1. Torsional Resistance of Standard Steel Shapes
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Torsional Resistance of Standard Steel Shapes
Background
Theory
Design Approaches
Worked Example
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

2. Torsional Resistance of Standard Steel Shapes by Carmen Chun
25/04/2017
Background
Limited guidance in the design for torsion in steel structures
CSA S16 has half a page dedicated to torsion
Torsion commonly a secondary effect to bending, shear, etc.
Situations where torsion plays a significant role in the design
Analysis for torsion covered in textbooks
Design for torsion addressed by publications by CISC and AISC
25/04/2017
Torsional Resistance of Standard Steel Shapes by Carmen Chun

3. Torsional Resistance of Standard Steel Shapes by Carmen Chun
25/04/2017
What is torsion?
The question should be how is torsion transferred?
Torsion is carried as shear stresses in a cross-section
Cross-section rotates through an angle
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

4. Torsional Resistance of Standard Steel Shapes by Carmen Chun
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Shear Centre
The shear centre is the location in a cross-section where no torsion occurs
Forces acting through the shear centre will not cause torsional stresses
Shear centre does not have to be in the same location as the centroid
Need to determine the shear centre to evaluate the torsional stress of a section
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

5. Pure Torsion (St. Venant Torsion)
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Pure Torsion (St. Venant Torsion)
Pure torsional shear stresses are always present in torsion
No out-of-plane warping
Torsional shear stress is proportional to the radial distance from the centre of twist
Resistance a function of:
Shear modulus (G)
Torsional constant of cross-section (J)
First derivative of the angle of twist with respect to the z-axis, i.e. change in angle of rotation per unit length (θ’)
T = GJθ’
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

6. Torsional Constant (J)
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Torsional Constant (J)
J is the torsional constant of cross-section with units of length to the 4th power, i.e. mm4
For open sections:
For closed sections:
Fillets are typically ignored in sections such as single angles and structural tees ds
Includes all section plates for open sections
Includes the area enclosed by the closed sections t Ao
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

7. Torsional Resistance of Standard Steel Shapes by Carmen Chun
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Warping Torsion
When the cross-section is prevented or restrained from warp freely, longitudinal bending results
This is typical in open sections, where the cross-section no longer remains plane after twisting
Axial forces are induced in the section
Resistance is a function of:
Modulus of elasticity of steel (E)
Warping constant for the cross-section (Cw)
Third derivative of the angle of twist with respect to the z-axis (θ’”)
Tw = ECwθ”’
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

8. Warping Torsional Constant (Cw)
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Warping Torsional Constant (Cw)
In HSS, warping deformations are small and is generally taken as zero
Warping constant is calculated differently for various shapes.
CSA S-16 includes the values
For example, Cw for a W-shape:
Cw = Ifh2/2
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

9. Torsional Resistance of Standard Steel Shapes by Carmen Chun
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Torsional Analysis
Pure torsional shear stresses
Shear stresses due to warping
Normal stresses due to warping
Bending stresses from plane bending
Shear stresses from plane bending
Axial stress from axial load
Combine all the above stresses, but pay attention to direction!
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

10. Torsional Resistance of Standard Steel Shapes by Carmen Chun
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Design Approaches
Used closed sections
Use diagonal bracing
Make rigid end connections
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

11. Examples of Torsional Loading
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Examples of Torsional Loading
Spandrel beams
Beam framing into a girder on one side only
Unequal reactions on either side of a girder
Crane runway girders
Situations where loading or reaction acts eccentrically to the shear centre
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

12. Torsional Resistance of Standard Steel Shapes by Carmen Chun
25/04/2017
Worked Example
MC 460 x 63.5
Span, L = 3658 mm (12 ft)
wf = 52.5 kN/m
Fixed-fixed boundary conditions
Load acts through centroid of channel
Resolve to torsional moment and load applied through the shear centre
25/04/2017
Torsional Resistance of Standard Steel Shapes by Carmen Chun

13. Worked Example (cont’d)
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Worked Example (cont’d)
Torsional properties:
J = 513 x 103 mm4
a = 1072 mm
Cw = 227 x 103 mm4
Wno = 14.2 x 103 mm2
Wn2 = 6.7 x 103 mm2
Sw1 = 7.24 x 106 mm4
Sw2 = 5.62 x 106 mm4
Sw3 = 2.81 x 106 mm4
e0 =24.6 mm
Qf = x 103 mm3
Qw = x 103 mm3
Flexural properties:
Ix = 231 x 106 mm4
Sx = 1010 x 103 mm3
tf = 15.9 mm
tw = 11.4 mm
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

14. Worked Example (cont’d)
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Worked Example (cont’d)
Calculate bending stresses:
At support:
Mf = wfL2/12 = 58.5 kNm
Vf = wfL/2 = 96 kN
σb = Mf/Sx = 58 MPa
τbf = Vf Qf/Ixtf = 8.4 MPa
τbw = Vf Qw/Ixtw = 22.6 MPa
At midspan:
Mf = 27.8 kNm
σb = 29 MPa
At z/l = 0.2:
Vf = 57.8 kN
τbf = 5.1 MPa
τbw = 13.7 MPa
25/04/2017
Torsional Resistance of Standard Steel Shapes by Carmen Chun

15. Worked Example (cont’d)
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Worked Example (cont’d)
Calculate torsional stresses:
t = wfe = 2.5 kNm per m
L/a = 3.40
z/L
0.2
0.5 θ
+0.07
0.15
taL
/2GJ θ'
+0.14
GJ/t * 2/L
θ"
-1.0
-0.20
GJ/t* 2a/L
θ‘”
+0.46
-0.46
GJ/t* 2a2/L
25/04/2017
Torsional Resistance of Standard Steel Shapes by Carmen Chun

16. Worked Example (cont’d)
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Worked Example (cont’d)
Shear stresses due to pure torsion:
At support and midspan, 0.
At z/L = 0.2
14.1 MPa in the web
20 MPa in the flange
Stresses due to warping:
z/L
0.2
0.5
τw1
9 MPa
4.1 MPa
τw2
7 MPa
3.2 MPa
τw3
4.8 MPa
2.2 MPa
σwo
139MPa
-60 MPa
σw2
65 MPa
-28 MPa
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

17. Worked Example (cont’d)
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Worked Example (cont’d)
Maximum normal stress occurs at support at Point 2 in the flange
Maximum shear stress occurs at z/L = 0.2 at Point 3 in the web
Maximum rotation:
θ = taL/2GJ
θ = rad (0.69 deg)
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Torsional Resistance of Standard Steel Shapes by Carmen Chun

18. Torsional Resistance of Standard Steel Shapes by Carmen Chun
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Thank you.
Thank you for your attention
Questions?
25/04/2017
Torsional Resistance of Standard Steel Shapes by Carmen Chun