CE 58A - PUSHOVER ANALYSIS
HANDOUT #11
INCREMENTAL PUSHOVER ANALYSIS FOR SEISMIC PERFORMANCE ASSESSMENT EXAMPLE OF INCREMENTAL PUSHOVER ANALYSIS TO ASSESS THE PERFORMANCE OF AN EXISTING REINFORCED CONCRETE FRAME: (TDY–2007 APPROACH)
B201
B202
C203
B101
3.5 m
C103
C202
C201
P2
3.5 m
B102
C102
C101
P1
7m
7m
1
• COLUMNS :
450 mm x 450 mm WITH
8-φ22
φ8 STIRRUPS AT A SPACING OF 200 mm PROVIDED ALONG THE ENTIRE LENGTH OF THE COLUMNS. • BEAMS
:
450 mm x 600 mm WITH
3-φ22 AT THE TOP 3-φ22 AT THE BOTTOM
φ8 STIRRUPS AT A SPACING OF 150 mm PROVIDED ALONG THE ENTIRE LENGTH OF THE BEAMS. • CONCRETE COVER
:
CLEAR COVER = 20 MM COVER TO THE CENTER OF LONGITUDINAL REINFORCEMENT = 20 MM + φ8 + (φ22)/2 = 39 MM TAKE COVER TO THE CENTER OF LONGITUDINAL REINFORCEMENT AS 40 mm FOR BOTH THE BEAMS AND THE COLUMNS
(NOTE: TOP AND BOTTOM REINFORCEMENT IN THE BEAMS IS ASSUMED EQUAL IN ORDER TO SIMPLIFY THE ANALYSIS (Mult+ = Mult-). TYPICALLY, THE AMOUNT OF TOP REINFORCEMENT WOULD BE LARGER SINCE NEGATIVE MOMENTS RESULTING FROM BOTH GRAVITY AND EARTHQUAKE LOADS NEED TO BE RESISTED AT THE BEAM/COLUMN JOINTS) • MATERIALS :
C20 CONCRETE S420 STEEL (FOR BOTH LONGITUDINAL AND TRANSVERSE REINFORCEMENT) 2
• BUILDING
:
OFFICE BUILDING LOCATED IN SEISMIC ZONE 1 LOCAL SOIL CLASS: Z2
• LOADS
:
DEAD LOAD (SELF WEIGHT) : G = 20 kN/m LIVE LOAD
: Q = 10 kN/m
(DISTRIBUTED LOAD ON THE BEAMS) (NOTE: IF GIVEN THE PLAN VIEW OF THE BUILDING, NEED TO CALCULATE THE DISTRIBUTED DEAD AND LIVE LOAD PER UNIT LENGTH ON THE BEAMS, BASED ON THE TRIBUTARY WIDTH OF EACH BEAM AND THE DEAD AND LIVE LOAD VALUES DISTRIBUTED OVER THE FLOOR AREA) • RESPONSE SPECTRUM VARIABLES (TDY-2007):
Spectral Acceleration Coefficient Spectral Acceleration
EQ Zone
Soil Type
3
• SEISMIC ZONE 1 LOCAL SOIL TYPE Z2
:
A0 = 0.40
:
TA = 0.15 sec
TB = 0.40 sec
BUILDING IMPORTANCE COEFFICIENT: ALWAYS TAKE BUILDING IMPORTANCE COEFFICIENT AS I = 1.0 FOR ASSESSMENT OF EXISTING BUILDINGS USING PUSHOVER ANALYSIS METHOD (SECTION 7.4.2 IN TDY-2007) • STORY MASSES (TDY-2007): SHALL BE DEFINED IN ACCORDANCE WITH STORY WEIGHTS DEFINED IN SECTION 2.7.1.2:
Live load participation factors (n) Building Type and Function Storage facilities Schools, dorms, sports facilities, theatres, concert halls, garages, restaurants, stores, etc. Residental buildings, office buildings, hotels, hospitals, etc.
THEREFORE,
wi = (20 kN/m)(14 m) + (0.30)(10 kN/m)(14 m) = (20 kN/m)(14 m) + (3 kN/m)(14 m) = (23 kN/m)(14 m) = 322 kN
STORY WEIGHTS :
wi = 322 kN (ON SINGLE FRAME)
STORY MASSES
mi = wi / g = (322 x 103 N) / (9.81 m/s2)
:
mi = 32,823 kg = 33 tons (ON SINGLE FRAME) 4
• SECTION STIFFNESS (TDY-2007):
(Beams) (Columns, Shear Walls)
¾ ¾ ¾ ¾
Use cracked section stiffness when modeling the structure ND : axial load under vertical loads only (service loads) Interpolation between 0.40EI0 – 0.80EI0 for intermediate values of axial load fcm : existing concrete compressive strength (no material factor)
FOR C20
: Ec 28 = 28,000 MPA
FOR ALL BEAMS
: I0 = (1/12)(0.450 m)(0.600 mm)3 = 0.0081 m4 EI0 = 226800 kN.m2 EI = 0.40(EI0)
FOR COLUMNS
: Acfcm = (0.45 m)(0.45 m)(20x103 kN/m2) = 4050 kN ASSUME INTERIOR COLUMNS RESIST 50% OF THE TOTAL VERTICAL LOAD, WHEREAS EACH EXTERIOR COLUMN RESISTS 25% OF THE TOTAL VERTICAL LOAD AT EACH STORY. (SINCE THE TRIBUTARY AREA OF THE INTERIOR COLUMNS ARE TWICE OF THE EXTERIOR COLUMNS) 5
B201
B202
C203
C202
C201
P2
B101
3.5 m
B102
C103
C102
C101
P1
7m
3.5 m
7m
STORY WEIGHT = 322 kN ( FROM g + 0.30q ) THEREFORE: C201 AND C203
:
(ND/ Acfcm) = [(0.25)(322 kN)]/(4050 kN)] = 0.02
C202
:
(ND/ Acfcm) = [(0.50)(322 kN)]/(4050 kN)] = 0.04
C101 AND C103
:
(ND/ Acfcm) = [(0.25)(2 x 322 kN)]/(4050 kN)] = 0.04
C102
:
(ND/ Acfcm) = [(0.50)(2 x 322 kN)]/(4050 kN)] = 0.08
SECTION STIFFNESS COEFFICIENTS: USE EI = 0.40(EI0) FOR ALL COLUMNS Ec 28 = 28,000 MPA I0 = (1/12)(0.450 m)(0.450 mm)3 = 0.00342 m4 EI0 = 95800 kN.m2
6
• ADDITIONAL CHECKS (TDY-2007):
Pushover analysis method can be used only if: Number of stories (not including basement) < 8 Torsional irregularity constant for the building < 1.4 The ratio of the effective mass corresponding to the first mode of vibration to the total mass of the building > 0.70
¾ ¾ ¾
M x1 N
∑ mi
> 0.70
i =1
• NUMBER OF STORIES = 2 • ASSUMING THE FRAMES IN THE BUILDING ARE SYMMETRICCALLY PLACED, THE TORSIONAL IRREGULARITY CONSTANT = 1.0 • NEED TO CALCULATE THE MODE SHAPES IN ORDER TO DETERMINE THE EFFECTIVE MASS RATIO CORRESPONDING TO THE FIRST MODE OF VIBRATION (COMPUTER MODEL)
¾
During the pushover analysis, the distribution (pattern) of lateral story forces acting on the building can be assumed to be constant
¾
The distribution of the lateral story forces shall be proportional to the product of the mass of each story and the amplitude of the first mode shape of vibration of that story. 7
•
PLASTIC MOMENT CAPACITIES FOR COMPUTER MODELING: CAN USE THE PROGRAM BETONARME TO DERIVE THE MOMENTCURVATURE RELATIONSHIPS FOR THE BEAMS AND THE P-M INTERACTION DIAGRAMS FOR THE COLUMNS: (http://www.ce.metu.edu.tr/betonarme) • BEAMS :
450 mm x 600 mm WITH
3-φ22 AT THE TOP 3-φ22 AT THE BOTTOM
φ8 STIRRUPS AT A SPACING OF 150 mm PROVIDED ALONG THE ENTIRE LENGTH OF THE BEAMS. COVER TO THE CENTER OF LONGITUDINAL REINFORCEMENT = 40 mm M o m e n t
MALZEME ÖZELLİKLERİ Eksenel Yük
Beton f ck
(Basınç +) (kN) 0,00
(MPa) 20,00
γmc
f yk
fu
γms
1,00
(MPa) 420
(MPa) 600
1
Alan 2
(mm ) 3 3
0,01
εsu
b
h
d
Cc
(mm)
(mm)
(mm)
(mm)
450
600
560
20
0,1
400
Kesit Merkezine
1140 1140
Uzaklık
350
(mm) -260 260
300
250
(kN.m)
(mm) 22 22
Adet
Moment
P r o g r a m ı
1 2 3 4 5 6 7 8 9 10
φA
εsh
(MPa) 200.000
BOYUNA DONATI DÜZENLEMESİ No.
E ğ r i l i k
BETON KESİT
Boyuna Donatı Es
200
150
100
ETRİYE BİLGİLERİ fyw
(MPa)
420
φe s bk
(mm) (mm) (mm)
8 150 402
hk
(mm)
552
b'k
(mm)
402
h'k ess
(mm)
552 1
50
0 0,0000
HESAPLA
0,0500
0,1000
0,1500
0,2000
0,2500
Eğrilik (rad/m)
YIELD MOMENT
My
= 250 kN.m
YIELD CURVATURE
φy
= 0.008 rad/m 8
• COLUMNS :
450 mm x 450 mm WITH
8-φ22
φ8 STIRRUPS AT A SPACING OF 200 mm PROVIDED ALONG THE ENTIRE LENGTH OF THE COLUMNS. COVER TO THE CENTER OF LONGITUDINAL REINFORCEMENT = 40 mm
K E S İ T A N A L İ Z İ
BETON ve ÇELİK MODELLERİ f ck
γ mc
f yk
γ ms
Es
(MPa) 20
1,00
(MPa) 420
1,00
(MPa) 200.000
BU PROGRAMDA: 1) Betonun çekme dayanı ihmal edilmektedir. 2) Beton basınç dağılımı dikdörtgen alınmaktadır. 3) Çelik modelinde pekleşme ihmal edilmektedir. 4) Sargı etkisi göz önüne alınmamaktadır.
KESİT GEOMETRİSİ Genişlik (b)
Yükseklik (h)
(mm)
(mm)
450
450
5000
(+) xi 4000
Eksenel Yük, P (kN)
D İ K D Ö R T G E N
(-) xi
DONATI DÜZENLEMESİ No.
Donatı Alanı (mm 2 )
1 2 3 4 5 6
1140 760 1140
Kesit Merkezinden Uzaklık (x i ) (mm) -185 0 185
N (kN): M (kN.m):
0,0 240,3
3000 2000 1000 0 -1000 -2000
0
50
100
150
200
250
300
350
400
Moment, M (kN.m)
NOTE THAT WE NEED THE P-M INTERACTION CURVES FOR THE COLUMNS SINCE THE AXIAL LOAD ON EACH COLUMN IS NOT THE SAME AND ALSO SINCE THE AXIAL LOAD WILL CHANGE WHEN LATERAL LOADS ARE APPLIED. CAN ALSO RUN MOMENT-CURVATURE ANALYSES FOR THE COLUMNS TO CHECK (TO COMPARE THE RESULTS OF THE P-M INTERACTION ANALYSES WITH THE RESULTS OF THE MOMENT-CURVATURE ANALYSES)
9
FOR EXAMPLE, FOR COLUMN C102, THE AXIAL LOAD DUE TO VERTICAL LOADS ONLY IS ND = [(0.50)(2x322 kN)] = 322 kN. D İ K D Ö R T G E N
BETON ve ÇELİK MODELLERİ f ck
γ mc
f yk
γ ms
Es
(MPa) 20
1,00
(MPa) 420
1,00
(MPa) 200.000
BU PROGRAMDA: 1) Betonun çekme dayanı ihmal edilmektedir. 2) Beton basınç dağılımı dikdörtgen alınmaktadır. 3) Çelik modelinde pekleşme ihmal edilmektedir. 4) Sargı etkisi göz önüne alınmamaktadır.
KESİT GEOMETRİSİ Yükseklik (h)
(mm)
(mm)
450
450
5000
(+) xi 4000
Eksenel Yük, P (kN)
Genişlik (b)
(-) xi
K E S İ T
DONATI DÜZENLEMESİ No.
Kesit Merkezinden Uzaklık (x i )
Donatı Alanı (mm 2 )
1 2 3 4 5 6
A N A L İ Z İ
(mm)
1140 760 1140
-185 0 185
N (kN): M (kN.m):
322,0 292,1
3000 2000 1000 0 -1000 -2000
0
50
100
150
200
250
300
350
400
Moment, M (kN.m)
FROM P-M INTERACTION ANALYSIS: Mult = 292 kN.m FOR N = 322 kN. M o m e n t
MALZEME ÖZELLİKLERİ Beton
Eksenel Yük
f ck
(Basınç +) (kN) 322,00
(MPa) 20,00
γmc
f yk
fu
γms
1,00
(MPa) 420
(MPa) 420
1
Alan 2
(mm ) 3 2 3
1140 760 1140
0,01
εsu
b
h
d
Cc
(mm)
(mm)
(mm)
(mm)
450
450
410
20
0,1
350
Kesit Merkezine Uzaklık
300
(mm) -185 0 185
250
(kN.m)
(mm) 22 22 22
Adet
Moment
P r o g r a m ı
1 2 3 4 5 6 7 8 9 10
φA
εsh
(MPa) 200.000
BOYUNA DONATI DÜZENLEMESİ No.
E ğ r i l i k
BETON KESİT
Boyuna Donatı Es
200
150
100
ETRİYE BİLGİLERİ fyw
(MPa)
420
φe s bk
(mm) (mm) (mm)
8 200 402
hk
(mm)
402
b'k
(mm)
402
h'k ess
(mm)
402 1
50
0
HESAPLA
0,0000
0,0500
0,1000
0,1500
Eğrilik (rad/m)
FROM MOMENT-CURVATURE ANALYSIS: My = 291 Kn.m for N=322 kN (CHECK OK) 10
NOTE: DO NOT USE MATERIAL FACTORS IN THE MOMENTCURVATURE OF P-M INTERACTION ANALYSES FOR ASSESSMENT OF EXISTING BUILDINGS. (ALWAYS TAKE γmc =1 AND γms = 1 IN THE PROGRAMS)
CAN LINEARIZE THE P-M INTERACTION DIAGRAM FOR THE COLUMNS (FOR EASY INPUT INTO THE STRUCTURAL ANALYSIS PROGRAM): 5000
Eksenel Yük, P (kN)
4000 3000 2000 1000 0 -1000 -2000
0
50
100
150
200
250
300
350
400
Moment, M (kN.m)
N = 4650 kN (COMPRESSION): M = 0 kN.m (PURE COMPRESSION) N = 1500 kN (COMPRESSION): M = 370 kN.m
(BALANCED POINT)
N = 0 kN
: M = 240 kN.m
(PURE BENDING)
N = 1275 kN (TENSION)
: M = 0 kN.m
(PURE TENSION)
11
•
COMPUTER MODELING USING SAP2000:
• NEED TO DETERMINE THE PERIODS OF VIBRATION, THE MODE SHAPES OF VIBRATION, AND THE PUSHOVER CURVE TO PROCEED WITH THE PUSHOVER ANALYSIS. • SET UP THE MODEL FOR THE FRAME USING CENTER-TO-CENTER DIMENSIONS BETWEEN THE BEAMS AND COLUMNS. • ASSIGN FIXED ENDS AT THE BOTTOM (RESTRAINTS) • DEFINE MATERIALS: o CONC: MASS AND WEIGHT PER UNIT VOLUME = 0 MODULUS OF ELASTICITY = 28000000 kN/m2 POISSON’S RATION = 0.2 • DEFINE FRAME SECTIONS: o ADD RECTANGULAR: BEAM (FOR ALL BEAMS) 0.45 m x 0.60 m MATERIAL: CONC SET MODIFIERS: 0.40 FOR MOMENT OF INERTIA ABOUT 2 AND 3 AXES 0 FOR MASS AND WEIGHT LARGE VALUE (E.G., 1000000) FOR CROSS-SECTIONAL AREA, SHEAR AREAS, AND TORSIONAL CONSTANT o ADD RECTANGULAR: COLUMN (FOR ALL COLUMNS) 0.45 m x 0.45 m MATERIAL: CONC SET MODIFIERS: 0.40 FOR MOMENT OF INERTIA ABOUT 2 AND 3 AXES 12
0 FOR MASS AND WEIGHT LARGE VALUE (E.G., 1000000) FOR CROSS-SECTIONAL AREA, SHEAR AREAS, AND TORSIONAL CONSTANT • ASSIGN FRAME SECTIONS • ASSIGN END (LENGTH) OFFSETS TO THE BEAMS AND COLUMNS: (FOR RIGID BEAM-COLUMN JOINTS) o FIRST STORY COLUMNS: END I
: 0
END J
: 0.3 m
RIGID ZONE FACTOR:
1
o SECOND STORY COLUMNS: END I
: 0.3 m
END J
: 0.3 m
RIGID ZONE FACTOR:
1
o ALL BEAMS: END I
: 0.225 m
END J
: 0.225 m
RIGID ZONE FACTOR:
1
• ASSIGN JOINT MASSES: MASSES WILL BE DEFINED ONLY IN LATERAL DIRECTIONS. SINCE THE BEAMS ARE AXIALLY RIGID (INFINITE CROSS-SECTIONAL AREA, IT DOES NOT MATTER WHICH JOINT YOU ASSIGN THE MASS IN A PARTICULAR STORY. THEREFORE, CAN ASSIGN THE MASSES AT THE INTERIOR BEAMCOLUMN JOINTS AT EACH STORY. o ASSIGN MASSES IN DIRECTION 2: (322 kN) / (9.81 m/sec2) = 33 kN.sec2/m (33 tons) 13
• AT THIS POINT, CAN RUN A MODAL ANALYSIS TO DETERMINE THE NATURAL PERIODS OF VIBRATION AND MODE SHAPES OF VIBRATION o DEFINE ANALYSIS CASE: MODAL o ANALYSIS CASE TYPE: MODAL o TYPES OF MODES: EIGEN VECTORS o START FROM UNSTRESSED STATE • ANALYZE: RUN ANALYSIS o CASE NAME: MODAL o RESULTS: MODE 1: T1 = 0.373 sec MODE SHAPE VECTOR: φ1 = {0.0841, 0.1524}T (CAN OBTAIN FROM DEFORMED SHAPE) MODE 2: T2 = 0.113 sec MODE SHAPE VECTOR: φ2 = {-0.1524, 0.0841}T o NOTE THAT THE MODE SHAPES GIVEN ARE MASS NORMAL:
⎡ 0.0841 −0.1524 ⎤ Φ = [φ1 φ2 ] = ⎢ ⎥ ⎣0.1524 0.0841 ⎦ 0 ⎤ ⎡33 0 ⎤ ⎡m m=⎢ 1 ⎥=⎢ ⎥ ⎣ 0 m2 ⎦ ⎣ 0 33⎦ ⎡1 0 ⎤ ΦT m Φ = ⎢ ⎥ ⎣0 1 ⎦
14
• RECALL (TDY-2007):
¾
During the pushover analysis, the distribution (pattern) of lateral story forces acting on the building can be assumed to be constant
¾
The distribution of the lateral story forces shall be proportional to the product of the mass of each story and the amplitude of the first mode shape of vibration of that story.
BACK TO THE SAP2000 MODEL
• DEFINE LOAD CASES: o LOAD NAME: GRAVITY TYPE: DEAD, SELF WEIGHT MULTIPLIER = 1 o LOAD NAME: LATERAL TYPE: QUAKE, SELF WEIGHT MULTIPLIER = 1, AUTO LATERAL LOAD: NONE
• ASSIGN FRAME LOADS ON ALL BEAMS: o DISTRIBUTED o LOAD CASE NAME: GRAVITY o COORD SYS: GLOBAL o DIRECTION: Z o UNIFORM LOAD = g + nq = -[(20 kN/m)+(0.3)(10 kN/m)]= -23 kN/m
• ASSIGN LATERAL LOADS AT STORY LEVELS o SINCE THE BEAMS ARE AXIALLY RIGID (INFINITE CROSSSECTIONALAREA, IT DOES NOT MATTER WHICH JOINT YOU ASSIGN THE LATERAL LOADS AT A PARTICULAR STORY. 15
THEREFORE, CAN ASSIGN THE LATERAL LOADS AT THE EXTERIOR BEAM-COLUMN JOINTS AT EACH STORY. o ASSIGN JOINT LOADS: FORCES o LOAD CASE NAME: LATERAL o COORDINATE SYSTEM: GLOBAL o LATERAL LOADS SHOULD BE PROPORTIONAL TO THE PRODUCT OF THE STORY MASS AND THE AMPLITUDE OF THE FIRST MODE SHAPE AT THAT STORY (TDY-2007). o STORY MASSES ARE EQUAL IN THE PRESENT EXAMPLE, THEREFORE, THE LATERAL LOADS WILL BE PROPORTIONAL TO THE FIRST MODE SHAPE. o FIRST MODE SHAPE: φ1 = {0.0841, 0.1524}T o 0.1524 / 0.0841 = 1.81 o THEREFORE, ASSIGN A FORCE GLOBAL Z OF 1 kN AT THE FIRST STORY EXTERIOR JOINT AND A FORCE FLOBAL Z OF 1.81 kN AT THE SECOND STORY EXTERIOR JOINT.
• DEFINE PLASTIC HINGES: o DEFINE: HINGE PROPERTIES FOR BEAMS: o ADD NEW PROPERTY o HINGE PROPERTY NAME: BEAMHINGE o DEFORMATION CONTROLLED o MOMENT M3 o MODIFY/SHOW PROPERTY o MOMENT SF = My = 250 kN.m 16
o CURVATURE = φy = 0.008 rad/m o TYPE: MOMENT-CURVATURE o HINGE LENGTH = 0.3 m (h/2 FOR THE BEAM) o DO NOT CHECK RELATIVE LENGTH o DISPLACEMENT CONTROL PARAMETERS: CHECK SYMMETRIC MOMENT/SF
CURVATURE/SF
1
0
1
50
(ANY LARGE VALUE)
0.2
50
(SUDDEN DROP)
0.2
60
(RESIDUAL)
LOAD CARRYING CAPACITY BEYOND POINT E DROPS TO ZERO FOR COLUMNS: o ADD NEW PROPERTY o HINGE PROPERTY NAME: COLUMNHINGE o DEFORMATION CONTROLLED o INTERACTING P-M3 o MODIFY/SHOW HINGE PROPERTY o MOMENT-CURVATURE TYPE o HINGE LENGTH = 0.225 m (h/2 FOR THE COLUMN) o DO NOT CHECK RELATIVE LENGTH o SCALE FACTOR FOR CURVATURE: USER SF=1 o LOAD CARRYING CAPACITY BEYOND POINT E DROPS TO ZERO o MOMENT-CURVATURE DEPENDENCE IS SYMMETRIC o MODIFY/SHOW P-M3 INTERACTION SURFACE DATA 17
o INTERACTION SURFACE: USER DEFINITION o AXIAL LOAD – DISPLACEMENT: ELASTIC-PERFECTLY PLASTIC o DEFINE/SHOW USER INTERACTION SURFACE INTERACTION CURVE IS SYMMETRIC NUMBER OF POINTS ON EACH CURVE = 4 SCALE FACTORS: kN.m C P
= 1500 kN (Pbalanced)
(REFERENCE POINT)
370 kN (Mbalanced)
(REFERENCE POINT)
M3 =
FIRST AND LAST POINTS: (FACTORS TO BE MULTIPLIED BY THE REFERENCE POINT) POINT 1: P = -3.1
M3 = 0 (PURE COMPRESSION)
POINT 4: P = 0.85
M3 = 0 (PURE TENSION)
INTERACTION CURVE DATA: (FACTORS TO BE MULTIPLIED BY THE REFERENCE POINT) POINT 2: P = -1
M3 = 1
(BALANCED POINT)
POINT 3: P= 0
M3 = 0.65 (PURE BENDING)
NOTE THAT IN SAP, TENSION IS POSITIVE WHEN DEFINING THE P-M INTERACTION CURVE. o MODIFY/SHOW MOMENT-CURVATURE CURVE DATA MOMENT/YIELD MOM
CURVATURE/SF
0
0
1
0
1
50
(ANY LARGE VALUE)
0.2
50
(SUDDEN DROP)
0.2
60
(RESIDUAL)
18
• ASSIGN PLASTIC HINGES: FOR BEAMS: o SELECT ALL BEAMS o ASSIGN/FRAME/HINGES o HINGE PROPERTY: BEAMHINGE o RELATIVE DISTANCE = 0
ADD
o RELATIVE DISTANCE = 1
ADD
o NOTE THAT SAP WILL ASSIGN THE HINGES OUTSIDE THE RIGID END OFFSETS FOR COLUMNS: o SELECT ALL COLUMNS o ASSIGN/FRAME/HINGES o HINGE PROPERTY: COLUMNHINGE o RELATIVE DISTANCE = 0
ADD
o RELATIVE DISTANCE = 1
ADD
o NOTE THAT SAP WILL ASSIGN THE HINGES OUTSIDE THE RIGID END OFFSETS
• DEFINE ANALYSIS CASES: NONLINEAR ANALYSIS UNDER GRAVITY LOADS: o ADD NEW CASE o ANALYSIS CASE NAME: PUSHOVER-GRAVITY o ANALYSIS CASE TYPE: STATIC o ANALYSIS TYPE: NONLINEAR o ZERO INITIAL CONDITIONS o MODAL ANALYSIS CASE: MODAL (IRRELEVANT) o LOAD TYPE: LOAD, LOAD NAME: GRAVITY, SCALE FACTOR: 1 19
o LOAD APPLICATION: FULL LOAD o RESULTS SAVED: FINAL STATE ONLY o NONLINEAR PARAMETERS: DEFAULT NONLINEAR ANALYSIS UNDER LATERAL LOADS: o ADD NEW CASE o ANALYSIS CASE NAME: PUSHOVER-LATERAL o ANALYSIS CASE TYPE: STATIC o ANALYSIS TYPE: NONLINEAR o CONTINUE FROM STATE AT END OF NONLINEAR CASE: PUSHOVER GRAVITY (REQUIRED BY TDY-2007) o MODAL ANALYSIS CASE: MODAL (IRRELEVANT) o LOAD TYPE: LOAD, LOAD NAME: LATERAL, SCALE FACTOR: 1 o LOAD APPLICATION: DISPLACEMENT CONTROL USE MONITORED DISPLACEMENT LOAD TO A MONITORED DISPLACEMENT MAGNITUDE OF 0.25 m (ARBITRARY – TAKE REASONABLY LARGE) o MONITORED DISPLACEMENT: DOF1 (HORIZONTAL) AT JOINT 3 (SECOND STORY EXTERIOR COLUMN JOINT) o RESULTS SAVED: MULTIPLE STATES MINIMUM NUMBER OF SAVED STEPS = 50 MAXIMUM NUMBER OF SAVED STEPS = 50 CHECK SAVE POSITIVE DISPLACEMENT INCREMENTS ONLY o NOTE THAT SAP WILL ASSIGN THE HINGES OUTSIDE THE RIGID END OFFSETS o NONLINEAR PARAMETERS: DEFAULT
20
• AT THIS POINT, CAN RUN THE PUSHOVER ANALYSIS • ANALYZE: RUN ANALYSIS o RUN THE PUSHOVER-GRAVITY AND THE PUSHOVERLATERAL CASES TOGETHER (PUSHOVER-GRAVITY SHOULD COME FIRST
• RESULTS: o DISPLAY: SHOW STATIC PUSHOVER CURVE: TOTAL BASE SHEAR VERSUS MONITORED DISPLACEMENT (LATERAL DISPLACEMENT AT THE TOP – DOF 1 AT JOINT 3)
21
o DISPLAY: SHOWHINGE RESULTS A BEAM HINGE IS SHOWN BELOW NOTE THAT THE PLASTIC ROTATION OF THE HINGE AT ANY POINT DURING THE ANALYSIS IS PROVIDED BY THE PROGRAM
22
A COLUMN HINGE IS SHOWN BELOW NOTE THAT THE PLASTIC ROTATION OF THE HINGE AT ANY POINT DURING THE ANALYSIS IS PROVIDED BY THE PROGRAM
23
CONVERSION OF THE PUSHOVER CURVE INTO THE MODAL CAPACITY CURVE (TDY-2007):
Pushover Curve
Modal spectral acc.
•
Skeleton curve
Modal Capacity Curve
Modal Hysteresis Modal spectral displacement
Effective mass defined for the first mode of vibration in the x-direction Amplitude of the first mode shape at the top of the building (N’th story) defined for the first mode of vibration in the x-direction Participation factor defined for the first mode of vibration in the x-direction
L2x1 M x1 = M1 N
Lx1 = ∑ mi Φ xi1 i =1 N
M 1 = ∑ mi Φ 2xi1 i =1
Γ x1 =
(Modal mass defined for the first mode of vibration)
Lx1 M1 24
Φ x11 = 0.0841 Φ x 21 = 0.1524 m1 = 33 m2 = 33 2
Lx1 = ∑ mi Φ xi1 = (33)(0.0841) + (33)(0.1524) = 7.8045 i =1 2
M1 = ∑ mi Φ 2xi1 = (33)(0.0841)2 + (33)(0.1524)2 = 1.0 i =1
Γ x1 =
Lx1 = 7.8045 M1
L2x1 M x1 = = 60.91 M1
• CAN GENERATE THE FOLLOWING TABLE: uxN1 (m) 0,000 0,005 0,010 0,015 0,025 0,030 0,038 0,045 0,050 0,080 0,100 0,170 0,210 0,250
Vx1 (kN) 0 73 146 219 328 371 409 426 434 454 462 464 464 464
Mx1 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91 60,91
φx21 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524 0,1524
Γx1 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805 7,805
d1 (m) 0,000 0,004 0,008 0,013 0,021 0,025 0,032 0,038 0,042 0,067 0,084 0,143 0,177 0,210
2
a1 (m/s ) 0,00 1,20 2,40 3,60 5,38 6,09 6,71 6,99 7,13 7,45 7,58 7,62 7,62 7,62
25
PUSHOVER CURVE 500
400 350 300 250 200 150 100 50 0 0,000
0,050
0,100
0,150
0,200
0,250
0,300
Top Displacement, u xN1 (m)
MODAL CAPACITY CURVE 8,00
Modal Spectral Acc., a1 (m/s2)
Total Base Shear, Vx1 (kN)
450
7,00 6,00 5,00 4,00 3,00 2,00 1,00 0,00 0,000
0,050
0,100
0,150
0,200
0,250
Modal Spectral Displacement, d 1 (m)
26
•
RECALL (TDY-2007):
Pushover analysis method can be used only if: ¾ ¾ ¾
Number of stories (not including basement) < 8 Torsional irregularity constant for the building < 1.4 The ratio of the effective mass corresponding to the first mode of vibration to the total mass of the building > 0.70
M x1 N
∑ mi
> 0.70
i =1
M x1 = 60.91 tons 2
∑ mi = 33 + 33 = 66 tons i =1
M x1 2
∑ mi
= 0.92 > 0.70
i =1
(OK – CHECKS)
27
•
DETERMINATION OF THE MODAL DISPLACEMENT DEMAND:
For flexible structures (high period of vibration) (T1>TB):
(TA)
(TB)
Sdi1 = Sde1 (linear elastic) (Equal Disp. Rule)
For rigid structures (low period of vibration) (T1
(TA)
(TB)
Sdi1 > Sde1 (linear elastic)
28
• FOR THE FRAME IN THIS EXAMPLE: T1 = 0.373 sec. TB = 0.40 sec.
LOCAL SOIL TYPE Z2:
CR1 =
1 + ( Ry1 − 1)TB / T1 Ry1
T1 < TB
≥1
Sae1 a y1
Ry1 =
FROM THE MODAL CAPACITY CURVE: ay1 = 7.62 m/s2
Spectral Acceleration Coefficient Spectral Acceleration
Sae1 = A0 IS (T ) g = (0.40)(1.0) [ 2.5] (9.81) = 9.81 m / s 2
Ry1 =
Sae1 9.81 = = 1.2874 a y1 7.62
CR1 =
1 + ( Ry1 − 1)TB / T1 Ry1
=
1 + (1.2874 − 1)(0.4 / 0.373) = 1.016 1.2874
29
Sdi1 = CR1Sde1
Sde1 =
Sae1 9.81 = = 0.035 m (2π / T1 ) 2 (2π / 0.373) 2
Sdi1 = CR1Sde1 = (1.016)(0.035) = 0.036 m
d1( p ) = Sdi1 = 0.036 m ( d1( p ) IS THE MODAL DISPLACEMENT DEMAND)
•
CONVERSION OF THE MODAL DISPLACEMENT DEMAND INTO DISPLACEMENT DEMAND (TARGET DISPLACEMENT):
Modal Capacity Curve
Pushover Curve
Back-conversion:
Modal displacement demand for the first mode Displacement demand (target displacement) at the top of the building (N’th story) in the x-direction (Ötelenme İstemi) p) u x( 21 = Φ x 21Γ x1d1( p ) = (0.1524)(7.8045)(0.036) = 0.043 m p ) IS THE DISPLACEMENT DEMAND (TARGET DISPLACEMENT u x( 21
AT THE TOP OF THE BUILDING)
30
•
THE COMPUTER MODEL OF THE FRAME (ALREADY DEVELOPED) NEEDS TO BE PUSHED UP TO THE DISPLACEMENT DEMAND:
o DEFINE: ANALYSIS CASES o MODIFY/SHOW CASE: PUSHOVER LATERAL o LOAD APPLICATION: MODIFY/SHOW o LOAD TO A MONITORED DISPLACEMENT MAGNITUDE OF 0.043 m
31
DETERMINATION OF TOTAL CURVATURE AND STRAIN DEMANDS:
B202
C103
B102
C102
B101
C203
C202
C201
B201
C101
•
o AT THE TARGET DISPLACEMENT (DISPLACEMENT DEMAND), PLASTIC HINGES HAVE FORMED ON:
C101, C102, C103, B101, B102
o NEED TO CALCULATE THE TOTAL CURVATURE DEMANDS AT THE SECTIONS WHERE THE PLASTIC HINGES HAVE DEVELOPED. 32
• FOR EXAMPLE, FOR THE PLASTIC HINGE ON BEAM B102, THE PLASTIC ROTATION DEMAND (AT THE TARGET DISPLACEMENT) IS
θP = 6.20x10-3 rad
o CONVERT THE PLASTIC ROTATION DEMAND TO PLASTIC CURVATURE DEMAND :
φP = θP / LP = (6.20x10-3 rad)/(0.6m / 2) = 0.021 rad/m o CONVERT THE PLASTIC CURVATURE DEMAND TO TOTAL CURVATURE DEMAND:
φT = φY + φP 33
o OBTAIN THE YIELD CURVATURE (φY) FROM THE RESULTS OF THE MOMENT-CURVATURE ANALYSIS (ALREADY PERFORMED) MY = 250 kN.m
RECALL FOR ALL BEAMS: YIELD MOMENT
YIELD CURVATURE φY = 0.008 rad/m
M o m e n t
MALZEME ÖZELLİKLERİ Eksenel Yük
Beton f ck
(Basınç +) (kN) 0,00
(MPa) 20,00
γmc
f yk
fu
γms
1,00
(MPa) 420
(MPa) 600
1
Adet
(mm) 22 22
Alan 2
(mm ) 3 3
1140 1140
0,01
εsu
b
h
d
Cc
(mm)
(mm)
(mm)
(mm)
450
600
560
20
0,1
400
Kesit Merkezine Uzaklık
350
(mm) -260 260
300
250
(kN.m)
1 2 3 4 5 6 7 8 9 10
φA
Moment
P r o g r a m ı
No.
εsh
(MPa) 200.000
BOYUNA DONATI DÜZENLEMESİ
E ğ r i l i k
BETON KESİT
Boyuna Donatı Es
200
150
100
ETRİYE BİLGİLERİ fyw
(MPa)
420
φe s bk
(mm) (mm) (mm)
8 150 402
hk
(mm)
552
b'k
(mm)
402
h'k ess
(mm)
552 1
50
0
HESAPLA
0,0000
0,0500
0,1000
0,1500
0,2000
0,2500
Eğrilik (rad/m)
o THEREFORE, THE TOTAL CURVATURE DEMAND ON BEAM 102 IS CALCULATED AS:
φT = φY + φP = 0.008 + 0.021 = 0.029 rad/m
34
o FROM THE RESULTS OF THE MOMENT-CURVATURE ANALYSIS, FOR φ = 0.029 rad/m,
εci
= 0.00183 (AT THE EXTREME FIBER IN COMPRESSION)
c
= 6.33 cm,
εs
= (εc)[(d-c)/c] = (0.00183)[(56 cm - 6.33 cm)/ 6.33 cm] = 0.0143
εs
= 0.0143
(AT THE OUTER LAYER OF TENSION STEEL)
• SIMILARLY, FOR THE PLASTIC HINGE ON COLUMN C102, THE PLASTIC ROTATION DEMAND (AT THE TARGET DISPLACEMENT) IS
θP = 3.25x10-3 rad
35
o CONVERT THE PLASTIC ROTATION DEMAND TO PLASTIC CURVATURE DEMAND :
φP = θP / LP = (3.25x10-3 rad)/(0.45m / 2) = 0.014 rad/m o CONVERT THE PLASTIC CURVATURE DEMAND TO TOTAL CURVATURE DEMAND:
φT = φY + φP o OBTAIN THE YIELD CURVATURE (φY) FROM THE RESULTS OF A MOMENT-CURVATURE ANALYSIS ON COLUMN C102 UNDER THE AXIAL LOAD THAT THE COLUMN EXPERIENCES AT THE TARGET DISPLACEMENT. o AT THE TARGET DISPLACEMENT, THE AXIAL LOAD ON COLUMN C102 IS 351 kN (FROM SAP2000 – AXIAL FORCE DIAGRAM) M o m e n t
MALZEME ÖZELLİKLERİ Eksenel Yük
Beton f ck
(Basınç +) (kN) 351,00
(MPa) 20,00
γmc
f yk
fu
γms
1,00
(MPa) 420
(MPa) 420
1
Alan 2
(mm ) 3 2 3
1140 760 1140
0,01
εsu
b
h
d
Cc
(mm)
(mm)
(mm)
(mm)
450
450
410
20
0,1
350
Kesit Merkezine Uzaklık
300
(mm) -185 0 185
250
(kN.m)
(mm) 22 22 22
Adet
Moment
P r o g r a m ı
1 2 3 4 5 6 7 8 9 10
φA
εsh
(MPa) 200.000
BOYUNA DONATI DÜZENLEMESİ No.
E ğ r i l i k
BETON KESİT
Boyuna Donatı Es
200
150
100
ETRİYE BİLGİLERİ fyw
(MPa)
420
φe s bk
(mm) (mm) (mm)
8 200 402
hk
(mm)
402
b'k
(mm)
402
h'k ess
(mm)
402 1
50
0
HESAPLA
0,0000
0,0500
0,1000
0,1500
Eğrilik (rad/m)
o APPROXIMATELY: φY = 0.01 rad/m 36
o THEREFORE, THE TOTAL CURVATURE DEMAND ON COUMN C102 IS CALCULATED AS:
φT = φY + φP = 0.01 + 0.014 = 0.024 rad/m o FROM THE RESULTS OF THE MOMENT-CURVATURE ANALYSIS FOR COLUMN C102 ABOVE, FOR φ = 0.024 rad/m,
εci
= 0.0027 (AT THE EXTREME FIBER IN COMPRESSION)
c
= 11.34 cm,
εs
= (εc)[(d-c)/c] = (0.0027)[(41 cm - 11.34 cm) / 11.34 cm] = 0.0071
εs
= 0.0071
(AT THE OUTER LAYER OF TENSION STEEL)
• (NEED TO REPEAT THESE CALCULATIONS FOR COLUMNS C101 AND C103, AND FOR BEAM B101 (AT EVERY SECTION WHERE A PLASTIC HINGE HAS FORMED)
• AT EACH SECTION WHERE A PLASTIC HINGE HAS FORMED, NEED TO CALCULATE THE STRAIN DEMANDS ON CONCRETE IN COMPRESSION AND STEEL IN TENSION (εci AND εs)
37
• STRAIN CAPACITIES GIVEN IN TDY-2007 FOR CONCRETE AND STEEL: (Unconfined concrete)
(Confined concrete)
For confined concrete:
(Volumetric ratio of existing confinement steel) (Volumetric ratio of confinement steel that is required for design of a new building)
o THE CALCULATED STRAIN DEMANDS ON CONCRETE AND STEEL (εci AND εs) NEED TO BE COMPARED WITH THE CODE DAMAGE LIMITS TO ASSESS THE LEVEL OF DAMAGE IN EACH MEMBER WHERE A PLASTIC HINGE HAS FORMED.
o FOR EXAMPLE, FOR BEAM B102: εci
= 0.00183
< (εcg)MN = 0.004
εs
= 0.0143
< (εsg)MN = 0.010
THEREFORE, BEAM B102 IS IN THE “BELİRGİN HASAR BÖLGESİ” (VISIBLE DAMAGE ZONE)
38
o
FOR COLUMN C102:
εci
= 0.0027 < (εcg)MN = 0.004
εs
= 0.0071 < (εsg)MN = 0.010
THEREFORE, BEAM B102 IS IN THE “MINIMUM HASAR BÖLGESİ” (MINIMUM DAMAGE ZONE)
• BASED ON THE DISTRIBUTION OF DAMAGE IN THE BEAMS AND COLUMNS AT EACH STORY, DETERMINE THE PERFORMANCE LEVEL OF THE STRUCTURE AND MAKE APPROPRIATE DECISIONS TOWARD REHABILITATION. (PER SPECIFICATIONS OF TDY-2007).
39
