McCombs Bruhn Supplement

2004·6.8 MB·English

by  William F. McCombs

A SUPPLEMENT TO ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES
William F. Mccombs
TAELE OF CONTENTS
Article and Main Topic Page Article and Main Topic Page
A5. 2Ja Beam-Columns 1 c·11. 29a Tension Field Beam Holes 49
A5,24a Beam-Column Formulas 1 Cll.24a Rivet Design 50
A5. 25a Beam-Column Deflections 1 Cll. Jla Stringer Construction 51
A5. 26a Beam-Column .Formulas 1 Cll. J2a Diagonal Tension - Stringers 51
A5.27a Margins of Safety 1 Cll,JJa Stringer System Allowables 54
A5,28a Truss Analyais 1 Cll,J4a Stringer Construction Example 57
A5,29a Tangent Moduius 1 ·c11.J5a Diagonal Tension - Longerons 60
A5.Jla Multispan·Beam~Columns 1 Cll,J6a Lorigeron System Analysis 61
A5. 32 Approximati! Buckling Formula 1 · Cll, J7a Longeron Construction Example 63
A6, 7a Plastic Torsion 1 Cll, J8a Diagonal Tension Summary 66
A7,la Beam Deflections 1 C11,J9a Problems for Part 2 67
A7,12a Numerical Analysis 1 C11,40a Problems for Part 1 67
All, 2a Moment Distribution· 2 ·CS, la Monocoque .Shell Buckling Data 67
All.5aa Stiffness/carry Over Factors 2 CS,2a Shell Axial Comp, Euckling 67
All,lJa Moment Distribution Factors 2 CS,5a Effect of Interna1.Pressure 67
Alf. 15a Truss Analysis 2 CS, 7a Shell Bending Buckling 67
A11, 15b Truss Analysis 5. CS, Sa.. Effect of Internal Pressure 67
AlJ,lla Curi(ed Beams 6 CS,9a External Hydrostatic Pressure 67
AlS, 5a Column Analysis 7 CS, lla Shell .~orsion Buckling 68
Al·B, Sa Colum.'l Analysis 7 cS, 12a Effect of Internal Pressure 68
A18, Sb Torsional Buckling 7 CB, 15a Combined Loads Buckling 6S
A1B,27a Column/Eeain-Column Data 8 CB,19a. Con.ical Shell Buckling 72
AlB, 28a Material Properties 8 CS, 20 Euckling of Spherical Caps 7J
A19,2J Rib Cruelling Loads 8 D1,2a Fitting Design 74
Cl, lJa Margins ot' Safety 8 Dl, Ja Fitting Margins of Safety 74
Cl,lJb Dealing With Tolerances 11 ·Di,4a Factors of Safety 74
C1,6a Combined Stresses 11 D1,5a Eolts 74
C2,la Column Cu):'Ve Construction 12 D1,6a Nuts 74
C2, 2a Free-Ended Columns 12 Dl, Baa Bushings . 74
C2,Ja Shear Effect on Buckling 14 Dl,12a Transversely Loaded Lugs 74
C2,Jb Multispan Columns 14 'Dl,lJa Obliquely Loaded Lugs 74
C2,6a Stepped Columns 16 Dl.2Ja Rive.ted Splices 74
C2,6b Numerical Column Analysis 17 Dl,27 Thread Design and.Strength 75
C2,6c Initially Bent Columns 21 DJ,5a Filler (Shim) Effects 76
C2, 7a Column Design Data 2J DJ, 7a Curved Beam Data 77
C2,8a Column Elastic Supports 23 D3,Ja Tension Clip Allowable Data 77
C2,10a. Need for Successive Trials 2J Dl,21a Flush Rivet Joints 77
C2,13a Column Elastic End Restraint 2J Dl,22a Elind Rivet Joints 77
c2,16 Tangent and Effective Moduli 25 Dl,7a Eolt Strengths 77
C2, 17 Buckling Load Data 26
C2,1B End Friction Effects 30 Design Check List At
References ·for Chapter C2 JO Margins of Safety Al
CJ,4a Plastic Bending Data 31 Tension Clips At
CJ,7a Plastic Bending Example 31 •Preload Torque Factors for Eolts A2
CJ, lOa Plastic Bending Procedure · Jl Rockwell and Erinell Hardness Data A2
CJ, lla Complex Plastic Bending 31 Minimum I for Stiffeners AJ
CJ,12a Shear Stresses in Plastic Range32 Stress-Strain Curves AJ
c3,15 Yield Stress Bending Modulus 33 .Sheet and Other Buckling Data A4
CJ,16 Residual Stress After Bending 3J .Material Propert~es A6
c3,1S Beam-Column Analyses 34 .Fastener Joint Design Notes AtO
c4, 20a Plastic Torsion 38 Fastener/J oint Allowable Data AlO
C4,2Ja Apparent Margins of Safety 39 Nut/Collar Tension Allowables A21
c5,7a Flat Plate Shear Buckling 39 Eolt Allowable Bending Moments A21
c5,Sa Flat Plate Bending Buckling 40 Beam Formulas A21
c7,lOa Unequal Angle Leg Thicknesses 41 Additional References A21
c7,JO Crippling Method Three 41 Bulkhead, Frame and Arch Analyses Bl
C7, Jl Torsional Buckling ···' er. 1;4 •Column Elastic Support De sigh ES
C9.13a Frame Stiffness Criteria 47 Operating Devices · E9
C10, 15b Thick-Web Beam Analysi.s 47 Doubler and Splice Desie;n .ElO.·
© 199B
William P. Mccombs
i
PREFACE
The widely used and.recommended'college/industry textbook
HAnalysis and Design of FlightVehtcle Structuresff by Dr, E.F.
Bruhn .has had only one revision .i:;ince its inception in. 1965,,
That was the 1973 ,ed1: tion in,. which Chapter A23 W!J..$., .. Iff!!Vi.sed and
expanded~ Chapter Cl) was .completely rewriJ;ten b;V' ~hother
author; .. and. a few minor ch~nge,s were made. i.n Chapte.r 1{1, .. Aside
from these the book· .remains in its original form. ··•o:.·r.
'.l'l'ie
purj)g13e .of tJ:i1,s supple111ent is to incre'as!! .t~~ ~'(5qpe
and usefulness, .of the textbook in .numerous .$peCifid. .. areas Of
analys,is • These inc;Lude. . columns, beam-column~,. ben4,h1g . .·
strenir.th, margins of.·safety, tension field analyses, fastener/
joint data; arches,:·bl1lkheads and numerous others •... The
Pl'a9J1:~~l.·· usf!! .. pf 'tl'ie ~.)tPP~emen-t is discusse.d, ~P ~?l" ,Introd.uc tton.
Only Ql'!e or. ;!;w9 ... aPPl:·i:;c;ia tiqrts of , the Supple.ment;'s, ;,<?,C>ntents ,can .·,
be· worth much more than its cost. ,, ,
'·, ·.· .,. Th,e 's~~pl:~,Jri'etl~· fri!;i.i be,: expanded in some futu~~.:.~ea,f;:c so ,any ';,
suggest. ioni;; f(>r. this., or for corrections or chaMes· in. .. J ts current
text wil. l. -b.... e;,-•.,/.'i l•.;'P ...P re. cia- ted."' 'T ' hqs: e readers who w1"-,;1 .,3 ,( !·1' ;j"t"p·";" b•. .. ;e.•;\,•,,1•'', 1:' 1fo' rl!l' ed' ·.
1
of any ·future.:.rev.isions or who ,have. suggest1ons.·.can/,c·ontact the:
p:.o.
author. at BO:i 763576, Dal1as, TX 7537·6-J57·6(TeTJ214'J37-5506).
. ,, - _h·" ,• '' .
The author is'. one'. Of the. coauthors Of the te'.Jitbook, His
career incluQ.es over f;orty ;years, of experience. Hi:,structural
analysis and design of·numerous aircraft and missiie'<projects in
the aerospace A1so included <are:1te9hn1c'al'' papers and the
industry~·
prep?,;i:·a t io!l .. an,d, t~;~~~i:tlg or p:l:jactical c.C>url\l.l:l~J"fi;l:'"~~ili€'tU,t~i design
and analysis .for .e_1:1gineer!l. work1,ng in the aerospa:Ce .. al:ld other
..i e's, · · · · · ·· ·· · ·· ·· · ·' · · ·· ·· ·· ·· ... · ····
industr 1 •·•· ·•
',·1 •'· ,T';· ·;;'·"[Xi:['._.'..:>·~
'· ··" · William F. Mccombs
. .
•' ~,
0 1> ···;
FOREWORD
I am please.~;;tq::11.~re ·.th~.jo,P1portuni.ty to fe~<;!i0¥~,n.c;l. ';tfiis
Supplement to my late, father's widely used c,ollegE!lind)lstrY
textbook "Analysis ai:liiibes ign 10'.f\ Flight YE! lii~l~ ;$ true ti.ires
H.
I hope its practical applicat<iorts will be a benefit to all who
have the textbook'. The e..aa1t'!o?l9.l da.ta contained in this ·
tO:
Supple111ent ca,n ''tie appf~~'4 iioth' the' ,S,k~lit,~ ~#,4, t)ie work of
structural desig?l anci ana].ysi,13. Thf;!! also be
:~upPJ,t;m~P,t,,131'lould
of interest· to, ·and eventually benefit, those.;.:W'ho may be.
considering purchasing the textbook.
-·
, .. .-··'
Patricia Bru'nh Beachler
ii
INTRODUCTION
The purpose of thls Supplement ls to lncrease the scope and
usefulness of the widely used and recommended college/industry
textbook "Analysis and Design of Flight Vehicle Structures" by
E.F.Bruhn. As such it ls by no means a revision of that book.
Rather, it ls, an expansion and clarification of numerous topics
and data in the book, along with the introduction of additional
topics and data.
For best coordination with the textbook the following has
been done. Where an existing article such as, for example,
Art. Cl.13 has been expanded or otherwise changed or corrected,
the Supplement includes it as Art. Cl.13a, the letter indicating
a change or addition. When a new topic is added to a chapter it
is given, an article number which is subsequent to the last article
number in that chapter and includes no letter. For example, the
last article in Chapter C3 ls Art. c3.14. Two additional topics,
"Yield Stress 'Bending Modulus" and "Residual Stresses Following
Plastic Bending" have been added, so they have been given article
numbers c3.15 and c3.16 ,respectively with no letters.
The same thing has been done with figure numbers and table
numbers. Where a figure has been changed or added to, it retains
the same figure number with. an added letter. For example,
Figure 2.27 has added information it has the designation
~o
F1g.2.27a in the Supplement. ,When a new figure, is added it ls
given a number which ls subsequent to the last figure number in
the chapter, so no letters are used with its number in the
Supplement. This procedtire also applies to tables.
Although the Supplement provides an Index, for most usefulness
the textbook must be marked in such a manner as to guide the reader
directly to revised, corrected and new topics, figures, tables and
references in the Supplement. , A highly recommended scheme for doing
this is provided later.
Structural design and analyses are based on theory, empirical
methods and data, various assumptions and individual judgement. The
assembled structure is a result of various specific manUfacturing
methods and procedures. Because of these things and also the possl
bili ty of inadvertent calculation errors, it is always necessary to
prove the adequacy and safety of the completed structure by means of
a sufficient test program before it ls put lnto use. Such tests must
demonstrate the structure's adequacy as to ultimate and yield ,
strength, fatigue life, fracture and stiffness. The test results
must be properly evaluated sin,ce the test article 1 a materials
usually have properties in excess of the minimum requlred values.
Procuring agencies such as the military, the airlines and other
government and private: organizations usually specify the desle;n and
te.st .,requirements and other crl teria which· must be·, used ''dr met. ,
iii
Coordination of the Supplement and the Te:x:tbook.*
In order to easily guide the reader from the te:x:tbook to
the Supplement, the following marking of the te:x:tbook is
recommended,
1.. On. the te:x:tbook' s Table of Contents place an asterisk, in
red ·ink, after. and add the following footnote
~Contents"
at the bottom of the page1
* A red asterisk preceding any articie or figure number
in.the te:x:tbook means that additional material is
available in the same-article or figure in the
Supplement,
2, t<or each· te:x:.tbook•. article listecf'iri''the Supple111ent • s Table ' ·
of Contents, place a large asterisk:; in red ink, just to
the left of eac.h .·corresponding article nUiilber in the
te:x:tbook (e,g,,,'*All,2)/ ··
J, The 12 articles i~the Supplement's Tabl~ of C:onterl.ts riot
having a letter in the article number a:r.e 11ew .a;:i;ic.les,
(e,g., CJ,18 Beam-column An9.1yses'),< Their nuriibe'fs and
titles should be written in at the. e!ld of. their chapters, .
with a red asterisk' Just to the left of their artic'le numbers,
' '• --<-·· ,' ,'","' _,.
4, The following figures ln ;the te:x:tb.o()k shoul_d have' .an
as:l;erlsk, in red ·ink·, placed' just ·to th~. left qf'tlieir
figure numbers, to indicfate that a revision:· or ·additional
data is in the Supplelnentii · · · ·
A5,1 c2.17 c2.26 c5.14 c8,l5 C8,28 Cll,47
.· .. A18•, 8. C2,18 C2;, 27• C8,8a c8,20 c8,29 Cl.1,48
Cl,8 C2,19 CJ,27 cs,11 c8~·25 c10.15 Dl,,15
c2.2 C2:, 20 · CJ,28 c8, 13 c8,26 d11.4J
c2.16 C2,25 cs.11 c8,14 c8.27 C11.44
5, The .. a9.ditional references shown on p. Jo (:for,Chi;i:pter C2)
and. . onip,A21 (for.Chapters CJ, c4, c7·· and.' C:11) shoul(i be
written in at the end Of the list .. Of referertces for.
these chapters• ·
'- ' '
6, Put. a. black asterisk after "RINGS" bn the title of p,A9.1 .
and , at the bottom of the ri~ht hand column add the footnote
* See Appendix B of the Supplement f9r alternative analyses ..
o~ bulkheads, frames,· arches and bents~ · · ·
*
To increase the scope· and usefulness of other such textbooks,
put appropriate asterisks in their texts with footnotes
saying which article to see in the Supple111ent,
iv
A SUPPLEMENT TO "ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES" l
for the beam material, the axial stress,
A5.2}a Introduot1on
P/A, is calculated, ·f c/Fo. 7 is calculated
Add the following at the end of Art. Fi~. C2.16 is entered with this value and
c5.23. For a discussion of all types of Et/Eis obtained. Then Et = E(Et/E). E'
beam-columns, including non-uniform mem in Art. A5.29 is actually Et.
bers, with numerous example problems see
"Engineering Column Analysis" described A5.3la Beam-Co~umna 1n aont1nuoua Structures
in Art.Al8.27a. Multispan beam-columns require a mo
ment distribution analysis to determine
A5.24a Et'teots ot Combined Ax1&1 and Lateral Loads the end acting on each span. Once
m~ents
Add the following at the end of Art. these are known the applicable single
c5.24. Formulas and example ·problems for span formulas in Table·A5,1 and A5.la can
beams in tension are available in the be used to determine the bending moments
book described in Art. Al8.27a, ·at any station within a span. This is
discussed in Art. C3.l8.
A5·25& Equat1ons tor a Compreaaive Axiallf Loaded
Strut w1th Unltormly D1at~1buted Load. A5.J2 App~ox1ma~e F~rmuJ.a. tor Beam.-Oolumna
Add the following at the end of Art. For preliminary sizing when there
A5.25. The deflection, y, at any beam are no end moments the following formula
station can be calculated. as follows.:¥ can be used to determine ·the final bend
ing moment at any station
14 = 14Q +- Mp = l4Q + Py
therefore
y = -(M 14Q)/P
where 14Q is the moment due to the trans
5 (MQ - M)/P verse loads only and Per is the critical
load as a column (Chapter 2). This is
The negative sign is introduced since a most accurate for a uniform lateral load
positive beam bending moment produces a and.least accurate for a concentrated
negative (downward) deflection. Mo is load. Per is calculated assuming the
the moment due to the lateral loadJ only member has a stable cross-section even
and 14 is the final moment;· if this is not the case. For a·beam in
tension the negative sign in the formula
To calculate the slope at any sta is replaced with a positive sign (ten
ticn calculate the deflection at a point sion makes the bending moment smaller).
very slightly to the left of the stat ion
and then very slightly to the right, the ¥i _• 7~ _T Ora1mi ,,r Soi.id. Jlon .. Q1r~ular Sb.ape•
same amount, AL. The slope is th.en All of the previous formulas are
based on the shear stresses being in the
slope = (YR -· ~L)/2AL elastic range. With ductile materials
failure (rupture) does not occur until
A5.26a Form.ulaa tor Otner Single Span Load.in&a<ft« the shear deformation has gone well in
Table A5.la presents numerous addi to the plastic range (similar to the
tional cases of single span 1oad.1ngs.""' plastic bending case). The torsional
moment at which rupture occurs can be
The case of. a beam-column with a predicted as discussed in Art. C4.20a.
varying EI or a varying axial loading re
quires a numerical analysis, just as does For the special case of tubes ha
a column of this nature. The numerical ving a circular cross-section the failing
analysis procedure is presented in .03.18. torque ean be calculated as discussed in
Art, C4.20 and its associated Figures
A5.27a Combinations ot Load Systems, K&ratna ot
Safety and Accuracy ot Calculations c4.17 to C4,30 and in the example prob
For a beam-column the true margin of lems of Art, C4.21.
safety must be calculated as discussed in
Art. C4.23a. The allowable stresses or A7.~a Intro4uat1on,
bending moments are calculated as dis For practical purposes the deflec
cussed in Art. C3.l8. tion and elopes of beams are calculated
as discussed in Art. A7.12a, standard
A5 .• 28& Example Problem11 formulas being used for uniform beams,
In Example Problem 2 member BD is and for varying section beams using
considered to be pinned to ABC at joint B tables such as Table 03.3.and the "End
This is why it does not pick up any ofthEI Fixity" discussion.
36 ,ODO in lb bending moment at joint B. •.
A7 .12& Detleat1on• and.-·~lar Changea _o t:
_Beama "by Jletnod. or "El.aatla 11e1gtltaA
AS.29& Str~asea Above :Proportional Limtt· St~eas For beams of uniform section def
----I-f a column curve is not available lections and slopes are most easily cal-
.. See footnote on -p,J7, L.H. column
•• Por other d1str1"buted loadln~s replace them w1th several sto,~1c•1.ty
equlvalent concentrated loade and ueo eu~rpoe1t1on tor these. Por
any csse of one end f1%ed one s1mpl:r su~DOrted not shown, use tho
both ends fl%od case and then multl"Pl:T the moment at the fl:s::l!ld end
by (l • COP}; ~ ls zero at tho other end. Tho •%eept1on to this ls
e~se 19 •hero (l - COP) must be used.
? SINGLE-SPAN BEAM-CO.LUMN FO.RMULAS
Table .A5.1a Cont1?)uat1on of Table A5. l
!'i'i:. C1s1nx/j + C2ci;>sx/j + f(w)
c, ......
LOAOllfO t (•) "°"'NT
..,, 0
2 De term.in•
•J ~ plattin9
Sin "J ..
0
0 -.".- \ 'O [.,·
;o -wJ2 CCIII a/J ... J~ •' J 2[1 . '"-" C?·Oo·i.'""1'I 7J 2J~ .
0
At M14.:pjln
... •'
ll
·'
.&ca;-!ai nCo Ie ;Jb /J 0 0 If b~41'j/2,
-1\t.u
JZ, •t. -·
0 If b>.,.j/2,
l\y:x at L- 'ftj/l
·~
'
' /J .. · ."w L J
2 .'.fa'n l./2J
P1i9d Erld.-B.Ur.-.. --.
.:c~
Un1Cor"lll Load~·
• <.L/2:· -1/<!., J
'p ·i ·-,.-t.12:
''\ ,~ ~:_: '1' 1;2·~·j'·-:(2Co• Ll2J
,Pl••d £nd, Bilaa_-c ,l'.'c•n_ ;c •. ".';
rated Loi.d. at Canter ........
iir.J[bnfr'.2J-W~JJ
iint;;'J - QJ
.... d'*•,r,tJj
'.
At :!t •.O
0 "-·... .
~ [JTa~lf;looW2J•l
)6 ' J . ~n ,J -. L
o· .. -'At x·· o' :
,- ~-'. ' ' ' .
···:Qr :u:¥ie CJt.,se __: :v ....
···17. w1 th· r,..2L; W"''-2W
,., -·
and a~.~t,, there
for same r.e.sul ts
·.,·-, '•!·.··'' '
Oi' use·· CSse III ,.-1\.a:..• •J [J_(l-!•c.L/J)~
18 with t.=2L'B.nd
same w there . . L Ta~ i../~
'
for same re~ults At .it • L '~ ··,
cai.CU1ate a P 'nln w21,
Or
' ·M=6Eia/L"-( 2{1-~)
See Art •.; 11.15a At it • 0
"'"' U;em,.,6. CG=></L)
'<'.°2•M1 (c&s• II) And - ....... At JI. • L
~·,· \ .
MA-WL(b/1cosL/ -s1nL/1+si:na/ 1+s1nb/ f~L/ 1oosb/ 1+a/ 1 l
O"•r ·•U1•1• .sz. ··2'. 0 ~'·''·''"<~OBLtJ~L1JsinL1J1·
M-= as above but s-t-tehina a and b
=
MMax Wjs1na/j ~()s~ij-~ii;b/j(Hco~L/j )/~irt,L!]
21 ..· · (O~_c;u:c-s .g?)de:c-... the ... J,oads .. W.) . , ..... ,
TM.s --1s a--supe-os1 tion,.,oJ\ Cases V ·
MOMENT DISTRIBUTION WITH AXIAL LOAD 3
culated by using beam deflection and vertent calculation errors it is best to
slope formulas widely available in the use the true value of the stiffness fac
structural literature. These formulas tor, SF, which is in in-lbs per radian
are based on bending stresses only and and is =
are accurate unless the web is very th·in SF 4SCEI/L
or otherwise quite flexible due to holes
eta .. , in which cases there may be signi where SC is the stiffness coefficient and
ficant additional deflection due to shea can be obtained from Fig. All.47 (as "a")
For beam-columns the deflections and the or calculated per Art. All.l3a. Doing
slopes at any station can be calculated this eliminates the need for introducing
as discussed in Art. A5.25a. the correction factors otherwise needed.
The carry over factor, COF, can be ob
For beams with a varying EI a num tained from Fig. All.Ji6 or calculated per
erical analysis is necessary to deter Art. Al1.l3a.
mine the deflections and slopes. This ca
be done as discussed and illustrated in In doing moment distribution .calcu
Art. c3.18 where Table 03.3 shows the de lations, unless the "far and" of a span
flections due to the transverse loads on is pinned (or free) it is assumed to be
ly and Tables c3.4 and 03.5 .show the ad fixed for determining the values of the
ditional deflections due to the axial stiffness and carry over factors.
loads, the final bending"moments being i
1'able C).o The deflections are obtained
All.l'a F1x•d End Moment•, St1ttneaa and
using the data in the tables and .the geo
carry OV'•r 7aator• tor Beam
'metric series, p.J5, as was done· in Table Column• ot Canatant Croaa-3ect1on
··cr;i • .s for moments. Tables for end fixity
are discussed in Art. c3.18. With ·any Sometimes it is necessary to use the
elastic end restraint a moment distribu values of the SC and the COF (see Art.
tion analysis is required to determine All.Saa) for larger values of L/J than
the end moments on the beam etc. as dis are given in Fig. All.46, All.47 and
cussed in Ref. 3 (Art. Al8.27a). All.56 (for 20-~). Those values can be
calculated as follows for compression
All.2& Det1n1t1ona and Der1V&t1ona of' Term.a
members:"
Add the followins at the end of the
article. In the moment distribution pro SC (far end pinned) = 3/4 ,!."" <!
cedure, where adjacent spans "meet" (at SC (far end fixed) = 3(31(...., - o( )
support) there is assumed to be a "Joint" COF :d./213 ·.
even thoughthe member may be continuous
across the Joint• ··The sketoh in Fig·. where o<_ = 6(-Jccsec; - l)/(L/J )a.
All.92 shows the direction of (+) and (-)
moments as they act upon the spans and f.3 3( l - jcot~/( L/ J ))a
a
also upon the Joint. As seen there (+)
moments act clockwise on the span and Fo.r members in tension the same formulas
counterclockwise on the Joint. This is can be used, but the ·trigonometric
different from conventional beam sisn functions are replaced with the hyperbo
conventlon where a (+) moment produces lic functions, cosech and coth. Exten
compression in the "upper" surface, sive table's of the SC and COF values (to
..___f>- M +M . 6 significant figures) from L/J c 0 to
(f) (J 217' for compression members and from 0 to
1
50 for tension members are in the book
'*
~11Jo1nt11 described in Art. Al8.27a.*
Fig. All.92 Siifl Convention* A11.l5a Secondary B•nd1n;; Koment1 1n
True••• with R1s1d Joints
All.5aa. Example Problems.
The stiftness factor, K, as used in Art. All.15 gives a procedure for
Art. All.Sa and subsequently,· is a "rel determining the secondary bending moments
ative" factor rather than a true one. It in such trusses but no illustrative exam
has the value EI/L for the far end fixed ple is provided. The following illus
and ,75 EI/L for the far end pinned. As trates the procedure except that in step
such it applies only when there is no ex one one finds the relative rotation of
ternal elastic restraint, k .. (in-lbs per each member usini the method of virtual
radian)at the ends or at any Joint, and work, step 2 is omitted and in step 3
when there is no axial load in the spans. "its relative rotation" is used in place
In such cases 11correctian fa.otors11 ·must of "these transverse displacements". That
be applied to it, as shown in Art.All.14 is, due to the applied loads the truss
for example. In general, to avoid inad- Joints move, and therefore the members
"Some books u~e an opposite sign conven_~ion. , •I#n(3 F-riig".'A ilL6l ./T56J" 2[/ 1 2 -o-tj L · c(a'ns inb+ Le c/coj asLlc/ui l~ia']t ed as follows~ d' ,.,. . .,
>l ~ /1-/J;;, ,pJo h,,M• ~)j" ' 7,J;- eJJ't. j>• j31J. .:lJ1 IV 1 / .
4 ~J;:CPNDARY. BEND.ING MOMENTS IN. TRUSSES
rotate relative to each other. These ro
Table All.·4 Bas1c Da.ta
tation• generate fixed end moments at ' '
their .. ends. (just as a beam with unequal-· Me•m;.: b,e.:r :· _J:L 111·L ;·~-.a. ,.·'oE-' 6-· .s~/AE· :· ~ 3:./ . ....!-;,, I./ J
ly deflecting supports will AB. I· 577.4 20··: ,.563' :.10:_·' ; .0205 o;,'0264' 6;76 2,91
undergo ·an angular rotation and develops AC -2887 20 ,563, .10 -.0103 ,0264 9.56 2.0,
fixed end momenta). The following illus BsaD· · -55777744 2200 .,556633 '1100 -..00220055 '..;00226644 66.,7166 22..9966
trates. the procedure using the truss CD 5774 20 ,563 -10. '.0205 • 0264 6.76. 2.96
shown in Fig.All.92. CE -8660 10 .360 10 -.0200 ..• 0100 3,53 2.83
"E. 0 l"'.""2 • O<;o 10 0 ·.ooi:;-~' """ 0
-5774
5774
c.
60'. ··. ·-21'17 , set,. cei;so E
5000~ c::::=:;:·i.~2~0·~·= ···=·i···~··=·:;·· ~·:C:•:::i.f'··t ~o~"=··=· ~,...,
861i660#
Fig. All.92 Applied and Internal Loads 05
l. For tlia aiit>fi'ed load.a the' resulting
internal loads. in the members· are de
F1g. ·All._~;, R~l.1.t.1ve _Rot~-_t;'l_on Loa:ds ..
termined as shown in the figure.
F.or, each of the above membera••·the
2. M!jml:)er BO ie a~b1trarily.1'eJ,ected as fixe<i encl,•moments are calculated as
the •ipase niem~'er,11 :f'r.o'm wl:i1cli t!ie ,ro"'.
tations of all other members are cal ~EM= ,-6EIS/L( 2/J - ot) .
culated.
where 2,13 -o( if< ol:r~ail\,ed. from .F is;.
3, For each other member ( onE1 'a.t ,a ,.,tl,mE!) AllS6 and accounts for the effect of
a clockwise l in-lb moment in the for the .·.a;><ial•.:loadf Si .1n .. the member, ·A
of couple loa.<is ( l/L) •at.. :+,t.1>. '!\'!d,'! 1,!! Po.s1:ti:v:e. {.clpckw:l;se,,J ·r.elative ·'.rotation,
applfeii. and·.reac.ted. V11tb, a, ,ciount'!r::. 8 •' .prodUC.E!S,'.;gegati:V.8 ,•(,o ount'.erclockW1,se)
clo.okwise couple at the en4EI' of the, ..FEM' .. s (per Example" 2 ·sketch on ·p•All.2)
base member BO. The resulting loads · l:<eno.e:· ..t he.·m1nua':sign·'.:1r,:'.the,..·f·ormula•'
in the tru~a. iUlembers are,: .• t.hen. ... dE!te.r Ifo'a memb.er. has •. :one end·,:pinned .a fixed
anii
minea. See Fig. All.93 note that end mcip1ent occurs only• at the other
only a fE!W (3 of 4) meml:)ers are loaii-. end. and is calculated as
in this procedure,
FEM=·-6EIS(l--.. COFJ/L(-2/3 - o{)
4. ·Then tor.each. ioadE!d.' meml:)~r.,;t.he, quan
tity SuL/J ).ji; 1 a ,.oalc;L!la ted and . the re where OOF is from the pinned end to .
,aults stimJllei!.Ao ,ot>,t.S:.in the, rel1>tiv:e th•i."fixed end. Table Atl,5 summarizes
ro,tat1on. • :·:ei o:t' .1'hE1.,member, to which the· calculations for' the' va·lues· of e.
tlie .c+ockV1ise. coup),E!, wa!! f,i,pplied.
5 ~ -Ste. !J'•''S .. ( 3., .f. .a.· n'""c :f·" "("''4,•·)·· 8,.-,~,.-,., e J;'epeB" .. .·t-·e·. d for ., ..·. 6. Tjohei ntsfi xaerde nerotwd ."kinn'oowm~nn ta anadt athlel mtroumsesn t
eB.ch 'of the 'renia1nin~ U1enibera ',to ,get distribµtion procedure can be carried
their relative l:'otationa, e; Table · out as 1llui!trated in Fig. All,43 to
All. 4 presents the b":sic lij.ata anlil. obta,in: ,the ,final moments. at the ends
Table All.5 si:iinriiari:Z:ee' ;t\).e ·calcu- of each member. Th<!:n sue A~t:. A5,3ia;..
lations. ·
Table All.5 Oalculation of Relative ~otat;ona, e
·Mam:. SLf.. A E Counle 'at AB Coirnle at AC Couole a:.t en· noun le 'at f!D Ooi.nole at CE
:i:• . :.:.:·- u uSLIAE u ''uSL/AE u uSLf AE u " ' uSL/AE u uSLlAE
AB .0205 -;0289 -593 ', ". 0577 --1183 0 0 0 o.· 0 0,
:Ac'. -.Ol03 .0577 -594 .0289, 2,89 0 .Q 0 0 0 0
BC - .0:205 -.0289 593 .0289 593 .0289 -592 -.0289 592 -.02~9 593
~
BD. .0205 0 0 0 0 .0289 592 .0577 -1183 .0577 1183
" ·OoD~' "' · I: ..' ..,0,o 2200 o5ff · 00 00 00 ·o0 ' -.005 77 -1180 3 -.02O8· 9 50 92 :...101 55 -236,0 7
·oE, 0 0 0 0 Q .. - 0 '.0 0 .1000 0
E;f& _n.r; ~""'LILL .- 'rA Ft : -.21 in5', e--= -llH'• . Q-- : . ll><'I 9,,,,,: -~ 91
Note: M»ltiply eacn e by 10~.: DE <ioean't rotate since it is held by the supports.
Seep.:.Bll. for the-critical (buckling) loading calculation and p.Bia-13 for gusset plate data.
SECONDARY BENDING MOMENTS IN TRUSSES 5
All.15b Th• Ettect1 ot Inoroaaed Internal
Load• on Secondary Bending Momenta Let AC be the base member, apply, a l in
Although the secondary bending mom lb couple on AB and react it with an op
ents in the truss of Art. All.15a were posite couple on AC as in Fig. All.95.
relatively small, they can become quite
l/17.3 /10
large as the internal axial loads in the
truss members increase. As the applied
loading on the truss approaches the crit
ical loading these moments will approach
infinity. This is why the theoretical
critical loading for a truss can never be
attained; bending failure will precede
i~
The following example, using the simplest l/.
l/ .3
rigid Joint truss illustrates this. Fig. All.95 Virtual Work Loads
C B
LAc= 17.311 EIAc= 5x1o!A=Z,0"1
, The relative rotation of member AB is
LAB= 30.011, EIAB = 5xl06..iA .. ,a1Ci then calculated as follows.
Membe
AC
AB
F1.~. All.94 Rigid Jo:tnt _Truss with Piruled Ende The only FEI.! is at A since end B is
A is a rigid Joint and ends B and C a.re pinned (and AC has no relative rotation).
pinned. Assuming A to be a pinned Joint
6 6
only to determine the axial loads in AC FEl.!AB:: -6( 5xl0 )(-5803xl0° )(1.,.. 2.323)/
and BC, 'they are as shown in the figure. 30(1.44)
With the axial loads known thia simple = -!Ss3i. in lbs
truss can be analyzed for stability
(buckling) as a two-span oa,lumn with Doing the moment distribution fo;i:,the
pinned ends, as •is illustrated and 'final moments at A
dis~
cussed in Art. C2.3b, Example 2. By suc
cessive trial calculations, when P .., .. (q>'\.(i:~"'
101700 lbs, PAB = 88072 and PAC= !)0850. lJ>. ~
(L/J )AB :-,o/;/5 x 10°/88072 =}.9816 SC ,8884/-1.394
SF 1027100 -932670
SF 94430
SCAB= -l.5428 DF 10,8771·9.877
FEM 0 ·5331
SFAB = 4(,.,,1.5428)(5 x 106 )/30 F1nal Mlle'o.m1 32'Z79I5D3" -• H$~~ .
=-1028500
(L/J )Ac= 17.3/V5 x lo'/50850 =l.7446 Fig. All,96 Mo~ent D1atr1but1on
The extremely final moments are, of
lar~e
SCAC : • 89059 coursa, unacceptable since bending fall
ure would occur (if P were 101700 the mo
SFAc .:4( .89059)(5 x l0')/17.'J ments would be infinite). Therefore,
when the applied leading on a truss is
=1029600 near the cv1tical loading bending failure
will eccur (and prevent the critical
*
Hence, at Joint AZSF = 1029600-1028500::!< 0 loading from being reached), due to the
so 101700 is the buckling (critical) load deflection of the truss Joints under
loa~
for the truss.* If A were assumed to be a
pinned Joint then an applied load of only Repeating the above calculat1ens for
63315 would cause AB to buckle as a pin suocesaively smaller valuea of the app
ended column, since for pinned ends PcR lied loading, P, results in the final mo
for AB is 54831 and .866 x 63315 ::. 54831. ments shown in Table All.6. Nete that as
the applied loading decreases from near
To illustrate the effects of applied the critioal loading the final moments
loadings near the critical loading on decrease very rapidly at first. When the
secondary bending momenta, assume that loading deoreasea to the value which is
P = 100000 and find the resulting momenta. the critical value assuming pin ·Joints
~.t' BI' at &n.J Joint £ O there 1• no re•1at.e.noe te rot.at10n
n an 1nt1n1teamal aomen't. will eauae Nt.at.10'1 and t&1lur•~ * The maxi?aum bending in AJ3 is ??.840 1n-lbs, occuring 18.3~
from A (using Case 2 in Table A5.1)
6 ' CURVED BEAMS
Table All.6 Variation "'o.r F·ins.l Moments E.or symmetrical. c.ross-sect1ons .the cir
a:t
with Applied Loading i:umfe,rentfs.l bendihg stress any point
' ,, on the sac.tlon c,an be ca:i:o·ulated as
Appl.ied % of Final
Loading Critico.l Momenta fb = ..A! 11+ 1 ( ~ 0.'
Loading At A : ARL Z R.,.Yi.j
=
101700 lOO.o <:><> wherEI .• • i.. ~rii.~ ot'. c,rosa.:'aeot'ron ,
10000'0 98.4 -57985 R:Ra,dius of ciUr\iature at the
99000 97.3 -24983 oentr.oidal axis ' '
98000 96,4 -20522 14 "'Applied momept; Positive for
95000 93.4 - 8876'.' '' t.eh,sfarl iri 't.he outer .:fibers
63315 62.3 0 and vice-versa
4000,,0 39,3 60 y: distance from centrciidal a.xis,
-
being -1- outw.ard from this a.xis
(63315) the final moments 'are reis.tively, and ·• if inward
sms.lJ;;- Tllia is why an a,s.sumed,,P,i)j'":J0int z ,,._1 n.zdA - ~ 1 (E/.S:L
analysis ,,which iSl'lores the:, .s:epcindary mom AJ R. .. y - A) R+ y
ents', a-· .. ~ellimon ·Preioedu;r:-e., +l.'s ·unl1keiy te
r~sult 'hi strength fa,ilufea. Fatigue where w =- cross-aeot1ons.l
life ni:r:ght -be a. ·cfotl'i:erri tor very' light . wi<ith at distance y
structure·&'. If a:' truss member is subject
z
to a lateral,: :j.qading, ths.t. as.uses an add Ts.bl& Al3.:1vprea·ents 'formulas for for
ition!"l. . type 1>.:f' secondary Il\Om!l?\t t,cf occur sEl'\l,ers.l f;poas-seiqttqns,.': WhE1,re, a, flange
and additional beaoi-co1umn·af'feots; width is,. . required, it i,s nqt the aot.ual
l'(,t~th, J1,· l:>.lJ.t. ,r~~re'r: &I\'affE1<:t.1ye. w1~t.~.
'AltH<;ugh thi~ 1li.'uatrativ e exafui;iie'' beff•'"due tq the, <i~flectiqn shown in' Fig.
use• only . the simple,st type qf rigid Al3.• 22S,,• This can oe ..o s.louls.ted s.s
joint truss, the results would be &imilar
fer a oonve!ltional type of. . ilrua,,.. The l:>err.,:: i:':Lb", .·
analysis would, of courae1· ;t>e mo.i;e t¥,,
dioua. Yore about truss'eii l"s 1h the book wh.ere d1 is ob}al.ned f~'!m Fig. Al3.22b. ·..
mentioned in Art.AlB.27s. (and Fig.All.43i beff is d:!,sr,.d ,t.pi; ii",t.'!1"1!\~!!.ing, A an\'.),,Z,,, ,
whan· .. :flangss: ar;.e,. present, .. , Being less:.·
Al3.lla Ourved Beam.• thari the -s.ctual b. it results iri' high~
er bending.a~tresjesl. f,.'h• '
For c0mpao.t cross-se'ctions the bes.m I.]' ' . 1 1 • 1'
is net aubject.te fl11nge instability ef l••• !•".. ":: -1:., ,·:::: ~~ :,·,·,·,. ,··.·.' ,.:·,.' . .: :. · ....., ·! ·vl=: d·1i ·..· ·i·11 :,f ..Ir
fects (Chs.pter 7}. •..' l'herefore, with duc
F1g.Al3.22b ~
tile materials,>.~l;\EI 1J.l'\..1ms.te bE1nding ·
strength can be' calculated s.s discussed
in Chapter 03, ignoring the curvs.ture. curved 'Beam E .: ~.o 1- -
Bending ... ~ ··t;: .!:. . :~ •• -
·Whan the' ·cross-section he.a rals.tiva- · ·.coefficients ;I l~ .1 =1: . ~"!:':~u •f -
ly ,thin .,fla,rig!'&.,, ,s.s W/.,th,, ,s.n ,I ,.,. .c hS.!lJ:la.l, .I .. ... . ·:.: ."''. . '·I·- ;c .: :
z etc. :there,i.s:another.' effect.or '.ourva-
·. t,u- r' e•. ,:.,,. ; ·-•:., ·I ·.,t,, •c-·;-;·,-;:ul s.!'>!"'l.i'..' ";-',''t''h e- · f..l. a,.,,n._ _g__ ...e, .. _. ,.,., !j'•;'"0;,.,,', .l,: >an.\.l•,•,, a-.. a. ·'• :: ·4o .• .1 INl.2R ,l;.it: 2.0 l.• 2.t l.2
sho~p ,,iq, ,:i;:ig. A,13 .•. 22a,, .. and, th9rafora l:>a,-:- The ,tranayerse bend,ing stress in
can
, come l<>e.s effect~v,e,, ,I'.e9ultlng in higl:\er the fls.nge, fpt• be os.lculated s.s
bend1n's 'iitre~saarcirthe l:>'ea.m; Its.lac''
gene.rs.;te,s, l:>Eil\41-iig.:~tfasse,~, \jl t,l;l~ flang9a
a
il\, .dii;,~oticih.noryial to. the:,,plii.ne. of the
. well w,h1oh a.re a maximum s.t '!:!he flange-to- where C2 is obtained from Fig.Al3. 22b
web int•l'rsection. ' ···· · · ' · . an.d fb ~.•ttf!e 9ti;~,s9 · c,a.lculat,ed previ-
/,'
'c;."'~~;;,:;::;;=;_,.._;:'~'.'.';,; -~~!:;~ :"'-> oµsly us~ng 91~:·: Jl.g":in,, .th.is d.1souss1on
s.ppl1e9 only,,;tCI '8,Ymmetr1qs.l oro.ss~
·_· -·c• • p.;'" 011-b.r // seot1a:ns. as .in Fig. Al3.22a.
l'l11.~6 11
"
· .. - ' \1- When weight is important and rela
l\
- --Ce..;tp. ;,.-1;.11•" tively thin flanges result in high
FJa119a \1,,
·atr1>sse&,t the stresses can be redu.oed .
..· i:.::.::::.~- -:-;..c.~ 4' -~.. \1 by using thin, closely; spaced,,,~acb,Lhed in
_;;~"'-"'°"'S·"""'-tr~;a''' pls.oe "bulkheads," between the· flanges, or
rig,Al3.2Zs. Curved Beam Section Bend:ir!g be1.111ath a. T'\member'.s .flan!!is• . This reduoaa