3-130801135420-phpapp01.pdf

(1)

Pushover Analysis

an

Inelastic Static Analysis Methods

(2)

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

(3)

Current Status

)

(

)

(

1

1 T

R

T

A

W

V

a

t



• Equivalent Lateral Force Procedure

- Assume global ductility (R

_a

)

- Detail accordingly

• Modal Superposition Procedure

- Include higher mode effects

• Time History Analysis

- Rarely used

(4)

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

(5)

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

(6)

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. A c c e le ra ti o n ( g )

(7)

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

(8)

Available Computer Programs

• Design Oriented:

SAP 2000, GTSTRUDL, RAM etc.

• Research Oriented:

Opensees, IDARC, SeismoStrut etc.

What is different?

• User interface capabilities

• Analysis options

(9)

Section Damage Levels

Damage levels are established based on concrete outermost

compressive fiber strain and steel strain (for nonlinear analysis

procedure).

(10)

Section Damage Levels

How should these values be decided?

- Construction practice

- Experience of engineers

- Input of academicians

(11)

Curvature demand at target curvatures

Φ

_p

= θ

_p

/ L

_p

Φ

_t

= Φ

_y

+ Φ

_p

0 100 200 300 400 500 600 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200

Eğrilik

(rad/m)

M

o

m

e

n

t

(k N.m ) AK GV _GÇ

(Φ

_t

)

(Φ

_y

)

(12)

How do we estimate strains from

a structural analysis?

Strain

Moment

Curvature

Moment

M

_y

ø

_y

ø

_u

Moment

Plastic

Rotations

M

_y

θ

_pu

θ

pu

=(ø

u

– ø

y

) L

p

OR

θ

p

=(ø – ø

y

) L

p

Where L

p

= 0.5h

Utilize this idealized

moment-rotation

response in inelastic

structural analysis

(13)

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 M

_y

(not

significantly more, hardening allowed) undergoing plastic

rotations

h

L

_p

Elastic

Beam-Column

Element

Plastic

Hinges

Rigid End

zones

(14)

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

Eğrilik

M

o

m

e

n

t

EI

_o

0.4-0.8EI

_o

Curvature

(15)

(16)

Steps of Pushover Analysis:

A Simple Incremental Procedure

1. Build a computational model of the structure

(17)

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

(18)

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 R

_a

value.

(19)

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 (V

1 _).

(20)

Steps of Pushover Analysis

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.

V

2 _{= V}

1 ₊

ΔV

F

2 _{= F}

1 ₊

ΔF

d

2 _{= d}

1 ₊

Δd

Results from Step 1 + Results from an

incremental analysis with a hinge placed at

first yield location = Second Hinge formation

(21)

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

V

i

_{= V}

i-1

₊

ΔV

F

i

_{= F}

i-1

₊

ΔF

d

i

_{= d}

i-1

₊

Δd

Results from Step i-1 + Results from an

incremental analysis with a hinge placed at i-1th

yield location = ith hinge formation

(22)

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

(23)

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 J4 J8 J7 J6 J5 _J9 J10 J11 J12

(24)

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

(25)

DATA

10-

f

10 60cm

60cm

Columns

3-

f

10 3-

f

10 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!

(26)

Section Capacities

Eğrilik

M

o

m

e

n

t

f

_y

M

_y

f

ult Eleman N My Φy Φ u lt 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<N

_b

)

Idealized member moment curvature

relations for estimated axial load level

(27)

Effect of Axial Force

• Compute the moment

capacity by accounting for

axial force variation

• Always remain on the yield

(28)

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 M

_y

±5%M

_y

)

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)=

(29)

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)=

(30)

Step 3 (Incremental)

Actual hinges at previously yielded

location for the incremental analysis

New location

at which yield

moment within

tolerance are

reached

ΔEQ=21kN

ΔEQ=14kN

ΔEQ=7kN

42 54 2.94 Total Lateral Disp. at J4 (mm)= 3.5212 Frame M ΔM 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)=

(31)

ΔEQ=3kN

ΔEQ=2kN

ΔEQ=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 ΔM 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)=

(32)

ΔEQ=18kN

ΔEQ=12kN

ΔEQ=6kN

Step 5 (Incremental)

36 96 3.41 Total Lateral Disp. at J4 (mm)= 7.4004 Frame M ΔM 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

(33)

Step 6 (Incremental)

ΔEQ=0.06kN

ΔEQ=0.04kN

ΔEQ=0.02kN

0.12 96.12 0.01277 Total Lateral Disp. at J4 (mm)= 7.41317 Frame M ΔM 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

(34)

Step 7 (Incremental)

ΔEQ=4.8kN

ΔEQ=3.2kN

ΔEQ=1.6kN

9.6 105.72 1.3 Total Lateral Disp. at J4 (mm)= 8.71317 Frame M ΔM 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

(35)

Step 9 (Incremental)

39 144.72

12.69 Total Lateral Disp. at J4 (mm)= 21.40317

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 Total Incremental Load (kN)=

Total Base Shear (kN) =

Inc. Lateral Disp. at J4 (mm)=

ΔEQ=19.5kN

ΔEQ=13kN

(36)

Step 9 (Incremental)

ElementFrame kNmM kNmΔM M + ∆M(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

ΔEQ=0.75kN

ΔEQ=0.50kN

ΔEQ=0.25kN

(37)

Step 10 (Incremental)

4.2 150.42

1.94 Total Lateral Disp. at J4 (mm)= 23.90917 Frame M ΔM 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

ΔEQ=2.1kN

ΔEQ=1.4kN

(38)

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) B a s e S h e a r (k N )

(39)

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) T o ta l B a s e S h e a r( k N )