2. Unit 2- Stresses in Beams
 Lecture -1 – Review of shear force and bending
moment diagram
 Lecture -2 – Bending stresses in beams
 Lecture -3 – Shear stresses in beams
 Lecture -4- Deflection in beams
 Lecture -5 – Torsion in solid and hollow shafts.
Topics Covered

3. Shear Stresses in Beams of
Rectangular Cross Section
 In the previous chapter we examined the case
of a beam subjected to pure bending i.e. a
constant moment along axis .
 When a beam is in pure bending, the only
stress resultants are the bending moments and
the only stresses are the normal stresses acting
on the cross sections.
 Most beams are subjected to loads that
produce both bending moments and shear
forces (non-uniform bending)

4. Shear Stresses in Beams of
Rectangular Cross Section
 In these cases, both
normal and shear forces
are developed in the
beam.
 Normal stresses are
calculated with the
Flexure Formula.
 We will now look at the
Shear Stresses

5. Vertical & Horizontal
Shear Stresses
 Consider a beam of rectangular cross section subjected
to a positive shear force.

6. Shear Stresses
τ = F ×
A y
−
I × b
x
σ +dσ
M M+dM
σ
b
y1
Area A
€
dM
I
× A × y
−
dx
€
y
−
B
A C
D
Shear forces and bending moments are
different across different sections.

7. Shear stress distribution
for different section
A is the area of the x-section cut off by a line
parallel to the neutral axis. is the distance of
the centroid of A from the neutral axis
Rectangular Section
Parabolic distribution of
shear stresses

8. Shear stress distribution
for different section
The maximum value of shear stress would
obviously beat the location y = 0.
Rectangular Section

9. Shear stress distribution
for different section
Rectangular Section

10. Section Shear stress Max shear
stress
Shear Stress
distribution
Shear stress distribution
€
τ =
F
2I
d2
4
− y2
⎛
⎝
⎜
⎞
⎠
⎟
€
τmax =1.5τavg
€
τmax =
4
3
τavg
€
τ =
F
3I
R2
− y2
( )