Demystifying Cold‐Formed SteelTorsion Analysis for Design

Bob Glauz, PERSG Software, Inc.

October 29, 2020

AnnouncementsDecember webinar – date/time TBD

2

Natasha Zamani, Ph.D., P.E.Code & Standards Senior Manager | Modular SystemsHilti North America

“Structural Considerations for Openings in Metal Deck”

3

4

Virtual Expo ‐ Upcoming

Virtual Expo – On Demand

5

[1]

[2]

[3]

[4]Watch whenever you want!Free for CFSEI members!

Register for the remaining Expo at:https://www.cfsei.org

Residential/Hospitality

Commercial

Municipal

Cold-Formed Steel Experts, LLC

Design Excellence

Innovative  Detail

Demystifying Cold‐Formed SteelTorsion Analysis for Design

Bob Glauz, PERSG Software, Inc.

October 29, 2020

PDH Certificate

8

The Steel Framing Alliance/CFSEI is a registered continuing education provider with several organizations. 

This program is registered for continuing professional education. As such, it does not include content that may be deemed or construed to be an approval or endorsement by the SFA/CFSEI /AISI of any material of construction or any method or manner of handling, using, distributing, or dealing in any material or product. Questions related to specific materials, methods, and services will be addressed at the conclusion of this presentation. 

Professional Development Hours earned upon completion of this program will be recorded for any participant that has indicated their attendance on the webinar sign in sheet and returned it to webinar@CFSEI.org at the conclusion of the webinar.

Certificates will be sent to participants within two weeks of the end of the webinar.

1.5 PDHFBPE  –1.5 PDHFBPE Provider # 0005013 

Copyright

9

Copyright Material

This presentation is protected by US and International copyright laws.

Reproduction, distribution, display and use of the presentation without written permission of the speaker and the Steel Framing Alliance is prohibited.

Steel Framing Alliance/CFSEI 2020

Outline

• Introduction• AISI Design Requirements• Pure Torsion• Warping Torsion• Warping & St. Venant Torsion• Design Examples• CFS® Software• Other Analysis Software

10

Introduction• Torsion is common in cold‐formed steel members• Try to reduce torsional loads and effects• Designing for torsion is unique

• Review torsion fundamentals• Learn about warping torsion characteristics• Provide tools to simplify torsion design• Develop a sense for torsion distribution based on knowledge of flexural behavior

Objectives

11

AISI Design Requirements

12

S100‐16 Section H4

M RM

Rf _

f f

fM y

I

fBwC

Mx Moment EIxν′′ varies along length of member

B Bimoment ECwϕ′′ varies along length of member

Bimoment Stresses

13

σBwC

B EC ϕ

M

M

d

B=MdB σ w dA

Bimoment Determination

14

AISC Design Guide 9

Bimoment Determination

15

Pure (St. Venant) Torsion

• Plane sections remain plane• Only occurs with closed or solid circular cross‐sections• Predominant for other closed shapes• Predominant for angles, tees, cruciform (Cw≈0)

𝜙𝑇𝐿𝐺𝐽

T

T

φ

L

16

Pure Torsion

𝐽4𝐴 𝑡

𝑆

Closed sections

t

𝐽 2𝜋𝑟 𝑡

Cylindrical tubes

Open sections

𝐽13 𝑡 𝑑𝑠 ⅓ 𝛴 𝑏𝑡

𝜏𝑇

2𝐴 𝑡

𝜏𝑇 𝑟

𝐽

𝜏𝑇 𝑡

𝐽

𝑇 𝐺𝐽𝜙 17

t

b

Pure Torsion Distribution

Area = ΔT

Slope = T′ = mtArea = −ΔGJφ

Slope = φ′ = −T/GJ

Torsion Intensity (k‐in/in)

Internal Torque (k‐in)

Twist Angle (rad)

mt = −GJφ″

T = −GJφ′

φ

18

Pure Torsion Diagrams

Torque

Twist

Mt Mt

−Mt

−MtL/GJ

Torque

Twist

−Mta/L

Mtb/L

a b

−Mtab/GJL

19

Pure Torsion Diagrams

Torque

Twist

mt

−mtL/2

−mtL²/8GJ

Torque

Twist

−mtL/3

mtL/6

−0.06415mtL²/GJ

mtL/2

mt0

20

Flexure – Pure Torsion AnalogyUsing Beam Tables with Pinned Supports

Flexure Pure Torsion

Pinned Support Twisting Restraint

Concentrated Load P Concentrated Torsion MtUniform Load w Uniform Torsion mtReaction Force R Reaction Torque RtInternal Shear V Internal Torque T

Internal Moment M Proportional to Twist −GJφ

21

22

Questions?

Warping Torsion

T

T

• Plane sections do not remain plane• Predominant for most open thin‐walled shapes

23

Warping PropertiesAISI S100‐16 Commentary

Normalized Unit Warping

w w1A w t ds S W t ds

Warping Statical Moment

(in2) (in4)

24

Warping Torsion Distribution

Area = ΔTw

Slope = Tw′ = mtArea = ΔB

Slope = φ′ = δ/Wn

Torsion Intensity (k‐in/in)

Internal Torque (k‐in)

Twist Angle (rad)

mt = ECwφ″″

Tw = ECwφ‴

φ

Bimoment (k‐in²)

B = ECwφ″

Warping Displacement (in)

δ = Wnφ′

Slope = B′ = TwArea = Δδ∙ECw/Wn

Slope = δ′ = BWn/ECw = εwArea = Δφ∙Wn/δ

25

Flexure – Warping AnalogyFlexure Warping Torsion

Differential equation EIxν″″ = w(z) Differential equation ECwφ″″ = mt(z)

Moment of inertia Ix = ∫ y² dA in⁴ Warping constant Cw = ∫ Wn² dA in⁶

First moment of area Q = ∫ y dA in³ First sectorial moment Sw = ∫ Wn dA in⁴

Section coordinate y in Sectorial coordinate Wn in²

Deflection ν in Twist φ rad

Load intensity w k/in Torsion intensity mt k‐in/in

Bending moment Mx = EIxν″ k‐in Warping bimoment B = ECwφ″ k‐in²

Bending stress σb = Mxy/Ix ksi Warping stress σw = BWn/Cw ksi

Stress resultant Mx = ∫ σby dA k‐in Stress resultant B = ∫ σwWn dA k‐in²

Shear force V = EIxν‴ k Warping torque Tw = ECwφ‴ k‐in

Shear stress τ = VQ/Ixt ksi Warping shear stress τw = TwSw/Cwt ksi

26

Flexure & Warping Diagrams

Deflection

P

−PL³/3EIx

Shear

−P

Moment

−PL

Twist

Mt

−MtL³/3ECw

Torque

−Mt

Bimoment

−MtL

27

Flexure & Warping Diagrams

Mt

Twist

L/2

Torque−Mt/2

Mt/2

Bimoment

MtL/4

L/2

−MtL³/48ECw

P

Deflection

L/2

Shear−P/2

P/2

Moment

PL/4

L/2

−PL³/48EIx

28

Flexure & Warping Diagrams

mt

Twist

Torque−mtL/2

mtL/2

Bimoment

mtL²/8

−5mtL⁴/384ECw

Deflection

Shear−wL/2

wL/2

Moment

wL²/8

−5wL⁴/384EIx

w

29

Flexure & Warping Diagrams

Twist

Torque

−5mtL/8

3mtL/8

Bimoment

9mtL²/128

−mtL⁴/185ECw

Deflection

Shear

−5wL/8

3wL/8

Moment

9wL²/128

−wL⁴/185EIx

wmt

−wL²/8 −mtL²/8

30

Flexure – Warping Analogy

Flexure Warping Torsion

Pinned Support Twisting Restraint

Fixed Rotation Warping Restraint

Concentrated Load P Concentrated Torsion MtUniform Load w Uniform Torsion mtReaction Force R Reaction Torque RtInternal Shear V Internal Torque TwInternal Moment M Internal Bimoment B

Moment of Inertia Ix Warping Constant CwDeflection Δ Twist Angle φ

Using Beam Tables

31

Warping & St. Venant Torsion

Small L/a – mostly warping torsionLarge L/a – mostly St. Venant torsion

Relative torsional stiffness:Warping torsion: ECw/L²St. Venant torsion: GJ

aEC

GJSection parameter(dimension of length)

32

Warping & St. Venant Torsion

aEC

GJ33

Tw+Tsv

−mtL/2

mtL/2

−5mtL⁴/384ECw

mtL²/8

Tsv @ L/a=5Tsv @ L/a=2

Tsv @ L/a=1

L/a

0.0

0.2

0.4

0.6

0.8

1.0

0.1 1 10

B / B

wo

L/a

Concentrated Torsion

Uniform Torsion

Warping & St. Venant TorsionSFIA Sections

L=10h L=20h

34

Typical Sections

Section Cw (in⁶) J (in⁴) a (in) L/a, L=80” L/a, L=160”

800S200‐27 2.448 0.000098 254.85 0.314 0.618

800S200‐33 2.971 0.000179 207.74 0.385 0.770

800S200‐43 3.797 0.000395 158.09 0.506 1.012

800S200‐54 4.663 0.000775 125.07 0.640 1.280

800S200‐68 5.712 0.001537 98.30 0.814 1.628

800S200‐97 7.684 0.004381 67.53 1.185 2.370

800S200‐118 8.981 0.007872 54.46 1.469 2.938

aEC

GJ

Cw α t L/a α tJ α t³ a α 1/t

35

36

Questions?

Warping & St. Venant TorsionDifferential equationECwφ″″ − GJφ″ = mt(z)

SolutionFor trapezoidal loading: mt(z) = mt + Δmtz/Lφ = C1 + C2z +   C3∙sinh(z/a)     +   C4∙cosh(z/a) − mtz2/2GJ − Δmtz3/6GJLφ′ =        C2 +  C3/a∙cosh(z/a) +  C4/a∙sinh(z/a) − mtz/GJ − Δmtz2/2GJLφ″ =                  C3/a2∙sinh(z/a) + C4/a2∙cosh(z/a) − mt/GJ − Δmtz/GJLφ‴ =                 C3/a3∙cosh(z/a) + C4/a3∙sinh(z/a) − Δmt/GJLφ″″ =                C3/a4∙sinh(z/a)  + C4/a4∙cosh(z/a)

Torque: T = Tw + Tsv = ECwφ‴ − GJφ′

Bimoment: B = ECwφ″

37

AISI Design Manual Example II‐11

38

w = 10 plf

25’ simple span with mid‐span brace 𝐺 11300 ksi 𝐽 0.00102 in 1.78

𝐸 29500 ksi 𝐶 11.15 in 𝑎 169 in

Bimoment using AISC DG9 Charts

39

Loading #2 𝑡 1.05 in 0.000875 k˗in/in

𝜃 0.30

𝜃 . 22.8 10 rad/in

𝐵 𝐸𝐶 𝜃 7.49 k˗in

Loading #3 𝑇 0.000875 300 0.164 k˗in

𝜃 0.35

𝜃 . 29.5 10 rad/in

𝐵 𝐸𝐶 𝜃 9.69 k˗in𝐵 7.49 9.69 2.20 k˗in

Bimoment using Beam Tables

40

𝑚 1.05 in 0.000875 k˗in/in

𝐵 2.46 k˗in

𝐿 150 in

𝜙

(slightly conservative, L/a=0.89)

0.1202 ˗ /

Bimoment using Proportioned Warping

41

𝑚 1.05 in 0.000875 k˗in/in

𝐿 150 in

𝐺𝐽𝜙

Pure Torsion (one span)

0.0041 ˗ /

0.1243 ˗ /

𝑚 𝑚 ..

0.000846 k˗in/in (96.7%) 

𝐵 2.38 k˗in

𝑚 𝑚 ..

0.000029 k˗in/in (3.3%) 

0.1202 ˗ /

Example II‐11 Bimoment Comparison

42

Method BimomentAISI Design Manual 2.46 k‐in²AISC Design Guide 9 Charts 2.20 k‐in²Beam Table (L/a=0.86) 2.46 k‐in²Proportioned Warping 2.38 k‐in²CFS® (theoretically correct) 2.37 k‐in²

Example – Strength Reduction

43

+ =

fM y

If

BwC f

M yI

BwC

f9.375 3.727

10.30 3.39 ksi f2.37 8.82

11.15 1.87 ksif 3.39 1.87 5.26 ksi

f9.375 4.254

10.30 3.87 ksi f2.37 6.99

11.15 1.49 ksif 3.87 1.49 5.36 ksi

f9.375 4.470

10.30 4.07 ksi f2.37 5.28

11.15 1.12 ksif 4.07 1.12 5.19 ksi

f9.375 4.470

10.30 4.07 ksi f2.37 3.69

11.15 0.78 ksif 4.07 0.78 3.29 ksi

𝑅4.07 ksi5.36 ksi 0.759

𝑀 62.3 k˗in

𝑀 𝑅𝑀 47.3 k˗in 9.375 k˗in (OK)

𝑤 10 plf ..

50.4 plf

𝑅f _

f f

Analysis Software• Finite Element Analysis

• Determine torsional stresses and displacements• Cannot obtain internal forces (B, Tw, Tsv) for design

• Frame Structural Analysis Software• Most handle pure torsion only (no warping torsion)• Most not sufficient for cold‐formed steel torsion response

• RISA‐3D• 6 DOF, 1st order analysis, warping for AISC sections only, warping‐fixed ends only

• VisualAnalysis• 6 DOF, 1st order analysis, isolated warping boundary conditions

• MASTAN2• 7 DOF, 2nd order analysis, cubic polynomial approximation

• RF‐/STEEL• 7 DOF, 2nd order analysis, solution method unclear

• CFS®• Multi‐span beams, 1st order analysis, rigorous solution, code checks

44

45

CFS® Software Examples

Torsion Reference Paper

46

Thank You!

Questions?

glauz@rsgsoftware.com

AnnouncementsDecember webinar – date/time TBD

48

Natasha Zamani, Ph.D., P.E.Code & Standards Senior Manager | Modular SystemsHilti North America

“Structural Considerations for Openings in Metal Deck”

49

50

Virtual Expo ‐ Upcoming

Virtual Expo – On Demand

51

[1]

[2]

[3]

[4]Watch whenever you want!Free for CFSEI members!

Register for the remaining Expo at:https://www.cfsei.org

Residential/Hospitality

Commercial

Municipal

Cold-Formed Steel Experts, LLC

Design Excellence

Innovative  Detail