CELE Steel & Timber Design — Steel Compression MembersCheat Sheet
A printable cheat sheet for Steel Compression Members, built for CELE reviewers who want one go-to reference in the final stretch. Covers formulas, key definitions, common question types, and the Professional Regulation Commission (PRC) — Board of Civil Engineering-specific twists you will see on CELE day.
Exam context
On the CELE 2026, the Steel & Timber Design subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Steel Compression Members lands at position 2nd out of 5 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Steel & Timber Design on a typical CELE paper.
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.
Sections
Formulas
Formula
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
Section Title
Elastic Buckling Stress & Euler Formula
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
Term
Radius of gyration (r)
Example
W-column r_xx = 150 mm, r_yy = 40 mm → use r = 40 mm (weak axis)
Definition
r = √(I/A); smallest value governs buckling; always use weak-axis radius unless bracing prevents weak-axis buckling.
Term
Effective-length factor (K)
Example
Typical building column: K = 1.0 (pinned top/bottom); braced frame may use K = 0.65–0.8
Definition
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.
Term
Slenderness ratio (KL/r)
Example
KL/r = 70 (inelastic); KL/r = 150 (elastic)
Definition
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)
Formulas
Formula
(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
Formula
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
Formula
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
Section Title
Transition Slenderness & Inelastic vs Elastic Buckling
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
Term
Inelastic buckling
Example
W-column KL/r = 70, Fy = 248 MPa → inelastic (transition = 133.7)
Definition
Buckling failure at stress below yield; stocky column (KL/r ≤ transition); governs most building columns.
Term
Elastic buckling
Example
Slender tube KL/r = 160, Fy = 248 MPa → elastic
Definition
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
Formulas
Formula
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
Formula
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)
Section Title
Design Axial Strength (Limit State Design)
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
Term
Design axial strength
Example
φ_c P_n = 0.90 × 200 MPa × 5000 mm² = 900 kN (design strength)
Definition
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)
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)
Section Title
Effective-Length Factor (K) & Buckling Modes
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
Term
Effective length (KL)
Example
Pinned column L = 4000 mm, K = 1.0 → KL = 4000 mm; fixed–fixed K = 0.65 → KL = 2600 mm
Definition
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)
Formulas
Formula
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)
Section Title
Local Buckling & Slender Elements
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
Term
Local buckling
Example
Thin flange (b/t > limit) buckles locally; effective area reduced by Q factor
Definition
Premature buckling of flange or web plate elements before overall flexural buckling; reduced by limiting width-thickness ratios.
Term
Compact section
Example
Most W-columns in NSCP 2015 Category 1 (compact)
Definition
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
Section Title
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
Term
Torsional buckling
Example
Single-angle column: torsional buckling may control instead of y-axis flexural
Definition
Twisting failure mode; governs singly symmetric and unsymmetric sections (angles, channels, tees) where flexural buckling stress exceeds torsional.
Term
Flexural-torsional buckling
Example
T-section: can fail by bending about major axis AND twisting simultaneously
Definition
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
Section Title
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
Rows
Values
- KL/r ≤ 4.71√(E/Fy)
- KL/r > 4.71√(E/Fy)
Property
Slenderness range
Values
- F_cr = [0.658^(Fy/Fe)] × Fy
- F_cr = 0.877 × F_e
Property
Buckling stress formula
Values
- ≤ 2.25
- > 2.25
Property
Fy/Fe ratio
Values
- Inelastic deformation before buckling; initial imperfections reduce capacity
- Linear elastic to buckling; initial out-of-straightness governs
Property
Physical behavior
Values
- Building columns, short–medium spans
- Very slender columns, tall frames, braced members
Property
Typical columns
Values
- 0.658^(Fy/Fe) ≈ 0.5–1.0
- 0.877 (constant)
Property
Correction factor in F_cr
Columns
- Criterion
- INELASTIC (Stocky)
- ELASTIC (Slender)
Table Title
Inelastic vs Elastic Buckling
Rows
Values
- 0.5
- 0.5L
- Rare; basement column fully restrained
Property
Fixed–Fixed
Values
- 0.70
- 0.70L
- One end fixed, one pinned (rare)
Property
Fixed–Pinned
Values
- 1.0
- L
- Simple span; typical building column
Property
Pinned–Pinned
Values
- 1.2–2.0
- 1.2L–2.0L
- Roof or parapet column; exposed top
Property
Fixed–Free (Cantilever)
Values
- 0.65–0.80
- 0.65L–0.80L
- Lateral loads resisted by shear walls/bracing
Property
Sidesway-Inhibited (braced frame)
Columns
- Boundary Condition
- K value
- Effective Length
- Typical Application
Table Title
Common K Values by Boundary Condition
Rows
Values
- 133.7
- Inelastic if KL/r < 133.7; elastic if > 133.7
Property
248 (Grade A36, older)
Values
- 113.4
- Inelastic if KL/r < 113.4; elastic if > 113.4
Property
345 (Grade 50, common)
Values
- 101.2
- Inelastic if KL/r < 101.2; elastic if > 101.2
Property
415 (Grade 60, high-strength)
Values
- 92.1
- Inelastic if KL/r < 92.1; elastic if > 92.1
Property
500 (Grade 70, very high-strength)
Columns
- Yield Strength Fy (MPa)
- Transition KL/r
- Short description
Table Title
Transition Slenderness for Common Fy Values
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