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MACHINE DESIGN AND CAD-II

LECTURER: ABDUL AHAD NOOHANI

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Shaft subjected to bending moment only

When the shaft is subjected to pure bending load, the principle stresses induced in the shaft are tension and compression.

In this case maximum stresses induced in the shaft is determined by the theory of simple bending.

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……………………….. Equation (1)

Putting these values in equation 1 We get

Putting these values in equation 1 We get

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Example no:14.4A pair of wheels of a railway wagon carries a load of 50 kN on each axle box,

acting at a distance of 100 mm outside the wheel base.

The gauge of the rails is 1.4 m.

Find the diameter of the axle between the wheels,

if the stress is not to exceed 100 Mpa.

Data:

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A little consideration will show that the maximum bending moment acts on the wheels at C and D.

Therefore maximum bending moment

Solution:

Putting this value of bending moment in equation (1)

……………eq. (1)

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Shafts Subjected to Combined Twisting Moment and Bending Moment

When the shaft is subjected to combined loading, the design is usually based on the maximum shear stress theory, since the shafts are usually made of ductile materials.

According to maximum shear stress theory, the following relation must be satisfied:

Simplifying, we have:

= Equivalent twisting moment

From this expression, diameter of the shaft ( d ) may be evaluated.

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According to normal stress theory (Rankine’s Theory: for brittle materials) :

Equivalent bending moment

From this expression, diameter of the shaft ( d ) may be evaluated.

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A shaft supported at the ends in ball bearingscarries a straight tooth spur gear at its mid span and is to transmit 7.5 kW at 300 rpm.

The pitch circle diameter of the gear is 150 mm.

The distances between the centre line of bearings and gear are 100 mm each.

If the shaft is made of steel and the allowable shear stress is 45 MPa, determine the diameter of the shaft.

Show in a sketch how the gear will be mounted on the shaft; also indicate the ends where the bearings will be mounted?

The pressure angle of the gear may be taken as 20°.

Example 14.6

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DATA:

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Since the gear is mounted at the middle of the shaft, therefore maximum bending moment at the centre of the gear,

normal load acting on the tooth of the gear

Tangential force on the gear,

We know that torque transmitted by the shaft,

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Shafts Subjected to Fluctuating Loads

line shafts

counter shafts

In the previous articles we have assumed that the shaft issubjected to constant torque and bending moment.

But in actual practice, the shafts are subjected to fluctuating torque and bending moments.

In order to design such shafts like line shafts and counter shafts, the combined shock and fatigue factors must be taken into account for the computed twistingmoment (T ) and bending moment (M ).

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Thus for a shaft subjected to combined fluctuating bending and torsion, the equivalent twisting moment,

and equivalent bending moment,

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Example 14.12.

A mild steel shaft transmits 20 kW at 200 rpm.

It carries a central load of 900N and is simply supported between the bearings 2.5 m apart.

Determine the size of the shaft, If the allowable shear stress is 42 MPa and the maximum tensile or compressive stress is not to exceed 56 MPa.

What size of the shaft will be required, if it is subjected to gradually applied loads.

DATA:

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maximum bending moment of a simply supported shaft carrying a central load,

Putting values in equation eq: 1

………… EQ: 1

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We know that the equivalent bending moment,

………… eq: 2

Putting values in equation eq: 2

Taking the larger of the two values, we have

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2.Size of the shaft when subjected to gradually applied load

Putting values of M, T, and other constants in above equation

Putting value of Te in equation no 3

…………equation no 3

for rotating shafts with gradually applied loads

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We know that the equivalent bending moment,

Putting values in above equation eq

Taking the larger of the two values, we have

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Shafts Subjected to Axial Load in addition to Combined Torsion and Bending Loads

1. Propeller shafts of ships

2. Shafts for driving worm gears

When the shaft is subjected to an axial load (F) in addition to torsion and bending loads , then the stress due to axial load must be added to the bending stress.

We know that

Bending equation:

Stress due to axial load: a

Resultant stress (tensile or compressive) for solid shaft,

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In case of a hollow shaft, the resultant stress,

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Slender shafts and column factor:

In case of long shafts (slender shafts) subjected to compressive loads,

a factor known as column factor (α) must be introduced .

to take the column effect into account.

Stress due to the compressive load:

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Value of column factor (α):

(when the slenderness ratio (L / K) is less than 115.)

(when the slenderness ratio (L / K) is more than 115).

The value of column factor (α) for compressive loads:

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Equivalent twisting moment and equivalent bending moment

In case of Shafts Subjected to Axial Load in addition to fluctuating Combined Torsion and Bending Loads

Equivalent twisting moment:

Equivalent bending moment:

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Example 14.18

A hollow shaft is subjected to a maximum torque of 1.5 kN-m and a maximum bending moment of 3 kN-m.

It is subjected, at the same time, to an axial load of 10 kN.

Assume that the load is applied gradually and the ratio of the inner diameter to the outer diameter is 0.5.

If the outer diameter of the shaft is 80 mm, find the shear stress induced in the shaft.

Given:

find:

load is applied gradually, therefore from Table 14.2

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We know that the equivalent twisting moment for a hollow shaft:

We also know that the equivalent twisting moment for a hollow shaft (Te),

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Example 14.19

A hollow shaft of 0.5 m outside diameter and 0.3 m inside diameter is used to drive a propeller of a marine vessel.

The shaft is mounted on bearings 6 metre apart and it transmits 5600 kW at 150 r.p.m.

The maximum axial propeller thrust is 500 kN and the shaft weighs 70 kN.

Determine :

1. The maximum shear stress developed in the shaft, and2. The angular twist between the bearings.

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Maximum shear stress developed in the shaft:

We know that the equivalent twisting moment for a hollow shaft,

We know that the torque transmitted by the shaft,

the maximum bending moment,

Now let us find out the column factor α.

We know that least radius of gyration