Mechanics of Materials MAE 243 (Section 002)

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Mechanics of Materials


MAE 243 (Section 002)

Spring 2008

Dr. Konstantinos A. Sierros

General info



M, W, F

8:00
-
8:50 A.M. at Room G
-
83 ESB



Office: Room G
-
19 ESB



E
-
mail: kostas.sierros@mail.wvu.edu



Tel: 304
-
293
-
3111 ext.2310


Course notes: http://www.mae.wvu.edu/~cairns/teaching.html


USER NAME: cairns PASSWORD: materials



Facebook : Konstantinos Sierros (using courses: Mechanics of
Materials)



Office hours:
M, W

9:00
-
10:30 A.M. or by appointment

Course textbook

Mechanics of Materials, 6
th

edition,
James M. Gere, Thomson,
Brooks/Cole, 2006

1.1: Introduction to Mechanics of Materials

Definition
:
Mechanics of materials
is a branch of applied
mechanics that deals with the behaviour of solid bodies
subjected to various types of loading


Compression

Tension (stretched)


Bending


Torsion (twisted)

Shearing


1.1: Introduction to Mechanics of Materials

Fundamental concepts



stress and strain



deformation and
displacement



elasticity and
inelasticity



load
-
carrying
capacity

Design and analysis of mechanical
and structural systems

1.2: Normal stress and strain



Most fundamental concepts in
Mechanics of Materials are
stress
and
strain



Prismatic bar:
Straight structural
member with the

same

cross
-
section throughout its length



Axial force:
Load directed along
the axis of the member



Axial force can be
tensile

or
compressive


Type of loading for landing gear
strut and for tow bar?



Examples

A truss bridge is a type of beam
bridge with a skeletal structure. The
forces of
tension
, or pulling, are
represented by red lines and the
forces of
compression
, or squeezing,
are represented by green lines.


The Howe Truss was originally
designed to combine diagonal
timber compression members and
vertical iron rod tension members


Normal stress



Continuously distributed
stresses

acting over the entire cross
-
section.
Axial force P is the resultant of those
stresses



Stress
(σ) has units of force per
unit area



If
stresses
acting on cross
-
section
are uniformly distributed then:

Units of stress in USCS: pounds per square inch (psi) or

kilopounds per square inch (ksi)

SI units: newtons per square meter (N/m
2
) which is equal to Pa

Limitations

The loads P are transmitted to the bar by pins that pass through
the holes

High localized stresses are produced
around the holes !!

Stress concentrations

Normal strain

A prismatic bar will change in length when under a uniaxial tensile
force…and obviously it will become longer…



Definition of elongation per unit
length or
strain (ε)



If bar is in tension, strain is
tensile and if in compression the
strain is compressive



Strain is a dimensionless
quantity (i.e. no units!!)

Line of action of the axial forces for a uniform stress
distribution

It can be demonstrated that in
order to have uniform tension
or compression in a prismatic
bar, the axial force must act
through the centroid of the
cross
-
sectional area.

*In geometry, the
centroid

or
barycenter

of an object
X

in
n
-
dimensional
space is the intersection of all hyperplanes that divide
X

into two parts of
equal moment about the hyperplane. Informally, it is the "average" of all
points of
X
.


*The geometric centroid of a physical object coincides with its center of mass if the object has
uniform density, or if the object's shape and density have a symmetry which fully determines the
centroid. These conditions are sufficient but not necessary.


Problem 1.2
-
4


A circular aluminum tube of length
L
=
400 mm is loaded in compression by
forces
P
(see figure). The outside and inside diameters are 60 mm and 50 mm,
respectively. A strain gage is placed on the outside of the bar to measure
normal strains in the longitudinal direction.

(a) If the measured strain is


550
x
10
-
6

, what is the shortening

of the bar?

(b) If the compressive stress in the bar is intended to be 40 MPa, what should
be the load
P
?

Problem 1.2
-
7

Two steel wires,
AB
and
BC
, support a lamp weighing 18 lb (see figure). Wire
AB
is at an angle α = 34
°

to the horizontal and wire
BC
is at an angle β = 48
°
.
Both wires have diameter 30 mils. (Wire diameters are often expressed in mils;
one mil equals 0.001 in.) Determine the tensile stresses
AB
and
BC
in the two
wires.

Problem 1.2
-
11

A reinforced concrete slab 8.0 ft square and 9.0 in. thick is lifted by four cables
attached to the corners, as shown in the figure. The cables are attached to a
hook at a point 5.0 ft above the top of the slab. Each cable has an effective
cross
-
sectional area
A

0.12 in2.

Determine the tensile stress σ
t

in the cables due to the weight of the concrete
slab. (See Table H
-
1, Appendix H, for the weight density of reinforced concrete.)