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SEISMIC ANALYSIS OF RC BUILDINGS

INTRODUCTION

• Since earthquake forces are random in nature and unpredictable, the static and dynamic analysis of the structures have become the primary concern of civil engineers.

• The main parameters of the seismic analysis of structures are load carrying capacity, ductility, stiffness, damping and mass.

• IS 1893-2002 is used to carryout the seismic analysis of multi-storey building.

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SEISMIC ANALYSIS OF STRUCTURES

• The seismic analysis type that should be used to analyse the structure depends upon :-

external action

the behavior of structure or structural materials

the type of structural model selected

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• The different analysis procedure are

Linear Static Analysis

Nonlinear Static Analysis

Linear Dynamic Analysis

Nonlinear Dynamic Analysis

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LINEAR STATIC ANALYSIS

• Also known as Equivalent Static method.

• Based on formulas given in the code of practice.

STEPS

• First, the design base shear is computed for the whole building.

• It is then distributed along the height of the building.

• The lateral forces at each floor levels thus obtained are distributed to individual lateral load resisting elements.

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Equivalent lateral shear force along two orthogonal axis

(Source: Nouredine Bourahla, "Equivalent Static Analysis of Structures Subjected to

Seismic Actions", Encyclopedia of Earthquake Engineering, Springer-Verlag Berlin

Heidelberg, 2013)

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Limitations

The use of this method is restricted with respect to

• High seismic zones and height of the structure

• Buildings having higher modes of vibration than the fundamental mode

• Structures having significant discontinuities in mass and stiffness along the height

PROCEDURE

• Calculation of the Design Seismic Base Shear, VB

• Vertical distribution of base shear along the height of the structure

• Horizontal distribution of the level forces across the width and breadth of the structure

• Determination of the drift, overturning moment, and P-Delta effect

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Design Seismic Base Shear, VB

From IS 1893- 2002, Clause 7.5.3, the design base shear

where, W - seismic weight of the buildingAh - horizontal seismic coefficient

Horizontal Seismic Coefficient, Ah

As per IS 1893(Part 1)-2002, Clause 6.4.2

Provided that for any structure with T < 0.1 s, the value of Ah will not be taken less than Z/2 whatever be the value of I/R.

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Ah =

VB= Ah W

Where,

Z - Zone factor

I - Importance factor

R- Response Reduction factor

Sa/g - Average response acceleration coefficient

T -Undamped Natural period of the structure

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Zone Factor ( Z)

• It is the indicator of the maximum seismic risk characterized by Maximum Considered Earthquake (MCE ) in the zone in which the structure is located.

• According to IS 1893(Part 1)-2002, Seismic Zones are classified into II, III, IV & V respectively.

Average response acceleration coefficient (Sa/g)

• It depends on the type of rock or soil sites and also the natural period and damping of the structure.

• It is obtained from, Clause 6.4.5, IS 1893-2002. 12

Importance Factor (I)

• It depends on the occupancy category of the building.

• It is obtained from table 6, Clause 6.4.2, IS 1893-2002.

Site Class

• Site Class is determined based on the average properties of the soil within a certain

depth (30 m) from the ground surface.

Response Reduction factor (R)

• It is determined by the type of lateral load resisting system used.

• It is a measure of the system’s ability to accommodate earthquake loads and absorb energy without collapse.

• It is obtained from table 7, IS 1893-2002. 13

Ta =

Fundamental Period • The approximate fundamental natural period of vibration ( Ta ),

of a MRF building from Clause 7.6, without brick infil panels,

with infil panels,

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where, h - height of the buildingd- Base dimension of the building at the plinth level

Ta = 0.075 h0.75 for RC frame building = 0.085 h0.75 for steel frame building

Vertical Distribution of Base Shear to Different Floor levelsThe lateral force induced at any level hi as per Clause 7.7.1, IS 1893-2002, can be determined by,

where,Qi - Design lateral force at floor iWi - Seismic weight of floor ihi - Height of floor i measured from base, andn - Number of storey's in the building is the number of levels

at which the masses are located.

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Horizontal Distribution of Base ShearThe horizontal distribution of base shear as per FEMA P749, can be determined by

where, Fij : force acting on the lateral force-resisting line j at a floor level i

nk : number of lateral force-resisting elements (lines)

Kij ,Kik : story stiffness of the lateral force-resisting element (line) k and j at level i

Fi : seismic force at floor (level) i 16

Drift Story• It is a measure of how much one floor or roof level displaces under

the lateral force relative to the floor level immediately below.

• It is the ratio of the difference in deflection between two adjacent floors divided by the height of the story that separates the floors.

Overturning Moment and P-Delta Effects• There is a tendency for the moment created by equivalent static

force acting above the base to overturn the structure.

• The dead weight of the building is sufficient to resist the overturning force, but it must be checked always.

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• The “stability coefficient” for each story as per FEMA P749, can be calculated as,

where, Pi - weight of the structure above the story being evaluated

i - is the design story drift determined

Vi - is the sum of the lateral seismic design forces above the story

hi - story height

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=

NONLINEAR STATIC ANALYSIS

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• Also known as Pushover Analysis

• Used to estimate the strength and drift capacity of existing structure and the seismic demand for this structure subjected to selected earthquake.

• It can be used for checking the adequacy of new structural design as well.

• It is an analysis in which, a mathematical model incorporates the nonlinear load-deformation characteristics of individual components and elements of the building which shall be subjected to increasing lateral loads representing inertia forces in an earthquake until a ‘target displacement’ is exceeded.

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• Response characteristics that can be obtained from the pushover analysis are

– Estimates of force and displacement capacities of the structure.

– Sequences of the failure of elements and the consequent effect on the overall structural stability.

– Identification of the critical regions, where the inelastic deformations are expected to be high and identification of strength irregularities of the building.

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PROCEDURE

In Pushover analysis the magnitude of the lateral load is increased monotonically maintaining a predefined distribution pattern along the height of the building.

Building is displaced till the ‘control node’ reaches ‘target displacement’ or building collapses.

The sequence of cracking, plastic hinging and failure of the structural components throughout the procedure is observed.

The relation between base shear and control node displacement is plotted for all the pushover analysis.

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Schematic representation of pushover analysis procedure

(Source: Jan, T.S.; Liu, M.W. and Kao, Y.C. (2004), “An upper-bond pushover analysis procedure for estimating the seismic demands of high-rise buildings”. Engineering

structures. 117-128)

• Pushover analysis may be carried out twice:

(a) first time till the collapse of the building to estimate target displacement.

(b) next time till the target displacement to estimate the seismic demand.

• The seismic demands for the selected earthquake are calculated at the target displacement level.

• The seismic demand is then compared with the corresponding structural capacity to know what performance the structure will exhibit.

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Lateral Load Patterns

FEMA 356 suggests the use of at least two different patterns for all pushover analysis.

Group – Ii) Code-based vertical distribution of lateral forces used in equivalent static analysis

ii) A vertical distribution proportional to the shape of the fundamental mode in the direction under consideration

iii) A vertical distribution proportional to the story shear distribution calculated by combining modal responses from a response spectrum analysis of the building

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Group – IIi) A uniform distribution consisting of lateral forces at each level proportional to the total mass at each level

ii) An adaptive load distribution that changes as the structure is displaced

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Lateral load pattern for pushover analysis as per FEMA 356(Source: Jan, T.S.; Liu, M.W. and Kao, Y.C. (2004), “An upper-bond pushover analysis procedure for estimating the seismic demands of

high-rise buildings”. Engineering structures. 117-128)

Target DisplacementTwo approaches to calculate target displacement:

(a) Displacement Coefficient Method (DCM) of FEMA 356

(b) Capacity Spectrum Method (CSM) of ATC 40

• Both of these approaches use pushover curve to calculate global displacement demand on the building.

• The only difference in these two methods is the technique used.

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Displacement Coefficient Method (FEMA 356)

• This method estimates the elastic displacement of an equivalent SDOF system assuming initial linear properties and damping for the ground motion excitation under consideration.

• Then it estimates the total maximum inelastic displacement response for the building at roof by multiplying with a set of displacement coefficients.

Capacity Spectrum Method (ATC 40)

• Uses the estimates of ductility to calculate effective period and damping.

• This procedure uses the pushover curve in an acceleration displacement response spectrum (ADRS) format.

• This can be obtained through simple conversion using the dynamic properties of the system.

• The pushover curve in an ADRS format is termed a ‘capacity spectrum’ for the structure.

• The seismic ground motion is represented by a response spectrum in the same ADRS format and it is termed as demand spectrum. 29

LINEAR DYNAMIC ANALYSIS

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• Response spectrum method is a linear dynamic analysis method.

• In this approach multiple mode shapes of the building are taken into account.

• For each mode, a response is read from the design spectrum, based on the modal frequency and the modal mass.

• They are then combined to provide an estimate of the total response of the structure using modal combination methods. 31

Combination methods include the following:

• Absolute Sum method• Square Root Sum of Squares (SRSS)• Complete Quadratic Combination (CQC)

• The design base shear calculated using the dynamic analysis procedure is compared with a base shear Vb , calculated using static analysis.

• If Vb is less than , all the response quantities, eg. member forces, displacements, storey forces, storey shears, and base reactions, should be multiplied by Vb /

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•Buildings with plan irregularities and with vertical

irregularities cannot be modelled for dynamic analysis by

this method.

•For irregular buildings, lesser than 40m in height in

zones II and III, dynamic analysis, though not mandatory,

is recommended.

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Modal Analysis

Modal Mass (clause 7.8.4.5(a))

Where, - mode shape coefficient at the floor i in the mode

k

- seismic weight of floor i

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Modal Participation Factor (Clause 7.8.4.5 (b))

Design lateral force at each floor level in each mode(clause7.8.4.5(c))

Where,Qik - peak lateral forceAk - design horizontal acceleration spectrum

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Storey shear forces in each mode (clause 7.8.4.5(d)) The peak storey shear, Vik

Lateral forces at each storey due to all modes considered(clause

7.8.4.5(f))

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The design lateral forces, Froof and Fi, at roof and at floor i are given by

Modal Combination• The peak response quantities should be combined as per the Complete Quadratic combinations (CQC) method

Where, r - number of modes being consider ρij - the cross-modal coefficient λi, - response quantity in mode i λj - response quantity in mode j ξ - model damping ratio β - frequency ratio

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Square Root Sum of Squares (SRSS)

Absolute Sum method• If the building has a few closely spaced modes the peak

response quantity λ* due to these modes should be obtained as

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Where λk is the absolute value of quantity in mode k, and r is the number of modes being considered.

NONLINEAR DYNAMIC ANALYSIS

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• Also known as Time History Analysis(THA)

• To perform such an analysis, a representative earthquake time history is required for a structure being evaluated.

• In this method, the mathematical model of the building is subjected to accelerations from earthquake records that represent the expected earthquake at the base of the structure.

• The method consists of a step- by- step direct integration over a time interval.

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• The time-history method is applicable to both elastic and inelastic analysis.

• In elastic analysis the stiffness characteristics of the structure are assumed to be constant for the whole duration of the earthquake.

• In the inelastic analysis, however, the stiffness is assumed to be constant through the incremental time only.

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PROCEDURE• An earthquake record representing the design earthquake is selected.

• The record is digitized as a series of small time intervals of about 1/40 to 1/25 of a second.

• A mathematical model of the building is set up, usually consisting of a lumped mass at each floor. Damping is considered proportional to the velocity in the computer formulation.

• The digitized record is applied to the model as accelerations at the base of the structure.

• The equations of motions are then investigated with the help of software program that gives a complete record of the acceleration, velocity, and displacement of each lumped mass at each interval.

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SAP2000• It is a finite-element-based structural program for the

analysis and design of civil structures.

• SAP2000 is object based, meaning that the models are created using members that represent the physical reality.

• All the seismic analysis procedures can be analysed effectively in SAP2000.

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CASE STUDY

Comparative Study of Static and Dynamic Analysis of Multi-Storey Regular & Irregular Building

• This study was carried out by Saurabh G. Lonkar, in the year 2015.

objectives of this paper were

To study the seismic behavior of RC building and to analyse the structure using equivalent static method, time history Method and response spectrum method followed by Pushover analysis.

Determination of storey displacements.

To check the accuracy and exactness of Time History analysis, Response Spectrum Analysis and Equivalent Static Analysis with respect to different conditions & aspects.

Also to check the seismic behavior and relative displacement of regular & irregular building in different seismic zone. 45

Structural Analysis and Modeling

• A 22 storey residential building was modelled for zone III in SAP2000.

• The storey plan was changing for irregular building & symmetric for regular building.

• The building had been analyzed by using equivalent static, response spectrum and time history analysis, based on IS codes.

• The maximum storey displacements result had been obtained by using all methods of analysis. 46

Results and Discussions

• Displacement values between static and dynamic analysis is insignificant for lower stories but the difference is increased in higher stories and static analysis given higher values than dynamic analysis.

• According to damage assessment of building, it was concluded that the damage percentage of building was different for each method of analysis.

• Static analysis is not sufficient for high rise building its necessary to provide dynamic analysis because of specific & non linear distribution of forces.

• Time history analysis should be performed as it predicts the structural response more accurately than other two methods based on damage assessment of building. 47

Comparative Study of Seismic Analysis of 3-Storey RC Frame on SAP2000

• This study was carried out by Akshay Mathane, Saurabh Hete, Tushar Kharabe, in the year 2016

The main Objectives were - • To analyze the building as per code IS 1893-2002 part I

• To study the response of the structure such as base shear and lateral displacement

• To study methods of earthquake analysis (Equivalent static and Response spectrum method)

• To study seismic analysis of frame by SAP200048

Modeling

• 3 storey building with storey height 3m having 4 bays of 5 m in X and 3 bays of 5m in Y directions for seismic zone V was modeled in SAP2000.

Results and Discussion

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Storey Level Displacement (Manual in mm) Displacement (SAP in mm) Displacement (%)

4 0.052469 0.050533 0.036897

3 0.044383 0.042554 0.041209

2 0.0131142 0.024788 -0.890164

1 0.015023 0.014306 0.0477268

Comparison of Storey Displacements

Storey Level Displacement by ESM in mm as per SAP

Displacement by RSM in mm as per SAP

4 0.050533 0.043112

3 0.042554 0.037057

2 0.029788 0.026739

1 0.014306 0.013248

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Sl. No. Manual shear( kN) Base shear in SAP (kN)

1 1269.64 1282.039

Comparison of Base reaction

Comparison of Storey Displacements in ESM & RSM

Sl.No. Base shear by ESM in SAP (kN)

Base shear by RSM in SAP (kN)

1 1282.039 1275.628

Comparison of Base reaction in ESM & RSM

• Equivalent static method was simpler than Response Spectrum method, but Static analysis was not sufficient for high-rise building.

• SAP results for Equivalent static and Response spectrum method were nearly same.

• The results obtained from static analysis method shows higher storey displacement values as compared to response spectrum analysis.

• Manual and SAP result of story displacement, base reaction of Equivalent Static method were approximately same.

• Response spectrum of irregular and multistory building was very tedious work but for the analysis of any type of building this method can be preferred to get better results.

• Response spectrum results were more accurate than Equivalent static method. 51

STRUCTRAL ANALYSIS AND MODELLING

• A 2D Frame of floor height 3m was modelled by SAP2000.

• Building has 2 bays of 3 m in X direction.

• The grade of concrete is M25.

• Pushover analysis procedure were carried out for 2D frame.

• Lateral load of 10kN and a Vertical load of 100kN was applied at the roof level.

• Hinge support was provided.

• P- Delta effects were included in analysis. 52

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2D Frame Model

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Pushover Curve

• Pushover analyses using uniform lateral load pattern yielded capacity curves with lower initial stiffness and base shear capacity but higher roof displacement

CONCLUSION• Dynamic analysis for simple structures can be carried out manually,

but for complex structures finite element analysis can be used to calculate the mode shapes and frequencies.

• Depending upon the accuracy of results needed and the importance of the building that should be analysed various seismic analysis procedures can be adopted like Linear Static Analysis, Nonlinear Static Analysis, Linear Dynamic Analysis and Nonlinear Dynamic Analysis.

• For smaller structures, response spectrum analysis or equivalent static analysis can be used with little effort.

• If accurate and precise result is wanted from the analysis, then we should carryout non-linear dynamic analysis. 55

• Nonlinear relationship between force and displacement in multi-storey building structures may be determined easy enough with the application of nonlinear static pushover analysis.

• SAP2000 provides almost accurate results when compared with manual calculations.

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REFERENCE

[1] Chopra AK (1995). “Dynamics of Structures Theory and Application to Earthquake Engineering”, University of California at Berkeley, USA.

[2] Duggal S K (2010). “Earthquake Resistance Design of Structure”, Fourth Edition, Oxford University Press, New Delhi.

[3] FEMA 356 (2000), “Pre-standard and Commentary for the Seismic Rehabilitation of Buildings”, American Society of Civil Engineers, USA.

[4] IS 1893 Part 1 (2002). “Indian Standard Criteria for Earthquake Resistant Design of Structures”, Bureau of Indian Standards, New Delhi.

[5] Jan. T.S, Liu. M.W. and Kao. Y.C. (2004), “An upper-bond pushover analysis procedure for estimating the seismic demands of high-rise buildings”, Engineering

structures. 117-128.

[6] Nouredine Bourahla (2013), "Equivalent Static Analysis of Structures Subjected to Seismic Actions", Encyclopedia of Earthquake Engineering, Springer-Verlag Berlin

Heidelberg.

[7] Pankaj Agarwal and Manish Shrikhande (2014)."Earthquake Resistant Design of Structures", PHI Learning Private Limited, Delhi.

[8] Prof. Sakshi Manchalwar, Akshay Mathane, Saurabh Hete and Tushar Kharabe "Comparative Study of Seismic Analysis of 3-Storey RC Frame", International Journal

of Science, Engineering and Technology Research (IJSETR), April 2016, ISSN: 2278- 7798 .

[9] Saurabh G Lonkar and Riyaz Sameer Shah, ''Comparative Study of Static and Dynamic Analysis of Multi-Storey Regular & Irregular Building-A Review", International

Journal of Research in Engineering, Science and Technologies (IJRESTs), ISSN 2395-6453. 57

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