Machine Design Databook Episode 3 part 11 pps

p pressure, MPa psiq load per unit length, kN/m lbf/in Qx, Qy shearing forces parallel to z-axis per unit length of sections of a plate perpendicular to x and y axis, N/m lbf/in Nr, N ra

p pressure, MPa (psi)

q load per unit length, kN/m (lbf/in)

Qx, Qy shearing forces parallel to z-axis per unit length of sections of a

plate perpendicular to x and y axis, N/m (lbf/in)

Nr, N radial and tangential shearing forces, N (lbf )

tension of a membrane, kN/m (lbf/in)

Mtxy twist of surface

u, v, w components or displacements, m (in)

x, y, z rectangular co-ordinates, m (in)

X, Y, Z body forces in x; y; z directions, N (lbf )

Z section modulus in bending, cm3(in3)

 density, kN/m3(lbf/in3)

x,y,z normal components of stress parallel to x, y, and z axis, MPa

(psi)

r, radial and tangential stress, MPa (psi)

r, z normal stress components in cylindrical co-ordinates, MPa

(psi)

 shearing stress, MPa (psi)

xy,yz,zx shearing stress components in rectangular co-ordinates, MPa

(psi)

" unit elongation, m/m (in/in)

"x,"y,"z unit elongation in x, y, and z direction, m/m (in/in)

"r," radial and tangential unit elongation in polar co-ordinates

xy,yz,zx shearing strain components in rectangular co-ordinate

r,z shearing strain in polar co-ordinate

r,z,rz shearing stress components in cylindrical co-ordinates, MPa

the design which will be used and observed throughout this Machine Design Data

Handbook

STRESS AT A POINT (Fig 27-1)

The stress at a point due to force
F acting normal to

an area dA (Fig 27-1b)

For stresses acting on the part II of solid body cut out

from main body in x, y and z directions, Fig 27-1b

Similarly the stress components in xy and xz planes

can be written and the nine stress components at the

point O in case of solid body made of homogeneous

and isotropic material

F ¼ force acting normal to the area
A

A ¼ an infinitesimal area of the body under theaction of F

dz a

(a) A solid body subject to action

of external forces (b) An infineticimal area ∆A of Part II of a solid body under the action of force

∆F at 0

(c) Stresses acting on the faces of a

small cube element cut out from the solid body

Summing moments about x, y and z axes, it can be

proved that the cross shears are equal

All nine components of stresses can be expressed by a

single equation

The FNx, FNy, and FNzunknown components of the

resultant stress on the plane KLM of elemental

tetra-hedron passing through point O (Fig 27-2)

The unknown components of resultant stress FNx, FNy

and FNz in terms of direction cosines l, m and n

FNx¼ xcos N; x þ xycos N; y þ xzcos N; z

FNy¼ yxcos N; x þ ycos N; y þ yzcos N; z

FNz¼ zxcos N; x þ zycos N; y þ zcos N; z ð27-6Þ

FNx¼ xl þ xym þ zxn

FNy¼ yzl þ ym þ yxn

FNz¼ zxl þ zym þ zn ð27-7Þwhere the direct cosines are

l ¼ cos  ¼ cos N; x; m ¼ cos ¼ cos N; y,

n ¼ cos  ¼ cos N; z,

lsþ m2þ n2¼ ðlÞ02þ ðm0Þ2þ ðn0Þ2¼ 1

τxy

Surface area KLM = A (normal to KLM)

x L

F z

z

y

K N

FIGURE 27-2 The state of stress at O of an elemental

tet-rahedron.

x

y y’

+

+

+ + +

z

dx

dz

dy dy

dz

dx dz

∂τyx

∂y dy

The resultant stress FNon the plane KLM

The normal stress which acts on the plane under

consideration

The shear stress which acts on the plane under

consideration

Equations (27-1), (27-2) and (27-7) to (27-8) can be

expressed in terms of resultant stress vector as follows

(Fig 27-2)

The resultant stress vector at a point

The resultant stress vector components in x, y and z

directions

The resultant stress vector

The normal stress which acts on the plane under

consideration

The shear stress which acts on the plane under

consideration

The angle between the resultant stress vector TN and

the normal to the plane N

cos ¼ l ¼ angle between x axis and Normal Ncos ¼ m ¼ angle between y axis and Normal Ncos ¼ n ¼ angle between z axis and Normal N

FN¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2

Nxþ F2

Nyþ F2 Nz

q

ð27-9eÞwhere the direction cosines are

cosðTN; xÞ ¼ TNx=jTNj, cosðTN; yÞ ¼ TNy=jTNj,cosðTN; zÞ ¼ TNz=jTNj

q

ð27-10bÞcosðTN; NÞ ¼ cosðTN; xÞ cosðN; xÞ

þ cosðTN; yÞ cosðN; yÞ

þ cosðTN; zÞ cosðN; zÞ ð27-10cÞ

EQUATIONS OF EQUILIBRIUM

The equations of equilibrium in Cartesian

coordi-nates which includes body forces in three dimensions

(Fig 27-3)

Stress equations of equilibrium in two dimensions

TRANSFORMATION OF STRESS

The vector form of equations for resultant-stress

vectors TN and TN0 for two different planes and the

outer normals N and N0in two different planes

The projections of the resultant-stress vector TNonto

the outer normals N and N0

Substituting Eqs (27-9b), (27-9c), (27-9d) and (27-9e)

in Eqs (27-13), equations for TN, N and TN, N0

The relation between TN, N0and TN0, N

By coinciding outer normal N with x0, N with y0,

and N with z0 individually respectively and using

Eqs (27-14a) to (27-14b), x0, y0 and z0 can be

x0¼ Tx0x0¼ xcos2ðx0; xÞ þ ycos2ðx0; yÞ

þ zcos2ðx0; zÞ þ 2xycosðx0; xÞ cosðx0; yÞ

þ 2yzcosðx0; yÞ cosðx0; zÞ

þ 2zxcosðx0; zÞ cosðx0; xÞ ð27-15aÞ

By selecting a plane having an outer normal N

co-incident with the x0 and a second plane having an

outer normal N0coincident with the y0and utilizing

Eq (27-14b) which was developed for determining

the magnitude of the projection of a resultant stress

vector on to an arbitrary normal can be used to

determine x0 y0 Following this procedure and by

selecting N and N0coincident with the y0and z0, and

z0 and x0 axes, the expression for y0 z0 and z0 x0 can

be obtained The expressions forx0y0,y0z0andz0x0are

y0¼ Ty0y0¼ ycos2ðy0; yÞ þ zcos2ðy0; zÞ

þ xcos2ðy0; xÞ þ 2yzcosðy0; yÞ cosðy0; zÞ

þ 2zxcosðy0; zÞ cosðz0; xÞ

þ 2xycosðy0; xÞ cosðy0; yÞ ð27-15bÞ

z0¼ Tz0z0¼ zcos2ðz0; zÞ þ xcos2ðz0; xÞ

þ ycos2ðz0; yÞ þ 2zxcosðz0; zÞ cosðz0; xÞ

þ 2xycosðz0; xÞ cosðz0; yÞ

þ 2yzcosðz0; yÞ cosðz0; zÞ ð27-15cÞ

x0y0¼ Tx0y0¼ xcosðx0; xÞ cosðy0; xÞ

þ ycosðx0; yÞ cosðy0; yÞ þ zcosðx0; zÞ cosðy0; zÞ

þ xy½cosðx0; xÞ cosðy0; yÞ þ cosðx0; yÞ cosðy0; xÞ

þ yz½cosðx0; yÞ cosðy0; zÞ þ cosðx0; zÞ cosðy0; yÞ

þ zx½cosðx0; zÞ cosðy0; xÞ þ cosðx0; xÞ cosðy0; zÞ

ð27-16aÞ

y0 z0¼ Ty0z0¼ ycosðy0; yÞ cosðz0; yÞ

þ zcosðy0; zÞ cosðz0; zÞ þ xcosðy0; xÞ cosðz0; xÞ

þ yz½cosðy0; yÞ cosðz0; zÞ þ cosðy0; zÞ cosðz0; yÞ

þ zx½cosðy0; zÞ cosðz0; xÞ þ cosðy0; xÞ cosðz0; zÞ

þ xy½cosðy0; xÞ cosðz0; yÞ þ cosðy0; yÞ cosðz0; xÞ

ð27-16bÞ

z0 x0¼ Tz0x0¼ zcosðz0; zÞ cosðx0; zÞ

þ xcosðz0; xÞ cosðx0; xÞ þ ycosðz0; yÞ cosðx0; yÞ

þ zx½cosðz0; zÞ cosðx0; xÞ þ cosðz0; xÞ cosðx0; zÞ

þ xy½cosðz0; xÞ cosðx0; yÞ þ cosðz0; yÞ cosðx0; xÞ

þ yz½cosðz0; yÞ cosðx0; zÞ þ cosðz0; zÞ cosðx0; yÞ

ð27-16cÞEquations (27-15a) to (27-15c) and Eqs (27-16a) to(27-16c) can be used to determine the six Cartesiancomponents of stress relative to the Oxyz coordinatesystem to be transformed into a different set of sixCartesian components of stress relative to an Ox0y0z0coordinate system

For two-dimensional stress fields, the Eqs (27-15a) to

(27-15c) and (27-16a) to (27-16c) reduce to, since

z¼ zx¼ yz¼ 0 z0 coincide with z and  is the

angle between x and x0, Eqs (27-15a) to (27-15c)

and Eqs (27-16a) to (27-16c)

FIGURE 27-5 The stress vector T N

PRINCIPAL STRESSES

By referring to Fig 27-5, where TN coincides with

outer normal N, it can be shown that the resultant

stress components of TN in x, y and z directions

Substituting Eqs (27-9b) to (27-9d) into (27-18), the

following equations are obtained

Eq (27-19) can be written as

From Eq (27-20), direction cosine (N, x) is obtained

and putting this in determinant form

Putting the determinator of determinant into zero,

the non-trivial solution for direction cosines of the

principal plane is

x0¼ xcos2 þ ysin2 þ 2xysin cos 

¼ xþ y

2 þx
y

2 cos 2 þ xysin 2 ð27-17aÞ

y0¼ ycos2 þ xsin2
2xysin cos 

Expanding the determinant after making use of Eqs.

(27-4) which gives three roots They are principal

stresses

For two-dimensional stress system the coordinating

system coinciding with the principal directions,

Eq (27-23) becomes

The three principal stresses from Eq (27-23a) are

The directions of the principal stresses can be found

from

From Eq (27-15)

From definition of principal stress

Substituting the values of TN1and TN2in Eq (27-15)

2

þ 2 xy

¼ x0y0þ y0z0þ z0x0
2

x0y0
2

y0z0
2

z0x0ð27-28bÞ

For the coordinating system coinciding with the

principal direction, the expression for invariants

from Eq (27-28)

STRAIN (Fig 27-6)

The normal strain or longitudinal strain by Hooke’s

law (Fig 27-6) in x-direction

The lateral strains in y and z-direction

The normal strains caused byyandz

THREE-DIMENSIONAL STRESS-STRAIN

SYSTEM

The general stress-strain relationships for a linear,

homogeneous and isotropic material when an element

subject tox,yandzstresses simultaneously

where I1¼ first invariant, I2¼ second invariantand I3¼ third invariant of stress

z

dx’

dy’

dx dz

K’ K k n

m I L L’

The expressions forx,yandzstresses in case of

three-dimensional stress system from Eqs (27-33)

BIAXIAL STRESS-STRAIN SYSTEM

The normal strain equations, when z¼ 0 from

Eq (27-33)

The normal stress equation, when z¼ 0 from

Eq (27.34)

SHEAR STRAINS

For a homogeneous, isotropic material subject to

shear force, the shear strain which is related to shear

stress as in case of normal strain

It has been proved that the shear modulus (G) is

related to Young’s modulus (E) and Poisson’s ratio

as

From Eqs (27-37), shear strain in terms of E and

STRAIN AND DISPLACEMENT

(Figs 27-8 and 27-9)

The normal strain in x-direction

The normal strain in y and z-directions

The shear strains xy, yz and zx planes

N’

w

v K

Unstrained element

FIGURE 27-8 Deformation of a cube element in a solid

body subject to loads.

dy K

K’

N u

X y

ν

Y

M’ L’

N’

ν+ dx∂ν∂x u+ dx ∂u ∂x

The amount of counterclockwise rotation of a line

segment located at R in xy, yz and zx planes

The strain"z and first strain invariant J1in case of

plane stress

The strains components"x0,"y0 and"z0, along x0, y0

and z0 axes line segments with reference to the

O0x0y0z0system

The shearing strain components (due to angular

changes) x0y0, y0z0 and z0x0 with reference to the

O0x0y0z0system

xy¼12

"x0¼ "xcos2ðx; x0Þ þ "ycos2ðy; x0Þ

þ "zcos2ðz; x0Þ þ xycosðx; x0Þ cosðy; x0Þ

þ yzcosðy; x0Þ cosðz; x0Þ þ zxcosðz; x0Þ cosðx; x0Þ

ð27-42aÞ

"y0¼ "ycos2ðy; y0Þ þ "zcos2ðz; y0Þ

þ "xcos2ðx; y0Þ þ yxcosðy; y0Þ cosðz; y0Þ

þ zxcosðz; y0Þ cosðx; y0Þ þ xycosðx; y0Þ cosðy; y0Þ

ð27-42bÞ

"z0¼ "zcos2ðz; z0Þ þ "xcos2ðx; z0Þ

þ "ycos2ðy; z0Þ þ zxcosðz; z0Þ cosðx; z0Þ

þ xycosðx; z0Þ cosðy; z0Þ þ yzcosðy; z0Þ cosðz; z0Þ

ð27-42cÞ

x0y0¼ 2"xcosðx; x0Þ cosðx; y0Þ þ 2"ycosðy; x0Þ cosðy; y0Þ

þ 2"zcosðz; x0Þ cosðz; y0Þ

þ xy½cosðx; x0Þ cosðy; y0Þ þ cosðx; y0Þ cosðy; x0Þ

þ yz½cosðy; x0Þ cosðz; y0Þ þ cosðy; y0Þ cosðz; x0Þ

þ zx½cosðz; x0Þ cosðx; y0Þ þ cosðz; y0Þ cosðx; x0Þ

ð27-43aÞ

For the case of two-dimensional state of stress when z0

coincides with z and zx¼ yz¼ 0, the angle between

x and x0coordinates

The cubic equation for principal strains whose three

roots give the distinct principal strains associated

with three principal directions, is

The three strain invariants analogous to the three

stress invariants

y0z0¼ 2"ycosðy; y0Þ cosðy; z0Þ þ 2"zcosðz; y0Þ cosðz; z0Þ

þ 2"xcosðx; y0Þ cosðx; z0Þ

þ yz½cosðy; y0Þ cosðz; z0Þ þ cosðy; z0Þ cosðz; y0Þ

þ zx½cosðz; y0Þ cosðx; z0Þ þ cosðz; z0Þ cosðx; y0Þ

þ xy½cosðx; y0Þ cosðy; z0Þ þ cosðx; z0Þ cosðy; y0Þ

ð27-43bÞ

z0 x0¼ 2"zcosðz; z0Þ cosðz; x0Þ þ 2"xcosðx; z0Þ cosðx; x0Þ

þ 2"ycosðy; z0Þ cosðy; x0Þ

þ zx½cosðz; z0Þ cosðx; x0Þ þ cosðz; x0Þ cosðx; z0Þ

þ xy½cosðx; z0Þ cosðy; x0Þ þ cosðx; x0Þ cosðy; z0Þ

þ yz½cosðy; z0Þ cosðz; x0Þ þ cosðy; x0Þ cosðz; z0Þ



"x"yþ "y"zþ "z"x


2 xy

4


2 yz

4
2zx4

4
"y zx2

4
"z

2 xy

4 þxyyzzx4



¼ 0ð27-45Þ

J1¼ "xþ "yþ "z¼ first invariant of strain

ð27-45aÞ

BOUNDARY CONDITIONS

The components of the surface forces Fsfxand Fsfyper

unit area of a small triangular prism pqr so that the

side qr coincides with the boundary of the plate ds

4


2 yz

4
zx24

¼ second invariant of strain ð27-45bÞ

y r

O

N ds

The volume dilatation of rectangular parallelopiped

element subject to hydrostatic pressure whose sides

are l1, l2and l3

The final dimensions of the element after straining

Substituting the values of l1f, l2f, l3f, l1, l2, l3 in

Eq (27-48) and after neglecting higher order terms

of strain

If hydrostatic pressure (0) or uniform compression

is applied from all sides of an element such thatx¼

y¼ z¼
0¼ 1¼ 2¼ 3,xy¼ yz¼ zx¼ 0,

Eq (27-48) becomes

The bulk modulus of elasticity

GENERAL HOOKE’S LAW

The general equation for strain in x-direction

accord-ing to general Hooke’s law in case of anisotropic or

non-homogeneous and non-isotropic materials such

as laminate, wood and fiber-filled-epoxy materials as

a linear function of each stress

For relationships between the elastic constants

The three-dimensional stress-strain state in anisotropic

or non-homogeneous and non-isotropic material such

as laminates, fiber filled epoxy material by using

generalized Hooke’s law which is useful in designing

machine elements made of composite material

(Fig 27-1c)

Note:½S matrix is the compliance matrix which gives

the strain-stress relations for the material The inverse

of the compliance matrix is the stiffness matrix and

the stress-strain relations If no symmetry is assumed,

there are 92¼ 81 independent elastic constants

pre-sent in the compliance matrix [Eq (27-55)]

ð1þ 2þ 3Þ ð27-50bÞ

3 7 7 7 7 7 7 7 7 7 7

3 7 7 7 7 7 7 7 7 7 7 5

3 7 7 7 7 7 7 7 7 7 7 5

ð27-55Þ

Equation (27-55) can be written as given here under

Eq (27-56) with the following use of change of

nota-tions and principle of symmetrical matrix in case of

3 7 7 7 7 7 7

3 7 7 7 7 7 7

3 7 7 7 7 7 7

3 7 7 7 7 7 7 ð27-56Þ

a Courtesy: Extracted from Ashton, J E., J C Halpin, and

P H Petit, Primer on Composite Materials—Analysis, Technomic Publishing Co., Inc., 750 Summer Street, Stamford, Conn 1969.

The general stress-strain equations under linear

The matrix expression from Eq (27-55) for

ortho-tropic material in a three-dimensional state of stress

The two-dimensional or a plane stress state matrix

expression after putting 3¼ 23¼ 13¼ 0 and

23¼ 13¼ 0 and "3¼ S131þ S232 in Eq (27-59)

for orthotropic material

The stress-strain relationship for homogenous

iso-tropic laminae of a laminated composite in the

matrix form, which is assumed to be in state of

plane stress

Alternatively Eqs (27-61) can be written for the nth

layer of laminated composite, which is assumed to

be in a state of plane stress

377777

377777

377777

ð27-59Þwhere there are 9 independent constants in theabove compliance matrix which is inverse ofstiffness matrix

"1

"2

12

26

37

3

7 12

12

26

37n

37n

"1

"2

12

26

37n

Substituting strain-displacement, Eqs (27-40) and

(27-41) into stress-strain Eqs (27-33) and (27-37) or

(27-39), displacement stress equation are obtained

with from 15 unknowns to 9 unknowns

Combining stress equation of equilibrium from Eqs

(27-11) with stress displacement Eqs (27-63) (from 9

to 3 unknowns)

Six stress equations of compatibility are obtained by

making use of stress strain relations of Eqs (27-33)

and (27-39), the stress equations of equilibrium

Eq (27-11) and the strain compatibility Eq (27-47)

in three dimension in Cartesian system of coordinates

AIRY’S STRESS FUNCTION

Differential equations of equilibrium for

two-dimensional problems taking only gravitational

force as body force

The stress components in terms of stress function

and body force

Substituting Eqs (27-66c) for stress components into

Eq (27-66b) that the stress function must satisfy the

equation

The stress compatibility equation for the case of plane

strain

If components of body forces in plane strain are

Substituting Eqs (27-68) into Eqs (27-11d), (27-11e)

and Eq (27-67) and taking2 ¼ 1

1
v

By assuming that the stress can be represented by a

stress function  such that x¼@2

@z þ

@Fbz

@y

ð27-65eÞ

@z þ

@Fbz

@x

ð27-65fÞ