Direct Analysis Method Tips

Lessons Learned regarding Appropriate use of Direct Analysis Method with RAM & ETABS

To quickly summarize, here is what we have learned:

• The Effective Length Method for the design of steel beams and columns has been used for

years, but this method has limitations on its applicability. Thus, in the past several years the

industry has migrated to another design methodology to check for strength and stability – The

Direct Analysis Method (DA). Attached herein are two papers that are must-reads:

o Background of DA obtained from AISC, and

o Recent paper published in Winter 2011 Engineering Journal by Shankar Nair, Jim Malley

and John Hooper on Design of Steel Buildings for EQ and Stability by AISC 360 (DA) and

ASCE 7.

Note: SidePlate President Henry Gallart personally spoke with Dr. Nair and got verbal

confirmation of their conclusions that the unreduced stiffness shall be used in the

calculation of building periods, associated base shears, and checking of building drifts.

A 1-hour free webinar that Dr. Nair gave through AISC is very educational and helpful for

those who are auditory and visual learners – highly recommend it:

http://www.aisc.org/content.aspx?id=4498

• Both the RAM Structural System and ETABS software have been programmed in such a way

that, unless the user really understands what the program is doing and is super careful with

their bookkeeping, an engineer selecting DA may inadvertently analyze a steel building with a

global 20% reduced stiffness, resulting in a softer structure (higher periods) than would

otherwise be calculated. For many buildings, this translates into a lower base shear, thus

resulting in un-conservative designs.

In an effort to assist structural engineers, we have assembled the following tips for RAM Structural

System and ETABS users:

• For RAM users, Allen Adams of RAM posted on the Bentley wiki an excellent write up that

describes the exact steps to follow.

o http://communities.bentley.com/products/structural/structural_analysis___design/w/s

tructural_analysis_and_design__wiki/6011.aspx

• For ETABS users, this issue was brought to their attention just recently. As such, a detailed write

up doesn’t exist today. However, using the same principles, here is one way to utilize ETABS

correctly if an engineer is using DA to design the beams and columns:

o When starting a new model in ETABS, the program utilizes an unreduced stiffness for

beams and columns. If you never hit DESIGN, all is well. However, if Direct Analysis was

selected under PREFERENCES as the Design Analysis Method, when a DESIGN is

performed it will prompt you to overwrite the analysis. This will trigger the 20%

stiffness reduction and the original building period calculated based on an unreduced

stiffness is lost.

� Recommendation:

• Create ETABS model with the usual preferences like DA. Alternately,

Effective Length Method could be selected to preclude the accidental

running of the model with the DA setting which will reduce the building

stiffness by 20%.

• Analyze the building to determine actual building periods (using the

unreduced stiffness model) in order to obtain the correct static base

shear to be used for calculating the seismic base shear for both strength

and drift. Perform all key ASCE 7 code checks, except strength checks

required by AISC 360 and AISC 341.

o If performing a dynamic analysis, be sure to create a load case

for both dynamic base shears (strength and drift) using the

unreduced stiffness model so that there is a basis with which to

scale later the dynamic base shear used for strength checks.

• Upon generating a design that is satisfactory and meets all ASCE 7 code

checks, save the model (without ever activating the DESIGN feature).

Now, to check strength (stress) per AISC 360 and AISC 341 seismic

provisions, follow the principles outlined in the Wiki for RAM. It is

anticipated that a similar document from CSI will be created in the near

future.

o Since some of the settings in ETABS may need to be different

(like overriding the period calculated by ETABS, etc.), we would

recommend that a separate model be used to check strength to

eliminate the possible confusion that can easily occur when

switching back and forth between a model with an unreduced

stiffness and a model with a reduced stiffness.

If you have ANY questions or would like some clarification on the content of this information, please feel

free to call our office at 949-305-7889.

ENGINEERING JOURNAL / THIRD QUARTER / 2011 / 199

Design of Steel Buildings for Earthquake and Stability by Application of ASCE 7 and AISC 360R. SHANKAR NAIR, JAMES O. MALLEY and JOHN D. HOOPER

Abstract

Design of steel buildings in the United States typically combines application of ASCE/SEI 7, Minimum Design Loads for Buildings and Other

Structures, and ANSI/AISC 360, Specification for Structural Steel Buildings. For buildings designed for seismic effects, ANSI/AISC 341, Seis-

mic Provisions for Structural Steel Buildings, may also be applicable. The ASCE 7 Minimum Design Loads standard includes specific design

provisions related to stability under seismic loading which overlap and, in some instances, appear to conflict with the stability design require-

ments of the AISC Specification. This paper explores the areas of overlap and apparent conflict between ASCE 7 and AISC 360 and offers

practical recommendations for seismic design incorporating the provisions of both.

Keywords: design loads, seismic design, structural stability.

Design of steel buildings in the United States typically

combines application of ASCE/SEI 7, Minimum Design Loads for Buildings and Other Structures (ASCE, 2005),

and ANSI/AISC 360, Specification for Structural Steel Buildings (AISC, 2005a; AISC, 2010). For buildings de-

signed for seismic effects, ANSI/AISC 341, Seismic Provi-sions for Structural Steel Buildings (AISC, 2005b), may be

applicable in conjunction with the Specification.

ASCE 7 is used, either directly or by reference from a

building code, to define the loads for which the structure

must be designed; AISC 360 and 341 are used to design the

steel structure for those loads.

The ASCE 7 Minimum Design Loads standard, though

generally focused on loads and not on design or the response

of the structure to those loads, includes specific design pro-

visions related to stability under seismic loading. These pro-

visions overlap and, in some instances, appear to conflict

with the stability design requirements of the AISC Specifi-cation. (The AISC Seismic Provisions do not include stabil-

ity design requirements.)

This paper explores the areas of overlap and apparent

conflict between ASCE 7 and AISC 360 and offers practi-

cal recommendations for seismic design incorporating the

provisions of both.

The paper does not attempt to correlate design with ex-

pected actual behavior beyond the degree of correlation im-

plied by compliance with ASCE 7 and AISC 360. This is

an important limitation. It is generally recognized that ac-

tual displacements in an earthquake could be much larger

than the elastic displacements due to code-specified design

loads, and the resulting second-order effects could be quite

different from those predicted by specification-compliant

analysis; exploration of this issue is beyond the scope of this

paper.

STABILITY DESIGN BY AISC 360

The Direct Analysis Method of design for stability was

introduced in the 2005 edition of the AISC Specification.

The 2010 edition makes that method the primary means of

design; alternative approaches have been moved to an ap-

pendix. The discussion in this paper will be limited to the

Direct Analysis Method and its application in seismic design

in conjunction with ASCE 7; other stability design methods

permitted in the AISC Specification are not considered.

The rational basis of the Direct Analysis Method is ex-

plained in AISC-SSRC (2003); a simple introduction to

the practical application of the method is provided in Nair

(2009a). Notes on the modeling of structures for design by

the Direct Analysis Method are provided in Nair (2009b).

Design for stability by the Direct Analysis Method in-

volves a second-order analysis, use of reduced stiffness in

the analysis, consideration of initial imperfections (either

by direct modeling of the imperfections or by application of

notional loads in the analysis) under certain circumstances,

and strength check of components using an effective length

factor, K, of unity for members subject to compression.

R. Shankar Nair, Ph.D., P.E., S.E., Senior Vice President, Teng & Associates,

Inc., Chicago, IL (corresponding). E-mail: [email protected]

James O. Malley, P.E., S.E., Senior Principal, Degenkolb Engineers, San Fran-

cisco, CA. E-mail: [email protected]

John D. Hooper, P.E., S.E., Principal and Director of Earthquake Engineering,

Magnusson Klemencic Associates, Seattle, WA. E-mail: [email protected]

199_204_EJ3Q_2011_2010_07R.indd 199199_204_EJ3Q_2011_2010_07R.indd 199 9/13/11 2:38 PM9/13/11 2:38 PM

200 / ENGINEERING JOURNAL / THIRD QUARTER / 2011

Second-Order Analysis

The analysis of the structure must be a second-order analy-

sis that, in the most general case, includes both P-Δ effects,

which are the effects of loads acting on the displaced lo-

cations of joints or nodes in the structure, and P-δ effects,

which are the effects of loads acting on the deformed shapes

of members. In tiered buildings, P-Δ effects are the effects

of loads acting on the laterally displaced locations of floors

and roofs.

The 2010 AISC Specification exempts most buildings

from the need to consider P-δ effects in the analysis of the

overall structure; a P-Δ–only analysis is sufficient in almost

all cases. This is an important simplification relative to the

2005 Specification, which does require inclusion of P-δ

effects in the analysis of most moment-frame buildings. Re-

gardless of whether P-δ effects need to be considered in the

analysis of the overall structure, the P-δ effects on individu-

al beam-columns must always be considered in the strength

check of those members.

The second-order analysis required by the Direct Analy-

sis Method of design may be performed using a computer

program formulated to provide a second-order solution.

Alternatively, a second-order solution may be obtained by

manipulating the results of a linear or first-order analysis to

account for second-order effects by application of the B1 and

B2 multipliers defined in the Specification.

In the B1 and B2 procedure, B2 alters the results of a first-

order analysis to account for P-Δ effects; the B2 multiplier

also accounts, in an approximate way (by application of the

factor RM in the calculation of B2), for the overall softening

of the structure’s response due to P-δ effects in individual

members. For a given vertical loading, there will be a sin-

gle value of B2 for each story and each direction of lateral

translation, applicable to the forces and moments caused by

lateral loading in all members and connections in that story.

In the unusual case where gravity load causes lateral transla-

tion, B2 is also applicable to the forces and moments caused

by the side-sway component of gravity load. While B2 is a

story parameter, B1 is a member parameter; a B1 multiplier

is applied to the moments in each beam-column to account

for P-δ effects in that member.

Another approach, usable in the typical case where P-δ effects do not need to be considered in the analysis of the

overall structure, is to obtain a P-Δ–only second-order solu-

tion from a computer program and then to apply B1 multipli-

ers to account for P-δ effects in individual members.

Given that in second-order analysis the effect of a load is

not proportional to its magnitude (and the principle of super-

position of loads does not apply), the second-order analysis

must be performed at the Load and Resistance Factor De-

sign (LRFD) load level: For design by LRFD, LRFD load

combinations must be applied in the analysis; for design by

Allowable Strength Design (ASD), ASD load combinations

increased by a factor of 1.6 must be applied in the analysis

and the results must be divided by 1.6 to get the forces and

moments for proportioning of members and connections.

Reduced Stiffness

In the second-order analysis, all stiffnesses in the model-

ing of the structure must be reduced by applying a factor

of 0.8. An additional reduction factor, τb, must be applied

to the flexural stiffnesses of all members whose flexural

stiffnesses are considered to contribute to the lateral stabil-

ity of the building. Factor τb is unity when the LRFD-level

required compression strength of the member is less than

half the yield strength.

When the compression is high and τb is not unity, the de-

signer may still avoid the complication of calculating and

applying τb by applying, instead, a small notional lateral

load (0.001 times the LRFD-level gravity load applied at

each floor) in the analysis. This notional load will typically

be so much smaller than seismic loads that it may reasonably

be neglected when seismic loads are present. Thus, it should

be possible to take τb as unity in all cases in seismic design;

what remains is the 0.8 factor, applied to all stiffnesses.

Initial Imperfections

The effect of initial imperfections must be considered in

the analysis, either by direct inclusion of the imperfections

in the analysis model or by application of notional loads, if

either (1) the load combination being considered is a gravity-

only loading with no applied lateral load or (2) the ratio of

second-order drift to first-order drift in any story of the

building is more than 1.7.

For seismic design, there will always be applied lateral

load and condition 1 will not apply; condition 2 will also

typically not apply for buildings that satisfy the limits on

stability coefficient, θ, specified in ASCE 7. Therefore, it

will not typically be necessary in seismic design to consider

initial imperfections, either explicitly or by application of

notional loads.

Component Strength Check

For design by the Direct Analysis Method, once the appro-

priate analysis has been performed, members and connec-

tions are checked for strength with no further consideration

of overall structure stability. The effective length factor, K,

for members subject to compression is taken as unity (unless

a lower value is justified by rational analysis).

SEISMIC DESIGN STABILITY PROVISIONS IN ASCE 7

Background and commentary on the seismic design provi-

sions of the ASCE 7 Minimum Design Loads standard may



be found in FEMA (2009). Specific requirements for stabil-

ity in conjunction with seismic design are presented in the

ASCE standard in a section titled “P-Delta Effects” (Sec-

tion 12.8.7)*; these requirements are included as part of the

“Equivalent Lateral Force Procedure” for seismic analysis

(Section 12.8).

The P-Delta Effects section of ASCE 7 defines a stability

coefficient, indicates when P-delta effects must be consid-

ered, places limits on the stability coefficient and specifies

methods of accounting for P-delta effects.

Stability Coefficient, θ

The stability coefficient, θ, is approximately the ratio of the

actual vertical force on a story of a building to the vertical

force that would cause elastic lateral buckling of the story.

An equation† is provided for the coefficient; the coefficient

is calculated with nominal, unreduced stiffnesses and for

load combinations with no individual load factor exceeding

unity.

When Must P-delta Effects Be Considered?

Under ASCE 7, P-delta effects need be considered only when

the stability coefficient, θ, is more than 0.10. This is roughly

equivalent to an AISC B2 multiplier (ratio of second-order

story drift to first-order story drift) of 1.2, after accounting

for the fact that θ is calculated with nominal stiffnesses and

for load combinations with no individual load factor exceed-

ing unity, while B2 is calculated with reduced stiffnesses

and for LRFD-level load combinations.

Limit on Stability Coefficient

The stability coefficient, θ, must never be higher than θmax,

which is variable (a function of β, the ratio of shear demand

to shear capacity of the story, and of Cd, the deflection am-

plification factor), but never higher than 0.25. A θ of 0.25

is roughly equivalent to an AISC B2 multiplier of 1.7 after

correction for the different stiffnesses and loadings in the θ

and B2 calculations.

Method of Analysis for P-delta Effects

ASCE 7 specifies that when P-delta effects are required to

be considered (i.e., when the stability coefficient, θ, exceeds

0.10), “the incremental factor related to P-delta effects on

* ASCE 7 does not explicitly differentiate between the P-Δ effects and P-δ effects recognized by the AISC Specification; the “P-delta” effects ad-

dressed in ASCE 7 are the P-Δ effects of AISC.

† There is an error in Equation 12.8-16, the equation for stability coef-

ficient, θ, in ASCE 7-05. There should be I (for importance factor) in the

numerator on the right-hand side of the equation. This has been corrected

in ASCE 7-10.

displacements and member forces shall be determined by

rational analysis.” Two types of rational analysis are envi-

sioned: (1) nonlinear static (pushover) analysis and (2) non-

linear response history analysis, both of which require

extensive, additional effort.

As an alternative to the rational analysis, “it is permit-

ted to multiply the displacements and member forces by

1.0/(1 − θ).” This is analogous to application of the AISC B2

multiplier with RM = 1 in the equation for B2, which amounts

to neglecting P-δ effects in the analysis. However, given that

θ is calculated with nominal stiffnesses and for load combi-

nations with no individual load factor exceeding unity, while

B2 is calculated with reduced stiffnesses and at LRFD-level

load combinations, the AISC approach will indicate signifi-

cantly higher second-order displacements and forces.

Use of an analysis that included P-Δ effects (but not P-δ

effects), and subsequent application of AISC B1 multipli-

ers to individual beam columns to account for P-δ effects,

would also satisfy the requirement of ASCE 7.

Seismic Design by Modal Response Spectrum Analysis

The preceding discussion of stability-related requirements

in the seismic design provisions of ASCE 7 was based on use

of the Equivalent Lateral Force procedure. An alternative

seismic analysis procedure prescribed in ASCE 7 (Section

12.9) is the Modal Response Spectrum Analysis approach.

Design by Modal Response Spectrum Analysis is applicable

to all structures of all Seismic Design Categories; the Equiv-

alent Lateral Force procedure is not permitted for certain

structures in Seismic Design Categories D through F.

Stability requirements are not presented independently in

the Modal Response Spectrum Analysis section; the same

P-delta requirements prescribed for the Equivalent Lateral

Force procedure are incorporated by reference in the Modal

Analysis section. There is an obvious complication here in

that the Modal Analysis approach involves combining the

results of analyses for different modes, but second-order

analyses (i.e., analyses incorporating P-delta effects) cannot

normally be combined. A means of overcoming this diffi-

culty is suggested in the following section.

COMPARISON AND RECOMMENDATIONS

Selected features of the Direct Analysis Method of design

for stability in the AISC Specification and provisions related

to seismic design and stability in the ASCE 7 Minimum De-sign Loads standard, as discussed in the preceding sections,

are summarized in Table 1. Clearly, there are areas of diver-

gence between the two sets of requirements. Nonetheless, as

outlined in the following, steel buildings may be designed

for seismic effects and stability in general conformance with

both the AISC Specification and ASCE 7.



General recommendations:

1. Observe the ASCE 7 limit on stability coefficient; θ must not exceed θmax. The nominal (unreduced) stiff-

ness of the structure and the ASCE 7 vertical load

(with no individual load factor greater than 1.0) may

be used in the calculation of coefficient θ.

2. In the analysis for assessing strengths, consider P-Δ

effects in the analysis for all structures; also consider

P-δ effects in the analysis where required by the AISC

Specification. (Do not observe the ASCE 7 provision

that exempts from P-delta considerations all buildings

with a stability coefficient, θ, less than 0.10.)

Recommendations specific to seismic design by the Equiva-

lent Lateral Force procedure:

1. Determine the fundamental period of the building ei-

ther by analysis or by use of the approximate meth-

ods prescribed in ASCE 7. If determined by analysis,

use first-order analysis (no second-order effects), with

nominal (unreduced) stiffnesses.

2. Consider second-order effects either by second-order

Table 1. Comparison of Analysis and Stability Provisions in ASCE 7 and AISC 360

SubjectASCE 7

Equivalent Lateral Force Procedurea

AISC 360Direct Analysis Method

Recommendation for Seismic Design

Limit on P-Δ effect Stability coefficient θ must

not exceed θmax.

No limit. Observe ASCE limit, which

corresponds to P-Δ multiplier of

1.33 or less.b

Must P-Δ effects be

considered?

Only when θ is greater

than 0.1.

Yes; in all cases. Always consider P-Δ effects.

P-Δ effects by rational

analysis?

Permitted. Permitted. Rational analysis may be used.

P-Δ effects by

approximate analysis?

Permitted.

[Multiply lateral load

effects by 1/(1 – θ).]

Permitted.

[Multiply lateral load

effects by B2.]

The AISC method may be used;

note that 1/(1 − θ) ≈ B2.

Must P-δ effects be

considered in the

analysis?

Not specified. Generally yes in 2005;

generally no in 2010.

Observe AISC 2010.c

Load in the stability

analysis

Not specified for rational

analysis. Load factor

not greater than 1.0 for θ

calculation.

LRFD load combinations

for LRFD; 1.6 times ASD

combinations for ASD.

Observe AISC: LRFD load

combinations for LRFD; 1.6 times

ASD load combinations for ASD.

Structure stiffness in

the stability analysis

Not specified. Apply factor of 0.8 τb

to EI; 0.8 to all other

stiffnesses.

Apply factor of 0.8 (no τb) to all

stiffnesses.

Must initial

imperfections be

considered?

Not specified. Yes, with exceptions;

either model directly or

apply notional loads.

Need not consider initial

imperfections (neither direct

modeling nor notional loads).

Analysis to assess

conformance to drift

limits

Elastic stiffness; same type

of analysis as for strength.

Not specified. Use of same analysis as for

strength is permissible but may

be too conservative; see text.

Analysis to determine

period

Not specified, although

upper limits are defined.

Not specified. Use linear analysis (no P-Δ

effects) with nominal, unreduced

stiffnesses.

a See text for additional considerations for Modal Response Spectrum Analysis procedure.b The stability coefficient, θ, is calculated at lower load than used in AISC stability analyses; therefore, P-Δ multipliers corresponding to the ASCE 7

thresholds are not strictly comparable to AISC 360 parameters.c Regardless of whether it is considered in the analysis of the overall structure, P-δ must always be considered in the strength check of individual

beam-columns.



analysis or by application of the AISC B1 and B2 mul-

tipliers to the results of first-order analysis. The op-

tions are:

a. Complete second-order analysis that considers

both P-Δ and P-δ effects.

b. P-Δ–only second-order analysis, followed by ap-

plication of B1 to individual beam columns (not

permissible in the unusual cases where inclusion of

P-δ effects in the overall analysis is required by the

AISC Specification).

c. First-order analysis, followed by application of B2

and B1 multipliers.

3. Perform the second-order analysis and/or calculate

the B1 and B2 multipliers at LRFD-level loads; that is,

use LRFD load combinations if design is by LRFD,

use 1.6 times ASD load combinations if design is by

ASD. (After second-order analysis under ASD, divide

the analysis results by 1.6 for member and connection

strength checks.)

4. Apply a factor of 0.8 to all stiffnesses in the second-

order analyses and in the calculation of B1 and B2 mul-

tipliers. For load combinations that include seismic

load, the additional stiffness reduction factor τb need

not be applied.

5. In the analysis for load combinations that include seis-

mic load, it is not necessary to model initial imper-

fections or to apply notional loads to account for the

imperfections.

6. Perform strength checks in accordance with the AISC

Specification, using an effective length factor, K, of

unity for members subject to compression (unless a

lower value is justified by rational analysis).

7. Conformance to ASCE 7 seismic drift limits may be

checked using the same analysis used for strength

checks (analysis at reduced stiffness, second-order

analysis, second-order effects determined at LRFD-

level loads). This may be excessively conservative,

however, and it is permissible to base drift checks on

an analysis using the full unreduced stiffness of the

structure, with second-order effects determined at the

lower loads specified by ASCE 7 for calculation of the

stability coefficient, θ. The deflection amplification

factor, Cd, should be applied in either case.

Recommendations specific to seismic design by the Modal

Response Spectrum Analysis approach:

1. Determine modes and frequencies using first-order

analysis, with nominal (unreduced) stiffnesses.

2. Determine member forces and moments due to gravity

load by first-order analysis, with a factor of 0.8 applied

to all stiffnesses.

3. Use the properties of each mode and the ASCE 7 de-

sign response spectrum to determine a set of lateral

forces for that mode; using these lateral forces, per-

form a first-order analysis, with a factor of 0.8 applied

to all stiffnesses, to determine member forces and mo-

ments. Repeat for all modes considered.

4. Calculate a single B2 multiplier, applicable to all

modes,‡ for each story and each direction of lateral

translation, based on reduced stiffness (0.8 factor) and

the full LRFD-level vertical load on the story.§

5. Combine first-order modal results (item 3) as speci-

fied in ASCE 7 Section 12.9.3. Apply B2 multipliers

to the combined results (member forces and moments

caused by lateral loading). Then scale the combined

results as specified in ASCE 7 Section 12.9.4. Alge-

braic signs will typically be lost in modal combina-

tions; the member forces and moments due to seismic

effects should, therefore, be considered reversible

(i.e., use absolute values of forces and moments in the

modal combinations and then consider the resulting

overall forces and moments due to seismic effects to

be fully reversible).

6. Combine the member forces and moment due to seis-

mic effects (item 5) with the member forces and mo-

ments due to gravity load (item 2), with load factors as

specified in ASCE 7.

7. Apply B1 multipliers to the moments in beam-

columns. The B1 multiplier should be based on the full

LRFD-level axial force in the member, including axial

forces due to lateral loading, but need be applied only

to that part of the moment in the beam-column that

is caused by gravity loading. (Designers may use the

conservative approximation of applying B1 to the full

moment to avoid the obvious bookkeeping difficulties

involved in this calculation.)

8. Perform strength checks in accordance with the AISC

Specification, using an effective length factor, K, of

unity for members subject to compression (unless a

lower value is justified by rational analysis).

‡ This is an approximation. If the B2 calculations for all stories are based

on story shears and drifts due to lateral load applied at the roof alone, the

resulting B2 values should be reasonably accurate or conservative (high)

for all modes.

§ It should also be possible (as an alternative to the B2 multiplier procedure

used herein) to adapt the modified geometric stiffness approach for use

with design by Modal Response Spectrum Analysis.



9. As in the Equivalent Lateral Force procedure, confor-

mance to ASCE 7 seismic drift limits may be checked

using either the same analysis used for strength checks

(convenient, but potentially very conservative) or an

analysis using the full unreduced stiffness of the struc-

ture, with second-order effects (B1 and B2 multipliers)

determined at the lower loads specified by ASCE 7 for

calculation of the stability coefficient, θ.

SUMMARY AND CONCLUSIONS

Provisions related to seismic design and stability in ASCE/

SEI 7, Minimum Design Loads for Buildings and Other Structures (ASCE, 2005), and features of the Direct Analy-

sis Method of design for stability in ANSI/AISC 360, Speci-fication for Structural Steel Buildings (AISC, 2005a; AISC,

2010), have been explored and compared. While there are

inconsistencies between the two sets of provisions, they are

not fundamentally incompatible. Recommendations are of-

fered for the design of steel buildings for seismic effects and

stability in general conformance with both the AISC Speci-fication and the ASCE 7 standard.

REFERENCESAISC (2005a), Specification for Structural Steel Buildings,

ANSI/AISC 360, American Institute of Steel Construc-

tion, Chicago, IL.

AISC (2005b), Seismic Provisions for Structural Steel Buildings, ANSI/AISC 341, American Institute of Steel

Construction, Chicago, IL.

AISC (2010), Specification for Structural Steel Buildings, ANSI/AISC 360, American Institute of Steel Construc-

tion, Chicago, IL.

AISC-SSRC (2003), “Background and Illustrative Examples

on Proposed Direct Analysis Method for Stability Design

of Moment Frames,” Technical White Paper, AISC Tech-

nical Committee 10, AISC-SSRC Ad Hoc Committee on

Frame Stability, American Institute of Steel Construction,

Chicago, IL.

ASCE (2005), Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-05, American Society of

Civil Engineers, Washington, DC.

FEMA (2009), NEHRP Recommended Seismic Provisions for New Buildings and Other Structures, FEMA P-750,

Federal Emergency Management Agency, Washington,

DC.

Nair, R.S. (2009a), “Simple and Direct,” Modern Steel Con-struction, AISC, January.

Nair, R.S. (2009b), “A Model Specification for Stability

Design by Direct Analysis,” Engineering Journal, AISC,

Vol. 46, No. 1, pp. 29–37.


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July 13, 2003

Background and Illustrative Examples on Proposed Direct Analysis Method for Stability Design of Moment Frames

Report 011 behalf of AISC TC 10 Prepared by: Gregory Deierleill

PREFACE

Summarized herein is background infonnation and illustrative examples for new frame stability design provisions proposed by AISC TC 10 for the 2005 AISC Standard. In the latest AlSC ballot (July 2003), most of the new provisions appear in a new Appendix 6, entitled "Direct Analysis Method for Moment Frames", which provides an alternative to the frame stability provisions in Section B6 of the Standard. The frame stability provisions of Section B6 are essentially identical to those in the 1999 (3"' edition) of the LRFD Specification for Structural Steel Buildings, except for the addition of a minimum moment requirement. The background and fundamental features of the standard (Section B6) and alternative (Appendix 6) provisions are described herein. Several illustrative example problems are presented to demonstrate and contrast the two stability design approaches.

ACKNOW LEDGM ENTS

The developments summarized in this report are the result of contributions by many individuals over the course of deliberations by AISC TC 10 and the AlSC·SSRC Ad Hoc Comrnittee on frame stabi lity. The AlSC·SSRC Ad Hoc Committee co·chairs (J. Yura and G. Deierlein) and members (W. Baker, J. Haliar, T. Galambos, R. Heinge, L. Lutz, K. Mueller, S. air, C. Rex, R. Tremb lay, D. White and R. Zieman) developed the first draft of the provisions. Other noteworthy contributions include those of D. White, A. Maleck, and R. Ziemian for their analysis of numerous benchmark validation studies and L. Lutz for suggestions of illustrative example problems. Portions of this report are excerpts from an earlier paper presented at the 2002 SSRC Annual Meeting, which introduced a preliminary (now superseded) draft of the proposed provisions (Deierlein, Hajjar, Yura, White, Baker, "Proposed new provisions for frame stability using secood·order analysis", SSRC 2002 Annual Meeting, Seattle, WA).

INTRODUCT ION

The proposed new provisions for frame stability represent the culmination of work by task committees in AlSC and SSRC over the past four years, which incorporate concepts of second·order analysis and design whose origins date back over twenty years. Concerted work on this began late in 1999, with the fonnation of a joint AISC·SSRC Ad·hoc Committee whose charge was to develop improved specification provisions for member and frame stability. The committee's goals were to develop design methods for stability that made more effective use of modem computer analysis methods, while reducing the over· reliance on effective buckling length procedures in the current AlSC Specifications. This ad·hoc committee was combined with AI SC TC lOin 200 I , and the combined group developed provisions, which are proposed for adoption in the 2005 AISC Standard. The new provisions were first balloted in March 2003 and have since been revised to address comments raised by the AISC Specification Committee. This report reflects the latest version of the proposed provisions for frame stability.

The July 2003 AlSC ballot outlines proposed provisions for the 2005 Standard, which will pernllt two alternative methods to design for stability effects in moment frames . For discussion purposes, the two approaches will be referred to as the "Effective Lellgth " and "Direct Allalysis" methods. Both approaches

" ,;0 ,z, ~

." ""

require evaluation of second-order effects and member force interaction equations. The methods differ in their specific requirements for calculating second-order effects and the axial strength tcrm, p., in the member interaction equation. Requirements for the Effective Length method are contained in the proposed Section B6 of the 2005 Standard. This method is essentially the same method as the approach used in Chapter C of the 1999 AISC-LRFD Specification. Requirements for the Direct Analysis method are specified in a newly proposed Appendix 6 to the 2005 Standard.

This report begins with a brief review of key behavioral effects and second-order analysis considerations, which are re levant to stability design. Next, the two proposed approaches to framc stability are summarized and contrasted through a design example of simplc cantilever column. This IS followed by highlights of validation studies to evaluate the accuracy of the two proposed methods. The report concludes with three design examples to illustrate practical application of the methods.

BEHAVIORAL EFFECTS

There are potentially many parameters and behavioral effects that influence stability of steel-framed structures. The extent to which these factors are modeled in analysis will affect the criteria that one applies in design of the (rame, its members and connecllons. Without repeating more complete presentations given elsewhere (Birnstiel and Imand, 1980; McGuire, 1992; White and Chen. 1993 ; ASCE, 1997; Deierlein & White 1998), it is helpful to review three basic aspects of behavior: geometric nonlinearities, inelastic spread-of-plasticity, and member limit states. These ultimately govern frame deformations under applied loads and the resulting internal load effects.

Geometric Nonlinearities and Imperfections: Modern stability design provIsions are based on the premise that the member forces are calculated by second-order elastic analyses, where equilibrium is satisfied on the defonned structure. When stability effects are significant, consideration must be given to initial geometric imperfections in the structure due to fabrication and erection tolerances. For the purpose of calibrating the stability requirements described later, initial geometric imperfection are conservatively assumed as equal to the maximum fabrication and erection tolerances pennitted by the AI Code oj Stalldard Practice (2000). For columns and frames, this implies a member out-of-straightness equal to UIOOO, where L is the member length between brace or framing points, and a frame out-of-plumb equal to H/500, where H is the story height. The out-of-plumb is also limited by the absolute bounds as speCified in the Code oj St{/l/{/ard Practice.

Inelastic Spread of Plasticity: The proposed analysis/design approaches are calibrated against inelastic distributed-plasticity analyses that account for spread of plasticity through the member cross-section and along the member length. Thermal residual stresses in W-shape members are assumed to have maximum values of 0.3Fy and are distributed according to the so-called Lehigh pattern - linearly varying across the flanges and unifornl tension in the web (Deierlein & White 1998).

Member Limit States: Member strength may be controlled by one or more of the following limit states: cross section yielding, local bUCkling, flexural buckling, and torsional-flexural buckling. For structural analyses envisioned for routine frame design, it is assumed that the analysis does not model local flange/web buckling or torsional-flexural buckling. Therefore, these limits must be considered in separate member design checks. For inelastic analyses, the member yield limit is incorporated directly in the analysis; and for elastic analyses, this limit can be checked by an interaction equation that approximates the P-M yield surface. Whether or not the analysis captures in-plane flexural buckling depends on the extent to which the maximum moments are affected by distributed plasticity and member str3lghtness. Concerns as to whether the analysis captures this effect suggest the need to apply a member check for lO

plane flexural buckling, even when an accurate second-order analysis is used. As will be addressed later,

Page2oJI7

a key consideration for the in-plane flexural buckling check relates to the assumed buckling length used in calculating the design compression strength, ¢P".

SECOND-ORDER ELASTIC ANALYSIS

The AISC stability design provisions are developed for use with second-order elastic analysis. In practice, there are alternative approaches one can employ for conducting second-order analyses, some of which are more rigorous than others. For the purpose of this discussion, second-order clastic analyses will be categorized as "rigorous" or "approximate". The difference between these two depends on the extent to which P-O effects are modeled and whether the problem is "linearized" to expedite the solution.

Rigorous second-order analyses are those that accurately model all significant second-order effects. Rigorous analyses include solution of the governing differential equation, either through stability functions or computer frame analysis programs that model these effects (McGuire 1992; Deierlein & White 1998). Many (but not all) modern commercial computer programs are capable of rigorous analyses, though users should verify this. Methods that modify first-order analysis results through second-order amplifiers (e.g. , B, and B: factors) are in some cases accurate enough to constitute a rigorous analysis, but this depends on the magnitude of second-order effects and other characteristics of the problem.

Approximate second-order analyses are any methods that do not meet the requirements of rigorous analyses. A common type of approximate analyses are those which only capture P-tJ due to member end translations (e.g., interstory drift) but fail to capture P-oeffects due to curvature of the member relative to its chord. Where P-O effects are significant, errors arise in approximate methods that do not accurately account for the effect of P-o moments on amplification of both local member moments and the calculated global (tJ)displacements. These errors can arise both with second-order computer analysis programs and with the B, and B, amplifiers. White and Maleck (2002) propose the following criteria to rule out cases where P-O effects can be safely ignored:

P, < 0.15 P,L = 0.15(,(£IIL') (I)

where P, is the required column strength and P<L is the elastic buckling load in the plane of bending. The alternative to this equation is to verify the accuracy of the second-order analysis by comparisons to known solutions for conditions similar to those in the structure. Examples of the errors one may encounter are discussed by LeMessurier (1977) and Deierlein & White (1998).

BEAM-COLUM INTERACTION EQUATIONS (SECTION HI OF THE 200S STANDARD)

80th the Effective Length (Section 86) and Direct Analysis (Appendix 6) stability procedures utilize the beam-column interaction equations of Chapter H, albeit with differences in how the required strengths (P, and M,) and the nominal compressive strength (p") are calculated. For reference in the later discussion, the interaction equations for members under combined axial compression and bending are briefly reviewed. For bi-symmetric beam-columns under combined axial compression and uniaxial bending, the 2005 Standard introduces a new interaction equation for checking out-of-plane (lateral-torsional) II1stability, which is separate from the check for in-plane (flexural buckling) instability. These separate equations are introduced since they provide more accurate predictions of in-plane and out-of-plane limit states, which tests and theory show are independent phenomena. The separate equations reduce the conservatism in the

Page J oj1 7

·;:1 • .::> '::J ,-.:0 -..J

current (I999 A1SC-LRFD) provisions, which combine the two limit state checks into one equation, by combining the most severe combinations of in-plane or out-of-plane limits for P ./;p" and M,/,pM".

Shown here for illustration are the interaction equations in LRFD fornlat for bi-symmetric beam-columns subjected to axial compression and uniaxial bending. For members subjected to compression and minor axis bending. only the in-plane check applies; whereas for columns under compression and major axis bending, both checks apply.

The limit state of in-plane flexural buckling is checked using the following equations, which have the same format as those in the 1999 A1SC-LRFD Specification:

p for -"- ;, 0.2

,p, p" (2a)

(2b)

where p" and M. are the required strengths, calculated from second-order analysis under the design loads; and p" and M" are Ole nominal compression and bending strengths, calculated in the plane of the frame. For the Effective Length method, p" is detemlined using the effective buckling length KL in the plane of bending, whereas in the proposed Direct Analysis method, p" is calculated using K= / (KL=L) in the plane of bending. For compact member sections, M" for the in-plane check is equal to M".

The out-of-plane lateral-torsional limit state is checked by the following equation:

(3)

Here the required strengths p. and M" are the same as for Eq. 2a and 2b, and p" and M" are calculated using the unbraced length in the out-of-plane direction. These out-of-plane nominal strengths would typically be evaluated on the same basis for the Effective Length and Direct Analysis methods.

EFFECTIVE LE GT H METHOD (SECTION 86 OF 2005 STANDARD)

The Effective Length (or critical load) approach for assessing member axial compressive strength has been used in various forms in the A1SC Specification since 1961. The provisions proposed for Section 86 of the 2005 Standard are essentially the same as those from the 3'd edition (1999) of the AISC-LRFD Specification, with the exception of a new minimum moment requirement. The approach is based on calculating effective column buckling lengths, KL, which have their basis in elastic (or inelastic) stability theory. The effective buckling length KL, or aiternatively the equivalent elastic column buckling load, P, = 1iEII(KL/, is used to calculate an axial compressive strength, p", through an empirical column curve that accounts for member geometric imperfections, yielding, and residual stresses. This column strength is then combined with the design moment strength, ,pM", and second-order member forces, p. and M., in the beam-column interaction equations.

Differences between the Effective Length and Direct Analysis approaches lie mainly in the in-plane check. Figure la shows a plot of the in-plane interaction equation for the Effective Length approach,

Page 4 of 17

where the anchor point on the vertical axis, p.KL , is determined using an effective buckling length factor. Also shown in this plot is the same interaction equation with the first term is based on the squash load. P,. The load-deformation response of a typical member, obtained from second-order spread-of-plasticity analysis and labeled "actual response," indicates the maximum axial force, p., that the member can sustain prior to the onset of instability. The load-deflection response of a second-order elastic analysIs, as would be done in design practice, is also shown. The "actual response" curve reveals larger moments than the second-order elastic curve due to the combined effects of partial yielding and geometric imperfections, which are not included in the second-order elastic analysis. The intersection of the second-order elastic curve with the p.KL interaction curve represents the design strength. The plots in Fig. 2a show how the effective length procedure has been calibrated to give a resultant axial strength, P" consistent with the actual response. For slender columns, accurate assessment of the effective length (and p.KL ) is critical to achieving an accurate solution.

While the effective length approach is calibrated to accurately predict the resultant member strength, one consequence of the procedure is that it under-estimates the actual internal moments under the factored loads (see Fig. I a). This is inconsequential for the beam-column (since the p. KL reduces the effective strength in the correct proportion), but the reduced moment can affect design of the beams and connections, which provide rotational restraint to the column. This is of greatest concern when the calculated moments are small and axial loads are large, where P-Ll moments induced by column out-ofplumb can be significant. As a safeguard for these cases, the Effective Length procedure in Section B6 includes a new minimum required moment strength for beams and connections, which restrain the column ends. This requirement is specified through the following equation (Eq. B6-3 in the July 2003 Ballot),

'fM. > O.OI'f.P.L (4)

where EMu is the minimum required strength, Pu is the required strength (axial compression force) in the columns being restrained, and L is the column length.

P P,

p ..

Pu

Pi P -'"R jWl P,

0 .. ': jWl elastic 2nd..order p.

elastic 2nd-order (D.A.) actual response actual response

(a) Mp M

(b) Mp M

Fig. I - Comparison of beam-column interaction checks for (a) the effective length approach and (b) direct analysis approach

DIRECT ANALYSIS METHOD (APPENDIX 6 TO 2005 STANDARD)

The Direct Analysis approach has been developed with the goal to more accurately model frame stability effects in analysis, and thereby. eliminate the need for calculating effective buckling length factors for

Page 5 of 17

columns. As summarized below, the new provIsions in Appendix 6 of the 2005 Standard Involve reducing the nominal elastic stiffness and applying a notional load to the frame. Some aspects of the proposed provisions are similar to so-called "notional load·' methods found in steel standards in olher countries, e.g., Canadian and Australian Standards and the Eurocode, however, many aspects of the proposed provisions are unique 10 the AISC Standard and address known shoncomings of conventional notional load approaches in other standards.

Like the Effective Length procedure, the Direct Analysis melhod begins with a basic requirement to calculale internal member forces using a second-order elaslic analysis. As will be shown later in the examples, the Direct Analysis method places a greater reliance on the second-order analysis (primarily in the accurate calculation of second-order moments, M.), and for this reason, the method stipulales requirements to ensure accuracy of the second order analysis . Analysis rigor is most important where second-order amplifications are large, one measure of which is given by the ratio of member aXial compression forces to their elastic buckling strengths (see Eq. I). Two additional requirements for Direct Analysis are as follows:

• A notional load of N, = 0.002 Y, is to be applied in combInalion with other factored loads. where N, is the notional lateral load applied at floor i and Y, IS the gravity load (from strength load combinations) acting at floor i . The notional load is applied to represent the destabilizing effect of a geometric imperfections and olher effects (yielding. non-ideal boundary and loading condItIons, elc.). The notional load magnitude of 0.002 corresponds to a frame out-of-plumb equal to HlSOO (where H is the story height).

• The nominal elastic flexural stiffness assumed in the second-order elastic analysis is equal 10

0.8tEI , where t is calculated as follows:

For members where p. S O.SPy: r = I For members where p. > O.SP, : r = 4[P/ P, (J -P/ P,)}

Alternatively, where p. > 0.5P, for any members in the frame. r = I provided that an additional notional load of N, = 0.001 Y, is applied to the frame.

There are two reasons for imposing the reduced stiffness for analysis. For frames with slender members, where the limit state is governed by elastic stability, the 0.8 factor on stiffness results in a system deSign strength equal to 0.8 times the elastic stability limit. This is roughly equivalent to the margin of safety implied by design of slender columns by the effective length procedure where the design strength ;p. 0.9(0.877)P, = 0.79P. where P, is the elastic critical load, 0.9 is the specified resistance factor, and 0.877 is a reduction factor in the column curve equation. For frames with inlermediate or stocky columns, the 0.8t factor reduces the stiffness to account for inelastic softening prior to the members reaching their design strength. The t is similar to the inelastic stiffness reduction factor implied in the column curve 10

account for loss of stiffness under high compression loads (P. > 0.5P,). and the 0.8 factor accounts for additional softening under combined axial compression and bending. It is a fonuitous coincidence that the reductions coefficients for the slender and stocky columns are close enough, such that the single reductIOn faclor ofO.8t works over the full range of slenderness.

The reduced stiffness and notional load requirements only penain 10 analysis of the strength lImit stale. and they do not apply to analysis of other serviceability conditions for excessive deflections, vibration , etc. For ease of application in design practice, the reduction on EI can be applied by modifying E In Ihe analysis; however, in doing so, one should consider whether the possible side-effects of reducing EA. Moreover, for computer programs that do semi-automated design, one should be sure that the reduced E IS

Page 6 of 17

'. ,;:p ,;:p

only applied for the second-order analysis, The elastic modulus should not be reduced in design equations, which involve E to evaluate the design strength (e.g., M, for laterally unbraced beams).

As shown in Fig. Ib, the net effect of modifying the analysis in the manner just described is to amplify the second-order moments to be closer to the actual internal moments in the member. It is for this reason that the beam-column interaction for in-plane flexural buckling is checked using an ax.ial strength P, calculated from the column curve using the actual unbraced member length L, i.e., with K ~ I . In fact, arguments have been made to use p.~ p). in the interaction equation. but this would require recalibration of the analysis adjustments, including additional adjustments to account for member out-of-straightness (sweep). After considering alternative strategies, TC 10 decided to use the proposed method (with p. based on L) as a pragmatic and conservative approach for practical design.

CANTILEVER EXAMPLE

To illustrate an application of the two stability design methods, consider the design of the cantilever beamcolumn shown in Fig. 2. The cantilever is subjected to the vertical and proportional horizontal load shown, such that the design is controlled by the combined p . and M. at the base of the column. Maximum strengths are calculated for three different column lengths, with slenderness ratios of Ur ~ 20, 40 and 60 (equivalent to KUr ~ 40, 80, and (20). Bending is about the major axis, and the column has full out-of-plane (lateral) restraint. The design checks are based on the in-plane interaction check (Eq. 2ab). ate that the checks were made using a resistance factor of 'A~0. 85

in compression (consistent with the 1999 AJSC-LRFD Specification), so the results would be slightly different if made with the revised value of IA~0. 9 as proposed for Chapter E of the 2005 Standard.

Shown Fig. 3a-c are plots of the axial load versus moment at the column base for the three column lengths, deternlined according to the Direct Analysis (DA) and Effective Length (EL) methods. Notice that the internal moments Mil increase much faster with PII for the Direct Analysis method, due to the reduced stiffness (0.8tEI) and added notional loads. Most of the stiffness adjustment is due to the 0.8 factor, since t only affects the column with Ur ~ 20 (Fig. 4a,) where the maximum load p. > 0.5P, ~ 440 kips. Overlaid on these force-point traces are the beam-column strength interaction diagrams, where the Pn

anchor point for the Effective Length method P,,KL is based on KL~2L

oJ

0 It)

.: Cl r:i <0 )(

0 ..... ~

and for the Direct Analysis method P,L is based on L. Fig. 2 - Canti lever Example

The calculated strengths, as determined by the two methods, are sununarized in Table I in tenns of the max.imum vertical load p. (shown in bold). Net strengths for the two methods are within 10%, even though the interim results are quite different. For example, as shown in Figs. 3a-c and summarized in Table 1, the maximum internal moments at the strength limit point are much larger for the Direct Analysis method; whereas the P ,/¢P" ratios, which indicated the relative significance of the axial load and moment terms in the governing interaction equation, are consistently larger for the Effective Length method. Moreover, the P ,/¢>P, ratios for the Effective Length procedure do not change much with increasingly slenderness, because this procedure relies to a much greater degree on capturing stability effects in the P, term. Conversely, in the Direct Analysis procedure the P '/¢P, contribution decreases and the moment term dominates the solution for cases with increasing slenderness.

Page 7 of 17

800

700

600

_ 500 a. ~ 400

" D. 300

200

100

0 0

700

600

500

E: 400 ~

" 300 D.

200

100

0 0

700

600

500 J

E: 400 ~

" 300 D.

200

100

0 0

1000

1000

(e)

1000

DALIr=20

.....-ELLIr·20

Interaction PnKL

~ Interaction PnL

2000

Mu (k-in)

3000

DALir='O

EL Lir='O

Interaction PnKl

_Interaction PnL

2000

Mu (k-in) 3000

DALir = 60

-M- EL Lir = 60

Interaction PnKL

Interaction PnL

2000

!tIu (k-in) 300D

Fig. 3 - Comparison of P-M interaction curves for cantilever column example (a) short Ur = 20, (b) medium Ur = 40, (c) long Ur = 60

Page 8 of 17

. .0 '.J

a e -T bl I R esu ts or anti ever ~ C '1 CI o umn E xam~le [frectin Len th Method Direc t Analys is "Iethod

Ur P.(kips) M.(k-ill) MI M, P/ (I', P.(kips) M,lk-iII) MI M, P / (1', 20 562 580 1.17 0.85 578 777 1.27 0.80 40 345 988 1.63 0.74 371 1680 2.16 0.56 60 193 1007 1.98 0.74 213 2376 3.53 0.37

This comparison highlighlS Ihe pros and cons of each method. Compared to Direct Analysis procedure, the Effective Length method has the advantage of being less sensitive to the accuracy of the second-order analysis. On the other hand, the method requires calculation of effective column buckling lengths (KL), which can be difficult for complicated structures. Direct Analysis eliminates the need to calculate effective buckling lengths and provides more accurate measures of the true second-order moments. This latter point is important for the design of members and connections, which restrain the beam-column. For example, in the cantilever column example, the base moments from the Direct Analysis procedure take into account initial out-or-plumb and inelastic second-order effects, which are not captured in the Effective Length procedure. Referring to the second-order moments reported in Table I, the difference in moments between the two methods can be quite large. Subject to the assumed geometric imperfections (out-of-plumb) and residual stresses, validation studies have shown that the momenlS calculated by the Direct Analysis procedure are generally conservative and closer to the true values. Observations of the type described here about the underestimation of design moments the Effective Length method, led to the new minimum moment requirement (Eq. 4) for the Effective Length method. One should recognize, however, that this minimum does not address cases such as shown in this cantilever example, where the calculated moments in the Effective Length procedure are above the minimum of O.O/PL, but still less than the actual values, which are calculated more accurately by the Direct Analysis procedure.

VA LIDATION STUDIES

Over the course of developing the proposed stability provisions, hundreds of validation analyses have been investigated by members of the SSRC-AISC Ad Hoc Committee. Some of the early investigations (e.g., Maleck 200 I, Maleck and White 2003) he lped guide development of the provisions, and two recent papers by Maleck and White (2002) and Martinez-Garcia (2002) provide selected case studies to validate the final version of the proposed design methods. These two studies investigated twenty-five frame configurations under multiple load cases, representing several hundred analyses with about 150 comparison points between the two design approaches and refined nonlinear analyses. These studies focus on the limit state of combined axial load and bending in the beam-colunms and do not specifically address design checks in restraining beams and connections.

Examples of the frame configurations considered in the benchmark srudies by Maleck and White (2002) are shown in Fig. 4. These two-colunm portal frames and individual colunm structures provide rigorous test cases of non-redundant systems of varying slenderness, levels of axial compression, and leaning column effects. Other multi-story and multi-bay frames investigated by Martinez-Garcia (2002) embody attributes of realistic structures that pose particular challenges in evaluating stability, three of which are presented in the next session of JIIustrative Examples. The problems investigated for the benchmark studies are ones where second-order effects are large and where errors between the stability design methods and more exact methods are accentuated. In this sense, these benchmark studies represent extreme cases, which tend to exaggerate the differences one would typically encounter in design practice.

Page 9 of 17

;0 .... .)

Symmetric Frame

Leaned-Column Frame

P w_+ 60

B. Eb• Ib ,Lb

Eel Ie ,Le , , ,

Lb » Lc

Pinned-Pinned Beam -Column it

~ , B. \

2W -;t

2P

+

f 2L,

(EI /L), = 0 and 3 (EI/L)b

(Llr), = 40

601L,= 0.002

BoiL, = 0 and 0.001

f" = 0 and 0.3F,

(EIIL), ---= 0 and 1 (EI/L)b

(Llr), = 40

601L,= 0.002

BoiL, = 0 and 0.001

f" = 0 and 0.3F,

(2L1r), = 80

BoI2L, = 0 and 0.001

f" = 0 and 0.3F,

straight line and sine curve

Fig. 4 - Test structures used for validation study (Maleck 2001)

Detailed analysis solutions based on second-order spread-of-plasticity analyses are used as benchmarks against which the proposed design methods were validated. These benchmark solutions incorporate the effects of gradual yielding, initial geometric imperfections, and residual stresses, as outlined previously in the section of this report on Behavioral Effects. Thus, they represent the state-of-art in simulating inelastic stability of beam-columns and frames . Material properties (E and Fy) in the spread-of-plasticity analyses were reduced using a resistance factor of 0.9, such that the maximum strength calculated in these analyses corresponds to the structural "design strength" - as opposed to a "nominal strength".

Overall, the two studies (Maleck and White 2002 and Martinez-Garcia 2002) confirm that both the Effective Length and Direct Analysis methods are sufficiently accurate (relative to current methods) for design and that the errors (relative to the spread-of-plasticity solutions) are comparable for the two methods. Maleck and White report that on average the two approaches give strengths within 2% to 7% of those obtained by refined analyses. The maximum discrepancies they observed for the Direct Analysis approach, relative to the refined analyses, range from - 6% (unconservative) to + 13% (conservative) for members subjected to strong-axis bending and - 13% to + 15% for members subjected to weak-axis bending. For the Effective Length approach, the maximum discrepancies range from - 8% to + 18% for strong-axis bending and - 17% to + 17% for weak-axis bending. These errors are based on design checks made with rigorous second-order elastic analyses. Maleck and White caution that the Direct Analysis design checks based on approximate P-6 analyses can be up to -23% (unconservative) for members subjected to weak axis bending. This is an example of why the Direct Analysis provisions specifY the need for a rigrous analysis when second-order effects are large . Maleck and White further note that for frames where Pu < 0.15 (71' ElfL' ), the maximum unconservative errors associated with approximate P-

Page 100JI7

'.0 J>.

analyses for the Direct Analysis approach are limited to the maximum errors present in existing Effective Length procedures.

ILLUSTRATIVE EXAMPLES

Three example problems are presented next to illustrate practical application of the proposed stability design methods. The first two examples are [rames with heavy gravity loads and large second-order amplification factors. The third example is a stiffer six-story frame, which is more representative of multi-story building frames. Each example includes a comparison of results from the Effective Length and Direct Analysis approaches and more rigorous spread-of-plasticity solutions. The spread-of-plasticity solutions have been independently reported by Maleck (2001) and Martinez-Garcia (2002), and the design solutions have been prepared by multiple members of A1SC TC 10 and the A1SC-SS RC Ad Hoc Committee. All design checks are based on the LRFD approach and load combinations. Resistance factors used in the design checks are ¢.=0.9 for bending and ¢,=0.85 for compression, the latter of which is slightly smaller than the proposed change to ¢,=0.9 in Chapter E of the 2005 Standard.

Low-Rise Industrial Example

The first example, see Figure 5, is a framing bent from a large floor plan single story industrial building, such as an automobile plant. With heavy material handling equipment hung from the roof and a small wind exposure, such structures are dominated by gravity loads wilh large second-order effects (Springfield, 199 1). Loading shown in Figure 5 represents an eleven bay configuration with ten leaning columns (only two of which are shown) and two lateral-load resisting columns. The concentrated load P has a tributary roof area of 35 ft x 35 ft , and the wind load W = 5.12 kips.

The member sizes satisfy a drift limit of Hl400 for the service load wind of 0.7W, and the design strength of the frame exceeds the minimum requirement of the LRFD strength load combinations. Based on refined spread-of-plasticiry analyses, the frame has a design strength ratio 17% larger than the required strength for gravity (¢A.UQ. /.6L= 1.I 7, where ~=0.9) and 20% larger than required under gravity plus wind (¢A/m .o.5L /.6,.,= 1.20). Using the equation B5-5 (from the July 2002 ballot of Chapter B for the 2005 Standard), the second-order amplification factor under design gravity loads is B,= 2.41. Under the design gravity plus wind loading combination, B,= 1.74.

4P 4P Frame spadng = 35'·0"

W-) W27 x 84 W27 x 84

Fy = 50 ksi '" 9 ..,. x E = 29,000 ksi 00 0 ~ ~

~

J \. ,

l 3 @ 35'-0" = 105'-0"

DL 80 psI Load Combinations: '1

LL 40 psI 1.2DL + 1.6LL Wind 16.25 psI 1.2DL + 0.5LL + 1.6WL

Figure 5 - Single-Story Industrial Building

Page II ofl7

• .£.' ' . .,

---------------------

Axial column forces and maximum moments under the factored load combinations are summarized in Table 2. The Effective Length results are from a second-order analysis of a model based on the ideal geometry of the frame under the factored load combinations (no geometric imperfections or notional loads are introduced). The Direct Analysis results incorporate initial geometric imperfections through the notional load of 0.2% times the factored gravity loads (1.20 + 1.6L for the first combination and 1.20 + 0.5L for the second combination); and stiffness degradation is incorporated by reducing the flexural stiffness of all framing members by to 0.8EI. Since the axial load ratio PIP, <0.5, no additional t-factor stiffness adjustments are required. The "spread of plasticity" results are from a second-order inelastic analysis, which models gradual yielding through the member cross sections and along their length due to the combined effects of thermal residual stresses and the applied loads.

Table 2: Member Effects Under Factored Load Combinations ~ L R' I d . I E I or ow- Ise n ustna -xample

~ l em bcr Analysis/Design l\ lelhod

Load Case Check SI)read of Effec ti ve Direct

Plasticitv Len~ lh Allah'sis p,. (kip) 215 216 218

1.2D+1.6L M"" (k-in) 930 407 1220 Mbm (k-in) 8660 8410 8690

P"" (kip) 154 158 160 1.2D+0.5L+1.6W M" (k-in) 1310 1040 1550

Mbm (k-in) 6490 6360 6630

Referring to Table 2, the Effective Length and Direct Analysis methods both predict the maximum beam moments and axial column forces within about 4% of those from the spread-of-plasticity analysis. On the other hand, there are significant differences in the column moments, particularly for the gravity load case (1.2D + 1.6L). The Direct Analysis method predicts the column moments on average about 25% higher than the spread-of-plasticity solution, and the Effective Length method predicts column moments on average about 40% smaller than the spread-of-plasticity solution. These differences are also reflected in the calculated displacements. This small moments calculated according to the Effective Length method illustrate the need for the newly proposed minimum connection moment requirement (Eq. 4, 'f.M. > 0.0I''i.P"L). Without this minimum requirement, the comlection would be under-designed for the secondorder moment induced by the combined effects of gravity load and column out-of-plumb.

Using the member forces from Table 2, the columns are checked using the interaction formula for in-plane or out-of-plane (torsional flexural) failure, and the resulting interaction ratios are summarized in Table 3. For the Effective Length method, the in-plane checks are based on a column strength of ¢P ~,.Kt = 236 kips, obtained with an effective length factor of K = 2.3 using Eq. C-C2-6 of A1SC (1999). In-plane checks for the modified stiffness and notional load methods are based on ¢p ... t = 511 kips, and out-of-plane checks

Table 3: Interaction Values for Low-Rise Industrial Example

Analysis/ Design 1\ l ethod

Load Case Member Check Effeclive Direcl

Length Analvsis

1.2D+1.6L In-plane 1.05 0.83

Out-of-plane 0.62 0.81

1.2D+0.5L+I.3W In-plane 1.0 I 0.82

Out-of-pJanc 0.58 0.77

Page 120[17

are all based on ;p .. L ~ 361 kips. The column design moment is ;Mp ~ 2718 k-in. In-plane interaction is checked using Eqs. 2a & 2b, and the out-of-plane check is made using Eq. 3.

Referring to Table 3, both the Effective Length and Direct Analysis checks are governed by the in-plane strength (shown shaded). The Effective Length method is more conservative, as evidenced by a larger interaction value as compared to the direct analysis method. The in-plane checks can be compared to inelastic limit load ratios of ;),."0 161.~ 1.I7 and ;),."D.0.J/.>1.61V ~ 1.20 , obtained from the spread-ofplasticity analyses. The inverse of these limits (0.85 and 0.84 for gravity and gravity+wind, respectively) help to gauge the conservatism in the methods, where larger interaction checks would be conservative and smaller checks unconservative. Compared to these values (0.85 and 0.84) the in-plane checks for the Effective Length method about 15% conservative, whereas the Direct Analysis results appear slightly unconservative (e.g., 0.83 < 0.85 and 0.82 < 0.84). However, since the member forces vary nonlinearly with load (due to second order effects), this simple comparison is approximate and a more accurate comparison would be obtained by scaling up the loads in the Direct Analysis to the point that the in-plane interaction check is equal to 1.0. Scaling the loads in this way results in a limit load of~AI.2"' 1.6L~ 1.I0 for the Direct Analysis which is about 6% smaller (conservative) as compared to the in-plane limit from the spread-of-plasticity solution. Thus, this case demonstrates that the Direct Analysis is conservative (safe) and provides the potential for a more efficient design as compared to the Effective Length method.

Exa mple Co nnection Design for Effective Length Method: For the gravity load case, the minimum beam-column connection strength provision of the Effective Length procedure (Eq. 4) provides for a minimum required connection strength of 1:M, > 2330 k-in . This is calculated using the vertical load in one lateral load resisting column and one of the leaning columns (a total of EP, ~ 1078 kips). Comparing this to the more exact required column moment from the spread-ofplasticity solution (M, ~ 930 k-in) indicates that the minimum required by the Effective Length method is quite conservative in this case. The required strength using the Direct Analysis method would be 1220 k-in. To help gauge the impact of these provisions, a connection designed for the Effective Length method moment of 2330 k-in is shown in Fig. 6. This connection would require eight - I inch diameter bolts, which is not excessive for the connected members.

Grain Storage Bin

_.

Figure 6: Example Connection Design (8-1" dia. bolts)

The second example is the support rack for a grain storage bin with the dimensions and loading shown in Figure 7. This is a case where calculation of the column effective lengths is not obvious, and where the Direct Analysis method offers a clear benefit. Colunms are assumed to be braced out-of-plane and the cross-beams and bracing are pin-connected to the colunms. For the diagonal bracing, one-inch diameter round bars are assumed. Using an elastic critical load analysis, the second-order amplification factors are B, ~ 2.75 and B, ~ 2.20 for the gravity and wind load combinations, respectively. The spread-of-plasticity analyses predict inelastic limit load ratios of ;),.I4G~ 1.13 and ;),.I1G- 16W ~ 1.07.

As in the previous example, results for the Effective Length method are calculated for the ideal geometry and stiffness; whereas the Direct Analysis method is based on a reduced stiffness (0.8EI) and with notional loads applied in combination with the design loads. Like the previous example, no additional t

adjustment of stiffness properties is required for Direct Analysis s ince the axial load ratio PIP, < 0.5.

Page 13 oj 17

'.0 --J

w

M X ~

WI2x26

"' iI"----""

H 12'-0" H

~slgn Loading:

~ IV - \.75 kops (10 psQ G - 360 kips

Load Comb inations:

1;> 1.4G

~ 1.2G + 1.6W

Materials:

1;> Fy - SOksi

'" E - 29.000 ksi

Figure 7 - Grain Bin Support Frame

Maximum column forces and moments (required strengths) are summarized in Table 4 and the interaction checks are summarized in Table 5. As in the previous examples, there is not much difference in axial loads between the methods, but there are large variations in the calculated moments. This is particularly prevalent for the gravity load case, where the calculated column moments for the Effective Length method are essentially zero, and the moments for the Direct Analysis method are about 36% larger than those in the spread-of-plasticity analysis. Under the lateral load case, the differences are less, with moments for the Effective Length method about 24% less (unconservative) than the spread-of-plasticity results and those for Direct Analysis about 38% larger (conservative).

a e em er eets T bl 4 M b Ef~ ~ G or ram S torage III xample

Member AnalYsis/ Desig n Method Load Case

C heck Spread of Effec tive Direct Plastici ty L.n~th Analysis

P,.,,,,,Jkip) 233 247 249 lAG P,.""dk-in) 255 252 257

M (k-in) 161 2 220 P , (kip) 203 217 220

1.2G+\.6W P. " k-in) 224 225 230 M, (k-in) 380 288 526

T bl a e 5: Interactton Va ues or ram ~ G . S torage B' E 10 ~ xampJe

Member Analysis/Design Method

Load Case Check Effec ti ve Direcl

Length Anal\'sis

\.4GL Top Column 1.07 0.84 Bot. Column 1.04 0.85

1.2GL+1.6W Top Column 1.12 0.96 Bol. Column 1.11 0.99

Page 14 of 17

• .:, .::> .:::> ..... . ..:> ,~o

The interaction checks, shown in Table 5, are based on the following in-plane column strengths: critical load method ;p.KL .. op = 232 kips (K = 2.4), ;P.K~ .... = 243 kips (K = 2.9); and the direct analysis method, ;p.L .• "" = 355 kips, ;P.~.bo' = 366 kips. The K factors for the Effective Length procedure are based on an elastic critical load analysis of the structure under gravity loads. The results in Table 5 show that strength interaction checks based on the Effective Length method are roughly 20% more conservative than the Direct Analysis method. Using the spread-of-plasticity analysis as a benchmark of the actual behavior, interaction values larger than 0.89 (for lAG) and 0.93 (for 1.2G+I.6W) are conservative. The Effective Length interaction values (1.07 and 1.12) exceed these and are about 20% conservative. The Direct Analysis method is slightly conservative for the gravity plus wind case (0.99 > 0.89) and appears slightly unconservative for the gravity load case (0.85 < 0.93). However, as mentioned in the industrial frame example these linear comparisons are only approximations. When the gravity loads are scaled in the direct analysis to provide an interaction value of 1.0 for the bottom column, the limit load ratio was ;)., ' Ii = 0.99, which exceeds the value of ;)."G= 1.07 from the spread-of-plasticity solution. This indicates that the Direct Analysis is, in fact, 9% lower (conservative) as compared to the spread of plasticity solution.

Multi-story Frame Example

The final example is the multi-story frame shown in Figure 8. One load case is investigated (J .OG + I .OW, where the specified loads are already factored), and member forces and interaction checks are presented for the three columns in the first story. Unlike the previous examples , this frame is fairly stiff with B1 = 1.10 for the first story. The second-order spread-of-plasticity analysis predicts an inelastic design strength ratio of ;)., ' Ii = 1.06 for this frame , which combined with the low B 1 indicates that it is dominated more by yielding than second-order effects. The center columns of the first two stories have high axial forces and are subject to the t - factor adjustment in the Direct Analysis method. As discussed earlier, when PIP, > 0.5 for any column, the t - factor adjustments can be used or an additional notional load of 0.001 Y, can be used. In this example, both approaches are presented and compared.

ell

IPf2"O

~ w x IPOOO

1ii ffi x

JPEJOO

~ N

ffi x

IPf330

~ ffi x IPOW

~ m w x IPf400

(il :;: CI2 w x

2 0 6.Om - 12m

CJ3

l.IwIl Gravity: "9.1 kN/m (11oor)

31.7 kN/m (root) Wind: 20 ..... kN (store 1 • 5)

10.23 kN (root)

F, _ 235 N/ mml' E • 205 kN/mml

Figure 8 - Multistory frame

Page 150[1 7

The first floor column forces, summarized in Table 6, reveal that differences between the three methods are much smaller than in the previous examples. This follows from the fact that the second-order amplification is smaller in this example, which is more typical of most multi-story frames than the prior examples.

T bl 6 M b Ef~ a e em er eelS or F u t,story rame E xample

Location and Ana l"sis/Desie" Method

[(ferr Spread of EffeClivt Direct Direct

Analysis "ilh Analysis with t (t.OG+ t.OW) Plasticity Length Notional Load Reduction P,,, (kN) 683 672 659 662 P " (kN) 1720 1770 t770 1770 P,,, (kN) 92t 884 897 894

M ,(kN-ml 67 48 58 59 M,,(kN-m) t t5 11 8 143 135 M u (kN·rn) 99 87 96 96

Results of the beam-column interaction checks (Table 7) show that the Effective Length method is slightly less conservative than the Direct Analysis method, which is in contrast to the previous two examples where the opposite was true. Based on the AlSC (1999) alignment charts, the effective buckling length factors for the first story columns are K = 1.35, assuming the AlSC suggested value of G = 1.0 for the foundation support . Accordingly, the in-plane interaction checks for the Effective Length procedure are based on design compression strengths of $P,KL" = $P,KL IJ = 1580 kN and $P,KL" - 2140 kN . The inplane Direct Analysis checks (with K = J) are based design strengths of ~P'L " - ~P'L IJ - 1680 kN and ,P,L 12 = 2230 kN. All out-of-plane checks are based on K = I, with ~P'L" = ~P'L IJ = 1460 kN and $P'L 12 = 2010 kN; and moment strengths of ~Mp', = ~MplJ = 175 kN-m and ~Mp" - 275 kN-m are used throughout. As summarized in Table 7, the resulting interaction checks were all close, with the Direct Analysis solutions about 2% to 3% conservative, relative to the Effective Length method . With reciprocal of the inelastic limit load factor equal to 0.94, the average interaction values ranging from 0.94 to 0.99 indicate that all of the stability design methods are conservative in this ca e.

Ta bl e 7: Interachon VI a ues ~ MI ' F or u ttstory ' rame E xampte

Location Anahsis/Design Method

(t.OGL+t.OW) Eerective Direct Analysis Direct Anll)sis Length with Notional Load with t Reduction

Ctl m·olane 0.67 0.69 0.69 CII - oUI-of-pt'ne 0.54 0.56 0.57 Ct2 In-plane I.2t 1.25 1.23 Ct2 OUI-or£~ane 1.06 t.t5 1.12 cn - In-ptane 1.00 1.02 1.02 Cl3 out-of-olanc 0.85 0.92 0.9t

A VCT3SlC in-olanc 0.96 0.99 0.98

Page 16 of 17

REFERENCES

A1SC (1999), Load alld Resistallce Factor Specificatioll for Structural Steel Buildillgs, Arner. Insl. of Steel Constr., Chicago, IL.

ASCE (1997), "Effective Length and Notional Load Approaches for Assessing Frame Stability", ASCE, New York, NY.

A1SC (2000), Code ofStalldard Practicefor Steel Bllildillgs alld Bridges, American Institute of Steel Construction, Chicago, IL.

Bimstiel, C., Imaod, 1. S. B. (1980), "Factors Influencing Frame Stability," JI. of the Struct. Div .. ASCE, I 06(ST2), 491-504.

Deieriein, G.G., White, D. W. (1988), "Chapter 16 - Frame Stability," Gllide to Stability Desigll Criteria for Metal Structures, Ed. T.V. Galambos, Fifth ed., Wiley, 1998.

Deierlein, Hajjar, Yura, White, Baker, "Proposed new provisions for frame stability using second-order analysis", Proceedillgs 0fSSRC 2002 Allllllal Meetillg, Seattle, WA

LeMessurier, W. 1., (1977), "A Practical Method of Second-Order Analysis - Part 2 Rigid Frames," Ellgr. JI ., A1SC, 14(2), 49-67.

Maleck, A. E. & White, D. W. (2002), "Direct Analysis Approach for the Assessment of Frame Stability: Verification Studies," SSRC Annual Technical Meeting, Baltimore, MD, SSRC, 17 pgs.

Maleck, A. E. & White, D. W. (2003), "Alternative Approaches for Elastic Analysis and Design of Steel Frames: Part I Overview, PartH Verification Studies, " JI. ofStruct. Ellgrg., ASCE, submitted for review.

Martinez-Garcia, J.M. (2002), "Benchmark Studies to Evaluate New Provisions for Frame Stability Using Second-Order Analysis," M.S. Tllesis, supervised by R.D. Ziemian, Bucknel1 University, 241 pgs.

McGuire, W. (1992), "Computer-aided analysis," COllst.1 Steel Desigll- Allillti. Gllide, Dowling, et al. (eds.), Elsevier, NY 915-932.

Springfield, J. (1991), "Limits on Second-Order Elastic Analysis," Proc. SSRC Annu. Tech. Sess., SSRC, Bethlehem, PA, 89-99.

White, D. W., Chen. W.-F. (eds.) (1993), Plastic Hillge Based Methodsfor Adv. Allalysis alld Desigll of Steel Frames, SSRC.

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