Highlights on Second order Analysis for Structural Steel

Information Paper

Highlights on Second order Analysis for Structural Steel

STRUCTURAL ENGINEERING BRANCH ARCHITECTURAL SERVICES DEPARTMENT September 2013

Structural Engineering Branch, ArchSD Information Paper on Second order Analysis Issue No./Revision No. : 1/-

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File code : 2nd-order Analysis.doc CTW/MKL/MFY/ First Issue Date : September 2013

Table of Contents 1.

Introduction ....................................................................................................................... 3

2.

Second Order Effects ........................................................................................................ 4

3.

First Order and Second Order Analysis ......................................................................... 7 3.1 First order analysis (linear analysis) for non-sway frames and sway frames (λcr≥5) ........................................................................................................................... 7 3.2 Second order analysis for ultra-sensitive sway frames (λcr<5) ............................. 10 3.3 Summary of HK Code ............................................................................................. 13

4.

Second Order Analysis Methods in Other National Codes ......................................... 14

5.

Overview of Softwares Available in SEB for Second Order Analysis........................ 24

6.

Comparison of Different Softwares with Examples ..................................................... 27

7.

Limitations and Recommended Settings ....................................................................... 31

8.

Conclusions ...................................................................................................................... 33

References

Copyright and Disclaimer of Liability This Information Paper or any part of it shall not be reproduced, copied or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without the written permission from Architectural Services Department. Moreover, this Information Paper represents the opinions of the author (s). It serves as an informational source to assist the internal use of the staff in Architectural Services Department, and should not be relied on by any third party. No liability is therefore undertaken to any third party. While every effort has been made to ensure the accuracy and completeness of the information contained in this Information Paper at the time of publication, no guarantee is given nor responsibility taken by the author(s) and Architectural Services Department for errors or omissions in it. Readers are advised to make sure that the information contained herein has not been affected or changed by recent developments. The author(s) further encourages readers to verify all relevant representation, statements and information with their own professional knowledge and by reviewing primary sources where appropriate. The author(s) and Architectural Services Department expressly disclaim all liability for any loss or damage arising from reliance upon any information in this Information Paper (including the formulae and data).


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1.

Introduction

1.1

Clause 6.3.3 of Code of Practice for Structural Use of Steel 2011 (“HK Code”) classifies frames into non-sway, sway, and ultra-sensitive sway, and Clause 6.3.5 of HK Code specifies that either second order P-Δ-δ elastic (direct) analysis in Clause 6.8 or advanced plastic analysis in Clause 6.9 can only be used for ultra-sensitive sway frames in their structural analysis and design. Though it is claimed that the second order analysis and design can achieve economy, such ultra-sensitive sway frames are usually very slender structures, and hence SEI No. 01/2012: Design Code for Structural Steel (available: http://asdiis/sebiis/2k/MAIN%20DOC/sei/index.htm) requires PSEs to consult his respective CSE via SSE in advance should ultra-sensitive sway frames is adopted.

1.2

Should ultra-sensitive sway frame be adopted, PSE is required to use either second order P-Δ-δ elastic (direct) analysis or advanced plastic analysis for its analysis and design. Currently, computer softwares providing full analysis functions for second order P--δ elastic (direct) analysis or advanced plastic analysis are limited in the market, and there are five softwares (QSE Space, SAP2000, OASYS GSA, SPACE GASS, and NIDA) in SEB that are capable for carrying out analysis with second order analysis functions. However, the second order analysis functions provided by those softwares differ.

1.3

The purposes of this information paper are: a) b) c) d)

to explain the differences between first order analysis and various types of second order analysis; to highlight the second order analysis methods specified in other national codes and to explain their differences with HK Code; to introduce features of commercial softwares on the second order analysis functions being included; and to highlight the limitations of second order computation by using commercial softwares and to recommend settings for different types of structures in using second order analysis functions in these softwares.


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2.

Second Order Effects

2.1

Euler’s buckling theory was first developed by Leonhard Euler, a Swiss mathematician, in 1757, and buckling is considered as a structural behaviour of any element under compression. A member will buckle in compression when it is subjected to a compressive force F exceeding: F

π 2 EI (KL) 2

where E is the modulus of elasticity of material; I is the moment inertia of the member; L is unsupported length of the member; and K is the effective length factor of the compressive length, whose value depends on the conditions of end support of the column as follows: for both ends pinned (hinged, free to rotate), K = 1.0; for both ends fixed, K = 0.50; for one end fixed and the other end pinned, K = 0.707; for one end fixed and the other end free to move laterally, K = 2.0. 2.2

Types of Buckling

2.2.1 There are two types of buckling: a) elastic buckling; and b) inelastic buckling. Instability due to elastic buckling (or the “elastic buckling instability”) is derived from Euler’s formula, which is applicable to a long member under compression. However, most compression members are governed by inelastic buckling, and their structural capacity is basically governed by its material strength instead of the elastic buckling instability. 2.2.2 These two types of buckling have been formulated into all design codes for structural steelwork and equations for the buckling resistance of structural sections with consideration of the effective length and the radius of gyration. This design practice has been familiar to structural engineers for nearly a century and is recognized as a usual method to cater structural members for compression. As the buckling resistance is defined by effective length and section properties, the section utilization of the member for a member of a given effective length is linearly and solely proportional to the magnitude of applied load.


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2.3

P-delta (P-Δ-δ) effect

2.3.1 Figure 1 shows the deformed shape of a frame with local deformation of the member δ and frame sway Δ under external load. Such local deformation and frame sway will magnifies the forces on a frame, and is called the “P-delta” (the “P-Δ-δ”) effect. The two sources of the P-Δ-δ effect are: (a) (b)

P-Δ effect (or P-“big-delta” effect), which arises due to the geometric change of the frame owing to applied loads to nodal displacement. P-δ effect (or P-“small-delta” effect), which may be due to the initial member imperfection δi (the “P-δi effect”) or the local deformation of a member δm (the “P-δm effect”). Initial member imperfection δi is due to the fact that it is extremely difficult (if not impossible) to manufacture perfectly straight members, while local deformation of a member δm is the deformation of the chord of a member between end nodes under the external load. The latter local deformation δm is, however, usually very small especially when compared with the P-Δ and Pδi effects.

Figure 1 Deformed shape of a frame in terms of Δ and δ (Source: Figure 6.1 of HK Code) 2.3.2 Both P-Δ and P-δ effects are not accounted for in the conventional analysis using stiffness matrix method to determine displacement, axial force, bending moment and shear force of members in a frame, as such secondary effects do not dominate the behaviour ordinary non-sway structures and can be neglected. Typically, the P-Δ-δ effect only becomes significant at unreasonably large displacement values, or in especially slender columns. Moreover, all national design codes now have included the P-δi effect in their design charts derived from buckling theories. Hence, designers traditionally do not include the P-Δ and P-δ effects in their analysis, since it may significantly increase computational time without providing the benefit of useful information.


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2.3.3 Classification of frame The above discussion concludes that the P-δi effect has been included in their design charts, and the P-Δ-δm effect will not usually dictate the structural design of a frame, especially if the frame has been braced. Nowadays, some commercial softwares are now able to provide an analysis function for the P-Δ and/or P-δ effects by using iteration technique to achieve the final result, which enables designers to decide whether such effects are to be included in their analysis. Clause 6.3.5 of HK Code further specifies that both effects should be included in the analysis for ultra-sensitive sway frames. In order to classify whether such effects should be included in the analysis, it is required to compute the elastic critical load factor (“λcr”) for the frame. λcr is the factor by which the design load must be increased in order to cause the first member of a structure to buckle elastically. λcr can be obtained by the eigenvalue analysis or deflection method. In order to explain in details of these two methods and to provide examples of the calculation, SEB has issued Information Paper on Determination of Elastic Critical Load Factor for Steel Structures (available: http://asdiis/sebiis/2k/resource_centre/). Clause 6.3.3 of HK Code provides the following classification: a)

b)

c)

λcr≥10: “non-sway frames”, where the P- effect is insignificant in both analysis and design, and P-δ effect can be included in the design chart provided in HK Code; 5≤ λcr<10: “sway frames”, where the P-δ effect is not required to be included in the analysis and can be included in the design chart provided in HK Code, and the P- may either be included in the analysis by using the second order P-Δ elastic (indirect) analysis or be included by using either the moment amplification method or effective length method; λcr<5: “ultra-sensitive sway frames”, where both the P- and P-δ effects should be included in both their analysis and design.


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3.

First Order and Second Order Analysis

3.1

First order analysis (linear analysis) for non-sway frames and sway frames (λcr≥5)

3.1.1 First order analysis is the conventional analysis approach in BS 5950 (BSI 2000) or design codes published earlier which assumes the deflection and stress of elements are linearly proportional to load. Neither buckling nor material yielding are considered in the analysis so that the internal forces of the original and undeformed structural elements are not amplified by overall structural deformation and member imperfections. Strength reduction owing to buckling behaviour based on Euler’s theory with PerryRobertson formula is then applied to compression members in their design according to various bucking curves which are related to section type, thickness and axis of buckling. Appendix 8.4 of HK Code specifies an equation for determining compressive strength of members in terms of Perry factor, which relates to Robertson constant according to selected type of member section. Alternatively, the magnitude of Perry factor can also be related to the magnitude of member imperfection. For HK Code, member imperfection (the P-δ effect) of a member with various Robertson constants is included for different design curves (Figure 2). Usually, the Roberson constants highly depend on the member type, axis of buckling considered and material grade. In addition to P-δ effect, these curves also include the residual stresses developed in the manufacturing process.

Notes: Table 8.7 of HK Code classifies the section types into curves a, b, c and d. For example, curve a is for hot-finished hollow section, and rolled I-beam with thickness not greater than 40mm about x-x axis. Curve d is for is for rolled Hsection or I-beam with thickness greater than 40mm about x-x axis.

Figure 2. Buckling curves (a) to (d) for various Robertson constants for Grade S275 steel (Source: HK Code)


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3.1.2 In Figure 2, slenderness ratio λ (determined by the effective length of the member and the radius of gyration) is related to pc, which is the reduced compressive strength of the member according to its slenderness ratio. Pc is then calculated and is used as a basis for checking member resistance. The following equations (which were referenced from the simplified method in BS 5950) for member checking in HK Code can then be used for design: My Fc M  x  1 (8.78) Ag p y M cx M cy

Fc m x M x m y M y   1 Pc M cx M cy

Fc Pc



(8.79)

mx M x m y M y  1 M cx M cy

(8.80)

Fc m LT M LT m y M y   1 Pcy Mb M cy in which: Fc Pc mx, my Mb Mx My

(8.81)

is the applied axial force; is the smaller of Pcx and Pcy; is the equivalent uniform moment factor for flexural buckling about xand y-axes; is the buckling resistance moment = pbSx for Class 1 and 2 sections; is the amplified maximum design moment about the major x-axis; is the amplified maximum design moment about the minor y-axis;

Mx

is the applied moment about the major x-axis;

My

is the applied moment about the minor y-axis;

Mcx

is the elastic moment capacity pyZx about the major principal x-axis;

Mcy MLT

is the elastic moment capacity pyZy about the minor principal y-axis; is the applied moment about major x-axis governing Mb (may usually be taken as Mx); is the smaller of the axial force resistance of the column about x- and y-

Pc

axes, taking member length as the effective length. The above method has been in use for years, and is still a valid and simple way for analysis and design of both non-sway and sway frames.


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3.1.3 Non-Sway Frames (λcr≥10) : For non-sway frames (i.e. with λcr≥10), HK Code does not require the inclusion of the P-Δ effect in the analysis; but requires the P-δ effect to be included in the design equations by using the design chart to determine compressive strength of member. HK Code further requires applied moments M x and M y to be multiplied by the P-δ amplification factor in Equation (8.83) as follows: 1

(8.83) 2 Fc LE 1 2  EI In SEB Guidelines on Design to the Code of Practice for Structural Use of Steel 2005 (available: http://asdiis/sebiis/2k/resource_centre/), it has been noted that the P-δ amplification factor in Equation (8.83) of HK Code will result in a rather conservative result, and is therefore considered not necessary. Clause E8.9.2 of Explanatory Materials to the Code of Practice for the Structural Use of Steel 2011 (BD 2013) also states that a more economic design may be obtained without including such amplification factor for plastic and compact steel sections. PSE should therefore refer to the modification as stated in SEB Guidelines on Design to the Code of Practice for Structural Use of Steel 2005. 3.1.4 Sway Frames (5λcr<10) For sway frames (i.e. with 5≤ λcr<10), Clause 6.6.1 of HK Code allows the use of the first order analysis followed by incorporating the P-Δ effect in the design using both the moment amplification and effective length methods. Clause E6.6.1 of Explanatory Materials to Code of Practice for the Structural Use of Steel 2011 clarifies Clause 6.6.1 of HK Code by stating that the P-Δ effect can be included in the design using either the moment amplification method or effective length method. SEB Guidelines on Design to the Code of Practice for Structural Use of Steel 2005, after reviewing the corresponding clauses in EuroCode 3, GB50017 and BS 5950, concludes that the effective length method will usually be more conservative than the moment amplification method for typical frame structures, and recommends that either method can be used. PSE may therefore refer to the discussion as stated in SEB Guidelines on Design to the Code of Practice for Structural Use of Steel 2005. The P-δ effect can be included in the design chart provided in HK Code.


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3.2

Second order analysis for ultra-sensitive sway frames (λcr<5)

3.2.1 HK Code specifies the following three methods to carry out the second order analysis: a) second order P-Δ elastic (indirect) analysis (Clause 6.7); b) second order P-Δ-δ elastic (direct) analysis (Clause 6.8); and c) advanced plastic analysis (Clause 6.9). These methods (which will be described in the paragraphs followed) are first introduced into HK Code in its 2005 edition. Clause 6.3.5 of HK Code further specifies that second order P-Δ elastic (indirect) analysis can only be used for sway or non-sway frames (i.e. with λcr5), and that either second order P-Δ-δ elastic (direct) analysis or advanced plastic analysis should be used for ultra-sensitive sway frames (i.e. with λcr<5). 3.2.2 Second order P-Δ elastic (indirect) analysis In addition to first order analysis for sway frames with 5≤ λcr<10, HK Code also permits the use of a second order P-Δ elastic (indirect) analysis, which only considers the P-Δ effect (but not P-δ effect) by including the global sway effect by an iterative process to cater for the geometrical non-linearity. After getting the forces, the design of the structural elements can then be carried out with consideration of the P-δ effect by using Equations (8.78) – (8.81) of HK Code using the member effective length factor of 1.0. 3.2.3 Second order P-Δ-δ elastic (direct) analysis Second order “elastic” analysis considers the P-Δ-δ effect by incorporation of frame imperfection, deformed nodal coordinates and initial member imperfection in the analysis. The required initial member imperfections e0 are related to the length of the member L, and are specified for different section types as follows: Buckling curve*

e0/L

a0 a b c d

1/550 1/500 1/400 1/300 1/200

Notes: * Table 8.7 of HK Code classifies the section types into curves a0, a, b, c and d. For example, curve a0 is for high strength hotfinished hollow sections with py460MPa, and curve a is for hotfinished hollow section with py460MPa or rolled I-beam with thickness not greater than 40mm about x-x axis.

(Source: Table 6.1 of HK Code)


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After iterations by elastic analysis with geometrical non-linearity, the forces in the members will be obtained, and no calculation for effective length of structural members is required. The design strength py instead of pc serves as a basis to assess the sectional capacity of the element under compression. HK Code then provides the following equations (Equations (6.12) and (6.13)) for member checking: Fc Mx My Fc M x  Fc ( x   x ) M y  Fc ( y   y ) Ag p y



M cx



M cy



Ag p y



M cx



M cy

1

(6.12)

in which (x+δx) and (x+δy) are the final member deflections through iteration process in consideration of the sway, bending moment on member and initial member imperfections about x- and y-axes, and M x  Fc ( x   x ) and M y  Fc ( y   y ) represent the total moments obtained from the second order analysis. HK Code also specifies that if moment equivalent factor mLT is less than 1, both Equation (6.12) and the following Equation (6.13) are required for member resistance check: My m [ M x  Fc ( x   x )] m y [ M y  Fc ( y   y )] Fc M Fc  x    LT   1 (6.13) Ag p y M cx M cy Ag p y Mb M cy

However, there are not many guidelines in HK Code, and the drafting committee of HK Code seems to rely on the commercial softwares to include the effects of P-Δ-δ effect in the analysis. Buildings Department would exercise control over the limitations of the softwares. Mb (the lateral torsional buckling of a member in bending), instead of Mcx, is used in Equation (6.13) because the softwares available in the market cannot account for member lateral torsional imperfections in both axes during computational analysis and hence, Mb is specified to cater for this software deficiency. 3.2.4 Advanced plastic analysis Advanced “plastic” analysis includes both P-Δ-δ effects and member plasticity in its analysis. Plastic design for structural steel has been introduced into BS 449 since 1948 and is also allowed in BS 5950, which has already allowed for material yielding and plasticity in a structure. The principle of plastic design is that when maximum stress of member exceeds yield point, a hinge is inserted to the member end close to the hinge position and analysis continues until the collapse load (or limit point) is reached. The collapse load is taken as the load level which does not allow further load increase indicated as a curve reaches plateau, descends or stagnates in the load versus deflection plot (Figure 3). In the design, this collapse load (or limit point) should be greater than or equal to the factored design load in all load cases. Again, there are few guidelines in HK Code, and the drafting committee of HK Code also relied on the commercial softwares to include the P-Δ-δ effects and member plasticity in the analysis.


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Figure 3 Load-Displacement Curve of Element under Compression


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3.3

Summary of HK Code

3.3.1 Table 1 summarizes features of first order and second order analysis in respect of initial member imperfection, geometric and material nonlinearity in HK Code: Table 1 Summary of various analysis methods in HK Code Geometric nonlinearity Member bowing Nodal (P-δ effect) Applicability Methods of displacement of to type of Effect from analysis Initial member frame structure bending imperfection (P-Δ effect) curvature (P-δi effect) (P-δm effect) included by Nonincluded by various non-sway sway X effective length Robertson (λcr >10) frame constants included by First included by various order effective length Robertson analysis Sway or moment constants or sway frame X frame amplification moment (10> λcr >5) factor (Equation amplification (8.82)) factor (Equation (8.82)) not mandatory included by Second order P-Δ but analysis various sway frame elastic (indirect) √ are provided Robertson (10> λcr >5) analysis in most constants commercial softwares Second order P-Δ-δ sway or ultraelastic (direct) √ √ √ sensitive analysis sway frame Advanced plastic sway or ultraanalysis (second √ √ √ sensitive order plastic sway frame analysis) 3.3.2 PSE should note that second order analysis is not a tool to make an “unsafe” structure to become a “safe” structure. Clause E6.1 of Explanatory Materials to Code of Practice for the Structural Use of Steel 2011 further remarks that second order analysis “has been mistakenly considered by many as a tool to reduce structural weight.”


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4.

Second Order Analysis Methods in Other National Codes

4.1

Other national codes for steelwork design such as EuroCode 3, AISC, AS 4100 and GB 50017 also specify second order methods and the conditions for applicability of such methods. The second order methods specified in these codes include: effective length method, moment amplification method and direct analysis method. The details of the methods in these codes will first be elaborated in the following paragraphs, and then their differences with HK Code are described.

4.2

EuroCode 3 (“EC3”)

4.2.1 EC3 allows first order analysis and second order analysis for steel structures. The first order analysis (which is similar to that in HK Code) is applicable to non-sway frame with a factor αcr 10 (i.e. λcr10 in HK Code) and for sway frames with αcr <10. For sway frames, two methods are specified: a) b)

indirect second order method, i.e. first order analysis with allowance for P-Δ effect; or direct second order method.

Design examples to EC3 can be found from Designers’ Guide to Eurocode 3: Design of Steel Buildings EN 1993-1-1, -1-3 and -1-8 (Gardner and Nethercot 2011) and Steel Designers’ Manual (Davison and Owens 2012). 4.2.2 Indirect second order method This “indirect second order method” is similar to the “first order analysis for sway frame” in HK Code. EC3 includes the P-Δ effect by multiplying all horizontal loads (e.g. wind) and equivalent loads due to initial sway imperfection, i.e. approximately 1/200 of the total design vertical load by following amplification factor:  cr  cr  1 where αcr is similar to λcr. The above amplification technique is similar to that in HK Code except of the amplification to horizontal loads instead of total moment, and is applicable for regular frames with αcr ≥3 (as compared with λcr≥5 in HK Code). The design can then be carried out by using the member length as the effective length. Hence, though HK Code considers that this method is only a first order analysis, EC3 already considers this method to be a second order analysis indirectly, as both P-Δ-δ effects have been included in the design.


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4.2.3 Direct second order method For frames with αcr<3 (which is ultra slender structure), EC3 requires the designer to adopt direct second order method that structure, which should include the effect of both global and member imperfections. The global initial sway imperfections (Figure 4) can be obtained by the following equation:

  0 h m where

0 is the basic value and is taken as 1/200;  h is the reduction factor for height h applicable to columns and is given by:

h 

h

2 h

but

2   h  1.0 ; 3

is the height of the frame ;

1 ); m m is the number of columns in a row including only those columns which carry a vertical load NEd not less than 50% of the average value of the column in the vertical plane considered.  m is the reduction factor for the number of columns in a row = 0.5(1 

Figure 4 Global Initial Imperfection (Source: EC3) Similar to HK Code, EC3 tabulates (Table 2) the initial member imperfections e0 for section types. To facilitate designers to model the imperfections in the structural analysis, EC3 offers a replacement of initial imperfections by equivalent horizontal forces as shown in Figure 5.


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Initial Sway Imperfections Initial Member Imperfections Figure 5 Equivalent Horizontal Forces for Global and Member Imperfection (Source: EC3) However, the specified initial member imperfection e0/L in EC3 is relatively stringent as compared with those specified in HK Code. The UK National Annex to EC3 modifies those values by requiring the initial member imperfections for an individual section about a particular axis to be back-calculated from the formula for the buckling curves given in EC3 using the elastic section modulus and full plastic section modulus. Brown (2011) gives some examples to back calculate the initial imperfection, and shows that the initial imperfections given in EC3 are very conservative (Table 3). Table 2 Initial Member Imperfections for Different Section Types in EC3 Elastic analysis Plastic analysis Buckling curve e0/L e0/L a0 1/350 1/300 a 1/300 1/250 b 1/250 1/200 c 1/200 1/150 d 1/150 1/100 (Source: EC3) Table 3 Comparison of initial member imperfections specified in the UK National Annex and EC3 for 203×203×46 S355 UC Effective Length (m)

The UK National Annex back-calculated e0/L

EC 3Curve b e0/L

1 2 4 6 8

1/4815 1/536 1/308 1/333 1/425

1/250


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4.3

AISC-360-10

4.3.1 There are three methods specified in AISC-360-10 (“AISC”) (AISC 2010): a) first order analysis in Appendix 7 of AISC; b) approximate second order analysis in Appendix 8 of AISC; and c) direct analysis in Chapter C of AISC. The 2010 edition of the AISC has moved the requirements of direct analysis into Chapter C as the preferred method of the steel frame analysis, while the other two methods were moved to Appendix 7 and Appendix 8 respectively. This move seems to encourage designers to adopt direct analysis in the steel frame analysis; but concurrently to permit designers to use approximation methods as an alternative to rigorous second order analysis for sway structures. AISC allows designers to adopt either Allowable Strength Design (ASD) method or Load and Resistance Factor Design (LRFD) method for the design. ASD method is an older method (similar to BS 449) which applies a safety factor to nominal strength of material to attain allowable strength. For LRFD, a partial safety factor is applied to each of load cases to calculate the ultimate limit state. LRFD method (which is similar to BS 5950) is to proportion structural components such that the design strength equals or exceeds the required strength of the component under action of the LRFD limit state. Design examples to AISC using LRFD method can be found in Structural Steel Design (LRFD Method) (Rokach 1991). 4.3.2 First order analysis The first-order analysis in AISC is similar to the first order analysis in HK Code and EC3 for non-sway frames (i.e. λcr>10). It is therefore applicable for frames that: a) support gravity loads primarily through nominally vertical columns, walls or frames; and b) the ratio of maximum second-order drift to maximum first-order drift in all stories is not greater than 1.5, and the ratio may be taken as the B2 multiplier. B2 may be calculated using the simplified equation (extracted in Section 4.3.3) given in Appendix 8 of AISC instead of undergoing a calculation using second order analysis. Trial calculations using this simplified equation show that the value of B2 for frames with λcr≈5 is about 1.5, and hence the first order analysis in AISC is applicable to frames with λcr>5. With the forces obtained from the first order analysis, the effective length method can be used for structural design. The effective length factor, K, of components of the braced frame is normally taken as 1.0, and for moment frames, a factor K>1.0 is normally adopted. Structural Engineering Branch, ArchSD Information Paper on Second order Analysis Issue No./Revision No. : 1/-

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4.3.3 Approximate second order analysis The “approximate second order analysis” is similar to the first order analysis followed by moment amplification method in HK Code and the indirect second order method in EC3. The P-Δ-δ effects are included in AISC by applying the amplification factors on the internal forces obtained in first order analysis through multipliers B1 and B2. The following equations are detailed in Appendix 8 of AISC to calculate the designed moment Mr and axial force Pr:

M r B1 M nt B 2 M lt

P r  P nt  B 2 Plt where B1 = multiplier to account for P-δ effect and may be taken as 1.0 for members not subject to compression; B2 = multiplier to account for P-Δ effect; Mlt = first order moment due to lateral translation of the structure only; Mnt = first order moment with the structure restrained against lateral translation; Plt = first order axial force due to lateral translation of the structure only; and Pnt = first order axial force with the structure restrained against lateral translation. AISC allows that the above equations are applicable to all members in all sway or nonsway structures (including ultra-sensitive sway frames). The multipliers B1 and B2 for Pδ and P-Δ effects respectively may be calculated as follows: 1 Cm 1 B1  1 and B 2  P story 1  Pr / Pel 1 Pe story where α = Cm = Pel =

1.0 (LRFD) or 1.6 (ASD); 0.6-0.4×(M1/M2) or conservatively taken as 1.0, and M1 and M2 are the smaller and larger moments in the member in the first order analysis; elastic critical buckling strength of the member in plane of bending and may be calculated as follows:

P el 

 2 EI * ( K1L) 2

Pstory= total vertical load supported by the member; Pestory=elastic critical buckling strength for the storey in the direction of translation being considered and may be calculated as follows: HL Pestory  RM H RM = 1- 0.15(Pmf/Pstory); L = height of storey; Pmf = total vertical load in columns in the storey (=0 for braced frame systems); ΔH = first order interstorey drift due to lateral forces; H= storey shear produced by lateral forces used to compute ΔH. Structural Engineering Branch, ArchSD Information Paper on Second order Analysis Issue No./Revision No. : 1/-

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The Commentary on AISC highlights that the P-δ effect will be significant when B1 is larger than 1.2. A rigorous second order elastic analysis is then advised to be carried out for obtaining the internal forces of the members. 4.3.4 Direct analysis The direct analysis in AISC is similar to the second order P-Δ-δ elastic (direct) elastic analysis in HK Code and the direct second order elastic method in EC, and is applicable for all types of frame. Brief summary of the analysis is as follows: 1)

Deformation of structures The analysis should take account of member deformations due to flexural, shear and axial behavior, and other component and connection deformations which affect the deformation of the global structure. P-Δ and P-δ effects should be reflected in the analysis.

2)

Initial Imperfections AISC specifies that either direct modelling of imperfections in the analysis or the application of notional loads can be adopted for inclusion of the effect. In general, the imperfections include: a) member out-of-straightness equal to L/1000, where L is the member length; and b) frame out-of-plumbness equal to H/500, where H is the storey height

3)

Adjustments to Stiffness AISC specifies that the reduction of stiffness of the all structures is required in the analysis. A factor of 0.8 shall be taken in the consideration of axial and flexural stiffness and applied to all members. Reduced stiffness, EI* and EA*, should be as follows: a) EI*=0.8τbEI; and b) EA*=0.8EA where τb =1.0 at Pr/Py ≤0.5 and τb =4(Pr/Py)[1-(Pr/Py)] at Pr/Py>0.5 in LRFD; but is usually taken conservatively as 1.0 in all cases.

Unlike HK Code (where full design strength can be used to assess the sectional capacity of the element), the design in AISC after analysis can be proceeded with the effective length of members taken as Le=1.0L.


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4.4

Australian Standard AS 4100: 1998 (Standards New Zealand NZS 3404-1: 2009)

4.4.1 AS 4100 (Standards Australia 1998) (and NZS 3404:1 (Standards New Zealand 2009)) requires all steel structures incorporating P-Δ effect by using one of the following methods: a) b) c)

a first order elastic analysis with moment amplification; or a second order elastic analysis; or a second order plastic analysis.

Design examples to AS 4100 can be found in Steel Designers’ Handbook (Gorenc et al 2005). 4.4.2 First order elastic analysis with moment amplification This method is generally applicable to non-sway and sway frames, and is similar to the first order analysis followed by moment amplification method in HK Code and indirect second order method in EC3. In this method, moment amplification factors δb and δm (defined as below) should be applied to both braced and non-braced frames respectively. For members in braced frames, the design moment of the members should be multiplied by δb as follows: cm b  1  N*   1    N omb   where Nomb is the elastic buckling load; and Cm = 0.6-0.4βm ≤ 1.0. If Cm is taken as 1.0, δb can be calculated as follows: 1 b  1  N * Le 2  1  2    EI    For members in unbraced frames, moment amplification factor δm should be multiplied to the design moment, which may be taken as the greater of – a) δb calculated above; b) δs calculated as follows: 1 s     N*   1   s *   hs  V  where Δs is the translational displacement of the top relative to the bottom in the storey of height (hs) by the design horizontal storey shears (V*) at the column ends, and N* is the design axial force in a column of the storey.


Page 20 of 35


The above amplification methods are almost identical to the methods required in HK Code. AS 4100 states that the moment amplification method is only applicable to the structure with δb and δs ≤1.4, which corresponds to λcr>3.5 for sway frames in HK Code. 4.4.3 Second order elastic/plastic analysis AS 4100 specifies that a second order elastic analysis should be adopted for frames with δb or δs1.4 which corresponds to ultra-sensitive sway frames with λcr <3.5 in HK Code. In such frames, the further deformation of the structures due to P-Δ should be directly modelled in the analysis. For frame with λc<5, changes in the effective stiffness of the members due to axial forces should be considered in the model. The design bending moment is directly obtained by the model accounting for the above considerations. There are two possible methods in the second order analysis - the elastic and plastic analysis. The main difference between these two methods is that material grade, type and characteristic, section type and loading condition should be limited in the plastic analysis. 4.5

Chinese National Standard GB 50017-2003

4.5.1 There are two methods specified in GB 50017-2003 (Ministry of Construction 2003): a) first order analysis for non-sway frames; b) second order P-Δ elastic (indirect) analysis for sway frames. In order to classify frames into non-sway or sway, GB 20017 specifies the calculation of the coefficient θi as follows:

 i 

N i  ui

H

i

 hi

where ∑Ni is the total axial forces of all columns at ith storey; ∑Hi is the total shear forces at ith storey; hi is the storey height above ith storey; and △ ui is the storey displacement due to ∑Hi This formula is actually an inverse of λcr in HK Code so that the formula can be rewritten as: 1 

 cr

GB 50017 states that first order analysis can be used for steel structures if θi  0.1 (i.e. λcr10 in HK Code), and that second order P-Δ elastic (indirect) analysis should be used if when θi  0.1. For ultra-sensitive sway frames, GB 50017, however, specifies the size of the steel members or the lateral resistance of the frame to be adjusted if θi  0.25 (i.e. λcr  4 in HK Code). This tallies with the requirement in SEI No. 01/2012: Design Code for Structural Steel (available: Structural Engineering Branch, ArchSD Information Paper on Second order Analysis Issue No./Revision No. : 1/-

Page 21 of 35


http://asdiis/sebiis/2k/MAIN%20DOC/sei/index.htm), which requires PSEs to consult his respective CSE via SSE in advance should ultra-sensitive sway frames with λcr 5 are adopted. 4.5.2 Second order P-Δ elastic (indirect) analysis For frames with θi >0.1, the P-Δ effect is calculated by a moment amplification factor (which is similar to the first order analysis followed by moment amplification method in HK Code), and the amplification factor  iII is given by:

 iII 

1 1  i

The design moment M II is then given by:

M II  M q   iII M H where M II is the design moment after the amplification; M q is the first order moment due to vertical load only; and M H is the first order moment due to lateral load only.

4.5.3 Draft revision of GB 50017 The draft revision of GB 50017 is now circulated to the industry for consultation, which includes a direct second order analysis. Similar to HK Code, such direct second order analysis requires designers to consider the effects of P-Δ-δ, imperfections of structural members, residual stress, joint stiffness, non-linearity of materials and geometry in the analysis. The initial member imperfection e0 can be taken as 1/750 while the initial frame imperfection cannot be less than hi/1000. Similar to EC3, equivalent notional force method putting UDL on the members considered can be used to simulate the initial member imperfection e0. When the design forces are obtained from the direct analysis method, the following equation should be satisfied: II N M xII M y    f A Wx Wy where M xII and M yII are the design moments about x-x and y-y axes obtained from the direct analysis; A is the cross sectional area; and W x and W y are the section modulus about x-x and y-y axes respectively.


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4.6

Summary of Codes on Structural Steel Design Table 3 summarizes the comparison among the national codes on the methods to be used in the analysis of structural steelworks with terminology used in HK Code. The following are noted: a) First order analysis without the P-Δ effect followed by design by effective length method is permitted in all national codes (except in AS 4100) for non-sway frames, and generally the limiting elastic critical factor λcr is taken as 10 as in HK Code. b) First order analysis followed by moment amplification method is available in all national codes, and can be used to include the P-Δ effect for sway frames and is termed as “second order analysis” in most of other national codes as it has included both P-Δ-δ effects in the design. The limiting elastic critical factor λcr of 5.0 in HK Code is the most stringent requirement, and EC3 requires λc of at least 3 only. c) Second order P-Δ-δ elastic (direct) analysis is required for ultra-sensitive frames, and GB 50017 recommends designers to re-size the members or re-frame the structures for frames with λcr  4. d) Second order P-Δ-δ elastic (direct) analysis is now allowed in all other national codes except GB 50017, and the draft revision of GB 50017 also suggests the introduction of such method. However, all these codes do not specify the detailed procedures in the analysis, and they differ in their specified member imperfections. In particular, AISC does not require P-δ effect due to initial member imperfection in the analysis, and hence commercial softwares in the market (such as SAP 2000) seldom include such features in their analysis. Table 3 Comparison among National Codes on Analysis Methods Codes Methods of analysis

First order analysis Second order P-Δ elastic (indirect) analysis

HK Code

EC3 (BS EN 1993:2005)

AISC 360-10

AS 4100:1998

GB 50017: 2003

√

√

√

√

√

√

√

√

√

√

√

X

√

X

Second order P-Δδ elastic (direct) analysis

√

√

Advanced analysis (Second order plastic analysis)

√

√


√ (member imperfection need not be included in the analysis, as it has been included in the design chart) √ (member imperfection need not be included, as this has been included in the design chart)

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5.

Overview of Softwares Available in SEB for Second Order Analysis

5.1

Five computer softwares are available in SEB with second order analysis functions. In the following paragraphs, the features of the second order analysis functions in these softwares will be highlighted.

5.2

QSE Space QSE Space is a pre-accepted software by Building Departments for the design of steel structures according to BS 5950, and is applicable for both non-sway and sway frames. It uses conventional stiffness matrix method. A second order P-Δ elastic (indirect) analysis with inclusion of P- can be included. Demand on resources is small and results are obtained quickly with results very comparable to those obtained using a more rigorous analysis. As only the P- effect will be included in the analysis, the P-δ effect has to be included by using design charts in BS 5950.

5.3

SAP 2000, OASYS GSA and SPACE GASS SAP 2000, OASYS GSA and SPACE GASS include the P- effect and P-δm (i.e. the member deformation under load) but not the initial member imperfection P-δi effect in the analysis, and can carry out more iteration in their calculation than that in QSE Space. Moreover, both geometric changes of the overall structures and individual member due to the internal forces can be included in the analysis. However, the initial member imperfections δi cannot be included in the software. Since the P- effect has been included in the analysis, amplification factors in Equations (8.78) – (8.81) of HK Code are not required to obtain the design moment. As the P-δi effect has not been included in the design using SAP 2000 or OASYS GSA, they have not yet been pre-accepted softwares by Buildings Department for the design of steel structures according to HK Code, although they are pre-accepted softwares by Building Departments for the design of steel structures according to BS 5950. SPACE GASS is a pre-accepted software by Buildings Department for the design of steel structures according to HK Code so that the industry for building projects has widely used this software. However, as SPACE GASS, same as SAP 2000 or OASYS GSA, does not include the P-δi effect due to initial member imperfection, SPACE GASS, rather than using design charts in HK Code, has a built-in analysis function to calculate the effective length of the member according to Clause E.6 of BS 5950 in order to obtain the compressive strength of the member.


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5.4

NIDA

5.4.1 The second order analysis of NIDA includes both second order elastic analysis and advanced analysis, and was tailor-made for HK Code. Hence, it is a pre-accepted software by Buildings Department for the design of steel structures according to HK Code, and hence is also widely used in the industry for building projects. Users are required to choose between these two options, i.e. whether they want to include material plasticity in the analysis. Both analyses include the P--δ effects under applied external load. Different initial member imperfections δi (from L/550 to L/200 specified in HK Code) depending on the section types are assumed in the software. Since initial member imperfection has been included in the analysis stage, Equations (6.12) and (6.13) of HK Code can be used in the design with the axial force and moments obtained after the analysis. 5.4.2 As initial member imperfections δi may occur in one of the two principal axes, NIDA claims that initial member imperfection will be chosen to the direction that is more likely to buckle. Alternatively, an option to specify initial imperfections (e.g.  L/500) to both principal axes is also provided in the program. However, in the latter case, the designer needs to choose an appropriate combination of imperfections in both axes to model the easiest buckling direction. 5.4.3 In addition, HK Code does not require designer to include imperfection for lateral torsional buckling. A recognized method to determine accurate position of member with correct imperfection magnitude for lateral torsional buckling is not available in HK Code or literatures. NIDA still relies on conventional design chart to determine Mb according to HK Code to satisfy Equation 6.13 of HK Code. 5.5

Summary of the key features of commercial softwares Table 4(a) and Table 4(b) summarizes the key features included in the commercial sofwares. It can be seen that NIDA is the only software available in SEB that can carry out second order P--δ elastic (direct) analysis and advanced plastic analysis, as it was tailor-made for HK Code. However, PSE may note that though SPACE GASS cannot carry out the exact second order P--δ elastic (direct) analysis and advanced plastic analysis according to HK Code, it is a pre-accepted software by Buildings Department for the design of steel structures according to HK Code. In Section 6, calculation of typical frames will be shown to show that the results of all these softwares are similar to each other, and that the results from the first order analysis are also comparable to those obtained by rigorous second order analysis.


Page 25 of 35


Table 4(a) Available Analysis functions of Softwares used for Second Order Analysis according to HK Code Method of analysis First order analysis Second order P-Δ elastic (indirect) analysis Second order P-Δ-δ elastic (direct) analysis (including initial member imperfection) Advanced plastic analysis Applicability

QSE Space Analysis Design

SAP2000 Analysis Design

OASYS GSA Analysis Design

SPACE GASS Analysis Design

NIDA Analysis Design

√

√

√

√

√

√

√

√

√

√

√

√

√

√

√

√

√

√

√

√

X

X

X

X

X

X

X

X

√

√

X

X

X

X

X

X

X

X

√

√

Non-sway or sway frame


Non-sway and sway frame

Page 26 of 35

Non-sway and sway frame

Non-sway, sway and ultrasensitive sway frame


Non-sway, sway and ultrasensitive sway frame

Table 4(b) Second order effects included in sofwares available in SEB Software

P-δi

P-

P-δm

Analysis

Design

Analysis

Design

Analysis

Design

QSE Space

√

√

X

√

X

X

SAP2000

√

√

X

√

√

√

OASYS GSA

√

√

X

√

√

√

SPACE GASS

√

√

X

√

√

√

NIDA

√

√

√

√

√

√

6.

Comparison of Different Softwares with Examples

6.1

This section will compare the results of using QSE Space, SAP2000, OASYS GSA, SPACE GASS and NIDA in the analysis and design of structural steel members in order to compare their respective accuracy. The following examples will be presented: a)

b) c)

6.2

a portal frame to study the differences among conventional first order analysis, first order analysis with moment amplification method by HK Code and EC3, and second order elastic (direct) method by NIDA (Example 1); a portal frame to study the differences among QSE Space, OASYS GSA, SAP2000 and NIDA (Example 1); and a portal frame to compare the section utilization determined by SPACE GASS and NIDA (since both of them are pre-accepted programs by Buildings Department) (Example 2).

Example 1: Frame for comparison among different methods and softwares

6.2.1 Consider an unbraced frame made of sections UC 356×368×153 kg/m as shown in Figure 11. This example serves to assess the differences of global analytical results determined by conventional first order analysis, first order analysis followed by moment amplification method in HK Code and EC3, and the second order P-Δ-δ elastic (direct) method in NIDA. The weak axis of all members in the portal frame is assumed fully restrained. Table 6(a) summarizes the results.


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Table 6(a)

Comparison among Different Methods for Second Order Effects Member CD

Wg (kN/m)

5 6 7 8 9 10 15 20

H (kN)

60 60 60 60 60 60 60 60

Moment Amplification Method (HK Code) P-Δ

First Order Method

λcr

Moment (kNm) 606.7 668 729.4 790.7 852.1 913.4 1220 1527

14.2 11.9 10.3 9.0 8.0 7.3 4.9 3.7

Moment (kNm) 612.8 675.3 738.9 889.3 973.4 1059.4 1535.3* 2100.2*

diff. to NIDA -3.8% -4.2% -4.5% -4.8% -5.1% -5.4% -6.6% -7.7%

Moment Amplification Method (EC3) P-Δ

diff. to NIDA -2.8% -3.1% -3.3% +7.0% +8.4% +9.7% +17.6%* +26.9%*

Moment (kNm) 606.8 668.1 729.5 828.2 894.8 961.5 1297.8 1639.6

diff. to NIDA -3.8% -4.2% -4.5% -0.3% -0.4% -0.4% -0.6% -0.9%

Second Order PΔ-δ Elastic (Direct) Method (NIDA) P-Δ-δ Moment (kNm) 630.6 697.1 763.9 830.9 898.0 965.5 1306.0 1655.0

Note: * denotes for reference only Table 6(b) Comparison among Softwares Member CD Wg (kN/m)

5 8 10 15 20

H (kN)

60 60 60 60 60

QSE Space P-Δ

λcr

14.2 9.0 7.3 4.9 3.7

Moment (kNm) 628 827 961 1301 1647

diff. to NIDA -0.1% -0.2% -0.2% -0.2% -0.2%

OASYS GSA P-Δ-δm Moment (kNm) 635 837 973 1317 1670

diff. to NIDA +1.0% +1.0% +1.1% +1.1% +1.2%

SAP 2000 P-Δ-δm Moment (kNm) 629 829 965 1306 1655

diff. to NIDA +0.0% +0.1% +0.2% +0.2% +0.3%

Note: OASYS GSA and SAP2000 include P-δm (effect of bending curvature) in the comparison. Structural Engineering Branch, ArchSD Information Paper on Second order Analysis Issue No./Revision No. : 1/-

Page 28 of 35


Second Order P-Δ-δ elastic (Direct) Method (NIDA) P-Δ-δ Moment (kNm) 628.7 828.5 962.8 1303 1650

Figure 11

Portal Frame made of UC 356x368x153kg/m

6.2.2 Table 6(a) shows that the differences between the moment determined by first order method and that by NIDA widen as cr decreases from non-sway to sway, and then to ultra-sensitive sway frame. Amplified moment method by EC3 accounting for second order effects is pretty similar to that by NIDA for cr ranges from 4 to 14. It is further noted that amplified moment by HK Code is very conservative, especially for the sway structure. The reason for such significant deviation is that HK Code requires designers to multiply the amplification factor on the total moment rather than the moment due horizontal action only. Thus, higher slender structures (i.e. smaller value of cr) results in a highly conservative solution for HK Code. The comparison shows that it is generally acceptable to multiply the amplification factor on the moment due to horizontal action only. 6.2.3 In the first order analysis using five different softwares, all obtained analytical results are exactly same, as they all base stiffness matrix method. For the second order analysis accounting P--δm effect, Table 6(b) shows that the differences among five analytical outputs are less than 1.2%. This further illustrates that the analytical result of the structure with λcr=3.7 given by program accounting for P-Δ is not far different from that by program providing second direct elastic analysis. Therefore, the comparison shows that HK Code is too stringent on limiting of first order and second order P-Δ elastic (indirect) method to non-sway and sway frames.


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6.3

Example 2: Members in a frame using SPACE GASS and NIDA The unbraced frame example given in Clause E6.12.1 of Explanatory Materials to Code of Practice for the Structural Use of Steel 2005 (Figure 12) is used for comparing the softwares, SPACE GASS and NIDA. Table 7 shows the result of the member utilization. The results calculated by SPACE GASS are similar to those by NIDA, although SPACE GASS designs the member by using effective length method followed by BS 5950 and NIDA bases on second order P-Δ-δ elastic (direct) analysis with all geometric imperfections included.

Figure 12

30×10m Height Portal Frame

Table 7 Comparison between SPACE GASS and NIDA in Design of Members Applied Applied Horizontal Vertical Force at Force at Point B Point C (kN) (kN) 60

1000

Fc of member CD (kN)

Mx of member CD (kNm)

Section Utilization of member CD

λcr

2.23

SPACE GASS

NIDA

SPACE GASS

NIDA

SPACE GASS

NIDA

1035

1034

529

518

0.90

0.93


Page 30 of 35


7.

Limitations and Recommended Settings

7.1

Unlike other national codes which only requires P-Δ effect and do not mandatorily require inclusion of initial member imperfection δi in the second order analysis, the HK Code is the only code that specifies such requirement in its second order P-Δ-δ elastic (direct) method. Given such requirement, NIDA is the most versatile, among the five softwares, capable of including the initial member imperfection in its analysis rather than using design charts to include such effect. Yet, NIDA still cannot include imperfections for lateral torsional buckling, and still relies on conventional design chart to determine Mb. Moreover, the following limitations and settings for NIDA are to be noted: (a)

Initial member imperfections The default setting of NIDA includes member imperfection on the major principal axis, which can accurately calculate the allowable compressive load of CHS in all range of slenderness ratio. For other sections with major and minor principal axes (e.g. I-sections), NIDA program sets the member imperfection on the major axis which may over-estimate the structural capacity as member tends to buckle about minor axis. Therefore, the section utilization of the member under pure axial force is not accurate, and PSE needs to adjust the default setting accordingly. That is, the initial member imperfection in some cases should be chosen to be set on the minor axis. This is the particular case for members of steel truss designed for pure axial load only, as the buckling direction cannot be effectively determined by means of imperfection of the major principal axis in NIDA. PSE may then note that this is a tedious (if not impracticable) task for frames with a large number of members. NIDA can enable setting the imperfection on either one principal axis or both principal axes in the model, and adjusting the default setting to imperfection on both principal axes would be a conservative way to handle this limitation.

(b)

Extra node in a member The initial member imperfection set in NIDA is based on a specified ratio of member length as Table 6.1 of HK Code, and is automatically included in the option of second order analysis. However, if an extra node is created between the ends of a member, the simulated member imperfection may be underestimated and the modelled curvature does not follow the actual deformation along the length. Although all extra nodes can be eliminated, the limitation of assigning initial member imperfection still exists in the case that a node is connected by a restraint element from only one direction. The continuous section would not be cut down into two pieces for connecting lateral restraining


Page 31 of 35


elements in practical situation. The actual imperfection shape is not reflected in the model analysis and the analytical result may be underestimated. To minimize the underestimation, assigning the imperfection should not only be limited to the displacement of individual member, but also consider the imperfection on the deformed shape obtained in eigen-buckling mode. In the latest version of NIDA, an option is now available to define the accurate effective length for consideration of initial member imperfection. For such case, the conventional method for checking buckling resistance of the member can be adopted to compare the result from the second order analysis. 7.2

Economy in the design The examples in Section 6 show that the second order analysis results from QSE Space, OASYS GSA, SAP2000, SPACE GASS or NIDA are approximately equal to each other for the cases studied. The maximum difference of outputs among these commercial softwares is less than 1.2%, which is considered negligible. Hence, it is widely believed that second order analysis can achieve a more economical design; but such claim may not be valid for typical frames, though such claim may be valid for ultra sensitive frames. Example 2 in Section 6 further shows that the moment amplification method for accounting second order effect specified in HK Code is quite conservative. The difference of design moment determined by EC3 which require designers to apply amplification factor to horizontal actions only is less than 1% and considered acceptable for all sway and ultra-sensitive sway frames. Thus, for typical frames, the effects due to P-Δ, P-δm and P-δi only results in negligible differences in the section utilization from these methods.

7.3

Lateral Torsional Buckling Resistance Lateral torsional buckling is more complex as compared with flexural buckling in view of its buckling direction and involved parameters. Currently, no available softwares in Hong Kong can include the lateral torsional buckling in the analysis stage. Mb is still required in Equation (6.14) of HK Code, even when the second order P-Δ-δ elastic (direct) analysis method is adopted. Members such as UB or UC under both bending moment and axial load would normally be governed by lateral torsional buckling. Therefore, second order P-Δ-δ elastic (direct) method is not a necessary tool to produce the most economical cost-saving design. PSE should note that in such case effective length factor under lateral torsional buckling should also be defined for each of members in NIDA.


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8.

Conclusions

8.1

Clause 6.3.5 of HK Code specifies that both P-Δ and P-δ effects should be included in the analysis for ultra-sensitive sway frames with λcr<5, and conventional first order analysis can still be used in non-sway frames with λcr10 and for sway frames with 5<λcr<10 by either moment amplification or effective length method. The comparison in Section 6 further shows that HK Code is too stringent on limiting of first order and second order P-Δ elastic (indirect) method to non-sway and sway frames.

8.2

Second order analysis cannot make an “unsafe” structure to become a “safe” structure. Examples in Section 6 show that for typical frames, the effects due to P-Δ, P-δm and Pδi only results in negligible differences in the section utilization from these methods.

8.3

Conventional analysis followed by moment amplification method, though termed as first order analysis, is called as “second order analysis” in most of other national codes, as it has included both P-Δ-δ effects in its analysis or design.

8.4

Most commercial softwares do not include the P-δi effect in their analysis, and it has to be included by designers using the design charts in their respective code. NIDA, which was tailor-made for HK Code, is the only one software available in SEB that is capable of including the P-δi effect in its analysis. However, PSE should set the initial member imperfections δi in the appropriate directions in order to obtain the correct result. Examples in Section 6 further show that for typical frames the differences by neglecting such imperfections in the analysis is negligible.

8.5

In view of the above, for frames with λcr5 all softwares available in SEB can be used. For frames with λcr<5, both SPACE GASS and NIDA are pre-accepted softwares by Buildings Department to carry out second order elastic P-Δ-δ (direct) analysis. When using NIDA, limitations and/or settings in Section 7 should be noted. PSE should particularly note that the initial member imperfection is set on the major principal axis, and in some cases (e.g. members subjected to pure axial load) such initial member imperfection should be chosen to be set on the minor axis.


Page 33 of 35


References AISC (2010), AISC 360-10: Specification for Structural Steel Buildings (Chicago: AISC) British Standard Institution (2005), BS EN 1993:2005 - Eurocode 3: Design of Steel Structures (London: BSI). British Standard Institution (2000), BS 5950-1:2000 - Structural Use of Steelwork Design of Steelwork in Building (London: BSI). Brown, D. (2011), “Member Imperfections”, New Steel Construction, 19(8), pp.28-30 (available: www.newsteelconstruction.com; accessed: 28 August 2013). Buildings Department (2005), Code of Practice for the Structural Use of Steel 2005 (Hong Kong: Buildings Department). Buildings Department (2011), Code of Practice for the Structural Use of Steel 2011 (Hong Kong: Buildings Department). Buildings Department (2013), Explanatory Materials to Code of Practice for the Structural Use of Steel 2011 (Hong Kong: Buildings Department). Davison, B. and Owens, G.W. (2012), Steel Designers' Manual (Chichester, West Sussex, UK; Hoboken, NJ: Wiley-Blackwell, 7th ed.). Chan, S.L. (1988), “Geometric and Material Non-linear Analysis of Beam-Columns and Frames using the Minimum Residual Displacement Method”, International Journal of Numerical Methods in Engineering, 26, pp. 2657-2699. Chan, S.L. (2001), “Non-Linear behaviour and design of steel structures”, Journal of Construction Steel Research, 57(12), pp.1217-1232. Chan, S.L. and Chan, S.T.P. (2005), “Proper second-order and advanced analysis of steel frames”, Proceedings for Annual Seminar 2005, Joint Structural Division, 7 June 2005, Hong Kong Convention & Exhibition Centre, pp.53-66. Chan, S.L. and Zhou, Z.H. (1995), “Second-order elastic analysis of frames using single imperfect element per member”, Journal of Structural Engineering, American Society of Civil Engineers, 121(6), pp.939-945. Chan, S.L. and Zhou, Z.H. (1998), “On the development of a robust element for secondorder ‘non-linear integrated design and analysis (NIDA)”, Journal of Constructional Steel Research, 47(1), pp.169-190. Chan, S.L. and Zhou, Z.H. (2000), ”Non-linear integrated design and analysis of skeletal structures by 1 element per member”, Engineering Structures, 22, pp.246-257.


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Highlights on Second order Analysis for Structural Steel

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