7. T.chhay
eKTTYl)antMélefr
Pe
− u
EI = −e e
B= =
Pu π 2
π EI Pe − 1
2
− 1− Pu
EI L2 Pu L2
Edl Pe = π
2
EI
= Euler buckling load
2
L
dUcenH y = B sin πLx = ⎡ (P / P ) − 1⎤ sin πLx
⎢
e
⎥
⎣ e u ⎦
M = Pu ( yo + y )
⎧
⎪ πx ⎡ e ⎤ πx ⎫⎪
= Pu ⎨e sin + ⎢ ⎥ sin ⎬
⎪
⎩ L ⎣ (Pe / Pu ) − 1⎦ L⎪
⎭
m:Um:g;GtibrmaekItenARtg; x = L / 2 ³
⎡ e ⎤
M max = Pu ⎢e + ⎥
⎣ (Pe / Pu ) − 1⎦
⎡ (P / P ) − 1 + 1 ⎤
= Pu e ⎢ e u ⎥
⎣ (Pe / Pu ) − 1 ⎦
⎡ 1 ⎤
= Mo ⎢ ⎥
⎣1 − (Pu / Pe )⎦
Edl M o minEmnCam:Um:g;bEnßmGtibrma (unampliflied maximum moment). kñúgkrNIenH
vaTTYl)anBI initial crookedness b:uEnþCaTUeTAvaGacCalTßplénbnÞúkTTwgGkS½ b¤m:Um:g;cug. dUcenHem
KuNm:Um:g;bEnßm (moment amplification factor) KW
1
1 − (Pu / Pe )
¬^>$¦
dUcEdl)anerobrab;mkehIy TMrg;emKuNm:Um:g;bEnßmrbs; AISC GacxusEbøkBIsmIkar ^>$
bnþic.
]TahrN_6>2³ eRbIsmIkar ^>$ edIm,IKNnaemKuNm:Um:g;bEnßmsMrab;Fñwm-ssrén]TahrN_ 6>1.
dMeNaHRsay³ edaysar Euler load Pe CaEpñkrbs;emKuNm:Um:g;bEnßm eKRtUvKNnavasMrab;GkS½én
karBt; EdlkñúgkrNIenHKWGkS½ x . eKGacsresr Euler load Pe edayeRbI effective length nig
slenderness ratio dUcxageRkam³
194 Fñwm -ssr
8. T.chhay
π 2 EAg
Pe =
(KL / r )2
¬emIlCMBUk 4 smIkar $>^ a¦. sMrab;GkS½énkarBt;
KL K x L 1.0(17 )(12 )
= = = 55.89
r rx 3.65
π 2 EAg π 2 (29000)(17.1)
Pe = = = 1567kips
(KL / r )2 (55.89)2
BIsmIkar ^>$
1 1
= = 1.15
1 − (Pu / Pe ) 1 − (200 / 1567 )
EdlbgðajkarekIneLIg 15% BIelIm:Um:g;Bt;. m:Um:g;bEnßmKW
1.15 × M u = 1.15(93.5) = 107.5 ft − kips
cMeLIy³ emKuNm:Um:g;bEnßm 1.15
6>4> Web Local Buckling in Beam-Columns
karkMNt;rbs; design moment tMrUveGayRtYtBinitümuxkat;sMrab; compactness . enAeBl
EdlKμanbnÞúktamGkS½ RTnugrbs;RKb;rUbragEdlmanenAkñúgtaragsuT§Et compact. RbsinebImanvtþ
manbnÞúktamGkS½ RTnugTaMgenaHGacnwgmin compact. enAeBlEdleyIgeGay λ = h / t w /
RbsinebI λ ≤ λ p rUbragKW compact.
RbsinebI λ p < λ ≤ λr rUbragKW noncompact.
RbsinebI λ > λr rUbragKW slender.
ASIC B5 enAkñúg Table B5.1 erobrab;nUvkarkMNt;xageRkam³
sMrab; φ PP ≤ 0.125 / λ p = 640 ⎛1 − 2φ.75Pu ⎞ ¬xñat US¦
u
F
⎜
⎜ P ⎟
⎟
b y y⎝ b y ⎠
1680 ⎛ 2.75 Pu ⎞
λp = ⎜1 −
Fy ⎜ φb Py ⎟
⎟ ¬xñat ¦
IS
⎝ ⎠
191 ⎛ ⎞
sMrab; φ PP
u > 0.125 λ p / = ⎜ 2.33 − Pu ⎟ ≥ 253
Fy ⎜ φb Py ⎟ Fy
¬xñat US¦
b y ⎝ ⎠
500 ⎛ ⎞
λp = ⎜ 2.33 − Pu ⎟ ≥ 665
Fy ⎜ φb Py ⎟ Fy
¬xñat IS¦
⎝ ⎠
195 Fñwm -ssr
9. T.chhay
sMrab;tMélepSg²rbs; φ PP / λr = 970 ⎛1 − 0.74 φ PP ⎞ ¬xñat US¦
u
F
⎜
⎜
u ⎟
⎟
b y y ⎝ b y ⎠
2550 ⎛ ⎞
λr =
Fy ⎜
⎜1 − 0.74 Pu
φb Py
⎟
⎟
¬xñat IS¦
⎝ ⎠
Edl Py = Ag Fy / bnÞúktamGkS½caM)ac;edIm,IeTAdl;sßanPaBkMNt; yielding.
edaysar Pu CaGBaØti eKminGacRtYtBinitü compactness rbs;RTnug nigminGacerobcMCata
ragTukCamun)aneT. b:uEnþ rolled shape xøHbMeBjnUvkrNId¾GaRkk;bMput 665 / Fy Edlmann½yfarUb
ragenaHmanRTnug compact edayminTak;TgnwgbnÞúktamGkS½. rUbragEdlmanenAkñúg column load
table in Part 3 of the Manual EdlminbMeBjlkçxNÐRtUv)ankMNt;bgðaj enaHeKRtUvRtYtBinitü
compactness rbs;RTnugrbs;va. rUbragEdlmansøabmin compact k¾RtUv)ankMNt;bgðaj dUcenHRKb;
rUbragTaMgGs;Edlmin)anbgðaj enaHmann½yfarUbragTaMgenaHKW compact.
]TahrN_6>3³ Edk A36 EdlmanrUbrag W12 × 65 RtUv)andak;eGayrgm:Um:g;Bt; nigbnÞúktamGkS½em
KuN 300kips . RtYtBinitü compactness rbs;RTnug.
dMeNaHRsay³ rUbragenHKW compact sMrab;RKb;tMélbnÞúktamGkS½ BIeRBaHminmankarkMNt;cMNaMNa
mYyenAkñúg column load table. b:uEnþ edIm,Ibgðaj eyIgRtYtBinitü width-thickness ratio rbs;RTnug
Pu Pu 300
= = = 0.4848 > 0.125
( )
φb Py φb Ag Fy 0.9(19.1)(36)
⎛ ⎞
dUcenH λp = ⎜ 2.33 − Pu ⎟ = 191 (2.33 − 0.4848) = 58.74
191
⎜
Fy φb Py ⎟ 36
⎝ ⎠
253 253
= = 42.17 < 58.74
Fy 36
dUcenH λ p = 58.74
BI dimensions and properties tables/
h
λ= = 24.9 < 58.74
tw
dUcenH RTnugKW compact. cMNaMfa sMrab;RKb;tMélrbs; Fy enaH th nwgmantMéltUcCag
w
253 / F y EdlCatMélEdltUcbMputrbs; λ p dUcenHRTnugrbs; W 12 × 65 nwgenAEtCa compact.
196 Fñwm -ssr
10. T.chhay
6>5> eRKagBRgwg nigeRKagGt;BRgwg Braced versus Unbraced Frame
AISC Specification erobrab;BI moment amplification in Chapter C, Frames and other
Structures”. eKmanemKuNbEnßmBIrEdleRbIenAkñúg LRFD: mYyedIm,IKitBIm:Um:g;bEnßmEdlCalT§plBI
PaBdabrbs;Ggát; nigmYyeTotsMrab;KitBIT§iBl sway enAeBlEdlGgát;CaEpñkrbs; unbraced
frame. viFIenHmanlkçN³RsedogKñaeTAnwgviFIEdleRbIenAkñúg ACI Building Code sMrab;ebtugBRgwg
edayEdk (ACI, 1995). rUbTI 6>5 nwgbgðajBIGgát;TaMgBIr. enAkñúg rUbTI 6>5 a Ggát;RtUv)anTb;Rb
qaMgnwg sidesway ehIym:Um:g;TIBIrGtibrmaKW Pδ EdlRtUvbEnßmeTAelIm:Um:g;GtibrmaenAkñúgGgát;enaH.
RbsinebIeRKagminRtUv)anBRgwg vanwgelceLIgnUvm:Um:g;TIBIr EdlbgðajenAkñúg rUbTI 6>5 b EdlbegáIt
eday sidesway. m:Um:g;TIBIrenHmantMélGtibrma PΔ EdlbgðajBIkarbEnßménm:Um:g;cug.
edIm,IKItBIT§iBlTaMgBIrenH emKuNm:Um:g;bEnßm B1 nig B2 RtUv)aneRbIsMrab;m:Um:g;BIrRbePT.
m:Um:g;bEnßmEdleRbIsMrab;KNnaRtUv)anKNnaBIbnÞúkemKuN nigm:Um:g;emKuNdUcxageRkam³
M u = B1M nt + B2 M lt (AISC Equation C1-1)
Edl M nt = m:Um:g;GtibrmaEdlsnμt;faminman sidesway ekIteLIg eTaHbICaeRKagBRgwgb¤minBRgwg
k¾eday ¬ nt mann½yfa no translation¦
M lt = m:Um:g;GtibrmaEdlekIteLIgeday sidesway ekIteLIg ¬ lt mann½yfa lateral
translation¦. m:Um:g;enHGacekItBI lateral load b¤edaysar unbalanced gravity
loads . bnÞúkTMnajGacbegáIt sidesway RbsinebIeRKagGt;sIuemRTI b¤k¾bnÞúkTMnaj
enaHRtUv)andak;edayminmanlkçN³sIuemRTI. M lt nwgmantMélesμIsUnüRbsinebIeRKag
RtUv)anBRgwg.
197 Fñwm -ssr
14. T.chhay
viFIsaRsþEdl)aneFVIeGayRbesIreLIgsMrab;Ggát;rgbnÞúkxagTTwgGkS½ ¬krNITIBIr¦ RtUv)anpþl;
eGayenAkñúg section C1 of the commentary to the Specification. emKuNkat;bnßyKW
P
Cm = 1 +ψ u
Pe1
sMrab;Ggát;TMrsamBaØ
π 2δ o EI
ψ= −1
M o L2
Edl δ o CaPaBdabGtibrmaEdlekItBIbnÞúkxagTTwgGkS½ ehIy M o Cam:Um:g;GtibrmaenA
cenøaHTMrEdl)anBIbnÞúkxagTTwgGkS½. emKuN ψ RtUv)anKNnaBIsßanPaBFmμtaCaeRcInehIyRtUv)an
pþl;eGayenAkñúg commentary Table C-C1.1.
]TahrN_6>4³ Ggát;EdlbgðajenAkñúg rUbTI 6>10 CaEpñkrbs; braced frame. bnÞúk nigm:Um:g;RtUv)an
KNnaCamYybnÞúkemKuN ehIykarBt;KWwFobnwgGkS½xøaMg. RbsinebIeKeRbI A572 Grade 50 etIGgát;
enHRKb;RKan;b¤eT? KL = KL y = 14 ft .
dMeNaHRsay³ kMNt;faetIRtUveRbIrUbmnþGnþrkmμmYyNa
KL K y L 14(12)
maximum = = = 55.63
r ry 3.02
BI AISC Table 3-50, φc Fcr = 33.89ksi dUcenH
φc Pn = Ag (φc Fcr ) = 19.1(33.89 ) = 647.4kips
Pu 420
= = 0.6487 > 0.2
φc Pn 647.4
201 Fñwm -ssr
15. T.chhay
dUcenHeRbI AISC Equation H1-1a.
enAkñúgbøg;énkarBt;
KL K x L 14(12 )
= = = 31.82
r rx 5.28
Ag F y π 2 EAg π 2 (29000 )(19.1)
Pe1 = = = = 5399kips
λc
2
(K x L / rx )2 (31.82)2
⎛M ⎞ ⎛ 70 ⎞
C m = 0.6 − 0.4⎜ 1 ⎟ = 0.6 − 0.4⎜ − ⎟ = 0.9415
⎜M ⎟
⎝ 2⎠ ⎝ 82 ⎠
Cm 0.9415
B1 = = = 1.021
1 − (Pu / Pe1 ) 1 − (420 / 5399 )
BI Beam design charts,CamYynwg Cb = 1.0 nig Lb = 14 ft. moment strength KW
φb M n = 347 ft − kips
sMrab;tMél Cb BitR)akd edayeyagtamdüaRkam:Um:g;enAkñúg rUbTI 6>10³
12.5M max 1.25(82 )
Cb = = = 1.06
2.5M max + 3M A + 4 M B + 3M C 2.5(82 ) + 3(73) + 4(76 ) + 3(79 )
dUcenH φb M n = Cb (347) = 1.06(347) = 368 ft − kips
b:uEnþ φb M p = 358 ft − kips ¬BItarag¦ < 368 ft − kips
dUcenHeRbI φb M n = 358 ft − kips
m:Um:g;emKuNKW M nt = 85 ft − kips M lt = 0
BI AISC Equation C1-1,
M u = B1M nt + B2 M lt = 1.021(82) + 0 = 83.72 ft − kips = M ux
BI AISC Equation H1-1a,
Pu 8 ⎛ M ux M uy ⎞
⎟ = 0.6487 + 8 ⎛ 83.72 ⎞ = 0.857 < 1.0
+ ⎜ + ⎜ ⎟ (OK)
φc Pn 9 ⎜ φb M nx φb M ny
⎝
⎟
⎠ 9 ⎝ 358 ⎠
cMeLIy³ Ggát;enHKWRKb;RKan;.
]TahrN_ 6>5³ Fñwm-ssredkEdlbgðajenAkñúgrUbTI 6>11 rgnUv service live loads dUcEdlbgðaj
kñúgrUb. Ggát;enHRtUv)anBRgwgxagenAxagcugrbs;vaTaMgBIr ehIykarBt;KWeFobnwgGkS½ x . RtYtBinitü
faetIGgát;enHRKb;RKan;tam AISC Specification.
202 Fñwm -ssr
16. T.chhay
dMeNaHRsay³ bnÞúkemKuNKW
Pu = 1.6(20) = 32.0kips
ehIym:Um:g;GtibrmaKW
M nt =
(1.6 × 20)(10) + (1.2 × 0.035)(10)2 = 80.52 ft − kips
4 8
Ggát;enHRtUv)anBRgwgTb;nwgkarbMlas;TIxagcug dUcenH M lt = 0 .
KNnaemKuNm:Um:g;bEnßm
sMrab;Ggát;rgbnÞúkxagEdlRtUv)anBRgwgTb;nwg sidesway ehIy unrestrained end enaH Cm = 1.0 .
tMélEdlsuRkitCagEdl)anBI AISC Commentary Table C-C1.1 KW
P
C m = 1 − 0 .2 u
Pe1
sMrab;GkS½énkarBt;
KL K x L 1.0(10 )(12 )
= = = 34.19
r rx 3.51
π 2 EAg π 2 (29000)(10.3)
Pe1 = = = 2522kips
(KL / r )2 (34.19)2
⎛ 32.0 ⎞
C m = 1 − 0.2⎜ ⎟ = 0.9975
⎝ 2522 ⎠
emKuNm:Um:g;bEnßm
Cm 0.9975
B1 = = = 1.010 > 1.0
1 − (Pu / Pe1 ) 1 − (32.0 / 2522 )
sMrab;GkS½énkarBt;
M u = B1M nt + B2 M lt = 1.010(80.52) + 0 = 81.33 ft − kips
edIm,ITTYl design strengths dMbUgemIleTA column load tables in Part 3 of the Manual Edl
eGay φc Pn = 262kips
BI beam design charts in Part 4 of the Manual sMrab; Lb = 10 ft nig Cb = 1.0
203 Fñwm -ssr
17. T.chhay
φb M n = 91.8 ft − kips
edaysarTMgn;FñwmtUcNas;ebIeRbobeFobnwgbnÞúkGefrcMcMnuc enaH Cb = 1.32 BI rUbTI 5>13 c.
φb M n = 1.32(91.8) = 121 ft − kips
m:Um:g;enHFMCag φb M p = 93.6 ft − kips EdlTTYl)anBI beam design chart dUcKña dUcenH design
strength RtUv)ankMNt;RtwmtMélenH. dUcenH
φb M n = 93.6 ft − kips
RtYtBinitürUbmnþGnþrkmμ³
Pu 32.0
= = 0.1221 < 0.2
φc Pn 262
dUcenHeRbI AISC Equation H1-1b³
Pu ⎛ M ux M uy ⎞ 0.1221 ⎛ 81.33 ⎞
+⎜ + ⎟= +⎜ + 0 ⎟ = 0.930 < 1.0 (OK)
2φc Pn ⎜ φb M nx φb M ny
⎝
⎟
⎠ 2 ⎝ 93.6 ⎠
cMeLIy³ W 8× 35 KWRKb;RKan;
]TahrN_ 6>6³ Ggát;EdlbgðajenAkñúg rUbTI6>12 eFVIBIEdk A242 EdlmanrUbrag W 12 × 65 ehIy
RtUvRTnUvbnÞúksgát;tamGkS½emKuN 300kips . enAcugTMenrmçagCa pinned nigcugmçageTotrgnUvm:Um:g;
emKuN 135 ft − kips eFobGkS½xøaMg nig 30 ft − kips eFobGkS½exSay. eRbII K x = K y = 1.0 cUreFVIkar
GegátBIGgát;enH.
dMeNaHRsay³ dMbUg kMNt; yield stress Fy . BI Table 1-2, Part 1 of the Manual, W 12 × 65
CarUbragRkumTIBIr. BI Table 1-1, Edk A242 manersIusþg;EtmYyKW Fy = 50ksi .
204 Fñwm -ssr
18. T.chhay
bnÞab;mkeTot rk compressive strength. sMrab; KL = 1.0(15) = 15 ft axial compressive design
strength BI column load table KW³
φc Pn = 626kips
cMNaMfa taragbgðajfasøabrbs; W 12 × 65 KW noncompact sMrab; Fy = 50ksi .
KNnam:Um:g;Bt;eFobGkS½xøaMg (strong axis bending moment).
= 0.6 − 0.4(0) = 0.6
M1
C mx = 0.6 − 0.4
M2
K x L 15(12 )
= = 34.09
rx 5.28
π 2 EAg π 2 (29000 )(19.1)
Pe1x = = = 4704kips
(K x L / rx )2 (34.09)2
C mx 0 .6
B1x = = = 0.641 < 1.0
1 − (Pu / Pe1x ) 1 − (300 / 4704 )
dUcenH eRbI B1x = 1.0
M ux = B1x M ntx + B2 x M ltx = 1.0(135) + 0 = 135 ft − kips
BI beam design charts CamYy Lb = 15 ft / φb M nx = 342 ft − kips sMrab; Cb = 1.0 ehIy
φb M px = 357.8 ft − kips . BI rUbTI 5>15 g, Cb = 1.67 ehIy
Cb × (φb M nx for Cb = 1.0) = 1.67(342) = 571 ft − kips
lT§plenHFMCag φb M px dUcenHeRbI φb M nx = φb M px = 357.8 ft − kips
KNna m:Um:g;Bt;eFobGkS½exSay (weak axis bending moment).
= 0.6 − 0.4(0) = 0.6
M1
C my = 0.6 − 0.4
M2
K yL 15(12)
= = 59.60
ry 3.02
π 2 EAg π 2 (29000)(19.1)
Pe1 y = = = 1539kips
(K y L / ry )2 (59.60)2
C mx 0.6
B1 y = = = 0.745 < 1.0
( )
1 − Pu / Pe1 y 1 − (300 / 1539 )
dUcenH eRbI B1y = 1.0
M uy = B1 y M nty + B2 y M lty = 1.0(30 ) + 0 = 30 ft − kips
205 Fñwm -ssr
22. T.chhay
rUbTI 6>14 bgðajbEnßmeTotBI superposition concept. rUbTI 6>14 a bgðajBI braced frame
rgnUvTaMgbnÞúkTMnaj (gravity load) nigbnÞúkxag (lateral load). m:Um:g;enA M nt enAkñúgGgát; AB
RtUv)anKNnaedayeRbIEt gravity load. edayPaBsIuemRTI eKminRtUvkar bracing edIm,IkarBar
sidesway BIbnÞúkenH. m:Um:g;enHRtUv)anbEnßmCamYyCamYynwgemKuN B1 edIm,IkarBarT§iBl Pδ .
M lt m:Um:g;EdlRtUvKñanwg sway ¬EdlbegáIteLIgedaybnÞúkedk H ¦ nwgRtUv)anbEnßmeday B2 edIm,I
karBarnwgT§iBl PΔ .
enAkñúg rUbTI 6>14 b unbraced frame RTEtbnÞúkbBaÄr. edaysarkardak;bnÞúkenHminsIuemRTI
vanwgman sidesway bnþic. m:Um:g; M nt RtUv)anKNnaedayBicarNafaeRKagRtUv)anBRgwg ¬kñúgkrNI
enH edaysarTMredkkkit nigkMlaMgRbtikmμRtUvKμaEdleKehAfa tMNTb;nimitþ (artificial joint restraint
AJR). edIm,IKNnam:Um:g; sidesway eKRtUvykTMrkkitecj ehIyCMnYsedaykMlaMgEdlmantMélesμInwg
artificial joint restraint b:uEnþmanTisedApÞúyKña. kñúgkrNIenH m:Um:g;TIBIr PΔ nwgmantMéltUcNas;
ehIyeKGacecal M lt )an.
RbsinebITaMgbnÞúkxag nigbnÞúkTMnaj minsIuemRTI eKGacbEnßmkMlaMg AJR eTAelIbnÞúkxagBit
R)akd enAeBlEdl M lt RtUv)ankMNt;.
]TahrN_ 6>7³ Edk W 12 × 65 RbePT A572 grade 50 RbEvg 15 ft sMrab;eRbICassrenAkñúg
unbraced frame. bnÞúkcMGkS½ nigm:Um:g;cugTTYl)anBI first-order analysis énbnÞúkTMnaj ¬bnÞúkefr
nigbnÞúkGefr¦ RtUv)anbgðajenAkñúg rUbTI 6>15 a . eRKagmanlkçN³sIuemRTI ehIybnÞúkTMnajk¾
RtUv)andak;sIuemRTIEdr. rUbTI 6>15 b bgðajBIm:Um:g;énbnÞúkxül;Edl)anBI first-order analysis. m:U
m:g;Bt;TaMgGs;KWeFobnwgGkS½xøaMg. emKuNRbEvgRbsiT§PaB K x = 1.2 sMrab;krNI sway nig
K x = 1.0 sMrab;krNI nonsway ehIy K y = 1.0 . kMNt;faetIGgát;enHeKarBtam AISC
Specification b¤eT?
dMeNaHRsay³ karbnSMbnÞúkTaMgGs;EdleGayenAkñúg AISC A4.1 suT§EtmanbnÞúkGefr ehIyelIk
ElgEtkarbnSMbnÞúkTImYyecj EdlkarbnSMbnÞúkTaMgGs;manbnÞúkxül; b¤bnÞúkGefr b¤TaMgBIr. Rbsin
ebIRbePTbnÞúk ¬ E, Lr , S , nig R ¦ enAkñúg]TahrN_enHminRtUv)anbgðaj lkçxNÐénkarbnSMbnÞúk
RtUv)ansegçbdUcxageRkam³
1 .4 D (A4-1)
1 .2 D + 1 .6 L (A4-2)
209 Fñwm -ssr
23. T.chhay
1.2 D + (0.5 L or 0.8W ) (A4-3)
1.2 D + 1.3W + 0.5 L (A4-4)
1 .2 D + 0 .5 L (A4-5)
0.9 D ± 1.3W (A4-5)
enAeBlEdlbnÞúkefrtUcCagbnÞúkGefrR)aMbIdg enaHbnSMbnÞúk (A4-1) GacminRtUvKit. bnSM
bnÞúk (A4-4) nwgmantMélFMCag (A4-3) dUcenH (A4-3) Gacdkecj)an. bnSMbnÞúk (A4-5) k¾Gac
ecal)anedaysarvanwgpþl;eRKaHfñak;tUcCag (A4-2). cugeRkay karbnSMbnÞúk (A4-6) nwgmineRKaH
fñak;dUc (A4-4) ehIyk¾Gacdkecj)anBIkarBicarNa EdlenAsl;EtbnSMbnÞúkBIrEdlRtUveFVIkarGegátKW
(A4-2)nig (A4-4) ³
1.2 D + 1.6 L nig 1.2 D + 1.3W + 0.5 L
rUbTI 6>16 bgðajBIbnÞúktamGkS½ nigm:Um:g;Bt;EdlKNnaecjBIbnSMbnÞúkTaMgBIrenH
kMNt;GkS½eRKaHfñak;sMrab;ersIusþg;kMlaMgsgát;tamGkS½
K y L = 15 ft
K x L 1.2(15)
= = 10.29 ft < 15 ft
rx / ry 1.75
dUcenHeRbI KL = 15 ft
BI column load tables CamYynwg KL = 15 ft / φc Pn = 626kips
sMrab;lkçxNÐbnÞúk (A4-2)/ Pu = 454kips / M nt = 104.8 ft − kips nig M lt = 0 ¬eday
sarEtsIuemRTI vaminmanm:Um:g; sidesway¦. emKuNm:Um:g;Bt;KW
210 Fñwm -ssr
24. T.chhay
⎛M ⎞ ⎛ 90 ⎞
C m = 0.6 − 0.4⎜ 1 ⎟ = 0.6 − 0.4⎜
⎜M ⎟ ⎟ = 0.2565
⎝ 2⎠ ⎝ 104.8 ⎠
sMrab;GkS½énkarBt;
KL K x L 1.0(15)(12 )
= = = 34.09
r rx 5.28
¬krNIenHKμan sidesway dUcenHeKeRbI K x sMrab; braced condition¦. enaH
π 2 EAg π 2 (29000)(19.1)
Pe1 = = = 4704kips
(KL / r )2 (34.09)2
emKuNm:Um:g;bEnßmsMrab;m:Um:g; nonsway KW
Cm 0.2565
B1 = = = 0.284 < 1.0
1 − (Pu / Pe1 ) 1 − (454 / 4704 )
211 Fñwm -ssr
25. T.chhay
dUcenHeRbI B1 = 1.0
M u = B1M nt + B2 M lt = 1.0(104.8) + 0 = 104.8 ft − kips
BI beam design charts CamYynwg Lb = 15 ft
φb M n = 343 ft − kips ¬sMrab; Cb = 1.0 ¦
φb M p = 358 ft − kips
rUbTI 6>17 bgðajBIdüaRkamm:Um:g;Bt;sMrab;m:Um:g;énbnÞúkTMnaj. ¬karKNna Cb KWQrelItMél
dac;xat dUcsBaØaenAkñúgdüaRkamenHminmansar³sMxan;eT¦. dUcenH
12.5M max
Cb =
2.5M max + 3M A + 4M B + 3M C
12.5 × (104.8)
= = 2.24
2.5(104.8) + 3(41.3) + 4(74) + 3(56.1)
sMrab; Cb = 2.24
φb M n = 2.24(343) > φb M p = 358 ft − kips
dUcenHeRbI φb M n = 358 ft − kips
kMNt;smIkarGnþrkmμEdlsmRsb
Pu 454
= = 0.7252 > 0.2
φc Pn 626
eRbIsmIkar AISC Equation H1-1a.
Pu 8 ⎛ M ux M uy ⎞
⎟ = 0.7252 + 8 ⎛ 104.8 + 0 ⎞ = 0.985 < 1.0
+ ⎜ + ⎜ ⎟ (OK)
φc Pn 9 ⎜ φb M nx φb M ny
⎝
⎟
⎠ 9 ⎝ 358 ⎠
sMrab;lkçxNÐbnÞúk (A4-4), Pu = 212kips / M nt = 47.6 ft − kips ehIy
M lt = 171.6 ft − kips . sMrab; unbraced condition/
212 Fñwm -ssr
26. T.chhay
⎛M ⎞ ⎛ 40.5 ⎞
C m = 0.6 − 0.4⎜ 1 ⎟ = 0.6 − 0.4⎜
⎜M ⎟ ⎟ = 0.2597
⎝ 2⎠ ⎝ 47.6 ⎠
Pe1 = 4704kips ¬ Pe1 minGaRs½ynwglkçxNÐbnÞúk¦
Cm 0.2597
B1 = = = 0.272 < 1.0
1 − (Pu / Pe1 ) 1 − (212 / 4704 )
dUcenH B1 = 1.0
eyIgminmanTinñy½nRKb;RKan;edIm,IKNnaemKuNm:Um:g;bEnßmeGay)ansuRkitsMrab; sway
moment B2 BI AISC Equation C1-4 b¤ C1-5. RbsinebIeyIgsnμt;fapleFobrvagbnÞúktamGkS½
EdlGnuvtþmkelIGgát; nig Euler load capacity mantMéldUcKñasMrab;RKb;ssrenAkñúgCan; nigsMrab;
ssrEdleyIgBicarNa enaHeyIgGacsresr Equation C1-5³
1 1
B2 = ≈
1 − (∑ Pu / ∑ Pe 2 ) 1 − (Pu / Pe 2 )
sMrab; Pe2 eRbI K x EdlRtUvnwg unbraced condition³
KL K x L 1.2(15)(12 )
= = = 40.91
r rx 5.28
π 2 EAg π 2 (29000)(19.1)
Pe 2 = = = 3266kips
(KL / r )2 (40.91)2
BI AISC Equation C1-5/
1 1
B2 ≈ = = 1.069
1 − (Pu / Pe 2 ) 1 − (212 / 3266)
m:Um:g;bEnßmsrubKW
M u = B1M nt + B2 M lt = 1.0(47.6 ) + 1.069(171.6) = 231.0 ft − kips
eTaHbICam:Um:g; M nt nig M lt mantMélxusKñak¾eday k¾BYkvaRtUv)anEbgEckdUcKña ehIy Cb
nwgenAdEdl . enARKb;GRtaTaMgGs; BYkvamantMélFRKb;RKan;Edl φb M p = 358 ft − kips Ca design
strength edayminKitBIm:Um:g;NamYyeLIy.
Pu 212
= = 0.3387 > 0.2
φc Pn 626
dUcenHeRbI AISC Epuation H1-1a³
Pu 8 ⎛ M ux M uy ⎞
⎟ = 0.3387 + 8 ⎛ 231.0 + 0 ⎞ = 0.912 < 1.0
+ ⎜ + ⎜ ⎟ (OK)
φc Pn 9 ⎜ φb M nx φb M ny
⎝
⎟
⎠ 9 ⎝ 358 ⎠
cMeLIy³ Ggát;enHbMeBjtMrUvkarrbs; AISC Specification.
213 Fñwm -ssr
27. T.chhay
6>8 KNnamuxkat;Fñwm-ssr Design of Beam-Column
edaysarenAkñúgrUbmnþGnþrkmμmanGBaØtiCaeRcIn enaHkarKNnamuxkat;Fñwm-ssrCadMeNIrkar
KNnaEdlRtUvkarCacaM)ac;nUv trial-and-error process. sMrab;kareRCIserIscugeRkay KWeKeRCIserIs
rUbragNakan;EtEk,r kan;Etl¥. muxkat;sakl,gRtUv)aneRCIserIs nigRtUv)anepÞógpÞat;eLIgvijeday
eRbIrUbmnþGnþrkmμ. dMeNIrkard¾manRbsiT§PaBbMputkñúgkareRCIserIsmuxkat;sakl,gRtUv)anbegáIteLIg
CadMbUgsMrab; allowable stress design (Burgett, 1973), ehIyRtUv)anTTYl ykmkeRbIsMrab; LRFD
Edlmanerobrab;enAkñúg part 3 of the Manual, “Column Design”. lkçN³sMxan;sMrab;viFIenHKWCa
karbMElgBIm:Um:g;Bt;eTACabnÞúktamGkS½smmUl. bnÞúkEdl)anBIkarbMElgRtUv)anyk eTAbEnßmelI
bnÞúkCak;Esþg ehIyrUbragEdlRtUvRTbnÞúksrubRtUv)aneRCIserIsBI column load tables. bnÞab;mk eK
RtUvBinitürUbragsakl,genHCamYy Equation H1-1a b¤ H1-1b. bnÞúktamGkS½RbsiT§PaBsrubRtUv)an
eGayeday
Pu eq = Pu + M ux m + M uy mu
Edl Pu = bnÞújktamGkS½emKuNCak;Esþg
M ux = m:Um:g;emKuNeFobGkS½ x
M uy = m:Um:g;emKuNeFobGkS½ y
m = tMélefrEdlmanenAkúñgtarag
n = tMélefrEdlmanenAkúñgtarag
eKalkarN_énkarviFIenHGacRtUv)anRtYtBinitüedaysresrsmIkar ^># eLIgvijdUcxageRkam.
dMbUgKuNGgÁTaMgBIreday φc Pn ³
φ PM φc Pn M uy
Pu + c n ux + ≤ φc Pn
φb M nx φb M ny
b¤ Pu + (M ux × a constant ) + (M uy × a constant ) ≤ φc Pn
GgÁxagsþaMénvismIkarCa design strength rbs;Ggát;EdlBicarNa ehIyGgÁxageqVgGacCa
bnÞúkemKuNxageRkAEdlRtUvTb;Tl;. tYnImYy²énGgÁxageqVgmanxñatkMlaMg dUcenHtMélefrCaGñkbM
Elgm:Um:g;Bt; M ux nig M uy eTACakMub:Usg;bnÞúktamGkS½.
tMélefrmFüm m RtUv)anKNnasMrab;RkumepSgKñarbs; W-shape ehIyRtUv)aneGayenAkñúg
Table 3-2 in Part 3 of the Manual. tMél u RtUv)aneGayenAkñúg column load table sMrab;rUbrag
214 Fñwm -ssr
28. T.chhay
nImYy²EdlmanenAkñúgtarag. edIm,IeRCIserIsrUbragsakl,gsMrab;Ggát;CamYynwgbnÞúktamGkS½ nigm:U
m:g;Bt;eFobGkS½TaMgBIr eKRtUvGnuvtþdUcxageRkam.
!> eRCIserIstMélsakl,g m edayQrelIRbEvgRbsiT§PaB KL . yk u = 2.0
@> KNnabnÞúksgát;tamGkS½RbsiT§PaB³
Pu eq = Pu + M ux m + M uy mu
eRbIbnÞúkenHedIm,IeRCIserIsrUbragBI column load tables.
#> eRbItMél u EdleGayenAkñúg column load tables nigtMélfμIrbs; m BI Table 3-2 edIm,I
KNnatMélfμIrbs; Pu eq . eRCIserIsrUbragepSgeTot.
$> eFVIeLIgvijrhUtdl;tMél Pu eq ElgERbRbYl.
]TahrN_ 6>8³ Ggát;eRKOgbgÁúMxøHenAkñúg braced frame RtUvRTbnÞúksgát;tamGkS½emKuN 150kips nig
m:Um:g;cugemKuN 75 ft − kips eFobnwgGkS½xøaMg ehIy 30 ft − kips eFobnwgGkS½exSay. m:Um:g;TaMgBIr
enHeFVIGMeBIenAelIcugmçag ÉcugmçageTotCaTMr pinned. RbEvgRbsiT§PaBeFobnwgGkS½nImYy²KW 15 ft .
minmanbnÞúkxageFVIGMeBIelIGgát;enHeT, eRbIEdk A36 nigeRCIserIs W-shape EdlRsalCageK.
dMeNaHRsay³ emKuNm:Um:g;bEnßm B1 Gacsnμt;esμInwg 1.0 edIm,IeFVIkareRCIserIsmuxkat;sakl,g.
sMrab;GkS½nImYy²
M ux = B1M ntx ≈ 1.0(75) = 75 ft − kips
M uy = B1M nty ≈ 1.0(30 ) = 30 ft − kips
BI Table 3-2, part 3 of the Manual, m = 1.75 edayeFVI interpolation
eRbItMéledIm u = 2.0
Pu eq = Pu + M ux m + M uy mu = 150 + 75(1.75) + 30(1.75)(2.0 ) = 386kips
cab;epþImCamYynwgrUbragtUcCageKenAkñúg column load tables, sakl,g W 8 × 67 ¬ φc Pn = 412kips /
u = 2.03 ¦³
m = 2 .1
Pu eq = 150 + 75(2.1) + 30(2.1)(2.03) = 435kips
tMélenHFMCag design strength= 412 ft − kips dUcenHeKRtUvsakl,gmuxkat;epSgeTot.
sakl,g W 10 × 60 ¬ φc Pn = 416kips / u = 2.0 ¦³
215 Fñwm -ssr
29. T.chhay
m = 1.85
Pu eq = 150 + 75(1.85) + 30(1.85)(2.00 ) = 400kips < 416kips (OK)
dUcenH W 10 × 60 CarUbragsakl,gEdlGaceRbIkar)an. RtYtBinitü W 12s nig W 14s . sakl,g
W 12 × 58 ¬ φc Pn = 397kips / u = 2.41 ¦³
m = 1.55
Pu eq = 150 + 75(1.55) + 30(1.55)(2.41) = 378kips < 397 kips (OK)
dUcenH W 12 × 58 CarUbragsakl,gEdlGaceRbIkar)an. W 14 EdlRsalCageKsMrab;eFVIkarCamYy
nwgbnÞúkxageRkAKW W 14 × 61 EtvaF¶n;Cag W 12 × 58 . dUcenHeRbI W 12 × 58 CarUbragsakl,g³
Pu 150
= 0.3778 > 0.2
φc Pn 397
dUcenHeRbI AISC Equatiom H-1-1a
KNnam:Um:g;Bt;eFobGkS½ x
K x L 15(12 )
= = 34.09
rx 5.28
π 2 EAg π 2 (29000)(17.0)
Pe1 = = = 4187kips
(KL / r )2 (34.09)2
C m = 0.6 − 0.4(M 1 / M 2 ) = 0.6 − 0.4(0 / M 2 ) = 0.6 ¬sMrab;GkS½TaMgBIr¦
Cm 0 .6
B1 = = = 0.622 < 1.0
1 − (Pu / Pe1 ) 1 − (150 / 4187 )
dUcenHeRbI B1 = 1.0
M ux = B1M ntx = 1.0(75) = 75 ft − kips
bnÞab;mk kMNt; design strength. BI beam designth curves, sMrab; Cb = 1 nig Lb = 15 ft /
φb M n = 220 ft − kips . BIrUbTI 5>15g, Cb = 1.67 . sMrab; Cb = 1.67 design strength KW
Cb × 220 = 1.67(220) = 367 ft − kips
m:Um:g;enHFMCag φb M p = 233 ft − kips
dUcenHeRbI φb M n = 233 ft − kips
KNnam:Um:g;Bt;eFobGkS½ y
K yL 15(12)
= = 71.71
ry 2.51
216 Fñwm -ssr
30. T.chhay
π 2 EAg π 2 (29000)(17.0)
Pe1 = = = 946.2kips
(KL / r )2 (71.71)2
Cm 0 .6
B1 = = = 0.713 < 1.0
1 − (Pu / Pe1 ) 1 − (150 / 946.2 )
dUcenHeRbI B1 = 1.0
M uy = B1M nty = 1.0(30 ) = 30 ft − kips
W 12 × 58CarUbrag compact sMrab;RKb;tMélrbs; Pu dUcenH nomical strength KW M py ≤ 1.5M yy .
Design strength KW
φb M ny = φb M py = φb Z y F y = 0.90(32.5)(36) = 1053in. − kips
= 87.75 ft − kips
b:uEnþ Z y / S y = 32.5 / 21.4 = 1.52 > 1.5 Edlmann½yfa φb M ny KYrEtykesμInwg
φb (1.5M yy ) = φb (1.5F y S y ) = 0.90(1.5)(36 )(21.4) = 1040in. − kips = 86.67 ft − kips
BI AISC Equation H1-1a,
Pu 8 ⎛ M ux M uy ⎞
⎟ = 0.3778 + 8 ⎛ 75 + 30 ⎞
+ ⎜ + ⎜ ⎟
φc Pn 9 ⎜ φb M nx φb M ny
⎝
⎟
⎠ 9 ⎝ 233 86.67 ⎠
= 0.972 < 1.0 (OK)
cMeLIy³ eRbI W 12 × 58 .
ebIeTaHbICaviFIEdleTIbnwgbgðajsMrab;eRCIserIsrUbragsakl,gqab;rkeXIjk¾eday k¾viFIEdl
manlkçN³smBaØCagenHRtUv)anesñIeLIgeday Yura (1988). bnÞúktamGkS½EdlsmmUlEdlRtUv)an
eRbIKW
2M x 7.5M y
Pequiv = P +
d
+
b
¬^>%¦
Edl P = bnÞúktamGkS½emKuN
M x = m:Um:g;emKuNeFobGkS½ x
M y = m:Um:g;emKuNeFobGkS½ y
d = kMBs;Fñwm
b = TTwgFñwm
tYTaMgGs;enAkñúgsmIkar ^>@ RtUvEtmanxñatRtUvKña.
217 Fñwm -ssr
31. T.chhay
]TahrN_ 6>9³ eRbI Yura’s method edIm,IeRCIserIsrUbragsakl,g W 12 sMrab;Fñwm-ssrén]TahrN_
6>8.
dMeNaHRsay³ BIsmIkar 6>5 bnÞúktamGkS½smmUlKW
2M x 7.5M y 2(75 × 12) 7.5(30 × 12)
Pequiv = P + + = 150 + + = 525kips
d b 12 12
EdlTTwg b RtUv)ansnμt;esμInwg 12inches . BI column load tables, sakl,g W 12 × 72
¬ φc Pn = 537kips ¦.
CamYynwg Yura’s method eKTTYl)anrUbragsakl,gFMCag Manual method Etvaminy:agdUcenH
rhUteT.
enAeBlEdltYm:Um:g;Bt;lub ¬]TahrN_ Ggát;manlkçN³CaFñwmCagssr¦ Yura ENnaMfa bnÞúk
tamGkS½RtUvbMElgeTACam:Um:g;Bt;smmUleFobGkS½GkS½ x . bnÞab;mkrUbragsakl,gRtUv)aneRCIserIs
BI beam design charts in part 3 of the Manual. m:Um:g;smmUlKW³
d
M equiv = M x + P
2
karKNnamuxkat;Fñwm-ssrEdlminBRgwg Design of Unbraced Beam-Column
karKNnamuxkat;dMbUgrbs;Fñwm-ssrenAkñúg braced frame RtUv)anbgðajrYcehIy. emKuNm:U
m:g;bEnßm B1 RtUv)ansnμt;esμI 1.0 edIm,IeRCIserIsmuxkat;sakl,g bnÞab;mk B1 RtUv)ankMNt;sMrab;
rUbragenaH. sMrab;Fñwm-ssrRbQmnwg sidesway emKuNm:Um:g;bEnßm B2 EdlQrelIGBaØtiCaeRcIn
EdlminsÁal;rhUtdl;ssrTaMgGs;enAkñúgeRKagRtUv)aneRCIserIs. RbsinebI AISC Equation C1-4
RtUv)aneRbIsMrab; B2 enaHeKminman sidesway deflection Δ oh sMrab;karKNnamuxkat;dMbUgeT. Rb
sinebIeKeRbI AISC Equation C1-5 enaHeKGacminsÁal; ∑ Pe2 . viFIxageRkamRtUv)anesñIeLIgedIm,Irk
B2 .
viFITI1> snμt; B2 = 1.0 . bnÞab;BIeRCiserIsrUbragsakl,g KNna B2 BI AISC Equation C1-5 eday
snμt;fa ∑ Pu / ∑ Pe2 KWdUcKñanwg Pu / Pe2 sMrab;Ggát;EdlBicarNa ¬dUcenAkñúg]TahrN_6>7¦.
viFITI2> eRbIkarkMNt;dMbUg (predetermined limit) sMrab; drift index Δ oh / L EdlCapleeFob story
drift elIkMBs;Can;. kareRbInUv drift index GnuBaØatGtibrmasMrab; serviceability
218 Fñwm -ssr
33. T.chhay
M nt = 1.2(50) + 1.6(94) + 0.8(− 32) = 184.8 ft − kips
M lt = 0.8(20) = 16.0 ft − kips
A4-4: 1.2 D + 0.5 Lr + 1.3W
Pu = 1.2(14) + 0.5(26 ) + 1.3(26) = 19.4kips
M nt = 1.2(50) + 0.5(94) + 1.3(− 32 ) = 65.4 ft − kips
M lt = 1.3(20) = 26 ft − kips
bnSMbnÞúk A4-3 pþl;nUvtMélFMCageK.
sMrab;eKalbMNgénkareRCIserIsrUbragsakl,g snμt;fa B1 = 1.0 . tMélrbs; B2 Gac
RtUv)anKNnaBI AISC Equation C1-4 nig design drift index³
1 1 1
B2 = = = = 1.107
1 − ∑ Pu (Δ oh / ∑ HL ) 1 − (∑ Pu / ∑ H )(Δ oh / L ) 1 − [2(52.0 ) / 2.7](1 / 400)
bnÞúkedkKμanemKuN ∑ H RtUv)aneRbIBIeRBaH drift index KWQrelI drift GtibrmaEdlbNþalmkBI
service load. dUcenH
M u = B1M nt + B2 M lt = 1.0(184.8) + 1.107(16) = 202.5 ft − kips
edayminsÁal;TMhMrbs;Ggát; eKminGaceRbI alignment chart sMrab;emKuNRbEvgRbsiT§PaB)aneT.
Table C-C2.1 enAkñúg Commentary to the Specification bgðajfakrNI (f) RtUvKñay:agxøaMgeTAnwg
lkçxNÐcugsMrab;krNI sidesway én]TahrN_enH ehIyEdl K x = 2.0 .
220 Fñwm -ssr
34. T.chhay
sMrab; braced condition, eKeRbI K x = 1.0 . edaysarEtGgát;TaMgGs;RtUv)anBRgwgTisedA
mYyeTotEdr enaHeKyk K y = 1.0 . bnÞab;mk eKGaceRCIserIsmuxkat;sakl,gEdlmaneGayenAkñúg
Part 3 of the Manual. BI Table 3-2 emKuNm:Um:g;Bt; m = 1.5 sMrab; W 12 CamYynwg KL = 15 ft .
Pu eq = Pu + M ux m + M uy mu = 52.0 + 202.5(1.5) + 0 = 356kips
sMrab; KL = K y L = 15 ft / W 12 × 53 man design strength φc Pn = 451kips . sMrab;GkS½ x
K x L 2.0(15)
= = 14.2 ft < 15 ft
rx / ry 2.11
dUcenH KL = 15 ft lub
sakl,g W 12 × 53 . sMrab; braced condtition
K x L 1.0(15)(15)
= = 34.42
rx 5.23
π 2 EAg π 2 (29000 )(15.6)
Pe1x = = = 3769
(K x L / rx )2 (34.42)2
⎛M ⎞ ⎛ 0 ⎞
C m = 0.6 − 0.4⎜ 1 ⎟ = 0.6 − 0.4⎜
⎜M ⎟ ⎜ M ⎟ = 0 .6
⎟
⎝ 2⎠ ⎝ 2⎠
BI AISC Equation C1-2
Cm 0 .6
B1 = = = 0.608 < 1.0
1 − (Pu / Pe1 ) 1 − (52.0 / 3769 )
dUcenHeRbI B1 = 1.0
cMNaMfa B1 = 1.0 CatMélsnμt;dMbUg ehIyedaysarEt B2 minRtUv)anpøas;bþÚr enaHtMél
M u = 202.5 ft − kips Edl)anKNnaBIdMbUgk¾minRtUv)anpøas;bþÚrEdr. BI beam design chart in Part
4 of the manual CamYynwg Lb = 15 ft design moment sMrab; W 12 × 53 CamYynwg Cb = 1.0 KW
φb M n = 262 ft − kips
sMrab;m:Um:g;Bt;EdlERbRbYlsmamaRtBIsUnüenAcugmçag eTAGtibrmaenAcugmçageTot tMélrbs;
Cb = 1.67 ¬emIlrUbTI 5>15 g¦. dUcenHtMélEdlEktMrUvén design moment KW
φb M n = 1.67(262) = 438 ft − kips
b:uEnþ m:Um:g;enHFMCag plastic moment capasity φb M p = 292 ft − kips / EdleKGacek)anenA
kñúg charts. dUcenH design strength RtUv)ankMNt;Rtwm
φb M n = φb M p = 292 ft − kips
221 Fñwm -ssr
35. T.chhay
kMNt;rUbmnþGnþrkmμEdlsmRsb
Pu 52
= = 0.1153 < 0.2
φc Pn 451
dUcenHeRbI AISC Equation H1-1b:
Pu ⎛ M ux M uy ⎞ 0.1153 ⎛ 202.5 ⎞
+⎜ + ⎟= +⎜ + 0 ⎟ = 0.751 < 1.0 (OK)
2φc Pn ⎜ φb M nx φb M ny
⎝
⎟
⎠ 2 ⎝ 292 ⎠
edaysarlT§plenHtUcCag 1.0 xøaMg dUcenHsakl,grUbragEdltUcCagenHBIrTMhM.
sakl,g W 12 × 45 . sMrab; KL = K y L = 15 ft, φc Pn = 299kips . sMrab;GkS½ x
K x L 2.0(15)
= = 11.3 ft < 15 ft
rx / ry 2.65
dUcenH KL = 15 ft lub
sMrab; braced condtition
K x L 1.0(15)(15)
= = 34.95
rx 5.15
π 2 EAg π 2 (29000 )(13.2)
Pe1x = = = 3093
(K x L / rx )2 (34.95)2
BI AISC Equation C1-2,
Cm 0 .6
B1 = = = 0.610 < 1.0
1 − (Pu / Pe1 ) 1 − (52.0 / 3093)
dUcenHeRbI B1 = 1.0
BI beam design charts CamYynwg Lb = 15 ft m:Um:g;KNnasMrab; W 12 × 45 CamYynwg
Cb = 1.0 KW
φb M n = 201 ft − kips
sMrab; Cb = 1.67
φb M n = 1.67(201) = 336 ft − kips > φb M p = 242.5 ft − kips
dUcenH design strength KW
φb M n = φb M p = 242.5 ft − kips
kMNt;rUbmnþGnþrkmμEdlsmRsb³
Pu 52.0
= = 0.1739 < 0.2
φc Pn 299
222 Fñwm -ssr
36. T.chhay
dUcenHeRbI AISC Equation H1-1b:
Pu ⎛ M ux M uy ⎞ 0.1739 ⎛ 202.5 ⎞
+⎜ + ⎟= +⎜ + 0 ⎟ = 0.922 < 1.0 (OK)
⎜
2φc Pn ⎝ φb M nx φb M ny ⎟ 2 ⎝ 242.5 ⎠
⎠
cMeLIy³ eRbI W 12 × 45 .
enA]TahrN_6>10 karkMNt; drift index CaviFIkñúgkarKNna ehIyeKminmanviFINaedIm,I
KNnaemKuNm:Um:g;bEnßm B2 . RbsinebIeKminR)ab; drift index tMélrbs; B2 GacRtUv)ankMNt;ecj
BI AISC Equation C1-5 dUcxageRkam ¬edayeRbIlkçN³rbs; W 12 × 45 ¦³
K x L 2.0(15)(12)
= = 69.90
rx 5.15
π 2 EAg π 2 (29000)(13.2)
Pe2 x = = = 773.2kips
(K x L / rx )2 (69.90)2
1 1
B2 = = = 1.072
1 − (∑ Pu / ∑ Pe 2 ) 1 − [2(52.0) / 2(773.2)]
6>9> Trusses With Top Chord Loads Between Joints
RbssinebIGgát;rgkarsgát;rbs; truss RtUvRTbnÞúkEdlmanGMeBIenAcenøaHcugsgçagrbs;va enaH
vanwgRtUvrgnUvm:Um:g;Bt; k¾dUcCabnÞúksgát;tamGkS½ dUcenHGgát;enHCa beam-colum. krNIenHGacekIt
manenAelI top chord of the roof truss edayédrEngsßitenAcenøaHtMN. eKk¾RtUvKNna top chord of
an open-web steel joist Ca beam-column Edr BIeRBaH open-web steel joist RtUvRTbnÞúkTMnajEdl
BRgayesμIenAelI top chord rbs;va. edIm,IkarBarbnÞúkenH eKRtUveFVIm:UEdl truss CakarpSMeLIgeday
man continuous chord member nig pin-connected web members. bnÞab;mkeKGacedaHRsayrk
bnÞúktamGkS½ nigm:Um:g;Bt;edayeRbIkarviPaKeRKOgbgÁúMdUcCag stiffness method. eK)anesñIeLIgnUvviFI
saRsþdUcxageRkam³
!> KitGgát;nImYy²rbs; top chord CaFñwmbgáb;cug. eRbIm:Um:g;bgáb;cugCam:Um:g;GtibrmaenAkñúg
Ggát;. Cak;Esþg top chord CaGgát;Cab; CagCaesrIénGgát;tMNsnøak; dUcenHkarcat;TukenH
manlkçN³suRkitCagkarEdlcat;TukGgát;nImYy²CaFñwmsmBaØ.
@> bEnßmkMlaMgRbtikmμBIFñwmbgáb;cugenHeTAbnÞúkenARtg;tMNedIm,ITTYl)anbnÞúkelItMNsrub.
223 Fñwm -ssr
37. T.chhay
#> viPaK truss CamYynwgbnÞúkRtg;tMNTaMgenH. bnÞúktamGkS½EdlCalT§plenAkñúg top chord
member CabnÞúksgát;tamGkS½EdlRtUvykeTAeRbIkñúgkarKNna.
viFIenHRtUv)anbgðajCalkçN³düaRkamenAkñúg rUbTI 6>20. müa:gvijeTot eKGacrkm:Um:g;Bt;
nigRbtikmμrbs;Fñwmedaycat;Tuk top chord CaFñwmCab;EdlmanTMrenARtg;tMNnImYy².
]TahrN_ 6>11³ rUbTI 6>21 bgðajBI parallel-chord roof trussEdl top chord RTédrENgenA
Rtg;tMN nigenARtg;cenøaHtMN. bnÞúkemKuNEdlbBa¢ÚnedayédrENgRtUv)anbgðaj. KNnamuxkat;
top chord. eRbIEdk A36 nigeRCIserIs structural tee Edlkat;ecjBI W-shape.
224 Fñwm -ssr
38. T.chhay
dMeNaHRsay³ m:Um:g;Bt; nigkMlaMgelItMNEdlbNþalmkBIbnÞúkEdlmanGMeBIenAcenøaHtMNRtUv)anrk
edaycat;Tuk top chord nImYy²CaFñwmbgáb;cug. BI Part 4 of the Manual, “Beam and girder
Design,”m:Um:g;bgáb;cugsMrab;Ggát; top chord nImYy²KW
PL 2.4(10)
M = M nt = = = 3.0 ft − kips
8 8
m:Um:g;cug nigkMlaMgRbtikmμTaMgenHRtUv)anbgðajenAkñúg rUbTI 6>22 a. enAeBlEdleKbEnßmkMlaMg
RbtikmμeTAelIbnÞúkelItMN enaHeKTTYl)ankardak;bnÞúkdUcbgðajenAkñúgrUbTI 6>22 b. kMlaMgsgát;
GtibrmatamGkS½nwgekItmanenAkñúgGgát; DE ¬nwgenAkñúgGgát;EdlenAEk,r EdlenAxagsþaMGkS½rbs;
ElVg¦ nigGacRtUv)anrkedayBicarNalMnwgrbs;GgÁesrIrbs;Epñkrbs; truss EdlenAxageqVgmuxkat;
a-a³
∑ M I = (19.2 − 2.4 )(30 ) − 4.8(10 + 20 ) + FDE (4 ) = 0
FDE = −90kips¬rgkarsgát;¦
KNnamuxkat;sMrab;bnÞúktamGkS½ 90kips nigm:Um:g;Bt; 3.0 ft − kips
225 Fñwm -ssr
39. T.chhay
Table 3-2 in Part 3 of the Manual min)anpþl;eGaysMrab; structural tee. eKGaceRbI Yura’s
method (Yura, 1988) Edl)anbegáIteLIgsMrab;Ggát; I- nig H-shape. eKRtUvkarrUbragtUc BIeRBaH
bnÞúktamGkS½tUc ehIym:Um:g;k¾tUcebIeFobnwgbnÞúktamGkS½. RbsinebIeKeRbI tee EdlmankMBs; 6in.
2M x 7.5M y 2(3)(12)
Pequiv = P + + = 90 + + 0 = 102kips
d b 6
BI column load table CamYynwg K x L = 10 ft nig K y L = 5 ft / sakl,g WT 6 ×17.5
¬ φc Pn = 124kips ¦. m:Um:g;Bt;KWeFobnwgGkS½ x ehIyGgát;RtUv)anBRgwgRbqaMgnwg sidesway³
M nt = 3.0 ft − kips M lt = 0/
edaysarmankMlaMgxagmanGMeBIelIGgát; ehIycugRtUv)anTb;enaH Cm = 0.85 ¬Commentary
approach minRtUv)aneRbIenATIenHeT¦. KNna B1 ³
KL K x L 10(12 )
= = = 68.18
r rx 1.76
π 2 EAg π 2 (29000)(5.17 )
Pe1 = = = 318.3kips
(KL / r )2 (68.18)2
Cm 0.85
B1 = = = 1.185
1 − (P / Pe1 ) 1 − (90 / 318.3)
1
m:Um:g;bEnßmKW
M u = B1M nt + B2 M lt = 1.185(3.0) + 0 = 3.555 ft − kips
RbsinebImuxkat;RtUv)ancat;fñak;Ca slender enaH nominal moment strength rbs; structural tee
nwgQrelI local buckling EtRbsinebImindUecñaHeT vanwgQrelI lateral-torsional buckling ¬emIl
AISC Equation F1.2c nig Epñk 5>14 kñúgesovePAenH¦. sMrab;søab
bf 6.560
λ= = = 6.308
2t f 2(0.520)
95 95
λr = = = 15.83 > λ
Fy 36
sMrab;RTnug
d 6.25
λ= = = 20.83
t w 0.300
127 127
λr = = = 21.17 > λ
Fy 26
226 Fñwm -ssr
40. T.chhay
edaysar λ < λr sMrab;TaMgsøab nigRTnug rUbragminEmnCa slender eT ehIy lateral-torsional
buckling lub. BI AISC Equation F1-15/
π EI y GJ ⎛ 2⎞
M n = M cr = ⎜ B + 1+ B ⎟ (AISC Equation F1-15)
Lb ⎝ ⎠
≤ 1 .5 M ysMrab;eCIg b¤tYxøÜnrgkarTaj
≤ 1.0 M y sMrab;eCIg b¤tYxøÜnrgkarsgát;
BI AISC Eqution F1-16,
⎡ 6.25 ⎤ 12.2
B = ±2.3(d / Lb ) I y / J = ±2.3⎢ ⎥ = ±1.378
⎣ 5(12 ) ⎦ 0.369
ehIy nominal strength BI AISC Equation F1-15 KW`
π 29000(12.2 )(11200)(0.369) ⎛ 2⎞
Mn = ⎜ ± 1.378 + 1 + (1.378) ⎟
5(12) ⎝ ⎠
= 2002(± 1.378 + 1.703) = 6168in. − kips b¤ 650.5in. − kips
tMélviC¢manrbs; B RtUvKñanwgkMlaMgTajenAkñúgtYxøÜnrbs; tee ehIysBaØaGviC¢manRtUv)aneRbIedIm,ITTYl
ersIusþg;enAeBltYxøÜnrgkMlaMgsgát;. sMrab;kardak;bnÞúkenAkñúg]TahrN_enH m:Um:g;GtibrmaekItmanenA
TaMgcugbgáb; nigkNþalElVg dUcenHersIusþg;RtUv)anRKb;RKgedaykMlaMgsgát;enAkñúgtYxøÜn ehIy
M n = 650.5in. − kips = 54.12 ft − kips
RbQmnwgtMélGtibrmaén
1.0(36)(3.23)
1.0 M y = 1.0 Fy S x = = 9.690 ft − kips < 54.21 ft − kips
12
dUcenHeRbI M n = 9.690 ft − kips
φb M n = 0.90(9.690) = 8.721 ft − kips
kMNt;rkrUbmnþGnþrkmμEdlRtUveRbI
Pu 90
= = 0.7258 > 0.2
φc Pn 124
dUcenHeRbI AISC Equation H1-1a:
Pu 8 ⎛ M ux M uy ⎞
⎟ = 0.7258 + 8 ⎛ 3.555 + 0 ⎞
+ ⎜ + ⎜ ⎟
φc Pn 9 ⎜ φb M nx φb M ny
⎝
⎟
⎠ 9 ⎝ 8.721 ⎠
= 1.09 > 1.0 (N>G)
227 Fñwm -ssr
![T.chhay
tMNrwg. karbBa¢Únm:Um:g;enHk¾ekIteLIgenARtg; C nig D ehIyvaBitsMrab;RKb; rigid frame eTaHbI
m:Um:g;TaMgenHtUcCagm:Um:g;Edl)anBIbnÞúkxagk¾eday. ssrCaeRcInenAkñúg rigid frames Ca beam-
columns ehIyT§iBlrbs;m:Um:g;Bt;minRtUv)anecal. b:uEnþ ssrrbs;GaKarmYyCan;EdlenAdac;BIeK
GacRtUv)anKitCaGgát;rgkMlaMgsgát;cMGkS½.
eBlxøH]TahrN_epSgeTotrbs; beam-columns GacCYbenAkñúg roof trusses. eTaHbICaFmμta
top chord RtUv)anKitCaGgát;rgkMlaMgsgát;tamGkS½k¾eday RbsinebI purlins RtUv)andak;enAcenøaH
tMN kMlaMgRbtikmμrbs;vanwgbegáItCa bending Edldac;xatRtUv)anKitkñúgkarKNna. krNIenHnwg
RtUv)anerobrab;enAkñúgCMBUkenH.
6>2> smIkarGnþrkmμ Interaction Formulas
vismPaBrbs;smIkar @># GacRtUv)ansresrkñúgTMrg;xageRkam³
∑ γ i Qi ≤ 1.0 ¬^>!¦
φRn
b¤ ∑resistance ≤ 1.0
load effects
RbsinebIman resistance eRcInRbePTBak;Bn§½ smIkar ^>! GacRtUv)ansresrkñúgTMrg;eKalrbs;
interaction formulas. dUcEdl)anerobrab;enAkñúgCMBUk 5 Rtg;Epñkm:Um:g;Bt;BIrTis plbUkénpleFob
load-to-resistance RtUv)ankMNt;RtwmmYyÉktþa. ]TahrN_ RbsinebIeKGnuvtþTaMgm:Um:g;Bt; nigkMlaMg
tamGkS½ interaction formulas GacsresrCa
Pu
+
Mu
φc Pn φb M n
≤ 1 .0 ¬^>@¦
Edl Pu = bnÞúksgát;tamGkS½emKuN
φc Pn = compressive design strength
Mu = m:Um:g;Bt;emKuN
φb M n = design moment
sMrab;m:Um:g;Bt;BIrTis vanwgmanpleFobm:Um:g;Bt;Bt;BIr
⎛ M ux M uy ⎞
Pu
+⎜ +
φc Pn ⎜ φb M nx φb M ny
⎟ ≤ 1.0
⎟
¬^>#¦
⎝ ⎠
Edl x nig y sMedAelIkarBt;eFobGkS½ x nigGkS½ y .
189 Fñwm -ssr](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)