A VIBRATION FINDING SURVEY OF EXISTING STEEL PENSTOCKS AND NATURAL FREQUENCY

CALCULATION FORMULAE

Hideharu NAKAMURA1 and Kosuke YAMAMOTO2

1Fellow of JSCE, Dr. Eng., Professor, Graduate School of Engineering, Hiroshima University

(1-4-1 Kagamiyama, Higashi-Hiroshima-Shi, Hiroshima, 739-8527, Japan) 2Member of JSCE, Dr. Eng., Central Research Institute of Electric Power Industry

(1646 Abiko, Abiko-Shi, Chiba, 270-1194, Japan)

When a momentary pressure pulsation takes place in a turbine, and its frequency coincides with the natural frequency of a steel penstock, resonance occurs resulting in noticeable vibration. While the current tendency is to increase the rotation frequency of turbines, penstock plates are gradually becoming thinner. As a result of these trends, the occurrence of oval vibration has been reported at some hydraulic power stations. Since rules on vibration prevention do not always include oval vibration, we sought out detailed information on existing penstocks constructed within the last 30 years. Based on this information and a numerical study, formulae to calculate the natural frequencies of steel penstocks including those with circumferential stiffeners were proposed.

Key Words : steel penstock, pressure pulsation, resonance, numerical analysis, design formula

1. INTRODUCTION Hydraulic power generation passed its golden age long ago and currently accounts for only about 10% of the Japan’s power supply. However, while it has been overshadowed by other forms of power generation in terms of total output, hydraulic power remains an indispensable source of clean energy for the nation. Maintaining this important power supply requires continuing operations for the maintenance, management and renewal of steel penstocks. Since dams are invariably located in mountain terrain, there is considerable demand to reduce the weight of the penstock by adopting high-tensile steel while maintaining pressure resistant performance. However, the use of high-tensile steel increases the possibility of vibration problems. A survey of conventional vibration problems occurring in existing penstocks highlights the three following major points: (i) Vibration of the straightening plate or

reinforcing plate at a branch (ii) Cross-sectional deformation of the pipe shell

caused by changes in hydraulic pressure (iii) Resonance caused by the hydraulic pressure

pulsation which occurs within the hydraulic turbine or draft tube and approaches the natural frequency of the pipe shell

The vibration described in (i) is a local phenomenon and can be resolved by reinforcing the effected sections. The cross-sectional deformation described in (ii) may be resolved by treating the columnar vibrations of the water within the pipe. However, this phenomenon does not result in vibration within the penstock, and out-of-plane displacement on the surface of the pipe shell is rarely large and is normally less than 1mm. The resonance described in (iii) occurs in relatively thin wall sections where comparatively low hydraulic pressure is applied, and generally at points close to the reservoir. In such cases, out-of-plane displacement on the surface of the pipe shell may be as great as 10mm, sufficient to cause fatigue damage in the pipe shell section. It has been shown that the effects of resonance cannot be avoided even when vibration-prevention measures are taken by applying circumferential stiffeners. This paper addresses the issue of resonance phenomena. The first study in Japan related to this issue, conducted in 1951 and 1952 by the Hydraulic Turbine Research Committee of the Japan Society of Mechanical Engineers, applied theoretical investigation to introduce a formula for calculating natural frequency of steel penstocks1)-6). Results of calculations obtained by the formula were said to show good agreement with the measurements and

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Structural Eng./Earthquake Eng., JSCE, Vol.20, No.2, 163s-178s, 2003 October

Circumferential mode: 2=n 3=n

Axial mode: 1=k 2=k 3=k

Fig.1 Vibration modes of penstocks the formula was eventually made available to the public in the 1973 edition of the Technical Standards for Gates and Penstocks7). These standards, including the 1981 and 1993 editions8), 9), however, do not provide a detailed description of vibration modes specific to penstocks. Precise prediction of the natural frequency and vibration modes for a specific penstock is difficult because the resonance phenomena of penstocks are frequently attributable to characteristics of the hydraulic turbine. A survey of past vibration problems shows vibration behavior with the circumferential mode number n = 2 and axial direction mode number k = 1 (Fig.1) occurring in the frequency range of 6－8Hz, which is close to the rotation speed of the hydraulic turbine. Analytical investigations factoring in the actual shape of the penstock, actual supporting and application conditions also show a primary vibration mode of n = 2 and k = 1, with a tendency for relatively larger diameters to show a primary vibration mode of n = 3 and k = 1. The use of hydraulic turbines with higher rotation speed is increasing, while at the same time, the thickness of the penstock wall has gradually been reduced and the support span has been extended. These trends are altering values for the primary natural frequency of penstocks and hydraulic turbine speed to the extent that they will eventually reach parity. This makes it necessary to address the resonance issue as a practical problem, and to attach increased attention to the importance of natural frequency calculation formulae that can ensure accurate prediction of actual phenomena. However, as previously mentioned, the formula specified in the current Technical Standards for Gates and

Penstocks is not sufficiently consistent in cases where n = 2. Further, no clear findings are available at present with regard to the effects of the rigidity of circumferential stiffeners on the vibration frequency and the determination of the fixing span for cases where one or more circumferential stiffeners are arranged between highly rigid ring girders. The natural frequency calculation formula is therefore most probably applied with a constant fixing span assumed for the circumferential stiffeners regardless of the stiffness, resulting in designs that are based on very unrealistic vibration frequency predictions. In order to clarify the actual effects of vibration phenomena specific to penstocks, this study first investigates penstocks that have been installed and renewed over the past 30 years. The parameters of the study incorporate current data on the conditions of penstock application and support systems, and include past examples of occurrences of vibration. Data from on site surveys of actual conditions and from many cases of analytical investigations are then applied to formulate simplified formulae which are proposed for calculation of the most important natural frequencies: (i) Less than 20Hz with n = 2 and k = 1 and (ii) Less than 30Hz with n = 3 and k = 1 Further analytical investigations are then applied to the effects of circumferential stiffeners on vibration frequencies, resulting in the proposal of a method for evaluating natural frequencies which allows values for n = 2 by introducing the idea of equivalent wall thickness for the purpose of design calculation. This study concludes that vibration frequencies can be reasonably predicted by applying the proposed natural frequency calculation method to existing records of actual vibration.

2. A SURVEY OF ACTUAL CONDITIONS AT EXISTING PENSTOCKS A survey of actual conditions at existing penstocks was carried out by the Research Committee for the Prevention of Penstock Vibration, established by the Japan Hydraulic Gate and Penstock Association, during the period of June to December 2000. The objectives of the survey were to obtain basic data indispensable for understanding the range of shapes and dimensions of existing penstocks, and for proposing formulae for natural frequencies compared to vibration examples. The survey was conducted by distributing questionnaires to power companies, public power

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Fig.2 Types of stiffeners taking into consideration

Fig.3 Penstock fixing span length

supply companies and other related organizations. Survey data was then compiled as follows. (1) Survey items a) Particulars of the facility, design conditions, type

of structure, basic shape, materials, etc. for exposed penstocks newly installed or renewed since 1970 (this does not include all of the penstock operations begun after 1970).

b) Items related to hydraulic turbines. This includes turbine type, rotation speed, number of runner blades, etc. (The survey includes reaction turbines

such as the Francis and Kaplan types, but does not include impulse turbines such as the Pelton which do not affect penstock vibration.)

S

Internaldiameter

Shellthickness

d1

t1

Single

S

Double

t1

b

t

DInternaldiameter

Shellthickness

t

D

d1

c) Items related to natural frequencies of penstocks. This includes penstock internal diameter, wall thickness (including corrosion allowance), static head (action hydraulic pressure), position and sectional profile of stiffeners (Fig.2), type of penstock support, and fixing span length (Fig.3).

d) Items related to past examples of vibration. This includes date of confirmation of vibration, place of occurrence, operating conditions, type of structure, vibration status confirmed on site, time, method and results of the vibration survey, possible causes derived from the results, vibration prevention measures, repair of facility (alteration of structure) and time of recovery.

Concretesaddle support

Fixing span

Steel Penstock

Ringgirder

Anchorblock

Steel Penstock

Fixing span

Concretesaddle support

Fixing span

Steel Penstock

(2) Questionnaire distribution and number of data collected Questionnaires were mailed to 11 power companies and 6 public power supply companies in Japan, and data was collected for 99 power stations. (3) Survey analysis a) Hydraulic turbine speed of rotation and number of runner blades Fig.4 shows the relation between the hydraulic turbine speed of rotation and number of blades on the runner. Though no evident correlation is observed between turbine speed and the number of runner blades, frequency is distributed between 250 － 1000rpm. The spectra show the highest frequency at 600rpm, with other peaks at 500rpm and 750rpm (Fig.5). While 70% of the runners reported in the survey consist of 13, 15 and 17 blades, only 15% consist of fewer than 10 blades (Fig.6). b) Internal diameter, wall thickness, span length and circumferential stress ratio Histograms showing frequencies of internal diameter, wall thickness, span lengths and circumferential stress ratios to the allowable tensile stress σ/σa (σ: tensile stress due to internal pressure and σa: allowable tensile stress) are given in Fig.7 －Fig.10, respectively. These data include the cases both with and without circumferential stiffeners. Frequencies for the internal diameter of penstocks are mainly distributed in the range of 60－300cm, forming a convex shape. The majority of data is concentrated in the range of 160cm－180cm, and a slight increase of frequencies is found in the range of 340－ 380cm. Furthermore, large size diameters for pumped-storage generation range up to 540cm.

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Fig.4 Relation between hydraulic turbine speed of rotation and number of runner blades Fig.5 Hydraulic turbine Fig.6 Number of runner speed of rotation blades Maximum frequencies for penstock wall thickness are in the range of 0.8－0.9cm. The majority of data is concentrated between the 0.6cm defined in the Technical Standards for Gates and Penstocks as the minimum wall thickness, and 1.1cm. Walls with a thickness greater than 1.2cm occur with gradually decreasing frequency, and those over 2.0cm are rarely seen. Maximum frequency for span lengths (distance between supports) occurs at 6m, and increase in the frequency is noticed at 9, 12, 15, 18 and 22m. For older penstocks, this commonly occurs at 2m on single pipes (minimum unit pipe free from circumferential joints) and 6m on unit pipes (factory built single unit block pipes). Recently manufactured penstocks show an increase in frequency at 3m intervals starting at a point 6m length wise. The circumferential stress ratio to the allowable tensile stress is evenly distributed in the range of σ/σa ≦ 0.65. This result is believed to be due in

Fig.7 Penstock internal diameter

Fig.8 Penstock wall thickness part to the phrasing of the questionnaire instructions which requested that data be compiled for all points from the start to the end of the penstock. Negligible distribution was seen in the range of σ/σa > 0.65. c) Internal diameter, wall thickness, span length and circumferential stress ratio with intermediate stiffeners Though no graph is shown for cases where intermediate stiffeners are applied, the following conditions are observed. The internal diameter of the penstock shows maximum frequencies in the range of 160－180cm, and distribution increases in the region where the diameter is relatively large. A wall thickness of 0.9cm is prominent. An increase in the frequency is seen in the span length at 12, 18 and 22m, and notable distribution is observed in the region where the span is relatively longer. The circumferential stress ratio shows a decrease in the frequency in the range of σ/σa = 0.3－0.5, which tends to occur clearly in two regions of stress occurrence, the lower (σ ≦ 0.3σa) and the higher (σ = 0.5σa).

4 6 8 10 12 14 16 18 20

200

300

400

500

600

700

800

900

1000H

ydra

ulic

turb

ine

spee

d of

rota

tion

(rpm

)

40 80 120 160 200 240 280 320 360 400 440 480 520 560

0

20

40

60 160～180cm

Frequency

Internal diameter (cm)Number of runner blades

200 400 600 800 1000

0

5

10

15

20

25

Frequency

Hydraulic turbine speed of rotation (rpm)

4 6 8 10 12 14 16 18 20

0

5

10

15

20

25

30

Frequency

Number of runner blades0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0

20

40

60

80

Frequency

0.9cm

0.8cm

Wall thichness (cm)

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Fig.9 Span length (Fixing span of penstocks)

Fig.10 Circumferential stress ratio

Fig.11 Number of stiffeners per span d) Number of stiffeners per span and their sectional profile The number of stiffeners per span is shown in Fig.11. A profile of stiffeners shows that some are constructed of concentrically cut steel plates welded to the pipe shell and others are equipped with flanges which form a T-shape configuration. The two types show no marked correlation in shape. For both types, remarkable distribution is observed at heights of 8－20cm and at wall thickness of 0.9－ 1.6cm. The majority of cases included in the

500 1000 1500 2000 2500 3000

0

20

40

60

80

22m

18m

15m

12m9m

6m

Frequency

Span length (Fixing span) (cm)

0 100 200 300 400 500 600

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Wal

l thi

chne

ss (c

m)

Internal diameter (cm)

without stiffener with stiffeners

Fig.12 Relation between internal diameter and wall thickness

0 100 200 300 400 500 600

0

500

1000

1500

2000

2500

3000

Span

leng

th (c

m)

Internal diameter (cm)

without stiffener with stiffeners

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

10

20

30

40

σ/σa

Frequency

Circumferential stress ratio

Fig.13 Relation between internal diameter and span length

0 500 1000 1500 2000 2500 3000

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Wal

l thi

chne

ss (c

m)

Span length (cm)

without stiffener with stiffeners

0 1 2 3 4 5 6 7 8 9 10 11

0

20

40

60Frequency

Number of stiffeners per span

Fig.14 Relation between span length and wall thickness histograms consist of one or two stiffeners per span, and units consisting of more than three stiffeners are rarely seen. Generally only one or two stiffeners are provided at the time of erection since it can be assumed that a number of stiffeners will be added after vibration occurs. The 112 cases in the study include only one consisting of a double stiffener (Fig.2) and 11 with T-shape configurations. e) Presence of stiffeners and shape parameters of penstocks Fig.12 shows the relation between the penstock internal diameter and the wall thickness for both

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Fig. 15 Relation between shape parameters of penstocks and examples of vibration occurrence cases with and without stiffeners. Cases with stiffeners are distributed in the relatively thin wall thickness and large diameter region. Fig.13, which shows the relation between the penstock internal diameter and the span length, indicates no marked tendencies associated with the presence of stiffeners. Fig.14 shows the relation between the penstock span length and the wall thickness. Cases with stiffeners are distributed in the relatively thin and long span region. f) Examples of vibration occurrence and shape parameters In Fig.15, which shows the shape parameter distribution for existing penstocks, the horizontal axis represents the ratio of the span length and radius (L/R) and the vertical axis represents the ratio of the radius and wall thickness (R/t). The graph shows four sites of past vibration occurrence (total of eight cases). It should be noted that the majority of vibration occurrence is concentrated in specific combination of shape parameters. Distribution is generally scattered between L/R = 3－32 and R/t = 40－170 while a downward sloping tendency is recognized in the distribution diagram. In general, stiffeners are not required for cases with thicker walls, smaller diameters and shorter spans, but they are required for cases with thinner walls, larger diameters and longer spans. Therefore, cases that include stiffeners should be expected to appear frequently at the upper left and lower right areas of the data region. While this assumption does hold true to a certain extent in Fig.15, no remarkable tendency is shown.

3 . N A T U R A L F R E Q U E N C Y O F PENSTOCKS The following matters can be cited as the primary

causes for hydraulic pressure variation in the penstock. Frequency of hydraulic pressure variation expressed by fw (Hz) is as follows according to the source specified:

0 5 10 15 20 25 30 35

40

60

80

100

120

140

160

180

Span length / radius ratio (R/t)

without stiffener with stiffeners vibration occurrence

(i) Rotation of Hydraulic turbine N (rpm): fw = N/60 (ii) Number of blades per runner Z: fw = NZ/60 (iii) Water vortex in draft tube: fw = N/(60×3.6) These data are determined by the specifications of the hydraulic turbine. Though amplitude of hydraulic pressure variation depends on the specific characteristics of each turbine, remarkable resonance will result if the natural frequency of penstock “f” meets the frequency of the hydraulic pressure variation “fw”. It is therefore necessary to be precise in calculating the natural frequency of the penstock. (1) Discussions on the conventional formula for calculating vibration frequency In linear vibration analyses of a cylindrical shell based on the small displacement theory, exact processing should be made in accordance with Flügge’s equations. However, analytical solution from Flügge’s equations cannot be derived by the explicit form, and this approach would interfere with understanding of the vibration characteristics of penstocks. The current Technical Standards for Gates and Penstocks incorporates the fluid coupled effects in the formula of a cylindrical shell that is equivalent to Flügge’s equations in order to establish the formula. Simplification is made in the course of the derivation of the analytical solution to induce a double Fourier solution based on the following assumptions: (i) Both ends of the cylindrical shell are simply

supported. (ii) Wall thickness is constant and no circum-

ferential stiffener is provided between the two ends.

(iii) Deformation without elongation is assumed at the center plane of the cylindrical shell

2

2 2mk r nLπ α⎛ ⎞ ≡ <<⎜ ⎟

⎝ ⎠ (1)

where: r : Radius to center of the wall m

L : Fixing span of the penstock k : Number of mode in axial modes n : Number of mode in circumferential sections (iv) Inertia force in the tangential and axial

directions of the cylindrical shell is neglected

Rad

ius /

wal

l thi

chne

ss ra

tio (R

/t)

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since it is considered too small in comparison to the internal pressure.

(v) Internal fluid is non-compressive and can be regarded as still state.

These assumptions are necessary to solve the double Fourier solution in which fluid is considered, and these are considered generally reasonable. However, care should be taken for assumption (iii), which is applicable to the case of a thin cylindrical shell where deflection in bending has the priority and is used for simplifying equations. If used for the penstock without careful consideration, reduction of accuracy will be caused for n = 2 which is the most important vibration mode. After theoretical investigation, the following formula was eventually incorporated in the current Technical Standards for Gates and Penstocks (the 4th edition, 1981 and the 5th edition, 1993).

( )( ) ( )11

1

1

11

12

1

22

22

222

4

2

2

2

2

−+−

−+

+

+

++

=

nEt

Prn

n

nn

nn

Er

f

m

m

νβ

α

α

ερπ

(2)

where: f: Characteristic frequency (1/sec) E: Modulus of elasticity of steel (206kN/mm2 → 2.06×108kg/mm･sec2) ν : Poisson's ratio of steel ρ: Density of steel (7.85×10-6kg/mm3) t: Wall thickness (mm) rm: Radius to center of the wall (mm) n: Number of mode in circumferential sections k: Number of mode in axial modes L: Fixing span length (mm) P: Internal pressure (MPa)

α : Lrk mπ , β : 22 12 mrh

ε: (Water density/steel density)nh

rm 1⋅⋅

In formula (2), in a manner similar to the introduction of a solution based on a simplified version of Donnell’s equation, the following components are given separately in the last radical of the right side: First term: Effect of in-plane rigidity Second term: Effect of bending rigidity Third term: Effect of internal pressure The details of an issue when a form analogous to a solution derived from Donnell’s equation is obtained during the course starting from the exact equations of a cylindrical shell till derivation of the final solution, cannot be known from the literature

disclosed to date. If an assumption of simplification as stated in (iii) is used, it is known that accuracy tends to drop remarkably for the following cases10), 11): (vi) Circumferential mode number less than n = 5 (vii) Span length relatively shorter than the radius (viii) Relatively thick wall In fact, calculation of vibration frequency in an empty penstock using formula (2) confirms that a relatively large number of errors are generated for the the smaller L/R and the relatively smaller R/t at n = 2 and n = 3. Though the number of errors decrease relatively when the penstock is filled with water due to the effects of additional mass and internal pressure, a significant number of errors are still observed when the span length of the penstock is as short as L/R < 10 as shown in Table 4. This tendency is considered to support the effect of assumption (iii) described above. (2) Parameter analysis of the natural penstock frequency a) Purpose of analysis This paper is not intended to be a derivation to a new analytical solution based on the exact equations of a cylindrical shell. Strict adherence to the application of theoretical equations may needlessly complicate vibration frequency calculation formulae and the overall aspect of vibration frequency characteristics of the penstock could easily be overlooked. As shown in the results of actual conditions described in 2, the range of the shape parameters and dimensions for existing penstocks are already known. Therefore, vibration frequency calculation formulae specifically designed for the case of vibration of n = 2, 3 and k = 1 can be proposed if numerical simulations by the finite element method covering dimensions and shape parameters of existing penstocks are made and approximate expressions for results obtained from these analyses are established. b) Analysis of fluid coupled vibrations by finite element method Static finite element analysis using an axi-symmetric thin shell element is a well-known technique12). The following improvements are made to the numerical method and computer programs described in reference 13) to allow effective analysis of fluid coupled vibrations. In the coupled analysis of the fluid and shell, equations of motion should be established and both should be coupled using boundary conditions at the water-contact surface14), 15). The equations of motion of the shell are:

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Fig.16 Shape of analysis model and supporting conditions

[ ]{ } [ ]{ } [ ]{ } { }psss fUKUCUM =++ &&& (3)

The equations of motion of the fluid, where the effects of compressibility, viscosity, effects of rotational flow and surface waves are neglected, are simply expressed as:

(4) [ ]{ } [ ] { } 0=+ USpK Tf

&&ρ

Boundary conditions at water-contact surface are expressed as:

(5) { } [ ] { }pSf p =

where: , , : Acceleration, velocity and { }U&& { }U& { }U displacement vectors : pressure vector { }p , [ ] : Mass, damping and stiffness [ ]sM sC [ ]sK, matrices of shell [ ]fK : Matrix introduced from the fluid { }pf , : Fluid force vector and fluid- [ ]S structure coupled matrix Equations (3) and (4) are then coupled using equation (5), and neglecting the term of damping, equation (6) is derived where the fluid is considered as an additional mass. This equation makes it possible to obtain the natural frequency of a penstock by eigen value analysis.

[ ] (6) { } [ ]{ } 01 =++ − UKUSKSM sT

fs&&ρ

c) Analytical conditions of penstocks without stiffeners In general, penstocks have a continuous beam structure that includes expansion joints. Therefore, a numerical model that assumes a constant wall thickness was used where one end is completely fixed while the other end is fixed for radial displacement (Fig.16). It was supposed that the penstock is filled with water, and shape parameters were set as shown in Table 1. Furthermore, in order to factor in the effects of

Table 1 List of shape parameters for numerical model t

R

L

a) Wall thickness and radius t(mm)

R (m) 6 9 12 18 21

0.50 83 56 42 0.75 125 83 63 42 1.00 167 111 83 55 48 1.50 167 125 83 71 2.00 167 111 95 2.50 139 119

b) Span length and radius (L/R) L(m)

R (m) 6 12 18 24

0.50 12 24 0.75 8 16 24 1.00 6 12 18 24 1.50 8 12 16 2.00 6 9 12 2.50 7.2 9.6

internal pressure and natural frequency was analyzed by adding in the conditions for generating circumferential membrane stress corresponding to 0.3σa and 0.6σa (allowable tensile stress σa = 135 N/mm2). A fine mesh was applied so that the results of the numerical analysis would not be dependent on mesh division. Mesh size of 7.5cm was selected from the results of a preliminary investigation. d) Analytical conditions of penstocks with stiffeners As previously stated, the formulae for calculating vibration frequency are not specified in the current Technical Standards for Gates and Penstocks for cases where circumferential stiffeners are provided between ring girders. In many cases, the formula for calculating vibration frequency is applied regardless of the rigidity of the stiffeners when the distance between the stiffeners is assumed to be the span length of the penstock. Especially when the circumferential modal number is n = 2, the effect of the stiffener is negligible. It should therefore be noted that results with the above assumption are apt to differ greatly from actual conditions. Based on the results of the survey described in 2, the typical range of parameters for penstocks equipped with stiffeners was set as shown in Table 2, and an analysis of the natural frequency was carried out. Table 3 shows changes in the local rigidity of the penstock (within effective widths around the stiffeners7)) which include four kinds of circumferential stiffeners (F100, F150, F200 and T250 shown in Fig.17). In order to consider the effects of internal pressure as studied for the case without stiffeners, analysis of the natural frequency

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w a s c o n d u c t e d u nde r c o n d i t i o n s w h e r e circumferential membrane stress corresponding to 0.3σa and 0.6σa is generated at the static head.

Table 2 Shape parameters with circumferential stiffeners a) Ratio of the radius and wall thickness (R/t)

t(mm)R(m)

9 12 18

1.00 111 83 2.00 167 111

e) Results of Numerical analyses Numerical results are shown in Table 4 for cases without stiffeners and in Table 5 for cases with stiffeners. In Table 4, the FEM column shows natural frequencies obtained by the finite element method, the columns of Eq.(9) and Eq.(10) show values obtained by the formulae proposed by the authors which will be dealt with later in this paper, and the column of Eq.(2) shows values obtained from the formula (2) specified in the current Technical Standards for Gates and Penstocks. In Table 5, only results of finite element analyses are shown.

b) Ratio of the span length and radius (L/R) L(m)

R (m) 6 12 18 24

Fig.17 Shapes of circumferential stiffeners

1.00 6 12 18 2.00 6 9 12

Table 3 Moment of inertia for shell walls (effective width) with circumferential stiffeners

R (m) 1 2 t (mm) 9 12 12 18

0I (cm4) 0.95 2.63 3.65 15.26

F100： I (cm4) 239 346 374 6410/ II 251 131 102 42

F150： I (cm4) 690 989 0/ II 724 375

F200： I (cm4) 1466 2091 2311 3828

0/ II 1536 794 633 251T250： I (cm4) 11941 20438

0/ II 3270 1339

NOTES: 0I : Moment of inertia for shell walls

(Calculation uses effective width with stiffeners) I : Moment of inertia for shell walls (effective width) in addition to circumferential stiffeners

Table 4 a) Results of numerical analyses and natural frequencies by proposed formulae

(L = 6m, without stiffener)

Span Radius Wall Tensile stress Internal R/t L/R Natural frequencies (Hz) k ：axial mode number，n ：circumferential mode numberlength thickness due to pressure k = 1, n = 2 k = 1, n = 3L (m) R (m) t (mm) internal press. P (Mpa) FEM (fe ) Eq.(9) (fs ) fs / fe Eq.(2) FEM (fe ) Eq.(10) (fs ) fs / fe Eq.(2)

6 0.50 6 0.3σa 0.476 83 12 22.1 21.4 0.97 19.9 36.8 36.3 0.99 37.49 0.714 56 12 27.5 27.1 0.99 25.0 51.3 50.8 0.99 51.8

12 0.953 42 12 32.8 32.7 1.00 30.2 67.5 67.6 1.00 67.70.75 6 0.3σa 0.318 125 8 21.0 20.2 0.96 16.0 21.1 20.4 0.97 20.1

9 0.476 83 8 25.1 24.8 0.99 19.3 27.2 26.5 0.98 26.012 0.635 63 8 28.5 28.9 1.01 22.1 33.6 33.0 0.98 32.318 0.953 42 8 34.3 36.1 1.05 27.1 47.7 47.6 1.00 46.2

1.00 6 0.3σa 0.238 167 6 21.8 22.0 1.01 16.3 16.9 16.1 0.95 14.69 0.357 111 6 26.1 26.9 1.03 19.5 20.8 20.2 0.97 18.1

12 0.476 83 6 29.5 31.2 1.06 22.1 24.5 24.0 0.98 21.618 0.714 56 6 34.9 38.3 1.10 26.2 32.0 31.9 1.00 28.721 0.833 48 6 37.1 41.5 1.12 28.0 35.9 36.1 1.00 32.6

6 0.50 6 0.6σa 0.953 83 12 25.8 25.7 0.99 25.1 46.3 47.4 1.02 47.79 1.429 56 12 31.6 32.2 1.02 30.8 60.7 63.0 1.04 62.1

12 1.905 42 12 37.0 38.4 1.04 36.1 76.3 80.1 1.05 77.50.75 6 0.6σa 0.635 125 8 22.3 21.6 0.97 18.2 26.4 26.3 0.99 26.2

9 0.953 83 8 26.6 26.6 1.00 21.9 33.0 33.4 1.01 32.712 1.270 63 8 30.2 30.9 1.02 24.9 39.5 40.5 1.02 39.118 1.905 42 8 36.2 38.5 1.06 30.2 53.3 55.6 1.04 52.6

1.00 6 0.6σa 0.476 167 6 22.4 22.6 1.01 17.2 20.0 19.3 0.97 18.49 0.714 111 6 26.8 27.7 1.03 20.6 24.3 24.1 0.99 22.5

12 0.953 83 6 30.3 32.0 1.06 23.4 28.3 28.4 1.00 26.218 1.429 56 6 35.8 39.3 1.10 27.7 35.9 36.9 1.03 33.521 1.667 48 6 38.0 42.5 1.12 29.6 39.8 41.3 1.04 37.2

171s

Table 4 b) Results of numerical analyses and natural frequencies by proposed formulae (L = 12m, without stiffeners )

Span Radius Wall Tensile stress Internal R/t L/R Natural frequencies (Hz) k ：axial mode number，n ：circumferential mode numberlength thickness due to pressure k = 1, n = 2 k = 1, n = 3L (m) R (m) t (mm) internal press. P (Mpa) FEM (fe ) Eq.(9) (fs ) fs / fe Eq.(2) FEM (fe ) Eq.(10) (fs ) fs / fe Eq.(2)

12 0.50 6 0.3σa 0.476 83 24 15.6 15.6 1.00 17.0 35.7 35.4 0.99 37.09 0.714 56 24 20.4 20.3 1.00 21.9 50.2 49.9 0.99 51.4

12 0.953 42 24 25.5 25.3 0.99 27.0 66.3 66.7 1.00 67.40.75 6 0.3σa 0.318 125 16 9.5 9.2 0.97 9.6 18.2 18.0 0.99 18.9

9 0.476 83 16 11.8 11.6 0.98 11.9 24.0 23.8 0.99 24.712 0.635 63 16 14.0 13.8 0.99 14.1 30.3 30.0 0.99 31.018 0.953 42 16 18.5 18.3 0.99 18.6 44.5 44.6 1.00 45.0

1.00 6 0.3σa 0.238 167 12 7.9 7.4 0.94 7.1 11.8 11.6 0.98 12.19 0.357 111 12 9.6 9.1 0.95 8.6 15.1 14.8 0.98 15.4

12 0.476 83 12 11.1 10.7 0.97 10.0 18.4 18.1 0.99 18.718 0.714 56 12 13.8 13.6 0.99 12.5 25.7 25.4 0.99 25.921 0.833 48 12 15.1 14.9 0.99 13.8 29.6 29.4 0.99 29.8

1.50 9 0.3σa 0.238 167 8 9.2 8.7 0.95 7.0 9.0 8.6 0.96 8.612 0.318 125 8 10.5 10.1 0.96 8.0 10.6 10.2 0.97 10.118 0.476 83 8 12.6 12.4 0.99 9.7 13.6 13.3 0.98 13.021 0.556 71 8 13.4 13.5 1.01 10.4 15.2 14.8 0.98 14.6

2.00 12 0.3σa 0.238 167 6 10.9 11.0 1.01 8.1 8.5 8.0 0.95 7.318 0.357 111 6 13.0 13.5 1.04 9.7 10.4 10.1 0.97 9.121 0.417 95 6 13.9 14.6 1.05 10.4 11.3 11.0 0.97 9.9

12 0.50 6 0.6σa 0.953 83 24 20.5 21.1 1.03 22.9 45.3 46.7 1.03 47.49 1.429 56 24 25.6 26.8 1.05 28.3 59.7 62.3 1.04 61.8

12 1.905 42 24 30.8 32.4 1.05 33.5 75.2 79.4 1.05 77.20.75 6 0.6σa 0.635 125 16 12.1 12.1 1.00 12.9 24.1 24.5 1.01 25.3

9 0.953 83 16 14.7 15.0 1.02 15.7 30.4 31.3 1.03 31.712 1.270 63 16 17.1 17.6 1.03 18.1 36.7 38.1 1.04 38.018 1.905 42 16 21.8 22.7 1.04 22.8 50.4 53.0 1.05 51.5

1.00 6 0.6σa 0.476 167 12 9.3 9.0 0.96 9.1 15.9 15.8 1.00 16.59 0.714 111 12 11.3 11.0 0.98 11.0 19.6 19.9 1.01 20.3

12 0.953 83 12 12.9 12.9 1.00 12.6 23.1 23.7 1.02 23.918 1.429 56 12 15.8 16.1 1.02 15.4 30.4 31.5 1.04 31.121 1.667 48 12 17.2 17.7 1.03 16.7 34.2 35.7 1.04 34.8

1.50 9 0.6σa 0.476 167 8 9.8 9.3 0.95 8.0 11.4 11.2 0.98 11.412 0.635 125 8 11.1 10.8 0.97 9.1 13.2 13.1 0.99 13.118 0.953 83 8 13.3 13.3 1.00 10.9 16.5 16.7 1.01 16.421 1.111 71 8 14.3 14.4 1.01 11.7 18.1 18.5 1.02 17.9

2.00 12 0.6σa 0.476 167 6 11.2 11.3 1.01 8.6 10.0 9.7 0.97 9.218 0.714 111 6 13.4 13.8 1.03 10.3 12.2 12.0 0.99 11.221 0.833 95 6 14.3 14.9 1.04 11.0 13.2 13.1 1.00 12.2

Table 4 c) Results of numerical analyses and natural frequencies by proposed formulae (L = 18m, without stiffeners )

Span Radius Wall Tensile stress Internal R/t L/R Natural frequencies (Hz) k ：axial mode number，n ：circumferential mode numberlength thickness due to pressure k = 1, n = 2 k = 1, n = 3L (m) R (m) t (mm) internal press. P (Mpa) FEM (fe ) Eq.(9) (fs ) fs / fe Eq.(2) FEM (fe ) Eq.(10) (fs ) fs / fe Eq.(2)

18 0.75 6 0.3σa 0.318 125 24 8.3 8.2 0.99 9.1 18.0 17.8 0.99 18.89 0.476 83 24 10.4 10.4 1.00 11.3 23.8 23.6 0.99 24.6

12 0.635 63 24 12.5 12.5 1.00 13.5 30.1 29.9 0.99 30.918 0.953 42 24 17.0 16.9 0.99 18.0 44.2 44.4 1.00 44.9

1.00 6 0.3σa 0.238 167 18 5.8 5.6 0.97 6.1 11.4 11.3 0.99 12.09 0.357 111 18 7.1 7.0 0.98 7.5 14.6 14.5 0.99 15.2

12 0.476 83 18 8.4 8.3 0.99 8.7 17.9 17.8 0.99 18.518 0.714 56 18 10.8 10.7 0.99 11.2 25.2 25.0 0.99 25.721 0.833 48 18 12.1 12.0 0.99 12.4 29.1 29.0 1.00 29.6

1.50 9 0.3σa 0.238 167 12 5.3 4.9 0.93 4.7 7.9 7.7 0.98 8.112 0.318 125 12 6.0 5.7 0.95 5.4 9.3 9.2 0.98 9.518 0.476 83 12 7.4 7.1 0.96 6.6 12.3 12.1 0.99 12.521 0.556 71 12 8.0 7.8 0.98 7.2 13.8 13.6 0.99 14.0

2.00 12 0.3σa 0.238 167 9 5.8 5.4 0.94 4.6 6.4 6.2 0.96 6.318 0.357 111 9 6.9 6.6 0.96 5.5 8.1 7.9 0.98 7.921 0.417 95 9 7.4 7.2 0.97 5.9 8.9 8.7 0.98 8.7

2.50 18 0.3σa 0.286 139 7 7.1 6.9 0.97 5.3 6.4 6.1 0.96 5.921 0.333 119 7 7.6 7.4 0.98 5.7 6.9 6.7 0.96 6.4

18 0.75 6 0.6σa 0.635 125 24 11.1 11.3 1.02 12.5 24.0 24.4 1.02 25.29 0.953 83 24 13.6 14.1 1.03 15.3 30.2 31.1 1.03 31.6

12 1.270 63 24 15.9 16.6 1.04 17.7 36.5 38.0 1.04 37.918 1.905 42 24 20.5 21.6 1.05 22.3 50.2 52.9 1.05 51.5

1.00 6 0.6σa 0.476 167 18 7.6 7.6 1.00 8.3 15.6 15.6 1.00 16.49 0.714 111 18 9.3 9.3 1.00 10.1 19.2 19.6 1.02 20.2

12 0.953 83 18 10.7 10.9 1.02 11.6 22.7 23.4 1.03 23.718 1.429 56 18 13.3 13.8 1.04 14.3 29.9 31.2 1.04 30.921 1.667 48 18 14.6 15.2 1.04 15.6 33.8 35.3 1.05 34.7

1.50 9 0.6σa 0.476 167 12 6.2 6.0 0.96 6.0 10.6 10.5 1.00 11.012 0.635 125 12 7.1 6.9 0.97 6.9 12.3 12.4 1.01 12.718 0.953 83 12 8.6 8.6 1.00 8.4 15.4 15.8 1.02 15.921 1.111 71 12 9.3 9.3 1.00 9.0 17.0 17.5 1.03 17.5

2.00 12 0.6σa 0.476 167 9 6.3 6.0 0.96 5.4 8.3 8.2 0.99 8.418 0.714 111 9 7.6 7.3 0.97 6.5 10.2 10.3 1.00 10.321 0.833 95 9 8.1 7.9 0.98 6.9 11.1 11.2 1.01 11.2

2.50 18 0.6σa 0.572 139 7 7.4 7.2 0.97 5.9 7.8 7.7 0.98 7.621 0.667 119 7 7.9 7.8 0.98 6.3 8.5 8.4 0.99 8.2

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Table 4 d) Results of numerical analyses and natural frequencies by proposed formulae (L = 24m, without stiffeners )

Span Radius Wall Tensile stress Internal R/t L/R Natural frequencies (Hz) k ：axial mode number，n ：circumferential mode numberlength thickness due to pressure k = 1, n = 2 k = 1, n = 3L (m) R (m) t (mm) internal press. P (Mpa) FEM (fe ) Eq.(9) (fs ) fs / fe Eq.(2) FEM (fe ) Eq.(10) (fs ) fs / fe Eq.(2)

24 1.00 6 0.3σa 0.238 167 24 5.3 5.3 0.99 5.9 11.4 11.2 0.99 11.99 0.357 111 24 6.6 6.6 1.00 7.3 14.6 14.4 0.99 15.2

12 0.476 83 24 7.8 7.8 1.00 8.5 17.8 17.7 0.99 18.518 0.714 56 24 10.2 10.2 1.00 10.9 25.1 25.0 1.00 25.721 0.833 48 24 11.3 11.4 1.01 12.2 28.8 29.0 1.01 29.6

1.50 9 0.3σa 0.238 167 16 4.1 4.0 0.97 4.2 7.7 7.6 0.99 8.012 0.318 125 16 4.8 4.6 0.97 4.8 9.1 9.0 0.99 9.418 0.476 83 16 5.9 5.8 0.98 6.0 12.0 11.9 0.99 12.421 0.556 71 16 6.4 6.3 0.98 6.5 13.6 13.4 0.99 13.9

2.00 12 0.3σa 0.238 167 12 4.0 3.7 0.93 3.5 5.9 5.8 0.98 6.118 0.357 111 12 4.8 4.6 0.96 4.3 7.5 7.4 0.98 7.721 0.417 95 12 5.2 5.0 0.97 4.7 8.4 8.2 0.99 8.5

2.50 18 0.3σa 0.286 139 10 4.6 4.3 0.94 3.7 5.5 5.4 0.97 5.521 0.333 139 10 4.9 4.7 0.96 4.0 6.1 5.9 0.98 6.0

24 1.00 6 0.6σa 0.476 167 24 7.3 7.3 1.00 8.2 15.5 15.6 1.00 16.49 0.714 111 24 8.8 9.0 1.02 10.0 19.2 19.6 1.02 20.2

12 0.953 83 24 10.2 10.6 1.04 11.4 22.7 23.3 1.03 23.718 1.429 56 24 12.8 13.4 1.05 14.1 29.9 31.1 1.04 30.921 1.667 48 24 14.1 14.8 1.05 15.4 33.7 35.3 1.05 34.7

1.50 9 0.6σa 0.476 167 16 5.3 5.2 0.99 5.6 10.4 10.4 1.00 10.912 0.635 125 16 6.0 6.0 1.00 6.5 12.1 12.2 1.01 12.618 0.953 83 16 7.4 7.5 1.02 7.8 15.2 15.6 1.03 15.821 1.111 71 16 8.0 8.2 1.03 8.5 16.8 17.3 1.03 17.4

2.00 12 0.6σa 0.476 167 12 4.7 4.5 0.96 4.5 7.9 7.9 1.00 8.218 0.714 111 12 5.6 5.5 0.98 5.5 9.8 9.9 1.01 10.221 0.833 95 12 6.1 6.0 0.99 5.9 10.7 10.9 1.02 11.0

2.50 18 0.6σa 0.572 139 10 5.1 4.8 0.95 4.4 7.2 7.2 1.00 7.321 0.667 139 10 5.4 5.2 0.96 4.8 7.8 7.8 1.00 7.9

Table 5 Results of numerical analyses (With stiffeners equally spaced)

Span Radius Wall Tensile stress Internal R/t L/R Shapes of I/Io Natural frequencies (Hz) k ：axial mode number，n ：circumferential mode numberlength thickness due to pressure stiffeners k = 1, n = 2 k = 1, n = 3

L R t internal press. P Number of stiffeners per span Number of stiffners per span(m) (m) (mm) (Mpa) 0 1 2 3 0 1 2 36 1.00 12 0.3σa 0.476 83 6 F100 131 29.5 30.4 30.8 24.5 32.3 35.8

F200 794 29.5 33.6 35.4 24.5 42.6 53.70.6σa 0.953 F100 131 30.3 31.1 31.5 24.5 35.3 38.5

F200 794 30.3 34.3 36.0 24.5 44.9 55.512 1.00 9 0.3σa 0.357 111 12 F100 251 9.6 10.7 11.1 11.6 15.1 18.1 21.5 23.7

F150 724 9.6 12.1 13.1 14.1 15.1 18.5 25.8 30.2F200 1536 9.6 13.6 15.4 17.0 15.1 18.6 26.9 35.0

12 0.3σa 0.476 83 12 F100 131 11.1 12.3 12.8 13.3 18.4 21.6 25.5 28.0F150 375 11.1 13.9 15.1 16.3 18.4 22.1 30.3 35.4F200 794 11.1 15.8 17.8 19.6 18.4 22.2 31.2 40.9

12 0.6σa 0.953 83 12 F100 131 12.9 14.0 14.4 14.9 23.1 25.8 29.1 31.3F150 375 12.9 15.4 16.5 17.6 23.1 26.2 33.4 38.0F200 794 12.9 17.1 19.0 20.7 23.1 26.3 34.2 43.3

2.00 12 0.3σa 0.238 167 6 F100 102 10.9 11.0 11.0 8.5 9.1 9.4F200 633 10.9 11.2 11.4 8.5 11.3 12.5T250 3270 10.9 12.0 8.5 14.2

18 0.3σa 0.357 111 6 F100 42 13.0 13.1 13.1 10.4 11.2 11.6F200 251 13.0 13.4 13.6 10.4 13.9 15.4T250 1339 13.0 14.5 10.4 17.6

18 0.6σa 0.714 111 6 F100 42 13.4 13.5 13.5 12.2 12.9 13.2F200 251 13.4 13.8 13.9 12.2 15.3 16.7T250 1339 13.4 14.8 12.2 18.7

18 1.00 9 0.3σa 0.357 111 18 F100 251 7.1 8.1 8.5 8.9 14.6 15.5 18.0 20.5F150 724 7.1 9.1 10.2 11.1 14.6 15.6 18.1 23.1F200 1536 7.1 10.1 12.0 13.4 14.6 15.6 18.2 23.3

12 0.3σa 0.476 83 18 F100 131 8.4 9.4 9.9 10.4 17.9 18.9 21.5 24.3F150 375 8.4 10.6 11.9 13.2 17.9 18.9 21.7 27.1F200 794 8.4 11.7 14.0 15.6 17.9 18.9 21.7 27.2

12 0.6σa 0.953 83 18 F100 131 10.7 11.5 11.9 12.3 22.7 23.6 25.7 28.1F150 375 10.7 12.5 13.6 14.5 22.7 23.6 25.9 30.5F200 794 10.7 13.4 15.5 17.0 22.7 23.6 25.9 30.7

2.00 12 0.3σa 0.238 167 9 F100 102 5.8 5.8 5.9 6.4 7.0 7.2F200 633 5.8 6.2 6.3 6.4 8.5 9.8T250 3270 5.8 7.1 7.6 6.4 9.7 13.2

18 0.3σa 0.357 111 9 F100 42 6.9 7.0 7.1 8.1 8.8 9.1F200 251 6.9 7.4 7.6 8.1 10.6 12.2T250 1339 6.9 8.6 9.3 8.1 11.9 16.6

18 0.6σa 0.714 111 9 F100 42 7.6 7.6 7.7 10.2 10.8 11.0F200 251 7.6 8.0 8.2 10.2 12.4 13.7T250 1339 7.6 9.1 9.8 10.2 13.5 17.8

24 2.00 12 0.3σa 0.238 167 12 F100 102 4.0 4.0 4.1 4.1 5.9 6.4 6.6 6.8F200 633 4.0 4.4 4.6 4.7 5.9 7.2 8.6 9.5T250 3270 4.0 5.3 5.8 6.3 5.9 7.5 10.9 13.1

18 0.3σa 0.357 111 12 F100 42 4.8 4.9 4.9 5.0 7.5 8.1 8.4 8.6F200 251 4.8 5.3 5.6 5.8 7.5 9.0 10.8 11.9T250 1339 4.8 6.5 7.2 7.9 7.5 9.3 13.3 16.5

18 0.6σa 0.714 111 12 F100 42 5.6 5.7 5.8 5.8 9.8 10.2 10.5 10.7F200 251 5.6 6.1 6.3 6.5 9.8 11.0 12.5 13.4T250 1339 5.6 7.2 7.8 8.4 9.8 11.2 14.8 17.6

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4 . P R O P O S A L O F S I M P L I F I E D FORMULAE FOR CALCULATING THE NATURAL FREQUENCY OF PENSTOCKS Formula (2) is derived from the exact equation of a cylindrical shell. This formula eventually assumes a form similar to that introduced from Donnell’s equation. This demonstrates that vibration frequency characteristics of a cylindrical shell can be well expressed in a simple form. The issue of accuracy is, however, not discussed at this point. Equation (7) is obtained only after adjusting the constants in formula (2). Any proposal for an approximate expression for calculating vibration frequency must be based on a form of this sort rather than on a high-order polynomial expression.

25

3

4 RP

Ec

Rtb

LtRaEf ++=

ρ (7)

where a, b, c are new constants and other symbols are identical to those for formula (2). Constants a, b and c were then adjusted according to the following procedure, and an approximation could be derived for the results of analysis of natural frequencies shown in Table 4. For the approximation, error Err should be determined by the following equation and a search made for ∂Err/∂a, ∂Err/∂b and ∂Err/∂c.

(8) 2)(∑ −= apperr ffE

where : fe: Natural frequency shown in Table 4 obtained by FEM f app : Vibration frequency calculated by

approximate formula (7) (Initially, temporary figures should be used for constants a, b and c.) Following this, various values should be assumed as the initial value of a, b and c, and the minimal value of Err should be found by the gradient descent method by trial and error to determine constants a, b and c which show the best approximation. The following formulae are thus obtained for calculation of f2 and f3 which are approximate vibration frequencies for n = 2, 3 obtained from a finite element analysis. For n = 2 and k = 1:

25

3

42 837.0142.077.3ER

PRt

LtREf ++=

ρ (9)

For n = 3 and k = 1:

25

3

43 82.378.109.1ER

PRt

LtREf ++=

ρ (10)

where : E: Modulus of elasticity of steel (206kN/mm2 → 2.06×108kg/mm･sec2) ρ : Density of steel (7.85×10-6kg/mm3) t : Wall thickness (mm) : Radius to center of the wall (mm) R : Fixing span length (mm) L P : Internal pressure (MPa)

ρE

(mm/sec) 7105120 ×= .

Results of the application of formulae (9) and (10) are shown in Table 4 in the column of simplified formulae (fs). In general, the application range for these formulae is as follows: (i) Radius of penstock: 0.50m ≦ R ≦ 2.50m (ii) Wall thickness: 6mm ≦ t ≦ 21mm (iii) Span length: 6m ≦ L ≦ 24m (iv) Pressure (hydraulic): 0.2MPa < P < 2MPa (v) Axial modal number: For k = 1 only (vi) Circumferential modal number: For n = 2: Generally less than 20Hz For n = 3: Generally less than 30Hz For other modal numbers: Not applicable It should be noted here that as evidenced from the boundary conditions of the finite element analysis, no circumferential stiffener is provided, both ends are assumed to be supported by ring girders, and accuracy will decrease with concrete saddle support. Since the density ratio of steel to water is considered as the constant, it should be remembered that these are not applicable when the density ratio differs significantly as in cases involving muddy water or pipe shell constructed of material other than steel. 5. INVESTIGATION OF THE EFFECTS OF CIRCUMFERENTIAL STIFFENERS ON NATURAL FREQUENCY There are considerable complications involved in any theoretical discussion on the effects that circumferential stiffeners have on natural frequency, and even the current Technical Standards for Gates and Penstocks fai ls to provide any clear understanding of the matter. In most cases the natural frequency calculation formula is applied at the design stage with the assumption that the distance between the ring girder and circumferential stiffener or between the circumferential stiffeners is equal to the fixing span length. However, a quantitative study on the effects of stiffeners is required to avoid evaluations that poorly reflect

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actual conditions. Table 6, which is based on Tables 4 and Table 5, provides a comparison of natural frequencies for three different cases; where the span length is short and no stiffener is provided, the span length is longer by a multiple of an integer number and stiffeners are provided, and where the length of the pipe shell without circumferential stiffeners equals that for the case without stiffeners. (1) In cases where n = 2 Discussion is based on the case where n = 2 and k = 1 as shown in Table 6. If the rigidity of the stiffener is infinite, the natural frequency should agree with the case where the span length is short and no stiffener is provided. However, it is revealed that the level of rigidity of the stiffeners during normal use contributes to a minor increase in the natural frequency. Therefore, the natural frequency calculation formula (9) should not be applied using the stiffener interval as the span length of the penstock while neglecting the difference in rigidity of the circumferential stiffeners. The values in Table 5 were then reconsidered from the perspective of equivalent wall thickness. It was assumed that there is general agreement in vibration frequency values for cases with stiffeners and for cases with a thicker wall but without stiffeners. The wall thickness was then calculated as equivalent wall thickness teq. Fig.18 shows a distribution of equivalent wall thickness ratio λ (ratio of teq to original wall thickness) for L/R = 12. The horizontal axis represents the common logarithm log (I/I0) for the ratio of the moment of inertia (I) for a pipe shell including a stiffener that is within the effective width7) to the moment of inertia (I0) of the pipe shell alone. The vertical axis represents λ. Distinctive features of this investigation are: (i) The effects of stiffeners on the natural frequency are very dependent on L/R. (ii) The effects of stiffeners can be negligible in the range of I/I0 < 100. Based on the results of this investigation, the equivalent wall thickness teq for evaluation of the vibration frequency in the case of one stiffener and n = 2 should practically be expressed as follows:

eqt tλ= ⋅ (11)

where: in the case where ( )( )2log 0 ≥II :

( ){ }01 log 2I Iλ ε= + − (12)

in the case where ( )( )2log 0 <II : 1λ = (13)

Table 6 Changes in natural frequency due to circumferential stiffeners n = 2, k = 1 n = 3, k = 1L=6m, R=1.00m, t=12mm,

aσσ 3.0= , without stiffener 29.5 Hz 24.5 Hz

L=12m, with a stiffener （F100, F150 and F200 are shown in Fig.17）

F100 : 12.3 F150 : 13.9 F200 : 15.8

21.6 22.1 22.2

L=18m, with two stiffeners F100 : 9.9 F150 : 11.9 F200 : 14.0

21.5 21.7 21.7

Table 7 Value of ε

RL / 6 9 12 18 ε 0.20 0.48 1.03 1.30

0 1 2 3 4

0

1

2

3

λ

log(I/Io)

FEM results with one stiffener FEM results with two stiffeners λ=1+1.03{(log(I/Io))-2} λ=1

Fig.18 Ratio of equivalent wall thickness for L/R = 12 Similar investigations were carried out for shapes other than L/R = 12, and a value of ε was obtained by the least square method. Results thus obtained are shown in Table 7. The relation between ε and L/R presents an S-shape curve indicating that presentation of a simplified equation is difficult. Therefore, it is determined that linear interpolation be made for the case where an exact value for L/R shown in Table 7 is not available. In rare cases, L/R over 18 is seen. Since it is considered that this is close to the upper limit for penstock shape, adoption of ε = 1.30 is considered appropriate even if L/R > 18. When two stiffeners are provided, (I/I0) should be doubled for evaluation. In Fig.18, data identified by ○ correspond to the example analysis for the case of two stiffeners. The results show no marked difference from the results for the case where one stiffener is applied. Fig. 11 shows the use of one or two stiffeners per span in most cases. The use of three or more stiffeners is extremely rare and it is assumed that the additional stiffeners were added only when a

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vibration problem occurred. Three or more stiffeners are practically never planned at the design stage. Therefore, no practical problems should be anticipated as long as the formulae provide for the evaluation of equivalent wall thickness dealing with up to two circumferential stiffeners. (2) In cases where n = 3 The values in Table 6 demonstrate that the effects of stiffeners are more remarkable for cases where n = 3 than for cases where n = 2. When ordinary circumferential stiffeners are used and intervals without circumferential stiffeners are the same, the natural frequencies show near parity regardless of the stiffness indicating that the stiffeners concerned are functioning satisfactorily. Further, no vibration is detected for cases where n = 3 in the survey on past vibration problems due to resonance or in the onsite survey described in 2. It is therefore judged that there is no reasonable justification for the use of complicated design formulae. For cases where n = 3, stiffener spacing should be replaced by fixing span length, and formula (10) should be applied from a design evaluation perspective. For cases where n ≧ 4, it is appropriate to use the conventional formula (2) specified in the current Technical Standards for Gates and Penstocks. 6. FEASIBILITY OF THE PROPOSED V I B R A T I O N F R E Q U E N C Y CALCULATION FORMULAE Fig.19 shows the relation between the vibration frequency obtained from the proposed formula (9) indicated on the horizontal axis and hydraulic turbine speed of rotation (Hz) indicated on the vertical axis. Vibration occurrence values (eight in four sites) were calculated from structural conditions at the point of vibration occurrence. Y=X denotes a state where there is agreement between the vibration frequency of the penstock (Y) and the hydraulic turbine speed (X). Y=X/3.6 denotes a condition where there is agreement between the vibration frequency of the penstock (Y) and the vibration frequency due to water vortex in the draft tube at partial load of the generator. Vibration occurrences in the four sites are distributed around Y=X, which indicates that the possibility of resonance is significant if a marked hydraulic pressure variation (pulsation) exists at the site. On the other hand, many points free from vibrations are noticed in the vicinity of Y=X (or Y=X/3.6). This suggests that as long as there is no

2 4 6 8 10 12 14 16 18

0

10

20

30

40

50

Y=X/3.6

Y=X

Cal

cula

ted

natu

ral f

requ

ency

of

pen

stoc

k (H

z)

Hydraulic turbine speed of rotation (Hz)

without stiffener with stiffeners vibration occurrence

Fig.19 Comparison of hydraulic turbine speed of rotation with calculated natural frequency of penstock

0 2 4 6 8 10 12

0

2

4

6

8

10

12

Y=1.4X Y=1.2XY=X

Fig.20 Changes in natural frequency caused by additional circumferential stiffeners remarkable hydraulic pressure variation, no problem will be caused even if the turbine speed of rotation is close to the penstock vibration frequency. All vibration occurrences shown here are based on penstocks with ring girder supports. Reinforcement measures were taken by mounting additional stiffeners at points where vibration occurred. A description of the degree of reinforcement effectiveness follows. In Fig.20, the horizontal axis represents the calculated natural frequency at the point of vibration occurrence and the vertical axis represents the calculated natural frequency after stiffeners are mounted. Increases of at least 20% were attained after mounting stiffeners. This fact should be useful in preparing future guidance for prevention of resonance. Fig.21 is based on the distribution of shape parameter for penstocks shown in Fig.15. The data for 44 cases (30 without stiffeners, 14 with stiffeners, as shown in Figure 19) is re-plotted here with an asterisk *. In each of these cases the

Cal

cula

ted

natu

ral f

requ

ency

afte

r stif

fene

rs m

ount

ed (H

z)

Calculated natural frequencyat vibration occurrence (Hz)

vibration occurrence

176s

calculated natural frequency for the penstock is within 90－110% of the hydraulic turbine speed. This data suggests that excessive hydraulic pressure variation is likely to incur resonance in the penstock. Data denoted by ▲ , indicating a vibration occurrence, are generally located at the upper part of * data distribution. Table 8 shows a comparison of vibration frequencies at four sites calculated by the proposed simplified formulae and the formula specified in the current Technical Standards for Gates and Penstocks (evaluation is made based on structural conditions at vibration occurrence). The hydraulic turbine speed is shown for reference. For calculation of frequencies, circumferential mode number n = 2 and axial direction mode number k = 1 are used in all cases. Further, the current Technical Standards for Gates and Penstocks does not specify the handling of stiffeners, and instead it calculates the frequency at site-1 supposing that no stiffeners are provided. Although the proposed calculation equation is specifically set for the case where n = 2 and n = 3, it agrees well with actual conditions. From a design viewpoint, it is important to involve the resonance region in the evaluation of frequencies. This range should be determined considering accuracy of the vibration frequency calculation formulae and damping characteristics. Judging from the results of the current investigation, the definition of the resonance region as ±20% of the specified vibration as reported in the ASCE design guideline for penstocks16), appears to be slightly low. 7. CONCLUSION Results of the current investigation are summarized as follows: (1) For penstocks newly installed or renewed over

the past 30 years, shape, size, shape parameters, operating conditions and occurrences of vibration were surveyed, and their actual conditions were clarified. It was revealed that vibration occurrences are concentrated in a specific region of the shape parameters and that careful attention should be given to the circumferential mode number n = 2.

(2) The natural frequency calculation formulae play a vital role in evaluating vibration characteristics of the penstock at the design stage. In this paper, efforts were made to determine simplified formulae that can best approximate analysis results for numerous parameters by the finite element method.

0 5 10 15 20 25 30 35

40

60

80

100

120

140

160

180

Span length / radius ratio (L/R)

without stiffener with stiffeners vibration occurrence Calculated natural frequency is within

90-110% of the hydraulic turbine speed

Fig. 21 Penstock shape parameters as possible cause of resonance under conditions of excessive hydraulic pressure variation

Table 8 Comparative analysis of frequencies for vibration occurrence (n = 2, k = 1)

Natural frequency calculation formulae were then proposed for cases without circumferential stiffeners. In applying the proposed formulae, it was shown that it is possible to estimate the frequency with good accuracy of less than 1Hz for frequencies under 20Hz for the circumferential mode number of n = 2 and under 30Hz for n = 3.

(3) The system is complicated by the fact that more parameters governing vibration phenomena are involved for cases with circumferential stiffeners more than for those without stiffeners. While ordinary stiffeners develop expected effects for cases of n = 3, it is proposed that for design criteria, stiffeners be replaced by an equivalent wall thickness for cases of n = 2.

(4) It is revealed that appropriate results can be achieved by applying the proposed natural frequency calculation formulae to actual vibration occurrences derived from surveys of actual onsite conditions.

ACKNOWLEDGEMENT: To solve the vibration problem of penstocks, the Japan Hydraulic Gate

Rad

ius /

wal

l thi

ckne

ss ra

tio (R

/t)Hydraulic Eq.(9) Eq.(2)

turbine (Hz) (Hz)speed of

Site rotation initial measuresNo. (Hz)1 6.3 6.2 5.2* 1 +22 6.7 6.7 7.1 0 +13 5.0 5.6 5.8 0 +104 4.3 4.3 4.4 0 +7

Number of stiffeners

177s

and Penstock Association established a committee (Chairman: Hideharu Nakamura; 2000-2002) and proceeded with the study, with the cooperation of the electric power companies. This paper is the result of the first half of the research program. The authors gratefully acknowledge the support of the members of the committee and the staff of the electric power companies.

8）Shiraki, K., Honma, T. and Nagata, O.: The penstock vibration characteristics - On the vibrations of simply supported and ring-stiffened cylindrical shells filled with pressurized water, Proc. of ASME Vibration Conference, 75-DET-114, 1975.

9）Shiraki, K., Honma, T., Takayama, H. and Kito, F.: Dynamic characteristics of penstocks, Hydraulic Gate and Penstock, No.126, 1981. (in Japanese)

10)Edited by Hayashi, T.: Mechanics for Light Weight Structures and Its Application, Vol.1, Nikka-Giren Publishing, pp.433-436, 1967. (in Japanese)

REFERENCES

11) Kobayashi, S.: Structural Dynamics, Maruzen, pp.208-221, 1994. (in Japanese)

1) Kito, F.: On deformation of a cylindrical shell which is suddenly subjected to a uniform external pressure, Transactions of JSME, Vol.18, No.66, 1952. (in Japanese) 12) Zienkiewicz, O.C.: The Finite Element Method (Third

Edition), McGraw Hill, pp.388-394, 1977. 2) Kito, F.: On the vibration of a penstock, Transactions of JSME, Vol.18, No.69, 1952. (in Japanese) 13) Nakamura, H.: Elastic-plastic dynamic buckling analysis of

a steel circular pier, Journal of Structural Mechanics and Earthquake Engineering, No.549/I-37, 1996. (in Japanese)

3）Kito, F.: Report on investigation of vibration of penstocks in hydro-electric power stations, Journal of JSME, Vol.55, No.403, 1952. (in Japanese) 14) Tong, P.: The finite element method for fluid flow,

Japan-U.S. Seminar on Matrix Methods of Structural Analysis and Design, 1969.

4）Kito, F.: On the vibration of a penstock - II, Transactions of JSME, Vol.19, No.79, 1953. (in Japanese)

5) Kito, F.: On vibration of a thin cylindrical shell, which is immersed in water and subjected to a combined stress, Transactions of JSME, Vol.22, No.115, 1956. (in Japanese)

15) Zienkiewicz, O.C. and Newton, R.E.: Coupled vibrations of a structure submerged in a compressible fluid, Proc. of the Symposium on Finite Element Techniques held at the University of Stuttgart, 1969. 6 ） Kito, F.: Some remarks on the penstock vibrations,

Hydraulic Gate and Penstock, No.20, 1961. (in Japanese) 16) ASCE: Steel Penstocks, ASCE Manuals and Reports on Engineering Practice, No.79, 1993. 7) Hydraulic Gate and Penstock Association: Technical

Standards for Gates and Penstocks, Third Edition, 1973, Fourth Edition, 1981, Fifth Edition, 1993. (in Japanese)

（Received December 9, 2002）

178s