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4.5 Pushover Analysis (Nonlinear Static Analysis) • Pushover Analysis is a static analysis of an inelastic structure subjected to monotonically increasing forces with an invariant force distribution, i.e., increasing load factor while fixing load pattern. • Its results tell the sequence and magnitudes of yielding (damage), internal forces, deformations, and failure mechanism

• Pushover analysis is similar to the plastic analysis, where failure mechanism and collapse load factor is determined and the moment-rotation relation of plastic hinge is rigidplastic. • But, pushover analysis also keeps track of structural response as the load factor increases incrementally and moment-rotation relation of plastic hinge can be other than rigid-plastic.

4-17

Structural Model for Pushover Analysis • Structural model consists of nonlinear elements • Nonlinear element can be for any type of forces, e.g., bending, shear, or axial force • Plastic hinge is a simple nonlinear element to model yielding in bending • Information on moment-rotation or moment-curvature relation of plastic hinge is required • SAP2000 is a program that is capable of performing pushover analysis as it has plastic hinge element

Plastic-hinge model

4-18

Force-Deformation Relation (or Moment-Rotation Relation) • Force-deformation relation is no longer linear elastic • Knowledge of cyclic behavior is not necessary in pushover analysis; only the first loading branch is required • Force-deformation relation can be elastoplastic, bilinear, degrading, etc • Moment-rotation relation is often rigid-plastic

M

M

θ

θ Elastoplastic

Bilinear

M

Degrading

θ

M

M

θ

θ Rigid-plastic with/without post-yield stiffness

4-19

Height-wise Force Distributions (Force Pattern)

1st Mode Shape

Uniform

4-20

Pushover Curve • Pushover curve is a plot of base shear versus roof displacement • It shows nonlinear behavior of the building • It is often idealized by bilinear curve to determine the yield base shear, which indicate the global lateral strength of the building • Note that global yield point not the same as first local yield point

Base shear Bilinear idealization Actual curve

Roof displacement

4-21

Seismic Evaluation using Pushover Analysis • Seismic evaluation is to assess the seismic performance of a structure by comparing the seismic demands to structure’s capacities • Pushover analysis is used to approximately determine the structural responses (seismic demands) due to an earthquake ground motion (or average response due to a set of earthquake ground motions)

Seismic Demands • Internal forces • Displacement ui of the ith story relative to the ground • Inter-story drift ∆ i of the ith story = ui - ui −1

u2

u1

∆2

ug

4-22

Pushover Analysis to Determine Seismic Demands • The seismic demands, which are the maximum responses during earthquakes, are estimated by the structural responses in pushover analysis when the roof displacement of the structure reaches a predetermined target. • The target roof displacement is determined from the deformation of an equivalent inelastic single-degree-offreedom (SDF) system due to the earthquake ground motion

Assumptions 1. The response of the multi-degree-of-freedom (MDF) structure can be related to the response of an equivalent SDF system, implying that the response is controlled by a single mode and this mode shape remains unchanged even after yielding occurs 2. The invariant lateral force distribution can represent and bound the distribution of inertia forces during an earthquake

4-23

Equivalent Inelastic Single-Degree-of-Freedom (SDF) System Force-displacement relation of SDF system is determined from pushover curve (base shear- roof displacement) V

(a) Idealized Pushover Curve

bn

Idealized V

bny

1

αnkn

Actual

kn 1

ur n

u

rny

Fsn / Ln

(b) Fsn / Ln − Dn Relationship

*

Vbny / Mn

1

α ω2 n n

2

1

ωn

D

ny

=u

rny

4-24

/Γ φ n

Dn rn

Response of SDF System to Earthquake Equation of motion is

+ 2ζω D + F ( D, D ) = −u ( t ) D s g

D

= Displacement of equivalent SDF system

ug (t ) = Earthquake ground acceleration Fs ( D, D ) is determined from pushover curve

Target Roof Displacement Target roof displacement is determined from displacement of equivalent SDF system uro = Γ1φ1r Do

where Do = peak value of D Seismic demands equal to response of structure from pushover analysis when roof displacement equal to target roof displacement.

4-25

FEMA (273 and 356) Nonlinear Static Procedure (NSP) FEMA-273 or the newer FEMA-356 describes in detail how to use do seismic evaluation using pushover analysis Specifies moment-rotation relationship, force patterns, how to determine target roof displacement (coefficient method), acceptance criteria, and limitation of the procedure (when NSP should not be used)

Moment-Rotation Relation of Plastic Hinges

Life safety performance level Collapse prevention performance level

M

C B

D

E θ

A

4-26

FEMA-273 Force Distributions (Force Patterns) Required • Uniform (acceleration) And choose one or more of the followings: • Equivalent lateral force (ELF) • SRSS pattern = Lateral force back calculated from story shear determined by response spectrum analysis (RSA) • First mode pattern (new in FEMA-356)

0.191

0.260

0.173

0.213

0.154 0.134 0.114 0.093 0.070 0.048 0.024

(a) 1st Mode

0.326 0.183

0.170

0.087

0.130 0.096 0.065

0.020

0.111 0.111

0.050

0.111 0.111

0.096

0.111

0.086

0.111

0.006

0.049

(b) ELF

(c) SRSS

4-27

0.111

0.044

0.079

0.040

0.111

0.111

(d) Uniform

Target Roof Displacement

Te2 δ t = C0C1C2C3Sa 2 g 4π δ t = Target roof displacement C0 = Factor to relate spectral displacement to roof disp. C1 = Factor to relate inelastic to elastic displacement C2 = Factor to include degradation of hysteresis loop C3 = Factor to include P-Delta effect Sa = Elastic spectral acceleration Te = Effective period

Limitation of NSP (FEMA-273: 2.9.2.1) • The NSP should not be used for structures in which higher mode effects are significant because it assumes that the response is controlled by a single (fundamental) mode • This leads to the development of modal pushover analysis (MPA), which includes the contribution of higher modes

4-28

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• Pushover analysis is similar to the plastic analysis, where failure mechanism and collapse load factor is determined and the moment-rotation relation of plastic hinge is rigidplastic. • But, pushover analysis also keeps track of structural response as the load factor increases incrementally and moment-rotation relation of plastic hinge can be other than rigid-plastic.

4-17

Structural Model for Pushover Analysis • Structural model consists of nonlinear elements • Nonlinear element can be for any type of forces, e.g., bending, shear, or axial force • Plastic hinge is a simple nonlinear element to model yielding in bending • Information on moment-rotation or moment-curvature relation of plastic hinge is required • SAP2000 is a program that is capable of performing pushover analysis as it has plastic hinge element

Plastic-hinge model

4-18

Force-Deformation Relation (or Moment-Rotation Relation) • Force-deformation relation is no longer linear elastic • Knowledge of cyclic behavior is not necessary in pushover analysis; only the first loading branch is required • Force-deformation relation can be elastoplastic, bilinear, degrading, etc • Moment-rotation relation is often rigid-plastic

M

M

θ

θ Elastoplastic

Bilinear

M

Degrading

θ

M

M

θ

θ Rigid-plastic with/without post-yield stiffness

4-19

Height-wise Force Distributions (Force Pattern)

1st Mode Shape

Uniform

4-20

Pushover Curve • Pushover curve is a plot of base shear versus roof displacement • It shows nonlinear behavior of the building • It is often idealized by bilinear curve to determine the yield base shear, which indicate the global lateral strength of the building • Note that global yield point not the same as first local yield point

Base shear Bilinear idealization Actual curve

Roof displacement

4-21

Seismic Evaluation using Pushover Analysis • Seismic evaluation is to assess the seismic performance of a structure by comparing the seismic demands to structure’s capacities • Pushover analysis is used to approximately determine the structural responses (seismic demands) due to an earthquake ground motion (or average response due to a set of earthquake ground motions)

Seismic Demands • Internal forces • Displacement ui of the ith story relative to the ground • Inter-story drift ∆ i of the ith story = ui - ui −1

u2

u1

∆2

ug

4-22

Pushover Analysis to Determine Seismic Demands • The seismic demands, which are the maximum responses during earthquakes, are estimated by the structural responses in pushover analysis when the roof displacement of the structure reaches a predetermined target. • The target roof displacement is determined from the deformation of an equivalent inelastic single-degree-offreedom (SDF) system due to the earthquake ground motion

Assumptions 1. The response of the multi-degree-of-freedom (MDF) structure can be related to the response of an equivalent SDF system, implying that the response is controlled by a single mode and this mode shape remains unchanged even after yielding occurs 2. The invariant lateral force distribution can represent and bound the distribution of inertia forces during an earthquake

4-23

Equivalent Inelastic Single-Degree-of-Freedom (SDF) System Force-displacement relation of SDF system is determined from pushover curve (base shear- roof displacement) V

(a) Idealized Pushover Curve

bn

Idealized V

bny

1

αnkn

Actual

kn 1

ur n

u

rny

Fsn / Ln

(b) Fsn / Ln − Dn Relationship

*

Vbny / Mn

1

α ω2 n n

2

1

ωn

D

ny

=u

rny

4-24

/Γ φ n

Dn rn

Response of SDF System to Earthquake Equation of motion is

+ 2ζω D + F ( D, D ) = −u ( t ) D s g

D

= Displacement of equivalent SDF system

ug (t ) = Earthquake ground acceleration Fs ( D, D ) is determined from pushover curve

Target Roof Displacement Target roof displacement is determined from displacement of equivalent SDF system uro = Γ1φ1r Do

where Do = peak value of D Seismic demands equal to response of structure from pushover analysis when roof displacement equal to target roof displacement.

4-25

FEMA (273 and 356) Nonlinear Static Procedure (NSP) FEMA-273 or the newer FEMA-356 describes in detail how to use do seismic evaluation using pushover analysis Specifies moment-rotation relationship, force patterns, how to determine target roof displacement (coefficient method), acceptance criteria, and limitation of the procedure (when NSP should not be used)

Moment-Rotation Relation of Plastic Hinges

Life safety performance level Collapse prevention performance level

M

C B

D

E θ

A

4-26

FEMA-273 Force Distributions (Force Patterns) Required • Uniform (acceleration) And choose one or more of the followings: • Equivalent lateral force (ELF) • SRSS pattern = Lateral force back calculated from story shear determined by response spectrum analysis (RSA) • First mode pattern (new in FEMA-356)

0.191

0.260

0.173

0.213

0.154 0.134 0.114 0.093 0.070 0.048 0.024

(a) 1st Mode

0.326 0.183

0.170

0.087

0.130 0.096 0.065

0.020

0.111 0.111

0.050

0.111 0.111

0.096

0.111

0.086

0.111

0.006

0.049

(b) ELF

(c) SRSS

4-27

0.111

0.044

0.079

0.040

0.111

0.111

(d) Uniform

Target Roof Displacement

Te2 δ t = C0C1C2C3Sa 2 g 4π δ t = Target roof displacement C0 = Factor to relate spectral displacement to roof disp. C1 = Factor to relate inelastic to elastic displacement C2 = Factor to include degradation of hysteresis loop C3 = Factor to include P-Delta effect Sa = Elastic spectral acceleration Te = Effective period

Limitation of NSP (FEMA-273: 2.9.2.1) • The NSP should not be used for structures in which higher mode effects are significant because it assumes that the response is controlled by a single (fundamental) mode • This leads to the development of modal pushover analysis (MPA), which includes the contribution of higher modes

4-28

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