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					CSA Design of Structural Members with Formatted Spreadsheets                                                                                                      Angus Chan



                                                                           Microsoft Excel spreadsheets with the use of pre-formatted macros
CSA Design of Structural Members                                           developed by former UBC students. The development of this
with Formatted Spreadsheets                                                project is used as a temporary substitute to CHEOPS due to
                                                                           difficulties from running the current version with Microsoft
                                                                           Windows Vista. Although the spreadsheet becomes a handy tool
                                                                           for the user to perform preliminary and quick checks of simple
                                                                           structural members, it lacks different section class checks and
                                                                           graphs provided from CHEOPS. In conclusion, the spreadsheet
                                                                           development is obviously lacking until a compatible update
                                                                           version of CHEOPS is readily available for Windows Vista users.



                                                                                                                   Table of Contents
                                                                           Abstract .......................................................................................... 1
                                                                           1.0 Introduction ............................................................................. 3
                                                                           2.0 Objectives................................................................................. 3
                                                                           3.0 Design of Compression Members .......................................... 3
                                                                                3.1 Euler Buckling .................................................................. 4
                                                                                     3.2 Effective Length Concept (Ref. Appendix F)............. 5
                                                                                3.3 Euler Buckling Stress ........................................................ 6
                                                                                3.4 Inelastic Buckling ............................................................. 6
                                                                                3.5 Effects of Geometric Imperfections .................................. 7
                                                                                3.7 Slenderness Ratio Limit .................................................... 9
                                                                           4.0 Design of Tension Members ................................................... 9
                                                                                4.1 General Design Steps (ref. Clause 13.2) ........................... 9
                                                                                4.2 Slenderness Ratio Limit (ref. Clause 10.4.2.2) ................. 9
Abstract                                                                        4.3 Effective Net Area, Ane (ref. Clause 12.3.3) ................... 10
Steel structure consists mainly of various types, including                5.0 Design of Bending Members ................................................ 11
compression, tension, bending, and bending-compression                          5.1 Modes of Failure for Bending ......................................... 11
members. These are essential components of any type of structures                    5.1.1 A. Local Section Requirement ............................. 11
and are designed by regulations such as the CSA. The design                          1. Local flange buckling ........................................... 11
method developed through this project will be implemented in                         2. Local torsional buckling of compression flange ... 11



angus_chan_CSA_Steel_Design                                         1/3/2008                                                                                   PAGE 1 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                                                                Angus Chan



           5.1.2 B. Overall Member Strength ................................ 11                                  12.         Biaxial Bending Interaction (Class 1&2 I- Shaped
      5.2 Lateral supports ............................................................... 11                    Sections Only)............................................................... 13
      5.3 Laterally Unsupported Members .................................... 11                                  6.3 Local Section Requirement ..................................... 13
      5.4 Modes of Failure for Shear ............................................. 12                      6.4 Combined Bending and Compression Stress Distribution
                                                                                                           for Prismatic Sections ........................................................... 14
                                                                                                           6.5 Combined Bending and Compression Stress Distribution
                                                                                                           for I-Shaped Sections ............................................................ 14
                                                                                                           6.7 Bending and Compression Interaction Curve for Class
                                                                                                           1&2 I-Shaped Sections (Strong Axis)................................... 15
                                                                                                           6.8 Bending and Compression Interaction Curve for Class
                                                                                                           1&2 I-Shaped Sections (Weak Axis) .................................... 15
                                                                                                           6.9 Moment Amplification: P-δ Effects............................... 16
                                                                                                           6.10 Overall Member Strength Requirement for Braced
                                                                                                           Frames vs. Unbraced Frames ................................................ 17
                                                                                                                 6.10.1 B1a Cross Sectional Strength Requirement for
                                                                                                                 Braced Frames .............................................................. 17
                                                                                                                 6.10.2 B1a.2 Cross-sectional strength for all classes
                                                                                                                 except for Class 1& 2 I-shaped members (braced frames)
     5.5 Limiting Shear Stress ...................................................... 12                         ....................................................................................... 18
     5.6 Shear Buckling ................................................................ 12                      6.10.3 B2a In-plane Strength & Stability Requirement
6.0 Combined Bending and Compression Members ............... 13                                                  for Braced Frames ......................................................... 18
     6.1 Modes of Failure for Combined Bending and                                                               6.10.4 B2a.2 In-plane member strength for all classes
     Compression ......................................................................... 13                    except for Class 1& 2 I-shaped members (braced frames)
          6.1.1 Local Section Requirement .................................. 13                                  ....................................................................................... 18
     6.2 Overall Member Strength Requirement .......................... 13                                       6.10.5 B3a Lateral Stability Requirement for Braced
          6.2.1 B1 Cross Sectional Strength Requirement ........... 13                                           Frames ........................................................................... 19
          9. Overall section yielding ........................................ 13                                6.10.6 B3a.2 Lateral torsional buckling strength for all
          6.2.2 B2. In-plane Strength& Stability Requirement ... 13                                              classes of sections except for Class 1&2 I-shaped
          10.    In-plane stability ............................................... 13                           members (braced frames) .............................................. 19
          6.2.4 B3. Lateral Stability Requirement ....................... 13                                     6.10.7 B4a Biaxial Bending Strength Requirement for
          11.    Lateral torsional buckling ................................. 13                                 Braced Frames (Class 1& 2 Sections only) .................. 19
          6.2.4 B4. Biaxial Bending Strength Requirement......... 13                                             6.10.8 B1b Cross Sectional Strength Requirement for
                                                                                                                 Unbraced Frames .......................................................... 20



angus_chan_CSA_Steel_Design                                                                     1/3/2008                                                                                 PAGE 2 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                                         Angus Chan



          6.10.9 B2b In-plane Strength & Stability Requirement                                            complying fully with CSA design standards with reference to
          for Unbraced Frames..................................................... 20                     structural development software, CHEOPS.
          6.10.10 B2b.2 In-plane member strength for all classes
          except for Class 1& 2 I-shaped members (unbraced                                                The design method developed through this project will be
          frames) .......................................................................... 20           implemented in Microsoft Excel spreadsheets with the use of pre-
          6.10.11 B3b Lateral Stability Requirement for Unbraced                                          formatted macros developed by former UBC students. The use of
          Frames ........................................................................... 20           automatic references to CISC shape tables is provides additional
          6.10.12 B3b.2 Lateral torsional buckling strength for all                                       advantages to design method, along with the flexibility for custom
          classes of sections except for Class 1&2 I-shaped                                               modification or code changes. The spreadsheet platform is
          members (unbraced frames) .......................................... 21                         intended for royalty-free means of distribution
          6.10.13 B4b Biaxial Bending Strength Requirement for
          Unbraced Frames (Class 1& 2 Sections only) .............. 21                                    The development of this project is used as a temporary substitute
     6.11 Second Order Effects due to Sway: P-Δ Effects ........... 22                                    to CHEOPS due to difficulties from running the current version
7.0 Conclusion ............................................................................. 22           with Microsoft Windows Vista.
8.0 Appendices ............................................................................. 23
     References ............................................................................. 23          3.0 Design of Compression Members
                                                                                                          All compression members as used in practical applications
                                                                                                          generally fail in buckling.
                                                                                                                a. Local buckling can occur in the elements of shapes, such
1.0 Introduction                                                                                                    as flanges or webs
Steel structure consists mainly of various types, including                                                     b. Overall buckling limits the load carrying capacity of the
compression, tension, bending, and bending-compression                                                              entire member.
members. These are essential components of any type of structures
                                                                                                          The most common failure of compression members happens in the
and are designed by regulations such as the CSA.
                                                                                                          inelastic buckling range. Only slender compression members
                                                                                                          behave linearly and fail according to the Euler theory. Slenderness
This report focuses on the design methods of the various types of
                                                                                                          is defined as the ratio of compression member length over
structural members to provide for a basic design tool for an
                                                                                                          minimum radius of gyration. Stresses are defined as the ratio of
everyday user.
                                                                                                          loads over the cross sectional area. The maximum load of a
                                                                                                          practical compression member can vary considerably depending on
2.0 Objectives                                                                                            the initial out-of-straightness. Other imperfections such as residual
The design procedure will comply with strength and stability                                              stresses or change in geometry of the cross section can influence
requirements as well as stiffness requirements to limit deflections                                       the capacity of compression members, too.



angus_chan_CSA_Steel_Design                                                                        1/3/2008                                                        PAGE 3 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                          Angus Chan




3.1 Euler Buckling                                                                                           2
                                                                                                               y
                                                                                                       − EI d 2 = Cy ( x )
                                                                                                            dx
                                                                                                              with
                                                                                                                    C
                                                                                                             λ2 =
                                                                                                                    EI
                                                                            we get
                                                                                                      d 2 y + 2 y( x) = 0
                                                                                                             λ
                                                                                                      d x2

                                                                            A solution for this differential equation is
           Compressive Element, Pinned Ends
                                                                                                 y(x) = A sin λ x + B cos λ x
At any point at the centreline of the column we find:
                                                                            The constant A and B can be solved by considering the conditions
                      EI
                         = Cy ( x )   with ρ = curvature.                   at the boundaries of the compression member:
                      ρ
                                                                                                at x = 0 → y (x) = 0 → B = 0
                                               2
                                            − d 2
                                                 y                          resulting in
                                   1          dx                                                         y = A sin λ x
                          Then,      =                  3
                                   ρ                                        and at the other end of the column we find for
                                         ⎡ ⎛ dy ⎞   2⎤ 2
                                         ⎢1+ ⎜ ⎟ ⎥                                                   at x = L → y (x) = 0
                                         ⎢ ⎝ dx ⎠ ⎥
                                         ⎣        ⎦
                                                                            which can be solved by either
                                                                                                    A = 0 or sin λ L = 0
                                            2
                                              y
                                         − d 2                              The first solution gives y (x) = 0 for any value of x, which means
                                           dx
                   Therefore, EI                    3
                                                        = Cy ( x )          the column would be straight.
                                ⎡ ⎛ dy ⎞         2⎤ 2                       The second yields
                                ⎢1+ ⎜ ⎟ ⎥
                                ⎢ ⎝ dx ⎠ ⎥                                                              λ L = π , 2π , ... , n π
                                ⎣        ⎦
This equation is not easy to solve. Therefore let’s assume the              or
deflections y(x) are small, then                                                                       2
                                                                                                         EI 4 2 EI         2 2
                                                                                                  C = π 2 , π 2 , . . . , n π2
                                                                                                                               EI
                                                                                                         L          L           L



angus_chan_CSA_Steel_Design                                          1/3/2008                                                       PAGE 4 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                    Angus Chan



The lowest load, the “Euler load”, at which the compression                            E = modulus of elasticity
member is                                                                              I = moment of inertia
failing by buckling is
                                      π 2 EI
                                Ce =                                          The effective length, kL, may be thought of as the actual unbraced
                                        L2
                                                                              length, L, multiplied by a factor, k, so that the product (kL) is
3.2 Effective Length Concept (Ref. Appendix F)                                equal to the length of a pin-ended compression member, Le, of the
                                                                              same capacity as the actual member.

Columns may be designed using a simplified analysis utilizing the
effective length concept. Pin-ended columns are employed as
reference for comparison with columns having various end
conditions.

Once the strength of a pin-ended column is known, the strength of
the same column with different end conditions can be obtained by                            KL
                                                                                                       L
                                                                                                                                 Le
using the effective length factor, k. This effective length factor k                                           equivalent to
is defined as:

                                        k =     Ce
                                                                                                                                      Ce
                                                C cr
                                                                                                 Ccr
where the Euler buckling load of a pin-ended column with same
                                                                                  Effective Length Concept for Axially Loaded Compression Members
length is
                                    π 2 EI
                              Ce =
                                    ( kL)2
and the critical load of end-restrained column
                                       2
                                  = π 2
                                         EI
                             C cr
                                    ( kL)                                                   Idealized Cases for Columns for which Joint Rotation and
                                                                                            Translation are either fully realized or nonexistent
with
       Le = length of pin-ended member
       L = member length


angus_chan_CSA_Steel_Design                                            1/3/2008                                                               PAGE 5 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                      Angus Chan




3.3 Euler Buckling Stress




3.4 Inelastic Buckling

Euler’s theory covers only buckling situations where compressive
stresses are below the elastic limit and are uniformly distributed
over the complete cross section.

Buckling of real compression members are influenced by
geometric imperfections (initial out-of-straightness) and residual
stresses.

Design strength formula based on testing
      Cr=Φ •A • Fy (1+λ2n) –(1/n)




angus_chan_CSA_Steel_Design                                          1/3/2008   PAGE 6 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                Angus Chan




3.5 Effects of Geometric Imperfections




angus_chan_CSA_Steel_Design                                    1/3/2008   PAGE 7 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                  Angus Chan




3.6 Effects of Residual Stresses
In hot rolled shapes we usually find residual stress caused by
uneven cooling speeds. Other sources of residual stresses in
structural shapes may be cold bending or welding during
fabrication and cutting operations or local plastic deformations
such as punching of holes.




                                                                                           20% Fy                            25% Fy
                                      ion                                                compression                       compression
                              pr   ess
                          m
                       Co                                                                                                                45% Fy
                                                                                                       30% Fy                            tension
                                                                                                       tension

                                                               +
                                                                                                                 30% Fy
                                                                                                                 tension
                +
                                       +
                                                                                                     30% Fy
                                                                                    Rolled W-shape compression             Welded box




                          Residual Stress in a Hot Rolled and Welded Structural Shapes




angus_chan_CSA_Steel_Design                                                     1/3/2008                                                    PAGE 8 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                               Angus Chan



                                                                                      1. select tentative cross section using approximate
    1. The separate effects of residual stress and initial curvature                     formulae and check slenderness ratio limit
       cannot be added.                                                                                      Tr = Φ Ag Fy
    2. Residual stress has little effect on the maximum strength of
       very slender columns, either straight or initially crooked,                     2. check design according to code requirements (least of
       which have strengths approaching the Euler load.                                    the following):
    3. Differences in maximum strength caused by variations in                                                 Tr = Φ Ag Fy or
       the shape of the residual-stress pattern are much smaller for                                           Tr = 0.85Φ An Fu or
       initially curved columns than for initially straight columns.                                           Tr = 0.85Φ Ane Fu
                                                                               •   failure modes:
3.7 Slenderness Ratio Limit                                                            1. excessive elongation         (Fy governs)
                                                                                       2. rupture in net area                 (Fu governs)
The value of slenderness ratio (KL/r) limit of 200 stems from
experience. (ref. Clause 10.4.2.1 and Table 4-4)                                       3. rupture in effective net area (considers shear lag effects)
                                                                               •   use “s2/4g - rule” for staggered bolt patterns

4.0 Design of Tension Members
                                                                               4.2 Slenderness Ratio Limit (ref. Clause 10.4.2.2)
Tension members act as principal structural members in trusses,
bridges, transmission towers, bracing systems. They also have                  The value of slenderness ratio (KL/r) limit of 300 stems from
applications as secondary members such as sag rods, bracing of                 experience and parallels values from foreign steel standards (i.e.
OWSJs. It is most efficient because limit of usefulness not reduced            German, Russian, American).
by stability problems. Tension single shapes are most economical
in applications, when doubling up use advantages of symmetry. All
being said, limiting slenderness ratio exists because of potential of
vibrations.

4.1 General Design Steps (ref. Clause 13.2)

•   design member in two steps:




angus_chan_CSA_Steel_Design                                             1/3/2008                                                         PAGE 9 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                Angus Chan



                                    Net Area, An                                         by a factor of 0.6.
                                     (ref. Clause 12.3)

                                                                                4.3 Effective Net Area, Ane (ref. Clause 12.3.3)

                                                                                This is applicable when critical net area includes the net area of
                                                                                unconnected elements

                                                                                For bolted connections:

                                                                                       Ane = 0.60 to 0.90 An (refer to S16.1)

                                                                                For welded connections:

                                                                                       Ane = An1 + An2 + An3
                                                                                        where
                                    bn = b - ∑D + ∑s2/(4g)                             An1 for elements connected by transverse welds
                                                                                       An2 for elements connected by longitudinal welds along
         where                                                                                 two parallel edges
                                                                                       An3 for elements connected by a single longitudinal weld
                  d = bolt diameter
                  D = d + 2mm for drilled holes
                  D = d + 4mm for punched holes

           Note: if shear block failure is involved, reduce shear path



                                                    g1
P                                                                          P
                            d   h                   g2




                    e       s2          s1
angus_chan_CSA_Steel_Design                                              1/3/2008                                                        PAGE 10 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                       Angus Chan




                                                                            5.2 Lateral supports
5.0 Design of Bending Members
Bending members have its application in being the principal
structural members in roof beams, floor beams, bridge decks, etc.
most space efficient because of low height to span ratio.
Furthermore, lateral torsional buckling is the most important mode
of failure for laterally unsupported bending members

5.1 Modes of Failure for Bending
5.1.1 A. Local Section Requirement
ref. Clause 11.2, Table 2, Table 5-1)
     1. Local flange buckling

    2. Local torsional buckling of compression flange                       5.3 Laterally Unsupported Members

    3. Local web buckling


5.1.2 B. Overall Member Strength
Laterally supported:
(ref. Clause 13.5)
    4. Yielding (full section vs. flange only)

Laterally unsupported:
(ref. Clause 13.6)
    5. Lateral torsional buckling




angus_chan_CSA_Steel_Design                                          1/3/2008                                   PAGE 11 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                      Angus Chan




                                                                      5.4 Modes of Failure for Shear
                                                                          1) Yielding (shear area)
                                                                          2) Shear buckling (web)




                                                                    5.5 Limiting Shear Stress

                                                                      Full yielding followed by strain-hardening:
                                                                                            Fs=0.66•Fy

                                                                      Yielding in shear in accordance with von Mises criterion:
                                                                                             Fvy=0.577•Fy

                                                                      5.6 Shear Buckling

                                                                      The modes of shear buckling includes 2 types:
                                                                         • Inelastic buckling: Fcri
                                                                         • Elastic buckling: Fcre


angus_chan_CSA_Steel_Design                                    1/3/2008                                                      PAGE 12 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                             Angus Chan




6.0 Combined Bending and Compression                                                9. Overall section yielding
Members                                                                       6.2.2 B2. In-plane Strength& Stability Requirement
These structural members include columns, top chord of trusses,               (ref. Clause 13.8.2(b), 13.8.3(b))
and principal members in vierendeel trusses and rigid frames. Its
                                                                                  10. In-plane stability
eccentricity of the applied load, or initial curvature of the column
axis always generate bending moments. The UDL on top chord of
                                                                              6.2.4 B3. Lateral Stability Requirement
a truss generate moments and compression loads in members.
Rigid frames and Vierendeel trusses generate bending moments                      (ref. Clause 13.8.3(c), 13.8.3(c))
and compression loads in members. The distinction of this                            11.    Lateral torsional buckling
structural member is made between braced and unbraced frames
                                                                              6.2.4 B4. Biaxial Bending Strength Requirement
                                                                                  (ref. Clause 13.8.2)
6.1 Modes of Failure for Combined Bending and                                        12.       Biaxial Bending Interaction (Class 1&2 I- Shaped
Compression                                                                                    Sections Only)

6.1.1 Local Section Requirement                                               6.3 Local Section Requirement
(ref. Clause 11.2, Table 2, Table 5-1)                                        (ref. Clause 11.2, Table 2, Table 5-1)
    6.      Local flange buckling
                                                                              Flanges:
  7.          Local torsional buckling of compression flange                  Same as for bending members

                                                                              Webs:
    8.        Local web buckling                                              Limiting h/w ratios depend on value of Cf/Cy

                                                                                        Where
6.2 Overall Member Strength Requirement                                                 Cf = the applied factored axial load
                                                                                        Cy = A· Fy
6.2.1 B1 Cross Sectional Strength Requirement
(ref. Clause 13.8.2(a), 13.8.3(a))                                            Note: see Table 4-1 for typical limiting ratio values




angus_chan_CSA_Steel_Design                                            1/3/2008                                                       PAGE 13 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                    Angus Chan




6.4 Combined Bending and Compression Stress                               6.5 Combined Bending and Compression Stress
Distribution for Prismatic Sections                                       Distribution for I-Shaped Sections




Bending and Compression Interaction Curve for Prismatic Sections




angus_chan_CSA_Steel_Design                                        1/3/2008                                  PAGE 14 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                 Angus Chan




6.7 Bending and Compression Interaction Curve
for Class 1&2 I-Shaped Sections (Strong Axis)




6.8 Bending and Compression Interaction Curve
for Class 1&2 I-Shaped Sections (Weak Axis)




angus_chan_CSA_Steel_Design                                    1/3/2008   PAGE 15 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                  Angus Chan




6.9 Moment Amplification: P-δ Effects
                                                                                        L2                Cf
                                                                                δ II = 2     ⋅ C f ⋅δ = δ
                                                                                      π ⋅ EI              Ce
                                                                      Where

                                                                                           π 2 ⋅ EI
                                                                                  Ce =
                                                                                              L2      , Euler Buckling load



                                                                                                δ = δo+δII

                                                                              Therefore,


                                                                                                                  Cf
                                                                                            δ o = δ ⋅ (1 −             )
                                  M=Mo+Cf·v                                                                       Ce
                                  v = vo+vII
                                                                                                             δo
                          2
                       d vII                 π ⋅z                                            δ =
            − EI             = C f ⋅ δ ⋅ sin                                                          ⎛ Cf         ⎞
                        dz 2                  L                                                       ⎜1 −
                                                                                                      ⎜ C          ⎟
                                                                                                                   ⎟
Where
                                                                                                      ⎝    e       ⎠
         δ is the maximum deflection at midspan z=L/2
                                                                                             Mmax=Momax+Cf·δ
                        2
                         L                     π ⋅z
           vII =               ⋅ C f ⋅ δ ⋅ sin
                      π 2 ⋅ EI                  L

angus_chan_CSA_Steel_Design                                    1/3/2008                                                    PAGE 16 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                           Angus Chan



                                                                               on Eurocode 3) Pδ effects important and moment amplification
                                  ⎛      Cf ⎞                                  factor, U1, applies
                                  ⎜1 + ϕ
                                  ⎜         ⎟
                                  ⎝      Ce ⎟
                                            ⎠M                                 Unbraced Frames
                   M max        =              o max                           frames without direct acting bracing, or with bracing of sway
                                   ⎛ Cf ⎞                                      stiffness less than 5 times that of the frames without direct acting
                                   ⎜1 −
                                   ⎜ C ⎟   ⎟                                   bracing. Pδ effects negligible and moment amplification factor,
                                   ⎝     e ⎠                                   U1, does not apply (U1=1). This is because the maximum second-
                                                                               order elastic moment, including P∆ effects, occurs at the ends of
                                                                               the beam-column.
                                        δ o ⋅ Ce
                               ϕ=                      −1
                                         M o max                               6.10.1 B1a Cross Sectional Strength Requirement
                                                                               for Braced Frames

                                                      Cf                           B1a.1 Cross-sectional strength for Class 1& 2 I-shaped
                                ω1 = 1 + ϕ                                                       members (braced frames)
                                                      Ce
                                                                                                Cf       0.85 ⋅ U1x M fx       0.6 ⋅ U1 y⋅ ⋅ M fy
                        ω1                                                                           +                     +                        ≤ 1.0
M max =                               M o max = U1 ⋅ M max                                      Cr            M rx                   M ry
                 ⎛ Cf             ⎞
                 ⎜1 −
                 ⎜ C              ⎟
                                  ⎟
                                                                                        Note:

                 ⎝    e           ⎠                                                              λ y = 0 , β = 0 .6 + 0 .4 ⋅ λ y = 0 .6
6.10 Overall Member Strength Requirement for                                                     Cr = φ ⋅ A ⋅ Fy       (Clause 13.3.1)
Braced Frames vs. Unbraced Frames
                                                                                                M r = φ ⋅ Z ⋅ Fy      (Clause 13.5)
Braced Frames
frames with direct acting bracing that provides sway stiffness at                                U1x & U1 y ≥ 1 (Clause 13.8.2a, 13.8.4)
least 5 times that of the frames without direct acting bracing (based



angus_chan_CSA_Steel_Design                                             1/3/2008                                                                    PAGE 17 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                             Angus Chan




6.10.2 B1a.2 Cross-sectional strength for all classes
                                                                                                             K ⋅L  Fy
except for Class 1& 2 I-shaped members (braced
                                                                                                    λy =
frames)
                                                                                                              ry  π2 ⋅E                  ,

          Cf
               +
                   U 1x M fx
                               +
                                   U 1 y⋅ ⋅ M fy
                                                    ≤ 1.0
                                                                                                    β = 0 .6 + 0 .4 ⋅ λ y
          Cr          M rx             M ry
                                                                                                                             (
                                                                                                    C r = φ ⋅ A ⋅ Fy ⋅ 1 + λ x               )
                                                                                                                                       2 n −1 / n
                                                                                                                                                    (Clause
         Note:                                                                                              13.3.1)
                   λ =0                                                                            M r = φ ⋅ Z ⋅ Fy       (Clause 13.5)

                   Cr = φ ⋅ A ⋅ Fy            (Clause 13.3.1)                                       U1x & U1 y        (Clause 13.8.2b, 13.8.4)
                 M r = φ ⋅ Z ⋅ Fy          or   φ ⋅ S ⋅ Fy     (Clause 13.5)
                                                                                       6.10.4 B2a.2 In-plane member strength for all
                   U1x & U1 y ≥ 1 (Clause 13.8.3a, 13.8.4)                            classes except for Class 1& 2 I-shaped members
                                                                                      (braced frames)
6.10.3 B2a In-plane Strength & Stability
Requirement for Braced Frames                                                                          Cf        U 1x M fx       U 1 y⋅ ⋅ M fy
                                                                                                             +               +                   ≤ 1.0
                                                                                                       Cr          M rx              M ry
B2a.1 In-plane member strength for Class 1& 2 I-shaped
             members (braced frames)                                                       Note:

                 Cf         0.85 ⋅U 1x M fx         β ⋅U 1 y⋅ ⋅ M fy                                K =1
                        +                       +                      ≤ 1.0
                   Cr              M rx                  M ry                                            K ⋅L  Fy
                                                                                                    λx =
         Note:
                                                                                                          rx  π2 ⋅E                      ,

                    K =1                                                                            Cr = φ ⋅ A ⋅ Fy ⋅ 1 + λ x(               )
                                                                                                                                       2 n −1 / n
                                                                                                                                                    (Clause
                                                                                                            13.3.1)


angus_chan_CSA_Steel_Design                                                    1/3/2008                                                               PAGE 18 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                                           Angus Chan




                 M r = φ ⋅ Z ⋅ Fy         or    φ ⋅ S ⋅ Fy       (Clause 13.5)
                                                                                               6.10.6 B3a.2 Lateral torsional buckling strength for
                                                                                               all classes of sections except for Class 1&2 I-
                  U1x & U1 y           (Clause 13.8.3b, 13.8.4)
                                                                                               shaped members (braced frames)

                                                                                                                  Cf       U 1x M fx       U 1 y⋅ ⋅ M fy
6.10.5 B3a Lateral Stability Requirement for Braced
                                                                                                                       +               +                       ≤ 1.0
                                                                                                                  Cr         M rx              M ry
Frames
                                                                                                    Note:
B3a.1 Lateral torsional buckling strength for Class 1& 2 I-
           shaped members (braced frames)                                                                       K ⋅L  Fy
                                                                                                             λ=
                 Cf        0.85 ⋅U 1x M fx          β ⋅U 1 y⋅ ⋅ M fy                                             r   π2 ⋅E                       ,
                       +                        +                        ≤ 1.0
                 Cr             M rx                     M ry
                                                                                                             Cr = φ ⋅ A ⋅ Fy ⋅ 1 + λ2 n(             )−1 / n
                                                                                                                                                                (Clause 13.3.1)
         Note:                                                                                                                                       0.28 ⋅ M P
                                                                                                            M r = 1.15 ⋅ φ ⋅ M P ⋅ (1 −                         ),
                        K ⋅L  Fy                                                                                                                        Mu
                   λy =
                         ry  π2 ⋅E                       ,                                                  or   φ ⋅ MU     (Clause 13.6)

                   β = 0 .6 + 0 .4 ⋅ λ y                                                                     U1x ≥ 1 & U1 y            (Clause 13.8.3c, 13.8.4)

                                            (
                   Cr = φ ⋅ A ⋅ Fy ⋅ 1 + λ2 n             )  −1 / n
                                                                      (Clause 13.3.1)          6.10.7 B4a Biaxial Bending Strength Requirement
                                                                                               for Braced Frames (Class 1& 2 Sections only)
                                                          0.28 ⋅ M P
                 M r = 1.15 ⋅ φ ⋅ M P ⋅ (1 −                         ),
                                                             Mu                                                        M fx       M fy
                                                                                                                              +            ≤ 1 .0
                 or   φ ⋅ MU      (Clause 13.6)
                                                                                                                       M rx       M ry

                  U1x ≥ 1 & U1 y               (Clause 13.8.2c, 13.8.4)


angus_chan_CSA_Steel_Design                                                             1/3/2008                                                                   PAGE 19 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                              Angus Chan




6.10.8 B1b Cross Sectional Strength Requirement                                      6.10.10 B2b.2 In-plane member strength for all
for Unbraced Frames                                                                  classes except for Class 1& 2 I-shaped members
                                                                                     (unbraced frames)
For this type of frames, strength Check not Required                                                          Cf        M fx         M fy
                                                                                                                    +            +          ≤ 1 .0
                                                                                                              Cr        M rx         M ry
6.10.9 B2b In-plane Strength & Stability
Requirement for Unbraced Frames                                                             Note:
                                                                                                     K =1
                                                                                                                             (               )
B2b.1 In-plane member strength for Class 1& 2 I-shaped
                                                                                                                                        2 n −1 / n
             members (unbraced frames)                                                               C r = φ ⋅ A ⋅ Fy ⋅ 1 + λ x                      (Clause
                       Cf         0.85 ⋅ M fx        β ⋅ M fy                                                13.3.1)
                              +                  +               ≤ 1.0                              M r = φ ⋅ Z ⋅ Fy        or   φ ⋅ S ⋅ Fy      (Clause 13.5)
                        Cr           M rx             M ry
                                                                                                     U1x & U1 y = 1 (Clause 13.8.3b)
         Note:
                   K =1                                                              6.10.11 B3b Lateral Stability Requirement for
                                                                                     Unbraced Frames
                        K ⋅L        Fy
                   λy =
                          rr     π2 ⋅E                   ,                           B3b.1 Lateral torsional buckling strength for Class 1& 2 I-
                                                                                                shaped members (unbraced frames)
                   β = 0 .6 + 0 .4 ⋅ λ y
                                                                                                        Cf        0.85 ⋅ M fx         β ⋅ M fy
                   C r = φ ⋅ A ⋅ Fy ⋅ 1 + λ x(               )
                                                       2 n −1 / n
                                                                    (Clause                             Cr
                                                                                                              +
                                                                                                                     M rx
                                                                                                                                  +
                                                                                                                                       M ry
                                                                                                                                                  ≤ 1.0

                             13.3.1)
                                                                                            Note:
                 M r = φ ⋅ Z ⋅ Fy           (Clause 13.5)

                  U1x & U1 y = 1 (Clause 13.8.2b)


angus_chan_CSA_Steel_Design                                                   1/3/2008                                                                 PAGE 20 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                                              Angus Chan




                                                                                                                                               0.28 ⋅ M P
                        K ⋅L        Fy                                                                     M r = 1.15 ⋅ φ ⋅ M P ⋅ (1 −                    ),
                   λy =                                                                                                                           Mu
                          rr     π2 ⋅E                     ,
                                                                                                           or   φ ⋅ MU   (Clause 13.6)
                   β = 0 .6 + 0 .4 ⋅ λ y
                                                                                                            U1x & U1 y = 1 (Clause 13.8.3c)
                   Cr = φ ⋅ A ⋅ Fy ⋅ 1 + λ  (              )
                                                      2 n −1 / n
                                                                        (Clause 13.3.1)
                                                                                                 6.10.13 B4b Biaxial Bending Strength Requirement
                                             0.28 ⋅ M P
                 M r = 1.15 ⋅ φ ⋅ M P ⋅ (1 −            ),                                       for Unbraced Frames (Class 1& 2 Sections only)
                                                Mu
                                                                                                                    M fx       M fy
                 or φ ⋅ M U (Clause 13.6)                                                                                  +          ≤ 1 .0
                                                                                                                    M rx       M ry
                  U1x & U1 y = 1 (Clause 13.8.2c)
6.10.12 B3b.2 Lateral torsional buckling strength for
all classes of sections except for Class 1&2 I-
shaped members (unbraced frames)

                              Cf       M fx         M fy
                                   +            +          ≤ 1 .0
                              Cr       M rx         M ry

         Note:

                             K ⋅L  Fy
                   λ =
                              r   π2 ⋅E                    ,

                                            (
                   Cr = φ ⋅ A ⋅ Fy ⋅ 1 + λ2 n              )   −1 / n
                                                                        (Clause 13.3.1)



angus_chan_CSA_Steel_Design                                                               1/3/2008                                                     PAGE 21 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                                                                         Angus Chan




6.11 Second Order Effects due to Sway: P-∆ Effects                              ∆f = the lateral displacement at the level considered relative
                                                                                        to the level below
                                                                                h = the height of the storey between these two levels
                                                                                ∑Cf = sum of the columns axial load at the storey
                                                                                ∑Vf = storey shear

                                                                               Mfg = factored first-order moment resulting from gravity
                                                                                           load
                                                                               Mft = factored first order moment resulting from lateral loads

                                                                          7.0 Conclusion
                                                                          The structural members are designed in almost every type of
                                                                          building ever existed. With the information provided in the design
                                                                          code, the designs can be implemented into Microsoft Excel
                                                                          formatted spreadsheets. The use of automatic reference of CISC
                                                                          shape tables enhances the flexibility for custom modification or
                                                                          code changes by the interface user. While the spreadsheet is
                                                                          designed for easy usage and meant to be royalty-free, this project
                                                                          can allow act as an temporary substitute for the CHEOPS software.
                                                                          The spreadsheet becomes a handy tool for the user to perform
                                                                          preliminary and quick checks of simple structural members. But
                                    Mf=Mfg+U2·Mfg                         lacking different section class checks and graphs provided from
                                                                          CHEOPS, the spreadsheet development is obviously lacking until a
        Where                                                             compatible update version of CHEOPS is readily available for
                                                                          Windows Vista users.
                                                 ω1
                        U2 =
                                    ⎛ ∑Cf Δ f                  ⎞
                                    ⎜1 −                       ⎟
                                    ⎜    ∑V f h                ⎟
                                    ⎝                          ⎠


angus_chan_CSA_Steel_Design                                        1/3/2008                                                       PAGE 22 OF 23
CSA Design of Structural Members with Formatted Spreadsheets                     Angus Chan




8.0 Appendices
References

[1] Cheops, Maplesoft Development Corp, 1998-2001

[2] Siegfried F. Stiemer, Dr.-Ing.(Ph.D.), University of British
Columbia




angus_chan_CSA_Steel_Design                                        1/3/2008   PAGE 23 OF 23

				
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