11 Ch4 5 Pushover Analysis

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 rigid-plastic.

• 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.

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

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

Elastoplastic Bilinear

Degrading

Rigid-plastic with/without post-yield stiffness

M

M

M

M

M

θ

θ

θ

θ

θ

Height-wise Force Distributions (Force Pattern)

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

Roof displacement Bilinear idealization

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

u

∆

u

u

u1 u2

2

∆

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-of-freedom (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

Equivalent Inelastic Single-Degree-of-Freedom (SDF) System

Force-displacement relation of SDF system is determined from pushover curve (base shear- roof displacement)

(a) Idealized Pushover Curve

u_{r n} V_bn u r n y V_bny Actual Idealized 1 k n 1 α_nk n (b) F_sn / L_n − D_n Relationship D_n F_sn / L_n V bny / M * n 1 ω_n2 1 αnωn 2

Response of SDF System to Earthquake Equation of motion is

( )

2 _s( , ) _g

D + ζωD + F D D = −u t

D = Displacement of equivalent SDF system

( ) g

u t = Earthquake ground acceleration

( , )

s

F D D is determined from pushover curve

Target Roof Displacement

Target roof displacement is determined from displacement of equivalent SDF system

1 1

ro r o

u = Γφ D

where D = peak value of _o D

Seismic demands equal to response of structure from pushover analysis when roof displacement equal to target roof displacement.

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

M θ A B C D E

Collapse prevention performance level

Life safety performance level

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)

0.024 0.048 0.070 0.093 0.114 0.134 0.154 0.173 0.191 (a) 1st Mode 0.006 0.020 0.040 0.065 0.096 0.130 0.170 0.213 0.260 (b) ELF 0.049 0.086 0.096 0.079 0.050 0.044 0.087 0.183 0.326 (c) SRSS 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 (d) Uniform

Target Roof Displacement 2 0 1 2 3 2

4

T

C C C C S

g

δ

π

=

δ = Target roof displacement 0

C = Factor to relate spectral displacement to roof disp.

1

C = Factor to relate inelastic to elastic displacement

2

C = Factor to include degradation of hysteresis loop

3

C = Factor to include P-Delta effect

a

S = Elastic spectral acceleration

e

T = 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