Mr=required flexural strength using LRFD or ASD load combinations, kip-in.
(N-mm)
Mc=available flexural strength, kip-in. (N-mm) x =subscript relating symbol to strong axisbending y =subscript relating symbol to weak axisbending For design according to Section B3.3 (LRFD):
Pr =required axial strength using LRFD load combinations, kips (N)
Pc = φcPn=design axial strength, determined in accordance with Chapter E, kips (N)
Mr =required flexural strength using LRFD load coMbinations, kip-in. (N-mm) Mc= φbMn=design flexural strengthdetermined in accordance with Chapter F,
kip-in. (N-mm)
φc =resistance factorfor compression =0.90 φb =resistance factor for flexure =0.90 For design according to Section B3.4 (ASD):
Pr =required axial strength using ASD load combinations, kips (N)
Pc =Pn/Ωc=allowable axial strength, determined in accordance with Chapter E, kips (N)
Mr =required flexural strength using ASD load combinations, kip-in. (N-mm) Mc=Mn/Ωb = allowable flexural strength determined in accordance with
Chapter F, kip-in. (N-mm)
Ωc =safety factorfor compression =1.67 Ωb=safety factor for flexure =1.67
2. Doubly and Singly Symmetric Members Subject to Flexure and Tension The interaction of flexure and tension in doubly symmetric members and singly sym- metric members constrained to bend about a geometric axis (x and/or y) shall be limited by Equations H1-1a and H1-1b
where
For design according to Section B3.3 (LRFD):
Pr =required axial strength using LRFD load combinations, kips (N)
Pc = φtPn=design axial strength, determined in accordance with Section D2, kips (N)
Mr =required flexural strength using LRFD load combinations, kip-in. (N-mm) Mc= φbMn=design flexural strength determined in accordance with Chapter F,
kip-in. (N-mm)
φt =resistance factorfor tension (see Section D2) φb =resistance factor for flexure =0.90
For design according to Section B3.4 (ASD):
Pr =required axial strength using ASD load combinations, kips (N)
Pc =Pn/Ωt= allowable axial strength, determined in accordance with Section D2, kips (N)
Mr =required flexural strength using ASD load combinations, kip-in. (N-mm) Mc=Mn/Ωb=allowable flexural strengthdetermined in accordance with
Chapter F, kip-in. (N-mm)
Ωt =safety factorfor tension (see Section D2) Ωb=safety factor for flexure =1.67
For doubly symmetric members, Cbin Chapter F may be multiplied by for axial tension that acts concurrently with flexure
where
and
α =1.0 (LRFD); α =1.6 (ASD)
A more detailed analysis of the interaction of flexure and tension is permitted in lieu of Equations H1-1a and H1-1b.
3. Doubly Symmetric Rolled Compact Members Subject to Single Axis Flexure and Compression
For doubly symmetric rolled compact members with 共KL兲z≤共KL兲ysubjected to flex- ure and compression with moments primarily about their major axis, it is permissible to consider the two independent limit states, in-plane instability and out-of-plane buckling or lateral-torsional buckling, separately in lieu of the combined approach provided in Section H1.1.
For members with Mry兾Mcy≥0.05, the provisions of Section H1.1 shall be followed.
(a) For the limit state of in-plane instability, Equations H1-1 shall be used with Pc, Mrxand Mcxdetermined in the plane of bending.
(b) For the limit state of out-of-plane buckling and lateral-torsional buckling:
(H1-2) where
Pcy =available compressive strengthout of the plane of bending, kips (N) Cb =lateral-torsional buckling modification factor determined from Section F1 Mcx=available lateral-torsional strength for strong axis flexure determined in
accordance with Chapter F using Cb= 1.0, kip-in. (N-mm)
User Note: In Equation H1-2, CbMcx may be larger than φbMpx in LRFD or Mpx/Ωb in ASD. The yielding resistance of the beam-column is captured by Equations H1-1.
1+αP P
r ey
P EI
ey L y
b
= π2
2
P P
P P
M C M
r cy
r cy
rx b cx
1 5 0 5 1 0
2
. − . .
⎛
⎝⎜
⎞
⎠⎟+ ⎛⎝⎜ ⎞
⎠⎟ ≤
H2. UNSYMMETRIC AND OTHER MEMBERS SUBJECT TO FLEXURE AND AXIAL FORCE
This section addresses the interaction of flexure and axial stressfor shapes not cov- ered in Section H1. It is permitted to use the provisions of this Section for any shape in lieu of the provisions of Section H1.
(H2-1) where
fra =required axial stress at the point of consideration using LRFD or ASD load combinations, ksi (MPa)
Fca =available axial stressat the point of consideration, ksi (MPa)
frbw, frbz =required flexural stress at the point of consideration using LRFD or ASD load combinations, ksi (MPa)
Fcbw ,Fcbz=available flexural stressat the point of consideration, ksi (MPa) w =subscript relating symbol to major principal axis bending z =subscript relating symbol to minor principal axis bending For design according to Section B3.3 (LRFD):
fra =required axial stress at the point of consideration using LRFD load combinations, ksi (MPa)
Fca = φcFcr=design axial stress, determined in accordance with Chapter E for compression or Section D2 for tension, ksi (MPa)
frbw, frbz =required flexural stress at the point of consideration using LRFD or ASD load combinations, ksi (MPa)
Fcbw, Fcbz= =design flexural stressdetermined in accordance with Chapter F, ksi (MPa). Use the section modulus for the specific location in the cross section and consider the sign of the stress.
φc =resistance factorfor compression =0.90 φt =resistance factor for tension (Section D2) φb =resistance factor for flexure =0.90 For design according to Section B3.4 (ASD):
fra =required axial stress at the point of consideration using ASD load combinations, ksi (MPa)
Fca = =allowable axial stressdetermined in accordance with Chapter E for compression, or Section D2 for tension, ksi (MPa)
frbw, frbz =required flexural stress at the point of consideration using LRFD or ASD load combinations, ksi (MPa)
Fcbw, Fcbz= =allowable flexural stressdetermined in accordance with Chapter F, ksi (MPa). Use the section modulus for the specific location in the cross section and consider the sign of the stress.
Ωc =safety factorfor compression =1.67 f
F f F
f F
ra ca
rbw cbw
rbz cbz
+ + ≤1 0.
φbMn
S
M S
n
Ωb
Fcr Ωc
Ωt =safety factor for tension (see Section D2) Ωb =safety factor for flexure =1.67
Equation H2-1 shall be evaluated using the principal bending axes by considering the sense of the flexural stresses at the critical points of the cross section. The flexural terms are either added to or subtracted from the axial term as appropriate. When the axial force is compression, second order effectsshall be included according to the provisions of Chapter C.
A more detailed analysis of the interaction of flexure and tension is permitted in lieu of Equation H2-1.
H3. MEMBERS SUBJECT TO TORSION AND COMBINED TORSION, FLEXURE, SHEAR AND/OR AXIAL FORCE
1. Round and Rectangular HSS Subject to Torsion
The design torsional strength, φTTn, and the allowable torsional strength, Tn/ΩT, for round and rectangular HSSaccording to the limit statesof torsional yieldingand tor- sional bucklingshall be determined as follows:
φT=0.90 (LRFD) ΩT=1.67 (ASD)
Tn= FcrC (H3-1)
where
C is the HSS torsional constant
The critical stress, Fcr, shall be determined as follows:
(a) For round HSS, Fcrshall be the larger of
(i) (H3-2a)
and
(ii) (H3-2b)
but shall not exceed 0.6Fy, where
L =length of the member, in. (mm) D=outside diameter, in. (mm) (b) For rectangular HSS
(i) When
F E
L D
D t
cr =
⎛⎝⎜ ⎞
⎠⎟
1 23
5 4
.
F E
D t
cr =
⎛⎝⎜ ⎞
⎠⎟
0 60
3 2
.
h t/ ≤2 45. E F/ y
Fcr=0.6Fy (H3-3) (ii) When
(H3-4)
(iii) When
(H3-5)
where
h=flat widthof longer side as defined in Section B4.1b(d), in. (mm) t =design wall thicknessdefined in Section B4.2, in. (mm)
User Note:The torsional constant, C, may be conservatively taken as:
For round HSS:
For rectangular HSS: C =2共B⫺t兲共H⫺t兲t⫺4.5共4⫺π兲t3
2. HSS Subject to Combined Torsion, Shear, Flexure and Axial Force
When the required torsional strength, Tr, is less than or equal to 20% of the avail- able torsional strength, Tc, the interaction of torsion, shear, flexure and/or axial force for HSS shall be determined by Section H1 and the torsional effects shall be neg- lected. When Trexceeds 20% of Tc, the interaction of torsion, shear, flexure and/or axial force shall be limited, at the point of consideration, by
(H3-6) where
For design according to Section B3.3 (LRFD):
Pr =required axial strengthusing LRFD load combinations, kips (N)
Pc = φPn=design tensile or compressive strengthin accordance with Chapter D or E, kips (N)
Mr =required flexural strengthusing LRFD load combinations, kip-in. (N-mm) Mc= φbMn = design flexural strength in accordance with Chapter F, kip-in.
(N-mm)
Vr =required shear strengthusing LRFD load combinations, kips (N) 2 45. E / 3 07.
F h t E
y < ≤ Fy
F F E F
h t
cr
y y
=
( )
⎛⎝⎜ ⎞
⎠⎟
0 6. 2 45. /
3 07. E / 260
F h t
y < ≤
C D t t
= π
(
−)
22
P P
M M
V V
T T
r c
r c
r c
r c
⎛ +
⎝⎜ ⎞
⎠⎟+⎛ +
⎝⎜ ⎞
⎠⎟ ≤
2
1 0.
F E
h t
cr =
⎛⎝⎜ ⎞
⎠⎟
0 458 2
2
. π
Vc = φvVn=design shear strengthin accordance with Chapter G, kips (N) Tr =required torsional strength using LRFD load combinations, kip-in.
(N-mm)
Tc = φTTn=design torsional strengthin accordance with Section H3.1, kip-in.
(N-mm)
For design according to Section B3.4 (ASD):
Pr =required axial strength using ASD load combinations, kips (N)
Pc =Pn/Ω = allowable tensile or compressive strength in accordance with Chapter D or E, kips (N)
Mr=required flexural strength using ASD load combinations, kip-in. (N-mm) Mc=Mn/Ωb=allowable flexural strengthin accordance with Chapter F, kip-in.
(N-mm)
Vr =required shear strength using ASD load combinations, kips (N)
Vc =Vn/Ωv=allowable shear strengthin accordance with Chapter G, kips (N) Tr =required torsional strength using ASD load combinations, kip-in. (N-mm) Tc =Tn/ΩT = allowable torsional strength in accordance with Section H3.1,
kip-in. (N-mm)
3. Non-HSS Members Subject to Torsion and Combined Stress
The available torsional strength for non-HSS members shall be the lowest value obtained according to the limit statesof yielding under normal stress, shear yielding under shear stress, or buckling, determined as follows:
φT =0.90 (LRFD) ΩT =1.67 (ASD) (a) For the limit state of yielding under normal stress
Fn=Fy (H3-7)
(b) For the limit state of shear yielding under shear stress
Fn=0.6Fy (H3-8)
(c) For the limit state of buckling
Fn=Fcr (H3-9)
where
Fcr=buckling stress for the section as determined by analysis, ksi (MPa) Some constrained local yieldingis permitted adjacent to areas that remain elastic.
H4. RUPTURE OF FLANGES WITH HOLES SUBJECT TO TENSION At locations of bolt holes in flanges subject to tension under combined axial force and major axis flexure, flange tensile rupture strengthshall be limited by Equation H4-1. Each flange subject to tension due to axial force and flexure shall be checked separately.
(H4-1) P
P M M
r c
rx cx
+ ≤1 0.
where
Pr =required axial strength of the member at the location of the bolt holes, pos- itive in tension, negative in compression, kips (N)
Pc =available axial strength for the limit state of tensile rupture of the net sec- tion at the location of bolt holes, kips (N)
Mrx=required flexural strength at the location of the bolt holes; positive for tension in the flange under consideration, negative for compression, kip-in.
(N-mm)
Mcx=available flexural strength about x-axis for the limit state of tensile rupture of the flange, determined according to Section F13.1. When the limit state of tensile rupture in flexure does not apply, use the plastic bending moment, Mp, determined with bolt holes not taken into consideration, kip-in. (N-mm) For design according to Section B3.3 (LRFD):
Pr =required axial strength using LRFD load combinations, kips (N)
Pc = φtPn= design axial strength for the limit state of tensile rupture, deter- mined in accordance with Section D2(b), kips (N)
Mrx=required flexural strength using LRFD load combinations, kip-in. (N- mm)
Mcx= φbMn =design flexural strength determined in accordance with Section F13.1 or the plastic bending moment, Mp, determined with bolt holes not taken into consideration, as applicable, kip-in. (N-mm)
φt =resistance factorfor tensile rupture =0.75 φb =resistance factor for flexure =0.90
For design according to Section B3.4 (ASD):
Pr =required axial strength using ASD load combinations, kips (N)
Pc = =allowable axial strength for the limit state of tensile rupture, deter- mined in accordance with Section D2(b), kips (N)
Mrx=required flexural strength using ASD load combinations, kip-in. (N-mm) Mcx= =allowable flexural strength determined in accordance with Section
F13.1, or the plastic bending moment, Mp, determined with bolt holes not taken into consideration, as applicable, kip-in. (N-mm)
Ωt =safety factorfor tensile rupture =2.00 Ωb =safety factor for flexure =1.67
Pn Ωt
Mn Ωb