Hafiz Kabeer Raza Research Associate

Faculty of Materials Science and Engineering, GIK Institute Contact: Office G13, Faculty Lobby

raza@giki.edu.pk, hkabeerraza@gmail.com, 03344025392

MM222

Strength of Materials

Lecture – 19

Spring 2015

The actual value of T is 420 lb.ft

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Design of Transmission Shafts • Principal transmission shaft

performance specifications are:

- power

- speed

• Determine torque applied to shaft at

specified power and speed,

f

PPT

fTTP

2

2

• Find shaft cross-section which will not

exceed the maximum allowable

shearing stress,

shafts hollow2

shafts solid2

max

41

42

22

max

3

max

Tcc

cc

J

Tc

c

J

J

Tc

• Designer must select shaft

material and cross-section to

meet performance specifications

without exceeding allowable

shearing stress.

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Example

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Problem 3.70

• Use T/τmax = J/c2

Hafiz Kabeer Raza Research Associate

Faculty of Materials Science and Engineering, GIK Institute Contact: Office G13, Faculty Lobby

raza@giki.edu.pk, hkabeerraza@gmail.com, 03344025392

MM222

Strength of Materials

Lecture – 20

Spring 2015

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Chapter 4

Pure Bending

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Pure Bending

Pure Bending: Prismatic members

subjected to equal and opposite couples

acting in the same longitudinal plane

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Other Loading Types

• Eccentric Loading: Axial loading which

does not pass through section centroid

produces internal forces equivalent to an

axial force and a couple

• Transverse Loading: Concentrated or

distributed transverse load produces

internal forces equivalent to a shear

force and a couple

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Symmetric Member in Pure Bending

• Internal forces in any cross section are equivalent

to a couple. The moment of the couple is equal

to the bending moment of the section.

• From statics, a couple M consists of two equal

and opposite forces.

• The sum of the components of the forces in any

direction is zero.

• The moment is the same about any axis

perpendicular to the plane of the couple and

zero about any axis contained in the plane.

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Bending Deformations Beam with a plane of symmetry in pure

bending:

• member remains symmetric

• bends uniformly to form a circular arc

• cross-sectional plane passes through arc center

and remains planar

• length of top decreases and length of bottom

increases

• a neutral surface must exist that is parallel to the

upper and lower surfaces and for which the length

does not change

• stresses and strains are negative (compressive)

above the neutral plane and positive (tension)

below it

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Tensile and Compression

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Strain Due to Bending Consider a beam segment of length L.

Where:

ρ = radius of curvature (length from center of

curvature to the neutral axis)

θ = the angle subtended by the entire length after

bending

y = the distance of the point where stress/strain is to

be computed from neutral axis (0, c)

After deformation, the length of the neutral surface

remains L. Length at other sections above or below,

mx

m

m

x

c

y

cρ

c

yy

L

yyLL

yL

or

linearly) ries(strain va

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Stress Due to Bending • For a linearly elastic material,

linearly) varies(stressm

mxx

c

y

Ec

yE

I

My

c

y

inertiaofmomenttionII

Mc

c

IdAy

cM

dAc

yydAyM

x

mx

m

mm

mx

ngSubstituti

sec,

2

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Beam Section Properties • The maximum normal stress due to bending,

modulussection

inertia ofmoment section

c

IS

I

S

M

I

Mcm

A beam section with a larger section modulus

will have a lower maximum stress

• Consider a rectangular beam cross section,

Ahbhh

bh

c

IS

613

61

3

121

2

Between two beams with the same cross

sectional area, the beam with the greater depth

will be more effective in resisting bending.

• Structural steel beams are designed to have a

large section modulus.