7. Shaft analysis for strength

Let us carry out the analysis of the shaft for strength if torque at the gear shaft T_g is equal to 209.192 N·m and from the previous analysis we have that turning (tangential) force F_t = 2134.612 N; F_r = 776.999 N; F_shaft = 1697.762 N; d_g = 196 mm; d = 55 mm (the shaft portion diameter where the gear is mounted), and from the sketch layouta = 63 mm; b = 63 mm; c = 82 mm.

7.1. Select the material of the shaft.

The main material of shafts is medium-carbon or alloy steels of grade Steel 40 (0.4C), Steel 45 (0.45C), Steel 40X (0.4C-Cr) and others with hardness H ≥ 200 BHN.

For example, let us choose Steel 40X (0.4C-Cr) heat treated by martempering to hardness ranged from 230 to 260 BHN, _ul = 850 MPa, _y = 550 MPa.

7.2. Analyze a shaft for static strength.

7.2.1. Plot the analytical model of a shaft and apply all acting forces In this case a shaft is considered as a beam mounted on two supports, in particular on one immovable hinge support and one movable hinge support.

Let us analyze forces which act on a shaft.

It is necessary to remember that in the engagement of straight spur gears turning force F_t and radial force F_r develop (radial force is always directed to the center of rotation of the gear).

On the cantilever portion of the shaft either a pulley, or a sprocket, or a pinion, or a half coupling may be mounted.

If a pulley or sprocket is installed on the shaft a force acting on the shaft from the side of the element is directed to side of the mating pulley (sprocket) along the center line. The value of this force was determined during analysis of the corresponding mechanical drive.

7.2.2. Plot the analytical model of the shaft in the vertical plane and transfer all forces to the shaft.

7.2.3. Determine vertical support reacting R_yA and R_yC. For this purpose we should consider the equilibrium of the beam and set up equations of moments with respect to points A and C:

, .

For checking we set up equation of forces that act in the vertical plane of the shaft. The sum of these forces should give zero ( ).

7.2.4. Plot the diagram of bending moments in the vertical plane ( )

7.2.5. Plot the analytical model of the shaft in the horizontal plane and transfer all forces to the shaft (Fig. 10.2). In this case as a result of parallel transferring force F_t the torque T develops

.

7.2.6. Determine horizontal support reacting forces R_XA and R_XC. For this purpose we should set up equations of moments with respect to points A and C:

, .

For checking we find the sum of all forces that acts in the horizontal plane of the shaft. This sum should be equal to zero ( ).

7.2.7. Plot the diagram of bending moments in the horizontal plane ( ).

7.2.8. Plot the diagram of total bending moments taking into account that

7.2.9. Plot the twisting moment diagram

7.2.10. Plot the reduced moments diagram taking into account that М_red

7.2.11. For the critical section of the shaft (where the reduced moment is maximum) we check the shaft for static strength

, ,

whered is the diameter of the shaft at the critical section; [σ_b] is the nallowable bending stress. For steels [σ_b] = 120 MPa.

Condition is satisfied.

7.3. Analyze the shaft for fatigue strength.

7.3.1. Determine the limit of endurance in bending and in torsion for the shaft material:

for carbon steels: _-1 = 0.43·_ul,

for alloy steels: _-1 = 0.35·_ul+ 120,

_-1 = (0.2…0.3)·_ul,

where_ulis the ultimate strength of the material.

_{In our case for Steel 40X (0.4C-Cr)}

10.3.2. Determine peak magnitudes of bending and torsion stresses (_p, τ_p) at the critical sections of the shaft.

Critical section is a section where the total moment M_Σ is maximum. It may be a section where a gear or a bearing is located.

, ,

whered is the diameter of the shaft at the critical section.

7.3.3. Determine mean components of the bending and torsion stresses (_m, τ_m).

In our case F_a= 346.7 N < 1000 N, so _m= 0 and τ_m= τ_p = 4.28 MPa.

7.3.4. Determine factors _ and _ of mean stress components

In our case _= 0.15 and _= 0.1.

7.3.5. Determine effective stress concentration factors K_and K_τ.

In our case the critical section of the shaft is a section where a bearing is mounted. So we will use as stress concentrator interference fit. Then

7.3.6. Determine the surface roughness factor K_F.

In our case the critical section is for the portion of the shaft where a bearing is installed. That is why we use curve 2 and K_F = 0.9.

7.3.8. Determine safety factors in terms of bending and torsion

;

.

7.3.9. Determine the safety factor of the shaft at the critical section

;

Allowable values of the safety factor [S] are given in table below.