# Strength of Materials

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15 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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CTC / MTC 222

Strength of Materials

Final Review

Final Exam

Tuesday, December 13,
3:00

5:00

Graded on the basis of 30 points in increments of ½ point

Open book

May use notes from first two tests plus two additional sheets
of notes

Equations, definitions, procedures, no worked examples

Also may use any photocopied material handed out in class

Work problems on separate sheets of engineering
paper

Hand in test paper, answer sheets and notes stapled to back

Course Objectives

To provide students with the necessary
tools and knowledge to analyze forces,
stresses, strains, and deformations in
mechanical and structural components.

To help students understand how the
properties of materials relate the
strains and deformations.

Chapter One

Basic Concepts

SI metric unit system and U.S. Customary
unit system

Unit conversions

Basic definitions

Mass and weight

Stress, direct normal stress, direct shear stress
and bearing stress

Single shear and double shear

Strain, normal strain and shearing strain

Poisson’s ratio, modulus of elasticity in tension
and modulus of elasticity in shear

Direct Stresses

Direct Normal Stress ,

σ

= Applied Force/Cross
-
sectional Area = F/A

Direct Shear Stress,

Shear force is resisted uniformly by the area of the part in
shear

= Applied Force/Shear Area = F/A
s

Single shear

applied shear force is resisted by a single cross
-
section of the member

Double shear

applied shear force is resisted by two cross
-
sections of the member

Direct Stresses

Bearing Stress,
σ
b

σ
b

= Applied Load/Bearing Area = F/A
b

Area A
b
is the area over which the load is transferred

For flat surfaces in contact, A
b

is the area of the smaller of the
two surfaces

For a pin in a close fitting hole, A
b

is the projected area,
A
b
= Diameter of pin x material thickness

Chapter Two

Design Properties

Basic Definitions

Yield point, ultimate strength, proportional
limit, and elastic limit

Modulus of elasticity and how it relates
strain to stress

Hooke’s Law

Ductility
-

ductile material, brittle material

Chapter Three

Direct Stress

Basic Definitions

Design stress and design factor

Understand the relationship between design stress,
allowable stress and working stress

Understand the relationship between design factor, factor of
safety and margin of safety

Design / analyze members subject to direct stress

Normal stress

tension or compression

Shear stress

shear stress on a surface, single shear and
double shear on fasteners

Bearing stress

bearing stress between two surfaces,
bearing stress on a fastener

Chapter Three

Axial
Deformation and Thermal Stress

Axial strain
ε
,

ε

=
δ

/ L , where
δ

= total deformation, and L = original
length

Axial deformation,
δ

δ

= F

L
/
A E

If unrestrained, thermal expansion will occur due to
temperature change

δ

=
α

x L x ∆T

If
restrained
, deformation due to temperature change
will be prevented, and stress will be developed

σ

= E

α

(∆T)

Chapter Four

Torsional
Shear Stress and Deformation

For a circular member,
τ
max
= Tc / J

T = applied torque, c = radius of cross section, J = polar moment
of inertia

Polar moment of Inertia, J

Solid circular section, J =
π

D
4
/ 32

Hollow circular section, J =
π

(D
o
4
-

D
i
4
)

/ 32

Expression can be simplified by defining the
polar
section modulus
, Z
p
= J / c, where c = r = D/2

Solid circular section,
Z
p

=
π

D
3
/ 16

Hollow circularsection,
Z
p

=
π

(D
o
4
-

D
i
4
)

/ (16D
o
)

Then,
τ
max
= T / Z
p

Chapter Five

Shear Forces and
Bending Moments in Beams

Sign Convention

Positive Moment M

Bends segment concave upward

compression on top

Shear and Moment

Shear Diagram

Application

causes a downward jump in the shear diagram.
An upward load causes an upward jump.

The slope of the shear diagram at a point
(dV/dx) is equal to the (negative) intensity of
the distributed load w(x) at the point.

The change in shear between any two points on
a beam equals the (negative) area under the

Shear and Moment

Moment Diagram

Application of a clockwise concentrated moment
causes an upward jump in the moment diagram.
A counter
-
clockwise moment causes a
downward jump.

The slope of the moment diagram at a point
(dM/dx) is equal to the intensity of the shear at
the point.

The change in moment between any two points
on a beam equals the area under the shear
diagram between the points.

Chapter Six

Centroids and
Moments of Inertia of Areas

Centroid of complex shapes can be calculated
using:

A
T

̅Y
̅

= ∑ (A
i
y
i
) where:

A
T
= total area of composite shape

̅Y
̅

= distance to centroid of composite shape from some
reference axis

A
i
= area of one component part of shape

y
i
= distance to centroid of the component part from the
reference axis

Solve for

̅Y
̅

= ∑ (A
i
y
i
) / A
T

Perform calculation in tabular form

See Examples
6
-
1
&
6
-
2

Moment of Inertia of

Composite Shapes

Perform calculation in tabular form

Divide the shape into component parts which are simple shapes

Locate the centroid of each component part, y
i

from some
reference axis

Calculate the centroid of the composite section,

̅Y
̅

from some
reference axis

Compute the moment of inertia of each part with respect to its
own centroidal axis, I
i

Compute the distance, d
i

=

̅Y
̅

-

y
i
of the centroid of each part
from the overall centroid

Compute the transfer term A
i
d
i
2

for each part

The overall moment of inertia I
T

, is then:

I
T
=
∑ (
I
i +
A
i
d
i
2
)

See Examples
6
-
5
through
6
-
7

Chapter Seven

Stress Due to
Bending

Positive moment

compression on top, bent concave
upward

Negative moment

compression on bottom, bent
concave downward

Maximum Stress due to bending (Flexure Formula)

σ
max

= M c / I

Where M = bending moment, I = moment of inertia, and c
= distance from centroidal axis of beam to outermost fiber

For a non
-
symmetric section distance to the top fiber,
c
t
, is different than distance to bottom fiber c
b

σ
top

= M c
t

/ I

σ
bot

= M c
b

/ I

Section Modulus, S

Maximum Stress due to bending

σ
max

= M c / I

Both I and c are geometric properties of the section

Define section modulus, S = I / c

Then
σ
max

= M c / I = M / S

Units for S

in
3

, mm
3

Use consistent units

Example: if stress,
σ
, is to be in ksi (kips / in
2

), moment, M, must be
in units of kip

inches

For a non
-
symmetric section S is different for the top and the
bottom of the section

S
top

= I / c
top

S
bot

= I / c
bot

Chapter Eight

Shear Stress in
Beams

The shear stress,

, at any point within a beams cross
-
section
can be calculated from the General Shear Formula:

= VQ / I t, where

V = Vertical shear force

I = Moment of inertia of the entire cross
-
axis

t = thickness of the cross
-
section at the axis where shear stress is to
be calculated

Q = Statical moment about the neutral axis of the area of the cross
-
section between the axis where the shear stress is calculated and the
top (or bottom) of the beam

Q is also called the first moment of the area

Mathematically, Q = A
P

̅y
̅

, where:

A
P

= area of theat part of the cross
-
section between the axis where the
shear stress is calculated and the top (or bottom) of the beam

̅y
̅

= distance to the centroid of A
P

from the overall centroidal axis

Units of Q are length cubed; in
3
, mm
3
, m
3
,

Shear Stress in Common Shapes

The General Shear Formula can be used to develop
formulas for the maximum shear stress in common
shapes.

Rectangular Cross
-
section

max

= 3V / 2A

Solid Circular Cross
-
section

max

= 4V / 3A

Approximate Value for Thin
-
Walled Tubular Section

max

≈ 2V / A

Approximate Value for Thin
-
Webbed Shape

max

≈ V / t h

t = thickness of web, h = depth of beam

Chapter Twelve

Pressure
Vessels

If R
m

/ t ≥ 10, pressure vessel is considered
thin
-
walled

Stress in wall of
thin
-
walled

sphere

σ

=
p D
m

/ 4 t

Longitudinal stress

in wall of
thin
-
walled

cylinder

σ

=
p D
m

/ 4 t

Longitudinal stress is
same

as stress in a sphere

Hoop

stress in wall of cylinder

σ

=
p D
m

/ 2 t

Hoop stress is twice the magnitude of longitudinal stress

Hoop stress in the cylinder is also twice the stress in a
sphere of the same diameter carrying the same pressure