67071_02.pdf

CHAPTER

2 STATIC STRESSES IN M A C H I N E ELEMENTS

S Y M B O L S 3,4,5

A Aw a

b

c

D C1 F Fc F,

Fcr e

Eo

G Mb M t i I

Ixx, Iyy

J k ko kt

area of cross section, m 2 ( in 2) area of web, m 2 (in 2) constant in Rankine's formula radius of area of contact, m (in) bandwidth of contact, m (in) width of beam, m (in) distance from neutral surface to extreme fiber, m (in) diameter of shaft, m (in) constant in straight-line formula load, kN (lbf) J compressive force, kN (lbf) tensile force, kN (lbf) shear force, kN (lbf) crushing load, kN (lbf) deformation, total, m (in) eccentricity, as of force equilibrium, m (in) unit volume change or volumetric strain thermal expansion, m (in) modulus of elasticity, direct (tension or compression), GPa

(Mpsi) combined or equivalent modulus of elasticity in case of

composite bars, GPa (Mpsi) modulus of rigidity, GPa (Mpsi) bending moment, N m (lbf ft) torque, torsional moment , N m (lbf ft) number of turns moment of inertia, area, m 4 o r c m 4 (in 4) mass moment of inertia, N s 2 m (lbf s 2 ft) moment of inertia of cross-sectional area around the respective

4 4 4 principal axes, m or cm ( i n ) 4 4 moment of inertia, polar, m or cm (in 4)

radius of gyration, m (in) polar radius of gyration, m (in) torsional spring constant, J/rad or N m/rad (lbf in/rad)

2.1

2.2 CHAPTER TWO

! n

l, m, n P

T AT r

q Q

V

AV Z oz

7xy, 7yz, %x ~5 c

C T

Cx , E y , C z

0

crb

O" c

'SC

O'cr

O" e

as o" t

Ost

O'x , O'y , Ot z

Or l , 0"2, 0" 3

cry O'sy

O" u

asu o" 0 .tt

length, m (in) length of rod, m (in) length, m (in) speed, rpm (revolutions per minute) coefficient of end condition speed, rps (revolutions per second) direction cosines (also with subscripts) power, kW (hp) pitch or threads per meter temperature, °C (°F) temperature difference, °C (°F) radius of the rod or bar subjected to torsion, m (in) (Fig. 2-18) shear flow first moment of the cross-sectional area outside the section at

which the shear flow is required velocity, m/s (ft/min or fpm) volume, m 3 (in 3) shear force, kN (lbf) volume change, m 3 (in 3) section modulus, m 3 (in 3) deformation of contact surfaces, m (in) coefficient of linear expansion, m/m/K or m/m/°C (in/in/°F) shearing strain, rad/rad shearing strain compOnents in x y z coordinates, rad/rad deformation or elongation, m (in) strain, gm/m (gin/in) thermal strain, gm/m (gin/in) strains in x, y, and z directions, gm/m (gin/in) angular distortion, rad angle, deg angular twist, tad (deg) angle made by normal to plane nn with the x axis, deg bulk modulus of elasticity, GPa (Mpsi) Poisson's ratio radius of curvature, m (in) stress, direct or normal, tensile or compressive (also with

subscripts), MPa (psi) bearing pressure, MPa (psi) bending stress, MPa (psi) compressive stress (also with subscripts), MPa (psi) hydrostatic pressure, MPa (psi) compressive strength, MPa (psi) stress at crushing load, MPa (psi) elastic limit, MPa (psi) strength, MPa (psi) tensile stress, MPa (psi) tensile strength, MPa (psi) stress in x, y, and z directions, MPa (psi) principal stresses, MPa (psi) yield stress, MPa (psi) yield strength, MPa (psi) ultimate stress, MPa (psi) ultimate strength, MPa (psi) principal direct stress, MPa (psi) normal stress which will produce the maximum strain, MPa (psi)

STATIC STRESSES IN MACHINE ELEMENTS 2.3

O" 0

7-

% 7-~y, 7-yz, 7-z~ 7-0 Cc2

normal stress on the plane nn at any angle 0 to x axis, MPa (psi) shear stress (also with subscripts), MPa (psi) shear strength, MPa (psi) shear stresses in x y , y z , and z x planes, respectively, MPa (psi) shear stress on the plane at any angle 0 with x axis, MPa (psi) angular speed, rad/s

Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage.

( N o t e : ~ and 7- with initial subscript s designates strength properties of material used in the design which will be used and observed throughout this M a c h i n e

D e s i g n D a t a H a n d b o o k . )

Particular Formula

SIMPLE STRESS AND STRAIN

The stress in simple tension or compression (Fig. 2-1 a, 2-1b)

The total elongation of a member of length l (Fig. 2-2a)

F, F~ cr t = -A ; Cr C = A

F/

A E

(2-1)

(2-2)

°, IlI[lllllllll %llll[lll[ll[I (a) ~ (c)

FIGURE 2-1

(~ O" C ~ k - - m

l E Strain, deformation per unit length (2-3)

2.4 CHAPTER TWO

Particular Formula

T I

l/Mill/Ill/ Z/////~ (a)

'/,t/

FIGURE 2-2

/ / / / , I / , "//~

Fc

I

¢///I ,////~/ (b)

F,~ ~ al*- ~ R' °[;°,

P (c) S

Young's modulus or modulus of elasticity

The shear stress (Fig. 2-1c)

Shear deformation due to torsion (Fig. 2-18)

Shear strain (Fig. 2-2c)

The shear modulus or modulus of rigidity from Eq. (2-7)

Poisson's ratio

Poisson's ratio may be computed with sufficient accuracy from the relation

The shear or torsional modulus or modulus of rigidity is also obtained from Eq. (2-10)

The bearing stress (Fig. 2-3c)

STRESSES

Unidirectional stress (Fig. 2-4)

The normal stress on the plane at any angle 0 with x axis

O" E = -

C

7" ---~ A

7"L 0----~

G

7" a

V=G=7 7"

G = - 7

v = lateral strain/axial strain = e__t ffa

E u = g - d - 1

G = 2(1 + u)

F

ah = bd2

O" 0 = O" x COS 2 0

(2-4)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(2-10)

(2-11)

(2-12)

(2-13)


Particular Formula

I ~'-d3 =i ]~- Pin

.., d .~__~ . ~ Fork

Eye ~ .

? N

FIGURE 2-3 Knuckle joint for round rods.

- d 3 -

I bl A hole .~

for a pin ~.

- s s S

F

F

n

(a) n ' (c) Stress acting on an

element at P

" ~ P l a A" ~e . . . . A = Area

A- % x

ne on which stresses (b) are considered

FIGURE 2-4 A bar in uniaxial tension. 3'4

(~w e

1;o

Go

(d) Stress acting on an element at Q

The shear stress on the p lane at any angle 0 wi th x axis

Pr inc ipa l stresses

Angles at which pr inc ipa l stresses act

M a x i m u m shear stress

Angles at which m a x i m u m shear stresses act

O" x "/-0 = -~- sin 20

0-1 -- 0-x and 0- 2 -- 0

O1 = 0 ° and 02 = 90 °

0 - x

"rma x =

01 = 45 ° and 02 = 135 °

(2-14)

(2-15)

(2-16)

(2-17)

(2-18)

2.6 CHAPTER TWO

Particular Formula

The normal stress on the plane at an angle 0 + (7r/2) (Fig. 2-4d)

The shear stress on the plane at an angle 0 + (7r/2) (Fig. 2-4d)

Therefore f rom Eqs. (2-13) and (2-19), (2-14), and (2-20)

0.0 = 0.x c0S2 0 q- ~ - - 0.x c0S2 0 (2-19)

r o = 0 . x s i n 0 + ~ cos 0 + ~ = 1 0 . x s i n 2 0 (2-20)

l l 0.0 = 0.o and To = -To (2-21)

PURE SHEAR (FIG. 2-5) The normal stress on the plane at any angle 0

The shear stress on the plane at any angle 0

The principal stress

Angles at which principal stresses act

M a x i m u m shear stresses

Angles at which m a x i m u m shear stress act

0.0 = %, sin 20 (2-22)

ro = rx), cos 20 (2-23)

0.1 = %y and 0" 2 = --3-xy (2-24)

01 = 45 ° and 02 = 135 ° (2-25)

rmax = r,.y = 0. (2-26)

Ol = 0 and 02 = 90 ° (2-27)

~xy

to Y (a) - ~ y = (~x = ~

n

"C I lOy \n

(b) 1:ma x = 1:xy = 0 (a) (b)

FIGURE 2-5 An element in pure shear. FIGURE 2-6 An element in biaxial tension.

BIAXIAL STRESSES (FIG. 2-6)

The normal stress on the plane at any angle 0

The shear stress on the plane at any angle 0

The shear stress ro at 0 = 0

The shear stress ro at 0 = 45 °

cT x + o" ,. 0.,. - cry ~o = ~ + 2 cos 20

0.x - % sin 20 t o - 2

~ - 0

"rmax = (Ox -- 0 . y ) / 2

(2-28)

(2-29)

(2-30)

(2-31)


Particular Formula

BIAXIAL STRESSES COMBINED WITH SHEAR (FIG. 2-7)

The normal stress on the plane at any angle 0 0.x - Cry

0.0 = 0.x +2 0.y + 2 cos 20 + "rxy sin 20 (2-32)

The shear stress in the plane at any angle 0 0.x - 0.y sin20 - rxy cos 20 (2-33) t o = 2

The maximum principal stress 0.1 + + r2y (2-34) 2 2

The minimum principal stress 0. 2 w 2 - ~ + 7xy (2-35)

Angles at which principal stresses act

Maximum shear stress

Angles at which maximum shear stress acts

2rxy 01, 2 --- ½ a r c t a n ~

0-x -- 0-y

where 01 and 02 are 180 ° apart

[ ( )2 ]1/2 °x - o-y +.rx2y rma,, = 2

0 - 1 - - 0 " 2

0 = ½ arctan O" x -- %

2%

(2-36)

(2-37)

(2-38)

The equation for the inclination of the principal planes in terms of the principal stress (Fig. 2-8)

tan 0 = 0 - 1 - - 0-x '7-x y

(2-39)

n\ t(TY~xY d,.. y',

L, .Ox ° oo x ~11 \ II ~x o

" I i

'lT×Y"~ ~Oy n

(a) (b)

.L,.i "q A I, / '~ \ .,

I 0 _ _ . _ . -

1;0= 0

C ' - , " - B

't'xY ~i ~ fly

FIGURE 2-7 An element in plane state of stress. FIGURE 2-8

2.8 CHAPTER TWO

Particular Formula

MOHR'S CIRCLE

Biaxial field combined with shear (Fig. 2-9)

M a x i m u m principal stress o.l

Min imum principal stress o'2

Max imum shear stress Tma x

al is the abscissa of point F

0 2 is the abscissa of point G

7-ma x is the ordinate of point H

0% 6"

Oy % / ¢ ,~,.~xy//" Op

Ox "7['-'" Ox

- ~ y

i

L, G K C F o2-<L i Z.x

~, ao 7

-'~xy, 0-----0)

(a)

FIGURE 2-9 Mohr's circle for biaxial state of stress.

TRIAXIAL STRESS (Figs. 2-10 and 2-11)

The normal stress on a plane nn, whose direction cosines are 1, m, n

The shear stress on a plane normal nn, whose direction cosines are 1, m, n

The principal stresses

The cubic equation for general state of stress in three dimensions from the theory of elasticity

The max imum shear stresses on planes parallel to x, y, and z which are designated as

9 (70 = O.X 12 + o'ym- + o.z n2 (2-40)

(2-41)

O"1,2,3 --- O.x, O'v, o." (2-42)

0 .3 - - ((7 x -Jr- o.y + O.z)O. 2 7 t- (o.xO'y + o.yO" z 7 t- o.zo.x

-- TAW . -

- (a . , . aya : + 2r,.yr>,:%x- o.xr2y- a . , .2~- a:rx 2) = 0 (2-43)

The three roots of this cubic equat ion give the magni- tude of the principal stresses o.1, o.2, and o.3-

° '2 -- ° '3 (7"1 -- ° '3 (rmax)l = 2 ; (rmax)2 = ~ ;

( r m a x ) 3 __ O'1 - - 0"2 (2-44) 2

STATIC STRESSES IN MACHINE ELEMENTS 2 .9

Particular Formula

MOHR'S CIRCLE

Triaxial field (Figs. 2-10 and 2-11)

N o r m a l stress at po in t (Fig. 2-11 b) on one oc t ahedra l p lane

Shear stress at po in t T (Fig. 2-11 b) on an oc t ahedra l p lane

or Crt is the abscissa o f po in t T

(2-45)

7_ t = ½ [(or x _ cry)2 + (Cry - Crz) 2 + (Crz - Crx) 2 (2-46a)

+ 6(7-2 + ~-y2 z + ~x)] 1/2

= ~ V/[(~, - ~ ) ~ + (~: - ~ ) : + ( ~ - ~1) ~]

or % is the o rd ina te o f po in t T

YL~ Oy

I %

' 2 O, x . . . . O'X . ~ ' ~ "CO, . ~ .

~[5" - " X ,,s SS . . . . . . . , ! \ ~

~ % (a) (o)

FIGURE 2-1t) An element in triaxial state of stress.

2 ~ G

D "c ax y

I / " ' " ' . N J ¢ "B ~o~-,I c A l l ]B ~ F o +o +o

i (a) (o) o3

/ x /z (c)

FIGURE 2-11 Mohr's circle for triaxial octahedral stress state.

2.10 CHAPTER TWO

Particular Formula

S T R E S S - S T R A I N R E L A T I O N S

Uniax ia l field

Strain in principal direction 1

The principal stress

The unit volume change in uniaxial stress

ry 1 o" 1 ry 1 6`1 = E ; 6`2 - - - - / ' I X ; 6`3 = - - / / X

Orl - - E6`l

V A (1 2u) ryl -___2_" = = el (1 - 2u) V E

(2-47)

(2-47a)

(2-48)

Biaxia l field




The principal stresses in terms of principal strains in a biaxial stress field

The unit volume change in biaxial stress

1 6`1 = 2 ( O " 1 - - / ' 0 " 2 )

1 ~2 = ~ (~2 - ~'¢~)

v 6`3 - - -- ~ (O'1 -+- 0"2)

E al = 1 - / /2 (6`1 q-" /Y6`2)

E O'2 "-" 1 - U 2 (6`2 -}- //6"1)

0" 3 = 0

AV (1 E2U) - 7 q- (O" 1 q-- 0"2)

(2-49)

(2-50)

(2-51)

(2-52)

(2-53)

(2-53a)

(2-54)

Triaxial field




The principal stresses in terms of principal strains in triaxial stress field

1 El - - 2 [O'1 -- //(0.2 -}- 0.3)]

1 C2 - - ~ [ °.2 -- /](0" 3 -ff rY 1 )]

1 C3 - - 2 [0.3 -- /j(Orl -}- 0.2)]

E °1 - (1 - t , , - 2t, ,2) [(1 - / , ' ) e l + t,'(c2 + ~3)]

E 0"2 = (1 -- t / - 2/,/2 ) [(1 -- P)82 + P(C3 q- CI)]

E a3 -- (1 - u - z ) " u 2" [(1 - u)e3 + U(el + 6`2)]

(2-55)

(2-56)

(2-57)

(2-58)

(2-59)

(2-60)


Particular Formula

The unit volume change or volumetric strain in terms of principal stresses for the general case of triaxial stress (Fig. 2-12)

dV (1 -- 2u) e = --~ = ~ ( 0 - x + cry + o-z) (2-61a)

(1 - 2u)

~"~x

z

Oy

~ ( 0 - 1 -+- 0"2 -t- 0-3) (2-61b)

F I G U R E 2-12 Uniform hydrostatic pressure.

The volumetric strain due to uniform hydrostatic A V pressure 0-c acting on an element (Fig. 2-12) V

-3(1 - 2u)0-c E

= 0-c (2 -62) /5;

The bulk modulus of elasticity E

3(1 - 2u)

The relationship between E, G and K 9KG

(3K + G)

STATISTICALLY INDETERMINATE MEMBERS (Fig. 2-13)

The reactions at supports of a constant cross-section bar due to load F acting on it as shown in Fig. 2-13 R a - - - ~

FLb FLb La + Lb L

R B - - - ~

FLa

La + Lb

FLa L

The elongation of left portion La of the bar RALa AE

FLaLb ....-

LAE

(2-63)

(2-63a)

(2-64a)

(2-64b)

(2-65)

2.12 CHAPTER TWO

Particular Formula

The shortening of right portion L b of the bar ~b - - RALa FLaLb

A E L A E (2-66)

CI "~F Lb B L a =I = (a)

RB

I I ~ F - - - -I ~ RA-~ AI ..... I "- ~ RB

C ~)

FIGURE 2-13

Fcb

B / / / / ) , P / / / / |

i

i

|

AT'---~ L I

!

|

I

..t A / / / /4 g'// / / IA

(a)

Feb

FIGURE 2-14

Y//// (o)

A T

Fcb

/ / / ~ ~/////

/ / / /A 9"//// (c)

THERMAL STRESS AND STRAIN

The normal strain due to free expansion of a bar or machine member when it is heated

The free linear deformation due to temperature change

The compressive force F,.h developed in the bar fixed at both ends due to increase in temperature (Fig. 2-14)

The compressive stress induced in the member due to thermal expansion (Fig. 2-14)

The relation between the extension of one member to the compression of another member in case of rigidly joined compound bars of the same length L made of different materials subjected to same temperature (Fig. 2-15)

The forces acting on each member due to temperature change in the compound bar

The relation between compression of the tube to the extension of the threaded member due to tightening of the nut on the threaded member (Fig. 2-16)

The forces acting on tube and threaded member due to tightening of the nut

eT" = a(AT)

6 = ~L(iX r)

Fcb = a A E ( A T )

Fcb = - a E ( A T ) O'cT: A

a~L crcL

crcAc = crsA s

crtL

Et

a~L + - -~ = [number of turns (i)

L s

x (threads/meter) or pitch (P)]

= iP

crsA s = o-tA t

(2-67)

(2-68)

(2-69)

(2-70)

(2-71)

(2-72)

(2-73)

(2-74)


Particular Formula

,•St " + Cu Copper St

(a) _l_ ~ ,- L _ Lt---~

Iioi t St I

Cu I St I ~

Extension of steel • ( b )

..... c u "

FIGURE 2-15

Difference in free lengths

Compression of copper

L L ~l

r I / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

~t + (5s = (5 = iP

//////,¢'/////¢'///¢'////////////J _1 Difference in

fit + (5s = (5 *- free lengths (a)

L ,, ~t L . . . . . . . [~ :~ . ~ ( s t = Et

(b) r I compression of tube =1 ~ Extension of screw ' 'n~ rod a L

/ / / / / / / / / / / / / / / / / / / / / / / / / ' ! . L ~ _ (5 - s ,' . ..... [ _l~Ym~ ' E,

P = pitch, m (c) i = number of turns

FIGURE 2-16

COMPOUND BARS The total load in the case of compound bars or columns or wires consisting of i members, each having different length and area of cross section and each made of different material subjected to an external load as shown in Fig. 2-17

An expression for common compression of each bar (Fig. 2-17)

E i A i 6 i __ EiAi 6 Z (2-75) F = Z L-'---i-- Li

6 = (2 -76 ) ~--](EiAi/ti)

Same compression i

I I I I_

First member ~

FIGURE 2-17

I

FI _. Original G. _--Z. - ~ level of G

IT;e'n ' . . . . . . . . . . . ' of G

~ I I - - l ] ~i thmember

2.14 CHAPTER TWO

Particular Formula

The load on first bar (Fig. 2-17)

The load on ith bar (Fig. 2-17)

F1 (E1AI/L1) = ~_ , (EA/L)F (2-77)

E i A i 6 F i = ~ (2-7S)

Li

EQUIVALENT OR COMBINED MODULUS OF ELASTICITY OF COMPOUND BARS

The equivalent or combined modulus of elasticity of a compound bar consisting of i members, each having a different length and area of cross section and each being made of different material

The stress in the equivalent bar due to external load F

The strain in the equivalent bar due to external load F

The common extension or compression due to external load F

E~ = E1A1 + E2A2 + E3A3 + " . + EnAn (2-79a) A1 + A2 + A3 + ' " + An

~-'~ EiA i

E i = 1,2 ..... n A i (2-79b)

F O" - - E i = 1,2,3 .... Ai (2-80)

F 6 E "-- E c E i = 1,2,3 .... A i = ~ (2-81)

FL 6 : = eL (2-82)

E,. E i = 1,2,3 ..... n A i

POWER

The relation between power, torque and speed P = Mta~ (2-83)

where Mt in N m (lbf ft), ~ in rad/s (rad/min), and P in W (hp)

_ - --Mtn SI (2-84a) 159

where Mt in kN m, n' in rps, and P in kW

Mrn

9550 SI (2-S4b)

where Mt in kN m, n in rpm, and P in kW

Mtn 63030

USCS (2-84c)

where M t in lbf in, n in rpm, and P in hp


Particular Formula

Another expression for power in terms of force F acting at velocity v

E L /

P = 100-------6 SI (2-85a)

where F in newtons (N), v in m/s, and P in kW

F / /

= 3300------6 USCS (2-85b)

where F in lbf, v in fpm (feet per minute), and P in hp (horsepower)

TORSION (FIG. 2-18)

The general equation for torsion (Fig. 2-18) M t GO T . . . . . (2-86) J L p

Torque m t -'- 159P

SI (2-87a)

where Mt in kN m, n t in rps, and P in kW

9550P . . . . SI

n

where Mi in kN m, n in rpm, and P in kW

- .

63030P =- USCS

n

where Mt in lbf in, n in rpm, and P in hp

The maximum shear stress at the maximum radius r of the solid shaft (Fig. 2-18) subjected to torque Mt

16Mt "rma x = 7rD 3

The torsional spring constant GJ k , = ~ =

0 L

(2-87b)

(2-87c)

(2-88)

(2-89)

Y Resisting

t°rque Mt t Mt Applied torque Xmax

, ::;r L

FIGURE 2-18 Cylindrical bar subjected to torque.

2.16 CHAPTER TWO

Particular Formula

BENDING (FIG. 2-19)

The general formula for bending (Fig. 2-19) M b o% E

I c p (2-90)

O

/ l \ \ 6~s o~ / ', \ ~ ~ ~-~ _ ,~e

i v G~w'."

M b " dy t s n"

i .. m

Neutral axis m n

F I G U R E 2-19 Bending of beam.

The maximum values of tensile and compressive bending stresses

The shear stresses developed in bending of a beam (Fig. 2-20)

m h c o- b = ~

I

vi' = y d A 7" -ib yo

The shear flow VQ q = ~ I

(2-91)

(2-92)

(2-93)

0

A

I F

Y

(b) (c)

FIGURE 2-20 Beam subjected to shear stress.


Particular Formula

The first moment of the cross-sectional area outside the section at which the shear flow is required

The maximum shear stress for a rectangular section (Figs. 2-20 and 2-21)

rn O'b n . .

Neutral.

o 7--I/. ax,

c ~_t / / F ~ ~ + dF /Mb+dMl: b ~ /~'-'~

Ob m'~.._ dx ~ n' "O'b + dOb (a)

-- b I "

i 0'

- ! -

g" / / / u ¢ / / / ~

ly (0)

FIGURE 2-21 Element cut out from a beam subjected to shear stress.

Q = y dA (2-94) 0

3V 3-max - - 2A (2-95)

~ : ~ x _t.Y :n

. . . . -wYlz,z /

(c)

For a solid circular section beam, the maximum shear stress

For a hollow circular section beam, the expression for maximum shear stress

An appropriate expression for "/-max for structural beams, columns and joists used in structural indus- tries

4V Tmax - - 3---A ( 2 - 9 6 )

2V "/ 'max--- A (2-97)

V :max = Aw

where Aw is the area of the web

(2-98)

E C C E N T R I C L O A D I N G

The maximum and minimum stresses which are induced at points of outer fibers on either side of a machine member loaded eccentrically (Figs. 2-22 and 2-23)

The resultant stress at any point of the cross section of an eccentrically loaded member (Fig. 2-24)

F Mb F Mb O'ma x - - ~ q-- - 7 and O'mi n - - A Z

F Mbxey Mbye x Crz=:k-~4- ix x 4- Iyy

(2-99)

(2-100)

C O L U M N F O R M U L A S (Fig . 2 - 2 5 )

Euler's formula (Fig. 2-26) for critical load nrr 2 EA nrr 2 E1 Fcr = (1/k)------- T . . . . . 12 (2-101)

2.18 CHAPTER TWO

Particular Formula

F

e

Jel_

(a)

I

!1! "~ " " W " ' ' "

!

v F

"~F

FIGURE 2-22 Eccentric loading.

F f

Centroid

z t Fzz / V Afey

x

7 Mb

( I = ( I a + (I b (I b =~- G a = F / A

FIGURE 2-23 Eccentrically loaded machine member. FIGURE 2-24

F

I

"/I,[/Iz ,'l/y/

(i) (iii)

F

"//X///,

(iO

F

"///, z//Xf "//,

(iv)

FIGURE 2-25 Column-end conditions. (i) One end is fixed and other is free. (ii) Both ends are rounded and guided or hinged. (iii) One end is fixed and other is rounded and guided or hinged. (iv) Fixed ends.


Particular Formula

Johnson's parabolic formula (Fig. 2-26) for critical load

4nTrZ E -k (2-~02)

500

13.. 400

300 w

I- _j 200

z

O 100 z

0 0

' , - 5 0 STEEL, Oy = 207 MPa (= 21.1 kgf/mm 2)

PIN ENDS, n=l E ., I , - 4 0

EULER

• J.B. JOHNSON REGION - z

I o 40 80 120 160 200

I SLENDERNESS RATIO, -~-

E

30 09" oo w n, F-

20 oo _J < z

FIGURE 2-26 Variation of critical stress with slenderness ratio.

Straight-line formula for critical load

Straight-line formula for short column of brittle material for critical load

Ritter 's formula for induced stress

Ritter's formula for eccentrically loaded column (Fig. 2-23) for combined induced stress

Rankine's formula for induced stress

The critical unit load from secant formula for a round-ended column

For = A IO-y - 2O y l

( ') Fcr = A c r - Cl-~

F (7" c -" -~ Oe I l l 2 C-~I

l + nTr2 E -k Jr-

~CF O" c ~ ' - - ~ /'7] l + a

F~r % A ,,,~ t

ec l v / (Fc;]4AE ) 1 +~-g sec

(2-103)

(2-104)

(2-105)

(2-106)

(2-107)

(2-108)

2.20 CHAPTER TWO

Particular Formula

HERTZ CONTACT STRESS

Contact of spherical surfaces

Sphere on a sphere (Fig. 2-27a) The radius of circular area of contact

a =0.721

l-v~ l-v2 2]

1 - 1 - - g+g

1/3

(2-109)

/: F

¢q

FIGURE 2-27 Hertz contact stress.

The maximum compressive stress

Combined deformation of both bodies in contact along the axis of load

Spherical surface in contact with a spherical socket (Fig. 2-27b)

The radius of circular area of contact


ffc(max) = 0.918

1 1 )2 1/3

F d-~l + d22 ( 1 - v ~ l - u 2 ) 2

El ..+ E2

a = 1.04

1 - u 1 - d 2

El " + E2 .... ) F2 ( dl d2

d I -+- d2 )

1/3

a =0.721

F ( l - u 2 l - r E ) E1 + E2 ' /

1 _ 1

1/3

O'c(max) = 0.918 (1-d l-d) E1 '+ E2 '

1/3

(2-110)

(2-111)

(2-112)

(2-113)


Particular Formula

Combined deformation of both bodies in contact along axis of load

Distribution of pressure over band of width of contact and stresses in contact zone along the line of symmetry of spheres

Sphere on a flat surface (Fig. 2-27c)

The radius of circular area of contact


1__/.,,2 1 - zj22)2 1/3

O~=1.04 2 E1 + E2 (2-114)

d 2 - dl

Refer to Fig. 2-28a.

a=0.721 Fdl E1 + E2 '

O_c(max)__0.918[d ( F ) 1 12 1 -u~ 1 -u~ 2 E1 ' + E 2

where d =dl (Fig. 2-27c).

1/3

(2-115)

(2-116)

Contact of cylindrical surfaces

Cylindrical surface on cylindrical surface, axis parallel (Fig. 2-27a and Fig. 2-28b)

The width of band of contact


Cylindrical surface in contact with a circular groove (Fig. 2-27b)



Distribution of pressure over band of width of contact and stresses in contact zone along the line of symmetry of cylinders

IF ( l - u 2 l - u 2 ..... E1 + E2 )

1/2

(' ') O'c(max) =0.798 d1+d22

E1 . . + E2 '

2b= 1.6 Ei + E2) I I

1/2

O'c(max) = 0.798 1 - u 1 - u 2

E1 .... + E2 /

Refer to Fig. 2-28b.

1/2

(2-117)

1/2

(2-118)

(2-119)

(2-120)

2.22 CHAPTER TWO

P a r t i c u l a r F o r m u l a

Cylindrical surface in contact with a f ia t surface (Fig. 2-27c)"



2 b = 1 . 6 [ - ~ ( 1 -~12E, + l - u ~ ) ] ' / 2 E 2

0"c(max) = 0.798 Ldl 1 - u21 1 - u~ "

E l + E2

where d = di (Fig. 2-27c).

(2-121)

(2-122)

Deformation of cylinder between two plates

The maximum shear stress occurs below contact surface for ductile materials

For sphere

For cylinders

The depth from contact surface to the point of the maximum shear

Stress ratio 0 x 0c (max) ' Gc (max)

0.2 0.4 0.6 0.8 1.0 ~ - a * - a - ~ ~" ~ ~ -

", O r = O 0 0.5a

N It~ ~ J

o a

° / 2 o 1.5a o / / O o F E 2a J

° T o e- ~ 2 . 5 a " al

ca. 3a I~ f / - 121

3.5a r IF "

4 a ' '

O,17

0.5b

4F (1 - Vl2) (1 2 d , ) Adl = --~ 7rE 3 + loge --if- (2-123)

NI..Q

b ¢:

1.5b t,.- 8 E g: 2b

o e- ~ 2.5b

e--

'~ ab 121

3.5b

"/'max "-- 0.310"c(max)

"/-max -- 0"2950"c(max)

h = 0.786b

4b /

Stress ratio o x Gc (max) ' Oc(max)

0.2 0.4 0.6 0"8 1 "0

/ / //

/ //

/ / / /-~,z

/

/ / F _

, dl

,17~, z fa) z (b)

(2-123a)

(2-123b)

(2-123c)

FIGURE 2-28 Distribution of pressure over bandwidth of contact and stresses in contact zone along line of symmetry of spheres and cylinders for u = 0.3.


Particular Formula

D E S I G N O F M A C H I N E E L E M E N T S A N D S T R U C T U R E S M A D E O F C O M P O S I T E

Honeycomb composite

For the components of composite materials which give high strength-weight ra t io combined with rigidity

For sandwich construction of honeycomb structure

Refer to Fig. 2-29.

Refer to Fig. 2-30.

Adhesive impragnated

sirlin cloth

I . . . . . . . ~ Face ,, shee t

111111111 I ! I I I I I I II-~~.one~com~ core

I I +

I lltlllltll:illl(Itl" Sandwich fabricated

panel

FIGURE 2-29 Sandwich fabricated panel.

Skin

Ho

.on , om , . . , \ Low density core

FIGURE 2-30 Honeycomb.

The moment of inertia of sandwich panel, Fig 2-30

Simplified Eq. (2-124) after neglecting powers of h

The flexural rigidity

The flexural rigidity of sandwich plate/panel

The flexural rigidity of sandwich construction for (I4c/h) > 5

The shear modulus of the core material as per Jones and Hersch

I = 2 ( Bh3

( i I = B h H c h+--~-

(2-124)

(2-125)

D = E1 (2-126)

where E = modulus of elasticity of the facing metal I is given by Eq. (2-125).

D = E(H3 - H3) (2-127) 12(1 - u 2)

o = Eh(/- /+/- /~)~ 8(1 - u 2) (2-128)

1.5FLc ac°re -- B(H .qL_ nc )2 (1164 _ 862) (2-129)

where 64 and 62 -- deflection at quarter-span and midspan respectively F -- force over a support span Lc

2.24 CHAPTER TWO

Particular Formula

The shear modulus G of isotropic material if the modulus of elasticity E is available

G = E (2-130) 2(1 - u)

The modulus of elasticity of the core material (Fig. 2-31)

1 - V 2/3 ) (2-131) E j = Em l _ VZ/3 + V

where V = (Hh/H) 3, Ef = modulus of elasticity of foam, GPa (psi), Em= modulus of elasticity of basic solid material, GPa (psi). Subscript

H J.,,''l F f stands for foam/filament, m stands for matrix, and c stands for composite.

FIGURE 2-31 A unit cube foam subject to a tensile load.

The deflection for a beam panel according to Casti- gliano's theorem = OF - O---F 2El + 2GA

F W F

Yl Y

. . . . . . . . . . . . . : "t2""

E 4 -- + -- Defl2;~o n

L "-I [ " W = Phanton Load

L

FIGURE 2-32 Phantom load. FIGURE 2-33

¢

. . . . . . . . . . . . - " ° "

L .-[

.... X

The deflection at midspan (Fig. 2-32) OU 6L/2 = O W

0 M~ dx O W ( J 2 E I

5FL 3 FL t

349EI 8GA

J V2dx I + 2GA w=0

(2-133a)

(2-133b)

where W is the phantom load (Fig. 2-32).


Particular Formula

The deflection per unit width for a sandwich panel at midspan (Fig. 2-32) under quarter-point loading

The deflection per unit width for a sandwich panel at quarter panel (Fig. 2-32) under quarter-point loading

The deflection/unit width for a sandwich panel at o,~ 'er loading (Fig. " """ , . ~ . . ,. L - O D )

The maximum normal stress (Fig. 2-32)

The minimum normal stress

T~,=, L,~ average stress often used in the composite panel design

The maximum shear stress in the core

The core shear strain

FILAMENT REINFORCED STRUCTURES (Fig. 2-34)

The strain in the filament is same a s t h e strain in the matrix of composite material if it has to have strain compatibility

5FL 3 FL 6L/2 = 349DB t 8DcB

where De = Gc°re(H(H+2H~lIe))

FL 3 FL 6L/4 - - - 96DB t 8DcB

FL 3 FL ~L/2 = 48DB I 4DeB

(F L) M ~x~-

°max = - z = ( BhH~(h + Hc/2)

FL O'min --- 8BhH

FL(L + He) FL O'av - - I g7 D I , , L I I T ~ ~ r l l . / T T T T . . . . t1"cll ~DnLr~ + ~c)

~ITI ax --- V 2V

[B(H + H~)]/2 B(H + Hc)

Tnlax

")/core : Gcore

e m - - C f

F ~ ¢

FIGURE 2-34

Filaments ~_. _~m=Matrix.,,

I - - _7 ~ - -

rn . _ _ _

Y Filaments

FL 8BhH~

(2-134)

(2-135)

(2-136)

(2-137)

(2-138)

(2-139)

(2-140)

(2-141)

(2-142)

r F

2.26 CHAPTER TWO

P a r t i c u l a r F o r m u l a

The relation between stress in matrix and stress in filament

cr._~m = cry (2-143) Em E,

For equilibrium

The stress in the filament

The stress in the matrix

The Young's modulus of composite

The Young's modulus of chopped-up glass filaments in resin matrix but still oriented longitudinally with respect to load as proposed by Outerwater

The relation between am and ay, which has to satisfy Eq. (2-142) at any location on the curves, Fig. 2-35

From Eq. (2-144), the expression for cr C

F = crmA m -Jr- crf Af -- o'cA c (2-144)

FEf (2-145) °'f -- Af Ef _k_ ZmEm

FErn (2-146) O'm Af gf._f_ Amgm

Ec = Ef Af (2-147) (Am + Af)

a p , . ) ] (2-148)

where cr = applied tensile stress, MPa (psi) O'yf = the strength of the fiber, MPa (psi) Df = diameter of fiber, mm (in) Pc = uniform distance of one fiber from another on circumference, mm (in) L = length of fiber, mm (in)

Subscript chpd-f stands for chopped-up fiber.

Crm = ~ (2-149)

(Eo)m (Eo)f

where E0 = secant modulus, GPa (Mpsi)

O-c=(--~c)((E°)ma~m--f-Af) (2-150) ~E0;j

(ffu)f ( (°'m)maX Am + Af ) (2-151)

For structure with all filament, A m -- 0 ~rc = ( c r . ) f A j . = (cru)f (2-152)

Ac

For structure with no filament, Af = 0 (°'u)f ( (°'m)maX Am) .__ (O.m)max __ (O.u) m

(2-153)


Particular Formula

03

(%)m

O" m

m

0 (%)t (eu)r~ "~

Strain, e

FIGURE 2-35 Stress-strain data for system shown in Fig. 2-34.

h

. . . . . ~ _ . _ . _ . _ . _ ~ . _ . ~ . ~ _ ~ _ . . . . . . . . . . .

I &

FIGURE 2-36 Filament wound cylindrical pressure vessel.

F I L A M E N T BINDER C O M P O S I T E (Fig. 2-36)

H o o p stress for a closed end vessel/cylinder made of filaments winding

• . 1 . - A Longitudinal/axial stress for a c, os,,u end filament wound vessel/cylinder

The force carried by a helical filament wound on a shell of width w subjected to internal pressure p in the a-direction

The force in helical filament wound on a shell of width w subjected to internal pressure p in the hoop direction

The hoop stress in the vessel wall due to the pressure p

The stress in the vessel wall in the longitudinal/axial- direction

F r o m Eq. (2-154) to (2-159) the op t imum winding angle for closed end cylinders

The opt imum winding angle for open end cylinders

pd cr 0 = ] - ~ (2-154)

pd (2-155) era = 4h

Fa = O~o wh

where %0 = strength of the filaments

(2-156)

Fo (2-158) or0 = -~ -= o0 sin 2 a

aa = cr0 cos2 a (2-159)

cr 0 tan2 o~ 55 ° / , ~ I z - g ' t \

= - - or a ~ tz-lou~ O" a

O- a O" C O S 2 0 z . . . . . . cot 2 a or a = 90 ° (2-161) or0 or0 sin 2 a

Fo -- F~ sin oz (2-157)

2.28 CHAPTER TWO

Particular Formula

The stress in the hoop/circumferential direction for the filament wound cylinder/vessel consisting windings in longitudinal, hoop and helical directions to satisfy equilibrium condition

The longitudinal stress for the case of winding under Eq. (2-162)

F rom Eqs. (2-159) and (2-158)

F rom Eqs. (2-154) and (2-155)

The sum of stresses cr o and a .

For the ideal vessel

O- 0 -- o/oho + o'oe, h~

(2-162)

where a~ = stress in the circumferential wound layer

~0~ = circumferential component of stress in the helical layer

ht = total thickness = ha + ho + ha

hu, ho and ha are the thicknesses in the preceding layers of filament windings

~/ah; + cr~h~ (2-163a) O'a "-- h

O's° ) w h e r e t o = y (ho+hasin 2a)

( Crs° ) (ha + h~ cos2 a) O- a -- __.ff-

Cr,o = uniform filament stress

aa. = longitudinal component of stress in

helical layer

(2-163b)

(2-163c)

c~ 0 + cr a -- ~r0(sin 2 ~ + COS 20~) = O" 0 (2-164)

h - pd 4o- a

(2-165)

pd O" 0 - - - - ~ (2-154)

o" 0 a 0 + c r , = 3 c r , = a 0 or cr a = - ~ - (2-166)

h = 3pd 4~r 0

(2-167)

h ha = ~ - h,~ COS 20~ (2-168a)

ho = 2 _ ha sin 2 c~ (2-168b)

2ha - ho = (2-168c)

ha 1 - 3 cos 2 c~


Particular Formula

The structural efficiency of the wound vessel/cylinder W

r I -- VenP i

where W = weight of the vessel, k N 0bf) Ven : enclosed volume, m 3 (in 3)

Pi = internal pressure, MPa (psi)

(2-169)

FILAMENT-OVERLAY COMPOSITE

The stress in the wire which is wound on thin walled shell/cylinder with a wire of the same material (Fig. 2-37)

Under equilibrium condition over the length of shell L, the hoop stress

T O'wr =

UW

• -'~'"""~,,,~, ~ T = tension, kN t~m) ~ . . . . uw = area of the element, m 2 (in 2)

(2-170)

T (~o) ,h = w h

(2-171)

~ , , , . % d

T P ~ Shell

FIGURE 2-37 Shell subjected to an internal pressure.

The tension in the wound wire on the shell under internal pressure

The tension in the shell under the above same condition

The yielding of shell due to internal pressure, i.e., due to plastic flow of material of the shell.

For the above same winding material under the tension equal to compression yield limits, the stress in the wire.

If the vessel material is different from the winding material then stress in the wire and vessel

p d T T w r = t

2(h + u) wu

Tcy = p d T

2(h + u) wh

T [ \

IaO)shy = w h = - %

O'WF ~ UW ~ O'y

O'cy = CshEsh

(2-172)

(2-173)

(2-174)

(2-175)

(2-176a)

Crwr = ewrEwr (2-176b)

2.30 CHAPTER TWO

Particular Formula

For uniform distribution of stress in the cylinder/shell and in the wire, strains are proportional to the mean radii

From Eq. (2-177), the stress in the cylinder and the wire

The total load on the cylinder and the winding

From Eq. (2-179), the stress in the cylinder (acy) and the winding (awr)

The stress in the cylinder is the sum of results of Eqs. (2-180a) and (2-171)

The resultant stress in the winding is the sum of results of Eq. (2-180b) and (2-170)

For advanced theory using Theory of elasticity and Plasticity construction on composite structures and materials

For representative properties for fiber reinforcement

Fc_..~y .__ ecy = CrcyE,'_________~r (2-177) Fwr Ewr °'wrEcy

Ecyrcy (2-178a) O'sh : O'cy : O'wr Ewr~wr

EwrFwr (2-178b) O-wr -- O-cy Ecy~cy

where subscripts cy stands for cylinder, sh for shell and wr for winding. ?cy and Fwr a r e mean radii of cylinder and winding respectively.

O~y(2Lh) + Owr(2Lu) = pdL (2-179)

~r,y = pd (2-180a) 2 (EwrrwrU -+ - h )

\ Ecyrcy

O'wr _.. p d (2-180b) ( E,,y~cy. u) 2 ,,Ewrfwr -+-

pd T (2-181) O'Rcy--( Ewrfwru2 +h)-W---h

\ E,.y ~,.y

pd T aRwr = ~- (2-182)

(Ecy~o,h ) wu 2 k Ewrrw~ + u

Refer to advanced books and handbooks on composites, structures, handbooks and design data for reinforced plastics and materials.

Refer to Table 2-1

FORMULAS AND DATA FOR VARIOUS CROSS SECTIONS OF MACHINE ELEMENTS

For further data on static stresses, properties and torsion of shafts of various cross-sections: shear, moments, and deflections of beams, strain rosettes, and singularity functions

For summary of stress and strain formulas under various types of loads

Refer to Tables 2-2 to 2-12

Refer to Table 2-13

.. ¢-t

~'~

I

a~

o °~

=1.

g~

m

I v-q

X

v-q ×

U'--

e..~.

¢,¢%

¢-~

tt%

to%

• "-~ ,"'~

~

U'-,

¢~

.

¢,e~

,,~-

c',l ,--.~

,-.~

~

"~1

',~"

•

w%

t"4

~. 7

~

J U'--

,_Q

.,..~

0

M

< C

2.31

2 . 3 2 C H A P T E R TWO

T A B L E 2-2 Torsion of shafts of various cross sections

Polar section modulus,

Cross section Zo = J/c Polar radius of gyration, ko

Angular deflection, 0

In terms of torsional moment, Mt In terms of maximum stress, ~-

7rD 3

16

D = 0.354D

v~

32l Mt 2l 7" 7rD 4 D D G

r at circumference

7r(D~ - D 4)

16Dl ~/ 02 + Da

8

32l M, 2l r

- 0 . 3 5 4 v / D 2 + D 2 7r(D 4 - D a)--G- D1 G

T at outer circumference

7rb2h a 16

h > b

¼ v/b2 + h2 16(b 2 + h2)l M__._ Z (b 2 + h2)l 7 7rb3 h 3 G bh 2 G

7 a t A b

~ ~ 2b2ha9 V/b2 12 + h2 m(b2+b3h3 h2)l M,G n(b2bh 2 + h2)l 7G

.~-N~\-,~ A h > b =0"289v/b2+h2 --h-1 2 4 8 - r a t A c

b

b - ~ m = 3 . 5 6 3.50 3.35 3.21 n = 0.79 0.78 0.74 0.71

~ 0 ° b 3 a 46.2l Mt 2.311 r 2---0 0.289b b 4 G b

I'*-- b - ~ ~- at center of side

Q 0.9671 M, 0.9l 7- 0.92b 3 a 0.645b b 4 G b G

I'*b"l ~- at center of side

" This value is not true value of Z0 but is the value of Z0 for a circular section of equal strength and may be used for determining the maximum stress by the formula T = Mt/Zo. b At B, shear stress = lOM,/Trbh 2. c At B, shear stress = 9M,/2bh 2. Source: V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.


T A B L E 2-3 Shear stress in beams, caused by bending

Section Shear stress at a distance y from neutral axis, 7", MPa (psi)

Maximum shear stress, 7" . . . . MPa (psi)

i~6~1

o t

17/2zA~'f

3F 2bh[1-(~--~Y) 2]

37rr 2 r

r V z yVA y b2 1 + - ~ - 4

3F

2bh -- 1.5 F (for y = 0)

4F _ 1 . 3 3 F (for 0) 37rr--- 5 -- ~ Y =

P ¢ 1.591 ~ fory = ~

3F [qlbc2-(b-a)d2] (for 0) 4a LJ~C3-(b-a-~ Y =

T A B L E 2-4 The values of constants a in Eq. ( 2 - 1 0 7 )

Yield stress in compression, ~yc

Material MPa kpsi

Value of a for various end-fixity coefficients

1 4 2 n

Timber 49 7

Cast iron 549 80

Mild steel 324 47

1 1 1 1 750 3000 1500 n x 750

1 1 1 1

1600

1 7500

6400 3200 n x 1600

1 1 1 30000 15000 n x 7500

T A B L E 2-5 End condition coefficient n (F ig . 2-25)

Particular

One end fixed and the other end free Both ends rounded and guided or hinged One end fixed, and the other end rounded and guided or hinged Both ends fixed rigidly Both ends flat

0.25 1 2

4 1 t o4

T A B L E 2-6 End-fixity coefficients for cast iron column to be used in Eq. ( 2 - 1 0 4 )

End conditions C 1 Maximum, l /k

Round 175 90 Fixed 88 160 One fixed, one round 116 115

2.34 CHAPTER TWO

T A B L E 2-7 Properties of cross sections

Section Area, A Moment of inertia, I

Distance to farthest point, c

Section modulus, Z =I/c

Radius of gyration, k = f f ' - i - / A

H-b"i

I i" b'\\\\~

¢ KX,\\\'X'I..L t"b~

bo, vbo,

A

T

0

[= b - - - - I

bh

( H - c)b

B H - bh

2b + bo 2 . ) h

7rD 2

4

r ( D ~ - D 2 )

7tab

bh

2

bh 3

12

b _ h3 T~( H3 )

B H 3 - bh 3

12

(6b 2 + 6bbo + b2)h 3

36(2b + bo)

,n-D 4

64

7r (D 4 _ D4 )

= -~ (R 4 - R~) 4

rrba 3

64

bh 3

36

H

2

H 2

(3b + 2bo)h 3(2b+bo)

DI __ R 1 2

bh 2

6

b ( H 3 - h a)

3H

B H 3 _ bh 3

6 H

(6b 2 + 6bbo + bZ)h 2

12(3b + bo)

7rD 3

32

r(D 4 - D 4)

32D~

7rba 2

32

bh 2

24

0.289h

~ H 3 _ h 3

1 2 ( H - h)

. B H 3 - bh 3

1 2 ( B H - b h )

D

4

vIDe, + D~ 4

v/R~ + R~ 2

0.236h

,.r

m~

<

+ I I II I II

I II

+

I +

~1~ ~1~

,'~

I~:~

,'-"

I',~

I I

II II

0 0 I

,...._,

II II

0 0

-t- I

+ II

II II ¢-,i

I I II

0

I II

.xx.\\\\\\\\\\\,~ ~ o -.~

0

a:

0'~

+ I I II I II II

+ II t.--i

I II

I I II

0

I II ;a,

I II

,-.,1~.1 + II 0

I

+ II

(; 0

+ II

I II

+ II

+ II

,.~lC~

,-~lCq

+ I

II II

0 0 ¢,,I

az ¢xl

,x t~

-~

.-

~

o t ~,

0 g~ r~

. "~

o~ ~

0

2.35

.

m @

~5

<

,-..--, e,i

e,i

t i

.,-'

I i

i i

..~

"~" ~

A I

~

,-,le, i

-F

I

,

o o

~ II

II

0%

~i ~ ~

-I~

-I~

~ o t

II II

eq

m,. ii

~I~

+ +

~ I

,-,ioo

II

II +

~ +

~ il

0 0

~

:1 ~ -

~ ~

~ ,

~ ~

~

~1~ r~

~1~ ~1~

-'" ~,1~

I I

- II

il II

il il

~ II

0 0

~ ~

~

+

r~ r~

4- I

.~ 12.. II

II

+ ~

II

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2.38 CHAPTER TWO

TABLE 2-9 Some equations for use with the Castigliano method

Type of load General energy equation Energy equation General deflection equation

Axial

Bending

Combined axial and bending

Torsion

Transverse shear

Transverse shear (rectangular section)

Open-coiled helical spring subjected to axial load F

Ii F2 F21 °2 Al U = ~ ds U = zAE = 2E

j' M~ t M~ t U = o - ~ ds U =~2EI

U = ds+ ds U = + ~ o 2--A-E o ~ - i ~ 2El

[i M2 M2l U = ~ ds U = 2a----J

IO V 2 l r 2 l V 2ds U = 2 G A - 2 G A I U = 2GA ~ -

Ii V2 U = -~-~ds U = 3 vz---J/5GA

U = ds+ ds U = + ~ o ~ - J o 2-E-i ~ 2El

LFR 2 [c0s2..0~ sin 2 a ]

2 L a J +---E--I -

Ii F(OF/OQ) ds 6= AE

Ii Mb(OMb/OQ) ds 6 = E1

Sum of axial and bending load

6 = it Mt(OMt/OQ) ds J o GJ

i t v ( o v / o a )

6= ds o GA

Ji V(OV/OQ) ds 6 = 5GA

[COS 2 a sin 2 a ] 6 = 27riFR 3 sec a [ GJ + E1 J

D where R = -~ = mean radius of coil

a = helix angle of spring i = number of coils or turns

STATIC STRESSES IN M A C H I N E E L E M E N T S 2 . 3 9

T A B L E 2-10 Mechanica l and physical constants of some materials 1'2

Modulus of Modulus of elasticity, E rigidity, G

Material GPa Mpsi GPa Mpsi

b Density, Unit weight, "7 Poisson's p a, ratio, v Mg/m 3 kfg/m 3 kN/m 3 lbf/in 3 lbf/ft 3

Aluminum 69 10.0 26 3.8 0.334 2.69 2,685 26.3 0.097 167 Aluminum cast 70 10.15 30 4.35 2,650 26.0 0.096 166 Aluminum (all alloys) 72 10.4 27 3.9 0.320 2.80 2,713 27.0 0.10 173 Beryllium copper 124 18.0 48 7.0 0.285 8.22 8,221 80.6 0.297 513 Carbon steel 206 30.0 79 11.5 0.292 7.81 7,806 76.6 0.282 487 Cast iron, gray 100 14.5 41 6.0 0.211 7.20 7,197 70.6 0.260 450 Malleable cast iron 170 24.6 90 13.0 7,200

Inconel 214 31.0 76 11.0 0.290 8.42 8,418 83.3 0.307 530 '7 Ch 1 ".6 " " " : M aenesium _ allov . 45 _ 6 . . . . S 16 _ 9.4 n,,..,,,,, "1 ~fl • 1 .o,,Qn I,, 9=, x, v.vu~ I 17

Molybdenum 331 48.0 117 17.0 0.307 10.19 10,186 100.0 0.368 636 Monet metal 179 26.0 65 9.5 0.320 8.83 8,830 86.6 0.319 551 Nickel-silver 127 18.5 48 7.0 0.332 8.75 8,747 85.80 0.316 546 Nickel alloy 207 30 79 11.5 0.30 8.3 0.300 518 Nickel steel 207 30.0 79 11.5 0.291 7.75 7,751 76.0 0.280 484 Phosphor bronze 111 16.0 41 6.0 0.349 8.17 8,166 80.1 0.295 510 Steel (18-8), stainless 190 27.5 73 10.6 0.305 7.75 7,750 76.0 0.280 484 Titanium (pure) 103 15.0 4.47 4,470 43.9 0.16 279 Titanium alloy 114 16.5 43 6.2 0.33 6.6

Brass 106 15.5 40 5.8 0.324 8.55 8,553 83.9 0.309 534 Bronze 96 14.0 38 5.5 0.349 8.30 8,304 81.4 Bronze cast 80 11.6 35 5.0 8,200

Copper 121 17.5 46 6.6 0.326 8.90 8,913 87,4 0.322 556 Tungsten 345 50.0 138 20.0 18.82 18,822 184.6 Douglas fir 11 1.6 4 0.6 0.330 4.43 443 4.3 0.016 28 Glass 46 6.7 19 2.7 0.245 2.60 2,602 25.5 0.094 162 Lead 36 5.3 13 1.9 0.431 11.38 11,377 111.6 0.411 710 Concrete (compression) 14-28 2.0-4.0 2.35 2,353 23.1 147 Wrought iron 190 27.5 70 10.2 7,700 Zinc alloy 83 12 31 4.5 0.33 6.6 0.24 415

a p __ mass density. b ,y = weight density; w is also the symbol used for unit weight of materials. Sources: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, and K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India. 1986.

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Function Graph off . (x) Meaning

Concentrated moment

Concentrated force

Unit step

Rump

Parabolic

< x _ a >-2

< x _ a >-1

< x - a > O

< x - a > 1

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REFERENCES

1. Maleev, V. L. and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.

2. Shigley, J. E., Mechanical Engineering Design, 3rd edition, McGraw-Hill Book Company, New York, 1977. 3. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. 1 (SI and Customary Metric

Units), Suma Publishers, Bangalore, India, 1986. 4. Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,

Bangalore, India, 1986. 5. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 6. Ashton, J. E, J. C. Halpin and P. H. Petit, Primer on Composite Materials-Analysis, Technomic Publishing

Co., Inc., 750 Summer St., Stanford, Conn. 06901, 1969. 7. Roark, R. J., and W. C. Young, Formulas for Stress and Strain, McGraw-Hill Publishing Company, New

York, 1975. 8. Hertz, H., On the Contact of Elastic Solids, J. Math. (Crelle's J.) Vol. 92, pp. 156-171, 1981. 9. Hertz, H., On Gesammelte werke, Vol I., p. 155, Leipzig, 1895.

10. Timoshenko, S., and J. N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, New York, 1951.

BIBLIOGRAPHY

1. Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1965. 2. Lingaiah, K, and B. R. Narayana Iyengar, Machine Design Data Handbook (fps units), Engineering College Co-

Operative Society, Bangalore, India, 1962. 3. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company,

New York, 1951. 4. Vallance, A. E., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York,

1951. 5. Timosheko, S., and J. N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, New York, 1951. 6. Timoshenko, S., and J. M. Gere, Mechanics of Materials, Van Nostrand Reinhold Company, New York, 1972. 7. George Lubin, Editor, Handbook of Composites, Van Nostrand Reinhold Company, New York, 1982. 8. John Murphy, Reinforced Plastic Handbook, 2nd edition, Elsevier, Advanced Technology, 1998. 9. Hamcox, N. L., and R. M. Mayer, Design Data for Reinforced Plastics, Chapman and Hall, 1994.

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