MP Beams in IDT, C_Moisiade_08!08!2009

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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 ensselaer Polytechnic Institute

in Partial Fulfillment of the

e!uirements for the degree of

MAS"E #F E$GI$EEI$G I$ MEC%A$ICA& E$GI$EEI$G

A''ro(ed)

*****************************************

Ernesto Gutierrez+Mira(ete, Project Ad(iser

ensselaer Polytechnic Institute%artford, Connecticut

August, -/



0 Co'yright -/

by

Cezar I. Moisiade

All ights eser(ed

ii



CONTENTS

&IS" #F "A1&ES.............................................................................................................(i

&IS" #F FIG2ES..........................................................................................................(ii

&IS" #F S3M1#&S......................................................................................................(iii

AC4$#5&E6GME$"...................................................................................................7i

A1S"AC".....................................................................................................................7ii

8. I$"#62C"I#$ 9 1AC4G#2$6.........................................................................8

-. ME"%#6#&#G3, 2&"IMA"E S"E$G"% #F 1EAMS I$ I$C#MP&E"E6IAG#$A& "E$SI#$...............................................................................................:

-.8 &imitations and Assum'tions of I6" "heory.....................................................;

-.- ecommended 6esign &imitations.....................................................................;

-.: 5eb, Post+1uc<ling Analysis..............................................................................=

-.:.8 Shear buc<ling coefficient for sim'ly su''orted 'anel >4 ss?..................=

-.:.- 5eb fi7ity coefficients > u @ f ?............................................................=

-.:.: Critical shear stress >Fscr ?.........................................................................

-.:.; 6iagonal+tension factor ><?.....................................................................B

-.:.= Angle of diagonal+tension >?.................................................................B

-.:.D Flange fle7ibility factor >d?.................................................................../

-.:. Angle and stress concentration factors >c8, c-, c:?.................................../

-.:.B 5eb 'ea< nominal stress >f s*ma7?............................................................8

-.:./ 5eb nominal stress alloable >Fs*all?....................................................8

-.:.8 5eb Margin of Safety >MSeb?..............................................................8

-.; 2'right Analysis...............................................................................................88

-.;.8 2'right column buc<ling......................................................................88

-.;.- 2'right forced cri''ling.......................................................................8:

-.= Analysis of Fasteners........................................................................................8;

-.=.8 5eb "o Flange Fasteners......................................................................8;

iii



-.=.- 2'right to Flange Fasteners..................................................................8;

-.=.: 2'right to 5eb Fasteners......................................................................8=

-.D Flange Analysis.................................................................................................8D

-.D.8 Com'ression Flange..............................................................................8D

-.D.- "ension Flange......................................................................................8D

-. 5eb Stress Com'onents....................................................................................8

:. $2MEICA& A$A&3SIS #F A 1EAM I$ I$C#MP&E"E 6IAG#$A&"E$SI#$...................................................................................................................8B

:.8 In'ut 6ata for I6" Analysis of beam III+-=+D6 ef. =, 'g. :DH...................8/

:.- &imitations I6" "heory, erification...............................................................-

:.: 5eb, Post+1uc<ling Analysis............................................................................-8

:.:.8 Shear buc<ling coefficient for sim'ly su''orted 'anel >4 ss?................-8

:.:.- 5eb fi7ity coefficients > u @ f ?..........................................................-8

:.:.: Critical shear stress >Fscr ?.......................................................................-:

:.:.; 6iagonal+tension factor ><?...................................................................-;

:.:.= Angle of diagonal+tension >?...............................................................-;

:.:.D Flange fle7ibility factor >d?.................................................................-=

:.:. Angle and stress concentration factors >c8, c-, c:?.................................-=

:.:.B 5eb 'ea< nominal stress >f s*ma7?............................................................-D

:.:./ 5eb nominal stress alloable >Fs*all?....................................................-D

:.:.8 5eb Margin of Safety >MSeb?..............................................................-D

:.; 2'right Analysis...............................................................................................-

:.;.8 2'right column buc<ling......................................................................-

:.;.- 2'right forced cri''ling.......................................................................-/

:.= Fasteners Analysis.............................................................................................:

:.=.8 5eb "o Flange Fasteners......................................................................:

:.=.- 2'right to Flange Fasteners..................................................................:

:.=.: 2'right to 5eb Fasteners......................................................................:8

i(



:.D Flange Analysis.................................................................................................:-

:.D.8 Com'ression Flange..............................................................................:-

:.D.- "ension Flange......................................................................................:-

:. 5eb Stress Com'onents....................................................................................::

;. ES2&"S A$6 C#MPAIS#$ 5I"% "ES" 6A"A..............................................:;

;.8 Margins of Safety Summary.............................................................................:;

;.- Analytical (s. "est esults, Com'arison..........................................................:=

=. C#$C&2SI#$S........................................................................................................:

EFEE$CES.................................................................................................................:B

APPE$6IJ A. A""AC%E6 E&EC"#$IC FI&ES......................................................:/

APPE$6IJ 1. FI$I"E E&EME$" A$A&3SIS + PE&IMI$A3.............................;

(



LIST OF TABLES

"able 8. Margin of Safety Summary.................................................................................:;

"able -. Analytical (s. "est esults, Com'arison............................................................:=

"able :. Current Methodology (s. $ACA Analytical Prediction.....................................:D

"able ;. Analytical Predictions (s. "est esults...............................................................:D

(i



LIST OF FIGURES

Figure 8. 1eam ith Stiffened 5ebs in I6", "ested by $ACA.........................................-

Figure -. 1eam ith "hin Stiffened 5ebs in Incom'lete 6iagonal "ension.....................:

Figure :. Finite Element Model and 1oundary Conditions.............................................;8

Figure ;. Eigen+1uc<ling esults, ;;th Eigen(alue, elati(e K 6is'lacement inH.........;-

Figure =. $onlinear+1uc<ling esults, K 6is'lacement inH............................................;:

Figure D. $onlinear+1uc<ling esults, 8st Princi'al Stress 'siH......................................;;

Figure . $onlinear+1uc<ling esults, Shear Stress 'siH................................................;;

(ii



LIST OF SYMBOLS

6" diagonal+tension

P6" 'ure diagonal+tension

I6" incom'lete diagonal+tension

Afc cress+sectional area of com'ression ca'+flange, in-

Aft cress+sectional area of tension ca'+flange, in-

Au cress+sectional area of u'right, in-

Aue effecti(e cress+sectional area of u'right, in-

bu idth of outstanding leg of u'right, in

c8 angle factor.

c-, c: stress concentration factors.

cc, ct distance from centroid of ca'+flange to e7treme fiber of flange, in

C u'right column buc<ling reduction factor.

d s'acing of u'rights, in

dc clear u'right s'acing, measured as shon in Figure -

E elastic modulus, 'si

eu distance from median 'lane of the eb to centroid of >single? u'right, in

ef distance from median 'lane of the eb to centroid of >single? u'right, in

f u u'right stress caused by diagonal+tension, 'si

f u*ma7 ma7imum u'right stress caused by diagonal+tension, 'si

f s shear stress a''lied to eb, 'si

f s*ma7 eb 'ea< nominal stress, 'si

f fc stress in com'ression flange caused by diagonal+tension effect, 'si

f ft stress in tension flange caused by diagonal+tension effect, 'si

Fs*all eb nominal stress alloable, 'si

Fc u'right column buc<ling alloable, 'siFcc u'right cri''ling alloable, 'si

Ffc u'right forced+cri''ling alloable, 'si

Fty yield tension alloable, 'si

Ftu ultimate tension alloable, 'si

Fsu ultimate shear alloable, 'si

(iii



Fscr*el elastic critical shear stress, 'si

Fscr critical shear stress corrected for 'lasticity effects, 'si

h de'th of beam, in

he effecti(e de'th of beam, measured beteen centroids of flanges, in

hc clear de't of eb, measured as shon in Figure -, in

hu u'right length, measured beteen controids of u'right+to+flange ri(et

'atterns, in

Ic com'ression ca'+flange cross+sectional moment of inertia about neutral

a7is, in;

It tension ca'+flange cross+sectional moment of inertia about neutral a7is,

in;

Iu u'right cross+sectional moment of inertia about neutral a7is, in;

< diagonal+tension factor

4 ss theoretical shear buc<ling coefficient for a sim'ly su''orted 'late

&e effecti(e u'right length, in

Mfc moment in com'ression ca' flange, not related to 6", in+lb

Mft moment in tension ca' flange, not related to 6", in+lb

Mf*ma7 ma7imum flange 'rimary bending moment caused by 6" effect, in+lb

matu flag, defining u'right material ty'e

mat flag, defining eb material ty'e

$u flag, defining number of u'rights

$uf u'right to flange, number of fasteners >one end only?

$gusset numbers of u'right fasteners reacting u'right load in gusset action.

Ps load a''lied to the beam that generates shear ! in the eb

Pu load in u'right, not related to 6", to u'right, lb

Pu*6" load in u'right, caused by 6", lb

Puf*all u'right to flange fasteners, total joint shear alloable, considering gusset

action, lb

Ptens*ult u'right fasteners, re!uired ultimate tension strength, lb

Pfc load in com'ression ca' flange, not related to 6", lb

Pft load in tension ca' flange, not related to 6", lb

i7



Pf*shear*ult eb to flange fasteners shear ultimate alloable, lb

Puf*shear*ult u'right to flange fasteners shear ultimate alloable, lb

Pu*shear*ult u'right to eb fasteners shear ultimate alloable, lb

Pu*tens*ult u'right to eb fasteners tensile ultimate alloable, lb

Puu*shear*ult u'right+to+u'right fasteners shear ultimate alloable >for double u'rights

only?, lb

! shear flo in eb, lb9in

!f shear flo reacted by the flange fasteners, lb9in

!u re!uired u'right fasteners shear flo to 're(ent 'remature column

buc<ling >for double fasteners only?.

!u*all u'right fasteners single shear alloable >for double fasteners only?.

Lu static moment about neutral a7is u'right >for double u'rights?, in:

f eb fi7ity coefficient at the flange

u eb fi7ity coefficient at the u'rights

sf eb to flange fasteners s'acing, in

su u'right to eb fasteners s'acing, in

tf thic<ness of flange, in

tu thic<ness of u'right, in

t eb thic<ness, in

d flange fle7ibility factor.

P6" angle of 'ure diagonal tension relati(e to natural a7is of the beam, deg.

angle of incom'lete diagonal tension relati(e to natural a7is of the beam,

deg.

ρu u'right cross+section centroidal radius of gyration about a7is 'arallel to

eb, in

7



ACKNOWLEDGMENT

I ish to dedicate my or< to my son that ill be born in fe months, and e7'ress

my lo(e and gratitude to my belo(ed ife for her understanding, 'atience and endless

lo(e, through the duration of my studies.

I ould li<e to con(ey than<s to Ernesto Gutierrez+Mira(ete, my 'roject ad(iser, for

his guidance and (aluable feedbac<, during the com'letion of my Master Project and

during my graduate studies at PI.

7i



ABSTRACT

In aeronautical a''lications, beams ith thin stiffened ebs are often designed

considering the 'ost+buc<ling ca'ability of the eb under shear load. 5eb buc<ling

under shear load does not re'resent failure. "he eb has additional 'ost+buc<ling

ca'ability to carry load in diagonal+tension.

"he analysis of 'ost+buc<ling ebs is tedious and time consuming, and the use of a

numerical 'rogram that incor'orates the methodology and 'erforms the calculations is

desired.

"he effort for the current 'roject as focused on de(elo'ing a numerical 'rogram using

MathCad, for analyzing beams in incom'lete diagonal tension.

"he current re'ort 'resents the methodology and a numerical analysis for 'redicting

ultimate failure of beams in incom'lete diagonal tension. "he numerical analysis as

'erformed for a beam that as tested by $ational Ad(isory Committee for Aeronautics

>$ACA? in reference =. An e(aluation of the analytical results and a com'arison ith

the test results from reference = as 'erformed in order to (alidate the methodology.

"he analytical 'rediction as different by only -.DN from the actual failure resulted

from test.

A MathCad file including the 'rogram that 'erforms the analysis of beams in

incom'lete diagonal tension is attached in A''endi7 A.

"he cur(e+fits for the charts from reference 8 ere com'leted in Microsoft E7cel

and a file including the resulted data is attached in A''endi7 A.

7ii



1. INTRODUCTION / BACKGROUND

"he de(elo'ment of the diagonal+tension ebs it as an outstanding ste' forard in the

structural aeronautical design. #riginal or< on beams in diagonal+tension as

'erformed by $ational Ad(isory Committee for Aeronautics >$ACA? in 8/-B and

documented in reference D. "he most com'lete theory of beams in incom'lete diagonal

tension as de(elo'ed by $ACA in 8/=-, and 'resented in references 8 and =.

Additional im'ro(ements ere de(elo'ed in 8/D/ by a $ASA funded 'rogram, and

'erformed by Grumman Aeros'ace, 'resented in reference -.

Post+buc<ling ca'ability of a beam ith stiffened thin ebs, under shear load, is far

greater then the load 'roducing buc<ling of the eb. "he structure does not fail hen the

eb buc<les the eb forms diagonal fold and functions as a series of tension diagonals,

hile the stiffeners act as com'ression 'osts. "he eb+stiffener system changes from a

structure ith shear resistant ebs toards a truss structure. 5hen the structure or<s

as a truss, the eb carries the entire load in diagonal+tension and none in shear, the eb

is in a state of O'ure diagonal+tension.

A Oshear+resistant eb carries the entire load in shear and none in diagonal+tension.

"ruly shear+resistant ebs are 'ossible but rare in aeronautical 'ractice. Practically, all

ebs fall into the intermediate region of Oincom'lete diagonal tension, here the eb

carries 'art of the load in shear, and the rest of it is carried in diagonal tension. "he state

of Oincom'lete diagonal tension is an inter'olation beteen the theoretical states of

Oshear+resistant and O'ure diagonal tension.

"he analysis of beams ith stiffened ebs, in incom'lete diagonal tension, is

tedious and time consuming, and the use of a numerical 'rogram that incor'orates the

methodology and 'erforms the calculations is desired. "he effort for the current 'roject

as focused on de(elo'ing a numerical 'rogram using MathCad, for analyzing beams in

incom'lete diagonal tension."he methodology used in the current re'ort, for 'redicting failure of beams ith

stiffened ebs in incom'lete diagonal tension is based on the theory and em'irical data

form references 8 to ;.

"he numerical analysis as 'erformed for a beam that as tested by $ACA in

reference =. An e(aluation of the analytical results and a com'arison ith the test

8



results from reference = as 'erformed in order to (alidate the methodology. "he

analytical 'rediction as different by only -.DN from the actual failure resulted from

test.

A MathCad file including the 'rogram that 'erforms the analysis of beams in

incom'lete diagonal tension is attached in A''endi7 A.

"he com'letion of a 'rogram that 'erforms I6" analysis, re!uired ha(ing a(ailable

e!uations for all the charts from reference 8. "he cur(e+fits ere com'leted in

Microsoft E7cel and a file including the resulted data is attached in A''endi7 A.

An e7am'le of a beam ith stiffened ebs in incom'lete diagonal tension, tested by

$ACA in reference =, is shon in Figure 8, here the diagonal eb rin<les can be

seen.

Figure 1. Be! "i#$ S#i%%e&e' We() i& IDT* Te)#e' (+ NACA1

.

8 eference 8, 'age 8:.

-



,. METHODOLOGY* ULTIMATE STRENGTH OF BEAMS

IN INCOMPLETE DIAGONAL TENSION

"he methodology 'resented belo is based on the theory de(elo'ed in references 8 to ;,and is a''licable to beams ith thin stiffened ebs, ha(ing single or double u'rights or

ca' flanges as shon in Figure -.

he

tf

tw

c

Cap

d

huhc h

Flange CapUpright

Web

Ps

dc

bu

Double Uprights

dc

eu

Single Uprights

tu

Figure ,. Be! "i#$ T$i& S#i%%e&e' We() i& I&!0e#e Dig&0 Te&)i&.

$ote) In Figure -, for both, u''er and loer ca', 'ositi(e moment is reacted by flange

ca' in com'ression.

"he theory of ebs Oincom'lete diagonal tension is a method for inter'olating beteen

the to limiting cases of Oshear+resistant and O'ure diagonal tension, the limiting cases

being included.

:



Failure modes for beams ith stiffened ebs in incom'lete diagonal+tension are defined

in four categories)

a? Sheet failure Q ru'turing of the sheet 'rior to any instability in the u'rights

>stiffeners?.

b? 2'right local failure by forced cri''ling Q local buc<ling of one or more

u'rights, causing a significant dro' in the u'rights sustained load, resulting in

sheet failure or total colla'se, due to redistribution of loads.

c? 2'right failure by column buc<ling Q long column buc<ling of one or more

stiffeners, that e(entually results in colla'se of the structure.

d? Fastener failure Q not common in a good design.

e? Flange failure Q not common in good design.

,.1 Li!i##i&) &' A))u!#i&) % IDT T$er+

"he folloing geometrical limitations shall be considered, due to limitation of test data)

88=hc

t

< 8=< .-dc

hc

< 8.<tu

t

.D>

Assum'tions that ere made)

+ 5eb and u'rights are made from the same material.

+ #'en section u'right ri(eted to the eb.

,., Re!!e&'e' De)ig& Li!i##i&)

"o 're(ent 'remature fatigue failure due to e7cessi(e rin<ling for ultimate loads, it is

recommended that,

4 4 limit< here) 4 limit .B tin

.8-−−:=

For fatigue critical ebs, it is recommended that,

f s

Fscr

=≤

;



,. We(* P)#-Bu20i&g A&0+)i)

,..1 S$er (u20i&g e%%iie&# %r )i!0+ )ur#e' &e0 3K ))4,

"he theoretical shear+buc<ling coefficient for a 'late ith sim'ly su''orted edges is

gi(en by)

4 ss -.BD=dc

hc

-

⋅ .Bdc

hc

⋅+ ;.B+:=

,.., We( %i5i#+ e%%iie&#) 3R u 6 R % 4

"he coefficients u and f are coefficients of eb edge restraint, ta<en as R 8 for sim'ly su''orted edges and R 8.D- for clam'ed edges. In actual beam ebs, the edge

su''orts are determined by the flanges and the u'rights the 'anel edges are thus neither

sim'ly su''orted nor clam'ed.

"he eb fi7ity coefficient at the u'rights, for single u'rights)

u8 .:-//tu

t

-

⋅ .-/;tu

t

⋅+ .8=+tu

t

.D≤if

.//tu

t

⋅ .:88− .D

tu

t

< 8.-=≤if

.-DDtu

t

:

⋅ 8.B-/tu

t

-

⋅− ;.;=tu

t

⋅+ 8.B:;:− 8.-=

tu

t

< -.=≤if

.;tu

t

⋅ 8.8/8+ otherise

:=

- ef. 8, 'g. 8D, fig. 8->a?. For the cur(e+fit of the chart see A''endi7 A.

: ef. 8, 'g. 8D, fig. 8->b?. For the cur(e+fit of the chart see A''endi7 A.

=



5eb fi7ity coefficient at the u'rights, for double u'rights)

u-

tu

t

tu

t

8≤if

.8=88

t

ut

:

⋅ 8.;D

t

ut

-

⋅− -.=-

t

ut

⋅+ .DB-− 8

t

ut

< -.=≤if

./tu

t

⋅ 8.:=B+ otherise

:=

5eb fi7ity coefficient at the u'rights, considering single or double u'rights is)

u u8 $u 8if

u- $u -if

:=

5eb fi7ity coefficient at the ca'+flanges, for single flange)

f8 .:-//tf

t

-

⋅ .-/;tf

t

⋅+ .8=+

tf

t

.D≤if

.//tf

t

⋅ .:88− .D

tf

t

< 8.-=≤if

.-DDtf

t

:

⋅ 8.B-/tf

t

-

⋅− ;.;=tf

t

⋅+ 8.B:;:− 8.-=

tf

t

< -.=≤if

.;tf

t

⋅ 8.8/8+ otherise

:=

5eb fi7ity coefficient at the ca'+flanges, for double flange)

f-

tf

t

tf

t

8≤if

.8=88tf

t

:

⋅ 8.;Dtf

t

-

⋅− -.=-tf

t

⋅+ .DB-− 8

tf

t

< -.=≤if

./tf

t

⋅ 8.:=B+ otherise

:=

D



5eb fi7ity coefficient at the ca'+flanges, considering single or double flanges is)

f f8 $f 8if

f- $f -if

:=

,.. Cri#i0 )$er )#re)) 3F)r4

"heoretical formulas for the critical shear stress are a(ailable for 'lates ith all edges

sim'ly su''orted, all edges clam'ed, or one 'air of edges sim'ly su''orted and the other

'air calmed. 5ith sufficient accuracy for 'ractical 'ur'oses, $ACA de(elo'ed a formula

for critical shear stress, hich includes the effect of eb fi7ity, by using the theoretical

formulas, su''lemented by em'irical restraint coefficient.

Elastic critical shear >Fscr*el?;, including the effect of eb fi7ity)

Fscr*el 4 ss E⋅t

dc

-

⋅ u8

- f u−( )⋅

dc

hc

:

⋅+

⋅

dc

hc

8≤if

4 ss E⋅t

hc

-

⋅ f 8

- u f −( )⋅

hc

dc

:

⋅+

⋅ otherise

:=

Critical shear stress corrected for 'lasticity effects >Fscr ?=)

For Clad A& =+"D)

Fscr*="D Fscr*el Fscr*el =<si≤if

.B=D Fscr*el⋅ ./--<si+ =<si Fscr*el< ;8< <if

.--88Fscr*el -;./:<si+ Fscr*el ;8<si≥if

:=

For Clad A& --;+":)

Fscr*--;": Fscr*el Fscr*el 8<si≤if

.:

<si-

Fscr*el:⋅

.:D

<siFscr*el

-⋅− 8.:= Fscr*el⋅+ :.BB8<si− Fscr*el 8<si>if

:=

; ef. 8, 'g. ;-, or 'g. -D formula :-.

= ef. 8, 'g. 8, figure 8-c. For the cur(e+fit of the chart see A''endi7 A.



Considering materials listed abo(e the critical shear corrected for 'lasticity effects is)

Fscr Fscr*="D mat =if

Fscr*--;": mat --;if

:=

,..7 Dig&0-#e&)i& %#r 3248

A diagonal+tension factor of 8. defines a eb in 'ure diagonal tension >no load carried

in shear?, and a diagonal tension factor of . defines a eb, carrying load in 'ure

shear.

< tanh .= logf s

Fscr

⋅

f s Fscr ≥if

otherise

:=

5here f s is the eb shear stress)

f s!

t

:=

,..9 A&g0e % 'ig&0-#e&)i& 3:4

"he effecti(e cross+sectional area of the stringer is)

Aue

Au

8eu

ρ u

-

+

$u 8if

Au $u -if

:=

here) ρ u

Iu

Au

:=

D ef. 8, 'g. 8B, formula -

ef. 8, 'g. ;8, section ;.8

B



"he buc<ling angle for 'ure diagonal+tensionB is)

αP6" atan

;

8h t⋅

Afc Aft++

8

d t⋅

Aue+

:=

"he buc<ling angle for incom'lete diagonal+tension as calculated by linear

inter'olation)

α ;=deg < ;=deg αP6"−( )⋅−:=

,..8 F0&ge %0e5i(i0i#+ %#r 3"'4;

d d s in α( )⋅

;8

It

8

Ic

+ t

;he

⋅:=

,..< A&g0e &' )#re)) &e&#r#i& %#r) 31* ,* 4

Angle factor 8)

c88

sin - α⋅( )8−:=

Stress concentration factors88)

c- d 8≤if

.=: d:⋅ .-; d

-⋅− .-//; d⋅+ .8=− 8 d< :≤if

.=; d⋅ 8.8/− otherise

:=

B ef. 8, 'g. 8, formula 8=

/ ef. 8, 'g. 88, formula 8/

8 er. 8, 'g. -

88 ef. 8, 'g. 88-, figure 8B. For the cur(e+fit of the chart see A''endi7 A.

/



c: 8 d 8≤if

.=: d:⋅ .D;B d

-⋅− .8-; d⋅+ ./:/8+ 8 d< =≤if

.=B otherise

:=

,..= We( e2 &!i&0 )#re)) 3% )>!541,

f s*ma7 f s 8 < -

c8⋅+ 8 < c-⋅+( )⋅:=

Stress in the eb is e7'ressed as nominal shear stress for < R and nominal diagonal+

tension stress for < R 8.

,..; We( &!i&0 )#re)) 00"(0e 3F)>0041

Fs*all ./ Fty*eb⋅ 88

-

Ftu*eb

Fty*eb

8−

-

⋅+

⋅

8

-8 < −> ?

:Fsu*eb

Ftu*eb

8

-−

⋅+

⋅:=

5eb nominal stress alloable satisfies shear failure for 'ure shear > < R ? and tensile

failure for 'ure diagonal+tension > < R 8 ?.

,..1? We( Mrgi& % S%e#+ 3MS"e(4

MSeb

Fs*all

f s*ma7

8−:=

8- ef. 8, 'g. -, formula >::a?.

8: ef) -, 'g. 8D

8



,.7 Urig$# A&0+)i)

,.7.1 Urig$# 0u!& (u20i&g

2'right effecti(e length8;)

&e

hu

8 < -

: -d

hu

⋅−

⋅+

d 8.= h⋅≤if

hu otherise

:=

2'right stress due to diagonal+tension8=)

f

u

< − f s⋅ tan α( )⋅

Aue

d t⋅.= 8 < −> ?⋅+

:=

2'right Euler column buc<ling stress alloable)

Fc π-

− Eu⋅&e

- ρ u⋅

-−

⋅ $u 8if

π-

− E

u⋅

&e

ρ u

-−

⋅ $

u

-if

:=

&imit the alloable to Fcy)

Fc min Fc Fcy*u'right,( )−:=

For double u'rights only, the fasteners holding the u'rights together need to be chec<ed

if they ha(e the ca'ability to transfer the folloing shear flo)

!u

-.= Fcy*u'right⋅ Lu⋅

bu &e⋅ $u -if

otherise

:=

8; ef. 8, 'g. ;D, formula :=

8= ef. 8, 'g. 8/, formula :a

>Single u'rights?

>6ouble u'rights?

88



2'rights fasteners single shear alloable)

!u*all

Puu*shear*ult

su

:=

If fastener shear alloable >!u*all? is less then a''lied shear, the alloable stress for

column failure must be multi'lied by the folloing reduction factor 8D)

C 8

8-=.D/−

!u*all

!u

-

⋅ =.B:8!u*all

!u

⋅+ :.=:+

!u*all< !u<if

8. otherise

:=

ecalculate u'right column buc<ling stress alloable > Fc ?)

Fc C Fc⋅:=

2'right margin of safety for column buc<ling)

MSu*col*bu<ling

Fc

Pu

Aue

f u+

8−:=

8D ef. 8, 'g. 88D, figure -8. For the cur(e+fit of the chart see A''endi7 A.

8-



,.7., Urig$# %re' ri0i&g

2'right ma7imum stress > f u*ma7 ?8)

f u*ma7 f u 8 < −> ?⋅ .-/ .D:DD

d

hu⋅−

⋅ f u+:=

2'right forced cri''ling stress alloable > Ffc ?8B)

Ffc :-=−

:

< -

tu

t

⋅ 's i⋅ matu =( ) $u 8( )∧if

-D−

:

< -

tu

t

⋅ 's i⋅ matu --;( ) $u 8( )∧if

-D−

:

< -

tu

t

⋅ 's i⋅ matu =( ) $u -( )∧if

-8−

:

< -

tu

t

⋅⋅ 's i matu --;( ) $u -( )∧if

:=

8 ef. 8, 'g. 88, figure 8=. For the cur(e+fit of the chart see A''endi7 A.

8B ef. 8, 'g. ;D+;, formulas :D and :

8:




Ffc min Ffc Fcy*u'right,( )−:=

2'right margin of safety for forced cri''ling)

MSu*forced*cri''ling

Ffc

f u*ma7

8−:=

$ote) $atural cri''ling is not a controlling factor in the design.8/

8/ ef. :, 'g. C88.8B

>Single u'rights, =+"D?

>Single u'rights, --;+":?

>6ouble u'rights, =+"D?

>6ouble u'rights, --;+":?

8;



,.9 A&0+)i) % F)#e&er)

,.9.1 We( T F0&ge F)#e&er)

"he fasteners that connect the eb to the ca'+flange are re!uired to ha(e the ca'ability

to carry the folloing shear flo-)

!f

! he⋅

hu

8 .;8; < ⋅+> ?⋅:=

5eb to flange fasteners margin of safety)

MSf*fasteners

Pf*shear*ult

sf !f ⋅:=

,.9., Urig$# # F0&ge F)#e&er)

"he fasteners that connect the u'right to the flange+ca' are re!uired to ha(e the

ca'ability to carry the u'right load into the flange.

2'right &oad due to 6" > Pu*6" ?-8)

Pu*6" f u Aue⋅:=

"otal fastener joint shear alloable considering gusset action)

Puf*all $uf Puf*shear*ult⋅ $gusset $uf −( ) Pu*shear*ult⋅+ $gusset $uf >if

$uf Puf*shear*ult⋅ otherise

:=

2'right to flange fasteners margin of safety)

MSuf*fasteners

Puf*all

Pu*6"

8−:=

- ef. 8, 'g. :;, formula :;.

-8 ef. 8, 'g. ;B, formula :/.

8=



,.9. Urig$# # We( F)#e&er)

2'right to eb fasteners re!uired to ha(e enough tension strength-- to 're(ent tension

failure caused by the eb rin<les)

Ptens*ult .-- t⋅ su⋅ Ftu*eb⋅ $u 8if

.8= t⋅ su⋅ Ftu*eb⋅ $u -if

:=

2'right to eb fasteners margin of safety)

MSu*tens*fasteners

Pu*tens*ult

Ptens*ult

8−:=

-- ef. 8, 'g. ;/, formulas ;8 and ;-

8D



,.8 F0&ge A&0+)i),

,.8.1 C!re))i& F0&ge

Com'ressi(e stress in flange caused by 6")

f fc

< − !⋅ he⋅ cot α( )⋅

- Afc⋅:=

Primary ma7imum bending moment in the flange >o(er an u'right? is)

Mf*ma7 < c:⋅! he⋅ d

-⋅ tan α( )⋅

8- h⋅⋅:=

Com'ression flange margin of safety)

MSc*flange

Fcc*flange

Pfc

Afc

f fc+ Mfc cc⋅

Ic

Mf*ma7 cc⋅

Ic

+

−8−:=

,.8., Te&)i& F0&ge

"ension stress in flange caused by 6")

f ft

< !⋅ he⋅ cot α( )⋅

- Aft⋅

:=

"ension flange margin of safety)

MSt*flange

Ftu*flange

Pft

Aft

f ft+ Mft ct⋅

It

Mf*ma7 ct⋅

It

+

+

8−:=

-: ef. 8, 'g. =, sec. ;.8D

8



,.< We( S#re)) C!&e&#),7

"ension in α direction)

f α

- < ⋅ f s⋅

sin - α⋅( ) 8 < −> ? f s⋅ sin - α⋅( )⋅+:=

Com'ression in α π9- direction)

f α±/° 8 < −> ?− f s⋅ sin - α⋅( )⋅:=

Shear in α 'lane)

f sα 8 < −> ? f s⋅ cos - α⋅( )⋅:=

Ma7imum 'rinci'al stress direction)

β8

-atan

tan - α⋅( )

<

⋅:=

Princi'al tension >in β direction?)

f 8

< f s⋅

sin - α⋅( )

f s 8 < - 8

sin - α⋅( ) -

8− ⋅+⋅+:=

Princi'al com'ression >in β π9- direction?)

f -

< f s⋅

sin - α⋅( )f s 8 <

- 8

sin - α⋅( )-

8− ⋅+⋅−:=

Princi'al shear >in β π9; 'lane?)

f : f s 8 < - 8

sin - α⋅( )-

8− ⋅+⋅:=

-; ef. -, 'g. A/, formulas A.8 to A.8D

8B



. NUMERICAL ANALYSIS OF A BEAM IN INCOMPLETE

DIAGONAL TENSION

"he analysis as 'erformed for beam III+-=+D6-=, from reference =, a''lying the

methodology 'resented in the 'rior cha'ter."he beam mentioned abo(e as tested, by $ACA >$ational Ad(isory Committee

for Aeronautics?, u' to failure. A com'arison beteen the analytical results and the test

data results from reference = is 'resented in ne7t cha'ter.

1eam III+-=+D6 as chosen to (alidate the methodology of the 're(ious cha'ter for

the folloing reason) $ACA analytical 'rediction for beam III+-=+D6 as one of the

most unconser(ati(e 'redictions from a set of ;/ beams-D. $ACA analytically 'redicted

failure at a load N higher then actual failure load resulted form test.

"he general built+u' structure of beam III+-=+D6 is as follo)

• beam height is -D.8

• eb is .-/=, =+"D A& Clad

• double u'rights) to bac<+to+bac< angles >.D-= 7 .D-=? fabricated for

.;/, =+"D A& Clad

• double flange) to bac<+to+bac< e7truded angles >-. 7 -. 7 .8BB?, =+

"D A& E7trusion.

&oading of the structure)

"he cantile(er beam III+-=+D6 as loaded at the free end ith a trans(ersal load

Ps R 88,D//lb, re'resenting the ultimate load at failure, based on methodology

from 're(ious cha'ter.

-= ef. =, 'g. :D, "able 8.

-D ef =, 'g. : and :/, "ables - and ;.

8/



.1 I&u# D# %r IDT A&0+)i) % (e! III-,9-8D @Re%. 9* g. 8

"he folloing data as used as in'ut for the MathCad Code from A''endi7 A.

A''lied &oads)

5eb shear flo)

! ;B8lb

in=

generated by a''lied trans(ersal load) Ps

R 88D//lb, here)

!Ps

he

:=

Internal stiffener and flange loads)

Pu lb:= Pfc lb:= Mfc in lb⋅:= Pft lb:= Mft in lb⋅:=

$ote) For both, u''er and loer ca', 'ositi(e moment is reacted by Flange Ca' in com'ression.

5eb Pro'erties)

t .-/=in:= he -;.: in= hc --.8in:= E 8='si:=

F

ty*eb

D:'si:= F

tu*eb

;'si:= F

su*eb

;;'si:=

mat =:= >matR= for material A& =+"D matR--; for material A& --;+":?

2'right Pro'erties)

d 8=.in:= dc 8;.:=in:= h -D.8in:= hu -:.:in:= tu .;/in:=

Au .8in-:= eu .in:= Iu .B=in

;:=

Lu .:Bin::= bu .D-=in:= >for double u'rights only?

$

u

-:= >$uR8 for single u'rights $uR- for double u'rights?

matu =:= >matuR= for material A& =+"D matuR--; for material A& --;+":?

Eu 8= 'si:= Fcy*u'right D:'si:=

-



Flange Pro'erties)

tf .8BBin:= Afc 8.=in-:= Aft 8.=in

-:= Ic .:;Bin;:= It .:;Bin

;:=

Fcc*flange − 'si:= Ftu*flange /'si:= cc .=;in:= ct .=;in:=

$f

-

:= ( $

f R8 for single flange $

f R- for double flange?

Fasteners Pro'erties)

5eb to flange fasteners ultimate joint alloable and s'acing)

PEf*shear*ult D8:lb:=>%&8B+= in .-/= A& Clad =+"D, double shear?.

sEf

.B=in:=

2'right to flange fasteners ultimate joint alloable, and number of fasteners

reacting the u'right load in gusset action)

Puf*shear*ult -;DDlb:= >8 7 %&8B+D in .;/ A& Clad =+"D, - 7 single shear?.

$uf 8:= $gusset -:=

2'right to eb fasteners ultimate joint alloable and s'acing)

PuE*shear*ult D8:lb:=>%&8B+= in .-/= A& Clad =+"D, double shear?.

PuE*tens*ult 8;;lb:= suE .B=in:=

For double u'rights only, u'right+to+u'right single shear fastener joint alloable.

Puu*shear*ult 8/Dlb:=>%&8B+= in .;/ A& Clad =+"D, single shear?.

., Li!i##i&) IDT T$er+* eri%i#i&

"he folloing geometrical limitations shall be considered, due to limitation of test data)

hc

t

;/= 88=hc

t

< 8=<

dc

hc

.D== .-dc

hc

< 8.<

tu

t

8.DD=tu

t

.D>

-8



"he beam meets all geometrical limitations shon abo(e.

--



. We(* P)#-Bu20i&g A&0+)i)

..1 S$er (u20i&g e%%iie&# %r )i!0+ )ur#e' &e0 3K ))4

From section -.:.8, theoretical shear buc<ling coefficient for sim'ly su''orted 'anel is)

4 ss -.BD=dc

hc

-

⋅ .Bdc

hc

⋅+ ;.B+:= 4 ss D.=:8=

.., We( %i5i#+ e%%iie&#) 3R u 6 R % 4

"he eb fi7ity coefficients are calculated based on the methodology from section -.:.-.

5eb fi7ity coefficient at the u'rights, for single u'rights)

u8 .:-//tu

t

-

⋅ .-/;tu

t

⋅+ .8=+

tu

t

.D≤if

.//tu

t

⋅ .:88− .D

tu

t

< 8.-=≤if

.-DDtu

t

:

⋅ 8.B-/tu

t

-

⋅− ;.;=tu

t

⋅+ 8.B:;:− 8.-=

tu

t

< -.=≤if

.;tu

t

⋅ 8.8/8+ otherise

:=

u8 8.8B=

5eb fi7ity coefficient at the u'rights, for double u'rights)

u-

tu

t

tu

t

8≤if

.8=88tu

t

:

⋅ 8.;Dtu

t

-

⋅− -.=-tu

t

⋅+ .DB-− 8

tu

t

< -.=≤if

./tu

t

⋅ 8.:=B+ otherise

:=

u- 8.;:=

-:



5eb fi7ity coefficient at the u'rights, considering single or double u'rights is)

u u8 $u 8if

u- $u -if

:= u 8.;:=

5eb fi7ity coefficient at the ca'+flanges, for single flange)

f8 .:-//tf

t

-

⋅ .-/;tf

t

⋅+ .8=+

tf

t

.D≤if

.//tf

t

⋅ .:88− .D

tf

t

< 8.-=≤if

.-DDtf

t

:

⋅ 8.B-/tf

t

-

⋅− ;.;=tf

t

⋅+ 8.B:;:− 8.-=tf

t

< -.=≤if

.;tf

t

⋅ 8.8/8+ otherise

:=

f8 8.;;=

5eb fi7ity coefficient at the ca'+flanges, for double flange)

f-

tf

t

tf

t

8≤if

.8=88tf

t

:

⋅ 8.;Dtf

t

-

⋅− -.=-tf

t

⋅+ .DB-− 8

tf

t

< -.=≤if

./tf

t

⋅ 8.:=B+ otherise

:=

f- 8./-;=

5eb fi7ity coefficient at the ca'+flanges, considering single or double flanges is)

f f8 $f 8if

f- $f -if

:= f 8./-;=

-;



.. Cri#i0 )$er )#re)) 3F)r4

"he folloing calculations are based on methodology from section -.:.:.

Elastic critical shear >Fscr*el?)

Fscr*el 4 ss E⋅ t

dc

-

⋅ u8

- f u−( )⋅ dc

hc

:

⋅+

⋅ dc

hc

8≤if

4 ss E⋅t

hc

-

⋅ f 8

- u f −( )⋅

hc

dc

:

⋅+

⋅ otherise

:=

Fscr*el ;-D'si=

Critical shear stress corrected for 'lasticity effects >Fscr ?)For Clad A& =+"D)

Fscr*="D Fscr*el Fscr*el =<si≤if

.B=D Fscr*el⋅ ./--<si+ =<si Fscr*el< ;8< <if

.--88Fscr*el -;./:<si+ Fscr*el ;8<si≥if

:=

Fscr*="D ;-D'si=

For Clad A& --;+":)

Fscr*--;": Fscr*el Fscr*el 8<si≤if

.:

<si-

Fscr*el:⋅

.:D

<siFscr*el

-⋅− 8.:= Fscr*el⋅+ :.BB8<si− Fscr*el 8<si>if

:=

Fscr*--;": ;-D'si=

Considering materials listed abo(e the critical shear corrected for 'lasticity effects is)

Fscr Fscr*="D mat =if

Fscr*--;": mat --;if

:=

Fscr ;-D'si=

-=



..7 Dig&0-#e&)i& %#r 324

From section -.:.;, the a''lied shear stress is)

f s!

t

:=

f s 8D:-'si=

From section -.:.;, the diagonal+tension factor is)

< tanh .= logf s

Fscr

⋅

f s Fscr ≥if

otherise

:=

< .D=/=

..9 A&g0e % 'ig&0-#e&)i& 3:4

"he folloing calculations are based on the methodology from section -.:.=.

Effecti(e cross+sectional area of the stringer is)

Aue

Au

8

e

uρ u

-

+

$u 8if

Au $u -if

:=

here) ρ u

Iu

Au:=

Aue .8 in-= ρ u .-B: in=

1uc<ling angle for 'ure diagonal+tension is)

αP6" atan

;

8

h t⋅

Afc Aft++

8d t⋅

Aue

+

:=

αP6" :=.8deg=

-D



1uc<ling angle for incom'lete diagonal+tension as calculated by linear inter'olation)

α ;=deg < ;=deg αP6"−( )⋅−:=

α :B.;/deg=

..8 F0&ge %0e5i(i0i#+ %#r 3"'4

From section -.:.D, the flange fle7ibility factor is)

d d s in α( )⋅

;8

It

8

Ic

+ t

;he

⋅:=

d 8./8=

..< A&g0e &' )#re)) &e&#r#i& %#r) 31* ,* 4

"he folloing calculations are based on the methodology from section -.:..

Angle factor)

c88

sin - α⋅( )8−:=

c8 .-D;=

Stress concentration factors)

c- d 8≤if

.=: d:⋅ .-; d

-⋅− .-//; d⋅+ .8=− 8 d< :≤if

.=; d⋅ 8.8/− otherise

:=

c- .D;=

c: 8 d 8≤if

.=: d:⋅ .D;B d

-⋅− .8-; d⋅+ ./:/8+ 8 d< =≤if

.=B otherise

:=

c: ./=

-



..= We( e2 &!i&0 )#re)) 3% )>!54

From section -.:.B, the eb 'ea< nominal stress is)

f s*ma7 f s 8 < -

c8⋅+ 8 < c-⋅+( )⋅:=

f s*ma7 8-D's=

Stress in the eb is e7'ressed as nominal shear stress for < R and nominal diagonal+

tension stress for < R 8.

..; We( &!i&0 )#re)) 00"(0e 3F)>004

From section -.:./, the eb nominal stress alloable is)

Fs*all ./ Fty*eb⋅ 88

-

Ftu*eb

Fty*eb

8− -

⋅+

⋅

8

-8 < −> ?

:Fsu*eb

Ftu*eb

8

-−

⋅+

⋅:=

Fs*all -B//'si=

5eb nominal stress alloable satisfies shear failure for 'ure shear > < R ? and tensile

failure for 'ure diagonal+tension > < R 8 ?.

..1? We( Mrgi& % S%e#+ 3MS"e(4

MSeb

Fs*all

f s*ma7

8−:=

MSeb .D/=

-B



.7 Urig$# A&0+)i)

.7.1 Urig$# 0u!& (u20i&g

"he folloing calculations are based on the methodology de(elo'ed in section -.;.8.

2'right effecti(e length)

&e

hu

8 < -

: -d

hu

⋅−

⋅+

d 8.= h⋅≤if

hu otherise

:=

&e 8.D; in=

2'right stress due to diagonal+tension)

f u

< − f s⋅ tan α( )⋅

Aue

d t⋅.= 8 < −> ?⋅+

:=

f u -D− 'si=

2'right Euler column buc<ling stress alloable)

Fc π-

− Eu⋅&e

- ρ u⋅

-−⋅ $u 8if

π-

− Eu⋅&e

ρ u

-−

⋅ $u -if

:=


Fc min Fc Fcy*u'right,( )−:=

Fc -DD-− 'si=

>Single u'rights?

>6ouble u'rights?

-/



For double u'rights only, the fasteners holding the u'rights together need to be chec<ed

if they ha(e the ca'ability to transfer the folloing shear flo)

!u

-.= Fcy*u'right⋅ Lu⋅

bu &e⋅ $u -if

otherise

:=

!u =;lb

in=

2'rights fasteners single shear alloable)

!u*all

Puu*shear*ult

su

:=

!u*all 8-B/

lb

in=

If fastener shear alloable >!u*all? is less than a''lied shear, the alloable stress for

column failure must be multi'lied by the folloing reduction factor)

C 8

8-=.D/−

!u*all

!u

-

⋅ =.B:8!u*all

!u

⋅+ :.=:+

!u*all< !u<if

8. otherise

:=

C 8.=

ecalculate u'right column buc<ling stress alloable > Fc ?)

Fc C Fc⋅:=

Fc -DD-− 'si=

2'right margin of safety for column buc<ling)

MSu*col*bu<ling

Fc

Pu

Aue

f u+8−:=

MSu*col*bu<ling .-B=

:



.7., Urig$# %re' ri0i&g

"he folloing calculations are based on the methodology de(elo'ed in section -.;.-.

2'right ma7imum com'ressi(e stress > f u*ma7 ?)

f u*ma7 f u 8 < −> ?⋅ .-/ .D:DD dhu

⋅− ⋅ f u+:=

f u*ma7 -::-− 's i=

2'right forced cri''ling stress alloable > Ffc ?)

Ffc :-=−

:

< -

tu

t

⋅ 's i⋅ matu =( ) $u 8( )∧if

-D−:

< -

tu

t

⋅ 's i⋅ matu --;( ) $u 8( )∧if

-D−

:

< -

tu

t

⋅ 's i⋅ matu =( ) $u -( )∧if

-8−

:

< -

tu

t

⋅⋅ 's i matu --;( ) $u -( )∧if

:=

:8




Ffc min Ffc Fcy*u'right,( )−:=

Ffc -::-− 'si=

2'right margin of safety for forced cri''ling)

MSu*forced*cri''ling

Ffc

f u*ma78−:=

MSu*forced*cri''ling .=

$ote) $atural cri''ling is not a controlling factor in the design.

>Single u'rights, =+"D?

>Single u'rights, --;+":?

>6ouble u'rights, =+"D?

>6ouble u'rights, --;+":?

:-



.9 F)#e&er) A&0+)i)

.9.1 We( T F0&ge F)#e&er)

As shon in section -.=.8, eb to flange fasteners react the folloing shear flo)

!f

! he⋅

hu

8 .;8; < ⋅+> ?⋅:=

!f D:/lb

in=

5eb to flange fasteners margin of safety)

MSf*fasteners

Pf*shear*ult

sf !f ⋅:=

MSf*fasteners 8.8:=

.9., Urig$# # F0&ge F)#e&er)

As shon in section -.=.-, u'right to flange fasteners react the load e7isting in the

u'right, due to 6" into the flange.

2'right &oad due to 6" > Pu*6" ?)

Pu*6" f u Aue⋅:=Pu*6" ---8− lb=

"otal fastener joint shear alloable considering gusset action)

Puf*all $uf Puf*shear*ult⋅ $gusset $uf −( ) Pu*shear*ult⋅+ $gusset $uf >if

$uf Puf*shear*ult⋅ otherise

:=

Puf*all :/ lb=

2'right to flange fasteners margin of safety)

MSuf*fastenersPuf*all

Pu*6"

8−:=

MSuf*fasteners .:/=

::



.9. Urig$# # We( F)#e&er)

As shon in section -.=.:, u'right to eb fasteners re!uired to ha(e enough tension

strength to 're(ent tension failure caused by the eb rin<les)

Ptens*ult .-- t⋅ su⋅ Ftu*eb⋅ $u 8if

.8= t⋅ su⋅ Ftu*eb⋅ $u -if

:=

Ptens*ult -B.::- lb=

2'right to eb fasteners margin of safety)

MSu*tens*fasteners

Pu*tens*ult

Ptens*ult

8−:=

MSu*tens*fasteners ;.8=

:;



.8 F0&ge A&0+)i)

.8.1 C!re))i& F0&ge

"he folloing calculations are based on the methodology 'resented in section -.D.8.

Com'ressi(e stress in flange caused by 6")

f fc

< − !⋅ he⋅ cot α( )⋅

- Afc⋅:=

f fc :-:;− 's i=

Primary ma7imum bending moment in the flange >o(er an u'right? is)

Mf*ma7 < c:⋅! he⋅ d

-⋅ tan α( )⋅

8- h⋅⋅:=

Mf*ma7 ;-: in lb⋅=

Com'ression flange margin of safety)

MSc*flange

Fcc*flange

Pfc

Afc

f fc+ Mfc cc⋅

Ic

Mf*ma7 cc⋅

Ic

+

−

8−:=

MSc*flange D.;=

.8., Te&)i& F0&ge

"he folloing calculations are based on the methodology 'resented in section -.D.-.

"ension stress in flange caused by 6")

f ft

< !⋅ he⋅ cot α( )⋅

- Aft⋅:=

f ft :-:;'si=

"ension flange margin of safety)

MSt*flange

Ftu*flange

Pft

Aft

f ft+ Mft ct⋅

It

Mf*ma7 ct⋅

It

+

+

8−:=

MSt*flange D./;=

:=



.< We( S#re)) C!&e&#)

"he folloing calculations are based on the methodology 'resented in section -..

"ension in α direction)

f α- < ⋅ f s⋅

sin - α⋅( )8 < −> ? f s⋅ sin - α⋅( )⋅+:= f α -='si=

Com'ression in α π9- direction)

f α±/° 8 < −> ?− f s⋅ sin - α⋅( )⋅:= f α±/° =;8D− 's i=

Shear in α 'lane)

f sα 8 < −> ? f s⋅ cos - α⋅( )⋅:=f sα 8-=:'si=

Ma7imum 'rinci'al stress direction)

β8

-atan

tan - α⋅( )

<

⋅:= β ;.deg=

Princi'al tension >in β direction?)

f 8

< f s⋅

sin - α⋅( )f s 8 <

- 8

sin - α⋅( )-

8− ⋅+⋅+:= f 8 -==='si=

Princi'al com'ression >in β π9- direction?)

f -

< f s⋅

sin - α⋅( )f s

8 < - 8

sin - α⋅( ) -8− ⋅+⋅−:= f

-=;D:− 's i=

Princi'al shear >in β π9; 'lane?)

f : f s 8 < - 8

sin - α⋅( )-

8− ⋅+⋅:= f : 8D=/'si=

:D



7. RESULTS AND COMPARISON WITH TEST DATA

7.1 Mrgi&) % S%e#+ Su!!r+

A summary of margins of safety calculated in 're(ious cha'ter are 'resented in "able 8

belo.

A''lied trans(ersal load, at the free end of the cantile(er beam III+-=+D6 as Ps R

88,D// lb.

Structure Critical Failure Mode MS

5eb Sheet Failure due to I6" .D/

2'right Column 1uc<ling

Forced Cri''ling

.-B

.

Fasteners + 5eb to Flange

+ 2'right to Flange

+ 5eb to 2'right

1earing in 5eb

1earing in 2'right

Fastener "ension

8.8:

.:/

;.8

Com'ression Flange $atural Cri''ling D.;

"ension Flange "ension Strength D./;

T(0e 1. Mrgi& % S%e#+ Su!!r+

As shon in "able 8, the failure mode of the beam is u'right forced cri''ling

>loest margin of safety?.

"he beam is e7'ected to fail at an a''lied trans(ersal load Ps R 88,D//lb, for

hich the u'right forced cri''ling margin of safety is zero.

:



7., A&0+#i0 ). Te)# Re)u0#)* C!ri)&

1eam III+-=+D6 as tested to failure by $ACA >$ational Ad(isory Committee for

Aeronautics? and the test results are documented in reference =.

A com'arison beteen the analytical results and the test results is 'resented in "able

-.

esult to Com'are S ym b ol

2nits $ACA

"est

esults

$ACA

Analytical

Prediction

Current

Methodology

Analytical

Prediction

5eb Critical Shear Stress Fscr Psi +++ ;8 ;-D

6" Factor < +++ +++ .DD- .D=/

2lt. Column 1uc<ling &oad Fc lb +++ 8;,B 8;,D8

2lt. Forced Cri''ling &oad Ffc lb 88,; 8-,- 88,D//

2lt. &oad Q 5eb Failure F lb +++ -,= 8/,=:

Failure Mode

+

++ +++ F.C. F.C. F.C

T(0e ,. A&0+#i0 ). Te)# Re)u0#)* C!ri)&

$ote) F.C. stands for u'right forced cri''ling.

:B



A com'arison of the current methodology to $ACA analytical 'rediction, based on the

results listed in "able -, is shon in "able :.

esult to Com'are Current Methodology (s. $ACA

Analytical Prediction

5eb Critical Shear Stress :./N

6" Factor .=N

2lt. Column 1uc<ling &oad +8.:N

2lt. Forced Cri''ling &oad +;.8N

2lt. &oad Q 5eb Failure +;.N

T(0e . Curre&# Me#$'0g+ ). NACA A&0+#i0 Pre'i#i&

A com'arison of the analytical 'redictions to the $ACA test results, based on the

results listed in "able -, is shon in "able ;.

esult to Com'are Current Methodology

Analytical Prediction (s.

$ACA "est esults

$ACA Analytical

Prediction (s. $ACA

"est esults

2lt. Forced Cri''ling &oad -.DN .N

T(0e 7. A&0+#i0 Pre'i#i&) ). Te)# Re)u0#)

:/



9. CONCLUSIONS

8? As can be seen in "able - and ;, $ACA analytical 'rediction for u'right forced

cri''ling >Ffc? as .N higher then the load at failure resulted from test. "he

current methodology analytical 'rediction as only -.DN higher then the actual

load at failure. "hat shos that the current methodology 'resented in Cha'ter -

of this re'ort is at least as accurate as $ACA analytical 'rediction.

-? 1oth analytical 'redictions >$ACA and current methodology form Cha'ter -?

shoed unconser(ati(e results for u'right forced cri''ling failure. Considering

that for ultimate failure analysis the loads ha(e a built in a factor of safety of 8.=,

the -.DN (ariation from the test failure is negligible.

:? Current methodology 'rediction of the u'rights ultimate column+buc<ling load is

8.:N more conser(ati(e than $ACA analytical 'rediction.

;? Current methodology 'rediction of the ultimate load for eb failure is ;.N more

conser(ati(e than $ACA analytical 'rediction.

=? 1ased on the com'arison shon in "ables -, : and ;, the methodology 'resented

in Cha'ter - is considered (alid and a''licable in 'ractice.

;



REFERENCES

8? T$ACA+"$+-DD8T A Summary of 6iagonal "ension, Part I Q Methods of Analysis,

$ACA, 5ashington, May 8/=-.

-? T$ASA+C+88B=;T In(estigation of 6iagonal+"ension 1eams ith ery "hin

Stiffened 5ebs, Grumman Aeros'ace Cor'oration, 1eth'age, $e 3or<, Uuly 8/D/

>Includes an im'ro(ement to the $ACA method. Study com'leted by Grumman

Aeros'ace for $ASA?.

:? TAnalysis and 6esign of Flight ehicle StructuresT >Cha'ter C88?, by E.F. 1ruhn,

Uacobs Publishing, Uune 8/:.

;? TAirframe Stress Analysis and SizingT -nd Edition, by Michael $iu >Cha'ter 8-?,

%ong 4ong Conmilit Press, 8//.

=? T$ACA+"$+-DD-T A Summary of 6iagonal "ension, Part II Q E7'erimental

E(idence, $ACA, 5ashington, May 8/=-.

D? T$ACA+"M+;/T Structures of "hin Sheet Metal, "heir 6esign and Construction,

$ACA, 6ecember 8/-B.

;8



APPENDI A. ATTACHED ELECTRONIC FILES

"he electronic files listed belo are com'ressed in file)

OA''endi7 A, MP 1eams in I6", C*Moisiade.zi'

File $ame File "y'e 6escri'tion

1eams*in*I6"*Cezar*Moisiade*

+88+-/.7mcdMathCad 8:

Includes the numerical

methodology to 'erform

analysis of beams in I6".

$ACA*Charts*Cezar*Moisiade*

+88+-/.7lsMicrosoft E7cel

Includes cur(e fits for $ACA+

"$+-DD8 charts used in I6"

analysis.

;-



APPENDI B. FINITE ELEMENT ANALYSIS -

PRELIMINARY

Additional efforts ha(e been done on 'erforming a finite element simulation of a

stiffened eb in incom'lete diagonal tension. "he efforts ha(e not been com'leted. "hesimulation got as far as de(elo'ing the methodology and getting 'reliminary results for a

test model that as used to (alidate the methodology.

"he nonlinear 'ost+buc<ling analysis as 'erformed in A$S3S B., folloing three

ste's)

8. Static &inear Analysis + of a 'anel under shear load.

-. Eigen+1uc<ling Analysis Q 'erformed for the 're+stress 'anel, using the

results from ste' 8.

:. $onlinear 1uc<ling Analysis Q the 'anel had the geometry 'erturbed

based on a s'ecific eigen+(alue resulted from ste' -, then a large

deflection analysis, using arc+length method as 'erformed.

"he 'anel geometry as .- 7 8-. 7 8-., and the material A& =+". For this

'reliminary run, the stiffeners and flanges ere defined by an area of .8 in-, area

moment of inertia of .8 in;, and elastic modulus of :eD 'si.

"he 'anel as modeled ith shell elements 8B8, and the stiffeners and flanges

ere modeled ith beam elements ;. Fasteners ere simulated using rigid cou'led

constrains.

All electronic files for A''endi7 1, are com'ressed in folder) OA''endi7 1, MP

1eams in I6", FEAnalysis, C*Moisiade.zi'

;:



"he finite element model including boundary conditions is shon in Figure :.

Figure . Fi&i#e E0e!e&# M'e0 &' Bu&'r+ C&'i#i&).

;;



Preliminary out of 'lane dis'lacement results from the eigen+buc<ling analysis are

shon in Figure ;.

Figure 7. Eige&-Bu20i&g Re)u0#)* 77#$ Eige&0ue* Re0#ie Di)0e!e&# @i&.

;=



Preliminary results from the nonlinear buc<ling analysis are shon in Figures =, D and .

"he a''lied shear load got u' to ;:.; lb9in. "he analysis ill ha(e to continue to get to

higher loads.

Figure 9. N&0i&er-Bu20i&g Re)u0#)* Di)0e!e&# @i&.

;D



Figure 8. N&0i&er-Bu20i&g Re)u0#)* 1)# Pri&i0 S#re)) @)i.

Figure <. N&0i&er-Bu20i&g Re)u0#)* S$er S#re)) @)i.

;



"he A$S3S in'ut file for the three analysis ste's, listed abo(e, is shon belo.

!**********************************!* Step 1, Static Linear Run *!**********************************/SOLU

!F=100 !shear force applied (lbs="00001 !side pressure applied (psifscale,Fsfscale,pres,!ant#pe,static$%SL&,S'R, ,0,L)$O,0S+R$S,O!SOL&$FS-!/OS+1FS-!!*********************************!* Step ., $ihen uclin *!*********************************/SOLU'+2$,bucleU3O+,L',40,0,05'6,40,0,0,#es,0"001,SOL&$FS-!/OS+1FS-V!*********************************!* Step 7, onlinear uclin *!*********************************

/R$8sfdele,all,all !delete pressureU)$O,0"01,1,99,:;s04<e:,:rst: !perturb eo;etr# per 99th buclin ;ode!F=>00 !shear force applied (lbsfscale,F!/SOLUnl?cntrl=1!'+2$,S+'+3L)$O,OOU+R$S,'LL,'LL,!*if,nl?cntrl,e@,0,then ti;e,F SOL3O+ROL,O

RO+,FULL SUS+,40,1e9,.4*elseif,nl?cntrl,e@,1 SOL3O+ROL,OFF

SUS+ 40 1e9 .4

Documents

MP Beams in IDT, C_Moisiade_08!08!2009