Shaft stress calculations

The sections to follow discuss determining keyway depth and width, keyway manufacturing tolerances, key stress calculations, and shaft stress calculations. 

The stress in the crank shaft is calculated approximately from the power and speed as follows. Bear in mind that approximate calculations of this sort may be in error by up to a factor of 2 - but this makes no difference to the conclusions reached below. Referring to Fig. 16.9 ... 

The calculation of tensile, compressive and shear stresses is relatively easy, but to obtain bending stresses it is necessary to know the moment of inertia of the section of the beam and in the case of torsion, although it is a shear stress, because it is spread along the length of the shaft its calculation is complex. With bending moments, a tensile stress is induced in the outer surface of the beam and an equal and opposite compressive stress occurs in the inner surface. 

The acquisition of analytical techniques and practical skills in the engineering sciences is important to the design system. Through a study of engineering of any label based on mathematics and physics applied through elemental studies, one acquires an all-round engineering competence. This enables, for example, one to calculate fatigue life, creep behavior, inertia forces, torsion and shaft stresses, vibration characteristics, etc. 

Example 2.7 A nylon ring with a nominal inside diameter of 30 mm, an outer diameter of SO mm and a width of S mm is to be made an interference fit on a metal shaft of 30 mm diameter as shown in Fig. 2.17. The design condition is that the initial separation force is to be 1 kN. Calculate (a) the interference on radius needed between the ring and the shaft and (b) the temperature to which the nylon must be heated to facilitate easy assembly. What will be the maximum stress in the nylon when it is in position on the shaft The coefficient of friction between nylon and steel is 0.2S. The short-term modulus of the nylon is 1 GN/m, its Poisson s ratio is 0.4 and its coefficient of thermal expansion is 100 X 10- °C- . 

Next we calculate the power input of a screw extruder. Equation E9.1-8 indicates that for calculating the total power we need to know the viscous energy dissipation and the pressure rise. To calculate the former according to Eq. E9.1-2, we need the complete velocity and temperature fields inside the machine. However, it is easier to calculate the total power input by multiplying the shear stress at any point on the barrel surface with the barrel velocity and integrating over the surface of the barrel. This will be equivalent to the total shaft power input. In tensor form, accounting for the direction of the shear stress and velocity, this is given by... 

Improved shaft-hub connections and 3-dimensional calculation of stress distribution (Fig. 14.1)... 

Determine the minimum shaft diameter for strength. Since the torque and bending moment may act simultaneously on the shaft, these loads must be combined and resolved into shear and tensile stresses on the shaft. The minimum shaft diameter must be the larger of the shaft diameters required by either shear- or tensile-stress limits. The shaft diameter for shear stress ds can be calculated as follows ... 

Shafting.—For large installations, it pays to calculate shafting sizes closely, considering both transmission and bending stresses. The accompanying table... 

Table 1. Calculation of principal stress in measuring points in No. 2 inclined shaft. 

In rotary shaft lip seals, internal stresses are an important factor when it comes to calculating useful product life and demolding. Stresses resulting from various types of shrinkage may also complicate matters. 

For piles in granular soil, a large portion of the load is supported by the pile tip (base). However, the contribution of the pile shaft must also be taken into account. As in the case of piles in clay, pile capacity is calculated using effective stress analysis From Equation 5.1... 

The Ingedrive MVlOO vector control loops and vector modulation techniques allow an accurate speed control with a very low noise and stress in the motor. The drive operates in speed control mode by receiving the speed reference from the hoist automation level through a fieldbus communication interface. A torque bias compensation (feed-forward action) is also applied for the drive during the shaft starting and acceleration process where the torque reference is calculated at the hoist automation level. The synchronized operation between the electrical drive and the hoist brake disc system is done at the hoist automation level too through the torque feed-forward compensation. 

In case of water-losing and compression of thick unconsolidated formation in eastern China, the side wall additional stress will be generated between the soil and coal mine vertical shaft side wall. Its results of calculation model (Fig. 2) (Li 2000, 2000 Li 2005). 

The maximum allowable diametral interference between a shaft and a hub for press-fitting assembly is a function of part geometry, stress, modulus of elasticity of shaft and hub, and Poisson s ratio for shaft and hub materials. When the shaft and hub are made of the same material, the moduli of elasticity are the same and Poisson s ratio is the same for the shaft and hub, simplifying the equation for maximum diametral interference. Calculating diametral interference is simplified when the shaft is made of high-modulus material. 

Table 11.1.10 is a summary of the interference for the solid and hollow shafts, which do not cause fracture of the inner ring, and the circumferential stress occurring in the bore surface of the silicon nitride inner ring at that time. The spline shaft is omitted because circumferential stress cannot be calculated... 

To design notched components, knowledge of Ki is required. Therefore, empirical formulae have been determined that can be used to calculate for different geometries and load cases. They are collected in tables e. g., Peterson s Stress Concentration Factors [109] or Dubbel [18]. One example, a shaft with a circumferential notch under tensile load, is shown in figure 4.3. The dimensions in the figure are the outer diameter D, the diameter at the notch root d, the notch depth t (with 2t = D — d), and the notch radius q. 

If the available materials to construct the shaft are a ceramic with i2m = 400 MPa or the aluminium alloy AlSilMgMn with Rpo.2 = 202 MPa and Rm = 237 MPa, we can expect the ceramic to fail because the stress at the notch root is much larger than the tensile strength. For the aluminium alloy, the tensile strength is also exceeded, and we thus might expect its failure as well. However, the calculation is not valid in the case of a ductile material, for equation (4.1) is valid only for a linear-elastic material, whereas the alloy AlSi 1 MgMn yields at Rpo.2 = 202 MPa. This increases the strain at the notch root and reduces the stress concentration. The actual stress at the notch root cannot be calculated with the tools introduced so far. In the next section, we will discuss Neuber s rule that allows to estimate the stresses. 

Shaft design must accommodate hydraulic and mechanical loads and must avoid vibration near the natural frequency. A typical overhung shaft arrangement with dimensional nomenclature is shown in Figure 21-32. Hydraulic loads on the shaft result from the torque required to turn the impeller(s) and random or systematic lateral hydraulic loads on the impeUer(s). Other sections of the book describe methods for determining impeller power. Shaft design will use impeller power to calculate torque and hydraulic forces and thus size a shaft within allowable stress limits. 

Calculate the minimum solid shaft size for an overhung shaft, which meets both shear [eq. (21-5)] and tensile [eq. (21-6)] stress limits. Then round up to the nearest in. increment (next-larger metric diameter) to obtain a size for commercially available bar stock. 

The minimum shaft diameter for the allowable tensile stress is calculated with a different equation ... 

A hollow shaft, made from pipe, can increase the stiffness and reduce the weight (mass) of a mixer shaft in critical speed calculations. Such changes will increase the natnral freqnency and extend the allowable shaft length or operating speed. When determining the appropriate shaft size for the strength of a hollow shaft, begin with the dimensions for standard available pipe or tnbe. Then compute the shear and tensile stress values and compare them with the allowable values. The equations for combined shear and tensile limits in hollow shafts are, respectively. 

The method for determining the deflection is very similar to calculating the critical speed. However, the hydraulic forces on the shaft are now taken into account. Also, the frequency is a known value. Hydraulic forces are determined based on the impeller torque. Because hydraulic forces used by mixer manufacturers already include the effect of dynamics, speed (frequency) is already included in the magnitnde. Consequently, a forced response at the shaft speed would not be appropriate because the results would reflect the effect of frequency twice. Therefore, determining the forced response for the static condition is necessary (frequency = 0). From the bending moment for each position along with the torque from the impeller(s), the tensile and shear stresses can be calculated for each position. The static condition can only be calcnlated where the forcing freqnency is effectively zero compared with the natnral frequency, and such an analysis requires a 4 x 5 matrix. 

---