Centers of Gravity - Centroids

Transcription

1 RCH Note Set 9. S205ab Ceters of Gravt - Cetrods Notato: C Fz L O Q Q t tw = ame for area = desgato for chael secto = ame for cetrod = force compoet the z drecto = ame for legth = ame for referece org = frst momet area about a as (usg dstaces) = frst momet area about a as (usg dstaces) = ame for thckess = 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 ˆ ŷ = 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 = smbol for tegrato = calculus smbol for small quatt z γ = dest of a materal (ut weght) Σ = 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 4 2 Resultat force: Over a bod of costat thckess ad F z = = = d Locato:, s the equvalet locato of the force from all s over all & locatos (wth respect to the momet from each force) from: M = = d = d = OR M = = d = d = OR = = ( ) ( ) 7

2 RCH Note Set 9. S205ab The cetrod of a area s the average ad locatos of the area partcles For a dscrete shape ( ) of a uform thckess ad materal, the weght ca be defed as: = γt where: γ s weght per ut volume (= specfc weght) wth uts of N/m or lb/ft t s the volume So f = γt : γ t = γtd = d OR ( ) = ad smlarl Smlarl, for a le wth costat cross secto, a ( = γ a L ): 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 = =. 72

3 RCH Note Set 9. S205ab Cetrods of Commo Shapes b b 7

4 RCH Note Set 9. S205ab 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. It does ot ecessarl have a as of smmetr. The ceter pot s the cetrod. - If the smmetr le s o a as, the cetrod locato s o that as (value of 0). th double smmetr, the cetrod s at the tersecto. - Smmetr ca also be defed b areas that match across a le, but are 80 to each other. Basc Steps. Draw a referece org. 2. 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 If we have a shape made up of basc shapes that we kow cetrod locatos for, we ca fd a average cetrod of the areas. ˆ = ˆ = ŷ = ŷ = OR ˆ Σ = Σ ŷ = 2 Cetrod values ca be egatve. rea values ca be egatve (holes) 74

5 RCH Note Set 9. S205ab Eample (pg 24) (.) (. ) (.) (. ) ˆ = = ŷ = = 2. 2 Eample 2 (pg 245)* 6 thck cocrete wall pael s precast to the dmesos as show. Usg the upper left corer of area II as the referece org, determe the ceter of gravt (cetrod) of the pael. Ref. org 75