Page 33 - Marks Calculation for Machine Design

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FUNDAMENTAL LOADINGS
Relationship among E, G, and ν.
The modulus of elasticity (E), shear modulus of
elasticity (G), and Poisson’s ratio (ν) are not independent but related by Eq. (1.16). 15
E
G = (1.16)
2 (1 + ν)
This is a remarkable relationship between material properties, and to the author’s knowl-
edge there is no other such relationship in engineering.
U.S. Customary SI/Metric
Example 2. Given the modulus of elasticity Example 2. Given the modulus of elasticity
(E) and Poisson’s ratio (ν), calculate the shear (E) and Poisson’s ratio (ν), calculate the shear
modulus of elasticity (G), where modulus of elasticity (G), where
2
6
2
9
E = 30 × 10 lb/in (steel) E = 207 × 10 N/m (steel)
ν = 0.28 (steel) ν = 0.28 (steel)
solution solution
Step 1. Substitute the modulus of elasticity Step 1. Substitute the modulus of elasticity
(E) and Poisson’s ratio (ν) into Eq. (1.16). (E) and Poisson’s ratio (ν) into Eq. (1.16).
9
6
E 30 × 10 lb/in 2 E 207 × 10 N/m 2
G = = G = =
2 (1 + ν) 2 (1 + 0.28) 2 (1 + ν) 2 (1 + 0.28)
6
9
30 × 10 lb/in 2 207 × 10 N/m 2
= =
2.56 2.56
6
9
= 11.7 × 10 lb/in 2 = 80.8 × 10 N/m 2
Punching Holes. One of the practical applications of direct shear is the punching of holes
in sheet metal as depicted in Fig. 1.18.
The holes punched are usually round so the shear area (A) is the surface area of the inside
of the hole, or the surface area of the edge of the circular plug that is removed. Therefore,
the shear area (A) is given by Eq. (1.17).
A = 2πrt (1.17)
where (r) is the radius of the hole and (t) is the thickness of the plate.
In order to punch a hole, the ultimate shear strength (S su ) of the material that is half the
ultimate tensile strength (S ut ) must be reached by the force (F) of the punch. Using the
definition of shear stress (τ) in Eq. (1.18)
V
τ = (1.18)
A
F

Punch
t
Plate

FIGURE 1.18 Hole punching.