Chapter 1-Strain Analysis
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Transcript of Chapter 1-Strain Analysis
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CHAPTER 1(week 1-3)Strain Analysis
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Plain Strain
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GENERAL EQUATION OF PLANE STRAINTRANSFORMATION
Normal strain ex and e y arepositive if cause elongationalong x and y axisShear strain gxy is positiveif the interior angle AOBbecome smaller than 90 0.q0 will be positive using theright-hand fingers.i.e counterclockwise
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Normal and Shear Strains
In Fig a :
Positive ex occurline d
x elongated e
xd x , which causeline dx toelongated ex d x cos
q.
q
q
sin
cos'
'
dxdy
dxdx
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Positive e y occur line d
y elongated e
yd y , which cause line d y to elongated
e y d y sin q. Assuming dx fix in position. Shear strain g xy changes in angle
between dx and dy. dy displaced g xy dy to the right.
Dx elongateg xy dy cos q
Normal and Shear Strains (cont.)
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Adding all the elongations
qqgqeqee
qqgqqeqqee
e
cossinsincosdx
cos)sindx(sin)sindx(cos)cosdx(dx
x
xy2
y2
xx
'
'xy
'y
'x
x
'
'
x
'
'
'
qgqeqe cosdysindycosdx xyyxx '
Normal and Shear Strains (cont.)
q q
sincos
'
'
dxdydxdx
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qqgqeqee cossinsincos x y2y2xx '
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Normal and Shear Strains (cont.)
qg
qeeee
e 2sin2
2cos22
xyyxyx
x '
qg
qeeee
e 2sin2
2cos22
x yyxyx
y '
Using trigonometric identities:
qg
q
eeg
2cos2
2sin22
xyyxyx ''
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Principal Strains
Direction axis of principlestrain:
Max in Plane shear strain
Ave shear strain
Direction axis of shearstrain
2
xy
2
yx planeinmax,
222
g
eeg
2
yx
ave
eee
)(2tan
yx
xy p ee
gq
g
eeqxy
yxs2tan
2
xy
2
yxyx2,1 222
g
eeeee
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Construction of the Mohrs Circle PROCEDURE FOR ANALYSIS
The procedure for drawing Mohr's circle for strainfollows the one established for stress.Construction of the Circle. Establish a coordinate system such that the
abscissa represents the normal strain e, withpositive to the right, and the ordinate representshalf the value of the shear strain, g/2, with positivedownward .
Using the positive sign convention for ex, e y, gxy asshown in the Fig. , determine the center of the circleC, which is located on the e axis at a distancee avg= (e x + e y)/2 from the origin.
Plot the reference point A having coordinatesA(e x, g xy/2). This point represents the case forwhich the x' axis coincides with the x axis. Henceq = q .
Connect point A with the center C of the circle andfrom the shaded triangle determine the radius R ofthe circle.
Once R has been determined, sketch the circle.
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The principal strains e1 and e2 are determinedfrom the circle as the coordinates of points Band D, that is where g/2 = 0, Fig. a.
The orientation of the plane on which e1 acts canbe determined from the circle by calculating 2 qP1,using trigonometry. Here this angle is measuredcounterclockwise from the radial reference lineCA to line CB, Fig. a. Remember that the rotationof qP1 must be in this same direction, from the
element's reference axis x to the x' axis, Fig. b.* When e1 and e2 are indicated as being positive as
in Fig. a, the element in Fig. b will elongate in thex' and y' directions as shown by the dashedoutline.
Principal Strain
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The average normal strain and half themaximum in-plane shear strain are determinedfrom the circle as the coordinates of points Eand F, Fig. a.
The orientation of the plane on which gmax in plane and eavg act can be determined from the circleby calculating 2 qs1, using trigonometry. Herethis angle is measured clockwise from the radialreference line CA to line CE, Fig. a. Rememberthat the rotation of qs1, must be in this same
direction, from the element's reference axis xto the x' axis, Fig. c.
Maximum In Plane Shear Strain
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The normal and shear strain components ex, and
g xy for a plane specified at an angle q, Fig. d, canbe obtained from the circle using trigonometry todetermine the coordinates of point P, Fig.a.
To locate P, the known angle q of the x' axis ismeasured on the circle as 2 q. This measurement ismade from the radial reference line CA to theradial line CP. Remember that measurements for2q on the circle must be in the same direction as q for the x' axis!
If the value of e y, is required, it can bedetermined by calculating the e coordinate ofpoint Q in Fig. a. The line CQ lies 180 o away from
CP and thus represents a rotation of 900
of the x'axis.
Strains on Arbitrary Plane
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We can not measure stresses within a structuralmember,
Instead we can measure strains and from them
the stresses can be computed Even so, we can only measure strains on the
surface Also in view of the very small changes in
dimensions, it is difficult to achieve accuracy inthe measurements
Strain Gauge and Rosette
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In practice, electrical strain gage provide a moreaccurate and convenient method of measuring strains.
A typical strain gauge is shown below. The gage shown can measure normal
strain in the local plane of the surface inthe direction of line PQ, which is parallelto the folds of paper.
This strain is an average value of forthe region covered by the gage, ratherthan a value at any particular point.
Strain Gauge and Rosette (cont)
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Strain Gauge and Rosette (cont)
The strain gauge is not sensitive to normal strain in the direction perpendicular to
PQ, nor does it respond to shear strain.
Therefore, in order to determine the state of strain ata particular small region of the surface, we need morethan one strain gage.
To define a general two dimensional state of strain, weneed to have three pieces of information, such as
ex , e y and gxy We therefore need to obtain measurements from threestrain gages.
These three gages must be arranged at differentorientations on the surface to from a strain rossette.
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Strain Gauge and Rosette (cont)
A group of three gages arranged in a particular fashion iscalled a strain rosette. Rosette is mounted on the surface of the body, where the
material is in plane stress, therefore, the transformationequations for plane strain to calculate the strains in variousdirections.
The axes of the three gauges are arranged at the angles of qa,qb, qc.
If the reading of ea, eb, ec taken, ex, e y, gxy can be defined. Value of ex, e y, gxy are determined by solving these equations.
ccxyc2
yc2
xc
b bxy b2
y b2
x b
aaxya2
ya2
xa
cossinsincos
cossinsincos
cossinsincos
qqgqeqeeqqgqeqeeqqgqeqee
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45 o or Rectangular Rosette
0
0
0
90
45
0
c
b
a
q
q
q
The equation become:
ca bxycy
ax
2 eeegeeee
Example of 45 o strain rosette
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60 0 Strain Rosette
c bxy
ac by
ax
32
2231
eeg
eeee
ee
0c
0 b
0a
120
60
0
q
q
q
The equation become:
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Example
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Exercise
Figure shows 60 strain rosette attached on the mechanicalcomponent to measure surface strains. The reading of thestrains measured by this gauge is as follows:
a = 1000,
b = 750,c = -650Determine:(i) the principal strains,
(ii) the maximum shearing strain and(iii) the principal angles
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Solution Applying Transformation equation
Solving the above equations , we get ex=1000 , e y=-266.7 and gxy=-1616.6
ccxyc2
yc2
xc
b bxy b2
y b2
x b
aaxya2
ya2
xa
cossinsincos
cossinsincos
cossinsincos
qqgqeqeeqqgqeqeeqqgqeqee
geegee
gee
60cos60sin60sin60cos650
120cos120sin120sin120cos750
0cos0sin0sin0cos1000
xy2
y2
x
xy2
y2
x
xy2
y2
x
l ( d )
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Solution (Contd.) Applying the principal strain equation or using Mohrs strain
circle, we get
e1=1394 and e2-660 and gxy=-1616.6 Max Shear Strain is
= 2050
Direction of principal planes
i.e., q1 =-25.9 or 64.1 andq2 =154.1(bcas 2 q2=180+2q1)
2xy
2yxyx
2,1 222
g
eeeee
2xy
2yxmax
222
g
eeg
)(2tan
yx
xy p ee
gq
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Stress Strain Relationship If a material subject to triaxial
stress ( s x, s y, s z), associatednormal stress( ex, e y, ez)developed in the material.
When s x is applied in x-direction, the element elongatedwith ex in x direction.
Application on s y cause theelement to contract with astrain e x in the x direction.
Application Of s z cause the
element to contract with astrain ex in the x direction. z
x
y x
x x
s e
s e
s e
'''
''
'
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The same result can bedeveloped for the normalstrain in the y and zdirection.
Final results can bewritten as..
yxzz
zxyy
zyxx
1
1
1
ssse
ssse
ssse
Stress Strain Relationship (cont.)
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Applying only shear stress, y to the element.If to apply shear stress, t y to the element.t xy will only cause deformation to gxy.t xy will not cause deformation to g yz.and gxzt yz and t xz will only cause deformation to
g yz and gxz respectively. Hooke Law for shear stress and shearstrain written as:
xz xz
yz yz
xy xy
G
G
G
t g
t g
t g
1
1
1
Element subjected to normalstresses only Shear stress applied to theelements
Stress Strain Relationship (cont.)
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Modulus of elasticity, E isrelated to shear modulus, G.
Dilatation (the change involume per unit volume orvolumetric strain,e .
Bulk Modulus (volumemodulus of elasticity), k .
12 E
G
zyxE21 sssue
Stress Strain Relationship (cont.)
213 E
k
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