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University College LondonDepartment of Civil, Environmental & Geomatic
Engineering
Advanced Seismic Designof Structures(CEGEG140/CEGEM140)
~ Coursework 4 ~
Student: Carmine Russo 1410!10"#nstru$tor: Carlos Molina %utt
&$ademi$ 'ear
01*1"
L+,-+,.S GL+&L U,#ERS#'
https://moodle.ucl.ac.uk/course/view.php?id=20463https://moodle.ucl.ac.uk/course/view.php?id=20463https://moodle.ucl.ac.uk/course/view.php?id=20463https://moodle.ucl.ac.uk/course/view.php?id=20463
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Sommario
#,R+-UC#+,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222!
C&LCUL&#+,S 3+R -#33ERE, ELEME,S2222222222222222222222222222222222222222222222222222222222222!
A) B1035.041: !E"#$!%!'DGE (*"B BEA+"C$+# -$'#% 2222222222222222222222222222222!
B) B1035.01A: EB* /EA! '#, #$ *$$! BEA+/, '# ( 100 *22222222222222222
C) C1011.001A: (A A!%'%'$#, %2E: G2/+ ('% +E%A /%D/, *E'G%, *'ED BE$(, *'ED AB$E2222222222222222222222222222222222222222222222222222222222222222222211
D) C303.001A: //E#DED CE''#G, /DC A, B6 A!EA 7A): A 50, E!% /$!% $#2 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221
5UES#+, & * 67at $7anges $ould you ma8e to $om9onents 1 t7roug7 4 to redu$edamage under an eual set o; engineering demand 9arameters
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Introduction
Using the results obtained from the previous analysis, performed on a moment resisting frame, in
the following steps we are going to calculate the damages and the economic losses that occur to
some particular structural components.
The aim of the exercise is to show the last part of a performance-based design procedure. Should be
noted that this could be either part of preliminary design, when the designer wants to calculate
different performances of alternative structural solutions to show to the (future) building owner or
can be part of a definitive design procedure when its necessary to assess the costs of eventual
damages or to label the building.
The damage levels have been evaluated by using the !"#T ( Performance Assessment Calculation
Tool ) database that gives some information about different Damage-State (DS) for various building
components. $ach damage state is associated with some degree of performance of the building.
%or each element, the probability of reaching a particular damage state has been calculated by using
the relative fragility curves. "lso, using the conse&uence functions, for each 'amage-State have
been obtained relevant repairing costs, the time to repair, and finally the total loss costs.
n the following table are reported the results obtained from the analysis performed
S9e$trummat$7ed
a$$elerograms
Linearly S$aleda$$elerograms
? @ ? @
Aea8 -is9la$ement Bm0.1521138
330.0405493
570.144319 0.040948
-is9la$ement Ratio Brad*#alculated considering that the height of the frame is +.m
0.043461095
0.011585531
0.0412340.0116994
29
Aea8 &$$eleration Bg0.4579471
630.0866762
260.485565
070.1003532
86
Rotation node Brad0.0364693
330.0096629
750.034548
060.0123033
39
Rotation node " Brad 0.042768367
0.013504671
0.03984613
0.011476659
Calculations for different elements
A) B1035.041: Pre-Northridge WUF-B beam-colum !oit
The fragility curves are derived in !"#T for the storey drift ratio and are showed in the following
image
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The mean values and the standard deviation for the analysis are
3or s9e$trally mat$7edeart7ua8es
3or linearly s$aled
μdrift =0.043461
σ drift =0.01159
¿ μdrift +σ drift =0.055047
μdrift −σ drift =0.031876
μdrift =0.041234
σ drift =0.011699505
¿ μdrift +σ drift =0 .052933451
μdrift −σ drift =0.029534441
This structural component consists of a post-/orthridge with a welded unreinforced flange-bolted
web and a beam on one side.
The database considers three damage states, named: DS1, DS2, DS3. The first two of these, are
subdivided into two mutually exclusive damage states the element can reach a damage, within the
particular damage-state considered, in only one of the two sub-damage state and both of these two
are associated with a certain probability.
-amageStates
-amageRe9resented
Mutually eD$lusivedamage state
Aroaility
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D S
Fracture of o!er"eam#ange
!ed andfaiure of !e" "ots$s%ear ta"
connection&' !it%
fracturescon(ned to
t%e !edregion.
D S1
Fracture of o!er "eam#ange !ed
and faiure of !e" "ots$s%ear ta"
connection&'!it% fractures
con(ned to t%e
!ed region.
0.75
D S1
Simiar to DS1'e)ce*t t%at
fracture*ro*agatesinto coumn
#anges.
0.25
D S
Fracture of u**er "eam #ange !ed'
!it%out DS1 t+*e damage. Fracture iscon(ned to "eam #ange region.
D S2
Fracture of u**er "eam#ange !ed'!it%out DS1
t+*e damage.Fracture iscon(ned to
"eam #angeregion.
0.75
D S2
Simiar to DS3'e)ce*t t%at
fracture*ro*agatesinto coumn
#anges.
0.25
D S
Fracture initiating at !ed access %oeand *ro*agating t%roug% "eam #ange'*ossi"+ accom*anied "+ oca "uc,ing
deformations of !e" and #ange.
none
0eading the values from the fragility curves we have
CasesS9e$trally mat$7ed
eart7ua8esLinearly s$aled
eart7ua8es
μdrift −σ drift
P DS3
=0.53
P DS2
=0.71−0.53=0.18
P DS1
=0.93−0.71=0.22
Pundmaged=1.0−0.93=0.07
P DS3
=0.49
P DS2
=0.68−0.49=0.19
P DS1
=0.92−0.68=0.24
Pundmaged=1.0−0.92=0.08
μdrift
P DS3=0.82
P DS2
=0.92−0.82=0.1
P DS1
=0.99−0.92=0.07
Pundmaged=1.0−0.99=0.01
P DS3=0.79
P DS2
=0.89−0.79=0.1
P DS1
=0.99−0.89=0.1
Pundmaged=1.0−0.99=0.01
μdrift +σ drift
P DS3
=0.93
P DS2
=0.96−0.93=0.03
P DS1
=1−0.96=0.04
Pundmaged=0
P DS3
=0.91
P DS2
=0.95−0.91=0.04
P DS1
=1−0.95=0.05
Pundmaged=0
The costs related to the different limit states considered have been extrapolated from the diagramsgiven in !"#T. The following figures show that costs decrease as a number of parts to be repaired
increases since fixed costs affect thousands more if the amounts are minor. %or this element costing
is on a per bay basis and does not include fireproofing removal or reapplication cost.
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Re9air Cost Conseuen$e ;or
DS1 I
Re9air Cost Conseuen$e ;or
DS1 II
Re9air Cost Conseuen$e ;or
DS2 I
Re9air Cost Conseuen$e ;or
DS2 II
Re9air Cost Conseuen$e ;or DS3
#onsidering that the cost is the higher possible, we have
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DS1
-e*air !i t+*ica+reuire gouging out andre/!eding of t%e "eam#ange !ed' re*air of
s%ear ta"' and re*acings%ear "ots.
Cost DS1 I =14472$
re*airs tocoumn !i "enecessar+ t%at
!i invovere*acing a
*ortion of t%e!eded *ates
Cost DS 1 II =16272 $
DS2
-e*airs !i "e simiar tot%ose reuired for DS1'e)ce*t t%at access to!ed !i i,e+ reuireremova of a *ortion of t%e #oor sa" a"ove t%e
!ed.
Cost DS2 I =20472$
n addition tocoumn
measures forDS3' re*airs tocoumn !i "enecessar+ t%at
!i invovere*acing a
*ortion of t%ecoumn #ange.
Cost DS2 II =21120$
DS3
-e*air is simiar to t%at for DS1 e)ce*t t%at a *ortion of t%e "eam !e"and #ange ma+ need to "e %eat straig%tened or re*aced.
Cost DS3=17472$
The total repairing cost can be calculated ta1ing into account the different probabilities related to
every damage state (also those related to the mutually exclusive damage states).Cost tot = P DS
1
∙ ( P DS1 I ∨ D S1
∙Cost DS1 I + P DS1 II ∨ D S 1∙Cost DS1 II )+¿+ P DS
2
∙ ( P DS2 I ∨ DS 2
∙Cost DS2 I + P DS2 II ∨ D S2 ∙Cost DS2 II )+ P DS3 ∙Cost DS3
2here we too1 in account also the conditional probabilities for mutual exclusive 'amage State
obtained from !"#T
-amage state 1 P DS1 I ∨ D S1
=0.75 P DS1 II ∨ D S1
=0.25
-amage State P DS2 I ∨ DS 2
=0.75 P DS2 II ∨ D S2
=0.25
Cases
analysedotal $ost ;or s9e$trally mat$7ed eart7ua8es otal
μdrift −σ drift Cost tot =0.22 ∙ (0.75 ∙14472$+0.25 ∙16272$ )+0.18 ∙ (0.75 ∙2047 16257$
μdrift Cost tot =0.07 ∙ (0.75 ∙14472 $+0.25 ∙16272$)+0.1 ∙ (0.75 ∙20472 17434$
μdrift +σ drift Cost tot =0.04 ∙ (0.75 ∙14472$+0.25 ∙16272$)+0.03 ∙ (0.75 ∙2047 17465$
Cases
analysed
otal $ost ;or linearly s$aled eart7ua8es otal
μdrift −σ drift Cost tot =0.24 ∙ (0.75 ∙14472$+0.25 ∙16272$)+0.19 ∙ (0.75 ∙2047 16063$
μdrift Cost tot =0.1∙ (0.75 ∙14472$+0.25 ∙16272$ )+0.2∙ (0.75 ∙20472$ 17358$
μdrift +σ drift Cost tot =0.05 ∙ (0.75 ∙14472$+0.25 ∙16272$ )+0.04 ∙ (0.75 ∙2047 17471$
The estimated repairing time has been derived from similar diagrams from !"#T database
considering the highest possible values
-amage State 1 T D S1 I =38.59days T D S
1 II =43.39days
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-amage State T D S2 I =54.59 days T D S
2 II =56.32days
-amage State ! T D S3
=46.6 days
2ith the same methodology used for the repairing costs, we have calculated the time
Cases
analysed
otal re9air time ;or s9e$trally mat$7ed eart7ua8es otal
μdrift −σ drift Timetot =0.22 ∙ (0.75 ∙38.59d+0.25 ∙43.39d )+0.18 ∙ (0.75 ∙54.59 43days
μdrift Timetot =0.07 ∙ (0.75 ∙38.59d+0.25∙43.39d )+0.1 ∙ (0.75 ∙54.59d 46.5days
μdrift +σ drift Timetot =0.04 ∙ (0.75 ∙38.59d+0.25 ∙43.39d )+0.03∙ (0.75 ∙54.59 46.6days
Casesanalysed otal re9air time ;or linearly s$aled eart7ua8es otal
μdrift −σ drift Timetot =0.24 ∙ (0.75 ∙38.59d+0.25 ∙43.39d )+0.18∙ (0.75 ∙54.59 42.3days
μdrift Timetot =0.1∙ (0.75 ∙38.59d+0.25 ∙43.39d )+0.2 ∙ (0.75 ∙54.59d+ 51.79days
μdrift +σ drift Timetot =0.05∙ (0.75 ∙38.59d+0.25 ∙43.39d )+0.04 ∙ (0.75 ∙54.59 46.6days
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B) B1035.0"1a: #BF $hear %i&' o (loor beam' li& * + 100 P%F
The component analysed is an eccentrically braced frame lin1. $ccentrically braced steel frames
($3%s) are very efficient structures for resisting earth&ua1es as they combine the ductility of that is
characteristic of moment frames and the stiffness associated with braced frames. n the $3%s
inelastic activity is confined to a small length of the floor beams which yields mostly in shear
(therefore called the shear lin1).
n this case, the fragility curves are derived for the plastic rotation hinges
3or s9e$trally mat$7edeart7ua8es
3or linearly s$aled
μrotation=0.036469333
σ rotation=0.009662975
¿ μrotation+σ rotation=0.046132
μrotation−σ rotation=0.026806
μrotation=0.034548067
σ rotation=0.012303339
¿ μrotation+σ rotation=0.046851
μrotation−σ rotation=0.022245
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-amageStates
-amage Re9resentedMutually eD$lusive damage
state
D S
Damage toconcrete
sa" a"ovet%e in,"eam.
none
D S
e" oca"uc,ing'
#ange oca"uc,ing.
none
D S
nitiation of fracture in
t%e in,"eam andin, #ange.
none
0eading the values from the fragility curves, we have
CasesS9e$trally mat$7ed
eart7ua8esLinearly s$aled
eart7ua8es
μrotation−σ rotation
P DS3
=0
P DS2
=0
P DS 1=0.05 Pundmaged=1.0−0.05=0.95
P DS3
=0
P DS2
=0
P DS1=0.045 Pundmaged=1.0−0.045=0.955
μrotation
P DS3
=0.005
P DS2
=0.025−0.005=0.002
P DS1
=0.33−0.025=0.305
Pundmaged=1.0−0.33=0.67
P DS3
=0.004
P DS2
=0.023−0.004=0.019
P DS1
=0.3−0.023=0.277
Pundmaged=1.0−0.3=0.7
μrotation+σ rotation
P DS3
=0.025
P DS2
=0.19−0.025=0.165
P DS1
=0.75−0.19=0.56
Pundmaged=1−0.75=0.25
P DS3
=0.03
P DS2
=0.19−0.03=0.16
P DS1
=0.76−0.19=0.57
Pundmaged=1−0.76=0.24
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#onsidering that the cost is the higher possible, from the we have
-amage State 1 Cost DS1=123900$ -e*ace concrete sa".
-amage State Cost DS2=132900 $eat straig%tening of "uc,edeements.
-amage State ! Cost DS3=188400 $ -e*ace F in,.
The total repairing cost can be calculated ta1ing into account the different probabilities related to
every damage state.
Cost tot = P DS1
∙Cost DS1+ P DS2
∙Cost DS2+ P DS3
∙Cost DS3
Cases analysed otal $ost ;or s9e$trally mat$7ed eart7ua8es otal
μrotation−σ rotation Cost tot =0.05∙123900$+0 ∙132900 $+0 ∙188400 $ 6195$
μrotation Cost tot =0.305 ∙123900$+0.002 ∙132900 $+0.005 ∙188400 38997$
μrotation+σ rotation Cost tot =0.56∙123900$+0.165 ∙132900$+0.025 ∙188400 $ 96022$
Cases analysed otal $ost ;or linearly s$aled eart7ua8es otal
μrotation−σ rotation Cost tot =0.045∙123900 $+0 ∙132900$+0 ∙188400$ 5576$
μrotation Cost tot =0.277 ∙123900$+0.019 ∙132900 $+0.004 ∙188400 37599$
μrotation+σ rotation Cost tot =0.57∙123900$+0.16 ∙132900 $+0.03 ∙188400$ 97539$
The estimated repairing for each damage state are
-amageState 1
T DS1=340.37 days -e*ace concrete sa".
-amageState
T DS2=364.61dayseat straig%tening of "uc,edeements.
-amageState !
T DS3=517.39days -e*ace F in,.
2ith the same procedure, we can calculate the total time repair
Cases analysedotal re9air time ;or s9e$trally mat$7ed
eart7ua8esotal
μrotation−σ rotation Timetot =0.05∙340.37d+0 ∙364.61d+0 ∙517.39 d 17days
μrotation Timetot =0.305 ∙340.37d+0.002 ∙364.61d+0.005 ∙517.39d 107days
μrotation+σ rotation Timetot =0.56 ∙340.37 d+0.165 ∙364.61d+0.025 ∙517.39d 264days
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Cases analysed otal re9air time ;or linearly s$aled eart7ua8es otal
μrotation−σ rotation Timetot =0.045 ∙340.37d+0 ∙364.61d+0 ∙517.39d 15days
μrotation Timetot =0.277 ∙340.37d+0.019 ∙364.61d+0.004 ∙517.39 d 103days
μrotation+σ rotation Timetot =0.57 ∙340.37 d+0.16 ∙364.61 d+0.03 ∙517.39d 268days
,) ,1011.001a: Wall Partitio' /e: /um *ith metal tud' Full eight' Fi2ed Belo*' Fi2ed
Aboe
4ypsum walls (illustrated in the picture below) are a type of lightweight, non-loadbearing,
partitions. 5etal framed partitions could be used in all types of new and existing buildings,
covering all applications, from simple space division, through to high-performance walls designed
to meet the most demanding fire resistance, sound insulation, impact and height re&uirements. This
1ind of wall partitions is easy to reconfigure with minimal impact to both building and occupants
resulting in less disruption, optimising the transformation process.
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The demand parameter for this (non-structural) component is the storey drift ratio
3or s9e$trally mat$7edeart7ua8es
3or linearly s$aled
μdrift =0.043461
σ drift =0.01159
¿ μdrift +σ drift =0.055047
μdrift −σ drift =0.031876
μdrift =0.041234
σ drift =0.011699505
¿ μdrift +σ drift =0.052933451
μdrift −σ drift =0.029534441
The database considers three damage states the second and the third of these are subdivided into
two mutually exclusive damage states.
-amageStates
-amageRe9resented
Mutually eD$lusivedamage state
Aroaility
D S Scre!s *o*/out' minor crac,ing of !a"oard' !ar*ing or crac,ing of ta*e.
none none
D S
oderate crac,ing or crus%ing of g+*sum !a "oards $t+*ica+ in cornersand in corners of o*enings&.
D S2
oderatecrac,ing orcrus%ing of
g+*sum !a"oards
$t+*ica+ incorners and in
corners of o*enings&.
0.8
D S2
oderatecrac,ing orcrus%ing of
g+*sum !a"oards
$t+*ica+ incorners and incorners of o*enings&..
0.2
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D S
Signi(cant crac,ing andor crus%ing of
g+*sum !a "oards/ "uc,ing of studsand tearing of trac,s.
D S3
Signi(cantcrac,ing andor
crus%ing of g+*sum !a
"oards/"uc,ing of studs andtearing of
trac,s.
0.8
D S3
Signi(cantcrac,ing andor
crus%ing of g+*sum !a
"oards/"uc,ing of studs andtearing of
trac,s.
0.2
0eading the values from the fragility curves, we have
CasesS9e$trally mat$7ed
eart7ua8esLinearly s$aled
eart7ua8es
μdrift −σ drift
P DS3
=0.98
P DS2
=1−0.98=0.02
P DS1
=0
Pundmaged=0
P DS3
=0.96
P DS2
=1−0.96=0.04
P DS1
=0
Pundmaged=0
μdrift
P DS3
=1
P DS2
=0
P DS1
=0
Pundmaged=0
P DS3
=1
P DS2
=0
P DS1
=0
Pundmaged=0
μdrift +σ drift
P DS3
=1
P DS2
=0
P DS1
=0
Pundmaged=0
P DS3
=1
P DS2
=0
P DS1
=0
Pundmaged=0
The costs related to the different limit states considered have been extrapolated from the diagrams
given in !"#T. #onsidering that the cost is the higher possible, we have
-S1
-eta*e oints' *aste and re*aint "ot% sides of t%e 50/foot engt% of !a"oard.
Cost DS3=2730$
-S
-emove 25/foot engt% of !a"oard$"ot% sides&' insta ne! !a "oard $"ot%
sides&' ta*e' *aste and re*aint.
-emove fu 100/foot engt% of !a"oard$"ot% sides&' insta ne! !a "oard $"ot%
sides&' ta*e' *aste and re*aint.Cost DS2 I =5190$ Cost DS 2 II =19800$
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-S!
-emove and re*ace a 25/foot engt% of t%e meta stud !a' "ot% sides of t%eg+*sum !a"oard and an+ em"eddedutiities' and ta*e' *aste and re*aint.
-emove and re*ace a fu 100/foot engt%of t%e meta stud !a' "ot% sides of t%eg+*sum !a"oard and an+ em"eddedutiities' and ta*e' *aste and re*aint.
Cost DS3 I =7940$ Cost DS3 II =31100$
The total repairing cost can be calculated ta1ing into account the different probabilities related toevery damage state (also those related to the mutually exclusive damage states).
Cost tot = P DS1
∙Cost DS1+ P DS2
∙( P DS2 I ∨ D S2
∙Cost DS2 I + P DS2 II ∨ D S2∙Cost DS2 II )+ P DS
3
∙ ( P DS3 I ∨ DS 3
∙Cost DS3 I + P DS3 II ∨ DS 1 ∙Cost DS3 II )
2here we too1 in account also the conditional probabilities for mutual exclusive 'amage State
obtained from !"#T
-amage state P DS2 I ∨ D S2
=0.80 P DS2 II ∨ D S2
=0.20
-amage State ! P DS3 I ∨ DS 3=0.80 P DS3 II ∨ DS 1=0.20
Casesanalysed
otal $ost ;or s9e$trally mat$7ed eart7ua8es otal
μdrift −σ drift Cost tot =0.98 ∙2730$+0.02∙ (0.8 ∙5190$+0.2∙19800$ )+¿
+0 ∙ (0.8∙7940$+0.2∙31100$ ) 2838$
μdrift Cost tot =1 ∙2730$+0∙ (0.8 ∙5190$+0.2∙19800$ )+¿
+0 ∙ (0.8∙7940$+0.2∙31100$ ) 2730$
μdrift +σ drift Cost tot =1 ∙2730$+0∙ (0.8 ∙5190$+0.2∙19800$ )+¿
+0 ∙ (0.8∙7940$+0.2∙31100$ ) 2730$
Casesanalysed
otal $ost ;or linearly s$aled eart7ua8es otal
μdrift −σ drift Cost tot =0.96∙2730$+0.04 ∙ (0.8 ∙5190 $+0.2 ∙19800 $ )+¿
+0 ∙ (0.8∙7940$+0.2∙31100$ ) 2945$
μdrift Cost tot =1 ∙2730$+0∙ (0.8 ∙5190$+0.2∙19800$ )+¿
+0 ∙ (0.8∙7940$+0.2∙31100$ ) 2730$
μdrift +σ drift Cost tot =1 ∙2730$+0∙ (0.8 ∙5190$+0.2∙19800$ )+¿
+0 ∙ (0.8∙7940$+0.2∙31100$ ) 2730$
The estimated repairing time has been derived from similar diagrams from !"#T database
considering the highest possible values
-amage State 1 T D S1
=8.04days
-amage State T D S2 I =15.3days T D S2 II =58.1days
-amage State ! T D S3 I =23.4 days T D S
3 II =91.5days
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2ith the same methodology used for the repairing costs, we have calculated the timeCases
analysedotal re9air time ;or s9e$trally mat$7ed eart7ua8es otal
μdrift −σ drift Timetot =0.98∙8.04 d+0.02 ∙ (0.8 ∙15.3d+0.2 ∙58.1d )+¿
+0 ∙ (0.8∙23.4d+0.2 ∙91.5d ) 8.35days
μdrift Timetot =1 ∙8.04d+0 ∙ (0.8∙15.3d+0.2 ∙58.1d )+¿
+0 ∙ (0.8∙23.4d+0.2 ∙91.5d ) 8.04days
μdrift +σ drift Timetot =1 ∙8.04d+0 ∙ (0.8∙15.3d+0.2 ∙58.1d )+¿
+0 ∙ (0.8∙23.4d+0.2 ∙91.5d ) 8.04days
Casesanalysed
otal re9air time ;or linearly s$aled eart7ua8es otal
μdrift −σ drift Timetot =0.96 ∙8.04 d+0.04 ∙ (0.8 ∙15.3d+0.2∙58.1d )+¿
+0 ∙ (0.8∙23.4d+0.2 ∙91.5d ) 8.67days
μdrift Timetot =1
∙8.04
d+0
∙ (0.8
∙15.3
d+0.2
∙58.1
d )+¿
+0 ∙ (0.8∙23.4d+0.2 ∙91.5d ) 8.04days
μdrift +σ drift Timetot =1 ∙8.04d+0 ∙ (0.8∙15.3d+0.2 ∙58.1d )+¿
+0 ∙ (0.8∙23.4d+0.2 ∙91.5d ) 8.04days
) ,303.001a: $u/eded ,eilig' $, A' B6 Area 7A): A + 50' 8ert u//ort ol
The ceiling structure is a non-structural element, in this case, supported by vertical hanging wires
only. The demand parameter is the acceleration and the values obtained from the analysis are
3or s9e$trally mat$7edeart7ua8es
3or linearly s$aled
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μacc=0.457947163
σ acc=0.086676226
¿ μacc+σ acc=0.544623
μacc−σ acc=0.371271
μacc=0.48556507
σ acc=0.100353286
¿ μacc+σ acc=0.58585
μacc−σ acc=0.38515
-amage
States-amage Re9resented
Mutually eD$lusive damage
state
D S1 5 of ties disodge and fa. none
D S2
30 of ties disodge and fa andt/"ar grid damaged.
none
D S3 ota ceiing coa*se. none
0eading the values from the fragility curves, we have
Cases S9e$trally mat$7edeart7ua8es
Linearly s$aledeart7ua8es
μdrift −σ drift
P DS3
=0
P DS2
=0
P DS1
=0
Pundmaged=1
P DS3
=0
P DS2
=0
P DS1
=0
Pundmaged=1
μdrift
P DS3
=0
P DS2
=0.01
P DS1
=0.03−0.01=0.02
Pundmaged=1−0.03=0.97
P DS3
=0
P DS2
=0.01
P DS1
=0.03−0.02=0.01
Pundmaged=1−0.03=0.97
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μdrift +σ drift
P DS3
=0.009
P DS2
=0.03−0.009=0.021
P DS1
=0.25−0.03=0.22
Pundmaged=1−0.25=0.75
P DS3
=0.0099
P DS2
=0.03−0.009=0.021
P DS1
=0.25−0.03=0.22
Pundmaged=1−0.25=0.75
#onsidering that the cost is the higher possible, we have
-amage State 1 Cost DS1=471.25 $ -einsta ne! acoustic tie for 5 or area.
-amage State Cost DS2=3688.75 $-einsta ne! acoustic tie and ceiing grids
for 30 of t%e area.
-amage State ! Cost DS3=7588.75 $ ntire+ re*ace ceiing and grid.
The total repairing cost can be calculated ta1ing into account the different probabilities related to
every damage state.Cost tot = P DS
1
∙Cost DS1+ P DS2
∙Cost DS2+ P DS3
∙Cost DS3
Cases analysed otal $ost ;or s9e$trally mat$7ed eart7ua8es otal
μrotation−σ rotation Cost tot =0∙471.25 $+0 ∙3688.75$+0 ∙7588.75 $ 0$
μrotation Cost tot =0.02 ∙471.25$+0.01 ∙3688.75 $+0 ∙7588.75$ 46.32$
μrotation+σ rotation Cost tot =0.22∙471.25 $+0.021 ∙3688.75$+0.009 ∙7588.75 249.44$
Cases analysed otal $ost ;or linearly s$aled eart7ua8es otal
μrotation−σ rotation Cost tot =0∙471.25 $+0 ∙3688.75$+0 ∙7588.75 $ 0$
μrotation Cost tot =0.02 ∙471.25$+0.01 ∙3688.75 $+0 ∙7588.75$ 46.32$
μrotation+σ rotation Cost tot =0.22∙471.25 $+0.021 ∙3688.75$+0.009 ∙7588.75 249.44$
The estimated repairing for each damage state are
-amageState 1
T DS1=1.5912 days-einsta ne! acoustic tie for 5 or
area.
-amageState
T DS2=11.6706 days-einsta ne! acoustic tie and ceiing
grids for 30 of t%e area.
-amageState !
T DS3=23.7441days ntire+ re*ace ceiing and grid.
2ith the same procedure, we can calculate the total time repair
Cases analysedotal re9air time ;or s9e$trally mat$7ed
eart7ua8esotal
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μrotation−σ rotation Timetot =0∙1.5912 d+0 ∙11.6706 d+0∙23.7441 d 0days
μrotation Timetot =0.02 ∙1.5912d+0.01∙11.6706d+0 ∙23.7441d 0.14853days
μrotation+σ rotation Timetot =0.22 ∙1.5912 d+0.021∙11.6706d+0.009 ∙23.7441 0.8088435da
Cases analysed otal re9air time ;or linearly s$aled eart7ua8es otal
μrotation−σ rotation Timetot =0∙1.5912 d+0 ∙11.6706 d+0∙23.7441 d 0days
μrotation Timetot =0.02 ∙1.5912d+0.01∙11.6706d+0 ∙23.7441d 0.14853days
μrotation+
σ rotation Timetot =0.22
∙1.5912
d+0.021
∙11.6706
d+0.009
∙23.7441 0.8088435
da
Question A - What changes could you mae to com!onents 1 through " to reduce damage
under an e#ual set of engineering demand !arameters$
f the set of demanding parameters is the same for all elements, a possible method to improve their
resistance to the effects of an earth&ua1e depends on the element considered an accurate analysis of
each one is necessary to find the wea1 points and to understand the way they reach the failure.
The first element analysed, a pre-Northridge WU-! "eam-column #oint$ the damages occur in the
weld or web bolts with a possible propagation of the crac1s. To improve the performance of the
connection we can select a different &uality of welding (li1e total penetration) or a different &uality
of bolts (increasing the class if possible). 6ther improvements could be a strengthening of thesection by using some supports under the beam (the fracture always begins from the bottom flange),
whether to change element completely and adopt a post-/orthridge beam connection, with a
reduced beam section, or, if the structure allows the adoption of hinged connection together with the
implementation of braces with the function of absorption of the hori7ontal forces.
The second elements is a shear lin1. n this case, damages occur to the concrete slab (in the first
damage state ), buc1ling of the web (in the second DS ) and finally the fracture in the web (third
DS ). " way to improve the section can be to increase the number of stiffeners of the lin1 so that the
connection would be more rigid. This change can give a better performance also with the respect to
the buc1ling of the web and increase the element resistance to the cyclic fatigue. "nother possible
way can be the selection of a lin1 with a thic1er web, flange and vertical stiffeners. ts also possibleto change the lin1 length (this also improves the ductility of the system) increasing the length is
possible to have a more flexible element with a better performance against brittle failure but in
charge of higher vertical displacements.
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The third element is a gypsum wall partition. Unfortunately, due to the brittle nature of the gypsum ,
is not easy to have better performances, also because the failure is mostly influenced by the
deformation of the metallic framing members that induce crac1 on the panel, especially in the
corners or in the openings border where most li1ely, there concentration of stresses (due to
singularity).
" possible solution can be the adoption of a material alternative to gypsum, with a more ductile
behaviour (li1e a fibred layered material) or to coat the partition with a net and then to cover it with
some special plaster (reinforced plaster). 6bviously, in this second case, the price of this
improvement has to be considered, against the cost of substitution of the partition wall.
The fourth element is the ceiling, made by tiles. n this case, for both sets of earth&ua1e records, the
damages that have been found are small same as the repairing costs. The fragility curves of this
element are particular because the ceiling is made to be uninstalled or inspected, and therefore has
all the tiles not connected to the supporting frame. The only possible improvement can be to have a
larger surface on which the tiles are bac1ed or to use some internally flexible connection that allows
the tiles to move but not to fall.
Question % - What changes could you mae to reduce damage to all com!onents$
" possible method to reduce the damage to all components could be the improvements in the
performance of the building against the seismic forces, li1e the adoption of braces to reduce the
drift or base isolation. n some cases, also, the adoption of a tuned mass absorber or dampers can
give an effective reduction of the seismic demand or a redistribution of masses (if possible) or a
change of layout (if possible) to improve regularity in plan and elevation.
Question C - &o' are fragility functions de(elo!ed$
n a damage analysis, whose input is the engineering demand parameters ($'!) calculated in the
structural analysis, and whose output is the damage measure ('5) of each damageable structural
and non-structural component in the facility, we ma1e use of fragility functions.
%ragility functions are probability distributions that are used to indicate the probability that a
component, element or system will be damaged to a given or more severe damage state as a
function of a single predictive demand parameter such as story drift or floor acceleration. %ragility
functions usually are in the form of lognormal cumulative distribution functions, having a median
value θ and logarithmic standard deviation, β . The mathematical expression for such afragility function is
F i ( D )=Φ (ln(
D
θi ) βi )
2here F i ( D ) is the conditional probability that the component will be damaged to damage state8 i 9 or a more severe damage state as a function of demand parameter, D : Φ denotes the
standard normal (4aussian) cumulative distribution function, θi denotes the median value of the
probability distribution, and β i denotes the logarithmic standard deviation. 3oth θ and β
are established for each component type and damage state using one of the methods presented later.
The probability that a component will be damaged to damage state 8 i 9 and not to a more or less
severe level given that it experiences demand, ' is given by P (i∨ D )= F i ( D )− F i+1 ( D )
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2here F i+1 ( D ) is the conditional probability that the component will be damaged to damage state“i+1” or a more severe state and F i ( D ) is as previously defined.
" typical form of fragility function plotted in the form of a cumulative distribution function is
showed in the picture below on the left-hand side: on the right is showed the calculation of the
probability that a component will be in damage state “i ” at a particular level of demand, D=d .
Question D - Why does )AC*+s loss assessment methodology follo' a onte Carlo simulation
a!!roach$
"ssessing uncertainty and explore variability in building performance is a process that would
ideally involve performing a large number of structural analyses, using a large suite of input ground
motions, and analytical models with properties randomly varied. 5odels would include all
structural and non-structural components and systems, and would be able to predict explicitly
damage to each component of each system as it occurs. The results of each single analysis would
represent one possible outcome, and the results of the large suite (thousands) of analyses would
produce a smoothed distribution for probabilistic evaluation of earth&ua1e conse&uences.
4iven the current state of modelling capability, such an approach would be impractical for
implementation in practice. nstead, a 5onte #arlo procedure is used to assess a range of possible
outcomes given a limited set of inputs.
0ather than re&uiring a large number of structural analyses to develop these demands, the results
from a limited suite of analyses are mathematically transformed, fictitiously, into a large series of simulated demand sets. %rom this limited number of analyses, one can derive a statistical
distribution of demands from a series of building response states for a particular intensity of motion.
%rom this distribution, statistically, consistent demand sets are generated representing a large
number of possible building response states. These demand sets, together with fragility and
conse&uence functions, are used to determine a building damage state and compute conse&uences
associated with that damage. (%$5" !-;