week 6 - 1 slides

Torsion of Circular Section

5-62 The two shafts are made of A-36 steel. Each has a diameter of 25 mm, and they are supported by bearings at A, B and C, which allow free rotation. If the support at D is fixed, Determine the angel of twist of ends B and A. G = 75 GPa (p. 211)

T [Nm]

L

-60 Nm

A F G

B

A G F B

60 Nm

D H C E

120 Nm

1. Internal Torque:

TE = 90 Nm TD = 30 Nm

T [Nm]

L

-90 Nm

D C H

E 30 Nm

F

E

FEF

F’FE

FEF = F’EF

TF /TE = RF/RE

H

G E

F

Torsion of Circular Section

5-62 The two shafts are made of A-36 steel. Each has a diameter of 25 mm, and they are supported by bearings at A, B and C, which allow free rotation. If the support at D is fixed, Determine the angel of twist of ends B and A. G = 75 GPa (p. 211)

T [Nm]

L

-60 Nm

A F G

B

A G F B

60 Nm

D H C E

120 Nm

1. Internal Torque:

TE = - 90 Nm TD = 30 Nm

T [Nm]

L

-90 Nm

D C H

E 30 Nm

H

G E

F

Segment DH: TDH = 30 Nm Shaft DE

Segment HE: THE = -90 Nm

Segments AG and FB: T = 0 Shaft AB

Segment GF: TGF = -60 Nm

Torsion of Circular Section

1. Internal Torque: Segment DH: TDH = 30 Nm Shaft DE

Segment HE: THE = -90 Nm

Segments AG and FB: T = 0 Shaft AB

Segment GF: TGF = -60 Nm 2. Angle of twist:

= - 0.02086

Ans.

Ans.

5-62 The two shafts are made of A-36 steel. Each has a diameter of 25 mm, and they are supported by bearings at A, B and C, which allow free rotation. If the support at D is fixed, Determine the angel of twist of ends B and A. G = 75 GPa (p. 211)

H

G E

F

F

E

ϕE

RE

Torsion

External Loading Types

Deformation

Strain

Shear Strain

τ = Gγ

τmax ≤ τallow

(Compatibility)

Internal Loading Diagram x

T

0

γ = π/2 – θ’

Axial Load

External Loading Types

Deformation

Strain

Average Normal Strain

σ = Eε

σ ≤ σallow

(Compatibility)

Internal Loading Diagram

N

x

F

0

Bending

Bending Moments and Shear Forces

BEAM SIGN CONVENTION

Where distributed load acts downward on the beam; internal shear force causes a clockwise rotation of the beam segment on which it acts; and the internal moment causes compression in the top fibers of the segment, or to bend the segment so that it holds water.

Bending Moments and Shear Forces

1. Determine the ground reactions:

2. Plotting the shear and moment diagrams:

Graphical Method for Constructing Shear and Moments Diagrams

; MA = 0 ; MB = 0

A B

A B

A B

Graphical Method for Constructing Shear and Moments Diagrams

1. Determine the ground reactions:

; MA = 0 ; MB = 0

2. Plotting the shear and moment diagrams:

A B

A B

A B

Graphical Method for Constructing Shear and Moments Diagrams

6-19 Draw the shear and moment diagrams for the beam. (p. 295)

30 kN/m 45 kN m

1.5 m 1.5 m 1.5 m