USAFA-TR-2005-4
Compression Buckling of Z-Stiffened Aluminum Panels, with and without Corrosion Grindouts James M. Greer, Jr., Daniel W. Hill, and Scott A. Fawaz
Center for Aircraft Structural Life Extension U.S. Air Force Academy CO 80840 (719) 333-3618, DSN 333-3618 Ron Logan
Northrop Grumman Corporation Integrated Systems AGS & BMS, Melbourne FL (321) 951-680 951-68033
Center for Aircraft Structural Life Extension Department of Engineering Mechanics United States Air Force Academy Colorado Springs, Colorado 80840 January 2005 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED
DEAN OF THE FACULTY UNITED STATES AIR FORCE ACADEMY COLORADO 80840
USAFA-TR-2005-4
USAFA-TR-2005-4 This article, “Compression Buckling of Z-Stiffened Aluminum Panels, with and without Corrosion Grindouts,” is presented as a competent treatment of the subject, worthy thy of public publicati ation. on. The United United States States Air Force orce Academ Academy y vouc vouches hes for the quality quality of the research, without necessarily endorsing the opinions and conclusions of the authors. Therefore, the views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government. This report has been cleared for open publication and public release by the appropriate Office of Information in accordance with AFI 61-202 and USAFA FOI 190-1. This report may have unlimited distribution.
i
USAFA-TR-2005-4
EXECUTIVE SUMMARY This effort is funded by the Aging Aircraft Squadron of the Aeronautical Enterprise Program Office (ASC/AAA, Col P.J. Clark). The prime contractor for the Air Vehicle Health Management program, of which this effort is a part, is S&K Technologies, Inc., Dayton, OH (Mr. Kevin Boyd). Twenty-seven Z-stiffened panels, intended to simulate upper wing skin panels of the Boeing 707, were tested to failure in compression to determine buckling strength. Pristine panels and panels with machined grindouts (with various depths up to 62.6% of the panel skin thickness) were tested to failure. Nine panels each of three configurations were fabricated for testing. The results showed a degradation of buckling strength with grindout depth that could be modeled with a modified Johnson-Euler method and a modified Gerard’s method for long and short panels respectively. The panels with the lower slenderness ratio (short panels) were degraded more by a given grindout depth than were their more slender counterparts. However, it was found that span-wise grindouts along the center stiffener—even deep ones—do not have a severe effect on strength. Even at over 60% grindout depth, the worst degradation was less than a 12% reduction in strength. A small number of panels were tested with deep chord-wise grindouts. These tests showed that the strength of the panel was dramatically reduced by these grindouts, which were transverse to the loading direction.
ii
USAFA-TR-2005-4
Table of Contents Executive Summary
ii
List of Figures
iv
List of Tables
v
List of Symbols
vi
1 Introduction
1
2 Background 2.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Previous Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 1 1
3 Specimen Fabrication and Instrumentation 3.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 5 6
4 Experimental Results 4.1 Test Set-Up . . . . . . . . . . 4.2 Test Procedure . . . . . . . . 4.3 Test Results . . . . . . . . . . 4.3.1 Span-Wise Grindouts . 4.3.2 Chord-Wise Grindouts 4.4 Errors in Experimental Values
. . . . . .
9 9 9 10 12 12 13
5 Modeling Panel Behavior 5.1 Long Panels (C1 and C2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Short Panels (C3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Modeling Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 14 15 17
6 Conclusions
18
Acknowledgements
19
Appendix: Sample Calculations
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
A–1
A.1 Johnson-Euler Method A–1 A.1.1 Damaged Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A–3 A.2 Method of Gerard A–3 A.2.1 Damaged Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A–4
iii
USAFA-TR-2005-4
List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14
iv
Approximate gage locations of Hickman and Dow. . . . . . . . . . . . . . Gage locations of Friedman. . . . . . . . . . . . . . . . . . . . . . . . . . Gage locations of Butler, et al. . . . . . . . . . . . . . . . . . . . . . . . . Cross section of stiffened panels. . . . . . . . . . . . . . . . . . . . . . . . Step-tapered grindout geometry. . . . . . . . . . . . . . . . . . . . . . . . Typical linear buckling analysis of panel. . . . . . . . . . . . . . . . . . . Initial and final strain gage locations. . . . . . . . . . . . . . . . . . . . . Typical long panel behavior of a gage pair. . . . . . . . . . . . . . . . . . Set up for panel buckling experiments. . . . . . . . . . . . . . . . . . . . Typical damage progression with load. . . . . . . . . . . . . . . . . . . . Panel strength as a function of grindout depth for 3 configurations with span-wise grindouts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of C1 panel results with modified Johnson-Euler method. . . Comparison of C2 panel results with modified Johnson-Euler method. . . Comparison of C3 panel results with modified method of Gerard. Dotted line indicates Johnson-Euler model for this panel. . . . . . . . . . . . . .
2 3 4 5 6 7 8 8 9 10 12 15 15 17
USAFA-TR-2005-4
List of Tables 1 2 3 4 A.1 A.2 A.3
Properties of panel materials. . . . . . . . . . . . . . . . . . . . . . . . Panel specifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Buckling test results for three panel configurations. . . . . . . . . . . . Buckling test results for chord-wise grindout panels. . . . . . . . . . . . Calculation of stiffener crippling load using the Johnson-Euler method. Calculation of segment section properties. . . . . . . . . . . . . . . . . Calculation of new segment section properties. . . . . . . . . . . . . . .
. 5 . 6 . 11 . 13 . A–1 . A–2 . A–3
v
USAFA-TR-2005-4
List of Symbols Aeff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . effective skin area As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . skin segment cross-sectional area Atot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . section cross-sectional area, A s + Aw Aw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . single stiffener cross-sectional a rea b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stiffener spacing C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . panel configuration #1 C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . panel configuration #2 C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . panel configuration #3 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . end-fixity condition d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . crosshead displacement E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Young’s modulus E c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Y o u n g ’ s m o d u l u s ( c o m p r e s s i v e ) H w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stiffener height I ox . . . . . . . . . . . . . . . . . . . . . . . . stiffener section second area moment about its own centroid K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . s l e n d e r n e s s r a t i o L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . original panel length L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equivalent length based on end-fixity condition m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerard equation parameter P cc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stiffener crippling load P co . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . segment Johnson-Euler allowable load P max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . max recorded load during test P s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . load at skin buckling ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . skin thickness tw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . s t i ff e n e r t h i c k n e s s t¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equivalent skin thickness, A tot/b W e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . effective skin width W f i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inboard flange width W f o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . outboard flange width Y c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . section centroid y . . . . . . . . . . . . . . . . . . . . . . . . distance referenced from outboard surface of outboard flange β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerard equation parameter δ g . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . depth of grindout as pct of skin thickness ∆L/L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . change in length per unit length at P max ∆P max . . . . . . . . . . . . . . . . . . . . . . . . . change in max load carrying capability due to damage εi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . axial strain in stiffener i ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radius of gyration σcc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . s t i ff e n e r c r i p p l i n g s t r e s s σco .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . segment Johnson-Euler allowable stress σcys .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compressive yield strength of the skin material σcyw .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compressive yield strength of the stiffener material σU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . material tensile ultimate strength σY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . m a t e r i a l t e n s i l e y i e l d s t r e n g t h
vi
USAFA-TR-2005-4
1
Introduction
The objective of this investigation was to determine the effect of corrosion grindouts on the compressive strength of B-707 upper wing skin panels. Three representative geometries were considered. One configuration represented the minimum strength wing panel (40 ksi, Configuration #1, or C1), and one represented the maximum strength wing panel (64 ksi, Configuration #3, or C3). A third configuration was also tested. These panels, denoted Configuration #2 (C2), were mistakenly manufactured with thicker stiffeners, but were tested nonetheless to provide additional data for this study. Two pristine panels in each configuration were tested, and damage was introduced into each of the other panels in the form of uniform spanwise grindouts. These grindouts ranged from 35% to 63% of the panel skin thickness, and were meant to simulate severe in-service corrosion grindouts.
2 2.1
Background The Problem
Current guidance for corrosion repairs to USAF aircraft includes the requirement to remove the visible products of corrosion. Removal of these products is often accomplished by grinding the surface of the material until pristine material is exposed. If the depth of the grindout exceeds a certain percentage of the thickness of the material, the material must be replaced. The limits specified in the Technical Orders for a particular aircraft are based on engineering judgement and analysis. The current effort attempts to quantify, through experiment, the degradation in buckling strength caused by a uniform grindout.
2.2
Approach
In this study, grindouts of known depth were made along the length of a panel stiffener, and the panel was tested to failure. Panels with no grindouts were are also tested for reference. The panels were manufactured by Valco Manufacturing, Inc. of Duncan, Oklahoma. Panels were received with four of five stiffeners attached (riveted) to the skin, with the center stiffener drilled, but not attached. These center stiffeners were attached after surface grindouts were machined into the specimen. The few panels meant for baseline testing (pristine panels) were received with all five stiffeners attached.
2.3
Previous Experimental Work
Significant relevant work has been done in the area of buckling of rib-stiffened panels as has some analytical work on corroded plates by Lakhote, Pandey, and Sherbourne [1], and by Roorda, Srivastava, Maslouhi, and Sherbourne [2]. However, none of this work has involved the sort of simulated grindout damage applied in the current effort. Still, it should be mentioned that Lakhote, et al. [1], in their analytical work on flat, unstiffened panels with deep, centrally-located square grindouts, found that a correct assessment of the pre-buckling redistribution of stresses is required to avoid seriously overestimating the reduction in buckling strength. As will be shown in the current work, even deep grindouts 1
USAFA-TR-2005-4
Figure 1: Approximate gage locations of Hickman and Dow [3], [4] based on text of references and two figures. (Six-stiffener panel shown; fastener pitch/size not to scale.)
caused relatively small reductions in buckling strength, and this stress redistribution in damaged panels could be an important consideration in any future analytical work. This section describes previous relevant work with a focus on the instrumentation and methods used to determine the initial panel buckling load. This skin buckling load is often much less than the maximum load a stiffened panel can ultimately carry. Hickman and Dow [3], [4] of NACA instrumented their six-stiffener panels with “...four 1 6 2 -inch resistance-type wire strain gages mounted on the quarter points along the length of the second and fifth stiffeners.” (They also performed tests on four- and five-stiffener panels.) These gages, which are not described further in their reports, were used to detect shortening per unit length of the panels. Although not described in the text, Figure 3 of Reference [3] and Figure 2 of Reference [4] both appear to show strain gage instrumentation on the panel skin (see Figure 1). These were likely used for detecting what the authors call the “local buckling load” using the strain-reversal method of Hu, Lundquist, and Batdorf [5]. In this method, local buckling is said to have occurred when a plot of the strains near the crest of a buckle first shows a decreasing strain with increasing load. (How the crest location was predicted or determined is not described.) These gages (if that is, in fact, what they are) are located on the stiffener side of the skin between stiffeners and near, but not always at, the mid-length line of the panels. 2
USAFA-TR-2005-4
Figure 2: Gage locations from Figure 2 of Reference [7]. White boxes indicate gages mounted on outer-most surface of Z-stiffener. Black boxes indicate gages mounted on flat side of panel. “(2)” indicates two gages in this location, one on each side of the skin. Fastener pitch/size not to scale.
Rothwell [6] measured the test panels for imperfections prior to testing, but collected no data other than maximum load carried during the displacement-controlled tests. Friedman, et al. [7] took strain and displacement measurements on their four-stiffener, Z-stiffened aluminum panels. Two strain gages were placed opposite each other on each side of the skin half-way between rivet rows and centered along the length. A strain gage was also placed on the top (outer-most) surface of each Z-stiffener, and these were also centered along the length (aligned with the skin gages). An additional gage was placed on the non-stiffener side of the skin, centered along the length, and directly underneath the vertical part of each Z-stiffener. Finally, two more gages were placed opposite each other on each side of the skin half-way between the central row of gages and the panel end1 , centered on the width of the panel. Therefore, a total of 16 gages were used to instrument a four-stiffener panel (see Figure 2). In their book, Singer, Arbocz, and Weller [8] suggest a number of methods to determine critical (buckling) load in plates under compression, including the aforementioned 1
Actually, these two gages were placed half-way between the central row of gages and the beginning of the potting for the panel end.
3
USAFA-TR-2005-4
Figure 3: Gage locations from Figures 2 and 3 of Reference [9]. Stiffener gages are located on the vertical flange of the L-stiffeners. Fastener pitch/size not to scale.
technique of Hu, et al. [5]. They also describe a method of detecting buckling by observing the inflection in the curve of the algebraic compressive strain average, εA = 12 (ε1 + ε2 ), versus axial load, where ε1 and ε2 are strains at the same location on opposite sides of the skin. Again, these strains are taken at a “crest” of a buckle. Butler, et al. [9] instrumented their panels at mid-length, with one gage on each stiffener and one gage centered on the stiffener side of the skin between stiffeners. Additional gages were placed to detect skin buckling (see Figure 3). The stiffeners were L-shaped, and the stiffener gages were placed 10 mm and 3 mm from the edge of the stiffener for the three-stiffener and four-stiffener panels, respectively. The gages served two purposes: (1) comparison with the optimization code used in their study and (2) monitoring the onset and advance of buckling. They found the strain gage information to be more accurate than load versus end displacement plots. Aalberg, et al. [10] did not use strain gages in their tests, but chose instead to monitor load versus displacement. They also measured out-of-plane displacement continuously throughout their displacement-controlled tests (displacement was approximately 1 mm per minute). After this review of the relevant literature, it was decided to instrument the panels as described in Section 3.2. 4
USAFA-TR-2005-4
3 3.1
Specimen Fabrication and Instrumentation Fabrication
Panels were manufactured by Valco Manufacturing, Inc. of Duncan, Oklahoma. Twentyseven panels were manufactured in three configurations. All panels consisted of a skin sheet with five evenly spaced Z-stiffeners. Panel configurations are shown in Figure 4. The skin was of 7075-T6 aluminum sheet material. The stiffeners were formed (bent) from 7075-0-BARE coil stock, then solution heat treated and aged per the SAE AMS2770G specification to the T62 temper before attaching them to the skin. The bend radii for the stiffeners was 4.3 mm (0.17 in). The material properties for the aluminum used to fabricate the panels are listed in Table 1. In addition to certifications provided by the aluminum manufacturers, Valco conducted in-house conductivity and hardness testing to confirm material properties.
Table 1: Properties of panel materials. Component Skin (7075-T6) Stiffener (7075-T62)
E
σU
σY
GPa (Msi) 71.7 (10.4) 71.7 (10.4)
MPa (ksi) 582.6–583.3 (84.5–84.6) 568.8 (82.5)
MPa (ksi) 515.8–517.1 (74.8–75.0) 508.9–509.5 (73.8–73.9)
Aluminum MS20470AD4-5 rivets, 2017 alloy, 1/8 in diameter, were used at 1/2 in pitch to attach the stiffeners. The close rivet spacing essentially eliminated inter-rivet buckling as a concern. The panels are approximately half-scale versions of the actual aircraft wing skin. They were scaled down to accommodate the capacity of the load frames used in the testing.
Figure 4: Cross section of stiffened panels. Measurements are in Table 2.
5
USAFA-TR-2005-4
Table 2: Panel specifications for Figure 4. Dimensions are mm (in). Config 1 2 3
ts
tw
W f i
W fo
H w
b
L
2.03 (0.080) 2.03 (0.080) 2.03 (0.080)
1.78 (0.070) 2.03 (0.080) 2.03 (0.080)
16.3 (0.64) 16.3 (0.64) 15.2 (0.60)
16.3 (0.64) 16.3 (0.64) 15.2 (0.60)
37.1 (1.46) 37.1 (1.46) 28.2 (1.11)
106 (4.19) 106 (4.19) 54.4 (2.14)
531 (20.9) 531 (20.9) 279 (11.0)
Panels were received assembled, except for those panels that were to receive grindouts. For these panels, the manufacturer drilled the skin and stiffeners with rivet holes, but did not attach the stiffeners. The grindouts were accomplished on the non-stiffener side in the form of a channel along the center rivet row. The center stiffener was then attached with the rivets. The grindouts and attachments were made using a flat end mill. The target grindout channel depths were of 30% and 50% skin thickness. The grindout edges were step-tapered on a slope of 25:1 (per Northrop-Grumman criterion) as shown in Figure 5. The actual depths of the grindouts are presented with the results in Section 4.
Figure 5: Step-tapered grindout geometry for simulated corrosion grindout. The grindout runs the entire length of the panel.
3.2
Instrumentation
Instrumentation for initial tests involved 13 strain gages: one on each of the five stiffeners, four between stiffeners, and four located in pairs at predicted skin buckling locations. After gaining some confidence in the loading scheme, the number of gages was reduced to five, applied to the panels as shown in Figure 7. For the short panels, skin buckling was more difficult to detect, so one additional pair of skin gages was used. The micromeasurements CEA-13-250UN-120 gages were used in this effort. The three stiffener gages were used to determine whether the loading on the panel was being evenly applied. Agreement of strains within 10% was taken as the criterion: max ε1
{| − ε |, |ε − ε |, |ε − ε |} × 100 ≤ 10 2
1
3
(ε1 + ε2 + ε3 ) /3
2
3
(1)
Typically, this check was performed at low load levels ( < 25% of the failure load), then minor adjustments were made to the fixed crosshead when necessary. Minor adjustments were made for almost every panel test. 6
USAFA-TR-2005-4
Figure 6: Typical linear buckling analysis of panel. Panel displacements (greatly exaggerated) are indicated by the deformed geometry and the colors.
The pair of gages on opposite sides of the skin was used to detect skin buckling. They were located through the use of a linear finite element buckling analysis, which indicated the locations of crests in the deformed geometry during skin buckling. A typical analysis result is shown in Figure 6.
The onset of skin buckling as measured by these gages was easily noticed in C1 and C2 (long) panels (see Figure 8 for typical behavior). The skin buckling load was recorded using the load reversal criterion of Hu, et al. [5]. On the short (C3) panels, the preferred skin gage location fell into the grindout taper area for some grindout cases. For these panels, the skin gage on the flat side of the panel had to be moved 1/4 in to keep it out of the grindout. Skin buckling occurred in these panels near the panel failure load, and was difficult to detect visually. Also, the strain gages used to detect skin buckling would, in some instances, give ambiguous readings due to the very high strains in the skin at these high loads. 7
USAFA-TR-2005-4
Figure 7: Initial (a) and final (b) Strain gage locations for buckling tests. Where two gage numbers are indicated, one gage is located on each side of the skin. The suffixes “B” and “C” on some gages indicate alternate locations tried at various stages of the program.
Figure 8: Typical long panel (C1 and C2) behavior of a gage pair located on opposite sides of the skin at the same location. These were used to detect skin buckling. Gages 1 and 2 are located on opposite sides of the skin at the same location.
8
USAFA-TR-2005-4
4
Experimental Results
4.1
Test Set-Up
The buckling tests were performed on a Satec model 120HVL electromechanical test machine having a capacity of 534 kN (120 kip). The load transducer was calibrated to ASTM E4-03 [11] prior to the start of testing. Data were recorded by the Satec computer controlling the machine (load and displacement) and by a LabVIEW2 system recording the strain gage output. The LabVIEW system also recorded the load and displacement by measuring the voltages on two channels from the Satec machine. The test set-up is shown in Figure 9.
Figure 9: Set up for panel buckling experiments.
4.2
Test Procedure
Panels were loaded in the stiffener direction as shown in Figure 9. The upper (fixed) crosshead had a 7.6 cm by 15.2 cm (3 in by 6 in) hole in it, so a top plate was fabricated from 12.7 mm- (0.500 in)-thick steel and bolted to the crosshead to cover the hole. The plate surface was milled flat to a tolerance of 0.025 mm ( 0.001 in). The bottom (moving) crosshead surface was flat and continuous. No potting or other modifications
±
2
±
LabVIEW is a registered trademark of National Instruments Corporation.
9
USAFA-TR-2005-4 to the panel ends (beyond milling flat) were performed. The panel ends rested directly on the flat metal supports (as in References [3] and [4]). The tests were run under displacement-controlled conditions at a commanded rate of 3.0 mm/min (0.12 in/min) to 10 kN, then 1.5 mm/min (0.06 in/min) to failure for C1 and C2 panels. For C3 panels, the loading rates were 1.5 mm/min (0.06 in/min) and 0.7 mm/min (0.028 in/min), respectively. These strain-rate based rates were chosen based on guidance in ASTM E9–89a [12]. Loading of the specimen was occasionally paused to inspect the specimen under load.
4.3
Test Results
The results are shown in Table 3 and Figure 11. The typical damage progression is shown in Figure 10. In two tests, the skin buckling strain gages were mislocated on the panel, so no skin buckling load is available.
Figure 10: Typical damage progression with load: (a) P = 0, (b) P > P s , (c) post-test. Foil in photo (b) is a shim used to equalize load between stiffeners. These are C1/C2 panels.
For most tests, agreement of results between similar configurations with similar damage was excellent. Of 25 tests, only 2 were rejected because of poor results. Panel 4 was rejected because is was realized after testing that it was mounted improperly in the test machine (it was not chord-wise flat at the beginning of the test). Panel 25 gave an anomalous result, and subsequent material tests indicated it had inferior material properties compared to other panels. The panel shortening was calculated as ∆L/L P
|
max
=
d P =P
|
max
− d| L
P =2.5 kip
(2)
By using 11.1 kN (2.5 kip) as the lower displacement bound, the initial settling response (which varies significantly from panel to panel) is removed. 10
USAFA-TR-2005-4
Table 3: Buckling test results for three panel configurations. All grindouts are span-wise. Panel Config ∆L/L P Strength Chga δ g P s P max No. No. Pct kN (kip) kN (kip) Pct µstrain 1 1 0% 154 (35) 405.7 (91.21) 7120 2 1 0% 166 (37) 412.5 (92.73) 7700 b 4 1 37.4% 139 (31) 354.9 (79.80) 8273 5 1 34.8% 154 (35) 395.4 (88.90) 6885 3% 7 1 56.1% 118 (27) 376.7 (84.70) 7062 8% 8 1 57.4% 119 (27) 382.4 (85.97) 7316 7% c 9 1 34.8% 150 (34) 396.6 (89.16) 7234 3% 10 2 Fixture Failure 11 2 0% 184 (41) 445.1 (100.07) 8180 12 2 0% 167 (38) 444.7 (99.97) 8270 13 2 36.1% 144 (32) 420.7 (94.58) 7750 5% 14 2 41.9% 148 (33) 425.2 (95.59) 7870 4% d 15 2 36.4% NR 437.6 (98.38) 7860 2% 16 2 40.6% 163 (37) 432.7 (97.28) 7710 3% 17 2 52.9% NR 423.4 (95.20) 7820 5% 18 2 Fixture Failure 19 3 0% 388 (87) 422.2 (94.93) 11,030 20 3 0% 409 (92) 425.8 (95.73) 10,430 22 3 41.3% 326 (73) 384.9 (86.53) 10,210 9% 23 3 41.3% 302 (68) 383.8 (86.29) 9640 9% b, e 25 3 56.1% 245 (55) 358.2 (80.52) 9782 16% 26 3 62.6% 251 (56) 375.7 (84.47) 10,680 11% 27 3 60.0% 234 (53) 370.1 (83.20) 10,227 13%
|
max
− − − − − − − − −
− − − − −
a
Compared to average of pristine results Result discarded. See text. c Approximate d “Not Recorded” (see text) e This panel had an unusual surface finish. Hardness testing revealed a lower strength (about 10%) than other panels. b
11
USAFA-TR-2005-4
Figure 11: Panel strength as a function of grindout depth for 3 configurations with span-wise grindouts.
4.3.1
Span-Wise Grindouts
Pristine C1, C2, and C3 panels were tested to establish baseline strength values. Panels with grindouts of various depths were then tested to determine the degradation in strength due to these grindouts. It must be remembered that this result assumes these important conditions: (1) the grindout runs parallel to and directly over the center stiffener and is symmetric with respect to the fastener row, (2) the loading is uniaxial and parallel to the stiffener, (3) the stiffener is securely reattached to the panel after the grindout is applied (i.e., no loose rivets), and (4) the stiffener is undamaged. The results indicate a slight degradation in buckling strength due to the grindouts, with the C3 panels being the most affected.
4.3.2
Chord-Wise Grindouts
A small number of extra panels were available for testing, and these were used to make an assessment of chord-wise grindouts on panel strength. The chord-wise grindouts were much more damaging to the panels than were the span-wise. Three reasons for this are (1) the span-wise grindout affects the integrity of only one stiffener bay, while the chord-wise grindout affects all stiffener bays, (2) the chord-wise grindout admits another failure mode, namely inter-rivet buckling, which can 12
USAFA-TR-2005-4 fail the rivets locally leading to zero effective skin width in that area, and (3) the chordwise damage has the effect of creating a hinge that leads to low-load out-of-plane skin buckling along the grindout, which helps destabilize the stiffeners. The results of the tests on panels with chord-wise grindouts are shown in Table 4. These panels with grindouts exhibited skin buckling at very low loading, so no skin buckling loads are shown for these panels. Table 4: Buckling test results for chord-wise grindout panels. Panel Config ∆L/L P δ g P s P max No. No. Pct kN (kip) kN (kip) µstrain 1 1 0% 154 (35) 405.7 (91.21) 7120 2 1 0% 166 (37) 412.5 (92.73) 7700 a 3 1 44.7% NR 202.5 (44.53) 4813 6 1 48.6% NR 209.8 (47.18) 4139 19 3 0% 388 (87) 422.2 (94.93) 11,030 20 3 0% 409 (92) 425.8 (95.73) 10,430 21 3 70% NR 259.1 (58.26) 8427 24 3 70% NR 241.4 (54.28) 9109
|
max
a
“Not Recorded” (see text)
4.4
Errors in Experimental Values
Sources of error in these experiments include load cell accuracy, strain gage accuracy, evenness of panel loading, grindout channel accuracy, and panel imperfections (other than grindouts). In addition, the detection of skin buckling is somewhat inexact. Because of the data collection rate, the specimen loading rate, the uncertainly in gage location, and the somewhat arbitrary choice of the “instant” of buckling (change in sign of strain slope) the skin buckling load should be considered to be no better than about 5 kN ( 1 kip) in accuracy (this represents approximately 1% of the buckling load). Load cell accuracy affects P s and P max data values. During calibration, an accuracy of 0.5% for both accuracy and repeatability was noted. However, this accuracy includes a coverage factor that provides a confidence level of 95%. During the actual calibration runs, the maximum absolute error seen was 21.547 lb at a load of 84 000 lb, or 0.0026%. Strain gages of the type used have a typical accuracy of 5%. This would affect load leveling prior to each test and the measurement of ∆ L/L P at the conclusion of each test. As was mentioned in Section 3.2, strain variation between stiffeners was used as a measure of load evenness. This was typically done at or below 100 kip at strains near 1000 µε. The strain gage error at these levels, at worst case, would widen the load leveling range from 10% to 15%. Quantifying the effect of the load unevenness is problematic, but the very high repeatability in pristine panel results indicates that the load leveling was successful. Even with the use of jigs specially made for the purpose, achieving the desired grindout depth to a close tolerance was difficult. However, based on measurements taken at the grindout ends, uniformity of the grindout along its length appeared to be good (i.e.,
± ±
±
±
±
±
± |
max
±
13
USAFA-TR-2005-4 within typical machine shop tolerance of 0.003 in). Future plans include using an NDI technique to better assess the uniformity of the grindout along its length. The panel specifications provided to Valco included requirements for sheet flatness ( 3/16 in), lateral bow ( 1/16 in), squareness ( 3/32 in), thickness variation ( 0.0025 in), width and length ( 1/16 in), and perpendicularity of stiffeners to the sheet surface ( 1 ). Panels were spot checked for compliance with these specifications and were found to be of very high quality and consistency (it was clear that they were manufactured on computer-controlled equipment). Furthermore, since parallel panel ends were of paramount importance, the panels were milled (in house) on the ends to ensure they were parallel prior to testing (0.003 in vs. the 1/16 in specified). Panel imperfections were, therefore, probably a small contributor to experimental errors. To summarize, the errors in load measurement were extremely small, and so the ultimate strength of each panel is very likely within 20 lb of that indicated in the results. Because these results are reported based on the actual measured grindout depths, the fact that the originally planned grindout depths of 30% and 50% of the skin thickness were not achieved introduces no error in the results. Since the evenness of loading (as determined by strain readings) could be off by as much as 15%, this becomes the most significant potential source of error in the results. However, the magnitude of this error is impossible to quantify without significant analytical or experimental work, and the repeatability of the process and the results indicates that this error is likely small as well.
≤ ±
≤
±
◦
≤
±
±
5
Modeling Panel Behavior
The ultimate strength of the pristine (no grindout) panels was calculated using standard analysis techniques. The Johnson-Euler method was used for C1 and C2 (long) panels, and the method of Gerard was used for C3 (short) panels. The Johnson-Euler method requires an end-fixity condition to be assumed for the calculation of a slenderness ratio. For the panels in this study, an end-fixity coefficient of 3.75 was used. For flat-ended specimens tested between rigid flat anvils, this value has been shown to be appropriate [13].
5.1
Long Panels (C1 and C2)
The Johnson-Euler method was used for the longer (C1 and C2) panels (for a description of this method, see, e.g., Ref [14]). This method calculates an allowable column stress, σco , as a function of the column cross sectional crippling strength, σ cc , as
σco = σ cc 1
σ cc (L /ρ)2 4π2 E
−
(3)
For C1 and C2 panels, the J-E strength is calculated to be 406.5 kN (91.4 kip) and 456.8 kN (102.7 kip), respectively. The actual strengths were found to be 409.2 kN (92.0 kip) and 444.8 kN (100.0 kip) for respective errors of +0.7% and 2.6%. For panels with grindouts, the analysis was identical, except that the skin thickness for the entire center section was reduced to the minimum skin thickness in the bottom of the grind-out. See Appendix 6 for some worked examples.
−
14
USAFA-TR-2005-4 Comparisons of these analyses to the experimental data are in Figures 12 and 13.
Figure 12: Comparison of C1 panel results with modified Johnson-Euler method.
Figure 13: Comparison of C2 panel results with modified Johnson-Euler method.
15
USAFA-TR-2005-4
5.2
Short Panels (C3)
The method of Gerard was used for the short (C3) panels (for a description of this method, see, e.g., Ref [15] or Ref [16]). This method calculates the stiffener yielding stress, σ cy , as
¯cy = σ
[σcys + σcyw (t¯/ts t¯/ts
− 1)]
(4)
and the section failure stress as
m
¯f = σ ¯cy β σ
g t t E s w
Atot
¯cy σ
(5)
and therefore the section failure load as
¯f Atot P f = σ
(6)
For the short panel, this leads to a failure strength of 478.2 kN (107.5 kip). The actual failure strength of this configuration was 424.0 kN (95.32 kip), so the error in this method for the pristine panel is +12.8% 3 . For C3 panels with grindouts, the Gerard method was used again, except (as in the long panel case) that the skin thickness for the entire center section was reduced to the minimum skin thickness in the bottom of the grind-out. See Appendix 6 for some worked examples. A comparison of this analysis to the experimental data is in Figure 14.
3
The J-E method calculates a pristine strength of 427.9 kN (96.2 kip), an error of +0.9%, but the J-E method does a very poor job modeling behavior of damaged short panels (see Section 5.3).
16
USAFA-TR-2005-4
Figure 14: Comparison of C3 panel results with modified method of Gerard. Dotted line indicates Johnson-Euler model for this panel.
5.3
Modeling Summary
The slenderness ratios for panel configurations 1, 2, and 3 are 22, 22, and 17, respectively. The behavior of the damaged longer panels (C1 and C2) is well described by Johnson-Euler theory, while the method of Gerard does a much better job describing the shorter panels’ behavior. Gerard’s method for stiffened panels is a modified version of his method for predicting the crippling stress of plates and is applicable when the interfastener buckling and wrinkling stresses are greater than the crippling stress [15, p. 488] (as is the case for all the panel configurations with span-wise grindouts tested in this study). Gerard’s method is for panels that buckle in the inelastic range. The C3 panels exhibit considerably more plastic deformation than do either the C1 or C2 panels (see column 6 in Table 3). This limited test program did not attempt to find where the “crossover point” is, that is, to find the slenderness ratio at which one theory describes the behavior better than the other. No attempt was made to model the behavior of panels with chord-wise grindouts. 17
USAFA-TR-2005-4
6
Conclusions
The test results indicate that uniform grindouts along the stiffener length, provided the middle stiffener (only) is affected, and the fastener is reattached snugly with new rivets, caused only minor degradation in panel buckling strength. (Assessing degradation to fatigue properties is the topic of ongoing work, but is not addressed here.) Moreover, by modifying the analysis methods of Johnson, Euler, and Gerard, curves can be generated that depict strength degradation as a function of grindout depth for these panels. The shorter (C3) panels experienced significantly more plasticity prior to collapse than did the longer (C1 and C2 panels). It is for this reason that the Gerard method was more appropriate, and better at modeling, the C3 panels. These curves give the structural engineer a new tool for assessing the degradation in strength to B-707 upper wing skin panels due to grindouts.
18
USAFA-TR-2005-4
Acknowledgements The authors would like to thank Lt Col P. J. Clark of ASC/AAA and Mr. Kevin Boyd of S&K Technologies, Inc. for their sponsorship of this effort. We also thank Mr. Stephan Verhoeven, Mr. Cornelis Guijt (Engineers), Mr. Chad Moon, and Mr. Jonathan Ingram (technicians) for their invaluable assistance on this project. Finally, we are indebted to Mr. John Lobdell and MSgt Michael Nero (Department of Civil and Environmental Engineering) for the generous use of their facilities for these tests.
19
USAFA-TR-2005-4
References [1] R. Lakhote, M. Pandy, and A. Sherbourne, “Buckling Behavior of Corroded Plates,” in Proceedings of the Fourteenth ASCE Engineering Mechanics Conference , (Austin, TX), May 2000. [2] J. Roorda, N. Srivastava, A. Maslouhi, and A. Sherbourne, “Residual Strength of Ship Structures with Corrosion-Induced Damage,” Technical Report, Defense Research Establishment Atlantic, Halifax, NS, Mar. 1996. [3] W. A. Hickman and N. F. Dow, “Data on the Compressive Strength of 75S-T6 Aluminum-Alloy Flat Panels with Longitudinal Extruded Z-Section Stiffeners,” Technical Note 1829, National Advisory Committee for Aeronautics, Langley Air Force Base, VA, Mar. 1949. [4] W. A. Hickman and N. F. Dow, “Data on the Compressive Strength of 75S-T6 Aluminum-Alloy Flat Panels Having Small, Thin, Widely Spaced, Longitudinal Extruded Z-Section Stiffeners,” Technical Note 1978, National Advisory Committee for Aeronautics, Langley Air Force Base, VA, Nov. 1949. [5] P. C. Hu, E. E. Lundquist, and S. Batdorf, “Effect of Small Deviations from Flatness on Effective Width and Buckling of Plates in Compression,” Technical Note 1124, National Advisory Committee for Aeronautics, Langley Air Force Base, VA, 1946. [6] A. Rothwell, “An Experimental Investigation of the Post-Buckled Efficiency of ZSection Stringer-Skin Panels,” Aeronautical Journal , pp. 29–33, Jan. 1981. [7] R. Friedman, J. Kennedy, and D. Royster, “Analysis and Compression Testing of 2024 and 8009 Aluminum Alloy Zee-Stiffened Panels,” Transactions of the ASME: Journal of Engineering Materials and Technology , vol. 116, pp. 238–243, Apr. 1994. [8] J. Singer, J. Arbocz, and T. Weller, Buckling Experiments: Experimental Methods in Buckling of Thin-Walled Structures; Basic Concepts, Columns, Beams, and Plates - Volume I . West Sussex, England: John Wiley & Sons, 1998.
[9] R. Butler, M. Lillico, H. G.W., and N. McDonald, “Experiments on Interactive Buckling in Optimized Stiffened Pnels,” Struct Multidisc Optim , vol. 23, pp. 40–48, 2001. [10] A. Aalberg, M. Langseth, and P. Larsen, “Stiffened Aluminum Panels Subjected to Axial Compression,” Thin-Walled Structures , vol. 39, pp. 861–885, 2001. [11] ASTM, “Standard Practices for Force Verification of Testing Machines,” Standard E4–03, American Society for Testing and Materials, West Conshohocken, PA, 2003. [12] ASTM, “Standard Test Methods of Compression Testing of Metallic Materials at Room Temperature,” Standard E9–89a, American Society for Testing and Materials, West Conshohocken, PA, 2000. 20
USAFA-TR-2005-4 [13] R. Papirno, “Inelastic Buckling of ASTM Standard E 9 Compression Specimens,” Journal of Testing and Evaluation, JTEVA , vol. 15, pp. 133–135, May 1987. [14] M. C.-Y. Niu, Airframe Structural Design . Hong Kong: Conmilit Press LTD, 1998. [15] R. M. Rivello, Theory of Analysis of Flight Structures . New York: McGraw-Hill, 1969. [16] G. Gerard, “Handbook of Structural Stability: Part V - Compressive Strength of Flat Stiffened Panels,” Technical Note 3785, National Advisory Committee for Aeronautics, Washington, Aug. 1957. [17] D. J. Peery and J. J. Azar, Aircraft Structures . New York: McGraw-Hill, second ed., 1982.
21
USAFA-TR-2005-4
This Page Intentionally Left Blank
22
USAFA-TR-2005-4
Appendix: Sample Calculations The following examples indicate how the Johnson-Euler method and the method of Gerard were used to calculate panel buckling strengths for pristine panels and panels with span-wise grindouts. English units are used for these examples.
A.1
Johnson-Euler Method
The Johnson-Euler method was employed as follows (this analysis is for a C1 panel). First, the stiffener crippling allowable stress, σ cc, is calculated. The stiffener is divided into 3 segments. Segments 1, 2, and 3 are the inboard (next to the skin), vertical, and outboard segments of the stiffener, respectively. Table A.1 shows how the crippling load is calculated [17].
Table A.1: Calculation of stiffener crippling load using the Johnson-Euler method. Edge bi ti bi ti σcci bi ti σcci σ cys b Seg Cond (in) (in) (b/t)i (in)2 (lb) σcci /σcys (psi) E c t 1 Free 0.604 0.072 8.389 0.043 2 Fixed 1.388 0.072 19.278 0.100 3 Free 0.604 0.072 8.389 0.043 Sum Aw = 0.187
0.707 1.624 0.707
0.7491 0.9591 0.7491
55,320 2,406 70,831 7,079 55,320 2,406 P cc = 11,890
The crippling stress is then σcc = P cc /Aw = 63, 613 psi
(7)
The effective skin area, Aeff , is found by calculating the effective skin width, W e , which is based on the stiffener crippling stress, the material modulus, and the skin thickness: 2W e = 1.70ts
E
c
σcc
= 1.685 in
(8)
and the effective skin area is then Aeff = 2W ets = 0.131 in2
(9)
The segment section allowables may now be calculated as shown in Table A.2, where Segment 4 is now the effective skin segment, and Segments 1–3 are the stiffener segments of Table A.1. A–1
USAFA-TR-2005-4
Table A.2: Calculation of segment section properties. Width Height A y Ay Ayy 2 3 Seg (in) (in) (in) (in) (in) (in)4 1 0.640 0.072 0.0461 0.0360 0.0017 0.0001 2 0.072 1.316 0.0948 0.7300 0.0692 0.0505 3 0.640 0.072 0.0461 1.4240 0.0656 0.0934 4 1.685 0.078 0.1306 1.4988 0.1957 0.2933 Totals 0.3175 0.3321 0.4373 Σ() ΣA ΣAy ΣAyy
Iox (in)4 0.0000 0.0137 0.0000 0.0001 0.0138 ΣI ox
The segment section centroid is Y c = ΣAy/ΣA = 1.046 in, and the segment area moment is I = ΣI ox + ΣAyy Y c2 ΣA = 0.104 in4 . To complete the segment allowable stress, we need the slenderness ratio, K , of the segment
−
L L = = 18.89 K = ρ ρ c
√
(10)
where L = 20.9 in, ρ = I/A = 0.571 in, and c, the end fixity coefficient, is 3.75. The Johnson-Euler stress, σco , and strength, P co , allowables are then given by recall σ cc = 63, 613 psi
σco = σ cc 1
σ cc (L /ρ)2 = 60, 095 psi 4π2 E c
−
P co = σ co A = 19, 078 lb
(11)
(12) (13)
This leads to a panel allowable of (P co )panel = σ co Apanel = 90, 526 lb
(14)
where A panel = (N 1)A + Aw + ts W f o = 1.506 in2 . This answer can be iterated upon by using σ co as σ cc in Eq (8), creating a refined W e:
−
2W e = 1.70ts
E
c
σco
= 1.733 in
(15)
and the new effective skin area is then Aeff = 2W ets = 0.134 in2
(16)
a 2.9% increase. The new segment section allowables may now be calculated as above (Table A.3). A–2
USAFA-TR-2005-4
Table A.3: Calculation of new segment section Width Height A y Ay 2 Seg (in) (in) (in) (in) (in)3 1 0.640 0.072 0.0461 0.0360 0.0017 2 0.072 1.316 0.0948 0.7300 0.0692 3 0.640 0.072 0.0461 1.4240 0.0656 4 1.733 0.078 0.1343 1.4988 0.2013 Totals 0.3212 0.3378 Σ() ΣA ΣAy
properties. Ayy Iox 4 (in) (in)4 0.0001 0.0000 0.0505 0.0137 0.0934 0.0000 0.3017 0.0001 0.4457 0.0138 ΣAyy ΣI ox
The new segment section centroid is Y c = ΣAy/ΣA = 1.051 in, and the new segment area moment is I = ΣI ox +ΣAyy Y c2 ΣA = 0.104 in4 . The new slenderness ratio becomes
−
L L = = 18.94 K = ρ ρ c
√
(17)
and the Johnson-Euler stress, σco , and strength, P co , allowables are now given by recall σ cc = 63, 613 psi
σco = σ cc 1
σ cc (L /ρ)2 = 60, 079 psi 4π2 E c
−
P co = σ co A = 19, 300 lb
(18) (19) (20)
This leads to a panel allowable of (P co )panel = σ co Apanel = 91, 408 lb
(21)
where Apanel = (N 1)A + A w + t s W f o = 1.521 in2 . Because the column allowables changed only 0.026% in this iteration, we can consider this answer satisfactory.
−
A.1.1
Damaged Panels
For the damaged panels, the calculation is the same, except that the skin thickness, t s is reduced to that corresponding to the maximum grindout depth, and is assumed to act over the entire (single) segment. That would need to first be accounted for in Eq (8) and carried through all calculations for the damaged segment (only). Other segments would be calculated as above. For a C1 panel with a 57% grindout, for example, these calculations would result in a panel allowable of P co = 84, 810 lb. ∗
A.2
Method of Gerard
The following calculation is for a C3 (short) panel, which represents the 64 ksi allowable panel on the B-707. A–3
