SKEM 2013
Mechanics of Materials
Topic 1: Introduction to Mechanics of
Materials
Information for this note was adapted from Hibbeler, R.C., Mechanics of Materials, Pearson Education, 2022 1
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SKEM 2013
Mechanics of Materials
Part 1
Equilibrium of a Deformable Body
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Introduction
Overview of this topic:
Review important
principles of statics.
Determine internal
resultant loadings in a
body.
Two areas
MECHANICS :
• Physical science that covers the
problems of bodies at rest or in mo
tion due to the forces acting on the
bodies
Introduce the concepts
of stress, strain, &
analysis.
Statics ‐ Study of bodies at
rest or with uniform velocity
Dynamics ‐ Study of
bodies in motion.
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Introduction
• Mechanics of materials (Solid Mechanics) is a study of the
relationship between the external loads on a body and the
intensity of the internal loads within the body.
• This subject also involves the deformations and stability of a body
when subjected to external forces.
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Equilibrium of a Deformable Body
Main Principles of Statics
Loads
Support
Reactions
Equations of
equilibrium
Normal force,
N
Internal Resultant
Loadings
Shear force, V
Torsional moment
or torque, T
Bending
moment, M
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Equilibrium of a Deformable Body
Loads
External Forces
Surface Forces
Concentrated
Body Forces
Distributed
1.
Surface Forces - caused by direct contact of other body’s surface
2.
Body Forces - other body exerts a force without contact
W1 (1)
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Equilibrium of a Deformable Body
Support Reactions
Type Of Supports
Connection & Reaction
Cable
Roller
Smooth
Support
Journal
Bearing
External
Pin
Internal
Pin
Fixed
Support
Thrust
Bearing
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Equilibrium of a Deformable Body
Equations of Equilibrium
Equations of Equilibrium
• Equilibrium of a body requires a balance of forces and a balance of moments
• For a body with x, y, z coordinate system with origin O,
• Best way to account for these forces is to draw the body’s freebody diagram (FBD).
Equilibrium of a body requires both a balance of forces, to prevent the body from translating or having accelerated motion along a straight or curved path, and a balance of moments, to prevent the body from rotating.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Internal Resultant Loadings
• Objective of FBD is to determine the resultant force and moment acting
within a body.
• In general, there are 4 different types of resultant loadings:
a) Normal force, N
b) Shear force, V
c) Torsional moment or torque, T
d) Bending moment, M
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 1: Question
• The hoist in the figure consists of the
beam AB and attached pulleys, the
cable, and the motor.
• Determine the resultant internal
loadings acting on the cross section
at C if the motor is lifting the 2000 N
(approximate 200 kg) load W with
constant velocity. Neglect the weight
of the pulleys and beam.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 2: Question
The metal stud punch is subjected to a
force of 120 N on the handle.
1. Determine the magnitude of the
reactive force at the pin A and in
the short link BC.
2. Determine the internal resultant
loadings acting on the cross section
passing through the handle arm at
D.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 3: Question
Determine the resultant internal
loading on the cross section through
point C of the pliers. There is a pin at A
and the jaws at B are smooth.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 4: Question
• The sky hook is used to support the
cable of a scaffold over the side of a
building. If it consists of a smooth rod
that contacts the parapet of a wall at
points A, B, and C, determine the
normal force, shear force, and
moment on the cross section at
points D and E.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 5: Question
• Determine the resultant internal
loadings acting on the cross section
at C of the beam.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 6: Question
Determine the resultant internal loadings acting on the cross section
at 2 m.
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Equilibrium of a Deformable Body
Internal Resultant Loadings
Example 7: Question
Determine the resultant internal loadings acting on the cross-section
AB & BC.
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SKEM 2013
Mechanics of Materials
Part 2
Stress:
Average Normal Stress in an Axially
Loaded Bar
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Stress
Introduction
• Distribution of internal loading is important in mechanics of
materials.
• We will consider the material to be continuous (uniform
distribution of matter).
• This intensity of internal force at a point is called stress.
NORMAL FORCE ‐
The force that acts perpendicular
to a certain surface and passes
through the symmetrical axis of
the body.
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Stress
Introduction
Normal Stress (σ) - sigma
• Force per unit area acting normal to ΔA
Shear Stress (τ) - tau
• Force per unit area acting tangent to ΔA
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Stress
Average Normal Stress in an Axially
Loaded Bar
• When a cross-sectional
area bar is subjected to
axial force through the
centroid, it is only
subjected to normal stress.
• Stress is assumed to be
averaged over the area.
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Stress
Average Normal Stress in an Axially
Loaded Bar
Average Normal Stress Distribution
• When a bar is subjected to a
constant deformation,
σ = average normal stress
P = resultant normal force
A = cross sectional area of bar
Equilibrium
• 2 normal stress components that are equal in magnitude
but opposite in direction (tension and compression).
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Stress
Average Normal Stress
Example 8: Question
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Stress
Average Normal Stress
Example 9: Question
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Stress
Average Normal Stress
Example 10: Question
The 80 kg lamp is supported by
two rods AB and BC as shown in
the figure. If AB has a diameter of
10 mm and BC has a diameter of
8 mm, determine the average
normal stress in each rod.
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Stress
Exercise: Average Normal Stress 1
Member AC shown in the figure is subjected to
a vertical force of 3 kN.
Determine the position x of this force so that
the average compressive stress at the support
C is equal to the average tensile stress in the
tie rod AB. The tie rod has a cross-sectional
area of 400 mm2 and the support C has a
cross-sectional area of 650 mm2.
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Stress
Exercise: Average Normal Stress 2
Determine the maximum safe value of axial load P. Given:
1. Maximum allowable stress for steel = 140 Mpa
2. Maximum allowable stress for aluminum = 90 Mpa
3. Maximum allowable stress for bronze = 100 MPa
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Stress
Exercise: Average Normal Stress 3
A hollow steel tube with an inside diameter
of 100 mm must carry a tensile load of 400
kN. Determine the outside diameter of the
tube if the stress is limited to 120 MN/m2.
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Stress
Exercise: Average Normal Stress 4
Determine the smallest diameter of bronze and
steel cables. Given;
Weight of bar = 800 kg
Maximum allowable stress for bronze = 90 MPa
Maximum allowable stress for steel = 120 MPa
Bronze
Steel
2m
4m
800 kg
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Stress
Exercise: Average Normal Stress 5
The rigid bar OAB is supported by link AC. Link
AC is made of aluminum (Eal = 70 GPa) and has a
cross-sectional area of 500 mm2. For the 10 kN
force shown, determine
1. reaction force at pin C
2. the average normal stress (σ) in link AC
OA = 600 mm, AB = 400 mm and AC = 400 mm
10 kN
O
C
A
B
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SKEM 2013
Mechanics of Materials
Part 3
Stress:
Average Shear Stress
Allowable Stress Design
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Stress
Average Shear Stress
• The average shear stress distributed over each
sectioned area that develops a shear force.
τ = average shear stress
V = internal resultant shear force
A = area at that section
• 2 different types of shear:
a) Single Shear
b) Double Shear
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Stress
Average Shear Stress
Single Shear
Shear force = P
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Stress
Average Shear Stress
Single Shear
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Stress
Average Shear Stress
Double Shear
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Stress
Average Shear Stress
Double Shear
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Stress
Average Shear Stress
Example 11 - Question
The supporting wheel on a scaffold
is held in place on the leg using a 4
mm diameter pin as shown. If the
wheel is subjected to a normal
force of 3 kN, determine the
average shear stress developed in
the pin. Neglect friction between
the inner scaffold puller leg and the
tube used on the wheel.
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Stress
Allowable Stress Design
• Many unknown factors that influence
the actual stress in a member.
• A factor of safety is needed to obtained
allowable load.
• The factor of safety (F.S.) is a ratio of
the failure load divided by the
allowable load
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SKEM 2013
Mechanics of Materials
Part 4
Strain:
Deformation
Strain
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Strain
Deformation
• When a force is applied to a body, it will change the body’s
shape and size.
• These changes are deformation.
• Can be highly visible or practically unnoticeable
• Can occur when temperature of a body is changed
Note the before and after positions of 3-line
segments where the material is subjected to
tension.
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Strain
Normal Strain
• Defined as the elongation / contraction of a line segment
per unit of length is referred to as normal strain.
• Average normal strain is defined as
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Strain
Units
• Normal strain is a dimensionless quantity since it is a ratio of two
lengths.
• But common practice to state it in terms of meter/meter (m/m)
• ɛ is small for most engineering applications, so is normally expressed as
micrometers per meter (µm / m)
• Also can expressed as a percentage, eg: 0.001 m/m = 0.1%
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Strain
If the normal strain is known, then the approximate final length is
+ε 🡺line elongate
-ε 🡺line contracts
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Strain
Example 12 - Question
A force acting on the grip of the lever arm shown in figure (a) causes the arm to rotate
clockwise through an angle of θ = 0.002 rad as shown in figure (b). Determine the
average normal strain developed in the wire BC.
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Strain
Shear Strain
• Change in angle between 2-line segments that were
perpendicular to one another refers to shear strain.
• Denote gamma and measured in radians (rad)
θ'<90 🡺+shear strain
θ'>90 🡺-shear strain
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Strain
Shear Strain
Example 13 - Question
The plate is deformed into the dashed shape as shown
in the figure. If in this deformed shape horizontal lines
on the plate remain horizontal and do not change their
length, determine:
(a) the average normal strain along the side AB, and
(b) the average shear strain in the plate relative to the x
and y axes.
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References
Hibbeler, R.C., Mechanics of Materials (11th and
SI edition), Pearson Education, 2022
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Thank You
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