Calculation and clear descriptions on the meaning of moment of inertiaFull description
Sheet Music
Full description
Descripción completa
bolt groupFull description
calculation of polar moment of inertiaFull description
Lab experiment
moment connectionFull description
asdDescripción completa
Full description
Full description
This essay constitutes a sustained reading of "The Impossible" by Georges Bataille, focusing upon the idea of the "lost moment." Drawing upon sources such as Freud, Kierkegaard and Proust, this ess...
Full description
Ultimate Moment Capacity of RC Beams
mr
Simplified Assessment of Bending Moment CapacityFull description
Sung Si Kyung - Every Moment of You piano sheetFull description
Bolted Moment Connection - Limit StatateDescripción completa
Descripción completa
uthm lab reportFull description
Full description
SATB and soprano soloFull description
Introduction to Statics .PDF Edition – Version 0.95
Unit 29
Moments of Inertia of Composite Areas Helen Margaret Lester Plants Late Professor Emerita
Wallace Starr Venable Emeritus Associate Professor
West Virginia University, Morgantown, West Virginia
Conditions o$ Use %is boo&, and related support 'aterials, 'ay be downloaded witout carge $or personal use $ro' www(Secrets)$*ngineering(net +ou +ou 'ay print one copy o$ tis docu'ent docu'ent $or personal use( use( +ou +ou 'ay install a copy copy o$ tis 'aterial 'aterial on a co'puter or oter oter electronic reader $or personal use( edistribution in any $or' is e-pressly proibited(
Unit 29 Moments of Inertia of Composite Areas Frame 29-1 Introduction This unit i!! teach "ou ho to com#ine the moments of inertia of sim$!e %eometric sha$es to o#tain the moment of inertia of a com$osite area. &t i!! a!so teach "ou ho not to com#ine them. 'oments of inertia of areas are used e(tensi)e!" in *stren%th* to ca!cu!ate stresses and def!ections in #eams. &n American +ustomar" ,nits e ca!cu!ate stress in $ounds $er suare inch $si/ so it is the common $ractice to use areas and deri)ed $ro$erties/ measured in inches. +on)enient!" +on)enient!" most American drain%s ha)e areas dimensioned in inches. &n "stem &nternationa! &/ stresses are ca!cu!ated in Pasca!s etons $er suare meter/ so it is the common $ractice to use area areas and deri)ed $ro$erties/ measured in meters. For #etter or orse cross-sections in most & en%ineerin% drain%s are !a#e!ed in centimeters or mi!imeters. &n this unit the *correct* units in ansers i!! #e in terms of inches or meters. The s"stem used for findin% the second moment for com$osite areas is )er" simi!ar to that used for findin% the first moments and centroids of com$osite areas. &t ou!d $ro#a#!" $a" "ou to re)ie "our note#oo3 for ,nit 12 #efore #e%innin% the ne or3. 4hen "ou ha)e done so %o to the ne(t frame.
+orrect res$onse to $recedin% frame
o res$onse
Frame 29-2 Co''on .-is econd moments of areas ma" #e added direct!" if the moments of the areas are ith res$ect to the same a(is.
Frame 29- Co''on .-is &t is a!so $ossi#!e to su#tract moments of inertia of )arious areas as !on% as the moments are ta3en a#out the same a(is. This a!!os us to com$ute the moment of inertia for an area ith a ho!e.
Find the moment of inertia of the ho!!o circ!e #" com$utin% the fo!!oin%7 For a circ!e of 20 mm radius
Ix = ___________________ Ix = ___________________
For a circ!e of 10 mm radius
For the rin%
Ix = ___________________
E($ress "our ansers in m
+orrect res$onse to $recedin% frame
Frame 29-5 Co''on .-is
Find the moment of inertia of the area shon a#out the -a(is and a#out the a(is.
Ix = ___________________ Iy = ___________________
+orrect res$onse to $recedin% frame
Frame 29-8 Co''on .-is The moment of inertia of a uarter-circ!e a#out its ed%e is
Find
Ix = ___________________ Iy = ___________________
and
for the semi-circ!e.
+orrect res$onse to $recedin% frame
Frame 29- Co''on .-is The moment of inertia of an" trian%!e ma" #e found #" com#inin% the moments of inertia of ri%ht trian%!es a#out a common a(is. Find for the isosce!es trian%!e shon. Loo3 u$ for a trian%!e in "our ta#!e if "ou ha)e for%otten.
Iy = ___________________
+orrect res$onse to $recedin% frame
Frame 29-: Co''on .-is
Find
for the isosce!es trian%!e shon.
+orrect res$onse to $recedin% frame
Frame 29-9 Co''on .-is Find and for the sca!ene trian%!e shon. &n ca!cu!atin% centroids of com$osite areas & find it usefu! to use a ta#u!ar a$$roach simi!ar to the one e used ith centroids.
+orrect res$onse to $recedin% frame
Frame 29-10 Co''on .-is +om$!ete $a%e 29-1 of "our note#oo3.
+orrect res$onse to $recedin% frame
Frame 29-11 %ransition ;" no "ou ha)e $ro#a#!" acuired an e(ce!!ent %ras$ of common a(is $ro#!ems. A!as7 &t is not a!a"s $ossi#!e to find a sin%!e a(is a#out hich the moments of inertia of a!! $arts are 3non. &n fact it can
+orrect res$onse to $recedin% frame
o res$onse
Frame 29-12 Parallel .-is %eore' The $ara!!e! a(is theorem is most sim$!" stated as an euation
tated in ords it sa"s that the moment of inertia a#out an" a(is / is eua! to the sum of the moment of inertia of the area a#out a $ara!!e! a(is throu%h its centroid / $!us the $roduct of the area and the suare of the distance #eteen ./ +om$arin% the euation and the statement does tend to ma3e one a$$reciate the euation./ =o to the ne(t frame.
+orrect res$onse to $recedin% frame
o res$onse
Frame 29-16 Parallel .-is %eore' The euation
im$!ies that the moment of inertia of an area a#out an a(is $assin% throu%h the centroid of the area is larger smaller/ than the moment of inertia a#out an" other $ara!!e! a(is.
+orrect res$onse to $recedin% frame
sma!!er
Frame 29-1 Parallel .-is %eore' The term
is often ca!!ed the *transfer term* and the *transfer distance*.
To transfer the moment of inertia from the centroida! a(is to an" other a(is the transfer term is added subtracted/.
+orrect res$onse to $recedin% frame
added
Frame 29-15 %rans$er */uation The Para!!e! A(is Theorem is strict!" s$ea3in% the ord )ersion %i)en in an ear!ier frame. The more con)enient a!%e#raic form is usua!!" ca!!ed the *Transfer Euation.* 4rite the transfer euation.
Ixa = __________________________
+orrect res$onse to $recedin% frame
Frame 29-18 Parallel .-is %eore' For the area shon the centroid is !ocated 6.5 in. from the a(is. o
Find
Ix = _______________________
+orrect res$onse to $recedin% frame
Frame 29-1 Parallel .-is %eore' Find
for the trian%!e shon. ;e%in #" drain% in the c entroida! -a(is.
+orrect res$onse to $recedin% frame
Frame 29-1: %rans$er */uation &t is a!so $ossi#!e to use the transfer euation in re)erse. To do so "ou must 3no the moment of inertia a#out a non-centroida! a(is and ant to find it a#out the centroida! a(is. &n this case the euation for
is
IxG = ________________________
+orrect res$onse to $recedin% frame
Frame 29-19 %rans$er */uation Find
for the uarter circ!e.
IxG = ______________________________
+orrect res$onse to $recedin% frame
Frame 29-20 0oteboo& Do $a%e 29-2 in "our note#oo3. Then com$!ete the Pro$erties of Areas Ta#!e on $a%e 2:- in "our note#oo3.
+orrect res$onse to $recedin% frame
Frame 29-21 Li'itation There is one im$ortant !imitation on the use of the transfer euation. >ne of the moments of inertia in)o!)ed must #e centroida!. &t is im$ossi#!e to transfer direct!" from one non-centroida! a(is to another. &n each of the fo!!oin% te!! hether the transfer euation a$$!ies. 1. A(is
to a(is
$ $ $ $ $ $ ?es
2. A(is
to a(is
?es
6. A(is
o
o
to a(is
?es
o
+orrect res$onse to $recedin% frame
1. ?es 2. ?es 6. o
Frame 29-22 %rans$er */uation >f course it is $ossi#!e to %o from an" a(is to an" a(is #" an indirect transfer.
For the trian%!e shon7
Ix = 180 in4 IG = ______________________ Ia-a = ______________________
First find
o find
+orrect res$onse to $recedin% frame
A!! of hich is rather si!!" since most $eo$!e ou!d sim$!" #e%in #" findin% @oe)er it can #e done this a" if one rea!!" needs to.
IxG
.
Frame 29-26 %ransition o "ou ha)e a!! the necessar" too!s for findin% moments of inertia of com$osite areas. A!! that remains is to !earn to use them on com$osite areas. The remainder of this unit i!! #e de)oted to some $rett" com$!e( areas and a hand" method for cuttin% the com$utations don to sie. This is a %ood $!ace for a #rea3. & su%%est "ou ta3e one #efore turnin% the $a%e.
+orrect res$onse to $recedin% frame
o res$onse
Frame 29-2 Co'posite .reas by %rans$er The moment of inertia of a com$osite area a#out an" a(is ma" #e found #" findin% the moments of inertia of a!! $arts a#out the a(is #" means of the transfer euation and then addin% them. &t is usua!!" a %ood idea to do this #" means of a t a#!e. As in ,nit 12 the su#scri$t * * means for the $art. +om$!ete the com$utation for findin% for the area shon.
+orrect res$onse to $recedin% frame
Frame 29-25 Co'posite .reas by %rans$er The most im$ortant sin%!e $oint to remem#er is that in the ta#!e means moment of inertia of the $art a#out its on centroid. Do not use an" other moment of inertia in this co!umn. Find
for the area shon.
+orrect res$onse to $recedin% frame
Frame 29-28 Holes ome areas ha)e ho!es in them. A ho!e has a ne%ati)e area and a ne%ati)e moment of inertia. Find
for the area shon
+orrect res$onse to $recedin% frame
Frame 29-2 Co'posite .reas
Find
for the area shon.
This time &
+orrect res$onse to $recedin% frame
Frame 29-2: %ransition o e come to the reason for a!! the or3 "ou ha)e done on com$osite areas. ?ou i!! find in a !ater course that the stren%th of a #eam is direct!" re!ated to the moment of inertia of its cross-section a#out a centroida! a(is. 'ost #eams used for hea)" !oads ha)e com$osite cross-sections so there "ou are. ?ou can no find the moment of inertia of a com$osite area a#out a s$ecified a(is. @oe)er "ou need to find it a#out a centroida! a(is. The transfer %i)es no trou#!e if "ou 3no here the centroid is #ut "ou must usua!!" !ocate the centroid. &n other ords if "ou ha)e not "et re)ieed ,nit 12 $!ease do so no. Then %o to the ne(t frame.
+orrect res$onse to $recedin% frame
o res$onse
Frame 29-29 eview et u$ a ta#!e and ca!cu!ate the -coordinate of the centroid of the area shon #e!o.
+orrect res$onse to $recedin% frame
Frame 29-60 Centroid and Mo'ent o$ Inertia &t is $ossi#!e to find the centroid and moment of inertia from the same ta#!e #" addin% Bust one co!umn to the moment of inertia ta#!e and findin% to more tota!s as shon #e!o.
4hat is the headin% of the ne co!umnC 4hate are the additiona! tota!sC ,se the ta#!e to find and the centroid for the area shon.
coordinate of the
+orrect res$onse to $recedin% frame
Frame 29-61 Centroidal Mo'ent o$ Inertia o ma3e use of the information from the so!ution a#o)e to find euation in the form
d = yG = ____________________ Ix =2 ________________________ Ad = ______________________ IxG = _______________________
Find for the area shon. The ho!e is hand!ed as a ne%ati)e area and a ne%ati)e moment of inertia Bust as "ou ha)e done #efore./
yG = _______________________ Ix =2 ________________________ Ad = ______________________ IxG = _______________________
+orrect res$onse to $recedin% frame
Frame 29-66 %ransition The a(is a#out hich the moment of inertia is first found is ca!!ed the *reference a(is.* &n the $ro#!ems "ou ha)e or3ed so far "our se!ection of a reference a(is has #een %uided. &n most $ro#!ems "ou i!! find that no such %uidance is $ro)ided and that a $oor choice of reference a(is ma3es a $ro#!em unnecessari!" hard. +onseuent!" the ne(t section of the unit i!! #e de)oted to he!$in% "ou a)oid $oor choices and ma3e %ood ones. =o to the ne(t frame.
+orrect res$onse to $recedin% frame
o res$onse
Frame 29-6 Selection o$ e$erence .-is &t is freuent!" ise to choose the reference a(is at one ed%e of the area so that ne%ati)e )a!ues for centroida! distances are a)oided. @oe)er this i!! often resu!t in rather !ar%e num#ers in the A"P= co!umn hich ma" #e a disad)anta%e if "ou don
+orrect res$onse to $recedin% frame
'" anser is or . ot &t introduces !ar%e num#ers and ne%ati)e coordinates. A(is i!! %i)e "ou eas" num#ers #ut a ne%ati)e coordinate. This cho$ice is $articu!ar!" %ood if "ou remem#er for an ed%e a(is of a rectan%!e. A(is i!! %i)e "ou a!! $ositi)e coordinates #ut !ar%er num#ers.
Frame 29-68 Coice o$ e$erence .-is A reference a(is hich $asses throu%h the centroid of the entire area is idea! if one is a)ai!a#!e. Another %ood choice is a reference a(is hich $asses throu%h the centroids of se)era! $arts. ;oth of these choices i!! ma3e man" terms in the ta#!e eua! ero. +hoose reference a(es for findin% for the areas #e!o. ho them on the fi%ures. A!so sho ho "ou ou!d di)ide the areas.
+orrect res$onse to $recedin% frame
'" choices ou!d #e those shon #e!o.
>ther %ood choices and di)isions cou!d #e made for the second area.
Frame 29-6 Coice o$ e$erences .-es A fourth %ood choice is an a(is $assin% throu%h the center of the one most difficu!t $art to com$ute. This is $articu!ar!" a$$!ica#!e to circ!es and reduces the num#er of terms "ou must com$ute. e!ect an a(is and find
for the area shon. Dra "our on ta#!e.
+orrect res$onse to $recedin% frame
Frame 29-6: Coice o$ e$erence .-es The !ast %ood choice is an a(is a#out hich "ou 3no the moment of inertia of se)era! $arts. This one is )er" ris3" for a #e%inner #ut is $ro#a#!" orth the ris3 if "ou must co$e ith a semi-circ!e.
+hoose the a(is as a reference a(is for the $ro#!em shon and com$!ete the ta#!e to find .
+orrect res$onse to $recedin% frame
Frame 29-69 0oteboo& +om$!ete $a%e 29-6 of "our note#oo3.