of inertia (I) was calculated for bending around the horizontal centroid
axis using the provided properties of the beam section. The moment about the maximum load at
the plane section was determined using a shear force and bending moment diagram. Stresses
were computed using the moment of inertia, moment about the plane surface, and distance from
the centroid axis. It was proven that the distributed loads on the top and bottom flanges were
very close to each other, thus confirming the theory.
During the beam experiment, loads were applied which caused sagging. As a result,
throughout the beam, there was a positive bending moment. Gauges on the plane section of the
beam indicated that a plane section within the beam before bending remains plane after bending
throughout a constant moment section of the beam. This proves that the beam bending theory is
accurate. The experiment concluded that the material is linearly elastic, as the relationships
between stress and strain were directly proportional. When a load was applied, the top of the
beam was in compression while the bottom of the beam was in tension. The lowest sagging point
of the beam showed the largest strains, and therefore had the largest stresses.
CEE 370L Mechanics of Materials
Laboratory 1
Fall 2023
1 Introduction
1.1 Background
A transverse load is applied to an aluminum beam in order to test the beam bending
theory. A number of electrical resistance strain gauges are attached at various points on the
beam to measure strain. The purpose of the testing is to confirm the relationship between
moment and bending stresses.
1.2 Reason for Experiment
The purpose of this lab was to test the deflection limits of beams that are installed in
buildings. The experiment provided valuable information to engineers regarding how beams
support and resist loads, which enables them to ensure the safety and stability of structures.
Finding the maximum values of quantities and their locations along the beam is essential for
beam design.
1.3 Theory
When a beam is subjected to loads that are transverse to its length, there is no axial load
applied. However, when a transverse load is applied, a relationship can be found between the
bending moment and the transverse deflection of the beam. The beam is made of a linearly
elastic material, which means it follows Hooke's Law. Therefore, a small deflection should cause
small angles due to the deflection.
The moment-curvature equation ( is determined by the bending moment divided by
the modulus of elasticity times the moment of inertia (also known as the flexural rigidity).
The equation given for the moment-curvature is:
CEE 370L Mechanics of Materials
Laboratory 1
Fall 2023
where M is the bending moment and EI is the flexural rigidity.
The flexure formula ( is determined by the bending moment multiplied by the distance from
the neutral distance divided by the inertia of the cross section. The equation given for the flexure
formula is:
where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of
the cross section
1.4 Objective
The goal of this laboratory experiment was to confirm the correlation between bending
moments and bending stresses and assess whether the stress distribution aligns with the
principles of beam bending theory.
CEE 370L Mechanics of Materials
Laboratory 1
Fall 2023
2 Approach
2.1 Test Setup and Instrumentation
The experiment was done with a load cell. There were a total of 20 strains on the steel,
five on the back, front, top, and bottom. Each of the strains were 6 inches apart from each other.
The strains are then hooked up to three different boxes. The load cell co impressed the aluminum
beam at 0 kips to 2 kips in 1 increment and back down to 0 kips
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I need assistance designing three structural models on Fusion 360 (bamboo-inspired, bone-inspired, and a control cylinder) for a mechanics project. These models will be tested under compression to compare their load-bearing capacity, energy absorption, and failure modes. Here are the specific details and requirements for each design:
General Requirements:
External Dimensions:
All models must have the same outer dimensions: height = 15 cm, diameter = 5 cm.
Material Volume:
Ensure each design uses approximately the same material volume (e.g., relative density = 30% of the total volume). If you have any other ideas on how to ensure the test is fair between all 3 samples please note it!
File Format:
Provide the designs as STL files compatible with 3D printing or a step-by-step process.
Design Specifications:
1. Control Cylinder:
A simple hollow cylinder with:
Wall Thickness: 2 mm.
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Further questions
4.The graphs show how the extension changes with stretching force for four different materials.
a. Which two graphs show materials that follow Hooke's law?
A
B
b. Which two graphs show materials that become less stiff as
they are stretched?
force
force
D
C. Which graph shows a material that maintains the same
stiffness throughout?
force
force
1. A force of 20N stretches a spring by 0.5m. The spring obeys Hooke's law.
a. Calculate the spring constant including a suitable unit.
b. How much force must a man use to stretch it by 1.5m?
5. A car has a weight of 240OON which is distributed equally over all four wheels. Each wheel has a spring of
spring constant 400N/cm.
a.
Determine the force through the spring on each wheel.
b. Calculate the compression of each spring when the car rests on the ground.
6.In a spring experiment the results were as follows:
Force (N)
Length (mpm
1
2
4
6.
7
50
58
70
74
82
9n
102
125
LELL
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O True
O False
QUESTION 3
Compute the expected properties of modulus of elasticity (Ec) of a composite made from S-glass fibers and a Polyester matrix. The volume fraction of fibers is
30%. (Unit is psi.)
Properties of Sample Matrix and Filler Materials
Tensile strength
Tఅబండి
Specific weight
bai
MPa
10 psi
GPa
Iblin
Matrix materials:
Polyester
Ероху
Aluminum
10
18
45
69
124
310
1170
0.40
0.56
10.0
165
2.76
3.86
69
114
0.047
12.7
0.047
12.7
0.100
0.160
27.1
Titanium
Filler materials:
170
43.4
S-glass
Carbon-PAN
Carbon-PAN (high-strength)
Carbon (high-modulus)
Aramid
600
470
4140
3240
5650
2200
3450
12.5
33.5
40
100
86.2
231
276
690
131
0.09
0.064
24.4
17.4
17.7
820
325
500
0రవ
0.078
0.052
21.2
19.0
14.1
Click Save and Submit to save and submit. Click Save All Answers to save all answers.
earch
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Question 2
You are a biomedical engineer working for a small orthopaedic firm that fabricates rectangular shaped fracture
fixation plates from titanium alloy (model = "Ti Fix-It") materials. A recent clinical report documents some problems with the plates
implanted into fractured limbs. Specifically, some plates have become permanently bent while patients are in rehab and doing partial
weight bearing activities.
Your boss asks you to review the technical report that was generated by the previous test engineer (whose job you now have!) and used to
verify the design. The brief report states the following... "Ti Fix-It plates were manufactured from Ti-6Al-4V (grade 5) and machined into
solid 150 mm long beams with a 4 mm thick and 15 mm wide cross section. Each Ti Fix-It plate was loaded in equilibrium in a 4-point bending
test (set-up configuration is provided in drawing below), with an applied load of 1000N. The maximum stress in this set-up was less than the
yield stress for the…
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. الإشارات الأخری
H التطبيقات
مطلوب
Section
5
3 m
2m
2 m
The rigid beam ABC is supported by pin A and wires BD and CE. If the load P on the beam
causes the end C to be displaced 6 mm downward. What is the normal strain developed in
* ?wire BD
)2 نقطة(
0.0015 C
2.57 O
0.00107
10:02 PM
AR
11/29/2020
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%YO l.
۱۳ ۱ ۱ من ۱۸ ۱ ۱
1 Semester Exam
Time: 90 Minutes
Date: 7/03 /2021
Code: CIV141
t force
irected applied loads and moments as shown, calculate
ne the
Q2/ A simply supported beam supports the
the support reactions?
12 kN
4 kN/m
8 kN-m
16 kN-m
2 kN/m
3 m
4 m
4 m
4 m
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I
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PHYS X PHYS X
印 PHYS X
PHYS X
POTPHYS X
PHYS X
E PHYS X E PHYS
top/semester2/physics%20for%20engineers/PHYS220_CH15_Lecture%20Notes_Problems%2015 19,15.29 S
(D Page view
A Read aloud
V Draw
Problem-15-19: page-475
A 0.500-kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion
with an amplitude of 10.0 cm. Calculate the maximum value of its
(a) speed, and acceleration.
(b) the speed and the acceleration when the object is 6.00 em from the equilibrium position, and
(c) the time interval required for the object to move from.r50 to r5 8.O0 cm.
Solution:
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Part 1 website: https://ophysics.com/r5.html
PArt 2 website: https://ophysics.com/r3.html
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Page3
Engineering Mechanics: Statics (EI13)
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Department of Civil Engineering
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Univensity of Misan
Course link on Moodle:
https://www.uomisan.edu.iq/moodle/course/view.php?id=597
Instructor: Dr. Murtada Abass
Assignment Due date:
February 18.2021 at 8:30
AM
F1
act on a brac
as shown in
Figure below.
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F = 100 N
30
20
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18 at 8:30 AM
Class time:
Page4
Tuesday 12:30PM-2:30PM
& Thursday 8:30AM-10:30AM
Lngineering Mechanics: Statics (E13)
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Modulus of Elasticity
106 psi
Material
MPа
psi
GPa
Ratio
30
0.30
120000
207
Steel alloy
Brass alloy
Aluminum alloy
Titanium alloy
830
0.34
0.33
380
55000
97
14
275
40000
69
10
690
100000
107
15.5
0.34
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6061 T6 Al
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25 ft-lb
12 ft-lb
27 ft-lb
Impact Strength (J/m
or ft-lb/in)
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63.452 ft-lb/in
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& SUSTAINABILITY, SAFETY & RELIABILITY:
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In Avengers 2, Captain America's improved
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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
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c. [25pts] A bar of unobtainium has a length of
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whesta fins dus al hoss
P = 375 KN
1.5 m
0.3
50 mm
100 mm
& [mm/mm]
Is the material within its…
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7490
*
اكتب اسمك الثلاثي
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Does the box will tip or not? and
Does the box will slip or not?
1 m
mass=50 kg
P = 100N
us=0.3
60
0.5 m
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