Claims:WE CLAIM:
1. A computer implemented method for estimating burst margin of rotating disc based on disc geometry and the disc’s material property comprising
receiving physical dimension and material characteristics of the disc in processor platform of a computer embedding finite element analysis process for estimation of the burst margin;
quantifying the rotating disc under rotation by said processor based on the received physical dimension and material characteristics of the disc and including elastic strain component therein corresponding to centrifugal load subjected on the rotating disc;
applying temperature field correction in the quantified rotating disc by the processor to include strain component therein corresponding to thermal expansion strain developed in the rotating disc when it subjected to temperature gradient field;
involving the embedded finite element analysis process for standard finite element discretization to radially divide the quantified rotating disc domain into plurality of identical discrete elements in said processor including
(i) determining average hoop stress and ultimate tensile stress of the quantified rotating disc through finite element based stress analysis of said identical discrete elements; and
(ii) calculating the burst margin involving the average hoop stress and the ultimate tensile stress.

2. The computer implemented method as claimed in claim 1, wherein the computer includes an input module configured to operate in conjunction with the finite element analysis process in the processor;
said input module enables feeding of the disc material parameters including Young’s Modulus, Poisson’s Ratio, density and like of the disc material and the disc physical dimension including inner and outer radius in the processor.

3. The computer implemented method as claimed in claim 1 or 2, wherein the processor is configured to mathematically quantify the rotating disc having uniform thickness under ( ) coordinate by
, here , = density of the disc material, = angular velocity of the rotating turbine disc, =Poisson’s ratio the disc material and is stress in the disc for the elastic strain component;
wherein the processor is further configured to mathematically quantify the rotating turbine disc having variable thickness under ( ) coordinate by

here disc thickness (h) is function of radius (r) of the disc.

4. The computer implemented method as claimed in anyone of the claims 1 to 3, wherein the processor is configured to apply the temperature field correction into the quantified rotating disc by modifying the mathematically quantified rotating disc having uniform thickness as

where is temperature variation in radial direction, E is the young modulus and is the linear temperature expansion coefficient.

5. The computer implemented method as claimed in anyone of the claims 1 to 4, wherein the processor is configured to apply the temperature field correction into the quantified rotating disc by modifying the mathematically quantified rotating disc having variable thickness as

6. The computer implemented method as claimed in anyone of the claims 1 to 5, wherein the processor involves steady state temperature distribution heat conduction equation operating under disc domain defined by its inner and outer radius ( ) to apply the temperature field correction.

7. The computer implemented method as claimed in anyone of the claims 1 to 6, wherein the processor embedding the finite element analysis process for standard finite element discretization is configured to compute the finite element equation as

where,
,

,
, and

e indicates the element number which is used to discretize the domain of the rotating disc.
8. The computer implemented method as claimed in anyone of the claims 1 to 7, wherein the processor computes the hoop stress and the radial stress based the value of F as calculated by the finite element analysis by

9. The computer implemented methodas claimed in anyone of the claims 1 to 8, the burst margin is further computed by the processor by


here UTS denotes ultimate tensile stress;
wherein the average hoop stress is calculated from the finite element based stress analysis of the rotating disc considering centrifugal and thermal loads which is multiplied with respective area of that element as and
the ultimate tensile stress which is temperature dependent, the temperature for each of the element is determined and is multiplied with respective area of that element to obtain the average UTS as

here, UTS of ith Element is given as equation is
= Temperature of ith Element
= Ultimate tensile stress at minimum disc temperature
= Ultimate tensile stress at maximum disc temperature.

10. A computer system for estimating burst margin of rotating disc based on disc geometry and the disc’s material property involving the method as claimed in anyone of the claims 1 to 9, comprising
an input module for receiving physical dimension and material characteristics of the disc in processor platform of a computer embedding finite element analysis process for estimation of the burst margin;
a processor for quantifying the rotating disc under rotation based on the received physical dimension and material characteristics of the disc and including elastic strain component therein corresponding to centrifugal load subjected on the rotating disc;
said processor applies temperature field correction in the quantified rotating turbine disc by the processor to include strain component therein corresponding to thermal expansion strain developed in the rotating disc when it subjected to temperature gradient field;
wherein the processor embodies finite element analysis process for standard finite element discretization to radially divide the quantified rotating disc domain into plurality of identical discrete elements in said processor including
(i) determining average hoop stress and ultimate tensile stress of the quantified rotating disc through finite element based stress analysis of saididentical discrete elements; and
(ii) calculating the burst margin involving the average hoop stress and the ultimate tensile stress.

Dated this the 18th day of December, 2018 Anjan Sen
(Applicants Agent)
IN/PA-199
, Description:FIELD OF THE INVENTION:
The present invention relates to estimation of burst speed of rotating disc. More specifically the present invention is directed to develop a method and a system for estimating burst margin i.e. the margin of safety beyond maximum operating speed within which the rotating disc such as turbine disc of any aero-engine will not fail. The present invention can be used to optimize design of turbine disc of the aero-engine for having a desired permissible gas turbine disc speed.

BACKGROUND OF THE INVENTION:
In a gas turbine of the aero-engine, the purpose of the turbine disc is to retain turbine blades in position and to transmit torque from turbine shaft to the turbine blades of compressor and from the blades to the shaft in the turbine.

The discs which rotate carrying the moving blades are highly stressed components in an aircraft gas turbine engine because they are required to operate at very high rotational speeds and therefore the discs are subjected to body forces and centrifugal forces owing to blades attachment at circumferences of the discs. Also, high temperature gradient exists between bore and rim of the disc, which combines with centrifugal stresses, and thereby produces severe conditions of operation. At normal operating condition, the disc behaves elastically, but circumstances may force pilot to exceed maximum operating speed. At certain speed, plastic yielding will initiate at the bore and gradually plasticity will take all parts of the disc and destabilizes the disc functionally and structurally [Ref: N. Eraslan, H. Argeso, Limit angular velocities of variable thickness rotating discs, Int. J. of Solids and Structures 39 (2002) 3109-3130]. Further increase in speed causes the yield zone to spread rapidly outward due to lost material properties. This results in large plastic strain in the disc which grows in diameter until burst takes place.

During operation, turbine disc possesses enormous energy and therefore outcome of a disc burst is ahead of imagination. During design of disc no failure of disc up to 122% of maximum operating speed has to be ensured as per military specification standard (MIL-E-5007E) [Ref: Research andtechnology organization report 28(2000)]. Generally, service life of a disc is determined by spinning a sample full size disc in a typically designed spinning facility under stress and temperature similar to those experienced during actual operation. The main objective of these tests is to determine the cycle life of the disc to the formation of an engineering crack of about 0.75 mm length at the surface [Ref: R.A. Claudio, C.M. Branco, E.C. Gomes, J. Byrne, Life prediction of a gas turbine disc using the finite element method, 8AS Jornadas de Fractura 8(2002)131-144].

In service, when disc reaches cycle life, they are retired from service without being inspected. Reuse of part beyond the safe-life is not considered [Ref: R.A. Claudio, C.M.Branco, E.C.Gomes, J. Byrne, Life prediction of a gas turbine disc using the finite element method, 8AS Jornadas de Fractura 8(2002)131-144]. The engineering crack of 0.75 mm surface length cannot give a constant margin of safety for all conditions, because the critical size associated with dysfunction depends on geometry, material, load state and many more factors. But due to service life criterion, large number of discs are retired from service with significant safe service life remaining, resulting in life cycle costs greater than necessary [Ref: K. U.Tschirne, W.Holzbecher, Qualification of Life Extension Schemes for Engine Components, NATO RTO-MP-17, 2 (1999)]. However, disc may also fail without reaching the service life because of critical working environment such as wind blow condition where operating condition is different and it is supposed to increase the speed beyond or more than maximum operating speed. Under such situation main failure mode is burst of the disc. Thus the conventional lifing methodology does not provide a measurable and constant safety factor for all conditions. This leads to introduction of some efficient method for the determination of bursting speed of a rotating disc.

The computation of the burst of a turbine disc for aeronautical engine is a problem of great interest to many researchers. Research works of rotating discs can be fundamentally classified depending on the state of stresses, i.e. elastic or elastic-plastic. The first study in this area was reported by Robinson [Ref: E.L.Robinson, Bursting test of steam turbine disk wheels. Trans. of the ASME 66 (1944)373-386] in 1943 and he proposed a criterion that a disc will burst when the average tangential stress equals the tensile strength of material.

Arthur and Jenkins [Ref: G. H. Arthur, J. E. Jenkins, Effect of strength and ductility on burst characteristics of rotating disks. National advisory committee for aeronauticsTechnical Note 1667 (1948)01-52] developed a solution for stress and strain based on von Mises criterion and Hencky deformation theory of plasticity and justified the case of solid disc of uniform thickness.

Weiss and Prager [Ref: H.J.Weiss, W.Prager, J.Aero. sci. 21(1954)196-200] assumed Tresca’s yield criterion and derived stress and strain distributions for hollow disc of uniform thickness.

In 1963, burst test of rotating disc was carried out by Waldren [Ref: N.E. Waldren, M.J. Percy, P.B.Mellor, Proc.Inst.Mech. Engnr.180 (1965)111-118] who compared theoretical and experimental results and made a prediction of instability strains using plasticity theory. The criterion proposed by the above mentioned researchers unfold a way for research in this field. Further advancement in aerospace technology gave attention of numerous researchers in last two decades. Many researchers studied the behavior of rotating discs with an aim to develop analytical methods to calculate stress and strain distributions on the basis of a series of hypotheses in order to simplify the problem. Very few works are reported on burst speed investigation of turbine disc under complex loading condition.

Toussiet al. [Ref: H. E. Toussi, M. R. Farimani, Elasto-plastic deformation analysis of rotating disc beyond its limit speed, Int. J. Pres. Ves. Pip. 89 (2012)170-177] investigated the concept of failure and limit speed of disc and the effect of different parameters including the cross section profiles and material properties upon the limit speed.

Hu [Ref: S. Hu, Study on the elastic– plastic interface and large deformation of axisymmetric disks under rotating status, Appl. Clay Sci. 79(2013)41–48] carried out a theoretical analysis of disc in elastic–plastic interface under rotating status and the burst speed of rotating disc was analysed.

Squarcella et al. [Ref: N. Squarcella, C. M. Firrone, M. Allara, M. Gola, The importance of the material properties on the burst speed of turbine disk s for aeronautical applications, Int. J. Mech. Sci. 84(2014)73–83] presented a numerical method to predict the burst speed, introducing the physical phenomenon at the base of the burst i.e. the inertial instability.

This review reveals that limited work is reported on prediction of over speed and burst margin of the aero engine disc and there has been a need for an improvement in the technique for estimating the burst margin of a rotating disc such as gas turbine disc under complex loading condition.

OBJECT OF THE INVENTION:
It is thus the basic object of the present invention method and a system for estimating burst margin of a rotating disc i.e. the margin of safety beyond maximum operating speed within which the disc will not fail involving the disc geometry and the disc’s temperature dependent material property.
Another object of the present invention is to develop a method and a system for estimating the burst margin which would be accurate, simple and easy to implement.
Another object of the present invention is to develop a method and a system for estimating burst margin of a rotating turbine disc of an aero-engine i.e. the margin of safety beyond maximum operating speed within which the rotating turbine disc will not fail involving the turbine disc geometry and the turbine disc’s material property.
Yet another object of the present invention is to develop a method and a system for estimating the burst margin which would be adapted to facilitate optimizing weight of the turbine disc by varying the thickness of the disc.

SUMMARY OF THE INVENTION:
Thus according to the basic aspect of the present invention there is provided a computer implemented method for estimating burst margin of rotating disc based on disc geometry and the disc’s material property comprising
receiving physical dimension and material characteristics of the disc in processor platform of a computer embedding finite element analysis process for estimation of the burst margin;
quantifying the rotating disc under rotation by said processor based on the received physical dimension and material characteristics of the disc and including elastic strain component therein corresponding to centrifugal load subjected on the rotating disc;
applying temperature field correction in the quantified rotating disc by the processor to include strain component therein corresponding to thermal expansion strain developed in the turbine disc when it subjected to temperature gradient field;
involving the embedded finite element analysis process for standard finite element discretization to radially divide the quantified rotating disc domain into plurality of identical discrete elements in said processor including
(i) determining average hoop stress and ultimate tensile stress of the quantified rotating disc through finite element based stress analysis of saididentical discrete elements; and
(ii) calculating the burst margin involving the average hoop stress and the ultimate tensile stress.
In the above computer implemented method, the computer includes an input module configured to operate in conjunction with the finite element analysis process in the processor;
said input module enables feeding of the disc material parameters including Young’s Modulus, Poisson’s Ratio, density and like of the disc material and the disc physical dimension including inner and outer radius in the processor.
In the above computer implemented method, the processor is configured to mathematically quantify the rotating disc having uniform thickness under ( ) coordinate by
, here , = density of the disc material, = angular velocity of the rotating disc, =Poisson’s ratio the disc material and is stress in the disc for the elastic strain component;
wherein the processor is further configured to mathematically quantify the rotating disc having variable thickness under ( ) coordinate by

here disc thickness (h) is function of radius (r) of the disc.

In the above computer implemented method, the processor is configured to apply the temperature field correction into the quantified rotating disc by modifying the mathematically quantified rotating disc having uniform thickness as

where is temperature variation in radial direction, E is the young modulus and is the linear temperature expansion coefficient.
In the above computer implemented method, the processor is configured to apply the temperature field correction into the quantified rotating disc by modifying the mathematically quantified rotating disc having variable thickness as

In the above computer implemented method, the processor involves steady state temperature distribution heat conduction equation operating under disc domain defined by its inner and outer radius ( ) to apply the temperature field correction.
In the above computer implemented method, the processor embedding the finite element analysis process for standard finite element discretization is configured to compute the finite element equation as

where,
,

,
, and

e indicates the element number which is used to discretize the domain of the disc.
In the above computer implemented method, the processor computes the hoop stress and the radial stress based the value of F as calculated by the finite element analysis by

In the above computer implemented method, the burst margin is further computed by the processor by


here UTS denotes ultimate tensile stress;
wherein the average hoop stress is calculated from the finite element based stress analysis of the rotating disc considering centrifugal and thermal loads which is multiplied with respective area of that element as and
the ultimate tensile stress which is temperature dependent, the temperature for each of the element is determined and is multiplied with respective area of that element to obtain the average UTS as

here, UTS of ith Element is given as equation is
= Temperature of ith Element
= Ultimate tensile stress at minimum disc temperature
= Ultimate tensile stress at maximum disc temperature.

According another important aspect of the present invention there is provided a computer system for estimating burst margin of a rotating disc based on disc geometry and the disc’s material property involving the above method comprising
an input module for receiving physical dimension and material characteristics of the disc in processor platform of a computer embedding finite element analysis process for estimation of the burst margin;
a processor for quantifying the rotating disc under rotation based on the received physical dimension and material characteristics of the disc and including elastic strain component therein corresponding to centrifugal load subjected on the rotating disc;
said processor applies temperature field correction in the quantified rotating disc by the processor to include strain component therein corresponding to thermal expansion strain developed in the rotating disc when it subjected to temperature gradient field;
wherein the processor embodies finite element analysis process for standard finite element discretization to radially divide the quantified rotating disc domain into plurality of identical discrete elements in said processor including
(i) determining average hoop stress and ultimate tensile stress of the quantified rotating disc through finite element based stress analysis of saididentical discrete elements; and
(ii) calculating the burst margin involving the average hoop stress and the ultimate tensile stress.

BRIEF DESCRIPTION OF THE ACCOMPANYING DRAWINGS:
Fig1 shows a preferred embodiment of a disc.
Fig 2 shows the typical sectional view of turbine disc of an aero-engine.
Fig 2a shows geometry and FE mesh for variable thickness rotating disc portion as identified in the turbine disc of Fig 2.
Fig 2b shows geometry and FE mesh for uniform thickness rotating disc portion of the turbine disc.
Fig 3 shows temperature distribution in the variable thickness rotating discportion.
Fig 4 shows hoop stress at speed of 22800 rpm for the variable thickness rotating discportion.

Fig 5 shows variation of maximum hoop stress with speed of the rotating disc.
Fig 6 shows burst margin for different speeds of the rotating disc.

DETAILED DESCRIPTION OF THE INVENTION WITH REFERENCE TO THE ACCOMPANYING DRAWINGS:
It is well known that, in any aero-engine, failure of the turbine disc is very dangerous and that may result into loss of human being. One of the main failure modes of the turbine disc is formation of yield zone at bore which leads to burst of the disc. Hence, disc burst is treated as a critical factor while designing the disc. The scope of present work is to carry-out the computer implemented elastic – plastic analysis and to find the burst margin of a typical model of a gas turbine disc by means of the finite-element technique for different speed of rotation. The additional goal of the analysis is to compare the burst margin of turbine disc having variable thickness and uniform thickness profile of the disc. The burst margin is estimated using area weighted mean hoop stress (AWMHS) approach. For given burst margin greater than 122%, the variable thickness disc provides lower operating speed in comparison to uniform thickness disc. This helps in reducing hoop stress, which is highly desirable in aero-engine.
The accompanying fig 1 shows a preferred embodiment of the disc while the accompanying fig 2 shows sectional view of turbine disc. A thin exemplary element portion of the turbine disc is shown in the fig 2. Plurality of such element portions can constitute the present turbine disc and the burst margin of the disc can be determined by analyzing each of the element portions. The present finite element analysis based burst margin estimating method is preferably implemented by a computer system. The computer system essentially includes aninput module, a processor and an output module. The processing module embodies the automated finite element analysis process while the input and the output modules are configured to operate in conjunction with thefinite element analysis process. The input module enables the user to model or design adisc geometry on a virtual platform provided by the finite element analysis process including feeding of the disc inner and outer radius in the processor. The input module also enables the user to feed the disc material parameters like Young’s Modulus, Poisson’s Ratio, density into the processor for calculation of the burst margin. The output module displays the calculated burst margin.
The processor involves different mathematical models for finite element analysis based burst margin calculation for disc with uniform thickness and disc with variable thickness.
According to a preferred embodiment of the present computer implemented method for estimating burst margin of rotating disc based on disc geometry and the disc’s material property, the processor of the computer after receiving the physical dimension and material characteristics of the disc first mathematically quantifies the disc under rotation including elastic strain component therein corresponding to centrifugal load subjected on the rotating disc e.g. centrifugal load subjected on the rotating turbine disc carrying turbine blades.
The mathematical model for quantification of the rotating disc with uniform thickness is developed assuming thin rotating disc in ( ) coordinate system. Problem formulation is based on axis symmetric hypothesis of disc i.e. stress and strain component are independent of coordinate. The axial stress is neglected and hence radial stress, and tangential stress, are the only principal stresses. Considering the inertia force due to rotation of the disc in radial direction, the equilibrium equation can be represented as

(1)

Where, = = density, = angular velocity, =Poisson’s ratio

The solution of above differential equation (1) is given by equation(2)

(2)
The tangential stress and radial stress are further given by equations (3, 4)

(3)
(4)

Where and are constant of integration and are determined by boundary condition. Considering a hollow disc with central hole of inner radius ‘a’ and outer radius ‘b’ and applying initial boundary condition of = 0 at = a and = 0at =b, the equation (3) and (4) reduces to equation (5) and (6) respectively.

Upon solving and making use of stress-strain relationship, maximum hoop stress and maximum radial stress can be represented as
(5)
(6)
The displacement along radial direction ( ) is given by equation (7)
(7)
The equations (5), (6) and (7) hold good to compute maximum radial stress, maximum tangential stress and deformation when the disc cross-section is uniform and are not exposed to temperature gradient field but in practical situation use of uniform cross-section disc is restricted for aircraft application due to constraint of weight optimization. Further the working condition of rotating disc such as the turbine disc is such that, the disc has to operate under temperature gradient field; hence this model needs to modified for thermal and variable thickness corrections.

Correction for rotating disc of variable thickness
The processor is also configured to mathematically quantify the rotating disc having variable thickness under ( ) coordinate system. According to the present invention, the equation of equilibrium of rotating disks with variable thickness can be given by equation (8)
(8)
For the variable thickness disc, thickness (h) is function of radius (r) of the disc. So the above equation can be solved further to find and if the relation for geometry variation profile is available.

Temperature field correction for uniform and variable thickness disc
In real working condition, the rotating disc is subjected to temperature gradient field. The processor is thus specifically configured to apply temperature field correction in the quantified rotating disc to include strain component therein corresponding to thermal expansion strain developed in the disc when it subjected to temperature gradient field.

It is assumed that the inner surface of the disc is fixed to the shaft and hence isothermal temperature may be assumed on it. The outer surface of the disc is maintained at uniform temperature gradient. Considering the hollow disk with central hole of inner radius ‘a’ and outer radius ‘b’, the disc domain boundary conditions can be given as:
At, = a, = 0, T= 0
At, = b, =0,
Now, if the material is exposed to temperature gradient field, it experiences a stress arising from an eigen-strain. These eigen strains are non-elastic strains that develop in a body due to temperature change or may be due to phase transformation. In the present case eigen strain is due to thermal expansion of the disc. Considering isotropic disc material, the thermal eigen strain at a point is same in all direction and hence relation can be given as,
(9)
where, is eigen strain is coefficient of thermal expansion in radial direction. is temperature variation in radial direction.
Total strain will be the sum of elastic strain and eigen strain. Therefore, components of total strain can be taken as,
(10)
(11)
where, and are radial and circumferential component of total strain and , are radial and circumferential component of elastic strain. Further, and can be given as


By considering the above relation of and , equation (1) i.e. equilibrium equation of uniform thickness disc can be written as,
(12)
Similarly equation (8) i.e. equilibrium equation for variable thickness disc will reduce to
(13)
The equations (12) and (13) constitute the mathematical model for uniform and variable thickness disc respectively, consisting of second order differential equations which provide us the function Fand the components of stress.

For steady state temperature distribution heat conduction equation is given by equation (14)
(14)
Upon solving the above equation one can obtain
(15)
where being uniform reference temperature.
(16)

The temperature distribution given by equation (16) holds in the domain .

Finite element formulation
Substituting the values of from equation (16) and following a standard finite element discretization approach, the domain of the rotating disk is divided radially into N number of elements of equal size then the equilibrium equation in (12) and (13) can be transformed to the following system of simultaneous equations and subsequently , , , & can be given by equation (23) to (27) respectively.

The finite element equation corresponding to the embedded finite element analysis and as computationally implemented by the processor can be given by equation (17)
(17)
where,
(18)
(19)
(20)
(21)
(22)
The symbol e used in the above equation indicates the element number which is used to discretize the domain of the disc. Once the value of F is calculated, various components of stress, strain and displacement can be easily calculated by the following relations:
(23)
(24) (25)
(26)
(27)
Burst margin
Gas turbine disc generally operates at very high rotational speed. The turbine disc possesses enormous energy during its operation. This leads to high hoop stress formation in the turbine disc, which is responsible for the failure. Failure of the disc in such a condition is very dangerous and it may result in loss of human being and damage to aircraft. Hence is treated as the most catastrophic mode of failure and therefore 122% of maximum operating speed has to be ensured during design of the disc. Generally maximum operating speed is restricted within the elastic limit of the disc material. But it is not necessary that disc will fail as soon as it crosses elastic limit. This means that even in plastic region disc may not fail up to its ultimate tensile stress. Area Weighted Mean Hoop Stress (AWMHS) is the criterion used to determine burst margin, which presuppose that regardless of rotor shape and material, burst will occur when average tangential stress in the rotor becomes equal to ultimate tensile strength of the material.

Area weighted mean hoop stress
Here burst margin is the margin of safety beyond maximum operating speed within which the rotating disc will not fail. Mathematically it is given by the processor by the equation (28)
(28)

where UTS denotes ultimate tensile stress.
The average hoop stress is calculated by the processor from finite element based non-linear stress analysis of the rotating disc while considering centrifugal and thermal loads which is multiplied with respective area of that element to obtain the numerator of the equation (29)
(29)
Now as ultimate tensile stress is temperature dependent, temperature for each element is determined and is multiplied with respective area of that element to obtain the numerator of equation (30)
(30)
where, UTS of ith Element is given as equation (31)
(31)

= Temperature of ith Element
= Ultimate tensile stress at minimum disc temperature e.g. 20°C
= Ultimate tensile stress at maximum disc temperaturee.g. 650°C

Finite element modeling in ANSYS:
According to a preferred embodiment, a typical parametric geometric model of rotating disc is developed using multi-purpose finite element analysis processing software package ANSYS. Because of the symmetry of the problem (loads and geometry), the disc is simulated by a two dimensional (2D) axis symmetric cross-section as shown in Figure 1 and Figure 2. The geometry of section is designed such that revolving it about an axis of symmetry produces the solid disc.

Nickel-based super alloy IN718 is the material used for the gas turbine discs. The relevant properties of IN718 are given in Table 1. Monotonic stress strain relations at different temperature is tabulated in Table 2 and temperature dependent material properties are listed in Table 3.

In Ansys couple field solid PLANE182 2-D element is used for meshing the geometric model of disc. The element is defined by four nodes having two degrees of freedom at each node: translations in the nodal x and y directions. A typical temperature of 481.5°C is applied at bore and a temperature of 510°C is applied at rim of the rotating disc. The temperature distribution throughout the disc is obtained which is shown in Figure 3. The obtained temperature distribution is coupled in structural analysis with temperature dependent material properties and temperature dependent stress-strain relation. The static non-linear structural stress analysis is carried out on the disc using variable thickness and uniform thickness to estimate first principal stress i.e. hoop stress at bore. For structural analysis, three major loads are applied on disc. They are centrifugal load due to self-weight of the disc, thermal load due to surrounded high temperature and additional centrifugal load due to blades attachment in terms of radially outward pressure at rim of the disc.

Table 1: Basic mechanical properties of disc material IN718
Parameter Values
Material INCO 718
Young’ modulus 209e3 N / mm2
Poisson’s ratio 0.3
Density 8.22e-9 tons/mm3
Shear Modulus 8.173e4 N / mm2
Thermal Conductivity 11.1e-3 W/mm°C

Table 2: Stress strain relationship
Strain
Stress(N / mm2)
At 20°C At 650°C
0.0058 1157.58 863.28
0.01 1245.87 922.14
0.012 1285.11 941.76
0.016 1314.54 1000.62
0.02 1334.16 1059.48

Table 3 Temperature Dependent material Properties of IN718
Property Temperature (°C)
20 100 400 500 600 650
Young’s Modulus
(KN/ mm2) 209.0 195.0 183.1 178.4 170.3 157.3
*CTE (10-6 / °C) 12.2 12.8 13.9 14.0 14.5 15.0
Thermal Cond.
(10-3W/mm °C) 11.11 12.41 17.95 19.48 21.21 23.09
*CTE denotes coefficient of thermal expansion

Validation of numerical results
For calculating burst margin of rotating turbine disc, the hoop stress is calculated. For validation of numerical results the hoop stress is compared at different speeds for the uniform thickness disc whose analytical solution are easily available as shown in Table 4. The number of element taken is approximately 90000 after convergence study.

Table 4 Analytical & numerical hoop stress at different speeds
Speed(rpm) Hoop stress (MPa)
(FEM) Hoop Stress(MPa)
(analytical)
10000 78.02 78.30
12000 112.35 112.76
14000 152.93 153.48
16000 199.74 200.47
18000 252.80 253.72
20000 312.10 313.24
22000 377.64 379.02
24000 449.43 451.07
26000 527.45 529.38
28000 611.72 613.95
Temperature distribution results (in ?c)
Temperature varies throughout the disc due to temperature difference between bore region and outer rim surface as shown in Figure 3. This variation in temperature is responsible for thermal stress which makes the structure weak. Hence this effect should be taken in to consideration for structural analysis. This temperature distribution is coupled in structural analysis to incorporate the effects of temperature load.

Stress distribution results (in MPa)
In this analysis stress distribution throughout the turbine disc is obtained. Hoop Stresses are determined at different speeds. These stress values are used to calculate the burst margin. Hoop stress distribution at 22800 rpm in a variable thickness disc is shown in Figure 4. At this speed of 22800 rpm, hoop stress approaches ultimate tensile strength of material and disc will eventually fail. A comparison of maximum hoop stress for different speeds in variable thickness disc and uniform thickness disc is shown in Figure 5.

Burst margin calculations
The burst margin of turbine disc is calculated based on Area Weighted Mean Hoop Stress Method (AWMHS). Here average hoop stress and average ultimate tensile stress values are used to determine Burst margin.

Present invention thus gives an insight understanding of the influence of stress components acting on the rotating disc which are very essential in determining the structural integrity of a rotating gas turbine disc. It is noteworthy that as speed increases the burst margin decreases. The reason behind this is, as rotational speed of the disc increases, the centrifugal force due to mass of the disc and blades attachment also increase which leads to development of high hoop stress at bore of the disc. As hoop stress increases at bore, it decreases the strength of the disc which in turn leads to the decrease in burst margin of disc. As per the MIL-STD-5007E, it is mandatory to ensure that gas turbine disc should have burst margin not less than 122% of maximum operating speed.From Figure 6, it is evident that for 122% of burst margin, the permissible speeds of gas turbine disc are 18000 rpm and 17800 rpm for variable thickness disc and uniform thickness disc respectively. For a given burst margin (greater than 122%), the permissible operating speed of variable thickness disc is lower than that of uniform thickness disc, resulting in lower hoop stress. For aero-engine, small reduction in mass due to variable thickness and hoop stress is desirable. From Figure 5, it is also observed that for lower operating speed the variable thickness disc will have lower value of hoop stress. However, beyond 18000 rpm, burst margin and maximum hoop stress became almost close to each other. When burst margin approaches 100%, the maximum hoop stress approaches UTS of disc material, and disc will eventually fail. This invention thus may be used as a basis for design application to optimize the weight of turbine disc by varying the thickness of the disc.