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Machine Design Lecture 4A: direct construtcion of Mohrs circle.
Page 1 2011 Politecnico di Torino
Machin e Design
Unit 2 Lecture 1St ate of st ress and str ain
2
St ate of st ress and str ain
State of stress in one and two dimensions Algebraic definition of the state of stressMohrs circles - advanced
Bi-vectorial relationship and quadratic formState of strainRelationship between stress and strain
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St at e of str ess and str ain
St ate of st ress in one and t w o dimensions
4
St ate of str ess in one and t w o dimensions
Normal stress on a surface at a pointOne-dimensional (uniaxial) state of stressMohrs circle I
Two-dimensional (plane) state of stressMohrs circle IIMohrs circles in three-dimensional state of stress
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St at e of str ess in one and tw o dimensions
Normal st ress on a surface at a point
6
Normal st ress on a surf ace at a point ( 1/ 4)
In the tensile test the average normal stress hasbeen defined as the ratio of the tensile force actingalong the axis of the specimen to the area of the
cross section whose normal unit vector is parallel tothe specimen axis (right cross-section).
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A specimen (prismatic or cylindrical) may be seen asa sheaf of specimens with infinitesimal cross-section
They all elongate in the same way;They do not transmit to each other any force throughthe lateral surfaces;
Normal st ress on a surf ace at a point ( 2/ 4)
8
Because of the uniform stress on the right cross-section we can deal with either the wholespecimen - A is the area of its cross section - orthe infinitesimal cross-section specimen withinfinitesimally small area dA.
dF dF
F F
dA
A
Normal st ress on a surf ace at a point ( 3/ 4)
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Machine Design Lecture 4A: direct construtcion of Mohrs circle.
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Then forces and stresses on the right cross-sectionare knownEven if this is a simple case it could be useful toanswer the following question:
w hat are th e forces and t he stresses on aninclin ed face?namely, w hat is the state of stress on th e
inclin ed face?
Normal st ress on a surf ace at a point ( 4/ 4)
St at e of str ess in one and tw o dimensions
St ate of st ress in on e dimension
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Machine Design Lecture 4A: direct construtcion of Mohrs circle.
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13
1
2
3
St ate of st ress in one dimension (3/ 6)
Fd
n
Fd
To restore equilibrium we must replace thediscarded portion by the internal forces it hadexerted on the kept portion.
14
1
2
3
St ate of st ress in one dimension ( 4/ 6)
Fd
Fd
nt
The force dF has two components: a normalcomponent, aligned with the cut-plane normal n,and a tangential component, lying on the cutplane, t-direction.
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St ate of st ress in one dimension (5 / 6)
The force equilibrium is given by the triangle of forces
nt NdTd
Fd
16
n= normal stress on the inclined area n = shear stress on the inclined area 1= normal stress on the area of the r ight cross-
section
St ate of st ress in one dimension (6 / 6)
=
cosdA dT n
=cos
dA dN n
dA dF 1=
dA
cosdA
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Machine Design Lecture 4A: direct construtcion of Mohrs circle.
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St at e of str ess in one and tw o dimensions
Mohr s circle I
18
Any right triangle is inscribed in a half-circlewhose diameter is the hypotenuse of thetriangle.
Mohr s circle I (1/ 5)
O A
C
B90-
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Given: ACB with right angle at CTaking the point O on AB such that:
OCB = ABC = , then OCB is isosceles OC = OB
moreover: ACO = ACB-OCB=90- CAB,then OCA is isosceles OC = OA It follows that OC = OA = OB: radius of the circle withcenter O
O A
C
B90-
Mohr s circle I (2/ 5)
20
Then: , , are inscribed in the half-circlewhose diameter is
=
cosdA dT n
=cos
dA dN n
dA dF 1=
TdFd NdFd
Mohr s circle I (3/ 5)
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Dividing all the segments (forces) by theinfinitesimal area dA and projecting on the verticaland horizontal direction
ndA cosdT =
ndA cosdN =
1dA dF
=
Mohr s circle I (4/ 5)
22
With this geometrical representation the stresscomponents n and n on the inclined surface, for anyangle , are determined once the principal stress 1 isgiven
The geometrical representation holds only for one-dimensional state of stress and it is possible because:
Mohr s circle I (5/ 5)
Fdthe force has been decomposed in twoorthogonal components and
A right triangle is inscribed in a half-circleTdNd
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St at e of str ess in one and tw o dimensions
St ate of str ess in t w o dim ensions
24
Reference cross-section:it s normal n outw ard theinfinit esimal prism ;Moreover the areas are:
M L
L
N
N
1
St ate of st ress in t w o dimensions (1/ 10)
Two-dimensional state of stress
n
dA n= NLLL dA 1= NMLL dA 2= MLLL
_ _ _ _ _ _
2
Preliminary remark 1:
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Value of the projectedareas:
M L
N 2
1
dA 1
Preliminary remark 2:n
_ _
dA n= NLLL dA 1= NMLL= NL cos LL dA 2= MLLL= NL sin LL
_ _ _ _
_ _ _ _
We think of the areas andtheir projections as vectorsand vector components
dA n
dA 2
St ate of st ress in t w o dimensions (2/ 10)
26
Preliminary remark 3:Assumption (it will be verified later) any state of stress is equivalent to three normal stress 1, 2 and 3 acting on three orthogonal surface dA 1, dA 2 anddA 3
This assumption also holds for the one-dimensionalcase where the two surfaces dA 2 and dA 3 are stressfree
If this assumption will not hold all our developmentfalls down
St ate of st ress in t w o dimensions (3/ 10)
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Within the Preliminary remark 3 the force 3dA 3,orthogonal to the plane (1,2), has no componentalong directions 1 and 2.
ds 1
ds 2
22dA
11dA
12
33dA
ds 3
3
St ate of st ress in t w o dimensions (4/ 10)
28
Force equilibrium of an infinitesimally smallprismatic body in plane (1,2) :
22dA
11dA 11dA
n
22dA
11dA
n
22dA
1
2
St ate of st ress in t w o dimensions (5/ 10)
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The two-dimensional state of stress, from now on (A+B),can be seen as the superimposition of the one-dimensional state of stress A and of the hydrostatic state of stress B
A One-dimensional
(Already developed)
Bhydrostatic
(A+B)
2 2
1 ( )21 2
0
St ate of st ress in t w o dimensions (6/ 10)
30
dA n NLdA 1 NMdA 2 ML
_
_
_
2 1 dA
n
22dA
M L
N
St ate of str ess B relationships among areasand equilibriu m
22dA
2 1 dA
St ate of st ress in t w o dimensions (7/ 10)
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31MNL similar to PQR
2dA 1:NM = 2dA 2:ML
dA 1:NM = dA 2:ML
R
Q P
St ate of stress B relationships areas/ forces
PQ :NM = PR :ML
n
M L
N
22dA
2 1 dA
St ate of st ress in t w o dimensions (8/ 10)
32
n
1
nn
12
n
1
dA dA
dA dA
dA dA
NLNM
QR PQ
=
==
St ate of st ress B hydrost atic st ate of st ress
As QR is parallel to n, thestress acting on dA n isperpendicular to thesurface;The equality n = 2 hold;
2n =
R
Q P
nn dA
St ate of st ress in t w o dimensions (9/ 10)
Then on any inclined cross-section the normal stress n equals 2 and the shearstress n is zero
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Closure:The state of stress A is one-dimensional andgives the Mohrs circle developed in the previoussection.
The state of stress B is a hydrostatic state of stress that is for any surface, independently onthe direction of its normal, 1) the shear stress is
nil and 2) the normal stress has the same value.In the next section the original two-dimensionalstate of stress A + B will be join together againand the related Mohrs circle sketched.
St ate of st ress in t w o dimensions (10/ 10)
St at e of str ess in one and tw o dimensions
Mohrs circle I I
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Case A one-dim ensional st ate of str ess
n A
n A
( 1- 2) A
Mohr s ci rcle I I (1 / 6)
n
n
36
nB = 2
Case B hydrostatic stat e of stress
nB =0
n
n
Mohr s ci rcle I I (2 / 6)
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Superposit ion A+ B
n A+B = n A + nB = n A + 2 n A+B = n A +0
1 dA 1
2 dA 2
n
Mohr s ci rcle I I (3 / 6)
1
n A+B
n A+B
2
38
If the angle is clockwise:
n A+B
1
2
n
Mohr s ci rcle I I (4 / 6)
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As to asses the strength of the material issufficient to known the modulus | n | a nd not thedirection of the shear stress.Then we use only the upper half-circle: we takethe direction through a counterclockwise angle| | and determine | n | on the ordinate.
Mohr s ci rcle I I (5 / 6)
40
1
| |
2
n
n
Mohr s ci rcle I I (6 / 6)
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St at e of str ess in one and tw o dimensions
Mohrs circle in t hree dim ensions
42
Mohrs circles in t hree dimensions ( 1/ 6)
ds 1
ds2
22dA
11dA
12
33dA
ds 3
3
If the three principal direction are given (theprincipal directions are orthogonal), for example
3= 0,
1>
2
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3 2 1 n( 3)
n
n( 3) n 3
3
12
3
Stress on planes whose normal lies on coordinateprincipal plane (1;2), that is rotation 3 is aboutprincipal direction 3
Mohrs circles in t hree dimensions ( 2/ 6)
n
3
44
1
3 2 1
n( 1)
n
n( 1) 1 n
1
2
3
Mohr s circles in t hree dimensions (3/ 6)
Stress on planes whose normal lies on coordinateprincipal plane (2;3), that is rotation 1 is aboutprincipal direction 1
n
1
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2
3 2 1
n( 2)
n
n( 2) 2 n
12
3
Mohr s circles in t hree dimensions (4/ 6)
Stress on planes whose normal lies on coordinateprincipal plane (1;3), that is rotation 2 is aboutprincipal direction 2
2
n
46
The Mohrs circle is a useful graphical tool todetermine the normal n and shear n stresses,both in magnitude and direction, at a point on a
surface for a given normal unit vector n.We developed the direct construction that isthe normal n and shear n stresses aredeterminate once the principal stresses 1, 2and 3 and the principal directions are given.
Mohr s circles in t hree dimensions (5/ 6)
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2
11
3 3
2
232 2
31
221
2
3
3 n
Mohr s circles in t hree dimensions (6/ 6)