3. Determination of rotor blade bending stress caused by centrifugal forces

1. The design scheme

The rotor blade bending by centrifugal forces occurs when the centers of gravity line of its cross sections do not coincide with a radial axis r (Fig. 3.6), which passes through the root section center of gravity. In this case being affected by centrifugal force dP_c the blade will be subject to tension as well as bending about root section if it is rigidly fixed in the disc. The values of a blade cross section gravity centers displacement about a radial axis r in axial (x_r) and circumferential (y_r) planes are called the offsets of gravity centers.

Gravity center offsets are usually made on purpose to relieve blades from bending stress caused by gas forces. This can also be caused either by inaccuracy in blades and their fastening or large flexions caused by gas forces.

The blade design scheme for the type of the loading under consideration is given in Fig. 3.6, where the same marks as in Fig. 3.2 are used. The dotted lines along the blade in Fig. 3.6 show projections of its cross section gravity center lines on axial and circumferential planes, respectively.

In this case just like at bending by gas forces, the blade can be considered as a rod, nipped in the disc, which is subject to load causing the blade bending about the disc. The difference is that the load intensity is not perpendicular to blade axis, but is applied along the radial plane to the line of cross section gravity centers, which are arbitrarly located about the radius going through the blade root section.

3.4.2. Equation of the bending momenta

The blade dr element of an infinitesimal length is located on some arbitrary radius r. It is acted on by a centrifugal force dP_c, whose value is determined by the formula

This force will create the elementary bending moment in an axial plane (see Fig. 3.6, a). It will act about the design section gravity center, located at some radius R

where x_r, x_R are the dr element center of gravity and design section offsets, respectively.

Fig. 3.6. Design scheme of the turbine rotor blade bending by the centrifugal

forces in axial (a) and circumferentiale (b) planes

The full bending moment in the design section is determined by integration

. (3.27)

It is necessary to add the cap bending moment if the blade has a shroud cap, whose centrifugal force can be calculated by the formula (3.6):

where x₀ is a blade tip section center of gravity offset.

The centrifugal force dP_c acts on dr element at angle  to a radial axis r in a circumferential direction (see Fig.3.6, a). It is determined by the offset of gravity center y_r. Therefore, to determine bending moment in a design section it is necessary that the force dP_c be decomposed:

It is possible to consider cos=1, sin=y_r /r with a high degree of accuracy for angle  is very small. Then the elementary bending moment in a circumferential plane about the design section gravity center will be equal to

The full bending moment is determined by integration

(3.28)

It is necessary to add the cap bending moment to M_b x for a shrouded blade taking into account, that the thickness of a cap is very small in comparison with the tip diameter of a rotor wheel

where y₀ is a gravity center offset of a blade tip section.

If linear laws are applied to gravity center offset change along the blade length:

and cross section areas are dependent on the radius, then the integration by the formulas (3.27) and (3.28) results in the following ratios:

(3.29)

(3.30)

where R* = (R – R_r) / (R₀ – R_r).

We will get for the blade root section from the formulas (3.29), (3.30):

(3.31)

(3.32)

To calculate the total bending stress in characteristic points A, B and C, momenta received from algebraic addition of bending stresses caused by gas and centrifugal forces, should be used in formula (3.25):

The degree of a blade relieving from bending by gas forces is characterized by the compensation coefficients:

It is theoretically possible to relieve the blade completely from bending loads. Nevertheless, in practice it can result in considerable technological difficulties because of the composite law of the section gravity center offsets changing along the blade length. Besides, the overbalance is possible at high-altitude engine operation, which results in considerable centrifugal force bending stresses. So, compensation coefficients are equal to 0,5...0,8 for a design rating.

The application of the blade hinge fastening to the disc (Fig. 3.7) is very effective for blade relieving from bending stress. In this case as the blade turns in a hinge being acted on by gas stress, a restoring moment of centrifugal force comes into action, which, in fact, completely relieves the blade from the bending moment in axial plane.

T he maximum values of total bending stresses of gas and centrifugal forces with relieving being taken into account make 50...80 MPa.

Fig. 3.7. Scheme of bending of blade with hinge fastening