cven214- lecture 2 strain -dr. wael alnahhal
TRANSCRIPT
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COLLEGE OF ENGINEERING
DEPARTMENT OF CIVIL & ARCHITECTURAL ENGINEERING
CVEN 214: STRENGTH OF MATERIALS
WAEL I. ALNAHHAL, Ph. D., P. Eng
Spring, 2015
Chapter 2: STRAIN
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Introduction
Applied
External Load
Internal
Loading DeformationsInternal
Stresses
Stability and Deformationsof a body depend on the intensity of
the external forces applied on the body as well as the load
carrying capacity of the body
Determine Using
Equilibrium
Equations
Strain
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Strain (deformation) terms
Deformationof the material is the change in
geometry when stress is applied
Deformation is expressed by the displacement
field of the material
Strainis the deformation per unit length
Deflectionis a term to describe the magnitudeto which a structural element bends under a
load
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Deformation
When a force is applied to a body, the internally
generated stress will change the bodys shape and size.
These changes are deformation.
Note the before and after
positions of 3 line segments
where the material is
subjected to tension.
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Strain
1. Normal Strain
The elongation / contraction of a line segment per unit of length is
referred to as normal strain.
Average normal strain is defined as
If the normal strain is known, then the approximate final length is
s
ssavg
='
( ) ss + 1'
+line elongate-line contracts
Units
Normal strain is a dimensionless quantitysince it is a ratio of two lengths
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Normal Strain
normalstress
normal strain
P
A
L
= =
= =
2
2
P P
A A
L
= =
=
2
2
P
A
L L
=
= =
Normal Strain tends to change the volume, Length or Sizeof a material
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Shear Strain
Change in angle between 2 line segmentsthat were perpendicular to one another
refers to shear strain.
'lim2
alongalong
tACnAB
nt
=
90-shear strain
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Shear Strain
Cubic element subjected to a shear stress will deform into a rhomboid. The
corresponding shearstrain is quantified in terms of the change in angle between
the sides
A
B
Note: Normal Strain tends to change the volume or Sizeof a material
While Shear Strain tends to change the Shapeof a material
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EXAMPLE 2.2
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EXAMPLE 2.2 (CONTINUED)
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EXAMPLE 2.2 (CONTINUED)
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Example 2.3The plate is deformed into the dashed shape. If, in this deformed shape, horizontal lines on
the plate remain horizontal and do not change their length, determine (a) the averagenormal strain along the sideAB, and (b) the average shear strain in the plate relative to the
x and y axes.
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Part (a)
LineAB, coincident with the y axis, becomes line after deformation, thus the length ofthis line is
The average normal strain forAB is therefore
The negative sign indicates the strain causes a contraction ofAB.
Solution:
( ) mm018.24832250' 22 =+=AB
( ) ( ) (Ans)mm/mm1093.7250
250018.248' 3=
=
=AB
ABABavgAB
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Part (b)
As noted, the once 90angle BAC between the sides of the plate, referenced from thex, y axes, changes to due to the displacement of B to B.
Since then is the angle shown in the figure. Thus,
Solution:
'2 =
xy xy
(Ans)rad0121.02250
3tan 1 =
= xy
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Example 2.4
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Solution Example 2.4
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CONCEPT QUIZ
1) The rectangular membrane has anunstretched length L1 and width L2. If the
sides are increased by small amounts L1
and L2, determine the normal strain along
the diagonal AB.
( )( )2221
2
2
2
1
21
21
2
2
2
1
2
2
2
1
2
2
1
1
D)C)
B)A)
LL
LL
LL
LL
LL
LL
L
L
L
L
+
+
+
+
+
++
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CONCEPT QUIZ (cont)
2) The rectangular plate is subjected to thedeformation shown by the dashed line.
Determine the average shear strain xyof the
plate.
( )22
200150
3D)
200
150C)
1200
17B)
150
3A)
+