VI.) Stresses in Beams

A.) Bending Stresses

1.) Introduction

When a beam is subjected to positive moment, the top shortens and the bottom lengthens.

The bending strains are zero at the neutral axis.

The bending strains increase proportionately with the distance from the neutral axis.

The maximum strains occur at the top and bottom of the beam (outer fibers)

Since stress is directly related to strain = E), bending stress is also zero at

the neutral axis and increases with the distance from the neutral axis.

The maximum bending stresses occur at the top and bottom of the beam (outer fibers)

2.) The Flexure Formula

b = My

I

b = Bending stress at a distance “y”

from the neutral axis (ksi).y = distance from the neutral axis (in).M = Moment at a given cross-section of

a beam (k-in).I = Moment of inertia of the cross-

section (in4).

N.A. y

3.) Maximum Bending Stress

b,max = Mc

I

b,max = Maximum bending stress at a

given cross-section (ksi).

c = Distance from N.A. to the outer fibers of cross-section (in)

M, I = as defined before.

N.A.ctop

cbot

y

4.) Section Modulus (S)

b,max = Mc = M

I SS = I_

c

N.A.

ctop

cbot

y

NA

ctop

cbot

5.) Strong Axis vs. Weak Axis

Strong Axis

N.A.

Weak Axis

B.) Shear Stress - Average Web Shear

v= V_

dtw

d twd

tw d

tw

C.) The Beam Shear Stress Formula

v= VQ

Ib

V = Shear force (lb)Q = First Moment of AreaI = 2nd Moment of Inertia(in4)b = thickness (in) at the location where shear is being computed

C.) The Beam Shear Stress Formula

Q = First Moment about the neutral axis of the area between the horizontal plane where the shear is to be calculated and the top or bottom of the beam.

Plane where shearIs being computed

Neutral axisyQ = Ay

C.) The Beam Shear Stress Formula

v= 1.5V for solid rectangular cross-sections

A

v= 4V for solid circular cross-sections

3A