Bending Stress & Shear Stress in beams

Bending (looking at radial sections)

Photographs

Normal Strain

Y measured from the neutral axis

Compress /Extend from Neutral Axis

Compress

Extend

Compress

Extend

Normal Bending Strain creates Normal Stress

E

y

y is measured from the neutral axis ρ is the radius of the curvature of the beam

The bending moment is all resisted by the sum of all normal bending stress. y

E

Maximum Bending Stress, Maximum farthest from centroid (neutral axis)

x

MAX

xMAXMAXMAX

R

MAX

MAX

I

Mc

Ic

dAyc

ydAyc

ydAM

yc

cE

2)()(

The bending moment is all resisted by the normal bending stress over x-sectional area.

Neutral axis is at centroid of cross sectional area

To find the maximum bending stress • Draw shear & bending moment diagrams

• Find maximum moment, M, from bending moment diagram

• Calculate cross-section properties

– Centroid (neutral axis)

– Calculate Area Moment of Inertia about x-axis, Ix

– Find the farthest distance from neutral axis for cross section, c

• Max Bending Normal Stress = x

MAXI

Mc

Shear Stress in Bending

Shear Load, V, is distributed on cross sectional area

Visualizing Shear Stress

Splitting due to Shear Stress

Shear Stress Distribution

Shear Stress is largest at

neutral axis

Shear Stress is 0 at the

extremes from neutral axis

Shear Stress definitions

yAdytyQ

y

y

max_

1

)'()(

centroiditsyaboveAreaQ

yAdytyQ

It

VQ

y

y

_*1__

)'()(

max_

1

V = shear force

I = area moment of inertia (2nd moment of area)

Q = first moment of area (above location y1)

t = thickness

y is measured from NA

Shear Stress at y1 above NA Shear Stress Max at y=0

Shear stress is maximum at Neutral Axis (NA)

What to remember about shear stress in bending?

• Shear stress is

– 0 at the points farthest from neutral axis

– maximum at the neutral axis

– It can be shown that :

A

V : sectioncross rrectangula a For

A

V : sectioncross circular a For

MAX

MAX

2

3

3

4

End