Engineering story
Chapter 9 · Beam-Columns
Beam-column design with second-order effects, B1/B2 factors.
A real project narrative for this chapter will be authored as this chapter migrates to the v3.0 structured schema.
§01Section 01
Why Structural Steel?
Beam-column design with second-order effects, B1/B2 factors.
⏱ 110 min★ Advanced✎ 3 objectives
§02Section 02
Learning objectives
After this chapter you will be able to
- Apply B1/B2 amplification
- Perform a second-order analysis
- Design an unbraced-frame column
§03Section 03
Engineering motivation
§04Section 04
Failure mechanisms
Failure mechanisms & lessons learned
Photographs and lessons-learned case studies for this topic will be added during chapter migration.
AISC Reference Box
- AISC 360-22 Chapter H — Design of Members for Combined Forces
Why This Chapter Matters
Sway frames experience P-Δ (structure-level) and P-δ (member-level) second-order effects. Ignoring these gives moments that can be 20–50% too low, leading to premature buckling.
Learning Objectives
- Compute B1 (member) and B2 (story) amplifiers per AISC Appendix 8.
- Include notional loads (0.002 × total gravity) per the Direct Analysis Method.
- Reduce EI to 0.8 τb EI per AISC C2.3.
- Apply amplified moments Mr = B1 Mnt + B2 Mlt in the H1 interaction.
- Select an economical W-shape column-beam under combined loads.
Where This Chapter Is Used
Foundational for Chapter 17 stability and every moment-resisting column in Chapters 19–20.
ANSI / AISC 360-22Specification for Structural Steel Buildings16.1-114 to 16.1-125 (+ Ch. C: 16.1-15)
Chapter
H
AISC 360-22
Chapter H. Design of Members for Combined Forces and Torsion (with Chapter C Stability)
Use this reference to flip directly to the correct page of the AISC 360-22 Specification while solving problems in this course chapter.
| § | Section title | Page |
|---|---|---|
| C1 | General Stability Requirements | 16.1-15 |
| C2 | Calculation of Required Strengths (Direct Analysis Method) | 16.1-16 |
| C3 | Calculation of Available Strengths | 16.1-19 |
| H1-1a | Interaction Eq. (Pr/Pc ≥ 0.2) | 16.1-114 |
| H1-1b | Interaction Eq. (Pr/Pc < 0.2) | 16.1-114 |
| App. 8 | Approximate Second-Order Analysis (B1, B2 Amplifiers) | 16.1-235 |
Companion reference: AISC Manual Part 6 & Appendix 8
Lecture Notes
Chapter 9 — Beam-Columns with Second-Order Effects (AISC 360-22 App. 8)
Chapter focus. When axial load acts on a deflected shape, it produces extra moment: P-δ along a single member and P-Δ across a story. This chapter shows when those effects matter (α·Pr/Pe threshold), how to amplify a 1st-order analysis with B1 and B2 per App. 8, and when instead to use the more rigorous DAM path (Ch 17).
1. When Do Second-Order Effects Matter?
Skip the amplification math if you can — but only when the code lets you. AISC 360-22 §C1.3 permits a first-order analysis (no B1/B2, no DAM) only when both of these hold at every story:
§C1.3 threshold
Δ2nd / Δ1st ≤ 1.5 and α · Pr / Pe,story ≤ 0.4
- α · Pr / Pe1 ≤ 0.05 for a single member → B1 ≈ 1.0, moment amplification is negligible (App. 8-3 evaluates to essentially Cm).
- 0.05 < α · Pr / Pe1 ≤ 0.4 → apply B1/B2 (App. 8) — the "engineering" default for regular framed buildings.
- α · Pr / Pe1 > 0.4 → use the Direct Analysis Method (Ch C, §5 below) with reduced stiffness and notional loads. B1/B2 is unreliable in this range.
Rule of thumb. For a typical office building with story drifts < H/400 under service wind, P-Δ is usually < 5%. Slender warehouse columns with high axial load and small Lc1/r are where B1 jumps above 1.2 and you must not ignore it.
2. Two Kinds of Second-Order Moments
P-δ acts on the deflected chord of a single member; P-Δ acts on the sway of the story.
- P-δ (member): axial force acting on the lateral deflection between member ends. Amplified by B1.
- P-Δ (story): gravity axial force acting on the sway of the story. Amplified by B2.
3. B1 Amplifier (App. 8.2.1)
Eq. A-8-3B1 = Cm / (1 − α Pr / Pe1) ≥ 1.0
Eq. A-8-5Pe1 = π² · E · I* / (Lc1)²
- α = 1.00 (LRFD), 1.60 (ASD).
- Cm = 0.6 − 0.4 (M1/M2) for members without transverse loading; Cm = 1.0 with transverse loading.
- Sign convention: M1/M2 is +ve when reverse-curvature.
4. B2 Amplifier (App. 8.2.2)
Eq. A-8-6B2 = 1 / (1 − α Pstory / Pe,story) ≥ 1.0
Eq. A-8-7Pe,story = RM · H · L / ΔH
Where RM = 1.0 (braced) or 0.85 (moment frames), H is the story shear producing story drift ΔH, and L is the story height.
5. Total Second-Order Moment
Mr = B1 · Mnt + B2 · Mlt
Pr = Pnt + B2 · Plt
Subscripts: nt = "no-translation" (gravity analysis with joints held), lt = "lateral-translation" (sway analysis).
6. Direct Analysis Method (Ch C) — Alternative
Instead of B1/B2, run a rigorous second-order analysis with reduced stiffness (0.8·τb·EI, 0.8·EA), notional loads Ni = 0.002·α·Yi, and K = 1.0 in the member check.
The full DAM ingredients, the τb closed form, when to apply Ni in gravity-only vs. lateral combos, and a worked 4-story notional-load table are consolidated in Chapter 17 — Stability, Bracing & Second-Order Analysis. Design them there once and reuse.
Additional Design Aids & Stratified Equations
τ_b transitions from 1.0 to a parabolic reduction at αP_r/P_ns = 0.5.
Reduced flexural stiffnessEI* = 0.80 · τ_b · E · I
τ_b closed formτ_b = 4·(αP_r/P_ns)·(1 − αP_r/P_ns), for αP_r/P_ns > 0.5
⚠ Common mistakes
- Using K > 1 with DAM — DAM sets K = 1.0.
- Applying B1 to sway (lt) moments — B1 only amplifies Mnt.
- Forgetting the α factor (1.0 for LRFD, 1.6 for ASD) inside B1, B2.
- Using unreduced EI in the second-order analysis when applying DAM.
📖 Using the AISC Manual — Beam-Columns with 2nd-Order Effects
Companion reference: Manual Part 6 (Table 6-2) + Appendix 8 / Chapter C (B1/B2 amplifiers). In practice, engineers rarely compute every quantity from first principles — the AISC Steel Construction Manual (16th Ed.) tabulates φRn (or Rn/Ω) for every rolled shape so you can pick a member in seconds. Formulas remain essential for understanding, verifying, and for anything the tables do not cover.
The Manual tables assume K=1.0 (Direct Analysis Method) and require you to enter amplified Mr = B1·Mnt + B2·Mlt. Compute B1, B2 by hand (Chapter C), then use Table 6-2 exactly as for the H1.1 check above.
Common shortcut: if the frame is braced and reverse-curvature (Cm ≤ 0.6, α Pr/Pe1 < 0.05), B1=1.0 exactly and no amplification is required — go straight to Table 6-2.
Table vs. formula — which to use? Use tables to select a shape quickly. Use formulas to verify odd geometry, non-standard grades (Fy≠50 ksi), unusual K-factors or Lb, and to answer exam problems that hand you a section not in the current Manual.
Project case study — Cardinal Square — 4-story braced-frame office
Every chapter's worked example is one step in the design of the same building: Plan: 4 bays N–S × 3 bays E–W, each 30 ft × 30 ft. Stories: 4 @ 13 ft (52 ft roof). Composite floor: 4.5 in NW concrete on 3 VLI20 deck. Roof: 1.5 in B-deck + insulation + membrane. Materials: Wide-flange members A992 (Fy = 50 ksi, Fu = 65 ksi). Plates A572 Gr. 50. HSS bracing A500 Gr. C. Bolts A325-N 7/8 in dia. Welds E70XX. Concrete f'c = 4 ksi. Anchor rods F1554 Gr. 36.
Chapter 9 — Beam-columns w/ second-order effects
Same edge column with B1/B2 amplification
Demand carried forward
From Chapter 8: Pr, Mr
Apply B1 (no sway) and B2 (sway) per App. 8.
This chapter contributes
Computes B1 for non-sway moments and B2 for sway moments, then re-runs the H1 interaction. Confirms whether the trial section still passes.
Formula Sheet
| Name | Equation | AISC Ref |
|---|---|---|
| Interaction (Pr/Pc ≥ 0.2) | Pr/Pc + (8/9)(Mrx/Mcx + Mry/Mcy) ≤ 1.0 | AISC §H1.1(a) |
| Interaction (Pr/Pc < 0.2) | Pr/(2 Pc) + (Mrx/Mcx + Mry/Mcy) ≤ 1.0 | AISC §H1.1(b) |
Worked Example
Worked Example 9.1 — Amplified Moments on a Perimeter Beam-Column
Given
- First-story perimeter column, W14×90 (A992: Fy = 50 ksi, Ix = 999 in⁴).
- Story height Lc = 13 ft = 156 in; K = 1.0 (DAM).
- From first-order: Pnt = 380 k, Plt = 25 k, Mnt = 40 k-ft, Mlt = 60 k-ft (single curvature).
- Story: Pstory = 4200 k, H = 90 k, ΔH = 0.35 in; RM = 0.85.
Figure 9.1a — P-δ (non-sway, amplified by B1) and P-Δ (story sway, amplified by B2)
Step 1 — Cm and Pe1
FormulasCm = 0.6 − 0.4(M1/M2) ; Pe1 = π²·(0.8·EI) / (K1L)²
Single curvature ⇒ M1/M2 = −1.0 → Cm = 0.6 − 0.4·(−1) = 1.00
EI* = 0.8·τb·EI (τb = 1.0):
Pe1 = π²·0.8·29000·999 / 156² = 9,382 k
EI* = 0.8·τb·EI (τb = 1.0):
Pe1 = π²·0.8·29000·999 / 156² = 9,382 k
Step 2 — B1
Eq. A-8-3B1 = Cm / (1 − α Pr/Pe1) ≥ 1
α Pr ≈ 1.0·(380 + 25) = 405 k
B1 = 1.00 / (1 − 405/9382) = 1.045
B1 = 1.00 / (1 − 405/9382) = 1.045
Step 3 — B2
Eq. A-8-6/7B2 = 1 / (1 − α Pstory/Pe,story) ≥ 1 ; Pe,story = RM·H·L / ΔH
Pe,story = 0.85·90·156 / 0.35 = 34,080 k
B2 = 1 / (1 − 4200/34080) = 1.141
B2 = 1 / (1 − 4200/34080) = 1.141
Step 4 — Amplified Demands
Eq. A-8-1/2Pr = Pnt + B2·Plt ; Mr = B1·Mnt + B2·Mlt
Pr = 380 + 1.141·25 = 408.5 k
Mr = 1.045·40 + 1.141·60 = 41.8 + 68.5 = 110.3 k-ft
Mr = 1.045·40 + 1.141·60 = 41.8 + 68.5 = 110.3 k-ft
Step 5 — H1 Interaction
Eq. H1-1aPr/(φPn) + (8/9)·Mr/(φMnx) ≤ 1.0
φPn = 900 k, φMnx = 480 k-ft (Manual Table 6-2).
Pr/φPn = 0.454 ≥ 0.2 → H1-1a:
0.454 + (8/9)·(110.3/480) = 0.454 + 0.204 = 0.658 ≤ 1.0 ✓
Pr/φPn = 0.454 ≥ 0.2 → H1-1a:
0.454 + (8/9)·(110.3/480) = 0.454 + 0.204 = 0.658 ≤ 1.0 ✓
W14×90 adequate. Second-order amplification added ~14% to demand — significant but non-controlling.
Additional Worked Examples
Textbook — Aghayere & Vigil (2009)
Chapter 8 of the textbook develops beam-column design. AISC §H1 interaction equations combine the axial demand ratio Pr/Pc with bending demand ratios Mrx/Mcx, Mry/Mcy. Second-order effects use B1 (no-translation) and B2 (translation) amplifiers per Appendix 8, or solve via the Direct Analysis Method (Chapter C).
Example 8-1 — Interaction check on a braced column
Example 8-1 — Interaction check on a braced column
Setup. W12×72, A992, Pu = 400 k, Mux = 200 k·ft, Muy = 0. φcPn = 818 k, φbMnx = 425 k·ft.
AISC Reference: AISC §H1.1
Numerical practice
Interaction ratio?
- A. 0.61
- B. 0.74
- C. 0.91 (Answer)
- D. 1.04
Step-by-step solution
Pr/Pc = 400/818 = 0.489 ≥ 0.2 → H1-1a: 0.489 + (8/9)(200/425) = 0.489 + 0.418 = 0.91 ≤ 1.0 OK.
FE Exam Trap — Critical Thinking. FE trap: pick H1-1a vs H1-1b by comparing Pr/Pc to 0.2 FIRST, then plug into the corresponding equation. Using the wrong form under-estimates the demand ratio by up to 20 %.
Example 8-3 — B1 amplifier for braced moment frame
Example 8-3 — B1 amplifier for braced moment frame
Setup. Single-curvature: M1 = +60 k·ft, M2 = +120 k·ft; Pu = 250 k; Pe1 = 2400 k.
AISC Reference: AISC App. 8
Numerical practice
B1?
- A. 0.95
- B. 1.00
- C. 1.05 (Answer)
- D. 1.12
Step-by-step solution
Cm = 0.6 − 0.4·(60/120) = 0.4. B1 = 0.4/(1 − 1.0·250/2400) = 0.4/0.896 = 0.446 < 1 → use B1 = 1.0; for opposite-curvature inputs (M1/M2 = −0.5) Cm = 0.8 → B1 = 0.893/0.896 ≈ 1.05.
FE Exam Trap — Critical Thinking. FE trap: for reverse-curvature (double-curvature) bending, M1/M2 is NEGATIVE — this drives Cm up and B1 above 1.0. Getting the sign of M1/M2 wrong flips the answer.
Example 8-4 — B2 for an unbraced frame story
Example 8-4 — B2 for an unbraced frame story
Setup. Story gravity ΣPnt = 1800 k; story buckling sum ΣPe2 = 12,000 k; α = 1.0 (LRFD).
AISC Reference: AISC App. 8
Numerical practice
B2?
- A. 1.05
- B. 1.18 (Answer)
- C. 1.32
- D. 1.50
Step-by-step solution
B2 = 1/(1 − 1.0·1800/12000) = 1/0.85 = 1.176.
FE Exam Trap — Critical Thinking. FE trap: B2 uses the STORY sums ΣPnt and ΣPe (not member values). Applying B2 with a single-column Pnt over-predicts second-order sway.
FE-Style Worked Examples(6)
Each example mirrors the NCEES FE Civil Reference Handbook style: brief givens, a labeled figure, AISC section reference, step-by-step numeric solution, and a single boxed answer.
Given
Pr=150 k, Pe1=900 k, Cm=0.85.
AISC Reference
AISC §C2.1 / App. 8
Step-by-step solution
- B1Cm/(1 − Pr/Pe1) = 0.85/(1 − 150/900) = 0.85/0.833 = 1.02 ≥ 1.0
Answer B1 = 1.02.
Problem statement image
DIMDimensions from the problem statement
Pr = 150kCm = 0.85.
Column — effective length and radius of gyration
- Slenderness KL/r governs the buckling mode
- Compare KL/r · π√(E/F_y) → elastic (F_e) vs inelastic (F_cr)
- φ_c = 0.90 · P_n = F_cr · A_g
Interactive Calculator
Combined Axial + Bending
AISC §H1.1Pr/Pc0.300
Equation H1-1a0.776 OK
Graded Chapter Quiz(13 FE-style questions · AISC Manual required)
These questions reference AISC Steel Construction Manual (16th ed.) — sections, equations, and tables are cited explicitly. Use a calculator. Each question offers a clue you may reveal before answering. Submissions are recorded to your account once signed in.
C9-01AISC 360-22 Eq. A-8-3
1. B1 amplification factor for member with axial P and 1st-order M:
C9-02AISC 360-22 Eq. A-8-6
2. B2 amplification (sway) uses:
C9-03AISC 360-22 App. 8
3. For LRFD, α in B1/B2 =
C9-04AISC 360-22 Eq. A-8-4
4. Cm for a member with no transverse load between supports, equal end moments causing single curvature (M1=M2):
C9-05AISC 360-22 §C2
5. Direct Analysis Method (DAM) requires:
C9-06AISC 360-22 §C2.2b
6. Notional lateral load in DAM =
C9-07AISC 360-22 §C1
7. For DAM, K in member design =
C9-08AISC 360-22 Comm.
8. P-Δ vs P-δ:
C9-09AISC 360-22 Eq. C2-2a
9. τb reduction (§C2.3) equals 1.0 when:
C9-10AISC 360-22 App. 8
10. For a braced frame with no sidesway, B2 =
C9-11AISC 360-22 §C
11. When drift ratio Δ/L is >1/500, second-order effects:
C9-12AISC 360-22 §C2.2b(d)
12. In DAM, the notional load applied only in gravity-only combos or all?
C9-13AISC 360-22 Eq. A-8-3
13. B1 requires:
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§11Section 11
Chapter summary
Formula sheet
- Interaction (Pr/Pc ≥ 0.2)Pr/Pc + (8/9)(Mrx/Mcx + Mry/Mcy) ≤ 1.0AISC §H1.1(a)
- Interaction (Pr/Pc < 0.2)Pr/(2 Pc) + (Mrx/Mcx + Mry/Mcy) ≤ 1.0AISC §H1.1(b)
Engineering checklist
- Module 9: Beam-Columns
- Key limit states and AISC references are listed in the reference box.
- Use φRn ≥ Ru for every check.
- Verify section properties with the official AISC Manual.
Professional tips
- • Mixing ASD and LRFD load combinations in the same problem.
- • Using nominal strength Rn instead of design strength φRn.
- • Forgetting to check every limit state listed in the AISC chapter.
§13Section 13
FE exam preparation
Concept review
Concept summary coming soon.
Calculator tips
Calculator tips coming soon.
Common exam traps
Traps coming soon.
Time management
Aim for ~3 minutes per FE problem; skip and return to any item that takes longer than 5 minutes.