BCN 240
5C
: Con
struction Mechanics I
I
(
Strength of Materials
).
Instructor: Dr Ian Flood
1
PROPERTIES OF MATERIALS
Notes
:
Tension Test
Strength of material problems can be divided into two broad categories:
analysis
and
design
.
Analysis is concerned with evaluating an existing design (built or proposed). For
example: (i) determining the
maximum load that can be imposed on a body so that it does
not
experience excessive stresses or strains; (ii) determining the stresses
and strains
induced by given loads to see if they exceed limiting values.
Design is concerned with determining the size
and shape of members and/or the type of
material that are required so that the loads do not induce excessive stresses and strains in
the body. It is more involved than analysis and may require several iterations to home

in
on an acceptable design.
Variou
s properties of materials, in particular the mechanical properties, are key factors in
analysis and design.
One of the most widely used and important test
s
of the mechanical properties of materials
is the tension test. Concrete is one notable exception a
nd will usually be tested in
compression. Tension test = static
,
axially
loaded
,
destructive test, standardized by
ASTM (American Society for Testing and Materials).
The tensile test applies a gradually increasing axial load to a sample such as that show
n
below, taking regular readings
of
load
and
gauge length
(distance between reference
points on sample
)
until failure of the sample. The test measures stress and strain with
referen
ce to the original gauge length
and original cross

sectional areas
. These
ar
e
termed the
nominal stress
and
nominal strain
; t
he term nominal is used because these
values
are not the same as the true stress and strain since the cross sectional area
and
length of the sample
will actually change with loading. At the end of the
test, the cross
sectional area at the po
int of failure is also measured, and the two pieces are carefully
placed back together
to determine the
final
gauge length.
reference points
measuring
gauge length
cross

sectional area,
A
, of sample
sample
P
P
BCN 240
5C
: Con
struction Mechanics I
I
(
Strength of Materials
).
Instructor: Dr Ian Flood
2
Stress
Strain
Diagram
s
After the test, the collected data
are converted to nominal stress (based on original cross

sectional area) and nominal strain (based on original gauge length)
and plotted to form a
stress

strain
diagram. A typical stress

strain diagram for a low

carbon
ductile
(opposite
of brittle) steel:
A =
Proportional Limit
:
Up to this point, the stress

strain curve is straight. The slope of
this line gives the Modulus of Elasticity.
B =
Elastic Limit
:
Up to this point, the material is in its elastic range. That is,
if the load
is completely removed,
the sample will return to its original length (that is, it will not
retain any stretching).
For most materials, the proportional limit and elastic limit a
re
almost the same thing. Past the elastic limit, the steel enters the plastic range. Here
the strain is not fully reversible when the load is removed, and the strain
increases at
a faster rate relative to the stress
.
C =
Upper Yield Limit
:
At this point
, the strain increases with little or no additional
stress. The stress at this point is termed the yield stress
s
Y
.
D =
Lower Yield Point
: For low carbon steel (more ductile steels) the stress

strain curve
can dip down after the
yield point. After this p
oint, the steel stretches fairly
constantly up to a strain of about 0.015 (1.5%), point D
2
.
D
2
= After this point, the steel begins to recover some of its strength and becomes
capable of resisting additional load. The steel has passed from the plastic ran
ge into
the
strain

hardening range.
Stress (s)
psi
Strain (
)
in/in
A
B
C
D
E
F
Plastic
Strain hardening
0.
015
0.0
0
0
0.
200
D
2
Elastic
BCN 240
5C
: Con
struction Mechanics I
I
(
Strength of Materials
).
Instructor: Dr Ian Flood
3
E =
This point is where the steel continues to stretch accompanied by a decreasing ability
to transmit load.
This represents the
ultimate strength
(or
tensile strength
) of the
steel.
It also develops a
neck
at this sta
ge
where locally the cross

section of the
sample decreases and its length increases
,
visibly.
Necking increases rapidly until
the steel suddenly ruptures
at point F
.
Different materials exhibit different shaped stress

strain curves.
For materials with a
less well

defined yield point, an artificial yield point may be
established, often by
a
0.
2% offset
(0.002 in/in)
in the strain dimension,
that
is, that
would represent a 0.2%
set (non

recovered strain). This can be illustrated
as follows:
The above discussions pertains to materials at normal temperatures. If temperatures are
significantly higher or lower than this then stress

strain curves for the material at those
temperatures should be used.
Allowab
le Stresses and Calculated Stresses
Once tests such as the tension test above have been completed, the results are used to
determine
what is considered to be a safe or limiting stress for a given situation, termed
the
allowable stress
. Appropriate values
for the allowable stress are determined on
several factors, including:
Mechanical properties (such as determined by the tension test).
Ductility of the material.
Strain
(
)
in/in
0.000
0.
2% offset
= 0.
002 in/in
Stress (s)
psi
Yield strength
BCN 240
5C
: Con
struction Mechanics I
I
(
Strength of Materials
).
Instructor: Dr Ian Flood
4
Confidence in the load predictions.
Type of loading: static (as above), cyclic, or impact.
Co
nfidence in the analysis and design methods.
Possible deterioration during the design life of the component (corrosion,
chemical attack, etc..).
Possible danger to life and property resulting from failure (risk).
Design life of the structure (temporary or
permanent).
Allowable stresses and design approaches are established by appropriate agencies, such
as AISC (American Institute of
Steel Construction) or ACI (American Concrete Institute).
These will then be incorporated into building codes of states or m
unicipalities.
Note, the
calculated stress
is the stress that is determined by calculation to be induced in
a member as a result of applied loads. This may vary, and normally will be a fraction of
the
allowable stress
, but should never exceed the
allowable stress
.
Factor of Safety
The
allowable stress
will always be significantly less than the
failure stress
, thus
providing a margin of safety.
The ratio of failure stress to allowable stress is called the
factor of safety
(FS):
FS = failure str
ess / allowable stress
Although reaching the yield point in a component does not mean the member will rupture,
it will
usually be considered to be a failure
render
ing
the component
unusable since it
will be on the verge of significant and unacceptable def
ormation.
Factors of safety may be based on a material
yield strength
for ductile materials, or on
ultimate strength
for brittle materials.
Factors of safety may vary from 1.5 (50% extra capacity) to 20.
Reading
:
Section
10

1, 10

2, 10

6, and 10

7
to 9

6
.
Worked Examples
:
Starting page
2
46
:
10

1,
and 10

3 to 10

6
.
BCN 240
5C
: Con
struction Mechanics I
I
(
Strength of Materials
).
Instructor: Dr Ian Flood
5
Problems
:
Starting page
2
65
:
10

1
to
10

1
1
(
odd numbered problems only,
answ
ers provided in
back of book).
Class Problem
s
:
Q
10

1
:
A 9/16 in
.
diameter steel rod is tested in tension and elongates 0.00715 in. in a
length of 8 in. under a tensile load of 6,500 lb.
The proportional limit of this steel is
34,000 psi. Based on this one reading, c
alculate:
(a)
the stress
,
(b)
the strain
,
(c)
the modulus of elasticity,
E
.
Q
10

8
:
A tension member in a roof truss is composed of two ASTM A36 structural steel
angles that together have a net cross

sectional area of 8.62 in
2
.
(a)
Compute the allowable total load if the allowable tensile stress
is
22,000 psi.
(b)
If the total tensile load in the member is 150,000 lb, compute the actual tensile
stress.
Q 10

10
:
A main cable in a large bridge is designed for a tensile force of 2,600,000 lb.
The cable consists of 1470 parallel wires, each 0.16
in. in diameter. The wires are cold

drawn steel with an average ultimate strength of 230,000 psi. What factor of safety was
used in the design?
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