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
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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
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
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
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 3-5: 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.
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
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 3-7: 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
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
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
NZS3404Proposed MethodActual CurveLay's Equation
(a) = 1 (b) = 0.5
alpha b = 0 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
NZS3404Proposed MethodActual CurveLay's Equation
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
(e) = -1
Figure 5-5: Comparison of proposed EYC procedure for αb = 0.5
35
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
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
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
NZS3404Proposed MethodActual CurveLay's Equation
(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
NZS3404Proposed MethodActual CurveLay's Equation
(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
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