Overview of Direct Analysis Method of Design for StabilityBy: Ryan BrothersonNeedham Consulting Engineerswww.needhamassoc.com913-385-5300

Presentation OverviewChanges & requirements of AISC 360 specificationOverview of current methods & limitationsOverview of Direct analysis methodDiscussion 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-arrangement2005Chapter CEffective Length Method (K factors)First Order analysisAppendix 7Direct Analysis Method

2010Chapter CDirect Analysis MethodAppendix 7Effective Length Method (K factors)First Order Analysis

Reference to Stability Requirements

AISC 360-10 Chapter CSection C1 - GENERAL STABILITY REQUIREMENTSStability 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- and P- 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 DesignThe 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 DesignThe 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 effectsP- effect. Effect of loads acting on the deflected shape of a member between joints or nodes.P- 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 issuesGeometric imperfectionsBeam sweep, camber, out of plumb, etc.Code of standard practice allows H/500 for column out of plumbResidual stressesUneven cooling of hot rolled shapesUncertainty in strength and stiffnessVariability in material properties

Strength vs. Resistance

Effective Length MethodCommonly called the K factor methodMost common method used at this timeIntroduced in 1963 (to some resistance)The K factor is a modification factor applied to the length of columns with defined restraint conditionsIt was used to account for 2nd order effects, geometric imperfections, stiffness reductions, and uncertainties.

Effective Length MethodLimitations 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 situationThe 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 methodEasy to understand & versatileThe 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 MethodProcedure is as follows:Perform analysis at strength levelApply notional loads at each floor levelModify stiffness of all members contributing to lateral stability of structurePerform 2nd order analysis for all load combinations to determine required strengthsDetermine available strengths of all members based on Chapters D through KVerify available strength is greater than required strength

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

Required StrengthsAnalysis 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- & P- effects. Permissible to ignore P- under following conditions.Columns are nominally verticalRatio of 2nd order to 1st order drift < 1.7One 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 ImperfectionsPermissible to account for imperfections by direct modeling of column out of plumbness, etc.More common to account for the imperfections with Notional LoadsNotional load is lateral load at each level as followsNi = 0.002**YiAlpha = 1.0 at LRFD & 1.6 at ASDYi is gravity load at level i0.002 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 StiffnessMembers 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 b reduction on the flexural stiffness where b is: 1.0 when Pr/Py 0.54(Pr/Py )[1- (Pr/Py)] otherwiseMay use b = 1.0 if an additional 0.001 notional load is added to all load cases

Adjustment to Stiffness contReduced stiffness (EI* = 0.8bEI 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, Pn = 0.9(0.877Pe) = 0.79Pe.For frames with intermediate or stocky columns, the 0.8b factor reduces the stiffness to account for inelastic softening prior to the members reaching their design strength. The b factor is similar to the inelastic stiffness reduction factor implied in the column curve to account for loss of stiffness under high compression loads (Pr > 0.5Py ), and the 0.8 factor accounts for additional softening under combined axial compression and bending.

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

Commentary to Chapter CRigorous second-order analyses are those that accurately model all significant second-order effects.Somebut not all, and possibly not even mostmodern commercial computer programs are capable of performing a rigorous second-order analysis, although this should be verified by the user for each particular program.

STAADDirect Analysis is available effective STAAD.Pro 2007See section 5.37.5 of technical reference manualGeneral Format PERFORM DIRECT ANALYSIS..(See sec. 5.37.5 and STAAD output)Use command in place of Perform Analysis or Pdelta ConvergeCommand directs the program to:Reduce axial & flexural stiffness as required by codeSolve static case w/ notional loads

STAAD Notional LoadDirect analysis must use Repeat Load or Reference Load specificationNotional loads need to be defined per section 5.31.7 and 5.32.14 of the reference manualSTAAD derives a lateral load from an existing vertical load caseExample:Load 1 Dead LoadJoint Load, Member Load, etc.Load 2 Dead Notional Load1 X 0.002 [Load case Direction Ratio]

STAAD P-deltaSTAAD Default for P-delta will include both P- & P-.Must be used with REPEAT LOAD commandBenchmark 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 AnalysisRAM version 14.05 implements the Direct Analysis method using a 2nd Order by Amplified 1st order elastic analysisThis is not necessarily considered a rigorous 2nd order analysisUses the B1 & B2 method per Appendix 8Allowed by section C.2.1(3)

RAM 2nd Order AnalysisFrom 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 AnalysisNotional 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 b, so the engineer either specifies 1.0 or some other value.Although technically b 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-DeltaRAM uses two methods to approximate P-Delta effectsBoth are based on the Geometric Stiffness MethodSmall, assumed deflections are used to create a Geometric Stiffness matrixThis matrix modifies the building stiffness matrix onceAccounts for P- only.

RAM P-DeltaNon-iterative P-Delta MethodUsed for Rigid diaphragmsPreliminary P-Delta AnalysisUsed for Semi-rigid diaphragms

SummaryDirect analysis is the preferred stability method of AISCDirect analysis directly accounts for the five issues contributing to stabilityDirect analysis appears relatively easy to implement in STAAD.RAM Frame uses approximate methods to account for stability

Questions?

Intro slide*