Lecture 4a - 2@Page

7/31/2019 Lecture 4a - 2@Page

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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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7

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;


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



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9

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?



St ate of st ress in on e dimension


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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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Mohr s circle I

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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-


20

Then: , , are inscribed in the half-circlewhose diameter is

=

cosdA dT n

=cos

dA dN n

dA dF 1=

TdFd NdFd



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21

Dividing all the segments (forces) by theinfinitesimal area dA and projecting on the verticaland horizontal direction

ndA cosdT =

ndA cosdN =

1dA dF

=


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:


Fdthe force has been decomposed in twoorthogonal components and

A right triangle is inscribed in a half-circleTdNd


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St ate of str ess in t w o dim ensions

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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


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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



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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


28

Force equilibrium of an infinitesimally smallprismatic body in plane (1,2) :

22dA

11dA 11dA

n

22dA

11dA

n

22dA

1

2



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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


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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



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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


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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


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.



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



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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


1

n A+B

n A+B

2

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If the angle is clockwise:

n A+B

1

2

n



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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.


40

1

| |

2

n

n



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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)


n

1


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2

3 2 1

n( 2)

n

n( 2) 2 n

12

3



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.



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47

2

11

3 3

2

232 2

31

221

2

3

3 n


Lecture 4a - 2@Page

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