DETERMINATION OF THE STRESS–STRAIN STATE OF A TURBINE DISK HUB

G. D. Del, B. P. Chebaevskii, V. D. Del, A. F. Pronkin

(Presented by the scientific seminar of the Laboratory of Plastic Deformations)

The method, developed in recent years, for determining the stress–strain state in the plastic region by measuring hardness [1—3] can be applied to determine the stresses and loads at the moment of fracture from the hardness of the fractured part. This may prove useful in clarifying the causes of failures, and in a number of cases makes it possible to avoid expensive full-scale tests, since information that is the result of such tests can be obtained by studying parts that have previously failed. If full-scale tests are nevertheless carried out, then by measuring the hardness of the fractured part it is possible to substantially expand the information obtained in this way.

This article presents the results of an investigation of the stress–strain state in the hub of a turbine disk made of steel EI437B, which failed during testing of a turbine installation. The dimensions of the hub were: outside diameter—130 mm, hole diameter—33 mm, height—60 mm.

From undeformed EI437B steel, having the same heat treatment as the disk material, prismatic specimens with dimensions \(20 \times 20 \times 15\) mm were made. The specimens were compressed with lubrication to various degrees of deformation. After deformation, on the lateral surface of the prisms the hardness HB was measured with a 10 mm ball under a load of 3000 kg. The stress intensity \(\sigma_i\) was determined from the deforming force \(P\) and the area \(F\) of the cross section of the specimen:

\[ \sigma_i = \frac{P}{F}. \]

The strain intensity \(e_i\) was determined by the formula

\[ e_i = \frac{h_0 - h}{h_0}, \]

where \(h_0, h\) are the height of the specimen before and after deformation. From the results of the compression tests, a calibration graph was constructed, relating the hardness HB to the stress and strain intensities. Then, at 41 points on the end face of the disk hub, the hardness was measured. From the hardness, using the calibration graph “\(\sigma_i\)—HB—\(e_i\),” the values of \(\sigma_i\), \(e_i\) were determined at various points along the radius. The variation of HB, \(\sigma_i\), \(e_i\) along the disk radius is shown in Fig. 1.

For axisymmetric deformation of an incompressible material, the radial strain \(e_r\) and the circumferential strain \(e_\varphi\) are related to \(e_i\) by the following relation:

\[ e_\varphi^2 + e_r^2 + e_\varphi e_r = \frac{3}{4} e_i^2 . \tag{1} \]

Solving the system consisting of equation (1) and the strain-compatibility equation

\[ \frac{d e_\varphi}{d r} = \frac{e_r - e_\varphi}{r}, \tag{2} \]

we determined \(e_r\) and \(e_\varphi\). The distributions obtained are given in Fig. 2.

The simultaneous solution of equations (1), (2) was carried out by the finite-difference method. The strains on the contour of the hole in the hub \((e_i = e_{\varphi0},\ e_{r0} = e_{z0} = -e_{\varphi0}/2)\) were used as boundary conditions.

The strains \(e_r, e_\varphi, e_z\) found from hardness were compared with the strains calculated from the change in the diameter of the hole. Their discrepancy was 8%. Such close agreement indicates that heating of the hub during its operation did not substantially affect the relation between hardness, stresses, and strains.

At point 1, located at a distance \(\Delta r\) from the contour of the hole, according to (2),

\[ e_{\varphi1} = e_{\varphi0} + \frac{1}{2} \left( \frac{e_{r0} - e_{\varphi0}}{r_0} + \frac{e_{r1} - e_{\varphi1}}{r_1} \right)\Delta r . \tag{3} \]

The numeral in the subscript indicates the number of the point to which the given quantity refers. From (3) we find

\[ e_{\varphi1} = B_1 + C_1 e_{r1}, \tag{4} \]

where

\[ B_1 = \frac{ r_1 \left[ 2 r_0 e_{\varphi0} + \Delta r \left(e_{r0} - e_{\varphi0}\right) \right] }{ r_0 \left(2r_1 + \Delta r\right) }, \]

\[ C_1 = \frac{\Delta r}{2r_1 + \Delta r}. \]

Solving equations (1), (4) simultaneously, we obtain

\[ e_{r1} = \frac{ -B_1(2C_1 + 1) \pm 1.73 \sqrt{(e_i)_1^2(1 + C_1 + C_1^2) - B_1^2} }{ 2(1 + C_1 + C_1^2) }. \]

The transverse strain \(e_z\) was determined from the incompressibility condition

\[ e_z = -e_r - e_\varphi . \]

The circumferential \(\sigma_\varphi\) and radial \(\sigma_r\) stresses were determined from the relations [3]:

\[ \sigma_\varphi = \sigma_i \left[ -\frac{e_z}{e_i} + \frac{1}{\sqrt{3}} \sqrt{1 - \left(\frac{e_z}{e_i}\right)^2} \right], \]

\[ \sigma_r = \sigma_i \left[ -\frac{e_z}{e_i} - \frac{1}{\sqrt{3}} \sqrt{1 - \left(\frac{e_z}{e_i}\right)^2} \right]. \]

The calculation of strains, as already indicated, begins from the hole. However, at the disk hole the strain gradient is large (Fig. 2); this explains the dependence of the calculation accuracy on the step size. Approximation of the initial data makes it possible to perform a series of calculations with a varying step. The strains and stresses obtained in the case when further reduction of \(\Delta r\) does not lead to a noticeable change in the stresses and strains are taken as reliable.

Fig. 1. Variation of \(HB\), \(\sigma_i\), \(e_i\) along the radius of the disk hub

Fig. 2. Strains and stresses along the radius of the disk hub

Figure 2 shows plots of the strains and stresses obtained with a step \(\Delta r = 0.5\) mm. The greatest stress was \(113\ \mathrm{kg/mm^2}\) at a distance of 1 mm from the edge of the hole. Farther on, the stresses decrease monotonically; here the ratio \(\sigma_r / \sigma_\varphi\) is approximately constant and equal to 0.8.

REFERENCES

1. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, Moscow–Leningrad, 1962.

2. G. D. Del. Hardness of a deformed metal. Izv. AN SSSR, “Metally”, No. 4, 1967.

3. G. D. Del. Determination of the stressed state of a material in the plastic region by hardness measurement. Transactions of the Metrological Institutes of the USSR, issue 91 (151), Moscow–Leningrad, 1967.