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

  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

TORSIONAL DEFORMATION OF A CIRCULAR SHAFT

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

5

Torsional Deformation

Φ

B

F

θ

F’

F’

R

€

φ is the shear strain, also remember that tanφ = φ,thus :

φ =F'FL

=RθL

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 increaseslinearly towards the outer periphery (see next slide)

L

F

Φ = shear strain

θ= angle of twist

Torsional Deformation

€

τR

=CθL

=qr

Shear stress at any point in the shaft is proportional to the distance of the point from the axis of the shaft.

Torque transmitted by shaft(solid)

r

R

€

total turning moment due to turning force= total force on the ring x Distance of the ring from the axis

= τR× 2πr3dr

Total turning moment (or total torque) is obtained by integratingthe above equation between the limits O and R

T = dT0

R∫ =

τR× 2πr3dr

0

R∫

=τR× 2π r3dr

0

R∫ =

τR× 2π r4

4⎡

⎣ ⎢

⎤

⎦ ⎥

0

R

= τ ×π2× R3

=π16τD3

Torque transmitted by shaft(hollow)

r

R

€

total turning moment due to turning force= total force on the ring x Distance of the ring from the axis

= τR0

× 2πr3dr

Total turning moment (or total torque) is obtained by integratingthe above equation between the limits O and R

T = dTR i

Ro∫ =

τR0

× 2πr3drRi

R0∫

=τR× 2π r3dr

R i

R 0∫ =τ

R0× 2π r4

4⎡

⎣ ⎢

⎤

⎦ ⎥ Ri

R0

= τ ×π2×

R04 −Ri

4

R0

⎡

⎣ ⎢

⎤

⎦ ⎥

=π16τ

D04 −Di

4

D0

⎡

⎣ ⎢

⎤

⎦ ⎥

Power transmitted by shaft

€

Power transmitted by the shaftsN = r.p.m of the shaftT = Mean torque transmitted ω = Angular speed of shaft

Power =2πNT*

60=ω × T

Torque in terms of polar moment of inertia

€

Moment dT on the circular ring

dT = τR× 2πr3dr =

τR× r2 × 2πrdr⇒ (dA = 2πrdr)

=τR× r2 × dA

Total Torque = dT0

R∫

T = dT0

R∫ =

τR× r2dA

0

R∫

=τR

r2dA0

R∫

r2dA = moment of elemnetary ring about an axis perpendicular to the plane and passing though the center of the circle

r2dA0

R∫ = moment of the circle about an axis perpendicular to the plane

and passing though the center of the circle

= Polar moment of inertia =π32

× D4

r

R

Torque in terms of polar moment of inertia

€

T =τR× J

TJ

=τR

τR

=CθL

TJ

=τR

=CθL

r

R

€

C = Modulus of rigidityθ = Angle of twistL = Length of the shaft

Polar Modulus

€

Zp =JR

For solid shaft => J = π32D4

Zp =

π32D4

D /2=π16D3

For hollow shaft => J = π32

D04 −Di

4[ ]

Zp =

π32

D04 −Di

4[ ]D0 /2

=π

16D0D0

4 −Di4[ ]

Polar modulus is defined as ration of polar moment of inertia to the radius of the shaft.

Torsional rigidity

€

= C * J

Torsional rigidity is also called strength of the shaft. It is defined as product of modulus of rigidity (C) and polar moment of inertia

Shaft in combined bending and Torsion stresses

€

Shear stress at any point due to torque Tqr

=TJ⇒ q =

T × rJ

Shear Stress at a point on the surface of the shaft r = D2

τc = T × rJ

=T

π32D4

×D2

=16TπD3

Bending stress at any point due to bending momentMI

=σy⇒σ =

M × yI

Bending Stress at a point on the surface of the shaft r = D2

σ b =M × yI

=Mπ64

D4×D2

=32MπD3

tanθ = 2τc

σ b

=2 × 16T

πD3

32MπD3

=TM

Shaft in combined bending and Torsion stresses

€

Major principal Stress

=σ b

2+

σ b

2⎛

⎝ ⎜

⎞

⎠ ⎟

2

+τc2

=32M

2 × πD3 +32M

2 × πD3

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+16TπD3

⎛

⎝ ⎜

⎞

⎠ ⎟

2

=16πD3 M + M 2 +T 2( )

Minor principal Stress

= 16πD3 M − M 2 +T 2( )

Max shear Stress

=Max principal Stress - Min principal Stress

2

=16πD3 M 2 +T 2( )

SOLID SHAFT

Shaft in combined bending and Torsion stresses

€

Major principal Stress

=16D0

π D04 −Di

4[ ]M + M 2 +T 2( )

Minor principal Stress

= 16D0

π D04 −Di

4[ ]M − M 2 +T 2( )

Max shear Stress

=16D0

π D04 −Di

4[ ]M 2 +T 2( )

HOLLOW SHAFT

Application to a Bar

Normal Force: Fn Fn

Shear Force: Ft Ft

Bending Moment:

Mt Mt

Torque or Twisting Moment:

Mn

Mn