3.4 pushover analysis

Pushover Analysis

an

Inelastic Static Analysis Methods

courtesy of Barış 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.

Düzce 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

Φp = θp / Lp

Φt = Φy + Φp

0

100

200

300

400

500

600

0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200

Eğrilik(rad/m)

Mo

men

t(k

N.m

)

AK

GVGÇ

(Φt) (Φy)

How do we estimate strains from

a structural analysis?

Strain

Moment

Curvature

Moment

My

øy øu

Moment

Plastic

Rotations

My

θpu

θpu =(øu – øy) Lp OR

θp =(ø – øy) 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) EIo

Eğrilik

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


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.


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).


Step 2: Beyond Step 1, yielded element’s 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 + ΔV

F2 = F1 + ΔF

d2 = d1 + Δd

Results from Step 1 + Results from an

incremental analysis with a hinge placed at

first yield location = Second Hinge formation


.

.

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 + ΔV

Fi = Fi-1 + ΔF

di = di-1 + Δd

Results from Step i-1 + Results from an

incremental analysis with a hinge placed at i-1th

yield location = ith hinge formation


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

SABİT YÜK HAREKETLİ YÜK YATAY YÜK

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

Eğrilik

Mo

men

t

fy

My

fult

Eleman N My ΦyΦ u 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

Elemnaların Moment-eğrilik ilişkileri

elasto-plastik, pekleşmesiz

To be conservative smaller axial load from two load

combinations can be selected (as long as N<Nb)

Idealized member moment curvature

relations for estimated axial load level Member

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 My±5%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)

ΔEQ=3kN

ΔEQ=2kN

ΔEQ=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 ΔM 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


Actual hinges at previously yielded


New location

at which yield

moment within

tolerance are

reached

ΔEQ=21kN

ΔEQ=14kN

ΔEQ=7kN

42

54

2.94




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



Condition


ΔEQ=3kN

ΔEQ=2kN

ΔEQ=1kN


Actual hinges at previously yielded


New location

at which yield

moment within

tolerance are

reached

6

60

0.4692




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




ΔEQ=18kN

ΔEQ=12kN

ΔEQ=6kN

Step 5 (Incremental) 36

96

3.41




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





ΔEQ=0.06kN

ΔEQ=0.04kN

ΔEQ=0.02kN

0.12

96.12

0.01277




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





ΔEQ=4.8kN

ΔEQ=3.2kN

ΔEQ=1.6kN

9.6

105.72

1.3




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



Condition



39

144.72

12.69


M ΔM 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




ΔEQ=19.5kN

ΔEQ=13kN

ΔEQ=6.5kN

Step 9 (Incremental) Frame M ΔM M + ∆M


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

ΔEQ=0.75kN

ΔEQ=0.50kN

ΔEQ=0.25kN


4.2

150.42

1.94




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




Condition

ΔEQ=2.1kN

ΔEQ=1.4kN

ΔEQ=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

Technology

3.4 pushover analysis