How Do You Determine Shaft Diameter Under Axial Load?

In ship propeller shafts, an additional axial load will act on the shaft in addition to the Torsion and Bending loads. Now, let us calculate the shaft diameter under the axial load and the torque and bending load.

How Do You Determine Shaft Diameter Under Axial Load?

Axial Load Propeller Shaft Diameter Calculation

As previously stated, axial load will occur in ship propeller shafts and shafts for driving worm gears; therefore, the stress due to axial load must be added to the bending stress (b).

Based on the bending equation

MI = by = ER

Where M is the Bending Moment

I = the shaft’s moment of inertia

Bending stress = σb

y = Distance from the neutral axis = d/2 

E = Material modulus of elasticity

R = Curvature radius

This allows us to consider the following part of the equation.

MI = by

Shafts Made of Solid Material

For the solid shafts

 We can rewrite the above equation as follows.

 a =F4d2

a =4Fd2

=We can write the stress due to axial load for the solid shaft as

b = M.yI

b = M d/264d4

b = 32Md3

We can write the resultant stress (tensile or compressive) for a solid shaft.

1 = 32 Md3 + 4Fd2

1 = 32Md3 M+F d8

1 = 32 M1d3

Where the M1 value given below has been substituted

M1= M + Fd8

Axial loads can be applied to shafts that are very long, such as propeller shafts and intermediate shafts. To account for the column effect in the case of long shafts (slender shafts) subjected to compressive loads, a factor known as column factor () must be introduced.

The stress caused by a compressive load,

c = 4Fd2

The column factor () for compressive loads can be calculated using the following equation:

= 11-0.0044 (L/K)

Where

 L is the distance between the bearings on the shaft.

 K = Minimum gyration radius

This expression is used when the slenderness ratio (L/K) is less than 115. The column factor can be calculated using the following equation when the slenderness ratio (L/K) is greater than 115.

The column factor formula will be used if the slenderness ratio (L/K) is greater than 115.

= y (L/K)2C2E

Where σy is the shaft material’s compressive yield point stress.

In Euler’s formula, C = Coefficient, depending on the end conditions

The following are the various values of C based on the end conditions.

The letter C

Shafts for Hallow

The bending stress from the bending equation will be calculated similarly to that of the solid shaft.

b = 32M (do)3 (1-k4)

Where k = di / do

do = shaft’s outer diameter

di = Internal Shaft Diameter

We can write the stress due to axial load for the hollow shaft as

a = F4(do)2- (di)2

a = 4F(do)2- (di)2

a = F (do)2(1-k2)    

We can write the resultant stress (tensile or compressive) for a hollow shaft.

1 = 32M(do)2 (1-k4) + 4F (do)2 (1-k2)

1 = 32(do)3 (1-k4) M+Fdo (1+k2)8

1 = 32M1(do)3 (1-k4)

Where the M1 value given below has been substituted

M1 = M+Fdo (1+ k2)8

In general, the equations for an equivalent twisting moment (Te) for a hollow shaft subjected to fluctuating torsional and bending loads, as well as an axial load, will be written as

Te = kmM+Fdo(1+k2)82+(kt+T)2   = 16(do)3  (1-k)4

How Do You Determine Shaft Diameter Under Axial Load?

and the equivalent Bending Moment (Me) with the additional Axial Load as

Me = 12 KmM+Fdo (1+k2)8+kmM+Fdo(1+K282+(K1T)2 = 32b(do)3(1-k4)

How Do You Determine Shaft Diameter Under Axial Load?

The above twisting moment and bending moment equations can also be written for solid shafts.

k = 0 and d0 = d for a solid shaft.

When there is no axial load on the shaft, F = 0.

When the shaft is subjected to an axial tensile load, α = 1

We can calculate the shaft diameter using the above relationship if the shaft is subjected to axial load in addition to the bending and twisting moments.

Let us consider an example problem in which we must calculate the shaft diameter under axial load and the bending and twisting moments.

Shaft Diameter Calculation Problem with Axial Load

A 1.5-meter-long solid shaft supported by a bearing is subjected to a maximum torque of 500 N-m and a maximum bending moment of 1 kN-m. It is subjected to an axial load of 10 kN at the same time. Assume that the load is gradually applied. The shaft material’s permissible shear stress is 40MPa. Determine the diameter of the propeller shaft.

Answer:

Twisting moment T = 500 N-m = 500 × 103 N-mm

Bending moment M = 1 kN-m = 1000 × 103 N-mm

Axial Load F = 10 kN = 10 × 103 N

Permissible shear stress τ = 40MPa = 40 N/mm2

Length of the shaft L =1.5m = 1500mm

Let d denote the diameter of the propeller shaft (solid shaft)

Because the load is applied gradually, we can deduce from the table that Km = 1.5 and Kt = 1.0. (In most cases, these values will be mentioned in the problem itself)

The equivalent twisting moment for a hollow shaft is known.

Te = kmM+Fdo(1+k2)82+(kt+T)2   = 16(do)3  (1-k)4

First, we must compute the Slenderness ratio.

= 11-0.0044 (L/K)

Where L is the length of the shaft between the bearings, which is 3 meters = 3000mm, and K is the least radius of gyration√(area moment of inertia/ Cross-sectional Area)

K = √((π/64)d4/√(π/4)d2 = √0.0625d2 = 0.25d

We substitute the value of K, L we get α = d / (d-0.0264)

For a solid shaft, k = 0 and d0 = d.

We substitute all of the values we have in the equivalent twisting moment equation from above.

Te = kmM+Fdo(1+k2)82+(kt+T)2   = 16(do)3  (1-k)4

  (1.51000103)+(dd-0.0264)10103d8 2+(1500103)2   = 1640d3

  (1.5106)+(dd-0.0264)1048 2+251010     = 7.854d3

How Do You Determine Shaft Diameter Under Axial Load?

Solving the above equation by trial and error yields an approximate diameter value of 61mm.

To simplify the above calculation, we can use the following formula for the shaft subjected to fluctuating, twisting, and bending together to find the least critical shaft diameter.

Te = (Km M)2 + (Kt+T)2

We can deduce Te = 1581139 from this. Equate this to the above 7.854 × d3

1581139 = 7.854 × d3

d3 = 201316.4

d = 58.60mm

Which is the least critical shaft diameter, and which is the next larger shaft diameter from the standard shaft diameters?

This process must also be checked with the bending moment formula, and the largest shaft diameter among the calculated shaft diameters must be chosen.

Read more: Why Is A Hollow Shaft Better Than A Solid Shaft?