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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, -/
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0 Co'yright -/
by
Cezar I. Moisiade
All ights eser(ed
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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;
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-.=.- 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(
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:.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.............................;
(
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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
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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................................................;;
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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
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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
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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
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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.
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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.
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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
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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:.
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,. 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&.
$ote) In Figure -, for both, u''er and loer ca', 'ositi(e moment is reacted by flange
ca' in com'ression.
"he theory of ebs Oincom'lete diagonal tension is a method for inter'olating beteen
the to limiting cases of Oshear+resistant and O'ure diagonal tension, the limiting cases
being included.
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Failure modes for beams ith stiffened ebs in incom'lete diagonal+tension are defined
in four categories)
a? Sheet failure Q ru'turing of the sheet 'rior to any instability in the u'rights
>stiffeners?.
b? 2'right local failure by forced cri''ling Q local buc<ling of one or more
u'rights, causing a significant dro' in the u'rights sustained load, resulting in
sheet failure or total colla'se, due to redistribution of loads.
c? 2'right failure by column buc<ling Q long column buc<ling of one or more
stiffeners, that e(entually results in colla'se of the structure.
d? Fastener failure Q not common in a good design.
e? Flange failure Q not common in good design.
,.1 Li!i##i&) &' A))u!#i&) % IDT T$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
=≤
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,. 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.
=
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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
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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.
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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 $u -if
:=
here) ρ u
Iu
Au
:=
D ef. 8, 'g. 8B, formula -
ef. 8, 'g. ;8, section ;.8
B
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"he buc<ling angle for 'ure diagonal+tensionB is)
αP6" atan
;
8h t⋅
Afc Aft++
8
d t⋅
Aue+
:=
"he buc<ling angle for incom'lete diagonal+tension as calculated by linear
inter'olation)
α ;=deg < ;=deg αP6"−( )⋅−:=
,..8 F0&ge %0e5i(i0i#+ %#r 3"'4;
d d s in α( )⋅
;8
It
8
Ic
+ t
;he
⋅:=
,..< A&g0e &' )#re)) &e&#r#i& %#r) 31* ,* 4
Angle factor 8)
c88
sin - α⋅( )8−:=
Stress concentration factors88)
c- d 8≤if
.=: d:⋅ .-; d
-⋅− .-//; d⋅+ .8=− 8 d< :≤if
.=; d⋅ 8.8/− otherise
:=
B ef. 8, 'g. 8, formula 8=
/ ef. 8, 'g. 88, formula 8/
8 er. 8, 'g. -
88 ef. 8, 'g. 88-, figure 8B. For the cur(e+fit of the chart see A''endi7 A.
/
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c: 8 d 8≤if
.=: d:⋅ .D;B d
-⋅− .8-; d⋅+ ./:/8+ 8 d< =≤if
.=B otherise
:=
,..= We( e2 &!i&0 )#re)) 3% )>!541,
f s*ma7 f s 8 < -
c8⋅+ 8 < c-⋅+( )⋅:=
Stress in the eb is e7'ressed as nominal shear stress for < R and nominal diagonal+
tension stress for < R 8.
,..; We( &!i&0 )#re)) 00"(0e 3F)>0041
Fs*all ./ Fty*eb⋅ 88
-
Ftu*eb
Fty*eb
8−
-
⋅+
⋅
8
-8 < −> ?
:Fsu*eb
Ftu*eb
8
-−
⋅+
⋅:=
5eb nominal stress alloable satisfies shear failure for 'ure shear > < R ? and tensile
failure for 'ure diagonal+tension > < R 8 ?.
,..1? We( Mrgi& % S%e#+ 3MS"e(4
MSeb
Fs*all
f s*ma7
8−:=
8- ef. 8, 'g. -, formula >::a?.
8: ef) -, 'g. 8D
8
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,.7 Urig$# 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⋅ $u -if
otherise
:=
8; ef. 8, 'g. ;D, formula :=
8= ef. 8, 'g. 8/, formula :a
>Single u'rights?
>6ouble u'rights?
88
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2'rights fasteners single shear alloable)
!u*all
Puu*shear*ult
su
:=
If fastener shear alloable >!u*all? is less then a''lied shear, the alloable stress for
column failure must be multi'lied by the folloing reduction factor 8D)
C 8
8-=.D/−
!u*all
!u
-
⋅ =.B:8!u*all
!u
⋅+ :.=:+
!u*all< !u<if
8. otherise
:=
ecalculate u'right column buc<ling stress alloable > Fc ?)
Fc C Fc⋅:=
2'right margin of safety for column buc<ling)
MSu*col*bu<ling
Fc
Pu
Aue
f u+
8−:=
8D ef. 8, 'g. 88D, figure -8. For the cur(e+fit of the chart see A''endi7 A.
8-
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,.7., Urig$# %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:
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&imit the alloable to Fcy)
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) $atural cri''ling is not a controlling factor in the design.8/
8/ ef. :, 'g. C88.8B
>Single u'rights, =+"D?
>Single u'rights, --;+":?
>6ouble u'rights, =+"D?
>6ouble u'rights, --;+":?
8;
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,.9 A&0+)i) % F)#e&er)
,.9.1 We( T F0&ge F)#e&er)
"he fasteners that connect the eb to the ca'+flange are re!uired to ha(e the ca'ability
to carry the folloing shear flo-)
!f
! he⋅
hu
8 .;8; < ⋅+> ?⋅:=
5eb to flange fasteners margin of safety)
MSf*fasteners
Pf*shear*ult
sf !f ⋅:=
,.9., Urig$# # 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
$uf Puf*shear*ult⋅ otherise
:=
2'right to flange fasteners margin of safety)
MSuf*fasteners
Puf*all
Pu*6"
8−:=
- ef. 8, 'g. :;, formula :;.
-8 ef. 8, 'g. ;B, formula :/.
8=
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,.9. Urig$# # 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
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,.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
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,.< 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
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. 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/
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.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:=
-
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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
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"he beam meets all geometrical limitations shon abo(e.
--
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. 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.;:=
-:
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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- $f -if
:= f 8./-;=
-;
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.. Cri#i0 )$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=
-=
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..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 $u -if
:=
here) ρ u
Iu
Au:=
Aue .8 in-= ρ u .-B: in=
1uc<ling angle for 'ure diagonal+tension is)
αP6" atan
;
8
h t⋅
Afc Aft++
8d t⋅
Aue
+
:=
αP6" :=.8deg=
-D
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1uc<ling angle for incom'lete diagonal+tension as calculated by linear inter'olation)
α ;=deg < ;=deg αP6"−( )⋅−:=
α :B.;/deg=
..8 F0&ge %0e5i(i0i#+ %#r 3"'4
From section -.:.D, the flange fle7ibility factor is)
d d s in α( )⋅
;8
It
8
Ic
+ t
;he
⋅:=
d 8./8=
..< A&g0e &' )#re)) &e&#r#i& %#r) 31* ,* 4
"he folloing calculations are based on the methodology from section -.:..
Angle factor)
c88
sin - α⋅( )8−:=
c8 .-D;=
Stress concentration factors)
c- d 8≤if
.=: d:⋅ .-; d
-⋅− .-//; d⋅+ .8=− 8 d< :≤if
.=; d⋅ 8.8/− otherise
:=
c- .D;=
c: 8 d 8≤if
.=: d:⋅ .D;B d
-⋅− .8-; d⋅+ ./:/8+ 8 d< =≤if
.=B otherise
:=
c: ./=
-
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..= We( e2 &!i&0 )#re)) 3% )>!54
From section -.:.B, the eb 'ea< nominal stress is)
f s*ma7 f s 8 < -
c8⋅+ 8 < c-⋅+( )⋅:=
f s*ma7 8-D's=
Stress in the eb is e7'ressed as nominal shear stress for < R and nominal diagonal+
tension stress for < R 8.
..; We( &!i&0 )#re)) 00"(0e 3F)>004
From section -.:./, the eb nominal stress alloable is)
Fs*all ./ Fty*eb⋅ 88
-
Ftu*eb
Fty*eb
8− -
⋅+
⋅
8
-8 < −> ?
:Fsu*eb
Ftu*eb
8
-−
⋅+
⋅:=
Fs*all -B//'si=
5eb nominal stress alloable satisfies shear failure for 'ure shear > < R ? and tensile
failure for 'ure diagonal+tension > < R 8 ?.
..1? We( Mrgi& % S%e#+ 3MS"e(4
MSeb
Fs*all
f s*ma7
8−:=
MSeb .D/=
-B
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.7 Urig$# 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
:=
&imit the alloable to Fcy)
Fc min Fc Fcy*u'right,( )−:=
Fc -DD-− 'si=
>Single u'rights?
>6ouble u'rights?
-/
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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⋅ $u -if
otherise
:=
!u =;lb
in=
2'rights fasteners single shear alloable)
!u*all
Puu*shear*ult
su
:=
!u*all 8-B/
lb
in=
If fastener shear alloable >!u*all? is less than a''lied shear, the alloable stress for
column failure must be multi'lied by the folloing reduction factor)
C 8
8-=.D/−
!u*all
!u
-
⋅ =.B:8!u*all
!u
⋅+ :.=:+
!u*all< !u<if
8. otherise
:=
C 8.=
ecalculate u'right column buc<ling stress alloable > Fc ?)
Fc C Fc⋅:=
Fc -DD-− 'si=
2'right margin of safety for column buc<ling)
MSu*col*bu<ling
Fc
Pu
Aue
f u+8−:=
MSu*col*bu<ling .-B=
:
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.7., Urig$# %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
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&imit the alloable to Fcy)
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) $atural cri''ling is not a controlling factor in the design.
>Single u'rights, =+"D?
>Single u'rights, --;+":?
>6ouble u'rights, =+"D?
>6ouble u'rights, --;+":?
:-
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.9 F)#e&er) A&0+)i)
.9.1 We( T F0&ge F)#e&er)
As shon in section -.=.8, eb to flange fasteners react the folloing shear flo)
!f
! he⋅
hu
8 .;8; < ⋅+> ?⋅:=
!f D:/lb
in=
5eb to flange fasteners margin of safety)
MSf*fasteners
Pf*shear*ult
sf !f ⋅:=
MSf*fasteners 8.8:=
.9., Urig$# # 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
$uf Puf*shear*ult⋅ otherise
:=
Puf*all :/ lb=
2'right to flange fasteners margin of safety)
MSuf*fastenersPuf*all
Pu*6"
8−:=
MSuf*fasteners .:/=
::
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.9. Urig$# # 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=
:;
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.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./;=
:=
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.< 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
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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.
:
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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
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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#)
:/
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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 $ACA analytical 'rediction.
=? 1ased on the com'arison shon in "ables -, : and ;, the methodology 'resented
in Cha'ter - is considered (alid and a''licable in 'ractice.
;
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REFERENCES
8? T$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
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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.
;-
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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.
:. $onlinear 1uc<ling Analysis Q the 'anel had the geometry 'erturbed
based on a s'ecific eigen+(alue resulted from ste' -, then a large
deflection analysis, using arc+length method as 'erformed.
"he 'anel geometry as .- 7 8-. 7 8-., and the material A& =+". For this
'reliminary run, the stiffeners and flanges ere defined by an area of .8 in-, area
moment of inertia of .8 in;, and elastic modulus of :eD 'si.
"he 'anel as modeled ith shell elements 8B8, and the stiffeners and flanges
ere modeled ith beam elements ;. Fasteners ere simulated using rigid cou'led
constrains.
All electronic files for A''endi7 1, are com'ressed in folder) OA''endi7 1, MP
1eams in I6", FEAnalysis, C*Moisiade.zi'
;:
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"he finite element model including boundary conditions is shon in Figure :.
Figure . Fi&i#e E0e!e&# M'e0 &' Bu&'r+ C&'i#i&).
;;
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Preliminary out of 'lane dis'lacement results from the eigen+buc<ling analysis are
shon in Figure ;.
Figure 7. Eige&-Bu20i&g Re)u0#)* 77#$ Eige&0ue* Re0#ie Di)0e!e&# @i&.
;=
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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
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Figure 8. N&0i&er-Bu20i&g Re)u0#)* 1)# Pri&i0 S#re)) @)i.
Figure <. N&0i&er-Bu20i&g Re)u0#)* S$er S#re)) @)i.
;
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"he A$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