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Overview of Direct Analysis Method of Design for Stability By: Ryan Brotherson Needham Consulting Engineers

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

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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.

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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

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Reference to Stability Requirements

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AISC 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.

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2 nd 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.

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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

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Strength vs. Resistance

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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 2 nd order effects, geometric imperfections, stiffness reductions, and uncertainties.

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Effective Length Method Limitations of the method include: It cannot be used for stability sensitive structures where the ratio of 2 nd order to 1 st 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

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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 2 nd order to 1 st order effects not exceed 2.5 to limit a runaway instability.

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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 2 nd 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

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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

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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 2 nd order to 1 st 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 2 nd order analysis

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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 N i = 0.002*α*Y i Alpha = 1.0 at LRFD & 1.6 at ASD Yi is gravity load at level i is based on H/500 out of plumbness (AISC COSP) Notional loads are applied to gravity cases only when Ratio of 2 nd order to 1 st order drift < 1.7

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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 τ b reduction on the flexural stiffness where τ b is: 1.0 when αP r /P y ≤0.5 4(αP r /P y )[1- (αP r /P y )] otherwise May use τ b = 1.0 if an additional notional load is added to all load cases

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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.

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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.

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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.

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STAAD Direct Analysis is available effective STAAD.Pro 2007 See section of technical reference manual General Format PERFORM DIRECT ANALYSIS……..(See sec 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

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STAAD Notional Load Direct analysis must use Repeat Load or Reference Load specification Notional loads need to be defined per section and 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 [Load case – Direction – Ratio]

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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 ) results were confirmed. Appears that STAAD meets a Rigorous 2 nd order analysis

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

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RAM 2 nd Order Analysis From Section 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.

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RAM 2 nd 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

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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.

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

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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

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Questions?

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