Alciatore D.G., Histand M.B. Introduction to Mechatronics and Measurement Systems

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541

APPENDIX

Mechanics of Materials

APPENDIX OBJECTIVES

After you read, discuss, study, and apply ideas in this appendix, you will:

1. Understand the basic relationships between stress and strain

2. Be able to determine the principal stress values and directions for a general

state of planar stress

3. Be able to construct Mohr’s circle for a state of planar stress

C.1 STRESS AND STRAIN RELATIONS

As shown in Figure C.1 , when a cylindrical rod is loaded axially, it will lengthen

by an amount Δ L and deform radially by an amount Δ D. The axial strain ( ε

axial

) is

defined by the change in length per unit length:

axial

ΔL

-------

(C.1)

Note that strain is a dimensionless quantity. The axial stress ( ␴

axial

) is related to axial

strain through Hooke’s law, which states that for a uniaxially loaded linear elastic

material the axial stress is directly proportional to the axial strain:

axial

Eε

axial

(C.2)

where E is the constant of proportionality called the modulus of elasticity or

Young’s modulus. The axial stress in the rod is

axial

FA⁄=

(C.3)

where F is the axial force and A is the cross-sectional area of the rod. Therefore, the

axial strain is related to the axial stress and load:

axial

-----------

FA⁄

-----------

(C.4)

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542 APPENDIX C Mechanics of Materials

The transverse strain is defined as the change in width divided by the original

width:

transverse

ΔD

--------

(C.5)

The ratio of the transverse and axial strain is defined as Poisson’s ratio ( ␯ ):

transverse

axial

------------------

–=

(C.6)

Note that for axial elongation (ε

axial

>0), ε

transverse

(from Equation C.6 ), and therefore

Δ D (from Equation C.5 ) are negative, implying contraction in the transverse radial

direction. Poisson’s ratio for most metals is approximately 0.3, implying the trans-

verse strain is ⫺ 30% of the axial strain.

A general state of planar stress at a point, acting on an infinitesimal square ele-

ment, is illustrated in Figure C.2a . It includes two normal stress components ( ␴

and

␴

) and a shear stress component ( ␶

) whose values depend on the orientation of

the element. At any point, there is always an orientation of the element that results

in the maximum normal stress magnitude and zero shear stress ( ␶

⫽ 0). The two

orthogonal normal stress directions corresponding to this orientation are called the

principal axes, and the normal stress magnitudes are referred to as the principal

stresses ( ␴

max

and ␴

min

). Figure C.2b illustrates this orientation and its corresponding

state of stress. The magnitude and direction of the principal stresses are related to the

stresses in any other orientation by

max

-----------------

⎝⎠

⎛⎞

–

-----------------

⎝⎠

⎛⎞

++=

(C.7)

Figure C.1 Axial and transverse

deformation of a cylindrical bar.

+ ΔL

D + ΔD

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C.1 Stress and Strain Relations 543

min

-----------------

⎝⎠

⎛⎞

–

-----------------

⎝⎠

⎛⎞

+–=

C.8)

2θ

()tan

2τ

–

-----------------

(C.9)

where ␪

is the angle from ␴

to ␴

max

, measured counterclockwise.

The principal stresses are important quantities when determining if a material

will yield or fail when loaded because they determine the maximum values of stress,

which can be compared to the yield strength of the material. The maximum shear

stress is also important when assessing failure and is given by

max

–

-----------------

⎝⎠

⎛⎞

max

min

–

--------------------------

(C.10)

This relation can be used to rewrite Equations C.7 and C.8 as

max

avg

max

(C.11)

min

avg

max

–=

(C.12)

where

avg

-----------------

(C.13)

The orientation of the element that results in ␶

max

is given by

tan 2θ

()

–

2τ

-----------------

–=

(C.14)

As with ␪

, ␪

is measured counterclockwise from the direction of ␴

. For the

cylindrical bar in Figure C.1 , with an element oriented in the axial ( y ) direction,

␴

max

⫽ ␴

⫽ F / A, ␴

⫽ 0, and ␪

⫽ 0 because the element is aligned in the direction

of the principal stress. Also, ␪

⫽ 45

⬚ and ␶

max

⫽ ␴

/2 ⫽ F /2 A.

The state of stress and its relation to the magnitude and direction of the princi-

pal stresses are often illustrated with Mohr’s circle, which displays the relationship

between the shear stress and the normal stresses in different directions (see Figure C.3 ).

Figure C.2 General state of planar stress and principal stresses.

max

min

(a)

eneral state of stress (b) principal stresses

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Figure C.3 Mohr’s circle of plane stresses.

max

min

(σ

, τ

)

(σ

, –τ

)

max

2θ

avg

2θ

(σ

avg

, τ

max

)

544 APPENDIX C Mechanics of Materials

Remember, tensile normal stresses are positive and compressive normal stresses are

negative. For the example shown in Figure C.3 , corresponding to the element shown

in Figure C.2 , both normal stresses are tensile. The sign of the shear stress is posi-

tive when it would cause the element to rotate clockwise about its center and nega-

tive when it would cause the element to rotate counterclockwise. For the element in

Figure C.2 , ␶

is negative on the ␴

side of the element since it would cause the ele-

ment to rotate counterclockwise, and ␶

is positive on the ␴

side for the opposite

reason. Note that the angle between the original stress directions and the principal

stresses ( ␪

) is measured in the same direction around the circle as with the actual ele-

ment, but angles on the circle are twice the actual angles (2 ␪

). Since ␪

is measured

counterclockwise from ␴

to ␴

max

in Figure C.2 , the angle between the ␴

point and

␴

max

is 2 ␪

counterclockwise in Figure C.3 . Also note that the orientation of the prin-

cipal stresses and the orientation of the maximum shear stress are always 90 ⬚ apart

on Mohr’s circle (45 ⬚ apart on the actual element). This is confirmed by the fact that

tan(2 ␪

) and tan(2 ␪

) are negative reciprocals of one another (see Equations C.9 and

C.14 ). For more information, Internet Link C.1 points to a derivation of the equation

for Mohr’s circle for uniaxial stress, and Video Demo C.1 discusses and illustrates the

results. Video Demo C.2 discusses how Mohr’s circle can help one understand why

brittle and ductile materials exhibit different fracture plans when they break.

Internet Lin

C.1 Mohr’s

circle equation

derivation for

uniaxial stress

Video Demo

C.1 Mohr’s circle

for uniaxial stress

C.2 Failure

theories for

brittle and ductile

materials

■ CLASS DISCUSSION ITEM C.1

Fracture Plane Orientation in a Tensile Failure

When a metal bar fails under axial tension, the resulting fracture planes are oriented

at 45

⬚

with respect to the bar

’

s axis. Why?

BIBLIOGRAPHY

Beer , F. and Johnston , E. , Mechanics of Materials, 5th Edition, McGraw-Hill, New York, 2008 .

Dally , J. and Riley , W. , Experimental Stress Analysis, 3rd Edition, McGraw-Hill,

New York, 1991 .

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INDEX

A/D converters and conversion, 6,

261, 263–264, 352–358,

373–374

absolute encoders, 383, 389

AC, 12, 37–44

AC coupling, 59

AC motors, 441

induction, 435

accelerometers, 414–424, 429

acceptors, 75

accumulator, 265

active devices, 165

active filters, 499

active low input, 218

active region, 93, 104

actuators, 431–477

definition of, 432

electric motors, 435–441

electromagnetic principles,

432–433

hydraulic systems, 473–474, 477

solenoids and relays, 433–435, 475

address lines, 260

air gaps, 437

aliasing, 348–350

alternating current, 12, 37–44, 70–71

ALU, 259–260

ammeters, 32

amp-hour capacity, 339

ampere, 12

amplifiers, 162–164, 352–353,

423, 447