# Shaft design Bending Moment

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```					                                   MACHINE DESIGN(1)

(1)Bending Moment

1- Find the reactions at the supports and plot the hear-force and bending-
moment diagrams for each of the beams shown in Figure.

300                     400                          100       100 N         400         200 N
1. 5kN

(b)
(a)
3 kN
50             100                 50                     150           150             200      5 kN
200 N                         200 N

(c)                                                            (d)

2- The shaft shown in figure is supported in bearings at O and C and is
subjected to bending loads due to the force components acting at A and B.
Sketch two diagrams for each plane of bending, and compute the moment
components at A and B. Also compute the maximum bending moment.

Y

150

100

A                           O
75                                                                    X
1.0 kN

B                                   3 kN

2 kN
4 kN
Z
C

2- The figure illustrates an over hanging shaft in bearings, assumed to be
self-aligning, at O and B. The shaft is loaded by external forces at A and
C. If the diameter of the shaft section at B is 30 mm, what is the bending
stress at this section? Is the stress likely to be higher at section A and B?
1
Y

2 kN
45º

O   X
0
A        15

0
B                 16

0
15
C
Z
1 kN

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(2) Fatigue Stress

1-The endurance limit of a steel member is 112 MPa and the tensile strength is
385 MPa. What is the fatigue strength corresponding to a life of 70(103) cycles?.

2-(a)Corresponding to a reliability of 99 percent, estimate the endurance limit of
a round cold drawn steel shaft 3o mm in diameter (Sut = 430 MPa and HB =
125).
(b)- What is the endurance limit of a non-rotating bar of the same material and
dimensions? Take ka = 0.82, kb = 1.189d-0.097 , kc = 0.814 and assume that kd = ke
= kf = 1

3-A 40mm round bar has been machined from AISI 1050 colddrawn round bar
(Sut= 400 MPa). This part is to withstand a fluctuating tensile load varying from
0 to 100 KN. Because of the design of the ends and the fillet radius, a fatigue
stress-concentration factor of 1.85 exists. The remaining Marin factors have
been worked out, and are ka=0.797, kb=kd=1, and kc=0.923. Find the factor of
safety using the Goodman interaction line.

2
4- Given:
Kt = 1.4 , Sut = 420 MPa notch sensitivity q = 0.3
Will the beam have infinite life?

30                     30
Pmax=10 kN
Pmin = 3 kN

2
1

50 mm dia
100 mm dia

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(3) Shaft Design
1- A section of commercial steel shafting 2.0 m long between bearings carries a
1000 N pulley at its midpoint, as shown in Fig.1. The pulley is keyed to the shaft
and receives 30 KW at 150 rev/min, which is transmitted to a flexible coupling
just outside the right bearing. The belt drive is horizontal and the sum of the
belt tensions is 8000 N. Assume Kt = Kb =1.5. Calculate the necessary shaft
diameter and determine the angle of twist between bearings. G = 80000 MN/m2.

100                 100

Fig.1
2-Fig.2 shows the forces acting on a steel shaft carrying two gears. The gears are
keyed to the shaft at B and D. A and C are journal bearing centers. Six KW is
transmitted at 650 rev/min of the shaft. The allowable stress for an unkeyed
section is 80 MN/m2, and Kb = Kt = 1.5. a)- Sketch horizontal, vertical and
resultant bending moment diagrams, shows values at change points. B)-
3
Determine the necessary shaft diameter to the nearest mm. Indicate the critical
section.
2000N
1500N
A
C

B                                        D
1000N                                                2500N

500N                           2200N

800N          100                          200                 75       1900N

Fig.2
3- A machine shaft turning at 600 rev/min is supported on bearings 750 mm
apart as shown in Fig.3 below. Fifteen KW is supplied to the shaft through a 450
mm pulley located 250 mm to the right of the right bearing. The power is
transmitted from the shaft through a 200 mm spur gear located 250 mm to the
right of the left bearing. The belt drive is at an angle of 60 0 above the horizontal.
The pulley weighs 800 N to provide some flywheel effect. The ratio of the belt
tensions is 3:1. The gear has a 20 0 tooth form and mates with another gear
located directly above the shaft. If the shaft material selected has an ultimate
strength of 500 MN/m2 and a yield point of 310 MN/m2 determine the necessary
shaft diameter using Kb = Kt = 1.0

250                 500             052

Fig.3

4- A hollow shaft, 500 mm outside diameter and 300 mm inside diameter, is
supported by two bearings 6 m apart. The shaft is driven by a flexible coupling
at one end and drives a ship propeller at 100 rev/min. The maximum trust on
the propeller is 5000 KN. The shaft transmits 6000 KW and it weighs 800 KN.
Determine the maximum shear stress in the shaft considering the weight of the
shaft and the column effect. Assume Kb= 1.5 and Kt= 1.0.

5- Fig.4 shows an arrangement for a motor and exciter with a pinion on the
same shaft. The pinion drives a gear with the gear directly below the pinion. The
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motor develops 55 KW at 200 revs/min. The exciter absorbs 5 KW, the
remainder goining to the pinion. The motor and the exciter are assembled to the
shaft by means of a force fit while the pinion is keyed to the shaft. What is the
required diameter of the shaft, if the shaft material selected has an ultimate
strength of 520 MN/m2 and a yield point of 330 MN/m2 determine the necessary
shaft diameter using Kb = 1.5 and Kt = 1.5, the pressure angle of the gear is 20 0 .

Motor rotor      Exciter motor                        pinion

500                    500                  500            250

Fig.4

6- A 600 mm diameter pulley driven by a horizontal belt transmits power
through a solid shaft to a 033 mm diameter pinion, which drives a mating gear.
1500N

800N

250
0000N

225          375
500N

Fig.5
The pulley weighs 1200 N. The arrangement of elements, the belt tensions, and
the components of the gear reactions on the pinion are indicated in Fig.5.
Determine the necessary shaft diameter using Kb =2.0, Kt = 1.5 and the
allowable torsional stress = 40 MN/m2.

5
7- The figure (6)is a schematic drawing of a shaft that supports two V-belt
pulleys. The loose belt tension on the pulley at A is 15% of the tension on
the tight side. The shaft material has a yield strength of 300 MN/m2 and an
ultimate tensile strength of 520 MN/m2. Calculate the shaft diameter.

Fig.6

8-Two bearings located 900 mm apart support a section of commercial
shafting. A 2000 N, 750 mm diameter, 20 degree involute gear is keyed to the
shaft 200 mm to the right of the right bearing. A 300 mm diameter pulley is
keyed to the shaft 500 mm to the right of the left bearing. The weight of the
pulley is 800 N and the belt tension ratio is 5:1. The gear receives 7 kW at
210 rev/min from a gear located above. Four kW is taken from the shaft at
the pulley and the remainder is taken from the shaft through a flexible
coupling located 150 mm to the left of the left bearing. Figure(7) shows an
end view of the arrangement as observed from the right.

a)- Draw the bending moment diagrams showing values at the change
points

6
b)- Calculate     the   diameter       of    the   steel   shafting   based   on
strength
Fr
Ft

T2

T1

Figure 7
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(4) Coupling Design

1- A rigid coupling has a bore diameter of 50 mm. Four machined bolts on a
bolt circle of 125 mm diameter and fitted in reamed holes. If the bolts are
made from the same material as the shaft, with an ultimate tensile
strength of 550 MN/m2 and a yield point in tension of 345 MN/m2,
determine the necessary size of bolts to have the same capacity as the shaft
in torsion.

2- Assume that a flange coupling has the following specifications:
Number of bolts, 6           Size of bolts, 12 mm diameter
Preloading of bolts, 22 KN in each bolt Inner diameter of contact, 175 mm
Outer diameter of contact, 200 mm Speed of rotation of coupling, 300 rpm
Coefficient of friction, 0.15     Shaft diameter, 50 mm
Shaft material, annealed steel with an ultimate tensile strength of 586 MN/m 2
and a yield point in tension of 310 MN/m2
The bolts are set in large clearance holes in the coupling. Determine:
1) The maximum power capacity based upon friction such that slip occurs
between faces of contact
2) Compare the shaft horsepower capacity with the friction horsepower
capacity. Assume steady load conditions and that the shaft is in torsion
only.

3- Set up the equations or relations necessary to determine:
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a) the hub diameter DH
b) the web thickness t
c) the flange thickness h

4- A flange coupling connects two 50 mm diameter lengths of commercial
shafting. The coupling webs are bolted together with four bolts of the
same material as the shaft. The bolts are set in clearance holes. The
diameter of the bolt circle is 240 mm and the web thickness is 22 mm.
Determine
a) the minimum bolt diameter required transmitting the same torque
that the shaft can transmit.
b) What power may be transmitted at 200 rev/min under steady state
conditions?

(5) Belt Drives

1- A shaft transmits maximum power from a pulley to a flexible coupling. The
shaft rotates at 900 rev /min, the pulley is 400 mm in diameter, the belt
stands horizontal, and the leather belt 50 mm wide and 6 mm thick.
Maximum stress in the belt is 2 MN/m2 and the coefficient of friction is 0.3.
If the shaft is to be checked for strength at section A-A, what bending
moment and what torque should be used? Leather has a density of p= 970
Kg/m3.
Pulley
Flexible coupling        A

250            250              250

A

2- A fan is driven by a belt from a motor which runs at 880 rev/min. a leather
belt 8 mm thick and 250 mm wide is used. The diameter of the motor pulley
and driven pulley are respectively 350 and 1370 mm. The center distance is
1370 mm, and both pulleys are made of cast iron. Coefficient of friction is
0.35. The allowable stress for the belt is 2.4 MPa. The belt mass is 970
Kg/m3. What is the power capacity of the belt?

3- A crossed belt drive is to transmit 7.5 KW at 1003rev/min of the smaller
pulley. The smaller pulley has diameter of 250 mm, the velocity ratio is 2,
and the center distance is 1.25 m. It is desired to use a flat belt 6 mm thick
with an expected coefficient of friction 0.3. If the maximum allowable stress
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in the belt is 1.7 MN/m2, determine the necessary belt width b. The leather
has a density of 970 Kg/m3.

4- A V-belt drive is to transmit 18.5 KW from a 250 mm pitch diameter pulley
at 1800 rev/min to a 900 mm diameter flat pulley. The center distance
between the input and the output shafts is 1.0 m. The groove angle is 40 0,
and the coefficient of friction between the belt and the pulley is 0.2. The
cross section of the belt is b2 = 38 mm wide at the top and b1 = 19 mm wide
at the bottom by d = 25 mm deep. Each belt weighs 11 KN/m3 and the
allowable tension per belt is 900 N. How many belts are required?

5- A 1.35 m diameter steel flywheel is to be connected to a 0.4 m diameter
rubber faced motor pulley by means of double ply leather belt which has a
thickness of 8 mm. The center distance is 3.0 m. The coefficient of friction for
leather on steel is 0.20 and for leather on rubber is 0.4. The leather has an
allowable stress of 2.75 MN/m2. Density of leather is 970 kg/m3. If 45 kW is
transmitted with a belt speed of 24.5 m/sec, determine:
a)- ef for the pulley which governs the design
b)- the necessary belt width
** Available widths of belt are:
2.5 mm increments from 10 to 25 mm
5.0 mm increments from 25 to 100 mm
10 mm increments from 100 to 200 mm
25 mm increments from 200 to 300 mm

6- An open belt drive delivers 15 KW when the motor pulley, which is 300
mm diameter, turns at 1750 rev/min. The belt is 10 mm thick and 150 mm
wide and has a density of 970 Kg/m3. The driven pulley, which is 1.2 m
diameter, has an angle of contact of 2000.What is the maximum stress in
the belt assuming a coefficient of friction 0.3 for both pulleys?
(1.37MPa)

7- A V-belt drive transmits 11 KW at 900 rev/min of the smaller sheave.
The sheaves pitch diameters are 173 mm and 346 mm. The center
distance is 0.76 m. If the maximum permissible working force per belt
is 560 N, determine the number of belts required if the coefficient of
friction is 0.15 and the groove angle of the sheaves is 34 0. The belt mass
is 0.194 Kg/m.
(3.16 use 4 belts)
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9
(6) POWER SCREWS AND THREADED FASTENERS

1- The screw shown in fig.(1) is operated by a torque applied to the lower end.
The nut is loaded and prevented from turning by guides. Assume friction in the
ball baring to be negligible. The screw has a 48 mm outside diameter and triple
ISO trapezoidal thread. The pitch is 8 mm. Thread coefficient of friction is 0.15.
Determine:
1)- The load which could be raised by a torque T of 40 Nm.
2)- Would the screw be overhauling.
3)- The average bearing pressure between the screw and the nut thread surfaces.

W

50

T
Figure (1)

2-The following data apply to the C-clamp shown in fig.(2).
Outside diameter = 12 mm, Root diameter = 9.85 mm, Root area = 76.25 mm2
Coefficient of thread friction f = 0.12, Coefficient of collar friction fc = 0.25
Mean collar radius rc = 6 mm, Load W = 4000 N
Operator can comfortably exert a force of 80 N at the n of the handle.
A)- What length of handle, L, is needed?
B)- What is the maximum shear stress in the body of the screw and where does
this exist?
C)- What is the bearing pressure P on the threads?

10
L

150
A          A
25
B          B

W

W

3-Estimate the maximum wrench torque which can be applied in tightening a 20
mm bolt if the shear stress in the body of the bolt is not to exceed 140 MN/m2.
Outside diameter = 20mm                         Root diameter = 16.72 mm
Thread angle  = 30  0
Pitch = 2.5 mm
Effective friction radius rc = 12 mm               Thread and collar coefficient of
friction = 0.10

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(7) CLUTCHES

1- Drive the torque capacity for one pair of surfaces pressed together with an
axial force F, assume uniform pressure.

2-Drive the torque capacity for one pair of surfaces pressed together with an
axial force F. Assume uniform wear.

3-Determine the maximum, minimum, and average pressure in a plate clutch
where the axial force is 4000 N, the inside radius of contact is 50 mm, the
outside radius of contact is 100 mm. Uniform wear is assumed.

4-A multiple disk clutch, steel and bronze, is to transmit 4 KW at 750 rev/min.
The inner radius of contact is 40 mm and the outer radius of contact is 70 mm.
The clutch operates in oil with an expected coefficient of friction 0.10. (Oil is

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used to give smoother engagement, better dissipation of heat, even though the
capacity is reduced). The average allowable pressure is 350 KN/m2 maximum.
1)- How many total disks of steel and bronze are required?
2)- What axial force required?
3)- What is the average pressure?
4)- What is the actual maximum pressure? Assume uniform wear.

5-A multiple disk clutch is composed of 5 steel and 4 bronze disks. The clutch is
required to transmit 16 Nm torque. If the coefficient of friction may be taken as
0.10, the average pressure is not to exceed 350 KN/m2 and the inner diameter is
restricted to 50 mm, assuming uniform wear, determine:
1-the necessary outer diameter of the disks
2-the necessary axial force

6-A soft cone clutch must handle 200 Nm of torque at 1250 rev/min. the large
diameter of the clutch is 350 mm, the cone pitch angle is 6.25 0, the face width b
is 65 mm, and the coefficient of friction is 0.20, determine:
1- the axial force F required to transmit the torque
2- the axial force Fe required to engage the clutch
3- the average normal pressure p on the contact surfaces
4- the maximum normal pressure, assuming uniform wear.

8- Consider the same clutch and conditions as the above problem, but assume
uniform pressure. Determine:
1- the axial force F required to transmit the torque
2- the axial force Fe required to engage the clutch
3- the average normal pressure p on the contact surfaces

9- A cone clutch is to transmit 100 Nm after engagement. If the maximum
force is 850 N, what is the required width of faces? The total included cone
angle is 240, and the maximum average pressure is limited to 100 KN/m2,
and the coefficient of friction is 0.20. Assume uniform wear.

10- A soft cone clutch has a cone pitch angle of 100, a mean diameter of 300
mm, and a face width of 100 mm. Using a coefficient of friction of 0.20, the
assumption that uniform wear exists, and the average pressure is 70 KN/m 2
for a speed of 500 rev/min, determine:
1- the force required to engage the clutch
2- the power that can be transmitted.

(8) Keys, Pins, and Splines

1- Determine the required length of a square key if the key and the shaft
are made of the same material and of equal length.

12
2- A feather key is 12 mm wide and 9 mm deep and is to transmit 680 Nm
of torque from a 38 mm diameter shaft. The steel key has an allowable
stress in tension and compression of 110 Mpa and an allowable stress in
shear of 57.5 Mpa. Determine the required length of the key.

3-A pin in knuckle joint is subjected to an axial load of 90 KN. Assume that
the thickness of the eye to be 1.5 times the diameter of the pin. The allowable
stress of the material in tension and compression due to bending is 60 MN/m 2
and the allowable stress in shear is 30 MN/m2 . The allowable stress in
bearing is 20 MN/m2 . Determine the required pin diameter.

4- A splined connection in an automobile transmission consists of 10 splines
cut in a 58 mm diameter shaft. The height of each spline is 5.5 mm and the
keyways in the hub are 45 mm long. Determine the power that may be
transmitted at 2500 rev/min, if the allowable normal pressure on he splines is
limited to 4.8 MN/m2 .
5- A flat key is used to key a gear on a 30 mm shaft made of 0.3%C cold drawn
steel. The key is made of the same material as the shaft (σu = 634 MPa and σy =
538 MPa). Determine the torque capacity of the shaft in accordance with the
ASME shafting code. Calculate the torque capacity of the key using a factor of
safety of 1.5 based upon the yield strength of the material, and assuming that σs
=0.6 σy . Shear and compression areas are 115 mm2 and 22 mm2.

5- If the key in Problem 2 had 9 mm wide and 12 mm deep, what would have
been the required length for the same load and material?

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