expended energy based damage assessment of rc bare...

Expended energy based damage assessment of RC bare frame using

nonlinear pushover analysis

by

Anthugari Vimala, Pradeep Kumar Ramancharla

in

Urban Safety of Mega Cities in Asia 2014(USMCA 2014)

Report No: IIIT/TR/2014/-1

Centre for Earthquake EngineeringInternational Institute of Information Technology

Hyderabad - 500 032, INDIANovember 2014

Expended energy based damage assessment of RC bare frame using nonlinear pushover analysis

Anthugari Vimala1 and Ramancharla Pradeep Kumar2 1Research Scholar, Earthquake Engineering Research Centre,

IIIT-Hyderabad, India [email protected]

2Professor and Head, Earthquake Engineering Research Centre, IIIT-Hyderabad, India.

ABSTRACT

Past response analyses of structures under earthquake excitations revealed that both the maximum displacement and the number of inelastic excursions cause higher damage to the structure. However, it is observed that the quantification of damage is a difficult task. In the paper, a new methodology is proposed for the quantification of damage of reinforced concrete framed structure. This method is based on the nonlinear energy dissipated by the structure along the complete displacement path. Three methods have been proposed to assess the global damage state of the structure, for any deformation level. In these proposed methods, the damage index is expressed as the ratio of nonlinear dissipated energy at an instant, to the total non-linear energy capacity of the structure. To calculate the energy, pushover curve is plotted between base shear versus displacement at C.G of external force profile, which provides consistent meaning for work done by external forces. The area under the curve represents the energy dissipated by the structure. To illustrate the proposed concept, two cases i.e., G+5 and G+9 story structures have been considered. Keywords: Pushover analysis, expended energy, total energy, damage index. 1. INTRODUCTION Many damage indices were proposed to quantify seismic damage sustained by complete RC frame structures, each storey in them or individual elements. Damage indices are classified broadly as local damage indices and global damage indices, the former quantify damage in individual members, at individual joints or at a particular cross-section, and the latter damage in the whole structure (Kappos, 1997). The indices used in literature are estimated combining local deformation quantities and/or some overall structure response quantities. A detailed review of damage indices is available in literature (Williams and Sexsmith, 1995; Ghobarah et al., 1999; and Padilla et al., 2009). The most widely used damage index (Park and Ang, 1985) is derived to estimate (a) local damage (member damage) using a damage function based on maximum displacement ductility and cumulative hysteresis energy, and (b) the global damage using an average of local indices, weighted by local energy absorption. This index D

November 2014, Yangon, Myanmar

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takes into account both maximum plastic displacement max and plastic dissipated

energy dE , given by:

dEQ

Duyu

max (1)

and is supported by a wide correlation with observed damage. But, experimental determination of parameter β is difficult. Another damage index globalD (Roufaiel and Meyer, 1987) expressed global damage

using deflections at roof level, as:

yf

ymglobalD

(2)

and yet another DI (Powell and Allahabadi, 1988) on plastic deformations and

ductility:

ymon

y

uu

uuDI

max

(3)

In another study (Bozorgnia and Bertero, 2001), two damage indices 1DI and 2DI were introduced for SDOF systems in terms of displacement ductility and energy capacity on the system, as

Hmon

H

mon

e

E

EDI 111 1

1

and (4)

Hmon

H

mon

e

E

EDI 221 1

1

(5)

where

ductilitynt Displacememax yu

u ,

behaviourelastic for

behaviourinelastic for1ndeformatio ofportion elastic Maximum

yy

elastice uuu

capacity ductilitynt displaceme Monotonicmon ,

motion ground earthquake bydemanded energy HystereticHE ,

ndeformatio lateral increasing llymonotonica under capacity energy HystereticHmonEand 0 ≤ α1 ≤ 1 and 0 ≤ α2 ≤ 1 are constants. These models which include energy term need knowledge of time history response of the structure during the earthquake. Another damage model proposed (Poljanšek and Fajfar, 2008) for seismic damage assessment of RC frame structures expressed damage index PFDI as a ratio of deformation demand to deformation capacity, given by:

equPF u

uDI (6)

In general, any damage in the structure is related to inelastic deformations or inelastic energy dissipated by member or structure. Inelastic energy dissipation capacity of a


structure depends on its structural configuration, yielding capacity of members and material properties of the structure. At any deformation, it is an important index that indicates the state of structural damage and reliability of the structure. Damage index for a general force-deformation relationship (Cosenza, et al, 2000) is expressed as the ratio of hysteretic energy capacity of the system under monotonically increasing deformation to irrecoverable hysteretic energy, given by

Hmon

HH E

EDI (7)

Though the damage model is energy based, to quantify the damage state of the structure, the dynamic time history response of the structure during earthquake event is needed, which is tedious and involve more calculations. Energy based damage model should be simple and require less calculations. 2. PROPOSED DAMAGE INDEX ESTIMATION APPROACH In the present study, a new global damage estimation approach is presented for seismic damage assessment of RC frame structures, using a simple static nonlinear procedure. Damage is represented as ratio of inelastic energy dissipated at any displacement to the total inelastic energy capacity of the structure. The energy dissipated is low when lateral deformation is small, than that when lateral deformation is large. For the sake of simplicity, the proposed damage estimation approach uses inelastic displacement excursion of the structure. Static pushover analysis provides a measure of nonlinear behaviour of the structure. Two example buildings are considered in a pilot study, namely one 6-storey and another 10-storey building. 2D frames are considered and the total energy capacity of the structure estimated by performing pushover analysis using SAP2000 software. The effectiveness of proposed damage models is discussed along with the global damage state of the structure at four different displacement excursions. In the proposed method, the total inelastic energy dissipated by frame in each incremental load step of the pushover analysis is calculated as the area under the pushover curve – base shear versus displacement of the frame at centre of gravity of external force system, which gives the real meaning of external work done or total energy dissipated by the structure. To represent the damage state of a structure in each incremental load step of the pushover analysis, a cumulative dissipated energy is used. Based on the capacity curve of a structure, the damage state of the structure can be represented in the four ranges as shown in Figure 1.



Roof displacement

Figure 1: Damage index estimation methods critical points

Point A indicates the elastic state of the structure, point B indicates the middle stage between elastic limit and ultimate point, point C indicates the ultimate strength of the structure, point D represents the stage of the structure between ultimate point and collapse point, point E indicates the collapse stage. In the load control region (points O to C), the strength carrying capacity of the structure increases in nonlinear state. In the displacement control region (points C to E), as the displacement increases, the strength of the structure reduces. Possible damage ranges are shown in Table 1.

Table 1: Description of the behaviour of a typical structure in the whole range of nonlinear action

Range of deformation Behaviour State OA Elastic No damage AB Strain hardening Light damage BC Ultimate strength Moderate damage CD Strength reduction Severe damage DE Imminent collapse Extreme damage and collapse

In current study, overall damage index is estimated as the ratio of dissipated energy to total energy capacity of the structure. In the present study, three methods are considered for damage estimation in RC structures, namely

Method 1: 1001

ieT

ie

EEEE

D (8)

Method 2: 1002

ieT

e

EEEE

D and (9)

Method 3: 1003

NLTLT

NLL

EEEE

D (10)

where E = Energy dissipated by structure at displacement level at which damage is being estimated;

Bas

e sh

ear


Eie = Initial yield energy of structure; ET = Total energy absorbed by structure; Ee = Instantaneous elastic energy at displacement level at which damage is being estimated; EL = Linear energy at displacement level at which damage is being estimated; ENL = Nonlinear energy at displacement level at which damage is being estimated; ELT = Linear energy at maximum displacement of structure; and ENLT = Nonlinear energy at maximum displacement of structure. All parameters of above methods are represented in figure 2.

(a)

(b)

(c)

Figure 2: Parameters used for damage estimation in (a) Method 1, (b) Method 2, and (c) Method 3



3. CASE STUDY To study the efficiency of the proposed methods, the energy dissipation is studied in two example structures, namely 6 and 10 storey RC frame buildings. 3.1 Details of the Structure The two structures considered are to represent shear dominated building and flexure dominated building for study. These 6 and 10 story buildings are designed in accordance with the Indian codes of practice for plain and reinforced concrete (IS: 456) and for earthquake resistant design (IS: 1893(1)) . The buildings are assumed to be situated in seismic zone V of IS: 1893–2002, with a zone factor of 0.36 ground acceleration. Material properties used are: 20 MPa for concrete compressive strength and 415 MPa for steel yield strength for both longitudinal and transverse reinforcement bars. Both 6 and 10 story buildings are 15 m by 15 m in plan (Figure 3). Typical floor-to-floor height is 3m. The interior frames as shown in Figure 3 is 2D model of these buildings.

Figure 3: Plan view of 6 and 10 story buildings

The 6-story building is 18 m in elevation. All columns are 300mm x 400 mm dimensions and the amount and arrangement of longitudinal reinforcement in columns and beams are shown in Figure 4. All beams are 230 mm × 330 mm in cross section. The 10-storey building is 30 m in elevation. Column dimension, location and the amount and arrangement of longitudinal reinforcement are shown in Figure 5. All beams are 300mm×400mm and the amounts of top and bottom reinforcement of beams are shown in mm2 in Figure 5. 3.2 Modeling Since there is no torsional effect in the selected structures, two-dimensional (2-D) modeling is employed. A two-dimensional model of each structure is created in SAP2000 to carry out nonlinear static analysis. Beam and column elements are modeled as nonlinear frame elements with lumped plasticity by defining plastic hinges at both ends of beams and columns. SAP2000 implements the plastic hinge properties described in FEMA-356 and ATC-40. The structure is subjected to incremental lateral


forces with uniform distribution along the height and the base shear versus displacement at centre of gravity of external force profile diagram is plotted to calculate the energy at any deformation. For any load pattern, if the curve is plotted with the displacement at C.G of external force system. The area under the curve represents the total seismic energy absorbed by the building which is equal to the work of seismic loads acting on the structure.

(a) (b)

Figure 4: Details of 6-storey frame (a) Longitudinal reinforcement in beams (mm2) (b) Column reinforcement

(a) (b)

Figure 5: Details of 10-storey frame (a) Longitudinal reinforcement in beams (mm2) (b) Column reinforcement



In the present analysis the pushover curve is considered up to 2/3 of ultimate strength or 4% drift of the structure, whichever reach first. At any of the condition, damage in the structure is assumed to be 100%, that means the structure is no longer in serviceable condition. Pushover curves for 6 and 10 storey structure are shown in Figure 6. Defined critical points for damage assessment are represented on the pushover curves. Using three damage assessment methods, damage of 6 and 10 storey structures is calculated. For damage estimation Method 1, the total non linear energy capacity ET, for the structure is calculated as the total area under the pushover curve up to maximum displacement where the pushover curve is stopped . the initial elastic energy Eie, is calculated as the area under the curve up to initial yield point of the structure. E is the energy dissipated by the structure up to a displacement where the damage to be calculated. For damage estimation Method 2, the instant elastic energy Ee, is energy restored in the structure when the structure is unloaded and it is assumed that the structure come back to static position by moving parallel to initial tangent to the curve. All other parameters are calculated as given in Method 1. For damage estimation Method 3, the damage at any deformation is estimated as the ratio of expended energy for damage to total expended energy capacity to sustain the damage of the structure. In this method, the actual non linear behavior pushover curve and imaginary linear elastic curve are drawn. At any deformation, it was assumed that if there is no damage, the structure should be in linear state. Based on this concept, at any deformation, damage causing energy which is named as expended energy is calculated and represented as a percentage of total capacity of the structure to know the damage status of the structure.

Displacement(mm)

(a) (b) (b)

Figure 6: Pushover curve up to 2/3 of Ultimate strength (a) 6 storey frame (b) 10 storey frame

Bas

e sh

ear

(kN

)


Table 2: Damage in % for 6 storey structure under flexure failure

Method of Estimating Damage

Damage at A B C D E

Method 1 0 7 70 87 100 Method 2 0 5 65 85 99.8 Method 3 0 0.5 44 71 100 Powell & Allahabadi 0 9 61 74 100

Table 3: Damage in % for 10 storey structure under flexure failure

Method of Estimating Damage

Damage at A B C D E

Method 1 0 5 93 94 100 Method 2 0 3 87 89 97 Method 3 0 0.4 84 87 100 Powell & Allahabadi 0 8 93 95 100

To study the efficiency of proposed methods, the damage calculated at each step of pushover analysis using three methods are presented in Figure 7. The damage profile in both frames are same. From the analysis it is understood that the damage estimation by using method 3 is more appropriate. The clear meaning of damage is that the amount of non linear energy dissipated by the structure, as the method 3 represents the clear meaning of damage estimation at any deformation, and estimates the damage state of the structure as 100% at its maximum deformation capacity. Method 1 has the limit that at any deformation, the elastic energy is assumed as the initial elastic energy, but the structure may not have the same. Method 2 is also based on the assumption that at any deformation, when the structure is unloaded, it moves to static position with initial stiffness. The damage calculated at defined critical points on the pushover curve is presented in table 2 and table 3 for 6 and 10 storey structures respectively. To validate the proposed methods, damage estimated at critical points is compared with Powel and Allahbadi model, which is based on displacement ductility.

Drift %

Figure 7: Damage Vs drift under flexure failure (a) 6 storey frame (b) 10 storey frame

Dam

age

%



4. CONCLUSIONS 1. The ultimate deformation capacity of the structure is found by using static nonlinear pushover analysis and for that deformation the energy capacity of the structure is calculated. 2. In the paper 3 new damage estimation methods are proposed based on energy concept. New damage estimation methods are expressed as a ratio of the nonlinear energy dissipated for any deformation to the total nonlinear energy capacity of the structure. 3. The proposed approach is very simple for quick assessment of damage state of the structure for any deformation. 4. Damage estimation method 3 is more appropriate to estimate the damage state of the structure. This method is based on the expended energy which is responsible for the damage. The deformation profile of the structure is clearly represented by damage method3. 5. Method1 and method2 lead to the less accurate result compared to method 3 because of assumptions in that approaches. REFERENCES Bozorgnia, Y. and Bertero, V. V., 2008. Improved shaking and damage parameters for post-earthquake applications. Proceedings of 14th World Conference on Earthquake Engineering. Comartin, C. D., Niewiarowski, R. W., Rojahn, C. and California Seismic Safety Commission. 1996. ATC-40 Seismic evaluation and retrofit of concrete buildings, Applied Technology Council, Applied Technology Council, Redwood City, California. Cosenza, E. and Manfredi, G., 2000. Damage indices and damage measures. Progress in Structural Engineering and Materials 2, 50-59. FEMA-356., 2000. Prestandard and commentary for the seismic rehabilitation of buildings, Federal Emergency Management Agency, Washington. Ghobarah, A., Abou-Elfath, H. and Buddha, A., 1999. Response-based damage assessment of structures. Earthquake Engineering and Structural Dynamics 28, 79-104. Kappos, A. J., 1997. Seismic damage indices for RC buildings: evaluation of concepts and procedures. Progress in Structural Engineering and Materials 1, 78-87. Kunnath, S. K., Reinhorn, A. M. and Park, Y. J., 1990. Analytical modeling of inelastic seismic response of r/c structures. Journal of Structural Engineering ASCE 116, 996-1017. Murty, C. V. R. and Pradeep Kumar, R., 2013. Critical Review of Indian Seismic Code- IS1893:2002. International colloquium on Architecture & Structure Interaction for Sustainable Development. Padilla, D. and Rodriguez, M., 2009. A damage index for the seismic analysis of reinforced concrete members. Journal of Earthquake Engineering 13, 364-383. Park, Y. J., Ang, A. H. S. and Wen, Y. K., 1985. Seismic damage analysis of reinforced concrete buildings. Journal of Structural Engineering ASCE 111,740-757. Powell, G. H. and Allahabadi, R., 1988. Seismic damage prediction by deterministic methods: Concept and procedure. Earthquake Engineering and Structural Dynamics 16, 719-734.


Roufaiel, M. S. L. and Meyer, C., 1987. Reliability of concrete frames damaged by earthquakes. Journal of Structural Engineering, ASCE 113, 445-457. Williams, M. S. and Sexsmith, R. G., 1995. Seismic damage indices for concrete structures: A State-of-the-Art Review. Earthquake Spectra 11, 740-757.

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