Rules of Thumb for Mechanical Engineers 2010 Part 10 ppsx

ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, 1983 Edition, American Society of Me- 2.. ASME Boiler and Pressure Vessel Code, Section VIII, Division 2, 1983 Edition, Am

Procedure 2

Stresses in Heads Due to Internal Pressure [3,4]

Notation

L = crown radius, in

r = knuckle radius, in

h = depth of head, in

RL = latitudinal radius of curvature, in

R, = meridional radius of curvature, in

q, = latitudinal stress, psi

ox = meridional stress, psi

P = internal pressure, psi

1 Latitudinal (hoop) stresses in the knuckle become

compressive when the R/h ratio exceeds 1.42 These

heads will fail by either elastic or plastic buckling, de-

pending on the R/t ratio

2 Head types fall into one of three general categories:

hemispherical, torispherical, and ellipsoidal Hemi-

spherical heads are analyzed as spheres and were cov-

ered in the previous section Tonspherical (also known

as flanged and dished heads) and ellipsoidal head for-

mulas for stress are outlined in the following form

Figure I Direction of stresses in a vessel head

Joint Efficiencies (ASME Code) [5]

See Note 3 S.O flange

Figure 1 Categories of welded joints in a pressure vessel

Table 1 Values of Joint Efficiency, E, and Allowable Stress, S*

*See Note 1

jSee Note 2

218 Rules of Thumb for Mechanical Engineers

Notes Table 2

Joint Efficiencies

X-Ray

Single and joints

1.0 85 .7

Single butt ing strip joint with back- 9 .8 .65

Single butt joint witho;; backing - - .6 Double full

joint Single full with plugs Single full joint

2 Combination radiography: Applies to vessels not fully radiographed where the designer wishes to apply a joint efficiency of 1.0 per ASME Code, Table UW-12, for only a specific part of a vessel Specifically for any part

to meet this requirement, you must perform the fol- lowing:

(ASME Code, Section UW-ll(5)): Fully x-ray any Cat A or D butt welds

(ASME Code, Section UW-l1(5)(b)): Spot x-ray any Category B or C butt welds attaching the part (ASME Code, Section UW-l1(5)(a)): All butt joints must be Type 1

3 Any Category B or C butt weld in a nozzle or com- municating chamber of a vessel or vessel part which

is to have a joint efficiency of 1 O and exceeds either

10 in nominal pipe size or 1% in in wall thickness shall

be fully radiographed See ASME Code, Section UW-

Figure 1 Dimensions of heads

Table 1 Partial Volumes

220 Rules of Thumb for Mechanical Engineers

3 Conversion factors

Multiply ft3 x 7.48 to get gallons

Multiply ft3 x 62.39 to get lb-water

Multiply gallons x 8.33 to get lb-water

1 = height of cone, depth of head, or length of cylinder

a = one-half apex angle of cone

D = large diameter of cone, diameter of head or cylinder

R = radius

r = knuckle radius of F & D head

L = crown radius of F & D head

h = partial depth of horizontal cylinder

cone

r e(D' + Dd + d2)

12

30° Truncated .227(D3 - d3) 1.57 (D' - d2) cone

Table 2 Values of c for Partial Volumes of a Horizontal Cylinder

.5

.55 6 65 7 75

.8 85

3735 A364

.5

5636 6265 6881 7477 b045 8576 9059

.9480

.9813

Maximum length of Unstiffened Shells

1 All values are in in

2 Values are for temperatures up to 500OF

3 Top value is for full vacuum, lower value is half vacuum

4 Values are for carbon or low alloy steel (Fy > 30,000 psi) based on Figure UCS 28.2 of ASME Code, Section VIII, Div

222 Rules of Thumb for Mechanical Engineers

Useful Formulas for Vessels C2,61

1 Properties of a circle (See Figure 1 .)

8xR2 - 180C(R - b)

A, =

360 xR2 a

Figure 1 Dimensions and areas of circular sections

Moment of inertia, I

I = n R i t xDkt

6 = -

Et Cone

(1 - S V )

PR2

Et cos a

6 =

where P = internal pressure, psi

R = inside radius, in

t = thickness, in

v = Poisson’s ratio (.3 for steel)

E = modulus of elasticity, psi

a = % apex angle of cone, degrees

o$ = circumferential stress, psi

ox = meridional stress, psi

4 Longitudinal stress in a cylinder due to longitudinal

where E =joint efficiency

R = inside radius, in

ML = bending moment, in.-lb

where L = crown radius, in

P, = external pressure, psi

E = modulus of elasticity, psi

6 Equivalent pressure of flanged connection under ex-

M = bending moment, in.-lb

G = gasket reaction diameter, in

7 Bending ratio of formed plates

% = lOOt [ 1 - 2)

R f

where Rf = finished radius, in

R,, = starting radius, in (= for flat plates)

8 Stress in nozzle neck subjected to external loads

t = thickness, in

PRm +-+- F MR, 2t, A I

ox =-

where R, = nozzle mean radius, in

t,, = nozzle neck thickness, in

A = metal cross-sectional area, in.2

I = moment of inertia, in?

F = radial load, lb

M = moment, in.-lb

P = internal pressure, psi

9 circumferential bending stress for out of round shells [2]

where D1 = maximum inside diameter, in

D2 = minimum inside diameter, in

P = internal pressure, psi

E = modulus of elasticity, psi

t = thickness, in

Figure 2 Typical nozzle configuration with internal baffle

References

1 ASME Boiler and Pressure Vessel Code, Section VIII,

Division 1, 1983 Edition, American Society of Me-

2 ASME Boiler and Pressure Vessel Code, Section VIII,

Division 2, 1983 Edition, American Society of Me-

chanical Engineers

3 Harvey, J E, Theory and Design of Modem Pressure Ves-

sels, 2nd Ed New York Van Nostrand Reinhold Co.,

1974

4 Bednar, H H., Pressure Vessel Design Handbook New

York Van Nostrand Reinhold Co., 1981

5 National Board Bulletin, Vol 32, No 4, April 1975

6 k k , R J., Formulas forStressandStmin, 5thEd New

Source

Moss, Dennis R., Pressure Vessel Design Manual, 2nd Ed Houston: Gulf Publishing Co., 1997

10

Tribology

Thomas N Farris Ph.D., Professor of Aeronautics and Astronautics Purdue University

Introduction 227 Friction 235

Contact Mechanics 227 Wear 235

Two-dimensional (Line) Hertz Contact of Cylinders 227 Lubrication 236

Three-dimensional (Point) lh-tz Contact 229 References 237

Effect of Friction on Contact Stress 232

Yield and Shakedown Criteria for Contacts 232

Topography of Engineering Surfaces 233

Definition of Surface Roughness 233

Contact of Rough Surfaces 234

Life Factors 234

226

I WTRODUCTIO W

Tribology is the science and technology of interacting sur-

faces in relative motion and the practices related thereto

The word was officially coined and defined by Jost [ll]

It is derived from the Greek root tribos which means rub-

bing Tribology includes fiction, wear, and lubrication

Tribology has several consequences in mechanical com-

ponents and modern-day life Most consequences of fric-

tion and wear are considered negative, such as power con-

sumption and the cause of mechanical failure However,

there are also some positive benefits of friction and wear

It is estimated that 20% of the power consumed in au-

tomobiles is used in overcoming friction, while friction ac-

counts for 10% of the power consumption in airplane pis-

ton engines and 1.5-2% in modern turb0jets:Friction also

leads to heat build-up which can cause the deterioration of

components due to thema-mechanical fatigue Under-

standing friction is the first step towards reducing friction

through clever design, ,the use of low-friction materials, and

the proper use of lubricating oils and greases

Friction has many benefits, such as the interaction be-

tween the tire and the road and the shoe and the floor with-

out which we would not be able to travel Friction serves

as the inherent connecting mechanism in knots, nails, and

the nut and bolt assembly It has some secondary benefits,

such as the interaction between the fiber and matrix in

composites and damping which may reduce deleterious

effects due to resonance

This chapter begins with a description of contact m e chanics and surface topography in sufficient detail to discuss friction, wear, and lubrication in the latter sections Tribolo-

gy is a rich subject that cannot be given justice in the space permitted, and the interested reader is encouraged to pursue the subject in greater depth in any of the following books: BowdenandTabor [3],Rabinowicz [13],Hall1ng PI, Suh [14], Bhushan and Gupta [l], and Hutchings [9] In addition, an enormous array of material properties is available in tribol-

ogy handbooks such as Peterson and Winer [ 121 and Blau [2]

The solution of contact problems can be reduced to the

solution of integral equations in which the known right-hand

sides relate to surface geometry and the unknown underneath

the integral sign relates to the unknown pressure distribu-

tion Details of the derivation and solution of these equa-

tions that highlight the necessary assumptions can be found

in Johnson (1985)

~

Two-dimensional (line) H k z Contactof Cylinders

In this section, the contact stress distribution for the fric-

tionless contact of two long cylinders along a line parallel to

their axes is derived Figure 1) This is a special case of the

contact of two ellipsoidal bodies first solved by Hertz in 1881

The origin is placed at the point where the cylinders

first come into contact At first contact the separation of the

cylinders is:

h = z, + z2

zI =R, -,/-

z2 = R, - 4

228 Rules of Thumb for Mechanical Engineers

Figure 1 Long cylinders brought into contact by a load

per unit length R

If the contact length is small compared to the size of the

cylinders, a << R, then Equation 1 can be approximated as:

where 1/R = l/R1 + I& The loads cause the cylinders to

approach each other a distance 6 = 61 + 6, The cylinders

must deform to cancel the interpenetration This is written

in equation form as:

or

w1 +w2 =6

2R Differentiating gives:

as, =* _ _ _ X

ax ax R

Using the equations of elasticity, the previous equation

can be written as:

a p(s)ds - xE*

I-a x - s 2R X

- a c x < a

where 1/E* = (1 - v?)/E, + (1 - v:)/E2

and the maximum contact pressure is:

It is interesting to note that the maximum pressure varies

as the square root of the load rather than linearly with the load This is because the contact length increases with the applied load, resulting in an increased area that bears the load From this perspective, hertzian contacts are very for- giving in the sense that overloads of a factor of two only increase the resulting contact pressure and subsurface stresses by a factor of a

Ductile materials will be subject to plastic deformation once the maximum shear stress in the material reaches the shear stress at yield in a tensile test The symmetry of the

problem requires that initial yielding occurs along the z-axis,

which is a line of symmetry Because T = 0 along the z- axis, the stresses ox and o, are principal stresses and the maximum shear stress is one-half of their difference These stresses along x = 0 are [ 101:

This equation can be inverted for p(x) and the constant

of integration is used to assure that the stress is continuous

at the edge of contact resulting in:

Position (x/a)

Figure 2 Hertz line contact pressure distribution

The maximum shear stress occurs below the surface at a

depth of z = 0.78a and has a value of 2- = 0 3 ~ ~ These

stress distributions are shown in Figure 3

Figure 3 Stress below the surface for Hertz contact

The Westergaard stress functions [ 171 can be used in con-

junction with a simple FORTRAN program to evaluate

the subsurface stress field induced by Hertz contact For fric-

(4)

where P is the contact force per unit length, a is the half con- tact length, and 2 = x + iz with i = cl This stress func- tion satisfies all of the traction boundary conditions along

z = 0 The branch cut on the radical in Equation 4 is cho- sen so that Z, + 0 as 2 + 0 Contours of the in-plane maximum shear stress:

( 5 )

are shown in Figure 4 The maximum in-plane shear stress

is about 0.3 po, where po is the maximum contact pressure,

and it occurs on the z-axis at a depth of about z - 0.78a

Figure 4 Stress contours of , T /po for Hertz contact

Three-dimensional (Point) Hertz Contact

In this section, the equations for three-dimensional or

point contacts are derived The approach taken is very sim-

ilar to that used in two-dimensional problems where the

point force is superposed to yield the distributed load so-

lutions The point load solution is derived using Love's ax-

isymmetric stress function, and more details can be found

in Chapter 12 of l'imoshenko and Goodier [ 161 and Chap-

ters 3 and 4 of Johnson [lo]

Contact of Spheres

For an ellipsoidal pressure distribution applied to a cir-

cle of radius, a, such that p(r) = po m2:

- n: 1-vz po u,= - (2a2 - r2), r e a

Consider the spheres being brought into contact in Fig-

ure 5 The load is P, the total approach is 6, and the radius

of contact is a Geometric considerations very similar to those for the contacting cylinders reveal that the sum of the displacements for the two spheres should satisfy:

230 Rules of Thumb for Mechanical Engineers

113

Po=(=) 6PE*2

These equations describe Hertz contact for spheres No- tice that these results are nonlinear and that the maximum pressure increases as the load is raised to the % power The corresponding surface stresses can be calculated as:

Figure 5 Hertz contact of spheres

- uzl + iiz2 = 6 - -1_r2

2R

[I - (1- r2/ a2 ) 3 1 2 ]

I-2v a2 where 1/R = l/R1 + 1/R2 That is, a pressure distribution 5, = - po

which gives a constant plus 1.2 term is needed to cancel the potential interpenetration of the spheres Comparing Equa- tions 6 and 7 reveals that the contacting spheres induce an ellipsoidal pressure distribution and

- - (1 - 2v)a2

(J z - 0 - This equation must be valid for any r < a requiring: e -Po 3r2

a= I T POR

2 E * and

Global equilibrium requires:

Finally:

outside the contact patch (r > a) These stresses are shown

in Figure 6 Notice that the radial stress is tensile outside the circle and that it reaches its maximum value at r = a This

is the maximum tensile stress in the whole body

0.25 0

5 0

1 -

-

-

The stresses along the z-axis (r = 0) can be calculated by

first evaluating the stress due to a ring of point force along

r = r and integrating from r = 0 to r = a For example:

bodies are aligned, the axes of the ellipse correspond to these directions Placing the x and y axes in the directions of prin-

cipal curvature, the equation comparable to Equation 7 is:

2R’ 2R” y2 where

Symmetry dictates that the maximum shear stress in the

body occurs along r = 0 Manipulation of the above equa-

tions leads to:

3 a’

=Po (1+v) 1 tan-’- [ ( E :I - 2 z z + a 2 ]

For v = 0.3, the maximum shear stress is about 0 3 1 ~ ~ and

occurs at a depth of approximately z = 0.48a The stresses

are plotted for v = 0.3 in Figure 7

and Ri are the curvatures in the x direction and Rr are the

curvatures in the y direction

The contact area is an ellipse, and the resulting pressure distribution is semi-ellipsoidal given by:

x2 y2 P(X¶ Y)“ PoJ - 2- 2

The actual calculation of a and b is cumbersome Howev-

er, for mildly elliptical contacts (Greenwood [5]), the con-

tact can be approximated as circular with:

stress (dpJ

with 6 and po given by Equations 9 and 10

R2, respectively, the effective radii are given by:

Note that for contact of crossed cylinders of radii R1 and

1 1 +- 1R’ R, = -=-

-=-+- R” = R,

Figure 7 Subsurface stresses induced by circular point

contact (v = 0.3)

so that if Rl = R2, the contact patch is circular and the equa- tions of the previous section (“Contact of Spheres”) hold