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Load Capacity of 4mm-Shafts

In metal construction sets mostly steel shafts with a diameter of 4 mm are used, which in heavy models often are considered as a weak point.

In some cases of course thatīs true, but sometimes also the to the shafts belonging bearings are too unstable and affect the proper operation of a model.
When driving heavy vehicles also components like breaker plates and gears reach their limits, because they are not suitable to transmit the heavy torques. Slipping shaft to collar connections can be fixed by flattening the shaft, but with a broken pressed in hub the concerned component is damaged.
Subsequent the load capacity of the shafts is examined and explained by means of practical applications.

The strength calculation of mechanical components is a very complex subject, which at this place canīt be applied detailed. Accurate forecasts are not possible anyway, because this would require detailed predictions of the real operating conditions.
Static loads mostly are proper to specify, for dynamic processes however at best is to make a rough estimate. But especially the number and the amplitude of the cycles affect the durability, and if a short operating time is sufficient, the components may be loaded stronger.

Dynamic loaded components mainly are designed for fatigue strength.
Basis for this examination therefore also are the current specific values and (simplified) calculation methods for the fatigue strength of axles resp. shafts with a circular cross section from steel S 235 (former ST 37).
Neither the surface condition (corrosion, notch effect etc.) nor the in engineering usual safety factors are considered.

The red illustrated values exceed the limits for fatigue strength; an operation in this area may damage the shafts resp. may cause early failures.



1. Case:   Static bending at one-sided bearing

Application:
Stub axles of stationary vehicles

A shaft which doesnīt transmit a torque is called an axle.
For the stub axles of a stationary vehicle a static bending can be assumed.


One-sided bearing
 

   ymax = F · l3 / 3 · E · I

   ymax = deflection in the distance l in mm
   F    = in the distance l affecting force in N
   l    = distance in mm
   E    = elastic modulus 210000 N/mm2
   I    = moment of an area p · d4 / 64
          (12,566 mm4 for d = 4 mm)

Tab 1:   Deflection ymax in mm with a force F in the distance l from the bearing

l [mm]
F [N]
 
10
20
30
40
50
60
70
80
90
100
10
20
30
40
50
60
70
80
190
100
0,001
0,010
0,034
0,081
0,158
0,273
0,433
0,647
0,921
1,263
0,003
0,020
0,068
0,162
0,316
0,546
0,866
1,293
1,842
2,526
0,004
0,030
0,102
0,243
0,474
0,818
1,300
1,940
2,762
3,789
0,005
0,040
0,136
0,323
0,632
1,091
1,733
2,587
3,683
5,052
0,006
0,051
0,171
0,404
0,789
1,364
2,166
3,234
4,604
6,316
0,008
0,061
0,205
0,485
0,947
1,637
2,599
3,880
5,525
7,579
0,009
0,071
0,239
0,566
1,105
1,910
3,033
4,527
6,446
8,842
0,010
0,081
0,273
0,647
1,263
2,183
3,466
5,174
7,366
10,10
0,011
0,091
0,307
0,728
1,421
2,455
3,899
5,820
8,287
11,37
0,013
0,101
0,341
0,803
1,579
2,728
4,332
6,467
9,208
12,63


As max. acceptable bending stress sb zul = k2 · Rp 0,2 = 330 N/mm2 is assumed (see Basis); the red illustrated values here indicate bending stresses above 330 N/mm2.

Example:
Assumed is a stationary vehicle with an evenly spreaded mass which is carried by 4 wheels.
If is also assumed that every stub axle is burdened in an average distance of l = 20 mm from its fixing it may carry a weight of approx. 100 N (corresponding to a mass of approx. 10 kg); the total mass of the model consequently reaches approx. 40 kg  (!).
The use of more wheels allows still higher weights, but the realization of a proper stiffness for the construction probably will be problematical.



2. Case:   Static bending at double sided movable bearing

This case is specified only for the sake of completeness.


Double sided movable bearing
 

ymax = F · l3 / 48 · E · I

ymax = deflection in the middle
       between the bearings in mm 
F    = in the middle affecting force in N 
l    = distance of the bearings in mm 
E    = elastic modulus 210000 N/mm2 
I    = moment of an area p · d4 / 64
       (12,566 mm4 for d = 4 mm)

Tab 2:   Deflection ymax in mm with a force F in the middle of the distance l

l [mm]
F [N]
 
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
180
200
0,000
0,005
0,017
0,040
0,079
0,136
0,217
0,323
0,460
0,631
0,001
0,010
0,034
0,081
0,158
0,273
0,433
0,647
0,921
1,263
0,002
0,015
0,051
0,121
0,237
0,409
0,650
0,970
1,381
1,894
0,003
0,020
0,068
0,162
0,316
0,546
0,866
1,293
1,841
2,526
0,003
0,025
0,085
0,202
0,395
0,682
1,083
1,616
2,301
3,157
0,004
0,030
0,102
0,242
0,474
0,818
1,299
1,940
2,762
3,788
0,004
0,035
0,119
0,283
0,552
0,955
1,516
2,263
3,222
4,420
0,005
0,040
0,136
0,323
0,631
1,091
1,733
2,586
3,682
5,051
0,006
0,045
0,153
0,364
0,710
1,227
1,949
2,910
4,143
5,683
0,006
0,051
0,170
0,404
0,789
1,364
2,166
3,233
4,603
6,314


Also here as max. acceptable bending stress sb zul = k2 · Rp 0,2 = 330 N/mm2 is assumed (see Basis), whose exceeding the red illustrated values characterize.



3. Case:   Dynamic (swelling) bending at one-sided bearing

Application:
Stub axles of non-driven wheels at radio controlled vehicles

Because of the driving on a rough bottom to the static load a dynamic component is added, which mainly acts into the same direction like the static load; this case is also called swelling bending.

The deflection of the stub axles of course is the same like in Tab. 1, but because of the wearing effect of the dynamic components the max. acceptable load is lower.

Tab 3:   Deflection ymax in mm with a force F in the distance l from the bearing

l [mm]
F [N]
 
10
20
30
40
50
60
70
80
90
100
10
20
30
40
50
60
70
80
190
100
0,001
0,010
0,034
0,081
0,158
0,273
0,433
0,647
0,921
1,263
0,003
0,020
0,068
0,162
0,316
0,546
0,866
1,293
1,842
2,526
0,004
0,030
0,102
0,243
0,474
0,818
1,300
1,940
2,762
3,789
0,005
0,040
0,136
0,323
0,632
1,091
1,733
2,587
3,683
5,052
0,006
0,051
0,171
0,404
0,789
1,364
2,166
3,234
4,604
6,316
0,008
0,061
0,205
0,485
0,947
1,637
2,599
3,880
5,525
7,579
0,009
0,071
0,239
0,566
1,105
1,910
3,033
4,527
6,446
8,842
0,010
0,081
0,273
0,647
1,263
2,183
3,466
5,174
7,366
10,10
0,011
0,091
0,307
0,728
1,421
2,455
3,899
5,820
8,287
11,37
0,013
0,101
0,341
0,803
1,579
2,728
4,332
6,467
9,208
12,63


The max. acceptable bending stress in this case is the limit for swelling bending sb zul = ssch = 255 N/mm2. This value is the sum of the static and dynamic components, from which an average (static) bending stress of only 127,5 N/mm2 resultiert (seeBasis). Here thus is assumed, that maximal the twice of the static load occurs.

Example:
If again is assumed, that every stub axle of a non-driven wheel of a radio controlled vehicle is burdened in an average distance of l = 20 mm from its fixing, then it may carry a static weight of only approx. 40 N (corresponding to approx. 4 kg mass).

This value certainly is very unexact, because the real dynamic load canīt be specified. But because metal kit models normally are operated on a flat bottom and shocks can be damped by the elasticity of the chassis, the above assumption seems to be passably realistic.



4. Case:   Static torsion

This case is only a calculation example.


Torsion of a shaft
 

tt = Mt / Wp (torsional stress)

Mt = torsional moment F · r

Wp = polar section modulus p · d3 / 16
     (12,566 mm3 for d = 4 mm)


The max. acceptable torsional stress for this load case is tt zul = k2 · Rp 0,2 = 135 N/mm2; according to this a 4mm-shaft is able to transmit a torque of approx. 1,7 Nm (see Basis).

Example:
Assumed is a radio controlled vehicle with 40 kg mass. By means of 2 driven wheels with a diameter of 150 mm (radius R = 75 mm) it shall be accelerated on a flat bottom from the zero speed to a velocity of 1 m/s (3,6 km/h) within 1 s (without consideration of the road resistance and the inertia of the drive components).
Every drive shaft is burdened with a torque of approx. 1,5 Nm; also in this case the drive shaft seems to be better than other drive components.



5. Case:   Dynamic bending + dynamic torsion

Application:
Drive shafts of radio controlled vehicles

This case is one of the most frequently load cases in engineering.
The because of the weight (and also because of shocks) bended drive shaft while rotation gets an alternating bending; simultaneous acts the torsional moment of the drive which can be assumed as a swelling torsion (thus mainly in one direction).

In this case the torsional stress can be converted and by means of superposition with the existing bending stress an equivalent stress can be calculated. Following the equivalent stress is to compare with the max. acceptable stress, which results from the limit of the dominating kind of load.

Example:
Assumed is a radio controlled vehicle with 4 wheels and an evenly spreaded mass of 10 kg. By means of 2 driven rear wheels with a diameter of 100 mm (radius R = 50 mm) it shall be accelerated on a flat bottom from the zero speed to a velocity of 1 m/s (3,6 km/h) within 1 s (without consideration of the road resistance and the inertia of the drive components).
Further at all stub axles and shafts the force shall act in an average distance l = 20 mm from the fixings resp. bearings.

Forecast for the stub axles of the front wheels:
For these a static resp. average bending stress of sb = 80 N/mm2 can be calculated; this value is significant lower than the acceptable bending stress of 127,5 N/mm2 (see 3. Case).

Forecast for the shafts of the rear wheels:
To the static resp. average bending stress of sb = 80 N/mm2 on every drive shaft an average torsional stress of tt = 10 N/mm2 is added (the shafts are not permanent loaded with the max. torque). From this values the equivalent stress can be calculated to

sv = Ö[sb2 + 3(0,7 · tt)2] = 81 N/mm2

In this example dominates the bending stress; therefore as limit the half of the acceptable alternating bending is determined: szul = 85 N/mm2.
The calculated equivalent stress is below the assumed limit and consequently acceptable.

Also the results of this example are very unexact, because both the bending and the torsion are dynamic and the real load canīt be specified.

The data for the acceleration of the model here are only mathematical values and in practice absolutely an overload of the shafts can occur in dependence of the power of the drive; also the drive has to be examined with regard to the torque (see Forecast for the 1½-Deck-Bus).


Example: Forecast for the 1½-Deck-Bus with detailed calculation.


The example "1½-Deck-Bus" shows clear, that with the drive of heavy models the 4mm-shafts indeed rapid reach their limits.
Of course the motor torque can be reduced by means of a lower supply voltage resp. a serial resistance in the supply line. But on the other hand a heavy model needs a strong drive for a proper operation, and under this aspect the use of thicker drive shafts probably would be the better solution.
An overload also can be ignored with the risk of an early fatigue break, which probably never occurs because of the relative short useful life of a metal kit vehicle.

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