Module 7: Steel Compression Members - Theory & Concepts

Module 7: Steel Compression Members

$Learning Objectives$

Classify compression elements as nonslender or slender using the applicable NSCP limits.
Determine controlling column slenderness from effective length and radius of gyration.
Evaluate flexural buckling using the NSCP compression curve.
Recognize when torsional or flexural-torsional buckling must also be checked.
Compare weak-axis and strong-axis stability and identify the governing mode.
Relate column proportions, bracing, end restraint, HSS/W-shape selection, and fire protection to architectural design.

$NSCP Code Basis$

Compression members are governed by NSCP 2015 Section 505 — Design of Members for Compression, together with the stability requirements established by the Chapter 5 design framework.

$Slenderness ratio$

The dimensionless ratio $KL/r$ that compares effective buckling length with the radius of gyration about the axis being checked.

Column Slenderness

Global member slenderness about a selected buckling axis.

\lambda_c=\frac{KL}{r}

Variables

Symbol	Description	Unit
$K$	Effective-length factor or equivalent stability parameter for the selected analysis method.	-
$L$	Unbraced member length.	-
$r$	Radius of gyration about the buckling axis.	-

$Compression Element Classification$

For axial compression, plate elements are evaluated as nonslender or slender according to the width-to-thickness limits applicable to the element and section type. Slender elements require reduced effective resistance because local buckling can occur before the full cross-section reaches the otherwise calculated column strength.

The terms compact, noncompact, and slender are primarily flexural classifications and should not be used as the general taxonomy for axial-compression elements.

$Common Axial-Compression Width-to-Thickness Limits$

For the AISC 14th Edition / ANSI/AISC 360-10 framework adapted by NSCP 2015, representative nonslender/slender limits include:

Compression element	Ratio	$\lambda_r$
Rolled I-shape/channel flange or tee flange	$b/t$	$0.56\sqrt{E/F_y}$
Single-angle leg / other unstiffened plate	$b/t$	$0.45\sqrt{E/F_y}$
Tee stem	$d/t$	$0.75\sqrt{E/F_y}$
I-shape/channel web	$h/t_w$	$1.49\sqrt{E/F_y}$
Rectangular HSS wall	$b/t$	$1.40\sqrt{E/F_y}$
Round HSS	$D/t$	$0.11E/F_y$

Use the exact element definition and width convention from the adopted NSCP table. Built-up flanges and special elements have additional cases and must not be forced into the simplified rows above.

$Flexural Buckling$

A concentrically loaded column can become unstable by lateral deflection about a principal axis. The axis with the larger $KL/r$ often governs, which is frequently the weak axis of a W-shape, but bracing and effective-length conditions can reverse that assumption.

$Compression-Member Design Workflow$

The workflow separates local-element slenderness from global member buckling, requires a frame-stability/effective-length basis, and branches to torsional or flexural-torsional buckling where the section requires it.

Steel Compression-Member Design Workflow

Local-element classification, global buckling-mode selection, frame stability, and available-strength sequence.

Define section, restraints, lengths, frame, and compression → Classify compression-element slenderness; Classify compression-element slenderness → Any slender compression elements?; Any slender compression elements? — Yes → Apply required effective-area or reduced-strength provisions; Any slender compression elements? — No → Establish frame stability, effective length, and second-order effects; Apply required effective-area or reduced-strength provisions → Establish frame stability, effective length, and second-order effects; Establish frame stability, effective length, and second-order effects → Determine effective slenderness about all required axes; Determine effective slenderness about all required axes → Can flexural buckling alone represent all required modes?; Can flexural buckling alone represent all required modes? — Yes → Evaluate flexural buckling Fe and Fcr about governing axis; Can flexural buckling alone represent all required modes? — No / special section → Evaluate required torsional/flexural-torsional buckling mode and compare; Evaluate flexural buckling Fe and Fcr about governing axis → Compute governing nominal and LRFD/ASD available compressive strength; Evaluate required torsional/flexural-torsional buckling mode and compare → Compute governing nominal and LRFD/ASD available compressive strength; Compute governing nominal and LRFD/ASD available compressive strength → Verify bracing, connections, built-up action, and serviceability; Verify bracing, connections, built-up action, and serviceability → Is required compression within governing available strength?; Is required compression within governing available strength? — Yes → Document governing buckling mode and utilization; Is required compression within governing available strength? — No → Revise section, bracing, effective length, or frame system; Revise section, bracing, effective length, or frame system → Define section, restraints, lengths, frame, and compression

Define section, restraints, lengths, frame, and compression: terminator
Classify compression-element slenderness: subprocess
Any slender compression elements?: decision
Apply required effective-area or reduced-strength provisions: process
Establish frame stability, effective length, and second-order effects: process
Determine effective slenderness about all required axes: process
Can flexural buckling alone represent all required modes?: decision
Evaluate flexural buckling Fe and Fcr about governing axis: process
Evaluate required torsional/flexural-torsional buckling mode and compare: subprocess
Compute governing nominal and LRFD/ASD available compressive strength: process
Verify bracing, connections, built-up action, and serviceability: process
Is required compression within governing available strength?: decision
Revise section, bracing, effective length, or frame system: process
Document governing buckling mode and utilization: terminator

$What Changes When a Compression Element Is Slender$

Passing the global $KL/r$ check does not restore the full resistance of a locally slender plate element. Under the AISC 360-10 E7 framework adapted by NSCP 2015, define the slender-element reduction

Q=Q_sQ_a,

where $Q_s$ is the reduction for applicable unstiffened slender elements and $Q_a$ accounts for applicable stiffened slender elements. For sections with only one category, the other factor is 1.0. For stiffened slender elements,

Q_a=\frac{A_e}{A_g},

where $A_e$ is formed using the code-defined effective widths.

The flexural-buckling stress for a slender-element member is then selected from

F_{cr} = Q\left(0.658^{QF_y/F_e}\right)F_y

when $QF_y/F_e\le2.25$ , and

F_{cr}=0.877F_e

when $QF_y/F_e>2.25$ . The nominal compressive strength remains

P_n=F_{cr}A_g

within this E7 formulation.

Therefore:

identify every slender element and its exact E7 $Q_s$ or effective-width/ $Q_a$ case;
form the section-level $Q$ ;
determine the applicable elastic buckling stress $F_e$ for flexural, torsional, or flexural-torsional buckling;
calculate the corresponding $F_{cr}$ and $P_n$ ; and
take the governing mode before applying LRFD/ASD available-strength factors.

Do not simply label the section "slender" and continue with the nonslender E3 curve using $Q=1.0$ .

$Frame Stability Must Be Established Before Choosing K$

The member equation does not determine the building stability method. Establish the permitted NSCP Chapter 5 stability-analysis procedure, sidesway condition, second-order effects, imperfections/notional-load requirements where applicable, and bracing assumptions before assigning effective length. Idealized $K$ values are explanatory boundary-condition models, not substitutes for the governing frame analysis.

Euler Elastic Buckling Stress

Elastic reference stress used by the NSCP steel compression curve.

F_e=\frac{\pi^2E}{(KL/r)^2}

Variables

Symbol	Description	Unit
$E$	Modulus of elasticity of steel.	-
$KL/r$	Member slenderness ratio for the axis being evaluated.	-
$F_e$	Euler elastic buckling stress.	-

$Select the Correct Column-Curve Branch$

For members governed by the basic NSCP 2015 / ANSI/AISC 360-10 flexural-buckling model without slender elements, the branch is selected from the elastic buckling stress:

use the inelastic expression when $F_y/F_e\le2.25$ ;
use the elastic expression when $F_y/F_e>2.25$ .

The equivalent slenderness boundary is $L_c/r=4.71\sqrt{E/F_y}$ for the flexural-buckling case. Evaluate the branch from the actual $F_e$ ; do not classify a column by visual slenderness.

NSCP Column Critical Stress

Compression-curve form for members within the applicable NSCP flexural-buckling provisions.

F_{cr}= \begin{cases} 0.658^{F_y/F_e}F_y, & \text{inelastic range} \\ 0.877F_e, & \text{elastic range} \end{cases}

Variables

Symbol	Description	Unit
$F_{cr}$	Critical compressive stress.	-
$F_y$	Specified yield strength.	-
$F_e$	Euler elastic buckling stress.	-

Nominal Compressive Strength

Nominal axial compression strength after the governing critical stress is established; slender-element effects are incorporated into Fcr through the applicable E7 reduction framework.

P_n=F_{cr}A_g

Variables

Symbol	Description	Unit
$F_{cr}$	Critical compressive stress for the governing buckling mode.	-
$A_g$	Gross cross-sectional area. In the AISC 360-10 E7 slender-element formulation, effective-width effects enter through Qa and Q, which modify Fcr before Pn=FcrAg is evaluated.	-

$Compression Available Strength$

After the governing nominal compression strength $P_n$ has been established from the applicable flexural, torsional, flexural-torsional, and slender-element provisions, convert it on the selected design basis:

P_c=0.90P_n \quad \text{(LRFD)}

P_c=\frac{P_n}{1.67} \quad \text{(ASD)}

Use the lowest applicable nominal compression strength before applying the corresponding design-basis factor. Do not apply the factor separately to one buckling mode and ignore another lower mode.

$Interactive Exploration$

Vary $F_y$ and the selected $KL/r$ to move through the inelastic and elastic branches. Compare the solid NSCP $F_{cr}$ curve with the dashed Euler $F_e$ reference and watch the transition move as $F_y$ changes. The Euler curve is clipped at the plot boundary rather than flattened, so values above the visible stress range do not create a false plateau. Use Reset to restore the reference case.

Controls

345 MPa

Transition KL/r113.4

Euler stress Fe548.3 MPa

Buckling regioninelastic

Critical stress Fcr265.1 MPa

This visual covers flexural buckling of nonslender compression members. Slender elements, torsional/flexural-torsional buckling, frame stability, and second-order analysis remain separate checks where required. The Euler reference is clipped only by the chart viewport; its calculated values are never flattened to fit the plot.

$Check More Than Flexural Buckling When Required$

Single angles, tees, cruciform members, built-up shapes, and other singly symmetric or unsymmetric sections may be controlled by torsional or flexural-torsional buckling. Do not use the flexural-buckling curve alone when Section 505 requires additional modes.

$Effective Length and Frame Stability$

The factor $K$ represents boundary restraint only within the assumptions of the chosen stability method. Real building stability depends on frame stiffness, bracing, connection behavior, member imperfections, gravity loads, and second-order effects.

Do not select $K$ by visual intuition alone. Establish whether the system is braced or moment-resisting and use the analysis/design procedure consistent with NSCP Chapter 5.

$Idealized Effective-Length K Reference$

Idealized condition	Idealized $K$	Comment
Fixed–fixed, sidesway inhibited	$0.50$	Strong rotational restraint at both ends
Fixed–pinned, sidesway inhibited	about $0.70$	One end rotationally restrained
Pinned–pinned	$1.00$	Euler reference condition
Fixed–free cantilever	$2.00$	Effective length doubles

These values are for idealized boundary conditions. Real steel frames should obtain effective length from the adopted stability method, frame stiffness, sidesway condition, and connection restraint; $K$ is not a cosmetic input chosen to improve capacity.

$Global Slenderness Is Not Local Slenderness$

$KL/r$ governs member/global flexural buckling. Ratios such as $b/t$ , $h/t_w$ , and $D/t$ govern local plate-element buckling. Both must be checked, and passing one does not imply passing the other.

$Practical Compression-Member Slenderness$

The AISC/NSCP design tradition recommends that effective slenderness $L_c/r$ preferably not exceed about $200$ for ordinary compression members. It is a practical user-note recommendation rather than the equation that defines compressive strength; the actual $F_{cr}$ calculation still governs.

$Built-Up and HSS Columns$

Built-up members must transfer shear between components so the assembly acts as intended. HSS columns can be architecturally efficient because of similar radii of gyration in two directions and clean exposed form, but their connections, internal corrosion protection, vent/drain holes, fire protection, and access for bolting/welding need early coordination.

$Architectural Column Selection$

A smaller-area section is not automatically the more efficient column. Radius of gyration, available bracing, floor-to-floor height, connection depth, façade alignment, enclosure thickness, and fireproofing can dominate.

For exposed columns, evaluate not only the governing $KL/r$ but also how the section terminates at base plates, beam connections, roof drainage interfaces, and fire-rated assemblies.

Key Takeaways

Axial-compression elements are classified as nonslender or slender under the compression provisions.
Global column strength depends on $KL/r$ , $F_e$ , $F_{cr}$ , and the governing buckling mode.
Check both principal axes and any required torsional or flexural-torsional modes.
Effective length belongs to a complete frame-stability model, not an isolated guess about end conditions.
Architectural column selection should coordinate structural efficiency, bracing, connections, enclosure, corrosion, and fire protection.

$References$

Association of Structural Engineers of the Philippines (ASEP) — National Structural Code of the Philippines, C101-15, Volume I, Seventh Edition (2015) — Governing course reference for Section 505 compression-member provisions; consult the official licensed code for complete local-slenderness, flexural, torsional, flexural-torsional, built-up-member, and stability requirements.
AISC — Historic Standards and Steel Construction Manuals — NSCP 2015 Chapter 5 adapts the AISC 14th Edition framework based on ANSI/AISC 360-10. Use the adopted NSCP text as governing and consult the corresponding historical AISC edition only as a technical cross-reference.

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