Designing a Cold-Formed Steel Beam Using AISI S100-16

Designing a Cold-Formed Steel
Beam Using AISI S100-16
Understanding the design process using the
Direct Strength Method
Brooks H. Smith, CPEng, PE, MIEAust, NER, RPEQ
brooks.smith@clearcalcs.com

Outline
• Introduction
• How CFS is Unique
• Designing a CFS Beam
• Flexural Capacity
• Shear Capacity
• Bearing Capacity
• Load Interactions
• Deflection
• Example Beam Calculations
• Conclusion & Questions
214 February 2019 ClearCalcs.com | FEA Structural Design in the Cloud

Introduction – About the Presenter
• Licensed Professional Engineer (TX)
• MSCE, CPEng
• Currently the lead engineering developer for ClearCalcs
• Launching in the US, recently released CFS beam and column/stud calculators
• 8 years of previous experience in:
• Structural engineering R&D consulting, specialising in cold-formed steel
• Research fellowship in system behaviour of thin-walled steel
• Forensic structural engineering, specialising in reinforced and PT concrete
3
Brooks H. Smith

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Intro Video Hyperlink

Introduction – Today’s Goals
• To be able to design a cold-formed steel beam to AISI S100-16
• Cee or Zee sections bent about strong axis
• Negligible holes in the cross-section
• Direct Strength Method, assuming prequalified
• Detailing will only be broadly addressed
• We’ll distribute this slide deck and video after the webinar
• Please ask quick questions as I go – best to answer while on the topic
• Please ask using the “Q&A” feature, NOT the chat/messaging feature
• I’ll save involved questions until the end
• Note: Everything today is based on the codes
• We are not on the AISI committee, are not communicating any special
knowledge

Outline
• Introduction
• Shear Capacity
• Deflection

How CFS is Unique
• Buckling is a major issue
• Most sections will buckle before yielding
• Bearing / web crippling can easily control
• Buckling of the web for either bottom supports or top point loads
• Design may require finite element/strip analysis
• But this only needs to be done once, and can be avoided
• Highly-customizable shapes
• So design methodology can be used
for any cross-section

Buckling in Cold-Formed Steel
• Hot-rolled steel classifies sections as compact, noncompact, or
slender – and requires extra equations for “slender”
• In cold-formed steel, “slender” checks always need to be done
• Local, distortional, or global buckling modes
• Global encompasses both lateral and lateral-torsional buckling
• Stiffeners function to mitigate buckling

Bearing / Web Crippling
• If the web isn’t directly restrained either at supports or under
point loads, web crippling must be checked
• In hot-rolled steel, checks are simple and rarely control
• But in CFS, they may commonly control and are highly-dependent upon
the precise cross-section and arrangement of forces
https://doi.org/10.1016/j.tws.2012.01.003

Finite Element / Strip Analysis
• The Direct Strength Method, which is the preferred method in
AISI S100-16, requires a rational analysis that usually takes the
form of the Finite Strip Method
• Generally only needs to be done once for a section, and
alternate methods do exist
https://dx.doi.org/10.1016/j.tws.2014.01.005
https://doi.org/10.1016/j.tws.2013.09.004

Highly-Customizable Shapes
• Standard sections available, but custom sections also economical
• SFIA (Steel Framing Industry Alliance); SSMA (Steel Stud Manufact’rs
Assoc.)
https://commons.wikimedia.org/wiki/File:Zg-prof.jpg

Outline
• Introduction
• Shear Capacity
• Deflection

Designing a Cold-Formed Steel Beam
• Calculate your demands by ASCE 7 or the IBC
• Limit states which must be checked:
• Positive moment flexural capacity (midspans)
• Negative moment flexural capacity (supports)
• Shear capacity
• Web crippling capacity
• Load interaction limits
• Deflection

Geometric Derivatives
• First, make sure you have some of the basic geometric
properties:
• 𝐼𝐼𝑥𝑥 = moment of inertia about the strong axis
• 𝐼𝐼𝑦𝑦 = moment of inertia about the weak axis
• 𝐴𝐴 = gross cross-sectional area
• 𝑥𝑥𝑜𝑜 = distance from centroid to shear center along x-axis
• 𝑟𝑟𝑥𝑥, 𝑟𝑟𝑦𝑦 = radii of gyration about centroidal principal axes
• 𝑟𝑟𝑜𝑜 = polar radius of gyration about the shear center = 𝑟𝑟𝑥𝑥
2
+ 𝑟𝑟𝑦𝑦
2
+ 𝑥𝑥𝑜𝑜
2
• 𝐽𝐽 = St Venant’s torsion constant
• 𝐶𝐶𝑤𝑤 = Torsional warping constant

Flexural Capacity – Global Buckling (F2.1)
• Yielding and global buckling considered in one equation:
• 𝑆𝑆𝑓𝑓 is relative to extreme compression fiber, 𝑆𝑆𝑓𝑓𝑓𝑓 is relative to first yield
fiber
• Usually the same. May differ, for example, in C-sections bent about the weak axis
• 𝐹𝐹𝑛𝑛 is where global buckling is considered, depends upon 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐 being
calculated
• May alternatively be determined via Finite Strip Analysis or Effective Width
Method

Flexural Capacity – Global Buckling (F2.1.x)
• Main global buckling parameter 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐 depends upon section:
• Cee sections:
• Zee sections:
where 𝑀𝑀𝐴𝐴, 𝑀𝑀𝐵𝐵, 𝑀𝑀𝐶𝐶 are moments at quarter points
where 𝐿𝐿𝑦𝑦, 𝐿𝐿𝑡𝑡 are unbraced lengths, and 𝐾𝐾𝑦𝑦, 𝐾𝐾𝑡𝑡 are usually both = 1

Flexural Capacity – Inelastic Reserve (F2.4.2)
• Allows small amounts of localized yielding that doesn’t affect
stability
• Optional provision; certain connections or member types may forbid it
• Only allowed if 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐 > 2.78𝑀𝑀𝑦𝑦, where 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑆𝑆𝑓𝑓 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐
• 𝑀𝑀𝑝𝑝 is the member plastic moment, equal to 𝑍𝑍𝑓𝑓 𝐹𝐹𝑦𝑦
• Generally not given in manufacturers’ information, but may be
calculated by setting compression area equal to tension area

Flexural Capacity – Finite Strip
• Local and distortional buckling critical buckling capacities (𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐
and 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐) most easily determined via Finite Strip Method:
• CUFSM (free), from Johns Hopkins University, or
• THIN-WALL (paid), from the University of Sydney
• Critical capacities generally do not depend upon length
• As long as the beam is longer than about 24 to 30 inches

Flexural Capacity – Finite Strip 2

Flexural Capacity – Local Buckling (F3.2)
• Local buckling involves the corners of the cross-section staying
still, while the flat portions bend
• Calculations account for local buckling’s interaction with global buckling
• Usually occurs with a half-wavelength of about 4-10 inches

Flexural Capacity – Local Buckling IR (F3.2.3)
• Inelastic reserve capacity also possible in local buckling, provided that
𝜆𝜆𝑙𝑙 ≤ 0.776 and 𝑀𝑀𝑛𝑛𝑛𝑛 ≥ 𝑀𝑀𝑦𝑦
• That is, provided global buckling occurs post-yield and local doesn’t control
• Calculation depends upon if first yield is in tension or compression:
• First yield in compression (or if theoretically simultaneous with tension):
• First yield in tension:
where 𝐶𝐶𝑦𝑦𝑦𝑦 = 3 and 𝑀𝑀𝑦𝑦𝑦𝑦 = 𝑆𝑆𝑓𝑓 𝐹𝐹𝑦𝑦 (i.e. yield in compression fiber)

Flexural Capacity – Distortional Buckling (F4)
• Distortional buckling involves movement of the corners of the
cross-section, but where not all corners move together
• Does not assume an interaction with global buckling
• Usually occurs with a half-wavelength of about 16-30 inches

Flexural Capacity – Dist. Buckling IR (F4.3)
• Distortional buckling may also include inelastic reserve, provided
that 𝜆𝜆𝑑𝑑 ≤ 0.673
• Again, calculation depends upon the nature of first yield:
• First yield in compression:
• First yield in tension:
where 𝐶𝐶𝑦𝑦𝑦𝑦 = 3 and 𝑀𝑀𝑦𝑦𝑦𝑦 = 𝑆𝑆𝑓𝑓 𝐹𝐹𝑦𝑦 (i.e. yield in compression fiber)

Flexural Capacity – Overall (Sec. F)
• Overall flexural capacity is minimum of local, distortional, and
global buckling capacities
• Note: AISI S100-16 lists 𝜙𝜙𝑏𝑏 in every clause, but 𝜙𝜙𝑏𝑏 always equals 0.90
here
𝜙𝜙𝑏𝑏 𝑀𝑀𝑛𝑛 = 0.90 ∗ min(𝑀𝑀𝑛𝑛𝑛𝑛, 𝑀𝑀𝑛𝑛𝑛𝑛, 𝑀𝑀𝑛𝑛𝑛𝑛)

• Based upon 𝐴𝐴 𝑤𝑤 = area of flat portion of web (i.e. without corner
radii)
• 𝑉𝑉𝑐𝑐𝑐𝑐 is comparable to 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐, 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐, 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐, but calculated analytically
• For unreinforced webs, 𝑘𝑘𝑣𝑣 = 5.34
• For reinforced webs:
Shear Capacity – Shear Buckling (G2.3)
https://dx.doi.org/10.1016/j.engstruct.2012.07.029
(𝜇𝜇 = 𝜐𝜐 = 0.3)

Shear Capacity – Without Stiffeners (G2.1)
• Based upon shear yield and buckling slenderness:
• Overall result: 𝜙𝜙𝑣𝑣 𝑉𝑉𝑛𝑛 = 0.95 ∗ 𝑉𝑉𝑛𝑛

Shear Capacity – With Stiffeners (G2.2)
• Assuming minimum shear web stiffeners, with spacing not
exceeding twice the web depth
• These equations are essentially identical to flexural local buckling!
• Overall result: 𝜙𝜙𝑣𝑣 𝑉𝑉𝑛𝑛 = 0.95 ∗ 𝑉𝑉𝑛𝑛

Web Crippling Capacity – Overview (G5)
• All based upon just one equation:
• Accounts for effects of web angle (𝜃𝜃), corner radius (𝑅𝑅), bearing length
(𝑁𝑁), and web height slenderness (ℎ)
• The key is in all those 𝐶𝐶𝑥𝑥 coefficients
• Different tables for Cee, Zee, built-up I-sections, hats, and steel decks
• Note that equation and tables are per web, so box sections, nested Zees,
etc would multiply 𝑃𝑃𝑛𝑛 by 2
• 𝜙𝜙𝑤𝑤 is not constant and also looked up in the tables!

Web Crippling Capacity – Cees (Table G5-2)

Web Crippling Capacity – Zees (Table G5-3)

Load Interaction – Flexure & Shear (H2)
• Calculation depends upon whether shear stiffeners exist or not:
• Without shear stiffeners:
• With shear stiffeners (only necessary if ⁄�𝑀𝑀 𝑀𝑀𝑎𝑎𝑎𝑎𝑎𝑎 > 0.5 and ⁄�𝑉𝑉 𝑉𝑉𝑎𝑎 > 0.7):
• Notes:
• �𝑀𝑀, �𝑉𝑉 are demands; 𝑉𝑉𝑎𝑎 = ϕ𝑣𝑣 𝑉𝑉𝑛𝑛; 𝑀𝑀𝑎𝑎𝑎𝑎𝑎𝑎 = 𝜙𝜙𝑏𝑏 𝑀𝑀𝑛𝑛𝑛𝑛𝑛𝑛 is local buckling without global
consideration:
𝜆𝜆𝑙𝑙 = ⁄𝑀𝑀𝑦𝑦 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐
• If 𝜆𝜆𝑙𝑙 ≤ 0.776: 𝑀𝑀𝑛𝑛𝑛𝑛𝑛𝑛 = 𝑀𝑀𝑦𝑦
• If 𝜆𝜆𝑙𝑙 > 0.776: 𝑀𝑀𝑛𝑛𝑛𝑛𝑛𝑛 = 1 − 0.15
𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐
𝑀𝑀𝑦𝑦
0.4
𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐
𝑀𝑀𝑦𝑦
0.4
𝑀𝑀𝑦𝑦
• Additionally, if there are web stiffeners, 𝑀𝑀𝑎𝑎𝑎𝑎𝑎𝑎 = 𝜙𝜙𝑏𝑏 ∗ min(𝑀𝑀𝑛𝑛𝑛𝑛𝑛𝑛, 𝑀𝑀𝑛𝑛𝑛𝑛)

Load Interaction – Flexure & Web Crip. (H3)
• Applies for both supports (negative moment) and point loads (usually
positive moment)
• 𝜙𝜙 = 0.9
• For unreinforced single webs:
• An exception exists for members spaced ≤ 10 inches o.c. with lateral bracing
• Back-to-back C-sections:
• Nested Z-sections:
• Note that a number of connection and geometric restrictions apply (see H3(c) )

Deflection
• Important difference between effective and gross moments of
inertia
• Conservatively, you may use the 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 values given by manufacturers
• There are long equations in the Cold-Formed Steel Design Manual
• More practically, only slightly more conservative is the following
equation:
• 𝑀𝑀 is the moment demand due to service loads being considered (max 𝑀𝑀𝑦𝑦)
• 𝑀𝑀𝑑𝑑 = 𝑀𝑀𝑛𝑛 except that 𝑀𝑀𝑛𝑛 is recalculated replacing all instances of 𝑀𝑀𝑦𝑦 with 𝑀𝑀
• Manufacturers’ 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 values are often equal to this equation with 𝑀𝑀 = maximum
reasonable DL+LL
• Note: 𝐸𝐸 = 29500 ksi

Beams - Wrapping It Up
• This represents the general requirements for cold-formed steel
beams
• However, there are a number of alternative equations, which
generally give a little more capacity, for specific types of systems:
• Beams with one flange through-fastened to deck or sheathing (I6.2.1)
• Beams with one flange through-fastened to standing-seam roof (I6.2.2)
• Steel racks (ANSI MH16.1)
• Diaphragm systems (AISI S310-deck, AISI S240-flat sheet, AISI S400-
seismic)
https://www.steelconstruction.info/File:L1_Fig9.png

Outline
• Introduction
• Shear Capacity
• Deflection

Example Beam #1 – Simply Supported
36
7’-2” = 86 inches
• Office building floor purlin
• 16” tributary width
• No transverse shear reinforcement
• Laterally unbraced at 24 inches o.c.
• Torsionally unbraced for full span
LL = 40 psf
DL = 15 psf
14 February 2019
Showing methods and formulas
using ClearCalcs’s new cold-formed
steel calculator
ClearCalcs.com | FEA Structural Design in the Cloud
362S137-54 [33ksi]
3.625”
1.375”
.054”

Example Beam #2 – Complex Beam
37
LL = 40 psf
DL = 15 psf
• Office building floor purlin
• No transverse shear reinforcement
• Tributary width of 16 inches o.c.
• Bottom flange and torsional bracing
at 48 inches o.c.
72 inches 120 inches 24 inches
14 February 2019
Ex #1 Beam @ 14 ft
362S162-68 [50ksi]
3.625”
.068”
1.625”

Outline
• Introduction
• Shear Capacity
• Deflection

Summing It Up
• CFS engineering design is unique because of:
Buckling • Web Crippling • Finite Strip Analysis • Customizable
Shapes
• Beam design checks at all sections include:
• Flexure: Global buckling → FSM → Local Buckling → Distortional
buckling
• Shear: Shear yield → Shear buckling → With or without stiffeners
• Bearing: Plug in coefficients, 𝜙𝜙𝑤𝑤 for end/interior and 1- / 2-flange
loading
• Load interaction: Flexure+Shear and Flexure+Web crippling
• Deflection: Effective 2nd moment of area → Long-term factor
• We performed examples with simply supported and complex
beams

Questions?
4014 February 2019
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Appendix
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14 February 2019

Designing a Cold-Formed Steel Beam Using AISI S100-16

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