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 21

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

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.

Important:

The direction of moment is denoted by an arrow perpendicular to the plane of moment

h is the dimension of cross-section which is along the plane of moment

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Section moment of inertia Important link to see

http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia

For regular shapes See the above link

General Formula

Class exercise

2dAII x

A

AyY

Section moment of inertia, also called area moment of inertia

Distance of centroid from a reference point

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Section moment of inertia

Depends upon the cross section of the member

Square, rectangular, circular, elliptical, semi-circular

Orientation of the moment Mostly symmetric Or acting along a plane passing through the centroid of the cross-section

In overall, we have to locate the position of centroid

http://en.wikipedia.org/wiki/List_of_centroids

Spring 2015 By Hafiz Kabeer Raza MM222 Strength of Materials

Deformations in a Transverse Cross Section Deformation due to bending moment M is

quantified by the curvature of the neutral surface

EI

M

I

Mc

EcEcc

mm

11

Although cross sectional planes remain planar

when subjected to bending moments, in-plane

deformations are nonzero,

yyxzxy

..

Expansion above the neutral surface and

contraction below it cause an in-plane curvature,

curvature canticlasti 1