ADVANCED MECHANICS

OF MATERIALS

Dr. KAMAL KUMAR B.Sc. Engg., M.E., Ph.D.

Professor in Mechanical Engineering

Gout. Engineering College, Jabalpur

Dr. R.e. GHAI B.E. (Hons.), M. Tech., Ph.D.

Professor in Mechanical Engineering Gout. Engineering College, Sagar

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PREFACE

The engineering curriculum in our country, in general, has advanced considerably in the last decade. Theory of elasticity once considered to the an advanced subject is now being taught at the undergraduate level at many institutions and universities in India.

Non-availability of a book suitable for our undergraduate and postgraduate students in this particular subject motivated the authors, with a long teaching experience, to write this book "Advanced Mechanics of Materials."

The first five chapters deal with Theories of Failure, Springs, Unsymmetrical Bending, Shear Centre and Curved Beams. Sixth and seventh chapters deals with Theory of Elasticity in Cartesian and Polar Coordinates respectively and the last chapter has been devoted to Bending ufAxi-symmetrical Plates.

At the end of each chapter several solved examples have been given to make the fundamentals properly understood by the students. Besides this, sufficient number of unsolved problems have also been included.

This edition of Advanced Mechanics of Materials adheres to the basic plan of earlier editions but with a considerable broadening of the chapter on Elastic Strain Energy and Energy Methods. As in the earlier editions, the subject matter is divided into chapters covering duly recognized areas of study. Theory, derivation of formulas and proofs of theorems are dealt in the beginning of each chapter followed by solution of problems and unsolved problems.

It is hoped that this book will prove to be of immense help to the students.

The authors thank Shri A.K. Jain and Shri Lamba for their help in the preparation of manuscript and to all those who have offered valuable suggestions and pointed out printing errors for the improvement of the book.

( iii )

KAMALKUMAR R.c. GHAI

Chapter

CONTENTS PART -I

Pages

1. Theories of Failure 1-19 1.1 Maximum Principal Stre55 Theory ........................................... 2 1.2 Maximum Shearing Stress Theory ............................................ 3 1.3 Maximum Strain Theory ............................................................. 5

1.4 Total Strain Energy Theory ........................................................ 6

1.S Maximum Distortion Energy Theory ...................................... 7

1.6 Octahedral Shearing Stress Theory ........................................ 10

1.7 Graphical Comparison of Theories of Failure ...................... 12

2. Elastic Strain Energy and Energy Methods 20-40 2.1 Elastic Strain Energy in a Uniaxial Stress System ............... 20

2.2 Elastic Strain Energy in Shear ................................................. 21 2.3 Strain Energy in Three Dimensional Stress System ............ 22 2.4 Castigliano's Theorem ............................................................... 25

2.5 Proof of Castigliano's Theorem ............................................... 26

2.6 Maxwell's Reciprocal Theorem ............................................... 27

2.7 Statically Indeterminate Beams ............................................... 29

3. Springs 41-79 3.1 Close Coiled Helical Springs ................................................... 41

3.2 Opel1 Coiled Helical Spring ..................................................... 44 3.3 Conical Spring .................................................................. , ......... 50

3.4 Flat Spiral Spring ....................................................................... 52

3.5 Leaf Springs ................................................................................. 53

4. Unsymmetrical Bending 80-107 4.1 Unsymmetrical and Symmetrical Bending ........................... 80 4.2 Stress at any Point in Cross-section ....................................... 82 4.3 Sign Convention ......................................................................... 83 4.4 Direction of Neutral Axis ......................................................... 83 4.5 Determination of Stress in Beams with

Unsymmetrical Sections ............................................................ 84 4.6 Formula for Stress Referred to Rectangular Axes

One of which is the Neutral Axes .......................................... 87 4.7 Deflection of Beam Subjected to

Unsymmetrical Bending ........................................................... 88

(V)

(vi)

Chapter Pages

5. Flexural Axis and Shear Centre 108-135 5.1 Shear Centre for Sections Symmetrical

About Both Axes ...................................................................... 109 5.2 Shear Centre for Sections Symmetrical

About Only One Axis .............................................................. 109 5.3 Location of Shear Centre for

Unsymmetrical Sections .......................................................... 128

6. Curved Beams 136-181 6.1 Bending of Beams with Small Initial Curvature ................ 136 6.2 Strain Energy of a Beam with

Small Initial Curvature ........................................................... 138 6.3 Deflection of Beams Having Small Initial Curvature ....... 139 6.4 Curved Beams with Large Initial Curvature ...................... 139 /l.5 Deflection of Curved Beams with

Large Initial Curvature ........................................................... 153 /l.6 Statically Indeterminate Curved beams .............................. 156

7. Thick Walled Cylinder 182-200 7.1 Lame's Theory of Thick-walled Cylinders .......................... 182 7.2 Special Cases ............................................................................ 185 7.3 Relation between Strain Components and

Radial Displacernents .............................................................. 186 7.4 Techniques for Effective Utilization of the

Material of Thick Cylinders ................................................... 187 7.5 External Loading by a Shrink

Fit-Compound Cylinders ....................................................... 187 7.6 Shrink Fit Assemblies .............................................................. 188 7.7 Shrink Fit Allowance ............................................................... 188

7. (A) Theory of Elasticity Problems in Cartesian Co-ordinates 201-249

7.1 Stress at a Point ........................................................................ 201 7.2 Notation for Stress ................................................................... 202 7.3 Sign· Convention of Stress ...................................................... 203 7.4 Differential Equations of Equilibrium ................................. 203 7.5 Strain Components .................................................................. 208 7.6 Compatibility Equations ........................................................ 210 7.7 Stress Function .......................................................................... 212

Chapter 7.8 7.9

7.10 7.11 7.12

( vii)

Pages St. Venant's Principle .............................................................. 220

Solution by Polynomials ......................................................... 221

Pure Bending of a Prismatic Bar ........................................... 224

Bending of a Narrow Cantilever Under an End Load ..... 229 Bending of a Simply Supported Narrow Beam by a Distributed Loading ............................................ 238

8. Problems in Polar Co-ordinates 250-297 8.1 Stress Equilibrium Equations in Polar Co-ordinates ........ 250 8.2 Strain Components in Polar Co-ordinates .......................... 253 8.3 Compatibility Equations in Cylindrical Co-ordinates ...... 255 8.4 Stress Function in Cylindrical Co-ordinates ....................... 256 8.5 Thick Cylinder Under Uniform Pressure ............................ 259 8.6 Stresses in Rotating Discs ....................................................... 263 8.7 Long Rotating Cylinder .......................................................... 267 8.8 Rotating Disc of Variable Thickness .................................... 269 8.9 Pure Bending of a Rectangular Section Curved

Beam by Stress Function ......................................................... 272 8.10 The Effect of Small Circular Hole in Standard Plate ........ 277 8.11 Semi Circular Beam Subjected to

End Shearing Forces ................................................................ 282

9. Bending ofAxi-Symmetrical Plates 298-316

9.1 General ....................................................................................... 298

9.2 Solid Circular Plate, Uniformly Loaded, Edge Freely Supported ........................................................... 303

9.3 Solid Circular Plate Uniformly Loaded and Edge Clamped .......................................................................... 305

9.4 Solid Circular Plate with Central Load P. Edge Freely Supported ....................................................... 308

10. Torsion of Non-circular Shafts 317-332 10.1 Saint Venant's Method ............................................................ 317 10.2 Stress-Function Method of Solution ..................................... 319 10.3 Boundary Conditions for Torsion Problems ....................... 320 10.4 Shaft of Elliptical Section ........................................................ 324 10.5 Membrane Analogy ................................................................. 326 10.6 Application of Membrane

Analogy-Circular Section Shafts ......................................... 329

CHAPTER 1 Theories of Failure

In practice, engineering materials have been observed to fail either by yielding or fracture. Yielding or permanent deformation is a pronounced sliding on planes through the crystalline grains of the material. It takes place without the actual rupture of the material. The functional utility for most machine parts is lost after a particular amount of yielding has taken place. Therefore, for all practical purposes yielding may be considered the criterion of failure for ductile materials. Fracture, on the other hand, is a failure in which separation occurs on a cross-section normal to the direction of tensile stress. The fracture criterion of failure is applicable to brittle materials. In practice, a limit of about 5 per cent elongation is usually taken as parting line between ductile and brittle materials.

For a machine part subjected to a uniaxial system of stress, the limiting allowable stress for design may be obtained from the mechanical testing of materials in simple tension. Usually the yield point stress is the deciding factor in such cases. But in majority of the cases, the parts are subjected to complex stress system and as such this simple approach is not applicable because the behaviour of material is greatly affected by the state of stress, type of loading, heat treatment process etc. Therefore, it is important to establish criterion for behaviour of materials under combined state of stress.

In a simple tensile test when specimen just starts yielding the following quantities are attained simultaneously:

(a) The principal stress a max reaches the yield point stress ayp or ault of the material.

2 ADVANCED MECHANICS OF MATERIALS

(b) The maximum shear stress [t = O"max] reaches the mm( 2

cr yield point stress in shear typ = :;.

(c) The tensile strain E reaches the yield point strain Eyp '

(d) The total strain energy U absorbed by the unit volume

of material reaches the value Uyp = 2~' 0";1"

(e) The strain energy of distortion Ud absorbed per unit

volume of material reaches a value Ud = 1 + 11 0"2 yp 3E YP'

(n The octahedral shearing stress reaches the value

'Oyp = c.J2i3)O"YP = 0.47 O"YP'

In case of multi-axial state of stress the above values will not be attained simultaneously and as such it is of utmost importance in design to choose anyone of the above quantities to calculate the limiting load which will not cause the inelastic action in the material with greatest possible econumy.

We will now discuss, one by one, the theories offailure based on the above quantities.

1.1 MAXIMUM PRINCIPAL STRESS THEORY

This theory put forward by Rankine asserts that failure or fracture of a material occurs when the maximum principal stress at a point in a complex system attain a critical value regardless of the other stresses. The critical value of stress O"ult is usually determined in a simple tensile test, where the failure of a specimen is defined to be due to either excessively large elongation or fracture, usually the latter is implied.

For complex stress system the major principal stress

O"x+O"y 1~ 2 2 0" = . + - (0" - 0" ) + 4.

1 2 2 x .Y

= O"ult in simple tension ... (1.1)

The maximum principal stress theory can be represented graphically by a square ABeD (Fig. 1.1) the sides being defined by

~ = ± 1 and 0" 2 = ± 1 ... (1.2) O"I/it O"I/it

THEORIES OF FAILURE

Failure occurs if point falls on the periphery of the square ABeD.

Experimental work

3

indicates that this theory gives quite good results for brittle materials in all ranges of stresses, provided that both the principal stresses are of tensile nature. Failure is by fracture in such cases.

Fig. 1.1. Graphical representation of maximum principal stress theory.

1.2 MAXIMUM SHEARING STRESS THEORY

This theory by Guest and Tresca is based on the observation that in ductile material slipping occurs during yielding along critically oriented planes. It is assumed in this theory that the maximum stress alone produce inelastic action and that the equal tensile stresses (0'1 = 0'2) have no influence in starting inelastic action. This implies that failure will occur when the maximum shearing stress 't'max in the complex system reaches the value of the maximum shearing stress in simple tension at the yield point.

Assuming biaxial stress system as shown in Fig. l.2 (a), we have

0' = ..2!!... in simple tension

2 ... (1.3a)

or 0'1 - 0'2 = O'yp when 0'1 and 0'2 are tensile ... (1.3b)

or - (crI - cr2) = cry!, when crI and cr2 lire compressive ... (l.3e) Equations l.3 (b) and (c) may be combined together as given

below

or

0'1 - 0'2 = ± O'yp

~_ 0'2 = ± 1 cr yp O'y!'

When 0'2 = 0, 0'1 = ± O'yp

and when 0'1 = 0, 0'2 = ± O'yp'

... (l.3d)

... (1.4)

Equation (1.4) represents straight lines in II and IV quadrants and may be expressed graphically as shown in Fig. l.3.

4

Fig. 1.2

ADVANCED MECHANICS OF MATERIALS

Fig. 1.3. Graphical representation of maximum shearing stress theory.

Now consider plane AEHD as shown in Fig. 1.2 (b). For yielding to occur in this plane

± 0'2 = O'yp ... (1.5a)

Similarly yielding to occur in plane ABFE

± 0'1 = O'yp ••. (1.5b)

Equations 1.5 (a) and (b) represent straight lines as shown graphically in quadrants I and III of Fig. 1.3.

If a point having co-ordinates as 0'1 and 0'2 lies inside the hexagon of Fig. 1.3, it may be presumed that no yielding of material has occurred. When this point falls on the periphery of hexa'gon, one should take it for granted that the material has undergone inelastic deformation.

If the state of stress consists of triaxial tensile stress of nearly same magnitude, shearing stress in such a case will be of very small magnitude and failure would be by fracture rather than by yielding and hence maximum principal stress theory should be applied.

The maximum shearing stress theory gives fairly good results for ductile materials and for state of stress in which comparatively large shearing stresses are developed. However, for the pure shear as in torsion test, where maximum shear stress is developed. the shearing elastic limit of ductile metals is on an

THEORIES OF FAILURE 5

average found to be 0.57 ofthe tensile elastic limit. Hence in such cases, the maximum shearing stress theory gives results on the positive side.

1.3 MAXIMUM STRAIN THEORY

This theory suggested by St. Venant states that yielding at a point in a material begins when the maximum strain corresponding to a particular complex state of stress exceeds the strain corresponding to the yield point. If crl and cr2 are the two principal stresses (crl > cr2), then the strain in the direction of crI is given by

crI jlcr2 £ - ----

2 - E E ... (1.6)

cr The limiting value of £1 should not be more than -2J!... In

E simple tension. Hence we may write

~ _ jlcr2 = cr YP E E E or crI - jlcr2 = crYP '

In Eq. (1.7) if crI and cr2 both are tensile then crI can be higher than cryp but if cr2 is compressive then crI will have a value smaller than crYJ)' Therefore, in the former case, according to this theory, crI can be increased beyond cr ) withuut causing yielding . h . I Yl III t e matena.

Maximum strain theory is an improvement over the maximum principal stress theory, even then it doesn't give satisfad-ory results for ductile materials. It is primarily used in cases where failure occurs by brittle fracture.

This theory is represented graphically as shown in Fig. 1.4, Fig. 1.4. Graphical r~presentation where the different portions of of maxImum stram theory.

the graphs are governed by the equations as given below: for ab

crI - jlcr2 = crYJI for ah crI - jlcr2 = - crYJI for ed cr2 - jlcrl = - crYJI for er

6

For unlike stresses, we have

0"2 + /10"1 = O"yp

0"1 + /10"2 =- - O"yp

0" 1 + ~l0"2 = 0" kp

0"2 + /10"1 = - O"yp

ADVANCED MECHANICS OF MATERIALS

for cb for cd

for gh for gf.

1.4 TOTAL STRAIN ENERGY THEORY

This theory proposed by Haigh states that inelastic action or yielding at a point in a material begins only when the energy per unit volume absorbed at a point is equal to the energy under uniaxial state of stress as in the case of simple tensile test. Thus in this case failure doesn't depend on the state of stress but governed by the energy stored in the material per unit volume.

Let us consider triaxial stress system (0"1 > 0"2 > 0"3)' For this state of stress, we have

get

1 £1 = E [0"1 - /1(0"2 + 0"8)]

£2 = ~[0"2 -/1(0"1 +0"8)J

1 fa = E [0"3 - /1(0" 1 + 0"2)]

Strain energy per unit volume can be expressed as

... (1.8)

v, 1 1 1 (1 9 ) = ""20"1£1 +""20"2£2 + ""20"3£3 '" . a

Substituting strain in terms of stresses from Eq. (1.8), we

... (1.9b)

For biaxial stress system put 0"3 = 0 and the strain energy expression is modified as under.

1 2 2 U = 2E [0"1 + 0"2 - 2/10"10"2] ... (1.9c)

For uniaxial stress system at yield point, we have

0"2 = 0"3 = 0 and 0"1 = O"yp

2 U = O"yp

Yl' 2E ... (1.9d)

Thus for failure by total strain energy theory, expressions (1.9b) or (1.9c), as the case may be, should be equated to (1.9d). For triaxial stress system, we have

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