1 - 2 BE-335-MOM-II-Week-1

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

1/33

College of

Electrical and MechanicalEngineering

BE 335 Mechanics of Materials -IIWeek 1-2

Plane Strain and Strain Transformation

Dr. Rizwan Saeed [email protected]

Dept of Mechanical Engineering,NUST, College of E&ME, Rawalpindi,

Pakistan
mailto:[email protected]:[email protected]:[email protected]

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

2/33

College of


Text books and Resources Main text book: Introduction to Solid Mechanics 3rd

Edition (Irwing H. Shames and James M. Pitarresi)

Engineering Mechanics of Solids (Egor P. Popov)

Mechanics of Materials (P. Beer, E. Russell SI

Edition) Mechanics of Materials: An Integrated Learning

System (Timothy A . Philpot)

Applied Strength of Materials 5th Edition (Robert L.

Mott) MIT OPENCOURSEWARE mechanical-

engineering/2.002

Lecture Notes Sussex University UK and The

University of Manchester UK

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

3/33

College of


About the Slide Notes Slide notes are not a substitute of lecturesand you

mustmake your own notes during lectures as manydetailed problems, derivations etc., will be covered on theboard.

Slide notes will usually have contents combined from thefollowing sources

Personal notes

Text books and reference books mentioned earlier

MecMovie Website (http://web.mst.edu/~mecmovie/) Lecture Notes of J. Walt Oler (Texas Tech University) for the

book Mechanics of Materials 3rd Edition Beer-Johnston-Dewolf, available athttp://www.mhhe.com/engcs/engmech/beerjohnston/mom/le

ctureppt.mhtml

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

4/33

College of


Grading , Procedure and schedules

Midterm 25%Course work (assignments, quizzes, project andclass participation (Homework and attendance)

35%

Final 40%

Rememberthat besides the 3 hours of weekly lectures

at least 6 hours of self study per week are required

Assignmentswill be due within 1 week unless specifiedotherwise.

The usual procedure for submission of assignments willbe same as for last semester.

Assignments classed as Homework will not be gradeddirectly but you should expect quizzes related toHomework

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

5/33

College of


Some general points Besides presentations/lectures I will try to have more of: Activity based learning

Come prepared for topic to be covered in next class

In the class we will have group discussion and problem

solving related to that topic The group discussion and problem solving may be graded

and will be counted towards your quizzes, assignments andclass participation.

For meetings longer than 5to10 minutes appointments

should be pre-booked (preferably through email) theschedule of available slots is displayed outside my officeand also available on the websitehttps://docs.google.com/document/d/1WgoFGxJ3aMZ-uiEtbLUdcyOYxsPkH4wOp5CAZI0BBAQ/edit?hl=en_US
https://docs.google.com/document/d/1WgoFGxJ3aMZ-uiEtbLUdcyOYxsPkH4wOp5CAZI0BBAQ/edit?hl=en_UShttps://docs.google.com/document/d/1WgoFGxJ3aMZ-uiEtbLUdcyOYxsPkH4wOp5CAZI0BBAQ/edit?hl=en_UShttps://docs.google.com/document/d/1WgoFGxJ3aMZ-uiEtbLUdcyOYxsPkH4wOp5CAZI0BBAQ/edit?hl=en_UShttps://docs.google.com/document/d/1WgoFGxJ3aMZ-uiEtbLUdcyOYxsPkH4wOp5CAZI0BBAQ/edit?hl=en_UShttps://docs.google.com/document/d/1WgoFGxJ3aMZ-uiEtbLUdcyOYxsPkH4wOp5CAZI0BBAQ/edit?hl=en_US

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

6/33

College of


4/22/2012

Topics Chap (I.HShames)

Week (Starting Monday)

Plane Strain and StrainTransformation

8 W1 -2 (12th Sep 19th Sep)

Failure Criteria 9 W2 (19th Sep)

DelPHE visit to UK W3-4 (26th Sep - 3rd Oct) classes to be rescheduled : 2 extraclasses in each of W 5, 6 and 7

Torsion 14 W5-6 (10th Oct -17th Oct)

Beams I 10-11 W 7-8 (24h Oct -31st Oct)

Midterm exam and Eid holidays W9-10 (7th Nov 14h Nov)

Beams II- III

IV 11-12-13 W11-15 (21

st

28

th

Nov

5

th

12

th

19th Dec)

Column Buckling and ElasticStability

17 W 16-17 (26th Dec 2nd Jan )

Final Exam W18 (9th Jan)

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

7/33

College of


Revisiting Solid Mechanics - I Stress and Strain

Equilibrium, compatibility and constitutive equations.

Equilibrium equations in differential form and dynamicequilibrium cases

Enforcing compatibility (displacement continuity and boundary

conditions) Materials Properties and Constitutive Equations.

The one dimensional problems Elastic response

Isotropic plasticity, residual strains and thermal strains for one

dimensional problems. Three dimensional stress state and stress strain relations

Introduction to Energy Methods and impact analysis

Stress transformation and Mohrs Circle for stress

Plane Stress and thin walled pressure vessels

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

8/33

College of


Plane Strain

The state of strain where the only permissible strains(non-zero) are those that lie within a plane i.e.

i.e. deformations of the material take place in parallelplanes and are the same in each of those planes.

In tensor notation :

0 zyzxzzxyy whilezero;nonarex

zzzyzx

yzyyyx

xzxyxx

ij

000

0

0

yyyx

xyxx

ij

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

9/33

College of


Plane Strain The Plane Strain approximation require that,

The load is uniformly distributed along the outof plane axis of the body and acts perpendicularto it

The body is not allowed to bend stretch or

shorten.

Examples: Dams, tunnelsand thick walled pressure

vessels

Image source: http://en.wikipedia.org/wiki/Stress_analysis

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

10/33

College of


Plane Strain

Examples:A plate subjected along its edgesto a uniformly distributed load andrestrained from expanding or

contracting laterally by smooth,rigid and fixed supports [1]

Consider a long bar subjected touniformly distributed transverse

loads. State of plane stress existsin any transverse section notlocated too close to the ends of thebar [1]

[1] Lecture Notes of J. Walt Oler (Texas Tech University) for the book Mechanics of Materials3rd Edition Beer-Johnston-Dewolf,

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

11/33

College of


Plain Strain - Thick Walled

Cylinders Reading Assignment (Homework): Popov : Section 5-11

to 5-13: optional module 5-14

Remember: In MOM-I we derived the relationships forprincipal stresses thin walled pressure vessels that was

plane stress

t

prt

t= hoop stress

2 = longitudinalstress

tl

lt

pr

2

2

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

12/33

College of



Cylinders Consider a cross section cut from a long cylinder

constrained at its ends and subjected only to pressureloading Any such section away from the ends will be ina state of plane strain*

Objective: To find the in-plane stress distribution

* Please see slide notes for a more generaltreatment

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

13/33

College of



Cylinders

At a distance rfrom the centre the normal radial stressacting on an element is r

At a distance r +drfrom the centre is

Hoop stress tvaries with rand is equal and opposite on

the two faces

drdr

d rr

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

14/33

College of



Cylinders

Equilibrium:Considering our element of unit depth (i.e. along

longitudinal axis) and making use of small angleapproximation for tangential stress

Simplifying and neglecting higher ordered terms

rdr 022 ddrt ddrrdrdrd

rr

0 dr

dr

rrt

1).........(..........0

rdr

d trr

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

15/33

College of



Cylinders Compatibility:

Requires that the displacement uof a point due to the

radial and tangential stresses is unique

Displacement at ris u

Displacement at r + dris

Hence

drdr

duu

(2)....

dr

du

dr

udrdr

duu

r

(3)....r

u

r

rurt

2

2)(2

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

16/33

College of



Cylinders

Constitutive Equations

For Plane Strain

Using this result in (4) and (5)

(4)..........)(1 xtrrE

(5)...........)(1

xrttE

(6)..........)(1 trxxE

0x (7)..........)( trx

(9)..........

(8)..........

trt

trr

E

E

)1(

)21)(1(

)1()21)(1(

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

17/33

College of


Constitutive relation for Plane

Strain

rt

t

r

rt

t

rE

.

)1(2)21(

00

01)1(

0)1(

1

)21)(1(

)1(

(10)..........rtrtrtE

G

)1(2

Taking note of the fact that

From eq. (8), (9) and (10) constitutive relation for ElasticPlane Strain can be expressed as

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

18/33

College of



Cylinders Formulation of differential Equation

Using Equations (2) and (3) in equations (8) and (9)

Back substituting in Equations (1)

(12)..........

(11)..........

r

u

dr

duEr

u

dr

duE

t

r

)1()21)(1(

)1(

)21)(1(

13).........(..........01

22

2

r

u

dr

du

rdr

ud

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

19/33

College of


Thick Walled Cylinders - Solution ofODE

The general solution for this equation which gives theradial displacement uof any point on the cylinder cross-section is given by,

A1 and A2 from boundary conditions

-ve compression

14).........(..........

drduand

r

22

1

21

rAA

r

AAu

15).........(..........-)(

-)(

oo

ii

pr

pr

r

r

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

20/33

College of


Thick Walled Cylinders - Solution ofODE

Using (14) and (15) in (11)

solving simultaneously

16).........(..........

22

22

2

22

22

1

1

211

io

oioi

io

ooii

rr

rrpp

E

A

rrrprp

E)(A

2

212

21

2

212

21

)1()21)(1(

)1()21)(1(

oo

o

ii

i

r

ArAr

AAEp

r

ArA

r

AA

Ep

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

21/33

College of



Cylinders Stress Distribution With equations (16) and (14) the required stress

distribution (i.e. Equation (11) and (12)) may be found.This is given as,

Where

(17)..........-

2

21

2

2

1

r

CC

r

C

C

t

r

18).........(..........

22

22

2

22

22

1

io

oioi

io

ooii

rr

rrppC

rr

rprpC

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

22/33

College of


Discussion of Results Example 5.6 Popov Sec-5-13 (old edition Sec-3-

13)

Make a comparison of the tangential stressdistribution caused by the pressure pias given by

the equation (17) (Lames formula) with thedistribution obtained in the plane stress case where

Po= 0 , Pi 0, when

a) ro

= 1.1ri

b) ro = 4ri

For Analytical Solution see book sec-5-13

Click to open MSExcel sheet with solution (Note: pi=

10MPa)
http://1%20-%202%20week/Example%205-6%20(3-6).pnghttp://mecahanics%20of%20materials.xlsx/http://mecahanics%20of%20materials.xlsx/http://mecahanics%20of%20materials.xlsx/http://mecahanics%20of%20materials.xlsx/http://mecahanics%20of%20materials.xlsx/http://1%20-%202%20week/Example%205-6%20(3-6).png

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

23/33

College of


Plane Strain In general does a Plane Stress situation equate to a

Plane Strain situation ???

No it is not ... as also clear from the previous example

4/22/2012

)(1

zzyyxxxx E

)(1 zzxxyyyyE

)(1 yyxxzzzzE

Gxyxy2

G

xzxz

2

G

yz

yz2

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

24/33

College of


Plane Strain vs. Plane Stress

zz

yyyx

xyxx

ij

yyyx

xyxx

ij

00

0

0

000

0

0

PlaneStrain

Plane Stress

zz

yyyx

xyxx

ij

yyyx

xyxx

ij

00

0

0

000

0

0

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

25/33

College of


Strain transformation Remember the Stress Camera Virtual experiment : (MOM-Ilecture notes week 14)

Just like the stresses the strains also vary with the orientationof element under consideration.

Assignment 1a: Submission Deadline 8th October

Note there are three parts of this assignment and youshould submit all parts together.

Prove that the form of transformation equations of planestress and plane strain is similar and for isotropicHookean materials the principal planes and plane ofmaximum shear are same for stress and strain.

(hint you may use an analytical formulation based on Taylorseries expansion of displacement field and directionalderivatives or you may use geometric relations directly using agraphical method)

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

26/33

College of


Strain transformation for plane

strain

2cos2sin2

2sin2cos

22

2sin2cos22

''

''

''

xy

xxyy

yx

xy

yyxxyyxx

yy

xy

yyxxyyxx

xx

2cos2sin2

2sin2cos22

xy

xxyy

ns

xy

yyxxyyxx

nn

x

yy

x

o

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

27/33

College of


Example 1 Strain transformation

MecMovie Example 13.1
http://1%20week/m13_01_xpl_one.swfhttp://1%20week/m13_01_xpl_one.swfhttp://1%20week/m13_01_xpl_one.swfhttp://1%20week/m13_01_xpl_one.swfhttp://1%20week/m13_01_xpl_one.swfhttp://1%20week/m13_01_xpl_one.swfhttp://1%20week/m13_01_xpl_one.swf

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

28/33

College of


Principal Strain Angle of Principal Axis

Principal Strains

Example 8-1 on the white board

Home work Example 8-2 (Shames): Construction ofMohrs Circle for plane stain please note you will usethe same procedure for Mohrs circle as you did for planestress in last semester.

yyxx

xy

2~2tan

22

,22

xy

yyxxyyxx

ba

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

29/33

College of


Strain Gages and Rosettes

Rectangular rosette

Delta rosette

Stacked delta rosette
http://1%20week/m13_05_rosette.swfhttp://1%20week/m13_05_rosette.swfhttp://1%20week/m13_05_rosette.swfhttp://1%20week/m13_05_rosette.swfhttp://1%20week/m13_05_rosette.swfhttp://1%20week/m13_05_rosette.swf

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

30/33

College of


Strain measurement with rossettes Strain Measurement with Rosettes: Click to open

example from MecMovie

Reading Assignments (Homework)Strain Gage Selection Criteria Case Study

Pressure Vessels (Page 29 of the document StrainGage Selection and measurement.pdf )

Measurement of principal stress using Rosettes:Click to open example from MecMovie
http://1%20-%203%20week/m13_05_rosette.swfhttp://1%20-%203%20week/m13_05_rosette.swfhttp://1%20-%203%20week/m13_06_genl_rosette.swfhttp://1%20-%203%20week/m13_06_genl_rosette.swfhttp://1%20-%203%20week/m13_06_genl_rosette.swfhttp://1%20-%203%20week/m13_06_genl_rosette.swfhttp://1%20-%203%20week/m13_06_genl_rosette.swfhttp://1%20-%203%20week/m13_05_rosette.swfhttp://1%20-%203%20week/m13_05_rosette.swfhttp://1%20-%203%20week/m13_05_rosette.swf

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

31/33

College of


Strain Gages for Principal Stresses Assignment: 1b As part of the product development process of for a new

pump, the designer has instrumented a critical area ofthe housing with a rectangular strain gage rosette (likethe one shown). Gage 1 is aligned with the horizontalcentreline of the pump inlet passage. During the testunder high capacity operating conditions the followingreadings were taken for the three arms of the gage:1 = 950; 2 = -375; 3 = 525.The material for pump housing is aluminium2014-T6. Calculate themaximum principal stress, the minimumprincipal stress and the maximum shear stressat the location where rosette is mounted.

1 2 3

Rectangular rosette

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

32/33

College of


Strain Gages for Principal Stresses Assignment 1c:

Create a spreadsheet solution that analyses the data fromstrain gage rosette and has the following features

Assumes a biaxial stress state and that the stressperpendicular to the plane of the measured strain is zero

Finds out Principal Strains and Principal Stresses Can calculate the solution for both the rectangular and delta

rosettes

Allows the user to differentiate between max in plane shearstress and absolute max shear stress (also called true max

shear stress) Allows for input of material data E and as variables

You may create the spreadsheet solution in Microsoft Excel,Calc (OpenOffice.org) or Google Spreadsheet.

Use this spread sheet to solve the assignment 1b andcompare the results.

8/3/2019 1 - 2 BE-335-MOM-II-Week-1

33/33

College of

Electrical and Mechanical

Quiz

30 Mech A 30 Mech B
http://quiz/Quiz%201%20-%2030MA-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MB-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MB-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MB-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MB-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MB-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MA-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MA-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MA-01-10-2010.docxhttp://quiz/Quiz%201%20-%2030MA-01-10-2010.docx

1 - 2 BE-335-MOM-II-Week-1

Documents

Transcript of 1 - 2 BE-335-MOM-II-Week-1