Page 47 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
12:26
Brown˙C01
Brown.cls
U.S. Customary FUNDAMENTAL LOADINGS SI/Metric 29
3
3
(2,000 lb)(9in ) (9,000 N)(0.00014 m )
VQ max VQ max
τ max = = τ max = =
4
4
Ib (36 in )(2in) Ib (0.000014 m )(0.05 m)
18,000 lb · in 3 1.26 N · m 3
= 5 = 5
72 in 0.00000007 m
2
2
= 250 lb/in = 250 psi = 1,800,000 N/m = 1.8MPa
Example 4. Determine the shear stress (τ) at Example 4. Determine the shear stress (τ) at
a distance (y = h/4) for the beam geometry of a distance (y = h/4) for the beam geometry of
Example 3, and where Example 3, and where
V = 2,000 lb V = 9,000 N
b = 2in b = 5cm = 0.05 m
h = 6in h = 15 cm = 0.15 m
4
4
I = 36 in (previously calculated) I = 0.000014 m (previously calculated)
solution solution
Step 1. Calculate the first moment (Q) for the Step 1. Calculate the first moment (Q) for the
rectangular cross section at a distance (y = rectangular cross section at a distance (y =
h/4) using Eq. (1.42). h/4) using Eq. (1.42).
3 2 3 2 3 2 3 2
Q = bh = (2in)(6in) Q = bh = (0.05 m)(0.15 m)
32 32 32 32
= 6.75 in 3 = 0.000105 m 3
Step 2. Substitute the shear force (V ), first Step 2. Substitute the shear force (V ), first
moment (Q) from Step 1, moment of inertia (I), moment (Q) from Step 1, moment of inertia (I),
and the width (b) into Eq. (1.39) to determine and the width (b) into Eq. (1.39) to determine
the shear stress (τ). the shear stress (τ).
3
3
VQ (2,000 lb)(6.75 in ) VQ (9,000 N)(0.000105 m )
τ = = τ = =
4
4
Ib (36 in )(2in) Ib (0.000014 m )(0.05 m)
13,500 lb · in 3 0.945 N · m 3
= = 5
72 in 5 0.0000007 m
2
2
= 187.5 lb/in = 188 psi = 1,350,000 N/m = 1.35 MPa
Notice that the maximum normal stress (σ max ) found in Example 1 is 80 to 90 times
greater than the maximum shear stress (τ max ) found in Example 3. This is typically the
case when the values for the bending moment (M) and the shear force (V ) are from the
middle of a beam. However, near a support the maximum shear stress will be greater than
the maximum normal stress, which may in fact be zero at a support.
As stated earlier, the maximum shear stress (τ max ) occurs at a distance (y = 0) that is
the neutral axis, and the shear stress is zero at the top and bottom of the beam that is at a
distance (y = h/2). Suppose the shear stress (τ) at an intermediate position was desired,
say at a distance (y = h/4). The only difference is the first moment (Q) that can be found
using the information shown in Fig. 1.30.
Based on the definition of the first moment (Q), its value is given by Eq. (1. 42)
bh 3h 3 2

Q = A y = = bh (1.42)
4 8 32