Axial Strain

cypriotcamelUrban and Civil

Nov 29, 2013 (4 years and 1 month ago)

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Axial Members


AXIAL MEMBERS, which support load only along their primary
axis, are the most basic of structural members.


Equilibrium requires that forces in Axial Members are always
Equal, Opposite, and Co
-
Linear.


In most cases, axial members have pinned ends.


Some examples of axial members include:


Bars;


Truss Members;


Ropes and Cables.

Axial Stress


If a cut is taken perpendicular to a bar's axis, exposing an
internal cross
-
section of area A, the
force per unit area

on the
face of this cut is termed
STRESS
.


The symbol used for normal or axial stress in most
engineering texts is
s
=
⡳楧浡(⸠
=

Stress in an axially loaded bar is:


s

= F/A



Stress is
positive

in
tension

(P>0) and
negative

in
compression

(P<0);


English units:
psi

(pounds per square inch), or
ksi

(kilopounds
per square inch);

Axial Stress


The axial stress of a member is determined by:


s
=
F / A


Where F = force applied along the longitudinal axis
of the member perpendicular to the cross
sectional area (A).

Axial Stress Example


A cylindrical steel bar has a diameter of ½”


The bar is attached at one end and a 5,000 lb
weight is hung from the steel bar
(axially
loaded)


What is the axial stress generated in the bar?


Area of a circle =
p
r
2


s =
F/A =5,000 lb/.20 in
2

=25,000 psi

Axial Strain


An axial bar of length L, and cross
-
sectional area A, subjected
to tensile force F, elongates by an amount, D.


The change in length divided by the initial length is termed
ENGINEERING STRAIN (or simply strain).


Strain is positive in tension and negative in compression


Strain is a non
-
dimensional length
-

a fraction.


Because strain is small, it is often given as a percentage by
multiplying by 100%: e.g., e = 0.003 = 0.3%.

Axial Strain


Axial strain is a measure of the
deformation to a member due to axial
stress.



e = d /
L


Where:


e
represents axial strain


d
r数e敳敮瑳

瑨攠捨慮c攠楮敮g瑨
=
=
䰠r数e敳敮瑳t瑨攠潲楧楮慬=汥湧瑨
=
Young's Modulus


Recall that all materials have a stiffness associated with them.


The stiffness of a material is defined through the relation:


s

= E
e

or E =
s / e



Where:


E is the YOUNG'S MODULUS or stiffness of the material.

e

is the axial strain

s
is the axial stress



Values of E for different materials are obtained experimentally from stress
-
strain curves.

Axial Strain Example


In the previous example, a cylindrical steel bar has a diameter
of ½” and a 5,000 lb weight is hung from the end (axially
loaded)


Young’s Modulus for steel is 29,000,000 psi


We found the axial stress to be 25,000 psi


What is the expected strain?


s

= E
e

or E =
s
/
e

or
e = s/
E


Where:


E is the YOUNG'S MODULUS or stiffness of the material.


e

is the axial strain


s
is the axial stress


e

= (25,000 psi)/(29,000,000 psi) = .00086 or .086%

Axial Strain Example (Cont.)


If the steel bar were 10 feet in length, what would the change
in length be when axially loaded?



e

=
d
=
⼠L



Where:



e
represents axial strain




d

represents the change in length (deformation)




L represents the original length


.00086 =
d /
10 feet


.0086 feet

Expected Deformation

d

=


Using Young’s Modulus, we can determine the expected
deformation of a member due to a constant force being
applied.





Where:

d =
Change in length

F = Force applied

L = Original length of member

A = Cross sectional area

E = Young’s Modulus

FL

AE

Expected Deformation


Using the Stress and Strain formulas, we found
the 10 foot steel rod is expected to change
length by .0086 feet when axially loaded with
5,000 pounds


Using the expected deformation formula, we
also find:


d

= FL/AE


=(5,000 lbs)(10 ft)/(.20 in
2
)(29,000,000 psi) = .0086 feet