analysis and design for stability-slides.pdf

Allen Adams, P.E., S.E.

Analysis and Design for Stability

Allen Adams, P.E., S.E.Chief Structural Engineer, RAM Group at Bentley Systems, Inc.

Senior Product Manager, RAM Structural System

AISC Committee on Specifications

AISC 360 Task Committee 10 – Stability

AISC Committee on Manuals – Seismic Design Manual

ASCE Committee on Design of Steel Building Structures

Strength

Stiffness

Stability

1st – Order vs 2nd – Order Analysis

1st – Order Analysis 2nd – Order Analysis

Stability and 2nd-Order Effects

• P-D

• P-d

• Out-of-Plumbness

• Member Out-of-Straightness

• Residual Stresses

• P-D

• P-d

• P-D

• P-d

• P-D

• P-d

• P-D

• P-d

Out-of-Plumbness

• P-D

• P-d

Member Out-of-Straightness

• P-D

• P-d

• P-D

• P-d

These effects can be handled by:

• Amplify the member forces obtained from a 1st-order analysis

• Reduce the available strength (the calculated allowable capacity) of the member

• Include the effects directly in the model used in the analysis

• A combination of the above

• Amplified 1st-Order Elastic Analysis

– AISC B2 Factor

• 2nd-Order Analysis

– Iterative

– Geometric Stiffness Method

Iterative Geometric Stiffness

– AISC B1 Factor

– Iterative

– AISC B1 Factor

– Iterative

• Reduced Capacity

– AISC 9th ASD (1989)

Out-of-Plumbness

• Direct Modeling of the displaced nodes

Out-of-Plumbness

Requires four separate

models, one for each

axis direction.

Out-of-Plumbness

• Notional Loads

Notional Loads

Notional Loads are a portion of the vertical gravity load applied horizontally.

Notional Loads

• Dead, Live and Roof

• Applied in each of the four directions (+/- X-axis, +/- Y-axis)

• AISC 360-10 specifies notional loads of 0.002 or 0.003 times the gravity loads

Load Combinations with Notional Loads

When B2 < 1.5:

1.4 D + 1.4 ND1

1.2 D + 1.2 ND1 + 1.6 L + 1.6 NL1

1.2 D + 1.6 W1

1.2 D + 0.5 Lp + 1.6 W1

When B2 > 1.5:

1.4 D + 1.4 ND1

1.2 D + 1.2 ND1 + 1.6 Lp + 1.6 NL1

1.2 D + 1.2 ND1 + 1.6 W1

1.2 D + 1.2 ND1 + 0.5 Lp + 1.2 NL1 + 1.6 W1

In AISC 360, if B2 is greater than 1.5, the notional loads must be included with all

combinations, otherwise they need only be included with the gravity load combinations:

Out-of-Plumbness

• Notional Loads

• Reduce Capacity by using the Effective Length (KL) in the strength equations

• Reduced Capacity from Strength equations that are calibrated to include effects

• Used reduced stiffnesses in the analysis

Residual Stresses

AISC LRFD 3rd

(1999)

Residual Stresses

• Used reduced stiffnesses in the analysis

• P-D

• P-d

Leaning Columns

Methods of 2nd Order Analysis

• Amplified 1st Order Elastic Analysis

– e.g., B1 and B2 Factors

• Iterative

• Geometric Stiffness Method (for P-D)

– e.g, B1 and B2 Factors

• Iterative

Limitations of the B2 method

• It was developed based on planar frames. It does not account for rotational effects of the 3-dimensional structure.

• May not be accurate for complex buildings.

Advantages of the B2 method

• Straightforward. The P-D affects are quantified (B2 is the ratio of the drift including P-D effects over the drift excluding P-D effects) so you can readily see the impact on design forces.

• Considers the effects of “Leaning Columns” if done correctly.

• Principle of Superposition applies. This allows Load Cases to be analyzed and then the results combined in combinations.

Principle of Superposition

Principle of Superposition says:

(A)results + (B)results + (C)results = (A + B + C)results

For example:

(Dead)results + (Live)results + (Seismic)results = (Dead + Live + Seismic)results

However, the 2nd order effects are nonlinear, so without accounting for those:

(Dead)results + (Live)results + (Seismic)results (Dead + Live + Seismic)results

It is possible, however, to modify the analysis results such that superposition can be used:

B2(Dead)results + B2(Live)results + B2(Seismic)results (Dead + Live + Seismic)results

Principle of Superposition applies when using the B2 method

The B2 method is a way of approximating a 2nd Order analysis by amplifying the results of a 1st Order analysis.

The B2 factors are calculated for each combination, based on the load case analysis results. This allows the analysis program to quickly analyze a small number of load cases (D, L, Rf, Wind, EQ, usually about 20 or so cases) and then comprehensively combine those results in the required load combinations (usually more than 160).

This is much more convenient for the user, it is quicker and it produces less analysis results to have to review.

• Iterative

Advantages of the Iterative P-delta method

• Theoretically most accurate of the three methods.

• Can be used for non-building structures.

Limitations of the Iterative P-delta method

• Live Load Reduction can not be easily applied (correctly). This can result in up to 60% error in Live Loads.

• Does not consider the effects of “Leaning Columns” unless they are included in the lateral analysis model. If not done properly the p-delta analysis could be off significantly.

• Requires that the Analysis be done on Load Combinations, not Load Cases. This can be very time-consuming and can produce an overwhelming amount of analysis data due to the large number of load combinations now required by Code.

• Impossible to do with Response Spectra?

• Iterative

Advantages of the Geometric Stiffness method (used by RAM Frame as an option)

• Live Load Reduction can be easily applied (correctly).

• Considers the effects of “Leaning Columns”.

• Principle of Superposition applies. This allows Load Cases to be analyzed and then the results combined in combinations.

• Valid with Response Spectra.

Principle of Superposition

Principle of Superposition says:

(A)results + (B)results + (C)results = (A + B + C)results

For example:

(Dead)results + (Live)results + (Seismic)results = (Dead + Live + Seismic)results

However, the 2nd order effects are nonlinear, so without accounting for that:

It is possible, however, to modify the analysis such that superposition can be used:

Principle of Superposition applies when using the Geometric Stiffness method

The Geometric Stiffness method is a way of modifying the analysis such that the 2nd Order analysis results can be obtained directly by performing a 1st Order analysis.

Theoretically the Geometric Stiffness method gives the same results as an iterative approach.

This allows RAM Frame to quickly analyze a small number of load cases (D, L, Rf, Wind, EQ, usually about 20 or so cases) and then comprehensively combine those results in the required load combinations (usually more than 160).

This is much more convenient for the user, it is quicker and it produces less analysis results to have to review.

• AISC 360-10 Specification for Structural Steel Buildings

• ACI 318-14 Building Code Requirements for Structural Concrete

• Eurocode 3 EN 1993-1-1:2005+A1:2014 Design of Steel Structures

• Australia AS 4100-1998 Steel Structures

Specification Requirements Related toStability and 2nd-Order Analysis

AISC 360-10

C1. GENERAL STABILITY REQUIREMENTS

Stability shall be provided for the structure as a whole and for each of its elements…

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.

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.

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.

• Purpose: to address the stability and 2nd-order effects

• Direct Analysis Method is not an analysis method!

– Finite Element Analysis, Response Spectra Analysis, Virtual Work, Moment Distribution, etc., are analysis methods

• It is a methodology consisting of several possible techniques for addressing the various stability effects, most notably directly modeling those effects for inclusion in the analysis

AISC 360 Direct Analysis Method

AISC 360-10 Chapter C

• P-D: Any valid method, including iterative, geometric stiffness, or B2

• P-d: Any valid method, including iterative or B1

• Out-of-Plumbness: Direct modeling of ‘leaning’ structure or Notional Loads (0.002)

• Member Out-of-Straightness: Stiffness reduction (0.8, tb)

• Residual Stresses: Stiffness reduction (0.8, tb)

RAM Frame:

K = 1.0

AISC 360-10 Stability Requirements

C1. GENERAL STABILITY REQUIREMENTS

Stability shall be provided for the structure as a whole and for each of its elements…

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.

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.

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.

Recommendation:

Use Direct Analysis Method for Steel Moment Frames (can use K=1)

Use Effective Length Method for Steel Braced Frames (K=1 for Braced Frames)

AISC Design Guide 28: Stability Design of Steel Buildings

Bentley Communities wiki:ASCE 7, AISC 360, and the Direct Analysis Method in the RAM Structural System

Go to: http://communities.bentley.com, and search for ASCE 7

AISC 360-10 Stability Requirements

ACI 318-14 Structural Concrete

Eurocode EN 1993-1-1:2005+A1:2014 Steel

If acr 10, 1st-order analysis may be used.

5.2.2 Structural Stability of Frames

This is a Notional Load

Australia AS 4100-98 Steel Structures

• Occupant Comfort

• Protect Nonstructural Elements

• Structural Stability

Bentley Systems, Inc. On-Demand Video:

Building Drift: Understanding and Satisfying Code Requirements

Go to: http://pages.info.bentley.com/videos/

or Google the title

Purpose of Drift Limits

ASCE 7 Stability Coefficient

Section 12.8.7 P-Delta Effects:

𝜃 =𝑃𝑥𝛥𝐼𝑒𝑉𝑥ℎ𝑠𝑥𝐶𝑑

(12.8-16)

“The stability coefficient (q) shall not exceed qmax determined as follows:

𝜃𝑚𝑎𝑥 =0.5

𝛽𝐶𝑑≤ 0.25 (12.8-17)

where b is the ratio of shear demand to shear capacity…. This ratio is

permitted to be conservatively taken as 1.0.

Where q is greater than qmax, the structure is potentially unstable and shall be

redesigned.”

Section 12.8.7 P-Delta Effects:

(12.8-16)

“The stability coefficient (q) shall not exceed qmax determined as follows:

𝛽𝐶𝑑≤ 0.25 (12.8-17)

Where the P-delta effect is included in an automated analysis, Eq. 12.8-17

shall still be satisfied, however, the value of q computed from Eq. 12.8-16

using the results of the P-delta analysis is permitted to be divided by (1+ q)

before checking 12.8-17.”

That is, compare qmax to q/(1+q).

(12.8-16)

𝛽𝐶𝑑≤ 0.25 (12.8-17)

𝜃 1 + 𝜃 when P-delta included in analysis

(12.8-16)

𝛽𝐶𝑑≤ 0.25 (12.8-17)

𝜃 1 + 𝜃 when P-delta included in analysis

For example, assuming b=1.0 and with Cd = 5½:

qmax = (0.5)/[(1.0)(5.5)] = 0.091.

Since P-delta included in analysis compare q/(1+q):

0.126 > 0.091 No Good

Stability Coefficient

ASCE 7-10:

ACI 318-14:

EN 1993:

AS 4100:

AISC 360-10:

Stability Coefficient

ASCE 7-10:

ACI 318-14:

EN 1994:

AS 4100:

AISC 360-10:

Thanks!

Questions?

allen.adams@bentley.com

analysis and design for stability-slides.pdf

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