S
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LecturesModule 8

Combined Axial Load and Bending

Interaction equations from AISC Chapter H.

AISC Reference Box
  • AISC 360-22 — Specification chapter governing this topic
  • AISC Manual 16th Ed. — Design tables and worked examples

Lecture Notes

This module introduces combined axial load and bending. Lecture content here covers the governing physics, LRFD philosophy, and how the relevant AISC 360-22 chapter organizes the limit states.

Instructors can replace this text in Admin Mode. Each section is structured around: (1) behavior, (2) failure modes, (3) AISC limit-state equations, (4) design workflow, (5) detailing requirements.

A short comparison to ASD is included only where the resistance factor / safety factor relationship clarifies the LRFD design check.

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 8 — Combined forces
Edge column receiving wind moment + gravity axial
Demand carried forward
Pu ≈ 380 k (gravity) + 25 k (wind axial) Muy ≈ 60 k-ft (cladding + wind moment).
This chapter contributes
Applies the AISC H1-1a / H1-1b interaction equation. Confirms ratio ≤ 1.0 governs the column size — typically driving the edge column one shape size up from the interior column.
Feeds into next chapter
Result sets the demand for the beam-column splice in Chapter 9 and base plate in Chapter 16.
Mr / McPr / Pc1.00.21.0H1-1a: Pr/Pc + (8/9)·ΣMr/Mc ≤ 1H1-1b: Pr/(2Pc) + ΣMr/Mc ≤ 1
AISC Chapter H interaction envelope: H1-1a for Pr/Pc ≥ 0.2, H1-1b for Pr/Pc < 0.2.

Formula Sheet

NameEquationAISC Ref
Design strengthφ Rn ≥ RuAISC 360-22 B3.1

Worked Example

Combined Axial Load and Bending

Given
Replace with project-specific given data (loads, geometry, material).
Load combination
Controlling LRFD load combination from ASCE 7.
Required strength
Compute required strength Ru from the controlling combination.
Limit states
  • Limit state 1
  • Limit state 2
AISC reference
AISC 360-22 — applicable chapter
Solution steps
  1. 1. Required strength
    Compute Ru.
  2. 2. Trial section
    Pick a trial from AISC shape tables Instructor should verify with official AISC Manual.
  3. 3. Check each limit state
    Apply φ Rn ≥ Ru for every governing limit state.
  4. 4. Iterate
    Resize until the most economical section satisfies all checks.
Final design decision
Select the lightest section that satisfies all LRFD limit states.
Common mistakes in this example
  • Skipping a limit state
  • Using the wrong φ factor
  • Forgetting serviceability checks

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=200 k, Pc=φPn=400 k, Mrx=80 k-ft, Mcx=φMnx=200 k-ft, no Mry.
AISC Reference
AISC §H1.1(a)
Step-by-step solution
  1. Pr/Pc
    200/400 = 0.50 ≥ 0.2 → use Eq H1-1a
  2. Interaction
    0.50 + (8/9)(80/200) = 0.50 + 0.356 = 0.856 ≤ 1.0 ✓
Answer Ratio = 0.86, OK.
PuLcAxially loaded column

Practice Problems

  1. [E] State H1-1a and H1-1b interaction equations.
  2. [E] At what Pr/Pc does the equation switch (0.2)?
  3. [E] Define Pr, Pc, Mrx, Mcx.
  4. [E] State φc and φb used inside H1.
  5. [E] Identify the AISC chapter for combined forces (Chapter H).
  6. [M] W14x90 with Pu = 480 k, Mux = 180 k-ft. Compute interaction ratio.
  7. [M] W12x72 with Pu = 320 k, Mux = 90 k-ft, Muy = 25 k-ft. Use H1-1a.
  8. [M] Verify W10x60 (A992, Lc = 14 ft) is adequate for Pu = 250 k, Mux = 110 k-ft.
  9. [M] At Pr/Pc = 0.15, demonstrate which equation governs.
  10. [M] Use Table 6-2 to size a W14 for Pu = 600 k, Mux = 240 k-ft, KL = 14 ft.
  11. [H] Design a W12 beam-column for Pu = 450 k, Mux,top = +220, Mux,bot = -100 k-ft, KL = 14 ft (compute Cm, B1).
  12. [H] Biaxial: W14x120, Pu = 700 k, Mux = 200, Muy = 80 k-ft, KL = 16 ft.
  13. [H] Tension + bending: WT9x27.5, Tu = 60 k, Mux = 25 k-ft. Apply §H1.2.
  14. [H] Use DAM to compute B1 and B2 for an unbraced W14x90 frame, 12 ft story, Pu = 400 k.
  15. [H] Compare H1-1a result for W12x96 with and without 80% Y-bracing; quantify capacity gain.
Structured Clues
  • Use H1-1a when Pr/Pc ≥ 0.2, otherwise H1-1b.
  • Compute Pc = φcPn at the controlling KL, NOT the unbraced length for flexure.
  • Table 6-2 lets you read interaction directly for W14 trial shapes.
Code References
  • AISC 360-22 §H1, H2
  • AISC Manual Part 6 — Tables 6-1, 6-2

Quiz

1. Which AISC 360-22 chapter primarily governs combined axial load and bending?
2. In LRFD, the basic design inequality is:

Common Student Mistakes

  • 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.

"Professor Explains" Script

Today we're talking about combined axial load and bending. Think of this topic as one step in the LRFD workflow: identify the demand, identify the limit states from the relevant AISC chapter, then check that φ·Rn is at least equal to Ru. We'll walk through the failure modes, the equations, and a worked example. Pay close attention to where the resistance factor changes — that's where students lose points on exams.