CIVL510
TORSIONAL RESISTANCE OF STANDARD STEEL SHAPES
Torsional Resistance of Standard Steel
Shapes – Background, Theory, and
Application
(Photo source: GES Tech Group, Inc.)
Prepared for:
Dr. Siegfried Stiemer
CIVL 510
Department of Civil Engineering
The University of British Columbia
Prepared by:
Carmen Chun, BASc EIT, LEED AP
64474042
MEng Student
Department of Civil Engineering
The University of British Columbia
Date:
April 7, 2011
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CARMEN CHUN
Abstract
In normal building construction, torsion is not typically a
governing failure load for which members are designed for.
However, the effects of torsional loading should not be neglected,
as torsion is induced when there is eccentric loading. This is
commonly seen at spandrel beams where the building façade is
supported off the side of the beam and applying a uniform
torsional load. Torsional effects also play an important role in
sizing of beams, as lateral torsional buckling is often a governing
failure mode in steel beams with no or minimal lateral support.
Torsion exists when a load is applied to member eccentrically from
its shear centre. Torsion is typically categorized into two forms:
St. Venant and warping. St. Venant (or pure torsion) is present in
all types of twisting effects as a shear stress. Warping is induced
when the cross-section does not remain plane, when the member is
restrained from warping at the supports. Normal stresses are
present in warping torsion. The effects of torsional loading are
determined by combining the stresses from bending and twisting.
This can then be compared to the factored strength to determine
the adequacy of the steel section in that application.
The resistance of standard steel shapes is determined by
calculating the torsional constants, J and Cw. These constants can
be calculated using torsional theory, however they are tabulated in
design guides, such as those provided by the CISC and AISC. The
resistance of standard steel shapes is typically governed by the
geometry.
Efficient design for torsion includes using closed cross-sections,
such as pipes and horizontal structural sections, and using bracing
to limit the amount of twisting in the section.
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CARMEN CHUN
Table of Contents
List of Figures
List of Figures ................................................................................. 2
Figure 1. Shear centre of common steel shapes (AISC, 2003) ....... 4
Figure 2. Warping torsion on I-beam (Salmon and Johnson, 1996) 5
Figure 3. Torsional loading from building facade (AISC, 2003) ... 7
Figure 4. Equations for lateral torsional resistance (CSA S16-09)10
Figure 5. W-shape (CISC, 2002) .................................................. 10
List of Equations ............................................................................. 2
1.0
Introduction .......................................................................... 3
2.0
Theory .................................................................................. 3
2.1
Shear Centre ..................................................................... 3
2.2
Pure Torsion (or St. Venant Torsion) ............................... 4
2.3
Warping Torsion .............................................................. 5
2.4
Asymmetry Parameter ...................................................... 6
2.5
Other Torsional Properties ............................................... 6
2.6
Torsional Functions .......................................................... 6
3.0
Analysis and Design Approaches ........................................ 7
3.1
Torsional Stress for Open Shapes .................................... 8
3.1.1.
Pure Torsional Shear Stresses ................................... 8
3.1.2.
Warping Shear Stresses............................................. 8
3.1.3.
Warping Normal Stresses ......................................... 8
3.1.4 Warping Stresses on Angles and Structural Tees .......... 8
3.2
Torsional Stress for Closed Shapes .................................. 9
3.3
Combining Stresses .......................................................... 9
4.0
Application of Theory – Lateral Torsional Buckling .......... 9
5.0
Conclusion ......................................................................... 11
List of Equations
Equation 1. Shear centre ................................................................. 4
Equation 2. y-coordinate of shear centre ........................................ 4
Equation 3. x-coordinate of shear centre ........................................ 4
Equation 4. Pure torsional resistance .............................................. 4
Equation 5. Torsional constant for open shapes ............................. 5
Equation 6. Torsional constant for closed shapes ........................... 5
Equation 7. Warping torsional resistance ...................................... 6
Equation 8. Warping torsional constant for a W-shape .................. 6
Equation 9. Asymmetry parameter ................................................. 6
Equation 10. Pure torsional shear stress for open shapes ............... 8
Equation 11. Warping shear stress for open shapes ........................ 8
Equation 12. Warping normal stress for open shapes ..................... 8
Equation 13. Combined normal stresses ......................................... 9
Equation 14. Combined shear stresses ............................................ 9
Equation 15. Typical resistance strength ........................................ 9
Equation 16. Torsional constant for W-shapes ............................. 10
Equation 17. Warping constant for W-shapes .............................. 10
References ..................................................................................... 12
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1.0 Introduction
Torsion is frequently considered a secondary effect that does not
significantly impact the design of a steel structure. There are very
few steel design standards that provide guidance for torsion design
of steel. However, there are occasions where torsion may be a
significant force for which it must be adequately resisted. It is also
important to be able recognize when torsion is present and
understand how torsional loads are resisted.
In the discussion of torsion, distinction must be made for strength
and stability analysis. Strength refers cross-sectional properties of
the member that offer resistance to the applied stresses; this is
commonly known as sectional analysis. A stability failure results
in a failure to maintain the initial configuration, such as lateral
torsional buckling in a beam.
This report will provide background and theory of torsion,
including analytical methods to determine torsional resistance,
worked examples, and a case study for torsional design.
CARMEN CHUN
To evaluate torsional stresses, the shear centre (the axis of rotation)
must be defined.
Torsion may be categorized into two types: pure torsion, often
called St. Venant torsion, and warping torsion, discussed below.
2.1
Shear Centre
The axis of which an object rotates about is defined as the shear
centre of a section. It is the point in the plane of a cross-section
where no twisting occurs. The shear centre does not necessarily
coincide with the centroid of a cross-section. However, if a shape
has a line of symmetry, the shear centre will lie on the line of
symmetry. For doubly symmetric cross-sections, such as round or
square hollow structural sections (HSS) and pipes, the shear centre
coincides with the centroid, as shown in Figure 1. For monosymmetric cross-sections, such as channels and tees, the shear
centre will lie on the axis of symmetry, as shown in Figure 1.
2.0 Theory
Torsion is the twisting of a solid object when subjected to torque
(turning force), such that the object rotates about an axis. Torque is
a moment created when a force is applied a distance away from the
axis of rotation. This is analogous to a bending moment, where a
bending forces are created from external forces are applied a
distance away from the point of interest. In this report, torsional
moment will mean the same as torque.
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TORSIONAL RESISTANCE OF STANDARD STEEL SHAPES
CARMEN CHUN
Then the shear centre is located such that the shear forces
counteract all the applied shear forces to produce equilibrium.
1 𝑛
𝑦0 = − ∫ (𝜏𝑡)𝑟 𝑑𝑠
𝑉𝑥 𝑜
Equation 2. y-coordinate of shear centre
𝑥0 = −
1 𝑛
∫ (𝜏𝑡)𝑟 𝑑𝑠
𝑉𝑦 𝑜
Equation 3. x-coordinate of shear centre
2.2
Figure 1. Shear centre of common steel shapes (AISC, 2003)
For common steel shapes, many design guides, such as the
American Institute of Steel Construction (AISC) Torsion Guide
will provide the torsional properties.
To locate the shear centre on an uncommon or asymmetric shape,
the following equations may be used.
𝑛
∫ (𝜏𝑡)𝑟 𝑑𝑠 = 𝑉𝑦 𝑥0 − 𝑉𝑥 𝑦0 = 0
Pure Torsion (or St. Venant Torsion)
Pure torsion assumes that “plane sections remain plane” principle;
only element rotation occurs. The torsional moment is carried as
torsional shear stresses in the element. The torsional resistance for
pure torsion is determined in Equation 4. This resistance is
analogous to bending moment resistance, which is comprised of
the bending rigidity (EI) multiplied by the curvature. For torsional
resistance, the torsional rigidity is defined by GJ and multipled by
the torsional curvature (rate of change of angle). Thus the stress
due to pure torsional stress is proportional to the distance from the
centre of twist.
𝑜
Equation 1. Shear centre
𝑀𝑠 = 𝐺𝐽
The location of the shear centre is independent of the magnitude or
type of loading, only on the geometrical configuration. Compute
the shear forces in each of the component elements by integrating
over the stress distribution, as shown in Equation 2 and Equation 3.
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𝜕𝜑
𝜕𝑧
Equation 4. Pure torsional resistance
where
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G = shear modulus of elasticity
J = torsional constant
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TORSIONAL RESISTANCE OF STANDARD STEEL SHAPES
CARMEN CHUN
The torsional constant J measures the resistance of a structural
member to pure torsion. This differs from open shapes (Equation
5) and closed shapes (Equation 6).
𝐽= ∑
𝑏′𝑡 3
3
Equation 5. Torsional constant for open shapes
b’ = plate lengths between points of intersection on
their axes
t = plate thicknesses
where
4𝐴20
𝐽=
∫𝑠 𝑑𝑠⁄𝑑𝑡
Equation 6. Torsional constant for closed shapes
where
2.3
A0 = enclosed area by the walls
t = wall thickness
ds = perimeter of shape
Warping Torsion
Warping torsion is present when plane sections do not remain
plane. The resulting translation produces lateral bending, or
warping. For example, a W-shape (or I-beam) subjected to
bending will have its compression flange bent in one direction
laterally while its tension flange is bent in another. When warping
is restrained, lateral flange bending causes flexural normal stresses
(tension and compression) along with shear stresses in the flange.
Warping stresses will not develop when a member is allowed to
warp freely. As a result, warping torsion results in torsional shear
stresses and normal stresses in a member.
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Figure 2. Warping torsion on I-beam (Salmon and Johnson, 1996)
The warping torsional constant Cw measures the resistance of a
structural member to the warping contribution of warping torsion.
The torsional resistance for restrained warping is defined in
Equation 7. For HSS, warping deformations are small and
generally taken to be zero. The warping torsional constant for
other sections can be found in most design guides; the calculation
for this constant can vary based on the geometry of the structural
section. The resistance is a function of the modulus of elasticity,
the shear resistance from lateral bending of one flange, and the
moment arm of the lateral bending. A general calculation for a Wshape is shown in Equation x.
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𝜕 3𝜑
𝑀𝑤 = −𝐸𝐶𝑤 3
𝜕𝑧
Equation 7. Warping torsional resistance
where
E = modulus of elasticity
Cw = warping torsional constant
𝐶𝑤 =
𝐼𝑓 ℎ2
2
Equation 8. Warping torsional constant for a W-shape
where
2.4
If = moment of inertia for one flange
h = moment arm (centre of flange to centroid)
Asymmetry Parameter
The asymmetry parameter (or monosymmetry constant) is used in
calculating the buckling moment resistance of laterally
unsupported monosymmetric beams loaded in the plane of
symmetry. The monosymmetry constant is defined as follows in
Equation 9. βx is zero for double-symmetric sections.
𝛽𝑥 =
1
∫ 𝑦(𝑥 2 + 𝑦 2 )𝑑𝐴 − 2𝑦𝑜
𝐼𝑥 𝐴
Equation 9. Asymmetry parameter
where
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Ix = moment of inertia about horizontal centroidal
axis
dA = area element
y0 = vertical location of shear centre with respect to
the centroid
2.5
CARMEN CHUN
Other Torsional Properties
For the solution of torsional analysis, the torsional constants
presented above (J, Cw) must be determined for open crosssections. Additional torsional properties are also required and are
dependent only on the geometry of the structural shape. These
have been tabulated in the AISC Torsion Guide.
These include:
𝐸𝐶
a:
√ 𝐺𝐽𝑤
Qs:
Statical moment at point s
Sws:
Warping statical moment at point s
Wns:
Normalized warping function at point s
2.6
Torsional Functions
In additional the geometric properties of the cross-section, the
torsional rotation and its derivatives are necessary to evaluate the
torsional response of a structural member.
Graphs are provided in AISC Torsion Guide for the torsional
functions, for corresponding values of the ratio of the span length
to the torsional property a. Equations have been evaluated for
common boundary conditions (fixed, pinned, and free) and loading
conditions.
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TORSIONAL RESISTANCE OF STANDARD STEEL SHAPES
CARMEN CHUN
3.0 Analysis and Design Approaches
The most commonly used structural shapes offer relatively poor
resistance to torsion, such as W-shapes and T-shapes. These
structural shapes are commonly used as they are efficient in
resisting bending moments, in beam and column situations.
However, as discussed previously, closed sections, such as HSS,
are the most effective in resisting torsional loads, however these
are typically inefficient and expensive for bending.
Therefore, it is practical to minimize torsional effects by detailing
loads and reactions to act through the shear centre of the member.
Any situation where the loading or reaction acts eccentrically to
the shear centre gives rise to torsion.
Torsion exists on spandrel beams, where such loading may be
uniformly distributed along the length, from the weight of the
building façade (see Figure 3). The uniform torsion load can be
quite significant, especially if the building cladding is curtain wall,
which is becoming increasingly prevalent in building construction.
To minimize twisting on the spandrel beam, diagonal bracing may
be used to transfer the vertical eccentric loading as a compression
force into the brace and to the floor diaphragm. However, this is
not always practicable as this may decrease the floor-to-ceiling
space to conceal the bracing, or may be aesthetically displeasing.
Figure 3. Torsional loading from building facade (AISC, 2003)
In design, it is practical to minimize torsional effects by reducing
eccentricity of loads or to add additional bracing to transfer loads
by other means. When this is not possible, the use of closed
sections will result in more effective resistance to torsional loads.
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CARMEN CHUN
3.1.2. Warping Shear Stresses
However, as discussed above, the most commonly used steel
shapes for building construction (W-, C-, L- shapes) offer
relatively poor resistances to torsion. Although torsion is not
usually the governing design case, secondary effects from torsion
should be considered in the design. Therefore, the method of
calculating torsional properties of standard steel shapes will be
provided.
In-plane shear stresses develop when warping is restrained that act
parallel to the edge of the element, and the magnitude is in
accordance to Equation 11.
𝜏𝑤𝑠 =
−𝐸𝑆𝑤𝑠 𝜃 ′′′
𝑡
Equation 11. Warping shear stress for open shapes
3.1
Torsional Stress for Open Shapes
The open shapes refer to I-shapes, channels, angles, and tees.
These open shapes tend to warp under torsional loading. When
warping is restrained, additional longitudinal stresses are added to
the shear stresses.
3.1.3. Warping Normal Stresses
These stresses are perpendicular to the surface of the element,
determined in accordance to Equation 12.
𝜎𝑤𝑠 = 𝐸𝑊𝑛𝑠 𝜃′′
Equation 12. Warping normal stress for open shapes
3.1.1. Pure Torsional Shear Stresses
Pure torsional shear stresses are in-plane shear stresses that act
parallel to the edge of the element. They are at maximum with
equal and opposite directions at the two edges, where the
maximum is calculated as shown in Equation 10.
𝜏 = 𝐺𝑡𝜃′
Equation 10. Pure torsional shear stress for open shapes
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3.1.4 Warping Stresses on Angles and Structural
Tees
Angles and structural tees tend to warp under torsional loading.
However, the warping stresses are relatively small and therefore
the shear and normal stresses due to warping typically are
considered negligible.
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3.2
TORSIONAL RESISTANCE OF STANDARD STEEL SHAPES
Torsional Stress for Closed Shapes
Torsion on a closed or solid circular shape is resisted by shear
stresses that vary directly with the distance from the centroid. Due
to its geometry, the cross-section remains plane and only pure
torsional shear stresses develop.
Non-circular closed shapes tend to warp under torsional loads, but
this warping is restricted as the longitudinal shear prevents relative
displacement of adjacent plate elements. Therefore, warping
stresses are typically neglected when closed shapes are used.
3.3
Combining Stresses
When torsional stresses are present, bending and shear stresses due
to plane bending are typically present as well. The normal and
shear stresses are combined with the stresses induced by torsion to
determine the resultant stresses from the factored loads (see
Equation 13 and Equation 14). Then the stress conditions can be
compared to the factored resistance (Equation 15) to determine if
the specified section is adequate in resisting the loads.
CARMEN CHUN
4.0 Application of Theory – Lateral
Torsional Buckling
The W-shape is arguably the most common steel shape used in
building construction. It is efficient in resisting bending moments
as the majority of the cross-section is located far from the centroid.
In terms of torsion, W-shapes are vulnerable to lateral torsional
buckling failures. The analysis for lateral torsional buckling
utilizes the concepts from torsional stress.
According to the CSA S16-09, where a continuous lateral support
is not provided to the compression flange of a member subjected to
uniaxial strong axis bending, the factored moment is determined as
per Clause 13.6 (Figure 4).
𝜎𝑓 = 𝜎𝑎𝑥𝑖𝑎𝑙 ± 𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔 ± 𝜎𝑤𝑎𝑟𝑝𝑖𝑛𝑔
Equation 13. Combined normal stresses
𝜏𝑓 = 𝜏𝑏𝑒𝑛𝑑𝑖𝑛𝑔 ± 𝜏𝑆𝑡.𝑉𝑒𝑛𝑎𝑛𝑡 ± 𝜏𝑤𝑎𝑟𝑝𝑖𝑛𝑔
Equation 14. Combined shear stresses
𝜎𝑟 = 𝜑𝑓𝑦
Equation 15. Typical resistance strength
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CARMEN CHUN
Figure 5. W-shape (CISC, 2002)
Equation 16. Torsional constant for W-shapes
Equation 17. Warping constant for W-shapes
The asymmetry parameter would not apply in this situation, as the
W-shape is doubly-symmetric.
Figure 4. Equations for lateral torsional resistance (CSA S16-09)
The torsional section properties for W-shapes (Figure 5) are
calculated in Equation 16 and Equation 17 below.
If the plane of bending is not in the strong axis, lateral torsional
buckling will not govern as the lateral bending strength is strong
than in the direction of uniaxial bending. Also, stability (lateral
torsional buckling) is not a governing failure mode, unless the
unbraced length is large.
Given this parameter, determining the unbraced length is critical in
determining the failure mode. This is greatly affected by the
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TORSIONAL RESISTANCE OF STANDARD STEEL SHAPES
boundary conditions – whether the ends are fixed or simply
supported. It may be beneficial to have rigid fixed ends to
minimize the unbraced length, thus reducing the likelihood for
lateral torsional buckling.
Finally, the type of loading can impact the load effects, whether
the loads are being transferred through the top or bottom flange
and if the loads are uniform throughout the length of the member.
However, lateral torsional buckling for beams should be
considered as this is normally a governing failure load unless
continuous lateral support can be provided.
CARMEN CHUN
all types of twisting effects as a shear stress. Warping is induced
when the cross-section does not remain plane, when the member is
restrained from warping at the supports. Normal stresses are
present in warping torsion. The effects of torsional loading are
determined by combining the stresses from bending and twisting.
This can then be compared to the factored strength to determine
the adequacy of the steel section in that application.
The resistance of standard steel shapes is determined by
calculating the torsional constants, J and Cw. These constants can
be calculated using torsional theory, however they are tabulated in
design guides, such as those provided by the CISC and AISC. The
resistance of standard steel shapes is typically governed by the
geometry.
5.0 Conclusion
In normal building construction, torsion is not typically a
governing failure load for which members are designed for.
However, the effects of torsional loading should not be neglected,
as torsion is induced when there is eccentric loading. This is
commonly seen at spandrel beams where the building façade is
supported off the side of the beam and applying a uniform
torsional load. As increasingly more buildings are designed with
curtain wall cladding, dead loading on the side of the beam,
consideration of torsion is more significant.
Efficient design for torsion includes using closed cross-sections,
such as pipes and horizontal structural sections, and using bracing
to limit the amount of twisting in the section.
Torsional effects also play an important role in sizing of beams, as
lateral torsional buckling is often a governing failure mode in steel
beams with no or minimal lateral support.
Torsion exists when a load is applied to member eccentrically from
its shear centre. Torsion is typically categorized into two forms:
St. Venant and warping. St. Venant (or pure torsion) is present in
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CARMEN CHUN
References
American Institute of Steel Construction. “Torsional Analysis of
Structural Steel Members.” Chicago, Illinois, 1997. Reprinted
October 2003.
Canadian Institute of Steel Construction. “Handbook of Steel
Construction.” 10th edition, Toronto, Ontario, March 2010.
Canadian Institute of Steel Construction. “Torsional Section
Properties of Steel Shapes.” 2002.
Salmon, C.G. and Johnson, J.E. “Steel Structures, Design and
Behavior”, 4th Edition. HarperCollins Publishers. New York,
N.Y. 1996.
Yao, C.C. Design of Monosymmetric Beams presentation. Read
Jones Christoffersen Ltd, Vancouver, 2010.
http://www.gestech.net/shaft_torsional_failure.htm (accessed April
5, 2011)
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