Final Report

Effect of Residual Stresses On the Design of Columns In Steel Frames

By Arthur Yen-Cheng, Lu

Third Professional Year Project

Department of Civil Engineering

University of Canterbury

Supervisors:

Associate Prof. Gregory MacRae

PhD Candidate Brian Peng

Co-Supervisors:

Prof. Ronald D Ziemian

Dr. Christopher E Hann

i

Abstracts

This project investigates the effect of residual stresses on the stiffness, EI, of beam-

column members in steel frames. The effective stiffness, (EI)eff, is quantified as a

function of the applied axial force from the column buckling curves in the NZ steel

code. To obtain the stiffness reduction factor (SRF) relationship, simple empirical

equations, suitable for design, were then developed to describe the SRF relationship.

Direct analysis, using the SRF equations, was described and implemented into

analysis software for design of steel frame. The proposed analysis method is

compared with the Appendix F method specified in the steel code. It was found that

the proposed procedure is simple to use and resulted in more economical section than

that from the Appendix F method.

Additionally, a procedure was developed to discourage column flexural yielding

occurring away from the member ends. It was carried out by considering stiffness

reduction effects and second-order effects separately. It was shown that the proposed

procedure is less conservative than the current code procedure; however, it produces

relatively accurate values when compared with the actual solutions.

ii

Acknowledgements

I would firstly thank to everyone who has made vital contributions to my project.

Without their timely help, the project could not have been a success. I would specially

thank my supervisors, Associate Professor Gregory MacRae and PhD candidate Brian

Peng. Thanks for your support and encouragement throughout this entire semester. I

really appreciate all the ideas and advice given to me.

I would also like to express my thanks to Dr. Christopher Hann and Professor Ronald

Ziemian. Thanks to your technical advises. This project went smoothly for me.

I also like to thank my classmate, Jwin-Hxen Tan, for the time to help on the general

matters.

iii

Table of Contents

Abstracts ........................................................................................................................ i

Acknowledgements ...................................................................................................... ii

Table of Contents ....................................................................................................... iii

List of Figures ............................................................................................................... v

List of Tables ............................................................................................................... vi

1. Introduction ............................................................................................................ 1

1.1 Statement of the Problem ................................................................................ 1

1.2 Objectives of the Project ................................................................................. 2

1.3 Report Outline ................................................................................................. 2

2. Literature Review .................................................................................................. 3

2.1 NZS3404 Steel Design Procedure .................................................................. 3

2.2 Direct Analysis Method .................................................................................. 6

2.3 End-Yielding-Criteria (EYC) Equation .......................................................... 8

2.4 Effect of Residual Stresses on Stiffness .......................................................... 9

2.5 Column Buckling Curves .............................................................................. 10

3. Stiffness Reduction Factor (SRF) ....................................................................... 12

3.1 Derivation of SRF ......................................................................................... 12

3.2 Actual SRF values by MATLAB® analysis ................................................. 13

3.3 Proposed Equation for SRF .......................................................................... 15

3.3.1 Comparison of Proposed Equations ........................................................ 15

3.3.2 Development Equation for Coefficient c ................................................ 18

3.3.3 Summary ................................................................................................. 19

4. Direct Analysis Procedure ................................................................................... 20

4.1 Description of Direct Analysis Procedure .................................................... 20

4.2 Proposed Direct Analysis Method ................................................................ 20

4.2.1 Constraints for Direct Analysis ............................................................... 20

4.2.2 General Analysis Procedure .................................................................... 21

4.3 Implementing into Analysis Software .......................................................... 21

4.3.1 Description of MASTAN2 ...................................................................... 22

4.3.2 Modifications made in MASTAN2 ........................................................ 22

4.3.3 Verification of MASTAN2 with Built in SRF value .............................. 23

4.4 Comparison with Current Analysis Method ................................................. 23

iv

4.4.1 Description of Example Models ............................................................. 23

4.4.2 Description of Comparison Techniques .................................................. 24

4.4.3 Results and Discussion ........................................................................... 25

5. End-Yielding-Criteria (EYC) Procedure ........................................................... 27

5.1 Description of Propose EYC Procedure ....................................................... 27

5.1.1 Background Derivation of Eq 5-1 ........................................................... 27

5.2 Proposed EYC Procedure ............................................................................. 28

5.3 Results and Comparison ............................................................................... 29

6. Conclusions ........................................................................................................... 36

7. Further Research Recommendations ................................................................. 37

8. References ............................................................................................................. 38

9. Appendices ............................................................................................................ 39

9.1 Appendix 1 – Matlab codes ............................................................................. 39

9.1.1 Matlab code for Stiffness Reduction Factor ............................................... 39

9.1.2 Matlab code for End-yielding-criteria Procedure ....................................... 41

9.2 Appendix 2 – Table of Stiffness Reduction Factor .......................................... 44

9.3 Appendix 3 – Derivation of SRF equation ...................................................... 46

9.4 Appendix 4 – Procedure of verifying MASTAN2 ........................................... 47

9.5 Appendix 5 – Spreadsheet of column check for Appendix F method ............. 49

v

List of Figures Figure 2-1: Graphical illustration of Appendix F method ............................................. 3

Figure 2-2: Stiffness Modification Factor given in US ................................................. 7

Figure 2-3: Determination of critical axial force for yielding to occur away from

member end. ................................................................................................................... 9

Figure 2-4: Typical residual stress pattern for I-Section ............................................. 10

Figure 2-5: Reduced effective of cross sectional area ................................................. 10

Figure 2-6: NZS3404 Column Buckling Curve ........................................................... 11

Figure 3-1: Illustration of SRF derivation ................................................................... 12

Figure 3-2: Stiffness Reduction Factor for NZS3404 .................................................. 14

Figure 3-3: SRF values obtained from different sections ............................................ 14

Figure 3-4: Comparison of SRF1 with actual SRF ...................................................... 16

Figure 3-5: Comparison of SRF2 with actual SRF ...................................................... 16

Figure 3-6: Differences between best-fit and approximate value of c ......................... 18

Figure 3-7: Comparison of SRF by final proposed equation and actual values .......... 19

Figure 4-1: Authors’ information and copyright for MASTAN2 ................................ 22

Figure 4-2: Modal uses for comparison: Rectangular Frame ...................................... 24

Figure 4-3: Modal uses for comparison: Rectangular Frame ...................................... 24

Figure 5-1: Deflection shape of a beam-column member ........................................... 28

Figure 5-2: Comparison of proposed EYC procedure for b = -1 .............................. 31

Figure 5-3: Comparison of proposed EYC procedure for b = -0.5 ........................... 32

Figure 5-4: Comparison of proposed EYC procedure for b = 0 ............................... 33

Figure 5-5: Comparison of proposed EYC procedure for αb = 0.5 ............................. 34

Figure 5-6: Comparison of proposed EYC procedure for αb = 1 ................................ 35

vi

List of Tables Table 2-1: Coefficients for each b cases ..................................................................... 8

Table 3-1: Best-fit coefficients for SRF1 (Eq 3-9) ....................................................... 16

Table 3-2: Best-fit coefficients for SRF2 (Eq 3-11) ..................................................... 16

Table 3-3: Difference of SRF1 and actual SRF over N*/Ns = 0 - 1 ............................. 17

Table 3-4: Difference of SRF2 and actual SRF over N*/Ns = 0 - 1 ............................. 18

Table 3-5: Difference of final equation and actual SRF over N*/Ns = 0 - 1 ............... 19

Table 4-1: Verification of Direct Analysis by MASTAN2 ......................................... 23

Table 4-2: Comparison of Appendix F and Direct Analysis Method for Test (i) ........ 25

Table 4-3: Comparison of Appendix F and Direct Analysis Method for Test (ii) ...... 26

1

1. Introduction

1.1 Statement of the Problem

The behaviour of steel moment type frames under non-seismic loading is affected by

both geometric and material non-linearity. These non-linearity effects generally

considered either by first order elastic analysis, second order elastic or plastic analysis

with corrections. In addition to the material non-linearity effects represented by

plastic hinges, residual stresses arising from differential cooling of the section

elements acting in conjunction with applied stresses can cause inelastic action which

decreases the stiffness of steel members. This residual stress effect is not generally

considered in analysis for design except in steel column axial strength equations. By

ignoring effects of residual stress or by not considering them properly, the computed

frame displacements may be underestimated and the distribution of forces throughout

the frame may not represent reality.

The American Institute of Steel Construction (AISC) has recently developed a

stiffness modification factor to consider effects of nonlinearities due to residual

stresses. This resulting modified stiffness is incorporated into a “direct analysis”

procedure. The procedure avoids empirical correction factors, and the need to

evaluate in-plane effective lengths, thereby resulting in a more transparent, logical

and rapid design approach. However, the modification factor developed based on the

US column design curve cannot be implemented directly into the Australian/NZ

standard as the design column curves are different. Therefore, there is a need to

develop a new set of reduced stiffness factors based on NZ standard.

Effects of residual stresses have recently been considered in the development of the

NZ steel code (NZS3404:1997 Amendment 2, 2007), in an End-yielding-criteria

(EYC) equation. This equation discourages yielding occurring away from the column

ends by providing limits on the axial forces. This equation is rather complex because

it considers second-order effects and residual stresses together. In addition, the

equation was derived empirically based on the analytical results which are quite

conservative in some cases. Therefore, it may be possible to obtain, a more

transparent and more accurate equation/procedure if these effects are considered

separately.

2

1.2 Objectives of the Project

From above description, it may be seen that there is a need to firstly develop a

rational analysis method for the design of frames in non-seismic regions considering

residual stresses effects on the member stiffness. Moreover, it is desired to develop a

more accurate and transparent equation or analysis procedure to discourage plastic

hinges forming along the member length.

In order to achieve these objectives, the project aims to answer the following issues:

1. Can a simple relationship for column stiffness reduction equation

considering residual stresses be developed for the NZS3404 standard?

2. Can a direct analysis procedure using the relationship obtained from (i) be

developed and implemented into a computer program for design of steel

frames?

3. How different are frame responses using NZS3404 Appendix F method and

the direct analysis procedure from (ii)?

4. Can the column stiffness (i) be used to improve and simplify the EYC

equations? If this is possible, what is the procedure and how good is it?

1.3 Report Outline

A general overview of the current steel frames analysis method in NZ steel code and

direct analysis method in the US steel code is described in the Chapter 2. The EYC

equation in the NZ code is also discussed.

The development of the stiffness reduction factor (SRF) considers residual stress

effect is described in Chapter 3.

Chapter 4 described the proposed procedure for steel frame analysis by using the

SRF values and how the procedure can be implemented into MASTAN2. The

comparison of the proposed procedure is also discussed in this chapter.

Chapter 5 presents the new EYC procedures as well as the comparison with the

current EYC equation. Chapter 6 gives an overall conclusion of this project

3

2. Literature Review

2.1 NZS3404 Steel Design Procedure

The code provides a simple analysis method, Appendix F method, based on first-

order elastic analysis with modification factors in an attempt to simulate the second

order effects (clause 4.4.3). This method will be used to compare with the direct

analysis stiffness approach. In the code specified approach, the maximum moments of

members (Mm*) is obtained using the first order elastic analysis that neglects the

effects of change in the geometry and reduction in stiffness due to axial force. This

effect of change in geometry is later accounted by amplifying the moments with

amplification factor. The amplified bending moments for the steel frame by Appendix

F method in the code are summarized below:

i. Find the bending moment for frame at braced, Mfb, at braced and sway states,

Mfs, using first-order elastic analysis. This is illustrated in Figure 1.

Figure 2-1: Graphical illustration of Appendix F method

ii. Determine the axial forces subjected to the columns for both braced and

sway cases and the lateral deflections for the sway case only.

iii. Calculate the sway amplification factor specified in clause 4.4.3.3 as

described below.

iv. Calculate the braced amplification factor specified in 4.4.3.2 as described

below.

v. The design amplified bending moment (M*) can be calculated as:

( )fssfbb MMM δδ +=* (2-1)

Original Frame

Braced Frame

For Mfb

Sway Frame For Mfs

= +

N*3

N*2

N*1

V*3

V*2

V*1

N*3

N*2

N*1

V*3

V*2

V*1

R3

R2

R1

R3

R2

R1

4

where

bδ = Moment amplification factor for braced frames.

sδ = Moment amplification factor for none braced frames.

fbM = Maximum bending moment from 1st order elastic analysis if

frame is braced.

fsM = Maximum bending moment from 1st order elastic analysis if

frame is none braced (sway).

The braced amplification factor specified in clause 4.4.3.2 is given in Eq 2-2.

( ) 1/1 * ≥

−=

omb

mb NN

cδ (2-2)

where

ombN = elastic buckling load 2

2

)( LkEIe

π=

E = elastic modulus of member.

I = second moment of area of member

ek = member effective length factor.

mc = mβ4.06.0 − for member subjects to end moment only.

tβ4.06.0 − for member subjects to loading along length.

mβ = the ratio of smaller to larger bending moments at the ends of

member, taken as positive when member is bent to reverse

curvature.

tβ = -1 or matching the distribution of bending moment of member to

Figure 4.4.3.2 in NZS3404

The sway amplification factors specified in clause 4.4.3.3 are different for different

types of frames. For rectangular frames that do not include earthquake load, the factor

is calculated using Eq 2-3.

( )( ) 1

//195.0

** ≥ΣΣΔ−

=VNhss

sδ (2-3)

5

where

sΔ = Lateral displacement from elastic analysis at top relative to the

bottom story

sih = Height of story which is considered.

*VΣ = Sum of all horizontal forces above considered story.

*iN = design axial force in a column at story i.

For non-rectangular frames that do not include the earthquake load, the factor is

calculated as:

( )

1/1195.0

≥−

=c

si λδ (2-4)

where

cλ = Elastic buckling load factor determine from clause 4.9.2.4 of

NZS3404.

Different equations are used to calculate cλ depending on the frame connection at

the base.

For pinned-based frames

[ ]rrecr

prc sNhNs

EI** 3.0

3+

=λ (2-5)

For fixed-based frames

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=

pc

ecf

pr

rr

fc

IhN

IsN

E2*2* 25

105ψ

ψλ (2-6)

where

fψ = epr

rpc

hIsI

*cN = maximum design axial compression force in the outer column.

*rN = Design axial compression force in the outer rafter into the

column carrying *cN .

6

pcI = Second moment of area of outer column.

prI = Second moment of area of rafter.

eh = Height to centre-line of rafter at the knee

rs = Centre-line length of rafter, from knee to apex.

2.2 Direct Analysis Method

Appendix 7 of American Institution of Steel Construction Specification addresses

the direct analysis method for the structural systems. Section 7.1, general

requirements, states that the required strengths for the members, connections and

other structural elements shall be determined using second-order elastic analysis with

constraints. In addition, all component and connection deformations that contribute to

the lateral displacement of the structure shall be also considered in the analysis.

Section 7.2 specifies that notional loads shall be applied to the lateral framing

system to account for the effects of geometric imperfections and any non-linearity

effects. The notional loads shall be applied at each framing level in the direction that

adds to the destabilizing effects under the specified load combination.

Section 7.3 gives the constraints requirement for the direct analysis method. Those

constraints are:

i) Any general second-order analysis methods are permitted but they shall consider

both the global and local P-delta effects. Analyses shall be conducted according

to the design and loading requirements specified in either section B3.3 (LRFD) or

section B3.4 (ASD).

ii) A notional load determined by Eq 2-7 should be applied independently in two

directions as a lateral load in all load combinations and in addition to other lateral

loads. The notional load coefficient, 0.002, is based on an assumed initial story

out-of-plumbness ratio of 1/500.

ii YN 002.0= (2-7)

7

where

iN = notional lateral load applied at level i, kN

iY = Gravity load from LRFD load combination or 1.6 times the ASD

load combination applied at level i.

iii) Reduced flexural stiffness, (EI)eff, must be applied to all the members whose

flexural stiffness is considered to contribute to the lateral stability of the structures.

The reduced flexural stiffness is obtained by multiplied a modification factor,

� to the unreduced flexural stiffness, EI. This factor is a function of axial

force and it is found by an exact equation given by equation C-C2-12 (Eq 2-8) or

an approximated equation in Appendix 7 (Eq 2-9) in the AISC Specification. It

should note that Eq 2-9 is less conservative than Eq 2-8. (See Figure 2-2)

( )[ ] 39.0/39.0/

/ln/724.21

>

≤

⎩⎨⎧

−=

yr

yr

yryra PP

PPPPPP

τ (2-8)

( )[ ] 5.0/5.0/

/1/41

>

≤

⎩⎨⎧

−=

yr

yr

yryrb PP

PPPPPP α

α

αατ (2-9)

where

rP = Required axial compressive strength under load combinations.

yP = Asfy, member yield strength.

α = 1.0 (LRFD) or 1.6 (ASD)

US Stiffness Modification Factor

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr/Py

Stiff

ness

mod

ifica

tion

fact

or (t

au)

tau = atau = b

Figure 2-2: Stiffness Modification Factor given in US

8

2.3 End-Yielding-Criteria (EYC) Equation

Clause 8.3.4.2 in the New Zealand Steel Structural Standard, NZS3404:1997,

specifies that a compact doubly symmetric I-section member which is assumed to

contain plastic hinges shall satisfy clause 8.4.3.2.1.

( )( )( )

λ

β

βφ ⎭

⎬⎫

⎩⎨⎧ +

≤+1/

*

*1

DC

B

s eeA

NN (2-10)

where

*N = design axial force.

sN = nominal section capacity for axial force.

φ = strength reduction factor

β = the ratio of the end moments (smaller over larger moments),

negative for single curvature.

λ = the slenderness factor, EF

rLor

NN y

ol

s

π

r = radius of gyration of member

olN = 22 / LEIπ

L = the actual length of the member

A, B, C and D are the constant coefficients which vary depending on the section

type, b as shown in the table 2-1.

Table 2-1: Coefficients for each b cases b A B C D

1 2472 0.95 9.26 0.21

0.5 2472 0.91 9.21 0.19

0 2472 0.88 9.15 0.19

-0.5 2472 0.92 9.14 0.17

-1 2472 0.87 9.10 0.19

The equation is based on research carried out by Peng (2006). The equation was

developed by curve fitting the analytical results. The analysis is based on stability

9

functions which consider second-order effects from the axial force and first-order

deformation; and the inelastic stiffness which considers loss in stiffness with a given

slenderness and load ratio. The analytical procedure is summarised below (See Figure

2-3).

Figure 2-3: Determination of critical axial force for yielding to occur away from

member end.

2.4 Effect of Residual Stresses on Stiffness

Residual stresses, r, are presented in the steel due to the uneven cooling. For an I-

section, a typical residual stress pattern is shown in Figure 2-4 with negative as

compression stress. With this effect, the flexural stiffness, EI, decreases as the axial

loading increases that causes by the early yielding of sections. This is due to the

effective area of a steel section decreased as the axial force increases as results in

decrease in the second moment of area, I, as illustrated in Figure 2-5. For an elasto-

plastic material, the effective flexure, (EI)eff, stiffness can be calculated by Eq 2-11. It

should note that dark lines indicates yielding on the section whereas the solid line

indicates the effective area and the stress diagrams in Figure 2-5 are for flange only

whereas web will yield at centre first instead of tips.

1) Choose a b and a member with following properties:

L, As, fy, E and I

2) Start with a smaller value of N*

3) Calculate the effective stiffness, EIeff

4) Calculate stability function and hence obtain MB for one member

5) Is MA ≤ MB

7) Obtain the critical N*

Yes

6) Add approximately 0.01% of axial capacity to N*

No

10

eeff EIEI = (2-11)

where

E = section elastic modulus.

eI = effective second moment of area.

Figure 2-4: Typical residual stress pattern for I-Section

Figure 2-5: Reduced effective of cross sectional area

2.5 Column Buckling Curves

The column buckling curves (Figure 2-6) specified in section 6.3 of NZS 3404 are

derived based on the tangent modulus theory and the probabilistic analyses of column

strength (Bjorhovde 1972), with an initial out-of straightness of 0.001 and accidental

+ + +

= = =

p = y � r

p = y

p > y

�

y y

y -2 r

y y

y - r

Applied Stress, p

Total Stress, t

�

�

�

�

y

�

Effective Area

Yielded Area

Residual Stress� r

― ― +

― ― +

― ― +

�r �r

�r

�r

�r

�r

�r

�r

�r

11

eccentric loading (Galambos and Structural Stability Research Council, 1988). The

five different curves represent the five different compression section constants, b,

depending on the initial residual stresses in the member. The values of b, rang from

-1 to 1, with 1 giving the lowest strength. The effective stiffness also depends on the

member slenderness ratio, �

Tangent modulus buckling theory indicates that the critical buckling strength, Ncr,

can be determined by Eq 2-12 when the flexure stiffness of the elastic section is

reduced due to the residual stresses to a level that will cause buckling according to

Euler’s Equation. Note that eL is the effective length and ( )effEI is from Eq 2-11.

( )

2e

effcr L

EIN

π= (2-12)

NZS 3404 Column Buckling Curve

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Red

uctio

n Fa

ctor

(N/N

s)

alpha b = 1alpha b = 0.5alpha b = 0alpha b = -0.5alpha b = -1

Figure 2-6: NZS3404 Column Buckling Curve

12

3. Stiffness Reduction Factor (SRF)

This chapter gives the detailed descriptions of analytical procedure of the SRF

equation. It firstly describes the sources of SRF and the background of derivation as

well as the analysis tool used. Results from the analysis and the final proposed SRF

equation is given in the end of the chapter.

3.1 Derivation of SRF

The stiffness reduction factor is the ratio of effective flexural stiffness, (EI)eff, to the

elastic flexural stiffness, EI. For a member with given effective length, Le, the

nominal column force, Ncr,, and the Euler buckling force, Nom, can be determined by

Eq 3-1 and 3-2.

2

2

eom L

EIN π= (3-1)

( )2

2

e

effcr L

EIN

π= (3-2)

Therefore, the SRF can be determined by the ratio of Ncr to Nom. This is illustrated

by Figure 3-1 below.

( )

om

creff

NN

EIEI

FactorSRF == (3-3)

Graphical illustration of SRF derivation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Red

uctio

n Fa

ctor

(N/N

s)

alpha b = 0Euler Buckling

Figure 3-1: Illustration of SRF derivation

Nom

Ncr

(=√(Ns/Nom)

13

For a given nominal column force, Ncr, and section member constant, b, the

relative slenderness, n, can be determined from the NZS 3404 column design curves

using an analytical iteration technique. This relative slenderness ratio is then used to

obtain the Euler buckling force, Nom, by Eq 3-4. The derivation of Eq 3-6 is shown

below:

2

22

22

2 /⎟⎠

⎞⎜⎝

⎛===LrEA

LAIEA

LEINom ππ

π (3-4)

250

22 yn

frL⎟⎠

⎞⎜⎝

⎛=λ (3-5)

Substituting Eq 3-5 into Eq 3-4

⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

2

250 n

yom

fEAN

λπ (3-6)

The iterative procedure for determination of the relative slenderness ratio is carried

out by MATLAB®. The Matlab codes are attached in the Appendix 1.

The above derivation shows that the SRF values can be obtained directly from the

NZS3404 column curves. It should be noted that these curves not only consider the

effects of residual stresses, but also initial out-of-straightness and accidental eccentric

loading.

3.2 Actual SRF values by MATLAB® analysis

The stiffness reduction factors (SRF) are given in Figure 3-1 and a table of list of

SRF values is attached in Appendix 2 for each member constant, b. The steel

section, 100UC14.8, was used as a dummy member for analysis. Figure 3-1 clearly

shows that:

i. When axial force ratio approaches 0, SRF values approach to 1 for all 5

cases. This means that there is no reduction in member stiffness.

ii. At very high axial force ratio, i.e. N* = Ns, the SRF values is 0 for all cases.

This because the section is yielded everywhere and flexural stiffness is zero.

14

iii. An axial force ratio, b = 1, gives the smallest SRF values while b = -1

gives the largest SRF value. This is agreed to that b = 1 gives the lowest

strength as it is governed by the residual stresses.

Another section, 150UB14, with different yielded strength is used to check the SRF

values do not affect by member size and material properties. This is shown in Figure

3-2. As all the lines and points of each b coincide with each other, the SRF values

are not affected by the section and material properties and they can be used for any

general sections. Note that the results were obtained by MATLAB®, but it was

plotted in Excel for better visualization.

Stiffness Reduction Factors for NZS3404 Column Curves

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

Axial Force Ratio, N*/N s

SR

F (=

EIt/

EI)

-1 -0.5

0 0.5

1

αb

Figure 3-2: Stiffness Reduction Factor for NZS3404

Verification for non-dimensionality of SRF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial force ratio, (N*/Ns)

SRF

(EIt/

EI)

100UC14.8 ab = -1100UC14.8 ab = 0.5100UC14.8 ab = 0100UC14.8 ab = -0.5100UC14.8 ab = 1150UB14 ab = -1150UB14 ab = -0.5150UB14 ab = 0150UB14 ab = 0.5150UB14 ab = 1

Figure 3-3: SRF values obtained from different sections

15

3.3 Proposed Equation for SRF

The form of equation to represent the SRF obtained from the column buckling

curves was developed by Dr Christopher Hann from the Mechanical Engineering

Department, University of Canterbury. The development procedure was carried out

using Maple® and it is attached in Appendix 3. The equation is expressed in the form

below:

( ) bxcxSRF p

p

+−−−

−=)4(1

4α (3-7)

44bhbchc −+−=α

(3-8)

where

x = axial force ratio, N*/Ns.

b = desired SRF value at x = 1, normally is 0.

h = desired SRF value at x = 0, normally is 1.

c = coefficient required to be found out.

p = power required to be found out.

Two different forms of SRF equation, with different complexity, are proposed

based on the initial equation, Eq3-7. The first proposed equation, Eq 3-9, depends on

both coefficient c and power, p. The second proposed equation, Eq 3-9, is only

depended on the coefficient c. For both equations, h in Eq 3-8 is set to be 1. Value b

for Eq 3-9 is initially as -0.02 and for Eq 3-11 is set as 0.

( )( ) 02.014114

1 −−+−

= p

p

xcxaSRF (3-9)

255.002.1 += ca (3-10)

and )1(1

12 xcxSRF−+

−= (3-11)

3.3.1 Comparison of Proposed Equations

Trial and error was used to determine the coefficients, c and p. Table 3-1 and 3-2

summarizes the results of each coefficient for Eq 3-10 and 3-11.

16

Table 3-1: Best-fit coefficients for SRF1 (Eq 3-9) SRF 1

alpha b -1 -0.5 0 0.5 1 Coefficient c 8.5 3 1.2 0.477 0.071

Power, p 0.25 0.32 0.44 0.57 0.74

Table 3-2: Best-fit coefficients for SRF2 (Eq 3-11) SRF 2

Alpha b -1 -0.5 0 0.5 1 Coefficient c 8.9 2.9 0.9 0.2 -0.1

Comparison of proposed equation 1 and actual SRF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axial Force Ratio, N*/Ns

SRF 1

NZS3404 alpha b = -1NZS3404 alpha b = -0.5NZS3404 alpha b = 0NZS3404 alpha b = 0.5NZS3404 alpha b = 1Proposed alpha b = -1Proposed alpha b = -0.5Proposed alpha b = 0Proposed alpha b = 0.5Proposed alpha b = 1

Figure 3-4: Comparison of SRF1 with actual SRF

Comparison of proposed equation 2 and actual SRF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axial Force Ratio, N*/Ns

SR

F (

=E

It/E

I)


Figure 3-5: Comparison of SRF2 with actual SRF

17

The following observation can be down from the comparison above:

For b = 1, both equations are non-conservative at high axial force ratios, N*/Ns

(N*/Ns > 0.75 for Eq 3-9 and N*/Ns > 0.7 for Eq 3-11). Comparing both equation at

low axial force ratio, say N*/Ns < 0.2, Eq 3-9 is better than Eq 3-11 as Eq 3-11 is

non-conservative. However, Eq 3-11 gives closer values to the actual SRF at rest of

axial force ratio. Note that the value is conservative (i.e. on the safe side) if the

proposed value is lower than the actual curve.

For b = 0.5, these two equations have opposite behaviour. For Eq 3-9, it gives

conservative values for axial force ratio less than 0.7 but Eq 3-11 gives conservative

values for axial force ratio greater than 0.3. The largest difference between Eq 3-11

and actual SRF values is about 1.71% where Eq 3-11 is bigger than the actual value.

For axial force ratios less than 0.7, Eq 3-9 is closer to the actual values; however, it is

non-conservative.

The behaviour for b = 0 and b = -0.5 is similar to b = 0.5 where Eq3-9 is not

conservative at high axial force ratio (say more than 0.7) but quite accurate for axial

force ratio below that. For Eq3-11, it does not give conservative values for axial force

ratio less than 0.3. However, it is quite conservative for N*/Ns > 0.3.

For b = 1, both equations are consistently conservative for all axial force ratio. But

Eq3-11 seems to be more accurate than Eq3-9 as the line and points are overlapped to

each other in Figure 3-4.

The table below shows the difference between the values from both equations and

actual SRF. It should note that positive value means that the actual value is higher

than the equation value which means that it is conservative.

Table 3-3: Difference of SRF1 and actual SRF over N*/Ns = 0 - 1 For Propose Equation 1, Eq3-10

b -1 -0.5 0 0.5 1 max 0.070 0.043 0.043 0.043 0.043 min -0.062 -0.050 -0.034 -0.014 0.001

18

Table 3-4: Difference of SRF2 and actual SRF over N*/Ns = 0 - 1

For Propose Equation 2, Eq3-12 b -1 -0.5 0 0.5 1 max 0.049 0.040 0.050 0.065 0.000 min -0.089 -0.016 -0.021 -0.012 0.000

From Table 3-3 and 3-4, it can be seems that Eq 3-11 is better than Eq 3-9 for b =

-0.5 to 1. In addition, Eq 3-11 is simpler and easy to use compared to Eq 3-9 since it

is only dependent on one variable, coefficient c.

3.3.2 Development Equation for Coefficient c

An equation is developed for coefficient c as a function of member constant, b

based on the values of c given in Table 3-2. The initial equation is shown given by Eq

3-12. It was then modified by trial and errors to find the most suitable coefficients for

the equation. The final equation for coefficient c is given by Eq 3-13 below. The

difference between the actual values and equation values are plotted in Figure 3-5.

( ) cbactcoefficien b += α*exp* (3-12) ( ) 35.08.1exp*5.1 −−= bctcoefficien α (3-13)

Relationship between coefficient c and alpha b

-1

0

1

2

3

4

5

6

7

8

9

10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Alpha b

Coe

ffici

ent c

Actual Value

Equation

Figure 3-6: Differences between best-fit and approximate value of c

From Figure 3-5, it clearly shows that there are small differences between the actual

values and the equation values for b = 0 to -1. Because of the differences, the

approximated SRF values will be different compared to use the best-fit coefficient c

values.

19

From Table 3-5, it is observed that the proposed equation is less conservative for b

= -1 and -0.5 (see Figure 3-7). When comparing with the SRF2 values, It can be seen

that the final proposed equation is more conservative for b = -0.5, 0 and 0.5.

Table 3-5: Difference of final equation and actual SRF over N*/Ns = 0 - 1 For Final Propose Equation

b -1 -0.5 0 0.5 1 max 0.050 0.023 0.050 0.053 0.023 min -0.085 -0.027 -0.007 0.000 -0.001

Comparison of Final Proposed Equation with Actual SRF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial Force Ratio, N*/Ns

SRF


Figure 3-7: Comparison of SRF by final proposed equation and actual values

3.3.3 Summary

The final proposed equation for stiffness reduction factor (SRF) is given as:

)*1(1

*

1

NsNc

NsN

SRF−+

−= (3-14)

And ( ) 35.08.1exp*5.1 −−= bc α (3-15)

where

*N = design axial force

Ns = section nominal axial strength

bα = member constant.

20

4. Direct Analysis Procedure

4.1 Description of Direct Analysis Procedure

The direct analysis procedure uses a realistic analysis of a realistic model of

structure. It is different from traditional analyses for design which often uses an

elastic and/or a first-order analysis with properties based on the gross nominal section

and straight members. These traditional approaches use empirical modification

factors to match with actual behaviour. If the frame in a direct analysis procedure

does not collapse under the applied loading, then, in general it is expected to behave

satisfactorily.

4.2 Proposed Direct Analysis Method

4.2.1 Constraints for Direct Analysis

The constraints requirements are same as the constraints specified for the US direct

analysis approach. These constraints are listed below.

i. The analysis software used in the direct analysis must consider the second-

order effects rigorously. It includes both P - and P - effects. Besides,

second-order analysis has to be used in the Direct Analysis approach.

ii. Notional loads must be applied independently in two directions as a lateral

load in all load combinations. The notional load is calculated independently

for each level by Eq 4-1 below. This is used to consider the initial out-of-

plumbness of the frame.

ii YN 002.0= (4-1)

where

iN = notional lateral load applied at level i, kN

iY = Gravity load from load combination applied at level i.

iii. The effective member stiffness, (EI)eff, must be applied to all the member that

may contributed to the frame lateral stability. The effective member stiffness

is obtained by multiplying the original member stiffness, EI, with SRF (Eq 4-

2).

( ) EISRFEI eff *= (4-2)

21

where

SRF = stiffness reduction factor, Eq 3-14

iv. To account for the possibility of undersize of section area, both the member

yield strength, fy, and elastic modulus, E, need to be multiplied by the safety

factor, φ.

4.2.2 General Analysis Procedure

The general procedure for direct analysis procedure is carried out in the following

steps:

i. Obtain the additional notional lateral loads for each level by Eq 4-1 and apply

them to the major direction of loading.

ii. Run the first-order elastic analysis with all the design forces including the

notional lateral loads. Obtaining the compression axial forces subject to each

member.

iii. Calculating the axial force ratio for each member and determine the reduction

stiffness factor (SRF) correspond to the axial force ratio by:

a. By graph or table – these give the actual SRF values.

b. By equations – SRF equation give approximated values.

iv. Multiply the SRF to the original stiffness value, EI, to determine the effective

stiffness, (EI)eff. The reduction factor, , should be applied to the yield

strength, fy, and elastic modulus, E.

v. Run the second-order analysis with the modified section and material

properties. Obtaining the bending moments, axial force and the deflections of

the frames.

vi. Repeat the step 2 to step 5 until sections are sufficient to carry the design

forces.

4.3 Implementing into Analysis Software

The SRF may be implemented into the second-order plastic analysis software quite

easily. This can be done by either inputting a table of actual SRF values or, even

simpler, just use the approximate SRF equations.

In this project, the direct analysis procedure has been implemented for us into a

computer programme, MASTAN2, by Associate Prof. Ronald D. Ziemian who is also

22

the developer of the programme. The strength reduction factor, , is still applied

manually to E and fy, but this can be easily done.

4.3.1 Description of MASTAN2

MASTAN2 is developed by Prof. Ronald D. Ziemian and Prof. William McGuire

(See Figure 4-1). It is an interactive graphics program that based on the MATLAB®.

It can perform first-order elastic analysis through to second-order inelastic analysis. It

is also capable of analysing a large variety of structures such as 2D, 3D frames or

trusses. The pre-processing options include definition of structural geometry, support

conditions, applied loads, and element properties. MASTAN2 also provides a range

of post-processing options such as interpretation of structural behaviour through the

deformation and force diagrams. In addition, it is also freely available. The most

special ability of MASTAN2 is that there is the opportunity to develop and

implement additional or alternative analysis routines for specific needs of project.

Figure 4-1: Authors’ information and copyright for MASTAN2

4.3.2 Modifications made in MASTAN2

From the information provides by Prof. Ronald D. Ziemian, only two files are

needed to be modified for implementing the direct analysis procedure into

MASTAN2.

Firstly, a file, NZ_tau.m, contains the table of actual SRF values is added to the

programme. Secondly, an additional routine is programmed into file called, el_stiff.m.

This sub-routine allows the programme to obtain the SRF value from the first file

corresponds to the current axial load ratio for each member automatically. While the

analysis is running, the new member stiffness is re-calculated for each load step.

23

4.3.3 Verification of MASTAN2 with Built in SRF value

The verification is done by testing a simply supported column subjects to the

compression axial force. The maximum compression axial forces obtained by hand

calculation from the column buckling curve is compared with MASTAN2. An

arbitrary section, 310UC137, is used for the testing. The member was subdivided into

eight elements to achieve accurate results. The detailed procedure is attached in

Appendix 4.

The results are given by the Table 4-1. Comparing the analytical values with the

actual values, it is observed that the differences between these two values are very

small for all six cases. As shown in the table, the maximum difference is about 0.21%,

so inclusion of the SRF into MASTAN2 is accurate.

Table 4-1: Verification of Direct Analysis by MASTAN2 Ncr (kN)

Elastic b = -1 b = -

0.5 b = 0 b =

0.5 b = 1 Actual 6494.2 4226.73 3918.19 3575.16 3216.30 2864.71

Analytical 6494 4230 3925.44 3580 3222 2870.80 Difference 0.2 3.27 7.24 4.84 5.70 6.09

% Difference 0.003 0.077 0.185 0.135 0.177 0.212

4.4 Comparison with Current Analysis Method

This section is aimed to answer the question “How different is the frame response

using NZS3404 Appendix F method and the proposed direct analysis method.” This

is done by comparing the response of two frames as shown in Figure 4-2 and 3-3.

These analyses are performed based on the MASTAN2 and an Excel spreadsheet,

Alternative Method, is used to check the section capacity for the Appendix F method

is attached in Appendix 5.

4.4.1 Description of Example Models

Two models frames, a rectangular frame (Figure 4-2) and a portal frame (Figure 4-2),

are used to check the difference, if any, by using Appendix F method and Direct

Analysis method. For both frames, all the members were assumed to be:

i. Oriented with their webs in the plane of the frame (subjects to major axis

bending only).

24

ii. Columns are fully restraint in weak axis against out-of-plane buckling.

iii. The member constant, b, is 0.5.

Figure 4-2: Modal uses for comparison: Rectangular Frame

Figure 4-3: Modal uses for comparison: Rectangular Frame

4.4.2 Description of Comparison Techniques

Two tests are conducted for the comparison:

i. Comparing the required section size – for each model, section sizes of

members are determined by analyses procedure, Appendix F and direct

analysis method. In this case, the models are subjected to the same set of

design loads as shown in the Figure 4-2 and 4-3.

fy = 300MPa E = 200 GPa

For Test 1 Wind, W1= 45kN

P = 1000kN

12m

5.5 m

P P w1

Load Combination: 1.2G + 1.5Q

For Test 2 Wind, W1 = unknown

P = 500kN

Column: 250UC72.9 Beam: 200UB18.2

G = 0.8 kN/m Q = 1.9 kN/m

E = 200 GPa Fy= 300 MPa Load Combination: 1.2G + 1.5Q

w1

w2

10 m 10 m

2.5 m

5 m

For Test 1 G = 3.65 kN/m Q = 7.3 kN/m

Wind, W1= 30kN Wind, W2= 20kN

For Test 2

G = 3 kN/m Q = 4 kN/m

Wind, W1= ? kN Wind, W2= ? kN

Column: 250UC89.5 Rafter: 410UB53.7

G, Q

25

ii. Comparing the maximum lateral loads can be carried with given section sizes.

In this case, the initial lateral wind loads are unknown. Test is carried out to

find the maximum lateral loads can be applied to the frames by both analysis

methods. For portal frames, the wind load 1 and wind load 2 are applied

proportional with each other at a 3:2 ratio to total lateral loads including the

notional load. The section sizes are given in the figure 4-2 and 4-3.

4.4.3 Results and Discussion

From the results of Test (i) (see Table 4-2), it is observed that the direct analysis

lateral displacement is larger than the values by Appendix F method. When

comparing the section size, both analyses give the same section size for the

rectangular frames but for the portal frame, direct analysis gives a smaller section.

For the design moment, the direct analysis gives larger moment for rectangular frame.

For portal frame, the design moment by direct analysis is smaller than the Appendix F

method. It may due larger displacements give larger P - forces. For axial forces,

both analysis methods give roughly the same values.

Table 4-2: Comparison of Appendix F and Direct Analysis Method for Test (i) Summary for Rectangular Frame

Direct Analysis Method Appendix F Method Section Size 310UC96.8 310UC96.8

Max. Moment 205 180.41 Axial Force 1024 1025

Lateral Sway (mm) 61.89 30.29

Summary for Portal Frame Direct Analysis Method Appendix F Method

Section Size 250UC89.5 310UC96.8 Max. Moment 330.2 339.05

Axial Force 156.9 159.7 Lateral Sway (mm) 81.8 31.09

Apex Vertical Deflection (mm) 157.1 80.33

The key features can be observed from results of Test (ii) (see Table 4-3) are:

i. Maximum allowable forces able to be carried by direct analysis are almost

twice the values by the applied Appendix F method.

ii. The resulting maximum bending moments by direct analysis is slightly larger

(approximately 10% for rectangular and 8% for portal) than the Appendix F

method.

26

iii. The design axial forces from these two methods are quite close in both cases.

iv. Frame deformation results for both frames show a significant difference

between the direct analysis and Appendix F methods.

Table 4-3: Comparison of Appendix F and Direct Analysis Method for Test (ii)

Summary for Rectangular Frame Direct Analysis Method Appendix F Method

Max. Lateral Force (kN) 132.5 70 Max. Moment 246.9 223.9

Axial Force 556.5 527 Lateral Sway (mm) 161.5 68.12

Summary for Portal Frame

Direct Analysis Method Appendix F Method Max. Lateral Force (kN) 190 100

Max. Moment 331.8 308.8 Axial Force 113.2 110

Lateral Sway (mm) 428.8 56.65 Apex Vertical Deflection (mm) 326.8 70.17

From the two examples studied in this project, the direct analysis predicts:

i. Larger frame deformation such as lateral displacements. This is due to the

consideration of effect of axial force that reduces the member stiffness

whereas Appendix F calculated the deformation based on the first-order

analysis.

ii. Higher design bending moments but similar axial forces

iii. Less time required to prepare and perform the analysis. It is because direct

analysis can be automatically processed by computer programme. On the

other hand, for the Appendix F method, it is needed to firstly obtain the

braced and sway moments by first-order analysis and calculate the amplify

moments. And at last, section must be checked by the beam-column

equations.

iv. Benefit of economy as requiring smaller section size and higher allowable

design forces than the Appendix F method.

It should be noted some of these benefits are due to the difference between plastic

and elastic analysis. Since the Appendix F uses superposition, it can only be used in

an elastic analysis sense. However, comparison is still valid since elastic, rather than,

plastic design is commonly used in NZ.

27

5. End-Yielding-Criteria (EYC) Procedure

5.1 Description of Propose EYC Procedure

End-Yielding-Criteria procedure is developed to improve the current EYC equations

(see Eq 2-10 in section 2.3). The main difference between the current equation and

the new procedure is that the way of considering the effects of material inelasticity

and the effect of end moments. As mentioned in the introduction, the current EYC

equation was developed the considering the effects together. On the contrary, the new

proposed EYC procedure considers these effects separately.

The two main equations used in the development of EYC procedure are the SRF

equation (Eq 3-14) in section 3-3 and Eq 5-1. The SRF equation is used to consider

the reduction in the member stiffness due to the axial force and residual stress.

Equation (Eq 5-1) is used to ensure the maximum moment occurs at the end of

member by limiting the applied axial force.

( ) 0cos ≤+ βθ (5-1)

EINL2

=θ (5-2)

5.1.1 Background Derivation of Eq 5-1

The derivative information of Eq 5-1 is extracted from section 160 in the Source

book for the Australian Steel Structures code, AS1250 by Lay. It states that the

horizontal deflection for a beam-column member (See Figure 5-1) subjects to an axial

compression force and end moments can be expressed as Eq 5-3 below.

( ) ⎥⎦

⎤⎢⎣

⎡+−⎟

⎠

⎞⎜⎝

⎛−+⎟⎠

⎞⎜⎝

⎛+=

Lz

Lz

Lz

PMv β

θθθθβ 1cos1sin

sincos (5-3)

Differentiate Eq 5-3 three times with respect to z. the maximum moment location

can be obtained by assuming d3v/dz3 equals to zero. The differentiation equation then

can be further simplified to Eq 5-4.

θ

θβθsincostan max

−−=⎟

⎠⎞

⎜⎝⎛

Lz (5-4)

28

To be able to ensure the maximum moments occurs at the end of column, the

location of maximum moment, zmax, must be zero. Substitute zmax = 0 into Eq 5-4, Eq

5-1 can be obtained.

Figure 5-1: Deflection shape of a beam-column member

5.2 Proposed EYC Procedure

To ensure that yielding will not occur away from the member end with a specified

flexural stiffness, EI, length, L, end moment ratio, β, which is subject to a factored

axial force demand, N*, where Ns is the section moment capacity, the following

criteria must be satisfied:

max** NN φ≤ (5-5)

where

max*N = maximum permitted axial compression force to prevent end

yields

φ = reduction factor

The maximum permitted axial compression force, max*N can be obtained by one of

the following method

a) Procedure method

i) Compute the stiffness reduction factor, SRF by Eq 5-5 or Table given in

appendix 2.

29

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ −+

−=

NsNc

NsN

SRF*11

*

1 (5-5)

Where

c = 35.0)8.1exp(5.1 −− bα

ii) Compute the reduced stiffness, effEI )( :

EISRFEI eff *)( = (5-6)

iii) Compute the θ as:

( )β−=θ −1cos (5-7)

iv) Instead of using the unreduced stiffness, EI, the reduced stiffness is used

to compute the maximum permitted value, N*max:

2

2

max* )(

LEI

N effθ= (5-8)

b) Directly by the equation

( )( ) ( )( )( ) ( )c

cNccNcNN sss

21411 2

max

+−++−++=

ωωω (5-9)

where

ω = ( )( ) ( )2

21cosL

EI effβ−−

Note that this gives exactly the same results as the method above.

5.3 Results and Comparison

To verify the new EYC procedure gives in previous section, it is compared with the

current NZS3404 EYC equation, the actual solution and Lay’s equation. The results

are presented in Figure 5-1 to 5-5 below.

30

For b = -1

From the Figure 5-1, it is shown that the new procedure gives accurate results when

it compares with the actual solutions. However, for axial force ratio greater than 0.7,

it is less conservative for all five cases. This may be due to the SRF equation that

gives larger SRF value than the true value. In addition, the differences between the

proposed procedure and actual results decrease as the end moment ratio decreases.

When comparing the proposed procedure with the current EYC equation, it shows

that the proposed procedures are much closer to the actual solutions for all

slenderness ratios for all axial force ratios. For = -1, as expected, no axial force can

be applied, N*/Ns = 0.

For b = -0.5

The results for b = -0.5 are pretty similar to b = -1 for all the cases (see Figure 5-

2). The only difference is that the proposed procedures are much accurate for high

axial force ratio. The differences between proposed and actual solutions are smaller

than the differences for b = -1. Comparing with the current EYC equation,

NZS3404, the new procedure are less restrictive except for = 0.5. In this case, the

proposed methods are more conservative than current equation.

For b = 0, 0.5 and 1

The results for these three cases are quite similar (see Figure 5-3 for b = 0, Figure

5-4 for b = 0.5 and Figure 5-5 for b = 1). When comparing the proposed

procedure with the current equation, the results of proposed method is less

conservative except case of = 0.5 when b = 0 and 0.5. For these two cases, the

proposed procedures are less restrictive than current equation.

Comparing the results of proposed procedure with the actual solutions, the

procedure method is quite accurate for most of cases except for = -0.5 of all three

member constant values. The results of these three cases are less conservative at very

high axial force ratio, N*/Ns > 0.9.

31

alpha b = -1 and beta = 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3lambda

N*/

ph

iNs

NZS3404Proposed MethodActual CurveLay's Equation

alpha b = -1 and beta = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(a) = 1 (b) = 0.5

alpha b = -1 and beta = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = -1 and beta = -0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(c) = 0 (d) = -0.5

alpha b = -1 and beta = -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(e) β= -1

Figure 5-2: Comparison of proposed EYC procedure for b = -1

32

alpha b = -0.5 and beta = 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = -0.5 and beta = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(a) = 1 (b) = 0.5

alpha b = -0.5 and beta = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3lambda

N*/

ph

iNs


alpha b = -0.5 and beta = -0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3lambda

N*/

ph

iNs


(c) = 0 (d) = -0.5

alpha b = -1 and beta = -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(e) = -1

Figure 5-3: Comparison of proposed EYC procedure for b = -0.5

33

alpha b = 0 and beta = 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = 0 and beta = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(a) = 1 (b) = 0.5


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = 0 and beta = -0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(c) = 0 (d) = -0.5

alpha b = 0 and beta = -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(e) = -1

Figure 5-4: Comparison of proposed EYC procedure for b = 0

34

alpha b = 0.5 and beta = 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = 0.5 and beta = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(a) = 1 (b) = 0.5

alpha b = 0.5 and beta = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = 0.5 and beta = -0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(c) = 0 (d) = -0.5

alpha b = 0.5 and beta = -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(e) = -1

Figure 5-5: Comparison of proposed EYC procedure for αb = 0.5

35


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = 1 and beta = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(a) = 1 (b) = 0.5


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


alpha b = 1 and beta = -0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

lambda

N*/

ph

iNs


(c) = 0 (d) = -0.5

alpha b = 1 and beta = -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3lambda

N*/

ph

iNs


(e) = -1

Figure 5-6: Comparison of proposed EYC procedure for αb = 1

36

6. Conclusions

The project was initiated with the aim of answering four questions given in the

introduction:

1. The Stiffness Reduction Factor (SRF) equation (Eq 3-14) proposed in this project

can be used to consider the reduced stiffness due to the axial force. As it is

derived based on the column buckling curves in NZS3404, the factor considered

the effect of residual stresses and initial out-of-straightness.

2. The direct analysis procedure developed in this project uses the SRF equations. It

is a simple analysis procedure that considers the both material and geometry non-

linearity effects by using the second-order analysis and reduced the member

stiffness by SRF. This procedure can be, and has been, implemented into the

computer programme MASTAN2 as part of this research project.

3. A comparison between frame designed by the direct analysis and by the Appendix

F method was carried out. It was found out that:

• The frame deformations by direct analysis are significantly larger than by

Appendix F method.

• When comparing the design actions between these two analysis procedures,

there is small difference in maximum bending moments but almost no

difference to the axial forces.

• Direct analysis is a lot faster than Appendix F method.

• Direct analysis is more economical than the Appendix F method as it leads

to smaller section sizes.

4. The proposed End-yielding-criteria procedure (EYC) to encourage flexural

yielding only at the member ends is more transparent and accurate than the

current EYC equation. It uses the SRF equation to consider the inelastic

behaviour of the member in conjunction with a simple closed form equation to

consider the second-order effects.

37

7. Further Research Recommendations

i. The direct analysis proposed in this project is only compared with Appendix F

method based on first-order elastic analysis. As direct analysis procedure is

classifies as second-order plastic analysis, it would be ideal to compare this

procedure with the plastic analysis specified in NZS3404.

ii. In this project, only two types of frames were considered by the proposed

direct analysis. Therefore, there is no guarantee that direct analysis procedure

is suitable for any frame types such as two-story frame. Hence, further

research can be focused on testing different types of frames to ensure the

direct analysis procedure provides consistent results.

38

8. References

Peng B. H. H., MacRae G. A., Walpole W. R., Moss P., and Dhakal R. 2006. “Plastic Hinge

Location in Columns of Steel Frames.” Civil Engineering Research Report,

Department of Civil Engineering, University of Canterbury, Christchurch, NZ.

Lay, M. G. (1975). Source book for the Australian steel structures code – AS1250, Australian

Institute of Steel Construction, Sydney.

Jose M. Martinez-Garcia. “Benchmark Studies to Evaluate New Provisions for Frame

Stability Using Second-Order Analysis,” Master Thesis, Bucknell University,

Lewisburg, Pennsylvania, USA.

American Institution of Steel Construction Inc (2007). Steel Construction Manual 13th edition.

16.1-196 - 16.1- 198, 16.1-247.

New Zealand Standard, NZS3404:1997, Steel Structure Standards. Standard New Zealand

McGuire, Gallagher and Ziemian, Matrix Structural Analysis, second edition. John Wiley and

Sons, New York, 2000.

Ziemian R. D., and McGuire W. “ Modified Tangent Modulus Approach, A contribution to Plastic Hinge Analysis.” American Society of Civil Engineers, Journal of Structural Engineering.

39

9. Appendices

9.1 Appendix 1 – Matlab codes

9.1.1 Matlab code for Stiffness Reduction Factor

function [EI_array, N_array] = reducek(As,fy,E) % This file is created by Arthur Lu % It is used to determine the reduced stiffness ratio (EIt/EI) % from NZS:3404 alpha c, versus slenderness ratio % Require to specify section area As(mm^2), yield stress fy(MPa),Young's % modulus E(MPa) and second moment of area I(mm^4) % Produce (EIt/EI) vs (N*/Ns) graph for alphab for -1,-0.5,0,0.5 and 1. clc; Ns =As*fy; col = 1; row = 1; for alphab = -1:0.5:1; % For 5 different alpha b curves for P = 0:(0.05/fy)*Ns:Ns; % For serious axial force for given alpha b if P == 0 rei = 1; N = 0; else [Nol]= elasticload(P,As,fy,E,alphab); % Find elastic load rei = P/Nol; % Calculate the ratio of reduced and initial stiffness N = P/Ns; if rei > 1 rei = 1; end end [EI_array(row,col)] = rei; [N_array(row,col)] = N; [P_array(row,col)] = P; row = row+1; end col = col+1; row = 1; end % Plot all the curves figure(1) plot(N_array,EI_array) xlabel('N*/Ns') ylabel('Reduce Stiffness Ratio, (EI)t/(EI)') title('Ratio of Reduce Member Stiffness Curve')

40

legend('alpha b = -1','alpha b = -0.5','alpha b = 0','alpha b = 0.5','alpha b = 1'); function [Nol]=elasticload(P,As,fy,E,alphab) % The file is written by Arthur Lu % This file was originally wriiten by Brian Peng % This is sub-function for reducek % It calculate EI effective by first calculating the lambda % by using iterative procedure Ns = As*fy; % Section Axial Force phab = alphab; lamn = 0.01; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; while Nc > P lamn = lamn+0.1; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; end lamn = lamn-0.1+0.01; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; while Nc > P lamn = lamn+0.01; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; end

41

lamn = lamn-0.01+0.001; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; while Nc > P lamn = lamn+0.001; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; end lamn = lamn-0.001+0.0001; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; while Nc > P lamn = lamn+0.0001; phaa = (2100*(lamn-13.5))/(lamn^2-15.3*lamn+2050); lam = lamn+phaa*phab; mui = max(0.00326*(lam-13.5),0); eph = ((lam/90)^2+1+mui)/(2*(lam/90)^2); phac = eph*(1-sqrt(1-(90/eph/lam)^2)); Nc = phac*Ns; end Nol = (pi^2)*E*As*fy/(250*lamn^2);

9.1.2 Matlab code for End-yielding-criteria Procedure

clc; clear; format long % This is the m-file for constructing the axial force and lambda curve % The member use to considering is 310UC137 % This EYC is obtained by the new method % This method uses the Proposed SRF equation 2!!!!! % Unit for Na is kN, Nb is kN % Section Property Input As = 17500; % Section area, mm2 Ix = 329e6; % x-axis second moment of area, mm4

42

fy = 300; % Reduced Yielding strength, phiNs, MPa Es = 200000; % Steel elastic modulus, MPa Ab = 0; % Section Calculation phi = 1; EI = Es*Ix/1000; % Section stiffness in kNmm2 Ns = As*fy/1000; % Section capacity in kN pNs = phi*Ns; % Reduced section capacity in kN c = 1.5*exp(-1.8*Ab)-0.35; % Other Inputs mlambda = 3; % maximum lambda value mL = mlambda*pi*sqrt(EI/Ns)/1000; % maximum member length in m row = 2; col = 1; % Obtain the maximum axial force for beta = 1:-0.5:-1 theta = acos(-beta); for L = 0.1:0.1:35; % Length in m Na = 0; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); while Na - Nb < 0.01 Na = Na + 0.01; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); end Na = Na - 0.01 + 1e-4; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); while Na - Nb < 1e-4 Na = Na + 1e-4; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); end Na = Na - 1e-4 + 1e-6; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); while Na - Nb < 1e-6 Na = Na + 1e-6; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); end

43

Na = Na - 1e-6 + 1e-10; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); while Na - Nb < 1e-8 Na = Na + 1e-10; x = Na/pNs; srfa = 1 - x/(1+c*(1-x)); Nb = (theta^2)*srfa*EI/((L*1000)^2); end [Na_array(row,col)]= Na; [Nb_array(row,col)]= Nb; [diff_array(row,col)]= Na-Nb; [maxN_array(row,col)]= Na/(Ns); [lambda_array(row,col)] = sqrt(Ns/((pi^2)*EI/((L*1000)^2))); [srfa_array(row,col)] = srfa; row = row + 1; end col = col + 1; row = 2; end [maxN_array(1,:)] = 1; plot(lambda_array,maxN_array) xlabel('Lambda') ylabel('N*/phiNs') legend('beta = 1','beta = 0.5','beta = 0','beta = -0.5','beta = -1') .

44

9.2 Appendix 2 – Table of Stiffness Reduction Factor

N*/Ns SRF for alpha,b of N*/Ns SRF for alpha,b of -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

0 1 1 1 1 1 0.005 1 1 1 1 1 0.505 0.9326 0.8235 0.712 0.5982 0.4827

0.01 1 0.999 0.9964 0.9937 0.991 0.51 0.9306 0.8207 0.7085 0.5938 0.4777 0.015 0.9978 0.9938 0.9898 0.9858 0.9818 0.515 0.9285 0.8179 0.7048 0.5894 0.4727

0.02 0.9949 0.9896 0.9843 0.9789 0.9736 0.52 0.9265 0.815 0.701 0.5849 0.4677 0.025 0.9926 0.986 0.9794 0.9727 0.9661 0.525 0.9244 0.812 0.6974 0.5804 0.4627

0.03 0.9907 0.9828 0.9749 0.9669 0.959 0.53 0.9222 0.809 0.6936 0.576 0.4576 0.035 0.9892 0.98 0.9708 0.9615 0.9522 0.535 0.92 0.8059 0.6897 0.5714 0.4526

0.04 0.9879 0.9774 0.9669 0.9563 0.9457 0.54 0.9176 0.8029 0.6858 0.5669 0.4476 0.045 0.9869 0.9751 0.9632 0.9513 0.9394 0.545 0.9152 0.7997 0.6819 0.5622 0.4425

0.05 0.9859 0.9729 0.9597 0.9465 0.9332 0.55 0.9128 0.7965 0.6779 0.5576 0.4375 0.055 0.9852 0.9708 0.9564 0.9419 0.9273 0.555 0.9103 0.7932 0.6738 0.5529 0.4325

0.06 0.9845 0.9689 0.9532 0.9373 0.9214 0.56 0.9077 0.7899 0.6698 0.5481 0.4274 0.065 0.9839 0.9671 0.95 0.9329 0.9156 0.565 0.905 0.7864 0.6657 0.5434 0.4223

0.07 0.9835 0.9653 0.947 0.9286 0.91 0.57 0.9023 0.7829 0.6614 0.5387 0.4173 0.075 0.983 0.9636 0.9441 0.9244 0.9044 0.575 0.8995 0.7794 0.6572 0.5337 0.4122

0.08 0.9826 0.962 0.9412 0.9202 0.8989 0.58 0.8966 0.7758 0.6529 0.529 0.4071 0.085 0.9823 0.9604 0.9384 0.9161 0.8935 0.585 0.8937 0.7722 0.6486 0.524 0.402

0.09 0.982 0.959 0.9356 0.912 0.8881 0.59 0.8907 0.7683 0.6441 0.5191 0.3969 0.095 0.9818 0.9575 0.9329 0.908 0.8828 0.595 0.8876 0.7646 0.6396 0.514 0.3918

0.1 0.9816 0.9561 0.9302 0.904 0.8775 0.6 0.8844 0.7607 0.6351 0.5091 0.3866 0.105 0.9815 0.9547 0.9276 0.9001 0.8723 0.605 0.8812 0.7568 0.6305 0.504 0.3816

0.11 0.9813 0.9533 0.925 0.8963 0.8671 0.61 0.8779 0.7526 0.6258 0.4989 0.3765 0.115 0.9812 0.952 0.9224 0.8924 0.862 0.615 0.8744 0.7485 0.621 0.4937 0.3714

0.12 0.9811 0.9507 0.9199 0.8886 0.8568 0.62 0.8708 0.7443 0.6163 0.4885 0.3662 0.125 0.981 0.9494 0.9174 0.8848 0.8517 0.625 0.8672 0.74 0.6113 0.4832 0.3611

0.13 0.981 0.9481 0.9148 0.8811 0.8467 0.63 0.8635 0.7357 0.6064 0.4779 0.3559 0.135 0.9809 0.9469 0.9124 0.8773 0.8416 0.635 0.8597 0.7312 0.6013 0.4725 0.3509

0.14 0.9808 0.9457 0.91 0.8736 0.8366 0.64 0.8558 0.7266 0.5963 0.4672 0.3456 0.145 0.9808 0.9445 0.9075 0.87 0.8316 0.645 0.8517 0.722 0.5912 0.4617 0.3405

0.15 0.9807 0.9432 0.9051 0.8663 0.8267 0.65 0.8478 0.7172 0.5859 0.4562 0.3354 0.155 0.9807 0.942 0.9026 0.8626 0.8217 0.655 0.8436 0.7124 0.5805 0.4507 0.3302

0.16 0.9807 0.9408 0.9002 0.8589 0.8167 0.66 0.8392 0.7075 0.5752 0.4451 0.3251 0.165 0.9806 0.9396 0.8979 0.8553 0.8118 0.665 0.8347 0.7025 0.5696 0.4394 0.3199

0.17 0.9806 0.9384 0.8955 0.8517 0.8069 0.67 0.8302 0.6973 0.5641 0.4337 0.3148 0.175 0.9805 0.9372 0.8931 0.8481 0.8021 0.675 0.8256 0.6922 0.5584 0.4279 0.3096

0.18 0.9805 0.936 0.8908 0.8445 0.7972 0.68 0.8207 0.6868 0.5527 0.4221 0.3045 0.185 0.9804 0.9348 0.8884 0.8409 0.7923 0.685 0.8158 0.6813 0.5468 0.4163 0.2993

0.19 0.9803 0.9336 0.886 0.8374 0.7875 0.69 0.8108 0.6757 0.5409 0.4104 0.2943 0.195 0.9803 0.9325 0.8836 0.8338 0.7826 0.695 0.8057 0.6701 0.5348 0.4044 0.2891

0.2 0.9803 0.9312 0.8812 0.8302 0.7778 0.7 0.8003 0.6643 0.5286 0.3985 0.284 0.205 0.9802 0.93 0.8789 0.8266 0.7729 0.705 0.7948 0.6584 0.5225 0.3924 0.2788

0.21 0.9801 0.9289 0.8765 0.823 0.7681 0.71 0.7893 0.6523 0.5161 0.3863 0.2737 0.215 0.98 0.9276 0.8741 0.8195 0.7633 0.715 0.7836 0.6461 0.5098 0.38 0.2686

0.22 0.9799 0.9264 0.8718 0.8159 0.7585 0.72 0.7777 0.6398 0.5033 0.3737 0.2635 0.225 0.9797 0.9251 0.8694 0.8124 0.7537 0.725 0.7718 0.6333 0.4966 0.3675 0.2584

0.23 0.9796 0.9239 0.8671 0.8088 0.7489 0.73 0.7656 0.6268 0.4899 0.3612 0.2533 0.235 0.9794 0.9227 0.8647 0.8053 0.7441 0.735 0.7592 0.6201 0.483 0.3547 0.2482

0.24 0.9793 0.9214 0.8623 0.8017 0.7393 0.74 0.7528 0.6132 0.4761 0.3483 0.2431 0.245 0.9791 0.9201 0.8599 0.7981 0.7346 0.745 0.7462 0.6063 0.469 0.3418 0.2381

0.25 0.9789 0.9189 0.8575 0.7945 0.7298 0.75 0.7394 0.5991 0.4619 0.3353 0.233 0.255 0.9787 0.9176 0.8551 0.791 0.725 0.755 0.7324 0.5918 0.4546 0.3286 0.228

0.26 0.9785 0.9163 0.8527 0.7875 0.7202 0.76 0.7252 0.5844 0.4472 0.322 0.223 0.265 0.9782 0.915 0.8503 0.7839 0.7154 0.765 0.7179 0.5768 0.4396 0.3152 0.2179

0.27 0.9779 0.9137 0.8478 0.7803 0.7107 0.77 0.7104 0.5691 0.4319 0.3085 0.2129 0.275 0.9776 0.9123 0.8454 0.7768 0.7059 0.775 0.7027 0.5611 0.4242 0.3017 0.2079

45

0.28 0.9773 0.9109 0.843 0.7731 0.7011 0.78 0.6947 0.553 0.4164 0.2948 0.2029 0.285 0.977 0.9096 0.8406 0.7695 0.6963 0.785 0.6867 0.5447 0.4084 0.288 0.198

0.29 0.9767 0.9082 0.838 0.766 0.6916 0.79 0.6784 0.5363 0.4001 0.2811 0.1931 0.295 0.9763 0.9068 0.8355 0.7623 0.6868 0.795 0.6698 0.5276 0.3919 0.274 0.1882

0.3 0.9758 0.9053 0.833 0.7587 0.682 0.8 0.6612 0.5188 0.3836 0.2671 0.1833 0.305 0.9755 0.9039 0.8305 0.7552 0.6773 0.805 0.6522 0.5099 0.375 0.26 0.1783

0.31 0.9751 0.9025 0.828 0.7515 0.6724 0.81 0.643 0.5007 0.3664 0.253 0.1735 0.315 0.9745 0.901 0.8254 0.7479 0.6677 0.815 0.6336 0.4913 0.3576 0.2458 0.1686

0.32 0.9741 0.8994 0.823 0.7442 0.6629 0.82 0.6239 0.4818 0.3488 0.2388 0.1638 0.325 0.9735 0.898 0.8203 0.7405 0.6581 0.825 0.614 0.472 0.3398 0.2316 0.1591

0.33 0.973 0.8964 0.8178 0.7369 0.6534 0.83 0.604 0.4619 0.3307 0.2245 0.1543 0.335 0.9724 0.8948 0.8152 0.7332 0.6485 0.835 0.5936 0.4518 0.3214 0.2173 0.1496

0.34 0.9719 0.8932 0.8126 0.7295 0.6438 0.84 0.583 0.4415 0.3121 0.2102 0.1449 0.345 0.9712 0.8916 0.8098 0.7259 0.639 0.845 0.5722 0.4309 0.3027 0.2031 0.1402

0.35 0.9706 0.8899 0.8073 0.7221 0.6342 0.85 0.561 0.4201 0.2931 0.1959 0.1356 0.355 0.9699 0.8883 0.8046 0.7183 0.6293 0.855 0.5497 0.409 0.2834 0.1888 0.131

0.36 0.9692 0.8866 0.8019 0.7147 0.6246 0.86 0.5381 0.3979 0.2736 0.1817 0.1264 0.365 0.9685 0.8848 0.7991 0.7109 0.6197 0.865 0.5261 0.3864 0.2638 0.1746 0.1218

0.37 0.9677 0.8831 0.7965 0.7071 0.6149 0.87 0.514 0.3748 0.2539 0.1676 0.1174 0.375 0.9669 0.8814 0.7937 0.7033 0.6101 0.875 0.5015 0.363 0.2439 0.1606 0.1129

0.38 0.9661 0.8796 0.7909 0.6995 0.6052 0.88 0.4887 0.3509 0.2338 0.1537 0.1085 0.385 0.9652 0.8778 0.7881 0.6957 0.6005 0.885 0.4758 0.3387 0.2237 0.1468 0.1041

0.39 0.9643 0.8758 0.7852 0.692 0.5956 0.89 0.4624 0.3261 0.2136 0.14 0.0998 0.395 0.9634 0.8741 0.7824 0.6881 0.5908 0.895 0.4489 0.3135 0.2034 0.1332 0.0956

0.4 0.9624 0.872 0.7795 0.6842 0.5859 0.9 0.4349 0.3007 0.1932 0.1266 0.0914 0.405 0.9613 0.8702 0.7765 0.6804 0.581 0.905 0.4208 0.2876 0.183 0.12 0.0872

0.41 0.9603 0.8681 0.7736 0.6764 0.5762 0.91 0.4062 0.2744 0.1729 0.1135 0.0831 0.415 0.9593 0.8661 0.7707 0.6726 0.5713 0.915 0.3916 0.2609 0.1628 0.1072 0.0791

0.42 0.9581 0.8642 0.7676 0.6686 0.5665 0.92 0.3764 0.2473 0.1528 0.1009 0.0751 0.425 0.957 0.862 0.7647 0.6646 0.5616 0.925 0.3611 0.2336 0.1429 0.0948 0.0712

0.43 0.9558 0.8599 0.7616 0.6606 0.5568 0.93 0.3454 0.2196 0.133 0.0888 0.0673 0.435 0.9546 0.8578 0.7586 0.6566 0.5518 0.935 0.3294 0.2055 0.1233 0.083 0.0636

0.44 0.9533 0.8556 0.7554 0.6526 0.5469 0.94 0.3131 0.1912 0.1137 0.0772 0.0599 0.445 0.9519 0.8534 0.7524 0.6485 0.5421 0.945 0.2965 0.1769 0.1044 0.0717 0.0563

0.45 0.9506 0.8511 0.7492 0.6445 0.5371 0.95 0.2794 0.1625 0.0953 0.0663 0.0528 0.455 0.9492 0.8489 0.746 0.6404 0.5322 0.955 0.262 0.1479 0.0865 0.0611 0.0494

0.46 0.9477 0.8465 0.7427 0.6363 0.5273 0.96 0.2441 0.1334 0.0779 0.0561 0.046 0.465 0.9463 0.8441 0.7395 0.6322 0.5224 0.965 0.2258 0.1187 0.0697 0.0513 0.0428

0.47 0.9447 0.8417 0.7362 0.628 0.5174 0.97 0.2069 0.104 0.0618 0.0466 0.0396 0.475 0.9432 0.8392 0.7328 0.6238 0.5125 0.975 0.1872 0.0895 0.0542 0.0422 0.0366

0.48 0.9415 0.8367 0.7295 0.6197 0.5075 0.98 0.1665 0.075 0.0471 0.0379 0.0337 0.485 0.9398 0.8341 0.7261 0.6154 0.5026 0.985 0.1446 0.0609 0.0404 0.0339 0.0309

0.49 0.938 0.8316 0.7227 0.6111 0.4977 0.99 0.1203 0.0472 0.0341 0.0301 0.0281 0.495 0.9363 0.829 0.7191 0.6068 0.4927 0.995 0.0915 0.0345 0.0284 0.0265 0.0256

0.5 0.9344 0.8263 0.7156 0.6025 0.4877 1 0 0 0 0 0

46

9.3 Appendix 3 – Derivation of SRF equation

47

9.4 Appendix 4 – Procedure of verifying MASTAN2

Checking procedure for MASTAN2_NZ with build in SRF

The checks is carried out by testing an simply supported column

1. Choose a dummy member for checking i.e. 310UC137.

a. Section Property: Depth Gross Area About x-axis

d Ag Ix Zx rx

mm mm2 mm4 mm3 mm

321 17500 3.29E+08 2.05E+06 137

About y-axis Member Length Material Property

Iy Zy ry L E G Fy

mm4 mm3 mm mm MPa MPa MPa

1.07E+08 6.91E+05 78.2 10000 2.00E+05 8.00E+04 300

2. Compare the elastic buckling load, Pcre, obtains from the MASTAN2 and

exact solution.

a. Calculate elastic buckling load by exact solution, ( )22

kLEIPcre

π=

b. Use Mastan2_NZ built in analysis “Elastic Critical Load” to find

elastic buckling load.

c. Results:

i. Exact Solution, Pcre = 6494 kN

ii. Mastan Solution:

No. of Elements Critical Load, Pcre (kN) Percentage of differences to

exact solution (%)

2 6543 0.75

4 6498 0.0585

6 6495 0.0154

8 6494 0

9 6494 0

From table above, Mastan gives accurate value when member is

divided into 8 elements.

3. Compare inelastic buckling load, Pcri, by exact solution and MASTAN2

a. Choose an alpha b value for testing ie b = -0.5

b. Use exact solution to obtain the inelastic critical load, Pcri.

48

i. Calculate lambda n value250y

n

frkL

=λ . Assume kf and k is 1

ii. Use column design curve to find alpha c value and Pcri = c Ns.

The Ns is section capacity which is equalled to Asfy.

c. Apply an axial compression force, N*, which requires to be greater

than Pcri. N* = 1.2*Pcri is used in this verification.

d. Analyse the column with “second order inelastic” and choose

“predictor-corrector” as solution type. Set incremental size to 0.001

and max number of increment to 1000.

e. Select the modulus, E, to the specified member constant, b, value

f. Run the analysis and obtain the applied load ratio. The failure load by

MASTAN2 can be calculated by equation below *NratioloadAppliedPfailure ×=

g. Replaced the previous applied axial force by the failure load and rerun

the analysis.

h. Repeat step f and g until the current Pfailure is same as the previous one.

This value is the failure load obtained by the MASTAN2.

i. Compare the value obtained from step h with the exact solution obtain

by the column curve.

49

9.5 Appendix 5 – Spreadsheet of column check for Appendix F method

Alternative Method According NZS3404 Input

Length of the beam Column L 5 m x axis

Effective buckling Length Le 5 Strength Reduction Factor 0.9

Design Moment M* 308.80 kNm

Design Axial Force N* 110 kN Check 1a - Major Principal Axis In-Plane Member Capacity

Plate element Slenderness Yield slenderness Plasticity slenderness limit

a)Flange 7.51 9 b)Web 22.72 45 Compact

kf 1

Nominal Section Moment Capacity Msx 344.40 kNm Section Moment Capacity phi Msx 309.96 kNm

Nominal Section Strength Nsx 3192 kN Section Strength phi Nsx 2872.8 kN Calculate buckling factor

Alpha b 0.5 Modified slenderness ratio 47.25

alpha a 19.91 lambda 57.20

Eta 0.14 Zeta 1.91

Alpha c 0.824

Nominal Ideal member strength Ncx 2630.31 Ideal member strength phi Ncx 2367.28

End Moment ratio m 1 Nominal Member Moment Capacity Mix 396.84 Member Moment Capacity phi Mix 357.15 Check for Major In-Plane Member Capacity (Mx* < phi Mix)

Results Major in-plane member capacity is Satisfied

Check 1b - Major Principal Axis In-Plane Section Capacity Nominal Section Moment

Capacity Mrx 344.40 Section Moment Capacity phi Mrx 309.96

Check for Major In-PlaneSection Capacity (Mx* < phi Mrx)

Results Major in-plane section capacity is Satisfied

Final Report

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Transcript of Final Report