Mechanics of Materials CIVL 3322 / MECH 3322

Transcription

1 Mechacs of Materals CVL / MECH Cetrods ad Momet of erta Calculatos Cetrods = A = = = A = = Cetrod ad Momet of erta Calculatos z= z A = =

2 Parallel As Theorem f ou kow the momet of erta about a cetrodal as of a fgure, ou ca calculate the momet of erta about a parallel as to the cetrodal as usg a smple formula z = z + A Cetrod ad Momet of erta Calculatos P07_045 4 Cetrod ad Momet of erta Calculatos

3 P07_045 5 Cetrod ad Momet of erta Calculatos A Eample Lets start wth a eample problem ad see how ths develops = A = 6 Cetrod ad Momet of erta Calculatos =

4 A Eample We wat to locate both the ad cetrods = A = = 7 Cetrod ad Momet of erta Calculatos A Eample There s t much of a chace of developg a fucto that s eas to tegrate ths case = A = 8 Cetrod ad Momet of erta Calculatos =

5 A Eample We ca break ths shape up to a seres of shapes that we ca fd the cetrod of = A = = Cetrod ad Momet of ertacalculatos A Eample There are multple was to do ths as log as ou are cosstet = A = 0 Cetrod ad Momet of erta Calculatos =

6 A Eample Frst, we ca develop a rectagle o the left sde of the dagram, we wll label that as area, = A = = Cetrod ad Momet of erta Calculatos A Eample A secod rectagle wll be placed the bottom of the fgure, we wll label t A = Cetrod ad Momet of erta Calculatos = A A =

7 A Eample A rght tragle wll complete the upper rght sde of the fgure, label t A = = = A A A Cetrod ad Momet of erta Calculatos A Eample Fall, we wll develop a egatve area to remove the quarter crcle the lower left had corer, label t = 4 Calculatos Cetrod ad Momet of erta = = A A A

8 A Eample We wll beg to buld a table so that keepg up wth thgs wll be easer The frst colum wll be the areas D () A A = A A = A = 5 Cetrod ad Momet of erta Calculatos A Eample Now we wll calculate the dstace to the local cetrods from the -as (we are calculatg a -cetrod) D ( ) () A.5.5 A = A A Cetrod ad Momet of erta Calculatos = = 6 A

9 A Eample To calculate the top term the epresso we eed to multpl the etres the last two colums b oe aother D * ( ) () ( ) A A.5 = = = A A A 7 Cetrod ad Momet of erta Calculatos A Eample f we sum the secod colum, we have the bottom term the dvso, the total area D * () () () A A = A 8 Cetrod ad Momet of erta Calculatos = = A A

10 A Eample D Ad f we sum the fourth colum, we have the top term, the area momet * ( ) () () A A.5 = A = = A A Cetrod ad Momet of erta Calculatos A Eample Dvdg the sum of the area momets b the total area we calculate the -cetrod D * () () () A A bar.54 = A 0 Cetrod ad Momet of erta Calculatos = = A A

11 A Eample You ca alwas remember whch to dvde b f ou look at the fal uts, remember that a cetrod s a dstace D * () () () A A bar.54 = A = = A A Cetrod ad Momet of erta Calculatos A Eample We ca do the same process wth the cetrod D * () () () A A bar.54 = A Cetrod ad Momet of erta Calculatos = = A A

12 A Eample Notce that the bottom term does t chage, the area of the fgure has t chaged D * () () () A A bar.54 = A = = A A Cetrod ad Momet of erta Calculatos A Eample We ol eed to add a colum of s D * () () () () A A bar.54 = A = = Cetrod ad Momet of erta Calculatos A A 4

13 A Eample Calculate the area momets about the as D * * () () () () () A A = bar.54 = = A A A 5 Cetrod ad Momet of erta Calculatos A Eample D Sum the area momets * * () () ( ) () ( ) A A bar.54 = A = = A A 6 Cetrod ad Momet of erta Calculatos

14 A Eample D Ad make the dvso of the area momets b the total area * * ( ) () ( ) () () A A = bar bar A = = A Cetrod ad Momet of erta Calculatos Parallel As Theorem f ou kow the momet of erta about a cetrodal as of a fgure, ou ca calculate the momet of erta about a parallel as to the cetrodal as usg a smple formula = + A = + A 8 Cetrod ad Momet of erta Calculatos A 7

15 Parallel As Theorem Sce we usuall use the bar over the cetrodal as, the momet of erta about a cetrodal as also uses the bar over the as desgato = + A = + A Cetrod ad Momet of erta Calculatos Parallel As Theorem f ou look carefull at the epresso, ou should otce that the momet of erta about a cetrodal as wll alwas be the mmum momet of erta about a as that s parallel to the cetrodal as. = + A = + A 0 Cetrod ad Momet of erta Calculatos

16 Aother Eample We ca use the parallel as theorem to fd the momet of erta of a composte fgure Cetrod ad Momet of erta Calculatos Aother Eample " Cetrod ad Momet of erta Calculatos

17 Aother Eample We ca dvde up the area to smaller areas wth shapes from the table " Cetrod ad Momet of erta Calculatos Aother Eample Sce the parallel as theorem wll requre the area for each secto, that s a reasoable place to start D () 6 7 " 4 Cetrod ad Momet of erta Calculatos

18 Aother Eample We ca locate the cetrod of each area wth respect the as. D bar () () " 5 Cetrod ad Momet of erta Calculatos Aother Eample From the table the back of the book we fd that the momet of erta of a rectagle about ts -cetrod as s " = bh D 6 bar () () Cetrod ad Momet of erta Calculatos

19 Aother Eample ths eample, for, b=6 ad h=6 ( 6 )( 6 ) = 08 4 = D 7 bar () () " 6 Cetrod ad Momet of erta Calculatos Aother Eample For the frst tragle, the momet of erta calculato s t as obvous " 8 Cetrod ad Momet of erta Calculatos

20 Aother Eample The wa t s preseted the tet, we ca ol fd the about the cetrod h " b Cetrod ad Momet of erta Calculatos Aother Eample The chage ma ot seem obvous but t s just how we oret our as. Remember a as s our decso. h " b b h 40 Cetrod ad Momet of erta Calculatos

21 Aother Eample So the momet of erta of the tragle ca be calculated usg the formula wth the correct oretato. bh 6 = ( 6 )( ) 6 = = 4 " Cetrod ad Momet of erta Calculatos Aother Eample The same s true for the tragle bh 6 = ( 6 )( ) 6 =.5 4 = 4 Cetrod ad Momet of erta Calculatos "

22 Aother Eample Now we ca eter the bar for each sub-area to the table Sub bar bar () () (4) " Cetrod ad Momet of erta Calculatos Aother Eample We ca the sum the ad the A(d) to get the momet of erta for each sub-area " Sub 44 bar bar A(d) bar + A(d) () () (4) (4) (4) Cetrod ad Momet of erta Calculatos

23 () 6 7 (4) () 7 6 (4) (4) Aother Eample Ad f we sum that last colum, we have the for the composte fgure " Sub bar bar A(d) bar + A(d) () () (4) (4) (4) Cetrod ad Momet of erta Calculatos Aother Eample We perform the same tpe aalss for the D 46 () 6 7 Cetrod ad Momet of erta Calculatos "

24 Aother Eample Locatg the -cetrods from the -as Sub- () bar () - " Cetrod ad Momet of erta Calculatos Aother Eample Determg the for each sub-area Sub- 48 bar bar () () (4) Cetrod ad Momet of erta Calculatos "

25 Aother Eample Makg the A(d) multplcatos Sub " bar bar () () (4) (4) ) A(d Cetrod ad Momet of erta Calculatos Aother Eample Summg ad calculatg Sub ) A(d bar + A(d) bar bar () () (4) (4) (4) Cetrod ad Momet of erta Calculatos "

26 Homework Problem Calculate the cetrod of the composte shape referece to the bottom of the shape ad calculate the momet of erta about the cetrod z-as. 5 Cetrod ad Momet of erta Calculatos Homework Problem Calculate the cetrod of the composte shape referece to the bottom of the shape ad calculate the momet of erta about the cetrod z-as. 5 Cetrod ad Momet of erta Calculatos

27 Homework Problem Calculate the cetrod of the composte shape referece to the bottom of the shape ad calculate the momet of erta about the cetrod z-as. 5 Cetrod ad Momet of erta Calculatos