2. Unit 2- Stresses in Beams
Topics Covered
 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.

3. TORSIONAL DEFORMATION
OF A CIRCULAR SHAFT
 Torsion is a moment that twists/deforms a member
about its longitudinal axis
 By observation, if angle of rotation is small, length of
shaft and its radius remain unchanged
3

4. Torsional Deformation of
Circular Bars
 Assumptions
 Plane sections remain plane and perpendicular to the
torsional axis
 Material of the shaft is uniform
 Twist along the shaft is uniform.
 Axis remains straight and inextensible
4

5. Torsional Deformation
L = angle of twist
B
F
F’
F R
F’
= shear strain
φ is the shear strain, also remember that tanφ = φ,thus :
F'F Rθ
φ= =
L L
Note that shear strain does not only change with the amount of twist, but also,
it varies along the radial direction such that it is zero at the center and increases
linearly towards the outer periphery (see next slide)
5
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6. Torsional Deformation
τ Cθ q
= =
R L r
Shear stress at any point in the shaft is proportional to
the distance of the point
from the axis of the shaft.
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7. Torque transmitted by
shaft(solid)
total turning moment due to turning force
= total force on the ring x Distance of the ring from the axis
r τ
= × 2πr 3 dr
R
Total turning moment (or total torque) is obtained by integrating
R
the above equation between the limits O and R
R R τ
T = ∫ 0 dT = ∫ 0 × 2πr 3 dr
R
τ R 3 τ ⎡ r 4 ⎤ R
= × 2π ∫ 0 r dr = × 2π ⎢ ⎥
R R ⎣ 4 ⎦ 0
π
=τ × × R3
2
π
= τD3
16
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8. Torque transmitted by
shaft(hollow)
total turning moment due to turning force
= total force on the ring x Distance of the ring from the axis
τ
= × 2πr 3 dr
R0
r
Total turning moment (or total torque) is obtained by integrating
R the above equation between the limits O and R
Ro R0 τ
T= ∫ Ri
dT = ∫ Ri R0
× 2πr 3 dr
τ R0 3 τ ⎡ r 4 ⎤ R 0
= × 2π
R
∫ Ri
r dr =
R0
× 2π ⎢ ⎥
⎣ 4 ⎦ R i
π ⎡ R 0 4 − R i 4 ⎤
= τ × × ⎢ ⎥
2 ⎣ R 0 ⎦
π ⎡ D0 4 − Di 4 ⎤
= τ ⎢ ⎥
16 ⎣ D0 ⎦
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9. Power transmitted by
shaft
Power transmitted by the shafts
N = r.p.m of the shaft
T = Mean torque transmitted
ω = Angular speed of shaft
2πNT *
Power =
60
=ω × T
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10. Torque in terms of polar
moment of inertia
Moment dT on the circular ring
τ τ
dT = × 2πr 3 dr = × r 2 × 2πrdr ⇒ (dA = 2πrdr)
R R
τ
r = × r 2 × dA
R
R
R
Total Torque = ∫ 0
dT
R R τ
T= ∫ 0
dT = ∫ 0 R
× r 2 dA
τ R 2
= ∫ r dA
R 0
r 2dA = moment of elemnetary ring about an axis perpendicular to the plane
and passing though the center of the circle
R 2
∫ 0
r dA = moment of the circle about an axis perpendicular to the plane
and passing though the center of the circle
π
= Polar moment of inertia = × D4
32
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11. Torque in terms of polar
moment of inertia
τ
T = ×J
R
r
T τ
R
=
J R
τ Cθ
=
R L
T τ Cθ C = Modulus of rigidity
= =
J R L θ = Angle of twist
L = Length of the shaft
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12. Polar Modulus
Polar modulus is defined as ration of polar moment of inertia to the radius
of the shaft.
J
Zp =
R
π 4
For solid shaft => J = D
32
π 4
D π
Z p = 32 = D3
D /2 16
π
For hollow shaft => J = [ D0 4 − Di 4 ]
32
π
[D04 − Di4 ] π 4 4
Z p = 32 = [D0 − Di ]
D0 /2 16D0

13. Torsional rigidity
Torsional rigidity is also called strength of the shaft. It is defined as product of
modulus of rigidity (C) and polar moment of inertia
=C*J
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14. Shaft in combined bending
and Torsion stresses
Shear stress at any point due to torque T
q T T×r
= ⇒q=
r J J
D
Shear Stress at a point on the surface of the shaft r =
2
T×r T D 16T
τc = = × =
J π 4 2 πD 3
D
32
Bending stress at any point due to bending moment
M σ M×y
= ⇒σ =
I y I
D
Bending Stress at a point on the surface of the shaft r =
2
M×y M D 32M
σb = = × =
I π 4 2 πD 3
D
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16T
2τ c 2×
tanθ = = πD3 = T
σb 32M M
3
πD

15. Shaft in combined bending
and Torsion stresses
Major principal Stress
σb ⎛ σ b ⎞ 2
= + ⎜ ⎟ + τ c 2
2 ⎝ 2 ⎠
32M ⎛ 32M ⎞ 2 ⎛ 16T ⎞ 2
= 3 + ⎜ ⎟ + ⎜ ⎟
2 × πD ⎝ 2 × πD 3 ⎠ ⎝ πD 3 ⎠
16
=
πD (
3 M + M2 + T2 )
SOLID SHAFT Minor principal Stress
16
=
πD 3 (M − M2 + T2 )
Max shear Stress
Max principal Stress - Min principal Stress
=
2
16
=
πD 3( M2 + T2 )

16. Shaft in combined bending
and Torsion stresses
Major principal Stress
16D0
=
[4
π D0 − Di 4
] (
M + M2 + T2 )
Minor principal Stress
16D0
]( )
HOLLOW SHAFT = M − M2 + T2
[4
π D0 − Di 4
Max shear Stress
16D0
]( )
= M2 + T2
π [ D0 − Di
4 4
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17. Application to a Bar
Normal Force:
Fn Fn
Bending Moment:
Mt Mt
Shear Force:
Ft Ft
Torque or Twisting Moment:
Mn
Mn