Permissible Width of Front Face Flaws GMT Primary ... - Carnegie

GMT Project 

Permissible Width of Front Face Flaws 

Doc # : 

Date: 2009-Feb-13 

Status: Rev. F 

Page: 1 of 9 

Permissible Width of Front Face Flaws 

GMT Primary Mirror Off-Axis Segment 

Prepared By: 

Name(s) and Signature(s) 

Brian Cuerden 

Approved By 

Date 

Name and Signature Title 

PI 

Technical Division Manager 

Responsible Engineer 

Date 

Optical Engineer 

NOTE: THIS DOCUMENT CONTAINS PROPRIETARY INFORMATION AND CANNOT BE 

DISCLOSED WITHOUT THE WRITTEN CONSENT OF STEWARD OBSERVATORY.

Revision History 


GMT, Permissible Width of Front Face Flaws 

Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


Issue Date Changes Responsible 

A 23-Jan-2009 Initial release Brian Cuerden 

B 26-Jan-2009 Standardize 

Factor 

on Blain’s Geometry Brian Cuerden 

C 27-Jan-2009 Corrected Y axis label of Fig. 4.2 

Corrected 3sigma a/w to 0.4 (section 

3.2, Table 4.1 and sections 5 and 6) 

Brian Cuerden 

D 28-Jan-2009 Incorporated numerous comments from 

Blain Olbert 

Brian Cuerden 

E 4-Feb-2009 Corrected a/2w to a/w in fig. 4.1 & 4.2 Brian Cuerden 

F 13-Feb-2009 Standardized phrasing Buddy Martin

Table Of Contents 



Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


1. Applicable Documents and Drawings ........................................................................ 4 

2. Acronyms and Abbreviations ..................................................................................... 4 

3. Introduction.................................................................................................................4 

3.1. Description of Front Face Flaws......................................................................... 4 

3.2. Calculation of the Permissible Flaw Width ........................................................ 5 

4. Discussion...................................................................................................................5 

4.1. Calculation Procedure......................................................................................... 5 

4.2. Use of Flaw Width as the Controlling Parameter ............................................... 7 

5. Results......................................................................................................................... 9 

5.1. Permissible Initial Flaw at 308 psi for One year life .......................................... 9 

5.2. Permissible Initial Flaw at 231 psi for One year life .......................................... 9 

5.3. Permissible Initial Flaw at 308 psi for Six Months............................................. 9 

6. Conclusions................................................................................................................. 9



1. Applicable Documents and Drawings 

Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


The following documents of the exact issue shown form a part of this Specification to the 

extent specified herein. In the event of conflict between the documents referenced herein 

and the present document, the terms of the Specification shall be considered as 

superseding requirements. 

Applicable Documents 

[AD1] Face Plate Flaws, Rev. D 

[AD2] Weiderhorn and Roberts, “Fracture Mechanics Study of 

Skylab Windows, NBS Project 3130450, NBS Report 10 892, 

05/31/1972 

[AD3] GMT Faceplate Life Final R00.pdf 

Drawings 

2. Acronyms and Abbreviations 

GMT Giant Magellan Telescope 

TBR To Be Reviewed 

a The depth of a flaw 

KI The flaw tip stress intensity 

KIC The critical flaw tip stress intensity 

w The full width of a flaw 

Y The flaw tip stress intensity geometry factor 

3. Introduction 

3.1. Description of Front Face Flaws 

Visible front face flaws are located near the center of the mirror. Most are 

arranged in bands 2.0 inches apart, the remainders are in bands 1.0 inch apart. 

Generating was performed with a 2.0” spiral cut with finish cuts being 1.0” 

apart. Etching apparently flaw free areas along tracks of visible flaws or 

stepped 2.0” laterally from these tracks has revealed additional flaws. 

Additional areas are being/have been etched and a separate report will 

describe the results of this flaw detection effort which will determine the



Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


width of the 99 th percentile flaw expected anywhere on the remaining, unetched, 

mirror face. 

3.2. Calculation of the Permissible Flaw Width 

4. Discussion 

This report defines the permissible flaw width on the front surface. It uses twice 

the estimate of the maximum front face stress defined in reference [AD1]. This is 

2*154 = 308 psi. This stress is assumed to be applied for an accumulated total of 

1 year in the 50 year operational life of the primary mirror. This time of exposure 

is intended to represent a worst case estimate of the exposure time. A naïve 

estimate of exposure time would be to assume that the worst case thermal stress 

occurs for 6 hours every month on those nights when the mirror is warmer than 

the night ambient (reported to occur after several days of downtime). This 

represents 0.83% of the time or 0.42 years in 50 years. 

Measurements of the geometry of revealed flaws give a mean depth, a, of 23% of 

the width, w. Three times the standard deviation of the depth to width ratio, a/w, 

is 16%, making the 99.9 th percentile flaw 39% as deep as it is wide. 

4.1. Calculation Procedure 

In reference [AD1] the flaw tip stress intensity was computed using equations 

for long surface flaws, a/w



Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


The differences are not significant since we are in good agreement on the 

initial dimensions of flaws which grow to failure in one year. In particular, 

the failure definition used by Olbert, [AD3], is when the flaw reaches critical 

size or grows through the faceplate. Cuerden allows the flaw to grow through 

the faceplate. In most cases this does not happen at 308 psi. When it does 

happen, it occurs very near the end of the one year period and has little effect 

on the result (the size of the initial flaw). One other difference is that Cuerden 

estimates the change in the aspect ratio of the flaw as it grows using the 

known flaw geometry parameters at the depth- and corner- points of the flaw. 

This has a small effect on the result, confirmed by freezing the aspect ratio 

and repeating the calculation. For an a/w=0.2 flaw with a growth-evolving 

aspect ratio, the permissible initial flaw size is 29.45 mm wide and 5.89 mm 

deep. When the aspect ratio is frozen at 0.2, the permissible initial flaw size is 

26.8 mm wide and 5.36 mm deep. Figure 4.1 shows the flaw depth and aspect 

ratio over time for the variable aspect ratio case. Figure 4.2 compares the flaw 

intensity geometry factor (the Y in KI = Y*σ√a)*. The reduction in the 

geometry factor as the flaw aspect ratio changes toward 0.4 over time gives a 

smaller geometry factor and therefore a reduced crack growth rate permitting 

a slightly larger initial flaw. Cuerden’s variable aspect ratio is based on 

evaluating crack growth at two points, one at the deepest point of the flaw 

and the other where the flaw meets the surface, assuming the flaw shape 

remains elliptical. It serves to demonstrate that the change in flaw geometry is 

beneficial but it may not be conservative. Olbert’s constant aspect ratio 

calculation is therefore used to establish permissible flaw widths. 

* Flaw propagation is a function of the flaw tip stress intensity, KI. This has 

the units force*sqrt(length)/unit area = stress*sqrt(length). The flaw tip stress 

intensity is generally proportional to the stress times the square root of the 

flaw length. Y is the proportionality constant which is a constant for some 

simple flaw configurations (for example a flaw in the edge of a semi-infinite 

plate) but varies with the flaw depth and other geometrical features in the 

general case.

crack depth, mm 



20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

depth 

a/w 

Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


An a/w=0.20 crack growing over time, 308 psi 

0 

0 

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 

time, days 

Figure 4.1 Flaw Depth and Evolving Aspect Ratio of an Initially a/w=0.2 

Flaw. 

crack depth, mm 

1.90 

1.85 

1.80 

1.75 

1.70 

1.65 

1.60 

1.55 

1.50 

1.45 

1.40 

An a/w=0.20 crack growing over time, 308 psi 

Y, a/w variable 

Y, a/w =0.2 (fixed) 

0.0 100.0 200.0 300.0 400.0 

time, days 

Figure 4.2 A Comparison of Flaw Intensity Geometry Factors from the 

Variable Aspect Ratio Case and the Constant Aspect Ratio Case. Since the 

flaw tip stress intensity, KI, is proportional to the geometry factor, Y, the 

reduced Y in the variable aspect ratio case means a lower KI and hence a 

lower crack growth rate. 

4.2. Use of Flaw Width as the Controlling Parameter 

0.4 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

Aspect Ratio, a/2w



Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


Since we are evaluating the permissible size of semi-elliptical surface 

flaws and since we can non-destructively measure the width but not the 

depth of the flaws we want to present our results in terms of the flaw 

width and not depth. It has been noted in section 3.2 that the average 

depth of front face flaws is 23% of the width with a 3σ variation of 19% 

of the width. It has been noted in section 4.1 that it is conservative to 

assume that the initial aspect ratio stays constant over time. It will now be 

shown that given a measured flaw width, the worst case assumption is that 

the depth is 40% of the width (rounded up from a mean plus 3σ value of 

39%) rather than 7% of the width (the mean minus 3σ aspect ratio). In 

other words, it will be shown that the deeper semi-circular flaw is worse 

than a shallow surface flaw of the same width. Table 4.1 lists the initial 

dimensions of semi-elliptical surface flaws that grow to failure in one year 

at 308 psi. Flaw aspect ratios are held constant as the flaw grows in Table 

4.1. The a/w=0.4 flaw has the smallest width so when determining the 

acceptable width of a flaw we should assume an aspect ratio of 0.4. 

Note the excellent agreement between Olbert’s results, scaled to 308 psi, 

and Cuerden’s. In this revision, Cuerden’s calculation of crack growth is 

using the more accurate geometry factor used by Olbert. Each of us is 

using the same crack growth law and are conservatively assuming a 

constant flaw aspect ratio. The close agreement therefore indicates that 

both computational procedures (Mathcad and Excel) are equivalent. 

Olbert’s results at 300 psi were scaled to 308 psi by multiplying the 

allowable flaw dimensions by (300/308) 2 since KI=Yσ√a the scaled flaw 

depth gives the same KI at 308 psi as the original depth gives at 300 psi. 

Cuerden’s calculation at 300 psi gives a flaw depth of 6.043 mm 

(a/w=0.25) compared to 5.74 mm at 308 psi. Applying the scaling factor 

to 6.043 mm gives 5.73 mm which is only off by 0.12%. 

Cuerden, 308 psi Olbert, 300 psi Olbert, 308 psi 

Design 

a/w w a w a w a w a 

mm mm mm mm mm mm mm mm 

0.4 21.5 8.6 21.5 8.6 

0.25 23.0 5.7 24.2 6.1 23.0 5.7 23.0 5.7 

0.2 25.1 5.0 26.4 5.3 25.0 5.0 25.0 5.0 

0.15 29.5 4.4 31.1 4.7 29.5 4.4 29.5 4.4 

0.1 39.1 3.9 41.1 4.1 39.0 3.9 39.0 3.9 

Table 4.1 Permissible Flaw Sizes for Various Aspect Ratios for One Year of Life 

at 308 psi Applied Stress. The design value is the smaller of the two independent 

calculations with Olbert’s result being multiplied by (300/308) 2 to scale the result 

to 308 psi. Aspect ratios are fixed as the flaw grows.

5. Results 



5.1. Permissible Initial Flaw at 308 psi for One year life 

Doc # : 

Date: 2009-Feb-13 

Status: Rev F 


For an elliptical surface flaw with a/w=0.4, Table 4.1 specifies a 

maximum allowed width of 21.5 mm at 308 psi for one year of exposure. 

5.2. Permissible Initial Flaw at 231 psi for One year life 

[AD1] specifies a maximum front surface stress of 154 psi. This was 

doubled to provide margin. To assess how conservative this is in terms of 

flaw size, the crack growth calculation has been repeated at 231 psi 

(150% of the maximum service stress). 

The result is a permissible flaw width of 36.2 mm which is 68% longer 

than the allowable flaw width at 308 psi. 

5.3. Permissible Initial Flaw at 308 psi for Six Months 

6. Conclusions 

It has been noted in section 3.2 that the accumulated exposure time is 

more likely to be 6 months than one year. Repeating the crack growth 

calculation for a six month life gives a permissible flaw width of 22.9 mm 

which is 7% longer than the allowable flaw width for one year of stress 

exposure. 

Flaws of up to 21.5 mm wide are permissible on the front face of the 

GMT primary segment. This value has been derived using a conservative 

calculation using 3σ crack growth data a 3σ aspect ratio value and twice 

the maximum expected stress for twice the estimated duration of exposure 

to maximum stress. The allowable flaw widths were independently 

calculated by Olbert and Cuerden. Small differences between the results 

of these two calculations were evaluated and traced to slightly different 

crack growth laws and flaw stress intensity geometry factors. When the 

same crack growth law and geometry factors were used in both 

calculations conservatively assuming a constant flaw aspect ration, the 

results, allowable flaw width, differ by less than 1%. The results 

presented in this report use the most conservative of the data and 

assumptions initially used by Olbert and Cuerden. 

These results apply only to flaws having the same characteristics as those 

demonstrating depths of less than 40% of the width.

Permissible Width of Front Face Flaws GMT Primary ... - Carnegie

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