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1 INRODUCTION
The design of steel structures by advanced analysisis emerging
as a practical design tool with the avail-ability of non-linear
analysis software, and U.S.technical committees are presently
considering the
adoption of advanced analysis techniques in upcom-ing
specifications. Advanced analysis captures im-portant non-linear
structural phenomena; foremostare geometric second-order effects,
frame stabilityand material yielding. The existing load and
resis-tance factor U.S. design (LRFD) specifications(AISC 1999)
approximate such effects through mo-ment amplification factors,
effective lengths andalignment charts. While the current
approximatemethods work well for a large class of steel
struc-tures, for others their application can be ambiguousand the
results over-conservative. Advanced analysis
directly captures system behavior and can thereforesimplify the
design process by eliminating the needfor individual member checks.
By more faithfullymodeling important structural phenomena,
advancedanalysis provides predictions of frame strength withgreater
accuracy than LRFD provisions. The im-proved accuracy of advanced
analysis can result inmore efficient structures with acceptable
reliability.
LRFD specifications enforce a target reliabilitythrough the use
of load factors and resistance fac-tors. Existing proposals for
design by advancedanalysis have used the load and resistance
factorsfrom the existing specifications with no explicit
probabilistic justification. Strength predictions byadvanced
analysis may have different means andvariances than those by LRFD,
thereby requiring adifferent value of to maintain the desired
target re-liability.
1.1 Existing advanced analysis design proposals
Ziemian et al. (1992a,b) analyzed a series of two-bay, two-story
planar frames and a 22-story, three-dimensional frame and showed
that design by ad-vanced analysis could save about 12% steel
byweight compared to design by the 1986 LRFD speci-fications. These
analyses captured discrete plastichinging and geometric
non-linearities. The resis-tance factor =0.90 was incorporated by
scaling theyield surface. A successful design required the
totalload at plastic collapse (frame strength) to equal or
exceed the total factored design load.Chen and Kim (1997) also
provide guidelines for
design with advanced analysis and present severalmodeling
approaches (e.g. notional load, reducedtangent modulus, semi-rigid
connections). No resis-tance factor is used, and again the design
conditionrequires the frame strength to exceed the
factoredloads.
Additional research on advanced analysis has fo-cused on the
development of analysis techniques andtools for frames which
exhibit complex non-linearbehavior. Galambos (1998) summarizes
variousanalysis and design techniques.
e a ty mp cat ons o a vance ana ys s n es gn o stee rames
S.G. Buonopane, B.W. Schafer & T. IgusaDept. of Civil
Engineering, Johns Hopkins Univ., Baltimore, MD 21218, USA
ABSTRACT: Advanced analysis methods are presently being
considered for adoption in U.S. design specifi-cations for steel
frames. Strength predictions by advanced analysis are expected to
be more accurate thanthose made by LRFD provisions, which can
ultimately lead to more efficient designs. This paper comparesthe
structural reliability of sixteen two-bay, two-story frames
designed by conventional LRFD methods tothose designed with
advanced analysis. Gravity loads and member yield stresses are
modeled as random vari-ables. Non-linear structural simulations are
used to generate strength distributions, from which reliabilities
are
estimated based on first-order methods and importance sampling.
Bias factors and resistance factors necessaryto achieve a specific
target reliability are calculated for each frame. The results
provide insight into the prob-abilistic differences between design
by advanced analysis and elastic-LRFD methods, and begin to
provideprobabilistic justification for resistance factors used with
advanced analysis.
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While general guidelines are available for the de-sign of steel
frames by advanced analysis, no previ-ous research has sought to
evaluate the structural re-liability of such frames or to
incorporate the effectsof randomness in design guidelines.
2 STRUCURAL RELIABILITY
For a structure with random strength (R) subjected to
random load (Q), the probability of failure is
( ) drdqqfrfqrIP QRf )()(,
= (1)
where fR and fQ are probability density functions(PDFs) of
strength and load and ( )qrI , is an indica-tor function which
takes the value 1 when the failurecondition is satisfied and 0
otherwise. Typically,Equation 1 cannot be solved in closed-form,
but isinstead estimated using analytical or numerical
tech-niques.
2.1 First-order LRFD format
The LRFD format enforces a target reliabilitythrough the load ()
and resistance factors () in thedesign equation
n
i
ii RQ (2)
The development of the LRFD code is presented indetail in
Ravindra and Galambos (1978) and Elling-wood et al. (1980). Using
the failure condition
( ) 0/ln QR , and assuming independent normal dis-tributions and
small variances forR and Q, results inthe first-order reliability
index
( ) 22ln QRmmFO VVQR += (3)
where the subscript m indicates a mean value; VR=coefficient of
variation (COV) ofR; and VQ=COV ofQ. For a given target
reliability, t, the resistancefactor is
( ) ( )Rtnm VRR 55.0exp = (4)
whereRm=mean of true resistance; andRn =nomi-nal resistance or
code strength.
2.2 Importance sampling
The probability of failure may be estimated throughMonte Carlo
sampling, thereby avoiding some of theassumptions of the
first-order method. However, forsmall Pf, a prohibitively large
number of samplesmay be required to achieve an estimate with
suffi-cient accuracy. Importance sampling can produce a
satisfactory estimate with fewer samples than naveMonte Carlo
sampling. Since the integral to be esti-
mated here has only two dimensions, importancesampling is an
appropriate technique. Its applicationin higher dimensional spaces
is more difficult andmay require additional refinements (Melchers
1999).
Equation 1 may be rewritten as
dxdyyxhyxh
yfxfyxIP XY
XY
QR
f ),(),(
)()(),(
= (5)
This integral may be estimated by sampling as
=
=
N
i iiXY
iQiRii
fyxh
yfxfyxI
NP
1 ),(
)()(),(1 (6)
where the samples (xi,yi) are drawn from the impor-tance
sampling density hXY. For this application weperform importance
sampling over the load dimen-sion only, hXY=fQ*. A normal
distribution is selectedfor Q
*with a mean equal to the average of the
means ofR and Q, and a COV equal to that ofQ.
3 FRAME STRUCTURES
The series of 16 steel frames analyzed is based onthose of
Ziemian (1990) typical of low-rise indus-trial structures. Figure 1
shows the geometry, sup-port conditions and loads for the 16 frames
ana-lyzed. The frames are labeled with the
followingnomenclature:
S, U: symmetric or unsymmetric geometry;P, F: pinned or fixed
base;50: 50 ksi (345 MPa) nominal yield strength
steel;
H, L: heavy or light gravity load;A, E: member sizes determined
by advanced
analysis or elastic LRFD.Member sizes for frames UP50HA and
UP50HE aregiven in Table 1; the member sizes of all frames
arelisted in Ziemian (1990).
U: 6.10 m
6.1
0m
4.5
7m
L: 16.42 kN/m H: 51.08 kN/m
FU: 14.63 m
C1 C2 C3
C4 C5 C6
B1 B2
B3 B4
S: 10.36 m S: 10.36 m
L: 32.84 kN/m H: 109.45 kN/m
P
Figure 1. Dimensions and loads of frames.
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Table 1. Member sizes for frames UP50HE and UP50HA.
UP50HE UP50HAMem-ber SI U.S. SI U.S.
C1 W310x28.3 W12x19 W310x21 W12x14C2 W360x196 W14x132 W360x147
W14x99C3 W360x162 W14x109 W360x122 W14x82C4 W250x17.9 W10x12
W250x17.9 W10x12C5 W360x162 W14x109 W360x162 W14x109C6 W360x162
W14x109 W360x162 W14x109B1 W690x125 W27x84 W690x125 W27x84B2
W920x201 W36x135 W920x201 W36x135B3 W460x60 W18x40 W460x60
W18x40
B4 W690x140 W27x94 W690x140 W27x94
The yield strengths of beams and columns, andgravity loads are
modeled as random variables.Other potential random effects, such as
residualstresses and geometric imperfections, are not con-sidered
in the current analyses.
3.1 Random yield strengths
Based on Galambos and Ravindra (1978) , the yieldstrengths are
modeled as a normal distribution
Fy~N(1.05 Fyn, 0.10) where the first parameter is themean; the
second, COV. More recent data reportedin FEMA (2000) indicates a
slightly smaller mean of1.04Fyn and COV of 0.08. Analyses were
performedwith the yield strength of all members both uncorre-lated
and perfectly correlated. Only the results forthe analyses with
uncorrelated Fy are presented here.
3.2 Random gravity loads
The design of these frames is controlled by gravityloading,
therefore only the load combination
nn LD 6.12.1 + is considered (Ziemian 1992a). Deadand live loads
are assumed to be equal. The totalnominal gravity load is Qn=1021
kN for the lightload case and Qn=3327 kN for the heavy. The
deadloads are normally distributed D~N(1.05 Dn, 0.10).The live
loads follow an extreme type I distributionL~ExI(Ln, 0.10)
(Ellingwood et al. 1980). The totalrandom gravity load, Q, is the
sum of four randomvariables, dead and live load on two stories. The
dis-tribution of Q cannot be expressed in closed formbut its PDF
can be computed through numerical in-
tegration. After normalizing by the total design loadQn, both
the light and heavy load cases have a meanof 1.026 and COV of
0.10.
3.3 Analysis details
Structural analyses were performed with OpenSees(McKenna &
Fenves 2001), including geometricnon-linear effects and an
elastic-perfectly-plasticmaterial model. Displacement-based
beam-columnelements with a cubic shape function were used.Columns
were subdivided into 4 elements, and
beams, 8. Cross-section yielding and axial-moment
interaction was captured with a fiber element model,integrated
at 4 points along the element length.
All members have their webs in the plane of theframe, and
out-of-plane behavior was restrained.Uniform gravity loads were
applied as equal concen-trated loads at all 9 nodes along the beam
lengths.Symmetric frames were given an initial out-of-plumb
imperfection of 1/400th of the building heightfor numerical
stability. No initial imperfection wasgiven to the unsymmetric
frames.
4 SIMULATION RESULTS
For each frame, 10,000 non-linear structural analy-ses were
performed with random yield strengths. Todetermine the distribution
of frame strength, theframes were loaded with an increasing gravity
loaduntil plastic collapse. For each simulation, the ap-plied load
at the occurrence of the first plastic hingewas also recorded. The
first plastic hinge strengthprovides an analog to the member-based
LRFD de-
sign criteria.Figure 2 shows the histogram of the frame
strength and first plastic hinge strength for UP50HE,both
normalized by the design load Qn, as well asthe PDF of the
normalized load. Table 2 lists themean and COV of the sampled
strengths for all 16frames with uncorrelated Fy. Those frames
designedby advanced analysis have smaller member sizes insome
locations, and therefore have smaller meanstrengths than the
corresponding elastically designedframe. The advanced analysis mean
frame strengthsrange from about 65-90% of mean strengths of the
LRFD designed frames; the first plastic hingestrengths range
from about 60-80%. The COVs ofthe first plastic hinge strength are
typically greater
0.5 1 1.5 2 2.50
1
2
3
4
5
Normalized Frame Strength
mean=1.927cov= 0.062
Q
UP50HE
0.5 1 1.5 2 2.50
1
2
3
4
5
Normalized Strength at 1st Plastic Hinge
mean=1.521cov= 0.08
Q
Figure 2. Strength distributions for frame UP50HE.
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Table 2. Parameters of strength distributions.
Frame Frame strength 1st plastic hinge
Mean COV Mean COV
UP50HA 1.575 0.051 1.133 0.080
UP50LA 1.679 0.059 1.189 0.079
UF50HA 1.709 0.055 1.179 0.084
UF50LA 1.739 0.050 1.178 0.077
SP50HA 1.716 0.061 1.237 0.088
SP50LA 1.673 0.069 1.178 0.086
SF50HA 1.744 0.064 1.239 0.085
SF50LA 1.654 0.064 1.136 0.077
UP50HE 1.927 0.062 1.521 0.080
UP50LE 2.011 0.066 1.620 0.075
UF50HE 1.942 0.053 1.505 0.075
UF50LE 2.075 0.065 1.560 0.077
SP50HE 2.027 0.058 1.614 0.077
SP50LE 2.653 0.071 1.892 0.078
SF50HE 1.875 0.079 1.556 0.072
SF50LE 2.219 0.062 1.578 0.075
1.4 1.6 1.8 2 2.2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frame Strength
St
rengthat1stPlasticHinge
UP50HE
Figure 3. Frame strength vs. 1st plastic hinge strength.
than those of frame strength. Also the design methodof the frame
does not greatly effect the COV of ei-
ther strength measure.Figure 3 plots frame strength against
first plastichinge strength for UP50HE, showing virtually
nocorrelation. A band of correlated samples is apparentnear the
upper left of the plot; it is likely that thissubset of samples has
a common failure mode. Nostrong correlation was observed for any of
theframes. The lack of correlation between these twoperformance
measures suggests that there is no sim-ple means of relating the
member-based failure cri-terion to the system-based criterion.
4.1 Reliability results
Table 3 lists the reliability for all 16 frames for bothframe
strength and first plastic hinge strength. Thereliability is
expressed as a probability of failure anda reliability index,
=-1(Pf) where is the stan-dard normal cumulative distribution
function. Thereliability by sampling is also compared to the
firstorder reliability, FO, computed from Equation 3.The
first-order estimate generally overestimates compared to the
sampling estimate, since the lower
tail of the strength distributions have a higher prob-ability
mass than the normal distribution assumedfor FO. The frames
designed by advanced analysishave a lower reliability index than
those designed byLRFD.
For the LRFD designed frames, FO for first plas-tic hinge is
greater than 3 for all frames. The target for steel beam-columns in
LRFD is approximately 3,and thus these analyses demonstrate that
the LRFDcode is effective in achieving this goal. For theseframes,
the sampling estimate of is also generallynear 3 as well.
For the frames designed by advanced analysis,the reliability
indices based on frame strength areabove 3 and thus could be
considered acceptable de-signs. The values offor first plastic
hinge are be-tween 0.84 and 1.44, corresponding to probabilitiesof
occurrence of up to 20%. This result suggests thatthe occurrence of
a plastic hinge in a frame designedby advanced analysis and
subjected to nominal grav-ity loads may not be an infrequent event.
The occur-rence of a plastic hinge may affect the
serviceabilityperformance, even though it does not compromise
overall strength and safety. This observation high-lights the
importance of serviceability issues inframes designed by advanced
analyses.
Table 3. Reliability by sampling and first-order methods.
Frame strength 1st plastic hinge
Frame Pf 10
-6
FO Pf 10
-3
FO
UP50HA 387. 3.36 3.79 200. 0.84 0.77
UP50LA 95.5 3.73 4.28 119. 1.18 1.17
UF50HA 83.9 3.76 4.48 137. 1.10 1.06
UF50LA 18.4 4.13 4.78 130. 1.13 1.10SP50HA 52.5 3.88 4.44 83.7
1.38 1.42
SP50LA 229. 3.50 4.01 140. 1.08 1.04
SF50HA 70.8 3.81 4.42 75.1 1.44 1.43
SF50LA 214. 3.52 4.07 196. 0.86 0.83
UP50HE 2.18 4.59 5.31 2.99 2.75 3.06
UP50LE 1.56 4.66 5.66 0.70 3.20 3.68
UF50HE 1.27 4.71 5.63 2.89 2.76 3.06
UF50LE 3.63 4.49 5.98 1.97 2.88 3.35
SP50HE 0.14 5.14 5.94 0.73 3.18 3.61
SP50LE 0.00 >8 7.90 0.03 4.05 4.90
SF50HE 14.6 4.18 4.76 1.52 2.96 3.40
SF50LE 0.00 6.16 6.62 1.20 3.03 3.48
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Elastic LRFD design controls the occurrence of thefirst plastic
hinge, while advanced analysis may al-low plastic hinging at
service load levels in caseswhere it does not compromise frame
strength. Theexistence of plastic hinges may also impact
struc-tural behavior for extreme load events, such as seis-mic,
since the structure cannot be assumed to beginin an elastic,
undamaged state.
5 RESISTANCE FACTORS FOR ADVANCEDANALYSIS
The general expression for the resistance factor hasbeen given
in Equation 4. The distribution of trueresistance is ideally based
on experimental data, andrelated to nominal resistance by
PMFRRn
= (7)
with mean and COV given by
mmmnmFMPRR = , 222
FMPRVVVV ++= (8)
where P, M and F are the random variables of pro-fessional,
material and fabrication factors, respec-tively; the subscript m
indicates a mean value andVR, VP, VM and VF are COVs of the random
vari-ables. The values used in the LRFD specificationswere
determined by experimental data, analyticalmodels and professional
judgment. For steel beam-columns, the LRFD values are Pm=1.02,
Mm=1.05,Fm=1.00, VP=0.10, VM=0.10, VF=0.05, VR=0.15. Us-ing these
values and t=3.00 in Equations 4 and 8,results in =0.84. The value
of =0.90 used in
LRFD corresponds to t=2.10.The goal of advanced analysis methods
is to pro-
vide predictions of strength which are closer to thetrue
strength than existing elastic-based code meth-ods. Since
experimental data are not available forthe frames analyzed here, we
assume the limitingcase that the distribution of true strength is
exactlypredicted by the advanced analysis with random ma-terial
properties. Practical design with advancedanalysis will be based on
a single analysis withnominal yield strengths, resulting in the
nominalstrength AA
nR . We relate the deterministic nominal
strength to the probability distribution of strength by
FBRRAAAA
n
AA= (9)
where BAA is the random variable bias factor and F isthe LRFD
fabrication factor, retained since ouranalyses do not consider
random geometric proper-ties. This equation is analogous to
Equation 7. With-out test data for frame strength it is not
possible de-termine individual bias factors equivalent to P, Mand
F.
Table 4 gives values of nominal strengths and
mean bias factors for frame strength and first plastic
hinge. The mean bias factor, AAm
B , is the meanstrength from Table 1 divided by AA
nR . Nearly all the
bias factors fall within the range of 0.95 to 1.05, andthe mean
of the bias factors is close to 1.0 indicatingthat AA
nR is a good predictor of AA
mR with little bias.
For comparison, the combined bias factor assumedin LRFD for
beam-columns is 1.07. Also important
Table 4. Nominal strengths and bias factors.
Frame Frame strength 1st plastic hingeAA
nR
AA
mB
AA
nR
AA
mB
UP50HA 1.566 1.006 1.137 0.997
UP50LA 1.625 1.033 1.185 1.004
UF50HA 1.674 1.021 1.173 1.005
UF50LA 1.717 1.013 1.211 0.973
SP50HA 1.686 1.018 1.244 0.994
SP50LA 1.671 1.001 1.202 0.980
SF50HA 1.767 0.987 1.268 0.977
SF50LA 1.672 0.989 1.220 0.931
UP50HE 1.884 1.023 1.550 0.981
UP50LE 1.945 1.034 1.708 0.948
UF50HE 1.928 1.007 1.557 0.966
UF50LE 2.042 1.016 1.594 0.979
SP50HE 1.960 1.034 1.654 0.976
SP50LE 2.654 1.000 1.937 0.977
SF50HE 1.798 1.043 1.664 0.935
SF50LE 2.244 0.989 1.697 0.930
min 0.987 0.930
mean 1.013 0.972
max 1.043 1.005
Table 5. COVs and resistance factors.
Frame Frame strength 1st plastic hinge
AA
RV
fort=3.00
fort=2.10
AA
RV
fort=3.00
fort=2.10
UP50HA 0.071 0.89 0.93 0.094 0.85 0.89
UP50LA 0.077 0.91 0.94 0.094 0.86 0.90
UF50HA 0.074 0.90 0.94 0.098 0.85 0.90
UF50LA 0.070 0.90 0.93 0.092 0.84 0.87
SP50HA 0.079 0.89 0.93 0.101 0.84 0.88
SP50LA 0.085 0.87 0.91 0.099 0.83 0.87
SF50HA 0.082 0.86 0.90 0.098 0.83 0.87
SF50LA 0.081 0.87 0.90 0.092 0.80 0.84UP50HE 0.080 0.90 0.93
0.094 0.84 0.88
UP50LE 0.083 0.90 0.94 0.090 0.82 0.85
UF50HE 0.073 0.89 0.93 0.090 0.83 0.87
UF50LE 0.082 0.89 0.92 0.092 0.84 0.88
SP50HE 0.076 0.91 0.95 0.092 0.84 0.88
SP50LE 0.086 0.87 0.90 0.093 0.84 0.88
SF50HE 0.094 0.89 0.94 0.088 0.81 0.84
SF50LE 0.080 0.87 0.90 0.090 0.80 0.84
min 0.86 0.90 0.80 0.84
mean 0.89 0.92 0.83 0.87
max 0.91 0.95 0.86 0.90
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is the observation that the bias factors appear inde-pendent of
the method by which the frame was de-signed.
Table 5 presents values of the variance of the ad-vanced
analysis strength prediction (includingVF=0.05) and values offor
both t=3.00 and 2.10.The COVs of the strengths are in the range of
0.07to 0.10, somewhat less than the LRFD value of 0.15.For a target
reliability of 3.00, the values ofrangefrom 0.80 to 0.91 including
both first plastic hinge
and frame strengths. For a target reliability of 2.10,the values
ofare slightly higher, 0.84 to 0.95.The resistance factor is
affected by the mean bias
and the variance of the strength distribution (Eq. 4).For the
frames analyzed here, the typical COV andbias factor of strength
was less than that assumed byLRFD. These differences offset one
another, result-ing in values ofwhich are approximately equal
tocurrent LRFD values. The incorporation of otherrandom variables
(e.g. residual stress, imperfec-tions), as well as professional
judgment, might jus-tify larger COVs of strength, in which case
smaller
values ofwould be necessary to maintain the sametarget
reliability.
6 CONCLUSIONS
Advanced analysis is emerging as the next-generation design tool
for steel structures. This pa-per summarized research into the
probabilistic char-acter of design by advanced analysis, due to
ran-domness in structural properties and loads. Based ona series of
16 steel frames with random yield
strengths and random applied gravity loads, non-linear
structural analysis simulations were per-formed to calculate
distributions of frame and firstplastic hinge strengths.
Frames deigned by advanced analysis had asmaller mean strength
than those designed byLRFD, since they contain smaller member
sizes.However, the calculated reliability indices of theframes
designed by advanced analysis were stillabove 3.0 based on the
failure condition of framestrength. Because advanced analysis
primarily con-trols frame strength, some frames exhibit
non-negligible probabilities of plastic hinging undernominal load
conditions. Occurrence of such hingesmay require greater attention
to serviceability crite-ria such as deflection and drift, as well
as considera-tion of a frames initial state when subjected to
ex-treme load events.
The resistance factors determined from thesesimulations are
generally in the range of 0.85 to0.95, suggesting that current
resistance factors maybe acceptable for design with advanced
analysis.However, these values depend on the variability of
the strength distributions, which may increase as ad-
ditional random effects are introduced into theanalysis.
Since advanced analysis is predicated on systembehavior, the
probabilistic results are dependent onthe peculiarities of a given
structures behavior. Theresults presented here are based on a group
of six-teen, closely-related steel frames, and the conclu-sions may
not be representative of other structures.Without requiring
explicit probabilistic analysis, oneof the greatest challenges of
design by advanced
analysis may be the selection of design coefficientswhich are
applicable to a wide range of structuralsystem behaviors.
ACKNOWLEDGEMENTS
This research has been supported in part by a Na-tional Science
Foundation Graduate Research Fel-lowship and National Science
Foundation Grant No.DMI-0087032.
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