Current guideline documents such as FEMA-356 (2000) utilize inter-story drift ratio and plastic rotation to establish building performance levels such as immediate occupancy, life-safety, and collapse prevention. While these measures provide information on the deformation of elements and the displaced profiles at critical states, they are inadequate in themselves to provide an assessment of the state of damage or proximity to collapse.

A review of the literature reveals that there are essentially five approaches to damage modeling: estimates based on measures of deformation and/or ductility; models based on the degradation of a selected structural parameter (typically stiffness); models developed from considerations of energy-dissipation demand and capacity; hybrid formulations combining some aspects of the aforementioned parameters; and more complex theories based on concepts derived from fatigue models. Comprehensive reviews of damage modeling techniques can be found in Williams and Sexsmith (1995) and Heo (2009).

Since the response of common structural engineering materials such as steel and RC from the elastic state to failure is represented by yielding, plastic or irreversible behavior, crack growth, and fatigue during monotonic and cyclic loading, it is possible to represent such deterioration phenomena by a numerical model which can be incorporated in fiber-based discretization of a section for material-based damage estimation at the element level.

### 2.1 Damage Modeling at Constitutive Level

In this section, a damage model is introduced at the material level that is related to the response of the section deformation. This deformation is characterized by the stress and strain in the fibers of the cross-section.

#### 2.1.1 Damage in Concrete Fiber

Strains at the threshold of damage initiation, attainment of compressive strength, and residual strength of crushed concrete are defined as damage parameters. Damage is considered only in the core concrete because it was determined that calibrating the damage state to compression damage in the core was a better indicator of section damage than incorporating deterioration in both core and cover concrete. Other measures of concrete damage such as tensile cracking was found to be unimportant since damage from tensile loading is better reflected in reinforcing steel. Moreover, the response in compression governs the section damage in the concrete core. The constitutive model proposed by Mander et al. (1988) is used to evaluate the stress–strain response of the confined concrete.

A simple bilinear model is proposed in Eq. (1) and (2) assuming that the damage index is 1.0 when the accumulated plastic strain reaches the strain at the residual strength:

{D}_{\mathit{ci}}=\frac{{D}_{\mathit{cu}}(f-{f}_{\mathit{cd}})}{({f}_{\mathit{cu}}-{f}_{\mathit{cd}})}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}\mathit{\epsilon}\le {\mathit{\epsilon}}_{\mathit{cu}}

(1)

{D}_{\mathit{ci}}=1+\frac{(1-{D}_{\mathit{cu}})(f-{f}_{\mathit{cf}})}{({f}_{\mathit{cf}}-{f}_{\mathit{cu}})}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}\mathit{\epsilon}>{\mathit{\epsilon}}_{\mathit{cu}}

(2)

where *D*_{
ci
} is the concrete damage index at the i\text{th} concrete fiber, *D*_{
cu
} denotes the damage index at the compressive strength of concrete, *f*_{
cd
} is the strength at damage initiation, *f*_{
cu
} is the concrete compressive strength, *f*_{
cf
} is the residual strength, and {\mathit{\epsilon}}_{\mathit{cu}}denotes strain at concrete compressive strength. As shown in Fig. 1, the damage rate changes at the peak compressive strength according to *D*_{
cu
} which can be determined by the ratio of the degraded strength at the failure (*f*_{
cu
}* − f*_{
cf
}) to the compressive strength (*f*_{
cu
}) denoted by *D*_{
cu
} as follows:

{D}_{\mathit{cu}}=\frac{{\mathit{\epsilon}}_{\mathit{cu}}-{\mathit{\epsilon}}_{\mathit{cd}}}{{\mathit{\epsilon}}_{\mathit{cf}}-{\mathit{\epsilon}}_{\mathit{cd}}}

(3)

#### 2.1.2 Damage in Reinforcing Steel Fibers

While the response of reinforcing steel beyond the elastic phase is described through yielding, hardening, softening, and fracture under monotonic loading, these monotonic parameters are inadequate to incorporate random cyclic effects such as strength degradation because steel is vulnerable to fatigue damage under seismic loads. It is more efficient to consider damage due to cyclic fatigue since it encompasses the combined effect of multiple damage parameters. Buckling of reinforcing bars is an important phenomenon that occurs under both monotonic and cyclic loading however, a cyclic fatigue model can also include buckling effects. Therefore Miner’s (1945) linear damage rule shown in Eq. (4) is applied to compute damage in reinforcing steel fiber:

{D}_{\mathit{si}}=\frac{1}{{\sum}_{j=1}^{n}{(2{N}_{f})}_{j}}

(4)

*D*_{
si
} denotes the damage index in the *i*th steel fiber, and (*2N*_{
f
})_{
j
} denotes the number of half-cycles to failure at the plastic strain amplitude corresponding cycle j which is described in Coffin (1954, 1971) and Manson (1953). *D*_{
si
} is initialized to zero until the cumulative plastic strain attains the damage initiation threshold and it reaches unity (ideally) when the rebar is fractured. It is implicitly assumed that the fatigue model is calibrated in a manner that incorporates buckling. This means that the strain is measured across a significant characteristic length that includes buckling. In experiments carried out by Brown and Kunnath (2004), the fatigue life of reinforcing bars is computed using an effective length that includes the buckling zone. Additional details to estimate (2*N*_{
f
})_{
j
} can be also found in Kunnath and Chai (2004).

#### 2.1.3 Structural Damage at Element Level

It is necessary to first generate damage at the element level from the section damage at the fiber level discussed in the previous section. It is reasonable to consider the damage index of the most critical fibers for concrete and reinforcing steel as representative damage indices for each section as defined below:

{D}_{\mathit{cx}}^{B}=\text{max}({D}_{\mathit{ci}}^{B}),\phantom{\rule{0.277778em}{0ex}}{D}_{\mathit{sx}}^{B}=\text{max}({D}_{\mathit{si}}^{B})

(5)

{D}_{\mathit{cx}}^{C}=\text{max}({D}_{\mathit{ci}}^{C}),\phantom{\rule{0.277778em}{0ex}}{D}_{\mathit{sx}}^{C}=\text{max}({D}_{\mathit{si}}^{C})

(6)

In the above equations, {D}_{\mathit{cx}} and {D}_{\mathit{sx}} denotes concrete and reinforcing steel damage index respectively on the *x*th element for each story and superscripts *B* and *C* denote beam and column elements. For a *n* story frame structure with *m* bays, *x* = 1, 2 ….*m* for beams and *x* = 1, 2… *m*+1 for columns for each story. It is assumed that the failure of any critical concrete or reinforcing steel fiber leads to section failure in the member. This assumption may be conservative if the concrete crushing strain is achieved prior to the fatigue failure of the reinforcing bar, however, the ultimate compressive strain in confined concrete is a severe damage state that also impacts the damage in the steel. Since the failure of a local member detected by the proposed damage model at the material level progressively affects adjacent members, it can lead to eventual collapse of the entire system. Hence the damage index at the material level can govern the damage index at the element as well as the system level.

The combination of individual section damage to compute the element damage requires the implementation of weighting factors. In the study, based on studying different weighting factors, it was determined that the damage index itself is quite effective to be regarded as the weighing factor in estimating section damage. This approach has also been used previously by Bracci et al. (1989). In Eq. (7) and (8), {w}_{\mathit{cx}} and {w}_{\mathit{sx}} denotes the weighting factor for the damage index of the extreme concrete and steel fiber for the *x*th element on each story respectively.

{w}_{\mathit{cx}}^{B}=\frac{{D}_{\mathit{cx}}^{B}}{({D}_{\mathit{sx}}^{B}+{D}_{\mathit{cx}}^{B})},\phantom{\rule{0.277778em}{0ex}}{w}_{\mathit{sx}}^{B}=\frac{{D}_{\mathit{sx}}^{B}}{({D}_{\mathit{sx}}^{B}+{D}_{\mathit{cx}}^{B})}

(7)

{w}_{\mathit{cx}}^{C}=\frac{{D}_{\mathit{cx}}^{C}}{({D}_{\mathit{sx}}^{C}+{D}_{\mathit{cx}}^{C})},\phantom{\rule{0.277778em}{0ex}}{w}_{\mathit{sx}}^{C}=\frac{{D}_{\mathit{sx}}^{C}}{({D}_{\mathit{sx}}^{C}+{D}_{\mathit{cx}}^{C})}

(8)

Finally, the damage index of *x*th beam and column element ({D}_{x}^{B},{D}_{x}^{C}) is estimated as follows:

{D}_{x}^{B}={D}_{\mathit{sx}}^{B}{w}_{\mathit{sx}}^{B}+{D}_{\mathit{cx}}^{B}{w}_{\mathit{cx}}^{B},\phantom{\rule{0.277778em}{0ex}}{D}_{x}^{C}={D}_{\mathit{sx}}^{C}{w}_{\mathit{sx}}^{C}+{D}_{\mathit{cx}}^{C}{w}_{\mathit{cx}}^{C}

(9)

#### 2.1.4 Structural Damage at Story Level

Damage is computed at each story level to facilitate the assessment of structural performance under earthquake loads. The same concept of using the damage index as weighting factor can be applied in computing the damage for each story as shown below:

{D}_{y}^{B}=\frac{{\sum}_{x=1}^{m}{\left({D}_{x}^{B}\right)}^{2}}{{\sum}_{x=1}^{m}{D}_{x}^{B}},\phantom{\rule{0.277778em}{0ex}}{D}_{y}^{C}=\frac{{\sum}_{x=1}^{m}{\left({D}_{x}^{C}\right)}^{2}}{{\sum}_{x=1}^{m}{D}_{x}^{C}}

(10)

{D}_{y}^{B} and {D}_{y}^{C} denote the damage indices for the *y*th story of a *n*-story frame structure. Since columns are more critical structural elements than beams, it is necessary to introduce the concept of an importance factor. In this study, this is achieved by imposing a higher weighting factor for the failure of column elements compared to beams. Assuming that a story fails when the combine damage index of the columns in that story reaches 0.5, the story damage index for columns ({D}_{y}^{C}) needs to be updated as given in Eq. (11):

{D}_{y}^{C}=\{\begin{array}{c}\hfill 1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}({D}_{y}^{C}\ge 0.5)\hfill \\ \hfill \text{adjusted by interpolation}\phantom{\rule{0.333333em}{0ex}}({D}_{y}^{C}<0.5)\hfill \end{array}

(11)

Finally, the damage index of the *y*th story using the weighted beam and column damage index of each story is computed as given below:

{D}_{y}=\frac{{\left({D}_{y}^{B}\right)}^{2}+{\left({D}_{y}^{C}\right)}^{2}}{{D}_{y}^{B}+{D}_{y}^{C}}

(12)