Overview of Direct Analysis Method of Design for

Overview of Direct
Analysis Method of
Design for Stability
By: Ryan Brotherson
Needham Consulting Engineers
www.needhamassoc.com
913-385-5300

Presentation Overview
•Changes & requirements of AISC
360 specification
•Overview of current methods &
limitations
•Overview of Direct analysis
method
•Discussion on NCE software

What is Direct Analysis?
•It is a stability design method that
that addresses the five factors
that affect stability through the
addition of ‘notional’ loads and
‘softening’ of the structure which
introduces P-Delta effects.

AISC 360 Re-arrangement
2005
• Chapter C
• Effective Length Method (K
factors)
• First Order analysis
• Appendix 7
• Direct Analysis Method
2010
• Chapter C
• Direct Analysis Method
• Appendix 7
• Effective Length Method (K
factors)
• First Order Analysis

Reference to Stability
Requirements
• AISC 360 Chapter H
• Eqn. H1-1a
•
𝑃𝑟
𝑃𝑐
+
8
9
∗ (
𝑀 𝑟𝑥
𝑀 𝑐𝑥
+
𝑀 𝑟𝑦
𝑀 𝑐𝑥
) ≤ 1.0
• Pr & Mr are the required axial & flexural strength using LRFD or ASD load
combinations
• AISC 360 Chapter B3 – Design Basis
• B.3.1 - Required Strength
• The required strength of structural members and connections shall be
determined by structural analysis for the appropriate load combinations.
• B.3.5 – Design for Stability
• Stability of the structure and its elements shall be determined in
accordance with Chapter C.

AISC 360-10 Chapter C
• Section C1 - GENERAL STABILITY REQUIREMENTS
• Stability shall be provided for the structure as a whole and for each of its
elements. The effects of all of the following on the stability of the
structure and its elements shall be considered: (1) flexural, shear and
axial member deformations, and all other deformations that contribute
to displacements of the structure; (2) second-order effects (both P-Δ and
P-δ effects); (3) geometric imperfections; (4) stiffness reductions due to
inelasticity; and (5) uncertainty in stiffness and strength. All load-
dependent effects shall be calculated at a level of loading corresponding
to LRFD load combinations or 1.6 times ASD load combinations.
• Any rational method of design for stability that considers all of the listed
effects is permitted; this includes the methods identified in Sections
C1.1 and C1.2.
• C1.1 - Direct Analysis Method of Design
• The direct analysis method of design, which consists of the calculation of
required strengths in accordance with Section C2 and the calculation of
available strengths in accordance with Section C3, is permitted for all
structures.
• C1.2 -Alternative Methods of Design
• The effective length method and the first-order analysis method, defined in
Appendix 7, are permitted as alternatives to the direct analysis method for
structures that satisfy the constraints specified in that appendix.

2nd Order effects
• P-δ effect. Effect of loads acting on the deflected shape of a
member between joints or nodes.
• P-Δ effect. Effect of loads acting on the displaced location of
joints or nodes in a structure. In tiered building structures, this
is the effect of loads acting on the laterally displaced location
of floors and roofs.

Other issues
• Geometric imperfections
• Beam sweep, camber, out of plumb, etc.
• Code of standard practice allows H/500 for
column out of plumb
• Residual stresses
• Uneven cooling of hot rolled shapes
• Uncertainty in strength and stiffness
• Variability in material properties

Strength vs. Resistance
• AISC 360 Chapter H
• Eqn. H1-1a
•
𝑃𝑟
𝑃𝑐
+
8
9
∗ (
𝑀 𝑟𝑥
𝑀 𝑐𝑥
+
𝑀 𝑟𝑦
𝑀 𝑐𝑥
) ≤ 1.0
• Pr is the required axial strength using LRFD or ASD
load combinations
• Pc is the available axial strength
• Effective length method modified the Resistance side of
the equation – Pc
• Direct Analysis method modifies the required strength
side of the equation – Pr & Mr

Effective Length Method
• Commonly called the K factor method
• Most common method used at this time
• Introduced in 1963 (to some resistance)
• The K factor is a modification factor applied to
the length of columns with defined restraint
conditions
• It was used to account for 2nd order effects,
geometric imperfections, stiffness reductions,
and uncertainties.

Effective Length Method
• Limitations of the method include:
• It cannot be used for stability sensitive
structures where the ratio of 2nd order to 1st
order effects is greater that 1.5.
• Requires determination of K factors for every
column situation
• The use of arbitrary lengths that are not based
on the real world is not direct or intuitive.
• K factor is technically load dependent

Direct Analysis method
• Easy to understand & versatile
• The method does not have the limitations – all
issues affecting global stability are accounted for
in the method.
• Eliminates the need to consider effective length
factors.
• AISC commentary recommends that ratio of 2nd
order to 1st order effects not exceed 2.5 to limit a
runaway instability.

Direct Analysis Method
• Procedure is as follows:
1. Perform analysis at strength level
2. Apply notional loads at each floor level
3. Modify stiffness of all members contributing to
lateral stability of structure
4. Perform 2nd order analysis for all load combinations
to determine required strengths
5. Determine available strengths of all members based
on Chapters D through K
6. Verify available strength is greater than required
strength

Chapter C – Direct Analysis
Method
•Required strengths are determined by
analysis by section C2.1
•Analysis shall include initial
imperfections per C2.2
•Analysis shall consider adjustments to
stiffness per C2.3

Required Strengths
• Analysis shall consider all deformations including connections that
contribute to the displacement of the structure.
• Analysis shall be performed at strength level (1.0*LRFD or 1.6*ASD
loadings)
• Analysis shall include both P-δ & P-Δ effects.
• Permissible to ignore P-δ under following conditions.
• Columns are nominally vertical
• Ratio of 2nd order to 1st order drift < 1.7
• One third or less of gravity load supported on frame columns.
• Use of approximate method provided in Appendix 8 is permitted as
an alternative to a rigorous 2nd order analysis

Initial Imperfections
• Permissible to account for imperfections by direct
modeling of column out of plumbness, etc.
• More common to account for the imperfections with
Notional Loads
• Notional load is lateral load at each level as follows
• Ni = 0.002*α*Yi
• Alpha = 1.0 at LRFD & 1.6 at ASD
• Yi is gravity load at level i
• 0.002 is based on H/500 out of plumbness (AISC COSP)
• Notional loads are applied to gravity cases only when
Ratio of 2nd order to 1st order drift < 1.7

Adjustment to Stiffness
• Members shall have a reduced stiffness on all members
that contribute to the stability of the structure.
• The reduction is 0.8 for axial & flexural stiffness & an
additional τb reduction on the flexural stiffness
• where τb is:
• 1.0 when αPr/Py ≤0.5
• 4(αPr/Py )[1- (αPr/Py)] otherwise
• May use τb = 1.0 if an additional 0.001 notional load is
added to all load cases

Adjustment to Stiffness con’t
• Reduced stiffness (EI* = 0.8τbEI and EA* = 0.8EA) is used in
the direct analysis method for two reasons.
• For frames with slender members, the 0.8 factor results in a
system available strength equal to 0.8 times the elastic stability
limit. This is roughly equivalent to the margin of safety implied for
slender columns by the effective length procedure where from
Equation E3-3, φPn = 0.9(0.877Pe) = 0.79Pe.
• For frames with intermediate or stocky columns, the 0.8τb factor
reduces the stiffness to account for inelastic softening prior to
the members reaching their design strength. The τb factor is
similar to the inelastic stiffness reduction factor implied in the
column curve to account for loss of stiffness under high
compression loads (αPr > 0.5Py ), and the 0.8 factor accounts for
additional softening under combined axial compression and
bending.

Available strength
• For direct analysis method – available strength is
calculated based on Chapters D, E, F, G, H, I, J, & K of the
specification
• Effective length factor = 1.0 in all cases.

Commentary to Chapter C
• Rigorous second-order analyses are those that accurately
model all significant second-order effects.
• Some—but not all, and possibly not even most—modern
commercial computer programs are capable of
performing a rigorous second-order analysis, although
this should be verified by the user for each particular
program.

STAAD
• Direct Analysis is available effective STAAD.Pro 2007
• See section 5.37.5 of technical reference manual
• General Format
• PERFORM DIRECT ANALYSIS……..(See sec. 5.37.5 and STAAD
output)
• Use command in place of Perform Analysis or Pdelta Converge
• Command directs the program to:
• Reduce axial & flexural stiffness as required by code
• Solve static case w/ notional loads

STAAD Notional Load
• Direct analysis must use Repeat Load or Reference Load
specification
• Notional loads need to be defined per section 5.31.7 and
5.32.14 of the reference manual
• STAAD derives a lateral load from an existing vertical load case
• Example:
• Load 1 Dead Load
• Joint Load, Member Load, etc.
• Load 2 Dead Notional Load
• 1 X 0.002 [Load case – Direction – Ratio]

STAAD P-delta
• STAAD Default for P-delta will include both P-δ &
P-Δ.
• Must be used with REPEAT LOAD command
• Benchmark problem Case 1 from commentary to
Chapter C of the Specification (page 16.1-276)
results were confirmed.
• Appears that STAAD meets a Rigorous 2nd order
analysis

RAM 2nd Order Analysis
• RAM version 14.05 implements the Direct
Analysis method using a 2nd Order by
Amplified 1st order elastic analysis
• This is not necessarily considered a
rigorous 2nd order analysis
• Uses the B1 & B2 method per Appendix 8
• Allowed by section C.2.1(3)

RAM 2nd Order Analysis
• From Section 5.1.3 of RAM manual (online)
• Second-Order Analysis - The requirements to perform
a second-order analysis is satisfied by performing a
first-order analysis and calculating and applying B1
and B2 factors to the design forces as outlined in
Section C2.1b of the Specification….It should be also
noted that the engineer is provided two options to
consider 2nd order (large P-delta) effects: either the
engineer use the current P-delta analysis implemented
or the engineer chooses B2 factors.

RAM 2nd order Analysis
• Notional loads - specified in the Loads – Load Cases
command in RAM Frame.
• Reduced Stiffness - An option to use the AISC 360
stiffness reduction is available (Criteria – General dialog).
• The program does not iterate to determine the correct
value of τb, so the engineer either specifies 1.0 or
some other value.
• Although technically τb is distinct for each load combo
& member, the program uses the specified value on all
members and does not vary stiffness for each load
combination

RAM P-Delta
• RAM uses two methods to approximate P-Delta
effects
• Both are based on the Geometric Stiffness
Method
• Small, assumed deflections are used to create a
Geometric Stiffness matrix
• This matrix modifies the building stiffness matrix
once
• Accounts for P-Δ only.

RAM P-Delta
• Non-iterative P-Delta Method
• Used for Rigid diaphragms
• Preliminary P-Delta Analysis
• Used for Semi-rigid diaphragms

Summary
• Direct analysis is the preferred stability method
of AISC
• Direct analysis directly accounts for the five
issues contributing to stability
• Direct analysis appears relatively easy to
implement in STAAD.
• RAM Frame uses approximate methods to
account for stability

Overview of Direct Analysis Method of Design for

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