Analysis of Members with Axial Loads and Moments. (Length effects Disregarded, Short Column )

Transcription

1 Analyi o emer wih Axial Loa an omen (Lengh ee Diregare, Shor Column ) A. Reaing Aignmen Chaper 9 o ex Chaper 10 o ACI B. reenaion o he INTERACTION DIAGRA or FAILURE ENVELO We have een ha a given eion an have a maximum apaiy in eiher axial loa or lexure. I i logial o uppoe ha or ome ominaion o axial loa an momen, a maximum apaiy an alo e oun. Conier a memer wih ome applie axial an ome applie momen. e = e. or e= / Shor Column Aumpion: e i onan along he memer. 1

2 Conier a eion a ulimae momen oniion an apply an axial loa. Coniion a - ure omen Coniion - omen, Axial loa ε Coniion - omen, Axial loa (Balane oniion) ε y a I we ake momen aou eion enroi in all ae: 1. omen o A y remain onan.. C inreae a we go rom rain oniion a o rain oniion a 3. omen o C inreae A y 4. Summaion inernal ore inreae wih C. Thi an e erme poiive ineraion, ine inreae in C on apaiy reul in inreae in oher apaiy. A y C (alane Coniion) A y C a

3 Conier a eion a ulimae momen oniion an apply an axial loa. Coniion e - ure Axial loa Coniion - omen, Axial loa e ε Coniion - omen, Axial loa (Balane oniion) ε y a e I we ake momen aou eion enroi in all ae: 1. omen o A y remain onan.. C ereae a we go rom rain oniion e o rain oniion 3. Σ inreae. A y C Thi an e erme negaive ineraion, ine inreae in on apaiy reul in ereae in oher apaiy. A y C e A y C e = mall (alane Coniion) a e = large 3

4 Colum 4

5 Srain Limi eho or Analyi an Deign (ACI ). In Srain Limi eho, omeime reerre o a he Uniie eho, he nominal lexural rengh o a onree memer i reahe when he ne ompreive rain in he exreme ompreion ier reahe he ACI oe-aume limi o in/in (ACI 10..3). I alo hypoheize ha when he ne enile rain in he exreme enion eel, ε = in/in, he ehavior i ully uile. The onree eam eion haraerize a Tenion-Conrolle, wih ample warning o ailure a enoe y exeive eleion an raking. I he ne enile rain in he exreme enion ier, ε, i mall, uh a in ompreion memer, eing equal or le han a Compreion-Conrolle rain limi, a rile moe o ailure i expee wih a uen an exploive ype o ailure. Flexural memer are uually enion-onrolle. However, ome eion uh a hoe ujee o mall axial loa, u large ening momen, he ne enile rain, ε, in he exreme enile ier, will have an inermeiae or raniional value eween he wo rain limi ae, namely, eween he ompreion-onrolle rain limi o ε = y 60ki 0.00 E = 9,000ki = (1.1) an he enion-onrolle rain limi ε = in/in. Figure 5.1 (ACI Figure. R9.3. page 100) how hee hree zone a well a he variaion in he rengh reuion aor appliale o he oal range o ehavior. Variaion o φ a a Funion o Srain Variaion o he value or he range o rain eween e = 0.00 in/in an e = in/in an e linearly inerpolae: ( ) 0.65 φ = ε 0.90 Tie Column ( ) 0.70 φ = ε 0.90 Spiral Column Variaion o φ a a Funion o Neural Axi Deph Raio / 5

6 φ = / φ = / Tie Column Spiral Column φ = / SIRAL OTHER φ = / Compreion Conrolle ε = 0.00 = Traniion ε = = Tenion Conrolle 6

7 ' 0.85 ε u 87, 000 = 87,000 + y T y = 60, 000 pi ε y = = ' 0.85 T ε y ' 0.85 ε u ε u = = 0.6 T ε = ε = 0.00 Tenion Failure Balane Coniion 7

8 Sree an Srain For enion eel ε = εu = ε E = ε E u y.c. or ompreion eel: ε = εu = ee u y where a = β1< h ee ACI an C = 0.85 a ε - ' ε A C A NOTE: In onra o eam we anno reri olumn eign uh ha yieling eore ailure raher han ruhing ailure alway e he reul o over loaing. Type o ailure o olumn epen on he value o eenriiy e. 8

9 C. Coe Allowale Ineraion Diagram (ACI 9.3.) 9

10 10

11 E. Deine lai Cenroi: oin a whih reulan o reiing ore a when rain i uniorm over he eion = 0 when e = 0, a meaure rom hi poin or ypial ymmerial reinore eion, plai enroi will e a eion geomeri enroi. For oher eion a mall alulaion i neee (ee he example elow). In oher wor, he plai enroi i he enroi o reiane o he eion i all he onree i ompree o he maximum re (0.85 ) an all he eel i ompree o he yiel re ( y ) wih uniorm rain over he eion. Example For he eion given elow eermine he loaion o he plai enroi. Given: = 4,000 ki y = 60,000 pi Tie Column Cener linear Soluion: C = 0.85 A ' 15" A = 10 in A' = 4 in C = 0.85 (4 ki) ( ) C = 180 kip 3" 3" No = =,10 kip 6" 10 = = 3,600 p = 1.7 in Coul alo ake momen aou C. = ( + 10) 600 (10 ) = 0 nroi: C A = 600 A = 40 y y = 1.7 in 11

12 F. Conruion o Ineraion Diagram Reall ha we have previouly analyze he iuaion in whih nominal momen or nominal axial loa exie. The alulaion aoiae wih hee oniion will no hange. Reall he alane rain oniion. The imporan hing o noe here i ha any eion an e pu uner appropriae momen an axial loa o aue alane rain: C.C. ε u ε a n ' 0.85 T C = A y A From geomery: where an = a = β 1 = ε E = 0.003( ) E y From ai: = 0.85 a + A - A n y ε an n = 0.85 a ( p a ) + A ( p ) + A y ( - p ) Noe ha, whenever axial loa i preen, momen mu e aken aou he plai enroi i onien reul are o e oaine. I mu e realle ha he momen value alulae aove i aually an inernal, reiing momen, whih i in equilirium wih an exernal applie momen o he ame magniue. We oul, i we wihe, alulae he inernal reiing momen aou any poin i we a he ame ime rememere o inlue he inluene o he exernal axial loa, aume applie a he plai enroi. I we hoe o ake momen aou ompreion eel in he aove ae, or example, hi alulaion o n woul eome: a n = A y( - ) 0.85 a( ) + n ( p ) I i generally eaier o ake momen aou he plai enroi. Thu, avoiing nee or onieraion o he exernal axial loa. 1

13 Example. Calulaion o oin on Ineraion Diagram Conier he ollowing eion or whih we have alulae he plai enroi loaion Given: Balane Coniion: A y =400 kip A = 160kip y = 4,000 ki y = 60,000 pi Tie Column in 6.C. 4 in = = 13.8 in a = = ε = εu in ε C = = > = 0.85 A C = ki [ ] 15 = 598. kip C = 60 4 = 40 kip 0.00 = T = = 600 kip T = A = 600 kip y C A = 40 y kip = C + C T = = 38. kip Take momen aou.c.: n = 598.(14.7 ) + 40(14.7 3) + 600(11.3 3) = 13,070 in kip when axial loa i preen, momen may e aken aoun.c. 13

14 Ulimae omen Coniion (No Axial Loa) Aume ompreion eel yiel: a + 40 = a = = = = inhe inhe ε = εu ε = = Aumpion ha ompreion eel yiel i 8.3 wrong. Thereore, he ompreion eel will no yiel. Sine he ompreion eel oe no yiel, C = A = a+ 4 = (0.85 ) + 4 εu E = = = = = = = 8.61 inhe

15 Axial loa = 0.0 n = in 4 in C = 348 = kip 15.C. C = 43.35= 43.35(8.61) = 373. kip Sine here i no axial loa, we an ake momen aou any poin an in he momen apaiy. To e onien, alway ake momen aou C = 14.7 = T = A = 600 kip y C 3 A = 348 kip n = 6.7(14.7 3) (14.7 ) + 600(11.3 3) = 11, 753 in kip 15

16 Cae ε = ε = u = 11.5 inhe a = 0.85= = 9.78 inhe 6 Chek o in ou wheher he ompreion 10 in 4 in eel yiel or no. 15.C. ε = εu ε = = Thereore, ompreion eel yiel C n = 0.85(4)(15)(9.78) = 499 = = 139 kip kip =11.5 ε T = A = 600 kip y C A 9.78 n = 499(14.7 ) + 40(14.7 3) + 600(11.3 3) = 1,683 in kip 16

17 Cae ε = 0 ε = u 6 = 3 inhe a = 0.85= = inhe Chek o in ou wheher he ompreion in.c. 4 in eel yiel or no. ε = εu ε = = Thereore, ompreion eel yiel =3 C = 0.85(4)(15)(19.55) = 997 kip ε n = = 137 kip T = 0 kip C A n = 997(14.7 ) + 40(14.7 3) = 7,70 in kip 17

18 Cae ε = ε = u = 17.5 inhe 6 a = 0.85= = inhe 10 in 4 in Chek o in ou wheher he ompreion eel yiel or no. 15.C. ε = εu ε = = Thereore, ompreion eel yiel =17.5 T = Aε E = 10(0.001)(9000) = 90 kip ε C = 0.85(4)(15)(14.66) = 748 kip n = = 698 kip T = Aε E kip C A n = 748(14.7 ) + 40(14.7 3) + 90(11.3 3) = 10,78 in kip 18

19 Ineraion Diagram Axial Loa (kip) omen (in-kip) ε u ε n φ n φ φ n ε = φ = ε = = 0.73 ε = φ = ε = = φ = 0.65 limi or pure axial loa 0.80φ = = 1100 kip 19