fundamentals of engineering supplied-reference handbook - Ventech!

The deflection θ and moment Fr are related by
Fr = kθ
where the spring rate k is given by
4
d E
k =
64DN
where k has units of N·m/rad and θ is in radians.
Spring Material: The strength of the spring wire may be
found as shown in the section on linear springs. The
allowable stress σ is then given by
Sy = σ = 0.78Sut cold-drawn carbon steel (A227,
A228, A229)
Sy = σ = 0.87Sut hardened and tempered carbon
and low-alloy steel (A232, A401)
Ball/Roller Bearing Selection
The minimum required basic load rating (load for which
90% of the bearings from a given population will survive 1
million revolutions) is given by
1
a
C = PL , where
C = minimum required basic load rating,
P = design radial load,
L = design life (in millions of revolutions), and
a = 3 for ball bearings, 10/3 for roller bearings.
When a ball bearing is subjected to both radial and axial
loads, an equivalent radial load must be used in the
equation above. The equivalent radial load is
Peq = XVFr + YFa, where
Peq = equivalent radial load,
Fr = applied constant radial load, and
Fa = applied constant axial (thrust) load.
For radial contact, deep-groove ball bearings:
V = 1 if inner ring rotating, 1.2 if outer ring rotating,
If Fa /(VFr) > e,
X = 0.
56,
where
and
⎛ Fa
⎞
e = 0.
513⎜
⎟
⎜ C ⎟
⎝ o ⎠
⎛ Fa
⎞
Y = 0.
840⎜
⎟
⎜ C ⎟
⎝ o ⎠
0.
236
,
and
−0.
247
Co = basic static load rating, from bearing catalog.
If Fa /(VFr) ≤ e, X = 1 and Y = 0.
204
MECHANICAL ENGINEERING (continued)
Intermediate- and Long-Length Columns
The slenderness ratio of a column is Sr = l/k, where l is the
length of the column and k is the radius of gyration. The
radius of gyration of a column cross-section is, k = I A
where I is the area moment of inertia and A is the crosssectional
area of the column. A column is considered to be
intermediate if its slenderness ratio is less than or equal to
(Sr)D, where
2E
S
( S ) = π , and
r
D
y
E = Young's modulus of respective member, and
Sy = yield strength of the column material.
For intermediate columns, the critical load is
⎡
2
1 ⎛ S ⎤
yS
r ⎞
P ⎢ ⎜ ⎟ ⎥
cr = A S y −
⎢
⎜ ⎟
, where
E ⎥
⎣ ⎝ 2π
⎠ ⎦
Pcr = critical buckling load,
A = cross-sectional area of the column,
Sy = yield strength of the column material,
E = Young's modulus of respective member, and
Sr = slenderness ratio.
For long columns, the critical load is
P
cr
2
π EA
=
S
2
r
where the variables are as defined above.
For both intermediate and long columns, the effective
column length depends on the end conditions. The AISC
recommended values for the effective lengths of columns
are, for: rounded-rounded or pinned-pinned ends, leff = l;
fixed-free, leff = 2.1l; fixed-pinned, leff = 0.80l; fixed-fixed,
leff = 0.65l. The effective column length should be used
when calculating the slenderness ratio.
Power Transmission
Shafts and Axles
Static Loading: The maximum shear stress and the von
Mises stress may be calculated in terms of the loads from
2
2 2 1 2
τ max = [ ( 8M
+ Fd ) + ( 8T
) ] ,
3
πd
[ ( ) ] 2 1
4
2 2
σ′ = 8M
+ Fd + 48T
, where
3
πd
M = the bending moment,
F = the axial load,
T = the torque, and
d = the diameter.