Centroids & Moments of Inertia of Beam Sections

Transcription

1 RCH 614 Note Set 8 S017ab Cetrods & Momets of erta of Beam Sectos Notato: b C d d d Fz h c Jo L O Q Q = ame for area = ame for a (base) wdth = desgato for chael secto = ame for cetrod = calculus smbol for dfferetato = ame for a dfferece = ame for a depth = dfferece the drecto betwee a area cetrod ( ) ad the cetrod of the composte shape ( ) = dfferece the drecto betwee a area cetrod ( ) ad the cetrod of the composte shape ( ) = force compoet the z drecto = ame for a heght = momet of erta about the cetrod = momet of erta about the cetrod = momet of erta wth respect to a -as = momet of erta wth respect to a -as = polar momet of erta, as s J = ame for legth = ame for referece org = frst momet area about a as (usg dstaces) = frst momet area about a as (usg dstaces) ˆ ŷ ro r r t tf tw W ˆ ŷ z P L = polar radus of grato = radus of grato wth respect to a -as = radus of grato wth respect to a -as = ame for thckess = thckess of a flage = thckess of web of wde flage = ame for force due to weght = desgato for wde flage secto = horzotal dstace = the dstace the drecto from a referece as to the cetrod of a shape = the dstace the drecto from a referece as to the cetrod of a composte shape = vertcal dstace = the dstace the drecto from a referece as to the cetrod of a shape = the dstace the drecto from a referece as to the cetrod of a composte shape = dstace perpedcular to - plae = plate smbol = smbol for tegrato = calculus smbol for small quatt = dest of a materal (ut weght) = summato smbol The cross secto shape ad how t ressts bedg ad twstg s mportat to uderstadg beam ad colum behavor. The ceter of gravt s the locato of the equvalet force represetg the total weght of a bod comprsed of partcles that each have a mass gravt acts upo. z W W 1 W 4 W W 19

2 RCH 614 Note Set 8 S017ab Resultat force: Over a bod of costat thckess ad F z W W W dw 1 Locato:, s the equvalet locato of the force W from all W s over all & locatos (wth respect to the momet from each force) from: M M 1 1 W W W W dw W dw OR W dw W dw OR W W W W W The cetrod of a area s the average ad locatos of the area partcles For a shape of a uform thckess ad materal: W t where: s weght per ut volume (= specfc weght) wth uts of N/m or lb/ft t s the volume So f W t : td t d OR ad smlarl Smlarl, for a le wth costat cross secto, a ( W al ): L dl OR L ad L L dl OR L L, wth respect to a, coordate sstem s the cetrod of a area ND the ceter of gravt for a bod of uform materal ad thckess. The frst momet of the area s lke a force momet: ad s the area multpled b the perpedcular dstace to a as. Q d Q d. 140

3 RCH 614 Note Set 8 S017ab Cetrods of Commo Shapes b b 141

4 RCH 614 Note Set 8 S017ab Smmetrc reas - area s smmetrc wth respect to a le whe ever pot o oe sde s mrrored o the other. The le dvdes the area to equal parts ad the cetrod wll be o that as. - area ca be smmetrc to a ceter pot whe ever (,) pot s matched b a (-,-) pot. t does ot ecessarl have a as of smmetr. The ceter pot s the cetrod. - f the smmetr le s o a as, the cetrod locato s o that as (value of 0). Wth double smmetr, the cetrod s at the tersecto. - Smmetr ca also be defed b areas that match across a le, but are 180 to each other. Basc Steps (Statcal Momet Method) 1. Draw a referece org.. Dvde the area to basc shapes. Label the basc shapes (compoets) 4. Draw a table wth headers of Compoet, rea,,,, 5. Fll the table value 6. Draw a summato le. Sum all the areas, all the terms, ad all the terms 7. Calculate ˆ ad ŷ Composte Shapes f we have a shape made up of basc shapes that we kow cetrod locatos for, we ca fd a average cetrod of the areas. 1 ˆ ˆ ŷ ŷ Cetrod values ca be egatve. rea values ca be egatve (holes) 1 14

5 RCH 614 Note Set 8 S017ab Defto: Momet of erta; the secod area momet d d ( or z a) We ca defe a sgle tegral usg a arrow strp: d = d for,, strp s parallel to for, strp s parallel to * ca be egatve f the area s egatve (a hole or subtracto). el d shape that has area at a greater dstace awa from a as through ts cetrod wll have a larger value of. Just lke for ceter of gravt of a area, the momet of erta ca be determed wth respect to a referece as. Defto: Polar Momet of erta; the secod area momet usg polar coordate aes J o J o r d d d Defto: Radus of Grato; the dstace from the momet of erta as for a area at whch the etre area could be cosdered as beg cocetrated at. r r radus of grato r radus of grato J o ro polar radus of grato, ad ro = r + r pole o r 14

6 RCH 614 Note Set 8 S017ab The Parallel-s Theorem The momet of erta of a area wth respect to a as ot through ts cetrod s equal to the momet of erta of that area wth respect to ts ow parallel cetrodal as plus the product of the area ad the square of the dstace betwee the two aes. d d d -d d d d d B d d B as through cetrod at a dstace d awa from the other as but d 0, because the cetrod s o ths as, resultg : as to fd momet of erta about z (tet otato) or d o where o (or ) s the momet of erta about the cetrod of the area about a as ad d s the dstace betwee the parallel aes Smlarl J d Momet of erta about a as o J c d Polar momet of erta ro rc d Polar radus of grato r Radus of grato r d * ca be egatve aga f the area s egatve (a hole or subtracto). ** f s ot gve a chart, but & are: YOU MUST CLCULTE WTH d Composte reas: where d Basc Steps s the momet of erta about the cetrod of the compoet area d s the dstace from the cetrod of the compoet area to the cetrod of the composte area (e. d = ŷ - ) 1. Draw a referece org.. Dvde the area to basc shapes. Label the basc shapes (compoets) 4. Draw a table wth headers of Compoet, rea,,,,,, d, d,, d, d 5. Fll the table values eeded to calculate ˆ ad ŷ for the composte 6. Fll the rest of the table values. 7. Sum the momet of erta ( s) ad d colums ad add together. 144

7 RCH 614 Note Set 8 S017ab Geometrc Propertes of reas about bottom left rea = bh = b/ = h/ Tragle b 1 ' 6b about bottom h rea = bh b h rea = r = 0 = 0 d 4 = = r 4 4 r 8 = r 4 = r 4 rea = r d = 0 = rea = r d 4 = 4r = 4r 8 4r 16 rea = = 0 = 0 ab = 16ah 175 = 4a h 15 rea = = 0 4ah = h 5 = 7ah 100 rea = ah = a h 80 = a = h

8 RCH 614 Note Set 8 S017ab Eample 1 rea ( ) (. ) (. ) (. ) ( (.).) 0. 5 ˆ ŷ Eample 6 thck cocrete wall pael s precast to the dmesos as show. Usg the lower left corer as the referece org, determe the ceter of gravt (cetrod) of the pael. 146

9 RCH 614 Note Set 8 S017ab 1 Eample ˆ.05" Fd the momets of erta ( ˆ =.05, ŷ = 1.05 ). 1 ˆ 1.05" Eample 4 147

10 RCH 614 Note Set 8 S017ab Eample 5 Determe the momets of erta about the cetrod of the shape. Soluto: There s o referece org suggested fgure (a), so the bottom left corer s good. fgure (b) area wll be a complete rectagle, whle areas C ad are "holes" wth egatve area ad egatve momet of ertas. o rea = 00 mm 100 mm = 0000 mm = (00 mm)(100 mm) /1 = mm 4 = (00 mm) (100 mm)/1 = mm 4 rea B = -(0 mm) = mm = = - (0 mm) 4 /4 = mm 4 rea C = -1/(50 mm) = 97.0 mm = - (50 mm) 4 /8 = mm 4 = (50 mm) 4 = mm 4 rea D = 100 mm 00 mm 1/ = mm = (00 mm)(100 mm) /6 = mm 4 = (00 mm) (100 mm)/6 = mm 4 shape (mm ) (mm) (mm ) (mm) (mm ) B C D mm ˆ 45.58mm mm ˆ mm 9. 9 mm mm shape (mm 4 ) d (mm) d (mm 4 ) (mm 4 ) d (mm) d (mm 4 ) B C D So, = = = mm 4 = = = mm 4 148

11 RCH 614 Note Set 8 S017ab Eample 6 11 W15 149