2015

Design of Penstock Pipe for a Hydro-electric Pumped Storage Station

MACE 61057: STRUCTURAL INTEGRITY (DR. ANDREY ZIVKOV)

AYMAN SIDDIQUE (ID: 7669311)

1 | P a g e

TABLE OF CONTENTS

TABLE OF FIGURES ........................................................................................................................................... 1

LIST OF TABLES ................................................................................................................................................ 1

1. INTRODUCTION ....................................................................................................................................... 2

2. STATIC DESIGN/ OVERLOAD ASSESSMENT ............................................................................................... 3

3. FRACTURE ASSESSMENT .............................................................................................................................. 4

3.1 Determining KIC for Steel A and Steel B .................................................................................................. 4

3.2 Critical Crack Sizes in Steel A and Steel B................................................................................................ 5

3.3 Failure Assessment Diagram (FAD) Analysis ........................................................................................... 5

3.4 Sensitivity Analysis ................................................................................................................................. 7

4. FATIGUE ASSESSMENT ............................................................................................................................... 10

5. DESIGN IMPROVEMENTS ........................................................................................................................... 11

6. CRITICAL DISCUSSION AND FINAL RECOMMENDATIONS ............................................................................ 12

7. REFERENCES............................................................................................................................................... 14

8. APPENDIX ................................................................................................................................................... 15

TABLE OF FIGURES Figure 1: Schematic diagram of a hydro-electric power station [3] .......................................................... 2

Figure 2: Fracture toughness data for Steel A .......................................................................................... 4

Figure 3: FAD for Steel A .......................................................................................................................... 6

Figure 4: FAD for Steel B .......................................................................................................................... 6

Figure 5: FAD for Steel A (Detectable crack length = 3mm) ...................................................................... 7

Figure 6: Pressurized cylinder with semi-elliptical crack ........................................................................... 8

Figure 7: Variation of Y with crack ratio a/c for Steel A and Steel B .......................................................... 9

Figure 8: FAD for Steel A using different values of a/c .............................................................................. 9

Figure 9: FAD for Steel B using different values of a/c ............................................................................ 10

Figure 10: Fatigue growth for Steel B, thickness = 38.5mm .................................................................... 11

Figure 11: FAD of Steel B with increased thickness of 85mm ................................................................. 11

Figure 12: Fatigue growth of Steel B, thickness = 85mm ........................................................................ 12

LIST OF TABLES

Table 1: Material Properties of Steel A and Steel B …........................................................................................2

Table 2: Individual Parameters.……………………………………………………………………………………………………………………2

Table 3: Charpy Data for Steel B…………………………………………………………………………………………………………………..4

2 | P a g e

1. INTRODUCTION Hydro-electric power stations utilise water from a source at a higher altitude, in order to convert

potential/kinetic energy into electrical energy. [1,2] The momentum of the water intake drives the turbines,

which in turn power the generators that produce electricity. Pumped storage stations have two reservoirs to

recirculate the water, and hence store electrical energy. Once the turbines are driven, the water is pumped

from the lower reservoir to the higher reservoir to create an immediate reserve of water. The implementation

of both the hydro-electric and pumped storage stations is quicker and more reliable than any other type of

station [1], and therefore, an informed design of the specified combined station and its components is of

utmost importance.

This design task is concerned with the

structural integrity of the intermediate

penstocks. Penstocks are the steel pipes

that convey water from the intake

reservoir to the turbine of the hydro-

electric power station. Their function can

be visually observed from Fig 1. The

intermediate penstocks are unsupported

(no presence of surrounding rock or

concrete) and are to be capable of

withstanding the full water pressure

(hydraulic load) throughout the entirety of

its design life. [2]

Two steels, A and B, are to be evaluated in this report as a suitable material to be used in the fabrication of the

intermediate penstock, using their respective given material properties in Table 1, and the assigned individual

parameters shown in Table 2:

Table 1: Material Properties of Steel A and Steel B

This report will assess and compare the two Steels A and B, via fracture and fatigue assessments, based on the

assigned individual parameters. Design improvements required to meet the assigned criteria for the penstocks

will be made, and evaluated, in order to provide a final recommendation, and suggest a suitable inspection

interval. All MATLAB codes used for analysis in the subsequent sections are included in the Appendix.

Detectable flaw size, mm

Pipe diameter, m

Static head of water, m

Max head of water during hammer, m

6 2.2 525 750

Table 2: Individual Parameters

Figure 1: Schematic diagram of a hydro-electric power station [3]

The penstocks are subjected to ultra-sonic non-

destructive testing (NDT), and a minimum detectable

crack length of 6mm has been determined. The

assigned maximum head of water or pressure transient

is 750m. The intermediate penstock is exposed to a

pressure change during the station’s change in modes,

i.e. changing from pumping to generating and vice

versa. [2] There is an additional residual stress of

70MPa, which is created by localised areas of cooling

and heating during welding.

3 | P a g e

2. STATIC DESIGN/ OVERLOAD ASSESSMENT The maximum allowable design stress is stated to be equal to 0.6 times the yield stress, and this value has

been determined by using the prescribed factor of safety. It is therefore necessary to estimate what wall

thickness will be needed in either steel in order to allow for safe operation under static conditions. The

maximum possible hoop stress occurs due to the 'water hammer' transient pressure. The penstocks are

fabricated like pressure vessels by longitudinally and circumferentially welding shaped steel plates. They are

modelled as thin-walled cylinders, and hence the hoop stress can be calculated:

𝜎ℎ = 𝑃𝑚𝑎𝑥 . 𝑟

𝑡⁄ = 0.6𝜎𝑦 ….(2.1)

where 𝜎ℎ is the hoop stress, 𝑃𝑚𝑎𝑥 represents the maximum pressure due to the water hammer, 𝜎𝑦 is the

tensile yield stress, whereas 𝑟 and 𝑡 represent the radius and thickness of the specified penstock, respectively. 𝑃𝑚𝑎𝑥 is calculated using the maximum head of water, ℎ𝑚𝑎𝑥 , which corresponds to:

𝑃𝑚𝑎𝑥 = 𝜌𝑔ℎ𝑚𝑎𝑥 ….(2.2)

where 𝜌 is the density of water (1000 𝑘𝑔/𝑚3) and 𝑔 represents acceleration due to gravity(9.81 𝑚/𝑠−2) Substituting Eq 2.2 into Eq 2.1 and rearranging for t gives:

𝑡 = 𝜌𝑔ℎ𝑚𝑎𝑥 . 𝑟

0.6𝜎𝑦⁄ … ..(2.3)

The Eq 2.3 allows the thickness, t to be calculated for both Steel A and Steel B. From Figure 1.1 the yield stresses for Steel A and Steel B are 750 MPa and 300 MPa, respectively. The radius, 𝑟 of the pipe is 1.1m.

𝑡𝐴 = 1000 ∗ 9.81 ∗ 750 ∗ 1.1

0.6 ∗ 700 ∗ 106= 19.3 𝑚𝑚, 𝑡𝐵 =

1000 ∗ 9.81 ∗ 750 ∗ 1.1

0.6 ∗ 350 ∗ 106= 38.5 𝑚𝑚

It can therefore be determined that the minimum thicknesses for Steel A and Steel B, are 19.3mm and 38.5mm, respectively. Additionally, the hoop stresses for the Steels A and B are calculated to be 420MPa and 210MPa, respectively. The material cost depends of the weight of steel used, which is a function of the wall thickness, t. Since the diameter and length of the vessel are fixed, the weight depends only on the wall thickness used. In the following equation, 𝑡𝐴, 𝑡𝐵 , £𝐴, £𝐵 are the thicknesses and prices of rolled plate/1000kg of Steels A and B, respectively . Thus, the ratio of the costs of Steels A and B, or the relative cost is:

𝐶𝑜𝑠𝑡𝐴

𝐶𝑜𝑠𝑡𝐵

= 𝑡𝐴£𝐴

𝑡𝐵£𝐵

= 0.0193 ∗ 965

0.0385 ∗ 525= 0.92

Therefore, the price of steel A is 92%, relative to that of steel B. One of the key objectives of during the design process is the minimization of costs, and the ratio given in Eq 2.4 will be used in the final evaluation.

4 | P a g e

3. FRACTURE ASSESSMENT

3.1 Determining KIC for Steel A and Steel B

Steel A:

The design brief includes fracture toughness data for Steel A, measured on a 40mm thick plate. This is

shown in Figure 2:

At 0℃ a fracture toughness of

approximately between 100 𝑀𝑁𝑚−3

2⁄

& 120 𝑀𝑁𝑚−3

2⁄ would be a

conservative choice. Figure 3.1 shows

that the steel would need to be at

roughly −50℃ for 100 𝑀𝑁𝑚−3

2⁄ to be

the correct value. The operating

temperature is not known and the

test-piece thickness of 40mm is greater

than the calculated thickness for Steel

A (19.3mm). The lower bound fracture

toughness of 100 𝑀𝑁𝑚−3

2⁄ will

therefore be used for analysis, as it is

assumed to be the worst case scenario.

Steel B:

Charpy test data is given for steel B, in Table 3 However, this data is deemed inaccurate/unreliable and could not lead to a very accurate value of fracture toughness.

Using the current SINTAP recommended equation 𝐾𝐼𝐶 = 12 √𝐶𝑣 , the corresponding calculated fracture

toughness value is 89 𝑀𝑁𝑚−3

2⁄ , at 0℃. It was expected of Steel B to have approximately the same or greater fracture toughness as Steel A.

It is therefore suitable to use an earlier and less convervative correlation, 𝐾𝐼𝐶 = 16 √𝐶𝑣 , which results in a

fracture toughness of approximately 120 𝑀𝑁𝑚−3

2⁄ at 0℃ for Steel B.

Fracture toughness for both Steels A and B at 0℃ have been determined. Hence:

𝐾𝐼𝐶𝐴 = 𝟏𝟎𝟎 𝑴𝑵𝒎

−𝟑𝟐⁄ ….(3.1.1) ; 𝐾𝐼𝐶

𝐵 = 𝟏𝟐𝟎 𝑴𝑵𝒎−𝟑

𝟐⁄ ……. (3.1.2)

Figure 2: Fracture toughness data for Steel A

Table 3: Charpy data for Steel B

5 | P a g e

3.2 Critical Crack Sizes in Steel A and Steel B

For simplicity, the growth of a long surface edge crack, lying parallel to the axis of the pipe is considered. This crack is influenced by the hoop stress generated when the vessel is filled with water. The calibration function for a similar geometry is given as an edge crack in a finite width plate in tension. A widely used equation to determine the calibration function is:

𝑌 = 1.12 − 0.231(𝑎𝑡⁄ ) + 10.55(𝑎

𝑡⁄ )2

− 21.72(𝑎𝑡⁄ )

3+ 30.39(𝑎

𝑡⁄ )4 ……….(3.2.1)

Since the geometry calibration function varies with crack size, 𝑎 , the critical crack size, 𝑎𝑐 is the solution to the

non-linear equation: 𝐾𝐼𝐶 = 𝑌𝜎 √𝜋𝑎𝑐 ……….(3.2.2)

The non-linear equation can be solved by using a simple iterative method like the Newton-Raphson technique. Initially, the equation 3.2.1 is substituted into equation 3.2 and rearranged in the form f(ac) = 0. The next iteration is calculated by the equation:

𝑎𝑐𝑖+1 =

𝑓(𝑎𝑐𝑖 )

𝑓′(𝑎𝑐𝑖 )

⁄ − 𝑎𝑐𝑖 (3.2.3)

The benefit of the Newton-Raphson method is that is quick to converge, and is stable. An initial estimation of the critical crack size is required for the first iteration, and in this case, a size of 1mm was chosen. The MATLAB program was used for the purpose of this calculation, and the relevant code is attached in the appendix. [11] The calculated critical crack lengths, 𝑎𝑐 for the two steels are: Steel A: 𝑎𝑐

𝐴 = 𝟔. 𝟏𝟎𝟒𝟔𝒎𝒎 ; Steel B: 𝑎𝑐𝐵 = 𝟏𝟕. 𝟒𝟒𝟏𝟓𝒎𝒎

It can be stated that Steel B is much more durable than Steel A, since the critical crack length in order to initiate catastrophic crack growth for Steel B is approximately 3.33 greater than that of Steel A. The critical crack size for Steel A is 6.1046mm, which is greater than 6mm, which is the assigned detectable flaw size. With a total crack propagation length of only 0.1046mm, it can be predicted that failure will occur very quickly and hence Steel A is deemed unsuitable.

3.3 Failure Assessment Diagram (FAD) Analysis

The geometry calibration function, Y has been calculated for both steel using Equation 3.2.1:

𝑌𝐴 = 𝟏. 𝟔𝟗𝟗𝟏 (𝑆𝑡𝑒𝑒𝑙 𝐴) 𝑌𝐵 = 𝟏. 𝟐𝟕𝟓𝟗 (𝑆𝑡𝑒𝑒𝑙 𝐵)

The 𝐾𝑟 curve used is the currently accepted curve in the R6 specification, given by:

𝐾𝑟 = (1 + 0.5𝐿𝑟2)−1

2⁄ ∗ [0.3 + 0.7 exp(−0.6𝐿𝑟6)] …… (3.3.1)

The type of failure occurring in each steel can be estimated by plotting two assessment points on the same

plot as the R6 𝐾𝑟 curve mentioned previously.

The first point considers the secondary stresses, or stresses which do not contribute to plastic collapse. The

secondary stress for this vessel is the residual stress due to welding, which is present even in the absence of an

applied primary (hoop) stress. Therefore it is plotted at 𝐿𝑟=0. Let this point be P1, and in the following Failure

Assessment Diagrams(FADs) for Steels A and B, is represented by a red marker.

The second point utilises both the primary and secondary stresses, where the maximum primary stress is due

to the pressure of the water hammer, as mentioned previously. Let this point be P2, represented by a green

marker in the following diagrams.

Values of 𝐿𝑟𝑝

, 𝐾𝑟𝑠, 𝐾𝑟

𝑝, 𝑎𝑛𝑑 𝐾𝑟

𝑝+𝑠were calculated using the following equations:

6 | P a g e

𝐿𝑟𝑝

= 𝜎𝑝

𝜎𝑦(1 −𝑎

𝑡)⁄ …(3.3.2) 𝐾𝑟

𝑝,𝑠= 𝑌 𝜎𝑝,𝑠√𝜋𝑎

𝐾𝐼𝐶⁄ …(3.3.3) 𝐾𝑟

𝑝+𝑠= (𝜎𝑝 + 𝜎𝑠)*𝑌 (√𝜋𝑎

𝐾𝐼𝐶⁄ )…(3.3.4)

𝑷𝟏 ∶ (0, 𝐾𝑟𝑠) , where for Steel A and Steel B, are (0, 0.1633) and (0, 0.1022), respectively. This point is signified

in the subsequent FAD plots as Lr(s), Kr(s). 𝐾𝑟𝑠 is calculated using Equation 3.3.3.

𝑷𝟐: (𝐿𝑟𝑝

, 𝐾𝑟𝑠 + 𝐾𝑟

𝑝), where for Steel A and Steel B, are (0,1.1431) and (0,0.4088), respectively. This point is

denoted in the subsequent FAD plots as Lr(p), Kr(p+s).

𝐾𝑟𝑠 + 𝐾𝑟

𝑝 is calculated using Equation 3.3.4.

STEEL A:

Figure 3: FAD for Steel A

Figure 3 shows a line of best fit is drawn between the two assessment points P1 (0,0.1633) and P2 (0.6193,

1.1431). The line crosses the boundary of the R6 Kr failure curve at approximately (0.5,0.9582). The relatively

steep gradient of the line indicates that Steel A is very likely to undergo brittle fracture, i.e. the crack will

spread rapidly with a very limited extent of plastic deformation in the structure that is ‘contained’ or ‘small

scale’ plasticity. The assessment point for Steel A, under maximum allowable hoop stress lies outside the

predicted R6 failure curve given by Equation 3.3.1, and thus failure will occur under the current conditions. The

critical crack length calculated for Steel A (6.1046mm) is also extremely close to the detectable flaw size,

(6mm) and this is a reason for why the material fails in a brittle manner. Hence, Steel A is not viable.

STEEL B:

Figure 4: FAD for Steel B

7 | P a g e

As per Figure 3, the line of best-fit is plotted in Figure 4 between the two assessment points, P1(0,0.1022) and

P2(0.6095,0.4088). For further investigation, the best-fit line is extrapolated until it intersects the failure curve.

It crosses the failure curve at approximately (0.98,0.6003), indicating a plastic-elastic collapse. The relatively

gradual gradient of this line typifies ductile failure, i.e. the crack will propagate slowly, and is accompanied by a

large amount of plastic deformation. The higher critical crack length for Steel B (17.4415mm) as compared to

Steel A(6.1045mm) is a contributing factor to this. However, in this case, both assessment lines lie within the

safe/acceptable threshold under the R6 Kr failure curve, signifying that Steel B will not fail under the current

conditions. Based on the FAD Assessment alone, it can be concluded that Steel B is the better, safer option.

Catastrophic failure is very unlikely to occur without signs of warning first, and under the given conditions,

Steel B will not undergo failure.

3.4 Sensitivity Analysis Accuracy of NDT Methods:

The minimum detectable crack depth is stated to be 6mm, i.e. the smallest detectable crack which could

be treated during the lifetime of the vessel. The geometry calibration factor, Y will decrease, whereas the

stress intensity ratio, Kr and load ratio, Lr both increase with crack size, as observed from their respective

equations. As a result, the assessment point will shift closer, within the failure curve of the material with

increasing crack size. Increasing the accuracy of the NDT methods will decrease chance of failure. For

example, reducing the detectable flaw size by half (i.e. 3mm) will significantly delay failure, as shown in

the following FAD:

Figure 5 shows that the second assessment point on the FAD for Steel A, which was previously seen to lie

outside the R6 specified failure curve, has now re-located to within the safe threshold. Hence, Steel A will

not fail under the current conditions. It is also observed that Steel A will now eventually undergo ductile

failure, instead of brittle failure. Therefore, a higher accuracy of NDT methods would have allowed a

more thorough and precise comparison of the two steels.

Figure 5: FAD for Steel A (Detectable crack length = 3mm)

8 | P a g e

Accounting for Residual Stress:

The welding process used for the steels induce residual tensile stress, which substantially decrease fatigue life.

Cracks are initiated in metals which are exposed to high tensile stresses over a significant number of load

cycles. []

The residual stress value is used explicitly to calculate the stress intensity factor, K, by the equation

𝐾𝐼𝐶 = 𝑌𝜎 √𝜋𝑎𝑐. The assessment points are displaced vertically along the y-axis of the FAD, by changing the

magnitude of the residual stress.

The residual stress is constantly applied to the penstock, and does not contribute to the yielding of the vessels

material. It is therefore concluded that an under-estimate of the stress could indicate that the vessel is indeed

safe from a brittle and sudden fracture due to crack propagation, where in practice it is more likely to do so.

More than 90% of failures occurring in mechanical components occur due to crack nucleation/propagation

arising from high tensile loading conditions, and as a result, adversely affects the load-bearing capacity of the

component. In this case, one of the key criteria for the intermediate penstock is to maximize its load-bearing

capabilities, and therefore reduce residual stresses [8]. This will be outlined in section ‘6. Critical Discussion

and Final Recommendations’

Shape of the defect:

The crack has been assumed to be a simple

edge crack until now, where its width was not

taken into consideration. In order to provide a

more accurate model, the crack is assumed to

be a shallow semi-elliptical crack which is

parallel to the axis of the pipe. Subsequently,

various crack shapes and their influence on

fracture analysis can be analyzed.The shape

and dimensions of the crack both have an

impact on both the internal collapse pressure,

𝑃𝑐 and the geometry calibration function, 𝑌.

Figure 3.5 shows the semi-elliptical crack and

its dimensions.

The dependent ratios or the geometry calibration function are:

𝑎𝑡⁄ (Crack length/Thickness) ; 𝑎 𝑐⁄ (Crack length/crack width)

The vessel thickness, t, and the crack depth, a, are known. The only unknown is the crack width, c. Using the following ratios:

𝑎

𝑡𝐴=

6

19.3= 𝟎. 𝟑𝟏𝟏 ,

𝑎

𝑡𝐵=

6

38.5= 𝟎. 𝟏𝟓

Figure 6: Pressurized cylinder with semi-elliptical crack

9 | P a g e

The corresponding a /t ratios

for Steel A and Steel B are used

in conjunction with the graph

shown in Fig, which is in the

appendix. It displays the

variation of the geometric

calibration factor Y with the

crack width, c, or more

specifically, the a/c ratios. [5]

The plot of the variation of Y

against ratio a/c for both Steels

A and B is shown in Figure 7:

The plot shows that as c tends to infinity, the geometry reverts to the initial, simplified model of a single edge-

crack under tension. Comparing the newly obtained values of Y to the original values in 3.3 FAD Analysis, it can

be seen that they are very similar. Current values of Y are 1.7 and 1.4 for Steels A and B, respectively.

The internal collapse hoop stress/collapse pressure for various crack widths are to be calculated. The internal

collapse pressure can be calculated using the following equation:

𝑃𝑐 =𝜂𝜎𝑦𝑡

(1 −1−𝜂

√1+1.05𝜌2)⁄ ∗ 𝑟….(3.4.1), where 𝜂 = 𝑡 − 𝑎

𝑡⁄ ….(3.4.2) and 𝜌 = √𝑐2

𝑟𝑡⁄ ……(3.4.3)

The collapse hoop stress can therefore be calculated using the following expression: 𝜎𝑐 =𝑃𝑐𝑟

𝑡⁄ ….(3.4.5)

Graphs showing the co-relation between collapse hoop stress and crack width for both Steel A and Steel B can

be found in the Appendix. They both show that as c tends to infinity, 𝑃𝑐 tends towards 𝜂𝜎𝑦𝑡. These subsequent

values of 𝑃𝑐 are 420 MPa and 210 MPa for Steels A and B, respectively.

In order to assess the extent to which crack widths affect the probability of failure of the two candidate steels,

new FADs are plotted, using different values of Y (which had been calculated from corresponding values of

a/c).

STEEL A:

Figure 7: Variation of Y with crack ratio a/c for Steel A and Steel B

Figure 8: FAD for Steel A using different values of a/c

10 | P a g e

STEEL B:

The assessment points for a/c = 0 are the same as those calculated when assuming a simplified edge-crack

geometry. Predictably, the crack is not expected to be seen, and the main method of plastic deformation will

be of the yielding of steel. The ratio Kr is found to decrease with decreasing value of c, and the lower stress

concentrations can be attributed to the smaller crack size.

From Figures 8 it is apparent that, for a straight edge crack, Steel A would fail under maximum stress. Shifting

to a more realistic value (that of a semi-elliptical crack) ensures that failure does not occur. Steel B does not

undergo failure This only provides further validation that Steel B is the more favourable candidate.

4. FATIGUE ASSESSMENT Following the previous analysis, Steel B has been determined to be a better-suited material for the design of

the penstocks. Steel A had been shown to have less toughness and be more likely to fracture due to unstable

crack propagation.

Fatigue crack growth rate using LEFM assumptions will follow. The crack propagation life can be estimated

using the relation between the stress intensity range, ∆𝐾 and the crack growth rate (Paris’ Law)

The stress cycle arises from the pressure change from 0m to 525m head of water within the pressure vessel.

Using this information, the range of hoop stresses acting on the vessel can be calculated.

∆𝜎 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛 = 𝑃525. 𝑟

𝑡𝐵⁄ − 0 =

𝜌𝑔ℎ. 𝑟𝑡𝐵

⁄ = 1000 ∗ 9.81 ∗ 525 ∗ 1.10.385⁄ = 𝟏𝟒. 𝟕 𝑴𝑷𝒂 …(4.1)

Likewise, for Steel A, ∆𝜎 is calculated to be 29.4 MPa.

The stress intensity range, ∆𝐾 can be calculated using the value of ∆𝜎, using:

∆𝐾 = 𝑌∆𝜎√𝜋𝑎…………………………………………. (4.2)

The change in crack sizes, ‘a’ can be calculated with respect to the number of cycles, ‘N’ by use of Paris’ Law. 𝐶,

𝑚 are material constants of the given steels, and the stated values are 10-11, 3 respectively.

𝑑𝑎

𝑑𝑁= 𝐶(∆𝐾)𝑚 …………………………………….. (4.3)

By utilising a small step size, (∆𝑎), an iterative numerical integration can be carried out to determine the

solution. The initial crack size, 𝑎𝑖 is equivalent to the given detectable crack/flaw size of 6mm. The final crack

size, 𝑎𝑓𝑛 , will be greater than the critical crack size, 𝑎𝑐 of the steel, as calculated previously:

Figure 9: FAD for Steel B using different values of a/c

11 | P a g e

The crack length propagation plot in Figure 10 shows that the time taken for the initial crack to propagate to

critical size within Steel B is 2.3 years, well below the required service life of 50 years. Design improvements

are to be made accordingly. The fatigue life plot for Steel A predictably shows that failure due to

fatigue will occur very quickly. This is included in the appendix.

5. DESIGN IMPROVEMENTS

As concluded in the section ‘4. Fatigue Assessment’, changes are required in the design of the penstock. An

appropriate design parameter to change would be the thickness of the pressure vessel. By increasing the

thickness to approximately 85mm, the critical crack length is increased to 27.6 mm, using the same procedure

in section ‘3.2 Critical Crack Sizes’.

Figure 11 shows the FAD for Steel B, incorporating the new specified thickness of 85mm. The probability of

fracture and yielding is reduced by increasing the thickness, which subsequently lowers the hoop stress. It can

be seen that Steel B also undergoes plastic collapse.

Figure 10: Fatigue growth for Steel B, thickness = 38.5mm

Figure 11: FAD of Steel B with increased thickness of 85mm

12 | P a g e

The fatigue life plot illustrated in Figure 11 shows that the time taken for the initial crack to grow to the critical

crack length size of 27.6mm is approximately 56 years. This is an acceptable value for the design life of the

penstock, with an appropriate 6 years of safety to accommodate for any load discrepancies.

6. CRITICAL DISCUSSION AND FINAL RECOMMENDATIONS

Generally, due to a hydraulic pressure P acting on the faces of the penstock pipe, the crack located at the

inside surface is additionally loaded. [1] The stress intensity due to this loading is given by

𝐾𝐼 = 𝑃√𝜋𝑎 . However, since the penstock pipe is treated as a thin-walled cylinder in ‘2. Overload Assessment’,

𝑃 ≪ 𝜎, and therefore, the additional loading effect has been not been considered.

It must be noted that only the upper estimates for the critical crack lengths have been calculated using the

MATLAB code. LEFM is not strictly applicable to the section thicknesses of Steels A and B, for values of K

approaching KIC , as the plastic zone at the crack tip is significant. Smaller values of the critical crack length

would have been obtained if a plastic zone correction factor would have been used. [1] The size of the crack tip

can either be calculated using Irwin’s model, which estimates the elastic-plastic boundary using elastic stress

analysis, and the strip-yield model [4]

In the section ‘3.4 Sensitivity Analysis’, it was shown that the risk of failure could be substantially reduced by

improving the accuracy of the Non-Destructive Testing (NDT) methods. The minimum detectable crack length

should be reduced to 3mm, as this ensures that both Steels A and B will not undergo failure under given

conditions. Steel A can therefore be compared with Steel B in a more critical and thorough manner, and a

more informed choice can be made. The reduction of minimum detectable crack length also allows a more

accurate estimation of crack propagation, so that risks of failure can be identified in advance and dealt with

accordingly.

Figure 12: Fatigue growth of Steel B, thickness = 85mm

13 | P a g e

As mentioned in sub-section ‘Accounting for Residual Stress’, the tensile residual stress problems can be

resolved by processes which induce compressive tensile stresses such as post weld heat treatment (PWHT),

shot peening and spot heating. [9] It must also be mentioned that the weld metal used in the welding

procedure decreases in toughness (COD toughness, or Crack Opening Displacement toughness). This

associated with the coarsening of the microstructure due to increases in temperature. This occurs in the sub-

critical Heat-Affected Zones (HAZs) [12]. The fracture toughness values for only the parent material (Steels A

and B) can be derived from the given information. This value is KIC = Kmat at fracture. Though the properties of

the weld material are not stated or given, it must have approximately the same fracture toughness values as

that of the parent material (steel). This would ensure safety, and a penstock fabrication of good quality, i.e.

lower risk of failure occurring at the weld joints.

The section ‘4. Fatigue Assessment’ does not take into account the damage caused by any variation in the

number of cycles, Nf. Variation in size, number and order of stress cycles will lead to cumulative fatigue

damage of the penstock. Therefore, fatigue damage must be evaluated by adding the detrimental effects of

each individual cycle, to ensure that any chance of failure is minimized. [10]

Cracks are to be repaired when they are approximately equal to 15mm, thus providing a safety window of

about 12 years to accommodate for the risk of failure. Maintenance of cracks of smaller lengths than 15mm

will increase the operational cost of the station, without any substantial increase in safety.

The inspection frequency for the intermediate penstock must be between 1 and 5 years, but no more than 5

years [5]. Therefore, the inspection interval should be co-ordinated with the maintenance interval, i.e.

stripping of the walls and repainting. [2]

14 | P a g e

7. REFERENCES

1. Hudson C, Rich T. Case histories involving fatigue and fracture mechanics. Philadelphia, PA: ASTM; 1986.

2. Jivkov A. The Design of a Penstock Pipe for a Hydro-electric Pumped Storage Station. 1st ed. 2015.

3. Saadat H. Power System Analysis [Internet]. Psapublishing.com. 2015 [cited 20 November 2015]. Available from: http://www.psapublishing.com/

4. Anderson T, Anderson T. Fracture Mechanics. Hoboken: CRC Press; 2005.

5. Plastic Collapse Handbook. 1st ed. 2002.

6. Bannister A. Structural integrity assessment procedures for European Industry. Swindon: British Steel plc; 1998.

7. McStraw B. Inspection of Steel Penstocks and Pressure Conduits. UNITED STATES DEPARTMENT OF THE INTERIOR BUREAU OF RECLAMATION DENVER, COLORADO; 1996.

8. Molzen M, Hornbach D. Evaluation of Welding Residual Stress Levels Through Shot Peening and Heat Treating. 1st ed. 2000.

9. O'Brien R. Welding handbook. Miami, Fla.: American Welding Society; 1991.

10. Kaechele L. Review and Analysis of Cumulative Fatigue-Damage Theories. 1963;

11. Deuflhard P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. 2011.

12. Neves J, Loureiro A. Fracture toughness of welds—effect of brittle zones and strength mismatch. Journal of

Materials Processing Technology. 2004;153-154:537-543.

15 | P a g e

8. APPENDIX

A. Ratio of Y with a/t

B. Plastic Limit Stresses for Steels A and B

16 | P a g e

C. Crack Length Propagation for Steel A

D. MATLAB Code: Critical Crack Length

% This script finds the solution to the non-linear crack size equation,

using the Newton-Raphson Method. %Number of iterations to be specified. In this case, 10 iterations will

suffice:

17 | P a g e

n=10;

%Critical Fracture Toughness for Steel A and Steel B, respectively K1c=[100*10^6;120*10^6];

%Maximum allowable hoop stress (60% of yield stress); %Steel A - 420MPa %Steel B - 210MPa sigmah =[420*10^6;210*10^6];

%Minimum thickness calculated for both Steel A(18mm) and Steel B(45mm); t=[0.0193;0.0385]; %An initial value of a=1mm for both Steels A and B a=[0.001;0.001];

%Newton-Raphson method iterations carried out using a for loop:

for i=1:n f=((1.12.*a.^0.5-(0.231.*(a.^1.5).*(t.^(-1)))+(10.55.*(a.^2.5).*(t.^(-2)))-

(21.72.*(a.^3.5).*(t.^(-3)))+(30.39.*(a.^4.5).*(t.^(-

4)))).*(pi^0.5).*sigmah)-K1c;

f1 =((0.56.*a.^-0.5-(0.3465.*(a.^0.5).*(t.^(-1)))+(26.375.*(a.^1.5).*(t.^(-

2)))-(76.02.*(a.^2.5).*(t.^(-3)))+(136.755.*(a.^3.5).*(t.^(-

4)))).*(pi^0.5).*sigmah);

a1=a-(f./f1); a=a1; end %The final output a is the critical crack length in m for both Steel A and

Steel B

%Critical crack length a is converted from m to mm: a=a*1000

a =

6.1046

17.4415

E. MATLAB Code: Crack Length Propagation

clear clc %Input known variables %Assigned parameter: Maximum head of water h=525; %Assigned parameter: Radius of the pipe r=1.1; %Assigned parameter: Material Constant C=10^-11; %Assigned parameter: Material Constant m=3; %Calculated parameter:Final crack size

18 | P a g e

af=0.085; %Initial Crack size ai=0.006; %Pipe thickness t=0.085; %Calculate stress range dsigma=(h*r*1000*9.81)/(t*10^6); %Input number of iterations Na=5000; %Calculate crack resolution (step-size) da=(af-ai)/Na; %Initialize the solution a=ai; N=0; Yr(1)=0; %Calculation Loop - Algorithm Shown in Figure 4.1 for i=1:Na-1 G=(a/t); Y=1.12-(0.231*G)+(10.55*G^2)-(21.72*G^3)+(30.39*G^4); dk=Y*dsigma*((pi*a)^0.5); dadn=C*(dk^m); dN=da/dadn; N=N+dN; Yr(i+1)=N/7500; a=a+da; end %Change a into millimeters a=1000*linspace(ai,af,Na); %Plot plot(Yr,a) grid on xlabel('Service Life (years)') ylabel('Crack Length (mm)') title('Crack Length Propagation vs Service Life (Steel A/B) ')

F. MATLAB Code: Plastic Collapse

clear;

%Enter your own input variables: a=0.006;%Detectable flaw size t=0.385;%Thickness c=(0:0.2:2);%Range of values of c r=1.1; %Radius

sigmay=350*10^6; n=(t-a)/t; rho=((c.^2)/(r*t)).^0.5;

%Plastic collapse hoop stress Pc=(n*sigmay*t)./((1-((1-n)./(1+(1.05.*(rho.^2))).^0.5)).*r);

sigmac=Pc.*r/(t*1000*(10^5));

plot(c,sigmac,'-bo','LineWidth', 2); xlabel('c (m)'); ylabel('Sigma_c (MPa)');

19 | P a g e

title('Plastic Limit Stress vs Crack Length (Steel B) ') grid on; hold on;

G. MATLAB Code: FAD

clear; Lr=[0:0.02:1.4]; kr=(1-0.14.*Lr.^2).*(0.3 + 0.7.*exp(-0.65*Lr.^6));

%SteelB Lr1=0; Lr2=0.6095; Kr1=0.092; Kr2=0.276;

plot(Lr,kr,'LineWidth',3); legend('Limit'); hold on; title('Level 2 Failure Assessment Diagram (FAD) for Steel B (Increased

Thickness to 85mm )'); xlabel('Lr'); ylabel('Kr'); hold on; scatter(Lr1,Kr1,'r','o','markerfacecolor','r' ) hold on; scatter(Lr2,Kr2,'g','o','markerfacecolor','g' ) hold on; grid on; legend('R6 Kr curve','Lr(s)=0 ,Kr(s)(Steel B)', 'Lr(p),Kr(p+s)(Steel B)')