Steel Compression Members - Cheat Sheet
Your 30-minute revision companion for steel column design, buckling stress, slenderness ratios, and design strength per NSCP 2015 (AISC 360-16). Focus on critical stress equations, transition detection, and common pitfalls.
Elastic Buckling Stress & Euler Formula
Formulas
F_e = π²E / (KL/r)²
Meaning
F_e = elastic buckling stress (MPa); E = modulus of elasticity (200,000 MPa for steel); K = effective-length factor (0.5–2.0); L = member length (mm); r = radius of gyration (mm)
Watch out
Use LARGEST KL/r (smallest r, weakest axis). Never use r for strong axis unless weak-axis is braced. Units must match: mm with mm, m with m.
When to use
Always calculate first; used in slender (elastic) range and to find transition point
Common values
Value
200,000 MPa
Symbol
E
Quantity
Young's modulus (steel)
Value
1.0
Symbol
K
Quantity
Effective-length factor (pinned–pinned)
Value
KL/r ≤ 200
Symbol
—
Quantity
Recommended slenderness limit
Important facts
- E = 200,000 MPa for all structural steel (constant in NSCP 2015)
- Elastic buckling occurs only in slender columns; most columns fail by inelastic buckling
- F_e alone never equals F_cr; always apply the correction factor (0.658^(Fy/Fe) inelastic, 0.877 elastic)
- KL/r must be calculated using LARGEST value (smallest r); compare both axes
Key definitions
Radius of gyration (r)
Example
W-column r_xx = 150 mm, r_yy = 40 mm → use r = 40 mm (weak axis)
r = √(I/A); smallest value governs buckling; always use weak-axis radius unless bracing prevents weak-axis buckling.
Effective-length factor (K)
Example
Typical building column: K = 1.0 (pinned top/bottom); braced frame may use K = 0.65–0.8
Empirical factor (0.5–2.0) accounting for boundary conditions; K = 1.0 for pinned, 0.65 fixed–fixed, 1.2 fixed–free, 2.0 free–free.
Slenderness ratio (KL/r)
Example
KL/r = 70 (inelastic); KL/r = 150 (elastic)
Non-dimensional measure of member slenderness; governs whether buckling is inelastic (stocky) or elastic (slender).
Diagrams to know
- Buckling mode shapes (single-curvature pinned, fixed–free cantilever, fixed–fixed, etc.)
- Critical stress vs. slenderness diagram (Perry-Robertson curve, smooth AISC curve)
Transition Slenderness & Inelastic vs Elastic Buckling
Formulas
(KL/r)_transition = 4.71√(E/Fy)
Meaning
Transition point between inelastic and elastic buckling; equivalently Fy/Fe = 2.25 at transition
Watch out
Common error: forgetting to calculate transition; assuming all columns are elastic. For Fy = 248 MPa → transition ≈ 133.7; for Fy = 345 MPa → transition ≈ 113.4
When to use
FIRST STEP: calculate this; if actual KL/r < transition → inelastic; if > transition → elastic
F_cr = [0.658^(Fy/Fe)] × Fy (INELASTIC, KL/r ≤ transition)
Meaning
Critical buckling stress in inelastic range; 0.658 is a calibrated constant from test data; exponent is Fy/Fe ratio
Watch out
Exponent is Fy/Fe (NOT Fe/Fy). If Fy/Fe > 2.25, formula is wrong—use elastic. Calculate F_e first, then the ratio.
When to use
When KL/r is below the transition value; most columns in practice
F_cr = 0.877 × F_e (ELASTIC, KL/r > transition)
Meaning
Critical buckling stress in elastic range; 0.877 factor accounts for initial out-of-straightness per AISC testing
Watch out
Do NOT use 0.877 × F_y; only multiply F_e by 0.877. Forgetting 0.877 → unconservative design.
When to use
Only when KL/r exceeds the transition value; slender columns
Common values
Value
4.71
Symbol
—
Quantity
Transition exponent constant
Value
0.658
Symbol
—
Quantity
Inelastic buckling base
Value
0.877
Symbol
—
Quantity
Elastic buckling correction factor
Important facts
- Transition always calculated as 4.71√(E/Fy); for steel E = 200,000 MPa (constant)
- At transition, Fy/Fe = 2.25 (AISC reference point)
- Inelastic formula reduces stress below yield due to initial imperfections; no 0.877 factor
- Elastic formula includes 0.877 factor to account for continued imperfection sensitivity in slender range
- CRITICAL: determine transition BEFORE choosing formula; most exam errors are formula-selection mistakes
Key definitions
Inelastic buckling
Example
W-column KL/r = 70, Fy = 248 MPa → inelastic (transition = 133.7)
Buckling failure at stress below yield; stocky column (KL/r ≤ transition); governs most building columns.
Elastic buckling
Example
Slender tube KL/r = 160, Fy = 248 MPa → elastic
Buckling at stress below proportional limit; slender column (KL/r > transition); stress below yield.
Diagrams to know
- AISC Column Curve: F_cr vs KL/r showing inelastic (parabolic) and elastic (hyperbolic) branches
- Fy/Fe ratio vs KL/r decision tree
Design Axial Strength (Limit State Design)
Formulas
P_n = F_cr × A_g
Meaning
P_n = nominal axial strength (kN); F_cr = critical buckling stress (MPa) from inelastic or elastic formula; A_g = gross cross-sectional area (mm²)
Watch out
Use GROSS area (A_g), not net area. For design, apply resistance factor: φ_c × P_n (LSD) or P_n / Ω_c (ASD)
When to use
After finding F_cr; multiply by gross area to get strength
φ_c × P_n = 0.90 × F_cr × A_g (LRFD, NSCP 2015 LSD-equivalent)
Meaning
Design axial strength; resistance factor φ_c = 0.90 for compression (constant for all steel grades & K values)
Watch out
φ_c = 0.90 (NOT 0.65 or 0.75 like RC). If using ASD: Ω_c = 1.67 (divide P_n by 1.67)
When to use
Limit State Design (LSD); find required strength ≤ design strength
Common values
Value
0.90
Symbol
φ_c
Quantity
Resistance factor (LSD compression)
Value
1.67
Symbol
Ω_c
Quantity
Safety factor (ASD compression)
Important facts
- φ_c = 0.90 (LSD/LRFD); Ω_c = 1.67 (ASD); NSCP 2015 adopts LSD terminology
- Design strength NEVER exceeds Fy × A_g (yield strength); realistic for most columns
- For compact columns, F_cr < Fy; for slender, F_cr << Fy (hence lower design strength)
- Recommended maximum slenderness KL/r ≤ 200 to avoid over-slender, uneconomical designs
Key definitions
Design axial strength
Example
φ_c P_n = 0.90 × 200 MPa × 5000 mm² = 900 kN (design strength)
φ_c × P_n or P_n / Ω_c; factored nominal strength available in column; must exceed factored loads.
Diagrams to know
- Design strength vs KL/r curve (shows 0.90 factor applied to F_cr curve)
Effective-Length Factor (K) & Buckling Modes
Common values
Value
1.0
Symbol
K
Quantity
K (pinned–pinned, typical building column)
Value
0.65
Symbol
K
Quantity
K (fixed–fixed, heavily restrained)
Value
1.2
Symbol
K
Quantity
K (fixed–free cantilever)
Important facts
- K = 0.5: both ends fixed (theoretical, rare in practice)
- K = 0.65–0.80: fixed–pinned or braced frame typical (NSCP 2015 nomograph method)
- K = 1.0: pinned–pinned (simple span, practical assumption)
- K = 1.2: fixed–free cantilever (roof-level columns)
- K = 2.0: free–free (unstable, not used)
- For building frames: use alignment chart (Jackson or Sidesway-Inhibited nomograph) if precise K needed; otherwise K = 1.0 is safe assumption
Key definitions
Effective length (KL)
Example
Pinned column L = 4000 mm, K = 1.0 → KL = 4000 mm; fixed–fixed K = 0.65 → KL = 2600 mm
Adjusted member length accounting for boundary conditions; KL = K × L where K empirically varies 0.5–2.0.
Diagrams to know
- Buckling mode shapes vs K (showing deflection profiles for each boundary condition)
- Jackson alignment chart (sidesway-inhibited and sidesway-uninhibited)
Local Buckling & Slender Elements
Formulas
Q = Q_s × Q_a (or simplified: Q = 1.0 for compact sections)
Meaning
Q = reduction factor for local buckling; Q_s = flange factor, Q_a = web factor; Q < 1.0 if section has slender flanges/web
Watch out
For most W-columns in building design, Q = 1.0 (compact). Thin-walled tubes, angles, channels may have Q < 1.0. Always verify flange & web slenderness.
When to use
Check AISC Table B4.1 width-thickness limits; if exceeded, Q < 1.0 and effective area A_eff = Q × A_g is used in lieu of A_g
Common values
Value
1.0
Symbol
Q
Quantity
Typical Q (compact W-column)
Important facts
- W-columns typically Q = 1.0 (compact) unless heavily tapered or unusually thin
- Built-up sections (welded box, channels) require flange & web slenderness check
- If Q < 1.0, use A_eff = Q × A_g in F_cr calculation (or in P_n = F_cr × A_eff)
- Local buckling is SECONDARY concern in most exams; focus on flexural buckling first
Key definitions
Local buckling
Example
Thin flange (b/t > limit) buckles locally; effective area reduced by Q factor
Premature buckling of flange or web plate elements before overall flexural buckling; reduced by limiting width-thickness ratios.
Compact section
Example
Most W-columns in NSCP 2015 Category 1 (compact)
Section with b/t ratios within AISC limits; Q = 1.0; controls by flexural buckling, not local buckling.
Diagrams to know
- Cross-section showing flange & web widths and thickness measurement for b/t limits
Torsional & Flexural-Torsional Buckling
Important facts
- ONLY relevant for singly or doubly non-symmetric sections (angles, channels, tees)
- Doubly symmetric columns (W, H, box) fail by FLEXURAL buckling only; ignore torsional modes
- For exam purposes: assume column is W-section (doubly symmetric) unless stated otherwise
- If singly symmetric: check torsional/flexural-torsional stress; AISC 360 Appendix E gives detailed formulas
Key definitions
Torsional buckling
Example
Single-angle column: torsional buckling may control instead of y-axis flexural
Twisting failure mode; governs singly symmetric and unsymmetric sections (angles, channels, tees) where flexural buckling stress exceeds torsional.
Flexural-torsional buckling
Example
T-section: can fail by bending about major axis AND twisting simultaneously
Combined bending + twisting failure; typical for singly symmetric shapes where both modes couple.
Diagrams to know
- Single-angle column showing axis of symmetry and rotation about minor principal axis
Step-by-Step Design Procedure
Important facts
- STEP 1: Identify member geometry (L, shape, A_g, r_x, r_y). Determine K from boundary conditions or alignment chart.
- STEP 2: Calculate KL/r for BOTH axes; use LARGEST value (weakest axis).
- STEP 3: Calculate transition: (KL/r)_trans = 4.71√(E/Fy). Compare actual KL/r to transition.
- STEP 4: Calculate F_e = π²E/(KL/r)². If inelastic: F_cr = [0.658^(Fy/Fe)] × Fy. If elastic: F_cr = 0.877 × F_e.
- STEP 5: Find nominal strength P_n = F_cr × A_g (or A_eff if Q < 1.0).
- STEP 6: Design strength = 0.90 × P_n (LSD) or P_n / 1.67 (ASD). Compare to required strength.
Diagrams to know
- Flowchart: given KL/r → transition check → formula selection → F_cr → P_n → design strength
Must remember
- ALWAYS use the LARGEST KL/r (smallest r) from both axes; weak-axis buckling governs unless bracing prevents it.
- CRITICAL: Calculate transition 4.71√(E/Fy) FIRST; it determines which F_cr formula (inelastic vs elastic) to use. Forgetting this → most exam failures.
- Inelastic formula: F_cr = [0.658^(Fy/Fe)] × Fy — exponent is Fy/Fe (NOT Fe/Fy). If Fy/Fe > 2.25, you are in elastic range — use F_cr = 0.877 × F_e instead.
- Elastic formula: F_cr = 0.877 × F_e (NOT 0.877 × Fy). The 0.877 factor ONLY multiplies F_e, not yield strength.
- Design strength: φ_c P_n = 0.90 × F_cr × A_g (LSD) or P_n/1.67 (ASD). φ_c = 0.90 for all compression (different from RC columns at 0.65/0.75).
- E = 200,000 MPa is constant for all steel grades in NSCP 2015. Do NOT vary E.
- If section is slender (local buckling check), use A_eff = Q × A_g instead of A_g in P_n formula. Most W-columns have Q = 1.0; check AISC Table B4.1 if unsure.
- Recommended maximum slenderness KL/r ≤ 200 to avoid uneconomical, overly flexible designs.
- K ≈ 1.0 is safe default for building columns (pinned–pinned assumption) unless alignment chart or problem states otherwise.
- Singly symmetric sections (angles, channels, tees) may fail by torsional or flexural-torsional buckling; assume doubly symmetric (W, box) unless noted.
Last minute tips
- EXAM STRATEGY: If given KL/r and Fy, IMMEDIATELY calculate 4.71√(E/Fy) and compare. This 10-second check prevents formula errors worth ~5 marks.
- COMMON TRAP: Students calculate F_e and forget to apply 0.658^(Fy/Fe) in inelastic range. Always ask: 'Is KL/r < transition?' → YES → use exponent formula (inelastic); NO → use 0.877 × F_e (elastic).
- UNIT CONSISTENCY: If L and r are both in mm, KL/r is dimensionless (good). If mixed (L in m, r in mm), convert one unit first. Wrong units → 10× error in F_e.
- QUICK CHECK: For Fy = 248 MPa, transition ≈ 134. Most building columns have KL/r = 60–120 (below transition, inelastic). If your KL/r > 150, likely elastic; double-check boundary conditions.
- PARTIAL CREDIT: If unsure of exact formula, always show P_n = F_cr × A_g structure. Examiners give marks for methodology even if F_cr constant is off.
Comparison tables
Inelastic vs Elastic Buckling
| Criterion | INELASTIC (Stocky) | ELASTIC (Slender) |
|---|---|---|
| Slenderness range | KL/r ≤ 4.71√(E/Fy) | KL/r > 4.71√(E/Fy) |
| Buckling stress formula | F_cr = [0.658^(Fy/Fe)] × Fy | F_cr = 0.877 × F_e |
| Fy/Fe ratio | ≤ 2.25 | > 2.25 |
| Physical behavior | Inelastic deformation before buckling; initial imperfections reduce capacity | Linear elastic to buckling; initial out-of-straightness governs |
| Typical columns | Building columns, short–medium spans | Very slender columns, tall frames, braced members |
| Correction factor in F_cr | 0.658^(Fy/Fe) ≈ 0.5–1.0 | 0.877 (constant) |
Common K Values by Boundary Condition
| Boundary Condition | K value | Effective Length | Typical Application |
|---|---|---|---|
| Fixed–Fixed | 0.5 | 0.5L | Rare; basement column fully restrained |
| Fixed–Pinned | 0.70 | 0.70L | One end fixed, one pinned (rare) |
| Pinned–Pinned | 1.0 | L | Simple span; typical building column |
| Fixed–Free (Cantilever) | 1.2–2.0 | 1.2L–2.0L | Roof or parapet column; exposed top |
| Sidesway-Inhibited (braced frame) | 0.65–0.80 | 0.65L–0.80L | Lateral loads resisted by shear walls/bracing |
Transition Slenderness for Common Fy Values
| Yield Strength Fy (MPa) | Transition KL/r | Short description |
|---|---|---|
| 248 (Grade A36, older) | 133.7 | Inelastic if KL/r < 133.7; elastic if > 133.7 |
| 345 (Grade 50, common) | 113.4 | Inelastic if KL/r < 113.4; elastic if > 113.4 |
| 415 (Grade 60, high-strength) | 101.2 | Inelastic if KL/r < 101.2; elastic if > 101.2 |
| 500 (Grade 70, very high-strength) | 92.1 | Inelastic if KL/r < 92.1; elastic if > 92.1 |
Diagrams
Steel Column Design Decision Tree
Step-by-step flowchart guiding selection of inelastic vs elastic formula based on slenderness ratio. Shows the critical transition comparison and subsequent calculations for design strength.
Steel Compression Member Design Formula Map
Hierarchical mind map organizing all key formulas, variables, and design steps from geometry through final design strength calculation. Shows relationships between slenderness, buckling stress determination, and design capacity.