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INTRODUCTION TO MECHANICS
OF MATERIALS
(INGE 4019)
Pablo G. Caceres-Valencia (B.Sc., Ph.D., U.K.)
Assessment
The course will be assessed in the following manner:
‰ 1st Exam
25%
‰ 2nd Exam
30%
‰ Quizzes*
30%
‰ Others**
15% (*)
(*) Date due WebCT or Moodle Quizzes and Pop‐Quizzes (max‐6).
Missed quizzes will be graded with zero. Lack of access to WebCT or
Moodle is not an excuse for not submitting your answers.
(**) Class participation and Attendance. After the third missed class,
one point will be deducted in the final grade for each missed class (up
to 15 points).
GENERAL INFORMATION
Course Number
INGE 4019
Course Title
Introduction to Mechanics of Materials
Credit Hours
4
Instructor
Dr. Pablo G. Caceres-Valencia
Office
Lucchetti L-212, Extension 2358
Office Hours
M-W from 9-12am and 1-3pm
e-mail
p[email protected]
Web-site
http://academic.uprm.edu/pcaceres
Grades
Final Grade Range
100 – 90
89 – 80
79 – 70
69 – 60
59 ‐ 0
Final Letter Grade
A
B
C
D
F
Attendance
Attendance and participation in the lecture are compulsory and will
be considered in the grading. Students should bring calculators,
rulers, pen and pencils to be used during the lectures. Students are
expected to keep up with the assigned reading and be prepared to
answer questions on these readings during lecture and for the pop‐
quizzes. Please refer to the Bulletin of Information for Undergraduate
Studies for the Department and Campus Policies.
Tentative Dates
Monday
Wednesday
Monday
Wednesday
08/12
Introduction
08/17
Mech.Prop.
08/19
Linear Elasticity
08/24
Axial Loads
08/26
Axial Loads
08/31
Statically
Indetermined
09/02
Thermal Effects
Tuesday 09/07
Stresses on Inclined
Planes
09/09
Stress Concentration
09/14
Torsion
09/16
Thin Walled Tubes
09/21
1st Exam
09/23
Holiday
09/28
Shear and Bending
09/30
Shear and Bending
10/05
Principal Stresses
10/07
Mohr’s Circle
10/12
Holiday
10/14
Triaxial Stresses
10/19
Plane Stresses
10/21
Plane Stresses
10/26
Hooke’s Law
10/28
Plane Strain
11/02
Plane Strain
11/04
Elasticity
11/09
Combined Loadings
11/11
Holiday
11/16
Combined Loadings
11/18
2nd Exam
11/23
Combined Loadings
11/25
Deflection of Beams
11/30
Deflection of Beams
12/02
Deflection of Beams
Outcomes
Upon the completion of the course the student should be
able to:
•
Calculate the principal stresses and strains in a loaded
component
•
Solve problems using stress transformation and Mohr’s
circle
•
Apply Hooke’s law for plane stress and plane strain
•
Calculate stresses in thin walled spherical or cylindrical
pressure vessels
•
Calculate the stresses produced by combined axial,
bending and torsional loads
Exams
All exams will be conducted during lecture periods on the specified
dates. There will be no final exam.
Neatness and order will be taking into consideration in the grading of
the exams. Up to ten points can be deducted for the lack of neatness
and order. You must bring calculators, class notes and blank pages to
the exams.
Texbooks
James M. Gere and Barry J. Goodno, Mechanics of Materials, 7th Edition, Cengage
Learning.
My lecture notes are available in the web at
http://academic.uprm.edu/pcaceres
See syllabus of the course for recommended books.
Review of Statics
Mechanics of materials is a branch of mechanics that
develops relationships between the external loads applied to
a deformable body and the intensity of internal forces acting
within the body as well as the deformations of the body.
Equations of equilibrium (i.e., statics) are mathematical
expressions of vector relationships showing that for a body not
to translate or move along a path then ΣF = 0 . For a body not
to rotate, ΣM = 0.
Stress
Stress has two components, one acting
perpendicular to the plane of the area and the
other acting parallel to the area. Mathematically,
the former component is expressed as a normal
stress which is the intensity of the internal force
acting normal to an incremental area such that:
where +σ = tensile stress = "pulling" stress and -σ = compressive
stress = "pushing“ stress.
The latter component is expressed as a shear
stress which is the intensity of the internal force
acting tangent to an incremental area such that:
COMPRESSION
(*squeezing)
-ve
TENSION
(*stretching)
+ve
Prismatic bar in tension: (a) free-body
diagram of a segment of the bar, (b)
segment of the bar before loading, (c)
segment of the bar after loading, and (d)
normal stresses in the bar.
Prismatic bar = Section does not
change in the length of the bar
Normal Stress and Normal Strain
σ=
ε=
P
= stress
A
δ
L
= normal- strain
2P P
=
σ=
2A A
ε=
δ
L
P
σ=
A
2δ δ
ε= =
2L L
Uniform Normal Stress
Stresses are constant over the cross
sectional area.
Uniform stress distribution in a prismatic bar: (a)
axial forces P, and (b) cross section of the bar.
To have uniform stresses (tension or
compression) over the cross-section
area (A), the axial force must act along
the centroid.
xδA
∫
x=
A
yδA
∫
y=
A
©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Does the material resist the applied force?
What is the relationship between the stress applied and the deformation?
Tensile Testing
• The
sample is pulled slowly
• The sample deforms and then fails
• The load and the deformation are measured
Force
Extension or change in length
E = 29 × 10− 6 psi
D = 1.07 in. d = 0.618 in.
Determine the deformation of the steel rod shown under the given
loads.
• Apply free-body analysis to each
component to determine internal forces,
P1 = 60 × 103 lb
P2 = −15 × 103 lb
P3 = 30 × 103 lb
• Evaluate total deflection,
Pi Li 1 ⎛ P1L1 P2 L2 P3 L3 ⎞
⎟⎟
= ⎜⎜
+
+
A
E
E
A
A
A
i i i
⎝ 1
2
3 ⎠
δ =∑
(
) (
) (
)
⎡ 60 × 103 12 − 15 × 103 12 30 × 103 16 ⎤
=
+
+
⎥
6⎢
0.9
0.9
0.3
29 × 10 ⎣⎢
⎦⎥
1
= 75.9 × 10−3 in.
L1 = L2 = 12 in.
L3 = 16 in.
A1 = A2 = 0.9 in 2
A3 = 0.3 in 2
δ = 75.9 ×10−3 in.
Find the stresses and strains:
(a) Change in length of the pipe
(b) Lateral strain
(c) Increase in outer and inner
diameters
(d) Increase in wall thickness
(*tearing)
Normal stress (σ) : the subscript identifies the face
on which the stress acts.
Tension is positive and compression is negative.
σx
σ xx
Shear stress (τ) : it has two subscripts. The first subscript denotes the
face on which the stress acts. The second subscript denotes the
direction on that face.
A shear stress is positive if it acts on a positive face and positive
direction or if it acts in a negative face and negative direction.
τ xy
3D
From equilibrium principles:
τxy = τyx , τxz = τzx , τzy = τyz
TORSION
(*twisting)
BENDING
(*flexure)
M
σM
Mc
=
I
For 2-D:
ΣFx = ma x
Where αz is the angular acceleration
ΣFy = ma y
ΣM z = I zα z
Moment of a Force about an axis
Free body Diagram
Change the steel for aluminum.
A steel strut S serving as a brace for a boat hoist
transmits a compressive force P to the deck of a pier.
Bearing pad in shear. Elastomer with Modulus of
Rigidity Ge
The connection shown in the figure consists of five steel
plates, each 2.5mm thick, to be joined by a single bolt.
Determine the required diameter of the bolt if the
allowable bearing stress, σb, is 180.0MPa and the
allowable shear stress, τallow, is 45.0MPa?
FBD
Determine the allowable load P based
on the following four considerations.
Two-bar truss ABC supporting a sign
of weight W. Determine the required
cross-sectional area of bar AB and
the required diameter of the pin at
support C
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