Machine Design Databook Episode 3 part 12 potx

27-158 after equating real and imaginary parts Rectangular plate under plane flexure Assume values of complex potentialsðzÞ and !ðzÞ as y ¼ b are stress free... 27-158 Thick cylinder unde

Specified displacement

From Eq (27-158) for D

If body force is absent Eq (27-161) becomes

FORCE AND COUPLE RESULTANTS

AROUND THE BOUNDARY (Fig 27-31)

The expression for force with components X and Y at

point O

The expression for couple at O

GENERALIZED PLANE STRESS

The average stress combinations assumingz¼ 0, a

stress free surface, i.e xz¼ yz¼ 0 at the surface

and body force potential Uðz; zzÞ is independent of z

ð3
4vÞðzÞ
z0ðzzÞ
!!ðzzÞ ¼ 2GD
1
2v

2ð1
vÞwð27-161Þð3
4vÞðzÞ
z0ðzzÞ
!!ðzzÞ ¼ 2Gðg1þ ig2Þ on C

ð27-162Þwhere g1and g2are functions of z only

where

o¼ 12h

FIGURE 27-31

y y

x x

z

2h O

FIGURE 27-32

The average complex displacement

The body force Eq.@

Taking into consideration the body force, Eq (27-167)

and other expression for F and obecome

The equations for generalized plane stress

CONDITIONS ALONG A STRESS-FREE

BOUNDARY, Fig 27-33

Adding Eqs (27-169) and (27-170) and putting F ¼ 0

along free boundary, i.e segment AB, the

displace-ment along AB

SOLUTION INVOLVING CIRCULAR

BOUNDARIES (Figs 27-33 and 27-34)

From stress strain transformation rules

The boundary conditions are

APPLICATION OF CONFORMAL

TRANSFORMATION (Fig 27-35)

The stress combinations after transformation

Eqs (27-178) are related stress combinations in

rectangular coordinates x and y as

0¼ F e
2i¼ r
þ 2i ¼zz

zwherer¼ , z ¼ r ei,zz ¼ r e
i

@w

@zð27-175Þ2GD0¼ e
i

An explanation for e
2i

Using Eqs (27-179a) and (27-179b), and Eqs (27-171)

and (27-172), when these are no body forces, letting

ðzÞ ¼ 1ðÞ and !ðzÞ ¼ !1ðÞ

The transformation of a given boundary in the

z-plane into the unit circle in the-plane

Using polar coordinates (, #), the stress components

become

Using polar coordinates Eqs (27-180a) and (27-181)

in terms of complex potentials become

where

e
2i¼ zz0ðÞ=z0ðÞ ð27-179cÞor

ð27-181aÞor

0¼
2

z0ðÞ

1zðÞ

ξ

FIGURE 27-35

Rectangular plate under all round tension

Value of complex potentialsðzÞ and !ðzÞ assumed

From stress combination Eqs (27-156c) and (27-157)

The stressxandyafter equating real and imaginary

parts

The displacement from Eq (27-158) after equating

real and imaginary parts

Rectangular plate under plane flexure

Assume values of complex potentialsðzÞ and !ðzÞ as

y ¼ b are stress free

y

T

x z y

y dy T

2b

2h T

Boundary conditions

From stress combinations Eqs (27-156) and (27-157)

boundary conditions

The bending moment

The values of complex potentials ðzÞ ¼ Az2 and

!ðzÞ ¼ Bz2

are

The displacement from Eq (27-158)

Thick cylinder under internal and external

pressure

Values of complex potentialsðzÞ and !ðzÞ assumed

using boundary conditions at r ¼ a or di=2 and

r ¼ b or do=2 with no body forces, assuming internal

pressure pi, external pressure po, values of A and B

in Eq (27-189), which are real, can be found From

Eqs (27-174) and (27-175)

The expressions forandrat any radius

Rotating solid disk and hollow disk of

uniform thickness rotating at ! rad/s

Values of complex potentialsðzÞ and !ðzÞ assumed

0ðzÞ þ 0ðzzÞ þ zz00ðzÞ þ !0ðzÞ ¼ yþ ixy ð27-156Þ

y¼ 0, xy¼ 0 throughout the plate

A ¼ iC and B ¼
iC where C is real

hð3
4vÞðzÞ
z0ðzzÞ
!!ðzzÞi¼ u þ ivwhen body forces are zero

Substituting the values ofðzÞ and !ðzÞ in the above,

u and v can be determined

ðzÞ ¼ Cz and !ðzÞ ¼B

where C and B are real

Using boundary conditions at ðrÞr ¼ b¼ 0 and

ðrÞr ¼ 06¼ 0 for solid disk ðrÞr ¼ a¼ 0 and

ðrÞr ¼ b¼ 0 for hollow disc taking into consideration

body forces, values of C and B in Eq (27-189c) which

are real can be found

The radial displacements at the boundaries

Large plate under uniform uniaxial tension

with a centrally located unstressed circular

hole

Values of complex potentialsðzÞ and !ðzÞ assumed

Using Eq (27-189b) and above complex potentials

Using boundary condition at r ¼ a

The new values ofðzÞ and !ðzÞ

Using Eqs (27-174), (27-175) and after equating the

real and imaginary parts, the stress components are

x T T

y a A

2b4E fð1
vÞb2þ ð3 þ vÞa2g ð27-189eÞ

z3zz ð27-190aÞðr
irÞr ¼ a¼

1

2T þB

a2

þ

1

2T þA

a2



e2iþ

3C

ðzÞ ¼Tz

4 þ12

r2
3a

4

r4

sin 2 ð27-193Þ

The randrat r ¼ a

The maximum tangential stress

The stress concentration factor

Large plate containing a circular hole under

uniform pressure

Values of complex potentialsðzÞ and !ðzÞ assumed

From Eqs (27-174) and (27-175) in the absence of

body forces

Boundary conditions are

The new complex potentials

The stress components are

The displacement from Eq (27-176)

Large plate containing a circular hole filled

by an oversize disk

1 Rigid Disk

The radius of disk rd

ðrÞr ¼ a¼ ðrÞr ¼ a¼ 0 ð27-194aÞðÞr ¼ a¼ Tð1
2 cos 2Þ ð27-194bÞ

From first of Eq (27-198), the radial displacement

The stress components

2 Elastic Disk

The complex potential for all round pressure on the

disk

The displacement from Eq (27-176)

The radial displacement of plate

The pressure between disc and plate

Elliptical hole in a large plate under tension

Transforms the outside of an ellipse of semiaxes a and

b in the z-plane into the outside of a unit circle r in the

or x þ iy ¼ a cos # þ ib sin # ð27-200dÞ

# ¼ eccentric angle around the ellipse

The points at which the transformation ceases to be

conformal are

The boundary condition at the stress free ellipse

The boundary condition in terms of

Eq (27-203) on unit circle becomes

The complex potentials for an infinite plate without a

hole acted upon by uniaxial tension at an angle to

the x-axis in the z-plane and -plane

The complex potentials for an infinite plate with stress

free elliptic hole subject to tension at an angle to the

zz0ðÞ1ðÞ þ zðÞ0 1ðÞ þ zz 0ðÞ þ zz0ðÞ!!1ðÞ ¼ 0

ð27-203Þ

zz0ðÞ1ðÞ þ zðÞ0 1ðÞ þ zz 0ðÞ!!0 1ðÞ ¼ 0or





þ zz01





!!1

1





¼ 0ð27-204Þwhere  ¼ ;  ¼  ¼1

on since  ¼ 1 on unitcircle

T

T

FIGURE 27-39

Using Eqs (27-206) in Eqs (27-204), after equating

coefficients of powers of  or  (since

E1¼ B ¼ C ¼ 0, D1¼ 1 þ m2
2Me2i, F1¼
e2i,

A ¼ 2e2i
mÞ

The tangential stress on the boundary of elliptical

hole from Eq (27-183) 00¼ #þ  where ¼ 0

after equating to real part of right hand side of

equation and simplification (Fig 27-39)

The tangential stress on the boundary of elliptical

hole for ¼ 0 (Fig 27-40)

The maximum tangential stress# maxon the contour

of any elliptical hole for any value of m ¼a
b

The stress concentration factor

By taking ¼ 458 with T ¼
S, and  ¼
458 with

T ¼ þS, and on adding these solutions, a solution

for pure shear S applied to an infinite flat plate with

an elliptical hole at infinity is obtained The shear

will be parallel to the axes of the ellipse with 

around the elliptical hole is given by

MUSKHELISHVILI’S DIRECT METHOD

In this method that a hole L can be transformed

conformally into a unit circle  in the -plane so

that outside of the hole is mopped on the inside of

(Fig 27-41)

The form of the conformal transformation will be

If the loading of the plate at infinity is given by the

complex potential ðÞ, !ðÞ, the full complex

potentials which will also satisfy the condition

around the hole, can be written as

þ e1 þ e22þ e33þ    þ enn

ð27-211Þ

The boundary condition around a stress free hole

assuming no body forces is given by (refer to Eqs

(27-203) and (27-204))

Substituting the complex potentials given by Eqs

(27-212), in Eq (27-214)

Using Harnack’s theorem, residue theorem and

Cauchy’s integral, multiplying by 1

2 i

d


 andintegrating around Eq (27-214) can be written as

The complex potentialoðÞ from Eq (27-216) is





þ !!

1



 0 o

1





þ !!o

1





¼ f1þ if2ð27-215aÞwhere





þ !!1



2

6

37ð27-215bÞ1



 0 o

1

oðÞ þ 1

2 i

ð

zðÞ 0o

1





zz01



ð
Þd ¼

S

S

S S

S

S

θ

S a

b

45 45

FIGURE 27-40

y

x L

Taking conjugate of Eq (27-215), remembering that

 ¼ 1 and multiplying by 1

2 i

d


and integratingaround

The complex potential !oðÞ can be found after

substituting the value of oðÞ Eq (27-218) which

can be evaluated from Eq (27-217)

Stress free square hole in a flat plate under

uniform uniaxial tension (Fig 27-42)

The form of the conformal transformation will be

The known complex potential in this case

After substitutingðÞ and !ðÞ from Eqs (27-221)

and (27-222) into Eq (27-215)

After substituting the value of f1þ if2 from

Eq (27-223) in Eq (21-217) and simplification

Substituting the value ofoðÞ from Eq (27-224) in

Eq (27-219) and after simplification

y

T T

x d

FIGURE 27-42

z ¼ C

1



36



36



36


2

3

3 þ313

ð27-223Þ

oðÞ ¼ TC

3

The full complete complex potentials after

simplifi-cation

The tangential stress around the square hole

By adding more terms to the expression for

trans-formation

The radius r will be rounded of

For graph of#=T versus # in degrees

Stress free square hole in a flat plate under

pure bending (Fig 27-44)

The conformal transformation for plate with a square

hole such that the diagonals along the coordinate axes

as shown in Fig 27-44

The known complex potentials from Eqs (27-188a)

ðÞ ¼ TC

3

7 þ14

z ¼ C

1

Refer to Fig 27-43

z ¼ C

1

I

II

II II

III

III III

The complete complex potentials in-plane will be of

the form

From Eq (27-217)

From Eq (27-219)

The full complex potentials become often simplifying

After knowing full complex potentials, the tangential

stress at various angles around the hole/cutout can be

calculated

For graphs ofv=ðMbc=IÞ versus # degree

Large plate containing an elliptical hole

subjected to uniform pressure (Fig 27-46)

The expression for transformation

The complex potential at infinity

The required complex potentials

Boundary conditions

ðÞ ¼
iMbC2

8I

1

4

32
1

34þ 1

166
3718

ð27-232Þ

ðÞ ¼
iMbC2

8I

1

bb

n¼ ¼
p; ns¼ 0 around the hole ð27-236Þ

remember that  is now a boundary to the region

external to the unit circle Thus it is necessary to

con-sider an integration around a contour consisting of

together with C circle 0of large radius R joined by

two close paths AB and CD, Fig 27-46]

Using Cauchy’s integral, Harnack’s theorem and

residue theorem, Eq (27-239) gives the expression





þ !!

1



d

I - Second moment of inertia

= Angle form z-axis to a point

on hole boundary

x

III II

I

FIGURE 27-45

b p

R A

γγ

FIGURE 27-46

Taking conjugate of Eq (27-239) and integrating

around, the expression for !ðÞ

The stress can be obtained by making use of

Eq (27-183) for00and (27-184b) for00and equating

real parts on both sides of equation

The tangential stress around by elliptical hole from

Eq (27-243)

Large flat plate under uniform uniaxial

tension with a circular hole whose edge is

rigidly fixed (Fig 27-49)

The edge of the hole r ¼ a is held fixed by a rigid

circular ring to which the material of the plate adheres

at all points

The boundary condition is given by T

The complex potential form of displacement for

generalized plane stress problem from Eqs (27-170)

when there are no body forces

KðzÞ
z 0ðzzÞ
!!ðzzÞ ¼ 0 on r ¼ a ð27-246cÞwhere

K ¼3
v

1þ vKðÞ
zðÞ0ðÞ

The conformal transformation for this problem can

Multiplying the conjugate of Eq (27-229) by

1

2 i

d


and integrating around and after

simplifi-cation, expression for!oðÞ

The full complex potentials are

From the Eqs (27-182a) and (27-182b) for00and00,

the following stress components are

TORSION (Fig 25-49)

The angle of twist , which is proportional to the

distance of cross-section from the fixed end

ðÞ ¼14



 0 o

1





!!o

1



2K

Pðx; y; zÞ is a point in a section of bar z-distance from

fixed end (Fig 27-48) and it is displaced to a new point

P0ðx þ u; y þ v; z þ wÞ after deformation due to twist

such that OP  OP0 r

The displacement of point P in x-direction assuming

that is small such that cos  ¼ 1 and sin   

The displacement of point P in y-direction

The warping of bar, which is invariant with z and is

defined by a function

The component of strains from Eqs (27-40) and

(27-41)

The stress components from Eqs (27-34) and (27-37)

The equations of equilibrium from Eqs (27-11)

u ¼ r cosð þ Þ
r cos 
y ¼
zy ð27-257Þ

v P(x, y) P’(x+u, y+ν)

α β

FIGURE 27-49 Shows a cross-section of twisted bar in plane.

xy-Neglecting body forces in z-direction, Eq (27-11)

yields after substituting the Eqs (27-261b) and

(27-261c) in it

From the equilibrium condition of the surface

Eq (27-7)

When surface forces are absent FNx¼ FNy¼ FNz¼ 0

and cosðNzÞ ¼ n ¼ 0, x¼ y¼ z¼ yz¼ 0 from

Eq (27-7c)

From the infinitesimal element pqr, if s increasing in

the direction from q to r then

Using Eqs (27-261b), (27-261c), (27-264a) (27-264b)

in Eq (27-263), an expression for boundary condition

is obtained (Fig 27-50)

In torsion problems involving in finding a function

which satisfy Eqs (27-262) and boundary condition

Eq (27-265)

Stress function 

From equation of equilibrium

A function  which satisfy the third equation of

Eq (27-266) is

From Eqs (27-267), (27-268), and Eqs (27-261),

equations involving and are:

By making use of Eqs (27-269) and after eliminating

from Eqs (27-269a) and (27-269b) by mathematical

method, a differential equation for stress function is

obtained

Boundary condition Eq (27-265) becomes

The total torque at the ends of the twisted bar due to

The stress function which satisfy Eq (27-270) and the

boundary condition Eq (27-271)

Substituting the expression for from Eq (27-274) in

Eq (27-270) and value of m can be found, and it is

Substituting the value of m from Eq (27-275) into

Eq (27-274) the stress function becomes

dy p

N

q r

x

FIGURE 27-51 Elliptical cross-section of bar under torsion.

The torque Mtis obtained after substituting this stress

function from Eq (27-276) into Eq (27-272) and

carrying out integration and simplification

M t Torque Convex(+ve)

FIGURE 27-52

After substituting the values of Ix, Iy and A into

Eq (27-277) and simplification, the expression for Mt

The expression for F from Eq (27-278)

The equation for stress function after substituting

the value of F from Eq (27-279) in Eq (27-276)

The stress componentsxzandyzfrom Eqs (27-267)

and (27-268) after substituting the value of  from

Eq (27-280)

The maximum shear stress which occurs at y ¼ b

The angle of twist after substituting the value of F from

Eq (27-279) into Eq (27-270a) and simplification

The torsional rigidity C which is defined as twist per

unit length

For various values of the angle of twist (0¼ ) and

thereby the values of C for various cross-sections

and built up beams

The expression for warping of elliptical cross-section

after substituting Eqs (27-280), (27-281) and

(27-282) into Eqs (29-260b) and (27-260c) and

integrating

For warping of elliptical cross-section

Note: The symbol is used for angle of twist here in

order to avoid confusion regarding which is used

as a stress function

Mt¼ a2b2F

a2þ b2

1

ðð

x2dx dy ¼ Iy¼ ba

34ðð

y2dx dy ¼ Ix¼ ab3

4ðð

where A ¼ ab, Ip¼ centroidal moment of inertia

of the cross-section¼ ð ab3Þ=4 þ ð a3

bÞ=4Refer to Tables 24-27 and 24-30 under Chapter 24

3< /small>6



3< /small>6



3< /small>6


2

3< /small>

3 ỵ3 13< /sub>...

real parts on both sides of equation

The tangential stress around by elliptical hole from

Eq (27-2 43)

Large flat plate under uniform uniaxial

tension... graph of#=T versus # in degrees

Stress free square hole in a flat plate under

pure bending (Fig 27-44)

The conformal transformation for plate with