studies on rotation ally periodic structures

8/8/2019 Studies on Rotation Ally Periodic Structures

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STUDIES ON ROTATIONALLY PERIODIC STRUCTURES

1. Introduction.

Many engineering structures have identical sub-structures in the circumferential direction. A rotationally

periodic structure consists of a finite number of identical sectors (sub-structures) coupled together in

identical ways to form a closed ring-type structure. These types of structures are met with in many

applications such as centrifugal compressors, gear wheels, turbomachinery bladings and frames. Each

sector is assumed to be loaded identically. Since the deflection shape of each sector is identical, only onesector is sufficient for the finite element analysis. In the present work, a structural frame (similar to the

one used in an aero-engine) is considered for the static sectorial analysis.

2. Uniform Thin Cylinder.

In order to get insight into the behaviour of rotationally or circumferencially periodic structures, an

uniform thin cylindrical shell has been studied in Cartesian (90o sector model) and cylindrical (30o sector

model) coordinate systems. Four noded shell element (Shell63 of ANSYS) model and eight noded solid

element (Solid45 of ANSYS) model are considered for a pressure load.

Fig. 2.1 shows the geometry of a thin cylinder with internal pressure. The cylinder is considered to bemade out of mild steel with the following material properties at room temperature of 150 C.

Young's Modulus, E = 21000 kgf/mm2

Poisson's ratio, V = 0.3

Mass density, = 8 x 10-10 kgf s2/mm4

Fig. 2.1 Thin Cylinder with Internal Pressure.

2.1 90o Sector Model of Thin Cylinder in Cartesian Co-ordinate System.

A 90o sector of the thin cylinder is studied in Cartesian coordinate system as shown in Fig. 2.2. Static

analysis is carried out with shell and solid finite element models given in Fig. 2.3 and Fig. 2.4. In Fig. 2.3 the

cylinder is discretised with four noded quadrilateral thin plate elements. Eleven elements along the length

and thirty elements around quarter circumference are used in the finite element model. Since the mesh is

very regular all the elements have rectangular shape of 3.19 mm x 5 mm size with an aspect ratio of 1.56.

Symmetric boundary conditions are imposed at the 0o and 90o sectorial planes (Plane -1 and Plane -2 of

Figs. 2.3 and 2.4) along the length of the cylinder. Symmetric boundary conditions at the plane ofsymmetry results in zero out of plane motion.

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To fix a body/object in space, all the six rigid motions(three translations-ux, uy, uz and three rotations x, y,

z) have to be restrained to zero. This condition make the stiffness matrix non-singular in the deflection-

force equation. In Fig. 2.3 rigid body rotations (x, y, z) are restrained to zero by applying the symmetric

boundary conditions at the Plane-1 and Plane-2. Also two of the rigid translations, ux, uy are restrained to

zero by the symmetric boundary conditions. uz translation is restrained to zero by the end fixity condition

at the edge-1. Hence, all the six rigid body motions are removed from the four noded shell element model

as shown in Fig. 2.3. With the above boundary conditions it is ensured that there is free radial expansion,

also a free axial expansion due to the internal pressure load.

In the solid element model as shown in Fig. 2.4, two of the rigid translation ux, uy are restrained to zero by

applying the symmetric boundary conditions at the Plane-2 and Plane-1 respectively. Rigid rotation x isrestrained to zero from the symmetric boundary condition of the Plane-1, that is rigid translations, uy in

more than one node is restrained to zero. Similarly rigid rotation, y=0 from the symmetric boundarycondition of the Plane-2, that is rigid translation, ux=0 in more than one node along the Plane-2. In Fig. 2.4,end fixity is applied by uz=0 at all the nodes along the Edge-1. The rigid translation, uy=0 along the

thickness of the shell (that is, at more than one node) at the Plane-1 make the rigid rotation, z=0. Alsofrom the rigid translation, ux=0 along the thickness of the shell (that is, at more than one node) at the

Plane-2 make the rigid rotation, z=0. Hence with the above conditions all the six rigid motions are

restrained to zero, making the stiffness matrix of the solid element model as shown in Fig. 2.4, non-singular. Similar to the shell element model (Fig. 2.3), in the solid element model also, it is ensured freeradial and axial expansions due to the internal pressure load.

The mass calculations for the quarter cylinder is given below:

Volume = x x (r12 - r2

2 ) x L

= x x (622 - 602) x 55

Volume = 10540.043 mm3

Mass = Volume x Mass Density = 10540.043 x 8 x 10-10

Mass = 8.432034 x 10-6 Kgf s2 / mm

The mass and DOF (Number of equations to be solved are listed in the Table 2.1 for the shell and solidelement models.

Fig. 2.2 90o Sector Model of Thin Cylinder.

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Fig. 2.3 Four Noded Shell Element Model (90o Sector).

Fig. 2.4 Eight Noded Solid Element Model (90o Sector).

Table 2.1 Mass and DOF for Shell and Solid Element Models (90o Sector).

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From the analysis of 90o sector model, the deflection shapes along the lines AB and CD as shown in Figs.

2.5 and 2.6 have been studied. This study clearly indicates from the Tables 2.2 (for Shell element model)

and Table 2.3 (for Solid element model) that the deflection shapes along the lines AB and CD are same,

though the symmetric boundary constraints are imposed along the line AB.

Fig. 2.5 Shell Element Model showing Line-AB and Line-CD (90o Sector).

Table 2.2 Radial and Axial Deflections for the Shell Element Model (90o Sector).

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Fig. 2.6 Solid Element Model showing Line-AB and Line-CD (90o Sector).

Table 2.3 Radial and Axial Deflections for the Solid Element Model (90o Sector).

Radial and axial deflections in the cylinder due to the pressure load have been calculated from the strain

equations.

Calculations.

Circumferential stress Axial (Longitudinal) stress

)..2.(..)....(.2 lxRxpxlxtx

;

2

...)...2( RxpxtxR a

Circumferential stress,t

Rxp ; Axial (Longitudinal) stress,

t

Rxpa

2

Circumferential strain,

t

Rxpx

t

Rxp

EEa

2

11

Circumferential strain,

2

2

x

Extx

Rxp

Where p=0.1 kgf/mm2

, R=60 mm, t=2 mm, E=21000 kgf/mm2, =0.3.

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3.02

2100022

601.0x

xx

x

6010121428.0

10121428.0

3

3

xxR

xRRherefore

R

R

x

Radial deflection, R=0.72857x10-2 mm

Axial (Longitudinal) strain,

t

Rxpx

t

Rxp

EEaa

2

11

Axial (Longitudinal) strain,

21x

Ext

Rxpa

3.0

2

1

210002

601.0x

x

xa

551057142.28

1057142.28

6

6

xxL

xLLherefore

L

L

x

a

a

a

Axial deflection, L=0.1571x10-2 mm

The radial and axial deflection distributions along the cylinder length have been plotted as given in Figs.

2.7 to 2.10. From Fig. 2.7 (Fig. 2.9), it is found that the finite element models (shell and solid elements

predict the radial deflection upper bound as compared to the calculation. It is noted that the solid

element model is more stiff than the shell element model as shown from Figs. 2.7 and 2.9. However, the

deflection from the solid element, model is close to the exact value as compared to the shell elementmodel. Axial constraint boundary condition applied at Edge-1 for the shell and solid element models are

reflected in the Figs. 2.8 and 2.10. It is found that the solid element model predict lower axial deflection as

compared to the shell element model when one observes the deflection away from the fixity (Edge-1).

From the 90o sector model (shell and solid elements) the procedure to obtain the deflection distribution

along the length and around the circumference by applying the symmetric boundary conditions at the

planes of symmetry is understood. The radial and axial deflections obtained from the 90o sector model are

comparable with the calculation. The radial deflection distribution of both the shell and solid element

models give upper bound solution for the calculation from the strain equations. In the next section, a 30o

sector model is considered in order to understand usage of symmetric boundary condition in cylindrical

coordinate system.

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Fig. 2.7 Comparative study of Radial deflection (Along Line-AB; 90o Sector) distribution between

Shell and Solid Element Models.

Fig. 2.8 Comparative study of Axial deflection (Along Line-AB; 90o Sector) distribution between Shell

and Solid Element Models.

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Fig. 2.9 Comparative study of Radial deflection (Along Line-CD; 90o Sector) distribution between

Shell and Solid Element Models.

Fig. 2.10 Comparative study of Axial deflection (Along Line-CD; 90o Sector) distribution between Shell

and Solid Element Models.

2.2 30o Sector Model of Thin Cylinder in Cylindrical Coordinate System.

A 30o sector of the thin cylinder (Fig. 2.1) is studied in cylindrical coordinate system as shown in Fig. 2.11.

Static analysis is carried out with Shell and Solid element models with different mesh density as given in

Figs. 2.12-2.15. Figs. 2.12 and 2.13 show the coarse and fine mesh discretisation with four noded

quadrilateral thin plate elements respectively. In the coarse mesh seven elements along the length and six

elements along the 30o circumference are used in the finite element model (Fig. 2.12). All the elements

have rectangular shape of 5.32 mm x 7.85 mm size with an aspect ratio of 1.47. Fine mesh of Fig 2.13 has

been arrived at with a discretisation of eleven elements along the length and ten elements around the 30o

circumference. These elements are rectangular in shape with 3.19 mm x 5 mm size and an aspect ratio of

1.59. Symmetric boundary conditions are imposed in cylindrical coordinate system at 0o and 30o sectorial

planes (Plane-1 and Plane-2 of Figs. 2.12, 2.13, 2.14 and 2.15) along the length of the cylinder. The massand DOF (number of equations to be solved) are listed in Table 2.4 for the shell and solid element models.

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Fig. 2.11 30o Sector Model of Thin Cylinder.

The mass calculations for the 30o sector cylinder is given below:

Volume = 1/12 x x (r12 - r2

2 ) x L

= 1/12 x x (622 - 602) x 55

Volume = 3513.3478 mm3

Mass = Volume x Mass Density = 3513.3478 x 8 x 10-10

Mass = 2.81067 x 10-6

Kgf s2

/ mm

Fig. 2.12 Four Noded Shell Element Model (Coarse Mesh; 30o Sector).

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Fig. 2.13 Four Noded Shell Element Model (Fine Mesh; 30o Sector).

Fig. 2.14 Eight Noded Solid Element Model (Coarse Mesh; 30o Sector).

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Fig. 2.15 Eight Noded Solid Element Model (Fine Mesh; 30o Sector).

Table 2.4 Mass and DOF for Shell and Solid Element Models (30o Sector).

From the static analysis of the 30o sector model due to the internal pressure load, the deflection shape

along the lines AB and CD as shown in the Figs. 2.16 and 2.17 have been studied. This study also indicates

from the Tables 2.5, 2.6, 2.7 and 2.8 that the deflection shapes along the lines AB and CD are same

(though the symmetric boundary constraints are imposed along the line AB) similar to the 90o sector

model given in section 2.1.

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Fig. 2.16 Shell Element Model showing Line-AB and Line-CD (30o Sector).

Table 2.5 Radial and Axial Deflections for the Shell Element Model (Coarse mesh; 30o Sector).

Table 2.6 Radial and Axial Deflections for the Shell Element Model (Fine mesh; 30o Sector).

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Fig. 2.17 Solid Element Model showing Line-AB and Line-CD (30o sector).

Table 2.7 Radial and Axial Deflections for the Solid Element Model (Coarse mesh; 30o

Sector).

** In the coarse mesh, since there is only one element across the thickness, deflections along the lines AB

and CD are obtained by taking average of the deflections at the top and bottom nodes.

Table 2.8 Radial and Axial Deflections for the Solid Element Model (Fine mesh; 30o Sector).

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From the 30o sector model, the radial and axial deflection distributions along the cylinder length have

been plotted as given in Figs. 2.18-2.21. These figures indicate the comparative study of distributions for

the shell and solid element models of both coarse and fine meshes. It is found that the 30o model (shell

and solid element models) also predict the radial deflection upper bound as compared to the calculation.

Fig. 2.18 (fig. 2.20) shows the convergence of radial deflection from the coarse solid element model to the

fine solid element model towards the calculated value. Whereas no significant change is obtained for the

radial deflection, from the coarse shell element model to fine shell element model. Similar trend is seen

for the axial deflection from Figs. 2.19 and 2.21 for both the shell and solid element models of coarse andfine meshes.

Conclusions. Studies of 90o sector model in Cartesian coordinate system and 30o sector model in

cylindrical coordinate system conclude that radial deflection will be the same in either model. This is

evident from the Figs. 2.7 (2.9) and 2.18 (2.20). Similar observation can be noted for the axial deflection

distribution from the Figs. 2.8 (2.10) and 2.19 (2.21).

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