# CVE Mechanics of Material Lecture No

Document Sample

```					       CVE 220
Mechanics of Material
Lecture No. 7 (S. 2002)

Gere: Section 3.1-3.3

James Hu
• pp. 187-203
Torques: right-hand rule
Torsion: twisting a straight bar when it is loaded by
moments (or torques) that tend to produce
rotation about the longitudinal axis of the bar.

right-hand rule
fingers curl in the direction of
the moment, and then thumb
will point in the direction of a vector

Torque T applied
to the handle
Notations used for torques/ couples

Torques or twisting moments

T1 = P1 d1   T2 = P2 d2

(moment vector)
direction: right-hand rule

Curved arrow: in the direction
of rotation
Deformation at Outer Surface
pure torsion

arc length qq’ = r f (from cross section)
= L gmax   (from outer surface)
gmax: shear strain at the outer surface of the bar

rf
gmax =
L
Deformation due to Torsion

arc length              (from cross section)
(from outer surface)
rate of twist

interior arc length =        (from cross section)
(from cylinder surface)

interior strain
Hooke’s Law in Shear

linear relationship

Hooke’s law in shear

G: shear modulus
of elasticity
Example: problem 3.2-2
Torsion Formula

Shear force on an elemental area at a radial distance r

Moment due to the shear force on an element area
Torsion formula

Torsion = the total moment due to shear force

(Polar moment of inertia)
Polar Moment of Inertia
Polar moment of inertia

For a circular cross section
Torsion / Angle of Twist
Example: problem 3.3-5
Prismatic bar                    normal stress: s = P / A

normal strain: e = d / L

Hooke’s law:     s=Ee

d = e L = (s / E ) L
= (P / A E ) L

d= PL
EA
EA: axial rigidity of the bar
Comparison Table: Axial Force/Torsion
pure tension pure torsion

relation

Hooke’s law

stress

strain
Circular Tube in Torsion

r = r1        g = gmin
r = r2        g = gmax
Homework Assignment 4
•   Problem 3.2-1 (page 251)
•   Problem 3.2-4 (page 251)
•   Problem 3.3-4 (page 252)
•   Problem 3.3-9 (page 253)

Due 2/21/2002