MP Beams in IDT, C_Moisiade_08-08-2009.pdf

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BEAMS WITH FLAT STIFFENED WEBS IN

INCOMPLETE DIAGONAL-TENSION

by

Cezar I. Moisiade

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING IN MECHANICAL ENGINEERING

Approved:

_________________________________________Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

August, 2009


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Copyright 2009

by

Cezar I. Moisiade

All Rights Reserved

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CONTENTS

LIST OF TABLES............................................................................................................ vi

LIST OF FIGURES ......................................................................................................... vii

LIST OF SYMBOLS...................................................................................................... viii

ACKNOWLEDGMENT .................................................................................................. xi

ABSTRACT .................................................................................................................... xii

1. INTRODUCTION / BACKGROUND........................................................................ 1

2. METHODOLOGY, ULTIMATE STRENGTH OF BEAMS IN INCOMPLETEDIAGONAL TENSION .............................................................................................. 3

2.1 Limitations and Assumptions of IDT Theory .................................................... 4

2.2 Recommended Design Limitations.................................................................... 4

2.3 Web, Post-Buckling Analysis ............................................................................ 5

2.3.1 Shear buckling coefficient for simply supported panel (Kss) ................. 5

2.3.2 Web fixity coefficients (Ru& Rf) .......................................................... 5

2.3.3 Critical shear stress (Fscr) ....................................................................... 7

2.3.4 Diagonal-tension factor (k) .................................................................... 8

2.3.5 Angle of diagonal-tension (

) ................................................................ 8

2.3.6 Flange flexibility factor (wd).................................................................. 9

2.3.7 Angle and stress concentration factors (c1, c2, c3).................................. 9

2.3.8 Web peak nominal stress (fs_max).......................................................... 10

2.3.9 Web nominal stress allowable (Fs_all)................................................... 10

2.3.10 Web Margin of Safety (MSweb) ............................................................ 10

2.4 Upright Analysis .............................................................................................. 11

2.4.1 Upright column buckling ..................................................................... 11

2.4.2 Upright forced crippling....................................................................... 13

2.5 Analysis of Fasteners ....................................................................................... 14

2.5.1 Web To Flange Fasteners..................................................................... 14

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2.5.2 Upright to Flange Fasteners ................................................................. 14

2.5.3 Upright to Web Fasteners .................................................................... 15

2.6 Flange Analysis................................................................................................ 16

2.6.1 Compression Flange............................................................................. 16

2.6.2 Tension Flange ..................................................................................... 16

2.7 Web Stress Components .................................................................................. 17

3. NUMERICAL ANALYSIS OF A BEAM IN INCOMPLETE DIAGONAL

TENSION .................................................................................................................. 18

3.1 Input Data for IDT Analysis of beam III-25-6D [Ref. # 5, pg. 36] ................. 19

3.2 Limitations IDT Theory, Verification.............................................................. 20

3.3 Web, Post-Buckling Analysis .......................................................................... 21

3.3.1 Shear buckling coefficient for simply supported panel (Kss) ............... 21

3.3.2 Web fixity coefficients (Ru& Rf) ........................................................ 21

3.3.3 Critical shear stress (Fscr) ..................................................................... 23

3.3.4 Diagonal-tension factor (k) .................................................................. 24

3.3.5 Angle of diagonal-tension () .............................................................. 24

3.3.6 Flange flexibility factor (wd)................................................................ 25

3.3.7 Angle and stress concentration factors (c1, c2, c3)................................ 25

3.3.8 Web peak nominal stress (fs_max).......................................................... 26

3.3.9 Web nominal stress allowable (Fs_all)................................................... 26

3.3.10 Web Margin of Safety (MSweb) ............................................................ 26

3.4 Upright Analysis .............................................................................................. 27

3.4.1 Upright column buckling ..................................................................... 27

3.4.2 Upright forced crippling....................................................................... 29

3.5 Fasteners Analysis............................................................................................ 30

3.5.1 Web To Flange Fasteners..................................................................... 30

3.5.2 Upright to Flange Fasteners ................................................................. 30

3.5.3 Upright to Web Fasteners .................................................................... 31

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3.6 Flange Analysis................................................................................................ 32

3.6.1 Compression Flange............................................................................. 32

3.6.2 Tension Flange ..................................................................................... 32

3.7 Web Stress Components .................................................................................. 33

4. RESULTS AND COMPARISON WITH TEST DATA........................................... 34

4.1 Margins of Safety Summary ............................................................................ 34

4.2 Analytical vs. Test Results, Comparison ......................................................... 35

5. CONCLUSIONS ....................................................................................................... 37

REFERENCES ................................................................................................................ 38

APPENDIX A. ATTACHED ELECTRONIC FILES.................................................... 39

APPENDIX B. FINITE ELEMENT ANALYSIS - PRELIMINARY........................... 40

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LIST OF TABLES

Table 1. Margin of Safety Summary ............................................................................... 34

Table 2. Analytical vs. Test Results, Comparison........................................................... 35

Table 3. Current Methodology vs. NACA Analytical Prediction ................................... 36

Table 4. Analytical Predictions vs. Test Results ............................................................. 36

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LIST OF FIGURES

Figure 1. Beam with Stiffened Webs in IDT, Tested by NACA....................................... 2

Figure 2. Beam with Thin Stiffened Webs in Incomplete Diagonal Tension. .................. 3

Figure 3. Finite Element Model and Boundary Conditions............................................. 41

Figure 4. Eigen-Buckling Results, 44th

Eigenvalue, Relative Z Displacement [in]. ....... 42

Figure 5. Nonlinear-Buckling Results, Z Displacement [in]. .......................................... 43

Figure 6. Nonlinear-Buckling Results, 1stPrincipal Stress [psi]. .................................... 44

Figure 7. Nonlinear-Buckling Results, Shear Stress [psi]. .............................................. 44

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LIST OF SYMBOLS

DT diagonal-tension

PDT pure diagonal-tension

IDT incomplete diagonal-tension

Afc cress-sectional area of compression cap-flange, in2

Aft cress-sectional area of tension cap-flange, in2

Au cress-sectional area of upright, in2

Aue effective cress-sectional area of upright, in2

bu width of outstanding leg of upright, in

c1 angle factor.

c2, c3 stress concentration factors.

cc, ct distance from centroid of cap-flange to extreme fiber of flange, in

CR upright column buckling reduction factor.

d spacing of uprights, in

dc clear upright spacing, measured as shown in Figure 2

E elastic modulus, psi

eu distance from median plane of the web to centroid of (single) upright, in

ef distance from median plane of the web to centroid of (single) upright, in

fu upright stress caused by diagonal-tension, psi

fu_max maximum upright stress caused by diagonal-tension, psi

fs shear stress applied to web, psi

fs_max web peak nominal stress, psi

ffc stress in compression flange caused by diagonal-tension effect, psi

fft stress in tension flange caused by diagonal-tension effect, psi

Fs_all web nominal stress allowable, psi

Fc upright column buckling allowable, psiFcc upright crippling allowable, psi

Ffc upright forced-crippling allowable, psi

Fty yield tension allowable, psi

Ftu ultimate tension allowable, psi

Fsu ultimate shear allowable, psi

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Fscr_el elastic critical shear stress, psi

Fscr critical shear stress corrected for plasticity effects, psi

h depth of beam, in

he effective depth of beam, measured between centroids of flanges, in

hc clear dept of web, measured as shown in Figure 2, in

hu upright length, measured between controids of upright-to-flange rivet

patterns, in

Ic compression cap-flange cross-sectional moment of inertia about neutral

axis, in4

It tension cap-flange cross-sectional moment of inertia about neutral axis,

in4

Iu upright cross-sectional moment of inertia about neutral axis, in4

k diagonal-tension factor

Kss theoretical shear buckling coefficient for a simply supported plate

Le effective upright length, in

Mfc moment in compression cap flange, not related to DT, in-lb

Mft moment in tension cap flange, not related to DT, in-lb

Mf_max maximum flange primary bending moment caused by DT effect, in-lb

matu flag, defining upright material type

matw flag, defining web material type

Nu flag, defining number of uprights

Nuf upright to flange, number of fasteners (one end only)

Ngusset numbers of upright fasteners reacting upright load in gusset action.

Ps load applied to the beam that generates shear q in the web

Pu load in upright, not related to DT, to upright, lb

Pu_DT load in upright, caused by DT, lb

Puf_all upright to flange fasteners, total joint shear allowable, considering gusset

action, lb

Ptens_ult upright fasteners, required ultimate tension strength, lb

Pfc load in compression cap flange, not related to DT, lb

Pft load in tension cap flange, not related to DT, lb

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Pwf_shear_ult web to flange fasteners shear ultimate allowable, lb

Puf_shear_ult upright to flange fasteners shear ultimate allowable, lb

Puw_shear_ult upright to web fasteners shear ultimate allowable, lb

Puw_tens_ult upright to web fasteners tensile ultimate allowable, lb

Puu_shear_ult upright-to-upright fasteners shear ultimate allowable (for double uprights

only), lb

q shear flow in web, lb/in

qf shear flow reacted by the flange fasteners, lb/in

qu required upright fasteners shear flow to prevent premature column

buckling (for double fasteners only).

qu_all upright fasteners single shear allowable (for double fasteners only).

Qu static moment about neutral axis upright (for double uprights), in3

Rf web fixity coefficient at the flange

Ru web fixity coefficient at the uprights

swf web to flange fasteners spacing, in

suw upright to web fasteners spacing, in

tf thickness of flange, in

tu thickness of upright, in

tw web thickness, in

wd flange flexibility factor.

PDT angle of pure diagonal tension relative to natural axis of the beam, deg.

angle of incomplete diagonal tension relative to natural axis of the beam,

deg.

u upright cross-section centroidal radius of gyration about axis parallel to

web, in

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ACKNOWLEDGMENT

I wish to dedicate my work to my son that will be born in few months, and express

my love and gratitude to my beloved wife; for her understanding, patience and endless

love, through the duration of my studies.

I would like to convey thanks to Ernesto Gutierrez-Miravete, my project adviser, for

his guidance and valuable feedback, during the completion of my Master Project and

during my graduate studies at RPI.

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ABSTRACT

In aeronautical applications, beams with thin stiffened webs are often designed

considering the post-buckling capability of the web under shear load. Web buckling

under shear load does not represent failure. The web has additional post-buckling

capability to carry load in diagonal-tension.

The analysis of post-buckling webs is tedious and time consuming, and the use of a

numerical program that incorporates the methodology and performs the calculations is

desired.

The effort for the current project was focused on developing a numerical program using

MathCad, for analyzing beams in incomplete diagonal tension.

The current report presents the methodology and a numerical analysis for predicting

ultimate failure of beams in incomplete diagonal tension. The numerical analysis was

performed for a beam that was tested by National Advisory Committee for Aeronautics

(NACA) in reference #5. An evaluation of the analytical results and a comparison with

the test results from reference #5 was performed in order to validate the methodology.

The analytical prediction was different by only 2.6% from the actual failure resulted

from test.

A MathCad file including the program that performs the analysis of beams in

incomplete diagonal tension is attached in Appendix A.

The curve-fits for the charts from reference #1 were completed in Microsoft Excel

and a file including the resulted data is attached in Appendix A.

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1.INTRODUCTION / BACKGROUNDThe development of the diagonal-tension webs it was an outstanding step forward in the

structural aeronautical design. Original work on beams in diagonal-tension was

performed by National Advisory Committee for Aeronautics (NACA) in 1928 and

documented in reference #6. The most complete theory of beams in incomplete diagonal

tension was developed by NACA in 1952, and presented in references #1 and #5.

Additional improvements were developed in 1969 by a NASA funded program, and

performed by Grumman Aerospace, presented in reference #2.

Post-buckling capability of a beam with stiffened thin webs, under shear load, is far

greater then the load producing buckling of the web. The structure does not fail when the

web buckles; the web forms diagonal fold and functions as a series of tension diagonals,

while the stiffeners act as compression posts. The web-stiffener system changes from a

structure with shear resistant webs towards a truss structure. When the structure works

as a truss, the web carries the entire load in diagonal-tension and none in shear, the web

is in a state of pure diagonal-tension.

A shear-resistant web carries the entire load in shear and none in diagonal-

tension. Truly shear-resistant webs are possible but rare in aeronautical practice.

Practically, all webs fall into the intermediate region of incomplete diagonal tension,

where the web carries part of the load in shear, and the rest of it is carried in diagonal

tension. The state of incomplete diagonal tension is an interpolation between the

theoretical states of shear-resistant and pure diagonal tension.

The analysis of beams with stiffened webs, in incomplete diagonal tension, is

tedious and time consuming, and the use of a numerical program that incorporates the

methodology and performs the calculations is desired. The effort for the current project

was focused on developing a numerical program using MathCad, for analyzing beams in

incomplete diagonal tension.The methodology used in the current report, for predicting failure of beams with

stiffened webs in incomplete diagonal tension is based on the theory and empirical data

form references #1 to 4.

The numerical analysis was performed for a beam that was tested by NACA in

reference #5. An evaluation of the analytical results and a comparison with the test

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results from reference #5 was performed in order to validate the methodology. The

analytical prediction was different by only 2.6% from the actual failure resulted from

test.

A MathCad file including the program that performs the analysis of beams in

incomplete diagonal tension is attached in Appendix A.

The completion of a program that performs IDT analysis, required having available

equations for all the charts from reference #1. The curve-fits were completed in

Microsoft Excel and a file including the resulted data is attached in Appendix A.

An example of a beam with stiffened webs in incomplete diagonal tension, tested by

NACA in reference #5, is shown in Figure 1, where the diagonal web wrinkles can be

seen.

Figure 1. Beam with Stiffened Webs in IDT, Tested by NACA1

.

1Reference #1, page 103.

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2.METHODOLOGY, ULTIMATE STRENGTH OF BEAMSIN INCOMPLETE DIAGONAL TENSION

The methodology presented below is based on the theory developed in references 1 to 4,and is applicable to beams with thin stiffened webs, having single or double uprights or

cap flanges as shown in Figure 2.

he

tf

tw

c

Cap

d

huhc h

Flange CapUpright

Web

Ps

dc

bu

Double Uprights

dceu

Single Uprights

tu

Figure 2. Beam with Thin Stiffened Webs in Incomplete Diagonal Tension.

Note: In Figure 2, for both, upper and lower cap, positive moment is reacted by flange

cap in compression.

The theory of webs incomplete diagonal tension is a method for interpolating between

the two limiting cases of shear-resistant and pure diagonal tension, the limiting cases

being included.

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Failure modes for beams with stiffened webs in incomplete diagonal-tension are defined

in four categories:

a) Sheet failure rupturing of the sheet prior to any instability in the uprights(stiffeners).

b) Upright local failure by forced crippling local buckling of one or moreuprights, causing a significant drop in the uprights sustained load, resulting in

sheet failure or total collapse, due to redistribution of loads.

c) Upright failure by column buckling long column buckling of one or morestiffeners, that eventually results in collapse of the structure.

d) Fastener failure not common in a good design.e) Flange failure not common in good design.

2.1 Limitations and Assumptions of IDT TheoryThe following geometrical limitations shall be considered, due to limitation of test data:

115hc

tw

< 1500if

Nuf Puf_shear_ult otherwise

:=

Upright to flange fasteners margin of safety:

MSuf_fasteners

Puf_all

Pu_DT

1:=

20Ref. 1, pg. 34, formula 34.

21Ref. 1, pg. 48, formula 39.

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2.5.3 Upright to Web FastenersUpright to web fasteners required to have enough tension strength

22 to prevent tension

failure caused by the web wrinkles:

Ptens_ult 0.22 tw suw Ftu_web Nu 1if

0.15 tw suw Ftu_web Nu 2if

:=

Upright to web fasteners margin of safety:

MSuw_tens_fasteners

Puw_tens_ult

Ptens_ult

1:=

22Ref. 1, pg. 49, formulas 41 and 42

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2.6 Flange Analysis232.6.1 Compression FlangeCompressive stress in flange caused by DT:

ffc

k q he cot ( )

2 Afc:=

Primary maximum bending moment in the flange (over an upright) is:

Mf_max k c3q he d

2 tan ( )

12 h:=

Compression flange margin of safety:

MSc_flange

Fcc_flange

Pfc

Afc

ffc+

Mfc cc

Ic

Mf_maxcc

Ic

+

1:=

2.6.2 Tension FlangeTension stress in flange caused by DT:

fft

k q he cot ( )

2 Aft

:=

Tension flange margin of safety:

MSt_flange

Ftu_flange

Pft

Aft

fft+

Mft ct

It

Mf_maxct

It

+

+

1:=

23Ref. 1, pg. 50, sec. 4.16

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2.7 Web Stress Components24Tension in direction:

f

2 k fs

sin 2( ) 1 k( ) fs sin 2( )+:=

Compression in + /2 direction:

f90 1 k( ) fs sin 2( ):=

Shear in plane:

fs 1 k( ) fs cos 2( ):=

Maximum principal stress direction:

1

2atan

tan 2( )

k:=

Principal tension (in direction):

f1

k fs

sin 2( )fs 1 k

2 1

sin 2( )2

1

++:=

Principal compression (in + /2 direction):

f2

k fs

sin 2( )fs 1 k

2 1

sin 2( )2

1

+:=

Principal shear (in + /4 plane):

f3 fs 1 k2 1

sin 2( )2

1

+:=

24Ref. 2, pg. A9, formulas A.10 to A.16

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3.NUMERICAL ANALYSIS OF A BEAM IN INCOMPLETEDIAGONAL TENSION

The analysis was performed for beam III-25-6D25

, from reference # 5, applying the

methodology presented in the prior chapter.The beam mentioned above was tested, by NACA (National Advisory Committee

for Aeronautics), up to failure. A comparison between the analytical results and the test

data results from reference #5 is presented in next chapter.

Beam III-25-6D was chosen to validate the methodology of the previous chapter for

the following reason: NACA analytical prediction for beam III-25-6D was one of the

most unconservative predictions from a set of 49 beams26

. NACA analytically predicted

failure at a load 7% higher then actual failure load resulted form test.

The general built-up structure of beam III-25-6D is as follow:

beam height is 26.1 web is 0.0295, 7075-T6 AL Clad double uprights: two back-to-back angles (0.625 x 0.625) fabricated for

0.049, 7075-T6 AL Clad

double flange: two back-to-back extruded angles (2.00 x 2.00 x 0.188), 7075-T6 AL Extrusion.

Loading of the structure:

The cantilever beam III-25-6D was loaded at the free end with a transversal load

Ps = 11,699lb, representing the ultimate load at failure, based on methodology

from previous chapter.

25Ref. # 5, pg. 36, Table 1.

26Ref # 5, pg. 37 and 39, Tables 2 and 4.

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3.1 Input Data for IDT Analysis of beam III-25-6D [Ref. # 5, pg. 36]The following data was used as input for the MathCad Code from Appendix A.

Applied Loads:

Web shear flow:

q 481lb

in=

generated by applied transversal load: Ps= 11699lb, where:

qPs

he

:=

Internal stiffener and flange loads:

Pu

0lb:= Pfc

0lb:= Mfc

0in lb:= Pft

0lb:= Mft

0in lb:=

Note: For both, upper and lower cap, positive moment is reacted by Flange Cap in compression.

Web Properties:

tw 0.0295in:= he 24.3 in= hc 22.1in:= Ew 10500000psi:=

Fty_web 63000psi:= Ftu_web 74000psi:= Fsu_web 44000psi:=

matw 7075:= (matw=7075 for material AL 7075-T6; matw=2024 for material AL 2024-T3)

Upright Properties:

d 15.00in:= dc 14.375in:= h 26.1 in:= hu 23.3in:= tu 0.049in:=

Au 0.107in2

:= eu 0.00in:= Iu 0.00857in4

:=

Qu 0.0038in3

:= bu 0.625in:= (for double uprights only)

Nu 2:= (Nu=1 for single uprights; Nu=2 for double uprights)

matu 7075:= (matu=7075 for material AL 7075-T6; matu=2024 for material AL 2024-T3)

Eu 10500000 psi:= Fcy_upright 63000psi:=

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Flange Properties:

tf 0.188in:= Afc 1.5in2

:= Aft 1.5in2

:= Ic 0.348in4

:= It 0.348in4

:=

Fcc_flange 70000 psi:= Ftu_flange 79000psi:= cc 0.547in:= ct 0.547in:=

Nf

2:= (Nf

=1 for single flange; Nf

=2 for double flange)

Fasteners Properties:

Web to flange fasteners ultimate joint allowable and spacing:

Pwf_shear_ult 613lb:= (HL18-5 in 0.0295 AL Clad 7075-T6, double shear).

swf 0.85in:=

Upright to flange fasteners ultimate joint allowable, and number of fasteners

reacting the upright load in gusset action:

Puf_shear_ult 2466lb:=

(1 x HL18-6 in 0.049 AL Clad 7075-T6, 2 x single shear).

Nuf 1:= Ngusset 2:=

Upright to web fasteners ultimate joint allowable and spacing:

Puw_shear_ult 613lb:= (HL18-5 in 0.0295 AL Clad 7075-T6, double shear).

Puw_tens_ult 1440lb:=

suw 0.85in:=

For double uprights only, upright-to-upright single shear fastener joint allowable.

Puu_shear_ult 1096lb:=

(HL18-5 in 0.049 AL Clad 7075-T6, single shear).

3.2 Limitations IDT Theory, VerificationThe following geometrical limitations shall be considered, due to limitation of test data:

hc

tw

749= 115hc

tw

< 1500if

Nuf Puf_shear_ult otherwise

:=

Puf_all 3079 lb=

Upright to flange fasteners margin of safety:

MSuf_fastenersPuf_allPu_DT

1:=

MSuf_fasteners 0.39=

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3.5.3 Upright to Web FastenersAs shown in section 2.5.3, upright to web fasteners required to have enough tension

strength to prevent tension failure caused by the web wrinkles:

Ptens_ult 0.22 tw suw Ftu_web Nu 1if

0.15 tw suw Ftu_web Nu 2if

:=

Ptens_ult 278.332lb=

Upright to web fasteners margin of safety:

MSuw_tens_fasteners

Puw_tens_ult

Ptens_ult

1:=

MSuw_tens_fasteners 4.17=

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3.6 Flange Analysis3.6.1 Compression FlangeThe following calculations are based on the methodology presented in section 2.6.1.

Compressive stress in flange caused by DT:

ffc

k q he cot ( )

2 Afc:=

ffc 3234 psi=

Primary maximum bending moment in the flange (over an upright) is:

Mf_max k c3q he d

2 tan ( )

12 h:=

Mf_max 4273in lb=

Compression flange margin of safety:

MSc_flange

Fcc_flange

Pfc

Afc

ffc+

Mfc cc

Ic

Mf_maxcc

Ic

+

1:=

MSc_flange 6.04=

3.6.2 Tension FlangeThe following calculations are based on the methodology presented in section 2.6.2.

Tension stress in flange caused by DT:

fft

k q he cot ( )

2 Aft:=

fft 3234psi=

Tension flange margin of safety:

MSt_flange

Ftu_flange

Pft

Aft

fft+

Mft ct

It

Mf_maxct

It

+

+

1:=

MSt_flange 6.94=

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3.7 Web Stress ComponentsThe following calculations are based on the methodology presented in section 2.7.

Tension in direction:

f

2 k fs

sin 2( )1 k( ) fs sin 2( )+:= f 27507psi=

Compression in + /2 direction:

f90 1 k( ) fs sin 2( ):= f90 5416 psi=

Shear in plane:

fs 1 k( ) fs cos 2( ):= fs 1253psi=

Maximum principal stress direction:

1

2atan

tan 2( )

k:=

40.7deg=

Principal tension (in direction):

f1

k fs

sin 2( )fs 1 k

2 1

sin 2( )2

1

++:=

f1 27555psi=

Principal compression (in + /2 direction):

f2

k fs

sin 2( )fs 1 k

2 1

sin 2( )2

1

+:=

f2 5463 psi=

Principal shear (in + /4 plane):

f3 fs 1 k2 1

sin 2( )2

1

+:= f3 16509psi=

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4.RESULTS AND COMPARISON WITH TEST DATA4.1 Margins of Safety SummaryA summary of margins of safety calculated in previous chapter are presented in Table 1

below.

Applied transversal load, at the free end of the cantilever beam III-25-6D was Ps =

11,699 lb.

Structure Critical Failure Mode MS

Web Sheet Failure due to IDT 0.69

Upright Column Buckling

Forced Crippling

0.28

0.00

Fasteners - Web to Flange

- Upright to Flange

- Web to Upright

Bearing in Web

Bearing in Upright

Fastener Tension

1.13

0.39

4.17

Compression Flange Natural Crippling 6.04

Tension Flange Tension Strength 6.94

Table 1. Margin of Safety Summary

As shown in Table 1, the failure mode of the beam is upright forced crippling

(lowest margin of safety).

The beam is expected to fail at an applied transversal load Ps = 11,699lb, for

which the upright forced crippling margin of safety is zero.

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4.2 Analytical vs. Test Results, ComparisonBeam III-25-6D was tested to failure by NACA (National Advisory Committee for

Aeronautics) and the test results are documented in reference # 5.

A comparison between the analytical results and the test results is presented in

Table 2.

Result to Compare Symbol

Units NACA

Test

Results

NACA

Analytical

Prediction

Current

Methodology

Analytical

Prediction

Web Critical Shear Stress Fscr

Psi --- 410 426

DT Factor k --- --- 0.662 0.659

Ult. Column Buckling Load Fc

lb --- 14,800 14,610

Ult. Forced Crippling Load Ffc

lb 11,400 12,200 11,699

Ult. Load Web Failure Fw

lb --- 20,500 19,530

Failure Mode

-

-- --- F.C. F.C. F.C

Table 2. Analytical vs. Test Results, Comparison

Note: F.C. stands for upright forced crippling.

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A comparison of the current methodology to NACA analytical prediction, based on the

results listed in Table 2, is shown in Table 3.

Result to Compare Current Methodology vs. NACA

Analytical Prediction

Web Critical Shear Stress +3.9%

DT Factor +0.5%

Ult. Column Buckling Load -1.3%

Ult. Forced Crippling Load -4.1%

Ult. Load Web Failure -4.7%

Table 3. Current Methodology vs. NACA Analytical Prediction

A comparison of the analytical predictions to the NACA test results, based on the

results listed in Table 2, is shown in Table 4.

Result to Compare Current Methodology

Analytical Prediction vs.

NACA Test Results

NACA Analytical

Prediction vs. NACA

Test Results

Ult. Forced Crippling Load +2.6% +7.0%

Table 4. Analytical Predictions vs. Test Results

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5.CONCLUSIONS1) As can be seen in Table 2 and 4, NACA analytical prediction for upright forced

crippling (Ffc) was 7.0% higher then the load at failure resulted from test. The

current methodology analytical prediction was only 2.6% higher then the actual

load at failure. That shows that the current methodology presented in Chapter 2

of this report is at least as accurate as NACA analytical prediction.

2) Both analytical predictions (NACA and current methodology form Chapter 2)showed unconservative results for upright forced crippling failure. Considering

that for ultimate failure analysis the loads have a built in a factor of safety of 1.5,

the 2.6% variation from the test failure is negligible.

3) Current methodology prediction of the uprights ultimate column-buckling load is1.3% more conservative than NACA analytical prediction.

4) Current methodology prediction of the ultimate load for web failure is 4.7% moreconservative than NACA analytical prediction.

5) Based on the comparison shown in Tables 2, 3 and 4, the methodology presentedin Chapter 2 is considered valid and applicable in practice.

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REFERENCES

1) "NACA-TN-2661" A Summary of Diagonal Tension, Part I Methods of Analysis,

NACA, Washington, May 1952.

2) "NASA-CR-101854" Investigation of Diagonal-Tension Beams with Very Thin

Stiffened Webs, Grumman Aerospace Corporation, Bethpage, New York, July 1969

(Includes an improvement to the NACA method. Study completed by Grumman

Aerospace for NASA).

3) "Analysis and Design of Flight Vehicle Structures" (Chapter C11), by E.F. Bruhn,

Jacobs Publishing, June 1973.

4) "Airframe Stress Analysis and Sizing" 2nd

Edition, by Michael Niu (Chapter 12),

Hong Kong Conmilit Press, 1997.

5) "NACA-TN-2662" A Summary of Diagonal Tension, Part II Experimental

Evidence, NACA, Washington, May 1952.

6) "NACA-TM-490" Structures of Thin Sheet Metal, Their Design and Construction,

NACA, December 1928.

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APPENDIX A. ATTACHED ELECTRONIC FILES

The electronic files listed below are compressed in file:

Appendix A, MP Beams in IDT, C_Moisiade.zip

File Name File Type Description

Beams_in_IDT_Cezar_Moisiade_

07-11-2009.xmcdMathCad 13

Includes the numerical

methodology to perform

analysis of beams in IDT.

NACA_Charts_Cezar_Moisiade_0

7-11-2009.xlsMicrosoft Excel

Includes curve fits for NACA-

TN-2661 charts used in IDT

analysis.

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APPENDIX B. FINITE ELEMENT ANALYSIS -

PRELIMINARY

Additional efforts have been done on performing a finite element simulation of a

stiffened web in incomplete diagonal tension. The efforts have not been completed. Thesimulation got as far as developing the methodology and getting preliminary results for a

test model that was used to validate the methodology.

The nonlinear post-buckling analysis was performed in ANSYS 8.0, following three

steps:

1. Static Linear Analysis - of a panel under shear load.2. Eigen-Buckling Analysis performed for the pre-stress panel, using the

results from step 1.

3. Nonlinear Buckling Analysis the panel had the geometry perturbedbased on a specific eigen-value resulted from step 2, then a large

deflection analysis, using arc-length method was performed.

The panel geometry was 0.020 x 12.0 x 12.0, and the material AL 7075-T0. For this

preliminary run, the stiffeners and flanges were defined by an area of 0.10 in2, area

moment of inertia of 0.010 in4, and elastic modulus of 30e6 psi.

The panel was modeled with shell elements # 181, and the stiffeners and flanges

were modeled with beam elements # 4. Fasteners were simulated using rigid coupled

constrains.

All electronic files for Appendix B, are compressed in folder: Appendix B, MP

Beams in IDT, FEAnalysis, C_Moisiade.zip

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The finite element model including boundary conditions is shown in Figure 3.

Figure 3. Finite Element Model and Boundary Conditions.

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Preliminary out of plane displacement results from the eigen-buckling analysis are

shown in Figure 4.

Figure 4. Eigen-Buckling Results, 44th

Eigenvalue, Relative Z Displacement [in].

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Preliminary results from the nonlinear buckling analysis are shown in Figures 5, 6 and 7.

The applied shear load got up to 43.4 lb/in. The analysis will have to continue to get to

higher loads.

Figure 5. Nonlinear-Buckling Results, Z Displacement [in].

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Figure 6. Nonlinear-Buckling Results, 1stPrincipal Stress [psi].

Figure 7. Nonlinear-Buckling Results, Shear Stress [psi].

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The ANSYS input file for the three analysis steps, listed above, is shown below.

!**********************************! * St ep 1, St ati c Li near Run *!**********************************/ SOLU

!F=100 ! shear f orce appl i ed ( l bs)P=. 00001 ! si de pr essur e appl i ed ( psi )f scal e, Fsf scal e, pr es, P!ant ype, stati cEQSLV, SPAR, , 0,NLGEOM, 0PSTRES, ON!SOLVEFI NI SH!/ POST1FI NI SH!

!*********************************! * St ep 2, Ei ghen Buckl i ng *!*********************************/ SOLUANTYPE, buckl eBUCOPT, LANB, 50, 0, 0MXPAND, 50, 0, 0, yes, 0. 001,SOLVEFI NI SH!/ POST1FI NI SH!!*********************************! * St ep 3, Nonl i near Buckl i ng *!*********************************/ PREP7sfdel e, al l , al l !del et e pressureUPGEOM, 0. 01, 1, 44, ' ms05- ev' , ' r st ' ! pert urb geomet r y per 44t h buckl i ng mode!F=600 ! shear f orce appl i ed ( l bs)f scal e, F!/ SOLUnl _cnt r l =1!ANTYPE, STATI CNLGEOM, ONOUTRES, ALL, ALL,!* i f , nl _cntr l , eq, 0, then

t i me, FSOLCONTROL, ON

NROPT, FULLNSUBST, 50, 1e4, 25*el sei f , nl _cntr l , eq, 1

SOLCONTROL, OFFNSUBST, 50, 1e4, 25AUTOTS OFF

Documents

MP Beams in IDT, C_Moisiade_08-08-2009.pdf