demystifying cold‐formed steel torsion analysis for design · 2020. 10. 30. · demystifying...
TRANSCRIPT
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Demystifying Cold‐Formed SteelTorsion Analysis for Design
Bob Glauz, PERSG Software, Inc.
October 29, 2020
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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
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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
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Residential/Hospitality
Commercial
Municipal
Cold-Formed Steel Experts, LLC
Design Excellence
Innovative Detail
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Demystifying Cold‐Formed SteelTorsion Analysis for Design
Bob Glauz, PERSG Software, Inc.
October 29, 2020
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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 [email protected] 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
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Copyright
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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
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Outline
• Introduction• AISI Design Requirements• Pure Torsion• Warping Torsion• Warping & St. Venant Torsion• Design Examples• CFS® Software• Other Analysis Software
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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
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AISI Design Requirements
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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
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Bimoment Stresses
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σBwC
B EC ϕ
M
M
d
B=MdB σ w dA
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Bimoment Determination
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AISC Design Guide 9
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Bimoment Determination
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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
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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
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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
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Pure Torsion Diagrams
Torque
Twist
Mt Mt
−Mt
−MtL/GJ
Torque
Twist
−Mta/L
Mtb/L
a b
−Mtab/GJL
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Pure Torsion Diagrams
Torque
Twist
mt
−mtL/2
−mtL²/8GJ
Torque
Twist
−mtL/3
mtL/6
−0.06415mtL²/GJ
mtL/2
mt0
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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
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Questions?
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Warping Torsion
T
T
• Plane sections do not remain plane• Predominant for most open thin‐walled shapes
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Warping PropertiesAISI S100‐16 Commentary
Normalized Unit Warping
w w1A w t ds S W t ds
Warping Statical Moment
(in2) (in4)
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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
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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
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Flexure & Warping Diagrams
Deflection
P
−PL³/3EIx
Shear
−P
Moment
−PL
Twist
Mt
−MtL³/3ECw
Torque
−Mt
Bimoment
−MtL
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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
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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
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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
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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
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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)
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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
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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
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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
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Questions?
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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″
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AISI Design Manual Example II‐11
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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
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Bimoment using AISC DG9 Charts
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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
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Bimoment using Beam Tables
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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 ˗ /
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Bimoment using Proportioned Warping
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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 ˗ /
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Example II‐11 Bimoment Comparison
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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
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Example – Strength Reduction
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+ =
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
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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
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CFS® Software Examples
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Torsion Reference Paper
46
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Thank You!
Questions?
-
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