Direct Analysis Method, B1/B2, appendix 6 bracing.
Direct Analysis Method, B1/B2, appendix 6 bracing.
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AISC's Direct Analysis Method (DAM, Chapter C) is the modern approach to frame stability. Missing notional loads, τb reductions, or B2 amplifiers under-predicts moments and can send an entire frame past its stability limit.
The lateral-load design of the capstone moment frames and any tall unbraced building.
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 | 16.1-16 |
| C2.2 | Consideration of Initial Imperfections (Notional Loads 0.002·Yi) | 16.1-17 |
| C2.3 | Adjustments to Stiffness (0.8·τb·EI, 0.8·EA) | 16.1-18 |
| C3 | Calculation of Available Strengths | 16.1-19 |
| App. 6 | Member Stability Bracing (Nodal, Relative, Continuous) | 16.1-215 |
| App. 7 | Alternative Methods (Effective Length, First-Order) | 16.1-225 |
| App. 8 | Approximate Second-Order Analysis (B1, B2) | 16.1-235 |
Companion reference: AISC Design Guide 28 — Stability Design of Steel Buildings
Every steel structure sways a little under load, and every column has some initial crookedness and out-of-plumbness from fabrication and erection tolerances. When axial load P acts through that lateral displacement, it creates additional bending moment that was not in the first-order (undeformed-geometry) analysis. This is called a second-order effect, and it comes in two flavors: P-δ, the amplification of moment along the length of a single bowed member, and P-Δ, the amplification of overturning moment caused by the sway (drift) of an entire story. A stability-adequate design must capture both.
Section C1 states that any analysis method used for stability design must account for four things simultaneously: (1) flexural, shear, and axial member deformations, and all other deformations that contribute to displacement; (2) second-order effects (both P-Δ and P-δ); (3) geometric imperfections (initial out-of-plumbness); and (4) stiffness reduction due to inelasticity (residual stress causing partial yielding before the nominal strength is reached). AISC gives you three ways to satisfy all four:
DAM modifies the analysis model itself, not the design equations. Three ingredients, applied together, satisfy all four §C1 requirements without any K-factor calculation.
Physically: a column near its squash load is softer than E predicts because parts of its cross-section have already yielded from residual stress; τb mimics that softening in an otherwise-elastic model, and a stiffer, all-purpose 0.80 knock-down covers the rest of the structure. Alternatively, §C2.3(b) permits using τb = 1.0 for all members if a supplemental notional load of 0.001αYi is added to every load combination (an added-stiffness-penalty trade for simplicity) — useful in software that can't vary τb by member.
Notional loads must be applied in every gravity-only combination, always. They must additionally be applied in combinations that already contain lateral load only if the structure is flexible enough that Δ2nd/Δ1st (computed with reduced stiffness) exceeds 1.7 — in stiffer structures the real wind/seismic load already envelopes the small imperfection effect.
Because DAM's reduced-stiffness second-order analysis already captures the true amplified moments and forces, there is no need — and no benefit — to computing an effective length factor. Every member in a DAM analysis uses K = 1.0, so Lc = L for flexural buckling in the plane of instability. This eliminates alignment-chart iteration entirely for DAM designs.
The ELM keeps the traditional K-factor approach that predates the 2005 Specification, but wraps it in the same second-order-analysis requirement as DAM. It uses full, un-reduced stiffness (no 0.80 or τb), applies notional loads only in gravity-only combinations, and requires the designer to compute K from a rational analysis such as the alignment chart for every column whose flexural buckling is sidesway-uninhibited.
When may ELM be used? Appendix 7 §7.2.1 requires both:
This second condition must be verified after the fact — you run the analysis, check the ratio, and if it's above 1.5 you must switch to DAM. In practice, stiff braced-frame and shear-wall buildings almost always satisfy it; slender moment frames often do not.
For sidesway-uninhibited (unbraced) frames, K is read from the AISC Commentary alignment chart using a stiffness ratio G computed at each end of the column:
With GA and GB known at the two ends of the column, a straightedge across the sidesway-uninhibited nomograph gives K directly (K ranges from 1.0 for both ends fixed to unbounded as the frame approaches a sway mechanism). Commentary Eq. C-A-7-5 (used in Example C.1B below) offers a closed-form "story-based" alternative to the chart that is often more accurate for irregular frames.
For very stiff, braced structures, AISC allows skipping second-order analysis software entirely. Instead of a true P-Δ analysis, the method amplifies the notional load coefficient itself and adds a supplemental notional load:
where Δ/L is the first-order drift ratio of the story under consideration. The method is only valid when: (1) the structure supports gravity load primarily through vertical columns/walls/frames, (2) Δ2nd/Δ1st ≤ 1.5, (3) Pr/Py ≤ 0.5 for all members whose flexural stiffness contributes to stability, and (4) K = 1.0 is used for all members. Member forces are then simply the first-order analysis results — no iteration, no second-order software — which makes this method attractive for hand calculations on braced low-rise structures.
When software performs only a first-order analysis, Appendix 8 gives a closed-form way to approximate second-order forces by amplifying first-order results with two multipliers: B1 for the P-δ effect within a member (no-sway component) and B2 for the P-Δ effect of story sway.
B1 amplifies moment along a single member due to its own axial load and curvature; B2 amplifies the sway moments and forces of an entire story due to the total gravity load riding on that story's drift. B2 ≥ 1.4 or so is a red flag that the frame may not qualify for ELM.
A brace only works if it is both strong enough to resist the small out-of-plumbness force delivered by the braced member, and stiff enough that it doesn't stretch or rotate so much that the buckling mode simply shifts past it. Appendix 6 classifies bracing into two geometric types.
Both types can brace either a column (against flexural buckling) or a beam (against lateral-torsional buckling); the beam case further splits into bracing that only restrains lateral movement of the compression flange ("lateral" bracing) versus bracing that restrains twist of the whole cross-section directly ("torsional" bracing, e.g., a stiff diaphragm attached near mid-depth).
What this means physically. The strength check ensures the brace can carry the small "kick" force generated by the member's assumed out-of-straightness (typically L/500) as it tries to buckle; the stiffness check ensures the brace doesn't act like a soft spring that lets the member buckle in a longer, lower-capacity mode through the brace point. A brace can pass strength and still fail stiffness, or vice versa — both must be checked independently.
A leaning column is a gravity-only column, typically pin-connected top and bottom, that carries substantial floor load but contributes essentially zero lateral stiffness of its own. Because it cannot resist sway, all of the P-Δ demand generated by the gravity load on a leaning column must be carried by the story's actual lateral system (moment frame, braced frame, or shear wall) through the diaphragm. In a second-order or B2 analysis, Pstory — the quantity that drives instability — must include the leaning-column loads, not just the loads directly on the frame columns. Ignoring leaning columns is a classic and dangerous underestimate of second-order demand, especially in buildings with a small braced core surrounded by large gravity bays.
| Name | Equation | AISC Ref |
|---|---|---|
| Design strength | φ Rn ≥ Ru | AISC 360-22 B3.1 |
Adapted from AISC Design Examples V16.0, Example C.1A.
Engineering interpretation. Even though the sway multiplier (drift ratio) is a substantial 1.60, the actual moment increase is modest because the only source of sway in this pure-gravity combination is the small notional load — real wind/seismic combinations with larger lateral loads will show a much bigger absolute moment amplification for the same ratio.
Adapted from AISC Design Examples V16.0, Example C.1B — repeats Example 17.1 with identical geometry and loads.
Engineering interpretation. DAM and ELM must give member designs of comparable adequacy for the same frame — but they get there differently. DAM folds all instability effects into the analysis and member forces, then uses the simplest possible K = 1.0. ELM keeps a first-order-like member-force philosophy but pushes all of the instability penalty into a large K-factor. Both are valid; DAM is preferred because it needs no applicability check up front.
Engineering interpretation. Stability braces are almost always governed by the stiffness requirement architecturally (the member needs to be stiff/short/steep enough), while the strength requirement is easily satisfied by nearly any reasonable section — but both must always be checked, because for very long or very shallow-angle braces the stiffness demand can dominate and force a larger member than strength alone would suggest.
Engineering interpretation. In a well-braced frame, B2 stays close to 1.0 (here 1.11) — most of the "second order" effect comes from the story sway captured by B2, while B1 commonly floors out at 1.0 for members with modest axial load relative to their Euler buckling load. This is the everyday hand-calculation alternative to running a full nonlinear second-order analysis when your software only performs first-order runs with and without sway.
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.
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.
Attach your handwritten or typed step-by-step solution for this chapter's graded quiz. The instructor can download every submission. PDF only, up to 25 MB.
Your file — PDF, Word document, scanned handwriting or a photo — is read page by page like an experienced structural engineering instructor would. The scan is validated first, then your reasoning, structural model, calculations, diagrams, code basis and final answers are graded on process, not just the final number. Design work is additionally reviewed against AISC 360-22 and ACI 318-19. Partial credit applies, and one early mistake carried correctly forward is only penalized once.
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