Compression Members

ANALYSIS OF COMPRESSION MEMBERS

For a short compression member: For a long member:

MODE OF FAILURE: YIELDING MODE OF FAILURE: FLEXURAL BUCKLING

Stress is governed by Fy Stress is governed by Euler’s buckling formula,

𝜋2 𝐸
𝐹𝑒 = 𝑘𝐿 2
𝑟
 𝑘 = 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑙𝑒𝑛𝑔𝑡𝑕 𝑓𝑎𝑐𝑡𝑜𝑟
𝑘𝐿
= 𝑠𝑙𝑒𝑛𝑑𝑒𝑟𝑛𝑒𝑠𝑠 𝑟𝑎𝑡𝑖𝑜
𝑟
Slenderness ratio for compression members
must not exceed 200

STRENGTH OF A COMPRESSION MEMBER

NSCP 2001 (Allowable Stress Design):

2𝜋 2 𝐸
𝐶𝑐 =
𝐹𝑦

Cc is the limiting slenderness ratio that separates short and intermediate members to long members

𝑘𝑙
1. 𝑊𝑕𝑒𝑛 < 𝐶𝑐 , 𝑚𝑒𝑚𝑏𝑒𝑟 𝑖𝑠 𝑖𝑛𝑡𝑒𝑟𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑎𝑛𝑑 𝑚𝑜𝑑𝑒 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑖𝑠 𝑖𝑛𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑏𝑢𝑐𝑘𝑙𝑖𝑛𝑔
𝑟𝑚𝑖𝑛
2
𝑘𝐿
1 𝑟𝑚𝑖𝑛
𝐹𝑛 = 1 − 𝐹𝑦
2 𝐶𝑐

𝐹𝑛
𝐹𝑎 =
Ω
3
𝑘𝐿 𝑘𝐿
5 3 𝑟 1 𝑟𝑚𝑖𝑛
Ω = + ( 𝑚𝑖𝑛 ) −
3 8 𝐶𝑐 8 𝐶𝑐
 𝑘𝑙
2. 𝑊𝑕𝑒𝑛 > 𝐶𝑐 , 𝑚𝑒𝑚𝑏𝑒𝑟 𝑖𝑠 𝑙𝑜𝑛𝑔 𝑎𝑛𝑑 𝑚𝑜𝑑𝑒 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑖𝑠 𝑏𝑢𝑐𝑘𝑙𝑖𝑛𝑔
𝑟𝑚𝑖𝑛

𝜋2𝐸
𝐹𝑛 = 2
𝑘𝐿
𝑟
𝐹𝑛
𝐹𝑎 =
Ω

23
Ω=
12

12𝜋 2 𝐸
𝐹𝑎 = 2
𝑘𝐿
23 𝑟
NSCP 2010/2015 (Allowable Stress Design and Load and Resistance Factor Design):

In NSCP 2010/2015, the flanges and web are classified as either compact, non-compact or slender based on limiting
width-thickness ratios for compression elements given in Table 502.4.1 (page 576). This must be identified first as there
is a different formula for each cases.

(3) Uniform compression in flanges of rolled I-shaped (10) Uniform compression in webs of doubly symmetric I-
sections, plates projecting from rolled I-shaped sections; shaped sections.
outstanding legs of pairs of angles in continuous contact λ𝑝 = 𝑁. 𝐴.
and flanges of the channels 𝐸
λ𝑟 = 1.49
λ𝑝 = 𝑁. 𝐴. 𝐹𝑦
𝐸 𝑕
λ𝑟 = 0.56 𝑤𝑕𝑒𝑛 ≤ λ𝑟
𝐹𝑦 𝑡𝑤
𝒄𝒐𝒎𝒑𝒂𝒄𝒕 𝒐𝒓 𝒏𝒐𝒏 − 𝒄𝒐𝒎𝒑𝒂𝒄𝒕
𝑏 𝑏𝑓
𝑤𝑕𝑒𝑛 = ≤ λ𝑟 𝑕
𝑡 2𝑡𝑓 𝑤𝑕𝑒𝑛 > λ𝑟
𝒄𝒐𝒎𝒑𝒂𝒄𝒕 𝒐𝒓 𝒏𝒐𝒏 − 𝒄𝒐𝒎𝒑𝒂𝒄𝒕 𝑡𝑤
𝒔𝒍𝒆𝒏𝒅𝒆𝒓
𝑏 𝑏𝑓
𝑤𝑕𝑒𝑛 = > λ𝑟
𝑡 2𝑡𝑓
𝒔𝒍𝒆𝒏𝒅𝒆𝒓
Compressive Strength for Flexural Buckling of Members Without Slender Elements

ALLOWABLE COMPRESSIVE STRENGTH (ASD)

𝑃𝑛
𝑃𝑎 = ; Ω = 1.67
Ω
ULTIMATE/DESIGN COMPRESSIVE STRENGTH (LRFD)

𝑃𝑢 = φ𝑃𝑛 ; φ = 0.90
Compressive Strength for Flexural Buckling of Members With Slender Elements
 Unstiffened Element

Stiffened Element

Take note: If flange is slender,

compute for Qs; If web is
slender, compute for Qa; If
both is slender, solve for both.
Take note: For hot-rolled W-shaped sections. Use
number 1.
Computing Aeff

𝐼𝑓 𝑏𝑒 < 𝑏

𝐴𝑒𝑓𝑓 = 2 𝑏𝑓 𝑡𝑓 + 𝑏𝑒 𝑡𝑓
Problem no 1
• A W12 x 50 is used as a column to support an axial compressive load
of 145 kips. The length is 20 feet, and the ends are pinned. Without
regard to load or resistance factors, investigate this member for
stability. (The grade of steel need not be known: The critical buckling
load is a function of the modulus of elasticity, not the yield stress or
ultimate tensile strength.)
Problem no 2
• A W14 x 74 of A992 steel has a length of 20 feet and pinned ends.
Investigate the column for local stability. Compute the design
compressive strength for LRFD and the allowable compressive
strength for ASD.
Problem no 3
• Determine the LRFD design strength and the ASD allowable strength
for the compression member shown with Fy = 50ksi
Problem no 4
• Determine the LRFD design strength and the ASD allowable strength
for the compression member shown with Fy = 50ksi