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  • Pushover Analysis

    an

    Inelastic Static Analysis Methods

    courtesy of Bar Binici

  • Target Performance

    Dictated by codes (DBYBHY 2007, Section 1.2.1):

    ....The objective of seismic resistant design is

    to have no structural/nonstructural damage

    in low magnitude earthquakes, limited and

    repairable damage in moderate earthquakes

    and life safety for extreme earthquakes...

  • Current Status

    )(

    )(

    1

    1

    TR

    TAWV

    a

    t

    Equivalent Lateral Force Procedure

    - Assume global ductility (Ra)

    - Detail accordingly

    Modal Superposition Procedure

    - Include higher mode effects

    Time History Analysis

    - Rarely used

    - Tedious and requires hysteretic models

  • Critique of Current Practice

    Advantages :

    - Simple to use

    - Have proven to work

    - Became a tradition all over the world

    - Uncertainty is lumped and easier to deal with

    Disadvantages :

    - No clear connection between capacity and demand

    - No option for interfering with the target performance

    - No possibility of having the owner involved in the decision process

    - Not easily applicable to seismic assessment of existing structures

  • DBYBHY 2007 (Chapter 7)

    - Evaluation and Strengthening of Existing Buildings

    is based on structural performances.

    - Steps:

    Collect information from an existing structure

    Assess whether info is dependable and penalize accordingly

    Conduct structural analysis

    - Linear static analysis

    - Nonlinear static analysis (Pushover analysis)

    - Incremental pushover analysis

    - Time history analysis

    Identify for each member the damage level

    Decision based on number of elements at certain damage levels

  • Time History? - Actual earthquake response is hard to predict anyways.

    - Closest estimate can be found using inelastic time-history analysis.

    - Difficulties with inelastic time history analysis:

    - Suitable set of ground motion (Description of demand)

    - hysteretic behavior models (Description of capacity)

    - Computation time (Time)

    - Post processing (Time and understanding)

    Alternative approach is pushover analysis.

    Dzce Ground Motion

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0 5 10 15 20 25 30

    Sec.

    Accele

    rati

    on

    (g

    )

  • Pushover Analysis

    Definition: Inelastic static analysis of a

    structure using a specified (constant or

    variable) force pattern from zero load to a

    prescribed ultimate displacement.

    Use of it dates back to 1960s to1970s to

    investigate stability of steel frames.

    Many computer programs were developed

    since then with many features and limitations.

  • Available Computer Programs Design Oriented:

    SAP 2000, GTSTRUDL, RAM etc.

    Research Oriented:

    Opensees, IDARC, SeismoStrut etc.

    What is different?

    User interface capabilities

    Analysis options

    Member behavior options

  • Section Damage Levels

    Damage levels are established based on concrete outermost

    compressive fiber strain and steel strain (for nonlinear analysis

    procedure).

  • Section Damage Levels

    How should these values be decided?

    - Construction practice

    - Experience of engineers

    - Input of academicians

  • Curvature demand at target curvatures

    p = p / Lp

    t = y + p

    0

    100

    200

    300

    400

    500

    600

    0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200

    Erilik(rad/m)

    Mo

    men

    t(k

    N.m

    )

    AK

    GVG

    (t) (y)

  • How do we estimate strains from

    a structural analysis?

    Strain

    Moment

    Curvature

    Moment

    My

    y u

    Moment

    Plastic

    Rotations

    My

    pu

    pu =(u y) Lp OR

    p =( y) Lp

    Where Lp = 0.5h

    Utilize this idealized

    moment-rotation

    response in inelastic

    structural analysis

  • Definition of Potential Plastic Hinges

    End regions of columns and beams (center for gravity loads) are the potential plastic hinges Plastic hinges are hinges capable of resisting My (not significantly more, hardening allowed) undergoing plastic rotations

    h

    Lp

    Elastic

    Beam-

    Column

    Element

    Plastic

    Hinges

    Rigid End

    zones

  • Elastic Parts For regions other than plastic hinging occurs, cracking is expected therefore use of cracked stiffness is customary (0.4 -0.8) EI o

    Erilik

    Mo

    men

    t

    EIo

    0.4-0.8EIo

    Curvature

  • Pushover Analysis

  • Steps of Pushover Analysis:

    A Simple Incremental Procedure

    1. Build a computational model of the structure

  • Steps of Pushover Analysis

    2. Define member behavior Beams: Moment-rotation relations

    Columns: Moment-rotation and Interaction Diagrams

    Beam-column joints: Assume rigid (DBYBHY 2007 )

    Walls: Model as beam columns but introduce a shear spring to model shear deformations

    Use cracked rigidities for elastic portions

  • Steps of Pushover Analysis

    3. Apply gravity loads

    1.0 G + n Q n=0.3 (live load reduction factor)

    (if the interaction diagrams will not be used a good

    estimate of the moment capacity of column hinges

    needs to be made)

    Possibilities:

    - Based on initial gravity load analysis

    - Based on a beam hinging mechanism

    - Based on elastic lateral force analysis with an

    assumed reasonable Ra value.

  • Steps of Pushover Analysis

    4. Specify a Lateral Load Profile: (Inverted triangular, constant, first mode shape are some of the

    possibilities)

    It is a good idea to have a spreadsheet page ready indicating all members, current load increment

    5. Lateral Load Incrementing:

    Step 1: Elastic analysis is valid up to the formation of the first hinge,

    i.e. when the first critical location reaches its moment capacity.

    Find the lateral loads that cause first hinge formation (V1).

    Record all member forces and deformations (F1, d1).

  • Steps of Pushover Analysis

    Step 2: Beyond Step 1, yielded elements critical location cannot

    take any further moment. Therefore place an actual hinge at that location. Conduct an analysis increment for this modified structure. This load increment should be selected such that upon summing the force resultant from this incremental step and previous step, second hinge formation is reached.

    V2 = V1 + V

    F2 = F1 + F

    d2 = d1 + d

    Results from Step 1 + Results from an

    incremental analysis with a hinge placed at

    first yield location = Second Hinge formation

  • Steps of Pushover Analysis

    .

    .

    Step i: Similar to step 2 but additional hinges form and

    incremental analysis steps are conducted for systems with more hinges. Results are added to those from the previous step

    Vi = Vi-1 + V

    Fi = Fi-1 + F

    di = di-1 + d

    Results from Step i-1 + Results from an

    incremental analysis with a hinge placed at i-1th

    yield location = ith hinge formation

  • Steps of Pushover Analysis

    Step n:

    Sufficient number of plastic hinges have formed and

    system has reached a plastic mechanism. Note that this

    could be a partial collapse mechanism as well. Beyond

    this point system rotates as a rigid body.

    ANALYSIS DONE

    - Plot Base Shear- Roof Displacement

    - Check member rotations and identify performance levels

  • Example Application: 3 Story- 2 Bay

    RC Frame (Courtesy of Ahmet Yakut)

    M O D E L

    3m

    3m

    3m

    1

    2

    3

    10

    11

    12

    13

    14

    15

    4

    5

    6

    7

    8

    9

    6m 6m

    J1

    J2

    J3

    J4J8

    J7

    J6

    J5 J9

    J10

    J11

    J12

  • Assumptions

    Assume

    Constant Axial Load on Columns for Analysis Steps

    Rigid-plastic with no hardening or softening moment-rotation behavior for columns and beams

    plastic hinging occurs when moment capacity is within 5% tolerance

    Load combinations 1.0 DL + 0.3 LL and 1.0 DL + 0.3 LL+1.0EQ to compute axial load levels

    DL=10kN/m

    DL=15kN/m

    DL=15kN/m

    LL=2kN/m

    LL=2kN/m

    LL=2kN/m

    EQ=60kN

    EQ=40kN

    EQ=20kN

    SABT YK HAREKETL YK YATAY YK

  • DATA

    10-f10

    60cm

    60cm

    Columns

    3-f10

    3-f10

    25cm

    50cm

    Beams

    Steel (fyd=495 Mpa)

    Concrete (fcd=25 Mpa)

    Clear cover=5 cm

    E=2.779E+4 MPa

    M+ is the same as M-

    Note that if this is a seismic evaluation problem strength values obtained

    at site should be used!

  • Section Capacities

    Erilik

    Mo

    men

    t

    fy

    My

    fult

    Eleman N Myu l t

    kN kNm rad/m rad/m

    1 -83,786 124 0,0055 0,111

    2 -51,347 115,5 0,0056 0,115

    3 -19,872 107,5 0,0056 0,119

    4 -253,392 166 0,0059 0,085

    5 -158,905 143 0,0060 0,099

    6 -64,797 119 0,0060 0,113

    7 -124,104 133,5 0,0056 0,105

    8 -77,747 122 0,0057 0,112

    9 -31,201 110 0,0054 0,118

    10 5,606 49 0,0073 0,103

    11 1,421 50 0,0069 0,102

    12 -17,233 53 0,0069 0,099

    13 5,606 49 0,0073 0,103

    14 1,421 50 0,0069 0,102

    15 -17,233 53 0,0069 0,099

    Elemnalarn Moment-erilik ilikileri

    elasto-plastik, peklemesiz

    To be conservative smaller axial load from two load

    combinations can be selected (as long as N

  • Effect of Axial Force

    Compute the moment

    capacity by accounting for

    axial force variation

    Always remain on the yield

    surface

  • Step 1

    DL=10kN/m

    DL=15kN/m

    DL=15kN/m

    LL=2kN/m

    LL=2kN/m

    LL=2kN/m

    EQ=3kN

    EQ=2kN

    EQ=1kN

    COMBO2: 1.0 DL + 0.3 LL + 1.0 EQ

    Detection of first yield (moment

    reaches My5%My )

    6

    Frame Joint Myield M

    Element Label kNm kNm

    J1 124.0 -4.33

    J2 124.0 20.60

    J2 115.5 -22.14

    J3 115.5 21.00

    J3 107.5 -22.23

    J4 107.5 27.35

    J5 166.0 6.23

    J6 166.0 -0.60

    J6 143.0 3.50

    J7 143.0 -2.94

    J7 119.0 1.52

    J8 119.0 -3.29

    J9 133.5 16.03

    J10 133.5 -20.07

    J10 122.0 26.88

    J11 122.0 -24.83

    J11 110.0 22.95

    J12 110.0 -30.82

    J2 49.0 -42.74

    J6 49.0 -49.58 YIELDED

    J3 50.0 -43.24

    J7 50.0 -49.28

    J4 53.0 -27.35

    J8 53.0 -34.34

    J6 49.0 -45.48

    J10 49.0 -46.95

    J7 50.0 -44.83

    J11 50.0 -47.79

    J8 53.0 -31.05

    J12 53.0 -30.82

    0.2947

    11

    12

    6

    7

    4

    14

    15

    Condition

    13

    5

    8

    9

    3

    10

    1

    2

    First yielding stage

    Total Base Shear (kN)=

    Lateral Disp. at J4 (mm)=

    J4 (monitored node )

  • Step 2 (Incremental)

    EQ=3kN

    EQ=2kN

    EQ=1kN

    Actual hinge at previously yielded

    location for the incremental analysis

    New

    locations at

    which yield

    moments

    within

    tolerance are

    reached

    6

    12

    0.2865

    Total Lateral Disp. at J4 (mm)= 0.5812

    Frame M M M + M

    Element kNm kNm (kNm)

    -4.33 6.39 2.06

    20.60 0.76 21.36

    -22.14 2.05 -20.10

    21.00 -2.18 18.82

    -22.23 0.24 -21.99

    27.35 -1.82 25.53

    6.23 6.47 12.71

    -0.60 0.39 -0.21

    3.50 2.79 6.29

    -2.94 -3.15 -6.09

    1.52 1.56 3.08

    -3.29 -3.43 -6.72

    16.03 6.48 22.51

    -20.07 0.20 -19.87

    26.88 2.57 29.45

    -24.83 -2.26 -27.09

    22.95 0.15 23.10

    -30.82 -1.80 -32.62

    -42.74 1.29 -41.46

    -49.58 0.00 -49.58 YIELDED

    -43.24 2.42 -40.82

    -49.28 -2.36 -51.64 YIELDED

    -27.35 1.82 -25.53

    -34.34 -1.73 -36.07

    -45.48 2.40 -43.08

    -46.95 -2.38 -49.33 YIELDED

    -44.83 2.35 -42.48

    -47.79 -2.41 -50.19 YIELDED

    -31.05 1.71 -29.34

    -30.82 -1.80 -32.62

    13

    14

    15

    9

    10

    11

    12

    5

    6

    7

    8

    1

    2

    3

    4

    Inc. Lateral Disp. at J4 (mm)=

    Total Base Shear (kN) =

    Total Incremental Load (kN)=

    Condition

  • Step 3 (Incremental)

    Actual hinges at previously yielded

    location for the incremental analysis

    New location

    at which yield

    moment within

    tolerance are

    reached

    EQ=21kN

    EQ=14kN

    EQ=7kN

    42

    54

    2.94

    Total Lateral Disp. at J4 (mm)= 3.5212

    Frame M M M + M

    Element kNm kNm (kNm)

    2.06 57.79 59.85

    21.36 12.12 33.48

    -20.10 24.68 4.58

    18.82 -16.19 2.64

    -21.99 -2.12 -24.11

    25.53 -18.94 6.58

    12.71 56.85 69.56

    -0.21 12.18 11.97

    6.29 24.58 30.87

    -6.09 -13.41 -19.49

    3.08 0.99 4.07

    -6.72 -34.94 -41.67

    22.51 53.65 76.16

    -19.87 18.00 -1.88

    29.45 18.00 47.45

    -27.09 -8.15 -35.24

    23.10 -8.15 14.95

    -32.62 -18.38 -51.00

    -41.46 12.56 -28.90

    -49.58 0.00 -49.58 YIELDED

    -40.82 14.07 -26.75

    -51.64 0.00 -51.64 YIELDED

    -25.53 18.94 -6.58

    -36.07 -17.61 -53.68 YIELDED

    -43.08 12.40 -30.68

    -49.33 0.00 -49.33 YIELDED

    -42.48 14.40 -28.08

    -50.19 0.00 -50.19 YIELDED

    -29.34 17.33 -12.01

    -32.62 -18.38 -51.00

    12

    13

    14

    15

    8

    9

    10

    11

    1

    2

    3

    4

    5

    6

    7

    Inc. Lateral Disp. at J4 (mm)=

    Total Base Shear (kN) =

    Condition

    Total Incremental Load (kN)=

  • EQ=3kN

    EQ=2kN

    EQ=1kN

    Step 4 (Incremental)

    Actual hinges at previously yielded

    location for the incremental analysis

    New location

    at which yield

    moment within

    tolerance are

    reached

    6

    60

    0.4692

    Total Lateral Disp. at J4 (mm)= 3.9904

    Frame M M M + M

    Element kNm kNm (kNm)

    59.85 8.59 68.44

    33.48 2.00 35.48

    4.58 3.91 8.49

    2.64 -1.96 0.67

    -24.11 0.29 -23.82

    6.58 -1.96 4.63

    69.56 8.43 77.99

    11.97 2.07 14.04

    30.87 3.95 34.82

    -19.49 -1.77 -21.26

    4.07 0.50 4.57

    -41.67 -3.40 -45.07

    76.16 7.95 84.12

    -1.88 2.90 1.02

    47.45 2.90 50.35

    -35.24 -0.50 -35.74

    14.95 -0.50 14.45

    -51.00 -3.35 -54.36

    -28.90 1.91 -26.99

    -49.58 0.00 -49.58 YIELDED

    -26.75 2.26 -24.49

    -51.64 0.00 -51.64 YIELDED

    -6.58 1.96 -4.63

    -53.68 0.00 -53.68 YIELDED

    -30.68 1.88 -28.79

    -49.33 0.00 -49.33 YIELDED

    -28.08 2.27 -25.81

    -50.19 0.00 -50.19 YIELDED

    -12.01 3.40 -8.61

    -51.00 -3.35 -54.36 YIELDED

    13

    14

    15

    9

    10

    11

    12

    5

    6

    7

    8

    1

    2

    3

    4

    Condition

    Inc. Lateral Disp. at J4 (mm)=

    Total Base Shear (kN) =

    Total Incremental Load (kN)=

  • EQ=18kN

    EQ=12kN

    EQ=6kN

    Step 5 (Incremental) 36

    96

    3.41

    Total Lateral Disp. at J4 (mm)= 7.4004

    Frame M M M + M

    Element kNm kNm (kNm)

    68.44 55.34 123.78

    35.48 15.86 51.34

    8.49 28.66 37.15

    0.67 -6.38 -5.71

    -23.82 10.42 -13.40

    4.63 -15.82 -11.19

    77.99 54.50 132.49

    14.04 16.03 30.06

    34.82 28.70 63.52

    -21.26 -6.00 -27.26

    4.57 10.75 15.33

    -45.07 -15.83 -60.90

    84.12 51.48 135.60 YIELDED

    1.02 21.43 22.45

    50.35 21.43 71.78

    -35.74 1.18 -34.57

    14.45 1.18 15.62

    -54.36 0.00 -54.36

    -26.99 12.80 -14.19

    -49.58 0.00 -49.58 YIELDED

    -24.49 16.80 -7.69

    -51.64 0.00 -51.64 YIELDED

    -4.63 15.82 11.19

    -53.68 0.00 -53.68 YIELDED

    -28.79 12.68 -16.12

    -49.33 0.00 -49.33 YIELDED

    -25.81 16.75 -9.05

    -50.19 0.00 -50.19 YIELDED

    -8.61 15.83 7.22

    -54.36 0.00 -54.36 YIELDED

    12

    13

    14

    15

    8

    9

    10

    11

    1

    2

    3

    4

    5

    6

    7

    Condition

    Inc. Lateral Disp. at J4 (mm)=

    Total Base Shear (kN) =

    Total Incremental Load (kN)=

  • Step 6 (Incremental)

    EQ=0.06kN

    EQ=0.04kN

    EQ=0.02kN

    0.12

    96.12

    0.01277

    Total Lateral Disp. at J4 (mm)= 7.41317

    Frame M M M + M

    Element kNm kNm (kNm)

    123.78 0.25 124.03 YIELDED

    51.34 0.03 51.38

    37.15 0.08 37.23

    -5.71 -0.03 -5.74

    -13.40 0.03 -13.37

    -11.19 -0.06 -11.25

    132.49 0.26 132.75

    30.06 0.02 30.09

    63.52 0.07 63.60

    -27.26 -0.02 -27.29

    15.33 0.04 15.36

    -60.90 -0.06 -60.96

    135.60 0.00 135.60 YIELDED

    22.45 0.09 22.54

    71.78 0.09 71.87

    -34.57 0.00 -34.57

    15.62 0.00 15.63

    -54.36 0.00 -54.36

    -14.19 0.05 -14.14

    -49.58 0.00 -49.58 YIELDED

    -7.69 0.06 -7.63

    -51.64 0.00 -51.64 YIELDED

    11.19 0.06 11.25

    -53.68 0.00 -53.68 YIELDED

    -16.12 0.05 -16.07

    -49.33 0.00 -49.33 YIELDED

    -9.05 0.06 -8.99

    -50.19 0.00 -50.19 YIELDED

    7.22 0.06 7.28

    -54.36 0.00 -54.36 YIELDED

    13

    14

    15

    9

    10

    11

    12

    5

    6

    7

    8

    1

    2

    3

    4

    Condition

    Inc. Lateral Disp. at J4 (mm)=

    Total Base Shear (kN) =

    Total Incremental Load (kN)=

  • Step 7 (Incremental)

    EQ=4.8kN

    EQ=3.2kN

    EQ=1.6kN

    9.6

    105.72

    1.3

    Total Lateral Disp. at J4 (mm)= 8.71317

    Frame M M M + M

    Element kNm kNm (kNm)

    124.03 0.00 124.03 YIELDED

    51.38 4.04 55.42

    37.23 8.81 46.05

    -5.74 -3.63 -9.37

    -13.37 2.07 -11.30

    -11.25 -5.15 -16.40

    132.75 35.16 167.90 YIELDED

    30.09 -3.63 26.45

    63.60 2.03 65.63

    -27.29 -2.56 -29.84

    15.36 3.01 18.38

    -60.96 -5.18 -66.14

    135.60 0.00 135.60 YIELDED

    22.54 5.95 28.49

    71.87 5.95 77.82

    -34.57 -1.02 -35.58

    15.63 -1.02 14.61

    -54.36 0.00 -54.36

    -14.14 4.77 -9.37

    -49.58 0.00 -49.58 YIELDED

    -7.63 5.70 -1.93

    -51.64 0.00 -51.64 YIELDED

    11.25 5.15 16.40

    -53.68 0.00 -53.68 YIELDED

    -16.07 5.67 -10.40

    -49.33 0.00 -49.33 YIELDED

    -8.99 5.57 -3.42

    -50.19 0.00 -50.19 YIELDED

    7.28 5.18 12.46

    -54.36 0.00 -54.36 YIELDED

    12

    13

    14

    15

    8

    9

    10

    11

    1

    2

    3

    4

    5

    6

    7

    Total Base Shear (kN) =

    Total Incremental Load (kN)=

    Condition

    Inc. Lateral Disp. at J4 (mm)=

  • Step 9 (Incremental)

    39

    144.72

    12.69

    Total Lateral Disp. at J4 (mm)= 21.40317

    M M M + M

    kNm kNm (kNm)

    124.03 0.00 124.03 YIELDED

    55.42 -46.64 8.78

    46.05 5.74 51.79

    -9.37 -44.15 -53.51

    -11.30 1.29 -10.01

    -16.40 -38.69 -55.09

    167.90 0.00 167.90 YIELDED

    26.45 -46.22 -19.76

    65.63 6.05 71.68

    -29.84 -43.74 -73.58

    18.38 1.72 20.10

    -66.14 -38.78 -104.91

    135.60 0.00 135.60 YIELDED

    28.49 -24.15 4.35

    77.82 -24.15 53.68

    -35.58 -21.98 -57.57

    14.61 -21.98 -7.37

    -54.36 0.00 -54.36

    -9.37 52.37 43.00

    -49.58 0.00 -49.58 YIELDED

    -1.93 45.43 43.51

    -51.64 0.00 -51.64 YIELDED

    16.40 38.69 55.09 YIELDED

    -53.68 0.00 -53.68 YIELDED

    -10.40 52.27 41.87

    -49.33 0.00 -49.33 YIELDED

    -3.42 45.46 42.03

    -50.19 0.00 -50.19 YIELDED

    12.46 38.78 51.24

    -54.36 0.00 -54.36 YIELDED

    Condition

    Total Incremental Load (kN)=

    Total Base Shear (kN) =

    Inc. Lateral Disp. at J4 (mm)=

    EQ=19.5kN

    EQ=13kN

    EQ=6.5kN

  • Step 9 (Incremental) Frame M M M + M

    Element kNm kNm (kNm)

    124.03 0.00 124.03 YIELDED

    8.78 -1.83 6.95

    51.79 0.44 52.22

    -53.51 -1.74 -55.25

    -10.01 0.30 -9.71

    -55.09 0.00 -55.09

    167.90 0.00 167.90 YIELDED

    -19.76 -1.82 -21.59

    71.68 0.44 72.12

    -73.58 -1.44 -75.02

    20.10 0.64 20.74

    -104.91 -1.86 -106.77

    135.60 0.00 135.60 YIELDED

    4.35 -0.84 3.50

    53.68 -0.84 52.83

    -57.57 -0.54 -58.11

    -7.37 -0.54 -7.91

    -54.36 0.00 -54.36

    43.00 2.27 45.27

    -49.58 0.00 -49.58 YIELDED

    43.51 2.03 45.54

    -51.64 0.00 -51.64 YIELDED

    55.09 0.00 55.09 YIELDED

    -53.68 0.00 -53.68 YIELDED

    41.87 2.26 44.13

    -49.33 0.00 -49.33 YIELDED

    42.03 2.08 44.11

    -50.19 0.00 -50.19 YIELDED

    51.24 1.86 53.10 YIELDED

    -54.36 0.00 -54.36 YIELDED

    12

    13

    14

    15

    8

    9

    10

    11

    1

    2

    3

    4

    5

    6

    7

    Condition

    EQ=0.75kN

    EQ=0.50kN

    EQ=0.25kN

  • Step 10 (Incremental)

    4.2

    150.42

    1.94

    Total Lateral Disp. at J4 (mm)= 23.90917

    Frame M M M + M

    Element kNm kNm (kNm)

    124.03 0.00 124.03 YIELDED

    6.95 -5.34 1.61

    52.22 2.18 54.40

    -55.25 -4.04 -59.29

    -9.71 3.14 -6.57

    -55.09 0.00 -55.09

    167.90 0.00 167.90 YIELDED

    -21.59 -5.17 -26.76

    72.12 2.35 74.47

    -75.02 -4.19 -79.21

    20.74 3.00 23.73

    -106.77 0.00 -106.77

    135.60 0.00 135.60 YIELDED

    3.50 -2.09 1.41

    52.83 -2.09 50.74

    -58.11 0.16 -57.95

    -7.91 0.16 -7.75

    -54.36 0.00 -54.36

    45.27 7.52 52.79 YIELDED

    -49.58 0.00 -49.58 YIELDED

    45.54 7.18 52.72 YIELDED

    -51.64 0.00 -51.64 YIELDED

    55.09 0.00 55.09 YIELDED

    -53.68 0.00 -53.68 YIELDED

    44.13 7.52 51.65 YIELDED

    -49.33 0.00 -49.33 YIELDED

    44.11 7.18 51.30 YIELDED

    -50.19 0.00 -50.19 YIELDED

    53.10 0.00 53.10 YIELDED

    -54.36 0.00 -54.36 YIELDED

    13

    14

    15

    9

    10

    11

    12

    5

    6

    7

    8

    1

    2

    3

    4

    Total Incremental Load (kN)=

    Total Base Shear (kN) =

    Inc. Lateral Disp. at J4 (mm)=

    Condition

    EQ=2.1kN

    EQ=1.4kN

    EQ=0.7kN

  • Collapse Mechanism

    S Y S T E M I S U N S T A B L E

    Beam sway mechanism is observed

    No further lateral load incrementing

    possible (only rigid body motion)

    0

    20

    40

    60

    80

    100

    120

    140

    160

    0 5 10 15 20 25 30

    Roof Displacement (mm)

    Ba

    se

    Sh

    ea

    r (k

    N)

  • What did we obtain?

    A simple representation of the capacity curve

    Plastic mechanism and sequence of hinge formation

    Lateral load and displacement capacity

    Ductility and plastic rotation demand

    0

    20

    40

    60

    80

    100

    120

    140

    160

    0 5 10 15 20 25 30

    Top Displacement (mm)

    To

    tal

    Ba

    se

    Sh

    ea

    r(k

    N) Incremental

    SAP2000

    SAP 2000 built in pushover

    analysis options include:

    hardening/loss of strength

    P-M interaction

    Systematic stiffness approach

  • Concluding Remarks

    Nonlinear analysis is becoming a part of

    the profession

    It gives us information on displacements

    which are indicators of damage

    Never forget that estimating deformations

    is harder compared to estimating strength

    Never replace engineering judgment with

    any analysis procedure