SFD AISC 360 10

User Manual: SFD-AISC-360-10

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Steel Frame Design Manual
AISC 360-10
Steel Frame
Design Manual
AISC 360-10
For SAP2000®
ISO SAP102816M9 Rev. 0
Proudly developed in the United States of America October 2016
Copyright
Copyright Computers and Structures, Inc., 1978-2016
All rights reserved.
The CSI Logo® and SAP2000® are registered trademarks of Computers and Structures,
Inc. Watch & LearnTM is a trademark of Computers and Structures, Inc.
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DISCLAIMER
CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE
DEVELOPMENT AND DOCUMENTATION OF THIS SOFTWARE. HOWEVER,
THE USER ACCEPTS AND UNDERSTANDS THAT NO WARRANTY IS
EXPRESSED OR IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ON
THE ACCURACY OR THE RELIABILITY OF THIS PRODUCT.
THIS PRODUCT IS A PRACTICAL AND POWERFUL TOOL FOR STRUCTURAL
DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC
ASSUMPTIONS OF THE SOFTWARE MODELING, ANALYSIS, AND DESIGN
ALGORITHMS AND COMPENSATE FOR THE ASPECTS THAT ARE NOT
ADDRESSED.
THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY
A QUALIFIED AND EXPERIENCED ENGINEER. THE ENGINEER MUST
INDEPENDENTLY VERIFY THE RESULTS AND TAKE PROFESSIONAL
RESPONSIBILITY FOR THE INFORMATION THAT IS USED.
Contents
1 Introduction
1.1 Load Combinations and Notional Loads 1-2
1.2 Stress Check 1-2
1.3 Direct Analysis Method vs. Effective Length Method 1-3
1.3.1 Effective Length Method 1-4
1.3.2 Direct Analysis Method 1-4
1.4 User Options 1-5
1.5 Non-Automated Items in Steel Frame Design 1-5
2 Design Algorithms
2.1 Check and Design Capability 2-1
2.2 Design and Check Stations 2-2
2.3 Demand/Capacity Ratios 2-3
i
Steel Frame Design AISC 360-10
2.4 Design Load Combinations 2-4
2.5 Second Order P-Delta Effects 2-5
2.6 Analysis Methods 2-6
2.7 Notional Load Patterns 2-10
2.8 Member Unsupported Lengths 2-11
2.9 Effects of Breaking a Member into Multiple Elements 2-12
2.10 Effective Length Factor (K) 2-14
2.11 Supported Framing Types 2-17
2.12 Continuity Plates 2-18
2.13 Doubler Plates 2-20
2.14 Choice of Units 2-21
3 Steel Frame Design Using ANSI/AISC 360-10
3.1 Notations 3-2
3.2 Design Loading Combinations 3-6
3.3 Classification of Sections for Local Buckling 3-9
3.4 Calculation of Factored Forces and Moments 3-18
3.5 Calculation of Nominal Strengths 3-22
3.5.1 Nominal Tensile Strength 3-23
ii
Contents
3.5.2 Nominal Compressive Strength 3-23
3.5.3 Nominal Flexure Strength 3-34
3.5.4 Nominal Shear Strength 3-65
3.5.5 Nominal Torsional Strength 3-71
3.6 Design of Members for Combined Forces 3-73
3.6.1 Doubly and Singly Symmetric Members
Subjected to Flexure and Axial Compression 3-74
3.6.2 Doubly and Singly Symmetric Members
Subjected to Flexure and Axial Tension 3-77
3.6.3 Unsymmetric Members Subjected to Flexure
and Axial Force 3-79
3.6.4 Members Subject to Torsion, Flexure, Shear
and Axial Force 3-81
4 Special Seismic Provisions (ANSI/AISC 341-10)
4.1 Notations 4-2
4.2 Design Preferences 4-2
4.3 Overwrites 4-3
4.4 Supported Framing Types 4-3
4.5 Applicability of the Seismic Requirements 4-4
4.6 Design Load Combinations 4-4
4.7 Classification of Sections for Local Buckling 4-7
4.8 Special Check for Column Strength 4-11
4.9 Member Design 4-12
4.9.1 Ordinary Moment Frames (OMF) 4-12
4.9.2 Intermediate Moment Frame (IMF) 4-13
4.9.3 Special Moment Frames (SMF) 4-13
iii
Steel Frame Design AISC 360-10
4.9.4 Special Truss Moment Frames (STMF) 4-14
4.9.5 Ordinary Concentrically Braced Frames (OCBF) 4-14
4.9.6 Ordinary Concentrically Braced Frames from
Isolated Structures (OCBFI) 4-15
4.9.7 Special Concentrically Braced Frames (SCBF) 4-16
4.9.8 Eccentrically Braced Frames (EBF) 4-17
4.9.9 Buckling Restrained Braced Frames (BRBF) 4-21
4.9.10 Special Plate Shear Walls 4-23
4.10 Joint Design 4-23
4.10.1 Design of Continuity Plates 4-23
4.10.2 Design of Doubler Plates 4-30
4.10.3 Weak Beam Strong Column Measure 4-34
4.10.4 Evaluation of Beam Connection Shears 4-37
4.10.5 Evaluation of Brace Connection Forces 4-40
Appendix A P-Delta Effects
Appendix B Steel Frame Design Preferences
Appendix C Steel Frame Design Procedure Overwrites
Appendix D Interactive Steel Frame Design
Appendix E Analysis Sections vs. Design Sections
Appendix F Error and Warning Messages
Bibliography
iv
Chapter 1
Introduction
The design/check of steel frames is seamlessly integrated within the program.
Initiation of the design process, along with control of various design parame-
ters, is accomplished using the Design menu. Automated design at the object
level is available for any one of a number of user-selected design codes, as
long as the structures have first been modeled and analyzed by the program.
Model and analysis data, such as material properties and member forces, are
recovered directly from the model database, and are used in the design process
in accordance with the user defined or default design settings. As with all de-
sign applications, the user should carefully review all of the user options and
default settings to ensure that the design process is consistent with the user’s
expectations. The AISC 360-10 steel frame design options include the use of
the Direct Analysis Method. The software is well suited to make use of the Di-
rect Analysis Method because it can capture the second-order P-Delta and P-δ
effects, provided the user specifies that a nonlinear P-Delta analysis be per-
formed.
Chapter 2 addresses prerequisites related to modeling and analysis for a suc-
cessful design in accordance with “AISC 360-10.” Chapter 3 provides detailed
descriptions of the specific requirements as implemented in “AISC 360-10.”
Chapter 4 provides detailed descriptions of the specific requirements for seis-
mic loading as required by the specification in ANSI/AISC 341-10 code. The
appendices provide details on various topics referenced in this manual. The us-
er also should review the AISC Direct Analysis Method Practical Guide.
1 - 1
Steel Frame Design AISC 360-10
1.1 Load Combinations and Notional Loads
The design is based on a set of user-specified loading combinations. However,
the program provides default load combinations for each supported design
code. If the default load combinations are acceptable, no definition of addition-
al load combinations is required. The Direct Analysis Method requires that a
notional load, N = 0.002Yi , where Yi is the gravity load acting at level i, be
applied to account for the destabilizing effects associated with the initial imper-
fections and other conditions that may induce sway not explicitly modeled in
the structure. The user must be aware that notional loads must be defined and
assigned by the user. Currently, the software creates design combinations that
include notional loads and gravity loads only. If the user needs notional loads
that include combinations containing lateral loads, the user must define such
combinations manually. The automation of combinations, including notional
loads, is currently limited to gravity loads only. Design load combinations of
notional loads acting together with lateral loads currently are NOT automated
by the software.
1.2 Stress Check
Steel frame design/check consists of calculating the flexural, axial, and shear
forces or stresses at several locations along the length of a member, and then
comparing those calculated values with acceptable limits. That comparison
produces a demand/capacity ratio, which typically should not exceed a value of
one if code requirements are to be satisfied. The program follows the same
review procedures whether it is checking a user-specified shape or a shape
selected by the program from a predefined list. The program also checks the
requirements for the beam-column capacity ratio, checks the capacity of the
panel zone, and calculates the doubler plate and continuity plate thickness, if
needed. The program does not do the connection design. However, it calculates
the design basis forces for connection design.
Program output can be presented graphically on the model, in tables for both
input and output data, or in calculation sheets prepared for each member. For
each presentation method, the output is in a format that allows the engineer to
quickly study the stress conditions that exist in the structure, and in the event
the member is not adequate, aid the engineer in taking appropriate remedial
1 - 2 Load Combinations and Notional Loads
Chapter 1 Introduction
measures, including altering the design member without re-running the entire
analysis.
The program supports a wide range of steel frame design codes, including
many national building codes. This manual is dedicated to the use of the menu
option “AISC 360-10.” This option covers the “ANSI/AISC 360-10 Specifica-
tion for Structural Steel Buildings” (AISC 2010a, b), and the “ANSI/ AISC
341-10 Seismic Provisions for Structural Steel Buildings” (AISC 2010c) codes.
The implementation covers loading and load combinations from “ASCE/SEI
7-10 Minimum Design Loads for Buildings and Other Structures” (ASCE
2010), and also special requirements from “IBC 2012 International Building
Code” (IBC 2012). Both LRFD (Load and Resistance Factor Design) and ASD
(Allowable Strength Design) codes are included in this implementation under
the same AISC 360-10 code name. The LRFD and ASD are available as two
options in the program’s preferences feature. In both cases, the strengths are
calculated in the nominal levels. The phi (LRFD) and Omega (ADS) factors
are applied during calculation of demand/capacity ratios only. The design
codes supported under “AISC 360-10” are written in kip-inch units. All the as-
sociated equations and requirements have been implemented in the program in
kip-in units. The program has been enabled with unit conversion capability.
This allows the users to enjoy the flexibility of choosing any set of consistent
units during creating and editing models, exporting and importing the model
components, and reviewing the design results.
1.3 Direct Analysis Method vs. Effective Length
Method
The Direct Analysis Method described in AISC 360-10, Chapter C, is
substantially different from previous design methods supported by AISC. The
user should be knowledgeable about the Design for Stability (Chapter C)
requirements and the requirements pertaining to consideration of the geometric
imperfections, stiffness reductions, and the P-Δ and P-δ effects. Several
methods for consideration of the second-order effects are available to the users.
Each of these are described in detail in a subsequent section (see User Options
in this chapter) and in the Steel Frame Design Preferences, Appendix B of this
manual. Alternatively, if the user desires to use a more traditional design
Direct Analysis Method vs. Effective Length Method 1 - 3
Steel Frame Design AISC 360-10
method, the Effective Length method can be specified using the Design
Preferences.
1.3.1 Effective Length Method
For structures exhibiting small second-order effects, the effective length
method may be suitable. The effective length approach relies on two main
assumptions, namely, that the structural response is elastic and that all columns
buckle simultaneously. The effective length method also relies on a calibrated
approach to account for the differences between the actual member response
and the 2nd-order elastic analysis results. The calibration is necessary because
the 2nd-order elastic analysis does not account for the effects of distributed
yielding and geometric imperfections. Since the interaction equations used in
the effective length approach rely on the calibration corresponding to a 2nd-
order elastic analysis of an idealized structure, the results are not likely
representative of the actual behavior of the structure. However, the results are
generally conservative. In the AISC 360-10 code, the effective length method
is allowed provided the member demands are determined using a second-order
analysis (either explicit or by amplified first-order analysis) and notional loads
are included in all gravity load combinations (AISC Appendix 7). K-factors
must be calculated to account for buckling (except for braced frames, or where
Δ2 /Δ1 ≤ 1.5, K = 1.0) (AISC App. 7.2).
1.3.2 Direct Analysis Method
The Direct Analysis Method is expected to more accurately determine the
internal forces of the structure, provided care is used in the selection of the
appropriate methods used to determine the second-order effects, notional load
effects and appropriate stiffness reduction factors as defined in AISC C2.
Additionally, the Direct Analysis Method does not use an effective length
factor other than K = 1.0. The rational behind the use of K = 1.0 is that proper
consideration of the second-order effects (P-∆ and P-δ), geometric
imperfections (using notional loads) and inelastic effects (applying stiffness
reductions) better accounts for the stability effects of a structure than the earlier
Effective Length methods.
1 - 4 Direct Analysis Method vs. Effective Length Method
Chapter 1 Introduction
1.4 User Options
In addition to offering ASD and LRFD design, the Design Options menu pro-
vides seven analysis methods for design, as follows:
General Second Order Elastic Analysis (AISC C1.2)
Second Order Analysis by Amplified First Order Analysis (AISC C1.2,
App. 7.2, App. 8.2)
Limited First Order Elastic Analysis (AISC C1.2, App. 7.3)
Direct Analysis Method with General Second Order Analysis and Variable
Factor Stiffness Reduction (AISC C1, C2)
Direct Analysis Method with General Second Order Analysis and Fixed
Factor Stiffness Reduction (AISC C1, C2)
Direct Analysis Method with Amplified First Order Analysis and Variable
Factor Stiffness Reduction (AISC C1, C2)
Direct Analysis Method with Amplified First Order Analysis and Fixed
Factor Stiffness Reduction (AISC C1, C2)
These options are explained in greater detail in Chapter 2. The first three op-
tions make use of the effective length approach to determine the effective
length factors, K. The four options available for the Direct Design Method dif-
fer in the use of a variable or fixed stiffness reduction factor and the method
used to capture the second-order effects. All four Direct Analysis Methods op-
tions use an effective length factor, K = 1.0.
1.5 Non-Automated Items in Steel Frame Design
Currently, the software does not automate the following:
Notional loads combinations that include lateral wind and quake loads
The validity of the analysis method. The user must verify the suitability of
the specified analysis method used under the User Options described in the
User Options 1 - 5
Steel Frame Design AISC 360-10
preceding sections. The AISC code requires, for instance, that the Direct
Analysis Method be used when a ratio of the second order displacements to
the first order displacements exceeds 1.5 (AISC C1.2, App. 7.2.1(2), App.
7.3.1(2)). This check currently must be performed by the user.
P-Δ analysis. Since many different codes are supported by the software and
not all require a P-Δ analysis, the user must specify that a P-Δ analysis be
performed during the analysis phase so that the proper member forces are
available for use in the design phase. See the AISC Direct Analysis Method
Practical Guide for additional information.
1 - 6 Non-Automated Items in Steel Frame Design
Chapter 2
Design Algorithms
This chapter provides an overview of the basic assumptions, design precondi-
tions, and some of the design parameters that affect the design of steel frames.
For referring to pertinent sections of the corresponding code, a unique prefix is
assigned for each code.
• Reference to the ANSI/AISC 360-10 code is identified with the prefix
“AISC.”
• Reference to the ANSI/AISC 341-10 code is identified with the prefix
“AISC SEISMIC” or sometimes “SEISMIC” only.
• Reference to the ASCE/SEI 7-10 code is identified with the prefix
“ASCE.”
• Reference to the IBC 2012 code is identified with the prefix “IBC.”
2.1 Check and Design Capability
The program has the ability to check adequacy of a section (shape) in accord-
ance with the requirements of the selected design code. Also the program can
automatically choose (i.e., design) the optimal (i.e., least weight) sections from
a predefined list that satisfies the design requirements.
2 - 1
Steel Frame Design AISC 360-10
To check adequacy of a section, the program checks the demand/capacity (D/C) ra-
tios at a predefined number of stations for each design load combination. It calcu-
lates the envelope of the D/C ratios. It also checks the other requirements on a pass
or fail basis. If the capacity ratio remains less than or equal to the D/C ratio limit,
which is a number close to 1.0, and if the section passes all the special require-
ments, the section is considered to be adequate, else the section is considered to be
failed. The D/C ratio limit is taken as 0.95 by default. However, this value can be
overwritten in the Preferences (see Chapter 3).
To choose (design) the optional section from a predefined list, the program first
orders the list of sections in increasing order of weight per unit length. Then it
starts checking each section from the ordered list, starting with the one with
least weight. The procedure of checking each section in this list is exactly the
same as described in the preceding paragraph. The program will evaluate each
section in the list until it finds the least weight section that passes the code
checks. If no section in the list is acceptable, the program will use the heaviest
section but flag it as being overstressed.
To check adequacy of an individual section, the user must assign the section
using the Assign menu. In that case, both the analysis and design sections will
be changed.
To choose the optimal section, the user must first define a list of steel sections,
the Auto Select sections list. The user must next assign this list, in the same
manner as any other section assignment, to the frame members to be opti-
mized. The program will use the median section by weight when doing the ini-
tial analysis. Check the program Help for more information about defining and
assigning Auto Select Section lists.
2.2 Design and Check Stations
For each design combination, steel frame members (beams, columns, and
braces) are designed (optimized) or checked at a number of locations (stations)
along the length of the object. The stations are located at equally spaced
segments along the clear length of the object. By default, at least three stations
will be located in a column or brace member, and the stations in a beam will be
spaced at most 2 feet apart (0.5 m if the model has been created in metric
units). The user can overwrite the number of stations in an object before the
2 - 2 Design and Check Stations
Chapter 2 Design Algorithms
analysis is run and refine the design along the length of a member by request-
ing more stations. Refer to the program Help for more information about
specifying the number of stations in an object.
2.3 Demand/Capacity Ratios
Determination of the controlling demand/capacity (D/C) ratios for each steel
frame member indicates the acceptability of the member for the given loading
conditions. The steps for calculating the D/C ratios are as follows:
The factored forces are calculated for axial, flexural, and shear at each de-
fined station for each design combination. The bending moments are calcu-
lated about the principal axes. For I-Shape, Box, Channel, T-Shape, Dou-
ble-Angle, Pipe, Circular, and Rectangular sections, the principal axes co-
incide with the geometric axes. For Single-Angle sections, the design con-
siders the principal properties. For General sections, it is assumed that all
section properties are given in terms of the principal directions.
For Single-Angle sections, the shear forces are calculated for directions
along the geometric axes. For all other sections, the program calculates the
shear forces along the geometric and principal axes.
The nominal strengths are calculated for compression, tension, bending
and shear based on the equations provided later in this manual. For flexure,
the nominal strengths are calculated based on the principal axes of bend-
ing. For the I-Shape, Box, Channel, Circular, Pipe, T-Shape, Double-Angle
and Rectangular sections, the principal axes coincide with their geometric
axes. For the Angle sections, the principal axes are determined and all
computations related to flexural stresses are based on that.
The nominal strength for shear is calculated along the geometric axes for
all sections. For I-Shape, Box, Channel, T-Shape, Double-Angle, Pipe,
Circular, and Rectangular sections, the principal axes coincide with their
geometric axes. For Single-Angle sections, principal axes do not coincide
with the geometric axes.
Factored forces are compared to nominal strengths to determine D/C ratios.
In either case, design codes typically require that the ratios not exceed a
Demand/Capacity Ratios 2 - 3
Steel Frame Design AISC 360-10
value of one. A capacity ratio greater than one indicates a member that has
exceeded a limit state.
2.4 Design Load Combinations
The design load combinations are the various combinations of the prescribed
load cases for which the structure needs to be checked. The program creates a
number of default design load combinations for steel frame design. Users can
add their own design combinations as well as modify or delete the program
default design load combinations. An unlimited number of design load combi-
nations can be specified.
To define a design load combination, simply specify one or more load cases,
each with its own scale factor. The scale factors are applied to the forces and
moments from the load cases to form the factored design forces and moments
for each design load combination.
For normal loading conditions involving static dead load (DL), live load (LL),
roof live load (RL), snow load (SL), wind load (WL), earthquake load (EL),
notional load (NL), and dynamic response spectrum load (EL), the program
has built-in default design combinations for the design code. These are based
on the code recommendations.
The default design combinations assume all load cases declared as dead or live to
be additive. However, each load case declared as wind, earthquake, or
response spectrum cases, is assumed to be non-additive with other loads and pro-
duces multiple lateral combinations. Also static wind, earthquake and
notional load responses produce separate design combinations with the sense (posi-
tive or negative) reversed. The notional load patterns are added to load combina-
tions involving gravity loads only. The user is free to modify the default design
preferences to include the notional loads for combinations involving lateral loads.
For other loading conditions involving moving load, time history, pattern live
load, separate consideration of roof live load, snow load, and the like, the user
must define the design load combinations in lieu of or in addition to the default
design load combinations. If notional loads are to be combined with other load
combinations involving wind or earthquake loads, the design load combina-
tions need to be defined in lieu of or in addition to the default design load com-
binations.
2 - 4 Design Load Combinations
Chapter 2 Design Algorithms
For multi-valued design combinations, such as those involving response spec-
trum, time history, moving loads and envelopes, where any correspondence
between forces is lost, the program automatically produces sub-combinations
using the maxima/minima values of the interacting forces. Separate combina-
tions with negative factors for response spectrum load cases are not required
because the program automatically takes the minima to be the negative of the
maxima response when preparing the sub-combinations described previously.
The program allows live load reduction factors to be applied to the member
forces of the reducible live load case on a member-by-member basis to reduce
the contribution of the live load to the factored responses.
2.5 Second Order P-Delta Effects
The AISC 360-10 steel frame design options include the use of the Direct
Analysis Method. The software is well suited to make us of the Direct Analysis
Method because each program can capture the second-order P-∆ and P-δ
effects, provided the user specifies that a nonlinear P-Delta analysis be
performed.
∆
Original position of frame
element shown by vertical
line
Position of frame element
as a result of global lateral
translation,
∆
, shown by
dashed line
Final deflected position of the
frame element that includes the
global lateral translation, ∆, and
the local deformation of the
element,
δ
δ
δ
P
∆
Original position of frame
element shown by vertical
line
Position of frame element
as a result of global lateral
translation,
∆
, shown by
dashed line
Final deflected position of the
frame element that includes the
global lateral translation, ∆, and
the local deformation of the
element,
δ
δ
δ
P
Figure 2-1 System sway and element order effects
For more details about the program capabilities and limitations, see Appendix A.
Second Order P-Delta Effects 2 - 5
Steel Frame Design AISC 360-10
2.6 Analysis Methods
The code requires that stability shall be provided for the structure as a whole
and for each of the elements. Any method of analysis that considers the influ-
ence of second order effects of
P-
∆
and
P-
δ
, geometric imperfections, out-of-
plumbness, and member stiffness reduction due to residual stresses are permit-
ted by the code. The effects of geometric imperfection and out-of-plumbness
generally are captured by the use of notional loads. The effect of axial, shear
and flexural deformations and the effects of residual stresses on the member
stiffness reduction has been considered in a specialized method called “Direct
Analysis Method.” This method can come in different incarnations (formats)
according to the choice of the engineer as allowed in the code.
The program offers the user seven analysis options for design:
Direct Analysis Method
• General Second Order Elastic Analysis with
τ
b variable (user option 1, Default)
τ
b fixed (user option 2)
• Amplified First Order Elastic Analysis with
τ
b variable (user option 3)
τ
b fixed (user option 4)
Equivalent Length Method
• General Second Order Elastic Analysis
(AISC C1.2, App. 7.2) (user option 5)
• Amplified First Order Elastic Analysis
(AISC C1.2, App. 8.2) (user option 6)
Limited First-Order Analysis (AISC C1.2, App. 7.3) (user option 7)
A summary of all of the user options and requirements is provided in
Table 2-1. The main difference between the various options concerns the use of
the Direct Analysis Method or the Equivalent Length Method. Within each of
the categories, the user can choose the method to calculate the second-order
2 - 6 Analysis Methods
Chapter 2 Design Algorithms
effects, namely, by a General Second Order Analysis or an Amplified First-
Order Analysis. When the amplified first-order analysis is used, the force
amplification factors,
1
B
and
(AISC App. 8.2), are needed. The
1
B
factor is
calculated by the program; however, the
2
B
factor is not. The user will need to
provide this value using the overwrite options that are described in Appendix
B.
When the user selects one of the options available under the Direct Analysis
Method, the user must further choose how the stiffness reduction factors for
EI
and
AE
are to be considered. For options 1 and 3, Table 2-1, the stiffness
reduction factors (
b
τ
) are variable because they are functions of the axial force
in the members, while for methods 2 and 4, the stiffness reduction factors are
fixed (0.8), and not a function of axial force. If the user desires, the stiffness
reduction factors (
b
τ
) can be overwritten. When options 2 and 4 are used, a
higher notional load coefficient (0.003) must be used compared to methods 1
and 3 for which the notional load coefficient is 0.002. Also, all the direct anal-
ysis methods (methods 1 through 4) allow use of
K
-factors for sway condition
(
2
K
) to be equal to 1, which is a drastic simplification over the other effective
length method.
The AISC requirements to include notional loads are also summarized in Table
2-1. The notional load coefficients (AISC C2.2b) are summarized as well. The
program automates creation of notional load combinations for all gravity loads
but does not automate the creation of notional load combinations that include
lateral wind or seismic loads. Combinations for notional loads with lateral
loads are required for the Direct Analysis Method when the
2nd 1st
∆∆
exceeds
1.7 (AISC E2.2b(4)). Additionally, combinations for notional loads with lateral
loads are required if the Limited First Order Analysis, option 7, is used (AISC
App. 7.3.2).
The Limited First Order Analysis, option 7, does not include the secondary
P-
∆
and
P-
δ
effects. This method has very limited applicability and might be
appropriate only when the axial forces in the columns are very small compared
to their Euler buckling capacities.
When using the LRFD provision, the actual load combinations are used for
second order P-∆ effects. When using the ASD provision, the load combina-
Analysis Methods 2 - 7
Steel Frame Design AISC 360-10
tions are first amplified by 1.6 before the P-∆ analysis and then the results are
reduced by a factor of
( )
1 1.6
(AISC C2.1(4)).
Table 2-1 The Essentials and Limitations of the Design Analysis Methods
Direct Analysis Method
Option Variable
Limitation or
Applicability
Essentials of the Method
General Second
Order Analysis
Variable
Factor Stiffness
Reduction
No limitation
2nd Order Analysis
Reduced stiffness
* 0.8 b
EI EI= τ
* 0.8EA EA=
1.0 for 0.5
4 1 for 0.5
r
y
b
rr r
yy y
P
P
PPP
PPP
α
≤
τ=
 
ααα
 
−≥
 
 
1
B
and
2
B
not used
21 (used for )=n
KP
Notional load with all combos, except for
21
1.7
nd st
∆ ∆≤
for
which notional load with gravity combos only
Notional load coefficient = 0.002 (typically)
Fixed Factor
Stiffness
Reduction
No limitation
2nd Order Analysis
Reduced stiffness
* 0.8
b
EI EI
= τ
* 0.8
EA EA=
τ
=1.0
b
1
B
and
2
B
not used
21 (used for )=n
KP
Notional load with all combos, except for
21
1.7
nd st
∆ ∆≤
for which notional load with gravity combos only
Notional load coefficient = 0.003 (typically)
Amplified First
Order Analysis
Variable
Factor Stiffness
Reduction
No limitation
1st Order Analysis
Reduced Stiffness
* 0.8 b
EI EI= τ
* 0.8EA EA=
1.0 for 0.5
4 1 for 0.5
r
y
b
rr r
yy y
P
P
PPP
PPP
α
≤
τ=
 
ααα
 
−≥
 
 
11
1 for =
KB
22
1 for and =
n
K PB
Notional load with all combos, except for
21
1.7
nd st
∆ ∆≤
for which notional load with gravity combos only
Notional load coefficient = 0.002 (typically)
2 - 8 Analysis Methods
Chapter 2 Design Algorithms
Table 2-1 The Essentials and Limitations of the Design Analysis Methods
Direct Analysis Method
Option Variable
Limitation or
Applicability
Essentials of the Method
Amplified First
Order Analysis
Fixed Factor
Stiffness
Reduction
No limitation
2nd Order Analysis
Reduced stiffness
* 0.8 b
EI EI= τ
* 0.8EA EA=
1.0
b
τ=
2
1 (used for )=
n
KP
Notional load with all combos, except for
21
1.7
nd st
∆ ∆≤
for which notional load with gravity combos only
Notional load coefficient = 0.003 (typically)
Effective Length Method
Option
Limitation or
Applicability
Essentials of the Method
General Second
Order Elastic
Analysis
2
1
1.5
(for all stories)
nd
st
∆≤
∆
α
=
r
y
Pany
P
(for all columns)
2nd Order Analysis
Unreduced Stiffness
2
=KK
(used for
n
P
)
Notional load with gravity combos only
Notional load coefficient = 0.002 (typically)
1
B
= 1
2
B
= 1
Amplified First
Order Analysis
2
1
1.5
(for all stories)
nd
st
∆≤
∆
α
=
r
y
Pany
P
(for all columns)
1st Order Analysis
Unreduced stiffness
1
K
for
1
B
2
K
for
2
B
2
=KK
(used for
n
P
)
Notional load with gravity combos only
Notional load with coefficient = 0.002 (typically)
Use of
1
B
and
2
B
Limited First Order Analysis
Limited First
Order Elastic
Analysis
2
1
1.5
(for all stories)
nd
st
∆≤
∆
α
≤0.5
r
y
P
P
(for all columns)
1st Order Analysis
Unreduced stiffness
2
K
for
n
P
(not
2
B
)
Notional load with all combos
Notional load with coefficient =
( )
2 0.0042
L
∆

α≥


The program has several limitations that have been stated in Section 1.5 and
the preceding paragraphs. Additionally, the user must be aware that it is possi-
ble to choose a design option that violates certain provisions of the AISC code
that will not be identified by the program. The limitation for the use of the
Analysis Methods 2 - 9
Steel Frame Design AISC 360-10
effective length method, namely, the requirement that
2
1
1.5
nd
st
∆≤
∆
and
α
r
e
P
P
must
be verified by the user. To assist users to in making validity checks, the ratio
α
r
e
P
P
and τ are now reported in tabular form for each member.
2.7 Notional Load Patterns
Notional loads are lateral loads that are applied at each framing level and are
specified as a percentage of the gravity loads applied at that level. They are
intended to account for the destabilizing effects of out-of-plumbness, geometric
imperfections, inelasticity in structural members, and any other effects that
could induce sway and that are not explicitly considered in the analysis.
The program allows the user to create a Notional Load pattern as a percentage
of the previously defined gravity load pattern to be applied in one of the global
lateral directions: X or Y. The user can define more than one notional load
pattern associated with one gravity load by considering different factors and
different directions. In the ANSI/AISC 360-10 code, the notional loads are
typically suggested to be 0.2% (or 0.002) (AISC C2.2b(3)), a factor referred to
as the notional load coefficient in this document. The notional load coefficient
can be 0.003 (AISC C2.3(3)). In some cases, it can be a function of second
order effects measured by relative story sway (AISC App. 7.3(2)). The code
also gives some flexibility to allow the engineer-of-record to apply judgment.
The notional load patterns should be considered in combination with appropri-
ate factors, appropriate directions, and appropriate senses. Some of the design
analysis methods need the notional loads to be considered only in gravity load
combinations (AISC C2.2b(4)), and some of the methods need the notional
loads to be considered in all the design load combinations (AISC C2.2b(4)).
For a complete list, see Table 2-1 in the preceding “Second Order Effects and
Analysis Methods” section of this chapter.
Currently, the notional loads are not automatically included in the default
design load combinations that include lateral loads. However, the user is free to
modify the default design load combinations to include the notional loads with
appropriate factors and in appropriate load combinations.
2 - 10 Notional Load Patterns
Chapter 2 Design Algorithms
2.8 Member Unsupported Lengths
The column unsupported lengths are required to account for column
slenderness effects for flexural buckling and for lateral-torsional buckling. The
program automatically determines the unsupported length ratios, which are
specified as a fraction of the frame object length. These ratios times the frame
object lengths give the unbraced lengths for the member. These ratios can also
be overwritten by the user on a member-by-member basis, if desired, using the
overwrite option.
Two unsupported lengths,
33
l
and
, as shown in Figure 2-2 are to be
considered for flexural buckling. These are the lengths between support points
of the member in the corresponding directions. The length 33
l
corresponds to
instability about the 3-3 axis (major axis), and
22
l
corresponds to instability
about the 2-2 axis (minor axis). The length
LTB
l,
not shown in the figure, is
also used for lateral-torsional buckling caused by major direction bending (i.e.,
about the 3-3 axis).
In determining the values for
22
l
and
of the members, the program recog-
nizes various aspects of the structure that have an effect on these lengths, such
as member connectivity, diaphragm constraints and support points. The pro-
gram automatically locates the member support points and evaluates the corre-
sponding unsupported length.
It is possible for the unsupported length of a frame object to be evaluated by
the program as greater than the corresponding member length. For example,
assume a column has a beam framing into it in one direction, but not the other,
at a floor level. In this case, the column is assumed to be supported in one
direction only at that story level, and its unsupported length in the other direc-
tion will exceed the story height.
By default, the unsupported length for lateral-torsional buckling,
LTB
l,
is taken
to be equal to the
22
l
factor. Similar to
22
l
and
33
l,
LTB
l
can be overwritten.
Member Unsupported Lengths 2 - 11
Steel Frame Design AISC 360-10
Figure 2-2 Unsupported lengths
and
22
l
2.9 Effects of Breaking a Member into Multiple
Elements
The preferred method is to model a beam, column or brace member as one sin-
gle element. However, the user can request that the program break a member
internally at framing intersections and at specified intervals. In this way, accu-
racy in modeling can be maintained, at the same time design/check specifica-
tions can be applied accurately. There is special emphasis on the end forces
(moments in particular) for many different aspects of beam, column and brace
design. If the member is manually meshed (broken) into segments, maintaining
the integrity of the design algorithm becomes difficult.
Manually, breaking a column member into several elements can affect many
things during design in the program.
1. The unbraced length: The unbraced length is really the unsupported length
between braces. If there is no intermediate brace in the member, the un-
braced length is typically calculated automatically by the program from the
top of the flange of the beam framing the column at bottom to the bottom
of the flange of the beam framing the column at the top. The automatically
2 - 12 Effects of Breaking a Member into Multiple Elements
Chapter 2 Design Algorithms
calculated length factor typically becomes less than 1. If there are interme-
diate bracing points, the user should overwrite the unbraced length factor in
the program. The user should choose the critical (larger) one. Even if the
user breaks the element, the program typically picks up the unbraced length
correctly, provided that there is no intermediate bracing point.
2. K-factor: Even if the user breaks the member into pieces, the program typi-
cally can pick up the
-factorsK
correctly. However, sometimes it can not.
The user should note the
-factorsK
. All segments of the member should
have the same
-factorK
and it should be calculated based on the entire
member. If the calculated
-factorK
is not reasonable, the user can over-
write the
-factorsK
for all the segments.
3.
m
C
factor: The
m
C
factor should be based on the end moments of
unbraced lengths of each segment and should not be based on the end
moments of the member. The program already calculates the m
C
factors
based on the end moments of unbraced lengths of each segment. If the
break-up points are the brace points, no action is required by the user. If
the broken segments do not represent the brace-to-brace unsupported
length, the program calculated
m
C
factor is conservative. If this
conservative value is acceptable, no action is required by the user. If it is
not acceptable, the user can calculate the
m
C
factor manually for the
critical combination and overwrite its value for that segment.
4.
b
C
factor: The logic is similar to that for the m
C
factor.
5.
1
B
factor: This factor amplifies the factored moments for the P-
δ
effect. In
its expression, there are the
m
C
factor and the Euler Buckling capacity
e
P
.
If the user keeps the unbraced length ratios (
33
l
and
22
l
) and the
-factorsK
( )
and
33 22
KK
correct, the
1
B
factor would be correct. If the
axial force is small, the
1
B
factor can be 1 and have no effect with respect
to modeling the single segment or multi-segment element.
6.
2
B
factor: The program does not calculate the
factor. The program
assumes that the user turns on the P-
∆
. In such cases,
2
B
can be taken as
equal to 1. That means the modeling with one or multiple segments has no
effect on this factor.
Effects of Breaking a Member into Multiple Elements 2 - 13
Steel Frame Design AISC 360-10
If the user models a column with a single element and makes sure that the
L
-
factors and
K
-factors are correct, the effect of
1
B
and
2
B
will be picked up
correctly. The factors
m
C
and
b
C
will be picked up correctly if there is no in-
termediate bracing point. The calculated
m
C
and
b
C
factors will be slightly
conservative if there are intermediate bracing points.
If the user models a column with multiple elements and makes sure that
L
-
factors and
-factorsK
are correct, the effect of
1
B
and
2
B
will be picked up
correctly. The factors m
C
and
b
C
will be picked up correctly if the member is
broken at the bracing points. The calculated m
C
and
b
C
factors will be con-
servative if the member is not broken at the bracing points.
2.10 Effective Length Factor (K)
The effective length method for calculating member axial compressive strength
has been used in various forms in several stability based design codes. The
method originates from calculating effective buckling lengths, KL, and is based
on elastic/inelastic stability theory. The effective buckling length is used to
calculate an axial compressive strength, Pn, through an empirical column curve
that accounts for geometric imperfections, distributed yielding, and residual
stresses present in the cross-section.
There are two types of
-factorsK
in the ANSI/AISC 360-10 code. The first
type of
-factor
K
is used for calculating the Euler axial capacity assuming that
all of the beam-column joints are held in place, i.e., no lateral translation is al-
lowed. The resulting axial capacity is used in calculation of the
1
B
factor. This
K
-factor is named as
1
K
in the code. This
1
K
factor is always less than 1 and
is not calculated. By default the program uses the value of 1 for
1
K
. The pro-
gram allows the user to overwrite
1
K
on a member-by-member basis.
The other
-factorK
is used for calculating the Euler axial capacity assuming
that all the beam-column joints are free to sway, i.e., lateral translation is al-
lowed. The resulting axial capacity is used in calculating
n
P
. This
-factorK
is
named as
2
K
in the code. This
2
K
is always greater than 1 if the frame is a
sway frame. The program calculates the
2
K
factor automatically based on
sway condition. The program also allows the user to overwrite
2
K
factors on a
2 - 14 Effective Length Factor (K)
Chapter 2 Design Algorithms
member-by-member basis. The same
2
K
factor is supposed to be used in cal-
culation of the
2
B
factor. However the program does not calculate
2
B
factors
and relies on the overwritten values. If the frame is not really a sway frame, the
user should overwrite the
2
K
factors.
Both
1
K
and
2
K
have two values: one for major direction and the other for
minor direction,
1minor
K
,
1major
K
,
2minor
K
,
2major
K
.
There is another
-factorK
.
ltb
K
for lateral torsional buckling. By default,
ltb
K
is taken as equal to
2minor
K
. However the user can overwrite this on a member-
by-member basis.
The rest of this section is dedicated to the determination of
2
K
factors.
The
-factorK
algorithm has been developed for building-type structures,
where the columns are vertical and the beams are horizontal, and the behavior
is basically that of a moment-resisting frame for which the
-factorK
calcula-
tion is relatively complex. For the purpose of calculating
-factorsK
, the ob-
jects are identified as columns, beam and braces. All frame objects parallel to
the
Z
-axis are classified as columns. All objects parallel to the
X
-
Y
plane are
classified as beams. The remainders are considered to be braces.
The beams and braces are assigned
-factorsK
of unity. In the calculation of the
-factorsK
for a column object, the program first makes the following four
stiffness summations for each joint in the structural model:

=

∑
cc
cx cx
EI
SL
bb
bx bx
EI
SL

=

∑
cc
cy cy
EI
SL

=

∑
bb
by by
EI
SL

=

∑
where the x and y subscripts correspond to the global X and Y directions and
the c and b subscripts refer to column and beam. The local 2-2 and 3-3 terms
22 22
EI L
and
33 33
EI L
are rotated to give components along the global X and
Y directions to form the
( )
x
EI L
and
( )
y
EI L
values. Then for each column,
the joint summations at END-I and the END-J of the member are transformed
back to the column local 1-2-3 coordinate system, and the
G
-values for END-I
Effective Length Factor (K) 2 - 15
Steel Frame Design AISC 360-10
and the END-J of the member are calculated about the 2-2 and 3-3 directions as
follows:
22
22
22
b
I
c
I
I
S
S
G=
22
22
22
b
J
c
J
J
S
S
G=
33
33
33
b
I
c
I
I
S
S
G=
33
33
33
b
J
c
J
J
S
S
G=
If a rotational release exists at a particular end (and direction) of an object, the
corresponding value of
G
is set to 10.0. If all degrees of freedom for a particu-
lar joint are deleted, the
G
-values for all members connecting to that joint will
be set to 1.0 for the end of the member connecting to that joint. Finally, if
I
G
and
J
G
are known for a particular direction, the column
-factorsK
for the cor-
responding direction is calculated by solving the following relationship for α:
α
α
α
tan
)(6
36
2=
+
−
JI
JI
GG
GG
from which
K
= π/α. This relationship is the mathematical formulation for the
evaluation of
-factorsK
for moment-resisting frames assuming sidesway to be
uninhibited. For other structures, such as braced frame structures, the
-factorsK
for all members are usually unity and should be set so by the user.
The following are some important aspects associated with the column
-factor
K
algorithm:
An object that has a pin at the joint under consideration will not enter the
stiffness summations calculated above. An object that has a pin at the far
end from the joint under consideration will contribute only 50% of the cal-
culated EI value. Also, beam members that have no column member at the
far end from the joint under consideration, such as cantilevers, will not en-
ter the stiffness summation.
If there are no beams framing into a particular direction of a column mem-
ber, the associated G-value will be infinity. If the G-value at any one end
of a column for a particular direction is infinity, the
K
-factor correspond-
ing to that direction is set equal to unity.
If rotational releases exist at both ends of an object for a particular direc-
tion, the corresponding
-factorK
is set to unity.
2 - 16 Effective Length Factor (K)
Chapter 2 Design Algorithms
The automated
-factor
K
calculation procedure can occasionally generate
artificially high
-factorsK
, specifically under circumstances involving
skewed beams, fixed support conditions, and under other conditions where
the program may have difficulty recognizing that the members are laterally
supported and
-factorsK
of unity are to be used.
All
-factorsK
produced by the program can be overwritten by the user.
These values should be reviewed and any unacceptable values should be
replaced.
The beams and braces are assigned
-factorsK
of unity.
When a steel frame design is performed in accordance with ANSI/AISC 360-
10 provision and the analysis method is chosen to be any of the Direct Analysis
Methods, the
ltb
K
and
2
K
factors (
2minor
K
and
2major
K
) are automatically taken
as 1 (AISC C.3). However, their overwritten values are considered in design
even if any of the Direct Analysis Methods is chosen.
2.11 Supported Framing Types
The code (ANSI/AISC 341-10) recognizes the following types of framing
systems.
Framing Type References
OMF (Ordinary Moment Frame) AISC SEISMIC E1
IMF (Intermediate Moment Frame) AISC SEISMIC E2
SMF (Special Moment Frame) AISC SEISMIC E3
STMF (Special Truss Moment Frame) AISC SEISMIC E4
OCBF (Ordinary Concentrically Braced Frame) AISC SEISMIC F1
SCBF (Special Concentrically Braced Frame) AISC SEISMIC F2
EBF (Eccentrically Braced Frame) AISC SEISMIC F3
BRBF (Buckling Restrained Braced Frame) AISC SEISMIC F4
SPSW (Special Plate Shear Wall) AISC SEISMIC F5
Supported Framing Types 2 - 17
Steel Frame Design AISC 360-10
With regard to these framing types, the program has implemented specifica-
tions for all types of framing systems, except STMF, BRBF, and SPSW. Im-
plementing those three types of framing require further information about
modeling.
The program recognizes the OCBF framing in its two separate incarnations:
OCBF for regular Ordinary Concentrically Braced Frames (AISC SEISMIC
F1.1) and OCBFI for (base) Isolated Ordinary Concentrically Braced Frames
(AISC SEISMIC F1.7).
See Chapter 4 Special Seismic Provisions (ANSI/AISC 341-10) for additional
requirements.
2.12 Continuity Plates
In a plan view of a beam/column connection, a steel beam can frame into a
column in the following ways:
The steel beam frames in a direction parallel to the column major direction,
i.e., the beam frames into the column flange.
The steel beam frames in a direction parallel to the column minor direc-
tion, i.e., the beam frames into the column web.
The steel beam frames in a direction that is at an angle to both of the prin-
cipal axes.
To achieve a beam/column moment connection, continuity plates, such as
shown in Figure 2-3, are usually placed on the column, in line with the top and
bottom flanges of the beam, to transfer the compression and tension flange
forces of the beam into the column.
For connection conditions described in the last two bullet items, the thickness
of such plates is usually set equal to the flange thickness of the corresponding
beam.
2 - 18 Continuity Plates
Chapter 2 Design Algorithms
Figure 2-3 Doubler Plates and Continuity Plates
Continuity Plates 2 - 19
Steel Frame Design AISC 360-10
However, for the connection condition described by the first bullet item, where
the beam frames into the flange of the column, such continuity plates are not
always needed. The requirement depends upon the magnitude of the beam
flange force and the properties of the column.
The program investigates whether the continuity plates are needed based on the
requirements of the selected code. Columns of I-sections supporting beams of
I-sections only are investigated. The program evaluates the continuity plate re-
quirements for each of the beams that frame into the column flange and reports
the maximum continuity plate area that is needed for each beam flange. The
continuity plate requirements are evaluated for moment frames only.
2.13 Doubler Plates
One aspect of the design of a steel framing system is an evaluation of the shear
forces that exist in the region of the beam column intersection known as the
panel zone. Shear stresses seldom control the design of a beam or column
member. However, in a moment resisting frame, the shear stress in the beam-
column joint can be critical, especially in framing systems when the column is
subjected to major direction bending and the web of the column resists the joint
shear forces. In minor direction bending, the joint shear is carried by the col-
umn flanges, in which case the shear stresses are seldom critical, and the pro-
gram does therefore not investigate this condition.
Shear stresses in the panel zone, due to major direction bending in the column,
may require additional plates to be welded onto the column web, depending
upon the loading and the geometry of the steel beams that frame into the col-
umn, either along the column major direction, or at an angle so that the beams
have components along the column major direction. See Figure 3-3. When
code appropriate, the program investigates such situations and reports the
thickness of any required doubler plates. Only columns with I-shapes and only
supporting beams with I-shapes are investigated for doubler plate requirements.
Also, doubler plate requirements are evaluated for moment frames only.
2 - 20 Doubler Plates
Chapter 2 Design Algorithms
2.14 Choice of Units
English as well as SI and MKS metric units can be used for input. The codes
are based on a specific system of units. All equations and descriptions present-
ed in the subsequent chapters correspond to that specific system of units unless
otherwise noted. However, any system of units can be used to define and de-
sign a structure in the program.
The Display Unit preferences allow the user to specify the units.
Choice of Units 2 - 21
Chapter 3
Design Using ANSI/AISC 360-10
This chapter provides a detailed description of the algorithms used by the pro-
grams in the design/check of structures in accordance with “ANSI/AISC 360-
10 — Specifications for Structural Steel Building” (AISC 2010a, b). The menu
option also covers the “ANSI/AISC 341-10 — Seismic Provisions for Struc-
tural Steel Building” (AISC 2010c), which is described in the next chapter. The
implementation covers load combinations from “ASCE/SEI 7-10,” which is
described in the section “Design Loading Combinations” in this chapter. The
loading based on “ASCE/SEI 7-10” has been described in a separate document
entitled “CSI Lateral Load Manual” (CSI 2012). References also are made to
IBC 2012 in this document.
For referring to pertinent sections of the corresponding code, a unique prefix is
assigned for each code.
• Reference to the ANSI/AISC 360-10 code is identified with the prefix
“AISC.”
• Reference to the ANSI/AISC 341-10 code is identified with the prefix
“AISC SEISMIC” or sometimes “SEISMIC” only.
• Reference to the ASCE/SEI 7-10 code is identified with the prefix
“ASCE.”
• Reference to the IBC 2012 code is identified with the prefix “IBC.”
3 - 1
Steel Frame Design AISC 360-10
3.1 Notations
The various notations used in this chapter are described herein.
A
Cross-sectional area, in2
A
e
Effective cross-sectional area for slender sections, in2
A
g
Gross cross-sectional area, in2
A
v2
,A
v3
Major and minor shear areas, in2
A
w
Shear area, equal dt
w
per web, in2
B
1
Moment magnification factor for moments not causing sidesway
B
2
Moment magnification factor for moments causing sidesway
C
b
Bending coefficient
C
m
Moment coefficient
C
w
Warping constant, in6
D
Outside diameter of pipes, in
E
Modulus of elasticity, ksi
F
cr
Critical compressive stress, ksi
F
r
Compressive residual stress in flange assumed 10.0 for rolled
sections and 16.5 for welded sections, ksi
F
y
Yield stress of material, ksi
G
Shear modulus, ksi
I
22
Minor moment of inertia, in4
I
33
Major moment of inertia, in4
J
Torsional constant for the section, in4
3 - 2 Notations
Chapter 3 - Design using ANSI/AISC 360-10
K
Effective length factor
K
1
Effective length factor for braced condition
K2
Effective length factor for unbraced condition
K
33
,K
22
Effective length K-factors in the major and minor directions for
appropriate braced (K1) and unbraced (K2) condition
L
b
Laterally unbraced length of member, in
L
p
Limiting laterally unbraced length for full plastic capacity, in
L
r
Limiting laterally unbraced length for inelastic lateral-torsional
buckling, in
Mcr
Elastic buckling moment, kip-in
Mlt
Factored moments causing sidesway, kip-in
Mnt
Factored moments not causing sidesway, kip-in
Mn33,Mn22
Nominal bending strength in major and minor directions, kip-in
Mob
Elastic lateral-torsional buckling moment for angle sections, kip-
in
M
r33
, M
r22
Major and minor limiting buckling moments, kip-in
Mu
Factored moment in member, kip-in
M
u33
, M
u22
Factored major and minor moments in member, kip-in
P
e
Euler buckling load, kips
Pn
Nominal axial load strength, kip
P
u
Factored axial force in member, kips
P
y
A
g
F
y
, kips
Q
Reduction factor for slender section, =
QaQs
Q
a
Reduction factor for stiffened slender elements
Notations 3 - 3
Steel Frame Design AISC 360-10
Q
s
Reduction factor for unstiffened slender elements
S
Section modulus, in3
S33,S22
Major and minor section moduli, in
3
S
eff,33
,S
eff,22
Effective major and minor section moduli for slender sections,
in3
S
c
Section modulus for compression in an angle section, in3
V
n2
,V
n3
Nominal major and minor shear strengths, kips
V
u2
,V
v3
Factored major and minor shear loads, kips
Z
Plastic modulus, in3
Z
33
,Z
22
Major and minor plastic moduli, in3
b
Nominal dimension of plate in a section, in
longer leg of angle sections, bf− 2tw for welded and bf− 3tw for
rolled box sections, and the like
be
Effective width of flange, in
b
f
Flange width, in
d
Overall depth of member, in
de
Effective depth of web, in
h
c
Clear distance between flanges less fillets, in
assumed d −2k for rolled sections, and d− 2tffor welded sections
k
Distance from outer face of flange to web toe of fillet, in
k
c
Parameter used for section classification
kc =
4
w
ht ,
0.35 ≤
c
k
≤ 0.763
l
33
,l
22
Major and minor directions unbraced member lengths, in
r
Radius of gyration, in
3 - 4 Notations
Chapter 3 - Design using ANSI/AISC 360-10
r
33
,r
22
Radii of gyration in the major and minor directions, in
t
Thickness, in
tf
Flange thickness, in
t
w
Thickness of web, in
β
w
Special section property for angles, in
λ
Slenderness parameter
λ
c
,λ
e
Column slenderness parameters
λ
p
Limiting slenderness parameter for compact element
λr
Limiting slenderness parameter for non-compact element
λ
s
Limiting slenderness parameter for seismic element
λ
slender
Limiting slenderness parameter for slender element
ϕb
Resistance factor for bending
ϕ
c
Resistance factor for compression
ϕ
t
Resistance factor for tension yielding
ϕT
Resistance factor for torsion
ϕ
v
Resistance factor for shear
Ω
b
Safety factor for bending
Ωc
Safety factor for compression
Ω
t
Safety factor for tension
Ω
T
Safety factor for torsion
Ωv
Safety factor for shear
Notations 3 - 5
Steel Frame Design AISC 360-10
3.2 Design Loading Combinations
The structure is to be designed so that its design strength equals or exceeds the
effects of factored loads stipulated by the applicable design code. The default
design combinations are the various combinations of the already defined load
cases, such as dead load (DL), live load (LL), roof live load (RL), snow load
(SL), wind load (WL), and horizontal earthquake load (EL).
AISC 360-10 refers to the applicable building code for the loads and load com-
binations to be considered in the design, and to ASCE 7-10 in the absence of
such a building code. Hence, the default design combinations used in the cur-
rent version are the ones stipulated in ASCE 7-10:
For design in accordance with LRFD provisions:
1.4 DL (ASCE 2.3.2-1)
1.2 DL + 1.6 LL + 0.5RL (ASCE 2.3.2-2)
1.2 DL + 1.0 LL + 1.6RL (ASCE 2.3.2-3)
1.2 DL + 1.6 LL + 0.5 SL (ASCE 2.3.2-2)
1.2 DL + 1.0 LL + 1.6 SL (ASCE 2.3.2-3)
0.9 DL ± 1.0WL (ASCE 2.3.2-6)
1.2 DL + 1.6 RL± 0.5WL (ASCE 2.3.2-3)
1.2 DL + 1.0LL+ 0.5RL± 1.0WL (ASCE 2.3.2-4)
1.2 DL + 1.6 SL± 0.5 WL (ASCE 2.3.2-3)
1.2 DL + 1.0LL+ 0.5SL± 1.0 WL (ASCE 2.3.2-4)
0.9 DL ± 1.0 EL (ASCE 2.3.2-7)
1.2 DL + 1.0 LL+ 0.2SL± 1.0EL (ASCE 2.3.2-5)
For design in accordance with ASD provisions:
1.0 DL (ASCE 2.4.1-1)
1.0 DL + 1.0 LL (ASCE 2.4.1-2)
1.0 DL + 1.0 RL (ASCE 2.4.1-3)
1.0 DL + 0.75 LL + 0.75 RL (ASCE 2.3.2-4)
1.0 DL + 1.0 SL (ASCE 2.4.1-3)
1.0 DL + 0.75 LL + 0.75 SL (ASCE 2.3.2-4)
3 - 6 Design Loading Combinations
Chapter 3 - Design using ANSI/AISC 360-10
1.0 DL ± 0.6 WL (ASCE 2.4.1-5)
1.0 DL + 0.75 LL+ 0.75 RL± 0.75 (0.6WL) (ASCE 2.4.1-6a)
1.0 DL + 0.75 LL+ 0.75 SL± 0.75 (0.6WL) (ASCE 2.4.1-6a)
0.6 DL ± 0.6 WL (ASCE 2.4.1-7)
1.0 DL ± 0.7 EL (ASCE 2.4.1-5)
1.0 DL + 0.75 LL+ 0.75 SL± 0.75(0.7 EL) (ASCE 2.4.1-6b)
0.6 DL ± 0.7 EL (ASCE 2.4.1-8)
Most of the analysis methods recognized by the code are required to consider
Notional Load in the design loading combinations for steel frame design. The
program allows the user to define and create notional loads as individual load
cases from a specified percentage of a given gravity load acting in a particular
lateral direction. These notional load patterns should be considered in the com-
binations with appropriate factors, appropriate directions, and appropriate
senses. Currently, the program automatically includes the notional loads in the
default design load combinations for gravity combinations only. The user is
free to modify the default design preferences to include the notional loads for
combinations involving lateral loads. For further information, refer to the “No-
tional Load Patterns” section in Chapter 2.
The program automatically considers seismic load effects, including over-
strength factors (ASCE 12.4.3), as special load combinations that are created
automatically from each load combination, involving seismic loads. In that
case, the horizontal component of the force is represented by Ehm and the verti-
cal component of the force is represented by Ev, where
Ehm = Ω0QE (ASCE 12.4.3.1)
Ev = 0.2SDSD (ASCE 12.4.2.2)
where, Ωo is the overstrength factor and it is taken from ASCE 7-10 Table
12.2-1. The factor SDS is described later in this section. Effectively, the special
seismic combinations that are considered for the LRFD provision are
(1.2 + 0.2SDS)DL ±Ω0QE (ASCE 2.3.2-5, 12.4.3.2)
(1.2 + 0.2SDS)DL ±Ω0QE+ 1.0LL (ASCE 2.3.2-5, 12.4.3.2)
(0.9 − 0.2SDS)DL ±Ω0QE (ASCE 2.3.2-7, 12.4.3.2)
and for the ASD provision the combinations are
Design Loading Combinations 3 - 7
Steel Frame Design AISC 360-10
(1.0 + 0.14SDS)DL ± 0.7Ω0QE (ASCE 2.4.1-5, 12.4.3.2)
(1.0 + 0.105SDS)DL ± 0.75(0.7Ω0)QE+ 0.75LL (ASCE 2.4.1-6b,12.4.3.2)
(0.6 − 0.14SDS)DL ± 0.7Ω0QE (ASCE 2.4.1-8,12.4.3.2)
The program assumes that the defined earthquake load is really the strength
level earthquake, which is equivalent to QE as defined in Section 12.4.2.1 of
the ASCE 7-10 code. For regular earthquake, load is considered to have two
components: horizontal, Eh and vertical Ev, which are taken as
Eh=ρQE (ASCE 12.4.2.1)
Ev= 0.2SDSD (ASCE 12.4.2.2)
where, ρ is the redundancy factor as defined in Section 12.3.4 of ASCE 7-10,
and the SDS is the design earthquake spectral response acceleration parameters
at short periods, as defined in Section 11.4.4 of ASCE 7-10 code.
Effectively, the seismic load combination for the LRFD provision becomes:
(1.2 + 0.2SDS)DL ±ρQE (ASCE 2.3.2-5, 12.4.2.3)
(1.2 + 0.2SDS)DL ±ρQE+ 1.0LL (ASCE 2.3.2-5, 12.4.2.3)
(0.9 − 0.2SDS)DL ±ρQE (ASCE 2.3.2-7, 12.4.2.3)
The seismic load combinations for the ASD provision become:
(1.0 + 0.14SDS)DL ± 0.7ρQE (ASCE 2.4.1-5, 12.4.2.3)
(1.0 + 0.105SDS)DL ± 0.75(0.7ρ)QE + 0.75LL (ASCE 2.4.1-6b, 12.4.2.3)
(0.6 − 0.14SDS)DL ± 0.7ρQE (ASCE 2.4.1-8, 12.4.2.3)
The program assumes that the seismic loads defined as the strength level load
is the program load case. Otherwise, the factors ρ, Ωo, and SDS will not be able
to scale the load to the desired level.
The combinations described herein are the default loading combinations only.
They can be deleted or edited as required by the design code or engineer-of-
record.
3 - 8 Design Loading Combinations
Chapter 3 - Design using ANSI/AISC 360-10
The program allows live load reduction factors to be applied to the member
forces of the reducible live load case on a member-by-member basis to reduce
the contribution of the live load to the factored responses.
3.3 Classification of Sections for Local Buckling
The nominal strengths for flexure are dependent on the classification of the
section as Seismically Compact, Compact, Noncompact, Slender, or Too
Slender. Compact or Seismically Compact sections are capable of developing
the full plastic strength before local buckling occurs. Non-compact sections can
develop partial yielding in compression, and buckle inelastically before reach-
ing to a fully plastic stress distribution. Slender sections buckle elastically
before any of the elements yield under compression. Seismically Compact
sections are capable of developing the full plastic strength before local
buckling occurs when the section goes through low cycle fatigue and
withstands reversal of load under seismic conditions.
Sections are classified as Compact, Noncompact, or Slender sections in
accordance with Section B4 of the code (AISC B4). For a section to qualify as
Compact, its flanges must be continuously connected to the web or webs and
the width-thickness ratios of its compression elements must not exceed the
limiting width-thickness ratios λp from Table B4.1b of the code. If the width-
thickness ratio of one or more compression elements exceeds λp, but does not
exceed λr from Table B4.1, the section is Noncompact. If the width-thickness
ratio of any element exceeds λr but does not exceed λs, the section is Slender.
If the width-thickness ratio of any element exceedλs, the section is considered
Too Slender. The expressions of λp, λr, andλs, as implemented in the program,
are reported in Table 3-1 (AISC Table B4.1b, B4, F8, F13.2). In that table all
expressions of λp and λr are taken from AISC section B4 and AISC Table
B4.1. The limit demarcating Slender and Too Slender has been identified as λs
in this document. The expressions of λs for I-Shape, Double Channel, Channel
and T-Shape sections are taken from AISC section F13.2. The expression of λs
for Pipe Sections is taken from AISC section F8. The expression of λp for
Angle and Double Angle sections is taken from AISC Seismic code
ANSI/AISC 341-10 Table D1.1.
Classification of Sections for Local Buckling 3 - 9
Steel Frame Design AISC 360-10
For compression, sections are classified as nonslender element or slender
element sections as reported in Table 3-2 (AISC B4.1, Table B4.1a). For a
nonslender element section, the width-to-thickness ratios of its compression
elements shall not exceed λr from Table 3-2. If the width-to-thickness ratio of
any compression element exceeds λr, the section is a slender element section.
The table uses the variables kc, FL, h, hp, hc, bf, tf, tw, b, t, D, d, and so on. The
variables b, d, D and t are explained in the respective figures inside the table.
The variables bf, tf, h, hp, hc, and tw are explained in Figure 3-1. For Doubly
Symmetric I-Shapes, h, hp, and hc are all equal to each other.
For unstiffened elements supported along only one edge parallel to the direc-
tion of compression force, the width shall be taken as follows:
(a) For flanges of I-shaped members and tees, the width b is one-half the full-
flange width, bf.
(b) For legs of angles and flanges of channels and zees, the width b is the full
nominal dimension.
(c) For plates, the width b is the distance from the free edge to the first row of
fasteners or line of welds.
(d) For stems of tees, d is taken as the full nominal depth of the section.
Refer to Table 3-1 (AISC Table B4.1) for the graphic representation of unstiff-
ened element dimensions.
For stiffness elements supported along two edges parallel to the direction of the
compression force, the width shall be taken as follows:
(a) For webs of rolled or formed sections, h is the clear distance between
flanges less the fillet or corner radius at each flange; hc is twice the dis-
tance from the centroid to the inside face of the compression flange less the
fillet or corner radius.
3 - 10 Classification of Sections for Local Buckling
Chapter 3 - Design using ANSI/AISC 360-10
Figure 3-1 AISC-360-10 Definition of Geometric Properties
Classification of Sections for Local Buckling 3 - 11
Steel Frame Design AISC 360-10
(b) For webs of built-up sections, h is the distance between adjacent lines of
fasteners or the clear distance between flanges when welds are used, and hc
is twice the distance from the centorid to the nearest line of fasteners at the
compression flange or the inside face of the compression flange when
welds are used; hp is twice the distance from the plastic neutral axis to the
nearest line of fasteners at the compression flange or the inside face of the
compression flange when welds are used.
(c) For flange or diaphragm plates in built-up sections, the width b is the dis-
tance between adjacent lines of fasteners or lines of welds.
Table 3-1 Limiting Width-Thickness Ratios of Compression Elements for Classifica-
tion Sections – Members Subjected to Flexure With or Without Axial Force
Section
Type
Description
of Element
Example
AISC
Case
No.
Width-
Thickness
Ratio,
()
λ
Limiting Width-Thickness Ratios for Compression Ele-
ment
Compact
( )
p
λ
NonCompact
( )
r
λ
Slender
( )
s
λ
Doubly Symmetric I-Shape
Flexural
compression
of flanges of
rolled
I-Shapes
10
2
ff
bt
0.38
y
EF
1.0 y
EF
No Limit
Flexural
compression
in flanges of
built-up
I-Shapes
11
2
ff
bt
0.38
y
EF
0.95
cL
kE F
No Limit
Flexure in web
15
w
ht
3.76
y
EF
5.70
y
EF
{}
min 0.40 ,260
y
EF
(beams)
No limit for columns
and braces
3 - 12 Classification of Sections for Local Buckling
Chapter 3 - Design using ANSI/AISC 360-10
Table 3-1 Limiting Width-Thickness Ratios of Compression Elements for Classifica-
tion Sections – Members Subjected to Flexure With or Without Axial Force
Section
Type
Description
of Element
Example
AISC
Case
No.
Width-
Thickness
Ratio,
()
λ
Limiting Width-Thickness Ratios for Compression Ele-
ment
Compact
( )
p
λ
NonCompact
( )
r
λ
Slender
( )
s
λ
Singly Symmetric I-Shapes
Flexural
Compression
of flanges of
rolled
I-Shapes
10
2
ff
bt
0.38 y
EF
1.0
y
EF
No Limit
Flexural
Compression
in flanges of
built-up
I-Shapes
11
2
ff
bt
0.38 y
EF
0.95
cL
kE F
No Limit
Flexure in
Web
16
cw
ht
2
0.54 0.09
c
p
p
y
r
hE
hF
M
M
≤λ
−




5.70 y
EF
No Limit
Flexure in
Web
w
ht
NA NA
{ }
min 0.40 ,260
y
EF
(beams)
No limit for columns
and braces
Channel
Flexural
compression
in flanges
10
ff
bt
0.38
y
EF
1.0 y
EF
No Limit
Flexure in web
15
w
ht
3.76 y
EF
5.70 y
EF
{ }
min 0.40 ,260
y
EF
(beams)
No limit for columns
and braces
Double Channel
Flexural
compression
in flanges
10
ff
bt
0.38
y
EF
1.0 y
EF
No Limit
Flexure in web
15
w
ht
3.76 y
EF
5.70 y
EF
{}
min 0.40 ,260
y
EF
(beams)
No limit for columns
and braces
Classification of Sections for Local Buckling 3 - 13
Steel Frame Design AISC 360-10
Table 3-1 Limiting Width-Thickness Ratios of Compression Elements for Classifica-
tion Sections – Members Subjected to Flexure With or Without Axial Force
Section
Type
Description
of Element
Example
AISC
Case
No.
Width-
Thickness
Ratio,
()
λ
Limiting Width-Thickness Ratios for Compression Ele-
ment
Compact
( )
p
λ
NonCompact
()
r
λ
Slender
( )
s
λ
Box
Flexural or
axial
compression
of flanges
under major
axis bending
17
bt
1.12 y
EF
1.40
y
EF
No Limit
Flexure in web
19
ht
2.42 y
EF
5.70 y
EF
No Limit
T-Shape
Flexural or
axial
compression
in flanges
10
2
ff
bt
0.38
y
EF
1.0 y
EF
No Limit
Compression
in stems
14
w
dt
0.84 y
EF
1.03
y
EF
No Limit
Double Angle
Any type of
compression
in leg
12
bt
0.54
y
EF
0.91
y
EF
No Limit
Any type of
compression
in leg
12
bt
0.54 y
EF
0.91 y
EF
No Limit
Angle
Flexural
compression
in any leg
12
bt
0.54
y
EF
0.91 y
EF
No Limit
Pipe
Flexural
compression
20
Dt
0.07
y
EF
0.31 y
EF
0.45 y
EF
Round
Bar
――― ――― ―― ――― Assumed Noncompact
Rectan-
gular
――― ――― ―― ――― Assumed Noncompact
General
―――
―――
――
―――
Assumed Noncompact
SD
Section
――― ――― ―― ――― Assumed Noncompact
3 - 14 Classification of Sections for Local Buckling
Chapter 3 - Design using ANSI/AISC 360-10
(d) For flanges of rectangular hollow structural sections (HSS), the width b is
the clear distance between webs less the inside corner radius on each side.
For webs of rectangular HSS, h is the clear distance between the flanges
less the inside corner radius on each side. If the corner radius is not known,
b and h shall be taken as the corresponding outside dimension minus three
times the thickness. The thickness, t, shall be taken as the design wall
thickness, in accordance with AISC Section B3.12.
Refer to Table 3-1 (AISC Table B4.1) for the graphic representation of stiff-
ened element dimensions.
Table 3-2 Limiting Width-Thickness Ratios of Compression Elements for
Classification Sections Subjected to Axial Compression
Section
Type
Description of
Element
Example
AISC Case
No.
Width-
Thickness
Ratio,
( )
λ
Limiting Width-Thickness Ratios
for Compression Element
NonCompact
( )
r
λ
Doubly Symmetric I-Shape
Axial only compres-
sion in flanges of
rolled
I-Shapes
1
2
ff
bt
0.56
y
EF
Axial only compres-
sion in flanges of
built-up
I-Shapes
2
2
ff
bt
0.64
cL
kE F
Web in axial only
compression
5
w
ht
1.49 y
EF
Singly Symmetric I-Shapes
Axial only compres-
sion in flanges of
rolled
I-Shapes
1
2
ff
bt
.0 56 y
EF
Axial only compres-
sion in flanges of
built-up
I-Shapes
2
2
ff
bt
0.64
cL
kE F
Classification of Sections for Local Buckling 3 - 15
Steel Frame Design AISC 360-10
Table 3-2 Limiting Width-Thickness Ratios of Compression Elements for
Classification Sections Subjected to Axial Compression
Section
Type
Description of
Element
Example
AISC Case
No.
Width-
Thickness
Ratio,
( )
λ
Limiting Width-Thickness Ratios
for Compression Element
NonCompact
( )
r
λ
Channel
Axial only compres-
sion in flanges
1
ff
bt
0.56
y
EF
Web in axial only
compression
5
w
ht
1.49 y
EF
Double Channel
Axial only compres-
sion in flanges
1
ff
bt
0.56 y
EF
Web in axial only
compression
5
w
ht
1.49 y
EF
Box
Axial
compression
6
bt
1.40
y
EF
T-Shape
Axial
compression in
flanges
2
2
ff
bt
0.56 y
EF
Compression in
stems
4
w
dt
0.75 y
EF
Double Angle
Any type of compres-
sion in leg
3
bt
0.45 y
EF
Any type of compres-
sion in leg
3
bt
0.45 y
EF
3 - 16 Classification of Sections for Local Buckling
Chapter 3 - Design using ANSI/AISC 360-10
Table 3-2 Limiting Width-Thickness Ratios of Compression Elements for
Classification Sections Subjected to Axial Compression
Section
Type
Description of
Element
Example
AISC Case
No.
Width-
Thickness
Ratio,
( )
λ
Limiting Width-Thickness Ratios
for Compression Element
NonCompact
( )
r
λ
Angle
Axial only compres-
sion in any leg
3
bt
0.45
y
EF
Pipe
Axial only compres-
sion
9
Dt
0.11 y
EF
Round Bar
―――
―――
――
―――
Assumed Noncompact
Rectan-
gular
――― ――― ―― ――― Assumed Noncompact
General
―――
―――
――
―――
Assumed Noncompact
SD
Section
――― ――― ―― ――― Assumed Noncompact
The design wall thickness, t, for hollow structural sections, such as Box and
Pipe sections, is modified for the welding process (AISC B4.2). If the welding
process is ERW (Electric-Resistance Welding), the thickness is reduced by a
factor of 0.93. However, if the welding process is SAW (Submerged Arc
Welded), the thickness is not reduced. The Overwrites can be used to choose if
the thickness of HSS sections should be reduced for ERW on a member-by-
member basis. The Overwrites can also be used to change the reduction factor.
The variable kc can be expressed as follows:
4,
c
w
kht
=
(AISC Table B4.1b Note a)
0.35 ≤kc≤ 0.76. (AISC Table B4.1b Note a)
For Doubly Symmetric I-Shapes, Channels, and Double Channels, FL can be
expressed as follows:
FL= 0.7Fy, (AISC Table B4.1b Note b)
and for Singly Symmetric I-Shape sections, FL can be expressed as follows:
Classification of Sections for Local Buckling 3 - 17
Steel Frame Design AISC 360-10
,
xt
Ly
xc
S
FF
S
=
where (AISC Table B4.1b Note b, F4-6)
0.5Fy≤FL≤ 0.7Fy. (AISC Table B4.1b Note b, F4-6)
Seismically Compact sections are compact sections that satisfy a more strin-
gent width-thickness ratio limit,
λ
md and
λ
hd. These limits are presented in Ta-
ble 4-1 in Chapter 4, which is dedicated to the seismic code.
In classifying web slenderness of I-Shapes, Box, Channel, Double Channel,
and all other sections, it is assumed that there are no intermediate stiffeners.
Double angles and channels are conservatively assumed to be separated.
Stress check of Too Slender sections is beyond the scope of this program.
3.4 Calculation of Factored Forces and Moments
The factored member loads that are calculated for each load combination are
Pr, Mr33, Mr22, Vr2, Vr3 and Tr corresponding to factored values of the axial load,
the major and minor moments and shears, and torsion, respectively. These fac-
tored loads are calculated at each of the previously defined stations.
The factored forces can be amplified to consider second order effects, depend-
ing on the choice of analysis method chosen in the Preferences. If the analysis
method is chosen to be General Second Order Elastic Analysis or any of the
Direct Analysis methods with General Second Order Analysis, it is assumed
that the analysis considers the influence of second-order effects (P-∆ and P-δ
effects); hence the analysis results are used without amplification (AISC C1).
Second-order effects due to overall sway of the structure can usually be ac-
counted for, conservatively, by considering the second-order effects on the
structure under one set of loads (usually the most severe gravity load case), and
performing all other analyses as linear using the stiffness matrix developed for
this one set of P-delta loads (see also White and Hajjar 1991). For a more accu-
rate analysis, it is always possible to define each loading combination as a non-
linear load case that considers only geometric nonlinearities. For both ap-
proaches, when P-δ effects are expected to be important, use more than one el-
ement per line object (accomplished using the automatic frame subdivide op-
tion; refer to the program Help for more information about automatic frame
subdivide).
3 - 18 Calculation of Factored Forces and Moments
Chapter 3 - Design using ANSI/AISC 360-10
If the analysis method is chosen to be Second Order Analysis by Amplified
First Order Analysis or any of the Direct Analysis Methods with Amplified
First Order Analysis (AISC C2.1(2), App.8.2), it is assumed that the analysis
does not consider the influence of second order effects (P-∆ and P-δ). Hence
the analysis results are amplified using B1 and B2 factors using the following
approximate second-order analysis for calculating the required flexural and ax-
ial strengths in members of lateral load resisting systems. The required second-
order flexural strength, Mr, and axial strength, Pr are determined as follows:
Mr=B1Mnt+B2Mlt (AISC A-8-1)
Pr=Pnt+ B2Plt (AISC A-8-1)
where,
1
1
1,
1
m
r
e
C
BP
P
= ≥
−α
and (AISC A-8-3)
2story
,story
11,
1
e
BP
P
= ≥
α
−
(AISC A-8-6)
where,
1.0 for LRFD,
= 1.6 for ASD,
α
Mr = required second-order flexural strength using LFRD and ASD
load combinations, kip-in (N-mm)
Mnt = first-order moment using LFRD and ASD load combinations,
assuming there is no lateral translation of the frame, kip-in. (N-
mm)
Mlt = first-order moment using LRFD or ASD load combinations
caused by lateral translation of the frame only, kip-in (N-mm)
Pr = required second-order axial strength using LRFD or ASD load
combinations, kip (N)
Calculation of Factored Forces and Moments 3 - 19
Steel Frame Design AISC 360-10
Pnt = first-order axial force using LRFD or ASD load combinations,
assuming there is no lateral translation of the frame, kips (N)
Pstory = total vertical load supported by the story using LRFD or ASD
load combinations, including gravity column loads, kips (N)
Plt = first-order axial force using LRFD or ASD load combinations
caused by lateral translation of the frame only, kips (N)
Cm = a coefficient assuming no lateral translation of the frame,
whose value is taken as follows:
(i) For beam-columns not subject to transverse loading be-
tween supports in the plane of bending,
()
0.6 0.4 ,
m ab
C MM
= −
(AISC A-8-4)
where, Ma and Mb, calculated from a first-order analysis,
are the smaller and larger moments, respectively, at the
ends of that portion of the member unbraced in the plane
of bending under consideration.
ab
MM
is positive when
the member is bent in reverse curvature, negative when
bent in single curvature.
(ii) For beam-columns subjected to transverse loading between
supports, the value of Cm is conservatively taken as 1.0 for
all cases.
When Mb is zero, Cm is taken as 1.0, the program defaults
Cm to 1.0, if the unbraced length is more than actual mem-
ber length. The user can overwrite the value of Cm for any
member. Cm can be expressed as follows:
1.00, if length is more than actual length,
1.00, if tension member,
1.00, if both ends unrestrained,
0.6 0.4 , if no transverse loading, and
1.00, if transverse loading is present.
m
a
b
C
M
M
=
−
3 - 20 Calculation of Factored Forces and Moments
Chapter 3 - Design using ANSI/AISC 360-10
(AISC A-8-4, App 8.2.1)
Pel = elastic critical buckling resistance of the member in the plane
of bending, calculated based on the assumption of zero side-
sway, kips (N)
( )
2
12
1
e
EI
PKL
π
=
(AISC A-8-5)
If any of the direct analysis methods are used, the reduced val-
ue of EI is used (AISC C3.3).
Pe,story = elastic critical buckling resistance for the story determined by
sidesway buckling analysis, kips (N)
For moment frames, where sidesway buckling effective length
factors K2 are determined for the columns, it is the elastic story
sidesway buckling resistance and calculated as
, story
,
eM
H
HL
PR=∆
(AISC A-8-7)
where,
E = modulus of elasticity of steel = 29,000 ksi
(200,000 MPa)
If any of the direct analysis methods are used,
the reduced value of EI is used (AISC App.
8.2.1).
I = moment of inertia in the plane of bending, in.4
(mm4)
L = story height, in. (mm)
K1 = effective length factor in the plane of bending,
calculated based on the assumption of no lateral
translation. It is taken to be equal to 1.0, conser-
vatively. The Overwrites can be used to change
Calculation of Factored Forces and Moments 3 - 21
Steel Frame Design AISC 360-10
the value of K1 for the major and minor direc-
tions.
K2 = effective length factor in the plane of bending,
calculated based on a sidesway buckling analy-
sis. The Overwrites can be used to change the
value of K2 for the major and minor directions.
In the expression of B1, the required axial force Pr is used based on its first or-
der value. The magnification factor B1 must be a positive number. Therefore,
αPr must be less than Pe1. If αPr is found to be greater than or equal to Pe1 a
failure condition is declared.
If the program assumptions are not satisfactory for a particular structural model
or member, the user has the choice to explicitly specify the values of B1 for any
member.
Currently, the program does not calculate the B2 factor. The user is required to
overwrite the values of B2 for the members.
3.5 Calculation of Nominal Strengths
The nominal strengths in compression, tension, bending, and shear are comput-
ed for Compact, Noncompact, and Slender members in accordance with the
following sections. The nominal flexural strengths for all shapes of sections
are calculated based on their principal axes of bending. For the Rectangular, I-
Shape, Box, Channel, Double Channel, Circular, Pipe, T-Shape, and Double
Angle sections, the principal axes coincide with their geometric axes. For the
Single Angle sections, the principal axes are determined and all computations
except shear are based on that.
For all sections, the nominal shear strengths are calculated for directions
aligned with the geometric axes, which typically coincide with the principal
axes. Again, the exception is the Single Angle section.
If the user specifies nonzero nominal capacities for one or more of the
members on the Steel Frame Overwrites form, those values will override
the calculated values for those members. The specified capacities should
be based on the principal axes of bending for flexure, and the geometric
axes for shear.
3 - 22 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
3.5.1 Nominal Tensile Strength
This section applies to the members subject to axial tension.
Although there is no maximum slenderness limit for members designed to re-
sist tension forces, the slenderness ratio preferably should not exceed 300
(AISC D1). A warning message to that effect is printed for such slender ele-
ments under tension.
The design tensile strength, φtPn, and the allowable tensile strength,
,
nt
PΩ
of
tension members is taken as the lower value obtained according to the limit
states of yielding of gross section under tension and tensile rupture in the net
section.
3.5.1.1 Tensile Yielding in the Gross Section
Pn=FyAg (AISC D2-1)
φ1= 0.90 (LRFD) (AISC D2)
Ωt= 1.67 (ASD) (AISC D2)
3.5.1.2 Tensile Rupture in the Net Section
Pn=Fu Ae (AISC D2-2)
φ1= 0.75 (LRFD) (AISC D2)
Ωt= 2.00 (ASD) (AISC D2)
The effective net area, Ae, is assumed to be equal to the gross cross-sectional
area, Ag, by default. For members that are connected with welds or members
with holes, the
eg
AA
ratio must be modified using the steel frame design
Overwrites to account for the effective area.
3.5.2 Nominal Compressive Strength
The design compressive strength, φcPn, and the allowable compressive
strength,
,
nc
PΩ
of members subject to axial compression are addressed in
this section. The resistance and safety factors used in calculation of design and
allowable compressive strengths are:
Calculation of Nominal Strengths 3 - 23
Steel Frame Design AISC 360-10
φc= 0.90 (LRFD) (AISC E1)
Ωc= 1.67 (ASD) (AISC E1)
In the determination, the effective length factor K2 is used as the K-factor. If
the chosen analysis method in the Preferences is the General Second Order
Elastic Analysis, the First Order Analysis using Amplified First Order Analy-
sis, or the Limited First Order Analysis, the calculated K2 factors are used. If
the user overwrites the K2 factors, the overwritten values are used. If the cho-
sen analysis method is one of any Direct Analysis Methods, the effective
length factor, K, for calculation of Pn is taken as one (AISC C3). The overwrit-
ten value of K2 will have no effect for the latter case.
The nominal axial compressive strength, Pn, depends on the slenderness ratio,
,Kl r
where
33 33 22 22
33 22
max , .
Kl Kl
Kl
r rr

=

For all sections except Single Angles, the principal radii of gyration r22 and r33
are used. For Single Angles, the minimum (principal) radius of gyration, rz, is
used instead of r22 and r33, conservatively, in computing
.Kl r
K33 and K22 are
two values of K2 for the major and minor axes of bending.
Although there is no maximum slenderness limit for members designed to re-
sist compression forces, the slenderness ratio preferably should not exceed 200
(AISC E2). A warning message to that effect is given for such slender elements
under compression.
The members with any slender element and without any slender elements are
handled separately.
The limit states of torsional and flexural-torsional buckling are ignored for
closed sections (Box and Pipe sections), solid sections, general sections, and
sections created using Section Designer.
3.5.2.1 Members without Slender Elements
The nominal compressive strength of members with compact and noncompact
sections, Pn, is the minimum value obtained according to the limit states of
flexural buckling, torsional and flexural-torsional buckling.
3 - 24 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
3.5.2.1.1 Flexural Buckling
For compression members with compact and noncompact sections, the nominal
compressive strength, Pn, based on the limit state of flexural buckling, is given
by
Pn=Fcr Ag. (AISC E3-1)
The flexural buckling stress, Fcr, is determined as follows:
  ≤


=
>
0.658 , if 4.71 ,
0.877 , if 4.71 ,
y
e
F
Fyy
cr
ey
KL E
FrF
FKL E
FrF
(AISC E3-2, E3-3)
where Fe is the elastic critical buckling stress given by
2
2.
eE
FKL
r
π
=


(AISC E3-4)
3.5.2.1.2 Torsional and Flexural-Torsional Buckling
For compression members with compact and noncompact sections, the nominal
compressive strength, Pn, based on the limit state of torsional and flexural-
torsional buckling is given by
Pn = Fcr Ag (AISC E4-1)
where Ag is the gross area of the member. The flexural buckling stress, Fcτ, is
determined as follows.
3.5.2.1.2.1 Box, Pipe, Circular, Rectangular, General and Section Design-
er Sections
The limit states of torsional and flexural-torsional buckling are ignored for
members with closed sections, such as Box and Pipe sections, solid sections
(Circular and Rectangular), General sections and sections created using the
Section Designer.
Calculation of Nominal Strengths 3 - 25
Steel Frame Design AISC 360-10
3.5.2.1.2.2 Double Angle and T-Shapes
( )
22 22
2
22
4
11 ,
2
cr crz cr crz
cr
cr crz
F F F FH
FHFF

+


= −−


 +

(AISC E4-2)
where,
22
22
22
22
22
(0.658 ) , if 4.71 ,
0.877 , if 4.71 ,
y
e
F
Fy
y
cr
e
y
KL E
FrF
FKL E
FrF
≤
=
>
(AISC E3-2, E3-3)
2
2
22
22
,
e
E
FKL
r
π
=


and (AISC E3-4)
2
0
.
crz
g
GJ
FAr
=
(AISC E4-3)
3.5.2.1.2.3 I-Shape, Double Channel, Channel, Single Angle Sections
For I-Shape, Double Channel, Channel, and Single Angle sections, Fcτ is calcu-
lated using the torsional or flexural-torsional elastic buckling stress, Fe, as fol-
lows:
  ≤


=
>
0.658 , if 4.71 ,
0.877 , if 4.71 .
y
e
F
Fyy
cr
ey
KL E
FrF
FKL E
FrF
(AISC E3-2, E3-3, E4b)
where Fe is calculated from the following equations:
3.5.2.1.2.3.1 I-Shapes and Double Channel Sections
( )
2
222 33
1
w
e
zz
EC
F GJ II
KL


π

= + 
+



(AISC E4-4)
3 - 26 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
3.5.2.1.2.3.2 Channel Sections
( )
33 33
2
33
4
11
2
e ez e ez
e
e ez
F F F FH
FHFF

+


= −−


 +

(AISC E4-5)
3.5.2.1.2.3.3 Single Angle Sections with Equal Legs
( )
33 33
2
33
4
11
2
e ez e ez
e
e ez
F F F FH
FHFF

+


= −−


+


(AISC E4-5)
3.5.2.1.2.3.4 Single Angle Sections with Unequal Legs
Fe is the lowest root of the cubic equation.
()( )
()
( ) ( )
2
22
33 22 22 33
00
0
oo
ee ee eez eee eee
xy
FF FF FF FFF FFF
rr
 
− − −−− −− =
 
 
(AISC E4-6)
In the preceding equations,
Cw is the warping constant, in6 (mm6)
x0, y0 are the coordinates of the shear center with respect to the cen-
troid, x0 = 0 for Double Angle and T-Shaped members (y-axis
symmetry)
0
r
=
2222 33
oo
g
II
xy A
+
++ =
polar radius of gyration about the shear
center (AISC E4-11)
H =
22
2
1
oo
xy
r

+
−

(AISC E4-10)
33e
F
=
( )
2
2
33 33 33
E
KL r
π
(AISC E4-7)
22e
F
=
( )
2
2
22 22 22
E
KL r
π
(AISC E4-8)
Calculation of Nominal Strengths 3 - 27
Steel Frame Design AISC 360-10
ez
F
=
( )
2
22
0
1
w
zz
EC GJ Ar
KL

π

+


(AISC E 4-9)
K22, K33 are effective length factors K2 in minor and major directions
Kz is the effective length factor for torsional buckling, and it is
taken equal to KLTB in this program; it can be overwritten
L22, L33 are effective lengths in the minor and major directions
r22, r33 are the radii of gyration about the principal axes
Lz is the effective length for torsional buckling and it is taken
equal to L22 by default, but it can be overwritten.
For angle sections, the principal moment of inertia and radii of gyration
are used for computing Fe. Also, the maximum value of KL, i.e.,
max(K22L22, K33L33), is used in place of K22L22 or K33L33 in calculating
Fe22 and Fe33 in this case. The principal maximum value rmax is used for
calculating Fe33, and the principal minimum value rmin is used in calculat-
ing Fe22.
3.5.2.2 Members with Slender Elements
The nominal compressive strength of members with slender sections, Pn, is the
minimum value obtained according to the limit states of flexural, torsional and
flexural-torsional buckling.
3.5.2.2.1 Flexural Buckling
For compression members with slender sections, the nominal compressive
strength, Pn, based on the limit state of flexural buckling, is given by
Pn= FcrAg. (AISC E7-1)
The flexural buckling stress, Fcr, is determined as follows:
3 - 28 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
0.658 , if 4.71 , and
0.877 , if 4.71 ,
y
e
QF
Fy
y
cr
e
y
KL E
QF
r QF
F
KL E
Fr QF

 ≤


=
>
(AISC E7-2, E7-3)
where Fe is the elastic critical buckling stress for flexural buckling limit state.
2
2
.
e
E
FKL
r
π
=


(AISC E3-4)
3.5.2.2.2 Torsional and Flexural-Torsional Buckling
For compression members with slender sections, the nominal compressive
strength, Pn, based on Torsional and Flexural-Torsional limit state is given by:
Pn= FcrAg, where (AISC E7-1)
Fcr is determined as follows:
0.658 if 4.71 , and
0.877 if 4.71 ,
y
e
QF
Fy
y
cr
e
y
KL E
QF
r QF
FKL E
Fr QF

 ≤


=
>
(AISC E7-2, E7-3)
where, Fe is the elastic critical buckling stress for torsional and flexural-
torsional limit states, which are given for different shapes as follows.
3.5.2.2.2.1 Box, Pipe, Circular, Rectangular, General and Section Design-
er Sections
The limit states of torsional and flexural-torsional buckling are ignored for
members with closed (Box and Pipe), solid (Circular and Rectangular), General
sections and sections created using the Section Designer.
Calculation of Nominal Strengths 3 - 29
Steel Frame Design AISC 360-10
3.5.2.2.2.2 I-Shape and Double Channel Sections
( )
2
222 33
22
1
w
e
EC
F GJ II
KL


π

= + 
+



(AISC E7, E4-4)
3.5.2.2.2.3 Channel Sections
( )
33 33
2
33
4
11
2
e ez e ez
e
e ez
F F F FH
FHFF

+


= −−


+


(AISC E7, E4-5)
3.5.2.2.2.4 Double Angle Sections and T-Shapes
( )
22 22
2
33
4
11
2
e ez e ez
e
e ez
F F F FH
FHFF

+


= −−


+


(AISC E7, E4-5)
3.5.2.2.2.5 Single Angle Sections with Equal Legs
( )
33 33 2
33
4
11
2
e ez e ez
e
e ez
F F F FH
FHFF

+

= −−


 +


(AISC E7, E4-5)
3.5.2.2.2.6 Single Angle Sections with Unequal Legs
Fe is the lowest root of the cubic equation.
( )( )
( )
( ) ( )
22
22
33 22 22 33
0
oo
ee ee eez eee eee
oo
xy
FF FF FF FFF FFF
rr
 
− − −−− −− =
 
 
(AISC E7, E4-6)
The variables used in the preceding expressions for Fe, such as Cw, x0, y0,
o
r
, H,
Fe33, Fe22, Fez, K22, K33, Kz, L22, L33, Lz,
,KL r
and so on, were explained in the
previous section.
3.5.2.2.3 Reduction Factor for Slenderness
The reduction factor for slender compression elements, Q, is computed as fol-
lows:
Q = QsQa, (AISC E7)
where Qs and Qa are reduction factors for slender unstiffened compression el-
ements (flanges of I-Shapes, T-Shapes, Double Angles, Channels, and Double
3 - 30 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
Channels; legs of angles; and stems of T-Shapes) and slender stiffened com-
pression elements (webs of I-Shapes, Channels, and Boxes; and Pipe sections),
respectively. For cross-sections composed of only unstiffened slender ele-
ments, Q = Qs(Qa = 1) and for cross-sections composed of only stiffened slen-
der elements, Q = Qa(Qs = 1).
The reduction factor, Qs, for slender unstiffened elements is defined as follows:
3.5.2.2.3.1 Flange of I-Shape, T-Shape, Channel and Double Channel Sec-
tions
Rolled:
2
1.0, if 0.56 ,
1.415 0.74 , if 0.56 1.03 , and
0.69 , if 1.03 .
y
y
s
yy
y
y
bE
tF
F
b Eb E
QtE Ft F
E bE
tF
b
Ft
≤

= − <≤


>



(AISC E7-4, E7-5, E7-6)
Built-Up:
2
1.0, if 0.64 ,
1.415 0.65 , if 0.64 1.17 ,
0.90 , if 1.17 ,
c
y
ycc
s
c yy
cc
y
y
Ek
b
tF
FEk Ek
bb
Qt Ek F t F
Ek Ek
b
tF
b
Ft
≤

= − <≤


>



(AISC E7-7, E7-8, E7-9)
where
4
c
w
kht
=
and 0.35 ≤kc≤ 0.76, (AISC E7.1b)
Calculation of Nominal Strengths 3 - 31
Steel Frame Design AISC 360-10
and
bt
is defined as
()
( )
2 for I Shapes,
2 for T Shapes,
for Channels,
for Double Channels.
tf
ff
ff
ff
bt
bt
b
tbt
bt
=
(AISC B4.1a, E7.1)
3.5.2.2.3.1.1 Legs of Single and Double Angle Sections
2
1.0, if 0.45 ,
1.34 0.76 , if 0.45 0.91 , and
0.53 , if 0.91 ,
y
y
s
yy
y
y
bE
tF
F
b Eb E
QtE Ft F
E bE
tF
b
Ft
≤

= − <≤


>



(AISC E7-10, E7-11, E7-12)
where b is the full width of the longest leg, and t is the corresponding thickness
(AISC B4.1a, E7.1c).
3.5.2.2.3.1.2 Stem of T-Sections
2
1.0, if 0.75 ,
1.908 1.22 , if 0.75 1.03 , and
0.69 , if 1.03 ,
y
y
s
yy
y
y
dE
tF
FEd E
QE Ft F
E dE
tF
d
Ft
≤
= − <≤
>



(AISC E7-13, E7-14, E7-15)
3 - 32 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
where d is the full nominal depth of the tee and t is the thickness of the element
(AISC B4.1b).
For T-Shapes, the Qs is calculated for the flange and web separately, and the
minimum of the two values is used as Qs. For Angle and Double Angle sec-
tions, Qs is calculated based on the leg that gives the largest
bt
and so the
smallest Qs.
The reduction factor, Qa, for slender stiffened elements is defined as follows:
,
eff
a
A
QA
=
(AISC E7-16)
where A is the total cross sectional area of the member, and Aeff is the summa-
tion of the effective areas of the cross-section,
Aeff=A−Σ(b−be)t,
based on the reduced effective width, be, which is determined as follows.
3.5.2.2.3.1.3 Webs of I -Shapes, Channels, and Double Channels
()
0.34
1.92 1 , if 1.49 , and
, if 1.49 ,
e
E E bE
tb
f bt F t f
bbE
btf

− ≤≥



=
<
(AISC E7-17)
where f is taken as Fcr with Q = 1.0 (AISC 7.2a), and b is taken for rolled
shapes as the clear distance between flanges less the corner radius, and is taken
for welded shapes as the clear distance between flanges.
3.5.2.2.3.2 Webs and Flanges of Box Sections
( )
0.38
1.92 1 , if 1.40 , and
, if 1.40 ,
e
y
E E bE
tb
f bt F t f
bbE
btF

− ≤≥



=
<
(AISC E7-18)
Calculation of Nominal Strengths 3 - 33
Steel Frame Design AISC 360-10
where f is conservatively taken as Fy (AISC 7.2b). The flange, b, is taken as
bf−3tw, and for webs, b is taken as h−3tf (AISC B4.1b). The design wall thick-
ness is modified for the welding process (AISC B4.2)
3.5.2.2.3.3 Pipe Sections
The reduction factor for slender stiffened elements is given directly by:
( )
1.0, if 0.11 ,
0.038E 2
+ , if 0.11 0.45 , and
3
1.0, if 0.45 ,
y
a
y yy
y
E
Dt F
ED E
QQ F Dt F t F
E
Dt F
<
== <<
>
(AISC E7-19)
where D is the outside diameter and t is the wall thickness. The design wall
thickness is modified for the welding process (AISC B4.2). If
D/t
exceeds
0.45
/,
y
EF
the section is considered to be too slender and it is not designed.
3.5.3 Nominal Flexure Strength
This section applies to members subject to simple bending about one principal
axis. The members are assumed to be loaded in a plane parallel to a principal
axis that passes through the shear center, or restrained against twisting.
The design flexural strength, φbMn, and the allowable flexural strength,
nb
M,
Ω
are determined using the following resistance and safety factors:
φb = 0.90 (LRFD) (AISC F1(1))
Ωb = 1.67 (ASD) (AISC F1(1))
When determining the nominal flexural strength about the major principal axis
for any sections for the limit state of lateral-torsional buckling, it is common to
use the term Cb, the lateral-torsional buckling modification factor for non-
uniform moment diagram. Cb is calculated as follows:
max
max
12.5 3.0,
2.5 3 4 3
= ≤
+++
b
ABc
M
CM MMM
(AISC F1-1, H1.2)
where,
3 - 34 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
Mmax = absolute value of maximum moment in unbraced segment, kip-in.
(N-mm)
MA = absolute value of moment at quarter point of the unbraced seg-
ment, kip-in. (N-mm)
MB = absolute value of moment at centerline of the unbraced segment,
kip-in. (N-mm)
MC = absolute value of moment at three-quarter point of the unbraced
segment, kip-in. (N-mm)
Cb should be taken as 1.0 for cantilevers. However, the program is unable to
detect whether the member is a cantilever. The user should overwrite Cb for
cantilevers. The program also defaults Cb to 1.0 if the minor unbraced length,
l22, is redefined to be more than the length of the member by the user or the
program, i.e., if the unbraced length is longer than the member length. The
Overwrites can be used to change the value of Cb for any member.
The nominal bending strength depends on the following criteria: the geometric
shape of the cross-section; the axis of bending; the compactness of the section;
and a slenderness parameter for lateral-torsional buckling. The nominal bend-
ing strength is the minimum value obtained according to the limit states of
yielding, lateral-torsional buckling, flange local buckling, web local buckling,
tension flange yielding as appropriate to different structural shapes. The fol-
lowing sections describe how different members are designed against flexure
in accordance with AISC Chapter F. AISC, in certain cases, gives options in
the applicability of its code section, ranging from F2 to F12. In most cases, the
program follows the path of the sections that gives more accurate results at the
expense of more detailed calculation. In some cases, the program follows a
simpler path. For an easy reference, Table 3-3 shows the AISC sections for the
various scenarios.
Table 3.3 Selection Table for the Application of Chapter F Sections
Section in
Chapter F
Cross Section
Flange
Slenderness
Web
Slenderness
Limit States
F2
C C Y, LTB
Calculation of Nominal Strengths 3 - 35
Steel Frame Design AISC 360-10
Table 3.3 Selection Table for the Application of Chapter F Sections
Section in
Chapter F
Cross Section
Flange
Slenderness
Web
Slenderness
Limit States
F3
NC, S C LTB, FLB
F4
C, NC, S NC Y, LTB, FLB
F5
C, NC, S S Y, LTB, FLB
F4
C, NC, S C, NC Y, LTB, FLB, TFY
F5
C, NC, S S Y, LTB, FLB, TFY
F6
C, NC, S Any Y, FLB
F7
C, NC, S C, NC Y, FLB, WLB
F8
N/A N/A Y, LB
F9
C, NC, S Any Y, LTB, FLB
F10
N/A N/A Y, LTB, LLB
F11
N/A Any Y, LTB
F12
Unsymmetrical shapes
N/A
N/A
All limit states
Y = yielding
LTB = lateral-torsional buckling
FLB = flange local buckling
WLB = web local buckling
TFY = tension flange yielding
LLB = leg local buckling
LB = local buckling
C = compact or seismically compact
NC = noncompact
S = slender
3 - 36 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
3.5.3.1 Doubly Symmetric I-Sections
3.5.3.1.1 Major Axis Bending
The nominal flexural strength for major axis bending depends on compactness
of the web and flanges.
3.5.3.1.1.1 Compact Webs with Compact Flanges
The nominal flexural strength is the lowest value obtained according to the
limit states of yielding (plastic moment) and lateral-torsional buckling.
3.5.3.1.1.1.1 Yielding
Mn=Mp=FyZ33 , (AISC F2-1)
where, Z33 is the plastic section modulus about the major axis.
3.5.3.1.1.1.2 Lateral-Torsional Buckling
()
33
33
, if ,
0.7 , if , and
, if ,
p bp
bp
n bp p y p pbr
rp
cr p p r
M LL
LL
M C M M FS M L L L
LL
FS M L L
≤


−
= − − ≤ <<



−



≤>
(AISC F2-1, F2-2, F2-3)
where, S33 is the elastic section modulus taken about the major axis, Lb is the
unbraced length, Lp and Lr are limiting lengths, and Fcr is the critical buckling
stress.Fcr, Lp, and Lr are given by:
2
2
2
33 0
1 0.078 ,
bb
cr
ts
b
ts
CE L
Jc
FSh r
L
r

π
= + 
 


(AISC F2-4)
1.76 ,
py
y
E
Lr
F
=
(AISC F2-5)
2
33 0
33 0
0.7
1.95 1 1 6.76 ,
0.7
y
r ts
y
FSh
E Jc
Lr
F S h E Jc

= ++ 

(AISC F2-6)
Calculation of Nominal Strengths 3 - 37
Steel Frame Design AISC 360-10
where,
2
33
,
yw
ts
IC
rS
=
(AISC F2-7)
c = 1, and (AISC F2-8a)
h0 is the distance between flange centroids.
3.5.3.1.1.2 Compact Webs with Noncompact or Slender Flanges
The nominal flexural strength is the lowest value obtained from the limit states
of lateral-torsional buckling and compression flange local buckling.
3.5.3.1.1.2.1 Lateral-Torsional Buckling
The provisions of lateral-torsional buckling for “Compact Web and Flanges” as
described in the provision pages also apply to the nominal flexural strength of
I-Shapes with compact webs and noncompact or slender flanges bent about
their major axis.
( )
33
33
, if ,
0.7 , if , and
, if .
p bp
bp
n bp p y p pbr
rp
cr p p r
M LL
LL
M C M M FS M L L L
LL
FS M L L
≤


−
= − − ≤ <<



−



≤>
(AISC F3.1, F2-1, F2-2, F2-3)
3.5.3.1.1.2.2 Compression Flange Local Buckling
( )
33
33
2
,
0.7 , for noncompact flanges,
0.9 , for slender flanges
pf
pp y
rf pf
n
c
M M FS
MEk S

λ−λ
−−


λ −λ

=
λ
(AISC F3-1, F3-2)
where λ,
λ
pf, and
λ
rf are the slenderness and limiting slenderness for compact
and noncompact flanges from Table 3.1, respectively,
3 - 38 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
,
2
f
f
b
t
λ=
0.38 ,
pf
y
E
F
λ=
(AISC Table B4.1b, F3.2)
1.0 (Rolled),
0.95 (Welded),
y
rf
c
L
E
F
kE
F
λ=
(AISC Table B4.1b, F3.2)
and kcis given by
4,
c
w
kht
=
0.35 ≤kc≤0.76. (AISC F.3.2)
3.5.3.1.1.3 Noncompact Webs with Compact, Noncompact and Slender
Flanges
The nominal flexural strength is the lowest values obtained from the limit
states of compression flange yielding, lateral-torsional buckling, and compres-
sion flange local buckling.
3.5.3.1.1.3.1 Compression Flange Yielding
Mn = RpcMy, (AISC F4-1)
where, Rpc is the web plasticity factor, which is determined as follows:
1 if 0.23,
, if ,and 0.23,
1 , if ,and 0.23,
yc y
p
pc pw yc y
y
p p pw p pw w rw yc y
y y rw pw y
II
M
R II
M
MM M II
MM M
≤
= λ≤λ >

 
λ−λ
− − ≤ λ <λ ≤λ >

 
 
λ −λ

 

(AISC F4-9a, F4-9b, F4-10)
where,
Calculation of Nominal Strengths 3 - 39
Steel Frame Design AISC 360-10
Mp = Z33Fy≤ 1.6S33Fy (AISC F4-2)
S33 = elastic section modulus for major axis bending
w
λ
=
c
w
h
t
(AISC F4.2, Table B4.1)
pw
λ
=
p
,
λ
the limiting slenderness for a compact web, as given in
Table 3-1 (AISC Table B4.1, F4.2)
λrw = λr, the limiting slenderness for a noncompact web, as given in
Table 3-1 (AISC Table B4.1, F4.2)
and My is the yield moment, which is determined as follows:
My = S33Fy (AISC F4-1)
3.5.3.1.1.3.2 Lateral-Torsional Buckling
()
33
33
, if ,
, if ,
, if ,
pc y b p
bp
n b pc y pc y L pc y p b r
rp
cr pc y b r
RM L L
LL
M C RM RM FS RM L L L
LL
FS R M L L
≤


−
= − − ≤ <≤



−



≤>
(AISC F4-1, F4-2, F4-3)
where,
2
2
233
1 0.078
bb
cr
ot
b
t
C E JC L
FSh r
L
r

π
= + 
 


(AISC F4-5)
2
0
0
1
12 6
f
t
w
b
rhh
a
d hd
=
+


(AISC F4-11)
10
cw
w
ff
ht
abt
= ≤
(AISC F4-12)
3 - 40 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
1, if 0.23
0, if 0.23
yc y
yc y
II
CII
>
=≤
(AISC F4.2)
.11
pt
y
E
Lr
F
=
(AISC F4-7)
2
33
33
1.95 1 1 6.76 Lo
rt
Lo
E J FSh
Lr
F Sh E J

= ++ 

(AISC F4-8)
FL = 0.7Fy (AISC F4-6a)
Rpc = web plastification factor, which is determined using a formula de-
scribe previously (AISC F4-9)
Iyc = moment of inertia of the compression flange about the minor axis
Iy = moment of inertia of the entire section about the minor axis.
3.5.3.1.1.3.3 Compression Flange Local Buckling
( )
33
33
2
, if flanges are compact,
, if flanges are noncompact, and
0.9 , if flanges are slender,
pc y
pt
n pc y pc y L
rf pt
c
RM
M R M R M FS
Ek S


λ−λ
= −−



λ −λ



λ
(AISC F4-1, F4-13, F4-14)
where,
FL = 0.5Fy (AISC F4-6a, F4.3)
Rpc = is the web plastification factor, which is determined using a formu-
la described previously (AISC F4-9, F4.3)
kc =
4
w
,
ht
35 0 76≤≤
c
k.
(AISC F4.3, Table B4.1)
Calculation of Nominal Strengths 3 - 41
Steel Frame Design AISC 360-10
λ =
2
f
f
b
t
λpf = λp, the limiting slenderness for compact flange, as given in Table
3-1 (AISC Table B4.1b, B4.3)
λrf = λr, the limiting slenderness for noncompact flange, as given in
Table 3-1 (AISC Table B4.1b, B4.3).
3.5.3.1.1.4 Slender Webs with Compact, Noncompact, and Slender
Flanges
The nominal flexural strength is the lowest value obtained from the limit states
of compression flange yielding, lateral-torsional buckling, and compression
flange local buckling.
3.5.3.1.1.4.1 Compression Flange Yielding
Mn = RpgFyS33 , (AISC F5-1)
where Rpg is the bending strength reduction factor given by
1 5.7 1.0,
1200 300
wc
pg
ww y
ahE
Rat F


=− −≤

+
(AISC F5-6)
10,
w
w
ff
ht
abt
= ≤
(AISC F5.2, F4-12)
where h0 is the distance between flange centroids (AISC F4.2).
3.5.3.1.1.4.2 Lateral-Torsional Buckling
Mn = RpgFcrS33 , (AISC F5-2)
where Fcr is the critical lateral-torsional buckling stress given by
3 - 42 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
( )
2
2
,
, if ,
0.3 , if , and
, if
y bp
bp
cr b y y y p b r
rp
by pr
b
t
F LL
LL
F CF F F L L L
LL
CEF LL
L
r
≤


−
= − ≤ <≤



−



π≤>



(AISC F5-1, F5-3, F5-4)
where,
1.1
pt
y
E
Lr
F
=
(AISC F5.2, 4-7)
0.7
rt
y
E
Lr F
= π
(AISC F5-5)
2
0
0
1
12 6
f
t
w
b
rhh
a
d hd
=
+


(AISC F5.2, F4-11)
Rpgis the bending strength reduction factor, which has been described in the
previous section.
3.5.3.1.1.4.3 Compression Flange Local Buckling
Mn = RpgFcrS33 , (AISC F5-7)
where Fcr is the critical buckling stress given by
Calculation of Nominal Strengths 3 - 43
Steel Frame Design AISC 360-10
( )
2
, if flanges are compact,
0.3 , if flanges are noncompact, and
0.9 , if flanges are slender,
2
y
pf
cr y y
rf pf
cy
f
f
F
FF F
Ek F
b
t


λ−λ
= −



λ −λ



≤




(AISC F5-1, F5-8, F5-9)
and λ, λpf, and λrf are the slenderness and the limiting slenderness ratios for
compact and noncompact flanges from Table 3-1, respectively, and kc is given
by
4
c
w
kht
=
where 0.35 ≤kc≤ 0.76. (AISC 5.3)
3.5.3.1.2 Minor Axis Bending
The nominal flexural strength is the lower value obtained according to the limit
states of yielding (plastic moment) and flange local buckling.
3.5.3.1.2.1 Yielding
Mn = Mp = FyZ22≤ 1.6FyS22, (AISC F6-1)
where S22 and Z22 are the section and plastic moduli about the minor axis, re-
spectively.
3.5.3.1.2.2 Flange Local Buckling
( )
22
22
, for compact flange,
0.7 , for noncompact flanges, and
, for slender flanges,
n
p
pf
pp y
rf pf
cr
M
M
M M FS
FS
=

λ−λ
−− 

λ −λ

(AISC F6-1, F6-2, F6-3)
where,
3 - 44 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
2
0.69
cr
E
F=λ
(AISC F6-4)
2
f
f
b
t
λ=
(AISC F6.2)
and λpf and λrf are the limiting slendernesses for compact and noncompact
flanges, respectively, as described in Table 3-1 (AISC B4.1b).
3.5.3.2 Singly Symmetric I-Sections
3.5.3.2.1 Major Axis Bending
The nominal of flexural strength for major axes bending depends on compact-
ness of the web and flanges.
3.5.3.2.1.1 Compact and Noncompact Webs with Compact, Noncompact
and Slender Flanges
The nominal flexural strength is the lowest values obtained from the limit sates
of compression flange yielding, lateral-torsional buckling, compression flange
local buckling, and tension flange yielding.
3.5.3.2.1.2 Compression Flange Yielding
Mn = RpcMyc, (AISC F4-1)
where, Rpc is the web plasticity factor, which is determined as follows:
1 if 0.23,
, if ,and 0.23,
1 , if ,and 0.23,
yc y
p
pc pw yc y
y
p p pw p pw w rw yc y
y y rw pw y
II
M
R II
M
MM M II
MM M
≤
= λ≤λ >

 
λ−λ
− − ≤ λ <λ ≤λ >

 
 
λ −λ

 

(AISC F4-9a, F4-9b)
where,
Mp = Z33Fy≤ 1.6S33cFy (AISC F4-2)
Calculation of Nominal Strengths 3 - 45
Steel Frame Design AISC 360-10
S33c = elastic section modulus for major axis bending referred to
compression flange
S33t = elastic section modulus for major axis bending referred to ten-
sion flange
λ =
c
w
h
t
(AISC F4.2, Table B4.1b)
λpw = λp, the limiting slenderness for a compact web, as given in
Table 3-1 (AISC Table B4.1b)
λrw = λr, the limiting slenderness for a noncompact web, as given in
Table 3-1 (AISC Table B4.1b)
and Myc is the yield moment for compression flange yielding, which is
determined as follows:
Myc = S33cFy. (AISC F4-1)
3.5.3.2.1.3 Lateral-Torsional Buckling
( )
33
33
, if
, if ,
, if ,
pc yc b p
bp
n b pc yc pc yc L c pc yc p b r
rp
cr c pc yc b r
RM L L
LL
M C RM RM FS RM L L L
LL
FS R M L L
≤


−
= − − ≤ <≤



−



≤>
(AISC F4-1, F4-2, F4-3)
where,
2
2
233
1 0.078
bb
cr
ot
b
t
C E JC L
FSh r
L
r

π
= + 
 


(AISC F4-5)
2
0
0
1
12 6
fc
t
w
b
rhh
a
d hd
=
+


(AISC F4-11)
3 - 46 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
10
cw
w
fe fc
ht
abt
= ≤
(AISC F4-12)
1, if 0.23
0, if 0.23
yc y
yc y
II
CII
>
=≤
(AISC F4.2)
1.1
pt
y
E
Lr
F
=
(AISC F4-7)
2
33
33
1.95 1 1 6.76 L co
rt
Lo
E J FS h
Lr
F Sh E J

= ++ 

(AISC F4-8)
33
33
33 33
33 33
0.7 , if 0.7
0.5 , if 0.7
t
y
c
L
tt
yy
cc
S
FS
FSS
FF
SS
≥
=
≥≤
(AISC F4-6a, F4-6b)
Rpc = web plastification factor, which is determined using a formula
describe previously (AISC F4-9)
Iyc = moment of inertia of the compression flange about the minor axis
Iy = moment of inertia of the section about the minor axis.
3.5.3.2.1.4 Compression Flange Local Buckling
( )
33
33
2
, if flanges are compact,
, if flanges are noncompact, and
0.9 , if flanges are slender,
pc yc
pt
n pc yc pc yc L c
rf pt
cc
RM
M RM RM FS
Ek S


λ−λ
= −−



λ −λ



λ
(AISC F4-1, F4-12, F4-13)
where,
FL = is a calculated stress, which has been defined previously
(AISC F4-6a, F4-6b, F4.3)
Calculation of Nominal Strengths 3 - 47
Steel Frame Design AISC 360-10
Rpc = is the web plastification factor, which is determined using a
formula described previously (AISC F4-9, F4.3)
kc =
4,
w
ht
35 ≤kc≤ 0.76 (AISC F4.3, Table B4.1)
λ =
2
fc
fc
b
t
λpf = λp, the limiting slenderness for compact flange, as given in
Table 3-1 (AISC Table B4.1b, B4.3)
λrf = λr, the limiting slenderness for noncompact flange, as given in
Table 3-1 (AISC Table B4.1b, B4.3).
3.5.3.2.1.5 Tension Flange Yielding
33 33
33 33
, if
, if
p tc
n
pt yt t c
M SS
MRM S S
≥
=<
(AISC F4-15)
where, Rpt is the web plastification factor corresponding to the tension flange
yielding limit state. It is determined as follows:
, if
1 , if
ppw
yt
pt
p p pw pw rw
yt yt rw pw
M
M
RMM
MM
λ≤λ
 
λ−λ
− − λ <λ≤λ
 
 
λ −λ
 
(AISC F4-16a, F4-16b)
where,
Mp = Z33Fy (AISC F2-1)
S33c = elastic section modulus for major axis bending referred to
compression flange
S33t = elastic section modulus for major axis bending referred to ten-
sion flange
3 - 48 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
λw =
c
w
h
t
(AISC F4.4, Table B4.1b)
λwp = λp, the limiting slenderness for a compact web, as given in
Table 3-1 (AISC Table B4.1b, F4.4)
λrw = λr, the limiting slenderness for a noncompact web, as given in
Table 3-1. (AISC Table B4.1b, F4.4)
3.5.3.2.1.6 Slender Webs with Compact, Noncompact and Slender Flang-
es
The nominal flexural strength is the lowest value obtained from the limit states
of compression flange yielding, lateral-torsional buckling, compression flange
local buckling, and tension flange yielding.
3.5.3.2.1.6.1 Compression Flange Yielding
Mn = RpgFyS33c, (AISC F5-1)
where, Rpg is the bending strength reduction factor given by
1 5.7 1.0
1200 300
wc
pg
ww y
ahE
Rat F


=− −≤

+
(AISC F5-6)
10
w
w
ff
ht
abt
= ≤
(AISC F5.2, F4-12)
where, h0 is the distance between flange centroids (AISC F2.2).
3.5.3.2.1.6.2 Lateral-Torsional Buckling
Mn = RpgFcrS33c , (AISC F5-2)
where, Fcr is the critical lateral-torsional buckling stress given by
Calculation of Nominal Strengths 3 - 49
Steel Frame Design AISC 360-10
( )
2
2
,
, if ,
0.3 , if , and
, if
y bp
bp
cr b y y y p b r
rp
by pr
b
t
F LL
LL
F CF F F L L L
LL
CE
F LL
L
r
≤


−
= − ≤ <≤



−



π≤>



(AISC F5-1, F5-3, F5-4)
where,
1.1
pt
y
E
Lr
F
=
(AISC F5.2, 4-7)
0.7
rt
y
E
Lr F
= π
(AISC F5-5)
2
0
0
1
12 6
=
+


fc
t
w
b
r
hh
a
d hd
(AISC F5.2, F4-11)
Rpg is the bending strength reduction factor, which has been described
in a previous section.
3.5.3.2.1.6.3 Compression Flange Local Buckling
Mn = RpgFcrS33c , (AISC F5-7)
where, Fcr is the critical buckling stress given by
3 - 50 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
()
2
, if flanges are compact,
0.3 , if flanges are noncompact, and
0.9 , if flanges are slender,
2
y
pf
cr y y
rf pf
cy
fc
fc
F
FF F
Ek F
b
t


λ−λ
= −



λ −λ



≤




(AISC F5-1, F5-8, F -9)
and λ, λpf, and λrf are the slenderness and the limiting slenderness ratios for
compact and noncompact flanges from Table 3-1, respectively, and kc is given
by
4,
c
w
kht
=
where 0.35 ≤kc≤ 0.76. (AISC 5.3)
3.5.3.2.1.6.4 Tension Flange Yielding
33 33
33 33 33
if ,
if .
p tc
n
yt t c
M SS
MFS S S
≥
=<
(AISC F5-10)
3.5.3.2.2 Minor Axis Bending
The nominal flexural strength is the lower value obtained according to the limit
states of yielding (plastic moment) and flange local buckling.
3.5.3.2.2.1 Yielding
Mn = Mp=FyZ22≤ 1.6Fy S22, (AISC F6-1)
where, S22 and Z22 are the section and plastic moduli about the minor axis, re-
spectively.
Calculation of Nominal Strengths 3 - 51
Steel Frame Design AISC 360-10
3.5.3.2.2.2 Flange Local Buckling
( )
22
22
, for compact flange,
0.7 , for noncompact flanges, and
, for slender flanges,
n
p
pf
pp y
rf pf
cr
M
M
M M FS
FS
=

λ−λ
−− 

λ −λ

(AISC F6-1, F6-2, F6-3)
where,
2
0.69
cr
E
F=λ
(AISC F6-4)
max ,
fb ft
tb ft
bb
tt


λ= 


(AISC F6.2)
and λpf and λrf are the limiting slendernesses for compact and noncompact
flanges, respectively, as described in Table 3-1 (AISC B4.1b).
3.5.3.3 Channel and Double Channel Sections
3.5.3.3.1 Major Axis Bending
The nominal flexural strength is the lowest value obtained according to the
limit states of yielding (plastic moment), lateral-torsional buckling, and com-
pression flange local buckling.
3.5.3.3.1.1 Yielding
Mn = Mp =FyZ33 , (AISC F2-1)
where Z33 is the plastic section modulus about the major axis.
3 - 52 Calculation of Nominal Strengths
Chapter 3 - Design using ANSI/AISC 360-10
3.5.3.3.1.2 Lateral-Torsional Buckling
()
33
33
, if ,
0.7 if , and
, if ,
p bp
bp
n bp p y p pbr
rp
cr p p r
M LL
LL
M C M M FS M L L L
LL
FS M L L
≤


−
= − − ≤ <<



−



≤>
(AISC F2-1, F2-2, F2-3)
where S33 is the elastic section modulus taken about the major axis, Lb is the
unbraced length, Lp and Lr are limiting lengths, and Fcr is the critical buckling
stress. Fcr, Lp and Lr are given by
2
2
2
33 0
1 0.078
bb
cr
ts
b
ts
CE L
Jc
FSh r
L
r

π
= + 
 


(AISC F2-4)
1.76
py
y
E
Lr
F
=
(AISC F2-5)
2
33 0
33 0
0.7
1.95 1 1 6.76
0.7
y
r ts
y
FSh
E Jc
Lr
F S h E Jc

= ++ 

(AISC F2-6)
where
2
33
yw
ts
IC
rS
=
(AISC F2-7)
1 for Double Channel sections
for Channel sections
2
y
o
w
CI
h
C
=
(AISC F2-8a, F2-8b)
and h0 is the distance between flange centroids.
Calculation of Nominal Strengths 3 - 53
Steel Frame Design AISC 360-10
3.5.3.3.1.3 Compression Flange Local Buckling
The nominal strength for compression flange local buckling is determined
based on whether the web is compact, noncompact, or slender.
If the web is compact,
( )
33
33
2
, for compact flanges,
0.7 , for noncompact flanges, and
0.9 , for slender flanges,
n
p
pf