3.2 Stress-strain state and the displacements of heated parts

In the finite element-based simulation, the heated parts are presented by their solid models illustrated by Figure 4. In its solid model, each HP is divided on elemental 3D simplex volumes. Each volume is presented in finite element calculations by four nodes.

After the determination of the temperature state of a heated part, the nonuniform distribution of its temperature t is known. In addition to the action of this temperature, the heated part undergoes the action of a surface force p x; y; z � � and a body force F x; y; z � �. The known temperature irregularity and the forces induce in each node a displacement u ¼ u<sup>х</sup> uy uz � �<sup>т</sup> , a strain h i<sup>т</sup>

ε ¼ ε<sup>х</sup> ε<sup>y</sup> ε<sup>z</sup> γ<sup>х</sup><sup>y</sup> γyz γz<sup>х</sup> , and a stress σ ¼ σ<sup>х</sup> σ<sup>y</sup> σ<sup>z</sup> σ<sup>х</sup><sup>y</sup> σyz σz<sup>х</sup> � �<sup>т</sup> that are described by the following linear equations of the elasticity theory (see [24]):

$$e = \mathbf{R}\mathbf{u};\tag{6}$$

σ ¼ Dð Þ ε � αt ; (7) <sup>R</sup><sup>T</sup><sup>σ</sup> <sup>þ</sup> <sup>F</sup> <sup>¼</sup> <sup>0</sup> (8)

p � Cσ ¼ 0 (9)

<sup>t</sup> and u0 F

and by the equation of boundary condition:

Solid models of the heated parts of HPT (a—disk, b—blade, c—casing).

Advanced Nonlinear Modeling of Gas Turbine Dynamics

DOI: http://dx.doi.org/10.5772/intechopen.82015

NDM.

161

Figure 4.

In these equations, R is a differential matrix operator, D presents a stiffness matrix depending on material elasticity and the Poisson's ratio, α denotes a linear expansion coefficient vector, and C stands for a rotation matrix. On the basis of Eqs. (6)–(9) of one elemental volume, a huge system of linear equations of a whole heated part is formed. The number of unknown variables in this system can be partly reduced because one volume node pertains to some adjacent elemental volumes. The system is solved by the least squares method. As a result, the displacements of the external surface of the heated element are determined separately for the action of thermal expansion and the force. In addition to the reference engine mode, the displacements of all the heated parts were determined at the idle regime. As mentioned in Section 3.1, the thermal boundary conditions are known at the reference mode and the variables necessary to determine mechanical loads at this mode are simply calculated by NDM. Using these data, the displacements u<sup>0</sup>

induced at this mode by temperature and force were firstly computed in ANSYS for the disk and the other heated parts. To know how these displacements vary during engine operation, let us firstly analyze how the thermal boundary conditions depend on an engine operating mode. It will be shown that the boundary conditions can be determined through actual and reference gas path variables known from

Advanced Nonlinear Modeling of Gas Turbine Dynamics DOI: http://dx.doi.org/10.5772/intechopen.82015

Figure 4. Solid models of the heated parts of HPT (a—disk, b—blade, c—casing).

$$\boldsymbol{\sigma} = \mathbf{D}(\boldsymbol{\varepsilon} - \boldsymbol{\alpha}\mathbf{t});\tag{7}$$

$$\mathbf{R}^{\mathsf{T}}\boldsymbol{\sigma} + \mathbf{F} = \mathbf{0} \tag{8}$$

and by the equation of boundary condition:

$$\mathbf{p} - \mathbf{C}\boldsymbol{\sigma} = \mathbf{0} \tag{9}$$

In these equations, R is a differential matrix operator, D presents a stiffness matrix depending on material elasticity and the Poisson's ratio, α denotes a linear expansion coefficient vector, and C stands for a rotation matrix. On the basis of Eqs. (6)–(9) of one elemental volume, a huge system of linear equations of a whole heated part is formed. The number of unknown variables in this system can be partly reduced because one volume node pertains to some adjacent elemental volumes. The system is solved by the least squares method. As a result, the displacements of the external surface of the heated element are determined separately for the action of thermal expansion and the force. In addition to the reference engine mode, the displacements of all the heated parts were determined at the idle regime.

As mentioned in Section 3.1, the thermal boundary conditions are known at the reference mode and the variables necessary to determine mechanical loads at this mode are simply calculated by NDM. Using these data, the displacements u<sup>0</sup> <sup>t</sup> and u0 F induced at this mode by temperature and force were firstly computed in ANSYS for the disk and the other heated parts. To know how these displacements vary during engine operation, let us firstly analyze how the thermal boundary conditions depend on an engine operating mode. It will be shown that the boundary conditions can be determined through actual and reference gas path variables known from NDM.

reference mode, the values T<sup>0</sup>

Figure 3.

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tions by four nodes.

ε ¼ ε<sup>х</sup> ε<sup>y</sup> ε<sup>z</sup> γ<sup>х</sup><sup>y</sup> γyz γz<sup>х</sup> h i<sup>т</sup>

160

<sup>i</sup> and α<sup>0</sup>

Design scheme of disk thermal boundary conditions and mechanical loads.

account by elevated values T1 and α<sup>1</sup> in Section 1 of the disk surface.

Section 1. The design schemes of the blade and the casing are similar.

3.2 Stress-strain state and the displacements of heated parts

induce in each node a displacement u ¼ u<sup>х</sup> uy uz

the experimental information. In a peripheral disk part, in addition to hot gases, heat is transmitted from the blades. This additional heat transfer is taken into

As to the mechanical loads, the centrifugal force acting on the disk is a body force that is applied to each elemental volume of the disk. The centrifugal force from the rotating blades is given as a surface force by a uniform distribution σ<sup>B</sup> in

In the finite element-based simulation, the heated parts are presented by their

� �<sup>т</sup>

, and a stress σ ¼ σ<sup>х</sup> σ<sup>y</sup> σ<sup>z</sup> σ<sup>х</sup><sup>y</sup> σyz σz<sup>х</sup>

, a strain

� �<sup>т</sup> that are

ε ¼ Ru; (6)

solid models illustrated by Figure 4. In its solid model, each HP is divided on elemental 3D simplex volumes. Each volume is presented in finite element calcula-

After the determination of the temperature state of a heated part, the nonuniform distribution of its temperature t is known. In addition to the action of this temperature, the heated part undergoes the action of a surface force p x; y; z � � and a body force F x; y; z � �. The known temperature irregularity and the forces

described by the following linear equations of the elasticity theory (see [24]):

<sup>i</sup> of these parameters are known on the basis of
