# 1.1 What is a Pushover Analysis?

Pushover analysis is a static, nonlinear procedure using a simplified, nonlinear technique to estimate seismic structural deformations. It is an incremental static analysis that is used to determine the force-displacement relationship – or the capacity curve – for a structure or structural element.

The analysis involves applying horizontal loads, in a prescribed pattern, to the structure incrementally, i.e., pushing the structure and plotting the total applied shear force and associated lateral displacement at each increment until the structure reaches a collapse condition or a prescribed limit.

This definition of a pushover analysis is in accordance with both the following references:

- FEMA 356 “Prestandard and Commentary for the Seismic Rehabilitation of Buildings” (2000, American Society of Civil Engineers for Federal Emergency Management Agency)
- ATC-40 “Seismic Evaluation and Retrofit of Concrete Buildings” (1996, Applied Technology Council).

# 1.2 Purpose of a Pushover Analysis

It is expected that most buildings rehabilitated in accordance with a standard, would perform within the desired levels when subjected to the design earthquakes. Structures designed according to the existing seismic codes provide minimum safety to preserve life and in a major earthquake, they assure at least gravity-load-bearing elements of non-essential facilities will still function and provide some margin of safety. However, compliance with the standard does not guarantee such performance. They typically do not address the performance of non-structural components neither provide differences in performance between different structural systems. This is because it cannot accurately estimate the inelastic strength and deformation of each member due to linear elastic analysis. Although an elastic analysis gives a good indication of the elastic capacity of structures and indicates where first yielding will occur, it cannot predict failure mechanisms and account for redistribution of forces during progressive yielding.

To overcome these disadvantages different nonlinear static analysis method is used to estimate the inelastic seismic performance of structures, and as the result, the structural safety can be secured against an earthquake. Inelastic analysis procedures help demonstrate how buildings really work by identifying modes of failure and the potential for progressive collapse. The use of inelastic procedures for design and evaluation is an attempt to help engineers better understand how structures will behave when subjected to major earthquakes, where it is assumed that the elastic capacity of the structure will be exceeded. This resolves some of the uncertainties associated with code and elastic procedures.

The practice of earthquake engineering is rapidly evolving to understand the behaviour of buildings subjected to strong earthquakes. In order to be able to predict such behaviour pushover analysis is performed. The overall capacity of a structure depends on the strength and deformation capacities of the individual components of the structure. In order to determine capacities beyond the elastic limit some form of nonlinear analysis, like Pushover Analysis, is required. It is a new performance-based seismic design to achieve analytically a structural design that will reliably perform in a prescribed manner under one or more seismic environments.

# 1.3 Objective of a Pushover Analysis

Pushover analysis is a performance-based analysis that refers to a methodology in which structural criteria are expressed in terms of achieving a performance objective. This is contrasted to the conventional method in which structural criteria are defined by limits on member forces resulting from a prescribed level of applied shear force.

A performance level describes a limiting damage condition that may be considered satisfactory for a given building and a given ground motion. The limiting condition is described by the physical damage within the building, the threat to the life safety of the building’s occupants created by the damage, and the post-earthquake serviceability of the building. The basic approach is to improve the probable seismic performance of the building or to otherwise reduce the existing risk to an acceptable level.

Two key elements of a performance-based design procedure are demand and capacity. Demand is the representation of earthquake ground motion or shaking that the building is subjected to. In nonlinear static analysis procedures, demand is represented by an estimation of the displacements or deformations that the structure is expected to undergo. Capacity is a representation of the structure’s ability to resist the seismic demand. The performance is dependent on the manner that the capacity is able to handle the demand. In other words, the structure must have the capacity to resist the demands of the earthquake such that the performance of the structure is compatible with the objectives of the design. The performance objective is to obtain the desired level of seismic performance of the building, generally described by specifying maximumly allowable (or acceptable) structural or nonstructural damage, for a specified level of seismic hazard.

There are two nonlinear procedures using pushover methods :

- Capacity Spectrum Method
- Displacement Coefficient Method

## 1.3.1 Capacity Spectrum Method

The objective of the Capacity Spectrum Method is to develop appropriate demand and capacity spectra for the structure and to determine their intersection point. During this process, the performance of each structural component is also evaluated.

The demand curve is based on the earthquake response spectrum and the capacity curves are based on static, nonlinear pushover analysis. In pushover analysis, the structure is subjected to increasing levels of load, and the base shear versus roof displacement of the structure is charted along the way. The capacity spectrum is obtained by transforming the base shear versus roof displacement spectrum into a spectral acceleration versus spectral displacement spectrum. The intersection of an appropriate demand curve with the capacity curve is called the performance point. The performance point defines the estimated base shear and displacement of the structure when subjected to the earthquake represented by the demand curve. The behaviour of the structure at the performance point is compared with predefined acceptance criteria to determine if the design objective is met.

### 1.3.1.1 Capacity (Pushover) Curve

The structure capacity is represented by a pushover curve, often termed as a capacity curve. This represents the lateral displacement as a function of the force applied to the structure. The most convenient way to plot the force-displacement curve is by tracking the base shear and roof displacement.

Figure 1-1: Roof deflection, Δ_{roof}, plotted versus base shear, V

### 1.3.1.2 Capacity Spectrum

It is the representation of spectral acceleration vs. spectral displacement derived from the capacity curve. The base shear and the roof displacement are converted to spectral acceleration and spectral displacement respectively by the procedure defined by FEMA to get the capacity spectrum from the capacity curve.

Figure 1-2: Spectral displacement, S_{d} plotted versus spectral acceleration, S_{a}

### 1.3.1.3 Conversion of Capacity Curve to Capacity Spectrum

The capacity curve, in terms of base shear and roof displacement, is converted to capacity spectrum, which is a representation of the capacity curve in Acceleration Displacement Response Spectra (ADRs) format (i.e., S_{a} versus S_{d}).

The required equations for conversion are the following:

…………………………………………….……… (1-3-1)

……………………………… (1-3-2)

S

_{a}= VW/α_{1}………………………………………………………………..…. (1-3-3)S

_{d}= Δ_{roof}/(PF_{1}φ_{roof,l}) …………………………………..……… (1-3-4)

Where:

PF

_{1}= Modal participation factor for the first natural mode.α

_{1}= Modal mass coefficient for the first natural mode.w

_{i}/g = Mass assigned to level i.φ

_{i1}= Amplitude of mode 1 at level i.N = Level N, the level which is the uppermost in the main portion of the structure.

V = Base shear.

W = Building dead weight plus likely live loads.

Δ

_{roof}= Roof displacement (V and the associated make up points on the capacity curve).S

_{a}= Spectral acceleration.S

_{d}= Spectral displacement ( and the associated make up points on the capacity spectrum).

### 1.3.1.4 Demand spectrum

This curve is obtained by redrawing the design earthquake response spectra as a curve of spectral acceleration vs spectral displacement.

Figure 1-3: Response spectrum

### 1.3.1.5 Performance

Once capacity and demand spectra are defined, a performance check can be one. A performance check verifies that structural and nonstructural components are not damaged beyond the acceptable limits of the performance objective for the forces and displacements implied by the displacement demand.

## 1.3.2 Displacement Coefficient Method

The objective of the Displacement Coefficient Method is to find the target displacement which is the maximum displacement that the structure is likely to be experienced during the design earthquake. This is equivalent to the performance point in the Capacity Spectrum method. It provides a numerical process for estimating the displacement demand on the structure, by using a bilinear representation of capacity curve and a series of modification factors, or coefficients, to calculate a target displacement.

The structure, directly incorporating the nonlinear load-deformation characteristics of individual components and elements of the building, is subjected to monotonically increasing lateral loads representing inertia forces in an earthquake until a target displacement is exceeded. The damage state comprises deformations for all elements in the structure. Comparison with acceptability criteria for the desired performance goal leads to the identification of deficiencies for individual elements. Performance check at the expected maximum displacement is done to verify whether the lateral force resistance has not degraded by more than the desired percentage (generally 20%) of the peak resistance and the lateral drifts satisfy limits are per standard code.

#### 1.3.2.1 Target Displacement, δt

The target displacement is calculated as per the procedure described in Section 3.3.3.3.2 of FEMA 356: 2000.

It is given by the following expression:

δ

_{t}= C_{0}C_{1}C_{2}C_{3}S_{a}[T_{e}^{2}/(4π^{2})]g

Where:

C

_{0}= Modification factor to relate spectral displacement to building roof displacement, as determined by Table 3-2 of FEMA 356.C

_{1}= Modification factor to relate expected maximum inelastic displacements to displacements calculated for linear elastic response= 1.5 for T e < 0.1 sec

= 1.0 for T e ≥ T s

= [ 1.0 + ( R – 1 ) T s / T e ] / R for T e < T s

Value of C1 should not be less than 1.0.

T

_{s}= Characteristic period of the response spectrum , defined as period associated with transition from const accleration segment of the spectrum to the constant velocity segment of the spectrum (to be calculated from demand spectrum)T

_{e}= Effective fundamental time period = T_{i}(K_{i}/K_{e})^{1/2}T

_{i}= Elastic fundamental periodK

_{i}= Elastic lateral stiffness of the buildingK

_{e}= Effective lateral stiffness of the building. Taken as equal to the secant stiffness calculated at a base shear force equal to 60% of the effective yield strength of the Structure obtained from bilinear representation of Capacity Curve.R = Ratio of elastic strength demand to calculated yield strength co-efficient = S

_{a}/(V_{y}/W)C_{m}V

_{y}= Effective yield strength calculated using the capacity curve . For larger elements or entire structural systems composed of many components, the effective yield point represents the point at which a sufficient number of individual components or elements have yielded and the global structure begins to experience inelastic deformation.S

_{a}= Response spectrum acceleration, at the effective fundamental period and damping ratio of the building (to be calculated from demand spectrum)W = Effective seismic weight .

C

_{m}= Effective mass factor as determined by Table 3-1 of FEMA 356.C

_{2}= Modification factor to represent the effect of pinched hysteretic shape , stiffness degradation and strength deterioration on maximum displacement response. Taken from Table 3-3 of FEMA 356 for different framing systems and structural performance levels. Alternatively, C_{2}may be taken as 1.0 for a nonlinear procedure.C

_{3}= Modification factor to represent increased displacement due to dynamic P-D effects= 1.0 for buildings with positive post – yield stiffness

= 1.0 + |α|(R – 1)

^{3/2}/T_{e}for buildings with negative post – yield stiffnessα = Ratio of post – yield stiffness to effective elastic stiffness , where the non-linear force-displacement relation shall be characterized by a bilinear relation .

Refer to Figure 3-1 of FEMA 356 for idealized force-displacement curves.

### 1.4 Types of Nonlinearity

Both geometric and material nonlinearities are considered in this static nonlinear pushover analysis.

### 1.5 Force controlled and deformation controlled actions

Force-controlled refers to components, elements, actions, or systems that are not permitted to exceed their elastic limits. This category of elements, generally referred to as brittle or nonductile, experiences significant degradation after only limited post-yield deformation.

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Deformation-controlled refers to components, elements, actions, or systems that can, and are permitted to, exceed their elastic limit in a ductile manner. Force or stress levels for these components are of lesser importance than the amount of deformation beyond the yield point.

Figure 1-4: Component force versus deformation curves (Ref. Fig. 2-3 FEMA 356)

Type 1 curve
Ductile | Type 2 curve
Semi-Ductile | Type 3 curve
Brittle |

(Deformation controlled) | <———————-> | (Force controlled) |

Refer to Section 2.4.4.3.1 of FEMA 356 for detailed information on these curves.

### 1.6 Frame element hinge properties

Discrete hinge properties for frame elements are usually based on FEMA-356 criteria. As per section 5.5.2.2.2 of FEMA 356, in lieu of relationships derived from experiment or analysis, the generalized load-deformation curve shown in figure 1.7.1, with parameters a, b, c, as defined in tables1.5.6 and 1.5.7, shall be used for components of steel moment frames. Modification of this curve shall be permitted to account for strain-hardening of components as follows:

- a strain-hardening slope of 3% of the elastic slope shall be permitted for beams and columns unless a greater strain-hardening slope is justified by test data; and
- where panel zone yielding occurs, a strain-hardening slope of 6% shall be used for the panel zone unless a greater strain-hardening slope is justified by test data.

Figure 1-5: Generalized Force-Deformation Relationship for Components

- Point A is the origin
- Point B represents yielding. No deformation occurs in the hinge up to point B, regardless of the deformation value specified for point B. The displacement (rotation or axial elongation as the case may be) will be subtracted from the displacements at points C, D and E. Only plastic deformation beyond point B will be exhibited by a hinge.
- Point C represents the ultimate capacity of the plastic hinge. At this point, hinge strength degradation begins (hinge starts shedding load) until it reaches point D.
- Point D represents the residual strength of the plastic hinge. Beyond point D the component responds with substantial strength to point E.
- Point E represents total failure. At deformation greater than point E the plastic hinge will drop the load to zero.

The parameters Q and Q_{CE} (Q_{y}) in Figure 1-6 are generalized component load and generalized component expected strength, respectively. For beams and columns, θ is the total elastic and plastic rotation of the beam or column, θ_{y} is the rotation at yield. For braces, Δ is totally elastic and plastic displacement, and Δ_{y} is yield displacement.

Use of equations (1-6-1) and (1-6-2) to calculate the yield rotation, θ_{y}, where the point of contra flexure is anticipated to occur at the mid-length of the beam or column, respectively, shall be permitted.

For beams:

θ

_{y}= Z·F_{ye}L_{b}/(6·EI_{b}) ……………… (1-6-1)For columns:

θ

_{y}= Z·F_{ye}L_{c}/(6·EI_{c}) (1 – P/P_{ye}) ……………… (1-6-2)

Q and Q_{CE} are the generalized component load and generalized component expected strength, respectively. For flexural actions of beams and columns, Q_{CE} refers to the plastic moment capacity, which shall be calculated using equations (1-6-3) and (1-6-4):

For beams:

Q

_{CE}= M_{CE}= Z·F_{ye}……………… (1-6-3)For columns:

Q

_{CE}= M_{CE}= 1.18·Z·F_{ye}(1 – P/P_{ye}) ……………… (1-6-4)

Where:

E = Modulus of elasticity

F

_{ye}= Expected yield strength of the materialI = Moment of inertia

L

_{b}= Beam lengthL

_{c}= Column lengthM

_{CE}= Expected flexural strength of a member or Joint, kip-in.P = Axial force in the member

P

_{ye}= Expected axial yield force of the member = A_{g}F_{ye}Q = Generalized component load

Q

_{CE}= Generalized component expected strength = Effective expected strength, which is defined as the statistical mean value of yield strengths, Q_{y}, for a population of similar components, and includes consideration of strain hardening and plastic section development .

### 1.7 Elements

Major horizontal or vertical portions of the building’s structural systems that act to resist lateral forces or support vertical gravity loads such as frames, shear walls, frame walls, diaphragms, and foundations.

### Primary Elements

These are structural components or elements that provide a significant portion of the structure’s lateral force resisting stiffness and strength at the performance point. These are the elements that are needed to resist lateral loads after several cycles of inelastic response to the earthquake ground motion.

### Secondary Elements

These are structural components or elements that are not, or are not needed to be, primary elements of the lateral load resisting system. However, secondary elements may be needed to support vertical gravity loads and may resist some lateral loads.

### 1.8 Lateral Load Distribution

Lateral loads can be applied by any one of the following three methods per Section 3.3.3.2.3 of FEMA 356.

#### 1.8.1 Method 1

The vertical distribution of the base shear shall be as specified in this section for all buildings. The lateral load applied at any floor level x shall be determined in accordance with equation (1-8-1) and equation (1-8-2):

F

_{x}= C_{vx}V ……………………..………. (1-8-1)

Where:

…………………………. (1-8-2)

#### 1.8.2 Method 2

A vertical distribution proportional to the shape of the fundamental mode in the direction under consideration is performed. Use of this distribution shall be permitted only when more than 75% of the total mass participates in the fundamental mode in the direction under consideration, and the uniform distribution is also used.

…………………….……. (1-8-3)

#### 1.8.3 Method 3

Vertical distribution is performed consisting of lateral forces at each level proportional to the total mass at each level.

………………………………. (1-8-4)

Where:

C

_{vx}= Vertical distribution factork = Linear interpolation shall be used to calculate values of for intermediate values of k for intermediate values of T.

= 2.0 for T ≥ 2.5 seconds

= 1.0 for T ≤ 0.5 seconds

V = User defined base shear

w

_{i}= Portion of the total building weight W located on or assigned to floor level iw

_{x}= Portion of the total building weight W located on or assigned to floor level xh

_{i}= Height (in ft) from the base to floor level ih

_{x}= Height (in ft) from the base to floor level xF

_{x}= Amplitude of mode a floor level x